Pierre Yannick

Frédéric Sanchez

Mickaël Battegay

Stéphanie Fourel

Marc Audran

Maurice Guillermet

Rivoire

Sonia Charles

Nathalie Sébastien

Aurélie Ainsi

Elena Benjamin

Guillaume Dmitri

Mon

Loic Alexia Gaspard

Thomas M Gabriel

Bien qu'un manuscript de these ne soit écrit que par une seule personne, la somme des résultats qu'il contient est généralement le fruit d'un travail d'équipe. Je tiens ainsi à remercier tout d'abord mes directeurs de these Orphée Cugat, Thomas Skotnicki et Jérôme Delamare, de même que mon encadrant Stéphane Monfray pour son expertise en récupération d'énergie, Jihane Boughaleb et Christophe Maître avec qui. Je souhaite aussi remercier les membres de l'équipe TSV de STMicrolectronics, Crolles, pour leur expertise technique qui a permis de concevoir des micro-bilames fonctionnels, plus particulièrement

Beam's initial amplitude ! ! , ! ! Heat capacitances ! ! Initial deflection ! !"# , ! !"# Cavity's thickness

! !"# Beam's pseudo-force ! Electric displacement ! Piezoelectric coefficient ! Damping coefficient ! Young's modulus ! Dielectric permittivity ℰ Electric field ! !" !"# Absolute thermo-mechanical efficiency ! !" !"#

Context, applications and advances in energy harvesting

The terms "energy harvesting" or "energy scavenging" define the action of converting ambient energy under whichever form (heat, mechanical vibrations, light, electromagnetic energy…) into usable electric energy (by means of a power management circuit), either by using a single conversion step (direct conversion) or by a sequence of conversion steps (indirect conversion). Given the variety of energy sources that can be found in our surroundings, the field of energy harvesting has widely developed by adapting a large range of transducer technologies to the specificities of the energy requirements of electronic devices. Even if the idea of producing electric energy from local energy sources is not a new concept, the various devices designed or fabricated in the mid-20 th century and reported in the literature were mainly intended to supply enough electric power to be classified as energy generators. Unlike outdoor photovoltaic plants or wind farms, energy harvesters only convert very small amounts of energy into electric energy, but quite sufficient to supply low-power electronics.

Indeed, the development of energy harvesting is strongly linked to the emergence of the Internet of Things (IoT). This notion of IoT was first introduced by Kevin Ashton [Shoenberger, 2002] to designate networks of devices such as sensors, actuators, RFID, industrial machines, computers, smart objects (connected cars, smart home…) connected with each other, mainly wirelessly. One of the aspect of the IoT is the development of Wireless Sensors Networks to respond to the growing needs for monitoring data in domains as diverse as industry, urban or natural environments, home, or even the human body (Fig. 1.1.a) [Yick et al, 2008;Zahid Kausar et al, 2014]. Beyond the simple interest aroused by the potential new applications of IoT (Fig.1.1.1b), the market of connected objects and smart sensors is growing and is expected to represent about 28 billion devices in 2020 against 9 billions in 2013 (Fig. 1.1.1c) with potential market revenues of more than 8900 billion USD by the end of 2020 [Bauer et al;2014;PwC, 2015]. The semiconductor industry would be a main actor and recipient of the development of this market. Networks. b. Projected adoption of some connected devices [START_REF] Bauer | The Internet of Things: Sizing up the opportunity[END_REF]. c. Evolution of the MEMS market until 2018 [McKinsey, 2014]. d. Simplified schematic explaining the architecture of a Wireless Sensing Node. e. Typical scenario for the power consumption of a Bluetooth Low Energy Sensor Node. The sensors market is expected to grow annualy by 40% [PwC, 2015] with potential revenues above 7 billion USD by 2019 [PwC, 2015], while MEMS market would grow up from 16 to 23 billion USD between [START_REF] Bauer | Photovoltaic Solar Energy Conversion[END_REF]2018 (Fig.1.1.d) mainly thanks to opportunities in consumer electronics and biomedical applications [McKinsey, 2014].

A short overview on Wireless Sensors Networks

The term Wireless Sensors Network (WSN) designates a group of Sensing Nodes spatially distributed and connected together by means of wireless connections, and whose main function is to collect data on their close surroundings and transmit them to an entity, either electronic or human, capable of interpretating them to take a decision. This definition enables to directly understand the generic architecture of a sensing node, which is necessary made of at least one sensor (pressure, temperature, humidity, light, movement, etc.), a radio transceiver, a microcontroller unit, all this powered by an energy source and its dedicated power management circuit (Fig. 1.1.d). Typically, the most powerhungry components of the wireless node are the radio, the Analog-Digital Converter and the sensor. Taking the example of electronic components fabricated at ST-Microelectronics (Tab.1), we give a simple representation of the operation of a Bluetooth Low Energy node. At a data rate of 1 Mb/s, the maximal duration of wireless communication is around 1 ms. During this period, the radio consumes about 30 µJ to receive and transmit data. The output data rate of the accelerometer being 100 Hz (Tab.1.1), the approximate time to measure data on an accelerometer is about 10 ms [ST, AN3182, 2010], during which the node consumes about 40 µJ. The total consumption of the node in active mode is thus about 100 µJ, corresponding to 8.5 mW. As we will see most energy harvesters are not able to supply such levels of electric power. The solution to get around this problem is to use duty cycling [START_REF] Anastasi | Energy conservation in wireless sensor networks: A survey[END_REF] to reduce the equivalent power supply, and a storage device to feed the communicating node. As a consequence, by sending data every minute, the equivalent power needed by the node in active node is 1.7 µW, whereas the power needed in standby mode is 4 µW. As a consequence, an energy harvester capable of delivering at least 5.7 µW of usable electric power would be able to feed a wireless sensor node. During the active mode of the wireless node, a micro-battery such as the EnFilm battery (Tab.1.1) is able to deliver up to 30 mW, enough to supply it [ST, AN3182, 2010]. As demonstrated with this example, the sustainability of WSNs is strongly linked to the possibility given to each sensing node of the Network to save energy by not communicating with the rest of the network over long periods of time. To allow this type of operation, new wireless communications standards were defined by industry consortiums like Zigbee Alliance and standardization organisms like IEEE: main communications standards used by Wireless Sensors Networks are IEEE 802.15.4 (used by Zigbee communication protocol), IEEE 802.15.1 (used by Bluetooth Low Energy devices) and IEEE 802.15.3 (defined for wireless sensors networks dedicated to wearable technologies (or Body Area Network)) [Yick et al, 2008]. As an example, to limit the power consumption of sensing nodes during data transmission, IEEE 802.15.4 limits the transmission power to 0.5 mW [START_REF] Adams | An Introduction to IEEE STD[END_REF]. As a consequence, the transmission range of wireless sensors is usually limited to around 10-50 meters.

An overview on energy harvesting technologies

The previous part showed that the development of energy harvesting devices generating at least more than the consumption of the node in standby mode (4 µW in our study case) is sufficient to power wireless sensor nodes, complemented by an energy storage buffer between the generator and the sensing node. Litterature on energy harvesting shows a large spectrum of generators and microgenerators demonstrating their capability to supply energy to these smart sensors. In this part, we present a short classification of generators according to the nature of the energy they scavenge as well as some examples of energy harvesters. Numerous synthesis works can be found in the literature providing more details on these technologies [Vullers et al 2009;Pryia and Inman, 2008;Briand et al, 2015].

Table 1. Performances of some electronic components fabricated by ST-Microelectronics [ST, LM135, 2008;ST, BlueNRG, 2015;ST, STM32, 2015;ST, AN3182, 2010].

Mode

Components Consumption/Supply Standby STM. BlueNRG ® STM32L (Microcontroller) 1,3 -2 µA @ 2 V 0,3 µA @ 2 V (without RTC) 0,9 µA @ 2 V (with RTC)

Sleep

BlueNRG ® STM32L (Microcontroller) 1,7 -3,5 µA @ 2 V 0,5 µA @ 2 V (without RTC) 1,2 µA @ 2 V (with RTC)

Active

BlueNRG ® STM32L (Microcontroller) AD Converter 12 bits DA Converter 12 bits H3LIS331DL (Accelerometer) LM135 (Temperature sensor) 1,3 -2 µA @ 2V @ 32 kHz 0,3 µA @ 2 V 0,9 µA @ 2 V 0,9 µA @ 2 V 300 µA @ 3V @ 100 Hz 0,45 -1 mA @ 3V

Reception

(2.4GHz)

BlueNRG ® Bluetooth LE radio 14,5 µA @ 2V

Transmission

(2.4GHz)

BlueNRG ® Bluetooth LE radio 0 dBm Bluetooth LE radio +5 dBm Bluetooth LE radio +5 dBm 10,6 -21 µA @ 2V 15,4 µA @ 2 V 21 µA @ 2 V 10,6 µA @ 2 V

Storage

Thin film Battery EnFilm EFL700A39

10 mA @ 3V 0,7 mAh

Vibration energy harvesting

Conversion of energy from mechanical vibrations into electrical energy is a technique perfectly suited to wireless sensors embedded on transport vehicles, industrial machines, clothing etc., even though the range of frequency and amplitude of mechanical energy is extremely variable between all these systems [Neri et al, 2013]. As a result, vibration energy harvesting may be the field of energy harvesting that knew the broadest diversification in terms of transducing technologies. All vibration energy harvesters work on the principle of the resonant vibrations of an oscillator caused by the inertia of a seismic mass [Mitcheson et al, 2008]. Three methods of conversion are commonly used to transform mechanical energy into electrical energy: either piezoelectric transduction to convert the mechanical stress of the oscillator into electric charges, electrostatic conversion based on capacitance variations induced by the displament of the oscillator, or electro-magnetic conversion based on the creation of an electromotive force induced by the temporal variation of the magnetic field due to the displacement of a permanent magnet.

Based on a generic spring-mass-damping lumped model, [Williams and Yates, 1996] established the equations giving the maximal energy that can be extracted from vibrations, and showed that the output power could be maximized by matching the resonant frequency of the linear oscillator to the frequency of the oscillations and by increasing the mechanical quality factor of the structure. This model owns the advantage of being common to every vibration energy harvester, enabling to define Figures of Merit based on the ratio between the generated electrical energy and the input mechanical energy calculated thanks to the acceleration of the oscillator. [Mitcheson et al, 2008] therefore report a comparison between electromagnetic, piezoelectric and electrostatic energy harvesters, showing that the conversion efficiency of electromagnetic devices is about 1%, whereas it can reach about 20% for electrostatic energy harvesters and more than 30% for piezoelectric energy harvesters. In the case of a generator using a seismic mass of 1 g submitted to an acceleration of 10 m.s -2 at 100 Hz, an efficiency of 20% yields an output electrical power of around 16 µW, which is enough to power a Wireless sensing node. In the case of piezoelectric energy harvesters, non-linear energy extraction techniques like SECE [START_REF] Lefeuvre | Buck-Boost Converter for Sensorless Power Optimization of Piezoelectric Energy Harvester[END_REF] or SSHI [Guyomar et al, 2005] can help increasing the efficiency of the transduction. Yet the drawback of using high-Q linear oscillators is a narrowing of the frequency bandwith of the generator which restricts access to the energy associated to frequencies other than the resonant frequency. This problem is even more relevant for miniaturized harvesters: their frequency grows with miniaturization and is inversely proportional to their length, while higher energy is generally associated to low frequencies [Neri et al, 2013]. Current research is oriented toward the design of non-linear oscillators (hardening or softening Duffing oscillators) to widen the oscillators' frequency bandwith, bistable oscillators [START_REF] Harne | A review of the recent research on vibration energy harvesting via bistable systems[END_REF], coupled oscillators [START_REF] Gu | Impact-driven, frequency up-converting coupled vibration energy harvesting device for low frequency operation[END_REF], frequency tuning [Challa et al, 2008], or arrays of oscillators working at different resonant frequency [START_REF] Zhu | Strategies for increasing the operating frequency range of vibration energy harvesters: a review[END_REF].

Thermal energy harvesting

Although the next chapter of this thesis manuscript is devoted to thermal energy harvesting technologies, we already present here a brief overview of thermal harvesters in order to have a global comparison between harvesting technologies. It is commonly admitted that the wide availability of low-grade heat flux makes heat a viable energy source to power WSN. Typically, heat flux around 5 mW.cm -2 can be generated from temperature difference of 5 o C with natural convection, allowing energy harvesting from heat generated by the human body. However, thermodynamics limits the efficiency of thermal energy harvesters which cannot exceed the Carnot limit of around 1.7 % for a hot source at 30 o C and a temperature difference of 5 o C. As a consequence, no more than 83 µW.cm -2 could be generated from such temperatures difference. Even though the efficiency of some conventional heat engines like Rankine generators can reach 60% of the Carnot limit, they generally work at very high temperatures which are less compatible with most WSN applications (around 565 o C with a temperature difference of 530 o C for the generators presented in [Lee and Frechette, 2006]). The main mature thermal generator technologies are based on thermoelectricity exploiting the Seebeck effect to directly convert heat into electrical energy. Theoretical efficiency of these devices generally reaches 14% of the Carnot limit at 30 o C, corresponding to a power of 14 µW.cm -2 , enough to power wireless sensing nodes. However, the need for thermal matching between Seebeck generators and heat sink to maintain the thermal gradient across the energy harvester limits practical applications of thermal energy harvesting applied to body area networks. Therefore, although [Leonov et al, 2009;Leonov et al, 2007] developed Seebeck generators adapted to human body heat energy recovery, technologies of thermal generators are generally developped to harvest heat at higher temperatures (around 100 o C), more adapted to industrial environments for instance. In such environment where the use of bulky heat sinks is less problematic, thermal energy harvesters can generate up to 10 mW.cm -2 , largely enough to power WSNs.

Photovoltaic energy harvesting

The advantage of harvesting energy from light to power WSNs is linked both to the ubiquity of these energy resources, and to the low-cost mature technologies, initially developed for solar energy production. Generally, we distinguish indoor from outdoor photovoltaic energy harvesting, given the difference of incident irradiance between these two cases (100 mW.cm -2 for outdoors direct sunlight, compared to 0.5 mW.cm -2 indoors) [Priya and Inman, 2008]. Moreover, the efficiency of solar cells depend not only on the cell technology but also on the irradiance of the light sources. Typically, monocrystalline silcon cells have an efficiency of about 14% for outdoor applications, but only 4% for indoor applications [Randall and Jacot, 2002]. Innovations like organic photovoltaic cells, even less efficient, are expected to lower the cost of energy harvesting devices but would also allow to power smart textiles [Wilson and Mather, 2015].

Table 2. Summary of some energy sources and potential energy density (for mechanical energy harvesters, we assume a mass of 10 g and a volume of 1cm 3 )

Source

Characteristics of the source Harvested energy

Light

Indoor Outdoor 10-100 mW.cm -2 0.5 mW.cm -2 1.5-15 mW.cm -3 @ η=15% 30 µW.cm -3 @ η=6% Mechanical vibrations Human [Feenstra et al, 2014] Industrial [Beeby et al, 2014] 1.4m.s -2 @ 10 Hz 7m.s -2 @ 110 Hz 32 µW.cm -3 @ η=20% 71 µW.cm -3 @ η=20% Heat Human Industrial losses 5 mW.cm -2 @ TH=30 oC @ TC=25 o C 375 mW.cm -2 @ TH=100 oC @ TC=25 o C 14 µW.cm -2 @ ZT=1 14 mW.cm -2 @ ZT=1 RF radiations 100 mW @ 2.4 GHz (WiFi) 0.1 µW.cm -2 @ 10 m

Others energy sources

Photovoltaic energy, mechanical vibrations and heat are the main technologies reviewed in this introduction because of the high levels of electric power that can be generated from these energy sources. However, other harvesting technologies exist, some of them exploiting the electro-magnetic radiations from the 900 MHz GSM band or the 2.4 GHz WiFi band, such as rectennas. Yet the amount of electric power generated by these devices is small: for a GSM signal of 2 W transmission power, the power received by a RFID tag at 10 m is around 2 µW, and for a 100 mW WiFi signal, the received power is 0.1 µW [Priya and Inman, 2008]. As a result, RF energy harvesting is more adapted to RFID recognition.

3 Objectives of this thesis

State of the project

Even if the working principle of the bimetallic strip heat engines is explained in deep detail in the next chapter, we give here a short description and we briefly recall the history of its development at STMicroelectronics. The idea of developing at STMicroelectrics a thermal energy harvester based on the thermal snap-through of bimetal thermostats (either shell-like thermostats or beam-like thermostats) was proposed by [START_REF] Puscasu | Dynamical formation of microlenses by the reflow method: numerical simulation and experimental study of the process fabrication[END_REF]. The basic idea is to place the bimetallic strip between a hot source and a cold source so as to create a constant instability of the bimetallic strip, switching from a heat source to the one without discontinuity (Fig. 3.1a). As a consequence, the bimetal thermostat realizes a thermo-mechanical transduction, generating kinetic energy from heat. A second conversion step is thus needed to transform the bimetallic strip's kinetic energy into electrical energy.

Up to now, two types of bimetallic strip heat engines were developed [START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF]Boisseau et al, 2013]. The first one is based on the mechanical energy transfer occurring during the shock between the bistable bimetallic strip and a piezoelectric oscillator and was jointly developed by STMicroelectronics, LGEF, CEA-Liten and Delta-concept. The current piezoelectric prototype and an exploded view reavealing its architecture are represented in Fig. 3.1c, whereas the bimetallic strip made of Invar-NC4 (Invar used as a thermo-mechanically passive material, and NC4 as an active material) is presented in Fig. 3.1b. This device is currently able to generate up to 3 µW of usable electric power with a bimetallic strip of hysteresis 67 o C-70 o C on a hot source at 100 o C (equivalent surface density 0.26 µW.cm -2 if it is normalized by the harvester's surface or 0,46 µW.cm -2 if it is normalized by the bimetallic strip surface). The advantage of this generator is its capability to maintain a thermal gradient inside the passive architecture's cavity without needing any bulky heat sink. This is mainly linked to the slow working frequency of the bimetallic strip (around 0.25 Hz), which allows the piezoelectric transducer, whose function is also to cool down the bimetallic strips, to returns to its equilibrium temperature once the bimetallic strip has snapped back. Thermal optimization of the passive architecture enabled to improve the thermal efficiency of the harvester by reducing thermal leakage, and to increase the temperature difference in the cavity, thus widening the operating window of the generator [Boughaleb et al, 2014]. Experimental measurements of the perfomances of these thermal generators are reported in [Puscasu et al, 2012, Arnaud et al, 2013[START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF], Boughaleb et al, 2014, Boughaleb et al, 2015[START_REF] Boughaleb | Design and performance benchmark of various architectures of a piezoelectric bimetallic strip heat engine[END_REF]. The latest experimental results (currently unpublished) demonstrated the capability of a single piezoelectric generator to power a WSN developed by STMicroelectronics and CEA-Leti in asynchronous mode with either Zigbee radio or Bluetooth Low-Energy, communicating acceleration or temperature data.

A second type of generator based on the use of bimetallic strip heat engines coupled with an electret-based capacitive transducer was jointly developed with CEA-Leti's R&D teams and Delta-Concept. In that case, the capability of generating electric energy from heat flux is not linked to the kinetic energy released by the switching bimetallic strips, but to the capacitance variations cause by the curvature reversal of the bistable membrane. The operation and performances of these electrostatic bimetallic strip heat engines are reported in [Boisseau et al, 2013, Arnaud et al, 2013]. Using Invar-B72M bimetallic strips (hysteresis temperatures of 47 o C and 43 o C) and FEP-based electrets (25 µm-thick layer) having a surface potential of 500 V, these devices are able to generate output power of around 5 µW at 2 Hz of working frequency. Experimental demonstration of their capability to power sensing nodes was demonstrate in [Arnaud et al, 2013]. Until now, difficulties were encountered to operate electrostatic generators without bulky heat sink, linked to the absence of thermal optimization of this kind of device.

Based on scaling rules predicting a linear improvement of the surface density of generated power at the microscale [Puscasu et al, 2012], two parallel thesis projects were impulsed in order to integrate micro bimetallic strip heat engines on silicon wafers using MEMS fabrication techniques. A first approach was to process thermo-mechanically bistable piezoelectric bimorphs in order to directly transform heat into electrical energy without decreasing the overall efficiency of the energy harvester, by multiplying the number of conversion steps. This work involved a collaboration between the Ceramics Laboratory LC of EPFL university at Lausanne and TIMA laboratory at Grenoble [START_REF] Trioux | Fabrication of bilayer plate for a micro thermal energy harvester[END_REF], demonstrating the generation of electrical charges due to the thermo-mechanical instability of Al-AlN bimorph beams.

The second approach, developped through a collaboration between STMicroelectronics, G2ELab and CEA-Leti consists in directly scaling down the electrostatic bimetallic strip heat engine to the sub-millimetric scale by developing a microelectronics process fully compatible with the facilities of the 300 mm clean room of STMicroelectronics in Crolles (near Grenoble).

Objectives and plan of the thesis

The aim of this PhD was initially to develop miniaturized bimetallic strip heat engines coupled with electret transducers using MEMS fabrication techniques, but also to contribute to the development of piezoelectric bimetallic strip heat engines at the macroscale. As few works were made on the accurate prediction the performances of the bimetallic strip heat engine and on the creation of design rules to guide the fabrication of microsystems, this PhD primarily focused on the modeling of the bimetallic strip heat engines.

This thesis manuscript thus is divided into five chapters starting from a state-of-the-art on thermal energy harvesting, to the design, fabrication and characterization of the bimetallic strip at the MEMS scale, including a comprehensive modeling of the various aspects of the energy harvester.

Chapter 1 presents an overview of the existing technologies dedicated to the conversion of thermal energy into electric energy by means of various effects such as the Seebeck effect, pyroelectricity, shape memory, thermomagnetism etc., enabling to compare these technologies with the bimetallic strip heat engines.

Chapter 2 sums up the modeling of the thermo-mechanical instability of composite beam, including bimetallic beams. The conditions of occurrence and the properties of the thermal snap-through are explicited. In this first chapter on the modeling of the device, we theoretically demonstrate the possibility to harvest heat by means of bistable composite beams thus justifying the name of "bimetallic strip heat engine". A reduced-order model is then developed to obtain approximate equations of the various performances of the energy harvester, such as the energy produced during a complete cycle, the thermal efficiency, and the Carnot efficiency of the device. This will enable us to evaluate the performances of the primary conversion step of the bimetallic strip heat engine.

Chapter 3 is devoted to the modeling of the conversion of the mechanical energy into electrical energy by means of piezoelectric materials, piezoelectric transducers or electrets-based transducers. For each of these converters, we developed models to explain the influence of the parameters of the system on the performances of the system. We finally compare each device in order to find which architecture is the most efficient one. This will enable us to find in which directions the future developments must be carried out in order to improve the performances of the current energy harvesters, but also to find the theoretical limits of our devices and to compare it with other technologies of thermal energy harvesters.

Chapter 4 extends the use of the models developed in Chapters 2 and 3 to the simulations of the performances of the bimetallic strip heat engine at the micrometric scale. We finally extract scaling rules from these models, enabling to make a comparison between the performances of the bimetallic strip heat engine at the microscale and at the macroscale.

Challa, V., Prasad, M., Shi, Y. and Fisher, F. (2008). A vibration energy harvesting device with bidirectional resonance frequency tunability. Smart Mater. Struct., 17(1), p.015035.

Feenstra, J., Granstrom, J. and Sodano, H. (2008). Energy harvesting through a backpack employing a mechanically amplified piezoelectric stack. Mechanical Systems and Signal Processing, 22(3), pp.721-734.

Chapter 1

Materials and devices for thermal energy harvesting 1 Introduction

In the introduction of this thesis, we briefly exposed some aspect of thermal energy harvesting, without giving a clear overview of the various technologies developed to transform heat into electrical energy. This first chapter thus is a state-of-the-art on thermal energy harvester. In a first time, we present some theoretical background on heat engines which enables to give to establish Figure-of-Merits that can be used to compare thermal generators with each other. Then we present main technologies of heat engines: thermo-electric generators, pyroelectric generators, fluid-based heat engines, solid-based heat engines, and thermomagnetic generators.

2 Maximal efficiency and maximal power of heat engines

Carnot heat engine

Thermodynamically speaking, heat is a form of energy that is differs from other form of energy qualified as "noble" like mechanical, electric, magnetic, chemical, nuclear, kinetic, radiative energies, in the sense that heat cannot be transform with an efficiency of 1 into another form of energy, whereas mechanical energy, electrical energy etc. can be fully converted in an other form of the energy, or degraded with an efficiency of 1 into heat. The other difference between thermal generators and other generators is also is that heat engines needs two heat sources at different temperatures to generate a quantity of work. This last observation hystorically is the first statement of thermodynamics' second principle by Sadi Carnot claiming that "la production de la puissance motrice est due dans les machines à vapeur, non à une consommation réel du calorique, mais à son transport d'un corps chaud à un corps froid" [START_REF] Carnot | Réflexions sur la puissance motrice du feu[END_REF]. This can be translated into equations by applying the first principle (1.1.1a) and second principle to the heat engine presented in Fig. 2.1.1a.

𝑄𝑄𝑄𝑄 ! = 𝑊𝑊𝑊𝑊 + 𝑄𝑄𝑄𝑄 ! (2.1.1a) dΣ = Q ! T ! - Q ! T ! > 0 (2.1.1b)
As a consequence we find that the heat engine's efficiency cannot exceed a value that is only function of the external temperatures, explaining why Carnot stated that "La puissance motrice de la chaleur est indépendante des agents mis en oeuvre pour la réaliser: sa quantité est fixée uniquement par les températures des corps entre lesquels se fait, en dernier résultat, le transport du calorique". Hence the Carnot efficiency can only be reached for a reversible transformation.

𝜂𝜂𝜂𝜂 = 𝑊𝑊𝑊𝑊 𝑄𝑄𝑄𝑄 ! < 𝜂𝜂𝜂𝜂 ! = 1 - 𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 ! (2.1.2a)
The strength of Carnot's model is to give a simple expression of the maximal theoretical efficiency for all heat engines without making assumption on the type of working medium. Efficiency of Carnot heat engines is plotted in Fig. 2.1.1a as function of the hot source's temperature and for various values of the cold source's temperature. Typically for a hot source at 100 o C and a cold source at 50 o C, the maximal theoretical efficiency of a heat engine is of 13,5% and reaches 58,2% at 500 o C. Yet the drawback of the Carnot's model is to assume a fully reversible transformation whereas thermal transfers are known to be an irreversible process. As a consequence, Carnot engine is a generating for which no thermal transfers occur, infinitely slowly cycled, and producing zero power. Carnot heat engine thus is not the ideal model to describe real thermal heat engines even if it furnishes a Figure of Merit to compare the performances of heat engines with each other.

Curzon-Ahlborn efficiency

Curzon-Ahlborn heat engine describes a generic model of reversible Carnot thermal generator taking into account thermal diffusive transfers between the hot source and the cold source. As a consequence this heat engine is qualified by the term "endoreversible" [START_REF] Chen | On the Curzon-Ahlborn efficiency and its connection with the efficiencies of real heat engines[END_REF]. With this model, Curzon and Ahlborn found an expression giving the efficiency at maximum power of the generator represented in Fig. 2.1.2a [START_REF] Curzon | Efficiency of a Carnot engine at maximum power output[END_REF]. For the detail, the demonstration of this formula is generally attributed to Curzon and Ahlborn, but is also named Chambadal-Novikov-Curzon-Ahlborn efficiency because of earlier independent demonstrations by Chambadal and by Novikov, both in 1957. However Vaudrey et al [START_REF] Vaudrey | Reitlinger and the origins of the efficiency at maximum power formula for heat engines[END_REF] found prior demonstration of the formula of the efficiency at maximal power by Yvon in 1955 and Reitlinger in 1929.

Curzon-Ahlborn model is a starting point for various models of heat engines [START_REF] Chen | On the Curzon-Ahlborn efficiency and its connection with the efficiencies of real heat engines[END_REF], including the model of the Stefan-Boltzmann engine [START_REF] Bauer | Photovoltaic Solar Energy Conversion[END_REF]. This engine is used to calculate the theoretical maximal efficiency of photovoltaic cells by replacing Fourier-type thermal transfers by Stefan-Boltzmann radiative transfers. It is based on the fact that incident flux of photons are either converted in electron-holes pairs or transformed into heat, elevating the temperature of the solar cell. The main difficulty in developing this model is linked to the non-linearity of radiative transfers. Similarly, the Müser engine assumes that radiative energy transfers only occur between the hot source and the solar cell [Muser, 1957].

To demonstrate the expression of the Curzon Ahlborn efficiency, we simply need to write the expressions of the input and output heat flux as a function of the temperatures of the system (2.1.3a and 2.1.3b) and the reversibility of the heat engine (2.1.3c) and the conservation of the energy (2.1.3d). The Carnot efficiency of the mover gives a link between the temperature T1 and T2 (2.1.3e).

𝑄𝑄𝑄𝑄 ! = 1 𝑅𝑅𝑅𝑅 ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! )

(2.1.3a)

𝑄𝑄𝑄𝑄 ! = 1 𝑅𝑅𝑅𝑅 !
. (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! )

(2.1.3b)

𝑄𝑄𝑄𝑄 ! 𝑇𝑇𝑇𝑇 ! -𝑄𝑄𝑄𝑄 ! 𝑇𝑇𝑇𝑇 ! = 0

(2.1.3c)

𝑄𝑄𝑄𝑄 ! = 𝑊𝑊𝑊𝑊 + 𝑄𝑄𝑄𝑄 ! (2.1.3d) 𝜂𝜂𝜂𝜂 = 1 + 𝑇𝑇𝑇𝑇 ! /𝑇𝑇𝑇𝑇 ! (2.1.3e)
Hence the expression of the temperatures T1 and T2 are simply given by (2.1.4a) and (2.1.4b), while the power is given by (2.1.4c Fig. 2.1.2b represents the evolution of the power as a function of the intrinsic efficiency of the reversible heat engine for various values of the ratio TC/TH showing that there exists an optimal value of the efficiency that maximizes the output power. Logically we observe that reducing the temperature difference between the two heat sources decreases the generator's output power. A simple derivation of this last equation enables to find that the power is maximized when the efficiency is a simple function of the external temperatures (2.1.5a). This value of the efficiency is the Curzon-Ahlborn bound [START_REF] Curzon | Efficiency of a Carnot engine at maximum power output[END_REF]. The maximal power generated by the heat engine is then given by (2.1.5b).

Thermal energy harvesters being generally in contact with the hot source and evacuating heat by means of a heat sink, we can rewrite this last equation to express the surface density of power generated by the Curzon-Ahlborn generator. Logically we see with this equation that the best way to produce energy is to use a bullky heat sink in order to evacuate more heat from the device.

𝜂𝜂𝜂𝜂 !"# !" = 1 - 𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 ! (2.1.5a) 𝑊𝑊𝑊𝑊 !"# !" = 𝜂𝜂𝜂𝜂 !"# !" ! . 𝑇𝑇𝑇𝑇 ! 𝑅𝑅𝑅𝑅 ! + 𝑅𝑅𝑅𝑅 ! (2.1.5b) 𝑊𝑊𝑊𝑊 !"#,!"#$ !" = 𝜂𝜂𝜂𝜂 !"# !" ! . 𝑇𝑇𝑇𝑇 ! . ℎ !"#$ (2.1.5c)
The evolution of the Curzon-Ahlborn efficiency is plotted against the ratio in Fig. 2.1.2c, showing that it is lower than the Carnot efficiency of around 30% for a ratio TC/TH equal to 0,4. In the same time, Fig. 2.1.2d represents the evolution of the maximal power generated by the Curzon Ahlborn efficiency, showing that the maximal power is of around 100mW.cm -2 for a heat engine working at 100 o C with a bulky heat sink. Equation (2.1.5) provides a useful metrics for the evaluation of the performances of various heat engines. Equation (2.1.6) defines the Curzon-Ahlborn relative efficiency of a thermal generator.

Carnot relative efficiency and Curzon-Ahlborn relative efficiency furnish two solutions to compare heat engines.

𝜂𝜂𝜂𝜂 !" = 𝜂𝜂𝜂𝜂 !"# 𝜂𝜂𝜂𝜂 !"# !"
(2.1.6)

Thermoelectric generators

Thermoelectric generators being the most widely used technology to convert heat into electrical energy, we first present an overview on the theory and the development of thermoelectric materials and thermoelectric generators.

Thermoelectric effect

Discovery of the thermoelectric effect is the results of three successive discoveries: the Seebeck effect, describing the creation of voltage drop across a junction of two materials, was first observed by Seebeck in 1821 [START_REF] Seebeck | Ueber den magnetismus der galvenische kette[END_REF], measuring the displacement of a needle placed on a thermocouple he heated. The second discovery is due to Peltier in 1834 [START_REF] Peltier | Nouvelles expériences sur la caloricité des courants électriques[END_REF] observing a temperature change of the two materials of Seebeck's thermocouple when the is is crossed by electric current. Finally, Thomson (also named Lord Kelvin) demonstrated in 1851 that heat rejection or heat absorption can also occurs in a single material, and that Peltier effect (characterized by the coefficient π), Seebeck effect (characterized by the parameter α) and Thomson effect (τ) are linked by two equations reproduced in (3.1.1a) and (3.1.1b) [START_REF] Thomson | On a mechanical theory of thermo-electric currents[END_REF].

𝜋𝜋𝜋𝜋 !" = 𝛼𝛼𝛼𝛼 !" . 𝑇𝑇𝑇𝑇 (3.1.1a) 𝑑𝑑𝑑𝑑𝛼𝛼𝛼𝛼 !" 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝜏𝜏𝜏𝜏 ! -𝜏𝜏𝜏𝜏 ! 𝑇𝑇𝑇𝑇 (3.1.1b)
Yet Thomson demonstrated the relation between the heat generated by a thermoelectric element and the current flowing through it, the rigorous demonstration of the link between thermal transfers and thermoelectricity is associated to the work of Onsager and to its development of the theory of irreversible thermodynamics in 1931 [START_REF] Onsager | Reciprocal Relations in Irreversible Processes[END_REF]. Applied to thermoelectricity, this theory shows that Seebeck coefficient simply expresses the coupling effect between the thermal flux JQ and the electric current JN, and that the thermal gradient is the driving force of heat flux and is equivalent to the by its nature to electric field [Callen, 1960]. Rewriting this last equation in short-circuit enables to define the adimmensionnal Figure of Merit ZT of the thermoelectric materials, as defined by Altenkirch in 1911 [Rowe, 1995].

𝒥𝒥𝒥𝒥 ! = -𝜅𝜅𝜅𝜅. 𝛼𝛼𝛼𝛼 ! . 𝜎𝜎𝜎𝜎 𝜅𝜅𝜅𝜅 . 𝑇𝑇𝑇𝑇 + 1 . 𝛻𝛻𝛻𝛻𝛻𝛻𝛻𝛻 (3.1.3a) 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 = 𝛼𝛼𝛼𝛼 ! . 𝜎𝜎𝜎𝜎 𝜅𝜅𝜅𝜅 . 𝑇𝑇𝑇𝑇 (3.1.3b)
As seen this factor increases as the electrical conductivity in the material, the square of the Seebeck coefficient and is inversely proportional to the thermal conductivity of the materials. A large part of the research on thermoelectricity is devoted to materials engineering in order to enhance this factor of Merit in order to boost the performances of thermoelectric generators. First techniques developed constisted in increasing the electrical conductivity of semiconductors by doping materials. Yet this technique is limited by the fact that Seebeck effect degrades with very strong doping. Current researchs mainly deals with fabrication of nanostructured materials or superlattices to increase phonons scattering given that phonons are known to be mainly responsible for the (a) (b) Figure 3.1.1. a. Typical ZT of some thermoelectric materials [START_REF] Dmitriev | Current trends in the physics of thermoelectric materials[END_REF]. b. Comparison between the performances of various sources of energy and thermoelectric ZT factor [START_REF] Vining | An inconvenient truth about thermoelectrics[END_REF].

thermal conductivities of semi-conductor materials. Extensive reviews of these techniques can be found in [START_REF] Dmitriev | Current trends in the physics of thermoelectric materials[END_REF] or [Alam and Ramakrishna, 2012]. Fig. 3.1.1a presents the evolution of ZT against the temperature for some thermoelectric materials. For applications at ambient temperatures or less than 100 o C, this figure shows that best materials are bismuth telluride (Bi2Te3 or similar compounds). Typically, a ZT factor equal to 4 is considered as the minimum value so that thermoelectric effect be used to produce energy and be competitive with solar energy as a exemple [START_REF] Vining | An inconvenient truth about thermoelectrics[END_REF]. Maximal values experimentally measured are between 2 -2,5 [START_REF] Ohta | Thermoelectrics based on strontium titanate[END_REF].

Optimization of the performances of thermoelectric generators

The first attempt to model the operation and efficiency of thermoelectric generators is due to Rayleigh in 1885 [Rowe, 1995] but was really achieved by Ioffe in 1957[Ioffe, 1957]. Considering the basic thermocouple made of two semiconductor legs (length l, and cross sections SN and SP, resistivities ρN and ρP, thermal diffusivity κP and κN, Seebeck coeffients αN and αP), he showed that the optimal design that maximized the intrinsic efficiency of the generator is obtained by matching the cross section according to (3.2.1a), and the load resistance according to (3.2.1b) where Tm is the average temperature of the heat engine and R the equivalent electrical resistance of the thermoelectric element. The optimization of the semiconductor rods' sections leads to the expression of the equivalent Z factor given by (3.2.1c). Hence the output voltage of a single thermocouple is then simply given by (3.2.1d), whereas the efficiency of the heat engine is given by (3.2.1e). To raise the output voltage up to some volts, generators are made of hundreds of series thermocouples. . c. Evolution of the output power with a bulky heat sink (h=1000W.K -1 .m -2 ). d. Evolution of the output power delivered by both generators with a planar heat sink (h=10W.K -1 .m -2 ).

𝑆𝑆𝑆𝑆 ! = 𝑆𝑆𝑆𝑆 ! . 𝜌𝜌𝜌𝜌 ! . 𝜅𝜅𝜅𝜅 ! 𝜌𝜌𝜌𝜌 ! . 𝜅𝜅𝜅𝜅 ! (3.2.1a) 𝑅𝑅𝑅𝑅 ! = 𝑅𝑅𝑅𝑅. 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! (3.2.1b) 0 T, K 200
𝑍𝑍𝑍𝑍 = 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! 𝜌𝜌𝜌𝜌 ! . 𝜅𝜅𝜅𝜅 ! + 𝜌𝜌𝜌𝜌 !" . 𝜅𝜅𝜅𝜅 ! ! (3.2.1c) 𝑉𝑉𝑉𝑉 !"# = 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! 1 + 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! . 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) (3.2.1d) 𝜂𝜂𝜂𝜂 !"# = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 ! . 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! -1 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! + 𝑇𝑇𝑇𝑇 ! /𝑇𝑇𝑇𝑇 ! (3.2.1e)
These expressions do not require any hypothesis on the way heat is evacuated from, or supplied to the thermal generator. Understanding that the Seebeck generator is equivalent to a simple thermal resistance, the maximal value of heat flux crossing the harvester is obtained by matching the thermal resistance of the heat exchangers to the generator's equivalent thermal resistance. Hence the temperature diffence across the [START_REF] Bottner | New thermoelectric components using microsystem technologies[END_REF].

b. Thermocouples made of Cu and Ni legs [Briand et al, 2015] generator is half the external temperature and the thermal flux flowing through the generator is simply given by (3.1.4a). The maximal power is then given by (3.1.4b) if a heat sink (either planar of bulky) is used.

𝑄𝑄𝑄𝑄 = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 2. 𝑅𝑅𝑅𝑅 !"#$ (3.1.4a) 𝑃𝑃𝑃𝑃 !"# = 𝜂𝜂𝜂𝜂 !"# . 𝑄𝑄𝑄𝑄 = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! 4. 𝑇𝑇𝑇𝑇 ! . 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! -1 1 + 𝑍𝑍𝑍𝑍. 𝑇𝑇𝑇𝑇 ! + 𝑇𝑇𝑇𝑇 ! /𝑇𝑇𝑇𝑇 ! . ℎ !"#$ . 𝑆𝑆𝑆𝑆 !"#! (3.1.4b)
As an example, we plot in Fig. 3.1.2 the evolution of the performances of two generators either made of Si-Ge thermocouples (Z=5.10 -4 K -1 ) or Bi2Te3 thermocouples (Z=3.10 -3 K -1 ). For a device working without heat sink (planar heat sink indeed), SiGe-based generator produces 160 µW.cm -2 whereas Bi2Te3-based generator produces 700 µW.cm -2 for a heat source at 100 o C and the threshold value of 10 µW.cm -2 necessary to power Wireless Sensing node is reached for a temperature of 33 o C for Bi2Te3-based generator and for 45 o C for SiGe-based generator. With a massive heat sink, the power can be multiplied by 1000 reaching 700 µW.cm -2 with Bi2Te3. For an effective temperature difference of 5 o C across the generator (hot source at 35 o C), the Bi2Te3-based generator is expected to deliver 2,1 mW.cm -2 , closed to the 1,8 mW.cm -2 produced in the same conditions by Micropelt's generators [Briand et al, 2015].

Once the thermoelectric materials are chosen, Ioffe's optimal design leaves only the legs' length as a variable parameter. [START_REF] Mayer | Optimization of Heat Sink-Limited Thermoelectric Generators[END_REF] proposed an optimization of the thickness of the energy harvester in order to match the thermal impedance of a given heat sink using the previous techniques. This approach is followed by Leonov in [START_REF] Leonov | Simulation of maximum power in the wearable thermoelectric generator with a small thermopile[END_REF] to design compact wearable thermo-electric generators: with doped poly-silicon-based thermocouples, its generator achieved generating up to 9 µW.cm -2 from the heat evacuated by a human body and the ambient air at 20 o C (in agreement with previous curves). Since industrialization of thermal generator is only possible for low-cost products, Rowe and Min [START_REF] Rowe | Design theory of thermoelectric modules for electrical power generation[END_REF] proposed a cost-driven optimization of the performances of thermoelectric generators consisting in scaling down the size of the thermocouple so that the volume of matter used is minimal and then to design a thermally matched heat sink so that the generated power be maximal. This strategy can be justified by the fact that heat sink are cheaper than thermoelectric materials. Rowe and Min found a minimal value of thermocouple's length around 500 µm explained by the relative increase of parasitic thermal resistance at the microscale that degrades the performances of the thermal generator. Yet the evolution of the microelectronics since the publication of Rowe and Min enabled reducing this minimum size. As an example, thermocouples produced by Micropelt based on bismuth telluride measure 36 µm and around 500 series thermocouples can generate voltages of around 0,5V with a fixed temperature difference of 5 o C. Taking into account the price of thermoelectric materials, heat sink, manufacturing costs, Leblanc et al [START_REF] Leblanc | Material and manufacturing cost considerations for thermoelectrics[END_REF] used Rowe and Min's optimization scheme to evaluate the price of thermoelectric energy, showing that that the mean cost of Bi2-Te3 modules was of around 80$/W for low-grade heat applications (heat source below 100 o C). Precise comparison of performances of thermoelectric generators can be found in [Briand et al, 2015].

Simple pyroelectric generators

Pyroelectricity refers to the property of some piezoelectric materials to polarize under a raise or a decrease of their temperature. Basically, pyroelectric materials can be described by means of the two constitutive equations exposed in (4.2.1), showing that pyroelectric effect represents the proportionality factor between the temperature and the polarization but also the proportionality factor between the electric field and the entropy. As a consequence, many works presented the generation of electricity from heat by means of pyroelectric materials. [START_REF] Mane | Cyclic energy harvesting from pyroelectric materials[END_REF] presented the cycling light heating and convective cooling of PMN-PT and PZT samples using a rotating wheel, enabling to generate power densities of about 8.6 µW.cm -3 (equivalent to a surface power density of 0.23 µW.cm -2 ) at a frequency of 0,1 Hz and a with a temperature change rate of 8 K.s -1 . Quite similar experiences with similar measured values of power densities are reported in [Zhang et al, 2011;[START_REF] Cuadras | Thermal energy harvesting through pyroelectricity[END_REF].

In [START_REF] Sebald | On thermoelectric and pyroelectric energy harvesting[END_REF], Sebald and co-workers presented the evaluation of the mean power generated by a Pb(MGNb)))3-PbTiO3 compound over a whole day, showing that the mean generated power along a day is of 1 µW.cm -3 . Similarities between pyroelectric generators and piezoelectric energy harvesters allowed Sebald coworkers to propose in [Sebald et al, 2008] various models of the performances of pyroelectric generators performing either resistive or SECE-like, SSHI-like or Carnot cycles, pointed out the matter of the electrothermal coupling factor as a determinant of pyroelectric generators' performances. For a SECE-like cycle (Fig. 4.1.1), the Carnot relative efficiency is given by (4.2.3a) and the power by (4.2.3b).

𝜂𝜂𝜂𝜂 !"# = 𝑘𝑘𝑘𝑘 !! ! 1 + 0.5. 𝑘𝑘𝑘𝑘 !! ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 ! (4.1.1d) 𝑊𝑊𝑊𝑊 = 𝑝𝑝𝑝𝑝 ! 𝜀𝜀𝜀𝜀 . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! . 𝑓𝑓𝑓𝑓 (4.1.1d)
From the observation that with values of electrothermal coupling factor of 10% and using SSHI power-management circuit, the theroretical efficiency of pyroelectric generators could reach 50% of the intrinsic Carnot efficiency, Sebald et al concluded by saying that pyroelectric generators could be more performant than thermoelectric generators. In fact, the conclusion must lightly qualified: In Fig. 4.1.1a we plot the power generated by a thermoelectric material working without heat sink and the frequency at which a pyroelectric generator made of a 200-µm thick PMN-PT layer and cycled in a temperature difference of 5 o C must operate so that it can generate the same power than the thermoelectric generator. As an exampled, to generate 700 µW.cm -2 (like a thermoelectric generator at 100 o C), the pyroelectric generator must be cycled at 3,7 Hz with a temperature change rate of 18,5 o C.s -1 . The characteristic time of propagation of heat into a 200-µm thick PMN-PT layer is of around 1,2s (0,8Hz). As a consequence, there is chance that the pyroelectric layer thermalizes when the frequency of the temperature variations raises. In the same time, Fig.4.1.1b shows that equivalent efficiencies between thermoelectric and pyroelectric generators can be obtained if pyroelectric materials having a electro-thermal coupling factor of 20% can be found. In the case of PMN-PT, the value of this coefficient is only of 0.4%.

Exploiting phase transitions of ferroelectric materials

Previous models assumed that the physical parameters of the pyroelectric materials were assumed to be constant whatever the temperature, they indeed depends on the temperature and are known to strongly vary around the Curie temperature for ferroelectric materials [Callen, 1960]. As a consequence, various thermodynamic cycles were developed to exploit the variation of these performances, notably the decrease of the dielectric permittivity near the Curie temperature [Clingman and Moore, 1961;[START_REF] Hoh | Conversion of thermal to electrical energy with ferroelectric materials[END_REF]. In [START_REF] Olsen | A pyroelectric energy converter which employs regeneration[END_REF] Olsen et al demonstrated the operation of Ericsson cycle producing 0,25 J.cm -3 with a Carnot efficiency of 3,4%. Several demonstrations of pyroelectric heat engines exploiting phase transitions are reported in the litterature [START_REF] Khodayari | Nonlinear pyroelectric energy harvesting from relaxor single crystals[END_REF]Guyomar et al, 2008;[START_REF] Navid | Towards optimization of a pyroelectric energy converter for harvesting waste heat[END_REF], showing the possibility to reach efficiency of 55,4% [START_REF] Navid | Towards optimization of a pyroelectric energy converter for harvesting waste heat[END_REF]. From these publications, only [START_REF] Navid | Towards optimization of a pyroelectric energy converter for harvesting waste heat[END_REF] and [START_REF] Nguyen | Pyroelectric energy converter using co-polymer P(VDF-TrFE) and Olsen cycle for waste heat energy harvesting[END_REF] report operation of real heat engines. The other demonstrations are made by placing ferroelectric sample in hot bath and cold bath to observe a thermal cycling. The comparison of data given in [Guyomar et al, 2008] and describing experimental performances of PMN-PT-based generators in the frame of the Sebald's model, and data reported in [START_REF] Khodayari | Nonlinear pyroelectric energy harvesting from relaxor single crystals[END_REF] which describes the performances of relaxor crystals realizing a ferroelectric-to-ferroelectric phase transition, enable to see that exploiting phase transitions would enable to gain at least one order of magnitude on the performances of pyroelectric generators. [Lee et al, 2011]. d. Rotor of a Brayton generator presented in [START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF].

Fluid-based heat engines

Downsizing of conventional heat engines

The comparison between fluid-based heat engines and thermoelectric and pyroelectric generators reveals that the performances of these last devices can be considered as weak. As an example, Fig. 3.1.1 shows that the equivalent ZTs of Stirling and Rankine engines are of around 20 [START_REF] Vining | An inconvenient truth about thermoelectrics[END_REF]. Because of these high theoretical efficiencies, many works reports the silicon integration of Rankine cycle [Lee and Frechette, 2011;Frechette et al, 2009]. Modeling of the operation of this heat engine and of its performances at the microscale are presented in [Frechette et al, 2009] and [Cui amd Brisson, 2008]. Fig. 5.1.1a represents a simplified schematic of the Rankine engine developed by Lee and coworkers [Lee et al, 2011] whereas Fig.5.1.1c and Fig.5.1.1d show some images of MEMS turbines. In order to understand the operation of the micro-Rankine engine, we represented in Fig. 5.1.1b the Rankine cycle. Basically, the fluid is vaporized in contact with the hot source. The expansion of the steam produced during the phase transition is then used to make the turbine turn (where comes the name of "steam engine"), producing a mechanical energy that can be converted into electrical energy by means of an electromagnetic converter. The vapor is then condensed in contact with the cold source in the condenser and then a pump drives the fluid back to the evaporator (Fig. 5.1.1b). Given that a phase transition occurs at a fixed temperature, and a fixed pressure, and that compression and expansion can be adiabatic (isentropic), the theoretical efficiency of Rankine engine is closed to Carnot efficiency. In [Frechette et al, 2008], Frechette and his coworkers estimated the efficiency of Rankine engines working either between 800 o C and -50 o C or between 400 o C and 250 o C, corresponding to conditions found in microcombustion as an example [START_REF] Fernandez-Pello | Micropower generation using combustion: Issues and approaches[END_REF]]. In the case of an engine working between 800 o C and -50 o C, they demonstrated the possibility to harvest up to 12,5W.cm -2 of electrical energy by means of an electromagnetic converter having an efficiency of 50%. These levels of power correspond to a Carnot efficiency of 14,7%, inferior to the theoretical efficiency of a Rankine engine (55,5% of Carnot efficiency in the same conditions). Similar results are found by Cui and Brisson (6,8W.cm -2 and Carnot efficiency of 18,8% for temperatures between 500 o C and 25 o C), even if they do not consider the transformation of shaft mechanical power into electrical power. The explanation of this performances degradation are explained in [Frechette et al, 2008;[START_REF] Deam | On Scaling Down Turbines to Millimeter Size[END_REF][START_REF] Fernandez-Pello | Micropower generation using combustion: Issues and approaches[END_REF]: As heat engines are scaled down to the microscale, surface effects like viscosity become important and are a source of irreversibility. As a consequence, a large part of the energy is lost by frictions in the turbine and in the bearing of the turbine's rotor. The other consequence of the size reduction, common to all types of thermal energy harvesters, is the increase of heat leakage due to the passive architectures, the change of the nature of thermal transfers occurring in working fluids, from a convective regim to a conductive regim, as explained in [START_REF] Deam | On Scaling Down Turbines to Millimeter Size[END_REF][START_REF] Fernandez-Pello | Micropower generation using combustion: Issues and approaches[END_REF]. Other heat engines based on the circulation of a working fluid like Stirling engines [Nakajima et al, 1989], Brayton engines [START_REF] Epstein | Millimeter-Scale, Micro-Electro-Mechanical Systems Gas Turbine Engines[END_REF] or Wrankel [START_REF] Lee | Design and fabrication of a micro Wankel engine using MEMS technology[END_REF] which have also proposed to generate electrical energy from heat are expected to face the same problems.

Heat engines using stationary fluids

Given the drawbacks of using circulating fluids in classical Rankine generators, another type of heat engive have been designed, whose design was directly thought to be functional at the microscale and which does not require that the fluid circulates from the hot source to the cold source to produce work. As a consequence, no viscous effect can degrades the performances of these heat engines. These generators are reffered to as "P 3 heat engines" in [START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF] or "Saturation Phase-change Internal Carnot Engine" in [START_REF] Lurie | Analysis of a microscale 'Saturation Phase-change Internal Carnot Engine[END_REF]. The simplified schematic of this energy harvester is presented in Fig. 5.2.1a whereas Fig. 5.2.1b shows a picture of the device fabricated by [START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF]. It basically consists in a cavity created by the sealing of two silicon wafers that contains a saturated fluid. In the design proposed by Whalen et al, the thin membranes delimiting the cavity are made of a piezoelectric layer whose function is to transform the mechanical pressure into electrical energy. The heat engine operates as following: in a first time, heat is provided to the saturated fluid through the bottom membrane. Being saturated, the fluid follows a phase transition (which is isothermal and isobaric like other phase transitions), which leads to the increase of steam volume in the cavity which applies pressure on the piezoelectric membrane. With the deformation of the piezoelectric membrane, the cavity comes into contact with the cold source, which reduces the part of the steam in the cavity which returns to its initial state. In order to guarantee the asymmetry of the displacement, the bottom membrane is smaller and thus more rigid. Therefore, the cycle described by P 3 heat engine is similar to the Rankine cycle but with a stationary fluid. Given that the vapor or the liquid phases are never saturated, Lurie and Kribus assumed that the thermal efficiency of the cycle is equal to the Carnot efficiency. Starting from this assumption, they developed a lumped thermal capacitance model they coupled with the equation of the piezoelectric transducer to evaluate the performances of the generators. With their model they estimated the possibility to produce up to 34W.cm -2 at a frequency of 1kHz with a cavity of 100µm of thickness and with a Carnot efficiency of 26,6%, and showing a linear increase of the performances when the dimensions are scaled down. [START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF]. c. Carnot cycle assumed in [START_REF] Lurie | Analysis of a microscale 'Saturation Phase-change Internal Carnot Engine[END_REF]. d. Cycle experimentally observed and reported in [START_REF] Bardaweel | Characterization of the thermodynamic cycle of a MEMS-based external combustion resonant engine[END_REF].

(a) (b) (a) (b)
Yet three arguments can qualify to these results. First, the assumption of Lurie that there is no performances degradations due to any heat losses linked to the passive architecture.

The second is on the fact that external temperatures are fixed whatever the scaling of the device. The last arguments comes from the comparison of this model with the experimental measurements made by Bardaweel et al, and reported in [Bardaweel et al, 2012] showing that the real thermodynamic cycle cannot be described by a Carnot engine since it looks like a loop (Fig. 5.2.1c). We can try to explain that by seeing that contrary to conventional Rankine engines, the various phases of the thermodynamic cycle are coupled: as a consequence, the compression and expansion steps are not adiabatic and cannot be represented by isentropic curves, since heat is supplied or rejected from the system at the time as mechanical energy is harvested by the piezoelectric transducer.

Similarly, phase transition steps are not isobaric because of the deformation of the piezoelectric membrane. As a consequence, the heat engine operation cannot be described by a Carnot cycle.

Measurements of the equivalent thermal resistance of the generator were performed by Cho and coworkers, showing that this resistance was very weak (around 1K.W -1 ) and without any strategy to actively evacuate heat from the generator, this last one rapidly thermalizes and stops working. Therefore they proposed to use liquid-metal microdropplet array either to supply heat to the device or to evacuate heat from it. In the first case of figure, the performances of the generators rapidly degrades if the thermal switch's frequency is increase because of the thermalization of the heat engine and the degradation of the efficiency due to the raise of the thermal switch's power consumption.

In the second case of figure, the improvement of heat evacuation linked to the raise of the thermal switch's operating frequency largely balances the raise of the switch's consumption. For a device working between 40 o C and 60 o C, mean power densities of around 1,4 mW.cm -2 are measured with an active heat supply at 5Hz (corresponding to a Carnot efficiency of 0,23%) and 10 mW.cm -2 with an active heat evacuation at 100 Hz (corresponding to a relative efficiency of 1,7%). The conclusions of that experiments first shows that Lurie and Kribus overestimated the efficiency of their devices and the thermal behavior of P 3 -heat engines (theoretical working frequency at 1kHz without active heat evacuation). This enables to conclude that despite a simple concept that can be directly scaled down to the microscale, stationary fluid-based heat engine are less efficient than thermoelectric generators and miniaturized Rankine generators. Typical performances of some fluid-based heat engines are then recalled in Table .5.

6 Solid-based heat engines

Observations on phase transitions

The two previous types of heat engines presented, pyroelectric generators and fluid-based heat engines, have shown the interest of exploiting phase transitions, that are non-linear thermodynamic phenomena, to transform heat into mechanical energy. In fluid-based heat engines, the phase transition that is exploited is the liquid-gas first-order phase transition whose order parameter is the density of the phase which strongly varies at the boiling temperature, enabling to produce large mechanical stress. In the case of pyroelectric heat engines, it is either the first-order phase transition of ferroelectric materials, from a ferroelectric state to a paraelectric state or from a ferroelectric state to another ferroelectric state, or the second-order phase transition form a ferroelectric state to a paraelectric state. In the first case, the order parameter that strongly evolves is the permanent polarization of ferroelectric materials at the Curie temperature (the polarization is a first partial derivative of the energy). In the second case, it is the strong variation variation of the pyroelectric coefficient (secondary partial derivative of the energy), and of the related electrocaloric effect near the Curie temperature that are exploited. The fact that most of the phase transitions occur at fixed temperatures and that transitions are characterized by a hysteretic behavior according to Landau's theory and Devonshire's theory [Uzunov, makes them perfectly suitable to the creation of heat engines.

Starting from this observation, a lot of phase transitions can be potentially exploited to transform heat into another energy. In the category of solid-based heat engines, we can class heat engines based on phase transitions associated to a mechanical order parameter (strain, stress displacement) even if some of them use pyroelectric effect to convert heat into electricity. In the category of thermo-magnetic heat engines, we will class thermal generators based on a phase transition associated to a magnetic order parameter (magnetization) even if a piezoelectric materials is used to finally convert heat into mechanical energy.

Shape-memory-alloy-based heat engines

Shape-memory-alloys are materials that are known to exhibit an hysteretic thermomechanical behavior associated to a first-order transition from a highly compliant phase (Martensitic phase) to a less compliant phase (Austenitic phase) resulting from the change of the crystal structure of the metallic alloy [START_REF] Mohd | A review of shape memory alloy research, applications and opportunities[END_REF]. As a consequence, when a pre-stressed martensite material is heated until becoming an austenite, it recovers a part of its initial strain by becoming stiffer. If it is cooled down to a temperature lower than the one of the martensite to austenite temperature, it comes back to its initial state. The hysteretic behavior of Shape-Memory-Alloys is represented in Fig. 6.2.1a. The term of Shape-Memory-Effect comes from the fact that between the transition temperatures, the shape of the alloy kept the memory of its last phase transition. Nitinol is one of the most used shape-memory alloy because of its important strain recovery which can reach 10% (corresponding to a mechanical energy density of 4J.cm -3 ) Yet, fatigue issues are often encountered with SMA. [START_REF] Kahn | The TiNi shape-memory alloy and its applications for MEMS[END_REF]] reports a decrease of 60% of the recovery strain after 10000 cycles (meaning that at a frequency of 0,01Hz, the performances degradation is observed after 11 days), and permanent elastic deformation of 2,5% after 800 cycles for TiNi wires submitted to a stress of 600 MPa. Contrary to other phase transitions that occur at a fixed temperature, Martensite phase transition is characterized by a progressive evolution from a phase to the other one. As a consequence, four temperatures are necessary to describe this thermodynamic cycle. From a thermodynamic point of view, this feature of martensite phase transition cannot be modeled with Landau's or Devonshire's theories but with Landau-Ginzburg theory [START_REF] Falk | Landau theory and martensitic phase transitions[END_REF].

As an hysteretic loop followed anti-clockwise is associated to the production of energy in thermodynamics, some works have addressed the issue of designing thermal heat engines exploiting Shape-Memory alloys [Nersessian, 2005;Namli, 2011;[START_REF] Zakharov | Thermal energy conversion by coupled shape memory and piezoelectric effects[END_REF]Saltzbrenner, 1984;[START_REF] Chang | Laminate composites with enhanced pyroelectric effects for energy harvesting[END_REF], or the issue of evaluating the theoretical efficiency of these heat engines [START_REF] Mohamed | Thermal conversion efficiency of an ideal thermoelastic marmem cycle[END_REF]Golestaneh, 1978;[START_REF] Liu | The work production of shape memory alloy[END_REF]]. Given the complexity of the description of the shape-memory alloys, evaluations of the performances strongly vary from a paper to anoter. In his paper, Liu evaluated the density of mechanical energy generated by Nitinol to 22 J.cm -3 with intrinsic efficiency of the conversion of 5,64%, quite in agreement with the data reported in [START_REF] Mohamed | Thermal conversion efficiency of an ideal thermoelastic marmem cycle[END_REF]] (27 J.cm -3 per cycle, thermal efficiency of 9,4%). In both case, the Carnot relative efficiency is between 40 and 50% assuming that the SMA is cycled between its austenic temperature Af and its martensitic temperature Mf. Yet none experimental measurements obtained such levels of efficiency. In [START_REF] Salzbrenner | Shape memory heat engines[END_REF], Saltzbrenner reported the performances [START_REF] Zakharov | Thermal energy conversion by coupled shape memory and piezoelectric effects[END_REF]. c. Bistable thermal switch presented in [START_REF] Ravindran | Performance improvement of a micro thermomechanical generator by incorporating Galinstan® micro droplet arrays[END_REF]. d. Thermodynamic cycle of the bimetallic strip heat engine.

of Nitinol realizing either a Stirling cycle or an Ericsson cycle. In the case of the Ericsson cycle, he obtained an energy density of 3,6 J.cm -3 and a Carnot efficiency of 3% by cycling a Nitinol sample between 25 o C and 190 o C. In order to finally produce electrical energy, Chang and Huang proposed to couple SMA materials with pyroelectric generators. In this way, by using sudden strain recovery of SMA, it would be possible to cycle a pyroelectric between two heat sources. In [START_REF] Chang | Laminate composites with enhanced pyroelectric effects for energy harvesting[END_REF] they thus simulated the behavior of a pyroelectric generator cycled between 27 and 37 o C and obtained energy density of around 1,2 mJ.cm -3 per cycle with a Carnot relative efficiency of 0,08%. Another proof of concept of SMA-based heat engine is reported in [START_REF] Zakharov | Thermal energy conversion by coupled shape memory and piezoelectric effects[END_REF] reporting energy densities of 3mJ.cm -3 by using the stress generated by a NiTiCu ribbon on a piezoelectric cantilever beam, and densities of 0,75mJ.cm -3 by laminating SMA with MFC pyroelectric material.

Energy harvesting from thermo-mechanical bistability

Other types of mechanical instabilities have been explored to realize thermal energy harvesters. In [Huesgen et al, 2010] and in [Ravindran et al, 2011], researchers from Imtek reported the fabrication and the operation of a heat engine based on the mechanical snap-through of an initially-curved membrane. A schematic of the heat engine is presented in Fig. 6.1.b. When the device is in contact with the hot source, thermal transfers occur between the heat source and the bistable actuator, causing the expansion of the fluid contained in the cavity delimited by the bistable membrane. As the fluid applies a pressure on the curved-membrane, this one can buckle if the pressure exceeds its crical load. Once it has snapped, the cavity comes in contract with the cold, source, discharging the heat of the fluid until the snap-back load is reached, causing the return of the cavity to its initial position. A similar device was already presented by Song and coworkers in [START_REF] Song | Simulation and experiments for a phase-change actuator with bistable membrane[END_REF] at the difference that the fluid in the cavity was biphasic. As we can see, the architecture and the operation of this thermal generator are quite similar to those of the P 3 heat engine, but the difference is that there is no phase transition of the fluid contained in the cavity. In [Ravindran et al, 2011], Ravindran et al demonstrated the use the bistable beam as a thermal switch, conveying heat for the hot source to a pyroelectric transducer located on the cold source. In that case of figure, the displacement of the bistable membrane enable to create temporal temperature variations inside the pyroelectric maerials. Their device was able to generate up to 1.33µW.cm -2 from a thermal gradient of 80 o C. In these experimental conditionsm the thermal efficiency is 1,8.10 -4 % and the working frequency of 7,04Hz, thanks to the use of liquid metal droplets to improve thermal transfer between the heat sources and the engine chamber.

Finally the other type of heat engine we can put in this category is the bimetallic strip heat engine. As we will demonstrate it in the next part, bimetallic strip heat engines are based on the first-order phase transition of a thermo-mechanically bistable membrane. The four different phases composing the thermodynamic cycle of the heat engine are recalled in Fig. 6.1d. With the experiments developed at STMicroelectronics and CEA-LETI only one paper [Ravindran et al, 2014] reports experimental measurements of the performances of the bimetallic strip heat engine. In this paper, they proposed a numerical model of the bistability of the bimetallic membrane, enabling to evaluate the efficiency of the thermomechanical transduction between 0,01% and 0,04%. In order to compare bimetallic strips with other thermal energy harvesters, we normalize the energy by their volume.

