List of Abbreviations

what it means to be a scientist. Then, I want to thank Nicolas Vaxelaire for the all work he did with me on the XRD study. For all the knowledge I got from him but also for his endless patience and enthusiasm. I would also like to thank Fabrice Domingues Dos Santos of Piezotech for providing the materials studied, for his support and his pragmatic advice along the way. Thanks to Sylvie Giraut and Francois Bargain of Arkema for their help and the fruitful discussions about polymers.

Thank you to all the people who helped me at any point during the thesis, whether by lending me a hand or giving insightful remarks: Sebastien Noel, Amelie Revaux, Julien Routin, Marine Galliari, Antoine Latour, Magalie Gasiglia, Mickael Charbonneau. For the good times and the memories we made in the laboratory I want to thank Clara Haddad, Alexandre gaitis, Julie Euvrard, Tamara Nunes-Domschke, Smail Amari, Jamal Tallal, Michael Bouvier and Alexandra Cantarano.

I am very grateful to you, Aurore, for having bore with me and supported me during both good and bad times. You made these past years so much happier and I cannot thank you enough.

And nally I want thank my beloved parents, to whom I owe everything.

Introduction

With the advent of new technologies, organic electronic is becoming an increasingly attractive research eld. It is already exploited to make OLED TVs as well as various sensors and actuators. Due to the general enthusiasm for these applications, a lot of prototypes and patents are presented every day. Yet, it is dicult to discern how organic materials fare against one another or even against 'traditional' ceramics. To this day the main advantage of polymeric materials is considered to be the possibility to produce exible devices, with cheaper fabrication processes based on printing techniques.

In partnership with Piezotech who provides the electroactive polymers, our laboratory at CEA-LITEN is focused on the development of fully printed organic devices, like sensors and actuators. Functional prototypes are developed as part of European projects or to meet the requests of industrial partners. Besides proofs of concept for new applications, the underlying objective of our research is to further improve the devices performances. There are several possible approaches to achieve that goal and among them, the selection of the most suitable polymer as active layer.

As part of this project, our rst objective is to evaluate which materials and compositions are best suited for the production of fully printed devices. The following step is trying to understand the physical strain mechanisms in our polymers, and thus provide directions for the development of new electroactive materials.

Chapter I is a general introduction to electroactive properties, their origin and their use in commercial applications. Dierent types of materials are presented and among them the focus of our study, polyvinylidene uoride (PVDF) derivatives. The comparison with ceramics provides a good opportunity to determine how polymers can be attractive in a market overwhelmingly dominated by inorganic materials. We identied poly(vinylidene uoride-triuoroethylene) P(VDFTrFE) and poly(vinylidene uoride-triuoroethylene-chlorotriuoroethylene) P(VDF-TrFE-CTFE) as polymers of particular interest for actuation based applications.

Chapter II is a comparative study of polymers performances once embedded in thin lms actuators. The inuence of CTFE content on the behavior of polymers is characterized for nine dierent compositions. The electromechanical responses are evaluated in regard of two gures of 7 CONTENTS merit: cantilever deection and energy cost. As it turns out, there is no ideal composition and the suitability of a given polymer depends on the prioritizing order between performance and energy consumption.

This study highlights the diculty of comparing polymers because of their dierences in electromechanical responses. To understand the origin of these disparities, Chapter III is focused on their strain-polarization relationship. Deviations from the standard electrostrictive formalism are approached as opportunities to isolate the physical mechanisms at hand. Based on the results, a phenomenological description of strain mechanisms is proposed for both copolymers and terpolymers. In order to strengthen the terpolymer strain model, the following chapter is focused on experimental validation, mostly with the help of in-situ XRD measurements.

In chapter 4 we start the study with copolymer, where the impacts of eld-induced phase transition and electrostriction can be observed separately. The case of terpolymers proved to be more complex with several eects taking place simultaneously. This study evidenced the link between the structural changes in the crystalline phase and the nonlinear response of terpolymers.

Confronting the XRD data to existent models from the literature, we proposed an improved description of the dipolar order inside the polymer chains.