Thermomagnetic heat engines

Thermomagnetic coupling is the last physical effect we have not studied yet. The operation of thermo-magnetic generators using phase transitions of ferromagnetic materials is quite similar to the operation of pyroelectric generators using phase transitions of ferroelectric materials. The basic idea is to thermally cycle a ferroelectric materials near its phase transitions to observe abrupt variations of its magnetization. Most of the phase transitions exploited are first-order phase transitions, either from a ferromagnetic phase to a paramagnetic phase, or from a ferromagnetic phase toan antiferromagnetic phase [START_REF] Hsu | Thermomagnetic conversion efficiencies for ferromagnetic materials[END_REF]]. Yet Ujihara reported the exploitation of a second-order phase transition [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF]. The possibility to generate electrical energy from heat by means of thermomagnetic coupling is generally first attributed to Edison and Tesla in the late 19 th centure [Edison, 1887;Tesla, 1890] whereas the first evaluation of the efficiency of thermamagnetic generators is due to Brillouin and Iskenderias [Brillouin and Iskanderias, 1948] finding values around 55% of Carnot. The basic principle of the thermomagnetic heat engine is presented in Fig. 7.1.1a. It consists in placing a ferromagnetic material in the air gap of magnet so that, when the temperature of the ferromagnetic evolves around the Curie temperature of the ferromagnetic material, the variation of the ferromagnetic material's magnetization induces the creation of a current [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF].

in the solenoid surrounding the magnet. Since the work of Brilouin and Iskenderias, many authors evaluated the performance of thermomagnetic generators performing various thermodynamic cycles, including Rankine cycle [START_REF] Song | Thermodynamics of energy conversion via first order phase transformation in low hysteresis magnetic materials[END_REF] and Ericsson cycle [START_REF] Song | Thermodynamics of energy conversion via first order phase transformation in low hysteresis magnetic materials[END_REF], Solomon 1988[START_REF] Kirol | Numerical analysis of thermomagnetic generators[END_REF]. The main part of these generators uses Gadolinium as a ferromagnet given its low Curie temperature (15 o C). In [START_REF] Elliott | Thermomagnetic Generator[END_REF], Elliot demonstrated the possibility to produce up to 3,2 J.cm -3 per cycle with a Carnot efficiency of 7,4% from a temperature variations (23 o C of temperature difference varying at 0,17 Hz). Simulating the performances of Gadolinium used with Ericsson cycle, Kirol and Mills reported lower electrical energy density (71mJ.cm -3 ) but higher Carnot efficiencies (47%) [START_REF] Kirol | Numerical analysis of thermomagnetic generators[END_REF].

Other works report the generation of electric energy from heat flowing in thermal gradients. As an example, Solomon reports the simulated performances of a rotary thermomagnetic heat engine, made of a rotor and a stator, the heat being supplied from outside of the engine, and evacuated from inside the engine. In a thermal gradient of 65 o C, he reported thermo-mechanical efficiencies of around 60% of Carnot and mechanical energy densities of 2,2mJ.cm -3 . The last possibility to harvest heat and to convert it into electrical energy is to use the generator whose architecture is presented in Fig. 7.1.1b. In this configuration, the ferromagnetic element is initially in a paramagnetic state and in contact with the cold source. As a consequence it is cooled down to a temperature lower than its curie temperature for which it becomes ferromagnetic. As a consequence, it is attracted by the permanent magnet generating a force that is higher than the restoring force of the springs. As it comes in contact with the hot source, its temperature increases, and exceeds the Curie temperature and cancelling the magnetic force. The ferromagnetic materials then comes back to its initial state. Modeling of this type of device was performed by [Hsu et al, 2013] and [START_REF] Post | Thermomagnetic energy harvesting with first order phase change materials[END_REF] whereas the first proof of concept is reported in [START_REF] Ujihara | Thermal energy harvesting device using ferromagnetic materials[END_REF] using piezoelectric membrane to realize the springs and piezoelectric transducers to convert the mechanical energy into electrical energy. In this paper, the authors claim the possibility to produce 1,57 mW.cm -2 by using piezoelectric membrane and up to 2,6mW.cm -2 by using piezoelectric stacks. They did not give any values for the thermo-mechanic transduction. In order to find complete modeling of these devices, we can report to the works of Hsu et al, or Post et al [Hsu et al, 2011;[START_REF] Post | Thermomagnetic energy harvesting with first order phase change materials[END_REF]. In the first of these papers, the authors indicate the possibility to reach Carnot efficiencies of 11,5% for a thermal gradient of 20 o C if energy is directly converted into magnetic energy. In [START_REF] Post | Thermomagnetic energy harvesting with first order phase change materials[END_REF] is reported the dynamic modeling of the same device than in [START_REF] Hsu | Thermomagnetic conversion efficiencies for ferromagnetic materials[END_REF], showing that a thermomagnetic generator made of Gadolinium can work at 0,3Hz and can produce up to 0,15mW.cm -2 with an efficiency of 0,65% of Carnot. By using a new category of materials called Heusler alloys whose excellent magneto-caloric properties were demonstrated in [START_REF] Srivastava | The Direct Conversion of Heat to Electricity Using Multiferroic Alloys[END_REF] and [START_REF] Song | Thermodynamics of energy conversion via first order phase transformation in low hysteresis magnetic materials[END_REF].

Synthesis

In this last part, we want to try to establish a comparison between every type of thermal energy harvester. In order to do so, for each harvester, we only retain the data concerning the density of energy, the thermal gradient used to produce this energy, the equivalent ZT of the thermal generator and the mean temperature at which the simulations or the experimental measurements are done. We do not take into account surface density of power since they depend on the working frequency of the device. Given that all technologies have not been developed, some theoretical works either choose an arbitrary value or present simulations of thermal transfers, whereas some experimental works boost the frequency by imposing a fixed thermal gradient, or slowly cycle by alternating cold baths and hot baths. Some others, like the piezoelectric bimetallic strip heat engines directly work without any heat sink. In our comparison, we compare directly the data of energy, even if some of them are of mechanical nature, whereas others are of electrical nature. A good solution would be to choose a typical value of piezoelectric conversion efficiency when there could be a deformation of a membrane (example of P 3 heat engine) to convert mechanical energy into electrical energy, or the efficiency of an electrostatic conversion when there is a rotatory motion, as for Rankine engines. As there is no direct notion of energy density for thermoelectric generator, we will take as a reference the electrical energy generated by a Micropelt generator during 1s when it works with a temperature difference of 5 o C. Given that the surface of the generator is 2,4mm x 3,3 mm x 1,1 mm (however we can argue that the volume to be taken into account is the volume of thermoelectric material since the other works used this volume to evaluate the volume density of generated energy), we find an energy density of around 30mJ.cm -3 at 5 o C. This enable us to evaluate how the working frequency of cyclic heat engines must evolve in order that it produces as much power as a Seebeck generator. Fig. 8.1.1a. compares the performances of the heat engines with respect to their efficiencies(equivalent ZT) and their working temperatures. As we can notice, most of the thermal generators have a ZT lower than 1, except for SMA-based heat engines (15 and 16), two thermomagnetic generators (29 and 31) and two fluid-based heat engines (14 and 10). Yet we can see that the corresponding works are simulation works. As we explained, 14 made a mistake on the thermodynamic cyle of P 3 heat engines and (31 and 29) correspond to first historical evalutions of thermomagnetic heat engines.

Except for these cases, we can observe that most of the heat engines have a ZT between 0.01 and 1. As the thermal gradient is the driving force associated to heat, we represent in Fig. 8.1.1b the density of mechanical energy as a function of the thermal hysteresis needed to produce this energy. In this figure we also plot the curve corresponding to the energy produced by a thermoelectric generator during a second, 10 seconds, 100 seconds... With these curves we can directly see at which frequency a thermal generator must be cycled in order to produce the same volume density of power than a thermoelectric generator. As an example, for the same volume of energy harvester and for the same temperature difference, a pyroelectric material needs only to be cycled at 0,1Hz to reach the same level of energy density than a thermoelectric generator. However as we observed before, it can be difficult to find practical situations where temperature can vary of 30 degres in 5 seconds. The working frequencies of Rankine engines being greater than 1kHz, these generators can reach output power densities 1000 times higher than thermoelectric generators. Except for these cases, we can observe that most of the heat engines have a ZT between 0.01 and 1. As the thermal gradient is the driving force associated to heat, we represent in Fig. 8.1.1b the density of mechanical energy as a function of the thermal hysteresis needed to produce this energy. In this figure we also plot the curve corresponding to the energy produced by a thermoelectric generator during a second, 10 seconds, 100 seconds... With these curves we can directly see at which frequency a thermal generator must be cycled in order to produce the same volume density of power than a thermoelectric generator. As an example, for the same volume of energy harvester and for the same temperature difference, a pyroelectric material needs only to be cycled at 0,1Hz to reach the same level of energy density than a thermoelectric generator. However as we observed before, it can be difficult to find practical situations where temperature can vary of 30 degres in 5 seconds. The working frequencies of Rankine engines being greater than 1kHz, these generators can reach output power densities 1000 times higher than thermoelectric generators.

From these two graphs we can observe some tendencies common to each categories of heat engines. First, Rankine engines and other similar heat engines using circulating working fluids are perfectly suited for high temperatures applications like microcombustion given their high power densities. Secondly, pyroelectric heat engines and thermomagnetic heat engines can be as efficient as thermoelectric generators. Finally, given the weak thermomechanical coupling, solid based heat engines (except Shape-Memory alloys) and stationary fluids heat engines are the less efficient heat engines especially if we take into account the needs to couple these heat engines with electro-mechanical transducers.

Conclusion

In this first chapter, we present an overview of various thermal coupling phenomenas that can be exploited to harvest heat and the various technologies exploiting these phenomena. We detailed the theoretical operation of thermoelectric generators, pyroelectric generators, thermomagnetic generators and heat engines using either working fluids or working solids. The comparison of these technologies showed that thermoelectric generators are the most efficient technology to harvest heat, followed by pyroelectric generators and thermomagnetic generators. Fluid-based heat engines can be particularly efficient to convert heat into mechanical energy at high temperatures, whereas stationary fluid-based heat engines and solid-based heat engines are the least efficient technologies, about one hundred times less efficient than conventional thermoelectric generators, for the conversion of heat into mechanical energy. The final efficiency of these generators would also be decreased because of the necessity of using a second conversion step to transform this mechanical energy into electrical energy. This first study on the state of the art of heat engines raises a first question on the interest of developing bimetallic strip heat engines to commercially compete with thermoelectric generators. Yet, looking for modeling this kind of devices and developing thermo-mechanically bistable structure stays interesting from a scientific point of view, especially if other types of applications than thermal energy harvesting can be found for miniaturized bimetallic strips. Zhang, Q., Agbossou, A., Feng, Z. and Cosnier, M. (2011). Solar micro-energy harvesting with pyroelectric effect and wind flow. Sensors and Actuators A: Physical, 168(2), pp.335-342.

Chapter 2

Thermo-mechanical modeling of the bimetallic strip heat engine 1 Introduction

Objectives of the chapter

As bimetallic strip heat engines' operation relies on the unstable behavior of bimetallic strips, the heart of the modeling of our devices is linked to the comprehension of thermal snap-through and of the energetical properties of beams and structures exhibiting thermally hysteretic behavior. As a consequence, this second chapter aims at providing a simple model thermo-mechanical instability of some geometries of composite membranes, among which the bimetallic beams. Given that the description of thermo-mechanical instabilities is not an isolated case in mechanics, we will first present an overview of some examples of mechanical instabilities of structures, either caused by thermal stress of external actions, and we will consider the effects of imperfection on the behavior of these structures. After that, we will establish and solve the accurate equilibrium equations and the reduced-order equations of laminated beams suffering both from thermal stress and from residual stress, enabling us to find the conditions under which thermal snap-through could occur. Our model will enable us to extend the demonstration to the case of initially curved laminated beams and to simplify the various results to the simplest laminated beams, which are the bimetallic beams. We will then focus on the evolution of the energy and the entropy of bistable beams to demonstrate the possibility to convert heat into mechanical energy and to evaluate the efficiency of this transduction. As the dynamic behavior of the energy harvester can only be deducted from the study of thermal transfers inside the harvester, we finally will then present a simple model of switched thermal capacitance enabling to express the power generated by the heat engine and to find the optimal device's design enabling to maximize the performances of the generator.

Researches on thermally-induced instabilities of structures

The research on these thermo-mechanical instabilities is at the crossroads between two domains of mechanics, treating in the first place the consequence of thermal stress on the mechanical response of solid materials, but also tackling the issue of mechanical instabilities, broadly described by non-linear mechanics [START_REF] Thompson | A general theory of elastic stability[END_REF]. Thermal stress is often associated with reliability issues, and the effect of temperature variations on the mechanical behavior of engineered structures has often been studied to find solutions to avoid their thermally-induced failures. As explained by Thornton in [START_REF] Thornton | Thermal Buckling of Plates and Shells[END_REF] this field of research has benefited from the emergence of aeronautics [START_REF] Schuman | Buckling of Laminated Composite Rectangular Plates Under Transient Thermal Loading[END_REF] and the development of supersonic flights: The rise of the flight speeds of aeroplanes emphasized the need to address the issues of aerodynamic heating and their consequences such as thermo-mechanical flexure or buckling of structures. It also led to the development of new materials such as Functionally Graded Materials (FGM) to prevent such phenomena [Ravichandran, 1995, Niino and[START_REF] Niino | Recent development status of functionally gradient materials[END_REF].

Before giving an overview on the research about thermo-mechanical stability issues, we need to explain what buckling is. Early modeled in 1744 by Euler in the case of perfectly straight beams (Fig. 1.1.1a) [Euler, 1744, Timoshenko and[START_REF] Timoshenko | Theory of elastic stability[END_REF], buckling describes the loss of stability of a structure submitted to a compressive in-plane load N, causing the bifurcation of the beam from an unstable equilibrium position to a stable one, which is accompanied by out-of-plane displacements. Euler demonstrated that the action of the axial load produces a strain in the straight beam (thickness t, length L) given by (1.1.1a), and that the equilibrium equation of the beam is described by equation (1.1.1b). In these equations, E is the Young modulus of the beam, A its cross-section, I its inertial moment and w its shape.

𝑁𝑁𝑁𝑁 = 𝐸𝐸𝐸𝐸. 𝐴𝐴𝐴𝐴. 𝜀𝜀𝜀𝜀

(1.1.1a) 𝐸𝐸𝐸𝐸. 𝐼𝐼𝐼𝐼. 𝑤𝑤𝑤𝑤 ,!! + 𝑁𝑁𝑁𝑁. 𝑤𝑤𝑤𝑤 = 0 (1.1.1b) With this equation, Euler showed that the simply-supported beam remains straight until the load N reaches the value, known as the first Euler load (1.1.2a), corresponding to the eigenvalue of (1.1.1). When the load p equals the Euler load, the beam can be curved, taking the shape of a half-sine wave (1.1.2b).

𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥 = 0 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑁𝑁𝑁𝑁 ≤ 𝐸𝐸𝐸𝐸. 𝐼𝐼𝐼𝐼.

𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! = 𝑁𝑁𝑁𝑁 !"#$%

(1.1.2a)

𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥 = 𝑤𝑤𝑤𝑤 ! . sin 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . 𝑥𝑥𝑥𝑥

( 1.1.2b) Other eigenvalues can be found, each time the compressive load equals a multiple of the first Euler buckling load, the odd multiples describing symmetrical beam shapes and the even multiples corresponding to asymmetrical beam shapes (1.1.3b).

𝑤𝑤𝑤𝑤(𝑥𝑥𝑥𝑥) = 𝑤𝑤𝑤𝑤 ! . sin 𝑛𝑛𝑛𝑛. 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . 𝑥𝑥𝑥𝑥

(1.1.3a)

𝑁𝑁𝑁𝑁 ! = 𝐸𝐸𝐸𝐸. 𝐼𝐼𝐼𝐼. 𝑛𝑛𝑛𝑛 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! (1.1.3b)
The difficulty arises when it is necessary to describe precisely the shape of the beam after it buckles. Equation (1.1.1) does not allow finding the amplitude of the beam's deflection.

It is then necessary to resort to non-linear mechanics to find the post-buckling behavior of the beam. As demonstrated in (Marguerre, 1967) when the beam buckles, the compressive axial load modifies the deflection of the beam, according to (1.1.4). The non-linear integral term then corresponds to the difference between the curved beam's length and its chord.

𝑁𝑁𝑁𝑁 = 𝐸𝐸𝐸𝐸. 𝐴𝐴𝐴𝐴 𝜀𝜀𝜀𝜀 - 1 2. 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ,! ! ! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (1.1.4)
The amplitude of the deflection after buckling is then a solution of the polynomial equation (1.1.5). Fig.1.1.1b and Fig.1.1.1c plot the evolution of the axial load p and the displacement of the beam's middle as a function of the normalized strain of the beam. It shows that a perfectly straight beam can buckle in the two opposite directions with the same probability, and that the straight position is unstable after the bifurcation point. The shape of the equilibrium diagram of the perfectly straight beam recalls the shape of a pitchfork, hence the name of pitchfork bifurcation.

𝑤𝑤𝑤𝑤 ! ! = 4. 𝐿𝐿𝐿𝐿 ! 𝜋𝜋𝜋𝜋 ! . 𝜀𝜀𝜀𝜀 - 𝑁𝑁𝑁𝑁 !"#$% 𝐸𝐸𝐸𝐸. 𝐴𝐴𝐴𝐴 (1.1.5)
The previous equations can be modified to predict the thermal buckling of a simply supported and axially-constrained beam. When temperature rises, it normally entails the thermal expansion of the beam, linked to its coefficient of thermal expansion α. Given that the beam is axially restrained, a thrust is created (1.1.6a). The thermal stress can then lead to the buckling of the beam if it reaches the Euler load. The temperature associated to the buckling of the beam is then (1.1.6b).

𝑁𝑁𝑁𝑁 ! = 𝐸𝐸𝐸𝐸. 𝐴𝐴𝐴𝐴. 𝛼𝛼𝛼𝛼. 𝑇𝑇𝑇𝑇 As beams represent a restrained class of mechanical structure, several researches dealt with the behavior of plates and shells. Bryan [START_REF] Bryan | On the Stability of a Plane Plate under Thrusts in its own Plane, with Applications to the "Buckling" of the Sides of a Ship[END_REF] first established the equation of buckling of an uniaxially compressed rectangular plate having its four edges simply supported, showing that it obeys equation (1.1.7a). The solution of the buckling equation is then given by (1.1.7b), and the critical charge is a function of the length a and the width b (1.1.7c) [START_REF] Reddy | Theory and analysis of elastic plates and shells[END_REF]. In these equations, D is the flexural stiffness of the beam and t its thickness. The lowest values of the critical load being obtained for n=1, it is common practive to plot the evolution of the buckling coefficient k as a function of the ratio a/b (Fig. 1.1.2b). This figure shows that the longer and narrower the beam is, the more k tends towards a constant value (k->4) but the higher the buckling mode is. For ratios a/b lower than unity, the increase of the plate's stiffness makes the buckling harder, explaining why the buckling coefficient increases.

𝐷𝐷𝐷𝐷. 𝛻𝛻𝛻𝛻

(1.1.9b)

As with Euler's equation in the case of beams, (1.1.8) does not describe the postbuckling behavior of plates. By using the Von Karman stress tensor, the Airy stress function F obeys the equations (1.1.10a-b).

𝛻𝛻𝛻𝛻 ! 𝐹𝐹𝐹𝐹 = 𝐸𝐸𝐸𝐸. 𝑡𝑡𝑡𝑡. (𝑤𝑤𝑤𝑤 ,!" ! -𝑤𝑤𝑤𝑤 ,!! . 𝑤𝑤𝑤𝑤 ,!! )

(1.1.10a)

𝐷𝐷𝐷𝐷. 𝛻𝛻𝛻𝛻 ! 𝑤𝑤𝑤𝑤 = 𝐹𝐹𝐹𝐹 ,!! . 𝑤𝑤𝑤𝑤 ,!! -2. 𝐹𝐹𝐹𝐹 !" . 𝑤𝑤𝑤𝑤 ,!" + 𝐹𝐹𝐹𝐹 ,!! . 𝑤𝑤𝑤𝑤 ,!!

(1.1.10b) [START_REF] Tauchert | Thermally Induced Flexure, Buckling, and Vibration of Plates[END_REF][START_REF] Thornton | Thermal Buckling of Plates and Shells[END_REF] give extensive bibliographies on the research works on the resolution of these equations for various types of plates and boundary conditions, using either numerical approaches or analytical simplifications like Ritz or Galerkin methods. Other methods based on the writing of the energy of the plate have been used to study the equilibrium of plates, such as Berger's method [START_REF] Berger | A new approach to the analysis of large deflections of plates[END_REF] or Banerjee's method [Bannerjee and Data, 1981].

Buckling of imperfect beams

In the previous part, we presented the behavior of perfectly straight beams and plates made of a single material. We explained that these structures follow a pitchfork bifurcation and that their buckling loads are associated to a single value of the temperature given by (1.1.6b) and (1.1.9a). However, the behaviors of composite or initially curved structures, or structures with heat conduction across their thickness, are different. The mechanical response of these structures to an external stress can be greatly affected by these properties, and their behavior tend to diverge from the behavior of perfectly straight and homogeneous structures. This explains why these properties are called imperfections. To take a simple example, in the case of an initially curved beam, .2.1b shows that the beam thus curves in the direction fixed by its initial imperfection before reaching the Euler load. It is particularly interesting to note the existence of an assymetric secondary equilibrium branch for high values of p. This branch shows that the beam can pass from one side to the other by a saddle-node bifurcation when it reaches the values represented by the gray dashed-line. The equilibrium positions located between the two gray dashed lines are unstable. These same curves can be obtained when studying the thermal buckling of imperfect axially-constrained isotropic beams, by noticing that the end-shortening can be replaced by thermal expansion α.T.

The description of the behavior of inhomogeneous structures under the effect of spatially homogeneous temperature variations, or isotropic structures with variations of the temperature through their thickness is quite similar. Given that the thermal expansion is not the same across the structure's thickness, in addition to a load N T , a thermal moment M T appears which behaves like an imperfection. For an axially constrained and simply supported beam, Euler's equation becomes (1.2.2a), whereas in the frame of the Van-Karman theory, the equilibrium equations of a plate become (1.2.2b-c) [START_REF] Boley | Theory of Thermal Stresses[END_REF]. We can directly observe in (1.2.2a) that the thermal stress moment plays directly the same part as the geometrical imperfection in (1.2.1a). This explains why the equilibrium paths described by (1.2.2) are similar to those plotted in Fig. 1.2.1b-d. Without seeking to be exhaustive, we can find various studies on the thermal buckling of composite beams. [Vosoughi et al, 2012], and [Vaz et al, 2010] explained the effect of the dependence of the mechanical properties on the buckling temperature of composite beams.

[Samsam [START_REF] Samsam Shariat | Thermal buckling of imperfect functionally graded plates[END_REF], [START_REF] Kiani | Thermal buckling analysis of functionally graded material beams[END_REF] and [START_REF] Schuman | Buckling of Laminated Composite Rectangular Plates Under Transient Thermal Loading[END_REF] also studied the evolution of the buckling temperature of FGM plates for various thermal profiles. Here again, [START_REF] Tauchert | Thermally Induced Flexure, Buckling, and Vibration of Plates[END_REF] gives a large overview on the research on the subject.

Thermal snap-through

Snap-through (or oil-canning) is a form of buckling that is a characterized by a sudden jump of a structure from an equilibrium position to another. Snap-trough can be observed when a load is applied on a curved structure or when two imperfections have antagonistic effects on the behavior of a structure. A classical problem to introduce the snap-through phenomenon is the behavior of the Mises truss [START_REF] Mises | Über die Stabilitätsprobleme der Elastizitätstheorie[END_REF], an arch made of two linear springs k, and initially curved (angle θo), submitted to the effect of a load Q. As demonstrated in [START_REF] Thompson | A general theory of elastic stability[END_REF], the potential energy of such a system can be approximated by a fourth-order polynomial (1.3.1a). In the same time, the potential energy of the load can be approximated by (1.3.1b) and the arch's deflection a by (1.3.1c).

Given that the equilibrium positions are the extrema of the total energy, the equilibrium equation is given by (1.3.1d).

𝒲𝒲𝒲𝒲 ! = 1 16 . 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 ! . (𝜃𝜃𝜃𝜃 ! -2. 𝜃𝜃𝜃𝜃 ! ! . 𝜃𝜃𝜃𝜃 ! + 𝜃𝜃𝜃𝜃 ! ! ) (1.3.1a) 𝒲𝒲𝒲𝒲 𝑄𝑄𝑄𝑄 = 𝑄𝑄𝑄𝑄. 𝐿𝐿𝐿𝐿 2 . (𝜃𝜃𝜃𝜃 -𝜃𝜃𝜃𝜃 ! ) (1.3.1b) 𝑎𝑎𝑎𝑎 = 𝐿𝐿𝐿𝐿 2 . (𝜃𝜃𝜃𝜃 -𝜃𝜃𝜃𝜃 ! ) (1.3.1c) 𝑄𝑄𝑄𝑄 = -𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 . 𝜃𝜃𝜃𝜃. (𝜃𝜃𝜃𝜃 ! -𝜃𝜃𝜃𝜃 ! ! ) (1.3.1d)
Fig. 1.3.1b plots the evolution of Q as a function of θ. Given that the equilibrium position is a third-order polynomial, three positions can coexists if the initial curvature is positive (θo>0). This condition corresponds to the criterion of bistability of the arch. To know which positions are allowed, it is necessary to derivate the equilibrium equation to study the stability of the equilibrium (1.3.2a). This equation shows that two equilibrium positions are critical (1.3.2b); the positions are called limit points. For these values of Q, the beam snaps from one position to the other.

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 . (3. 𝜃𝜃𝜃𝜃 ! -𝜃𝜃𝜃𝜃 ! ! ) (1.3.2a) 𝜃𝜃𝜃𝜃 = ± 𝜃𝜃𝜃𝜃 ! 3 (1.3.2b)
Snap-through can also be a consequence of the thermal stress acting on curved beam. The first explanation of this type of thermal snap-through is due to Timoshenko who modeled the instability of bimetallic thermostats in [START_REF] Timoshenko | Analysis of bi-metal thermostat[END_REF]. In this article, he demonstrated that the curvature caused by the difference of thermal expansion of the two layers of a bimetallic beam produces a curvature ρ verifying (1.3.3a). Under this assumption, the axial thrust p due to the reduction of the curvature of a bimetallic beam having an initial deflection wo is given by (1.3.3d).

𝑡𝑡𝑡𝑡 𝜌𝜌𝜌𝜌 = 3 2 . 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! . 𝑇𝑇𝑇𝑇 (1.3.3a) 𝑤𝑤𝑤𝑤 ! 𝑥𝑥𝑥𝑥 = 𝑎𝑎𝑎𝑎 ! . sin 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 (1.3.3b) 𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥 = 𝑎𝑎𝑎𝑎 ! -𝑎𝑎𝑎𝑎(𝑇𝑇𝑇𝑇) 1 -𝑁𝑁𝑁𝑁/𝑁𝑁𝑁𝑁 !"#$%
. sin 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 .

(1.3.3c)

1 - 𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 !"#$% ! . 1 - 4. 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎 ! ! . 𝐴𝐴𝐴𝐴 . 𝑁𝑁𝑁𝑁 𝑁𝑁𝑁𝑁 !"#$% = 1 - 𝑎𝑎𝑎𝑎 𝑇𝑇𝑇𝑇 𝑎𝑎𝑎𝑎 ! (1.3.3d)
Hence he found that the bimetallic beam can be bistable if its initial deflection is superior to its thickness according to (1.3.4a). In that case, the temperatures at which the beam snaps and snaps back are given by (1.3.4b-c) and plotted in Fig. 1.3.1d.

-4. 𝐼𝐼𝐼𝐼 𝑎𝑎𝑎𝑎

! ! . 𝐴𝐴𝐴𝐴 > 0 ⟺ 𝑎𝑎𝑎𝑎 ! > 𝑡𝑡𝑡𝑡 3 (1.3.4a) 𝑇𝑇𝑇𝑇 ! = 16 3 . 𝑎𝑎𝑎𝑎 ! . 𝑡𝑡𝑡𝑡 𝐿𝐿𝐿𝐿 ! . 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! 1 + 2. 3𝑎𝑎𝑎𝑎 ! ! 𝑡𝑡𝑡𝑡 ! . 1 - 𝑡𝑡𝑡𝑡 ! 3. 𝑎𝑎𝑎𝑎 ! ! ! (1.3.4b) 𝑇𝑇𝑇𝑇 !" = 16 3 . 𝑎𝑎𝑎𝑎 ! . 𝑡𝑡𝑡𝑡 𝐿𝐿𝐿𝐿 ! . 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! 1 - 2. 3𝑎𝑎𝑎𝑎 ! ! 𝑡𝑡𝑡𝑡 ! . 1 - 𝑡𝑡𝑡𝑡 ! 3. 𝑎𝑎𝑎𝑎 ! ! ! (1.3.4c)
The strength of Timoshenko's model is to give the conditions of occurrence of the bistability and the temperatures associated to the instability of the beam. However, a closer examination of these equations shows that the mean thermal expansion of the beam is not taken into account, which can be a problem when the difference of thermal expansion between the two materials constituting the bimetallic beam is weak. In that case, the effect of the mean thermal expansion of the beam cannot be neglected and Timoshenko's model loses its validity. [START_REF] Burgreen | Higher mode buckling of bimetallic beam[END_REF]) solved this problem and showed that in certain conditions, the approximation of a first order buckling mode was false and that the beam can snap with a third-order buckling mode, but at very high temperature. [START_REF] Rutgerson | Thermo-elastic buckling of layered shell segments[END_REF], [Karlson and Bottega, 2000] also studied extreme behaviors of bimetallic beams. [START_REF] Orthwein | Thermal Buckling of a Bimetallic Strip[END_REF] also studied the thermal snapthrough of bimetallic beams from a manufacturing point of view and showed the difficulty to obtain weak variabilitise around the limit of stability. [Michael and Kwok, 2006] and [Cabal and Ross, 2002] studied the thermal instability of bimorph beams at the microscale. In a similar approach, [Moghadassie and Stanciulescu, 2013] studied the snap-through of shallow arches under the combined effects of a static load and thermal stress.

The phenomenon of thermal snap-through has also been studied for other structures than beams. Wittrick studied the instability of free bimetallic shells and found the conditions on the initial curvature to observe thermal snap-through [Wittrick, 1951]. This work was extended by Aggarwala and Saibel to the case of simply-supported spherical shells [START_REF] Bibliography Aggarwala | Thermal stability of bimetallic shallow spherical shells[END_REF] and by Kosel and Battista in the case of free open spherical shellss, conical shells and cubic shells [Kosel and Battista, 2007]. The thermal snapthrough of plates has also been studied: [START_REF] Irschik | Large thermoelastic deflections and stability of simply supported polygonal panels[END_REF]] used Berger's approximation of the strain energy of a simply-supported plate to demonstrate the snap-through of initially curved plates, and [START_REF] Heuer | Thermoelastic stability of layered shallow shells[END_REF]] used it to model the thermal snap-buckling (snap-through associated to a second order buckling mode) of composite plates

Plan of the chapter

The continuation of this chapter is divided into six parts. The first part presents the first model we developed to explain the instability of composite beams. Starting from a basic recall on mechanics, it presents the demonstration of the non-linear Euler equation given in (1.1.4), and solves it in the case of a simply supported beam. This enables to find the criterion of bistability of composite beams, and to compare it with the criterion given by [START_REF] Timoshenko | Analysis of bi-metal thermostat[END_REF]. The second part is devoted to the study of the energetic properties of bistable beams, to the demonstration of the phase transition associated to thermal snapthrough, and to the evaluation of the performances of bimetallic strips as a prime mover of our heat engine. The third part presents a simplified model of thermo-mechanical bistability with the final objective of providing analytical expressions of the performances of the heat engine. Because it is necessary to understand the role of thermal transfers in the occurrence of bistability, and their impact on the performances of the harvester, the fourth part of this chapter treats the modeling of these thermal transfers, but also the way they modify the dynamic behavior of bimetallic strips.

A first model of the instability of composite beams

A short course on the theory of elasticity

The goal of mechanics of the elastic body is to predict how a system deforms and moves under the action of external forces. In order to link causes (external forces fext, heat fluxes qext) to final effects (displacement fields u, v and w), four different kinds of equations are needed: thermodynamic, constitutive, kinematic, and kinetic equations. Thermodynamics explains how a system exchanges energy with its surroundings, and is useful to determine equilibrium positions of a system by using the principle of virtual displacement. The first principle (2.1.1a) states that the variations of the internal energy 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 and the kinetic energy 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 of a system are equal to the work of external forces dWext and the external flux dQext. The second principle (2.1.1b) describes the evolution of systems and will be useful to model thermal transfers inside the harvesters.

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝛿𝛿𝛿𝛿𝑄𝑄𝑄𝑄 !"# + 𝛿𝛿𝛿𝛿𝒲𝒲𝒲𝒲 !"# (2.1.1a) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝛿𝛿𝛿𝛿𝑄𝑄𝑄𝑄 !"# 𝜃𝜃𝜃𝜃 !"# ≥ 0 (2.1.1b)
The expression of the internal energy as a function of the extensive parameters of the systems, or the expression of one of its Legendre transformations enables to find the constitutive equations of a material, which link the body's extensive parameters to its intensive parameters by means of physical properties of the body. Particularly, by means of the Gibbs energy of the body ℊ (2.1.2a), the strain tensor and the entropy can be approximated as functions of the body's temperature and the stress tensor. These last two equations are the constitutive equations of the body. In the case of a lowstressed material, at constant temperature, the relation between the stress and the strain tensor is linear (Hooke's law). In that case, the variation symbol can be suppressed in (2.1.2c). Similarly, at constant stress, the thermal expansion can be linearized. For orthotropic materials, each coefficient of the strain tensor (see Fig. 2.1.1a) can be written as functions of the body's physical parameters E (Young's modulus), ν (Poisson's ratio) and α (coefficient of thermal expansion). (2.1.4e)

𝒮𝒮𝒮𝒮 !! = 1 𝐸𝐸𝐸𝐸 . 𝒯𝒯𝒯𝒯 !! -𝜈𝜈𝜈𝜈. 𝒯𝒯𝒯𝒯 !! + 𝒯𝒯𝒯𝒯 !! + 𝛼𝛼𝛼𝛼. 𝑇𝑇𝑇𝑇 (
𝒮𝒮𝒮𝒮 !" = 𝒮𝒮𝒮𝒮 "! = 1 + 𝜈𝜈𝜈𝜈 𝐸𝐸𝐸𝐸 . 𝒯𝒯𝒯𝒯 !" (2.1.4f)
As a matter of convenience, the Voigt-Kelvin notation is often used (2.1.5) [START_REF] Reddy | Theory and analysis of elastic plates and shells[END_REF].

𝒮𝒮𝒮𝒮 !! = 𝒮𝒮𝒮𝒮 ! ; 𝒮𝒮𝒮𝒮 !! = 𝒮𝒮𝒮𝒮 ! ; 𝒮𝒮𝒮𝒮 !! = 𝒮𝒮𝒮𝒮 ! ; 𝒮𝒮𝒮𝒮 !" = 𝒮𝒮𝒮𝒮 ! ; 𝒮𝒮𝒮𝒮 !" = 𝒮𝒮𝒮𝒮 ! ; 𝒮𝒮𝒮𝒮 !" = 𝒮𝒮𝒮𝒮 ! (2.1.5)
With the thermodynamic principles and constitutive equations, two final equations are needed to describe the behavior of the elastic body. The kinetic equation directly links the stress tensor's components to the body force N and to its inertia (2.1.6a.).

𝒮𝒮𝒮𝒮 !",! + 𝒮𝒮𝒮𝒮 !!,! + 𝒮𝒮𝒮𝒮 !",! + 𝜌𝜌𝜌𝜌. 𝑓𝑓𝑓𝑓 ! = 𝜌𝜌𝜌𝜌. 𝜕𝜕𝜕𝜕 ! 𝑣𝑣𝑣𝑣 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 !

(2.1.6c) 𝒮𝒮𝒮𝒮 !",! + 𝒮𝒮𝒮𝒮 !",! + 𝒮𝒮𝒮𝒮 !!,! + 𝜌𝜌𝜌𝜌. 𝑓𝑓𝑓𝑓 ! = 𝜌𝜌𝜌𝜌. 𝜕𝜕𝜕𝜕 ! 𝑤𝑤𝑤𝑤 𝜕𝜕𝜕𝜕𝑡𝑡𝑡𝑡 !

(2.1.6d)

The last relation to be considered is on the kinematics of the elastic body. Once the components of the strain tensor have been determined by means of the kinetic equations and constitutive laws, it is necessary to find how the strain is related to the displacement of the elastic body. Based upon geometrical considerations (see Fig. 2.1.1b), it can be demonstrated [START_REF] Reddy | Theory and analysis of elastic plates and shells[END_REF] that the strain tensor's components obey the Green-Lagrange equation (2.1.7a). When deformations are small, the quadratic terms of the Green-Lagrange strain tensor are neglected (2.1.7b). However, to study the behavior of unstable membranes, this term will play a crucial role.

𝒮𝒮𝒮𝒮 !" = 1 2 . 𝑢𝑢𝑢𝑢 !,! + 𝑢𝑢𝑢𝑢 !,! + 𝑢𝑢𝑢𝑢 !,! . 𝑢𝑢𝑢𝑢 !,! (2.1.7a) 𝒮𝒮𝒮𝒮 !" ≈ 1 2 . 𝑢𝑢𝑢𝑢 !,! + 𝑢𝑢𝑢𝑢 !,! (2.1.7b)
Once the various equations of mechanics have been given, it is necessary to see which assumptions can be made to simplify them in order to explain the behavior of thermomechanically bistable beams.

Equilibrium equations of composite beams suffering from residual stress

In this second part, we seek to study the static behavior of the composite beams submitted to the effect of the residual stress and to homogeneous variations of the temperature. Fig. 2.2.1a represents the composite beam of dimensions (L, v, t) Each material layer of the beam being stressed by a residual stress σo measured at the reference temperature θo (expressed in kelvin degrees), two other parameters are defined to take into account the effect of the residual stress on the behavior of the beam. Fo is defined by (2.2.3a) and represents the resultant force created by the residual stress of each layer. Similarly to mα, Mo is mathematically defined as the barycenter of the residual stress, and physically as the moment associated to the force Fo (2.2.3b).

𝐹𝐹𝐹𝐹 ! = 𝑣𝑣𝑣𝑣. 𝒯𝒯𝒯𝒯 ! (𝑧𝑧𝑧𝑧) ! ! . 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 (2.2.3a) 𝑀𝑀𝑀𝑀 ! = 𝑣𝑣𝑣𝑣. 𝒯𝒯𝒯𝒯 ! (𝑧𝑧𝑧𝑧). 𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! . 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ! ! (2.2.3b)
In the case of a beam, the equations presented in the previous part can be simplified since the strain and stress tensors' components depending on the y-axis are zero (2.2.3a). Most beam deformations can be studied in the frame of the Euler-Bernoulli beam theory [START_REF] Reddy | Theory and analysis of elastic plates and shells[END_REF]: this theory assumes that the cross sections stay straight (Hyp.1), inextensible (Hyp.2), and perpendicular to the bending line (Hyp. 3). The first two assumptions imply that the transverse stress is also null (2.2.3b) whereas the third one leads to neglecting the transverse shear stress (2.2.3c).

𝒮𝒮𝒮𝒮 !! = 𝒮𝒮𝒮𝒮 !" = 𝒮𝒮𝒮𝒮 !" = 0 (2.2.3a) 𝒮𝒮𝒮𝒮 !! = 𝑤𝑤𝑤𝑤 ,! = 0 ⇒ 𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 = 𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥 (2.2.3b) 𝒮𝒮𝒮𝒮 !" = 𝑢𝑢𝑢𝑢 ,! + 𝑤𝑤𝑤𝑤 ,! = 0 ⇒ 𝑢𝑢𝑢𝑢 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 = 𝑢𝑢𝑢𝑢 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧. 𝑤𝑤𝑤𝑤 ,! 𝑥𝑥𝑥𝑥 (2.2.3c)
The axial displacement of each point of the beam can then be known if the displacement of the neutral axis is known for instance. Hence, the axial strain can be expressed by means of the Green-Lagrange tensor (2.1.6a), leading to the expression of the Van Karman strain (2.2.4c). The stress tensor is given by (2.2.4b).

𝒮𝒮𝒮𝒮 !! = 𝑢𝑢𝑢𝑢 ,! 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 ! + 1 2 . 𝑤𝑤𝑤𝑤 ,! ! 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ,!! 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 ! (2.2.4a) 𝒯𝒯𝒯𝒯 !! = 𝐸𝐸𝐸𝐸. 𝒮𝒮𝒮𝒮 !! -𝛼𝛼𝛼𝛼. 𝑇𝑇𝑇𝑇 + 𝒯𝒯𝒯𝒯 ! (2.2.4b)
To determine the equilibrium equations of the composite beam, the strain energy's expression is integrated over the whole beam volume Ω in order to apply the principle of virtual displacement. In that case, the variation of the beam's strain energy and kinetic energy is equal to the work of the external forces (2.2.5a). Equation (2.2.5a) can then be developed in (2.2.5b), neglecting the kinetic energy variation linked to the axial displacement u.

𝑑𝑑𝑑𝑑𝒲𝒲𝒲𝒲 ! + 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑑𝑑𝑑𝑑𝒲𝒲𝒲𝒲 !"# (2.2.5a) 𝒯𝒯𝒯𝒯 !! . 𝑑𝑑𝑑𝑑𝒮𝒮𝒮𝒮 !! + 1 2 . 𝜌𝜌𝜌𝜌. 𝑤𝑤𝑤𝑤 ,!! ! . 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ! = 𝑑𝑑𝑑𝑑𝒲𝒲𝒲𝒲 !"# (2.2.5b)
In the case of an axially constrained beam, no external force works since axial displacement is forbidden (2.2.6a By making a double integration of this last expression, we can find that the axial displacement is constant and is given by (2.2.9a), where λ represents the difference between the length of the curved beam and the length L of the straight beam, already presented in the introduction of this chapter. The beam stretching is directly equal to the ratio of λ over the length of the clamp (2.2.9b). Hence, the axial strain can be rewritten in (2.2.9c).

𝑢𝑢𝑢𝑢 ,! + 1 2 . 𝑤𝑤𝑤𝑤 ,! ! = 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 (2.2.9a) 𝜆𝜆𝜆𝜆 = 1 2 . 𝑤𝑤𝑤𝑤 ,! ! !/! !!/! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (2.2.9b) 𝒮𝒮𝒮𝒮 !! = 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 -𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ,!! 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 ! (2.2.9c)
Similarly, we treat the formulation of the equilibrium given by (2.2.6c). Taking the case of a simply supported beam, the beam's edges cannot move (2.2.10a), but can rotate in order to cancel the moment created by the variation of the temperature and the residual stress (2.2.10b). It is interesting to note in (2.2.10b) the antagonistic effects caused by the residual stress asymmetry and the asymmetry of the thermal expansion Mα.T. If both effects are positive, Mo gives a convexe shape to the beam whereas Mα.T tends to give a concave shape to the beam. The fact that both effects have the same sign is a necessary condition to observe any bistable phenomenon.

𝑤𝑤𝑤𝑤 - 𝐿𝐿𝐿𝐿 2 = 𝑤𝑤𝑤𝑤 𝐿𝐿𝐿𝐿 2 = 0 (2.2.10a) 𝑖𝑖𝑖𝑖 ! . 𝑤𝑤𝑤𝑤 ,!! 𝐿𝐿𝐿𝐿 2 = 𝑖𝑖𝑖𝑖 ! . 𝑤𝑤𝑤𝑤 ,!! - 𝐿𝐿𝐿𝐿 2 = 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 (2.2.10b)
Hence the boundary equation resulting from a double integration by parts of (2.2.6c) vanishes (2.2.11a). The equilibrium is then described by (2.2.11b).

𝒯𝒯𝒯𝒯 !! . 𝑤𝑤𝑤𝑤 ,! + 𝑧𝑧𝑧𝑧. 𝜎𝜎𝜎𝜎 ,! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 -𝑧𝑧𝑧𝑧. 𝒯𝒯𝒯𝒯 !! . 𝑑𝑑𝑑𝑑𝑤𝑤𝑤𝑤 ,! ! = 0 (2.2.11a) 𝜌𝜌𝜌𝜌. 𝑤𝑤𝑤𝑤 ,!! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝒯𝒯𝒯𝒯 !! . 𝑤𝑤𝑤𝑤 ,! . 𝑑𝑑𝑑𝑑𝑤𝑤𝑤𝑤 ,! -𝑧𝑧𝑧𝑧. 𝑑𝑑𝑑𝑑𝑤𝑤𝑤𝑤 ,!! . 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ! = - 𝜌𝜌𝜌𝜌. 𝑤𝑤𝑤𝑤 ,!! ! + 𝒯𝒯𝒯𝒯 !! . 𝑤𝑤𝑤𝑤 ,! ,! + 𝑧𝑧𝑧𝑧. 𝒯𝒯𝒯𝒯 !!,!! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑. 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ! = 0 (2.2.11b)
The strong formulation of the equilibrium is then (2.2.12).

𝜌𝜌𝜌𝜌. 𝑤𝑤𝑤𝑤 ,!!

! + 𝒯𝒯𝒯𝒯 !! . 𝑤𝑤𝑤𝑤 ,! ,! + 𝑧𝑧𝑧𝑧. 𝒯𝒯𝒯𝒯 !!,!! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! = 0
(2.2.12)

By developping this equation, we finally establish the non-linear Euler equation, which describes the equilibrium of the beam (2.2.13a). The terms between brackets correspond to the axial thrust applied by the clamp on the beam (2.2.13b): as the beam cannot move, a thrust ie.k 2 tends to appear to balance the effect of the residual stress force Fo, the beam's mean thermal expansion and the beam's stretching force Ne.λ/L. That way, the static equilibrium (omitting inertia) is described by (2.2.13c).

𝓂𝓂𝓂𝓂 ! 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ,!! ! + 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 -𝑁𝑁𝑁𝑁 ! . 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 -𝐹𝐹𝐹𝐹 ! . 𝑤𝑤𝑤𝑤 ,!! + 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!!!! = 0 (2.2.13a) 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! = 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 -𝑁𝑁𝑁𝑁 ! . 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 -𝐹𝐹𝐹𝐹 ! (2.2.13b) 𝑘𝑘𝑘𝑘 ! . 𝑤𝑤𝑤𝑤 ,!! + 𝑤𝑤𝑤𝑤 ,!!!! = 0 (2.2.13c)
The equilibrium equation (2.2.13d) can be rewritten by defining two forces N and No by (2.2.14a) and (2.2.14b): N describes the effect of the beam's internal forces and No is an invariant of the transformation, representing the effective residual force acting on the beam.

𝑁𝑁𝑁𝑁 = 𝑁𝑁𝑁𝑁 ! . 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 - 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! -𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! (2.2.14a) 𝑁𝑁𝑁𝑁 ! = 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! -𝐹𝐹𝐹𝐹 ! (2.2.14b)
Hence the beam's equilibrium is simply described by the equality (2.2.15).

𝑁𝑁𝑁𝑁 = 𝑁𝑁𝑁𝑁 !

(2.2.15)

Criterion of bistability

In order to study the non-linear Euler equation (2.2.13a), we first find the general solution of (2.2.13d) and we then determine the amplitude of the beam deflexion using (2.2.15). The general forms of the solution to (2.2.13d) are the functions sine and cosine and the linear functions of the position x (2.3.1a-b). These functions can be regrouped according to their parity (2.3.1c).

𝑤𝑤𝑤𝑤 ! 𝑥𝑥𝑥𝑥 = 𝑎𝑎𝑎𝑎. cos 𝑘𝑘𝑘𝑘. 𝑥𝑥𝑥𝑥 + 𝑏𝑏𝑏𝑏 (2.3.1a) 𝑤𝑤𝑤𝑤 ! 𝑥𝑥𝑥𝑥 = 𝑐𝑐𝑐𝑐. sin 𝑘𝑘𝑘𝑘. 𝑥𝑥𝑥𝑥 + 𝑑𝑑𝑑𝑑. 𝑥𝑥𝑥𝑥 (2.3.1b) 𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥 = 𝑤𝑤𝑤𝑤 ! 𝑥𝑥𝑥𝑥 + 𝑤𝑤𝑤𝑤 ! 𝑥𝑥𝑥𝑥 (2.3.1c)
The two boundary conditions on the displacement field w given in (2.2.10a) and ( 2 These equations show that the beam shape associated to symmetrical buckling mode w1 can be non-zero even when k is not an odd multiple of π/L, because of the effect of the asymmetries mα and Mo. However, assymetrical modes only occur when k is a multiple of π/L (2.3.3d).

𝑤𝑤𝑤𝑤 ! (𝑥𝑥𝑥𝑥) = 𝑎𝑎𝑎𝑎. cos 𝑘𝑘𝑘𝑘. 𝑥𝑥𝑥𝑥 -𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2

(2.3.3a) 𝑎𝑎𝑎𝑎 = 𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 -𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! . cos 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤ℎ 𝑘𝑘𝑘𝑘 ≠ (2. 𝑛𝑛𝑛𝑛 + 1). 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤ℎ 𝑛𝑛𝑛𝑛 ∈ ℕ (2.3.3b) 𝑤𝑤𝑤𝑤 ! (𝑥𝑥𝑥𝑥) = 𝑐𝑐𝑐𝑐. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝑥𝑥𝑥𝑥 - 2 𝐿𝐿𝐿𝐿 . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 (2.3.3c) 𝑐𝑐𝑐𝑐 ≠ 0 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 𝑘𝑘𝑘𝑘 = 2. 𝑛𝑛𝑛𝑛. 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 𝑤𝑤𝑤𝑤𝑤𝑤𝑤𝑤 𝑤𝑤𝑤𝑤ℎ 𝑛𝑛𝑛𝑛 ∈ ℕ (2.3.3d)
A particular solution is obtained for the equivalent relative temperature To =Mo/Mα. At this point either the wave vector k is equal to the first buckling load, or the amplitude is zero and the beam is straight (2.3.4).

𝑇𝑇𝑇𝑇 = 𝑀𝑀𝑀𝑀

! 𝑀𝑀𝑀𝑀 ! ⇒ 𝑘𝑘𝑘𝑘 = 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 𝑎𝑎𝑎𝑎 = 0 (2.3.4)
Equation (3.3.3b) enables to calculate the length λ (2.3.5).

𝜆𝜆𝜆𝜆 = 1 4 . 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 -𝑘𝑘𝑘𝑘. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 . 𝑎𝑎𝑎𝑎 ! (2.3.5)
Hence the equilibrium of the beam is described by a second order polynomial of a (2.3.6a).

As a matter of convenience, we define two functions of k, φ1 and φ2. We also define do representing the deflexion of the middle of the beam's centre line (2.3.6d).

𝑁𝑁𝑁𝑁 ! 4. 𝐿𝐿𝐿𝐿 . 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 . 𝑎𝑎𝑎𝑎 ! - 𝑁𝑁𝑁𝑁 ! . 𝐼𝐼𝐼𝐼 ! 𝑀𝑀𝑀𝑀 ! . 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 . 𝑎𝑎𝑎𝑎 + 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! -𝑁𝑁𝑁𝑁 ! = 0 (2.3.6a) 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 = 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 -𝑘𝑘𝑘𝑘. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 (2.3.6b) 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 = 𝑘𝑘𝑘𝑘 ! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 (2.3.6c) 𝑑𝑑𝑑𝑑 ! = 𝑤𝑤𝑤𝑤 0 = 𝑎𝑎𝑎𝑎. 1 -𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 (2.3.6d)
In Fig. thrust k 2 and the length λ/L as a function of temperature. These figures show that for monostable beams, only one equilibrium position is authorized whatever the beam's temperature. As No progressively increases, the sensitivity of the beam to temperature decreases and the beam amplitude varies less rapidly. When No exceeds a critical value, the beam becomes bistable and up to three possible equilibrium positions exist simultaneously. The central position is unstable and the beam trajectory exhibits a hysteretic behavior characterized by two temperature TS and TSB at which the beams snaps from an unstable position (which is a limit point) to a stable position. In Fig. 2.3.1b and 2.3.1c, the occurrence of the bistability is characterized by a loop when the axial thrust exceeds the Euler load. Fig. 2.3.1d represents the deformation of a bistable beam for the four values of amplitude that characterize the hysteresis cycle. Necessary, if the force No exceeds the first Euler buckling mode, (2.3.7) has two other solutions at this temperature and the beam is de facto bistable. The criterion of bistability is then (2.3.8a). The criterion of stability can also be rewritten under the form (2.3.8b). This equation means that the temperature defined by the two imperfections must be higher than the buckling temperature of the straight beam in order to observe a bistable behavior. The stability of the equilibrium is associated to the sign of the second partial derivative of the strain energy and to the convexity/concavity of the strain energy. The beam can snap only when the stability disappears. Hence we find that there is equivalence between the nullity of the strain energy's second derivative and the nullity of the first derivative of the force N (2.3.10).

Dimensions Ne Ie M α M α Cv NEuler Mo (mm²) (N) (N.m²) (N.K -1 ) (N.m.K -1 ) (J.K -1 ) (N) (N.
𝑁𝑁𝑁𝑁 ! = 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! -𝐹𝐹𝐹𝐹 ! ≥ 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! = 𝑁𝑁𝑁𝑁 !"#$% (2.3.8a) 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! ≥ 𝐼𝐼𝐼𝐼 ! 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! + 𝐹𝐹𝐹𝐹 ! 𝑁𝑁𝑁𝑁 ! (2.3.8b)
𝑑𝑑𝑑𝑑 ! 𝒲𝒲𝒲𝒲 !"# = 0 ⇔ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ! = 0 ⇔ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ! = 0 ⇔ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ! = 0 (2.3.10)
Hence, the limit points are characterized by (2.3.11).

𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 ! = 0 ⇔ 𝑛𝑛𝑛𝑛 ! 4. 𝐿𝐿𝐿𝐿 . 𝑎𝑎𝑎𝑎 ! . 𝜙𝜙𝜙𝜙 ! ! 𝑘𝑘𝑘𝑘 -2. 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 . 𝜙𝜙𝜙𝜙 ! ! 𝑘𝑘𝑘𝑘 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 + 2. 𝑖𝑖𝑖𝑖 ! . 𝑘𝑘𝑘𝑘 = 0 (2.3.11)
Computing (2.3.6a) and (2.3.11) enables to find the values of the beam's amplitude associated to the limit points and then to deduct the critical temperatures. Logically, the rise of No widens the thermal hysteresis. As a consequence, the displacement of the beam during the snap and the snap-back is increased. The difference between these displacements is linked to the effect of the mean thermal expansion, which increases the thrust of the clamp on the beam when the beam is heated and snaps, and then decreases this thrust when the beam snaps back.

Simplification of the equations for a bimetallic beam

The bimetallic beam represents the simplest case among beams presenting an asymmetry of the coefficients of thermal expansion. Considering the stack presented in Fig. 2.2.1a, we can simplify the expression of the beam's parameters (2.3.2b-h) by introducing the ratio of the layers' thicknesses (2.4.1a).

𝑡𝑡𝑡𝑡 !" = 𝑡𝑡𝑡𝑡 ! + 𝑡𝑡𝑡𝑡 ! ; 𝑟𝑟𝑟𝑟 !" = 𝑡𝑡𝑡𝑡 ! 𝑡𝑡𝑡𝑡 ! (2.4.1a) 𝑧𝑧𝑧𝑧 ! = 𝐸𝐸𝐸𝐸 ! -𝐸𝐸𝐸𝐸 ! . 𝑟𝑟𝑟𝑟 !" ! + 𝐸𝐸𝐸𝐸 ! 2. ( 𝐸𝐸𝐸𝐸 ! -𝐸𝐸𝐸𝐸 ! . 𝑟𝑟𝑟𝑟 !" + 𝐸𝐸𝐸𝐸 ! ) (2.4.1b) 𝑁𝑁𝑁𝑁 ! = 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 !" . 𝐸𝐸𝐸𝐸 ! . 𝑟𝑟𝑟𝑟 !" + 𝐸𝐸𝐸𝐸 ! . 1 -𝑟𝑟𝑟𝑟 !" (2.4.1c) 𝐼𝐼𝐼𝐼 ! = 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 !" ! 3 . 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! . 𝑟𝑟𝑟𝑟 !" + 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! . 1 -𝑟𝑟𝑟𝑟 !" (2.4.1d) 𝑁𝑁𝑁𝑁 ! = 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 !" . 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! . 𝑟𝑟𝑟𝑟 !" + 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! . 1 -𝑟𝑟𝑟𝑟 !" (2.4.1e) 𝑀𝑀𝑀𝑀 ! = 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 ! 2 . 2. 𝑧𝑧𝑧𝑧 ! . 𝑟𝑟𝑟𝑟 !" -𝑟𝑟𝑟𝑟 !" ! . 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! (2.4.1f) 𝐹𝐹𝐹𝐹 ! = 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡. 𝜎𝜎𝜎𝜎 ! . 𝑥𝑥𝑥𝑥 + 𝜎𝜎𝜎𝜎 ! . 1 -𝑥𝑥𝑥𝑥 (2.4.1g) 𝑀𝑀𝑀𝑀 ! = 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 ! 2 . 2. 𝑧𝑧𝑧𝑧 ! . 𝑥𝑥𝑥𝑥 -𝑥𝑥𝑥𝑥 ! . 𝐸𝐸𝐸𝐸 ! 𝐸𝐸𝐸𝐸 ! . 𝒯𝒯𝒯𝒯 ! -𝒯𝒯𝒯𝒯 ! (2.4.1h)
In that case, the temperature To and the force No can be simplified into (2.4.2a) and

(2.4.2b).

𝑇𝑇𝑇𝑇 ! = 𝒯𝒯𝒯𝒯 ! 𝐸𝐸𝐸𝐸 ! - 𝒯𝒯𝒯𝒯 ! 𝐸𝐸𝐸𝐸 ! (𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! ) + 𝜃𝜃𝜃𝜃 ! (2.4.2a) 𝑁𝑁𝑁𝑁 ! = 𝑁𝑁𝑁𝑁 ! . 𝒯𝒯𝒯𝒯 ! 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! - 𝒯𝒯𝒯𝒯 ! 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! (𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! ) (2.4.2b)
The criterion of bistability cannot be simplified further. To illustrate the role of each parameter, we take the case of a bimetallic beam where the neutral axis is placed at the interface between the two materials (zo=x). To obtain this case, x must verify (2.4.3a). The criterion of stability thus becomes (2.4.3b).

𝑟𝑟𝑟𝑟 !" = 𝐸𝐸𝐸𝐸 ! 𝐸𝐸𝐸𝐸 ! + 𝐸𝐸𝐸𝐸 ! ; 𝑡𝑡𝑡𝑡 ! 𝑡𝑡𝑡𝑡 ! = 𝐸𝐸𝐸𝐸 ! 𝐸𝐸𝐸𝐸 ! (2.4.3a) 𝐸𝐸𝐸𝐸 ! + 𝐸𝐸𝐸𝐸 ! ! 𝐸𝐸𝐸𝐸 ! . 𝐸𝐸𝐸𝐸 ! . (𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! ) . 𝜎𝜎𝜎𝜎 ! 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! - 𝜎𝜎𝜎𝜎 ! 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! ≥ 𝜋𝜋𝜋𝜋 ! 3 . 𝑡𝑡𝑡𝑡 !" ! 𝐿𝐿𝐿𝐿 ! (2.4.3b)
Equation (2.4.3b) then shows that bimetallic beams with a low thermal asymmetry and a small equivalent Young modulus are more sensitive to the effect of the residual stress; in these conditions the Euler load is reached for smaller values of the residual stress.

Similarly, beams with a high aspect ratio (thin and long beams) have a smaller Euler load and easily become unstable. The bistability equation also shows that if material 2 is used as an active layer and material 1 as a passive layer (�2> �1), the best solution to produce a bistable beam is to create a compressive stress in the passive layer (σ1<0) (Fig. 2.4.1b) and a tensile stress in the active layer (σ2>0) (Fig. 2.4.1a). In this case, both thermal asymmetry and residual asymmetry are positive, so the beam initially has a convex shape.

Experimentally, it can be difficult to process simply-supported bimetallic beams with a different values of residual stress in either material. The simplest solution is to apply an initial thrust Ni to the beam in order to buckle it. By using (2.3.9), we can express this force (2.4.4a) and deduce the criterion of bistability (2.4.4b). The temperature To associated to this force is then the temperature at which the thrust is applied (To=0).

𝑁𝑁𝑁𝑁 ! = 𝑁𝑁𝑁𝑁 ! . 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 (2.4.4a) 𝑁𝑁𝑁𝑁 ! ≥ 𝜋𝜋𝜋𝜋 ! 3 . 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 !" ! 𝐿𝐿𝐿𝐿 ! . 𝐸𝐸𝐸𝐸 ! . 𝐸𝐸𝐸𝐸 ! 𝐸𝐸𝐸𝐸 ! + 𝐸𝐸𝐸𝐸 ! ! (2.4.4b)

Equilibrium of composite beams with an initial curvature

The previous part presented the modeling of the instability of initially-stressed beams, contrary to the models of [START_REF] Timoshenko | Analysis of bi-metal thermostat[END_REF], [START_REF] Burgreen | Higher mode buckling of bimetallic beam[END_REF], and [Cabal and Ross, 2002] where the beams are initially curved. In this part, we show that initiallycurved beams can be modeled with the previous equations, assuming that the beams have an initial polynomial shape. In the case of beams with an initial curvature wo, the axial strain is given by (3.6.1).

𝒮𝒮𝒮𝒮 !! = 𝑢𝑢𝑢𝑢 ,! + 1 2 . 𝑤𝑤𝑤𝑤 ,! ! -𝑤𝑤𝑤𝑤 !,! ! -𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! . (𝑤𝑤𝑤𝑤 ,!! -𝑤𝑤𝑤𝑤 !,!! ) (2.6.1)
Without detailing the demonstration of the principle of virtual displacement, we can find that the equilibrium is described by the non-linear Euler equation (2.6.2a) with the boundary equations (2.6.2c).

𝓂𝓂𝓂𝓂 ! 𝐿𝐿𝐿𝐿 . 𝑑𝑑𝑑𝑑 ! 𝑤𝑤𝑤𝑤 𝑑𝑑𝑑𝑑𝑡𝑡𝑡𝑡 ! + 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 - 𝑁𝑁𝑁𝑁 ! 2. 𝐿𝐿𝐿𝐿 . (𝑤𝑤𝑤𝑤 ,! ! -𝑤𝑤𝑤𝑤 !,! ! ! ! ! ! ! ). 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 . 𝑤𝑤𝑤𝑤 ,!! + 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!!!! -𝑤𝑤𝑤𝑤 !,!!!! = 0 (2.6.2a) 𝑖𝑖𝑖𝑖 𝑖𝑖𝑖𝑖 ! . 𝑘𝑘𝑘𝑘 ! = 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 - 𝑛𝑛𝑛𝑛𝑛𝑛𝑛𝑛 ! 2. 𝐿𝐿𝐿𝐿 . (𝑤𝑤𝑤𝑤 ,! ! -𝑤𝑤𝑤𝑤 !,! ! ! ! ! ! !
). 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

(2.6.2b)

𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!! ± 𝐿𝐿𝐿𝐿 2 -𝑤𝑤𝑤𝑤 !,!! ± 𝐿𝐿𝐿𝐿 2 = -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 ; 𝑤𝑤𝑤𝑤 ± 𝐿𝐿𝐿𝐿 2 = 0 ; 𝑢𝑢𝑢𝑢 ± 𝐿𝐿𝐿𝐿 2 = 0 (2.6.2c)
In [START_REF] Timoshenko | Analysis of bi-metal thermostat[END_REF], [START_REF] Burgreen | Higher mode buckling of bimetallic beam[END_REF]] and [Cabal and Ross, 2002], the beam's initial shape is assumed to be a half-sine function (2.6.3a) because this solution is ideal when Galerkin's method is used to approximate the beam shape, since it is the general solution of the Euler equation. In that case, the equilibrium equation is (2.6.3d). This equation has been numerically solved in [Cabal and Ross, 2002]. However the complexity of this equation did not allow to find a criterion of stability for the composite beams. Instead of using (2.6.3a) as the beam's initial shape, we use the polynomial function given in (2.6.4a). Hence the equilibrium is described by (2.6.4b). The equation for initially curved beams is then the same as for initially-stressed beams if Fo and Mo are defined by (2.6.5a) and (2.6.5b). Whatever the sign of the initial curvature, initially-curved beams behave as if a compressive stress was applied to them. Fig. 2.6.1a shows the equivalent residual stress in the layers of a bimetallic beam as a function of the initial curvature. The thicker the beam, the more compressive the stress in the bimetallic beam. Fig. 2.6.1b-d represent the equilibrium path of curved beams for various values of the normalized deflection ao/t.