In order to go further and build on the results presented here, chapter 5 discusses dierent paths of interest. The study on P(VDF-TrFE-CTFE) can be continued with both XRD and electromechanical follow-up measurements. We also present the results of preliminary measurements made on another class of polymers: P(VDF-TrFE-CFE). By relying on the framework laid out here, it would be possible to produce similar results on these polymers and further extend the comparison.

Chapter 1

PVDF based polymers and their applications 9 This rst chapter aims at explaining the interest for polyvinylidene-uoride (PVDF) and its derivatives. This introduction is a mere overview of the polymers nature and applications, their properties will be detailed in the following chapters.

PVDF

PVDF is a semi-crystalline polymer of formula (C 2 F 2 H 2 ) n . It possesses remarkable properties such as a strong resistance to corrosion, low reactivity to most chemical products and a high mechanical resistance. For those reasons, it is commonly found in industry to coat chemical tanks and pipes. The pie chart displayed below is taken from [1] and represents the annual production of uorine based polymer as of year 2012. Figure 1.1: Production of uorinated polymers in 2012 [1] PTFE, often know as TEFLON is by far the most commercialized uorinated polymer. PVDF comes in second and its production volume is expected to double by the year 2022. PVDF mechanical and chemical features are the main reasons behind these numbers but its electroactive properties are what we are interested in here. Indeed, PVDF can be ferroelectric depending on which phase it crystallizes on. Whether PVDF possesses electroactive properties or not is entirely dependent on the polymer chains conformation inside the crystalline phase. The three most common phases, called α, β and γ are pictured in gure 1.2. This is taken from a review on the dierent PVDF fabrication and characterization methods, [2]. There exists a fourth phase called δ that is not represented in gure 1.2, most likely because it is the less common form of PVDF. Figure 1.2: Representation of the α, β and γ phases of PVDF [2] The α phase is the most stable one when the polymer is processed at ambient pressure. Unfortunately, this is the one phase that does not possesse ferroelectric properties. This can be inferred from gure 1.2 by looking at the positioning of carbon and uorine atoms. The CF 2 strong dipolar moments cancel each other in the α chain because of their opposite orientations.

In other phases, these moments add up and grant its ferroelectric properties to PVDF. This eect is maximized in the β phase which is the phase sought after in electroactive devices. displays the available paths to obtain the dierent phases of PVDF. Among the possibilities, the most common way is the mechanical drawing of the PVDF lm, several times its initial length. The major drawback of this technique is its incompatibility with thin lm tech-nologies due to the presence of a substrate. A solution to that particular issue is to copolymerise PVDF with triuoroethylene (TrFE) to favour the β form directly. The structure of this polymer will be detailed in chapter 4 and before anything else we need to understand why obtaining ferroelectric properties is worth the eort. A dielectric material is an insulator, its quality as a dielectric depends on its insulation properties but also on its polarizability. Electrically speaking, this means a good dielectric makes a capacitor with a high capacitance value and no leakage current.

Electroactive materials

As charges are displaced with the application of an electrical eld, the material will be strained ever so slightly; this property is referred to as electrostriction. Electrostriction is negligible in the vast majority of dielectrics with a few exceptions such as P b(M n 1/3 N b 2/3 )O 3 or poly(vinylidene uoride-triuoroethylene-chlorotriuoroethylene) P(VDF-TrFE-CTFE).

Among the dierent materials, a few ones such as quartz display piezoelectric properties. The piezoelectric eect arises from the crystalline lattice symmetry and its occurrence can be inferred based on the crystal classes [3]. In these particular structures, the application of a stress will sep-arate the positive and negative charge centroids and therefore generate a dipolar moment. The apparition of those dipoles in the crystal will result in the generation of charges at the electrode and this mechanism is called the direct piezoelectric eect. The opposite mechanism, referred to as converse eect corresponds to a lattice strain induced by the application of an electrical eld.

There are twenty crystal classes with piezolectric properties and half of them are also pyroelectric. A pyroelectric material possesses a dipolar moment even before the application of any electrical bias or mechanical stress. As a consequence, a temperature variation will result in the generation of electrical charges, the so-called pyroelectric eect. This is represented in g 1. 4 with an example where the cation is not located in the lattice centre and thus is separated from the negative charge centroid. Such material is also piezoelectric because the application of a mechanical stress will necessarily change the dipole amplitude and therefore induce a charge displacement.

With a pyroelectric material, the dipolar moments might be oriented through the application of an electrical eld. This is not always the case because dipoles are not isolated but constrained in a crystalline structure. If they can align with the electrical eld, the material is referred to as ferroelectric. Naturally, a ferroelectric material will possess both piezoelectric and pyroelectric properties.

There are two main categories of materials falling into the classication described in gure 1.4. The rst family is comprised of ceramics which are inorganic compounds and often contain heavy metals. The other category is that of eleactroactive polymers (EAP), separated into two sub-classes: the ionic and the electronic families. We summed up in table 1 The ceramics are capable of producing the highest levels stress when submitted to an electrical eld. The ionic EAPs can reach the highest levels of strain, often at the cost of a very low rigidity.

The electronic family to which PVDF belongs, does not reach the highest strain nor stress values and their main advantage is the fabrication process. They can be printed quickly and cheaply, onto large surfaces and on exible substrates. These are just a few elements of comparison, a thorough analysis between dierent EAPs can be found in [4].

Besides strain and stress, other parameters such as the time response or acoustic impedance can be determining factors in the choice of a given electroactive material. In that regard, their respective features can be seen as complementary, each material being better suited to one particular application. This is a recurring argument in most comparative studies found in the literature.

Yet, the industrial market for actuators is overwhelmingly dominated by piezoceramics despite the promising potential of EAPs in general, and PVDF in particular. In order to assess if PVDF is still a "promising" material 50 years after its discovery [5], we conducted a bibliometric study on the last three decades.

PVDF in the scientic landscape

Interest over the years The publication volume on PVDF increased dramatically over the past 30 years and this trend is still in the ascending phase. To put this evolution into perspective, we represented the number of annual publications on PZT, the most common piezoceramics in industry. If we just consider these data, there seems to be a real paradigm shift, with PVDF potentially revealing itself as the piezoelectric material of the future. However as mentioned before, PVDF also has remarkable chemical properties which could also be the reason behind that renewed interest. To uncover the actual motivations, 4000 publications between 2016 and 2018 were analyzed with

VOSviewer. This software analyses results from a scientic database and forms clusters to highlight the dierent topics of interest. The resulting graph is displayed in gure 1.6.

It appears that PVDF has three main areas of interest with few links to one another. The red cluster corresponds to publications focused on the use of PVDF as a membrane, it is the most prolic topic at the moment. The main application is water treatment and the key properties are chemical resistance and mechanical strength [6].

The green cluster corresponds to the use of PVDF in batteries. For that application, PVDF is used as a solid electrolyte and the key property is its ionic conductivity. To improve its eciency, PVDF is often copolymerized with hexauoropropylene to form P(VDF-HFP).

The third cluster is divided into two sub-clusters: yellow and blue. The yellow one corresponds to PVDF electrical properties and the related applications. For an organic material PVDF has a high permittivity, which makes it an interesting candidate for organic capacitors and transistors. The algorithm displays clusters based on occurrence of terms in title and abstract

The notions of llers, matrix and composite materials occur frequently in that cluster because the inclusion of conducting particles in a PVDF matrix is a promising road towards organic materials with a high dielectric constant.

The blue cluster corresponds to the piezoelectric properties and their applications, such as sensor, transducer, or energy harvester. It is close to the yellow cluster because piezoelectricity is intrinsically linked to the material electrical properties. This is the area of interest for us but, as displayed in g 1. 6, only a minor topic among the dierent elds of application.

It turns out the infatuation over PVDF is largely driven by its use as a membrane and not its electroactive properties. In order to make the comparison with PZT more relevant we narrowed down the query to P(VDF-TrFE). The advantage of P(VDF-TrFE) over the pristine PVDF is that it is ferroelectric without the need for mechanical stretching. It is more expensive than PVDF and not of any use in membranes or batteries. The only reason to use it is as a ferroelectric material in thin lms devices. Figure 1.7 displays the corresponding publication volumes.