𝑤𝑤𝑤𝑤 ! 𝑥𝑥𝑥𝑥 = 𝑎𝑎𝑎𝑎 ! . 1 - 2.
𝐹𝐹𝐹𝐹 ! = - 8. 𝑁𝑁𝑁𝑁 ! 3. 𝐿𝐿𝐿𝐿 ! . 𝑎𝑎𝑎𝑎 ! ! (2.6.5a) 𝑀𝑀𝑀𝑀 ! = 8. 𝐼𝐼𝐼𝐼 ! 𝐿𝐿𝐿𝐿 ! . 𝑎𝑎𝑎𝑎 ! (2.6.5b)
Thus the bistability criterion of an initially-curved beam is (2.6.6).

𝑁𝑁𝑁𝑁

! 3. 𝐿𝐿𝐿𝐿 ! . 𝑎𝑎𝑎𝑎 ! ! - 8. 𝑁𝑁𝑁𝑁 ! . 𝐼𝐼𝐼𝐼 ! 𝑀𝑀𝑀𝑀 ! . 𝐿𝐿𝐿𝐿 ! . 𝑎𝑎𝑎𝑎 ! ≥ 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! (2.6.6)
If we develop this criterion for bimetallic beams with the neutral axis at the interface between both layers, the equation becomes (2.6.7).

𝐸𝐸𝐸𝐸 ! + 𝐸𝐸𝐸𝐸

! ! 𝑡𝑡𝑡𝑡 ! . 𝐸𝐸𝐸𝐸 ! . 𝐸𝐸𝐸𝐸 ! . 𝑎𝑎𝑎𝑎 ! ! + 2. 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! + 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! 𝑡𝑡𝑡𝑡. 𝐸𝐸𝐸𝐸 ! . 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! . 𝑎𝑎𝑎𝑎 ! ≥ 𝜋𝜋𝜋𝜋 ! (2.6.7)
In Fig. 2.6.1f, we compare the occurrence of the bistability and the snap and snap-back temperatures predicted by the previous model, by the previous model with the assumption that n α =0, and by Timoshenko's model. This shows that Timoshenko's model cannot be used to predict the temperature of an axially-restrained beam whose mean thermal expansion is not negligible. However, if it is possible to create a clamp that can move to balance the mean thermal dilatation, as assumed by Timoshenko, its model gives good accuracy with the current model. In order to find its criterion of bistability, [START_REF] Timoshenko | Analysis of bi-metal thermostat[END_REF] neglected the influence of the mean thermal expansion on the instability of composite beams, and assumed the same Young modulus for both materials. In that case, the criterion of bistability only depends on the ratio of the beam's thickness over its deflexion (2.6.8a). With the first assumption, (2.6.7) becomes (2.6.8b) and by assuming that the bimetallic beam is made of materials with very different coefficients of thermal expansion. In that case the solution of the polynomial (2.6.7) tends towards the value (2.8c).

𝑎𝑎𝑎𝑎 ! 𝑡𝑡𝑡𝑡 ≥ 1 3 ≈ 0.577 … (2.6.8a) 32. 𝑎𝑎𝑎𝑎 ! ! 𝑡𝑡𝑡𝑡 ! + 2. 𝛼𝛼𝛼𝛼 ! + 𝛼𝛼𝛼𝛼 ! 𝑡𝑡𝑡𝑡. 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! . 𝑎𝑎𝑎𝑎 ! ≥ 𝜋𝜋𝜋𝜋 ! (2.6.8b) 𝑎𝑎𝑎𝑎 ! 𝑡𝑡𝑡𝑡 ≥ 1 + 32. 𝜋𝜋𝜋𝜋 -1 32 ≈ 0.525 … (2.6.8c)
Finally, Fig. 2.6.1g gives the critical value of initial deflection associated with the occurrence of the bistability as a function of the coefficient of thermal expansion (CTE) of the active layer, and for various CTE of the passive layer. This figure logically shows that for bimetallic beams with a small CTE mismatch, the bistability is observed for small values of the initial deflection. It also shows that for such beams Timoshenko's model cannot be applied, since the error made is important (Fig. 2.6.1h). As shown by Orthwein, small variations of the initial curvatures can result in large variations of the beam's thermal hysteresis.

Curvature and initial stress

Given the previous equations, the criterion of bistability of beams having an initial curvature and an initial residual stress depends on the sum of the forces No associated to both imperfections (2.7.1). The equation of equilibrium thus stays unchanged (3.21).

𝑁𝑁𝑁𝑁 ! = 𝑁𝑁𝑁𝑁 ! 𝑁𝑁𝑁𝑁 ! . 8. 𝐼𝐼𝐼𝐼 ! . 𝑎𝑎𝑎𝑎 ! 𝐿𝐿𝐿𝐿 ! + 𝑀𝑀𝑀𝑀 ! + 8. 𝑁𝑁𝑁𝑁 ! 3. 𝐿𝐿𝐿𝐿 . 𝑎𝑎𝑎𝑎 ! ! -𝐹𝐹𝐹𝐹 ! ≥ 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 !
(2.7.1)

Influence of the architecture

In the previous equations, we did not take into account the fact that the beam must be in contact with the hot source until it snaps, and in contact with the cold source until it snaps back. In order to model the influence of the architecture and to see how it can impact the properties of the bistable beams, we study the behavior of the system represented in Fig. 2.8.1. The height of the cavity measured from the beam's clamp to the bottom of the cold source is dtop, and dbot from the clamp to the top of the hot source. Given that the beam has a thickness t, the dimensions to be considered are the beam's center deflection do and the dtop-t+zo and dbot-zo. The miminal conditions on these dimensions to observe the working of the device are given by (2.8.1a) and (2.8.1b) (Fig. 2.8.1a).

𝑑𝑑𝑑𝑑 ! 𝑇𝑇𝑇𝑇 !" ≥ 𝑑𝑑𝑑𝑑 !"# + 𝑡𝑡𝑡𝑡 -𝑧𝑧𝑧𝑧 ! = 𝑑𝑑𝑑𝑑 !"# !"# (2.8.1a) 𝑑𝑑𝑑𝑑 ! 𝑇𝑇𝑇𝑇 ! ≤ 𝑑𝑑𝑑𝑑 !"# -𝑧𝑧𝑧𝑧 ! = 𝑑𝑑𝑑𝑑 !"# !"# (2.8.1b)
We can assume that the contacts between the beam and the two heat sources are punctual. In that case, by constraining the beam, the heat sources create a force Ftop (or Fbot) on the beam's middle:the equilibrium equation of the beam in contact with the top surface thus becomes (2.8.2).

𝓂𝓂𝓂𝓂 ! 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ,!! ! + 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 -𝑁𝑁𝑁𝑁 ! . 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 -𝐹𝐹𝐹𝐹 ! . 𝑤𝑤𝑤𝑤 ,!! + 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!!!! + 𝐹𝐹𝐹𝐹 !"# . 𝛿𝛿𝛿𝛿(𝑥𝑥𝑥𝑥) = 0 (2.8.2)
And the boundary conditions are given by (2.8.3).

𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!! 𝐿𝐿𝐿𝐿 2 = 𝑀𝑀𝑀𝑀 ! + 𝐹𝐹𝐹𝐹 !"# . 𝐿𝐿𝐿𝐿 2 -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 (2.8.3a) 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!! - 𝐿𝐿𝐿𝐿 2 = 𝑀𝑀𝑀𝑀 ! + 𝐹𝐹𝐹𝐹 !"# . 𝐿𝐿𝐿𝐿 2 -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 (2.8.3b)
Following the same approach as in Section 2.3, the equilibrium equation (2.8.2) can be rewritten into (2.8.4).

𝐹𝐹𝐹𝐹 !"# = 2. 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 . 𝑁𝑁𝑁𝑁 ! 4. 𝐿𝐿𝐿𝐿 . 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 . 𝑎𝑎𝑎𝑎 ! -𝑁𝑁𝑁𝑁 ! . (𝑇𝑇𝑇𝑇 -𝑇𝑇𝑇𝑇 ! ) + 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! -𝑁𝑁𝑁𝑁 ! (2.8.4)
During the heating and cooling phases, the evolution of Ftop can be plotted as a function of the temperature by using the fact that the beam's maximal deflection is constant. During heating, a must be replaced by (2.8.5a), and by (2.8.5b) during cooling.

𝑎𝑎𝑎𝑎 !"# = 𝑑𝑑𝑑𝑑 !"# -𝑧𝑧𝑧𝑧 ! 1 -cos 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 (2.8.5a) 𝑎𝑎𝑎𝑎 !"# = 𝑑𝑑𝑑𝑑 !"# + 𝑡𝑡𝑡𝑡 -𝑧𝑧𝑧𝑧 ! 1 -cos 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 2 (2.8.5b)
The beam thus snaps and snaps back when the load Ftop vanishes (2.8.6a). The snap back temperature is then given by (2.8.6b) and the snap temperature by (2.8.6c). Here 100% represents the ratio between the cavity's height and the maximal allowed cavity's height given by (2.8.1). This figure clearly shows that, by reducing the cavity's thickness, the thermal hysteresis is modified and decreases. This phenomenon was numerically shown in [Ravindran et al, 2014] but without physical explanations.

𝑁𝑁𝑁𝑁 ! 4. 𝐿𝐿𝐿𝐿 . 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 !"# . 𝑎𝑎𝑎𝑎 !"# ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 !"# ! -𝑁𝑁𝑁𝑁 ! = 0 (2.8.6a) 𝑇𝑇𝑇𝑇 !" = 𝑇𝑇𝑇𝑇 ! + 1 𝑀𝑀𝑀𝑀 ! . 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 !"# ! . cos 𝑘𝑘𝑘𝑘 !"# . 𝐿𝐿𝐿𝐿 2 . 𝑑𝑑𝑑𝑑 !"# -𝑡𝑡𝑡𝑡 + 𝑧𝑧𝑧𝑧 ! 1 -cos 𝑘𝑘𝑘𝑘 !"# . 𝐿𝐿𝐿𝐿 2 (2.8.6b) (a) (b) (c) (d)
𝑇𝑇𝑇𝑇 ! = 𝑇𝑇𝑇𝑇 ! + 1 𝑀𝑀𝑀𝑀 ! . 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 !"# ! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘 !"# . 𝐿𝐿𝐿𝐿 2 . 𝑑𝑑𝑑𝑑 !"# -𝑧𝑧𝑧𝑧 ! 1 -𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝑘𝑘𝑘𝑘 !"# . 𝐿𝐿𝐿𝐿 2 (2.8.6c)

b. Comparison between the thermal hysteresis of a free beam and the thermal hysteresis of a beam in a cavity, for various ratios of the cavity's thickness-to-deflexion. C. Evolution of the forces developed by the cavity on the beam. d. Evolution of the clamp thrust as as function of temperature.

For the aforementioned beam, the Euler load is 112 N. This shows that it can be more interesting to attempt to harvest the strain energy associated to the clamp's thrust instead of harvesting the strain energy associated to the forces Fbot and Ftop. To quantify the influence of the cavity's height reduction on the properties of the bistable beam, the forces developed by the cavity on the beam, the clamp's thrust and the thermal hysteresis are represented as functions of the cavity's relative height in Fig. 2.8.2. This figure shows that the reduction of the cavity's thickness is accompanied by a reduction of all the parameters of the composite beam, due to the impact of the architecture on the bistability of the beam. Fig. 2.8.2d shows a quasi-linear proportionality relation between the decrease of the relative hysteresis and the reduction of the cavity's height.

Performances of the harvester using composite beams

Evolution of the strain energy of the composite beams

Studying the evolution of the beam's strain energy along an equilibrium trajectory is a good solution to deeply understand the origins and the signification of the bistability. The expression of the beam's potential energy can be found according to two methods. Since the equilibrium positions are the extrema of the strain energy and the equilibrium equation its first derivative, the integration of this last equation enables to find the strain energy function. Yet the best solution is to integrate the local expression of the strain energy on the whole beam's volume.

𝓌𝓌𝓌𝓌 ! = 1 2 . 𝐸𝐸𝐸𝐸. 𝒮𝒮𝒮𝒮 !! ! + 𝒯𝒯𝒯𝒯 ! -𝐸𝐸𝐸𝐸. 𝛼𝛼𝛼𝛼. 𝑇𝑇𝑇𝑇 . 𝒮𝒮𝒮𝒮 !! (3.1.1)
When this last expression is developed with the help of the axial strain expression and integrated, the strain energy can be written as the sum of the stretching energy, the bending energy, and the energies due to the residual stress and the thermal stress, as defined by (3.1.2a) to (3.1.2d).

𝒲𝒲𝒲𝒲 ! = 𝒲𝒲𝒲𝒲 + 𝒲𝒲𝒲𝒲 ! -𝒲𝒲𝒲𝒲 !! (3.1.2a) 𝒲𝒲𝒲𝒲 = 𝑁𝑁𝑁𝑁 ! 2. 𝐿𝐿𝐿𝐿 . 𝜆𝜆𝜆𝜆 ! + 𝐼𝐼𝐼𝐼 ! 2 . 𝑤𝑤𝑤𝑤 ,!! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! ! ! ! (3.1.2b) 𝒲𝒲𝒲𝒲 ! = 𝐹𝐹𝐹𝐹 ! . 𝜆𝜆𝜆𝜆 -𝑀𝑀𝑀𝑀 ! . 𝑤𝑤𝑤𝑤 ,!! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! ! ! ! (3.1.2c) 𝒲𝒲𝒲𝒲 !! = 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇. 𝜆𝜆𝜆𝜆 -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇. 𝑤𝑤𝑤𝑤 ,!! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! ! ! ! (3.1.2d)
Knowing that the solutions of the non-linear equations are the harmonic functions, it is interesting to study how the energy evoles for the cosine functions that verify the boundary equations (2.3.3c). In that case, the energy can be written under the form (3.1.3).

𝒲𝒲𝒲𝒲 ! = 𝑁𝑁𝑁𝑁 ! 2. 𝐿𝐿𝐿𝐿 . 𝜆𝜆𝜆𝜆 ! + 𝐼𝐼𝐼𝐼 ! 𝑘𝑘𝑘𝑘 ! -2. 𝑘𝑘𝑘𝑘 ! . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 𝜑𝜑𝜑𝜑 ! (𝑘𝑘𝑘𝑘) . 𝜆𝜆𝜆𝜆 + 𝐹𝐹𝐹𝐹 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 . 𝜆𝜆𝜆𝜆 (3.1.3)
The shape of the curve of the beam's free work is difficult to guess with this equation. However it must be noted that λ is a quadratic function of the beam's amplitude. Therefore the beam's free work is a fourth-order polynomial of the strain energy.

According to the sign of the function of k between the brackets, the strain energy will evolve from a double potential well to a single potential well. The residual stress force and mean thermal stress act as external fields that desymmetrize the shape of the free work. Fig. 3.1.1a shows the action of these fields for a bistable beam. It shows that when the external field vanishes at To, three equilibrium positions coexist, the central one being unstable and playing the role of a potential barrier. When the external field is non-null, one of the stable positions becomes a global stable position (or minimum minimorum) and the other one a local stable position. However, the beam cannot switch from one position to the other because of the potential barrier. When either the snap temperature or snapback temperature is reached, the local stable position and the unstable position merge to form a limit point (or inflexion point). The beam thus slides along the potential well to reach the unique stable position and releases its strain energy. represents the strain energy as a function of temperature. The asymmetry of the butterfly shape is due to the effect of the mean thermal expansion which increases the kinetic energy release during the beam snap and reduces it during snap-back.

Evolution of the entropy of the composite beams

The local expression of the entropy being given by (3.2.1a), we can deduce the expression of the beam's entropy by integrating (3.2.1a) on the whole beam's volume. The result is given in (3.2.1b).

𝜎𝜎𝜎𝜎 = 𝑐𝑐𝑐𝑐 ! 𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# + 𝐸𝐸𝐸𝐸. 𝛼𝛼𝛼𝛼. 𝒮𝒮𝒮𝒮 !! (3.2.1a) 𝛴𝛴𝛴𝛴 = 4. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 -𝑘𝑘𝑘𝑘. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 + 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 -𝑇𝑇𝑇𝑇 ! . 𝜆𝜆𝜆𝜆 𝑇𝑇𝑇𝑇 -𝑇𝑇𝑇𝑇 ! + 𝐶𝐶𝐶𝐶 ! . 𝑙𝑙𝑙𝑙 𝑛𝑛𝑛𝑛 𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# 𝑇𝑇𝑇𝑇 ! (3.2.1b)
The other way to find the expression of the beam's entropy is to write the differential of the beam's Helmholtz free energy. Given (3.1.8), the Helmholtz free energy is a function of the privileged variables λ and T.

𝑑𝑑𝑑𝑑ℱ = -𝛴𝛴𝛴𝛴. 𝑑𝑑𝑑𝑑𝑇𝑇𝑇𝑇 + (𝑁𝑁𝑁𝑁 -𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 ). 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (3.2.2)
This last equation shows the beam entropy and force N as the first partial derivatives of the Helmholtz free energy. They can be written in function of T and λ.

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝑁𝑁𝑁𝑁 ,! ! . 𝑑𝑑𝑑𝑑𝑇𝑇𝑇𝑇 + 𝑁𝑁𝑁𝑁 ,! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (3.2.3a) 𝑑𝑑𝑑𝑑𝛴𝛴𝛴𝛴 = 𝛴𝛴𝛴𝛴 ,! ! . 𝑑𝑑𝑑𝑑𝑇𝑇𝑇𝑇 + 𝛴𝛴𝛴𝛴 ,! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑) (3.2.3b)
Applying the Schwartz theorem on the partial derivatives of the beam's free energy and strain energy is a first step needed to find the entropy's expression. Equation (3.2.5a) can hence be interpreted as a Maxwell relation linking 𝛴𝛴𝛴𝛴 ,! ! the partial derivative of the entropy with respect to λ, to 𝑁𝑁𝑁𝑁 ,! ! the partial derivative of the force N with respect to the temperature.

𝛴𝛴𝛴𝛴 ,! ! = -𝑁𝑁𝑁𝑁 ,! ! (3.2.5a) 𝛴𝛴𝛴𝛴 ,! ! = 𝜆𝜆𝜆𝜆 ,! ! . 𝑁𝑁𝑁𝑁 ,! ! -𝜆𝜆𝜆𝜆 ,! ! . 𝑁𝑁𝑁𝑁 ,! ! (3.2.5b)
Knowing the expression of the coefficients (𝜆𝜆𝜆𝜆 ,! ! , 𝜆𝜆𝜆𝜆 ,! ! , 𝑁𝑁𝑁𝑁 ,! ! , 𝑁𝑁𝑁𝑁 ,! ! , 𝛴𝛴𝛴𝛴 ,! ! ) allows deducing the expression of each coefficient. The first four coefficients can be easily found, by differentiating the expressions of the force N and beam length excess at the equilibrium.

𝑁𝑁𝑁𝑁 ,! ! = 𝑛𝑛𝑛𝑛 ! 𝐿𝐿𝐿𝐿 . 𝜆𝜆𝜆𝜆 ,! ! -𝑁𝑁𝑁𝑁 ! (3.2.6a) 𝑁𝑁𝑁𝑁 ,! ! = 𝑛𝑛𝑛𝑛 ! 𝐿𝐿𝐿𝐿 . 𝜆𝜆𝜆𝜆 ,! ! + 2. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 (3.2.6b) 𝜆𝜆𝜆𝜆 ,! ! = 2. 𝜆𝜆𝜆𝜆 𝑇𝑇𝑇𝑇 (3.2.6c) 𝜆𝜆𝜆𝜆 ,! ! = 1 4 . 𝜑𝜑𝜑𝜑 ! ! -2. 𝜑𝜑𝜑𝜑 ! . 𝜑𝜑𝜑𝜑 ! ! 𝜑𝜑𝜑𝜑 ! . 𝑎𝑎𝑎𝑎 ! (3.2.6d)
In particular, the dependance of the entropy with respect to k is found using (3.2.5b). By integrating (3.2.7), we find the expression of the beam's entropy given in

𝛴𝛴𝛴𝛴 ,! ! = 2. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘. 𝜆𝜆𝜆𝜆 ,! ! + 𝑁𝑁𝑁𝑁 ! . 𝜆𝜆𝜆𝜆 ,! ! (3.2.7)
The advantage of this last method is that it enables to describe the beam as a classical thermodynamic body, whose constitutive equations are given by (3.2.8a) and (3.2.8b). In these matrices, the diagonal terms represent direct effects whereas antidiagonal terms represent coupling terms.

𝑑𝑑𝑑𝑑 𝛴𝛴𝛴𝛴 𝑁𝑁𝑁𝑁 = 𝐶𝐶𝐶𝐶 ! 𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# -𝑁𝑁𝑁𝑁 ,! ! 𝑁𝑁𝑁𝑁 ,! ! 𝑁𝑁𝑁𝑁 ,! ! . 𝑑𝑑𝑑𝑑 𝑇𝑇𝑇𝑇 𝜆𝜆𝜆𝜆 (3.2.8a) 𝑑𝑑𝑑𝑑 𝛴𝛴𝛴𝛴 𝑁𝑁𝑁𝑁 = 𝐶𝐶𝐶𝐶 ! 𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# 𝜆𝜆𝜆𝜆 ,! ! . 𝑁𝑁𝑁𝑁 ,! ! -𝜆𝜆𝜆𝜆 ,! ! . 𝑁𝑁𝑁𝑁 ,! ! 𝑁𝑁𝑁𝑁 ,! ! 𝑁𝑁𝑁𝑁 ,! ! . 𝑑𝑑𝑑𝑑 𝑇𝑇𝑇𝑇 𝑘𝑘𝑘𝑘 (3.2.8b)
We can especially express the heat capacitance of the beam as

𝐶𝐶𝐶𝐶 ! = 𝐶𝐶𝐶𝐶 ! + 4. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 -𝑘𝑘𝑘𝑘. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 + 2. 𝑁𝑁𝑁𝑁 ! . (𝑇𝑇𝑇𝑇 -𝑇𝑇𝑇𝑇 ! ) . 𝜆𝜆𝜆𝜆. (𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# ) (𝑇𝑇𝑇𝑇 -𝑇𝑇𝑇𝑇 ! ) ! (3.2.9a) 𝐶𝐶𝐶𝐶 ! = 𝐶𝐶𝐶𝐶 ! + 4. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 -𝑘𝑘𝑘𝑘. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑘𝑘𝑘𝑘. 𝐿𝐿𝐿𝐿 -8. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘. 𝜆𝜆𝜆𝜆 𝜆𝜆𝜆𝜆 ,! ! . 𝜆𝜆𝜆𝜆. (𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# ) (𝑇𝑇𝑇𝑇 -𝑇𝑇𝑇𝑇 ! ) ! (3.2.9b)
What is interesting for us it to plot the evolution of the beam's entropy and the derivative of the entropy as a function of the temperature (Fig. 3.2.1a). These two figures show that the beam's entropy presents two discontinuities at TS and TSB. The discontinuities are also visible on the plot of the beam's heat capacitance that becomes infinite for these temperatures. In fact these discontinuities correspond to the thermal hysteressis observed on the evolution of N. N and Σ being the first derivatives of the Helmholtz free energy, these discontinuities reveal that the thermal snap-through corresponds to a first-order phase change, which stable phase is the unsnapped state of the beam, and the paraphase is the snapped beam [START_REF] Uzunov | Introduction to the theory of critical phenomena[END_REF], [Callen, 1960]. The existence of this hysteresis on the (T,Σ) diagram is particularly interesting since the area enclosed by the loop corresponds to a work. It shows that if the unsnapped beam is heated until it snaps, then cools down until it snaps back, the loop is followed counter-clockwise. The area is then negative, meaning that the beam behaves like a heat engine and produces work from two heat sources. The different steps of the thermodynamic cycle of the bimetallic strip heat engine are presented in Fig. 3.2.1c. The work supplied by our heat engine can then be compared to the work produced by a Carnot engine in the same conditions (Fig. 3.2.1.b).

By comparing the two areas, it can be observed that the effiency of the heat engine is reduced, which can be attributed to the weak thermo-mechanical coupling effect inside solid materials.

Two last features associated to the first order transition must be cleared. As with other first-order transitions, the thermal snap-through is characterized by a Curie temperature at which the bistability disappears. This temperature is none other that the temperature given by the bistability criterion.

𝑇𝑇𝑇𝑇 ! = 𝐼𝐼𝐼𝐼 ! 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! + 𝐹𝐹𝐹𝐹 ! 𝑁𝑁𝑁𝑁 ! (3.2.11)
As the beam must increase its entropy to snap and to decrease it to snap back, an extra amount of energy must be supplied to the beam at TS and rejected at TSB. These two latent heats are proportional to the entropy's discontinuities and are given by (3.2.12).

ℒ ! = 𝑇𝑇𝑇𝑇 ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (𝑇𝑇𝑇𝑇 ! ) (3.2.12)

Intrinsic mechanical performances of the harvester

Now that we have established all the equations necessary to describe the bimetallic strip heat engine, we are able to evaluate its performances. Taking the example of the Al-Invar Chapter 2|Thermo-mechanical modeling of the bimetallic strip heat engine beam, we first evaluate the latent heat associated to the beam's snap and snap-back (Fig. 3.3.1a) and the kinetic energy released in the same time (Fig.3.3.1b). In the configuration where the kinetic energy is transferred to a piezoelectric transducer, the sum of these two quantities represents the maximal thermo-mechanical efficiency of the harvester. For a 3K-hysteresis beam, the available mechanical energy is of 115 µJ whereas for a 20 K-hysteresis beam it is 1.52 mJ.

𝒲𝒲𝒲𝒲 ! = 𝒦𝒦𝒦𝒦 𝑇𝑇𝑇𝑇 ! + 𝒦𝒦𝒦𝒦(𝑇𝑇𝑇𝑇 !" ) (3.3.1)
Another important parameter of the heat engine is its thermo-mechanical efficiency. This efficiency is obtained by dividing the total amount of energy by sum of Qhot the heat provided by the hot source to heat the bimetal plus the latent heat associated to the beam's snap (3.3.1). This effiency can be compared to the Carnot efficiency representing the maximal theoretical efficiency of a heat engine. The formula of this relative efficiency is given by (3.3.2b).

𝜂𝜂𝜂𝜂 !"

!"# = 𝒦𝒦𝒦𝒦 𝑇𝑇𝑇𝑇 ! + 𝒦𝒦𝒦𝒦 𝑇𝑇𝑇𝑇 !" 𝑄𝑄𝑄𝑄 !!" + ℒ ! (𝑇𝑇𝑇𝑇 ! )

(3.3.2a) 𝜂𝜂𝜂𝜂 !" ! = 𝑇𝑇𝑇𝑇 ! + 𝜃𝜃𝜃𝜃 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !" . 𝜂𝜂𝜂𝜂 !" !"# (3.3.2b) Fig.3.3.1c
shows the evolution of the thermal efficiency of the harvester as a function of the thermal hysteresis of the bistable beams. This figure must be compared with Fig.3.3.1d showing the dependence of the Carnot efficiency on the thermal hysteresis. The thermal efficiency is better for highly unstable beams than for beams with weak hystereses, but the Carnot efficiency rapidly falls under one percent when the thermal hysteresis widens. This means that large thermal hysteresis enables to harvest more energy but the transformation followed by the beam in these conditions is less reversible.

As an example, the thermal efficiency of a 3K-hysteresis beam is of 0.01% and its relative efficiency is 1.2%. In the same time, the efficiency of a 20K-hysteresis beam is 0.02% for a Carnot efficiency of 0.38%.

We can now study the performances of nine different bimetals in order to evaluate the impact of the physical properties of materials. These bimetals are made of a passive metallic layer with a low coefficient of thermal expansion (Invar, tungsten, titanium) and an active metallic layer highly sensitive to temperature variations. (copper, brass, aluminum). The bimetals are designed to have thermal hystereses of 1 K, 2 K and 3 K. The choice of these low values of thermal hysteresis makes sense when the aim of these heat engines is to harvest low-grade heat energy: as explained in the first chapter, small hystereses allow to harvest heat in a wider range of temperatures and potentially to reach higher switching frequencies, in order to increase the power generated by the bimetals.

The nine bimetals all have the same dimensions (20x2x0.5 mm 3 ). The thickness ratio of both materials is chosen so that the neutral axis is located at the limit between the metallic layers, and the residual stress asymmetry is adjusted so that the hystereses be centered around 75°C (348 K). Tab.3.3.1 reports the bimetals' main properties: the bending stiffness ie serves to calculate the buckling load (6), whereas the ratio of the thermal stress coefficients describes the bimetal's thermal behavior: a bimetal with a high value of m α /n α is more sensitive to the temperature and will bend easily. The evolution of m α is slightly different and is higher for tungsten-based bimetals than for Invar-based bimetals because of the stiffness of tungsten: the active layer can thus be thicker than the passive tungsten layer, then causing an increase of the thermal asymmetry. Fig. 3.3.2b reports the residual stress measured for the 2 K-hysteresis beams and calculated by using Po and the snap temperature. Given the high sensitivity of the bimetals to residual stress, the stress levels for 1 K and 3 K are almost equal. This figure shows that the residual stress is more important for Al-W bimetals since they have the greatest value of M α . To obtain hysteresis around 75°C, the asymmetry of the residual stress must be more important for these bimetals.

Fig. 3.3.2c shows that the energy density is directly correlated to the thermal stress asymmetry m α : this density is highest for W/Al bimetals. By normalizing the surface density by the total deflection, we obtain a volume energy density independent of the size of the beam if the beam is proportionally scaled down. We can compare the values of surface densities with those calculated with the data reported in [Ravindran et al, 2014] and [START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF]. [Ravindran et al, 2014] gave a simulated density of around 83 µJ.cm-2 (200 µJ; 40x6x0.28 mm 3 ; 5 K hysteresis) while [START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF] reports experimental mechanical energy densities of around 60 µJ.cm -2 (Invar/NC4; 36x18x0.3 mm 3 ; 3 K hysteresis). This first comparision validates the coherence of our mode (in Section 4.4 we go further in comparing our model with experiments). Table 3.3

Synthesis on the first model

In this part, we developed a first model to describe the properties of bistable composite beams that are simply supported and submitted to the effect of residual stress. By extension, we studied the case of initially-curved beams and extended the criterion of bistability demonstrated by Timoshenko who neglected the effect of the mean thermal expansion. This first model enabled us to demonstrate the first-order phase transition associated to the thermal snap-through of composite beams and to evaluate the performances of the bimetallic strip heat engines for the first time. However, we have seen that this model does not allow to simply predict the influence of each physical parameters of the beams. For this reason we propose in the next part to approximate the solution of the problem by using the Galerkin method. This second model hereunder will enable to find analytical expressions describing the performances of the heat engine and to create Figures of Merit to compare various composite beams.

A second model of the instability of composite beams

Ritz approximation, Galerkin method, Cardan formula

There are two way to proceed in order to obtain the simplified equilibrium equations of the simply supported beams: the first consists in looking for solutions of the non-linear Euler equation under the form of a function series verifying the linear equations, and the second to write the strain energy function associated to the solutions of the linear Euler Euqation and to derivate it in order to obtain the equilibrium equation. Both methods give the same results, and we expose here the first method which is widely used for the study of the instability of mechanical structures. The static equilibrium equation of composite beams is recalled below. In order to simplify the equations, the beam is now assumed to be located between x = 0 and x = L. This assumption does not modify the problem.

𝑘𝑘𝑘𝑘 ! . 𝑤𝑤𝑤𝑤 ,!! + 𝑤𝑤𝑤𝑤 ,!!!! = 0 (4.1.1a) 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! = 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 - 𝑁𝑁𝑁𝑁 ! 2. 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ,! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! ! ! ! -𝐹𝐹𝐹𝐹 ! (4.1.1b) 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!! 0 = 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!! 𝐿𝐿𝐿𝐿 = 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 ; 𝑤𝑤𝑤𝑤 0 = 𝑤𝑤𝑤𝑤 𝐿𝐿𝐿𝐿 = 0 (4.1.1c)
The general solution of the Euler equation ( 4 Galerkin's method leads to equation (4.1.3a). When writing Euler's equation, we see that the assymmetrical modes are independent of temperature and can only appear when the axial thrust equals the even multiple of the first Euler buckling load.

𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑛𝑛𝑛𝑛 ! . 𝑤𝑤𝑤𝑤 ! ! !! !!! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! + 𝐹𝐹𝐹𝐹 ! = 0 (4.1.3a) 𝑘𝑘𝑘𝑘 ! -(2. 𝑖𝑖𝑖𝑖 + 1) ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 !.!!! = 4 2. 𝑖𝑖𝑖𝑖 + 1 . 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 -𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! (4.1.3b) 𝑘𝑘𝑘𝑘 ! -(2. 𝑖𝑖𝑖𝑖) ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 !.! = 0 (4.1.3c)
If the study is limited to the first two buckling modes, the equations become

𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . (𝑤𝑤𝑤𝑤 ! ! + 4. 𝑤𝑤𝑤𝑤 ! ! ) -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! + 𝐹𝐹𝐹𝐹 ! = 0 (4.1.4a) 𝑘𝑘𝑘𝑘 ! - 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! = 4 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 -𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! (4.1.4b) 𝑘𝑘𝑘𝑘 ! -4. 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! = 0 (4.1.4c)
In that case, as long as the second buckling mode does not appear, the equilibrium equation is given by a third-polynomial of the beam's amplitude. Following the same reasoning than before, the criterion of bistability can be deduced by noticing that for T = To, three real solutions exist if the force Po exceeds the first Euler buckling load. In order to find the temperatures associated to the limit points, two methods can be used. First, the equilibrium equation can be derivated in order to obtain a second equation, which gives the critical value of the beam's deflection amplitude at this point, once combined with the critical equation.

𝑁𝑁𝑁𝑁

! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! + 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹 ! = 0 (4.1.7)
The second method consists in using Cardan's formula for the resolution of third-order polynomials. This way, we directly find that the limit points are characterized by a critical amplitude verifying (4.1.8a). When the beam switches, its deflexion is finally equal to twice this critical value. The snap and snap-back temperatures are then given by the nullity of the discriminant (4.1.8c). This last equation enables to directly plot the (No,T) diagram of equilibrium of the composite beam showing the occurrence of the bistability, the area where the beam is unstable, and the areas where the snapped position (paraphase) is stable and where the unsnapped state is stable.

𝑤𝑤𝑤𝑤 ! ! = 6 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 ! -𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 ! + 𝐹𝐹𝐹𝐹 ! ; 𝑤𝑤𝑤𝑤 !" ! = 6 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 !" -𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 !" + 𝐹𝐹𝐹𝐹 ! (4.1.8a)
𝑤𝑤𝑤𝑤 ! ! = -2. 𝑤𝑤𝑤𝑤 ! ! ; 𝑤𝑤𝑤𝑤 !" ! = -2. 𝑤𝑤𝑤𝑤 !" ! (4.1.8b) 𝛥𝛥𝛥𝛥 = 27. 𝑁𝑁𝑁𝑁 ! . 𝑚𝑚𝑚𝑚 ! ! . 𝑇𝑇𝑇𝑇 - 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! ! + 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹 ! ! = 0 (4.1.8c)
This discriminant must be studied to explain when bistability occurs and to find the expression of the snap and snap-back temperatures as a function of the force Po. By defining the parameters p and q by (4.1.9a) and (4.1.9b) and X by (4.1.9c), we find that the study of Δ is equivalent to the study of (4.1.9d

𝑋𝑋𝑋𝑋 ! = 𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 -𝐹𝐹𝐹𝐹 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! (4.1.9c) 1 27. 𝑁𝑁𝑁𝑁 ! . 𝑋𝑋𝑋𝑋 ! - 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 . 𝑋𝑋𝑋𝑋 ! + 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 . 𝑖𝑖𝑖𝑖 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! = 0 (4.1.9d)
The discriminant of this polynomial is then given by (4.1.10a). It shows that if X is located between the values given by (4.1.10a), the beam is bistable. The values of X associated to the snap and snap-back temperatures are given by (4.1.10b-d). The temperatures are then found by using the definition of X given by (4.1.9c). These approximated formulae enable to depict the evolution of the hysteresis temperature with very good accuracy (see Fig. 4.1.1).

Δ = 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! - 4. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! ! . 𝐿𝐿𝐿𝐿 ! (4.1.9a) 𝑋𝑋𝑋𝑋 ! = 3. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 . 1 + 2. cos 𝜙𝜙𝜙𝜙 -𝜋𝜋𝜋𝜋 3 (4.1.9b) 𝑋𝑋𝑋𝑋 !" = 3. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 1 + 2. cos 𝜙𝜙𝜙𝜙 + 𝜋𝜋𝜋𝜋 3 (4.1.9c) cos 𝜙𝜙𝜙𝜙 = 𝑁𝑁𝑁𝑁 ! ! . 𝐿𝐿𝐿𝐿 ! 2. 𝑀𝑀𝑀𝑀 ! ! . 𝑛𝑛𝑛𝑛 ! . 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! + 2. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! ! . 𝐿𝐿𝐿𝐿 ! (4.1.9d)

Simplification of the model

Solving the previous equations in order to find analytical expressions of the performances of the heat engine is not easy. The difficulty arises mainly from the cross term of the beam's deflexion amplitude and the temperature due to the mean thermal expansion represented by n α . Great simplifications can be derived from the assumption that thermal expansion has a weak effect on the beam's mechanical behaviour if the temperature is between the snap and the snap-back temperatures. The equilibrium equation given in (4.1.1) can thus be simplified into (4.2.1)

𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! + 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 ! + 4 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 = 0 (4.2.1)
Hence by derivating this equation we directly conclude that the limit points are located in

𝑤𝑤𝑤𝑤 !" ! = -𝑤𝑤𝑤𝑤 ! ! = 2. 𝐿𝐿𝐿𝐿 𝜋𝜋𝜋𝜋 1 3. 𝑁𝑁𝑁𝑁 ! . 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! (4.2.2)
By injecting this equation in (4.2.1), we show that the thermal hysteresis is centered around the value To and verifies (4.2.3a). The snap and snap-back temperatures are then given by (4.2.3b) and (4.2.3c), and the beam's deflexion at the limit points can directly be expressed as a function of the thermal hysteresis. Logically with these equations we observe the stress force No is, the more unstable the beam becomes and its thermal hysteresis widens and the displacement of the beam increases.

Δ𝑇𝑇𝑇𝑇 = 2. 𝐿𝐿𝐿𝐿 𝑀𝑀𝑀𝑀 ! . 1 27. 𝑁𝑁𝑁𝑁 ! . 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! ! (4.2.3a) 𝑇𝑇𝑇𝑇 ! = 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! + Δ𝑇𝑇𝑇𝑇 2 (4.2.3b) 𝑇𝑇𝑇𝑇 !" = 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! - Δ𝑇𝑇𝑇𝑇 2 (4.2.3c) 𝑤𝑤𝑤𝑤 ! ! = - 1 𝜋𝜋𝜋𝜋 . 4. 𝑀𝑀𝑀𝑀 ! . 𝐿𝐿𝐿𝐿 ! 𝑁𝑁𝑁𝑁 ! . Δ𝑇𝑇𝑇𝑇 ! (4.2.3d)

Mechanical performances of the energy harvester

Now that we have validated the approximation on the effect of the mean thermal expansion, we establish the expression of the performance of the harvester. The main physical parameters are represented in Fig. 4.3.1a. At the limit points the shape of the strain energy well can be developped by taking into account the expression of the snap and snap-back temperatures. Equation (4.3.1) shows that the system oscillates around a position between ws c and -3.ws c , representing the positions having the same potential energy.

𝒲𝒲𝒲𝒲 ! = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 32. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! -𝑤𝑤𝑤𝑤 ! ! ! . 𝑤𝑤𝑤𝑤 ! + 3. 𝑤𝑤𝑤𝑤 ! ! + 3. 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 32. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! ! (4.3.1)
When the beam snaps, it releases an amount of kinetic energy verifying (4.3.2).

Δ𝒦𝒦𝒦𝒦 𝑤𝑤𝑤𝑤 ! = 𝒲𝒲𝒲𝒲 ! 𝑤𝑤𝑤𝑤 ! -𝒲𝒲𝒲𝒲 ! 𝑤𝑤𝑤𝑤 ! ! = 27. 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 32. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! ! (4.3.2)
The kinetic energy release also has (4.3.2) for expression when the beam snaps back, since the strain energy well is spatially symmetrical because the thermal expansion is neglected. The available mechanical energy produced by the composite beam is then twice this value. By using the expression (4.2.3d), it can be expressed as a function of the thermal hysteris, showing a4/3-power dependence. Its evolution is plotted against the beam's thermal hysteresis in Fig. 4.3.1e, showing the relatively good agreement between the reduced model and the exact model exposed in Section 2. The underestimation of the mechanical energy is due to the effect of the mean thermal expansion that increases the snap energy more than it decreases the snap-back energy. Globally, the sum of these two quantities is higher than the energy quantity predicted by the reduced model

Δ𝒲𝒲𝒲𝒲 ! = 2. Δ𝒦𝒦𝒦𝒦 𝑤𝑤𝑤𝑤 ! = 27. 𝑀𝑀𝑀𝑀 ! . Δ𝑇𝑇𝑇𝑇 ! 16. 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 ! (4.3.3)
The maximal force produced by the switching beam is obtained at the positionws c . This force is a linear function of the thermal hysteresis, as shown in Fig.4.3.1b. Values between 1 and 2 N can be measured when the thermal hysteresis ranges between 10 and 20°C.

𝐹𝐹𝐹𝐹 !"# = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! ! = 2. 𝑀𝑀𝑀𝑀 ! . 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (4.3.4)
In the frame of the Ritz approximation, the kinetic energy is given by (4.3.5).

𝒦𝒦𝒦𝒦 = 1 2 . 𝓂𝓂𝓂𝓂 ! 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! = 1 4 . 𝓂𝓂𝓂𝓂 ! . 𝑤𝑤𝑤𝑤 ! ! (4.3.5)
This expression enables to extract some dynamic parameters associated to the snapthrough. Assuming that the beam has a null speed a null acceleration at the limit point, we can express the formula of the speed, the power, and the natural frequency of the switching beam. However, the switching duration cannot be found since the integral (4.3.6) is indefinite. We will further study the beam's dynamic behaviour later.

Δ𝑡𝑡𝑡𝑡 = 1 𝑤𝑤𝑤𝑤 ! . 𝑑𝑑𝑑𝑑𝑤𝑤𝑤𝑤 ! ! ! ! ! ! ! = 2. 𝓂𝓂𝓂𝓂 ! . 𝐿𝐿𝐿𝐿 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 𝑤𝑤𝑤𝑤 ! ! . 𝑤𝑤𝑤𝑤 ! + 3. 𝑤𝑤𝑤𝑤 ! ! 𝑤𝑤𝑤𝑤 ! ! -𝑤𝑤𝑤𝑤 ! ! ! ! ! ! ! (4.3.6)
By approximating the potential well as a second-order polynomial in the vicinity of the stable position, the natural frequency of vibration of the composite beam can be expressed by (4.3.7). Its evolution is plotted against the thermal hysteresis in Fig. 4.3.2a. The fact that the frequency ranges between 500 and 1500 Hz for a hysteresis between 1 and 30°C explains why the jumping poppers can produce audible sound when it snaps, and is in agreement with the results of [START_REF] Pandey | Dynamics of snapping beams and jumping poppers[END_REF].

𝑓𝑓𝑓𝑓 ! = 3. 𝜋𝜋𝜋𝜋 4 . 𝑤𝑤𝑤𝑤 ! ! 𝐿𝐿𝐿𝐿 . 𝑁𝑁𝑁𝑁 ! 𝓂𝓂𝓂𝓂 ! . 𝐿𝐿𝐿𝐿 (4.3.7)
The switching speed is given by (4.3.8). We find that the maximal speed is reached when the strain energy is minimal. The maximal speed is about 1.3 m.s -1 for a 10 K-hysteresis beam (Fig.4.3.1c).

𝑣𝑣𝑣𝑣 !"# = 𝜋𝜋𝜋𝜋 ! . 27. 𝑁𝑁𝑁𝑁 ! 8. 𝓂𝓂𝓂𝓂 ! . 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! ! (4.3.8)
The power generated by the beam is the product of the speed by the force of the bimetal. By derivating (4.3.1), we find that the power is maximal for two values of the beam's deflexion amplitude. For these values, the power is given by (4.3.9d) and (4.3.9e) and is plotted in Fig. 4.3.1d.

𝑃𝑃𝑃𝑃 ! = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8 8. 𝐿𝐿𝐿𝐿 ! . 𝑁𝑁𝑁𝑁 ! 𝓂𝓂𝓂𝓂 ! . 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ! ! -𝑤𝑤𝑤𝑤 ! ! . 𝑤𝑤𝑤𝑤 ! + 2. 𝑤𝑤𝑤𝑤 ! ! . (𝑤𝑤𝑤𝑤 ! ! -𝑤𝑤𝑤𝑤 ! ). 𝑤𝑤𝑤𝑤 ! + 3. 𝑤𝑤𝑤𝑤 ! ! (4.3.9a) 𝑤𝑤𝑤𝑤 !! = -2 + 15 5 . 𝑤𝑤𝑤𝑤 ! ! ≈ -2,78. 𝑤𝑤𝑤𝑤 ! ! (4.3.9b) 𝑤𝑤𝑤𝑤 !! = -2 - 15 5 . 𝑤𝑤𝑤𝑤 ! ! ≈ -1,23. 𝑤𝑤𝑤𝑤 ! ! (4.3.9c) 𝑃𝑃𝑃𝑃 𝑤𝑤𝑤𝑤 !! ≈ -1,70. 𝜋𝜋𝜋𝜋 ! . 𝑁𝑁𝑁𝑁 ! 𝑤𝑤𝑤𝑤 ! ! ! 𝐿𝐿𝐿𝐿 ! . 𝑁𝑁𝑁𝑁 ! 𝓂𝓂𝓂𝓂 ! . 𝐿𝐿𝐿𝐿 (4.3.9d) 𝑃𝑃𝑃𝑃 𝑤𝑤𝑤𝑤 !! ≈ 0,75. 𝜋𝜋𝜋𝜋 ! . 𝑁𝑁𝑁𝑁 ! 𝑤𝑤𝑤𝑤 ! ! ! 𝐿𝐿𝐿𝐿 ! . 𝑁𝑁𝑁𝑁 ! 𝓂𝓂𝓂𝓂 ! . 𝐿𝐿𝐿𝐿 (4.3.9e)
To finish, we study the thermal properties of the bistable beams and the efficiency of the transduction. The entropy discontinuity is given by (4.3.10a). This equation leads to the analytical solutions of the latent heats of snap and snap-back.

𝐿𝐿𝐿𝐿 ! = 6. 4 ! . 𝜃𝜃𝜃𝜃 !"# + 𝑀𝑀𝑀𝑀 ! 𝑚𝑚𝑚𝑚 ! + Δ𝑇𝑇𝑇𝑇 2 . 𝑚𝑚𝑚𝑚 ! ! 𝐿𝐿𝐿𝐿. 𝑛𝑛𝑛𝑛 ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ! (4.3.10b) 𝐿𝐿𝐿𝐿 !" = -6. 4 ! . 𝜃𝜃𝜃𝜃 !"# + 𝑀𝑀𝑀𝑀 ! 𝑚𝑚𝑚𝑚 ! - Δ𝑇𝑇𝑇𝑇 2 . 𝑚𝑚𝑚𝑚 ! ! 𝐿𝐿𝐿𝐿. 𝑛𝑛𝑛𝑛 ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ! (4.3.10c)
In order to evaluate the efficiency of the transduction, we first evaluate the heat provided to the beam by the hot source during the heating phase. The total heat supplied by the hot source is the sum of the heat provided to the beam to increase its temperature from TSB to TS, and the latent heat needed for the beam to snap.

𝑄𝑄𝑄𝑄 !"! = 𝐶𝐶𝐶𝐶 ! . Δ𝑇𝑇𝑇𝑇 + 8. 4. 𝑚𝑚𝑚𝑚 ! ! 𝐿𝐿𝐿𝐿. 𝑛𝑛𝑛𝑛 ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ! . 𝜃𝜃𝜃𝜃 !"# + 𝑀𝑀𝑀𝑀 ! 𝑚𝑚𝑚𝑚 ! (4.3.11) (c) (d)
In this case, the efficiency of our heat engine, which is the ratio between the mechanical energy generated and the heat supplied by the hot source, shows a cube-root dependence on the thermal hysteresis for large values of this thermal hysteresis. The Carnot efficiency is obtained by normalizing the efficiency of the heat engine by the Carnot efficiency in the same conditions.

𝜂𝜂𝜂𝜂 !" !"# = Δ𝒲𝒲𝒲𝒲 ! 𝑄𝑄𝑄𝑄 ! = 27. 𝑀𝑀𝑀𝑀 ! ! 16. 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 ! (4.3.12a) 𝜂𝜂𝜂𝜂 !" !"# = 𝑇𝑇𝑇𝑇 ! + 𝜃𝜃𝜃𝜃 ! Δ𝑇𝑇𝑇𝑇 . 𝜂𝜂𝜂𝜂 !" !"# = 27. 𝑀𝑀𝑀𝑀 ! ! 16. 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! + 𝜃𝜃𝜃𝜃 ! 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! (4.3.12b)
In the case of a beam with small thermal hysteresis, and as long as the latent heat can be neglected, the thermal hysteresis can also be neglected with respect to the mean temperature Mo/m α . The relative efficiency then shows a -2/3 exponential dependence on the thermal hysteresis (4.3.13), linked to the fact that the Carnot efficiency increases linearly with the difference in temperature between the hot source and the cold source.

𝜂𝜂𝜂𝜂 !" !!" ≈ 27. 𝑀𝑀𝑀𝑀 ! ! 16. 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! + 𝜃𝜃𝜃𝜃 !"# 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! (4.3.12)

Experimental studies and validity of the reduced-order model

In order to attempt to explain the properties of the bimetals used in the current harvester, we realized a series of measurement on bimetallic shells (Fig. 4.4.1a). The main problem with these membranes is that they do not behave like simply-supported curved beams, but like free curved shells. The model we developed in this chapter would not normally fit the experimental data concerning the performances of the harvesters. However, by assuming that the buckled shape of these membranes is cut into a square membrane (Fig. 4.4.1b), the deflection of the beam can be described by (4.4.1) according to [START_REF] Bryan | On the Stability of a Plane Plate under Thrusts in its own Plane, with Applications to the "Buckling" of the Sides of a Ship[END_REF].

𝑤𝑤𝑤𝑤 𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦 = 𝑤𝑤𝑤𝑤 ! (𝑡𝑡𝑡𝑡 The bimetallic shells studied in this part are made of NC4 alloy as active layer and Invar as passive layer. The bistability of these shells comes from the dent stamped on their center which gives them an initial curvature. The dimensions of the shells are 3.6x1.8x0.03 cm 3 . Shells of various hystereses have been chosen with typical values centered around 3, 6, 9, 12 and 15°C (Fig. 4.4.1c). For all these bimetal thermostats, we measured the switching speed using a velocimeter (Polytec OFV-505). Once the switching speed is measured, it is possible to find the value of the mechanical energy produced by the harvester by using thermostat. In Figures 4.4.1 (next page), we superimpose the various values predicted by the reduced model in the same conditions. The comparison between these curves and the experimental data shows that it is possible to explain the tendencies observed on the harvester in operation. Fig.4.4.1d shows that by adjusting the speed given by the model by the factor calculated with the expression of the kinetic energy (4.4.2b), it is possible to fit with a relatively good precision the speed measured on the harvester prototypes. With these results we can conclude that by adding fitting parameters to the reduced model, it would be possible to create a compact model able to explain the performances of the future devices.

Figures of Merit

It is also important to compare various bimetallic couples used as prime mover in the bimetallic strip heat engine in order to see the impact of the materials properties on the performances of the heat engine. To do so, we observe that the harvested energy and the efficiencies are functions of the beam's parameters and geometry. We thus define two Figures of Merit, 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 describing the potential energy generated by the bimetallic beams (4.5.1), and 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ! the efficiency of the thermo-mechanical transduction (31). Using (4.5.1) and (4.5.2), we compare sixteen different bimetal couples, made of an active layer (aluminum, brass (CuZn), copper (Cu) or gold (Au)) and a passive layer (Invar (FeNi), tungsten (W), titanum (Ti), platinum (Pt)). Even if the choice of these materials is not always realistic since some materials like Gold would not be used, it enables to describe a wide range of values of physical properties. To compare these bimetals, we choose the same beam geometrie (20 x 2 x 0.5 mm) with the same thickness for each material layer. Fig. 4.5.1 represents both Figures of Merit for these bimetals. As we can see, there is a strong correlation between both parameters. For these bimetals, their heat capacity is very similar, the difference between each bimetal is mainly linked to their axial stiffness and above all to their thermal asymmetry m α . Logically, we observe that bimetallic beams with high thermal asymmetry can generate more energy because they are more easily bistable.

Similarly, they convert a larger part of heat into strain energy and are thus more efficient in fine. Among these seventeen bimetals, we find that the best bimetals are the couples CuZn-FeNi for harvested energy and CuZn-W for efficiency. Because bimetallic beams made of FeNi or W give the best efficiencies, it could be concluded that the choice of the passive layer has a great influence on the performance of the bimetallic strip heat engine

Study of the impact of the architecture on the performances

The use of the Ritz approximation of the beam deflexion shape and of the Galerkin method is very useful to study the impact of the harvester's architecture on the composite beam's bistability. We define the two deflections wbot and wtop by (4.6.1a-b), which must verify (4.7.1c) so that the system works.

𝑤𝑤𝑤𝑤 !"# = 𝑑𝑑𝑑𝑑 !"# -𝑧𝑧𝑧𝑧 ! (4.6.1a) 𝑤𝑤𝑤𝑤 !"# = 𝑑𝑑𝑑𝑑 !"# + 𝑡𝑡𝑡𝑡 -𝑧𝑧𝑧𝑧 ! (4.6.1b) 𝑤𝑤𝑤𝑤 !"# ≤ 𝑤𝑤𝑤𝑤 ! ! ; 𝑤𝑤𝑤𝑤 !"# ≤ |𝑤𝑤𝑤𝑤 ! ! | (4.6.1b)
The existence of the force Q applied by the architecture on the beam modifies the equilibrium equation which becomes (4.6.2a) when the beam is snapped, and (4.6.2b) when the beam is unsnapped

𝐹𝐹𝐹𝐹 !"# = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 !"# ! + 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 !"# + 2𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . (𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇) (4.6.2a) -𝐹𝐹𝐹𝐹 !"# = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 !"# ! + 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 !"# + 2𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . (𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇) (4.6.2b)
The snap and snap-back temperatures are then given by the nullity of these two equations, and the thermal hysteresis by (4.6.3)

Δ𝑇𝑇𝑇𝑇 !"# = 𝐿𝐿𝐿𝐿 2. 𝜋𝜋𝜋𝜋. 𝑀𝑀𝑀𝑀 ! . 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . (𝑤𝑤𝑤𝑤 !"# ! -𝑤𝑤𝑤𝑤 !"# ! ) + 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . (𝑤𝑤𝑤𝑤 !"# -𝑤𝑤𝑤𝑤 !"# ) (4.6.3c)
Once the beam snaps or snaps back, the architecture applies a maximal force on the beam, which can be found by combining (4.6.2a) and (4.6.2b).

𝐹𝐹𝐹𝐹 !"# !"# = -𝐹𝐹𝐹𝐹 !"# !"# = 2𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . 𝑀𝑀𝑀𝑀 ! . Δ𝑇𝑇𝑇𝑇 !"# (4.6.4)
When the beam is in contact with the hot source or the cold source, its strain energy is a simple linear function of the temperature. The kinetic energy released during the snap is then given by (4.6.5). The kinetic energy of snap-back is obtained by replacing wbot by wtop and reversely.

𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥(𝑇𝑇𝑇𝑇 ! ) = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 32. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 !"# ! + 3. 𝑤𝑤𝑤𝑤 !"# ! -4. 𝑤𝑤𝑤𝑤 !"# ! . 𝑤𝑤𝑤𝑤 !"# + 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 !"# -𝑤𝑤𝑤𝑤 !"# ! (4.6.5)
The total mechanical energy produced by the harvester is then given by (4.6.6a). When the deflections of the beam in contact with the heat sources are equal to a percentage of the critical deflection ws c (4.6.6b), the expression of the energy can be simplified into (4.6.6c).

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! !"# = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 !"# + 𝑤𝑤𝑤𝑤 !"# ! + 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 !"# -𝑤𝑤𝑤𝑤 !"# ! (4.6.6a) 𝑤𝑤𝑤𝑤 !"# = 𝑤𝑤𝑤𝑤 !"# = 𝑟𝑟𝑟𝑟 ! . |𝑤𝑤𝑤𝑤 ! ! | (4.6.6b) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! !"# = 3. 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 ! . 𝑟𝑟𝑟𝑟 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! + 2. 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑟𝑟𝑟𝑟 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! (4.6.6c)
With this last hypothesis, the thermal hysteresis becomes (4.6.7a) (Fig. 4.6.1a). We directly see that the kinetic energy release can be simply expressed as a function of the thermal hysteresis (4.6.7b).

Δ𝑇𝑇𝑇𝑇 !"# = 3 2 . 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 . Δ𝑇𝑇𝑇𝑇 (4.6.7a) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! !"# = 2. 𝐹𝐹𝐹𝐹 !"# !"# . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! = 8 9 . 𝑟𝑟𝑟𝑟 ! ! - 𝑟𝑟𝑟𝑟 ! ! 3 . 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! (4.6.7b)
The thermal efficiency can be approximated by (4.7.9) by neglecting the latent heat of snap, and by replacing the critical deflexion by its expression (4.2.3d). Using the expression of the thermal efficiency of the harvester outside de cavity (4.3.11a), we can find a linear relation between the two efficiencies. It shows that the best efficiency can be reached when the cavity thickness equals the deflection of the beam before it snaps and snaps back (rw=1). In that case, the efficiency of the harvester represents 59.3% of the efficiency of the efficiency of a beam that is not constrained by any architecture.

𝜂𝜂𝜂𝜂 !" !"#,!"# = 4. 𝑟𝑟𝑟𝑟 ! . 4. 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . Δ𝑇𝑇𝑇𝑇 ! = 16 27 . 𝑟𝑟𝑟𝑟 ! . 𝜂𝜂𝜂𝜂 !" !"# (4.6.8)
The Carnot efficiency is obtained by normalizing the thermal efficiency by the efficiency of the Carnot engine working between the snap and snap-back temperatures. By using the expression of the thermal hysteresis (4.6.7a), the Carnot efficiency of the transduction of the beam inside the cavity can be expressed as a function of the Carnot efficiency of the beam without the cavity (4.6.9b). This last equation shows that the Carnot efficiency of the harvester with rw=1 is of 59.3% of the efficiency of the not constrained beam and if rw tends to 0, it becomes equal to 39.5% of this maximal efficiency (by assuming that the latent heat is still negligible) (Fig. 4.6.1d).

𝜂𝜂𝜂𝜂 !" !"#,!"# = 𝜂𝜂𝜂𝜂 !"#,!"# Δ𝑇𝑇𝑇𝑇 !"# . 𝑇𝑇𝑇𝑇 ! + Δ𝑇𝑇𝑇𝑇 !"# 2 (4.6.9a) 𝜂𝜂𝜂𝜂 !" !"#,!"# = 32 27(3 -𝑟𝑟𝑟𝑟 ! ! ) . 𝜂𝜂𝜂𝜂 !" !"# (4.6.9b)

Synthesis on the second model

In this second model, we used a strong method to simplify the equations describing the state of the simply-supported composite beams. Using the Ritz approximation of the beam's shape, we proposed laws on the available mechanical energy and the efficiency of the harvester, and we created Figures of Merit to compare materials. However, we can observe that this model first assumed that the temperature was homogeneous through the whole beam. This leads to forget the role played by thermal transfers on the bistability, and to neglect the influence of the thermal conductivity of materials on the duration of the heating and cooling phases. In order to correct this mistake, the next part is devoted to the modeling of the thermal transfers inside the composite beams, and to the study of the dynamic behavior of the harvester.

5 Thermal transfers and dynamic behavior of the energy harvester

Equations of the thermal transfers

In the previous parts, the analysis of the equilibrium of the system was based on the principle of virtual displacement which was derived from the first thermodynamic principle. In order to study the thermal transfers and how they affect the bistability of the composite beams, we need to use the second thermodynamic principle which states that the system evolves so that the entropy of the systems is the sum of the entropies carried into the system by heat transfers (5.1.1a). The phenomenological law of Fourier [START_REF] Boley | Theory of Thermal Stresses[END_REF] gives a direct link between the heat flux and the thermal gradient for conductive heat transfers (5.1.1b), where κ represents the thermal conductivity of the material.

𝜎𝜎𝜎𝜎 ,! = 𝑞𝑞𝑞𝑞 ,! 𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# (5.1.1a) 𝑞𝑞𝑞𝑞 ,! = 𝛻𝛻𝛻𝛻𝑗𝑗𝑗𝑗 ! = -𝜅𝜅𝜅𝜅. 𝛻𝛻𝛻𝛻 ! 𝑇𝑇𝑇𝑇 (5.1.1b)
By locally writing the entropy as a function of temperature and the parameters of the beam, the equation of the second principle becomes (5.1.2). Because of the weak thermomechanical coupling inside solid materials, the variation of the entropy associated to the strain can be neglected, and the classical law of heat transfer manque la fin. In this way, the study of the thermal transfers in the harvesters can be decoupled from their consequences on the instability of the beam. Consideration of the influence of the strain on the thermal transfer has been treated in numerous publications among which (Novacki, 1970) and [START_REF] Fox | Generalised thermoelasticity[END_REF].

𝑐𝑐𝑐𝑐 𝒮𝒮𝒮𝒮 . 𝑇𝑇𝑇𝑇 ,! + 𝛼𝛼𝛼𝛼. (𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# ). 𝒮𝒮𝒮𝒮 !!,! = -𝜅𝜅𝜅𝜅. 𝛻𝛻𝛻𝛻 ! 𝑇𝑇𝑇𝑇 (5.1.2a) 𝑐𝑐𝑐𝑐 𝒮𝒮𝒮𝒮 . 𝑇𝑇𝑇𝑇 ,! = -𝜅𝜅𝜅𝜅. 𝛻𝛻𝛻𝛻 ! 𝑇𝑇𝑇𝑇 (5.1.2b)
In this part, conduction will be considered as the main form of heat transfer occurring in the heat engine. In order to obtain a model that is simple yet accurate, we represent the harvester as as beam punctually heated in its center and thermally insulated at its edges (Fig. 5.1.a). The boundary conditions of the problem are then given by the Robin condition at the interface between the heat sources and the beam (5.1.3a), and the Neuman condition (5.1.3b) at the beam's edges. In this equation h is the coefficient of thermal exchange of the medium.

𝛻𝛻𝛻𝛻𝛻𝛻𝛻𝛻 0 = ℎ 𝑘𝑘𝑘𝑘 . 𝑇𝑇𝑇𝑇 !"# -𝑇𝑇𝑇𝑇 0 (5.1.3a) 𝛻𝛻𝛻𝛻𝛻𝛻𝛻𝛻 𝐿𝐿𝐿𝐿 2 = 𝛻𝛻𝛻𝛻𝛻𝛻𝛻𝛻 - 𝐿𝐿𝐿𝐿 2 = 0
(5.1.3b)

Switched thermal capacitance model

In this first model, we assume that the thermal transfers are limited by the contact resistances between the heat sources and the bistable beam, so that the thermal gradient in the beam is negligible and the temperature is homogeneous across the beam. This hypothesis can be justified if the Biot number obtained with (5.1.2a) is weak. Given that the heat can spread in two directions, both through the beam's thickness and along the beam's length, we define two Biot numbers for the thermal transfer. The first Biot number compares the propagation of the heat through the thickness of the beam with the conduction at the interface of the heat source and the beam. Given the thickness of the beam, it can easily be assumed that the temperature is homogenous through the beam's thickness (Bi1<0.2). In these conditions, the error is inferior to 5% [START_REF] Rohsenow | Handbook of heat transfer[END_REF]. The second Biot number compares the propagation of the heat through the half length of the beam with the external heat supply. In this part, the difference of propogation in the beam due to the variation of the thermal conductivity of the layers composing the beam is neglected, and the thermal resistivity (1/κb) of the beam is taken as the average value of the resistivities of the materials. If we can assume that the second Biot number is also small, the system can simply be modeled with the equivalent lumped capacitance model of Fig. 5.2.1b where RCC and RCH represent the contact resistances between the cold source and the beam at the upper position and between the hot source and the beam at the lower position. The composite beam then behaves like a simple switched thermal capacitance that can switch at TS and snaps back at TSB.

𝐵𝐵𝐵𝐵𝑖𝑖𝑖𝑖 ! = ℎ. 𝑡𝑡𝑡𝑡 𝜆𝜆𝜆𝜆 ! (5.2.1a) 𝐵𝐵𝐵𝐵𝑖𝑖𝑖𝑖 ! = ℎ. 𝐿𝐿𝐿𝐿. 𝑙𝑙𝑙𝑙 !"#$%!$ 𝜅𝜅𝜅𝜅 ! . 𝑡𝑡𝑡𝑡 (5.2.1b) 1 𝜅𝜅𝜅𝜅 ! = 1 𝑡𝑡𝑡𝑡 . 1 𝜅𝜅𝜅𝜅 𝑧𝑧𝑧𝑧 ! ! . 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 (5.2.1c)
In that case, the mechanical equation of the reduced model is still valid (5.2.2a), and the equation of the thermal transfers is given by (5.2.2b).