The amount of publications on P(VDF-TrFE) is about 10% that of PVDF and the increase scopus over time is milder. However, there is also a growing interest for this material and therefore the use of PVDF electroactive properties. This is not the piezoelectric revolution hinted by the trend in gure 1.5 but it most likely means that PVDF can be used in applications where the current piezoceramics have found their limits. A few possibilities regarding the advantages of polymers over ceramics are listed below.

The rst one is on the side of environment and health issues as PZT contains toxic metals such as lead. Over the last few years, lead-free ceramics have been the focus of several scientic studies and PVDF is even more environmentally friendly than most of the other alternatives. This trend in eco-friendly materials grew during the mid-00's, suggesting that it is a possible factor behind the renewed interest in P(VDF-TrFE).

The second edge PVDF based device have over ceramics is the possibility of making fully printed exible devices. The corresponding fabrication processes are usually quick and cheap [7]. Furthermore, the exibility of the overall device can help generate strains that could not be achieved on rigid substrates such as glass or silicon. The need for exible devices is also related to the emergence of new technologies such as 'smart' textiles [8] or articial muscles [9].

These advantages can help P(VDF-TrFE) nding markets in the future, but in order to increase those odds the main drawback of copolymers must also be addressed. This issue is, as displayed in table 1.1, the lower levels of stress polymers can generate compared to ceramics. In attempts to bridge that gap, new polymers have been developed over the past two decades.

Most of these polymers are P(VDF-TrFE) derivatives and they appear as promising candidates to replace the copolymer but their electromechanical properties are still largely undocumented.

P(VDF-TrFE) derivatives

The rst copolymer derivative to be studied was an irradiated sample [10]. After receiving the proper dosage, P(VDF-TrFE) exhibits a large electrostrictive strain although it comes at the cost of its ferroelectric properties [11]. In order to reach similar levels of strain but without the need to irradiate polymers, a third monomer was introduced in the polymer chain. This third monomer can be either chlorouoroethylene (CFE) [12] or chlorotriuoroethylene (CTFE) [13].

Other attempts with dierent monomers proved to be unsuccessful [14].

The raw formula of P(VDF-TrFE-CTFE) is represented g 1.8. This is just a brief introduction on these terpolymers, and their properties will be detailed in the following chapters. enhanced the electromechanical performances at low frequencies [15]. P(VDF-TrFE-CFE) has also been blended with P(VDF-CTFE) [16] or used as a matrix for carbon black nanocomposite [17]. Overall, there are several proofs of concept and theoretical models promising new improved performances for electroactive polymers. However, their interest once embedded in an actual device is still unclear, mostly because their high strain comes with a lower rigidity. Besides questions about their performances, the physical understanding of strain mechanisms in these polymers is lacking and it seems unlikely that the behaviour of blends or nanocomposites can be fully explained if the polymer matrix itself remains a mystery. For that reason we chose P(VDF-TrFE-CTFE) as the focus of this work. The other polymer family P(VDF-TrFE-CFE) also deserves its own study but we focused our eorts on CTFE because its fabrication process is less expensive, making it a potentially better material from an industrial perspective.

Applications

In this section we present a few examples to illustrate the dierent uses of thin lm PVDF based devices.

Sensors

With the ability to generate charges under stress, a piezoelectric material can be used as strain gauge or pressure sensor. If it is pyroelectric, it can also be used to detect temperature variations.

Figure 1.9 displays two functional sensor prototypes exploiting these properties. The two devices in g 1.9 were fabricated using a screen printing technique (g 2.4) and P(VDF-TrFE) as active layer. The white lines are made of conducting silver ink and the blue color is due to poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS), a conducting polymer used to make the electrodes. The left picture prototypes are soles with piezoelectric and piezoresistive captors, to detect and quantify the distribution of weight inside a foot. The right picture device was designed to reproduce a piano keyboard pattern. When pressing a key, charges will be generated and detected by an electronic circuit that will play the corresponding sound. This sensor exploits both the piezoelectric and the pyroelectric eect with the combined heat and pressure of the user's nger. The pyroelectric properties of PVDF and copolymers also allow for the detection of heat sources without direct contact and they can be used as infrared sensors [18].