𝓂𝓂𝓂𝓂 ! 2 . 𝑤𝑤𝑤𝑤 ,!! + 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! + 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 + 2. 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 = 0 (5.2.2a) 𝑞𝑞𝑞𝑞 ,! = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 𝑅𝑅𝑅𝑅 !" (5.2.2b)
Given that the beam is initially at TSB and switches at TS, the temporal evolution of the temperature is given by (5.2.3a) when the beam is unsnapped. Similarly, the evolution of the temperature when the beam cools down is given by (5.2.3b).

𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡 = 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! . 𝑒𝑒𝑒𝑒 ! ! ! !" .! ! + 𝑇𝑇𝑇𝑇 ! (5.2.3a) 𝑇𝑇𝑇𝑇 𝑡𝑡𝑡𝑡 = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . 𝑒𝑒𝑒𝑒 ! ! ! !! .! ! + 𝑇𝑇𝑇𝑇 ! (5.2.3b)
These formulae give the duration of the heating phase (5.2.4a) and cooling phase (5.2.4b) of a beam constrained in a cavity. Logically, we can observe that the harvester cannot work if the hot source is colder than the snap temperature, nor if the cold source is hotter than the snap-back temperature.

𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 !!"#$%& = 𝑅𝑅𝑅𝑅 !" . 𝐶𝐶𝐶𝐶 ! . ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !",!"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !,!"# (5.2.4b) 𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 !""#$%& = 𝑅𝑅𝑅𝑅 !! . 𝐶𝐶𝐶𝐶 ! . ln 𝑇𝑇𝑇𝑇 !,!"# -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !",!"# -𝑇𝑇𝑇𝑇 ! (5.2.4c)
Hence the cycling frequency of the system is

𝑓𝑓𝑓𝑓 = 1 𝐶𝐶𝐶𝐶 ! . 𝑅𝑅𝑅𝑅 !! . ln 𝑇𝑇𝑇𝑇 !,!"# -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !",!"# -𝑇𝑇𝑇𝑇 ! + 𝑅𝑅𝑅𝑅 !" . ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !",!"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !,!"# (5.2.5)
If the thermal hysteresis of the beam is weak compared to the external gradient TH-TC, it can be approximated at the first order by (5.2.6).

𝑓𝑓𝑓𝑓 ≈ 1 𝐶𝐶𝐶𝐶 ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) 𝑅𝑅𝑅𝑅 !! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! + 𝑅𝑅𝑅𝑅 !" . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) (5.2.6)
This expression clearly shows that for a given set of temperatures (TC, TH, To), the switching frequency is higher for weakly unstable beams than for beams with large thermal hystereses. Moreover, it directly expresses the frequency as a function of the equivalent resistance of the switched capacitance RCv (5.2.7a) and the heat flux qeff (5.2.7b). Assuming that the mean thermal hysteresis temperature To is equal to the average of the heat sources' temperatures, and that the contact resistances are equal, the thermal flux is given by (5.2.7c) and is a decreasing function of the thermal hysteresis.

𝑅𝑅𝑅𝑅 ! ! = 1 𝐶𝐶𝐶𝐶 ! . 𝑓𝑓𝑓𝑓 (5.2.7a) 𝑞𝑞𝑞𝑞 !"" = 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑅𝑅𝑅𝑅 !! . 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑇𝑇𝑇𝑇 !,!"# -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !",!"# -𝑇𝑇𝑇𝑇 ! + 𝑅𝑅𝑅𝑅 !" . 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !",!"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !,!"# (5.2.7b) 𝑞𝑞𝑞𝑞 !"" = 1 2. 𝑅𝑅𝑅𝑅 !! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.2.7c)
With this expression, we can deduce the expression of the mean mechanical power generated by a harvester, that is the product of the strain energy by the frequency of the cycle (5.2.8a). By using (5.2.7c), the available power (5.2.8b) and thermal efficiency (5.2.8c) can be expressed as a function of the beam's thermal hysteresis in the cavity. A Carnotrelative efficiency can also be defined as a function of the device's working conditions.

𝑃𝑃𝑃𝑃 ! = 𝑓𝑓𝑓𝑓. 𝛥𝛥𝛥𝛥𝐸𝐸𝐸𝐸 !"# (5.2.8a) 𝑃𝑃𝑃𝑃 ! = 𝑟𝑟𝑟𝑟. 72. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# ! ! 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.2.8b) 𝜂𝜂𝜂𝜂 !" !"#,!"# = 2. 𝑟𝑟𝑟𝑟. 72. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 !"# 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! (5.2.8c) 𝜂𝜂𝜂𝜂 !" !"#,!"# = 2. 𝑟𝑟𝑟𝑟. 72. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 !"# 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! + 273.15 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.2.8d)
It is interesting to plot the evolution of the output power of the generator (Fig. 5.2.2c) to see that it is not a monotonic function of the thermal hysteresis, but that it presents an optimal value for a given ratio of the beam's thermal hystereses. Clearly, for small thermal hystereses (either associated to naturally weakly unstable beams or to the cavity's thinning), the small amount of kinetic energy produced by the beam is counterbalanced by the increase of the beam's operating frequency (Fig. 5.2.2a) and the thermal flux (Fig. 5.2.2b). However, the thermal efficiency (Fig. 5.2.2d) and the Carnot efficiency remain small (Fig. 5.2.2e). For a contact resistance of 10 K.W -1 , the simulation of the operating frequency agrees with the experimental measurements (around 0.5 Hz). Closer simulated values can be obtained with a contact resistance of 35 K.W -1 . In these conditions, the average surface power density is around 20-30 µW.cm -2 for an efficiency around 0.005~0.01%. Here again, the simulated values are in agreement with experimental data. Logically, by decreasing this contact resistance, the device's performances are boosted by the increase of the switching frequency of the prime mover. For large thermal hystereses, the output power is degraded by the reduction of the operating frequency, whereas the efficiency of the transduction is maximized. When the beam's hysteresis equals the temperature difference between the two heat sources, the Carnot efficiency is maximal and is given by (4.7.10a). As with the Carnot heat engine, this efficiency means that the harvester does not supply output power, since the transformation must be infinitely slow to be reversible, and the operating frequency is then null. In order to optimize the harvester, one should not first focus on the transduction efficiency but on the output power. The analysis of equation (5.2.8b) shows that the output power is maximal when the beam's thermal hysteresis approximately verifies (5.2.9a). This optimum is clearly visible in Fig. 5.2.2f. Equation (5.2.9a) can be inverted in order to obtain the optimal beam's thermal hysteresis (5.2.9b) (see Fig. 5.2.3a). For this optimal point, the maximal output power is given by (5.2.9c) and the thermal efficiency by (5.2.9d).

𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"#

!"# ≈ 0.589. (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! )

(5.2.9a)

𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# ≈ 1.178. 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! (5.2.9b) 𝑃𝑃𝑃𝑃 ! !"# ≈ 1.52. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! (5.2.9c) 𝜂𝜂𝜂𝜂 !" !"#,!"#,!"# ≈ 2.45. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! (5.2.9d)
Instead of using the Carnot efficiency to compare the efficiency of the transduction with other harvesters, we define the Curzon-Ahlborn relative efficiency by (5.2.10a). Hence the maximal efficiency of the transduction is given by (5.2.10b).

𝜂𝜂𝜂𝜂 !" !",!"# = 2. 𝑟𝑟𝑟𝑟. 72. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 !"# 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! + 273.15 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.2.10a) 𝜂𝜂𝜂𝜂 !" !",!"#,!"# = 2.45. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! + 273.15 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.2.9b)
The evolution of the device's performances is plotted in Fig. 5.2.3 as a function of the external thermal gradient. This figures show that by increasing the temperature differences between the two heat sources, the output power and the thermal efficiency increase but the Curzon-Ahlborn efficiency decreases. For a hot and cold sources at 50°C and 25°C respectively, the energy harvester's efficiency and the output power represent around 0.25% of the maximal efficiency and maximal power that can be generated by a perfect endoreversible heat engine.

Leakage paths and heat sink

At this stage it becomes necessary to take into account the real conditions of the harvester and to include thermal leakages, as well as imperfect thermal transfers with the ambient surrounding, if the boundary conditions are Robin conditions instead of Dirichlet conditions (Temperature prescriped at the interface) , as in the case of a heat sink. In the electrical model presented in Fig. 5.2.1b, we have not taken into account the effect of the leakage path modeled by the resistance Rleak. As explained in Chapter 1, this resistance impacts the efficiency of the harvester since a part of the heat flowing from the hot source to the cold source travels through the leakage path and is not used to produce energy. A simple application of Kirchhoff's law gives the real efficiency of the device (5.3.1a). Most of the time, the leakage flux is dominant and the efficiency becomes (5.3.1b). The optimum thermal hysteresis is conserved but its efficiency decreases (5.3.1c). 𝜂𝜂𝜂𝜂 !" !"#,!"# = 𝑃𝑃𝑃𝑃 ! 𝑞𝑞𝑞𝑞 !"#$ + 𝑞𝑞𝑞𝑞 !""

(5.3.1a) 𝜂𝜂𝜂𝜂 !" !"#,!"# = 𝑞𝑞𝑞𝑞 !"" 𝑞𝑞𝑞𝑞 !"#$

. 𝜂𝜂𝜂𝜂 !"#,!

(5.3.1b)

𝜂𝜂𝜂𝜂 !" !"#,!"#,!"# ≈ 1.52. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! . 𝑀𝑀𝑀𝑀 ! ! . 𝑅𝑅𝑅𝑅 !"#$ ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! (5.3.1c)
The effect of the heat sink on thermal transfers can be modeled as a thermal resistance in series with the harvester: Boughaleb, 2015] proved that this assumption leads to a description of the harvester with a relatively good accuracy. The thermal gradient inside the harvester is simply given by the voltage divider equation (5.3.2).

𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! = 𝑅𝑅𝑅𝑅 !"#$ 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# (5.3.2a) 𝑃𝑃𝑃𝑃 ! !"# ≈ 1.52. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! 𝐿𝐿𝐿𝐿 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐶𝐶𝐶𝐶 ! ! ! . 𝑅𝑅𝑅𝑅 !"#$ ! 𝑅𝑅𝑅𝑅 !! ! . 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ ! ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ! ! (5.3.2b) 𝜂𝜂𝜂𝜂 !" !"#,!"#,!"# ≈ 1.52. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! 𝐿𝐿𝐿𝐿 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐶𝐶𝐶𝐶 ! ! ! . 𝑅𝑅𝑅𝑅 !"#$ ! 𝑅𝑅𝑅𝑅 !! ! . 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ! (5.3.2c)
In these conditions, the Curzon-Ahlborn relative efficiency of the energy harvester is given by (5.3.3c).

𝜂𝜂𝜂𝜂 !" !",!"#,!"# ≈ 1.52. 𝑟𝑟𝑟𝑟.

𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! 𝐿𝐿𝐿𝐿 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐶𝐶𝐶𝐶 ! ! ! . 𝑅𝑅𝑅𝑅 !"#$ ! 𝑅𝑅𝑅𝑅 !! ! . 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ! . 𝑇𝑇𝑇𝑇 ! + 273.15 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# (5.3.3c)
To clarify these equations, Fig. 5.3.1 presents the evolution of the parameters and performances of the energy harvester as a function of the hot source's temperature and for various values of the heat transfert coefficient of the heat sink used as a cold source of the system, assuming an ambient temperature of 25 o C. The plane heat sink has a heat transfer coefficient of 10 W.m -2 .K -1 , representing a thermal resistance of 625 K.W -1 if the heat sink has the same surface as the bistable beams. The leakage resistance of the harvester is taken equal to this value in order to be closer to the experimental conditions presented in [Boughaleb et al, 2015 1]. Fig. 5.3.1a demonstrates that the presence of a heat sink with a great heat transfer coefficient allows the use of a prime mover with larger thermal hysteresis, and thus enables to generate more power with greater efficiency. For instance, at 100 o C, a plane heat sink (10 W.m -2 .K -1 ) limits the output power density to 16 µW.cm -2 whereas this power can be more than doubled (35 µW.cm -2 ) by using a heat sink with cooling fins and with a heat transfer coefficient of 100 W.m -2 .K -1 . Whatever the type of heat sink, the operating frequency of the device is constant and independent of the working temperature (0.027 Hz). This observation was made on the current harvester without being explained: although the harvester was thermally optimized, no enhancement of the working frequency was observed. For the three types of heat sink, the Curzon-Ahlborn efficiency is around 0.13% and is a decreasing function of the hot source temperature. From these results, it can be concluded that it is better to use a heat sink with cooling fins to increase the thermal gradient in the cavity and to improve the performances of the energy harvester, even if this heat sink would decrease the compactness of the device.

Thermal transfers inside the bimetallic strip

To see the influence of the thermal transfers inside the prime mover on the cycling frequency of the system, we will treat the opposite case and assume that the beam experiences a thermal shock each time it comes in contact with the hot source or the cold source. First we need to establish the equation of the mechanical equilibrium with a spatially inhomogeneous temperature. Assuming that the temperature can be written as a function of separable variables (5.4.1a), the thermal properties of the beam can be modified into (5.4.1b) and (5.4.1c).

𝑇𝑇𝑇𝑇 𝑥𝑥𝑥𝑥, 𝑧𝑧𝑧𝑧 = 𝑇𝑇𝑇𝑇 ! 𝑥𝑥𝑥𝑥 . 𝑇𝑇𝑇𝑇 ! (𝑧𝑧𝑧𝑧) In this case, by applying the principle of virtual displacement, the expression of the axial strain is modified into (5.4.2a), with the definition of the mean temperature (5.4.2b). The non-linear Euler equation becomes (5.3.2c). Once the mechanical equations are established, we need to treat the propagation of heat along the half-length of the beam, assuming perfect contact with the hot source in x = L/2 and perfect isolation in x = 0. By using the separation of variables and the expression (5.4.3a) for the temperature distribution [START_REF] Boley | Theory of Thermal Stresses[END_REF], the heat conduction problem can be easily treated and the Fourier heat transfer law becomes (5.3.3b). The boundary conditions of the problem are then given by (5.4.3c) and (5.4.3d). In these equations, κ represents the thermal diffusivity of the materials. To model the behavior of the composite beam, we take a mean value µb defined by (5.4.3e).

𝑇𝑇𝑇𝑇 𝑥𝑥𝑥𝑥, 𝑡𝑡𝑡𝑡 = 𝜒𝜒𝜒𝜒 𝑥𝑥𝑥𝑥 . 𝜏𝜏𝜏𝜏 𝑡𝑡𝑡𝑡 + 𝑇𝑇𝑇𝑇 !

(5.4.3a)

𝜒𝜒𝜒𝜒. 𝜏𝜏𝜏𝜏 ,! = 𝜇𝜇𝜇𝜇. 𝜏𝜏𝜏𝜏. 𝜒𝜒𝜒𝜒 ,!!

(5.4.3b)

𝜒𝜒𝜒𝜒 𝐿𝐿𝐿𝐿/2 = 0

(5.4.3c)

𝜒𝜒𝜒𝜒 ,! 0 = 𝜒𝜒𝜒𝜒 ,! 𝐿𝐿𝐿𝐿 = 0

(5.4.3d)

1 𝜇𝜇𝜇𝜇 ! = 1 𝑡𝑡𝑡𝑡 . 𝜌𝜌𝜌𝜌. 𝐶𝐶𝐶𝐶 ! 𝜅𝜅𝜅𝜅 ! ! . 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 (5.4.3e)
The two differential equations to solve are then (5.4.4a) and (5.4.4b), introducing the constant λ.

Chapter 2|Thermo-mechanical modeling of the bimetallic strip heat engine 110 𝜒𝜒𝜒𝜒 𝑥𝑥𝑥𝑥 = 𝑎𝑎𝑎𝑎. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜆𝜆𝜆𝜆. 𝑥𝑥𝑥𝑥 + 𝑏𝑏𝑏𝑏. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝜆𝜆𝜆𝜆. 𝑥𝑥𝑥𝑥

(5.4.4a)

𝜏𝜏𝜏𝜏 𝑡𝑡𝑡𝑡 = 𝑐𝑐𝑐𝑐. 𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 (-𝜆𝜆𝜆𝜆 ! . 𝜇𝜇𝜇𝜇 ! . 𝑡𝑡𝑡𝑡)

(5.4.4b)

With the boundary conditions, it can be shown that � is quantified and inversely proportional to the beam length (5.4.5b). The function � is then a Fourier serie as well as the temperature distribution. During the heating of the beam (initial homogeneous temperature Ti), the temperature profile thus obeys (5.4.5d).

𝜒𝜒𝜒𝜒 ,! 0 = 0 ⇒ 𝑏𝑏𝑏𝑏 = 0

(5.4.5a)

𝜒𝜒𝜒𝜒 𝐿𝐿𝐿𝐿 2 = 0 ⇒ 𝜆𝜆𝜆𝜆. 𝐿𝐿𝐿𝐿 = 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑛𝑛𝑛𝑛 > 0

(5.4.5b)

𝜒𝜒𝜒𝜒 𝑥𝑥𝑥𝑥 = 𝑎𝑎𝑎𝑎 !.!!! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑥𝑥𝑥𝑥 ∈ 0, 𝐿𝐿𝐿𝐿 2 !! !!! (5.4.5c) 𝑇𝑇𝑇𝑇 𝑥𝑥𝑥𝑥, 𝑡𝑡𝑡𝑡 = 𝑇𝑇𝑇𝑇 ! + 2. 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . -1 ! 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋 . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 !! !!! (5.4.5d)
Once the initial heating of the beam is known, we can treat the periodic heating and cooling of the beam with the same procedure. The temperature profile on the beam's halflength during heating is given by (5.4.6a), and by (5.4.6b) during cooling. The working period of the system is given by (5.4.6c).

𝑇𝑇𝑇𝑇 ! ! 𝑥𝑥𝑥𝑥, 𝑡𝑡𝑡𝑡 = 𝑇𝑇𝑇𝑇 ! + 𝑎𝑎𝑎𝑎 !.!!! . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .(!!! ! !!! ! ) . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 !! !!! (5.4.6a) 𝑇𝑇𝑇𝑇 ! ! 𝑥𝑥𝑥𝑥, 𝑡𝑡𝑡𝑡 = 𝑇𝑇𝑇𝑇 ! + 𝑏𝑏𝑏𝑏 !.!!! . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .(!!! ! ) . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 !! !!!
(5.4.6b)

𝜙𝜙𝜙𝜙 ! = 𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 ! + (𝑖𝑖𝑖𝑖 -1). 𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 ! + 𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 ! (5.4.6c)
By assuming the continuity of the thermal profile between the heating andcooling phases, we can deduce the expression of the Fourier serie's coefficients by solving the equatio system (5.4.7a-b) along the interval [0, L/2].

𝑎𝑎𝑎𝑎 !.!!! . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .!! ! -1 = 𝑏𝑏𝑏𝑏 !.!!! . 1 -𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .!! ! (5.4.7a) 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! = 𝑎𝑎𝑎𝑎 !.!!! . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .(!! ! !!! ! ) -1 1 -𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .!! ! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 2. 𝑛𝑛𝑛𝑛 + 1 . 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 !! !!! (5.4.7b)
In order to apply the Galerkin method to the mechanical equilibrium equation, the thermal profile must be developed in Fourier series along the whole beam's length during heating (5.4.8a) and cooling. The coefficients cn during heating can be calculated as functions of the an coefficients in (5.4.8b).

𝑓𝑓𝑓𝑓 𝑥𝑥𝑥𝑥 = 𝑐𝑐𝑐𝑐 ! . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .(!!! ! !!! ! ) . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠 𝑛𝑛𝑛𝑛. 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 !! !!! (5.4.8a) 𝑐𝑐𝑐𝑐 !.!!! = 2 𝜋𝜋𝜋𝜋 . 𝑎𝑎𝑎𝑎 !.!!! 2. 𝑖𝑖𝑖𝑖 + 1 . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .(!!! ! !!! ! ) !! !!! + 1 𝜋𝜋𝜋𝜋 . 𝑎𝑎𝑎𝑎 !.!!! . 2. 𝑛𝑛𝑛𝑛 -2. 𝑖𝑖𝑖𝑖. -1 !!! + 1 --1 !!! 𝑛𝑛𝑛𝑛 -𝑖𝑖𝑖𝑖 . 𝑖𝑖𝑖𝑖 + 𝑛𝑛𝑛𝑛 + 1 !! !!! !!! . 𝑒𝑒𝑒𝑒 ! !.!!! ! . ! ! ! ! .! ! .(!!! ! !!! ! )
(5.4.8b)

Given that we only study the first-order buckling mode, only the first Fourier coefficient c1 matters. In its expression, we can neglect other terms after the first terms a1. This could be partly justified by the n-decay of the an parameters. With this assumption, the first term a1 is given by (5.4.9a) and the equilibrium equation during the beam's heating becomes (5.4.9b).

𝑎𝑎𝑎𝑎 ! = 2. 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . 1 -𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! .!! ! 𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! . !! ! !!! ! -1 (5.4.9a) 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! + 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 ! + 4. (𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 ! ) 𝜋𝜋𝜋𝜋 - 2. 𝑀𝑀𝑀𝑀 ! 𝜋𝜋𝜋𝜋 . 𝑎𝑎𝑎𝑎 ! . 𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! .(!!! ! !!! ! ) = 0 (5.4.9b)
In this case, the beam switches when its temperature profile verifies (5.3.10a) during heating and (5.4.10b) during cooling.

𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . 1 -𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! .!! ! 𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! . !! ! !!! ! -1 . 𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! .!! ! = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.4.10a) 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . 1 -𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! .!! ! 𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! . !! ! !!! ! -1 . 𝑒𝑒𝑒𝑒 ! ! ! ! ! .! ! .!! ! = 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! (5.4.10b)
By solving this equation system, we finally find solutions close to (5.2.4b) and (5.2.4c) obtained with the lumped capacitance model (5.4.11). The evaluation of the operating frequency of the system shows that this value is too high to explain the frequencies measured experimentally. This shows that the thermal transfers are limited by the contact resistance between the heat sources and the bistable beam.

𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 ! = 𝐿𝐿𝐿𝐿 ! 𝜋𝜋𝜋𝜋 ! . 𝜇𝜇𝜇𝜇 ! . ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.4.11a) 𝛥𝛥𝛥𝛥𝑡𝑡𝑡𝑡 ! = 𝐿𝐿𝐿𝐿 ! 𝜋𝜋𝜋𝜋 ! . 𝜇𝜇𝜇𝜇 ! . ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! (5.4.11b) 𝑓𝑓𝑓𝑓 = 𝜋𝜋𝜋𝜋 ! . 𝜇𝜇𝜇𝜇 ! 2. 𝐿𝐿𝐿𝐿 ! . 1 ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! + ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.4.11c)
A simple calculation of the heat supplied to the beam during the heating phase shows that the quantity of heat is preserved. A more accurate model would have taken into account the fact that the Biot number is not infinite, leading to a result similar to the one presented in [START_REF] Boley | Theory of Thermal Stresses[END_REF], by taking into account the difference of propagation in both layers and the influence of the contact size on the heat transfers [Incropera and DeWitt, 2002], or by taking into account higher-mode Fourier coefficients. However, these models would lose in comprehension what they would attempt to gain in precision.

𝑄𝑄𝑄𝑄 ! = 𝐶𝐶𝐶𝐶 ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"

(5.4.12)

The previous calculations are still valid, and the optimal power and the thermal efficiency of the optimal point are given by (5.4.13). These formulae are written in a way to separate geometric parameters from physical parameters of the composite beams, and to define new

𝑃𝑃𝑃𝑃 ! !"# ≈ 1.52. 𝜋𝜋𝜋𝜋 ! . 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! 𝐿𝐿𝐿𝐿 ! ! . 𝑀𝑀𝑀𝑀 ! ! . 𝜇𝜇𝜇𝜇 ! ! 𝑁𝑁𝑁𝑁 ! ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! ! (5.4.13a) 𝜂𝜂𝜂𝜂 !" !"#,!"# ≈ 2.45. 𝑟𝑟𝑟𝑟. 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 ! 𝐿𝐿𝐿𝐿 ! . 𝑚𝑚𝑚𝑚 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! (

Conclusion

In this first chapter about the working principle of our device, we presented the modeling of the thermo-mechanical instability of composite beams using the principle of virtual displacement. This model helped us explain the origins of the thermal snap-through of simply-supported beams either initially stressed or initially curved. For both cases, we found the criterion of bistability and described the evolution of the critical load, of the characteristical temperatures of hysteresis, and of the beam displacement associated to the first and second order buckling modes. For initially-curved bimetallic beams, we proved that our model was more accurate than Timoshenko's model since it takes into account the mean thermal expansion of beams.

Contrary to a simple study of the thermal snap-through of composite beams, we also clarified the thermodynamic properties of bistable beams, and studied the evolution of the strain energy and the entropy of such structures. This enabled us to show the equivalence between thermal snap-through of thermo-mechanically bistable beams and first-order phase transition. With these results, we explained the thermodynamic cycle of the bimetallic strip heat engine and we theoretically demonstrated the possibility to harvest heat with bistable beams. By using these results, we first evaluated the performances of our heat engine for various types of bimetallic couples. We found that in the best cases, the efficiency of the thermo-mechanical transduction does not exceed few percents of Carnot when the thermal hysteresis is below 1 o C, and is generally of around 0,1% of the Carnot efficiency. In agreement with the data reported in the first chapter and with experimental data, these various results also support the idea that bimetallic strip heat engines are less efficient than thermoelectric generator to harvest heat fluxes, and thus contradict the estimations reported in Puscasu's thesis [START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF].

Because of the complexity of this first model based on non-linear mechanics, we developed in a second time a reduced order model based on a Ritz approximation of the beam's deflection and the application of the Galerkin method. This second model enabled us to analytically extract the properties of bistable beams, and to describe the performances of the heat engine (available energy, maximal power, switching force, switching speed, latent heat of snap, thermal efficiency, Carnot efficiency) as functions of the physical parameters of the mechanical structures. The most important formula found with this model are reported in Table 6. We finally established Figures of Merit to compare the materials, showing that performances are improved by the use of materials with large difference of thermal expansion coefficients, justifying the use of Invar as a material for the passive layer.

By modeling the double role played by the architecture which must ensure thermal transfers between the heat sources and the bimetallic strip by applying mechanical action on this strip, we show how it modifies the performances of the energy harvester but also how the switching frequency of the bimetallic strip evolves as function of the physical properties of the bimetallic strip's materials. We showed that thermal transfers are mainly driven by the contact resistance between the heat sources and the prime mover and not by the thermal transfers inside the prime mover, explaining why the switching frequency of the device has never exceed few tenths of hertz so far, and the generated power few microwatts. As a consequence, future investigations to increase the performances of the generator must be carried on the improvement of thermal transfers

In the last part of this chapter, we studied and modeled the thermal transfers inside the harvester in order to understand their role in the dynamic behavior of the system. This led us to find expressions of the device's operating frequency, maximal power and efficiency as functions of the external working conditions. A first lumped-capacitance model helped us to find leakage paths inside the energy harvester and to quantify their impact on the device's efficiency. A second model was established to characterize the propagation of heat in the bistable beam and to see its impact on the working frequency of the device.

Chapter 3

Conversion of the mechanical energy into electrical energy 1 Introduction

The previous chapter addressed the issue of modeling the thermo-mechanical behavior of composite beams, and demonstrated the capability of a bistable beam to behave like a heat engine and to convert thermal energy into mechanical energy when it is placed between two heat sources. The aim of a thermal energy harvester being to create electrical charges from heat, it is necessary to study the various ways to convert mechanical energy into electric energy. The literature on the harvesting of mechanical energy shows that two solutions are mainly conceivable due to their ease of implementation and their efficiency: piezoelectric conversion (and pyroelectric conversion) and electret-based capacitive transduction.

Four different types of harvesters will be studied in this chapter. Fig. 1a represents the architecture of a thermo-mechanically bistable piezoelectric composite membrane: when the composite membrane deforms due to thermal stress, part of the mechanical energy is directly converted into electric charges stored in the piezoelectric capacitance. This first architecture presents numerous advantages: first, even if the variation of the strain energy of the switching membrane is quite large during a complete cycle, this variation does not reflect the strain energy variations seen by each layer of the composite beam which can experience a large amount of stress, either tensile or compressive. However this stress, especially when it is tensile, can speed up aging of piezoelectric materials and ceramics [START_REF] Chaplya | Durability properties of piezoelectric stack actuators under combined electromechanical loading[END_REF][START_REF] Tsujiura | Reliability of vibration energy harvesters of metal-based PZT thin films[END_REF]. Another advantage of this architecture is the possibility to exploit the pyroelectric effect of some materials to significantly improve the performances of the harvester. Contrary to the third architecture, the thermal optimization of the harvester is decorrelated from the optimization of the electro-mechanical conversion. Yet the advantages of thermally cycling a piezoelectric material deposited on the bistable membrane can be counterbalanced by the increase of the leakage current at high temperatures [START_REF] Lipscomb | The effect of relative humidity, temperature and electrical field on leakage currents in piezo-ceramic actuators under dc bias[END_REF] and by the operating limit imposed by the Curie point and the loss of polarization in the case of ferroelectric materials [START_REF] Olsen | Pyroelectric energy conversion: Hysteresis loss and temperature sensitivity of a ferroelectric material[END_REF]. Finally, another advantage of the bistable piezoelectric composite membrane is the possibility to fabricate either at the macroscale by depositing piezoelectric layer on a bistable membrane [START_REF] Capsal | Piezoelectric sensing coating for real time impact detection and location on aircraft structures[END_REF][START_REF] Savelli | Systeme de conversion d'energie thermique en energie electrique[END_REF], or to process them at the microscale by using conventional MEMS fabrication techniques [Michael and Kwok, 2006, Arnaud et al, 2013[START_REF] Trioux | Fabrication of bilayer plate for a micro thermal energy harvester[END_REF] and the availability of methods to deposit piezoelectric materials (ZnO, AlN, PZT, BaTiO3 etc.). In that case, the piezoelectric layer can also play the role of the thermally passive layer of the bistable membrane. However, the size reduction of the piezoelectric harvester is accompanied by an increase of the impedance of the system and the reduction of the output voltages of the piezoelectric generators, which represents a challenge for the development of adapted power management circuits.

Fig. 1b represents the coupling between a bistable composite beam and two piezoelectric stacks. In this case, the piezoelectric stacks harvest the work of the thrust exerced by the bistable beam on its clamps. Contrary to the first architecture, the temperature variations inside the piezoelectric stacks would be negligible and the pyroelectric effect would not have any impact on the performances of the harvester. This architecture has been conceived to exploit the high piezoelectric coupling factor of the 3-3 mode. However, the high compressive stiffness of piezoelectric stacks can reduce the efficiency of the transduction, if the bistable beam is not able to deform the stacks sufficiently. Here again, contrary to the third architecture, the fact that the thermal optimization of the harvester is decoupled from its electromechanical optimization can represent a benefit for the improvement in performances of the harvester.

Fig. 1c represents the coupling between a bistable composite beam and oscillating piezoelectric transducer. This solution represents the first architecture that has experimentally been developed at ST-Microelectronics [Puscasu et al, 2012, Arnaud et al, 2013, Boughaleb et al, 2015]. In this configuration, the switching membrane hits a clamped-clamped bimorph beam that converts the energy due to the impact of the bistable beam into electric energy by using conventional power management circuits dedicated to piezoelectric mechanical energy harvesters. The initial idea which lead to this architecture was the possibility to use the piezoelectric oscillating beam both as an electromechanical transducer and a heat exchanger. This way, the thermal energy harvester would be able to work without any heat sink, thus increasing the compactness of the device. However this strategy presents the drawback to intimately link the optimization of the mechanical energy transfer between the bistable membrane and the piezoelectric transducer, and the optimization of the thermal gradient inside the cavity of the harvester [Boughaleb et al, 2015]. Moreover, this type of system recquires an additional conversion step (mechanical shock between the transducer and the bistable membrane) which efficiency impacts the global efficiency of the harvester. It has also been observed that the type of material used for the fabrication of the passive part of the system has an impact on the mechanical damping of the piezoelectric oscillator and reduces the efficiency of the electromechanical transduction. Yet these mechanical losses can be reduced by the design of adapted piezoelectric oscillators [Arnaud et al, 2013].

Finally Fig. 1d represents the coupling between a bistable composite beam and an electret. This type of device was jointly developed with the CEA-Leti [Boisseau et al, 2013, Arnaud et al, 2013]. In this configuration, the switching bistable beam displaces electrical charges in the electrical field induced by an electret, causing at the same time a variation of the capacitance formed by the bistable membrane and a counter-electrode placed on the opposite side of the electret. Contrary to other electrostatic converters, electret-based transducers are passive devices that do not require any external voltage source to harvest energy. When electrets are fabricated using a Corona discharge, their surface potential can easily be controlled and adjusted [Sessler, 1987], which allows to control and optimize the coupling between the bistable membrane and the electret. The high output voltages of the electret-based transducer ease the implementation of dedicated power management circuits [START_REF] Boisseau | Récupération d'énergie vibratoire à électrets[END_REF] even if their performances can be impacted by the small variations of capacitance of the bistable beam and the presence of parasitic capacitances.

Largely used for the fabrication of microphones [START_REF] Kestelman | Electrets in engineering[END_REF]] the integration of electrets in MEMS devices is perfectly controlled, making it possible to consider the size reduction of the electrostatic bimetallic strip heat engine. Moreover reliability tests reported in the litterature [START_REF] Suzuki | Recent progress in MEMS electret generator for energy harvesting[END_REF] or performed at the CEA on our devices have shown the high stability of Corona electrets. From an industrial point of view, electretbased transducers thus represent low-cost energy harvesting solutions that are effective at the macroscale as well as at the microscale.

This chapter is divided as follows: in Section 2 we explain the origins of piezoelectricity. Sections 3 and 4 are devoted to the modeling of the pyroelectric bimetallic strip heat engine, Section 5 to the coupling between an oscillating piezoelectric transducer and a bistable beam, and Section 6 to the electret-based capacitive transduction. For these two architectures, theoretical results will be compared with experimental data. Finally, Section 7 presents a synthesis of the device performances and a comparison of the various architectures.

An overview on piezoelectricity

The history of piezoelectricity

Piezoelectricity refers to the property of some dielectric materials to produce electrical charges in response to a mechanical stress or, reversely, to deform when they are placed in an electric field. Although the direct effect and the reverse effect were scientifically explained at the end of the 19 th century (the direct effect was studied by Jacques and Pierre Curie in 1880 [START_REF] Curie | Développement, par pression, de l'électricité polaire dans les cristaux hémièdres à faces inclinées[END_REF] and the reverse effect suggested by Gabriel Lippmann in 1881 [START_REF] Lippmann | Principle of the conservation of electricity[END_REF] and demonstrated by the Curie brothers [START_REF] Curie | Contractions et dilatations produites par des tensions électriques dans les cristaux hémièdres à faces inclinées[END_REF], first applications of the piezoelectricity appeared at the beginning of the 20 th century [Brissaud and Baland, 2007] mainly in the field of acoustics (Langevin's ultrasonic transducer in 1919 (Sonar)) and in the field of electro-mechanical resonators and filters (Quartz oscillators by Cady in 1921). Piezoelectricity is currently used in sensors, mechanical actuators, ultrasonic transducers, microphones, SAW filters, delay lines, clocks, gas lighters... [START_REF] Tichy | Fundamentals of piezoelectric sensorics[END_REF]. Applications in energy harvesting only appear in the middle of the 1980s and really emerge in the 1990: [START_REF] Häsler | Implantable physiological power supply with PVDF film[END_REF] probably reported the first use of piezoelectric materials (PVDF) for the generation of electrical energy.

Pyroelectricity is the property of some classes of piezoelectric materials that polarize under a variation of their temperature. The study of the pyroelectric effect by William Thomson (Lord Kelvin) is prior to the study of piezoelectricity [START_REF] Thomson | On the piezoelectric property of quartz[END_REF] and the explanation of the links between pyroelectricity and piezoelectricity is due to Woldemar Voigt's work, who thermodynamically described both effects in 1890. Pyroelectricity is currently exploited in infrared photodetectors and thermal imaging [START_REF] Lang | Pyroelectricity: From ancient curiosity to modern imaging tool[END_REF]. First investigations on the use of the pyroelectric effect to harvest heat are linked to [Clingman and Moore, 1961].

The following paragraphs address basic knowledge about piezoelectricity and pyroelectricity. First we come back to the electrostatic behavior of dielectric materials and then we explain the constitutive equations of piezoelectric and pyroelectric materials by means of the expression of the internal energy, which allows to introduce a classification of piezoelectric and pyroelectric materials. We finally introduce piezoelectric and pyroelectric coupling factors and we simplify the constitutive equations of piezoelectric materials for the study of piezoelectric beams and for 3-1 and 3-3 coupling.

Thermodynamics of piezoelectric materials

In order to understand the origins of the piezoelectric, pyroelectric and electrocaloric effects, we go back to the local expression of the internal energy u of a dielectric material placed in a stress field and an electrostatic field. The internal energy being a function of extensive parameters it is a privileged function of the entropy σ, the strain tensor Si and the electrical displacement matrix Di (2.2.1).

An overview on piezoelectricity 123

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝜃𝜃𝜃𝜃. 𝑑𝑑𝑑𝑑𝜎𝜎𝜎𝜎 + 𝒯𝒯𝒯𝒯 ! . 𝑑𝑑𝑑𝑑𝒮𝒮𝒮𝒮 ! + ℰ ! . 𝑑𝑑𝑑𝑑𝒟𝒟𝒟𝒟 ! (2.2.1)
The absolute temperature θ, the stress tensor Ti and the electrical field matrix Ei are then defined as the first partial derivatives of the internal energy (2.2.2).

𝜃𝜃𝜃𝜃 = 𝜕𝜕𝜕𝜕𝓊𝓊𝓊𝓊 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝒮𝒮𝒮𝒮 ! ,𝒟𝒟𝒟𝒟 ! = 𝓊𝓊𝓊𝓊 ,! 𝒮𝒮𝒮𝒮,𝒟𝒟𝒟𝒟
(2.2.2a)

𝒯𝒯𝒯𝒯 ! = 𝜕𝜕𝜕𝜕𝓊𝓊𝓊𝓊 𝜕𝜕𝜕𝜕𝒮𝒮𝒮𝒮 ! !,𝒟𝒟𝒟𝒟 ! = 𝓊𝓊𝓊𝓊 ,𝒮𝒮𝒮𝒮 !,𝒟𝒟𝒟𝒟 (2.2.2b) ℰ ! = 𝜕𝜕𝜕𝜕𝓊𝓊𝓊𝓊 𝜕𝜕𝜕𝜕𝒟𝒟𝒟𝒟 ! !,𝒮𝒮𝒮𝒮 !" = 𝓊𝓊𝓊𝓊 ,𝒟𝒟𝒟𝒟 !,𝒮𝒮𝒮𝒮 (2.2.2c)
Since it is difficult to guarantee a constant entropy than a constant temperature during a transformation, it is more adapted to define new thermodynamic potentials that are functions of intensive parameters instead of extensive parameters. These potentials are usually defined with the help of the Legendre transformation [Callen, 1960]. For example, the Helmholtz free energy 𝒻𝒻𝒻𝒻 is the potential that uses the temperature θ as a variable instead of the entropy σ. The entropy is then the partial derivative of 𝒻𝒻𝒻𝒻 when Si and Di are kept constant. The Helmholtz free energy is defined by (2.2.3).

𝒻𝒻𝒻𝒻 ≡ 𝓊𝓊𝓊𝓊 𝜃𝜃𝜃𝜃 = 𝓊𝓊𝓊𝓊 -𝜃𝜃𝜃𝜃. 𝜎𝜎𝜎𝜎 (2.2.3a) 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = -𝜎𝜎𝜎𝜎. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝒯𝒯𝒯𝒯 ! . 𝑑𝑑𝑑𝑑𝒮𝒮𝒮𝒮 ! + ℰ ! . 𝑑𝑑𝑑𝑑𝒟𝒟𝒟𝒟 ! (2.2.3b)
Other Legendre transformations can be defined such as enthalpy h(σ,Ti,Ei), elastic enthalpy he(σ,Ti,Di), electric enthalpy he(σ,Si,Ei), elastic Gibbs function gs(θ,Ti,Di) etc. [START_REF] Tichy | Fundamentals of piezoelectric sensorics[END_REF]. The best suitable transformation for the description of piezoelectric materials is the Gibbs function g which is a privileged function of the three intensive parameters θ, Ti and Ei. Hence the extensive parameters become the first partial derivatives of the Gibbs function (2.2.4).

ℊ ≡ 𝓊𝓊𝓊𝓊 𝜃𝜃𝜃𝜃, 𝒯𝒯𝒯𝒯 ! , ℰ ! = 𝓊𝓊𝓊𝓊 -𝜃𝜃𝜃𝜃. 𝜎𝜎𝜎𝜎 -𝒯𝒯𝒯𝒯 ! . 𝒮𝒮𝒮𝒮 ! -ℰ ! . 𝒟𝒟𝒟𝒟 ! (2.2.4a) 𝑑𝑑𝑑𝑑ℊ = -𝜎𝜎𝜎𝜎. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 -𝒮𝒮𝒮𝒮 ! . 𝑑𝑑𝑑𝑑𝒯𝒯𝒯𝒯 ! -𝒟𝒟𝒟𝒟 ! . 𝑑𝑑𝑑𝑑ℰ ! (2.2.4b) 𝜎𝜎𝜎𝜎 = 𝜕𝜕𝜕𝜕ℊ 𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕 𝒯𝒯𝒯𝒯 ! ,ℰ ! = ℊ ,! 𝒯𝒯𝒯𝒯ℰ (2.2.4c) 𝑆𝑆𝑆𝑆 !" = 𝜕𝜕𝜕𝜕ℊ 𝜕𝜕𝜕𝜕𝒯𝒯𝒯𝒯 ! !,ℰ ! = ℊ ,𝒯𝒯𝒯𝒯 !ℰ (2.2.4d) 𝐷𝐷𝐷𝐷 ! = 𝜕𝜕𝜕𝜕ℊ 𝜕𝜕𝜕𝜕ℰ ! !,𝒯𝒯𝒯𝒯 ! = ℊ ,ℰ !𝒯𝒯𝒯𝒯 (2.2.4e)
Following the same approach used in Chapter 3, we can linearize the expression of the extensive parameters as a function of the intensive parameters (2.2.5).

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = 𝜎𝜎𝜎𝜎

𝒮𝒮𝒮𝒮 ,! !" = 𝑠𝑠𝑠𝑠 !" (2.2.6c) 𝒟𝒟𝒟𝒟 ,ℰ !𝒯𝒯𝒯𝒯 = 𝜀𝜀𝜀𝜀 !" 𝒯𝒯𝒯𝒯 (2.2.6d)
In (2.2.6), S ,θ T,E and σ ,T θ,E correspond to the thermo-mechanical coupling (thermal expansion), σ ,E θ,T and D ,θ T,E , if they are non-zero, represent the thermo-electrical coupling for a pyroelectric material (respectively the electrocaloric effect and pyroelectric effect), and D ,T θ,E and S ,E θ,T the electro-mechanical coupling for a piezoelectric material (respectively the direct piezoelectric effect and the reverse piezoelectric effect). As these matrices are the first partial derivatives of the extensive parameters, they are also the second partial derivatives of the Gibbs function. It is then possible to apply the Shwarz theorem (2.2.7).

𝜕𝜕𝜕𝜕 ! ℊ 𝜕𝜕𝜕𝜕𝑋𝑋𝑋𝑋 ! 𝜕𝜕𝜕𝜕𝑋𝑋𝑋𝑋 ! ! ! ,! ! = 𝜕𝜕𝜕𝜕 ! ℊ 𝜕𝜕𝜕𝜕𝑋𝑋𝑋𝑋 ! 𝜕𝜕𝜕𝜕𝑋𝑋𝑋𝑋 ! ! ! ,! ! ! (2.2.7)
This theorem enables to express the Maxwell relations for a dielectric body which show that the direct piezoelectric effect and the reverse piezoelectric effect are described by the same matrix dij (2.2.8a): the proportionality relation between Di and Tij is the same as between Sij and Ei. Similarly, the electrocaloric and pyroelectric effects are described by the same matrix p (4.8b), and the thermal expansion by a single matrix α (2.2.8c).

𝒮𝒮𝒮𝒮 ,!

!ℰ ! = 𝒟𝒟𝒟𝒟 ,𝒯𝒯𝒯𝒯 !ℰ = 𝑑𝑑𝑑𝑑 !"

(2.2.8a)

𝜎𝜎𝜎𝜎 ,ℰ !𝒯𝒯𝒯𝒯 ! = 𝒟𝒟𝒟𝒟 ,! 𝒯𝒯𝒯𝒯ℰ = 𝑝𝑝𝑝𝑝 ! 𝒯𝒯𝒯𝒯 (2.2.8b) 𝜎𝜎𝜎𝜎 ,𝒯𝒯𝒯𝒯 !ℰ ! = 𝑆𝑆𝑆𝑆 ,! 𝒯𝒯𝒯𝒯ℰ = 𝛼𝛼𝛼𝛼 !" (2.2.8c)
Thus the constitutive equations of the dielectric body (2.2.5) can be rewritten with the newly defined parameters (2.2.9). We now introduce a matrix notation similarly to the one presented in [IEEE, 1988].

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

Instead of limited this matrix to the piezoelectric effect only, we add the thermal effects.

We know from the mechanics that the link between the compliance matrix is a 6x6 matrix. Since there are three coordinates to describe the polarization and the electric field, the permittivity matrix is a 3x3 matrix. The piezoelectric effect is then described by a 6x6 matrix. To obtain a square matrix by adding the thermal effects, we define a 6x3 matrix for the thermal expansion, a 3x3 matrix for the heat capacity (2.2.10). Given that this matrix is associated to the properties of crystallographic structures, 32 different matrices with various symmetry properties can be written [Brissaud and Baland, 2007].

The most complex matrix corresponds to the triclinic class of crystals and is represented in Fig. 2.2.2. In this figure, each point represents an independent value of parameters.

𝑑𝑑𝑑𝑑 𝜎𝜎𝜎𝜎 𝒮𝒮𝒮𝒮 𝒟𝒟𝒟𝒟 = 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 !! 𝒯𝒯𝒯𝒯 𝜃𝜃𝜃𝜃 𝛼𝛼𝛼𝛼 ! 𝑝𝑝𝑝𝑝 ! 𝛼𝛼𝛼𝛼 𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑 ! 𝑝𝑝𝑝𝑝 𝑑𝑑𝑑𝑑 𝜀𝜀𝜀𝜀 . 𝑑𝑑𝑑𝑑 𝜃𝜃𝜃𝜃 𝒯𝒯𝒯𝒯 ℰ (2.2.10)
Thanks to symmetries, it can be shown that only 20 of these matrices describe pyroelectric materials, and 10 of these matrices possess non-zero pyroelectric term. To illustrate these properties of symmetry, we present in Fig. 2.2.2 the matrices corresponding to the crystallophic class 4mm to which belong perovskite crystals (BaTiO3, PbTiO3, PZN-PT, PMN-PT) [START_REF] Tichy | Fundamentals of piezoelectric sensorics[END_REF] and 6mm to which belong wurtzite crystals (AlN, ZnO, GaN). For these materials, the number of physical parameters is reduced and only three parameters are needed to describe the piezoelectric effect. To reduce the number of constitutive equations to get the simplest description of the piezoelectric beam, it is necessary to take into account the configuration of the electrodes of the capacitor. Being deposited on the surface of the beam, the only component of the electrical field to be considered is the term E3 associated to the relative permittivity ε3. In the frame of the Euler-Bernoulli assumptions, the only term of the stress tensor is the axial strain σ1. This implies that the piezoelectric coupling inside the piezoelectric layer will be described by the coefficient d31 only. Finally, there are only three constitutive equations for the description of the piezoelectric capacitor. The previous matrix can be rewritten in (2.2.11). [Bowen et al, 2014]) and figures of merit of pyroelectric generators 3 Equations of the pyroelectric bimetallic strip heat engine

𝑑𝑑𝑑𝑑

Tertiary pyroelectric effect

The modeling of the thermo-mechanical instability of composite beams in the previous chapter enabled to establish the expression of the strain tensor in the case of a buckled beam. In this chapter, we will only focus on the thermo-mechanical instability described by a first-order buckling mode. The corresponding model has been presented in the previous chapter. In this case, (4.14a) is the expression of the strain tensor.

𝒮𝒮𝒮𝒮 ! = 𝜆𝜆𝜆𝜆 𝐿𝐿𝐿𝐿 + 𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! . cos 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 (3.1.1a) 𝜆𝜆𝜆𝜆 = 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ! ! (3.1.1b)
Since the expression of the axial strain is known, it is better to use constitutive equations where the axial strain is used as a parameter. By using the electric Gibbs function, the axial stress T1, the entropy σ and the electrical displacement D3 are simply expressed as functions of the electrical field, the temperature and the axial strain (3.1.2). These equations introduce new parameters such as the pyroelectric coefficient p 3 S , the relative dielectric permittivity ε 33 S , and the piezoelectric modulus e 31 . By combining the equations of (3.1.2), these parameters can be expressed as functions of those given in (2.2.9).

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 !! 𝒮𝒮𝒮𝒮 = 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 !! 𝒯𝒯𝒯𝒯 -𝐸𝐸𝐸𝐸. 𝛼𝛼𝛼𝛼 ! . 𝜃𝜃𝜃𝜃 = 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 !! 𝒯𝒯𝒯𝒯 (1 -𝛾𝛾𝛾𝛾 ! . 𝛼𝛼𝛼𝛼. 𝜃𝜃𝜃𝜃)

(3.1.3a) 𝑒𝑒𝑒𝑒 !" = 𝐸𝐸𝐸𝐸. 𝑑𝑑𝑑𝑑 !" (3.1.3b) 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 = 𝑝𝑝𝑝𝑝 ! 𝒯𝒯𝒯𝒯 -𝐸𝐸𝐸𝐸. 𝛼𝛼𝛼𝛼. 𝑑𝑑𝑑𝑑 !" (3.1.3c) 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 = 𝜀𝜀𝜀𝜀 !! 𝒯𝒯𝒯𝒯 -𝐸𝐸𝐸𝐸. 𝑑𝑑𝑑𝑑 !" ! (3.1.3d)
Equation (3.1.3c) shows that the pyroelectric effect can be decomposed into two terms, the first term being the pyroelectric coefficient at constant strain is known as the primary effect [START_REF] Bhalla | Primary and secondary pyroelectricity[END_REF], since it directly links the variation of the polarization to the variation of the temperature. The secondary effect represents the contribution of the thermal expansion and the piezoelectric effect. As explained in [Korotosov et al, 1994[START_REF] Xu-Sheng | Tertiary pyroelectric effect on thick ferroelectric crystal plates with partially uniform heating[END_REF], a tertiary effect also exists due to the spatial variation of the temperature entailing a thermal shear stress inside the material.

The beam modeled in this part is represented in Fig. 3.1.1. In this model, the piezoelectric layer has a thickness noted tp and its middle line is located in zp and is polarized with two parallel electrodes only. In the model, the impact of the electrodes on the properties of the composite beams is neglected because of their weak thickness. To model the pyroelectric/piezoelectric composite beam, the equations (4.15) are integrated on the whole beam volume. In weak coupling, the impact of the polarization of the piezoelectric material on the value of zo can be neglected. In that case, we can simplify the expression of some integrals:

𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! = 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝑡𝑡𝑡𝑡 ! (3.1.4a) 𝑒𝑒𝑒𝑒 !" . (𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! ). 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! = 𝑒𝑒𝑒𝑒 !" . 𝑡𝑡𝑡𝑡 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! (3.1.4b)
Hence the electrical displacement can be integrated on the whole volume, leading to the expression of the variation of the electrical charge Q e stored on the piezoelectric capacitance (3.1.5).

𝑑𝑑𝑑𝑑𝒬𝒬𝒬𝒬 This last equation demonstrates that the charge carried by the piezoelectric capacitor is directly a function of its voltage, and that the variation of the beam's temperature during the heating and cooling phases creates electrical charges due to the pyroelectric effect. The last terms show the real contribution of the thermo-mechanical bistability to the increase of the charge generated. As the beam's displacement is a function of temperature, it is possible to describe this contribution as a kind of tertiary pyroelectric effect. By derivating the displacement with respect to the temperature at equilibrium yields (3.1.7). This tertiary effect thus is given by (3.1.8). It shows that the sign of this effect depends of the sign of the beam's curvature, the position of the piezoelectric layer with respect to the neutral fiber and the value of the thermal asymmetry. Logically it can be observed that the increase of the thermal asymmetry, representing the thermal sensitivity of the bistable beam, and the use of highly piezoelectric materials enhance the beam's pyroelectric effect. We will see in the next sections that this effect boosts the performances of the heat engine when the beam thermal hysteresis is small.

𝑑𝑑𝑑𝑑𝒬𝒬𝒬𝒬

𝑝𝑝𝑝𝑝 ! !!! =

Equilibrium equations of the piezoelectric bistable composite beam

We now treat the constitutive equation describing the mechanical behavior of the composite beam. The fact that the piezoelectric material is polarized by an electric field E modifies the stress in the beam. The impact of this stress can be modeled using two parameters FE and ME, representing the piezoelectric force (3.2.1a) and the piezoelectric moment (3.2.1b).

𝐹𝐹𝐹𝐹 ℰ = -𝑣𝑣𝑣𝑣. 𝑒𝑒𝑒𝑒 !" 𝑧𝑧𝑧𝑧 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 Three cases can be treated: in the general case (3.2.2) is solved with the help of (3.1.6b) to see the influence of the piezoelectric capacitor's voltage on the instability of the beam. In the second case, a mean value of the output voltage is taken into account. The results of the reduced-order model are then modified, mainly by replacing the residual stress force No by No elec (3.2.3a). One of the main consequences is the modification of the criterion of instability (3.2.3b). The last solution is the weak electro-mechanical coupling; in this case we recall that the equilibrium equation is simply (3.2.3c). This last approach will mainly be followed in the rest of the chapter.

𝑁𝑁𝑁𝑁 ! !"!# = 𝑁𝑁𝑁𝑁 ! + 𝑁𝑁𝑁𝑁 ! 𝑀𝑀𝑀𝑀 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! -1 . 𝐹𝐹𝐹𝐹 ℰ (3.2.3a) 𝑁𝑁𝑁𝑁 ! !"!# ≥ 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! (3.2.3b) 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! + 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑤𝑤𝑤𝑤 ! + 2. 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿 . 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 = 0 (3.2.3c)
The last equation to be treated is the variation of entropy of the beam. As demonstrated in the previous chapter, knowing the expression of the beam entropy and using the Fourier law enable to study the thermal exchanges between the beam and its surroundings and to find the temporal evolution of the temperature.

𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = ! !! ! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑁𝑁𝑁𝑁 ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 2. ! ! . 𝑀𝑀𝑀𝑀 ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 + 𝒞𝒞𝒞𝒞 ! . ! ! .! ! 𝒮𝒮𝒮𝒮 ! !! 𝒯𝒯𝒯𝒯 . 𝑑𝑑𝑑𝑑𝒰𝒰𝒰𝒰 ! (3.2.4)

Pyroelectric conversion using a Stirling cycle

Stirling cycle

Even if the simplest solution to harvest the pyroelectric charges is to connect a resistance to the piezoelectric capacitor, this type of cycle is not the most efficient. Moreover, the optimization of the load resistance depends on the frequency of the heating and cooling phases, which depends on the external thermal gradient. In these conditions, looking for other thermodynamic cycles can lead to a solution to increase the efficiency of the harvester. In this part we study the implementation of the Stirling cycle to harvest heat. As explained in the Chapter 2, this cycle is originally divided in four phases: isochoric heating, isothermal expansion, isochoric cooling and isothermal compression. The volume being an extensive parameter, the equivalent parameter in the case of the electric Stirling cycle is the electric displacement. The isochoric transformations are then replaced by heating at constant electric displacement or at constant electrical charge. The Stirling cycle is represented in Fig. 4.1.1a. During the heating phase, the elevation of the voltage difference across the pyroelectric generator increases the potential energy of the electrical charges. These charges are then harvested by closing the circuit once the maximal potential is reached. By opening the circuit during the cooling phase, the generator voltage decreases while the charge is kept constant. The circuit is finally closed to recharge the capacitor. The analysis of the theoretical performances of the Stirling cycle was presented in [Sebald et al, 2008]. This work defined two figures of Merit for the performances of pyroelectric generators. The first FOM recalled in (4.1.1a) is related to the output energy of the generator (4.1.1b). The second one is also called electro-thermal coupling factor and is given by (4.1.1c). The Carnot efficiency of the Stirling cycle (temperatures θh and θc) is then given by (4.1.1d), which can be simplified into (4.1.1e) in the case of low electro-thermal coupling.

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 !"#$ = 𝑝𝑝𝑝𝑝 ! 𝜀𝜀𝜀𝜀 !! ! (4.1.1a) 𝒲𝒲𝒲𝒲 ! = 𝑝𝑝𝑝𝑝 2 𝜀𝜀𝜀𝜀 33 𝜃𝜃𝜃𝜃 . 𝜃𝜃𝜃𝜃 ℎ -𝜃𝜃𝜃𝜃 𝑐𝑐𝑐𝑐 2 (4.1.1b) 𝑘𝑘𝑘𝑘 ! = 𝑝𝑝𝑝𝑝 ! 𝜀𝜀𝜀𝜀 !! ! . 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 !! 𝒯𝒯𝒯𝒯 . 𝜃𝜃𝜃𝜃 ! (4.1.1c) 𝜂𝜂𝜂𝜂 !"# = 𝑘𝑘𝑘𝑘 ! 1 + 0.5. 𝑘𝑘𝑘𝑘 ! . 𝜃𝜃𝜃𝜃 ! -𝜃𝜃𝜃𝜃 ! 𝜃𝜃𝜃𝜃 ! (4.1.1d) 𝜂𝜂𝜂𝜂 !"# ≈ 𝑘𝑘𝑘𝑘 ! (4.1.1e)
Another Figure of Merit for the energy has also been defined in [START_REF] Bowen | A modified figure of merit for pyroelectric energy harvesting[END_REF]. This FOM just comes from the expression of the heat stored in the heat capacitance during the whole cycling of the pyroelectric generator (4.1.2a) and is quite similar to the expression of the electro-thermal coupling factor, apart from the fact that it is independent from the working temperature (4.1.2b). The typical values of these FOM are reported in Tab.2.2.1, showing that PMN-PT is the best material to harvest heat, followed by PZT ceramics.

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 !"#$ ! = 𝑝𝑝𝑝𝑝 ! 𝜀𝜀𝜀𝜀 !! ! . 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 !! 𝒯𝒯𝒯𝒯 ! (4.1.2b) 𝒲𝒲𝒲𝒲 ! = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 !"#$ ! . 𝑄𝑄𝑄𝑄 !!" ! (4.1.2c)
Even if the Stirling cycle is less efficient than the Olsen-Ericsson cycle [START_REF] Olsen | Cascaded pyroelectric energy converter[END_REF][START_REF] Olsen | Pyroelectric conversion cycles[END_REF], it has the advantage of not requiring any external polarization voltage to work and it is similar to the SECE cycle currently used to harvest piezoelectric energy from mechanical vibrations. As a consequence, the circuits devoted to mechanical energy harvesting can be used with the Stirling generator, such as SECE [Sebald et al, 2008] or PSCE circuits [START_REF] Hehn | A Fully Autonomous Integrated Interface Circuit for Piezoelectric Harvesters[END_REF]. As explained in [Sebald et al, 2008], a way to boost up the efficiency of the pyroelectric generator would be to use SSHI architectures instead of SECE circuits.

Implementation of the Stirling cycle

In the case of the bimetallic strip heat engine, several types of Stirling cycles can be implemented given the possibility to deposit the piezoelectric layer on the active layer of the bimetallic beam or on its passive layer, and given the possibility to polarize the piezoelectric capacitor at different steps of the hysteretic cycle of the bistable bimetallic beam. In this study, we will consider three different scenarios of operation.

a. Stirling cycle I

In this first scenario, the piezoelectric capacitor is charged and discharged each time the beam is about to switch from a critical equilibrium point to a stable equilibrium point (Fig. 4.2.1a). To describe every step of the Stirling cycle, we write the evolution of the electrical charges carried by the piezoelectric capacitor and its voltage for each phase of the cycle. The first step is the initial heating of the system from the reference temperature to the beam's snap temperature. During this first step, the voltage increases from 0 to Vo (4.2.1a). The circuit is then closed, leading to the creation of a charge Qo (4.2.1b).

𝒱𝒱𝒱𝒱 ! = 𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# + 𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 2. 𝐿𝐿𝐿𝐿 ! . 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝜋𝜋𝜋𝜋 2 . 𝑤𝑤𝑤𝑤 ! ! ! -𝑤𝑤𝑤𝑤 !"# ! + 4. 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! ! -𝑤𝑤𝑤𝑤 !"# (4.2.1a) 𝒬𝒬𝒬𝒬 ! ! = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! (4.2.1b)
𝒬𝒬𝒬𝒬 ! ! -𝒬𝒬𝒬𝒬 !" ! = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (4.2.3b)
The two voltages being of opposite sign, we can define a maximal output voltage by taking the absolute values of these voltages.

𝒱𝒱𝒱𝒱 ! = 𝒱𝒱𝒱𝒱 ! ! = 𝒱𝒱𝒱𝒱 !" ! = 𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 + 4. 𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿 ! . 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! ! (4.2.4)
With these equations, it is possible to plot the thermodynamic cycle of the Stirling heat engine in the plan (Q,V), (Fig. 4.2.1b), when the piezoelectric layer is deposited on the active layer or on the passive layer. Two differences can be observed between these two configurations. First, the areas enclosed by the two loops are different, meaning that the two types of beams do not produce the same energy. The fact that the thermo-mechanical properties of the piezoelectric layer modify the value of the thermal asymmetry, plays an important role in the modification of the generated energy. Generally, as the thermal expansion coefficient of the piezoelectric material is lower than the active layer's one, the thermal sensitivity of the beam is better if the piezoelectric layer is deposited on the passive layer, partly explaining the increase of the generated energy by the increase of the critical curvature wS C . The second effect is linked to the position of the piezoelectric layer with respect to the neutral axis. When the piezoelectric layer is deposited on the active layer, the term (zp-zo) is negative, and given that the critical curvature wS C and the piezoelectric modulus e31 are also negative, the charges created by the pyroelectric effect are added to the charges created by the curvature of the piezoelectric layer. In the other case, when the piezoelectric layer is deposited on the passive layer, the pyroelectric charges are substracted to the charges linked to the piezoelectric effect. The position of the neutral axis also modifies the way the capacitor is charged or discharged when the piezoelectric layer is deposited on the passive layer. If the pyroelectric effect is predominant, the capacitor charges when the beam is at TS, and discharges at TSB, like when the piezoelectric layer is on the active layer. In the other case, the capacitor charges at TSB and discharges at TS. ). These figures first show that for weak thermal hystereses, the tertiary pyroelectric effect is predominant over the primary pyroelectric effect, explaining why the bimetallic strip heat engine helps harvesting more energy than a simple PZT layer used as pyroelectric generator. As an example, for a hysteresis of 5 o C, the pyroelectric bimetallic strip heat engine produces 54 µJ instead of 5 µJ for the pyroelectric generator. However, when the piezoelectric layer is on the passive layer of the bimetallic strip, the piezoelectric charges and the pyroelectric charges neutralize each other for a thermal hysteresis of 13 o C. For wider hystereses, the charges are mainly generated by the pyroelectric effect.

b. Stirling cycle II

In that second case, the "isochoric" phases begin once the beam snaps (or snaps back) and finish after the beam snaps back (or snaps). The charge and discharge of the piezoelectric capacitor happen when the beam's equilibrium is stable (in 2 and 4). The principle of this second thermodynamic cycle is presented in Fig. 4.2.3a. Following the same approach as for the first type of Stirling cycle, we can express the main variations of the piezoelectric capacitor's voltage and charge. During the beam's heating and the beam's snap, the voltage evolves according to (4.2.6).

𝒱𝒱𝒱𝒱 ! ! = 𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 -8. 𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿 ! . 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! ! (4.2.6a) 𝒬𝒬𝒬𝒬 ! ! -𝒬𝒬𝒬𝒬 !" ! = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (4.2.6b)
Similarly, during the beam's cooling and the beam's snap-back, the voltage evolves according to (4.2.7). We can thus define a maximal voltage by (4.2.7c). This last expression directly shows that the output voltage generated by this type of Stirling cycle is greater because of the wider variations of the beam's curvature during the two "isochoric" phases of the cycle (twice more curvature). The expression of the output energy is unchanged and is given by (4.2.7d). With these expressions, we can plot the evolution of the electrical charges and the capacitor's voltage along the whole cycle (Fig4.2.3b). Fig. 4.2.3c and Fig. 4.2.3d compare the evolution of the output voltage and available energy when the piezoelectric layer is deposited on the active or passive layer of the bimetallic beams, whereas Fig. 4.2.3b represents the thermodynamic cycle of the heat engine for a beam having a hysteresis of 3 o C. Contrary to the previous cycle, the change of the sign of the piezoelectric effect in the expression of the output voltage has for consequence that the pyroelectric and piezoelectric effects are summed when the piezoelectric layer is deposited on the passive layer and substracted when it is on the active layer. As a result, the first case generates much more energy. For a hysteresis of 5 o C, the heat engine generates 120 µJ instead of 63 µJ when the piezoelectric layer is on the active layer and 5 µJ for the simple pyroelectric generator. Once again, it can be observed that for hysteresis around 16 o C, the pyroelectric generator produces more energy than the bimetallic strip heat engine with PZT on the active layer, because of the antagonistism between the pyroelectric and piezoelectric effects.

𝒱𝒱𝒱𝒱 !" ! = - 𝑡𝑡𝑡𝑡 ! .

c. Stirling cycle III

The last Stirling cycle enables to separate the charges created during the heating and cooling phases of the thermodynamic cycle of the bimetallic strip, from the charges created during the beam's two switching events. This cycle is then made of two charges, two discharges and four iso-charges (Fig. 4.2.4a). The bimetal is first heated up to the snap temperature and is charged before it snaps. The quantities associated to these two steps are given by (4.2.8). The same operations happen when the bimetal is cooled down to its snap-back temperature and snaps back to return to its initial configuration Following the same reasoning as for the other two cycles, we consider the two possible cases for the position of the piezoelectric layer on the bimetallic beam, and we plot the evolution of the electrical charge and capacitor's voltage along the thermodynamic cycle in Fig. 4.2.4b. The area enclosed by the cycle's loop is smaller when the piezoelectric layer is deposited on the beam's active layer, which can be understood thanks to 4.2.11a giving the device's output voltage when the beam is heated or cooled. Equation 4.2.11d enables to plot the evolution of the available energy as a function of the thermal instability of the beam, showing that, even if the PZT on the passive layer produces more energy than the PZT on the active layer, the fact of decoupling the beam's switching from the beam's heating and cooling reduces the difference of performance between the two configurations.

𝒱𝒱𝒱𝒱

𝒱𝒱𝒱𝒱 !" ! = - 𝑡𝑡𝑡𝑡 ! .

d. Comparison of the three cycles and enhancement factors

To ease the comparison between these three different cycles, Fig. 4.2.6a presents the evolutions of the output energy for the heat engine for the best configuration (PZT on the active layer for the first cycle, PZT on the passive layer for the other two), emphasizing the fact that the second type of Stirling cycle produces more energy than the other two cycles. To complete the study, we also plot in Fig. 4.2.5b the evolution of the Carnot efficiency of these engines obtained by dividing the electrical energy by the heat provided by the hot source (and assuming that the latent heat of snap is still negligible). It is important to note that for small thermal hysteresis, the bimetallic strip heat engine yields higher conversion efficiencies than a simple pyroelectric generator. Yet, whatever the cycle, this thermal efficiency rapidly falls below the efficiency of the pyroelectric generator, mainly because of the higher heat capacitance of the bimetallic beams which need more heat per cycle. Whatever the value of the thermal hysteresis, the efficiency of the heat engine stays low, generally below one percent of Carnot.

Using the expression of the critical beam's amplitude (4.2.12a), it is possible to re-express the output energy and the thermal efficiency of the device. This way, the maximal output voltage generated by the second Stirling cycle is given by ( 4 Hence the performances of the systems are given by (4.2.14). The analysis of the new FOM and the new electro-thermal coupling factor reveals two important facts explaining the observations made on Fig. 4.2.5. First the tertiary pyroelectric effect is efficient for low thermal hystereses and its impact on the energy becomes less important as the thermal hysteresis widens, because of its 2/3 power dependence on the thermal hysteresis. The relative increase of the FOM is plotted in Fig. 4.2.6a showing a FOM enhanced by a factor x23 for a hysteresis of 5 o C using thermomechanical bistability. The second observation is the degradation of the electro-thermal coupling factor for wide hystereses. Because of the ratio of heat capacitances, the advantage of using the bistability is lost for very large hystereses: for a hysteresis of 3 o C, the electro-thermal coupling factor is improved by a factor 5, while it falls below 1 above 30 o C of hysteresis.

𝒲𝒲𝒲𝒲 ! !"#$% = 𝐹𝐹𝐹𝐹 ! !"#$ . 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! (4.2.14a) 𝜂𝜂𝜂𝜂 !" !"#,!"#$% = 𝑘𝑘𝑘𝑘 !"#$ ! . 𝜂𝜂𝜂𝜂 !" !"#,!"#$

e. Comparing piezoelectric materials

Now that we found the best type of Stirling cycle, we compare the performances of the heat engine when the PZT layer is replaced by another pyroelectric material. Using the data from Tab.2.2.1, we calculate the FOM and the electro-thermal coupling factor (also corresponding to the intrinsic Carnot efficiency of the beam) obtained with a beam having a hysteresis of 3 o C. The data obtained are represented in Fig. 4.2.7a where the FOM of the pyroelectric bimetallic strip heat engine is plotted against the FOM of the pyroelectric generator, and in Fig. 4.2.7b which compares the Carnot efficiency of the two heat engines. Because of the weak stiffness of PVDF, the performance of the heat engine based on PVDF is weakly increased by the use of the bimetallic strip. In contrary for all other materials, the improvement of the FOM is significant and PMN-PT stays the best material to harvest heat, even if the greatest improvement is observed for PZT soft ceramics. Fig. 4.2.7b shows that for PMN-PT and PVDF, with a beam having a hysteresis of 3 o C, the use of the bimetallic strip heat engine does not increase the Carnot efficiency of the transduction nor the electro-thermal coupling factor of the pyroelectric material. However, for the other materials, the improvement of the efficiency is important, passing from 0.17% to 0.58% for PZT-5H. PMN-PT is the best material with an efficiency of 3.6%.

Influence of the architecture, extrinsic performances of the energy harvester

In the previous part, we considered the intrinsic performances of the pyroelectric bimetallic strip heat engine when it is polarized according to the Stirling cycle. The modeled performances thus correspond to the case of a bimetallic strip not constrained into an architecture, and they could be reached if the heat engine is used to harvest slow variations of the temperature. To obtain the generated power, it would be necessary to multiply the previous formula of the available energy by the frequency of the temperature variations. In the previous chapter we treated both the modeling of the thermal transfers inside the energy harvester (bimetallic strip and architecture), and the modeling of the architure's impact on the mechanical bistability of the bimetallic beam. We especially demonstrated that the architecture modifies the bimetal thermostat's thermal hysteresis, thereby increasing the operation frequency of the device, but also that it degrades the energetical performances of the devices. The consequence of these two results is the occurrence of an optimal design that maximizes the output energy of the system. In this part, we will describe the main effects of the architecture on the dynamic performances of the energy harvester when a pyroelectric material is used to convert the beam's mechanical strain into electrical energy.

a. Stirling cycles

We follow the same reasoning than in the previous chapter and we consider a beam constrained in an architecture characterized by wtop and wbot, which verify (4.3.1a) and (4.3.1b), and by a total thickness given by (4.3.1c). Fig. 4.4.1a recalls the scheme of the harvester's architecture whereas Fig. 4.4.1b gives the evolution of the thickness of the device's cavity as a function of the beam's intrinsic thermal hysteresis when a PZT-5H-Al-Invar beam is used as a prime mover of the heat engine. We recall that the thermal hysteresis of the beam in the cavity is a function of the ratio rw and the intrinsic thermal hysteresis and is given by (4.3.1d).

𝑤𝑤𝑤𝑤 !"# = 𝑤𝑤𝑤𝑤 !"# = 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! (4.3.1a) 𝑟𝑟𝑟𝑟 ! ≤ 1 (4.3.1b) 𝑡𝑡𝑡𝑡 !"# = 𝑤𝑤𝑤𝑤 !"# + 𝑤𝑤𝑤𝑤 !"# + 𝑡𝑡𝑡𝑡 (4.3.1c) 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# = 3 2 . 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 . Δ𝑇𝑇𝑇𝑇 (4.3.1d)
As for the description of the Stirling cycles for a non-constrained beam, we can write the evolution of the electrical charge carried by the pyroelectric capacitor as a function of its output voltage. In brief, we can demonstrate that the maximal output voltage for the first Stirling cycle is given by (4.3.2a). This expression leads to find (4.3.2b) as an expression of the electrical energy produced by the energy harvester during a complete cycle. It can be seen that the Stirling cycle of the constrained beam produces as much energy as the nonconstrained beam when the thickness ratio is equal to 1. The fact of constraining the beam in the architecture reduces the performances of the second Stirling cycle. For that cycle, the performances are given by (4.3.3a) and (4.3.3b), and differ from the previous equations only by the sign of the tertiary pyroelectric effect. The difference of performances between these two cycles is mainly due to the value of the thermal asymmetry linked to the position of the pyroelectric layer with respect to the bimetallic beam. Finally, the third Stirling cycle is characterized by the voltage drop due to the beam's heating and the voltage of the beam's snap. These two quantities are given by (4.3.5a) and (4.3.5b). As a consequence, the energy generated by the third type of Stirling cycle is given by (4.3.5c). This last equation shows that by decoupling the heating and cooling phases from the beam's snap and snap-back, the energy associated to the cross-term of the primary and tertiary pyroelectric effect is lost. If we plot the evolution of these energies as a function of the thermal hysteresis (Fig.4.3.2a and Fig.4.3.2b), we can see that depositing PZT on the active layer produces slightly more energy with the first Stirling cycle than PZT on the passive layer with the second Stirling cycle. This is mainly due to the position of the neutral axis that is closer to the piezoelectric layer position for the beam with PZT on the passive layer, making this type of beam more sensitive to the temperature. Because this difference is weak, in the rest of this chapter we will keep focusing on the performances of the second Stirling cycle.

𝒱𝒱𝒱𝒱 ! !"# = 𝒱𝒱𝒱𝒱 ! ! = 𝒱𝒱𝒱𝒱 !" ! = 𝑡𝑡𝑡𝑡 ! .
𝒱𝒱𝒱𝒱 !! !"# = 𝒱𝒱𝒱𝒱 ! ! = 𝒱𝒱𝒱𝒱 !" ! = 𝑡𝑡𝑡𝑡 ! .
𝒰𝒰𝒰𝒰 !!!,!!"#$!" !"# = 𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# (4.3.5a) 𝒰𝒰𝒰𝒰 !!!,!"#$ !"# = 4
To see how the cavity's thinning impacts the performances of the heat engine, we extract from the previous equations the Figure of Merit of the energy per cycle and the electrothermal coupling factor of the pyroelectric layer. Hence, the FOM becomes (4.3.6.a) whereas the electro-thermal coupling factor is given by (4.3.6b).

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 !"#$%,!"# = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 !"#$ . 3 2 . 𝑟𝑟𝑟𝑟 ! - 𝑟𝑟𝑟𝑟 ! ! 3 + 4 4 ! . 𝑟𝑟𝑟𝑟 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝑒𝑒𝑒𝑒 !" 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 ! . 1 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! ! (4.3.6a) 𝑘𝑘𝑘𝑘 !"#$%,!"# ! = 𝑘𝑘𝑘𝑘 !"#$ ! . 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 ! . 𝜌𝜌𝜌𝜌 !"#$ . 𝑐𝑐𝑐𝑐 !"#$ 𝐶𝐶𝐶𝐶 ! . 1 + 8. 4 ! 3. 1 - 𝑟𝑟𝑟𝑟 ! ! 3 . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝑒𝑒𝑒𝑒 !" 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁. 𝐿𝐿𝐿𝐿 ! . 1 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! ! (4.3.6b)
These two parameters are increasing functions of the factor rw and are represented in Fig.4.3.2c and Fig.4.3.2d. Fig.4.3.2d shows that the advantage of using a bimetallic strip heat engine rapidly disappears: the Carnot efficiency of the device becomes lower than the efficiency of a simple pyroelectric generator when the beam's natural thermal hysteresis increases or when the cavity's thickness becomes smaller. For a beam with a hysteresis of 3 o C, the efficiency of the system stays above the Carnot efficiency of the pyroelectric generator (around 0.2%). However, it should be noted that, contrary to the pyroelectric generator, the system now harvests heat from thermal gradients instead of harvesting heat from temperature temporal variations. Thus it is possible to define an equivalent ZT for the system in order to compare it with Seebeck generators. By inverting the expression of the Carnot efficiency of a Seebeck generator we can find the equivalent ZT of the generator (4.3.7a). Given that the electro-thermal coupling is weak, the heat engine's ZT can be linearly approximated by (4.3.7b).

𝑍𝑍𝑍𝑍𝑇𝑇𝑇𝑇 !"#$%,!"# = 1 + 𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! . 𝑘𝑘𝑘𝑘 !"#$%,!"# ! 1 -𝑘𝑘𝑘𝑘 !"#$%,!"# ! ! -1 (4.3.7a) 𝑍𝑍𝑍𝑍𝑇𝑇𝑇𝑇 !"#$%,!"# ≈ 2. 1 + 𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! . 𝑘𝑘𝑘𝑘 !"#$%,!"# ! (4.3.7b)
The Carnot efficiency and the equivalent ZT of various pyroelectric materials are represented in Fig.4.3.3a and Fig.4.3.3b. It shows that PMN-PT is still the best materials with an efficiency of 2% and an equivalent ZT of 0.081. The use of PZT ceramics enables to reach a Carnot efficiency of 0.22% with a ZT of 0.0089.

b. Dynamic performance of the harvester for fixed external temperature and no architectural leakage

Now that we determined the static performances of the harvester, we need to combine the previous equations with the lumped capacitance model developed in the previous chapter in order to extract the expression of the dynamic performances of the heat engine. Previously, we showed that the working frequency of the bimetallic strip between two heat sources depends on the two hot sources temperatures and the beam's thermal hysteresis in the cavity according to (4.3.8a), where RCC and RCH were the contact resistances between the heat sources and the beam. The comparison between the experimental measurements and the theoretical model showed that it is possible to get good results by taking them both equal to 10 K.W -1 . Moreover, because the frequency is maximized if the median temperature of the hysteresis is equal to the average value of the hot and cold sources temperatures, the equation of the frequency can be simplified to (4.3.8b).

𝑓𝑓𝑓𝑓 = 1 𝐶𝐶𝐶𝐶 ! . 𝑅𝑅𝑅𝑅 !! . ln 𝑇𝑇𝑇𝑇 !,!"# -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !",!"# -𝑇𝑇𝑇𝑇 ! + 𝑅𝑅𝑅𝑅 !" . ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !",!"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !,!"# (4.3.8a) 𝑓𝑓𝑓𝑓 = 1 2. 𝑅𝑅𝑅𝑅 !! . 𝐶𝐶𝐶𝐶 ! . 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (4.3.8b)
With these equations, the average power produced by the harvester is simply the product of the working frequency by the electrical energy produced during a complete cycle, which means in terms of equation (4.3.9a). Developping this last equation as a function of the thermal hysteresis in the cavity leads to (4.3.9b).

𝑃𝑃𝑃𝑃 ! = 𝒲𝒲𝒲𝒲 !! !"# . 𝑓𝑓𝑓𝑓 (4.3.9a) 𝑃𝑃𝑃𝑃 ! = 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡 ! . 𝐹𝐹𝐹𝐹 𝒲𝒲𝒲𝒲 !"#$ . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# + 8. 𝑟𝑟𝑟𝑟 ! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝑒𝑒𝑒𝑒 !" 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# ! ! 2. 𝑅𝑅𝑅𝑅 !! . 𝐶𝐶𝐶𝐶 ! . 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (4.3.9b)
This equation is quite complex to understand. In order to simplify the study of this function, we introduce the quantity x corresponding to the ratio between the thermal hysteresis of the beam in the cavity and the temperature difference TH-TC across the device (4.3.10a). In the same way, we define βpyro the non-linear term coming from the tertiary pyroelectric effect (4.3.10b).

𝑥𝑥𝑥𝑥 = 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (4.3.10a) 𝛽𝛽𝛽𝛽 !"#$ = 8. 𝑟𝑟𝑟𝑟 ! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝑒𝑒𝑒𝑒 !" 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 ! . 1 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! ! (4.3.10b)
This way, the power can be re-expressed as (4.3.11). In the same time, to complete the description of the dynamic performances of the system, Fig. 4.3.5b draws the evolution of the working frequency of the prime mover as a function of the ratio x. The first figure shows that for very small values of x (hystereses smaller than 2 o C for an external temperature of 30 o C), the output power is a decreasing function of the ratio x. However, it rapidly increases for values of x between 0.1 and 0.8. This means that the increase of the electrical energy per cycle for highly unstable beams outweighs the reduction of the working frequency for large values of the beam's thermal hysteresis. After reaching an optimal value, the output power decreases when the beam's thermal hysteresis in the cavity becomes closer to the external temperature difference.

The optimal value of the beam's thermal hysteresis is 21 o C for a ratio rw of 1 and a temperature difference of 30 o C. For this hysteresis value, the output power is of 25.7 µW for a working frequency of 0.045 Hz. For an external temperature of 15 o C, the optimal hysteresis value is of 9.6 o C, producing 8.9 µW. When they exist, the extrema of power, either minimum or maximum, are solutions of (4.3.12a). The study of this function presented in Fig. 4.3.5c shows that the optimum values of x appear only for small values of βpyro. These values must be lower than the limit value βpyro max given by (4.3.12b). The values of x maximizing the output power are then the extrema that are higher than 0.34 (4.3.12c). Indeed, for values of βpyro higher than βpyro max , the power is a monotonic and decreasing function of x: the smallest the hysteresis, the higher the output power generated by the device.

𝛽𝛽𝛽𝛽 !"#$ = -3. 𝑥𝑥𝑥𝑥 ! ! . 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝑥𝑥𝑥𝑥 1 -𝑥𝑥𝑥𝑥 . 𝑥𝑥𝑥𝑥 ! -1 + 𝑥𝑥𝑥𝑥 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝑥𝑥𝑥𝑥 1 -𝑥𝑥𝑥𝑥 . 𝑥𝑥𝑥𝑥 ! -1 + 3. 𝑥𝑥𝑥𝑥 (4.3.12a) 𝛽𝛽𝛽𝛽 !"#$ ≤ 𝛽𝛽𝛽𝛽 !"#$ !"# = 1,06 (4.3.12b)
𝑥𝑥𝑥𝑥 !"# ≥ 0,34

(4.3.12c)
These equations thus enable to draw the dependence of the optimal natural thermal hysteresis of the beam on the external temperature difference (Fig. 4.3.5d), the evolution of the maximal output power (Fig. 4.3.5e), and the efficiency associated to the use of the beam having an optimal thermal hysteresis (Fig. 4.3.5f). For a temperature difference of 40 o C, the maximal output power is 40 µW (corresponding to a density of 10 µW.cm -2 ) for a Carnot efficiency of 0.048%.

c. Dynamic performance of the harvester with Robin conditions

Now that we determined the performances of the harvester when the temperature across the harvester are fixed, we need to evaluate these same performances when the heat flowing through the system is evacuated by means of a heat sink, either planar or with fans, and when the harvester architecture is leaky. The equations describing this case were presented in the previous chapter. Assuming that the heat sink has a thermal resistance Rsink and exchanges heat with its surroundings at ambient temperature Tamb, and that the harvester architecture has a thermal resistance Rleak, the temperature difference across the bistable beam is given by (4.3.13a), whereas the thermal efficiency of the device is given by (4.3.13b), and the Curzon-Ahlborn efficiency of the device by (4.3.13c). In Fig4.3.7a, we plot the evolution of the evolution of the maximal power obtained with the optimal bimetallic beam as a function of the hot source temperature and when the system works with a plane heat sink (hsink = 10 W.m -2 .K -1 or Rsink = 250 K.W -1 if it has the same surface area as the bimetallic beam) or with a bulk heat sink (hsink = 1000 W.m -2 .K - 1 ). In both cases, the thermal resistance of the architecture is taken equal to 625 K.W -1 . The first observation, already made in the previous chapter, is the necessity to use a heat sink with fans to boost the performance of the harvester.

𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! = 𝑅𝑅𝑅𝑅 !"#$ 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# (4.3.13a) 𝜂𝜂𝜂𝜂 !" !"#,!"# ≈ 𝑅𝑅𝑅𝑅 !"#$ . 𝑃𝑃𝑃𝑃 ! (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) (4.3.13b) 𝜂𝜂𝜂𝜂 !" !",!"# ≈ 𝑅𝑅𝑅𝑅 !"#$ . 𝑃𝑃𝑃𝑃 ! (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) . 𝑇𝑇𝑇𝑇 ! +
As a consequence, with a hot source temperature fixed at 100 o C, the surface density of power can be increased by a factor 3.3, from 2.6 µW.cm -2 to 8.5 µW.cm -2 . Moreover, the using a heat sink improves the efficiency of the harvester by increasing the effective temperature difference across the energy harvester. The consequence of having a thermal conducting architecture can be mainly seen on the reduction of the thermo-electric efficiency of the system which is about 1.2. Thermo-electric efficiency of the system working with or without heat sink, when output power is maximal. c. Curzon-Ahlborn efficiency of the system as a function of the hot source temperature when output power is maximal. d. Comparison of the surface density of output power of a beam with hysteresis of 3 o C working with our without heat sink, with the maximal surface density of output power produced with the optimal design of bimetallic beam.

Coupling with an oscillating piezoelectric transducer

A short review of energy harvesting devices based on mechanical shock

The second type of thermal energy harvester using thermo-mechanically bistable beams is based on the use of a simple oscillating piezoelectric transducer that is impacted by the bistable beam each time the beam switches from the hot source to the cold source. Although it is the first time this technique is used in a thermal energy harvester, designing impact-based energy harvesters is not a new technique in the context of harvesting energy from mechanical vibrations. The first shock systems proposed were based on the impact of balls on piezoelectric transducers in order to harvest the energy of random vibrations, whereas classical piezoelectric harvesters used to harvest energy from vibrations close to their vibration frequency. Such devices were first proposed by [START_REF] Umeda | Analysis of the Transformation of Mechanical Impact Energy to Electric Energy Using Piezoelectric Vibrator[END_REF] and lately exposed in [START_REF] Renaud | Harvesting energy from the motion of human limbs: the design and analysis of an impact-based piezoelectric generator[END_REF] and [START_REF] Alghisi | Triaxial ball-impact piezoelectric converter for autonomous sensors exploiting energy harvesting from vibrations and human motion[END_REF]. These three teams also report the modeling of the shock between the harvester and the ball, hovewer this model is not directly applicable to our system since the ball is not an oscillator. The use of impact as also been presented in [START_REF] Gu | Impact-driven, frequency up-converting coupled vibration energy harvesting device for low frequency operation[END_REF][START_REF] Gu | Low-frequency piezoelectric energy harvesting prototype suitable for the MEMS implementation[END_REF][START_REF] Liu | A compact architecture for passively-switched energy harvesters[END_REF] in order to transfer the energy of a slowly-actuated, high-inertia oscillator to a higher-frequency piezoelectric oscillator. This kind of devices also enables to widen the bandwith of the mechanical energy harvesters.

Modeling the shock between two oscillators a. Mason model of the piezoelectric oscillator

In this part, we study three architectures represented in Fig. 5.2.1a-b-c. In the first case (1a), the bimetallic strip is coupled with a clamped-clamped piezoelectric bimorph beam, which represents the current architecture of the piezoelectric system. The second architecture (1b) represents the coupling between the bimetallic strip and a simply supported beam. The last case (1c) corresponds to two cantilever beams having a length corresponding to half the length of the bimetallic beam. By using the constitutive equations of a piezoelectric layer of thickness tp (properties d31, ε3, Ep, located at a distance zp of the neutral axis zo,p) given in (2.2.9), and the local expression of the electric Gibbs energy (5.2.1a), its is possible to find the general expression of the piezoelectric oscillator's energy (5.2.1b) as a function of the mass mp of the oscillator (or sum of the masses for the two cantilever beams), the equivalent bending stiffness of the beam ie,p, the capacity of the piezoelectric layer Cp and the capacitor's voltage Up. It must be noticed that the absolute value in the coupled term is used to define the use of one capacitor in the case of a simply supported beam or a cantilever beam, or three capacitors in the case of a clamped-clamped beam.

b. Modeling of the shock

The bistable beam is modeled by a non-linear Duffing oscillator with negative linear stiffness, whose equivalent stiffness is deduced from the equation of equilibrium of the beam at the snap temperature (5.2.4a) (see Chapter 2, section 4). To model the losses due to the damping of the bimetallic strip, we define a damping coefficient δb.

1 2 . 𝓂𝓂𝓂𝓂 ! . 𝑤𝑤𝑤𝑤 ! -𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! + 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! + 2. 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! ! + 2. 𝛿𝛿𝛿𝛿 ! . 𝑤𝑤𝑤𝑤 ! = 0 (5.2.4a) 𝑘𝑘𝑘𝑘 !! = 3. 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! ! = 𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿 . (𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! ) (5.2.4b) 𝑘𝑘𝑘𝑘 !! = 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! (5.2.4c)
We assume that the neutral axis zo p of the piezoelectric oscillator is initially located at a distance rw.ws c of the neutral axis of the bimetallic strip, so that when they are in contact, we have the equality (5.2.5). Naturally the ratio rw still needs to be lower than 1 for the beam to be cycled between the two heat sources.

𝑤𝑤𝑤𝑤 ! = 𝑤𝑤𝑤𝑤 ! -𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! (5.2.5)
The shock between the two oscillators is quite complex to model mainly because of the non-linearity brought by the equilibrium equation of the bimetallic strip. Otherwise, if the bimetallic strip was a linear oscillator, (5.2.3.b) and (5.2.4a) could be easily solved independently when the two oscillators are not in contact and the coupling solved when they are in contact. If we assume that the initial speeds of the bimetallic strip and the piezoelectric oscillator are equal and that the capacitor is initialy discharged, the oscillators stay stuck together because the piezoelectric transducer slows down due to the increase of its strain energy, and the bimetallic strip accelerates for the inverse reason. In this case, the equilibrium equation of the coupled oscillators is simply given by (5.2.6a), whereas the final position of equilibrium w∞ is given by (5.2.6b). This equation can simply be solved with Cardan's formula.

𝓂𝓂𝓂𝓂 ! 2 + 𝓂𝓂𝓂𝓂 ! * . 𝑤𝑤𝑤𝑤 ! + 𝑘𝑘𝑘𝑘 ! * -𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! + 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! + 𝑘𝑘𝑘𝑘 ! * . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! + 2. 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! ! + 𝛿𝛿𝛿𝛿 ! + 𝛿𝛿𝛿𝛿 ! . 𝑤𝑤𝑤𝑤 ! + 𝛤𝛤𝛤𝛤 ! * . 𝒱𝒱𝒱𝒱 ! = 0 (5.2.6a) 𝑟𝑟𝑟𝑟 ! ! + 3. 𝑘𝑘𝑘𝑘 ! * 𝑘𝑘𝑘𝑘 !! -1 . 𝑟𝑟𝑟𝑟 ! + 3. 𝑘𝑘𝑘𝑘 ! * 𝑘𝑘𝑘𝑘 !! . 𝑟𝑟𝑟𝑟 ! + 2 = 0 (5.2.6b) 𝑤𝑤𝑤𝑤 ! = -𝑟𝑟𝑟𝑟 ∞ . 𝑤𝑤𝑤𝑤 ! ! (5.2.6c)
Assuming that the damping of the bimetallic strip is null before the impact, the conservation of energy enables to write that the initial kinetic energy of the coupled oscillators is given by (5.2.7a), whereas the maximal mechanical energy that can be harvested is given by (5.2.7b). Using the formula developed in the previous chapter, we can also express the efficiency of the thermo-mechanical transduction (5.2.7c), and the efficiency of the energy transfer during the shock (5.2.5e) by dividing the incident mechanical energy of the piezoelectric transducer by the kinetic energy of snap of the beam (5.2.7d).

(c) (d) 1 2 . 𝓂𝓂𝓂𝓂 ! 2 + 𝓂𝓂𝓂𝓂 ! * . 𝑤𝑤𝑤𝑤 ! ! = 4 3 . 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! !
(5.2.7a)

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! = 𝑘𝑘𝑘𝑘 !! 12 . 𝑤𝑤𝑤𝑤 ! ! ! . 𝑟𝑟𝑟𝑟 ! -1 ! . 𝑟𝑟𝑟𝑟 ! + 3 + 𝑟𝑟𝑟𝑟 ∞ + 1 ! . 3 -𝑟𝑟𝑟𝑟 ∞ (5.2.7b) 𝜂𝜂𝜂𝜂 !"" !"# = 2. 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! 𝐶𝐶𝐶𝐶 ! . 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (5.2.7c) 𝛥𝛥𝛥𝛥𝒦𝒦𝒦𝒦 !"#$ = 27. 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 32. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! ! = 27 12 . 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! !
(5.2.7d)

𝜂𝜂𝜂𝜂 !! !"# = 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! 𝛥𝛥𝛥𝛥𝒦𝒦𝒦𝒦 !"#$ = 𝑟𝑟𝑟𝑟 ! -1 ! . 𝑟𝑟𝑟𝑟 ! + 3 + 𝑟𝑟𝑟𝑟 ∞ + 1 ! . 3 -𝑟𝑟𝑟𝑟 ∞ 27
(5.2.7e)

In Fig. 5.2.2a, we first plot the dependence of the final position on the ratio of the linear stiffness. Logically, we observe that the final position of the joined oscillators is a decreasing function of the piezoelectric transducer's stiffness and the ratio rw: increasing the transducer's stiffness or bringing the transducer closer to the bimetallic strip opposes more resistance and the final position is reduced. The two extreme values of the transducers stiffness were already modeled in the previous chapter: if the transducer's rigidity is null, the bimetal can reach its natural equilibrium position corresponding to twice its critical amplitude before its snap. If the rigidity is infinite, we found that the value of r∞ tends to rw, as modeled in the previous chapter. These phenomena also explain the tendencies observed on the evolution of the shock's efficiency in Fig. 5.2.2b. Reducing the transducer's thickness or increasing the gap between the two oscillators enables to let the bimetallic strip releases its kinetic energy more easily and so to transfer a maximal amount of this energy to the piezoelectric transducer.

In order to give the performances of a real device we consider the performances of an Al-Invar bimetallic beam having a thermal hysteresis of 3 o C and impacting a brass-PZT5H beam whose properties are given in Table 5.2.1a. The PZT layer on the oscillator has the same thickness as the layer deposited on the bimetallic beam in section 3. The thickness of the brass is chosen so that the neutral axis is at the interface between the brass layer and the PZT layer in order to preserve electrical charges due to the inversion of the stress (tensile to compressive stress) in the PZT layer. In Table 5.2.1b, we give the lumped parameters of the Mason model of the piezoelectric tranducers for the three types of boundary conditions. Logically, we observe that the effective mass and stiffness of the two clamped free beams are smaller than the clamped-clamped beam's effective mass and stiffness, however because of the reduction of the curvature of the beam, the piezoelectric coupling of the cantilever beams is weaker. The comparison between the simply supported beam and the cantilever beams shows that the effective mass of the cantilever beams is smaller, thus reducing its inertia.

In Fig. 5.2.2a, we reported the values of the ratio between the effective stiffness of the piezoelectric transducer and the bimetallic strip's stiffness for the three types of boundary conditions. This clearly shows that, from a mechanical point of view, using two cantilever beams or a simply-supported transducer yields to better mechanical conversion efficiency than aclamped-clamped oscillator. Fig. 5.2.2c represents the surface density of energy that can be generated with an Al-Invar beam and a PZT5H-brass transducer. The stiffness ratio is 16.3% for the SS and CF beams, and 65.3% for the CC beam. The maximal surface density of mechanical energy received by the piezoelectric transducer (if rw=1) is 6.9 µJ.cm -2 (99% efficiency) in the first case and 6.8 µJ.cm -2 (97.5 % efficiency) in the second case. The difference between the three systems is more visible if thicker transducers are used or if the device's cavity is thinner. Fig. 5.2.2f gives the thermal efficiency of the conversion chain made of comprising the thermo-mechanical transduction by the bimetallic beam and the shock between the bimetallic beam and the piezoelectric transducer. Even if the shock is less efficient when the cavity is thin, when the transducer's stiffness is weak, the fact that the thermo-mechanical transduction is better for small thermal hysteresis has for consequence that the overall thermo-mechanical efficiency is better when the cavity's thickness is small. If the transducer stiffness becomes predominant, the device is more efficient when the cavity is thick.

c. Optimal mechanical energy

Even if it is not the final efficiency, we can define the maximal mechanical power that can be transferred to the piezoelectric transducer by following the same approach as in the previous chapter (Section 5.2). If the harvester works between two thermostatic heat sources, the mechanical power generated by the device is given by (5.2.8a). The study of a similar function was already presented in the previous chapter. The optimal thermal hysteresis that maximizes the output power is given by (5.2.8b). The equation of the maximal power is then given by (5.2.8c), whereas the mechanical efficiency associated to this optimum is given by (5.2.8d) (for a non-leaky architecture).

𝑃𝑃𝑃𝑃 ! !"# =. 27 4 . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! . 𝜂𝜂𝜂𝜂 !! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! ! 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# ! ! 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.2.8a) 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# !"# = 0,589. (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) (5.2.8b) 𝑃𝑃𝑃𝑃 ! !"#,!"# = 2,46. 𝜂𝜂𝜂𝜂 !! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) ! ! (5.2.8c) 𝜂𝜂𝜂𝜂 !"" !"#,!"#,!"# = 11,3. 𝜂𝜂𝜂𝜂 !! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ! (5.2.8e)
Similarly, if the heat is evacuated with a heat sink (resistance Rsink) and if the architecture conducts heat from the hot source to the heat sink (resistance Rleak lower than the equivalent resistance of the bimetallic strip), the maximal power is given by (5.2.9a) and the efficiency associated to this optimum by (5.2.9b).

𝑃𝑃𝑃𝑃 ! !"#,!"# = 2,46. 𝜂𝜂𝜂𝜂 !! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! . 𝑅𝑅𝑅𝑅 !"#$ ! (𝑅𝑅𝑅𝑅 !"!" + 𝑅𝑅𝑅𝑅 !"#$ ) ! ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) ! ! (5.2.9a) 𝜂𝜂𝜂𝜂 !"" !"#,!"#,!"# = 2,46. 𝜂𝜂𝜂𝜂 !! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! ! . 𝑀𝑀𝑀𝑀 ! ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! ! . 𝑅𝑅𝑅𝑅 !! ! ! . 𝑅𝑅𝑅𝑅 !"#$ ! 𝑅𝑅𝑅𝑅 !"#$ ! . (𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ ) ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ! (5.2.9b)
Since in the current energy harvester the piezoelectric transducer is intended to be used as a planar heat sink, we focus our analysis on the performances of the harvester working without bulk heat sink. Fig. 5.2.3a represents the maximal surface density mechanical power (corresponding to the maximal output electrical power if the efficiency of the electro-mechanical transduction is 100%) that is transferred to a clamped-clamped piezoelectric transducer with a bimetallic strip having an optimized thermal hysteresis or a thermal hysteresis of 3 o C. At 100 o C, for a bimetal thermostat 80 o C-83 o C, the available output power is 1.0 µW.cm -2 , corresponding to 59% of the maximal power that can be obtained with the bimetal thermostat 70 o C-99 o C (1.7 µW.cm -2 ). At that temperature, the thermal efficiency of the energy harvester using the optimal bimetallic strip is 4.10 -3 %, corresponding to 4.3.10 -2 % of the Curzon-Ahlborn efficiency.

Electrical performances of the system a. Open circuit voltage

Now that we studied the mechanical performances of the harvester, we can look at the evolution of the output voltage generated by the harvester. In open circuit, the stiffness of the piezoelectric transducer is modified and the output voltage is given by (5.3.1a), the equilibrium equation becomes (5.3.1b), whereas the final position is given by (5.3.1c). The initial conditions are given by (5.3.1d) and (5.3.1e).

𝒱𝒱𝒱𝒱 !

!" = 𝛤𝛤𝛤𝛤 !

(5.3.1d)

𝑤𝑤𝑤𝑤 ! 0 = 𝑣𝑣𝑣𝑣 ! = -4. 𝑤𝑤𝑤𝑤 ! ! . 𝑘𝑘𝑘𝑘 !! 𝓂𝓂𝓂𝓂 ! + 2. 𝓂𝓂𝓂𝓂 ! * (5.3.1e)
We can solve the previous equation by using the Runge-Kutta method. In Fig. 5.3.1, we plot the temporal evolution of the output voltage in open-circuit, the oscillator displacement amplitude, and the phase portrait obtained with the three possible configurations of piezoelectric transducers, assuming a mechanical quality factor of 14. These figures show that the output voltage is higher for the clamp-clamped transducer because of its piezoelectric coupling that is higher than those of the other configurations. As a consequence, this type of piezoelectric transducer will be the only one to be studied in the next part of this chapter. The value of 16.9 V of the first peak is close to the values experimentally measured (15 to 20 V), which validates the model.

An important fact that can be noticed is the asymmetry of the phase portrait, which is more pronounced in the case of clamped-free beams. When the stiffness of the piezoelectric oscillator is smaller than the linear stiffness of the bimetallic strip, (5.3.1b) describes the behavior a damped asymmetric bistable Duffing oscillator. Numerous examples of symmetric bistable Duffing oscillators (or twin-well oscillator) are reported in the literature [START_REF] Tseng | Nonlinear Vibrations of a Buckled Beam Under Harmonic Excitation[END_REF], Moon and Homes, 1979, Arrieta et al, 2010] and have extensively been studied, mainly because of the possible occurrence of chaotic behavior in forced regime [Holmes, 1979, Kovacic and[START_REF] Kovacic | The Duffing Equation[END_REF]. Using bistable Duffing oscillators to widen the bandwith of vibrations energy harvester was first suggested by [START_REF] Ramlan | Potential benefits of a nonlinear stiffness in an energy harvesting device[END_REF] and is now an important field of research in energy harvesting given the large number of techniques to induce mechanical bistability (magnetic repulsion/attraction, buckling…). Extensive reviews of these applications are exposed in [START_REF] Harne | A review of the recent research on vibration energy harvesting via bistable systems[END_REF] and [START_REF] Zhu | Strategies for increasing the operating frequency range of vibration energy harvesters: a review[END_REF]. In this chapter, we will only present two methods to approximate the solutions of this equation. Exhaustive overviews of other mathematical methods are reported in [START_REF] Kovacic | The Duffing Equation[END_REF][START_REF] Nayfeh | Nonlinear oscillations[END_REF], Jordan and Simith, 2007].

The first method uses Jacobi elliptic functions to approximate the behavior of the oscillator. When the piezoelectric oscillator's stiffness is greater than the bimetallic strip's stiffness, (5.3.1b) describes a hardening Duffing oscillator with a positive linear stiffness and a positive cubic stiffness in damped free regim. The beam's shape can be approximated by ( 5

𝛿𝛿𝛿𝛿 = 𝛿𝛿𝛿𝛿 ! + 𝛿𝛿𝛿𝛿 ! 𝓂𝓂𝓂𝓂 ! + 2. 𝓂𝓂𝓂𝓂 ! * assuming 𝛿𝛿𝛿𝛿 ! = 𝛿𝛿𝛿𝛿 ! (5.3.2c) 𝛼𝛼𝛼𝛼 = 2. 𝑘𝑘𝑘𝑘 ! * -𝑘𝑘𝑘𝑘 !! + 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! 𝓂𝓂𝓂𝓂 ! + 2. 𝓂𝓂𝓂𝓂 ! * (5.3.2d) 𝛾𝛾𝛾𝛾 = 2. 𝑘𝑘𝑘𝑘 !! 𝓂𝓂𝓂𝓂 ! + 2. 𝓂𝓂𝓂𝓂 ! * (5.3.2e)
The solution of (5.4.2a) is a damped elliptic Jacobi function (5.3.3a), where the amplitude and the initial phase shift must be solved with the initial conditions. (5.3.3c) Injecting this function in the differential equation leads to find the expression of the pulsation at the resonance, and the elliptic modulus k as a function of the initial amplitude and the physical parameters of the system. Finding the amplitude and phase shift is quite hard and this problem is generally solved with the assumption that the oscillator's initial speed is equal to zero [START_REF] Kovacic | The Duffing Equation[END_REF]]. In our case, we found that the vibration amplitude is a solution of the polynomial (5.3.4c).

𝜔𝜔𝜔𝜔 ! = 𝛼𝛼𝛼𝛼 + 𝛾𝛾𝛾𝛾. 𝑤𝑤𝑤𝑤 ! ! -𝛿𝛿𝛿𝛿 ! (5.3.4a) 𝑘𝑘𝑘𝑘 ! = 𝛾𝛾𝛾𝛾. 𝑤𝑤𝑤𝑤 ! ! 2. 𝜔𝜔𝜔𝜔 ! (5.3.4b) 𝑤𝑤𝑤𝑤 ! ! + 2. 𝛼𝛼𝛼𝛼 -𝛿𝛿𝛿𝛿 ! 𝛾𝛾𝛾𝛾 . 𝑤𝑤𝑤𝑤 ! ! -𝐵𝐵𝐵𝐵 ! . 2. 𝛼𝛼𝛼𝛼 -𝛿𝛿𝛿𝛿 ! 𝛾𝛾𝛾𝛾 + 𝐵𝐵𝐵𝐵 ! - 2. 𝐴𝐴𝐴𝐴 ! 𝛾𝛾𝛾𝛾 = 0 (5.3.4c)
Elliptic functions being difficult to use and to compute, we can also approximate the solution of the equation with harmonic functions (5.3.5a) verifying the exact equation (5.3.5b), by defining the new parameters of the equation by (5.3.5c) and (5.3.5d).

𝑤𝑤𝑤𝑤 ! 𝑡𝑡𝑡𝑡 = 𝑤𝑤𝑤𝑤 ! + 𝑤𝑤𝑤𝑤 𝑡𝑡𝑡𝑡 = 𝑤𝑤𝑤𝑤 ! + 𝑤𝑤𝑤𝑤 ! . 𝑒𝑒𝑒𝑒 !!.! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(𝜔𝜔𝜔𝜔. 𝑡𝑡𝑡𝑡 -𝜃𝜃𝜃𝜃)

(5.3.5a) 𝑤𝑤𝑤𝑤 + 2. 𝛿𝛿𝛿𝛿. 𝑤𝑤𝑤𝑤 + 𝛼𝛼𝛼𝛼. 𝑤𝑤𝑤𝑤 + 𝛽𝛽𝛽𝛽. 𝑤𝑤𝑤𝑤 ! + 𝛾𝛾𝛾𝛾. 𝑤𝑤𝑤𝑤 ! = 0

(5.3.5b) 𝛼𝛼𝛼𝛼 = 2. 𝑘𝑘𝑘𝑘 ! * + (𝑟𝑟𝑟𝑟 ! ! -1). 𝑘𝑘𝑘𝑘 !! + 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! 𝓂𝓂𝓂𝓂 ! + 2. 𝓂𝓂𝓂𝓂 ! * (5.3.5c) 𝛽𝛽𝛽𝛽 = -6. 𝑘𝑘𝑘𝑘 !! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! 𝓂𝓂𝓂𝓂 ! + 2. 𝓂𝓂𝓂𝓂 ! * (5.3.5d)
Hence, by simply injecting (5.3.5a) in (5.3.5b), we can see that the quadratic term has no influence at the first order. By neglecting the second and third harmonics created by the non-linearity, we find an equation similar to (5.3.4a), showing the dependence of the frequency on the initial conditions. By writing the initial conditions, we demonstrate that the beam's shape amplitude also verifyies a polynomial equation (5.3.6b), whereas the initial phase shift is given by (5.3.6c).

𝜔𝜔𝜔𝜔 ! = 1 4 . 4. 𝛼𝛼𝛼𝛼 + 3. 𝛾𝛾𝛾𝛾. 𝑤𝑤𝑤𝑤 ! ! -2. 𝛿𝛿𝛿𝛿 ! (5.3.6a) 𝑤𝑤𝑤𝑤 ! ! + 4. 𝛼𝛼𝛼𝛼 -2. 𝛿𝛿𝛿𝛿 ! 3. 𝛾𝛾𝛾𝛾 -𝐵𝐵𝐵𝐵 ! . 𝑤𝑤𝑤𝑤 ! ! - 1 3. 𝛾𝛾𝛾𝛾 . ( 4. 𝛼𝛼𝛼𝛼 -2. 𝛿𝛿𝛿𝛿 ! . 𝐵𝐵𝐵𝐵 ! + 4. 𝐴𝐴𝐴𝐴 ! ) = 0 (5.3.6b) 𝜃𝜃𝜃𝜃 = 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝜔𝜔𝜔𝜔. 𝐵𝐵𝐵𝐵 𝐴𝐴𝐴𝐴 (5.3.6c)
To evaluate the accuracy of these models, we represent in Fig. (obtained with the Runge Kutta method). Finally, we plot the evolution of the voltage of the first vibration peak as a function of the thermal hysteresis and the configuration of the piezoelectric transducer. Logically we observe that it is an increasing function of the thermal hysteresis, varying from 12 V for a hysteresis of 3 o C to 18 V for a hysteresis of 15 o C. We also see that clamped-clamped transducers produce greater output voltage peaks than simply-supported oscillators, mainly because of their better piezoelectric coupling. b. Energy harvested with a SECE circuit.

In order to estimate the maximal energy that can be harvested with a SECE circuit, we assume that the oscillator has a quasi-linear behavior, meaning that harvesting charges does not modify its oscillation frequency contrary to what is predicted by (5.3.6a). The principle of SECE circuits consists in discharging the piezoelectric capacitor once the maximal output voltage is reached [START_REF] Lefeuvre | Buck-Boost Converter for Sensorless Power Optimization of Piezoelectric Energy Harvester[END_REF]. In order to find the law giving the energy we will make a mathematical induction. The n-th relaxed position of the beam (wn') after the piezoelectric capacitor is discharged can be obtained from the expression of the deflection peak wn before discharge by writing the conservation of the energy (5.3.7a).

In order to obtain a linear expression, we only keep the quadratic term of the strain energy. Hence, we obtain a simple arithmetic progression (5.3.7b).

1 2 . 𝑘𝑘𝑘𝑘 ! * !" + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! ! + 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 !!"#$%& = 1 2 . 𝑘𝑘𝑘𝑘 ! * !" + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! (5.3.7a) 𝑤𝑤𝑤𝑤 ! ! = 1 - 𝛤𝛤𝛤𝛤 !
At the n+1-th voltage peak, the energy extracted from the capacitor is a function of the n'th beam's deflection peak (5.3.8a). As a consequence the deflection also follows an arithmetic (5.3.8b). We can deduce that the n-th maximal deflection is given by (5.3.8c) by defining a ratio depending on the piezoelectric coupling factor and the damping ratio (5.3.8d).

𝑤𝑤𝑤𝑤 !!! = -𝑤𝑤𝑤𝑤 ! ! . 𝑒𝑒𝑒𝑒 ! !.! ! 𝑤𝑤𝑤𝑤 !!! = -1 - 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! . 𝑘𝑘𝑘𝑘 ! * + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! + 𝛤𝛤𝛤𝛤 ! * ! . 𝑒𝑒𝑒𝑒 ! !.! ! . 𝑤𝑤𝑤𝑤 ! (5.3.8b) 𝑤𝑤𝑤𝑤 ! = --𝑎𝑎𝑎𝑎 !" ! . 𝑤𝑤𝑤𝑤 ! 𝑎𝑎𝑎𝑎 !"
. 𝑒𝑒𝑒𝑒 ! !.! !

(5.3.8c)

𝑎𝑎𝑎𝑎 !" = 1 - 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! . 𝑘𝑘𝑘𝑘 ! * + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! + 𝛤𝛤𝛤𝛤 ! * ! . 𝑒𝑒𝑒𝑒 ! !.! ! (5.3.8d)
The energy extracted from the piezoelectric transducer is then given by (5.3.9a). Using the same hypothesis on the quadratic dependence of the strain energy on the beam's deflection, we can approximate the mechanical losses by (5.3.9b), leading to the expression of the transduction efficiency give by (5.3.9c). By defining the equivalent electromechanical coupling factor of the coupled oscillator by (5.3.9d) and the equivalent mechanical quality factor by (5.3.9e), the efficiency is given by (5.3.9f).

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! = 1 2 . 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! . 𝑤𝑤𝑤𝑤 ! ! . 𝑒𝑒𝑒𝑒 ! !.!.! ! . 𝑎𝑎𝑎𝑎 !" ! ! !! !!! = 1 2. 1 -𝑎𝑎𝑎𝑎 !" ! . 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! . 𝑤𝑤𝑤𝑤 ! ! . 𝑒𝑒𝑒𝑒 ! !.!.! ! (5.3.9a) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 !"## = 1 -𝑒𝑒𝑒𝑒 !!.!.! ! 2. (1 -𝑎𝑎𝑎𝑎 !" ! ) . 𝑘𝑘𝑘𝑘 ! * + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! . 𝑤𝑤𝑤𝑤 ! ! . 𝑒𝑒𝑒𝑒 ! !.!.! ! (5.3.9b) 𝜂𝜂𝜂𝜂 !" = 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! . 𝑘𝑘𝑘𝑘 ! * + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! . 1 -𝑒𝑒𝑒𝑒 !!.!.! ! + 𝛤𝛤𝛤𝛤 ! * ! (5.3.9c) 𝑘𝑘𝑘𝑘 !" ! = 𝛤𝛤𝛤𝛤 ! * ! 𝒞𝒞𝒞𝒞 ! . 𝑘𝑘𝑘𝑘 ! * + 𝑟𝑟𝑟𝑟 ! ! -1 . 𝑘𝑘𝑘𝑘 !! (5.3.9d) 𝑄𝑄𝑄𝑄 !" = 𝑒𝑒𝑒𝑒 ! !.!.! ! 1 -𝑄𝑄𝑄𝑄 !" + 𝑘𝑘𝑘𝑘 !" ! (5.3.9f)
In Fig. 5.3.3a, we show the electrical energy that can be harvested by using SECE techniques each time the bimetallic strip heats the piezoelectric transducer. We still observe that the clamped-clamped oscillator is the best configuration: with a bistable beam having a thermal hysteresis of 5 o C, this configuration generates 40 µJ when the mechanical quality factor is 15. In the same conditions, the simply-supported transducer only generates 20 µJ per snap. Fig. 5.3.3b also shows the advantage of using a clampedclamped piezoelectric oscillator to boost the efficiency of the transduction. With a 5 o C hysteresis, the conversion efficiency is around 45%. The fact that the efficiency is a decreasing function of the thermal hysteresis is linked to the increase of the bistable beam's stiffness. To improve the conversion efficiency, the stiffness of both beams must be matched.

c. Overall performances of the energy harvester

Now that we have modeled the full chain of conversion of the current energy harvester, we can evaluate its performances when it is cycled between a hot source and a cold source, assuming that the piezoelectric transducer is used as a planar heat sink. First, we give the intrinsic Carnot efficiency of the coupled oscillators (5.3.10a), by normalizing the electrical energy by the thermal hysteresis of the bimetallic strip in the cavity. This enables to give the equivalent ZT of the energy harvester in order to compare it with Seebeck generators (5.3.10b), whose evolution is presented in Fig. 5.3.3c.

𝜂𝜂𝜂𝜂 !"

!"# = 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! 𝐶𝐶𝐶𝐶 ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# ! . 𝑇𝑇𝑇𝑇 ! + 𝜃𝜃𝜃𝜃 !"# + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 2

(5.3.10a)

𝑍𝑍𝑍𝑍𝑇𝑇𝑇𝑇 !" ≈ 2. 1 + 2. 𝑇𝑇𝑇𝑇 ! -𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 2. 𝑇𝑇𝑇𝑇 ! + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# . 𝜂𝜂𝜂𝜂 !"! (5.3.10b)
By multiplying the electrical energy generated per snap, we can obtain the mean power generated by the harvester. We can also deduce all the various expressions of the performances of the system.

𝑃𝑃𝑃𝑃 ! = 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! 2. 𝐶𝐶𝐶𝐶 ! . 𝑅𝑅𝑅𝑅 !! . 1 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.3.10a) 𝜂𝜂𝜂𝜂 !" !"#,!"! = 𝑃𝑃𝑃𝑃 ! 𝑅𝑅𝑅𝑅 !"#$ . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) (5.3.10b)
In Fig. 5.3.3d we represent the evolution of the surface density of output power for various values of the bimetallic strip's thermal hysteresis. This shows that the output power is maximized by the use of the smallest possible thermal hysteresis. For a hot source temperature of 100 o C, the mean output power is 0.5 µW.cm -2 . The output power of the current energy harvester is generally between 3 and 4 µW for a bimetallic strip surface of 6.48 cm 2 , corresponding to a power density of 0.6 µW.cm -2 . The fact that both values are of the same order of magnitude validates the complete model of the harvester. To finish, we plot in Fig. 5.3.3e the evolution of the thermal efficiency of the harvester and in Fig. 5.3.3f the evolution of the Curzon-Ahlborn efficiency. This also demonstrates that the system is more efficient by choosing bimetal thermostats having weak thermal hystereses.

d. Ways to improve the efficiency of the transduction

Equation (5.3.9) shows that the efficiency depends both on the equivalent mechanical quality factor and the equivalent electro-mechanical coupling factor. There are thus two ways to improve the performances of the energy harvester. The first way is to enhance the electro-mechanical coupling factor of the piezoelectric transducer by two means: either by thinning the thickness of the oscillator to reduce its stiffness and thus increasing the efficiency of the shock, or using piezoelectric materials with better piezoelectric properties. The first solution is limited: the thickness of the metallic membrane cannot be decreased less than the thickness imposed by the position of the neutral axis in order to preserve the quantity of strain energy in the piezoelectric layer. The second solution is also limited by the current properties of piezoelectric materials. A second way to enhance the efficiency of the conversion is to improve the mechanical quality factor of the piezoelectric transducer. This second solution is the one that brings more results. In [Boughaleb et al, 2015], changing the materials used for the device's architecture (Teflon replaced by PEEK) and changing the type of clamp for the piezoelectric transducer (four metal screws instead of two metal flanges) enabled to improve the quality factor from 6 to 20 and the energy stored on the piezoelectric capacitor from 9 µJ to 90 µJ. Fig. 5.3.4 presents another strategy we tried in order to improve the efficiency of the transduction without changing the material nor the shape of the cavity. To optimize the transducer design, numerical simulations of the oscillator are performed with Comsol software. The current transducer (45 x 2 x (0.25+0.25) m 3 ) is first modeled with its clamp on an architecture made of PTFE, and the parameters representing the absorption of the vibration energy by the architecture are modified in order to fit the experimental data. The value of the PZT loss factor is fixed to 0.989 in order to represent a piezoelectric coupling coefficient of 0.3. The PTFE loss factor is taken equal to 3, so that the clamp diffuses 75% of its mechanical energy. With these values, we obtained an transduction efficiency of 28% and a mechanical quality factor of about 9.6.

Image of the optimized energy harvester and experimental measurement of the snap signal with a load resistance of 10 MOhms

The basic idea of this new harvester is to reduce the mechanical losses associated to the fact that the piezoelectric transducer is clamped on a leaky architecture, by reducing the mechanical efforts on the clamp. Fig.5.3.5b and Fig.5.3.5c compare the repartition of the mechanical stress inside the oscillator and the PTFE architecture, showing the stress reduction. In order to reduce this effort, four beams are cut in the brass membrane and their positions are lightly inclined and shifted from the clamp in order to reduce the mechanical efforts acting on the clamp. In the same time, reducing the width of the beams and increasing their length enables to reduce the transducer's stiffness in order to improve the efficicency of the mechanical energy transfer during the shock between the bimetallic strip and the piezoelectric oscillator. Fig. 5.3.5d and Fig. 5.3.5e show the evolution of the efficiency of the transduction linked to the optimization of the angle between the branches and the clamp and the length of the branches. As can be seen, the optimal shape of the harvester enables to increase the efficiency from 28% up to 70%.

The optimized piezoelectric transducer was processed with a milling machine at G2ELab. The results of the experimental measurements are presented in Fig. 5.3.4. As can be seen, the reduction of the mechanical stiffness of the oscillator and the increase of its mechanical quality factor helped improving the performances of the thermal energy harvester. During a snap, the energy stored in the piezoelectric capacitor is now 75 µJ instead of 9 µJ and the quality factor increases from 6 to 14. These very positive results validate the possibility of improving the performances of the system without changing the type of piezoelectric materials used on the piezoelectric transducer.

Coupling with an electret transducer

The last physical coupling to be studied is the coupling between an electret and a bistable beam. Before explaining the modeling of this system, we give a short overview on electrets.

A short overview on electrets and electrostatic energy harvesting a. A history of electrets

Even if properties of electrets were first reported in the mid-18 th century, by Gray in 1732 studying the properties of waxes such as beeswax and creating the first thermoelectrets, by Franklin trapping charges in glass by means of Leyden jars, and first theorized by Faraday in 1839, the name of electret was first given by Heaviside in 1892 [START_REF] Heaviside | Electrical papers[END_REF], hesitating between the term "electret" and "electric" even if this name was already an adjective. According to Heaviside, electrets are some dielectric materials for which electrization is not "perfectly elastic", but for which there is "residual electrization when absorption occurs" when they are electrized. "Electret" thus is to electricity what "magnet" is to magnetism: "an intrinsically electrized body".

A wider definition of "electret" could include piezoelectric materials [Sessler, 1987]. The first controlled fabrication of electrets is generally attributed to Eguchi in 1919, fabricating thermoelectrets by heating carnauba wax and applying an external electric field to orientate molecular dipoles in the wax. Among other significant advances, we can cite the fabrication of electrets by charge injection by means of electron beams by Selenyi in 1928, the fabrication of photoelectrets by Nadjakoff in 1938, or the discover of the piezoelectric effect of PVDF by [Kawai, 1969].

Electrets are commonly used as sensors and transducers, finding applications in the design of microphones [START_REF] Scheeper | A review of silicon microphones[END_REF], light modulators, dosimeters [START_REF] Sessler | Electret transducers: a review[END_REF], etc. The applications of electrets in energy harvesting are more recent. A first electret motor was proposed by [Wieder and Kauffman, 1953], but the first theoretical modeling of their operation was proposed later [START_REF] Jefimenko | Electrostatic Current Generator Having a Disk Electret as an Active Element[END_REF]. Later still, Tada improved the efficiency of the motor by using PTFE-based electrets [START_REF] Tada | Experimental Characteristics of Electret Generator, Using Polymer Film Electrets[END_REF].

b. Types of electret and production

Classification of electrets is generally based on their way of fabrication [Sessler, 1987[START_REF] Kestelman | Electrets in engineering[END_REF], the origins of their electrization, or the type of dielectric materials used for the electrets. With the first classification, we can distinguish three main categories: "thermoelectrets", fabricated by heating the dielectric sample and applying strong external electric fields to orientate dipoles and, similarly, "electroelectrets" which do not need any heat sources (but the electric field must be higher), "radioelectrets" based on the use of X-rays, β-rays, γ-rays or monoenergetic electron beams to charge the dielectric layers, "corona-electrets" based on the use of Corona discharge in order to implant ionized air molecules in the dielectric layers. The methods for obtaining the three types of electrets are represented in Fig. 6.1.1 where the typical values of the applied electric fields are given. Other categories also exist: "magnetoelectrets" obtained by heating polymers in an external magnetic field, "photoelectrets" where light is used as a heat source but which also require an external electric field, "mechanoelectrets" whose polarization appears by heating and compressing the dielectric materials, "metal-polymer electrets" quite similar to thermoelectrets obtained by triboelectricity or by liquid charging. Electrets can also be classified according to the source of their permanent polarization. We thus can distinguish a first category regrouping dipolar electrets whose polarization comes from the alignment of dipoles inside the dielectric layers. They are generally obtained by means of thermal charging method with strong electric fields (mainly thermoelectrets): as an example, piezoelectric films are obtained with electric fields of around 800 kV.cm -1 at high temperature or electric fields up to 4 MV.cm -1 at room temperature [Sessler, 1987]. Spacecharge electrets are electrets that are globally neutral (like dipolar electrets) but whose polarization comes either from homocharges (charges generated by the electrodes) or heterocharges (charges directly generated in the dielectric layer). In the case of thermoelectret, for low electric fields (around 1-2 kV.cm -1 ), the predominant type of charge is heterocharges. Increasing the electric field (around 10 kV.cm -1 ) leads to the creation of homocharges by Schottky effect or air gap breakdown.

The last type of electrets is designated by the term of charge excess electrets. This type of electret is generally obtained by Corona discharge (Coronaelectrets), ion implantation or radiations (radioelectrets), for which the charge does not come from the dielectric layer nor the electrodes but are implanted into the dielectric layer. The advantage of Corona is its great simplicity and the speed of realization that makes it one of the most used fabrication techniques for electrets. Ion implantation has the avantage of being well mastered, the depth of implantation and the quantity of charges being easily controlled.

Classification of electrets based on dielectric materials distinguishes six categories [START_REF] Kestelman | Electrets in engineering[END_REF]: natural waxes or resins (Carnauba, paraffin…), polymers (PTFE, PDVF, Parylene, Cytop…), biopolymers, low-melting organic dielectrics (benzoic acid), ceramic dielectrics (SiO2, Si3N4, Al2O3…) and dielectric monocrystals if we take into account piezoelectric materials. Generally, ceramic dielectrics have better electrical properties since they trap more electrical charges than organic polymers, but they can encounter problems of long-term stability [START_REF] Suzuki | Recent progress in MEMS electret generator for energy harvesting[END_REF].

c. Energy harvesters exploiting electrets

As explained, the first electrets motors were proposed in the mid-20 th century but realistic proofs of concept are due to [START_REF] Jefimenko | Electrostatic Current Generator Having a Disk Electret as an Active Element[END_REF] and [START_REF] Tada | Experimental Characteristics of Electret Generator, Using Polymer Film Electrets[END_REF]. These first harvesters were based on the variation of capacity of the transduced due to the change of the overlapping area between the electret and the movable electrode (Fig. 6.1.2a). This strategy can be exploited to harvest energy from rotational motions by using either textured electrets and textured electrodes [START_REF] Jefimenko | Electrostatic Current Generator Having a Disk Electret as an Active Element[END_REF]Walker, 1978, Tada, 1992, [Boland, 2003]. a'''. Rotary MEMS with interdigitated electrodes and overlapping-area change strategy [START_REF] Yang | A MEMS rotary comb mechanism for harvesting the kinetic energy of planar vibrations[END_REF]. b'. Principle of Gap-closing topology. b''. Out-of plane energy harvester [START_REF] Chiu | Flat and robust out-of-plane vibrational electret energy harvester[END_REF]. b'''. Device with interdigitated capacitors for in-plane vibration energy harvesting with gap-closing topology [START_REF] Le | Wideband excitation of an electrostatic vibration energy harvester with power-extracting end-stops[END_REF]. Boland et al, 2003[START_REF] Nakano | MEMS Rotational Electret Energy Harvester for Human Motion[END_REF], rotary-comb generator [START_REF] Yang | A MEMS rotary comb mechanism for harvesting the kinetic energy of planar vibrations[END_REF], or unidirectional in-plane vibrations by using either plane textured electrets [Basset et al, 2009[START_REF] Tao | Sandwich-structured two-dimensional MEMS electret power generator for low-level ambient vibrational energy harvesting[END_REF] or interdigitated capacitors [START_REF] Le | Wideband excitation of an electrostatic vibration energy harvester with power-extracting end-stops[END_REF] (Fig. 6.2.1a''').

A second possibility to observe capacity variations of the electret transducers is to reduce the gap separating the electret from the counter-electrode, as in our case (Fig. 6.1.2b). This strategy can be used wih full-sheet electrets to harvest energy from out-of-plane motion [START_REF] Mizuno | Investigation of a resonance microgenerator[END_REF][START_REF] Chiu | Flat and robust out-of-plane vibrational electret energy harvester[END_REF][START_REF] Perez | An electret-based aeroelastic flutter energy harvester[END_REF], or energy from inplane motion using interdigitated capacitors [START_REF] Oxaal | Investigation of gap-closing interdigitated capacitors for electrostatic vibration energy harvesting[END_REF]. Other less frequent types of energy harvesters use the variation of capacity of an electret transducer due to the variation of the permittivity in the gap separating the electret from the counter-electrode [START_REF] Nakano | MEMS Rotational Electret Energy Harvester for Human Motion[END_REF].

Modeling the electret a. Electric field and electrostatic pressure induced by an electret with a charge layer

In order to model the electrical field inside the harvester and calculate the charge induced by the electret on the electrode and the counter-electrode, we take the example of the simplified device's architecture presented in Fig. 6.2.1a, approximately corresponding to the energy harvester presented in [Boisseau et al, 2013]. Given that the ratio of the beam's deflection over the beam's length is weak, we can consider that the electric field is perperdicular to the electret's surface and the bimetallic strip is equivalent to a parallel infinitesimal small capacity. We thus consider the electric field acting on a surface v.dx and the equivalent diagram represented in Fig. 6.2.1b, first assuming that the electric charge of the electret is implanted at a depth de. By simply writing the Gauss law, we can express the electric fields E1 and E2 inside the electret layer and the field in the cavity by (6.2.1a) and (6.2.1b).