This application of electractive polymers as sensors is only accessible to PVDF and P(VDFTrFE) copolymers. Terpolymers are not ferroelectric nor do they possess piezoelectric properties, this makes them unable to gather charges in a similar fashion.

Energy harvesting

The piezoelectric eect can also be used for energy harvesting. The aim is to exploit mechanical motions such as vibrations to convert them into electrical energy. The left picture in gure 1.11 is taken from [19], recently published. The structure was designed to use PVDF as the active layer to harvest energy. In that domain, PVDF could hope to compete with piezoceramics because large strains can be easily reached in the active layer. This is illustrated with the picture of one of our samples displayed in the right panel. A large strain of the entire device can be achieved with a mild movement of the two ngers. This is a much more ecient approach than applying pressure vertically, as we should have done had the substrate been made of silicon.

Despite the lack of piezoelectricity, it is also possible to harvest energy with terpolymers. The sample must be stressed electrically and mechanically according to the following cycle, taken from [20]. P(VDF-TrFE-CFE) terpolymers can be used that way and have proven to be good candidates for such applications [21]. The theoretical gures of merit of these generators are higher than those of piezoelectric materials, but the need to function with an applied bias can be a signicant drawback. Because of the mechanical stretch step, terpolymers cannot be deposited onto a substrate and this application is not compatible with thin lm devices.

Actuation

Actuation uses the converse piezoelectric eect or the electrostrictive eect. The strain responses are dierent but in both cases it comes down to an electric potential changing the polymer dimensions. If the layer is deposited onto a substrate, the entire device will be bent, as displayed in gure 1.12.

The device on the right part of gure 1.12 is a 1.3 cm long cantilever under an electric potential of 100V. The substrate is 125µm thick and the active layer is 2µm thick. To maximize the deection, up to 20 capacitors (40 layers) were stacked and the resulting deection is visible with the naked eye. Among the three domains of applications presented above, actuation is the only one where copolymers and terpolymers can be compared directly. For this reason, actuation was chosen as the application of reference in the following study.

Other applications

The P(VDF-TrFE) based terpolymers can also be employed in others applications with no link to their electrostrictive properties. Two main domains of applications fall into that category, their use as gate dielectric in organic transistors [22] and as electrocaloric material in cooling devices.

In The variations of temperature in this polymer can reach 12K for an electrical eld of 300M V.m -1 .

Recently a functional device using ECE has been fabricated with P(VDF-TrFE-CFE) as active material. Fig 1 .13 taken from [24], displays an experiment in which the test battery was cooled by 8°C under an applied electrical eld of 68M V.m -1 . In the future, terpolymers might very well nd their use as electrocaloric materials and not for what they were invented for in the rst place.

Conclusion

PVDF is a ferroelectric polymer and therefore possesses piezoelectric and pyroelectric properties.

The ferroelectric order is not occurring naturally and the polymer lms need to be stretched.This is not compatible with thin lm devices and fully printed technologies. The most practical way around this limitation is the addition of TrFE in the polymer chain. This makes of P(VDFTrFE) the ideal material for thin lm applications such as sensors, energy harvesters or actuators.

The ferroelectric properties of PVDF have been known since 1969 but ceramics have supplanted electroactive polymers in industry ever since. Interestingly, PVDF is now the focus of more academic research than PZT, the most common piezoceramic. It turns out that most of the activity in this eld is focused on the use of PVDF as a membrane for water treatment. Still, the interest for electroactive applications is also growing. The most likely explanation for this new-found enthusiasm is the search for lead-free alternative to PZT as well as the possibility of printing exible devices.

The main issue with polymers remains their lower instrinsic performances compared to piezoceramics. In eorts to reduce that gap, new formulations have been developed such as P(VDFTrFE-CTFE) terpolymers. These materials lose their ferroelectric properties with the addition of the third monomer but becomes signicantly more electrostrictive in return. There are several proofs of concepts and even derivated formulations making this material potentially

interesting. Yet, it is still unclear whether terpolymers are more suited than P(VDF-TrFE) for practical purposes.

Actuation is the one application where the polymers performances can be directly compared to one another. The following chapter is a comparative study between P(VDF-TrFE) and P(VDF-TrFE-CTFE) based devices, to assess the interest of using terpolymers in thin lm actuators. The emphasis will be put on the inuence of CTFE content on the actuator response.