𝜀𝜀𝜀𝜀 ! . ℰ ! -ℰ ! = 𝜎𝜎𝜎𝜎 ! (6.2.1a) 𝜀𝜀𝜀𝜀 ! . ℰ ! -𝜀𝜀𝜀𝜀 ! . ℰ ! = 0 (6.2.1b) ℰ ! . 𝑡𝑡𝑡𝑡 ! -𝑑𝑑𝑑𝑑 ! + ℰ ! . 𝑑𝑑𝑑𝑑 ! + ℰ ! . 𝑔𝑔𝑔𝑔 ! + 𝑤𝑤𝑤𝑤 = -𝒱𝒱𝒱𝒱 (6.2.1c)
From these three equations, we can deduce the expression of the electrical field seen by the electric charges carried by the bimetallic strip (6.2.2a). We can thus define the surface potential of the electret by (6.2.2b), and the electret's capacitance by (6.2.2c). The electrical charge trapped in the dielectric layer is then simply related to other parameters by (6.2.2d)

ℰ ! . 𝑡𝑡𝑡𝑡 ! 𝜀𝜀𝜀𝜀 ! + 𝑔𝑔𝑔𝑔 ! + 𝑤𝑤𝑤𝑤 𝜀𝜀𝜀𝜀 ! = - 𝒱𝒱𝒱𝒱 𝜀𝜀𝜀𝜀 ! + 𝑡𝑡𝑡𝑡 ! -𝑑𝑑𝑑𝑑 ! 𝜀𝜀𝜀𝜀 ! . 𝜀𝜀𝜀𝜀 ! . 𝜎𝜎𝜎𝜎 ! (6.2.2a) 𝒱𝒱𝒱𝒱 ! = 𝑡𝑡𝑡𝑡 ! -𝑑𝑑𝑑𝑑 ! 𝜀𝜀𝜀𝜀 ! . 𝜎𝜎𝜎𝜎 ! (6.2.2b) 𝒞𝒞𝒞𝒞 ! = 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝜀𝜀𝜀𝜀 ! 𝑡𝑡𝑡𝑡 ! (6.2.2c) 𝒬𝒬𝒬𝒬 ! = 𝐶𝐶𝐶𝐶 ! . 𝑡𝑡𝑡𝑡 ! 𝑡𝑡𝑡𝑡 ! -𝑑𝑑𝑑𝑑 ! . 𝒱𝒱𝒱𝒱 ! (6.2.2b)
With all these parameters definition, we can simply express the electric field seen by the bimetallic strip by (6.2.3a). The density of electric charges induced by the electric field is then given by (6.2.3b). The electric pressure exerced by the electret on the beam is finally given by (6.2.3c).

ℰ ! 𝑥𝑥𝑥𝑥 = - 𝒱𝒱𝒱𝒱 -𝒱𝒱𝒱𝒱 ! 𝜀𝜀𝜀𝜀 ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝑔𝑔𝑔𝑔 ! + 𝑤𝑤𝑤𝑤 (6.2.3a) 𝜎𝜎𝜎𝜎 ! 𝑥𝑥𝑥𝑥 = 𝜀𝜀𝜀𝜀 ! . ℰ ! 𝑥𝑥𝑥𝑥 (6.2.3b) 𝑝𝑝𝑝𝑝 ! 𝑥𝑥𝑥𝑥 = 1 2 . 𝜀𝜀𝜀𝜀 ! . ℰ ! 𝑥𝑥𝑥𝑥 ! (6.2.3c)
By using these three equations, we will be able to describe the electrical behavior of the system as well as the influence of the electrets and the polarization of the bimetallic strip on its mechanical behavior b. Electric equations of the thermally bistable electrode By simply integrating the local expression of the surface charge density over the whole surface of the bimetallic strip, we obtain the electric charge carried by the bimetallic strip (6.2.4a). Once integrated this equation becomes (6.2.4b).

𝒬𝒬𝒬𝒬 ! = 𝑣𝑣𝑣𝑣. 𝜎𝜎𝜎𝜎 ! 𝑥𝑥𝑥𝑥 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! ! ! ! = -𝑣𝑣𝑣𝑣. 𝜀𝜀𝜀𝜀 ! . 𝒱𝒱𝒱𝒱 -𝒱𝒱𝒱𝒱 ! . 1 𝜀𝜀𝜀𝜀 ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝑔𝑔𝑔𝑔 ! + 𝑤𝑤𝑤𝑤 ! . cos 𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! 𝒞𝒞𝒞𝒞 ! = 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝜀𝜀𝜀𝜀 ! 𝑔𝑔𝑔𝑔 ! (6.2.4d)
This equation being quite complicated to use, we exploit the fact that the deflection is necessarily smaller than the sum of the gap and the electret's thickess since it equals the gap between the beam and the hot source only when there is a contact between the two domains. Hence, we can obtain a simplified expression of the charge with (6.2.5a). By defining the capacitance of the movable electrode by (6.2.5b), this equation becomes (6.2.5c). We also define the charge Q* by (6.2.5d), which will be important when we will study the performance of the energy harvester.

𝒬𝒬𝒬𝒬 ! = - 4. 𝑔𝑔𝑔𝑔 ! 𝜋𝜋𝜋𝜋. 𝑔𝑔𝑔𝑔 ! + 𝑤𝑤𝑤𝑤 ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 -𝒱𝒱𝒱𝒱 ! (6.2.5a) 𝒞𝒞𝒞𝒞 ! = 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝜀𝜀𝜀𝜀 ! 𝑔𝑔𝑔𝑔 ! + 𝑤𝑤𝑤𝑤 ! (6.2.5b) 𝒬𝒬𝒬𝒬 ! = 4. 𝒞𝒞𝒞𝒞 ! . 𝒞𝒞𝒞𝒞 ! 𝜋𝜋𝜋𝜋. (𝐶𝐶𝐶𝐶 ! + 𝐶𝐶𝐶𝐶 ! )
. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 (6.2.5c)

𝒬𝒬𝒬𝒬 * = 4. (𝑡𝑡𝑡𝑡 ! -𝑑𝑑𝑑𝑑 ! ) 𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑄𝑄𝑄𝑄 ! = 4 𝜋𝜋𝜋𝜋 . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! (6.2.5d)

`Mechanical equations of the bistable bimetallic electrode a. Equilibrium of bistable beams under electrostatic pressure.

The main difficulty posed by the existence of the electrostatic pressure is that it adds an additional source of non-linearity to the problem of the thermal snap-through which is already non-linear. Given the expression of the electrostatic pressure, the equilibrium equation of the beam is given by (6.3.1).

𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 - 𝑁𝑁𝑁𝑁 ! 2. 𝐿𝐿𝐿𝐿 . 𝑤𝑤𝑤𝑤 ,! ! ! ! ! ! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 -𝐹𝐹𝐹𝐹 ! . 𝑤𝑤𝑤𝑤 ,!! + 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!!!! = 1 2 . 𝜀𝜀𝜀𝜀 ! . 𝑣𝑣𝑣𝑣. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! 𝑔𝑔𝑔𝑔 ! -𝑤𝑤𝑤𝑤 ! (6.3.1)
One of the consequences of the non-linearity brought by the electrostatic pressure is the occurrence of pull-in phenomenon. This can simply be understood by the fact that the equivalent stiffness of the electrostatic pressure tends to increase when the beam gets closer to the electret. As a consequence, the beam can lose its stability if the electrostatic force is always stronger than the restoring force of the beam. This phenomenon can be easily understood if we neglect the effect of the clamp's thrust on the mechanical equilibrium. As a consequence, the equilibrium equation becomes (6.3.2a). Using the cosine harmonic function to approximate the beam's shape, the equilibrium equation becomes (6.3.2b).

𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!!!! . 𝑔𝑔𝑔𝑔 ! -𝑤𝑤𝑤𝑤 ! = 1 2 . 𝜀𝜀𝜀𝜀 ! . 𝑣𝑣𝑣𝑣. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! (6.3.2a) 𝐼𝐼𝐼𝐼 ! . 3. 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! . 4. 𝑔𝑔𝑔𝑔 ! 𝜋𝜋𝜋𝜋 -𝑤𝑤𝑤𝑤 ! ! = 2 𝜋𝜋𝜋𝜋 . 𝜀𝜀𝜀𝜀 ! . 𝑣𝑣𝑣𝑣. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! (6.3.2b)
By studying the stability of this equilibrium equation, we see that unstable positions can appear when (6.3.3a) is verified. The corresponding voltage is given by (6.3.3b) and (6.3.3c). This voltage is known as the pull-in voltage of the beam.

𝑤𝑤𝑤𝑤 ! = 4. 𝑔𝑔𝑔𝑔 ! 3. 𝜋𝜋𝜋𝜋 (6.3.3a) 1 2 . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! = 64 27 . 𝑔𝑔𝑔𝑔 ! ! 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . (6.3.3b) 𝒱𝒱𝒱𝒱 !" = 𝒱𝒱𝒱𝒱 ! + 8 2 3 . 𝑔𝑔𝑔𝑔 ! 𝐿𝐿𝐿𝐿 . 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! . 𝐿𝐿𝐿𝐿 𝐶𝐶𝐶𝐶 ! (6.3.3c)
Another possible consequence of the electrostatic pressure is the snap of the thermallybistable beam before it thermally snaps. The corresponding value of the electret voltage must be considered as the limit value of acceptable voltage. The complete study of equation (6.3.1a) has been the subject of several studies, mainly in the field of MEMS. As an example, [Krylov et al, 2008, Krylov and Dick, 2010, Das and Batra, 2009] proposed numerical solutions of this equation for initially-curved beams using the Galerkin method, showing the conditions of occurrence of electrostatic snap-through and pull-in.

b. Moderate electrostatic coupling

A solution to get around the non-linearity can be to approximate the electrostatic pressure, which is a function of x, by a constant value (6.3.4a). Hence integrating this equation twice yields equation (6.3.4b). Using the Galerkin method, this equation is equivalent to (6.3.4c). With this approximation, the simplest thermodynamic cycle to model is the Ericsson cycle (constant voltage during heating) since the electrostatic forces can be treated as a simple increase of the residual stress moment. As a consequence, the thermal hysteresis and the average temperature of the hysteresis are modified according to (6.3.5a) and (6.3.5b). The position of the electret with respect to the beam's initial shape is of great importance: if the electret is placed at the interface between the prime mover and the cold source, it will tends to lower the snap and snap-back temperatures, and to reduce the beam's thermal hysteresis by reducing the effective mechanical moment acting on the beam. Reversely, if the electret is placed between the hot source and the electret, the snap and snap-back temperatures will increase as well as the beam's thermal hysteresis.

𝑝𝑝𝑝𝑝 ! ≈ 1 2 . 𝜀𝜀𝜀𝜀 ! . 𝑣𝑣𝑣𝑣. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! 𝑔𝑔𝑔𝑔 ! ! (6.3.4a) 𝑘𝑘𝑘𝑘 ! . 𝑤𝑤𝑤𝑤 + 𝑤𝑤𝑤𝑤 ,!! = 1 4 . 𝜀𝜀𝜀𝜀 ! . 𝑣𝑣𝑣𝑣. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! 𝐼𝐼𝐼𝐼 ! . 𝑔𝑔𝑔𝑔 ! ! . 𝑥𝑥𝑥𝑥 ! + 𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! . 𝑇𝑇𝑇𝑇 - 𝑀𝑀𝑀𝑀 ! 𝐼𝐼𝐼𝐼 ! (6.3.4b) 𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! + 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹 ! . 𝑤𝑤𝑤𝑤 ! + 4 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 = - 𝐿𝐿𝐿𝐿 𝜋𝜋𝜋𝜋 ! . 𝑔𝑔𝑔𝑔 ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! (6.3.4c) (a) (b)
𝑇𝑇𝑇𝑇 ! ! = 𝑀𝑀𝑀𝑀 ! + 𝐿𝐿𝐿𝐿. 𝐶𝐶𝐶𝐶 ! . 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! ! 4. 𝜋𝜋𝜋𝜋. 𝑔𝑔𝑔𝑔 ! 𝑀𝑀𝑀𝑀 = 𝑀𝑀𝑀𝑀 ! + 𝑀𝑀𝑀𝑀 ! 𝑀𝑀𝑀𝑀 ! (6.3.5a) 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! = 2. 𝐿𝐿𝐿𝐿 𝑀𝑀𝑀𝑀 ! . 1 27. 𝑁𝑁𝑁𝑁 ! . 𝑁𝑁𝑁𝑁 ! + 𝑛𝑛𝑛𝑛 ! 𝑚𝑚𝑚𝑚 ! . 𝑀𝑀𝑀𝑀 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! ! (6.3.5b)
The case of the Stirling cycle is more complicated since the charge of the bistable electrode is kept constant during the displacement of the bistable beam. Hence, using (6.2.4) as an expression of the electric charge, the equilibrium equation is modified into (6.3.6a).

𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 4. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! + 4. 𝑀𝑀𝑀𝑀 ! 𝜋𝜋𝜋𝜋. 𝑔𝑔𝑔𝑔 ! ! . 𝑤𝑤𝑤𝑤 ! ! + 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐹𝐹𝐹𝐹 ! + 8. 𝑀𝑀𝑀𝑀 ! 𝜋𝜋𝜋𝜋. 𝑔𝑔𝑔𝑔 ! . 𝑤𝑤𝑤𝑤 ! + 4 𝜋𝜋𝜋𝜋 . 𝑀𝑀𝑀𝑀 ! + 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 = 0 (6.3.6a) 𝑀𝑀𝑀𝑀 ! = 𝜋𝜋𝜋𝜋. 𝐿𝐿𝐿𝐿 64. 𝑔𝑔𝑔𝑔 ! . 𝒬𝒬𝒬𝒬 ! ! 𝒞𝒞𝒞𝒞 ! (6.3.6b)
As a consequence, the criterion of bistability is modified (6.3.7a), and the hysteresis curve of the bistable beam becomes asymmetric. Hence, the beam's thermal hysteresis is given by (6.3.7c).

𝑁𝑁𝑁𝑁 ! + 𝑁𝑁𝑁𝑁 ! ≥ 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! - 18 𝜋𝜋𝜋𝜋 ! . 𝐿𝐿𝐿𝐿 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑔𝑔𝑔𝑔 ! ! . 𝑁𝑁𝑁𝑁 ! (6.3.7a) 𝑃𝑃𝑃𝑃 ! = 𝑛𝑛𝑛𝑛 ! 𝑚𝑚𝑚𝑚 ! . 𝑀𝑀𝑀𝑀 ! - 8. 𝑀𝑀𝑀𝑀 ! 𝜋𝜋𝜋𝜋. 𝑔𝑔𝑔𝑔 ! (6.3.7b) 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! = 2. 𝐿𝐿𝐿𝐿 𝑚𝑚𝑚𝑚 ! . 1 27. 𝑛𝑛𝑛𝑛 ! . 𝑃𝑃𝑃𝑃 ! + 𝑃𝑃𝑃𝑃 ! + 18 𝜋𝜋𝜋𝜋 ! . 𝐿𝐿𝐿𝐿 ! . 𝑀𝑀𝑀𝑀 ! ! 𝑔𝑔𝑔𝑔 ! ! . 𝑛𝑛𝑛𝑛 ! -𝑖𝑖𝑖𝑖 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! ! (6.3.7c)
In order to illustrate the influence of the electrical charges carried by the bimetallic strip, Fig. 6.3.1a and Fig. 6.3.1b plot the evolution of the beam's thermal hysteresis as a function of the electrical charge, taking the example of an Al-Invar beam (40 x 10 x 0.4 mm 3 ) coupled with a 10 µm thick Cytop electret with -400 V surface potential. By setting the electret on the hot source, the thermal hysteresis of the bimetallic strip is widened by a factor 10 if the charges correspond to half the charge of the electret. On the opposite, for such values of electrical charges, the beam stability is lost if the electret is placed between the cold source and the bimetallic strip.

Harvesting heat with electret transduction a. Possible thermodynamic cycles

In the case of common electrostatic heat engines (Fig. 6.4.1a), two simple thermodynamic cycles can be implemented. The first is the charge-constrained cycle, which is quite similar to SECE and Stirling cycles, whereas the second is a voltage-constrained cycle [Meninger et al, 2001]. Given that the electrical charge is simply given by the product of the capacitance by the capacitor's voltage, the electric cycle is necessary delimited by the curves of maximal and minimum capacitances (Fig. 6.4.1b). In the case of the chargeconstraint cycle, the electric charge is injected when the capacitance of the sytem is maximal. As a consequence, the displacement of the movable electrode causes an increase of the potential of the electrical charges that is maximal when the capacitance becomes smaller. The discharge of the circuit enables to harvest charges at a potential higher than their initial potential. Hence the energy generated by this first type of cycle is given by (6.4.1a). In the case of a voltage-constrained cycle, charges of high potential are injected during the initial step, when the capacitance is maximal. As a consequence, the deformation of the beam enables to harvest charges at high potentials. The comparison of both cycles shows that the energy harvested by the voltage-constrainted cycle is greater than the ernegy harvested by SECE-like cycle. Similar observations can be made by comparing Ericsson and Stirling cycles.

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 1 2 . 1 𝒞𝒞𝒞𝒞 !"# - 1 𝒞𝒞𝒞𝒞 !"# . 𝒬𝒬𝒬𝒬 ! ! (6.4.1a) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 1 2 . 𝒞𝒞𝒞𝒞 !"# -𝒞𝒞𝒞𝒞 !"# . 𝑉𝑉𝑉𝑉 ! ! (6.4.1b)
Electrets-based energy harvesters have been introduced to bypass the problem of using an external voltage source to harvest energy, since the polarization of the movable electrode is ensured by the electret. Given that a counter-electrode is used, the equivalent circuit of the electret transducer is made of a variable capacitance representing the electrostatic effects of the out-of-plane movement of the electrode, and a fixed charge over the capacitance of the electret representing the permanent polarization of the device. Hence, [START_REF] Nakano | MEMS Rotational Electret Energy Harvester for Human Motion[END_REF].

the natural cycle of the electrets energy harvester is composed of a first phase charging the moving electrode when its capacitance is maximal. During the displacement of the movable electrode, the charge and the voltage evolve in such manner that there is a global neutrality of the charge distributed between the movable electrode, the counter-electrode and the charge trapped in the dielectric layer. Finally, the movable electrode is discharged and returns to its initial state. The equivalent electric diagram of the harvester and its cycle are represented in Fig. 6.4.1.It can thus be demonstrated that the energy harvested by this cycle is given by (6.4.12a) [START_REF] Nakano | MEMS Rotational Electret Energy Harvester for Human Motion[END_REF].

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 1 2 . 𝒞𝒞𝒞𝒞 !"# 𝒞𝒞𝒞𝒞 !"# + 𝒞𝒞𝒞𝒞 ! ! - 𝒞𝒞𝒞𝒞 !"# 𝒞𝒞𝒞𝒞 !"# + 𝒞𝒞𝒞𝒞 ! ! . 𝒬𝒬𝒬𝒬 ! (6.4.2a)
b. Intrinsic performances of the bimetallic strip heat engine.

We consider the implementation of the charge-constraint cycle with the bimetallic strip heat engine by assuming a weak electrostatic coupling. The bimetallic strip is free to move and its maximal excursion goes from 2.ws C to -2.ws C . We first recall the expression of the charge carried by the bimetallic beams as a function of its voltage Ve-V (6.4.3a). We can see that the maximal charge is obtained when the harvester is short-circuited (V = 0) and when the capacitance is maximal (6.4.3b), whereas the minimal charge is measured once the bimetal has snapped (6.4.3c) (it should be noticed that ws C is negative). As the bimetal is heated and snaps in open-circuit, the maximal voltage measured after it has snapped is given by (6.4.3.c). Similarly, when it snaps back, the minimal voltage measured is given by (6.4.3e).

𝒬𝒬𝒬𝒬 ! = 4. 𝒞𝒞𝒞𝒞 ! . 𝒞𝒞𝒞𝒞 ! 𝜋𝜋𝜋𝜋. (𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! )

. 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! (6.4.3a)

𝒬𝒬𝒬𝒬 ! !"# = 𝒞𝒞𝒞𝒞 ! !"# 𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! !"# . 𝒬𝒬𝒬𝒬 * = 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝒬𝒬𝒬𝒬 * 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! + 2. 𝑤𝑤𝑤𝑤 ! ! (6.4.3b) 𝒬𝒬𝒬𝒬 ! !"# = 𝒞𝒞𝒞𝒞 ! !"# 𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! !"# . 𝒬𝒬𝒬𝒬 * = 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝒬𝒬𝒬𝒬 * 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! -2. 𝑤𝑤𝑤𝑤 ! ! (6.4.3c) 𝒱𝒱𝒱𝒱 ! !"# = 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# 𝒬𝒬𝒬𝒬 ! !"# . 𝒱𝒱𝒱𝒱 ! (6.4.3d) 𝒱𝒱𝒱𝒱 ! !"# = 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# 𝒬𝒬𝒬𝒬 ! !"# . 𝒱𝒱𝒱𝒱 ! (6.4.3e)
As a consequence, the energy that is generated during a cycle is given by (6.4.4a). The negative sign of this expression shows that the cycle is that of a generator. This expression can be rewritten as a function of the output voltages, in order to extract the theroretical value of the maximal available energy during the characterization of device.

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 1 2 . 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# . 𝒱𝒱𝒱𝒱 ! !"# -𝒱𝒱𝒱𝒱 ! !"# = 1 2 . 𝒱𝒱𝒱𝒱 ! . 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# ! 𝒬𝒬𝒬𝒬 ! !"# . 𝒬𝒬𝒬𝒬 ! !"# < 0 (6.4.3c) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = - 2 𝜋𝜋𝜋𝜋 . 𝒞𝒞𝒞𝒞 ! !"# . 𝒞𝒞𝒞𝒞 ! 𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! !"# . 𝒱𝒱𝒱𝒱 ! !"# . 𝒱𝒱𝒱𝒱 ! !"# -𝒱𝒱𝒱𝒱 ! !"# (6.4.3c)
Given that we need to see how the performance of the energy harvester depends on the properties of the bimetallic strip, we develop the expressions of the energy and output voltages as functions of the maximal deflection amplitude.

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = - 128 𝜋𝜋𝜋𝜋 . 𝜀𝜀𝜀𝜀 ! . 𝜀𝜀𝜀𝜀 ! ! . 𝑡𝑡𝑡𝑡 ! . 𝑤𝑤𝑤𝑤 ! ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! ! -4. 𝜀𝜀𝜀𝜀 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (6.4.4a) 𝒱𝒱𝒱𝒱 ! !"# = 4. 𝜀𝜀𝜀𝜀 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! 𝑤𝑤𝑤𝑤 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! ! -4. 𝜀𝜀𝜀𝜀 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝒱𝒱𝒱𝒱 ! (6.4.4b) 𝒱𝒱𝒱𝒱 ! !"# = 4. 𝜀𝜀𝜀𝜀 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! 𝑤𝑤𝑤𝑤 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! ! + 4. 𝜀𝜀𝜀𝜀 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝒱𝒱𝒱𝒱 ! (6.4.4c)
To avoid making any hypothesis on the value of the gap go, we directly study the performances of a bimetallic strip constrained in a cavity. As a consequence, the gap go is given by (6.4.5a), enabling to simplify the expression of the device's performances. 𝑔𝑔𝑔𝑔 ! = 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! (6.4.5a)

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 16 𝜋𝜋𝜋𝜋 . 𝜀𝜀𝜀𝜀 ! ! . 𝑟𝑟𝑟𝑟 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (6.4.5b) 𝒱𝒱𝒱𝒱 ! !"# = 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! . 𝒱𝒱𝒱𝒱 ! (6.4.5c) 𝒱𝒱𝒱𝒱 ! !"# = 2. 𝜀𝜀𝜀𝜀 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! ! + 𝜀𝜀𝜀𝜀 ! ! . 𝑟𝑟𝑟𝑟 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝒱𝒱𝒱𝒱 ! (6.4.5d)
Assuming that the electret thickness is smaller that the beam deflexion, we can express the electrical energy as a function of the beam's thermal hysteresis, enabling to give the expression of the thermal efficiency of the harvester in the cavity (6.4.6a).

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 4 4 ! 𝜋𝜋𝜋𝜋 ! . 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝑀𝑀𝑀𝑀 ! . 𝐿𝐿𝐿𝐿 ! 𝑁𝑁𝑁𝑁 ! . Δ𝑇𝑇𝑇𝑇 ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (6.4.6a) 𝜂𝜂𝜂𝜂 !" !"# = 8 4 ! 𝜋𝜋𝜋𝜋 ! . 3 -𝑟𝑟𝑟𝑟 ! ! . 𝜀𝜀𝜀𝜀 ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝑀𝑀𝑀𝑀 ! . 𝐿𝐿𝐿𝐿 ! 𝑁𝑁𝑁𝑁 ! . 𝐶𝐶𝐶𝐶 ! ! . Δ𝑇𝑇𝑇𝑇 ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (6.4.6b)
In order to present the performances of the heat engine predicted by the model, we study the performance of an Al-Invar beam (40 x 10 x 0.4 mm 3 ) coupled with an electret made of a 20 µm thick Cytop electret with surface potential of -400 V. Fig. 6.4.1a shows the evolution of the peak voltages generated by the beam when it snaps and when it snaps back. We can observe that the peaks are higher when the beam snaps, since the initial amount of charges carried by the bimetallic beam is more important as the capacitance is higher when the beam is in contact with the hot source. We observe that by increasing the beam's thermal hysteresis, the snap voltage converges toward the value of the electret's surface potential. The evolution of the electrical energy is plotted in Fig. 6.4.2b. It is important to compare this figure with Fig. 6.4.2.c. Given that the energy evolves as the cubic root of the thermal hysteresis (6.4.6a), we observe that the thermal efficiency of the system is a decreasing function of the thermal hysteresis, contrary to other energy harvesters presented in the previous parts. Using an electret with surface potential of 150 V, the available mechanical energy per cycle is of around 100 µJ for a thermal efficiency of 4.10 -3 % and a Carnot efficiency of 0,12%.

c. Dynamic performances of the heat engine

As for other energy harvesters, we can use the thermal lumped capacitance model to evaluate the maximal power that is generated by the electrostatic bimetallic strip heat engine. The main performances are given below: the mean output power (6.4.7a), the thermal efficiency of the harvester working between two heat sources (6.4.7b), and the Curzon-Ahlborn efficiency of the device (6.4.7c). As a reminder, the temperature difference across the energy harvester can be expressed as a function of the external temperature and the thermal properties of the device (Rleak) and of the heat sink (Rsink).

𝜋𝜋𝜋𝜋 ! . 𝑟𝑟𝑟𝑟 ! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! . 𝜀𝜀𝜀𝜀 ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝑀𝑀𝑀𝑀 ! . 𝐿𝐿𝐿𝐿 ! 𝑁𝑁𝑁𝑁 ! . 𝑅𝑅𝑅𝑅 !! ! . 𝐶𝐶𝐶𝐶 ! ! ! . Δ𝑇𝑇𝑇𝑇 !"# ! 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (6.4.7a) 𝜂𝜂𝜂𝜂 !" !"#,!"! = 𝑃𝑃𝑃𝑃 ! 𝑅𝑅𝑅𝑅 !"#$ . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ) (6.4.7b) 𝜂𝜂𝜂𝜂 !" !",!"! = 𝑃𝑃𝑃𝑃 ! 𝑅𝑅𝑅𝑅 !"#$ . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! . 𝑇𝑇𝑇𝑇 ! + 273 𝑇𝑇𝑇𝑇 ! + 273 -𝑇𝑇𝑇𝑇 !"# + 273 (6.4.7c) 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! = 𝑅𝑅𝑅𝑅 !"#$ 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) (6.4.7d)
The study of the performances is plotted in Fig. 6.4.3. The first figure presents the dependence of the mean power on the beam's thermal hysteresis and on the electret's surface potential when a fixed temperature difference of 30 o C is maintained in the device's cavity. We already saw that the working frequency of the bimetallic strip is a decreasing function of the thermal hysteresis. As the amount of electric energy generated during a complete cycle increases as the cubic root of the thermal hysteresis, we observe that the bimetallic strips with the smallest thermal hysteresis generates more energy because of their high working frequency. Fig. 6.4.3b represents the evolution of the output power when the hot source temperature evolves from 25 o C to 200 o C. We observe that at 100 o C, the device generates up to 3 µW.cm -2 when the bimetallic strip has a thermal hysteresis of 1 o C. Logically we can observe that the power can be boosted by the use of a heat sink with fins (up to 6 µW.cm -2 for a thermal hysteresis of 3 o C). The other figures represent the evolution of the efficiency of the device for the same working conditions. Logically, the fact that the device's architecture is a path for thermal leakages explains why the efficiency of the energy harvester is degraded.

d. Effect of the moderate electrostatic coupling on the performances of the heat engine

In the previous part, we modelled the performances of the electrostatic bimetallic strip heat engine by assuming that we could neglect the effect of the electrostatic coupling on the mechanical behavior of the bimetallic strip. However, we can see that by using the previous equations, increasing the electret's polarization increases all the performances of the heat engine. In Section 6.3.c, we presented a simple approximation of the electrostatic pressure in order to determine its influence on the thermal hysteresis of the bimetallic strip. We now use this approximation to see the influence of the electric field on the output power. We assume that the electrostatic moment is equal to (6.4.8a), corresponding to a fixed charge on the bimetallic strip given by (6.4.8b). Hence, we need to modify the values of the beam's displacement by (6.4.8e) and the beam's thermal hysteresis by (6.4.8f), by introducing the corrective factor (6.4.8e).

𝑀𝑀𝑀𝑀 ! = 𝜋𝜋𝜋𝜋 4 . 𝜀𝜀𝜀𝜀 ! . 𝐿𝐿𝐿𝐿 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (6.4.8a) 1 𝒬𝒬𝒬𝒬 = 1 𝒬𝒬𝒬𝒬 ! !"# + 1 𝒬𝒬𝒬𝒬 ! !"# (6.4.8b) 𝑃𝑃𝑃𝑃 ! = 𝑁𝑁𝑁𝑁 ! 𝑀𝑀𝑀𝑀 ! . 𝑀𝑀𝑀𝑀 ! + 8. 𝜀𝜀𝜀𝜀 ! . 𝑀𝑀𝑀𝑀 ! 𝜋𝜋𝜋𝜋. (𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! ) (6.4.8c) 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! = 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥. 𝐹𝐹𝐹𝐹(𝒱𝒱𝒱𝒱 ! ) ! (6.4.8d) 𝑤𝑤𝑤𝑤 ! ! ! = 𝑤𝑤𝑤𝑤 ! ! . 𝐹𝐹𝐹𝐹(𝒱𝒱𝒱𝒱 ! ) (6.4.8e) 𝐹𝐹𝐹𝐹 𝒱𝒱𝒱𝒱 ! = 1 + 𝑁𝑁𝑁𝑁 ! + 𝑁𝑁𝑁𝑁 ! + 18. 𝑀𝑀𝑀𝑀 ! ! 𝜋𝜋𝜋𝜋 ! . 𝑁𝑁𝑁𝑁 ! . 𝜀𝜀𝜀𝜀 ! ! . 𝐿𝐿𝐿𝐿 ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! ! 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! (6.4.8f)
Hence, the performances of the heat engine can be re-written in order to include the effect of the moderate coupling (6.4.9).

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 16 𝜋𝜋𝜋𝜋 . 𝜀𝜀𝜀𝜀 ! ! . 𝑟𝑟𝑟𝑟 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝐹𝐹𝐹𝐹(𝒱𝒱𝒱𝒱 ! ) ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! . 𝐹𝐹𝐹𝐹(𝒱𝒱𝒱𝒱 ! ) ! . 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! ! (6.4.9a) 𝑓𝑓𝑓𝑓 = 1 2. 𝑅𝑅𝑅𝑅 !! . 𝐶𝐶𝐶𝐶 ! . 1 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 + 𝐹𝐹𝐹𝐹(𝒱𝒱𝒱𝒱 ! ) ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 1 - 𝐹𝐹𝐹𝐹(𝒱𝒱𝒱𝒱 ! ) ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (6.4.9b) 𝑃𝑃𝑃𝑃 ! = 𝑓𝑓𝑓𝑓. 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! (6.4.9c)
In order to see the influence of the electrostatic coupling on the performance of the electrostatic bimetallic strip heat engine, we plot in Fig. 6.4.4 the evolution of the working frequency and the mean output power generated by the device. Fig. 6.4.4a and Fig. 6.4.4b represent their evolution when the electret is put on the cold source (electret made of 20 µm of CYTOP). As it reduces the beam's thermal hysteresis, the working frequency is first improved by the increase of the electric field, thus increasing the output power for small electric fields. After reaching a maximal value, the power decreases because of the reduction of the generated energy per cycle. The system stops when the beam becomes monostable. We can observe that the optimal value of the electret's polarization is a function of the thermal hysteresis. For a beam having a thermal hysteresis of 5 o C, the maximal power is 0.4 µW.cm -2 for a working frequency of around 0.023 Hz. The shape of the curve indicates that we can find an optimal thermal hysteresis and an optimal electret potential surface that maximize the output power of the generator. In our case, the optimal hysteresis is 5 o C and the optimal surface potential 110 V.

Fig. 6.4.4c and Fig. 6.4.4d represent the evolution of the performances when the electret is placed on the hot source. In this case we can observe that the frequency is a decreasing function of the electric field since the electric field increases the effective thermal hysteresis of the bimetallic strip. Even if the increase of the electric field improves the transduction of the thermal energy into electric energy during a complete cycle, decreasing the working frequency leads to an optimal value of the electret's surface potential that maximizes the output power generated by the device. This has already been observed during experimental measurements. Here again we can observe that the optimal value of the surface potential is a function of the thermal hysteresis and that the best performances are obtained for an optimal value of the thermal hysteresis. In our case, the optimal thermal hysteresis if 3 o C and the optimal surface potential is about 80 V. In that case, the maximal output power is around 0.24 µW.cm -2 when a temperature difference of 30 o C is maintained in the device's cavity. We also observe that the system stops working when the effective thermal hysteresis becomes equal to the temperature difference in the cavity.

To finish, we plot the performances predicted by the model for an Al-Invar bimetallic strip coupled with a Cytop electret (surface potential 100V) placed on the cold source, and assuming that the device works without heat sink. As we can see, the best thermal hysteresis is still 5 o C. With this bimetallic strip, the energy harvester generates 0.5 µW.cm -2 at 100 o C with a thermal efficiency of 1,4.10 -3 %.

Synthesis

Now that we have established all the laws describing the operations of the pyroelectric, piezoelectric and electrostatic bimetallic strip heat engines, we are able to compare their performances in order to choose the best architecture of bimetallic strip heat engines, but also to sum up the various strategies to optimize their relative performances.

a. Design rules common to all harvesters

From chapter 2, we can say that to optimize the thermo-mechanical transduction, we should:

- -Design a passive architecture that minizes the mechanical stress applied to the bimetallic strip (rw = 1); -Improve thermal transfers but reduce the thermal resistance of the contact between the heat sources and the bimetallic strip (decreasing RCC);

-Improve the evacuation of heat by using a heat sink with fins. This enables to maintain a large temperature difference inside the cavity and thus to boost the working frequency of the bimetallic strip and the output power (gain factor of around 3), to use bistable beams with larger thermal hysteresess and to widen the temperature operating range of the heat engine.

b. Design rules for pyroelectric bimetallic strip heat engines

In the case of the pyroelectric bimetallic strip heat engines, we should:

-Use pyroelectric bimetallic strips with the largest electro-thermal coupling factor

k pyro 2 . L.v

2

. If the temperature difference inside the cavity is weak, the best pyroelectric material is the one that has both the best electro-thermal and electro-mechanical coupling factors. If the temperature difference is large, the tertiary pyroelectric effect becomes negligible and the best materials are those with the best electro-thermal coupling factors. As a consequence, we should use either PZT ceramics or MPM-PN ceramics; -Implement the Stirling cycle by using a SECE power management circuit; -Choose a bimetallic strip having a thermal hysteresis corresponding to 70% of the temperature difference in the cavity.

c. Design rules for piezoelectric bimetallic strip heat engines

In the case of the piezoelectric bimetallic strip heat engines, we should:

-Use bimetallic strips with a thermal hysteresis corresponding to 60% of the temperature difference in the cavity to maximize the mechanical power generated by the bimetallic strip (chapter 2);

-Use piezoelectric transducers with the smallest bending stiffness in order to maximize the energy transfer during the impact of the bistable bimetallic strip, and with a piezoelectric material having the largest electro-mechanical coupling factor. As a consequence, soft PZT ceramics are the best piezoelectric materials for this usage;

-Use a clamped-clamped piezoelectric transducer with three piezoelectric capacitors to maximize the piezoelectric coupling of the oscillator;

-Improve the mechanical quality factor of the piezoelectric transducer in order to reduce the mechanical losses. This can be done by improving the quality of the transducer's clamp or by changing the shape of the piezoelectric transducer;

-Use SECE or SSHI power management circuits.

d. Design rules for electrostatic bimetallic strip heat engines

In the case of the electrostatic bimetallic strip heat engines, we should:

-Choose the optimal thermal hysteresis that optimize the output power;

-Determine the value of the electret's surface potential that optimizes the power generated by the bimetallic strip heat engine.

e. Comparison of the various heat engines

To finish we recall in Fig. 7.1.1 the performances of the three configurations of heat engines modelled in this chapter. The three bimetallic strip heat engines are made of Al-Invar beams (40 x 10 x 0.4 mm 3 ) with either 0.1 mm of PZT deposited on the surface of the bistable beam, or a piezoelectric transducer made of 0.2 mm of brass and 0.1 mm of PZT, or coupled with a 20 µm thick CYTOP electret with surface potential 100 V.

As we can see by comparing the images in the first column, the piezoelectric engine is intrinsically the most efficient system for small thermal hystereses with an equivalent ZT around 5.10 -3 for a thermal hysteresis of 5 o C (with a clamped-clamped piezoelectric transducer having a quality factor of 15), followed by the pyroelectric engine (ZT of 3.10 -3 for a thermal hysteresis of 5 o C) and by the electrostatic engine (ZT of 1.10 -3 for a thermal hysteresis of 5 o C). However the reduction of the ZT for greater values of thermal hysteresis is larger for the piezoelectric engine, and the pyroelectric engine finally becomes the most efficient device (ZT of 2.10 -3 for a thermal hysteresis of 15 o C), ahead piezoelectric devices (ZT of 1,7.10 -3 ) and electrostatic devices (ZT of 1,8.10 -4 ).

The second and third columns present the dynamic performances of the three devices, working without heat sink between a cold source at 25 o C and a hot source whose temperature varies. The output power is plotted for five different bimetallic strips. We can see that the pyroelectric energy harvester is the one that generates the most power, because at a given hot source temperature, the bimetallic strip that maximizes the output power has a greater thermal hysteresis. At 100 o C, using a bimetallic strip having a thermal hysteresis of 15 o C, we can generate up to 2.8 µW.cm -2 with a thermal efficiency of 7.5. 10 -3 %. With a piezoelectric device, the maximal power is obtained with a bimetallic strip of 10 o C of hysteresis, which generates 1.28 µW.cm -2 . Finally, with the electrostatic device, the maximal power is 0.51 µW.cm -2 , obtained with a thermal hysteresis of 5 o C. As a consequence, we can see that the system that must be preferably developed is the pyroelectric bimetallic strip heat engine since it is twice as efficient than the piezoelectric bimetallic strip heat engine, and over 5 times more efficient than the electrostactic bimetallic strip heat engine.

Conclusion

In this chapter, we presented and modelled three different energy harvesters: the pyroelectric, piezoelectric and electrostatic bimetallic strip heat engines. To model the pyroelectric generator, we came back to the local constitutive equations of pyroelectric materials and applied it to the case of a pyroelectric layer deposited either on the passive layer or the active layer of the bistable beams. We described three configurations of Stirling cycles and established the expressions of the generated power and the thermoelectric efficiency, enabling to compare the performances of these cycles, as a function of the geometry of the bimetallic strip, the pyroelectric materials, and the bimetallic strip's thermal hysteresis. We finally found that the power levels generated by the pyroelectric generators are of around 1 to 3 uW.cm -2 for an equivalent ZT of around 0,01.

In the case of the piezoelectric bimetallic strip heat engines, we determined the maximal theoretical performances by assuming a perfect mechanical coupling between the piezoelectric oscillator and the bimetallic strip. In that case of figure, we found that the perfect coupling is described by a non-linear Duffing equation we solved using either harmonic balance or Jacobi's elliptic functions. We thus found analytic expressions of the output voltages generated by piezoelectric oscillators for three types of oscillators: clamped-clamped oscillators, double supported oscillators, clamped-free oscillators. We found that the best performances are obtained with clamped-clamped oscillators whose bending stiffness is weaker than the bimetallic strip's stiffness in order to maximize stress and mechanical energy transfer. We then simulated the case of a SECE strategy to extract electrical energy from the piezoelectric oscillators. The analytical expressions we found showed us the interest of improving the quality factor of the piezoelectric transducer's mechanical oscillations in order to improve the electro-mechanical conversion's efficiency. From this observation, we designed new geometries of oscillators with better mechanical quality factors and smaller bending stiffness, enabling to increase the generators' performances by a factor 5. We find that the best performances are of around 1 uW.cm -2 for a generator working without any heat sink with fins. The fact that these levels of performances are simulated by assuming a perfect mechanical coupling between the piezoelectric transducer and the bimetallic strip explains why we cannot obtain them experimentally: most of the time, only one or two schocks between the bimetallic strip and the piezoelectric transducers occur, limiting the energy transfer between the two oscillators.

The last device we modeled consisted in coupling a bimetallic strip with an electret transducer. In that case of figure, we modeled both the weak coupling and strong coupling between the electret and the bimetallic strip. We evaluated the performances of the generator in the case of a SECE energy harvesting strategy. This enabled us to show that there exists an optimal value of electret's surface potential that maximizes the power generated by the bimetallic strip heat engine. Typically, we found that this type of generators can generate up to 0,5 uW.cm -2 without a heat sink with fins.

From this theoretical study, we can conclude that efforts must be made on developing pyroelectric bimetallic heat engine if we want to improve the performances of current energy harversters. As a final conclusion, we can say than in the best case, bimetallic strip heat engines are far from reaching the performances of thermoelectric generators. Typically, the maximal efficiency obtained with this generator is less than 1% of the thermoelectric generators' efficiency, and the slowness of thermal transfers ininside the generator's architecture explained why the power levels generated by the bimetallic strip heat engines are far from being competitive with thermoelectric generators. Now that we modeled the performances of the generator at the macroscale, we need to se how they evolve when the system is miniaturized using MEMS fabrication techniques. This is the topic of the next chapter.

Chapter 4

Theoretical performances of the energy harvester at the microscale

Introduction

The previous chapter proposed a demonstration of the operation of three types of thermal energy harvesters and the evaluation of their performances at the macroscopic scale. In this part we will focus on the evolution of these performances when the device is scaled down to the sub-millimetric scale and fabricated by conventional MEMS techniques. This chapter is divided in five parts. The first is devoted to the modeling of the bistable beams at the microscale, and the evaluation of the thermo-mechanical performances when reducing the size and when varying the type of beam clamp. The second and third parts explain the modeling of the pyroelectric generator and the electrostatic converter at the microscale. The fourth part synthesizes the various results obtained before and gives a description of the thermal transfers at the microscale.

Thermo-mechanical instability at the microscale

Equilibrium equation

In Chapter 2 and Chapter 3, we assumed that, at the macroscale, the energy harvester was based on the use of pinned-pinned bimetallic beams instead of free bimetallic shells in order to obtain a simple description of the thermal snap-through. Since miniaturized beams are fabricated by the successive deposition of the material's layers, the beams' edges are generally assumed to be clamped to the wafer substrate. As a consequence, a simple approach to model the behavior of beams would be to assume a perfect clamp of the beam's edges, meaning that no displacement or rotation of the edge are authorized. For instance, this type of boundary conditions suits the problem of the evaluation of the vibration frequencies of beams at the microscale. However, for problems involving the buckling of beams, the use of perfect clamp conditions leads to inadequacies between theoretical models and experimental data, for example in the case of Guckel structures used to locally determine the stress in thin films [Guckel et al, 1985, Fang and[START_REF] Fang | Determining mean and gradient residual stresses in thin films using micromachined cantilevers[END_REF]. As a consequence, many works [START_REF] Mullen | Theoretical modeling of boundary conditions in microfabricated beams[END_REF][START_REF] Meng | Theoretical modeling of microfabricated beams with elastically restrained supports[END_REF], Kobrinski et al, 2000, ] proposed to model micro-structures by using imperfect clamped conditions by means of finite torsional stiffness and compression stiffness at the beam's edges (see Fig. 2.1.c). For example, [START_REF] Alkharabsheh | Dynamics of MEMS Arches of Flexible Supports[END_REF] compared the theoretical modeling of the dynamic snap-through of buckled beams with experimental data. In our model, we will only use finite torsional stiffness as in [Tung et al, 2013, Cabal andRoss, 2002]. As a recall, the strong formulation of the equilibrium projected on the axial displacement field u leads to the boundary equation (2.1.1a) and to the equilibrium equation (2.1.1b). Similarly, projected on the lateral displacement field w, it gives the equations (2.1.1c) and (2.1.1d). The evolution of the Euler load is plotted in Fig. 2.2.1a as a function of the product γ.L, which is a non-dimensional number. When the torsional stiffness γ tends to infinity, the beam is double clamped and the Euler load is equal to 2π. When γ = 0, the beam is simply supported and k γ = π. The observations made by [START_REF] Fachin | Analytical extraction of residual stresses and gradients in MEMS structures with application to CMOS-layered materials[END_REF] are verified: γ does not need to be null or infinite for the beam to behave like a simply-supported beam or a double-clamped beam. If γ.L = 36.7, the beam's Euler load represents 90% of the Euler load of a perfectly clamped beam, and for γ.L = 396.9, the beam's Euler load reaches 99% of this limit value. This means that by choosing one of these values, we are able to describe the mechanical behavior of clamped beams but with the possibility to predict the effect of the residual stress asymmetry on the initial curvature of the beam. The torsional stiffness also determines the beam's shape, since the wavelength of is proportional to the square root of the Euler load. Fig.2.1.3a and Fig.2.1.3b represent two beams which snap and snap back at the same temperatures, one being simply supported and the other double clamped. The comparison of both figures reveals that the deflection of the clampedclamped beam is about half the deflection of the simply-supported beam because the beam's edges cannot rotate because of the clamp's rigidity. In Fig. 2.13c and Fig. 2.1.3b we compare the evolution of the beam's deflection for a macroscopic Al-Invar beam (simply supported beam) and the deflection of a microscopic Al-Si beam. For 5 K hysteresis, the beam's snap is associated with a displacement of 619 nm, representing 0.6% of the length of the beam. This value can be compared to the relative displacement of the Al-Invar beam (1%). This confirms the fact that the rigidity of the clamp limits the beam's displacement. These figures show that the reduction of the relative deflection of the beam at the microscale could be a problem for the electrostatic converter using the displacement of the beam to create a capacitance variation.

𝒯𝒯𝒯𝒯 !! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! = 0 (2.1.1a) 𝒯𝒯𝒯𝒯 !!,! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! = 0 (2.1.1b)

Thermo-mechanical performances of bimetallic strips

Now that we have established the equation giving the dependence of the beam's shape on the torsional stiffness, we can express the energy of the clamped beam. As the clamp is modeled by a torsional stiffness, it stores some energy due to the rotation of the beam's edges. As a consequence, the expression of the energy becomes (2.2.1a). Using the equation (2.1.4), it can be developed into (2.2.1b).

𝒲𝒲𝒲𝒲 ! = 𝑁𝑁𝑁𝑁 ! 2. 𝐿𝐿𝐿𝐿 . 𝜆𝜆𝜆𝜆 ! + 𝑖𝑖𝑖𝑖 ! 2 . 𝑤𝑤𝑤𝑤 ,!! ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑
From the expression of the beam's energy, we can deduce the expression of the beam's entropy by using the Maxwell relations. The beam's entropy is then given by (2.2.2). As a consequence, we can plot the evolution of the total amount of energy released by the beam during its snap and snap-back (2.2.3b), as well as the efficiency of the transduction (2.2.3b) by calculating the amount of heat provided by the hot source to the beam (2.2.3c). 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! = 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑇𝑇𝑇𝑇 ! + 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑇𝑇𝑇𝑇 !"

(2.2.2a)

𝜂𝜂𝜂𝜂 ! !"# = 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! 𝑄𝑄𝑄𝑄 ! + ℒ ! (2.2.2b)
𝑄𝑄𝑄𝑄 ! + ℒ ! = 𝑇𝑇𝑇𝑇 + 𝜃𝜃𝜃𝜃 !"# . 𝒮𝒮𝒮𝒮 𝑘𝑘𝑘𝑘, 𝑇𝑇𝑇𝑇 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! ! ,! ! ! ! !" ,! !" ! (2.2.2c)
The performances of the heat engine at the microscale and the macroscale are compared by taking the example of a macro-device using a 20 mm longAl-Invar simply-supported beam, and a micro-device using a 100 µm long Al-Si beam. The scaling factor of the length between these two devices is 200. Proportionally to the size of the beams, the properties are slightly different: the thermal asymmetries are similar but the bimorph Al-Si is lightly more rigid. The mean thermal expansion and heat capacity are higher for Al-Invar than for Al-Si. The experimental data reported in [START_REF] Fachin | Analytical extraction of residual stresses and gradients in MEMS structures with application to CMOS-layered materials[END_REF] show that the boundary conditions are such that the Euler load of these beams often represents around 90% of the Euler load of a clamped-clamped beam. As a consequence, we assume a torsional stiffness verifiying γ.L = 36.7 for our simulations.

The energy being an extensive parameter depending on the volume, in order to compare the energy produced by the macro-and microscopic bimetallic strips, we normalize the energy by the strip's volume to obtain adimensional data. Fig. 2.2.1a shows that by clamping the beam to the substrate, the energy of the double-clamped Al-Si beam at the micro-scale is reduced by a factor of 8.6 compared to the simply-supported Al-Invar beam (1.2 mJ.cm -3 instead of 10.3 mJ.cm -3 for a thermal hysteresis of 15 o C). Since the thermal efficiency and Carnot efficiency are dimensionless, we can directly compare energy harvesters at the macro-scale and at the micro-scale. As shown in Fig. 2.2.1b, the heat engine's efficiency is strongly impacted by the boundary conditions. As the Al-Invar bimorph has a greater volume heat capacity than Al-Si (3.68 J.cm -3 .K -1 for Al-Invar and 1.64.J.cm -3 .K -1 for Al-Si), Al-Invar beams should be less efficient than Al-Si beams at converting heat into mechanical energy. Fig. 2.2.1b shows that this is not the case: because of the rigidity of the clamp, for a hysteresis of 5 o C, the efficiency of the double-clamped Al-Si beams is 6 times smaller than the efficiency of the simply-supported Al-Invar beams (0.14% instead of 0.8%), and 9 times smaller than the simply-supported Al-Invar beams (1.3%).

Reduced-order model with the Galerkin method a. Intrinsic performances

For both clamped-clamped and pinned-pinned beams, the use of the accurate model is quite complex if one wants to easily understand how physical parameters influence the performances and the mechanical behavior of bistable beams. A reduced-order model would enable to compute more quickly equilibrium equations, but also to couple more easily mechanical equations with piezoelectric or electrostatic equations in order to describe the performances of the complete heat engine. As the wave vector is close to the Euler load when the beam has a weak thermal hysteresis, we assume that the beam's shape at the equilibrium is given by (2.3.1a).

𝑤𝑤𝑤𝑤 Again, we make the assumption that the thermal hysteresis is smaller than the difference between the average values of the snap and snap-back temperatures, so that the effect of the mean thermal expansion vanishes. The equilibrium equation is obtained by looking for the extrema of the energy, leading to (2.3.3a). The stability of the equilibrium is then given by (2.3.3b 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 ∶ 𝜑𝜑𝜑𝜑 ! = 1 ; 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶: 𝜑𝜑𝜑𝜑 ! = 1,99

(2.3.3e)

The performances of the bimetallic strip can also be rewritten. The sum of the kinetic energy of snap and the kinetic energy of snap-back is given by (2.3.4a). Equation (2.3.4b) gives the value of the adimensionnal number depending on the boundary conditions. This shows that the energy produced by a clamped-clamped beam corresponds to 8.3% of the energy provided by a simply-supported beam, agreeing with the observation made in Section 2.2. The correcting factor also appears in the expression of the thermo-mechanical efficiency and the Carnot efficiency of the thermodynamic cycle (2.3.4c). Finally the maximal force generated by the beam when it switches is given by (2.3.4e). Beside the complexity of the energy harvester using the shock between the bimetallic strip heat engine and a piezoelectric transducer, the decrease of the efficiency of the thermo-mechanical conversion, and the decrease of the force and the increase of the stiffness of clamped-clamped beams enables to discard this type of architecture for the MEMS integration of the heat engine.

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! = 𝜑𝜑𝜑𝜑 ! ! .

b. Extrinsic performances

The results presented in Chapter 2 can be directly adapted to the use of the torsional stiffness, just by taking into account that the crucial parameter is the initial deflection of the beam and not its amplitude. The cavity's thickness ratio rw must be defined by (2.3.5a) so that the equations are not changed. As a reminder, the thermal hysteresis in the cavity is given by ( 2

c. Performances of the materials used in microelectronics

To complete the thermo-mechanical description of the bimetallic strip, we use (2.3.4a) and (2.3.4d) to compare the performances of some couples of materials that are available in microelectronics (Si,SiO2,Si3N4,Ti,AlN,ZnO,PZT,Al,Cu,Ni,Au and Pt). We choose a reference bimetal thermostat having a length of 400 μm,a total thickness of 4 µm (2 µm +2 µm) and a 3 K thermal hysteresis (98,5 o C-101,5 o C). The evolution of the Carnot efficiency against the surface density of energy for these couples is represented in Fig. 2.3.1. This figure shows that the most efficient couples are Al-Si and Al-Si3N4 with an efficiency of 0.056% of Carnot for an energy density of 11 nJ.cm -2 , followed by Cu-Si3N4 and Al-SiO2.

The couples using Ti as a passive layer are the least efficient since Ti has a large coefficient of thermal expansion. If we look at the performances of piezoelectric materials, we can see that the best couples are Al-PZT and Al-AlN.

d. Impact of the variability of the residual stress on the performances of the bistable beams

The reduced order model also enables to study how the variability of the residual stress can impact the behavior of the bistable beams. As demonstrated by [START_REF] Orthwein | Thermal Buckling of a Bimetallic Strip[END_REF], bistable systems are very sensitive to the variability of manufacturing process. Variability of residual stress in thin films could have many origins: variability between wafers, variability of stress along the wafer surface, variability of stress through the thickness of the deposited thin films etc.

For thermally bistable beams, the impact of the residual stress variability is especially visible on the evolution of the snap and snap-back temperatures. In the case of a bimorph beam, if we only consider local variations of the stress, by using (2.3.3b), we can simply write the dispersion of the thermal hysteresis (2.3.6a) and the dispersion of the mean temperature of the hysteresis (2.3.6b) as a function of the variabilities of the stress in both layers, (simplified expression when both layer have the same thickness and Young modulus).

𝜎𝜎𝜎𝜎 !" = 3. 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑃𝑃𝑃𝑃 ! -𝑖𝑖𝑖𝑖 ! . 𝑘𝑘𝑘𝑘 ! ! . 𝛼𝛼𝛼𝛼 ! + 𝛼𝛼𝛼𝛼 ! 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! . 𝜎𝜎𝜎𝜎 ! (2.3.6a) 𝜎𝜎𝜎𝜎 ! ! = 2. 𝑇𝑇𝑇𝑇 ! 𝛼𝛼𝛼𝛼 ! -𝛼𝛼𝛼𝛼 ! . 𝜎𝜎𝜎𝜎 ! (2.3.6b)
The formulae show that the impact of the residual stress variability is all the more important when the difference in thermal expansion coefficients is small, the residual stress force is large, and when the beam is clamped-clamped (sensibility enhanced by a factor 6). To illustrate this, we consider the impact of the variability of the residual stress in the Aluminum layer of a SiO2-Al bimorph beam of dimensions 200 x 50 x (2+2) µm 3 , assuming a initial tensile stress of 100 MPa in Al. Around the value of -80 MPa, with a deviation of 2 MPa, we can observe that some beams are monostable whereas others have thermal hysteresis greater than 30 K. The variability is worse when the residual stress variability increases (see Fig.2.3.3b). Fig. 2.3.3c represents the evolution of the standard deviation of the thermal hysteresis as a function of the mean thermal hysteresis as predicted by (2.3.6a). Assuming that the variability of the stress follows a Gaussian distribution, we can simply give an expression of the beams that are monostable due to the residual stress dispersion. Its evolution as a function of the thermal hysteresis is represented in Fig. 3.3.2d. For example for a targeted hysteresis value of 3 o C, 30% of the beams cannot harvest heat if the stress deviation is of 2 MPa and 40% for a deviation of 4 MPa. A similar study would have to be done to determine the part of bistable beams that would not work because their snap and snap-back temperatures are not framed by the heat sources' temperature.

% !"#"$%&'() = 50. erf 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝜎𝜎𝜎𝜎 !" . 2 + 1

(2.3.6a)

3 Pyroelectric coupling

Equations of the piezoelectricity

In this part we assume that the bistable beam is either a bimorph beam whose passive layer is piezoelectric, or is made of a stack of materials that is already bistable and onto which we deposit a piezoelectric layer. From the point of view of the modeling, there is no difference between the two cases. We only assume that the piezoelectric layer is directly deposited on a metallic layer so that the capacity is that of the piezoelectric layer only.

The beam being clamped, we will consider two configurations represented on Fig. 3.1.1a: in the first case, the piezoelectric layer is connected to a single electrode. In the second case, the piezoelectric layer is sandwiched between three electrodes, which positions are defined by the positions of the nodes delimitating areas of compressive and tensile stress. . In order to establish the equations describing the tertiary pyroelectric effect inside clampedclamped beam, we come back to the expression of the electric displacement of the piezoelectric layer. In the case of a clamped-clamped beam, the strain tensor is given by (3. By integrating this equation, we obtain equation (3.1.2a) when a single piezoelectric capacitor is used, and (3.1.2b) when three piezoelectric capacitors are used and have a common electrode. The comparison of both equations shows that by using an architecture with three electrodes, the electrical charges created by the primary pyroelectric effect cannot be harvested. In the other case, by using a single electrode, the electrical charges stored in the piezoelectric capacitor are mainly those due to the primary pyroelectric effect and to clamp thrust, and weakly to the curvature of the beam because the term φ2 is small for clamped-clamped beams.

Performances of the Stirling cycle

In this part as in the previous chapter, we neglect the effect of the piezoelectric coupling on the thermo-mechanical behavior of the bistable beam. Without detailing the operation of the Stirling cycle (already presented in the previous chapter), we give the expression of the maximal output voltage generated when the pyroelectric bistable beam is heated and snaps in open-circuit then is discharged once it has snapped (Fig. Following the same approach as in the previous chapter, we define the figures of Merit for the electrical energy and the efficiency of the thermo-electrical transduction, as functions of the performances of a pyroelectric generator. For the device with three electrodes, they can be expressed as a function of the electromechanical coupling factor k31 2 . In order to quantify the performances of the pyroelectric bimetallic strip heat engine at the microscale, we take the example of a bistable Al-SiO2-PZT beam (length 400 µm, width 100 µm) where each layer has a thickness of 2 µm. We assume that the PZT layer is sandwiched between two very thin metallic layers not taken into account in the model. The properties of the beam are given in Table3.2.1. Fig. 3.2.1a plots the evolution of the output voltage of the device when the beam is clamped and when the two configurations of electrodes are used. They are then compared to the output voltage of a simple pyroelectric generator and to the output voltage of a simply-supported beam. With a thermal hysteresis of 5 o C, the simply-supported beam is more effective than the clamped-clamped beam, generating voltages of around 1.1 V. For a clamped beam with 3 electrodes, the output voltage is of 0.77 V and of 0.41 V for the clamped beam with a single electrode, similar to the output voltage of the pyroelectric generator. These values can be compared with twice the value of the threshold voltage of silicon and germanium diodes: Below 5 o C, it is difficult to reach the voltage imposed by the use of germanium diode bridges diodes. This limitation must be taken into account when designing the system since it imposes a minimum size for the bistable beams. Fig. 3.2.1b and Fig. 3.2.1c respectively represent the evolution of the electrical energy produced per cycle, and the Carnot efficiency associated to the thermal cycling of the pyroelectric bistable beam. As observed previously, clampedclamped beams generate less energy than pinned-pinned beams, and for small thermal hysteresis, preference should be given to the architecture with three electrodes: for a thermal hysteresis of 5 o C, the clamped-clamped beam with three electrodes produces 150 pJ of energy with a Carnot efficiency of 0.35%, whereas the beam with a single electrode produces 43 pJ with an efficiency of 0.1%.

𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ! !" = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ! !"#$ . 1 + 8 4 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝑒𝑒𝑒𝑒 !" 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝜑𝜑𝜑𝜑 ! ! . 𝐿𝐿𝐿𝐿 𝜑𝜑𝜑𝜑 ! ! ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! ! . 1 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! ! (3.2.3a) 𝑘𝑘𝑘𝑘 !" ! = 𝑘𝑘𝑘𝑘 !"#$ ! . 𝑡𝑡𝑡𝑡 ! . 𝜌𝜌𝜌𝜌 !"#$ . 𝑐𝑐𝑐𝑐 !"#$ 𝑡𝑡𝑡𝑡. 𝜌𝜌𝜌𝜌 ! . 𝑐𝑐𝑐𝑐 ! . 1 + 8 4 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝑒𝑒𝑒𝑒 !" 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 . 𝜑𝜑𝜑𝜑 ! ! . 𝐿𝐿𝐿𝐿 𝜑𝜑𝜑𝜑 ! ! ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! ! . 1 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! ! (3.2.3b) 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 ! !!!" = 𝑘𝑘𝑘𝑘 !" ! . 8 4 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! 𝐿𝐿𝐿𝐿 . 𝜑𝜑𝜑𝜑 ! . 2. 𝑘𝑘𝑘𝑘 ! -𝜑𝜑𝜑𝜑 ! ! 𝐿𝐿𝐿𝐿 𝜑𝜑𝜑𝜑 ! ! ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! ! . 1 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 ! ! ! (
To finish, we compare the performances of various pyroelectric bistable beams made of Al-Si when PZT is replaced by another piezoelectric material (hysteresis of 3 o C). Fig. 3.2.1d shows that PZT is the most efficient material, with a density of energy of 273 nJ.cm -2 and an equivalent ZT of 0.03. To compare, AlN-based bistable beams produce 14 pJ of energy per cycle, corresponding to a surface density of 14 nJ.cm -2 for an equivalent ZT of 0.0017. These values of the surface density of electrical energy can be compared with the values reported in [Trioux, 2015], showing that during the snap, densities of 43 pJ.cm -2 can be obtained with 1.4 mm long AlN bistable beams. The difference can be partly attributed to the fact that there was no attempt to implement a Stirling cycle to maximize the energy harvested.

Dynamic behavior of the bimetallic strip at the microscale a. Scaling law on the operation frequency

In Chapter 2, we modeled two types of thermal transfers. In the first case, the limitating mode of transfer is linked to the thermal contact between the heat sources and the prime mover. In that case, the device's operation frequency is given by (3.3.1a), where hC is the equivalent coefficient of thermal exchange associated to the thermal contact. The second mode of heat tranfer assumes that the contact between the heat sources and the bistable beam is perfect and that the operation frequency of the device is associated to the propagation of heat in the beam, leading to (3.3.1b). Thus, at any time, we can consider that the operation frequency of the device is given by (3.3.1c). The key hypothesis of the scaling law developed in [Puscasu et al, 2012;[START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF], is that the operation frequency of the heat engine is associated to the thermal transfers in the beam. However, we showed in Chapter 2 that the characteristic time associated to the beam's thermal diffusivity is very small and the operation frequency of the device is limited by the thermal contact between the heat sources and the bistable beam (3.3.1c). The dimensional analysis of both frequencies shows that, if the temperature difference in the cavity is kept constant despite the reduction of the cavity's thickness at the microscale, the frequency associated to first type of thermal transfer is inversely proportional to the characteristic length of the system (3.3.1d), whereas the frequency of the second type of thermal transfer is inversely proportional to the square of the length (3.3.1d). As a consequence, even at the microscale, the limiting mode of thermal transfer remains associated to the thermal contact between the heat sources and the bistable beam.

𝑓𝑓𝑓𝑓 ! = ℎ ! . 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣 𝐶𝐶𝐶𝐶 ! . 1 ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! + ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (3.3.1a) 𝑓𝑓𝑓𝑓 ! = 𝜋𝜋𝜋𝜋 ! . 𝜅𝜅𝜅𝜅 ! 2. 𝐿𝐿𝐿𝐿 ! . 1 ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! + ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (3.3.1b) 𝑓𝑓𝑓𝑓 = 𝑓𝑓𝑓𝑓 ! . 𝑓𝑓𝑓𝑓 ! 𝑓𝑓𝑓𝑓 ! + 𝑓𝑓𝑓𝑓 ! = 𝑓𝑓𝑓𝑓 ! = ℎ ! . 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣 𝐶𝐶𝐶𝐶 ! . 1 ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝑇𝑇𝑇𝑇 !" -𝑇𝑇𝑇𝑇 ! + ln 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !" 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (3.3.1c) 𝑓𝑓𝑓𝑓 ! ∝ 1 𝐿𝐿𝐿𝐿 (3.3.1d) 𝑓𝑓𝑓𝑓 ! ∝ 1 𝐿𝐿𝐿𝐿 ! (3.3.1e)
For the rest of this section, we choose hc = 2500 W.m -2 .K -1 corresponding to a metallic contact resistance of 10 K.W -1 at the macroscale, as we found in Chapter 2.

b. Simplified model of the harvester at the microscale

In order to see the evolution of temperature in the cavity as a function of external conditions, we define a reference architecture presented in Fig. 3.3.2. In this simple configuration, the harvester is made of two bonded wafers, separated by a polymer resist ring used to maintain the cavity under vacuum but also to limit the thermal transfers between the bottom wafer used as a hot source and the top wafer used as a plane heat sink, or on which it is possible to put a heat sink with fan. This system can be simply modeled as two series thermal resistances since the thermal resistance of the silicon wafers is negligible. The thermal gradient in the cavity is then given by (3.3.3a).

𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇

! = 𝑅𝑅𝑅𝑅 !"#$ 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# (3.3.2a)
Without any bulk heat sink, the thermal resistance of the top wafer is taken equal to 1000 K.W -1 (hsink = 10 W.m -2 .K -1 ), and the thermal resistance of the architecture to 2 K.W -1 (kpolymer = 0.5 W.m -1 .K -1 ). This limits the temperature difference in the cavity to 0.16 o C when the hot source temperature is 100 o C and the cold source 25 o C.

Even if other values of conductivity can be taken, the main way to increase the temperature difference is to use a bulk heat sink. With a heat sink of resistance 10 K.W -1 (hsink = 1000 W.m -2 .K -1 ), the temperature difference across the cavity can reach 13 o C for a hot source at 100 o C.

c. Performances of a bimetallic strip in a cavity

As with the thermo-mechanical performances of the bimetallic strip in the cavity of the energy harvester, we can easily demonstrate how the cavity influences the electrical behavior of the bimetallic strip. In brief, the performances of a clamped-clamped beam (γ.L = 36.

𝒲𝒲𝒲𝒲 ! = 59,3. 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝑡𝑡𝑡𝑡. 𝑟𝑟𝑟𝑟 ! ! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! ! ! ! . 𝑡𝑡𝑡𝑡 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! ! 𝑡𝑡𝑡𝑡. 𝐿𝐿𝐿𝐿 ! . 𝑘𝑘𝑘𝑘 !" ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 !"# ! ! (3.3.3b) 𝑃𝑃𝑃𝑃 ! = 29,7. 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣. 𝑟𝑟𝑟𝑟 ! ! 3. 𝑟𝑟𝑟𝑟 ! . 𝑟𝑟𝑟𝑟 ! ! ! ! . 𝑡𝑡𝑡𝑡 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! ! 𝑡𝑡𝑡𝑡. 𝐿𝐿𝐿𝐿 ! . ℎ ! . 𝑘𝑘𝑘𝑘 !" ! 𝜌𝜌𝜌𝜌 ! . 𝑐𝑐𝑐𝑐 ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿 ! ! . 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 !"# ! ! ln 1 + 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -ln 1 - 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (3.3.3c) 𝜂𝜂𝜂𝜂 !" !"#,!"! ≈ 𝑅𝑅𝑅𝑅 !"#$ . 𝑃𝑃𝑃𝑃 ! 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# (3.3.3d) 𝜂𝜂𝜂𝜂 !" !",!"! = 𝜂𝜂𝜂𝜂 !" !"#,!"! . 𝑇𝑇𝑇𝑇 ! + 273 ! 𝑇𝑇𝑇𝑇 ! + 273 ! -𝑇𝑇𝑇𝑇 !"# + 273 ! (3.3.3e)
A closer look at the expression of the power shows that it is a decreasing function of the beam's thermal hysteresis. As a consequence, the performances are maximized for small thermal hystereses. In Fig. 3.3.2, we present some projections about the performances of the heat engine at the microscale. These projections assume that the energy harvester uses a massive heat sink to evacuate heat from the bistable beams. Without this heat sink, the size reduction of the cavity drastically increases the thermal leakage, impeding its ability to maintain a thermal gradient inside which a bistable beam can oscillate. We choose a thermal hysteresis of at least 5 o C for which the output voltage is higher than 0,4V. For this thermal hysteresis, the operating frequency of the device lies in the 50-200 Hz range when the hot source's temperature varies from 75 o C to 15 o C. The maximal density of output power generated by the heat engine is then of about 4 µW.cm-2, for a maximal Curzon-Ahlborn efficiency of 1,3.10 -4 %. Logically, bistable beams with higher thermal hysteresis can only work at higher temperatures for the temperature difference in the cavity to be large enough to create self-sustaining oscillations of the prime movers.

We then observe a degradation of the performances, notably a decrease of the Curzon-Ahlborn efficiency.

Electrostatic coupling

Performances in weak coupling

By using the various results of Chapter 3, we can also study how the performances of the electrostatic bimetallic strip heat engine evolve when we scale the device down to the microscale. Fig. 4.1.1a shows a simplified diagram of the harvester at the microscale, where the electret is a SiO2 layer deposited on the upper wafer of the device. In order to model the system, we first give the expression (4.1.1a) of the electric charge carried by the bimetallic strip when it is clamped-clamped, which can be approximated with (4.1.1b).

𝒬𝒬𝒬𝒬 ! = -𝑣𝑣𝑣𝑣. 𝜀𝜀𝜀𝜀 ! . 𝒱𝒱𝒱𝒱 -𝒱𝒱𝒱𝒱 ! . 1 𝜀𝜀𝜀𝜀 ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝑔𝑔𝑔𝑔 ! -𝑤𝑤𝑤𝑤 ! . cos 𝑘𝑘𝑘𝑘 ! . 𝑥𝑥𝑥𝑥 -cos 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 2
. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 ! Fig. 4.1.1b recalls the way to polarize the bimetallic strip in order to realize a chargeconstrained cycle: when the bimetallic strip is unsnapped and stable at the snap-back temperature, the capacitance of the bimetallic strip is maximal, the circuit is closed until the maximal charge Qb max is reached. Then the beam is heated and snaps in open-circuit, increasing the potential of the electric charges. The beam is then discharged, until Qb min , and it cools down and snaps back in open-circuit. Applying the equations demonstrated in Chapter 3, we find the following expressions when the bimetallic strip is not constrained in a cavity:

𝒬𝒬𝒬𝒬 ! = 4. 𝒞𝒞𝒞𝒞 ! . 𝒞𝒞𝒞𝒞 ! 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿. (𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! ) . 𝒱𝒱𝒱𝒱 ! -𝒱𝒱𝒱𝒱 ! (4.1.3a) 𝒬𝒬𝒬𝒬 ! !"# = 𝒞𝒞𝒞𝒞 ! !"# 𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! !"# . 𝒬𝒬𝒬𝒬 * = 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝒬𝒬𝒬𝒬 * 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! + 2. 𝜑𝜑𝜑𝜑 ! . 𝑤𝑤𝑤𝑤 ! ! (4.1.3b) 𝒬𝒬𝒬𝒬 ! !"# = 𝒞𝒞𝒞𝒞 ! !"# 𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! !"# . 𝒬𝒬𝒬𝒬 * = 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝒬𝒬𝒬𝒬 * 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑔𝑔𝑔𝑔 ! -2. 𝜑𝜑𝜑𝜑 ! . 𝑤𝑤𝑤𝑤 ! ! (4.1.3c) 𝒱𝒱𝒱𝒱 ! !"# = 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# 𝒬𝒬𝒬𝒬 ! !"# . 𝑉𝑉𝑉𝑉 ! (4.1.3d) 𝒱𝒱𝒱𝒱 ! !"# = 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# 𝒬𝒬𝒬𝒬 ! !"# . 𝑉𝑉𝑉𝑉 ! (4.1.3e)
The expression of the energy is unchanged and is recalled in (4.1.4a). It can be developed as a function of the maximal and minimal voltages. If we assume that the gap is at least equal to the maximal beam's deflection to ensure a contact between the cold source and the bimetallic strip (4.1.4c), the final expression of the electrical energy is given by (4.1.4d).

𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 1 2 . 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# . 𝑉𝑉𝑉𝑉 ! !"# -𝑉𝑉𝑉𝑉 ! !"# = 1 2 . 𝒱𝒱𝒱𝒱 ! . 𝒬𝒬𝒬𝒬 ! !"# -𝒬𝒬𝒬𝒬 ! !"# ! 𝒬𝒬𝒬𝒬 ! !"# . 𝒬𝒬𝒬𝒬 ! !"# < 0 (4.1.4a) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = - 2 𝜋𝜋𝜋𝜋 . 𝒞𝒞𝒞𝒞 ! !"# . 𝒞𝒞𝒞𝒞 ! 𝒞𝒞𝒞𝒞 ! + 𝒞𝒞𝒞𝒞 ! !"# . 𝒱𝒱𝒱𝒱 ! !"# . 𝒱𝒱𝒱𝒱 ! !"# -𝒱𝒱𝒱𝒱 ! !"# (4.1.4b) 𝑔𝑔𝑔𝑔 ! = 𝜑𝜑𝜑𝜑 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! (4.1.4c) 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! = 16 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 . 𝜀𝜀𝜀𝜀 ! ! . 𝑟𝑟𝑟𝑟 ! ! . 𝜑𝜑𝜑𝜑 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝜑𝜑𝜑𝜑 ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝐶𝐶𝐶𝐶 ! . 𝒱𝒱𝒱𝒱 ! ! (4.1.4d)
Similarly, the expressions of the characteristic voltages are given by (4.1.5).

𝒱𝒱𝒱𝒱 ! !"# = 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝜑𝜑𝜑𝜑 ! . 𝑤𝑤𝑤𝑤 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 2. 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝜑𝜑𝜑𝜑 ! . 𝑤𝑤𝑤𝑤 ! ! . 𝒱𝒱𝒱𝒱 ! (4.1.5a) 𝒱𝒱𝒱𝒱 ! !"# = - 2. 𝜀𝜀𝜀𝜀 ! . 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! . 𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! 𝜀𝜀𝜀𝜀 ! . 𝑡𝑡𝑡𝑡 ! + 𝜀𝜀𝜀𝜀 ! . 𝑟𝑟𝑟𝑟 ! . 𝜑𝜑𝜑𝜑 ! , 𝑤𝑤𝑤𝑤 ! ! ! + 𝜀𝜀𝜀𝜀 ! ! . 𝑟𝑟𝑟𝑟 ! ! . 𝜑𝜑𝜑𝜑 ! ! . 𝑤𝑤𝑤𝑤 ! ! ! . 𝒱𝒱𝒱𝒱 ! (4.1.5b)
In order to have a first idea of the performances of the energy harvester at the microscale, we plot in Fig. 4.1.1c the evolution of the beam's displacement amplitude, in Fig. 4.1.1d the maximal and minimal values of the bimetallic strip's capacitance and in Fig. 4.1.1d the absolute values of the peak voltages Vb min and Vb max . The available electric energy is represented in Fig. 4.1.1e as a function of the thermal hysteresis. The simulations are performed for an Al-Si beam (400 µm x 100 µm x (2+2 µm)) coupled with an electret made of a 2 µm thick SiO2 layer. With an electret surface potential of 400 V, this can generate up to 20 nJ per cycle with output voltage of 280 and 180 V. However, the difficulty to harvest energy with the electrostatic harvester is due to the small values of the capacitances. For a thermal hysteresis of 5 o C, the minimal capacitance is 26.5 fF, whereas the maximal capacitance is 460 fF. This highlights the problem of the influence of the parasitic capacitances on the efficiency of the electric conversion.

Performances in moderate coupling

Now that we have determined the performances in weak coupling, we can establish the equation describing the mechanical equilibrium of the beam. We simulate the performances of an Al-Si beam coupled with an electret made of a thick SiO2 layer (20 µm) whose surface potential varies. Figs.4.2.1a-f present some of the performances: (a) working frequency, (b) energy per cycle, (c) mean power, (d) mean surface power, (e) thermal efficiency and finally (f) Carnot efficiency of the bistable beam. These simulations are performed assuming a temperature difference of 15 o C in the cavity, corresponding to a device working between a hot source at 100 o C and a cold source at 25 o C and evacuating heat via a bulk heat sink. These figures first show that the optimal value of the surface potential is very low, around 15 V. Because of the size reduction of the device, the electric field increases like the scaling factor 1/L of the device, whereas the pull-in voltage decreases like the length L. For our model beam this voltage is 20 V for a beam measuring 400 µm, and 40 V for 200 µm [START_REF] Hu | An approximate analytical solution to the pullin voltage of a micro bridge with an elastic boundary[END_REF][START_REF] Pamidighantam | Pull-in voltage analysis of electrostatically actuated beam structures with fixed-fixed and fixed-free end conditions[END_REF][START_REF] Chowdhury | Pull-In Voltage Study of Electrostatically Actuated Fixed-Fixed Beams Using a VLSI On-Chip Interconnect Capacitance Model[END_REF]]. Thus we cannot use an electret with high surface potential. Fig. 4.2.1b shows that the energy generated for a surface potential does not exceed 1 pJ.cm -2 , which can be quite difficult to harvest. Moreover, for a bimetallic strip having a thermal hysteresis of 1 o C, the corresponding output power is about 60 pW, and the thermal efficiency and Carnot effiency are 0.2 ppm and 9 ppm respectively for a surface potential of 15 V. Finally, Fig. 4.2.2 presents the evolutions of the dynamic performances of the electrostatic energy harvester when the external temperature changes. The thermal hysteresis of the bistable beam must be as small as possible to maximize the output power. The quantity of energy that can be harvested with a single bimetallic beam is very weak but is compensated by the increase of the switching frequency (around 1 kHz for a thermal hysteresis of 1 o C). Normalized by the device's surface, the mean output power is 120 nW.cm -2 for an energy harvester made of beams having a thermal hysteresis of 1 o C working at 100 o C. The thermal efficiency of the transduction is also weak, about 10 -9 %. These observations would tend to point to downscaling limits beyond which there might be little interest in scaling down the electrostatic energy harvester any further.

Synthesis

This last section is devoted to a synthesis on the performances of the miniaturized bimetallic strip heat engines where we recall all the previous results, but also where we refine the scaling rules that predict the evolution of these performances. In this part, we assume that the miniaturized energy harvesters are fabricated at the microscale when their length is lower than 1 mm.

a. Synthesis on the mechanical properties of the bimetallic strip heat engine

We first focus on the evolution of the mechanical performances of the bistable beams and we look at their evolution as a function of the scaling factor 1/L. Equation (5.1.1a) describes the evolution of the maximal deflection, (5.1.1b) the maximal force, and (5.1.1c) the energy produced during a complete cycle. As a consequence we can see that these parameters follow the following scaling rules. Logically, we observe that the energy is a volume parameter and the force a surface parameter. Their evolutions are represented in Fig. b. Synthesis on the dynamic behavior of the system To see the evolutions of the dynamic performances with downscaling, we must study how the frequency of the device evolves with the scaling parameter. We choose to focus on two different strategies. The first strategy consists in maintaining a fixed thermal gradient in the cavity whatever the size of the prime mover. We recall first the expression of the frequency associated to the thermal transfer at the interface between a heat source and the prime mover (5.1.3a), and the frequency associated to the internal heat transfers (5.1.3b) and linked to the thermal diffusivity of the materials. We can approximate the real working period of the system as the sum of both periods. As a consequence, we can approximate the working frequency by (5.1.3c). For small values of the thermal hysteresis, we can linearize the expression of both frequencies, leading to (5.1.3d) and (5.1.3e).

𝑓𝑓𝑓𝑓 !"#$%!$ = ℎ ! . 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣 2. 𝐶𝐶𝐶𝐶 ! . 1 ln 1 + 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -ln 1 - 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.1.3a) 𝑓𝑓𝑓𝑓 !"#$%"&' = 𝜋𝜋𝜋𝜋 ! . 𝜅𝜅𝜅𝜅 ! 4. 𝐿𝐿𝐿𝐿 ! . 1 ln 1 + 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! -ln 1 - 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! (5.1.3b) 1 𝑓𝑓𝑓𝑓 !"! = 1 𝑓𝑓𝑓𝑓 !"#$%"&' + 1 𝑓𝑓𝑓𝑓 !"#$%!$ (5.1.3c) 𝑓𝑓𝑓𝑓 !"#$%!$ = ℎ ! . 𝐿𝐿𝐿𝐿. 𝑣𝑣𝑣𝑣 𝐶𝐶𝐶𝐶 ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (5.1.3d) 𝑓𝑓𝑓𝑓 !"#$%"&' = 𝜋𝜋𝜋𝜋 ! . 𝜅𝜅𝜅𝜅 ! 2. 𝐿𝐿𝐿𝐿 ! .
𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥

(5.1.3e)

The temperature difference in the cavity is given by the law (5.1.4a), which can be rewritten as (5.1.4b) by using an equivalent thermal resitivity to model the architecture's leakage paths and the heat sink by an equivalent thermal exchange coefficient.

𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! = 𝑅𝑅𝑅𝑅 !"#$ 𝑅𝑅𝑅𝑅 !"#$ + 𝑅𝑅𝑅𝑅 !"#$ . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) (5.1.4a) 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! ≈ 2. 𝑤𝑤𝑤𝑤 ! 2. 𝑤𝑤𝑤𝑤 ! + 𝜌𝜌𝜌𝜌 !"#$ ℎ !"#$ . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) (5.1.4b)
The only possible strategy to maximize the output energy is to maintain a fixed thermal gradient in the cavity in order to increase the frequency of the device (in the other case, the temperature difference in the cavity is so small that no bimetallic strip can switch from a hot source to the other). The scaling rules corresponding to this case are given in (5.1.5), and represented in Fig. 5.1.2. As the equivalent resistance of the device decreases with the beam's maximal deflection, it is necessary to use a heat sink whose effective coefficient of thermal transfer must be inversely proportional to the length of the prime mover in order to evacuate heat flowing through the energy harvester. As a consequence, we cannot maintain a constant temperature difference in the cavity by using a planar heat sink as was the case for the piezoelectric bimetallic strip heat engine.

ℎ !"#$ ∝ 1 𝐿𝐿𝐿𝐿 . 𝜌𝜌𝜌𝜌 !"!" Δ𝛼𝛼𝛼𝛼. Δ𝑇𝑇𝑇𝑇 ! (5.1.5a) 𝑓𝑓𝑓𝑓 !"#$%!$ ∝ 1 𝐿𝐿𝐿𝐿 . ℎ ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (5.1.5b) 𝑓𝑓𝑓𝑓 !"#$%"&' ∝ 1 𝐿𝐿𝐿𝐿 ! . 𝜅𝜅𝜅𝜅 ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (5.1.5c) 𝑓𝑓𝑓𝑓 !"! ≈ 𝑓𝑓𝑓𝑓 !"#$%!$ ∝ 1 𝐿𝐿𝐿𝐿 . ℎ ! . (𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# ) 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 (5.1.5d) 𝑃𝑃𝑃𝑃 !"#$ = 𝛥𝛥𝛥𝛥𝑊𝑊𝑊𝑊 !"# . 𝑓𝑓𝑓𝑓 !"! ∝ 𝐿𝐿𝐿𝐿 ! . ℎ ! . 𝐸𝐸𝐸𝐸 Δ𝛼𝛼𝛼𝛼 ! ! 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 . Δ𝑇𝑇𝑇𝑇 ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# (5.1.5e) 𝑃𝑃𝑃𝑃 !"#$ !"#$ = 𝑃𝑃𝑃𝑃 !"#$ 𝑆𝑆𝑆𝑆 ∝ ℎ ! . 𝐸𝐸𝐸𝐸 Δ𝛼𝛼𝛼𝛼 ! ! 𝜌𝜌𝜌𝜌. 𝑐𝑐𝑐𝑐 . Δ𝑇𝑇𝑇𝑇 ! . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !"# (5.1.5f) 𝑃𝑃𝑃𝑃 !! !"#$ = 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 2. 𝑤𝑤𝑤𝑤 ! . 𝜌𝜌𝜌𝜌 !"#$ ∝ 1 𝐿𝐿𝐿𝐿 . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 ! 𝜌𝜌𝜌𝜌 !"#$ . Δ𝛼𝛼𝛼𝛼. Δ𝑇𝑇𝑇𝑇 ! (5.1.5g) 𝜂𝜂𝜂𝜂 !"# = 𝑃𝑃𝑃𝑃 !"#$ !"#$ 𝑃𝑃𝑃𝑃 !! ∝ 𝐿𝐿𝐿𝐿 (5.1.5h)
If we look at the evolution of the switching frequency, we can see that the thermal transfers inside the prime mover are faster than the thermal transfers between the prime mover and the heat sources at the microscale. As a consequence, the prime mover's working frequency is determined by the thermal transfers between the heat sources and the bistable beam, which are proportional to the beam's length. Hence we found that the mechanical power generated by a single bimetallic strip is a function of the square of the length instead of being a linear function of the bimetallic strip's length. As a consequence, if the bimetallic strips were simply-supported at the micrometric scale, we would be able to generate as much power at this scale as at the macrometric scale if we used two devices having the same surfaces.

Finally the consequence of the linear increase of the frequency is visible on the thermal efficiency of the energy harvester since more heat flows through the leaky architecture than through the bimetallic strips at the microscale. The thermal efficiency thus decreases as the size factor of the device. Solutions to avoid these issues would be to use highly insulating materials like aerogels for separating the upper wafer from the bottom wafer. Aerogels are materials having very low values of thermal conductivity (around 0.24.W.m -1 .K -1 [START_REF] Bauer | Thin-film aerogel thermal conductivity measurements via 3ω[END_REF][START_REF] He | Ultra-low thermal conductivity and high strength of aerogels/fibrous ceramic composites[END_REF]), which can be fabricated using standard microelectronics process [Yokokawa et al, 2004].

c. Synthesis on the pyroelectric bimetallic strip heat engines

We start with the performances of the pyroelectric bimetallic strip heat engines and we recall the formula giving the expression of the available energy and the output voltage obtained with a Stirling cycle at the macroscale (5.1.6a-b) and at the microscale (5.1.6c-d).

Logically we observe that the energy is always linked to the volume of matter, and that the voltage is a linear function of the length as the capacitance.

𝒱𝒱𝒱𝒱 !! !"# = 𝒱𝒱𝒱𝒱 ! ! = 𝒱𝒱𝒱𝒱 !" ! = 𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# + 4. 𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿 ! . 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 !"# (5.1.6a) 𝒲𝒲𝒲𝒲 !! !"# = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 !! ! = 𝒞𝒞𝒞𝒞 ! . 3
𝒰𝒰𝒰𝒰 !! !"# ∝ 𝐿𝐿𝐿𝐿. 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# (5.1.6f)
As we can see on Fig. 5.1.1, the main challenges for piezoelectric energy harvester is the reduction of both the piezoelectric capacitance and output voltage. The break on the curves is due to the use of a configuration with three piezoelectric capacitors at the microscale.

d. Synthesis on the electrostratic bimetallic strip heat engines

In the electrostatic case, it is important to see how the bimetallic strip capacitance evolves and how the pull-in voltage depends on the dimensions of the bistable beam, since this gives an idea of the maximal value of the electret's potential surface at the microscale.

𝒱𝒱𝒱𝒱 !" = 8 2 3 . 𝑔𝑔𝑔𝑔 ! 𝐿𝐿𝐿𝐿 𝐼𝐼𝐼𝐼 ! . 𝜋𝜋𝜋𝜋 ! 𝜀𝜀𝜀𝜀 ! . 𝒞𝒞𝒞𝒞 ! . 𝐿𝐿𝐿𝐿 ∝ 𝐿𝐿𝐿𝐿 (5.1.7)
The evolution of the mean value of the bimetallic strip capacitance, the pull-in voltage and the mean power are represented in Fig. 5.1.1 as functions of the length scaling factor.

e. Aging of the device

A last consequence of the device's size reduction is the reduction of the lifetime of the device. Assuming a constant number of cycles before failure whatever the size of the device, the linear increase of the working frequency as a function of the length factor causes a linear reduction of the lifetime of the devices. As a consequence, it is expected that the miniaturized devices would work no more than a hundred days.

Conclusion

In this last theoretical chapter we developed new models to describe the mechanical behavior of bistable beams at the microscale. These models add a torsional stiffness and an axial stiffness to model the beam's clamp: we simulated bimetallic strip using either the non-linear Euler equation or an equivalent reduced-order model, and we showed that by clamping beams on the substrate, the thermo-mechanical performances of the harvester are reduced (down to 8.3% of the mechanical energy produced by a simplysupported beam). We also study the sensitivity of the properties of the bistable beams to the variability of the residual stress used to buckle the beams, showing that the change of the clamp conditions from supported beams to clampled clamped beams sensibly increase this sensitivity. As a consequence, it appears quite impossible to consider an industrial fabrication of the miniaturized energy harvesters since variability issues may be a problem to create a reliable technology.

We then modelled the performances of the pyroelectric bimetallic strip heat engine at the microscale, proposing an expression of the power generated by a bistable beam made of three piezoelectric capacitor. We showed that such anarchitecture does not allow harvesting the electrical charges created by the pyroelectric effect, thus reducing the performances of the device. Moreover, by reducing the size of the device, the output voltages, electrical capacitance and electrical charges are strongly reduced, creating difficulties to harvest them.

We also extended the model of the electrostatic bimetallic strip heat engine to describe the performances of a miniaturized energy harvester. We showed that an important parameter to be taken into account is the value of the pull-in voltage that can act as a limit value for the electret's surface potential. The reduction of the pull-in voltage for MEMS devices thus degrades the performances of the energy harvester at the microscale.

Finally we demonstrated that it is necessary to use a bulk heat sink to maintain a thermal gradient in the device's cavity. In these conditions, the reduction of the amount of heat converted into electrical energy by each bimetallic strip is counterbalanced by the increase of the switching frequency of the device, but the increase of the frequency is slower than predicted in Puscasu's work. As a consequence, the mean output power of miniaturized generators is not higher than the mean output power of macro energy harvester, but smaller. This last fact also justifies the fact of forgetting the idea of fabricating miniaturized energy harvesters. Contrariwise, it seems worth considering other opportunities for miniaturized thermo-mechanically bistable beams such as thermal sensing applications for example. Chapter 5

Silicon integration of thermo-mechanically bistable actuators

Introduction

In the second chapter, we explained that the beams' thermo-mechanical instability was induced b the antagonistic effects of the thermal expansion asymmetry and of the residual stress asymmetry or the initial curvature of the beam. As a consequence, two kinds of fabrication process have been considered and realized on 300mm facilities of STMicroelectronics, Crolles. This chapter separately presents these two manufacturing methods in the order they were created. As a result, the first part of this chapter is devoted to the fabrication of microstructures by means of the microlenses arrays, whereas the second part of this chapter presents the implementation of a second process based on the use of residual to create thermo-mechanically bistable beams.

A first fabrication process

A short overview on microlenses arrays

Microlens arrays are present in a lot of microelectronic technologies, being used to increase the light sensitivity of CCD and CMOS image sensors, to collimate the light from light-emitting diodes (LED), LCD displays, or lasers [START_REF] Dal Zilio | Microlens arrays on large area UV transparent hybrid sol-gel materials for optical tools[END_REF]. Several fabrication methods have been developed over the twenty last years like localized doping of glass, mechanical polishing, LIGA process, hydrophobic surfaces… [START_REF] Daly | Microlens arrays[END_REF]. The main technique, first developed by [START_REF] Popovic | Technique for monolithic fabrication of microlens arrays[END_REF] consists in melting photoresist spots by heating resist above its glass transition temperature. During this reaction, surface tensions can be in competition with resist crosslinking reaction that can hinder the formation of the microlens by increasing the resist's glass transition temperature (Fig. 2.1.1a). As a consequence, the formation process of microlenses can be stopped to an intermediary state where the microlens has a low curvature on its edges and a concave shape in the midde [Audran et al, 2010]. With the heating temperature, numerous parameters influence the final shape of the microlenses: the substrate type, more or less hydrophilic which modifies the contact angle between the microlenses and the substrate [START_REF] Wei | Fabrication of a transparent and self-assembled microlens array using hydrophilic effect and electric field pulling[END_REF], the resist layer's thickness, the resist spots' diameter. Extensive review on these techniques can be found in [START_REF] Daly | Microlens arrays[END_REF].

Fabrication process

The first fabrication consists in using microlenses as a sacrificial layer to give an initial curvature to the bimetallic strips. The process can be divided in three major steps: the fabrication of the microlenses, the deposition of the layers constituting the bimetallic strips and finally the etching of the bimetallic strips and of the microlenses to release the microstructures. Each step of this process is presented in Fig. 2.2.1b. Details on these steps performed by the TSV team are presented below.

a. Microlenses process -Spin-coating of a resist TOK TMR-TS01 on the substrate, using HMDS as an adhesion promoter at 900 rpm during 60s (3,8µm-thick layer).

-

b. Bimetallic layers deposition

The main parameters to be taken into account are the raise of the surface temperature during PVD deposition that must not exceed 210 o C to avoid resist collapsing and the deposition time that must stay acceptable. As a consequence, deposition power must be adjusted to find a tradeoff between these two constraints. We give a first recipe that was initially used for Ti-Cu bimetallic strips. Similar prodedures are followed for PVD deposition of other materials.

-Three-step PVD deposition of Cu at 5000W (300 s + 300 s in idle mode) in order to create a 3µm-thick layer.

c. Etching of bimetallic strips and photoresist's removal

-Ti layer is etched using diluted HF 0,5% -Copper layer and gold layer are etched with potassium iodie (KI,I2) -Wet etching of the microlenses using TMAH 2.38% and DMSO

Characterization of microstructures

Bimetallic strips were successively made of Ti-Cu, Ti-Au and Ti-Au-Ti. First bimetallic strips made of Ti-Cu suffered from oxidation when they were heated (Fig. 2.1.1g). As a consequence, copper layer is replaced by a gold layer that has a lower coefficient of thermal expansion but that does not oxidize. First tests were realized wit a stack of Ti-Au in order to validate the possibility to deposit materials without observing collapse of the resist even though Ti-Au bimetallic strips cannot be functional given that Au layer must be on the inner side of the curved strips. Au-Ti bimetallic strips are then fabricated to create functional bimetallic strips. Given delamination issues due to the use of gold layer, a first thin layer of titanium is first deposited on the substrate. Several wafers are processed either with various values of resist thickness (2 -4 -8 µm) and various values of metallic layers thickness.

In order to characterize bimetallic strips, a heating bench is fabricated, made of a DBK HP03-09-240 heating resistance, whose temperature is either measured with a TI LM35 temperature sensors supplied by and communicating with an Arduino Uno board (Fig. 2.3.1d), or calculated by measuring the input electrical power. Interferometric measurements are performed with a Polytec interferometer. Difficulties encountered with interferometric measurements are linked to the important curvature of bimetallic strips, which hinders the reflection of coherent light to the sensor of the interferometer. As a result, most of the bimetallic strip's surface is not visible on interferometric measurements. Fig. 2.2.1f shows that for 210 µm bimetallic strips, no significant variation of the deflection is observed when the temperature raises until 187 o C. This absence of deformation for the shortest bimetallic strips is due to their excessive curvature. To simply explain this, we take the example of simply supported beams modeled in Chapter 2. For bimetallic strips like Ti-Au, the minimum initial curvature must represent about 40% of the total thickness. In the case of the bimetallic strips presented in Fig. 2.2.1, it is equal to 16 times the beam's thickness. As a result the expected thermal hysteresis is too much important so that bimetallic strips be functional.

To illustrate this, we plot in Fig. 2.2.1f the evolution of the mean temperature of the hysteresis as a function of the beam's length and the initial curvature for Ti-Au beams having a total thickness of 4 µm. We also plot two limits found by Daly and Nussbaum defining the limit ratio between the microlens' diameter (which is also the beam's length) and the microlens' height: below these limit values, the diameter of the microlens is too much important and the microlens is not spherical but concave. As a results, bimetallic strip are not functional. As we can see on this figure, to obtain 200 µm-long bimetallic strips with small thermal hysteresis, the initial deflection must be of 0,88 µm. Yet for this value of deflection, the ratio between the beam's length and its initial deflection is far from the limit value experimentally found by Daly [START_REF] Daly | Microlens arrays[END_REF]. For bimetallic strips that are not clamped on their whole width, an irreversible mechanical deformation is observed.

As the effect of residual stress and thermal stress has also an influence on the transverse curvature of bimetallic strips, transverse stress can cause the irreversible buckling of the membranes (Fig. 2.2.2a). As a result, the initial curvature conferred by the microlens finally disappears and the beam only transversally curves when it is heated again (Fig. 2.2.2b). The effect of this transverse stress can also break the membranes' clamp (Fig. 2.2.3b). As a consequence, microlenses-based fabrication process has many limitations. First, all materials currently used in microelectronics cannot be used to process bistable membranes because the deposition temperature cannot exceed the resist outgassing temperature. Second, the minimal curvature needed to produce beams with small thermal hystereses cannot be obtained with microlenses (Daly's limit or Nussbaum's limit). Third, the membrane's thermal hysteresis depends on the microlenses' curvature but also on the residual stress of the deposited materials layers. For all these reasons, the fabrication process based on the use of resist microlenses is finally abandoned in favor of a fabrication process using the residual stress to create initially buckled beams.

In order to avoid the occurrence of transverse buckling, a new set of lithography masks is also designed. [START_REF] Moulton | Micromechanical devices with embedded electro-thermal-compliant actuation[END_REF]. b. Image of V-Shape thermal actuators presented in [START_REF] Baker | Final Report: Compliant Thermomechanical MEMS Actuators LDRD #52553[END_REF]. c. Image of a thermally actuated micro-mirror presented in [START_REF] Schweizer | Thermally actuated optical microscanner with large angle and low consumption[END_REF] based on bimorph/bimetal effect. d. Image of an out-of-plane micropositionners presented in [START_REF] Kim | Creating large out-of-plane displacement electrothermal motion stage by incorporating beams with step features[END_REF] and based on bimorph effect. e. Example of microgrippers presented in [Andersen et al, 2008]. f. Architecture of bistable actuators developed by Matoba and Coworkers in [Matoba et al, 1994]. g. Schematic of bistable actuactors proposed in [START_REF] Lin | Vertically driven microactuators by electrothermal buckling effects[END_REF]]. h. Bistable out-of-plane bimorph actuator developed by [START_REF] Michael | Thermo-mechanical behavior of buckled multi-layered micro-bridges[END_REF]. i. Out-of-plane piezoelectric bimorph membrane presented in [Trioux, 2015].

A second fabrication process

Short review on bistable thermal actuators

Even though the development of thermo-mechanically bistable microactuators is less frequent, numerous examples of electro-thermal microactuators, also called heatuators [START_REF] Comtois | Applications for surface-micromachined polysilicon thermal actuators and arrays[END_REF], are reported in the literature, heated by Joule effect and employing either difference of thermal expansion of two materials to produce lateral inplane or out-of-plane displacements [START_REF] Riethmuller | Thermally excited silicon microactuators[END_REF][START_REF] Schweizer | Thermally actuated optical microscanner with large angle and low consumption[END_REF][START_REF] Kim | Creating large out-of-plane displacement electrothermal motion stage by incorporating beams with step features[END_REF], thermal expansion difference of two legs made of a same materials Pull-ring for probe tip Support and guide springs having different lengths [Andersen et al, 2008], different cross section [START_REF] Guckel | Thermo-magnetic metal flexure actuators[END_REF]Comptois and Bright, 1997], or an initial inclination of legs like V-shape thermal actuators [START_REF] Baker | Final Report: Compliant Thermomechanical MEMS Actuators LDRD #52553[END_REF].

Designing out-of-plane thermally buckled structures was first proposed by Seki and coworkers in [START_REF] Seki | Thermal buckling actuator for micro relays[END_REF]. Their goal was to increase thermal stress sensitivity of a Si beam near its snap temperature in order to create large displacements and important forces with small temperatures variations. They measured displacement of about 30 µm for a 2,5 mm-long, 20 µm-thick Si beam. Other examples of bistable thermal actuators are reported in [Hoffman et al, 1999] demonstrating an out-of-plane displacement of 50 µm of a 20mm long, 50 µm-thick silicon membrane supplied with 100mW of electrical power (temperature variation of 150 o C), [START_REF] Lin | Vertically driven microactuators by electrothermal buckling effects[END_REF]] describing the operation of in-plane actuators (620 µm-long poly-Si structure), displacing from around 4 µm between 200 and 400 o C, or [START_REF] Matoba | A bistable snapping microactuator[END_REF] presented a buckled structure displacing of 6 µm when it supplied with 166 mW of electrical power. Closer to the properties of our own system, we can cite the works of Michael and Kwok [Michael and Kwok, 2006;[START_REF] Michael | Thermo-mechanical behavior of buckled multi-layered micro-bridges[END_REF] on the thermal snap-through of multi-layered beams made of Silicon and Silicon oxide heated by Joule effect and snapping at 800 o C, or bistable piezoelectric Al-AlN beams developed by Trioux in [Trioux, 2015]. Electro-thermal actuators, either continuous or bistable can be used as micropositionners [START_REF] Kim | Creating large out-of-plane displacement electrothermal motion stage by incorporating beams with step features[END_REF]] micropumps or microvalves, micro-mirrors [START_REF] Schweizer | Thermally actuated optical microscanner with large angle and low consumption[END_REF], optical fiber switches [Hoffman et al, 1999] or in the case of thermo-mechanically bistable MEMS as mechanical memory device by exploiting the Shape-Memory Effect of these structures. Applications as relays [Hoffman et al, 1999] are less frequent because of the their slow response time compared with electrostatic switches [Lopez-Walle et al, 2010].

Design of new lithography masks

In order to avoid transverse buckling of microstructures and to allow electrical characterization/ Joule heating of thermo-mechanical bistable beams, a new set of lithography masks was created, initially thought and designed for the integration of bimetallic strip heat engines fabricated with the microlenses-based fabrication process but finally used to design initially-buckled beams. Three masks are drawn for the whole fabrication process. A first mask is used to open trenches in the substrate that will be filled of sacrificial material. The second mask is used to design the bimetallic strip and the third mask is used for the creation of the polymer ring whose function is to allow the bonding of the upper wafer and the bottom wafer and to hold the cavity under vacuum. Fig. 3.2.1a shows the structure of a single test chip for the characterization of the bimetallic strip heat engine, whereas Fig. 3.2.1b is a top view of the three masks and Fig. 3.2.1c is a top view of the pattern repeated on the whole surface of the 1X 300 mmdiameter lithography mask. This pattern is made of thirty chips having the same footprint even if each chip is dedicated to a single geometry of bimetallic strips (rectangular, trapezoidal or circular bimetallic strips). Fifteen different dimensions of bimetallic strips are defined, the beams' length varying from 60 µm to 480 µm and the beams' width from 30 µm to 120 µm, and the length-towidth ratio varying from 2 to 6 (Fig. 3.2.1d). Fig. 3.2.1e presents the global fabrication process chosen for the electrostatic bimetallic strip heat engines. Two wafers are separately process: bimetallic strips are fabricated on the bottom wafer, and an oxide layer is created on the top wafer, either by thermal oxidation or by deposition. This oxide layer is then used to create a Corona electret. After having created the sealing ring on the top wafer, the two wafers are bonded together. The top wafer is then sawed in order to reveal the external electrical pads used for the characterization of the bimetallic strip heat engine. Chips are finally separated.

Design rules of the bimetallic strips

Before presenting the new fabrication process of bistable actuators, we first simulate the behavior of Al-SiO2 microstructures. Given that the Young Moduli of Aluminum and Silicon oxide are quite equivalent, SiO2 layer and Al layer will have the same thickness.

In that case, we describe a simple procedure to directly process functional bimetallic strips having small thermal hysteresis. We first need to choose a value for the mean working temperature. Knowing the properties of the materials, we can simply find the relation between the mean working temperature and the residual stress (3.3.1a). We then use the criterion of bistability (3.3.1b) to find a relation between the beam's thickness, the beam's length and the residual stress (3.3.1c). In the case of Al-SiO2 bimetallic strip, we can approximate these equations by the set of equations (3.3.2) where the stress is directly expressed in MPa. Given the high coefficient of thermal expansion of Aluminum, we can neglect the influence of the Aluminum film's residual stress in the expression of the bistability criterion. Finally,Fig.3.3.1c and Fig.3.3.1d show the evolution of the thermal hysteresis of bistable beams as a function of the residual stress of each material layer.

Possible strategies for the development of the second process

A long and complicated step of the creation of the new fabrication process was the definition of the performances requirements for the bimetallic strip heat engines but also of the constraints on the availability of materials, manufacturing techniques and machines on Back-End-Of-Lines (BEOL) facilities of Crolles' 300 mm clean room. Yet a large range of manufacturing machines and process solutions are present in such industrial clean rooms, access to these facilities is generally limited in the case of the development of "exotic" technologies.

In the case of bimetallic strip heat engines, performances requirements are of two types. First, materials composing the thermo-mechanically bistable actuators must have excellent thermo-mechanical properties whose associated metrics are the thermal expansion coefficients difference, but also the Figure-of-Merit we defined in the previous chapter, giving the thermo-mechanical transduction efficiency and the mechanical energy produced during a complete cycle. From that figure, we can see that best bimorphs are mainly made of silicon compounds, like polysilicon, silicon oxide, silicon nitride and of aluminum, copper and gold. Even though polysilicon has very good mechanical properties, (explaining why it is often used as a materials for mechanical parts of MEMS) [START_REF] Ghodssi | MEMS materials and processes handbook[END_REF][START_REF] French | Polysilicon: a versatile material for microsystems[END_REF], and though residual stress in polysilicon thin films can be easily controlled, this material is absent from TSV facilities and finally cannot be used. As a consequence, bistable actuators will be made of silicon oxide or silicon nitride used as passive materials. Because of its high coefficient of thermal expansion, aluminum is in the first instance preferred to copper that faced oxidation issues with the microlenses-based fabrication process and to gold which has a smaller thermal expansion coefficient.

The second requirement of the fabrication process is the capacity to control residual stress in thin films in order to obtain functional bistable actuators, meaning that their thermal hystereses must be small (few degrees) and their snap and snap-back temperature lower than 200 o C so that the bistable membranes could harvest low grade heat flux flowing through the heat engine. It may be noted that, in the case where bistable membranes would be used as thermal sensors to disconnect circuits, the condition of weak thermal gradient would not be necessary. RF sputtering is a perfectly suitable deposition technique since it offers many parameters to adjust and control the residual stress in metallic thin films. The influence of RF sputtering process conditions on the residual stress of thin metallic films was highlighted by the successive works (among others) of Hoffman and Thornton [START_REF] Hoffman | The compressive stress transition in Al, V, Zr, Nb and W metal films sputtered at low working pressures[END_REF], Kim and his coworkers [START_REF] Kim | A study on the crystallographic orientation with residual stress and electrical property of Al films deposited by sputtering[END_REF]], Chinmulgund and his coworkers [START_REF] Chinmulgund | Effect of Ar gas pressure on growth, structure, and mechanical properties of sputtered Ti, Al, TiAl, and Ti3Al films[END_REF]. It is generally observed that for low temperatures, low substrate bias voltages and low Argon pressures, residual stress in metallic thin films is tensile whereas an increase of the RF power and Argon pressure creates a change from a tensile stress to a compressive stress. sputtering conditions [START_REF] Detor | Stress and microstructure evolution in thick sputtered films[END_REF]. b. Dependence of metallic thin films' residual stress on substrate bias and working pressure according to [START_REF] Kim | A study on the crystallographic orientation with residual stress and electrical property of Al films deposited by sputtering[END_REF]]. c. Evolution of PECVD dielectric thin films' residual stress on RF duty cycle according to [START_REF] Van De Ven | Advantages of dual frequency PECVD for deposition of ILD and passivation films[END_REF]. d. Evolution of dielectric thin films' residual stress on layer thickness [START_REF] Temple | Mechanical-Stress in Sio2-Films Obtained by Remote Plasma-Enhanced Chemical Vapor-Deposition[END_REF]]. e. Dependence of thin films' residual stress on PECVD deposition temperature [START_REF] Temple | Mechanical-Stress in Sio2-Films Obtained by Remote Plasma-Enhanced Chemical Vapor-Deposition[END_REF]. f. Reduction of thermal stress by thermal annealing [START_REF] Zhang | Residual stress and fracture in thick tetraethylorthosilicate (TEOS) and silane-based PECVD oxide films[END_REF].

As explained by Windischmann [Windischmann, 1992], this evolution from a tensile stress to a compressive stress is mainly due to atomic peening: as the energy of Argon or metal particles increases with the raise of the substrate bias or temperature, they can cause a densification of thin films. [START_REF] Hoffman | The compressive stress transition in Al, V, Zr, Nb and W metal films sputtered at low working pressures[END_REF] explained the link between the film microstructure and the type of residual stress. For more informations, we can report to [Windischmann, 1992] and [START_REF] Detor | Stress and microstructure evolution in thick sputtered films[END_REF] presenting various modeling attempts to predict the evolution of the residual stress as a function of the RF sputtering conditions like substrate bias, temperature, film thickness, temperature etc.

Several techniques can be used to deposit SiO2 or Si3N4 thin films, like LPCVD (low pressure chemical vapor deposition), PECVD (Plasma-enhanced chemical vapor deposition), sputtering etc. Details on these various deposition techniques can be found in [START_REF] Ghodssi | MEMS materials and processes handbook[END_REF][START_REF] Gad-El-Hak | MEMS[END_REF]. As PECVD was the main available technique in our case, we only investigated techniques reported in the literature to control the residual stress in dielectric thin films. Residual stress in PECVD thin films is generally known to be a result of the incorporation of hydrogen atoms into the materials layer. Hydrogen atoms come from the reaction precursor SiH4. Incorporation of hydrogen is more important in the case of PECVD silicon nitride because of the use of NH3 as a precursor instead of N2O for PECVD silicon oxide [START_REF] Hess | Plasma-enhanced CVD: Oxides, nitrides, transition metals, and transition metal silicides[END_REF]. As a result, hydrogen atoms cause the densification of the dielectric layer leading to excessive compressive stress. Yet, incorporation of hydrogen atom depends on power supply frequency. Low-frequency plasmas are known to ease diffusion of hydrogen in thin films whereas high frequency leads to porous tensile films. As a result, [ [START_REF] Van De Ven | Advantages of dual frequency PECVD for deposition of ILD and passivation films[END_REF][START_REF] Guan | Stress control of plasma enhanced chemical vapor deposited silicon oxide film from tetraethoxysilane[END_REF][START_REF] Tarraf | Stress investigation of PECVD dielectric layers for advanced optical MEMS[END_REF] proposed to use a duty cycle between low-frequency plasma and highfrequency plasma to alternate deposition of compressive films and tensile porous films.

Van de Ven and his coworkers demonstrated a linear relation between the duty cycle and the residual stress [ Van de Ven et al, 1990] (Fig.3.4.1c). Other parameters that can be adjusted are the Argon flow rate, the ratio between SiH4 flow and O2 flow, or the power supply. Raise of power supply, deposition temperature and Argon flow enhances compressive stress because of atomic peening [START_REF] Temple | Mechanical-Stress in Sio2-Films Obtained by Remote Plasma-Enhanced Chemical Vapor-Deposition[END_REF], whereas the increase of the SiH4/O2 flow ratio reduces the compressive stress [START_REF] Kim | The origin of intrinsic stress and its relaxation for SiOF thin films deposited by electron cyclotron resonance plasma-enhanced chemical vapor deposition[END_REF]. If residual stress is too much compressive, thermal annealing allows thin films outgassing hydrogen atoms in order to reduce compressive stress (Fig. 3.4.1f). Such techniques are reported in [START_REF] Zhang | Residual stress and fracture in thick tetraethylorthosilicate (TEOS) and silane-based PECVD oxide films[END_REF][START_REF] Windischmann | Intrinsic stress and mechanical properties of hydrogenated silicon carbide produced by plasma-enhanced chemical vapor deposition[END_REF][START_REF] Guan | Stress control of plasma enhanced chemical vapor deposited silicon oxide film from tetraethoxysilane[END_REF]. Now that we found materials and fabrication techniques adapted to the design of lowstress thermo-mechanically bistable actuators, we need to define the type of sacrificial materials adapted to the fabrication of bistable actuators. Sacrificial materials are chosen according their availability on TSV facilities, their etching recipes and etching techniques and the etching selectivity with respect to the materials composing the bistable actuators. We first examined various possible etching materials: Si3N4, Si (bulk), polyimide, photoresist. Si3N4 is generally etched with the same liquid etchant than silicon oxide like Buffered Oxide Etch (NH4F+HF 60 nm/min), or with H3PO4 (20 nm/min at 180 o C). Yet BOE etches faster silicon oxide (490 nm/min) and H3PO4 etches aluminum (530 nm/min). As a consequence Si3N4 is not the best sacrificial material. Another solution can be to directly use silicon substrate as a sacrificial layer. Isotropic etching of silicon can be realized by using either liquid etchants KOH (1100 nm/min at 80 o C), NH4F+HNO3 (150 nm/min), SF6 plasma [START_REF] Larsen | Study of the Roughness in a Photoresist Masked, Isotropic, SF6-Based ICP Silicon Etch[END_REF], plasmaless dry etching using XeF2, BrF3 or ClF3, or by realizing a backside deep-RIE (SF6-based RIE with passivation steps using C4F8, around 2000 nm/min). However, KOH rapidly etches aluminum (around 13000 nm/min) and etching selectivity of NH4F+HNO3 with silicon and SiO2 is not important. Moreover, as the lithography mask was initially designed for microlenses, no holes were made in the membranes to ease silicon etching. In our case we did not want to release microstructure by using a backside RIE in order to keep the energy harvester's cavity under vacuum. As a result, we abandoned the idea of etching silicon. As we explained in the previous part, photoresists presents the drawback of outgassing at high temperature and to not be compatible with SiO2 PECVD. Another solution, often applied to the fabrication of RF MEMS switches [START_REF] Rebeiz | RF MEMS[END_REF], is to use polyimides as a sacrificial layer, which can be easily etched by using O2 plasma with very good etching selectivity against SiO2 and aluminum. Using O2 plasma also present the advantage of preventing stiction issue that can happen during wet etching.

Fabrication process developed at STMicroelectronics

Now that we have presented the various constraints and requirements that have led the design of the new architecture of thermo-mechanically bistable actuator, we detail each step of its fabrication process. The complete fabrication process is presented in Fig. 3.5.1. A first lithography is realized using the microlenses' mask in order to define the position of the trenches (CT-3050 negative resist, coating at 800 rpm and softbake at 100 o C during 120 s, development with TMAH 90s, and hardbake at 3230 o C during 5 min). In order to validate the first fabrication step, etching tests are realized with (CF4, O2) RIE (Fig. 3.5.1b), and DRIE. (Fig. 3.5.1c). Given the greater consumption of the resist and the over-etching observed with RIE than with DRIE and the better aspect ratio obtained with DRIE, DRIE technique is chosen for the next wafers. A Ti-W seed layer is then sputtered in the cavity to improve the adhesion of copper layer. In order to obtain a plane surface, a chemical-mechanical planarization is performed on the wafers. As the quality and the duration of a CMP step depends on the quantity of materials to be planarized, two solutions are tested: a counter-mask with the microlenses mask is first deposited on the copper layer and a wet etching of the copper layer is performed in order to reduce the surface of copper to be planarized (step 5 of Fig. 3.5.1a). For other wafers, planarization is directly realized, without using a counter mask. Results of these two procedures are presented in Fig. 3.5.1d and Fig. 3.5.1e. The comparison of both methods reveals the presence of a copper protrusion of around 20 nm for the cavities located near the middle of the wafer and no protrusion for cavities near the edges of the wafer when no partial copper layer etching is realized before CMP. When a partial etching step is performed, we observe a better planarity near the center of the wafer and a light overetching at the periphery of the wafer. Moreover the curvature is reduced. In both case, near the edges of the cavities we observed an overetching of about 30 nm, which can be attributed to the scallops created by the DRIE and to the accumulation of slurries during the CMP. Yet it can be noted that the dimensions of protrusion and overteching are of tens nanometers whereas the cavity's length varies between 60 and 480 µm. Given the better results obtained when a partial etching is performed before CMP, this extra step is added to the fabrication process.

As we explained in the previous part, SiON is deposited by means of a PECVD step. The main parameters of these steps are: HF RF power: 245W LF RF load power: 120 W, SiH4: 150 sccm, N2O: 450 sccm N2: 1000 mTorr. Measurements of the wafers' curvature indicated for the first series of wafers (thickness of 1.5 µm) a residual stress of -97 MPa.

Aluminum is then deposited by PVD (thickness of 1.5 µm). Residual stress in Al layer is of 105 MPa. Al layer's etching is then realized using a RIE with a CL2 plasma and SiON layer's etching with a RIE with SF6, C4F8 (Pressure between 10 and 48mTorr, source power of 2500 Watts, platen power at 675 Watts). First attempts to selectively etch copper layer with respect to Al layer were made using sodium iodide (NaI), an etching solution generally used to etch gold layers similar to potassium iodide (KI) [START_REF] Green | Gold etching for microfabrication[END_REF], but which can also etch copper with a good selectivity against aluminum. This etching step enabled to fully etch copper layer in the trenches. Yet we observed by destructive methods that NaI-based etching reaction only partially etched copper layer located below SiON-Al stack (Fig. 3.5.1h). According to a CEA expert, this could be also linked to the electrochemical potential of Al layer during the etching that hinders the reaction the reaction between NaI and Cu. As a consequence, we tried other etching solutions at PTA clean room (CEA). The first solution was to use FeCl3, which a common etching recipe for copper (at room temperature Cu: 3900 nm/min, Al: 35 nm/min, no significant etching of SiO2 [START_REF] Williams | Etch rates for micromachining processingpart II[END_REF]). The second etching recipe is based on the use of (NH4)2S2O8 (at 30 o C Cu: 2500 nm/min, Al: 0,3 nm/min, no significant etching of SiO2 [START_REF] Williams | Etch rates for micromachining processingpart II[END_REF]). In both cases, we observe that structures were released (Fig. 3.5.1i). As a consequence, we will provide more informations on the measurements in the final manuscript.

Alternative fabrication process

In addition to the fabrication process we developed with TSV team, we developed an alternative fabrication process at CEA's experimental clean room. Instead of sputtering copper to form a sacrificial layer, we proposed to release microstructures by selectively etching silicon under microstructures with XeF2 (Fig. 3.7.1a). As a result, the fabrication process begins by the PECVD of a SiON layer and the PVD of an Al layer, and is followed by the etching of the Al-layer using a mixture of H3PO4 and HNO3. SiON layer is then etched using a CHF3-based RIE. Finally, silicon is etched using XeF2. Compared with the previous fabrication process, this one presents the advantage of being simpler, with fewer steps. The fact of using holes to ease releasing of the micro-structures plays also a role on the reduction of the residual stressMoreover, the masks designed for this fabrication process integrate more test structures for the measurement of residual stress in SiON and Al layers. As SiON layer's residual stress is expected to be compressive, we designed series of beams of various lengths so that their buckling allows locally determining the average value of the compressive residual stress. In order to determine stress gradient in SiON layer and in Al layer, we also designed cantilever beams whose curvature is directly proportional to this residual stress. By designing bimorph cantilever beams we can finally deduct the average tensile stress in Al layer. Another solution to determine this stress would have been to design Guckle rings on the lithography masks. As for the previous part, more details on these process and on structures' characterization will be provided later.

Conclusion

In this last chapter, we presented the work done on the fabrication of thermomechanically bistable actuators at the micro-scale. Three fabrication processes were presented. The first one adapted the microlenses' technology developed at STMicroelectronics for CMOS sensors to create initially curved membranes. Yet characterizations of bimetallic structures and bibliography work have shown the inability of this fabrication method to produce thermo-mechanically bistable structures operating with small thermal hysteresis and at low temperature (<200 o C). As a consequence, this fabrication process was finally abandoned and two new fabrication process were developed in order to exploit residual stress to fabricate initially buckled membranes. The first one, enabled to manufacture membranes exhibiting thermo-mechanical bistability even if few characterizations were made up to now. The second fabrication is in course of development in CEA's experimental clean room. Characterizations of microstructures are in progress to determine if there is an agreement between the experimental data and the predictive model.

Bibliography

Andersen, K., Carlson, K., Petersen, D., Mølhave, K., Eichhorn, V., Fatikow, S. and Bøggild, P. (2008). Electrothermal microgrippers for pick-and-place operations. Microelectronic Engineering, 85(5-6), pp.1128-1130.

General conclusion

The work presented in this PhD manuscript thesis dealt with the modeling and the silicon integration of miniaturized electrostatic bimetallic strip heat engines whose function is to generate usable electrical power from heat flux created by thermal gradients.

In a first time, this thesis presented an overview of the various physical coupling phenomena that can be used to transform heat into electrical energy: among these, we presented heat engines using thermoelectric materials, pyroelectric/ferroelectric materials, ferromagnetic materials, but SMA and bimetallic strip. This first study allowed to highlight the interest of exploiting phase transitions to maximize thermal coupling. Finally, we compared the performances of these different technologies and we showed that thermoelectric materials are the most efficient materials to transform heat into electricity, followed by ferroelectric and ferromagnetic materials, Shape Memory Alloys and then fluids and solid materials, these three last categories needing a second conversion step to convert mechanical energy into electrical energy.

In order to provide design rules for the system and to theoretically evaluate the performances of the bimetallic strip heat engines, great efforts were made to understand the origins of the thermo-mechanical instability of bimetallic beams. We explained how initial curvature (or residual stress) and thermal expansion modify the mechanical behavior of composite beams. Thanks to this work, we were able to model the thermal snap-through of composite beams, to find the conditions on the bimetallic strips geometry and on the choice of the materials which must be filled to observe the occurrence of any hysteretic behavior. This enabled to theoretically demonstrate the possibility to convert heat into mechanical energy by placing a bistable bimetallic strip between two heat sources at different temperature, but also to evaluate the energetical performances of these heat engines. We then coupled this mechanical model with a thermal model of the device's architecture in order to predict the dynamic performances of the generator.

In a second time, we proposed several models to explain the performances of the current generators that are based either on pyroelectric coupling, piezoelectric coupling and electrostratic coupling. For each device, we gave laws describing the evolution of the energetical performances, thus allowing a comparison of between each generator. These models finally showed that the best solution to convert heat into electrical energy is to laminate a bimetallic strip with a pyroelectric material in order to create a tertiary pyrolectric effect that enhances the performances of the heat engine for small values of the bimetallic strip's thermal hysteresis.

Finally, we applied these various models to the prediction of the performances of the miniaturized bimetallic strip heat engines. This allowed us to establish scaling rules of the performances of the generators when they are miniaturized down to the submillimetric scale, and to give design rules for the silicon integration of the electrostatic generators. From these scaling laws, we deduced that there are limitations to the miniaturization of the bimetallic strip heat engines. Main issues are: important reduction of the energy generated during a complete cycle, reduction of the output voltages of pyroelectric bimetallic strip heat engines, reduction of the pull-in voltage limiting the maximal value of the electret potential surfaces, small capacitances, reduction of the conversion efficiency due to the increase of the thermal leakage. However, it is possible to find other applications to thermo-mechanically bistable beams: for instance, bistable beams could be used as electro-thermal switches integrated in CMOS circuits.

In the last part of this thesis, we presented three fabrication process developed either in the 300-mm facilities of STMicroelectronics, Crolles, or in CEA's experimental clean room. The first fabrication process was an innovative solution based on the use of microlenses arrays. Yet we demonstrated that this type of fabrication process was unable to produce functional thermo-mechanically bistable beams. As a consequence, we developed two other fabrication processes exploiting residual stress in materials in order to buckle membrane and to control the thermo-mechanical behavior of these bistable membrane. Characterization of these structures is ongoing but we already qualitatively demonstrated the possibility to produce thermo-mechanically bistable membrane with these fabrication processes, opening the way to the development of bistable thermal actuators.

MODELING AND DESIGN OF THERMOMECHANICAL CONVERSION FOR THERMAL ENERGY HARVESTING APPLICATIONS RESUME

Le développement de systèmes de récupération d'énergie (ou energy harvesting systems en anglais) va de pair avec l'émergence de l'Internet des Objets et notamment la prolifération de réseaux de capteurs devant répondre aux besoins croissants en informations, que ce soit dans le domaine de l'industrie, de la sante, de la domotique ou de l'environnement qu'il soit urbain ou naturel. Les progrès réalisés ces dernières années dans le domaine des Technologies de l'Information et de la Communication ont permis de lever certains verrous technologiques au développement de ces réseaux de capteurs intelligents et autonomes, notamment grâce a l'amélioration des performances intrinsèques des composants microélectroniques (vitesse, consommation), la conception de circuits plus économes en énergie, ou bien la mise en place de standards de communications radio adaptes a ces contraintes énergies. Etant donné l'ubiquité des sources d'énergie, la fabrication de générateurs permettant d'alimenter directement ces capteurs à partir de ces sources représente une alternative viable à l'utilisation de batteries pour prolonger la durée de vie de ces capteurs communicants. Diverses technologies de générateurs ont ainsi été proposes pour s'adapter aux différentes formes que peut prendre l'énergie, qu'elle soit d'origine thermique, mécanique, solaire ou électromagnétique.

Le présent travail est une contribution au développement de certains dispositifs de récupération thermiques basés sur l'exploitation des propriétés thermiques et mécaniques de bilames thermostatique. Ce type de générateurs, proposé et développé au sein de STMicroelectronics à Crolles, se veut être une alternative fiable et bas cout a l'utilisation de matériaux thermoélectriques exploitant l'effet Seebeck pour générer de l'énergie électrique. Divers dispositifs ont déjà été fabriqués, démontrant la capacité des moteurs thermiques à base de bilames thermostatiques à alimenter des capteurs autonomes en fonctionnement synchrone et asynchrone. L'objectif de cette thèse est alors de démontrer la possibilité de miniaturiser de tels moteurs thermiques grâce aux techniques de fabrications utilisées en microélectronique. Afin de garantir le fonctionnement de ces systèmes a micro-échelle, un important travail de fond a d'abord été effectue sur la compréhension et la modélisation des phénomènes de couplages thermomécaniques a l'origine du comportement bistable des membranes bimétalliques. Ce travail a débouché sur la démonstration théorique du fonctionnement des moteurs thermiques a base de bilames et sur l'évaluation de leur performances énergétiques (énergie disponible, efficacité thermique, efficacité de Carnot relative). Dans la continuité de ce premier modèle, d'autres travaux ont été menés pour évaluer les performances de moteurs thermiques exploitant différents phénomènes de couplage électromécanique en vue de convertir l'énergie mécanique générée par les bilames thermostatiques en énergie électrique exploitable par les capteurs autonomes. La simulation du comportement des micro-générateurs à l'aide de ces divers modèles a débouché sur des lois d'échelles sur les performances des moteurs thermiques. Finalement, divers procédés de fabrications ont été développés pour permettre la fabrication de microstructures thermiquement bistables !

SUMMARY

The development of energy harvesting systems is linked to the emergence of the Internet of Things (IoT) and especially the proliferation of Wireless Sensors Networks that should respond to the growing needs for monitoring data in domains as diverse as the industry, the urban or natural environments, the home, or the human body etc. Recent progress in the field of information technologies have enabled to remove some of the technical obstables to the deployment of these smart and autonom devices, in particular thanks to the improvement of the performances of microelectronic components, the design of ultra-low-power circuits, or the creation of wireless communications standards adapted to the energy needs of wireless sensors. Given the great availability of energy sources, energy harvesters are reliable alternatives to batteries in order to extend the autonomy of these sensors. Various technologies of generators have been developped to adapt to the type of local energy sources (heat, vibration, light, radio-frequencies).

The present work is a contribution to the development of thermal energy harvesters exploiting the thermal and mechanical properties of bimetal thermostats. This type of technology developped at STMicroelectronics are intended to be a reliable and low-cost alternative to the use of thermoelectric materials exploing Seebeck effect to generate electricity from heat. Various devices were already fabricated at the macro-scale, demonstrating their ability to power wireless sensor nodes. In the continuity of these works, this PhD thesis aims to demonstrate the operation of these generators at the sub-millimetric scale. As a consequence, an important work on the modeling of the thermo-mechanical instability of bimetallic strips was made to understand the operation of bimetallic strip heat engines. This work enabled to theoretically demonstrate the capability of bimetallic to transform heat into mechanical energy and to evaluate the performances of such heat engines. Coupling between bimetallic strip heat engines and electro-mechanical transducers was also modeled to compare the performances of the current prototypes of generators. We then modeled the thermo-mechanical behavior of composite beams at the microscale and established scaling rules of the performances of the bimetallic strip heat engines, We finally developped microlectronic fabrication process to manufacture thermo-mechanically bistable beams at the microscale !