Bibliography

State of the art

In this chapter we will study dierent types of P(VDF-TrFE-CTFE) terpolymers and assess their interest as active material in thin lm actuators.

Electromechanical formalism

The constitutive equations of electroactive materials link the electrical displacement and eld to other physical quantities such as stress, strain or temperature. Their notation and units are regrouped in table 2.1.

Pyhsical quantity Notation Units

Strain

S ij

Stress

T ij P a

Electrical eld

E i V.m -1 Electrical displacement D i C.m -2 Polarization P i C.m -2
d ijk m.V -1 Piezoelectric coecients (T,D) g ijk m 2 .C -1 Electrostrictive coecients (T,E) M ijkl m 2 .V -2
Electrostrictive coecients (T,D)

Q ijkl m 4 .C -2
S ij = s ijkl .T ij + d kij .E k + M klij .E k E l D ij = ij .E j + d ij .T j (2.1) S i = s ij .T j + g ij .D j + Q klij .D k D l E i = β ij .D j -g ij .T j (2.2)
The rst terms of each equation in 2.1 and 2.2 are used to describe all dielectric materials.

The strain-stress relationship made with the s ij coecients is nothing more than Hooke's law for elastic materials. The relationship between electrical displacement and electrical eld is made using permittivitty ( ij ), which is the description of a linear dielectric. In the case of electroactive materials, other coecients are added to account for the interdependence of mechanical and electrical quantities. That is why equations 2.1 and 2.2 contains the piezoelectric and electrostrictive coecients d, g, Q and M . The electromechanical equations have several forms depending on which quantity we want to express. We only present here the equations with an explicit strain formulation. The other forms and coecients are described in [1].

A fourth order tensor such as M or Q contains 81 coecients but based on symmetry considerations this number can be reduced to 36. Among these 36 coecients, only a few are actually relevant for practical calculations. We do not apply any external stress to the sample so T ij = 0 in equation, additionally the electrical eld is applied in the principal direction 3 so E 1 = E 2 = 0. If the material is piezoelectric, the second order terms are negligible behind the rst order ones. The strains along thickness and length directions are then expressed with equation 2.3.

S 1 = d 311 .E 3 S 3 = d 333 .E 3 +M 3333 .E 2 3 (2.3)
With simplied Einstein notations, d 3ii and M 33ii become d 3i and M 3i . A few more coecients will be used in chapter 4 but these are the most important ones.

Here we conveniently forget about the direction 2. However, in a clamped device blocking one direction (here direction 2) will aect the two others. This issue will is not developed here nor its it in most studies, but we will keep that in mind when comparing dierent geometries in chapter 5.

Copolymer

The material to which we will compare P(VDF-TrFE-CTFE) terpolymers is the better-known P(VDF-TrFE) copolymer. Several grades of copolymers are commercially available with dierent piezoelectric and pyroelectric properties. (polarization at zero volt) observed in gure 2.2. This ferroelectric response is especially relevant for actuation because in PVDF and P(VDF-TrFE) the piezoelectric eect is proportional to remanent polarization [3]. Based on these considerations and the compositions available at the time, we chose as our reference the copolymer with 75% VDF and 25 % TrFE.

In terms of processing conditions, several studies have been dedicated to the impact of annealing on copolymers structure and properties [4][5][6] [7]. To maximize the piezoelectric performances, copolymer must be annealed between their Curie temperature and their melting point.

Based on Piezotech data, we chose an annealing temperature of 130 °C.

The electromechanical performances of copolymers depend on their composition and fabrication process but the values are in the same range throughout the dierent studies. TrFE). The stress output of an actuator is a function of both rigidity and strain so ceramics are overwhelmingly dominant in that regard.

To compensate the discrepancy in d values, a PVDF energy harvester would have to be strained at least ten times more than a PZT one. As we discussed in the previous chapter this might not be an issue and can be achieved with something as simple as folding a plastic sheet (i.e the substrate). This is the same with actuation where a thin exible substrate can help compensate the lower strain values of PVDF.

Terpolymer Electrical behaviour

The high strain of terpolymers is associated with their relaxor ferroelectric nature. This denomination of relaxor ferroelectric has been taken from the electrostrictive formalism commonly used with ceramics. The relaxor behaviour is associated to a reduction of the crystalline domains size which in turn leads to a frequency dependent Curie temperature.

In gure 2.3 left panel displays the permittivity versus temperature at dierent frequencies.

The peak position denes the Curie temperature and its frequency dependence is a characteristic terpolymer. Right: strain versus electrical eld [14] feature of relaxor ferroelectric materials. The curve in the right panel is the rst report of the important strain in P(VDF-TrFE-CTFE).

To achieve this transition from ferroelectric to relaxor ferroelectric and improve the strain response, a certain amount of CTFE has to be introduced. The inuence of composition on terpolymers electrical properties was mostly investigated by Z.Zhang [15] [16]. The permittivity and also the polarization response were measured for more than twenty terpolymers with dierent compositions and fabrication processes. The polymers were evaluated on their viability as dielectric material for high permittivity capacitors and no elelectromechanical data were collected.

Mechanical

Published data are scarce on the mechanical performances of terpolymers. In terms of rigidity, terpolymers are softer than copolymers and this value is heavily dependent on their composition, ranging from 0.1 to 0.6 GPa [17]. To compare the terpolymers strain and stress to that of copolymers we regrouped in table 2.4 values of Y and M found in dierent reports. The terpolymers in these studies contain between 7 and 9 of% CTFE.

Value Ref

M 31 (nm 2 .V -2 )
3, 2.3, 2.4 [18], [19] Y (GPa) 0.4, 0.1, 0.16 [20], [18], [19] Table 2 We can see that while the strain is superior in terpolymers, the stress in twice smaller. This is for an electrical eld of 20 V.µm -1 and terpolymers are likely to be more ecient at higher electrical elds where they can reach strain up to 4 %, as shown in gure 2.3. Because of non linearities in the strain-electrical eld relationship (g 2.3, [21]), the electrostrictive coecients are never used at high electrical elds and it becomes impossible to reproduce the comparison in table 2.5.

From the overall lack of data it is impossible to conclude properly on the interest of terpolymers over copolymers and even more so on the inuence of CTFE. In particular, there are almost no reports on terpolymers with a low CTFE content, below 7% [17]. It is likely that these polymers have been dismissed as promising materials because of their incomplete transition toward relaxor materials. Nevertheless, this lack of interest seems unwarranted and they will be incorporated in our study.

Despite these uncertainties on their performances, terpolymers are used in demonstrators at both low [22] and high electrical elds [23], or in patented devices [24][25].

Printing

The rst tryouts were spin coated polymers with sputtered gold electrodes, but too few devices were functional after the process. It could be the consequence of manually positioning the electrodes masks or the gold deposition itself. Either way we chose to use a dierent process and more specically a screen printing technique. A survey of the other possible methods (slot-dye, inkjet, engraving...) and their features can be found in [26]. Screen printing ts our needs and is quite easy to use, making it an adapted technique for the following study. The principle is similar to that of a stencil but the hole is replaced by a thin mesh. The paste is spread out by a blade with programmed speed and pressure instructions. There are several parameters that can be tuned to obtain the desired layer thickness: blade rigidity, paste viscosity, speed, pressure and mesh dimensions. Even so, the thickness range we get is quite small and limited, from 0.8 to 2.5 µm thick polymer layers. The diculty to reach smaller layers is seen as a drawback of this technique for devices such as organic transistors. In this study oriented toward actuation, there are competing interests regarding the ideal thickness.

A thick polymer layer requires a very large electrical potential and cannot be actuated at high electrical elds. On the other hand thin devices have higher chances of failure and increased leakage currents. A thickness of 2 µm is a decent compromise given the 200V generator at our disposal.

With screen printing it is possible to deposit dierent patterns onto a single substrate. Taking advantage of that, we designed a mask with 12 cantilevers and 4 disks of dierent sizes. The cantilever width is either 2, 4 or 10 mm and the length 5, 10, 15 or 20 mm. A picture of a sample is shown in g 2.5.