  . Optimization of the performances of thermoelectric generators 4. Pyroelectric generators 4.1. Simple pyroelectric generators Content ii 4.2. Exploiting phase transitions of ferroelectric materials 5. Fluid-based heat engines 5.1. Downsizing of conventional heat engines 5.2. Heat engines using stationary fluids 6. Solid-based heat engines 6.1. Observations on phase transitions 6.2. Shape-memory-alloy-based heat engines 6.2. Energy harvesting from thermo-mechanical bistability 7Plan of the chapter 2. A first model of the instability of composite beams 2.1. A short couse on the theory of elasticity 2.2. Equilibrium of composite beams suffering from residual stress 2.3. Criterion of bistability 2.4. Simplification of the equations for a bimetallic beam 2.5. Equilibrium of composite beams with an initial curvature 2.6. Curvature and initial stress 2.7. Influence of the architecture 3. Performances of the harvester using composite beams 3.1. Evolution of the strain energy of the composite beams 3.2. Evolution of the entropy of the composite beams 3.3. Intrinsic mechanical performances of the harvester 3.3. Intrinsic mechanical performances of the harvester Content iii 4. A second model of the instability of composite beams 4.1. Ritz approximation, Galerkin method, Cardan formula 4.2. Simplification of the model 4.3. Mechanical performances of the energy harvester 4.4. Experimental studies and validity of the reduced-order model 4.5. Figures of Merit 4.6. Study of the impact of the architecture on the performances 4.7. Synthesis on the second model 5. Thermal transfers and mechanical instability 5.1. Equations of the thermal transfers 5.2. Switched thermal capacitance model 5.3. Leakage paths and heat sink 5.4. Thermal transfers inside the bimetallic strip of the three cycles and enhancement factors e. Comparing piezoelectric materials 4.3. Influence of the architecture, extrinsic performances Content iv of the energy harvester a. Stirling cyles b. Dynamic performances of the harvester for fixed external temperature and no architectural leakage c. Dynamic performances of the harvester with Robin conditions and thermal conducting architecture 5. Coupling with an oscillating piezoelectric transducer 5.1. A short review of energy harvesting devices based on mechanical shock 130 5.2. Modeling the shock between two oscillators a. Mason model of the piezoelectric oscillator b. Modeling of the shock c. Optimal mechanical energy 5.3. Electrical performances of the system a. Open circuit voltage b. Energy harvested with a SECE circuit c. Overall performances of the energy harvester d. Ways to improve the efficiency of the transduction 6. Coupling with an electret transducer 6.1. A short overview on electrets and electrostatic energy harvesting a. A history of electrets b. Types of electret and production c. Energy harvesters exploiting electrets 6.2. Modeling the electret a. Electric field and electrostatic pressure induced by an electret with a charge layer b. Electric equations of the thermally bistable electrode 6.3. Mechanical equations of the bistable bimetallic electrode a. Equilibrium of bistable beams under electrostatic pressure b. Moderate electrostatic coupling 6.4. Harvesting heat with electret transduction a. Possible thermodynamic cycles b. Intrinsic performances of the bimetallic strip heat engine c. Dynamic performances of the heat engine d. Effect of the moderate electrostatic coupling Content v on the performances of the heat engine 7. Synthesis a. Design rules common to all harvesters b. Design rules for pyroelectric bimetallic strip heat engines c. Design rules for piezoelectric bimetallic strip heat engines d. Design rules for pelectrostaticbimetallic strip heat engines e. Comparison of the various heat engines of the materials used in microelectronics d. Impact of the variability of the residual stress on the performances of the bistable beams 3. Pyroelectric coupling 3.1. Equations of the piezoelectricity 3.2. Performances of the Stirling cycle 3.3. Dynamic behavior of the bimetallic strip at the microscale a. Scaling law on the operation frequency b. Simplified model of the harvester at the microscale c. Performances of a bimetallic strip in a cavity 4. Electrostatic coupling 4.1. Performances in weak coupling 4.2. Performances in moderate coupling 5. Synthesis a. Synthesis on the mechanical properties of the bimetallic strip Content vi heat engine b. Synthesis on the dynamic behavior of the system c. Synthesis on the pyroelectric bimetallic strip heat engines d. Synthesis on the electrostatic bimetallic strip heat engines e. Aging of the device Short review on bistable thermal actuators 3.2. Design of new lithography masks 3.3. Design rules of bimetallic strips 3.4. Possible strategies for the development of the second process 3.5. Fabrication process developed at STMicroelectronics 3

  a. Some applications of energy harvesting devices and Wireless Sensors

Figure 2 . 1 . 1 .

 211 Figure 2.1.1. Schematic vibration harvesters: a. piezoelectric, b. electrostatic, c. electromagnetic.

  Simplified representation of: a. Thermoelectric PN junction with bulk heat sink for thermal energy harvesting. b. Solar cell for indoor/outdoor photovoltaic energy harvesting.

Figure 3

 3 Figure 3.1. a. Principle of the bimetallic strip heat engine. b. Image of Invar-NC4 bimetallic strip fabricated by DeltaConcept and used in the piezoelectric generator. c. Architecture of the piezoelectric bimetallic strip heat engine.

  1.1. a. Representation of the Carnot heat engine. b. Efficiency of the Carnot heat engine as a function of the hot source's temperature.

  Figure 2.1.2. a. Representation of the Curzon-Ahlborn heat engine. b. Evolution of the normalized power against the intrinsic Carnot efficiency for various values of the ratio TC/TH. c. Comparison of the Curzon-Ahlborn efficiency and Carnot efficiency for various values of TC/TH. d. Surface density of output power generated by a heat engine working with a bulky heat sink or with a planar heat sink.

  a. Representation of the thermoelectric generator. b. Evolution of the thermal efficiency of two thermoelectric generators using either SiGe-based thermocouples or Bi2Te3-based thermocouples (cold source at room temperature

  a. Architecture of thermoelectric generator presented in

  . d. Technologies developed by Micropelt [Micropelt]. d. Wireless sensing node developed by Laird [Laird].

  1.1. a. Performances of a thermoelectric generator as a function of the temperature and cycling frequency of a pyroelectric generator. b. Comparison between the efficiencies of a 200-µm thick PMN-PT layer and a thermoelectric generator working with a planar heat sink.

  a. Representation of the Rankine cycle. b. Simplified architecture of a miniaturized Rankine generator. c. Rotor of a Rankine generator presented in

Figure 5

 5 Figure 5.2.1. a. Simplified schematic of a P 3 heat engine. b. Image of the thermal generator fabricated by[START_REF] Whalen | Design, fabrication and testing of the P3 micro heat engine[END_REF]. c. Carnot cycle assumed in[START_REF] Lurie | Analysis of a microscale 'Saturation Phase-change Internal Carnot Engine[END_REF]. d. Cycle experimentally observed and reported in[START_REF] Bardaweel | Characterization of the thermodynamic cycle of a MEMS-based external combustion resonant engine[END_REF].

  a. Natural cycle of Shape-Memory-Alloy heat engine b. Architecture of the SMA-based generator proposed in

  a. Representation of a basic thermomagnetic heat engine (temperature fluctuations). b. Principle of the thermomagnetic heat engine presented in

Figure 8

 8 Figure 8.1.1. a. Comparison of the ZT of various heat engines with respect to the mean operation temperature. b. Density of energy versus temperature difference needed to produce it

  a. Illustration of the buckling of a straight beam. b. Evolution of the compressive load p as a function of the strain ε. c. Evolution of the beam's deflection and pitchfork bifurcation of the straight beam.

  a. Representation of the first buckling load of a simply-supported beam axially loaded. b. Evolution of the buckling coefficient as a function of the shape ratio a/b for n=1.

Fig. 1

 1  the beam deflection amplitude against the beam's shortening and axial load, for various values of the imperfection ao/t. Fig.1.2.1b shows that the beam thus curves in the direction fixed by its initial imperfection before reaching the Euler load. It is particularly interesting to note the existence of an assymetric secondary equilibrium branch for high values of p. This branch shows that the beam can pass from one side to the other by a saddle-node bifurcation when it reaches the values represented by the gray dashed-line. The equilibrium positions located between the two gray dashed lines are unstable. These same curves can be obtained when studying the thermal buckling of

Figure 1 .

 1 Figure 1.2.1. a. Imperfect simply-supported beam. b. Relative beam deflection amplitude vs. normalized beam shortening for various values of imperfection ao/t with a ratio t/L=5%. c. Normalized axial load vs. beam amplitude. d. Normalized axial load vs. normalized beam shortening.

Figure 1 .

 1 Figure 1.3.1. a. Arch submitted to the effect of a static load. b. Equilibrium path of the arch. c. Stability of the equilibrium positions. d. Bimetallic beam studied by Timoshenko. e. Equilibrium of bimetallic beam according to Timoshenko for an Al-Invar beam (L=40mm, t=0.1mm). Snap and Snap-Back temperatures of the beam as a function of its thickness.

  a. Representation of the strain tensor components. b. Representation of the link between the displacement fields and the strain tensor.

  a. Representation of the composite beam studied in this chapter b. Simplification in the case of a bimetallic beam made of an active layer and a passive layer (α2>α1). 𝑀𝑀𝑀𝑀 ! = 𝑣𝑣𝑣𝑣. 𝐸𝐸𝐸𝐸(𝑧𝑧𝑧𝑧). 𝛼𝛼𝛼𝛼(𝑧𝑧𝑧𝑧). (𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! ).

  2.3.1, the equilibrium paths predicted by (2.3.6a) are plotted for various values of the normalized force No/NEuler for an Al-Invar beam (L = 40 mm, t = 150 + 150 µm) and for To=100°C. Fig.2.3.1a is the typical figure associated to bistability: it represents the relative displacement of the beam's middle do/L as a function of the temperature T/To. Fig.2.3.1b and Fig.2.3.1c respectively represent the evolution of the normalized axial

Figure 2 .

 2 Figure 2.3.1. a. Equilibrium paths in the (do,T) plane for various values of No/NEuler. b. Equilibrium paths in the plane (pt,T). c. Equilibrium paths in the plane (λ,T). d. Beam's shape for No/NEuler =1.1. e-f-g-h. Properties of an Al-Invar beam.

  Fig.2.3.1e shows the evolution of the critical load as a function of the normalized force Po/pE for an Al-Invar beam (L = 40 mm, t = 400 + 400 µm). The evolution of the snap and snap-back temperatures for the same beam is represented in Fig.2.3.1f, and the displacement of the beam in Fig.2.3.1g. Fig.2.3.1h plots the dependence of the beam amplitude on the thermal hysteresis.

Figure 2

 2 Figure 2.4.1. Dependence of the hysteresis temperature of an Al-Invar beam: a. on the tensile stress in the Al layer. b. on the compressive stress in the Invar layer.

Figure 2 .

 2 Figure 2.6.1. a. Equivalence between initial curvature and residual stress. b-c. Equilibrium path for various values of initial deflection. d. Evolution of the critical charges of the beam as a function of its initial curvature. e. Critical thrust of the beam against initial curvature. f. Snap and snap-back temperatures according to Timoshenko's model and the current model, for an initial curvature of 300 µm. g. Comparison of the criteria of bistability of both models. h. Relative error of Timoshenko's model.

Figure 2 .

 2 Figure 2.8.1. a. Schematics of the beam in the cavity. b. Comparison between the thermal hysteresis of a free beam and the thermal hysteresis of a beam in a cavity, for various ratios of the cavity's thickness-to-deflexion. C. Evolution of the forces developed by the cavity on the beam. d. Evolution of the clamp thrust developed by the beam.

Fig. 2

 2 Fig.2.8.1b. represents the equilibrium paths of an Al-Invar beam (L = 40 mm, t = 400 + 400 µm) for various values of the cavity's thickness and for No/NEuler=1.28. Here 100% represents the ratio between the cavity's height and the maximal allowed cavity's height given by (2.8.1). This figure clearly shows that, by reducing the cavity's thickness, the thermal hysteresis is modified and decreases. This phenomenon was numerically shown in[Ravindran et al, 2014] but without physical explanations. Fig.2.8.1c represents the evolution of the force developed by the architecture on the beam to reduce its curvature. Typical values of the force are around 1 N during the snap and 0.3 N at snap-back, in agreement with the experimental data measured on the harvester prototypes. The force is greater during the snap than at snap-back because of the asymmetry of the thermal hysteresis due to the effect of the beam's mean thermal expansion. This figure can be compared with Fig.2.8.1d representing the evolution of the axial thrust.

  Fig.2.8.1b. represents the equilibrium paths of an Al-Invar beam (L = 40 mm, t = 400 + 400 µm) for various values of the cavity's thickness and for No/NEuler=1.28. Here 100% represents the ratio between the cavity's height and the maximal allowed cavity's height given by (2.8.1). This figure clearly shows that, by reducing the cavity's thickness, the thermal hysteresis is modified and decreases. This phenomenon was numerically shown in[Ravindran et al, 2014] but without physical explanations. Fig.2.8.1c represents the evolution of the force developed by the architecture on the beam to reduce its curvature. Typical values of the force are around 1 N during the snap and 0.3 N at snap-back, in agreement with the experimental data measured on the harvester prototypes. The force is greater during the snap than at snap-back because of the asymmetry of the thermal hysteresis due to the effect of the beam's mean thermal expansion. This figure can be compared with Fig.2.8.1d representing the evolution of the axial thrust.

  Figure 3.1.1. a. Evolution of the strain energy of a bistable beam (in black) and part of the energy due to the thermal stress in blue. b. Thermodynamic cycle of the heat engine in the (do,𝒲𝒲𝒲𝒲 ! ) diagram. c. Thermodynamic cycle of the heat engine in the (T,WM) diagram.

  a. Evolution of the beam's entropy as a function of temperature. b. Thermodynamic cycle of the heat engine and comparison with the Carnot cycle. c. Illustration of the four phases of the heat engine's cycle. d. Details of the energetic transfers inside the harvester.

  Fig.3.3.2c shows that the energy density is directly correlated to the thermal stress asymmetry m α : this density is highest for W/Al bimetals. By normalizing the surface density by the total deflection, we obtain a volume energy density independent of the size of the beam if the beam is proportionally scaled down. We can compare the values of surface densities with those calculated with the data reported in[Ravindran et al, 2014] and[START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF].[Ravindran et al, 2014] gave a simulated density of around 83 µJ.cm-2 (200 µJ; 40x6x0.28 mm 3 ; 5 K hysteresis) while[START_REF] Puscasu | Dispositifs innovants pour la récupération de l'énergie thermique[END_REF] reports experimental mechanical energy densities of around 60 µJ.cm -2 (Invar/NC4; 36x18x0.3 mm 3 ; 3 K hysteresis). This first comparision validates the coherence of our mode (in Section 4.4 we go further in comparing our model with experiments). Table3.3.1 presents the Carnot efficiency of the nine bimetals

Figure 3

 3 Figure 3.3.2. a. Total deflexion during the snap of the beam. b. Residual stress in the layers of the bimetallic beams. c. Density of mechanical energy provided by the bimetals. d. Theoretical Carnot efficiency of the heat engines.

  Figure 4.1.1. a. Comparisons between the exact model, the model with Cardan's formula and the reduced model. b. Evolution of the Al-Invar beam's deflection as a function of the thermal hysteresis predicted by the three models.

Figure 4 Figure 4

 44 Figure 4.3.1. a. Total deflexion during the snap of the beam. b. Residual stress in the layers of the bimetallic beams. c. Density of mechanical energy provided by the bimetals. d. Carnot efficiency of the heat engines

  (4.4.2d), the thermal efficiency of the transduction by knowing the heat capacitance of the bimetallic shell, and the Carnot efficiency of the transduction (Fig.4.4.1e) if the hot source is assumed to be equal to the snap temperature of the bimetal a. Image of the bimetallic shells characterized in this part. b. Schematized hypothesis for the calculation of the kinetic energy. c-e. Comparison between experimental data and the reduced model.

  Figure 4.5.1. Figures of Merit for sixteen different bimetals.

Figure 4

 4 Figure 4.6.1. a. Thermal hysteresis of the beam in the cavity as a function of its true hysteresis and for various values of the ratio rw. b. Evolution of the force generated by the architecture of the harvester as a function of the hysteresis of the beam. c. Comparison of the available energies produced by the harvester for various value of the cavity's thicknessto-deflexion ratio. d. Evolution of the Carnot efficiency of the harvester.

Figure 5

 5 Figure 5.2.1. a. Sheme of the harvester and thermal transfers. b. Equivalent lumped capacitance model of the harvester. c. Simplified representation of the evolution of some parameters of the harvester in the frame of the lumped capacitance model.

  a. Evolution of the frequency of an Al-Invar beam as a function of the ratio of the thermal gradients and for various values of the contact resistance RCC. b. Evolution of the thermal flux as a function of the ratio of the thermal gradients and for various resistance's values and for TH -TC = 30 K. c. Evolution of the output power for various values of RCC and of the cavity's size factor rw. d. Evolution of the thermal efficiency for Rcc = 10 K.W -1 and various values of the cavity's size factor. e. Carnot efficiency of the harvester in the same conditions. f. Surface power density against Carnot efficiency in the same conditions.

  Figure 5.2.3. a. Evolution of the optimal bimetal's thermal hysteresis as a function of the cavity thinning and the thermal gradient. b. Evolution of the maximal output power as a function of the heat sources' temperatures. c. Evolution of the optimized thermal efficiency as a function of the heat sources' temperatures. d. Evolution of the Curzon-Ahlborn efficiency of the device.

Finally

  we compare the performances of bimetallic beams with a fixed thermal hysteris value of 3 o C, with or without heat sink. The first curves in Fig.5.3.2a give the values of the snap and snap-back temperatures of the beams as a function of the hot source temperature. As an example, for a hot source at 100 o C, without any heat sink (plane heat sink in fact), the best bimetal with a hysteresis of 3 o C must snap at 83 o C and snap back at 80 o C. Experimentally, 80 o C-77 o C bimetals are used in the current harvester. When a heat sink is used, these values are lower because of the increased thermal gradient inside the cavity (65 o C-68 o C). At this temperature, the first harvester works at 0.22 Hz (in agreement with experimental data), whereas the heat sink boosts the frequency up to 0.41 Hz (Fig.5.3.2b). Fig.5.3.2c shows the evolution of the output power generated by the 3 o Chysteresis beam with and without heat sink. The presence of a heat sink almost doubles the surface density of power generated by the harvester whatever the hot source temperature (from 9.3 to 17.3 µW.cm -2 ).

Figure 5 .

 5 Figure 5.3.1. a. Optimal value of the thermal hysteresis as a function of the hot source temperature. b. Surface density of output power of the generator against hot source temperature. c. Thermal efficiency of the system. d. Curzon-Ahlborn efficiency of the device.

  𝑣𝑣𝑣𝑣. 𝐸𝐸𝐸𝐸 𝑧𝑧𝑧𝑧 . 𝛼𝛼𝛼𝛼 𝑧𝑧𝑧𝑧 . 𝑇𝑇𝑇𝑇 ! (𝑧𝑧𝑧𝑧). 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ! ! (5.4.1b) 𝑀𝑀𝑀𝑀 ! = 𝑣𝑣𝑣𝑣. 𝐸𝐸𝐸𝐸 𝑧𝑧𝑧𝑧 . 𝛼𝛼𝛼𝛼 𝑧𝑧𝑧𝑧 . 𝑇𝑇𝑇𝑇 ! 𝑧𝑧𝑧𝑧 . (𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! ). 𝛿𝛿𝛿𝛿𝛿𝛿𝛿𝛿 ! ! (5.4.1c)

  . 𝑤𝑤𝑤𝑤 ,!! + 𝐼𝐼𝐼𝐼 ! . 𝑤𝑤𝑤𝑤 ,!!!! = 𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 !,!!(5.4.2c)

  Figures of Merit taking into account the thermal diffusive properties of materials: FOMf is directly equal to the thermal diffusivity of the beam and is the Figure of Merit of the device's operating frequency (5.4.13c), FOMP the Figure of Merit of the maximal output power (5.4.13d), and FOMη the Figure of Merit of the efficiency associated to the optimal thermal gradient (5.4.13e).

  Figure 5.4.1. a. Figure of merit of the output power versus thermal diffusivity. b. Figure of merit of the output power vs thermal efficiency.

  a. Piezoelectric bimetallic strip heat engine. b. Coupling between a piezoelectric transducer and a bimetallic strip heat engine. c. Coupling between a bimetallic strip and a 3-3 piezoelectric stack. d. Coupling between a bimetallic strip and an electret transducer.

Figure 2 . 2 . 2 .

 222 Figure 2.2.2. Matrix representation of the thermal, mechanical and electrical properties of solid materials, and matrix and lattices associated to Perovskites and Wurtzites.

  Figure 3.1.1. Schematic of the beam with the piezoelectric layer deposited a. on the active layer. b. on the passive layer.

Figure 4 .

 4 Figure 4.1.1. a. Representation of the Stirling cycle in the plan (E,D). b. Representation of the Stirling cycle in the plan (S,T)

Figure 4

 4 Figure 4.2.1. a. Principle of the first Stirling cycle. b. Stirling cycles of the heat engine for a thermal hysteresis of 3 o C in the (Q,V) plane, when the piezoelectric layer is deposited on the active layer or on the passive layer.

Fig. 4

 4 Fig.4.2.2a and Fig.4.2.2b respectively represent the evolutions of the maximal output voltage and the electrical energy generated by the device as functions of the beam's thermal hysteresis for a beam made of Aluminum, Invar and PZT-5H (properties given in Tab.4.2.1). These figures first show that for weak thermal hystereses, the tertiary pyroelectric effect is predominant over the primary pyroelectric effect, explaining why the bimetallic strip heat engine helps harvesting more energy than a simple PZT layer used as pyroelectric generator. As an example, for a hysteresis of 5 o C, the pyroelectric bimetallic strip heat engine produces 54 µJ instead of 5 µJ for the pyroelectric generator. However, when the piezoelectric layer is on the passive layer of the bimetallic strip, the piezoelectric charges and the pyroelectric charges neutralize each other for a thermal hysteresis of 13 o C. For wider hystereses, the charges are mainly generated by the pyroelectric effect.

Figure 4 .

 4 Figure 4.2.2. a. Evolution of the output voltage generated by the bimetallic strip heat engine. b. Evolution of the available electrical energy per cycle.

Figure 4 .

 4 Figure 4.2.3. a. Principle of the second Stirling cycle. b. Stirling cycles of the heat engine for a thermal hysteresis of 3 o C in the (Q,V) plane when the piezoelectric layer is deposited on the active layer or on the passive layer. c. Output voltage as a function of thermal hysteresis. d. Available energy vs. thermal hysteresis.

Figure 4 .

 4 Figure 4.2.4. a. Principle of the third Stirling cycle. b. Stirling cycles of the heat engine for a thermal hysteresis of 3 o C in the (Q,V) plane when the piezoelectric layer is deposited on the active layer or on the passive layer. c. Output voltage as a function of the thermal hysteresis. d. Available energy vs thermal hysteresis

  a. Comparison of the electrical energy produced by each cycle. b. Comparison Carnot efficiency of each cycle.

  a. Evolution of the improvement factor of the Figure of Merit of the available energy for the second Stirling cycle. b. Improvement factor of the electro-thermal coupling factor. a. Comparison of the Figures of Merit of the pyroelectric bimetallic strip heat engine and the pyroelectric generators using various pyroelectric materials. b. Comparison of the electro-thermal coupling factor of the pyroelectric materials using the two types of heat engine.

Figure 4

 4 Figure 4.3.1. a. Schematic of the harvester. b. Cavity thickness as a function of the intrinsic thermal hysteresis of the beam (thermal hysteresis not modified by the mechanical action of the cavity on the beam).

Figure 4 .

 4 Figure 4.3.2. a. Comparison of the output energy per cycle when the piezoelectric layer is deposited on the active layer with rw=1. b. Comparison of the output energy per cycle when the piezoelectric layer is deposited on the passive layer. c. Evolution of the Figure of Merit as a function of the beam's thermal hysteresis and the cavity's thinning ratio and comparison with the intrinsic figure of Merit of the PBSHE and the pyroelectric generator. d. Evolution of the Carnot efficiency of the device.

Figure 4

 4 Figure 4.3.3. a. Carnot efficiency of the harvester with various pyroelectric materials, assuming a mean temperature of 100 o C. b. Equivalent ZT of various pyroelectric materials.

  Fig.4.3.5a then represents the evolution of the output power generated by a PZT-Al-Invar beam as a function of the ratio x and for various values of the thermal gradient across the harvester. In the same time, to complete the description of the dynamic performances of the system, Fig.4.3.5b draws the evolution of the working frequency of the prime mover as a function of the ratio x. The first figure shows that for very small values of x (hystereses smaller than 2 o C for an external temperature of 30 o C), the output power is a decreasing function of the ratio x. However, it rapidly increases for values of x between 0.1 and 0.8. This means that the increase of the electrical energy per cycle for highly unstable beams outweighs the reduction of the working frequency for large values of the beam's thermal hysteresis. After reaching an optimal value, the output power decreases when the beam's thermal hysteresis in the cavity becomes closer to the external temperature difference. The optimal value of the beam's thermal hysteresis is 21 o C for a ratio rw of 1 and a temperature difference of 30 o C. For this hysteresis value, the output power is of 25.7 µW for a working frequency of 0.045 Hz. For an external temperature of 15 o C, the optimal hysteresis value is of 9.6 o C, producing 8.9 µW.

  a. Carnot efficiency of the harvester with various pyroelectric materials, assuming a mean temperature of 100 o C. b. Equivalent ZT of various pyroelectric materials.

  Figure 4.3.5. a. Output power generated by the PZT-Al-Invar beam as a function of the ratio x, for three values of the cavity's thickness ratio and two values of the external temperature difference. b. Working frequency of the device. c. Optimal values of x as a function of the parameter βpyro. d. Choice of the hysteresis of the bistable beam as a function of external temperatures. e. Maximal power generated by the harvester when the thermal hysteresis is optimized. f. Carnot efficiency of the optimized system assuming that the meam temperature of the two heat sources is of 100 o C.

  a. Maximal output power generated by the PZT-Al-Invar beam as a function of the hot source temperature and for an ambient temperature of 25 o C. b.

  a. Architecture of the CC oscillator. b. Architecture of the SS oscillator. c. Architecture of the CF oscillator. d. Shape of the potential energy of both oscillators. e. Mason's model of the coupled oscillators.

Figure 5 .

 5 Figure 5.2.2. a. Evolution of the final equilibrium position as a function of the cavity's thickness ratio and the oscillators' stiffness ratio. b. Efficiency of the shock as a function of the oscillators' stiffnesses. c. Surface density of mechanical energy for the shock between a bistable Al-Invar beam and a piezoelectric oscillator. d. Thermo-mechanical efficiency of the mechanical transduction.

  a. Mechanical energy released by the bimetallic strip to a clamped-clamped piezoelectric transducer as a function of the hot source temperature, for rw=1 and Tamb=25 o C. b. Thermal efficiency of the transduction. c. Curzon-Ahborn efficiency of the transduction. d. Evolution of the snap and snap-back temperatures of the optimal bimetallic strip as a function of the hot source temperature.

  𝑤𝑤𝑤𝑤 ! 𝑡𝑡𝑡𝑡 = 𝑤𝑤𝑤𝑤 ! + 𝑤𝑤𝑤𝑤 ! . 𝑒𝑒𝑒𝑒 !!.! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(𝜔𝜔𝜔𝜔. 𝑡𝑡𝑡𝑡 + 𝜃𝜃𝜃𝜃, 𝑘𝑘𝑘𝑘 ! ) (5.3.3a) 𝑤𝑤𝑤𝑤 ! . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃𝜃𝜃, 𝑘𝑘𝑘𝑘 ! = 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! . 𝑤𝑤𝑤𝑤 ! ! = 𝐵𝐵𝐵𝐵 (5.3.3b) 𝑤𝑤𝑤𝑤 ! . 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃𝜃𝜃, 𝑘𝑘𝑘𝑘 ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝜃𝜃𝜃𝜃, 𝑘𝑘𝑘𝑘 ! = -1 𝜔𝜔𝜔𝜔 . (𝑣𝑣𝑣𝑣 ! + 𝐵𝐵𝐵𝐵. 𝛿𝛿𝛿𝛿) = -𝐴𝐴𝐴𝐴 𝜔𝜔𝜔𝜔

  Figure 5.3.1. Phase portrait, Open-Circuit voltage and displacementof a clampedclamped oscillator (a), simply supported oscillator (b) and two clamped free oscillators (c).

  5.3.2a and Fig.5.3.2b the transient evolution of the transducer deflection and the output voltage generated by the piezoelectric oscillator, and in Fig.5.3.2c the Fast Fourier Transform of each signal. Here we assume an Al-Invar bimetallic strip (40 x 10 x (0.2+0.2) mm 3 ) impacting a clampedclamped PZT-brass oscillator (40x 10x (0.2+0.1) mm 3 ), and a quality factor of 30 (see Table5.3.1a). We can see inFig.5.3.2a and Fig.5.3.2b the relatively good agreement between the three simulations. Fig.5.3.2c shows that the oscillation frequency is around 590 Hz, and that the energy stored in the capacity is 200 µJ assuming a load resistance of 10 MOhms a. Detail of the transient evolution of the clamped-clamped PZT brass piezoelectric transducer when it is shocked by a bimetallic strip of 3 o C. b. Detail of the transient evolution of the open-circuit output voltage of the piezoelectric transducer. c. FFT of the three signals and equivalent energy when using a load resistance of 10 MOhms. d. Evolution of the output voltage as a function of the bimetallic strip thermal hysteresis and the type of piezoelectric transducer (clamped-clamped, simply supported, clamped-free).

  a. Evolution of the available electrical energy per cycle as a function of the type of piezoelectric transducer, the mechanical quality factor of the system and the thermal hysteresis of the bistable beam. b. Transduction efficiency associated to the shock. c. Equivalent ZT of the harvester. d. Surface density of power of the bimetallic strip heat engine (clamped-clamped transducer) as a function of the hot source temperature and for various values of thermal hysteresis. e. Thermal efficiency of the energy harvester as a function of the hot source temperature. f. Curzon-Ahlborn efficiency of the system.

  a. Map of the efficiency of the piezoelectric transduction and efficiency of the PZT-Brass oscillator for some values of mechanical quality factor. b. Dependence of the efficiency on the equivalent mechanical quality factor, and way of improvement of the efficiency of the PZT-Brass oscillator. c. Image of the non-optimized energy harvester and experimental measurement of the snap signal with a load resistance of 10 MOhms. d.

  a. Modeling of the mechanical energy losses in the current piezoelectric oscillator. b. Repartition of the stress in the current piezoelectric oscillator. c. Repartition of the stress in the optimized oscillator. d. Top view of the oscillator and optimized parameters. e. Evolution of the efficiency as a function of the angle of the beams. f. Evolution of the efficiency as a function of the length of the beams.

  a. Fabrication of thermoelectrets. b. Fabrication of coronaelectrets c. fabrication of radioelectrets. d. Dipolar electret with surface charge. e. Space-charge electret with two domains (also known as Maxwell-Wagner effect). f. Charge excess electret.

  a'. Principle of overlapping-area change topology. a''. Textured stator of a rotational energy harvester

  a. Simplified schematic thermal energy harvester. b. Detail of a portion dx.

Figure 6

 6 Figure 6.3.1. a. Modification of the thermal hysteresis of a beam in open circuit as a function of the electrical charges carried by the bimetallic strip. b. Improvement factor of the beam's thermal hysteresis due to the movable electrode's charge.

Figure 6

 6 Figure 6.4.1 a. Thermodynamic cycles of an electrostatic energy harvester. b. Equivalent circuit of the electret energy harvester. c. Natural cycle of an electret energy harvester[START_REF] Nakano | MEMS Rotational Electret Energy Harvester for Human Motion[END_REF].

  Figure 6.4.2. a. Evolution of the snap snap-back voltages as a function of the beam's thermal hysteresis (assuming a ratio rw = 1) and for various values of the surface potential of a Cytop electret. b. Evolution of the electrical energy per cycle as a function of the thermal hysteresis. c. Thermal efficiency of the energy harvester. d. Evolution of the Carnot efficiency of the device as a function of the thermal hysteresis.

  a. Mean power generated with a bimetallic strip placed in a cavity where a temperature difference of 30 o C is maintained. b. Evolution of the mean power generated by three different bimetallic strips (hysteresis of 1 o C, 5 o C, 10 o C) as a function of the hot source temperature (ambient temperature 25 o C) and the type of heat sink (heat sink with fins represented by dashed lines, planar heat sink by continuous lines). The electret's surface potential is 100 V. c. Evolution of the thermal efficiency for various values of thermal hysteresis (device with a planar heat sink). d. Curzon-Ahlborn efficiency of the device as a function of the hot source temperate (device with planar heat sink).

  a. Evolution of the working frequency as a function of the electret's surface potential when the electret is placed on the cold source (assuming a temperature difference in the cavity of 30 o C). b. Output power of the energy harvester when the electret is placed on the cold source for various thermal hystereses. c. Working frequency as a function of the electret's surface potential when the electret is placed on the hot source. d. Output power in the same conditions. e. Output power generated by an Al-Invar bimetallic strip coupled with a Cytop electret having a surface potential of 100 V and placed on the cold surface. f. Evolution of the thermal efficiency of the device as a function of the hot source temperature.

  Choose the bimetallic couple having the best Figure of Merit F W = 27. heat capacitance is weak since it has quite the same value whatever the metal used. As a consequence, bimetallic strips made of Al-Invar or CuZn-Invar are the best materials since the difference between their coefficients of thermal expansion is the largest possible;

  Performances of the a. pyroelectric, b. piezoelectric, and c. electrostatic bimetallic strip heat engines (equivalent ZT, mean power and thermal efficiency).

Figure 2

 2 Figure 2.1.1. Various types of boundary conditions: a. Pinned-pinned beam. b. Clamped-clamped beam. c. Imperfectly clamped beam with compression stiffness and torsional stiffness. d. Beam with torsional stiffness only.

  a. Shape of a pinned-pinned Al-Si beam with thermal hysteresis of 3 o C. B. Shape of a clamped-clamped Al-Si beam with hysteresis of 3 o C. c. Evolution of the beam amplitude of a macroscopic Al-Invar beam.

  2.1. a. Comparison of the density of energy produced by the three beams. c. Evolution of the Carnot efficiency of the thermo-mechanical transduction for each beam.

Figure 2 . 3 . 1 .

 231 Figure 2.3.1. Evolution of the Euler load as a function of torsional stiffness.

  Figure 2.3.3. a. Dispersion of the snap and snap-back temperatures for an Al-SiO2 clamped-clamped beam assuming a deviation of 2 MPa. b. Dispersions of the temperatures assuming a deviation of 4 MPa. c. Evolution of the standard deviation of the thermal hysteresis as a function of the mean value of the thermal hysteresis and for various deviations of the residual stress. d. Part of monostable beams linked to the variability of residual stress.

  Figure 3.1.1. a. Schematics of the pyroelectric bistable beam with a single electrode and w three electrodes. b. Stirling cycle implemented with the beam at the microscale.

Figure 3

 3 Figure 3.3.1. Simplified representation of the energy harvester at the microscale.

  a. Evolution of the operation frequency of some Al-Si-PZT bistable beams as a function of external conditions. b. Output power of the energy harvester vs hot source temperature. c. Thermal efficiency of the device. d. Curzon-Ahlborn efficiency of the device.

  a. Simplified schematic of the electrostatic harvester at the microscale. b. Strategy for the polarization of the electrostatic energy harvester. c. Maximal displacement variation of an Al-Si beam as a function of its thermal hysteresis. d. Value of the minimal capacitance as a function of the thermal hysteresis. e. Maximal charge Qb max for various values of the electric field. f. Evolution of the output voltage as a function of the thermal hysteresis. g. Electric energy generated during a complete cycle for various values of the electret's potential.

  a. Evolution of the frequency of various bimetallic strips against the electret's potential surface. b. Available energy per cycle. c. Mean power per bimetallic strip. d. Mean surface power. e. Thermal efficiency of the energy harvester. f. Evolution of the Carnot efficiency as a function of the electret's potential surface. 𝜂𝜂𝜂𝜂 !" !"# = 𝛥𝛥𝛥𝛥𝒲𝒲𝒲𝒲 ! ! 𝐶𝐶𝐶𝐶 ! . 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# . 𝐹𝐹𝐹𝐹 𝒱𝒱𝒱𝒱 ! (4.2.4d) a. Evolution of the power generated by a single bimetallic strip as a function of the hot source temperature and for an electret having a potential of 20 V. b. Available energy per cycle. c. Mean power per bimetallic strip. d. Mean surface power. e. Thermal efficiency of the energy harvester. f. Evolution of the Carnot efficiency as a function of the electret's potential surface.

  5.1.1a for a thermal hysteresis of about 3 o C. The number given on each figure indicates the reduction of the parameter due to the change of configuration.

z

  Scaling rule of the deflection Scaling rule of the beam's force Scaling rule on the mechanical energy Scaling rule of the heat flow through the device Scaling rule of the heat sink's convective transfer coefficient Scaling rule of the working frequency of the prime mover Scaling rule of the available mechanical power Scaling rule of the mean surface density of mechanical power Scaling rule of the thermal efficiency of the thermo-mechanical conversion Scaling rule of the piezoelectric capacitance Scaling rule of the output voltage of the piezoelectric device Scaling rule of the output power per bimetallic strip Scaling rule of surface power of the piezoelectric device Scaling rule of the electrostatic capacitance Scaling rule of the pull-in voltage of the electrostatic device Scaling rule of the generated power Scaling rule of surface power of the electrostatic device Scaling rule of the lifetime of the device (10 8 cycles before failure)

  Post-Apply Bake of the resist at 90 o C during 90 s to eliminate residual solvent and stabilize the film -UV-exposition at 365 nm (I-line) (radiation dose between 175 mJ and 330 mJ) -Development using TMAH 2,38%, two steps of 1 min -UV-exposition (400mJ) to generate acid molecules used as a catalyzer of the resist crosslinking -Melting of the resist at 140 o C during 5 min -Annealing at 200 o C during 5 min to crosslink the resist polymer.Fig.2.1.1c and Fig.2.1.1d shows some images of the microlenses obtained with this fabrication process.

  Figure 2.1.1. a. AFM profiles of microlenses fabricated at STMicroelectronics with differe times of heating [Audran et al, 2010]. b. Fabrication process of the bimetallic strips. c. Images microlenses developed at STMicroelectronics. d. Image of square microlenses developed for th bimetallic strips fabrication. e. SEM images of the Ti-Cu bimetallic strips (1 µm for each layer f. Image of Ti-Cu bimetallic strips after releasing. g. Image of heated Ti-Cu bimetallic strips.

  a. Tilted image of 210 µm-long Au-Ti bimetallic strips (0,25 µm -0,25 µm) fabricated with 8 µm-high microlenses b. Top view of the same bimetallic strips. c. Image of the heating bench. d. Interferometric measurement of the curvature of 210 µm-long Au-Ti bimetallic strips at ambient temperature. e. Profile of bimetallic strips at various temperatures. f. Limit of the use of microlenses.

Figure 2 .

 2 Figure 2.2.2. a. Image of heated 150 µm-long Au-Ti bimetallic membranes. b. Image of the same bimetallic strips after buckling. c. Tilted image of bimetallic strips suffering from excessive residual stress. d. Effect of thermal stress on partially clamped membranes. e. Post-buckling behavior of bimetallic membranes.

  Figure 3.2.1. a. Architecture of the prototype of bimetallic strip heat engine. b. Basic structure of a test chip c. Basic pattern repeated on the set of lithography masks. d. Choice of the dimensions of the bimetallic strips. e. Silicon integration scheme of the bimetallic strip heat engines.

  Fig.3.3.1a represents the evolution of the residual stress value in the SiO2 layer associated to the bistability criterion. It shows that first-order mode buckling occurs for higher values of residual stress if the beam is thicker. For a 240 µm-long 4 µm-thick beam, bistability appears for -63 MPa, and for a 180 µm-long 4 µm-thick beam, bistability appears for a stress of -112 MPa. As we are limited by the thickness of deposited thin films, we will target a value of stress between 80 MPa and 100MPa in Aluminum layers (tensile stress) and between -60 and -100 MPa of compressive stress in SiO2 layers.

  Figure 3.3.1. a. Dependence of the residual stress in SiO2 on the beam's thickness and beam's length. b. Dependence of the average temperature of the thermal hysteresis on the residual stress in Al and SiO2 layers. c. and d. Snap and Snap-back temperatures of Al-SiO2 bistable beams as a function of the stress in materials. e. Figure of Merits of the performances of bimetallic strips.

  Figure 3.4.1. a. Image showing the dependence of residual stress of metallic thin films on RF

  Figure 3.4.2. a. Fabrication process based on etching of a sacrificial layer directly deposited on silicon substrate b. Fabrication process based on etching of a planarized sacrificial layer in a trench. c. Principle of a fabrication process based on isotropic etching of silicon substrate. d. Principle of a fabrication process based on a backside DRIE.

  Figure 3.5.1. a. Flow of the second fabrication process. B. Images of trenches realized by RIE. c. Images of trenches realized by DRIE. d. Image and profiles of cavities after CMP. e. Image and profiles of cavities after CMP when a counter-mask is used. f. Image of the wafer after RIE of Al layer. g. Wafer after SiON etching. h. Destructive test on Wafer after NaI2-based copper etching. i. Test after FeCl3 etching.

  Figure 3.6.1. a. Unsnapped shape of a 360 µm-long membrane. b. Snapped shape of a heated 360 µm-long membrane. c. Interferometric measurements of a 360 µm-long membranes. e. Longitudinal cut of a 360 µm-long membrane

  Figure 3.7.1. a. Third fabrication process. b. Example of alternative structures. c. Buckled beams to measure residual stress. d. Cantilevers to measure stress gradients. e. Detail of the lithography masks. f. Complete set of masks.

  ).

	𝑇𝑇𝑇𝑇 ! =	𝑅𝑅𝑅𝑅 ! 𝑅𝑅𝑅𝑅 ! + 𝑅𝑅𝑅𝑅 !	. 𝑇𝑇𝑇𝑇 ! +	𝑅𝑅𝑅𝑅 ! 𝑅𝑅𝑅𝑅 ! + 𝑅𝑅𝑅𝑅 !	.	1 1 -𝜂𝜂𝜂𝜂	. 𝑇𝑇𝑇𝑇 !	(2.1.4a)
	𝑇𝑇𝑇𝑇 ! =	𝑅𝑅𝑅𝑅 ! 𝑅𝑅𝑅𝑅 ! + 𝑅𝑅𝑅𝑅 !	. 1 -𝜂𝜂𝜂𝜂 . 𝑇𝑇𝑇𝑇 ! +	𝑅𝑅𝑅𝑅 ! 𝑅𝑅𝑅𝑅 ! + 𝑅𝑅𝑅𝑅 !	. 𝑇𝑇𝑇𝑇 !	(2.1.4b)
	𝑊𝑊𝑊𝑊 =	1 𝑅𝑅𝑅𝑅 ! + 𝑅𝑅𝑅𝑅 !	.	𝜂𝜂𝜂𝜂 1 -𝜂𝜂𝜂𝜂	. 1 -𝜂𝜂𝜂𝜂 . 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !	(2.1.4c)

Table 3 .

 3 Performances of some thermoelectric generators

	Generator	ΔT ; TH ( o C)	Vout (V)	Vout (1 TC) (mV)	S (cm 2 )	Psurf (mW.cm -2 )	Comment
	Micropelt	5 ; -	0,55	0,95	0,14	1,77	(Bi,Sb)2(Te,Se)3 Bulk heat sink
	Nextreme/Laird	10 ; -	0,26	-	0,1	15	Bi2Te3 Bulk heat sink
	Perpetua	5 ; -	3,66	-	6,4	0,078	Bi2Te3 Bulk heat sink
	Bottner et al, 2004	5 ; -	0,043	3,6	0,01	0,056	(Bi,Sb)2Te3
	Leonov et al, 2011	14; 37	0,9	0,52	12	0.009	SiGe Planar heat sink

Table 4 .

 4 Performances

	N o	Work	ΔT ; TH ( o C)	3 )	Wvol (mJ.cm -	f (Hz)	Psurf (mW.cm -2 )	ηrel (%)	ZT eq T m ( o C)	Comment
	1	Sebald et al, 2009	16 ; 28		-	0,03	1.10 -5	2,4	0,1 20	Resistive No trans. (Exp.)
	2	Guyomar et al, 2008	35 ; 140		56	-	-	-	-	Stirling FE-FE (Exp.)
	3	Khodayari et al, 2009	10 ; 110		30	0,1	0,3	5,5	0,24 105	Ericsson FE-FE (Exp.)
	4	Khodayari et al, 2009	60 ; 160		250	0,1	2,5	1	0,04 130	Ericsson FE-FE (Exp.)
	5	Guyomar et al, 2008	7 ; -		-	0,2	0,016	0,05	-.	SSHI No trans. (Exp.)
	6	Olsen et al, 1981	33 ; 178		250	0,128	15	3,4	0,14 161,5	Ericsson FE-PA (Exp.)
	7	Nguyen et al, 2010	15 ; 85		0,058	0,12	0,043	1,26	0,05 73	Ericsson FE-PA (Exp.)
	8	Navid et al, 2010	40 ; 185		38	0,5		55,4	10,4 165	Ericsson FE-PA (Theory)

of some pyroelectric generators reported in the literature. FE-FE designates a Ferromagnetic phase to Ferromagnetic phase transition whereas FE-PA designate a Ferroelectric-paraelectric phase transition.

Table 5 .

 5 Performances of some fluid-based heat engines.

	N o	Work	ΔT ; TH ( o C)	Wvol (mJ.cm -3 )	f (Hz)	Psurf (mW.cm -2 )	ηrel (%)	ZT eq ( o C) T m	Engine
	9	Frechette et al, 2009	375 ; 400	93.6	64000	1600 T→M	9,4	0,32 213	Rankine (theory)
	10	Frechette et al, 2009	850 ; 800	1308	64000	25000 T→M	28	1,16 375	Rankine (theory)
	11	Cui and Brisson, 2008	470 ; 500	-	-	6800 T→M	18,8	0,75 265	Rankine (theory)
	12	Cho et al, 2007	40 ; 60	28	5	1,4 T→M	0,23	0,008 7 40	P-3 active (Exper.)
	13	Cho et al, 2007	40 ; 60	10	100	10 T→M	1,7	0,066 40	P-3 active (Exper.)
	14	Lurie and Kribus, 2010	110 ; 140	3,3	1090	34000 T→E	26,6	1,65 85	P-3 passive (theory)

Table 5 .

 5 Performances of some solid-based heat engines

	N o	Work	ΔT ; TH ( o C)	Wvol (mJ.cm -3 )	f (Hz)	Pvol (µW.cm -2 )	ηrel (%)	ZT eq ( o C) T m	Engine
	15	Liu, 2004	40 ; 70	22,4.10 3	-	-	48,3	6,6 40	SMA (theory) T→M
	16	Mohamed, 1979	28 ; 33	27.10 3	-	-	45	5,56 19	SMA (theory) T→M
	17	Saltzbrenn er, 2008	165 ; 190	3,6.10 3	0,001	1310 T→M	3,1	0,11 108	SMA (Exper.) Ericsson
	18	Chang and Huang, 2010	10 ; 37	1,2	0,01	1,2 if t=100µm	0.08	3,2.1 0 -3 32	SMA + Pyro (Theory)
	19	Chang and Huang, 2010	10 ; 37	4,2	0,07	3 if t=100µm	0,02	8.10 -4 32	SMA + Pyro (Exper.)
	20	Huesgen et al, 2012	37 ; 97	4.10 -3	0,72	0,58 T→M	0,18	6,9.1 0 -3 78,5	Bistable membrane
	21	Ravindran et al, 2012	80 ; 140 0,95.10 -3	7,04	1,34 T→E	0,093	3,4.1 0 -3 110	Bist. Memb. + Pyro
	22	Boisseau et al, 2012	24 ; 60	7,3.10 -3	3	3,3 T→E	0,5	0,02 48	BSHE + electret
	23	Bimetallic strip alone	75 ; 100	0,2	0,22	2,2 T→M	0,2	7,2.1 67,5 0 -3	BSHE

Table 7 .

 7 Performances of some thermomagnetic heat engines

	N o	Work	ΔT ; TH ( o C)	Wvol (mJ.cm -3 )	f (Hz)	Pvol (mW.cm -3 )	ηrel (%)	ZT eq T m ( o C)	Comments
	24	Elliott, 1959	23 ; 27	3,2.10 3	0,17	550	7,43	0,33 16	Temp.fluct. (theory)
	25	Hsu et al, 2011	50 ; 50	106	-	-	0,63	0,024 25	Therm. Grad (theory)
	26	Hsu et al, 2011	5 ; 15	-	-	-	11,5	0,58 12,5	Therm. Grad (theory)
	27	Post et al, 2013	20 ; 16	0,5	0,3	0,15 T→E	0,65	0,025 6	Therm. Grad FE-PA (Theory)
	28	Post et al, 2013	2; 151	1,5	0,8	1,2 T→E	0.65	0,026 150	Therm. Grad FE-AF (Theory)
	29	Solomon, 1989	65 ; 90	2,2	0,5	4,4 T→M	60	12,9 57,5	Therm. Grad FE-PA (Theory)
	30	Kirol and Mills, 1984	24 ; 60	24,8	0,25	6,2 T→E	7,5	0,34 42	Temp.fluct. (theory)
	31	Kirol and Mills, 1984	24 ; 60	71	0,1	7,1 T→E	47	6,34 42	Temp.fluct. (theory)

Table 2

 2

.3.1. Properties of the bimorph in Al-Invar and value chosen of Mo so that To=100°C.

  Equation (2.3.7a) is derivated to study the stability of the equilibrium position (2.3.9).

	𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 =	𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 2. 𝐼𝐼𝐼𝐼 ! . 𝐿𝐿𝐿𝐿	.	𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 +	. 𝑎𝑎𝑎𝑎 -𝑁𝑁𝑁𝑁 ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝑁𝑁𝑁𝑁 ! . 𝑎𝑎𝑎𝑎 ! 4. 𝐿𝐿𝐿𝐿 . 𝜙𝜙𝜙𝜙 ! ! 𝑘𝑘𝑘𝑘 -2.	𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 . 𝜙𝜙𝜙𝜙 ! 𝜙𝜙𝜙𝜙 ! 𝑘𝑘𝑘𝑘 ! 𝑘𝑘𝑘𝑘	+ 2. 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑	(2.3.9)

  ).

	𝑝𝑝𝑝𝑝 = -	! 9. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! ! . 𝐿𝐿𝐿𝐿 !				(4.1.9a)
	𝑞𝑞𝑞𝑞 = -	27. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿	. 𝑁𝑁𝑁𝑁 ! -𝐼𝐼𝐼𝐼 ! .	𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! +	! 2. 𝑁𝑁𝑁𝑁 ! . 𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! ! . 𝐿𝐿𝐿𝐿 !	(4.1.9b)

  With this equation, it is possible to calculate the expression of the kinetic energy of the membrane as a function of the speed of the shell's center (4.4.2b). This last equation give us a corrective factor that will be used to first adjust the kinetic energy of snap calculated with the experimental measurement of the speed of the bistable shell's center.

							). 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠	𝜋𝜋𝜋𝜋. 𝑥𝑥𝑥𝑥 𝐿𝐿𝐿𝐿	. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠𝑠𝑠	𝜋𝜋𝜋𝜋. 𝑦𝑦𝑦𝑦 𝐿𝐿𝐿𝐿	(4.4.1)
	𝒦𝒦𝒦𝒦 =	1 2	.	!.! ! ! !	! !	! !	𝜌𝜌𝜌𝜌 𝑧𝑧𝑧𝑧 . 𝑤𝑤𝑤𝑤 ,! ! 𝑥𝑥𝑥𝑥, 𝑦𝑦𝑦𝑦 . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑. 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑	(4.4.2a)
	𝒦𝒦𝒦𝒦 = 𝓂𝓂𝓂𝓂 ! .	𝜋𝜋𝜋𝜋 + 2 16. 𝜋𝜋𝜋𝜋	. 𝑤𝑤𝑤𝑤 ,! ! 𝐿𝐿𝐿𝐿 2	,	𝐿𝐿𝐿𝐿 2	(4.4.2b)

Table 2

 2

.2.1. Physical properties of some pyroelectric materials (based on

  𝑀𝑀𝑀𝑀 ℰ = -𝑣𝑣𝑣𝑣. 𝑒𝑒𝑒𝑒 !" 𝑧𝑧𝑧𝑧 . 𝑧𝑧𝑧𝑧 -𝑧𝑧𝑧𝑧 ! . 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

	!
	!			. ℰ = 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝐹𝐹𝐹𝐹 ℰ			(3.2.1b)
	This way, the thermo-mechanical equilibrium of the beam is described by equation (3.2.2).
	𝑁𝑁𝑁𝑁 ! . 𝜋𝜋𝜋𝜋 ! 8. 𝐿𝐿𝐿𝐿 ! . 𝑤𝑤𝑤𝑤 ! ! +	𝜋𝜋𝜋𝜋 ! 2. 𝐿𝐿𝐿𝐿	. 𝐼𝐼𝐼𝐼 ! .	𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿 ! + 𝐹𝐹𝐹𝐹 ℰ + 𝐹𝐹𝐹𝐹 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 . 𝑤𝑤𝑤𝑤 ! +	2. 𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿	. 𝑀𝑀𝑀𝑀 ! + 𝑀𝑀𝑀𝑀 ℰ -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 = 0	(3.2.2)

Table 4 .2.1.

 4 Properties of the bistable beam made of

	Position of the piezoelectric layer	Ne (N)	Ie (N.m 2 )	N α (N.K -1 )	M α (N.m.K -1 )	zp (mm)
	On the passive layer	4,90.10 5	7,76.10 -3	3,72	4,77.10 -4	0,05
	On the active layer	4,90.10 5	9,58.10 -3	3,72	3,37.10 -4	0,45

Then the bimetallic beam snaps and is cooled in open circuit. During this phase, the voltage of the piezoelectric capacitor reaches a maximal value when the beam snaps (4.2.2a). The final voltage of the piezoelectric layer is given by (4.2.2b). By closing the circuit, the variation of the charge carried by the capacitor is given by (4.2.2c).

  From the previous equations, it is possible to calculate the energy generated by the Stirling cycle (4.2.5).

	𝒲𝒲𝒲𝒲 ! = 𝒞𝒞𝒞𝒞 𝑝𝑝𝑝𝑝 . 𝒱𝒱𝒱𝒱 𝐼𝐼𝐼𝐼 2 = 𝒞𝒞𝒞𝒞 𝑝𝑝𝑝𝑝 .	𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝 . 𝑝𝑝𝑝𝑝 3 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 33 𝒮𝒮𝒮𝒮	. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 + 4.	𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 𝑝𝑝𝑝𝑝 . 𝑒𝑒𝑒𝑒 31 𝐿𝐿𝐿𝐿 2 . 𝜀𝜀𝜀𝜀 33 𝒮𝒮𝒮𝒮	. 𝑧𝑧𝑧𝑧 𝑝𝑝𝑝𝑝 -𝑧𝑧𝑧𝑧 𝑜𝑜𝑜𝑜 . 𝑤𝑤𝑤𝑤 𝑆𝑆𝑆𝑆 𝐶𝐶𝐶𝐶	2	(4.2.5)

  The bimetal then snaps in open-circuit and is discharged once it has snapped.

	! =	𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮	. 𝑇𝑇𝑇𝑇 ! -𝑇𝑇𝑇𝑇 !!" +	𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 2. 𝐿𝐿𝐿𝐿 ! . 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 .	𝜋𝜋𝜋𝜋 2	. 𝑤𝑤𝑤𝑤 ! ! ! -𝑤𝑤𝑤𝑤 !"# !	+ 4. 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! ! -𝑤𝑤𝑤𝑤 !"#	(4.2.8a)
	𝒬𝒬𝒬𝒬 ! ! = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 !						(4.2.8b)
	𝒱𝒱𝒱𝒱 ! ! =	𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿 ! . 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 .	3. 𝜋𝜋𝜋𝜋 4	. 𝑤𝑤𝑤𝑤 ! ! ! -6. 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! !	(4.2.9a)
	𝒬𝒬𝒬𝒬 ! ! -𝒬𝒬𝒬𝒬 ! ! = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 ! !					(4.2.9b)

Table 5 .

 5 2.1. a. Properties of the two oscillators of

		L x v x t	tlayers	zo(mm)	Ne (N)	Ie (N.m 2 )	Nα (N.K -1 )	Mα	mb (g)
		(mm)	(mm)					(N.m.K -1 )
	Al-Invar	40x10x0,4	0,2 + 0,2	0,17	4,29.10 5	5,22.10 -3	3,54	4,17.10 -4	0,86
	Brass-PZT	40x10x0,1 8	0,08+0,1	0,08	1,41.10 5	3,74.10 -4	-	-	0,57

Table 5 .

 5 2.1. b. Lumped parameters of the oscillators for a thermal hysteresis of 3 o C. The last parameter is the fundamental frequency of the piezoelectric oscillator.

		Boundary	kb1	kb3	kp *	Γp*	Cp	mp *	fo
		conditions	(N.m -1 )	(N.m -3 )	(N.m -1 )	(N.V -1 )	(nF)	(g)	(Hz)
	Al-Invar	SS	1,75.10 3	8,17.10 10	-	-	52,1	0,86	-
	Brass-PZT	CC	-	-	1,14.10 3	3,41.10 -3	52,1	0,21	367
	Brass-PZT	SS	-	-	2,84.10 2	1,71.10 -3	52,1	0,28	159
	Brass-PZT	2CF	-	-	2,84.10 2	1,71.10 -3	52,1	0,13	236

Table 5 .3.1.

 5 Lumped parameters of the oscillators of the thick piezoelectric transducer

		Boundary	kb1	kb3	kp *	Γp*	Cp	mp 8	fo (Hz)
		conditions	(N.m -1 )	(N.m -3 )	(N.m -1 )	(N.V -1 )	(nF)	(g)
	Brass-PZT	CC	-	-	5,38.10 5	-7,68.10 -3	52,1	0,37	609
	Brass-PZT	SS	-	-	1,35.10 3	-3,84.10 -3	52,1	0,49	263
	Brass-PZT	2CF	-	-	1,35.10 3	-3,84.10 -3	52,1	0,22	391

  .t p .ρ pyro .c pyro

	C v	. 1+	8. 4 3 3. 1-r w 2 3	.	z p -z o L	.	e 31 p 3 S .	3	m α n e .L	.	3	1 ΔT	2

Table 2 .

 2 2.1. a. Properties of the bimorph made ofAl-Invar.

	Dimensions	tAl/t	Ne	Ie	N α	M α	Cv
	(mm 3 )	(1)	(N)	(N.m²)	(N.K -1 )	(N.m.K -1 )	(J.K -1 )
	20 x 2 x 1	0.2	2,59x10 5	1.8x10 -2	8,84 x10 -1	2.78x10 -4	0.14
	Table 2.2.1. b. Properties of the bimorph made of Al-Si.
	Dimensions	tAl/t	Ne	Ie	N α	M α	Cv
	(µm 3 )	(1)	(N)	(N.m²)	(N.K -1 )	(N.m.K -1 )	(J.K -1 )
	100 x 10 x 5	0.2	2,59x10 5	1.8x10 -2	8,84 x10 -1	2.78x10 -4	0.14

  ! 𝑥𝑥𝑥𝑥 = 𝑤𝑤𝑤𝑤 ! . cos 𝑘𝑘𝑘𝑘 ! . 𝑥𝑥𝑥𝑥 -cos 𝑘𝑘𝑘𝑘 ! .

												𝐿𝐿𝐿𝐿 2	(2.3.1a)
	𝑘𝑘𝑘𝑘 ! . cos	𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 2	+ 𝛾𝛾𝛾𝛾. sin	𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 2	= 0	(2.3.1b)
	As a consequence, the strain energy of the beam (2.2.1a) can be simplified into (2.3.2a),
	where the functions φ1 and φ2 are defined by (2.3.2b) and (2.3.2c). In the case of a simply
	supported beam (γ.L = 0), the values of this coefficient are given by (2.3.2d), whereas for a
	clamped-clamped beam (γ.L = 36.7), they are given by (2.3.2e).
	𝒲𝒲𝒲𝒲 ! =	𝑁𝑁𝑁𝑁 ! 32. 𝐿𝐿𝐿𝐿	. 𝜑𝜑𝜑𝜑 ! ! . 𝑤𝑤𝑤𝑤 ! ! + 𝐹𝐹𝐹𝐹 ! -𝑁𝑁𝑁𝑁 ! . 𝑇𝑇𝑇𝑇 + 𝐼𝐼𝐼𝐼 ! . 𝑘𝑘𝑘𝑘 ! ! .	𝜑𝜑𝜑𝜑 ! 4	. 𝑤𝑤𝑤𝑤 ! ! + 2. 𝑀𝑀𝑀𝑀 ! -𝑀𝑀𝑀𝑀 ! . 𝑇𝑇𝑇𝑇 . 𝜑𝜑𝜑𝜑 ! . 𝑤𝑤𝑤𝑤 !	(2.3.2a)
	𝜑𝜑𝜑𝜑 ! = 𝑘𝑘𝑘𝑘 ! ! . 𝐿𝐿𝐿𝐿 -𝑘𝑘𝑘𝑘 ! . sin 𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿				(2.3.2b)
	𝜑𝜑𝜑𝜑 ! = 𝑘𝑘𝑘𝑘 ! . sin		𝑘𝑘𝑘𝑘 ! . 𝐿𝐿𝐿𝐿 2							(2.3.2c)
	𝜑𝜑𝜑𝜑 ! 𝛾𝛾𝛾𝛾 = 0 =	𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿	; 𝜑𝜑𝜑𝜑 ! 𝛾𝛾𝛾𝛾 = 0 =	𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿	;	𝜑𝜑𝜑𝜑 ! . 𝐿𝐿𝐿𝐿 𝜑𝜑𝜑𝜑 !	= 𝐿𝐿𝐿𝐿 ;	!	𝜑𝜑𝜑𝜑 ! 𝜑𝜑𝜑𝜑 ! ! . 𝐿𝐿𝐿𝐿	=	1 𝜋𝜋𝜋𝜋	(2.3.2d)
	𝜑𝜑𝜑𝜑 ! 𝛾𝛾𝛾𝛾 = 0 = 3,79.	𝜋𝜋𝜋𝜋 ! 𝐿𝐿𝐿𝐿	; 𝜑𝜑𝜑𝜑 ! 𝛾𝛾𝛾𝛾 = 0 = 0,3.	𝜋𝜋𝜋𝜋 𝐿𝐿𝐿𝐿	;	𝜑𝜑𝜑𝜑 ! . 𝐿𝐿𝐿𝐿 𝜑𝜑𝜑𝜑 !	= 6,49. 𝐿𝐿𝐿𝐿 ;	!	𝜑𝜑𝜑𝜑 ! 𝜑𝜑𝜑𝜑 ! ! . 𝐿𝐿𝐿𝐿	=	0,28 𝜋𝜋𝜋𝜋	(2.3.2e)

  In the case of the pyroelectric beam with three electrodes, the maximal output voltage is given by (3.2.2a), leading to the electrical energy (3.2.2b).

	𝒱𝒱𝒱𝒱 !!!" = 8.	𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿. 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 2. 𝑘𝑘𝑘𝑘 ! -𝜑𝜑𝜑𝜑 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! !	(3.2.2a)
	𝒲𝒲𝒲𝒲 !!!" = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 !!!" ! = 𝒞𝒞𝒞𝒞 ! . 8.	𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿. 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 2. 𝑘𝑘𝑘𝑘 ! -𝜑𝜑𝜑𝜑 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! !	!	(3.2.2b)
								3.1.1b). If the
	pyroelectric bistable beam is polarized by a single plane electrode, the maximal output
	voltage is given by (3.2.1a). Hence we can give the expression of the electrical energy
	generated by the energy harvester (3.2.1b).
		𝒱𝒱𝒱𝒱 !" =	𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮	. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 -8.	𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿. 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 . 𝜑𝜑𝜑𝜑 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! !	(3.2.1a)
		𝒲𝒲𝒲𝒲 !" = 𝒞𝒞𝒞𝒞 ! . 𝒱𝒱𝒱𝒱 !" ! = 𝒞𝒞𝒞𝒞 ! .	𝑡𝑡𝑡𝑡 ! . 𝑝𝑝𝑝𝑝 ! 𝒮𝒮𝒮𝒮 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮	. 𝛥𝛥𝛥𝛥𝛥𝛥𝛥𝛥 -8.	𝜋𝜋𝜋𝜋. 𝑡𝑡𝑡𝑡 ! . 𝑒𝑒𝑒𝑒 !" 𝐿𝐿𝐿𝐿. 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮	. 𝜑𝜑𝜑𝜑 ! . 𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! . 𝑤𝑤𝑤𝑤 ! !	!	(3.2.1b)

Table 3 .

 3

2.1. Properties of the bistable beam made of Al-Si and PZT-5H.

  [START_REF] Bibliography Aggarwala | Thermal stability of bimetallic shallow spherical shells[END_REF] with three electrodes are given by (3.3.3). Taking into account thermal leakages, the efficiency of the device is described by (3.3.3c), and the Curzon-Ahlborn efficiency of the harvester by (3.3.3d).

	𝒱𝒱𝒱𝒱 !"!# = 7,7. 𝐿𝐿𝐿𝐿.	!	𝑟𝑟𝑟𝑟 ! 3. 𝑟𝑟𝑟𝑟 ! -𝑟𝑟𝑟𝑟 ! !	.	𝑡𝑡𝑡𝑡 ! . (𝑧𝑧𝑧𝑧 ! -𝑧𝑧𝑧𝑧 ! ) 𝐿𝐿𝐿𝐿 !	.	𝑒𝑒𝑒𝑒 !" 𝜀𝜀𝜀𝜀 !! 𝒮𝒮𝒮𝒮 .	!	𝑀𝑀𝑀𝑀 ! 𝑁𝑁𝑁𝑁 ! . 𝐿𝐿𝐿𝐿	. 𝛥𝛥𝛥𝛥𝑇𝑇𝑇𝑇 !"# !	(3.3.3a)

(a) (b) (c) (d) (e) (f)

(a) (b) (c) (d) (e) (f) (g) (h)

& & & & & & & & & & & & && & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &

& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &

& & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & &

Remerciements