The presence of several geometries makes of a single printed sheet a versatile tool for the dierent studies. The rst objective of this design is to verify if the analytical model explains correctly the experimental deections so we can use the analytical model to extract electromechanical coecients. Secondly, it allows to change the substrate, the frequency or any other parameters and always have a few geometries adapted to the new framework. As an example, long cantilevers have larger deections but higher vibration modes appear at lower frequencies (annex). Lastly, some devices are not ideally suited for mechanical studies but are appropriate for other experiments. The upper row (10mm width) was principally used for diraction measurements where a large surface allows for faster acquisition. The disks were supposed to be used for charge generation although this path was not pursued eventually.

Characterization

Once the 3 layers (PEDOT/P(VDF-TrFE-CTFE)/PEDOT) have been printed onto the PEN sheet, the cantilevers are cut out using a laser with programmed pattern in order to have a cut as reproducible as possible. At that point the fabrication process is over and we move on to the electric and mechanical characterization. The polarization cycles are acquired by a Padiant premier precision ferroelectric tester from Radiant inc. The tester is commanded with the dedicated Vision software and the measurement unit is a virtual ground circuit.

The cantilever deection is measured using a chromatic confocal sensor (STIL Initial) with a resolution of a few hundred nanometers. The radiant tester is linked to the STIL sensor and sends it a trigger when the polarization measurements starts. That way we have the electrical eld, the polarization and the strain as a function of time. Unfortunately, the radiant tester does not accept any interaction other than through its dedicated software. Thus, the bench automation was done using a python macro to deal with the laser directly and simulate the user behaviour in the vision software when the radiant tester was needed. This allows for lengthy measurements with dierent waveforms, frequencies and amplitudes without the presence of an operator. A picture of the characterization bench is displayed g 2.6. The link between measured deection and polymer properties is made using beam theory.

Considering the equilibrium at the layers interface and writing the sum of forces and the sum of moments as null yields the relation 2.4 [1]. T : stress inside the polymer layer

In the case of thin lms, the cantilever deection is directly proportional to the polymer strain times its rigidity i.e. its stress. This makes a comparative study actually transferable without the need to measure the polymer Young's modulus beforehand.

In this discussion we neglected the inuence of electrodes. Being only a micron thick (against 125 µm of PEN) and with a Young's modulus of the same order as the substrate, their inuence is negligible in the present case. This was veried with a more complete model of the multilayered cantilever that can be found in [28].

First experiments

Cantilever length

Before moving on to the polymers comparative study we checked how our processed samples relates to beam theory (eq 2.5). Fig 2.7 shows the deection at 120 V of terpolymer based cantilevers with 1.6 µm thickness, 4mm width and 4 dierent lengths (0.5-1-1.5-2 cm).

The blue dots are deections of 4 cantilevers measured experimentally and plotted as a function of their squared length. The measurements follow closely a linear evolution with L 2 , as predicted by equation 2.5. Based on this observation, all the deections in the following work will be normalized to that of a 1cm long cantilever (eq 2.6).

δ N = ( L N L ) 2 δ (2.6)
δ N : normalized deection δ: measured deection

Cantilever width

After the eect of length we study the inuence of width. Following cantilever equation 2.4, the devices width should not aect the results. However, this equation obtained analytically supposes that the cantilever length is much larger than its width, a condition hardly respected. We used a nite element software to estimate how this can impact our measurements (g 2.8). As we can see, the end is also bent in the width direction. In a 10mm wide cantilever the height variation reaches 20% of the deection in the length direction. This eect is reduced in narrower polymers as it represents 7% and 3% respectively for a 4mm and 2mm wide cantilevers. In order to stay as close as possible to the theory we should use the narrowest cantilevers.

However, this is not the only place where we deviate from beam theory. The other and more problematic dierence with an ideal beam is the presence of additional substrate around the cantilever after the cut. There is about 1mm of PEN on each side which is not covered by the active stack and is likely to hinder the cantilever deection. To estimate the deviation induced by the additional substrate we can take a look at the ratio of the actual stack width over the total width. The deection of an ideally cut device and the deection measured experimentally are linked through equation 2.7.

δ = w P V DF w P V DF + w P EN δ th (2.7)
δ th : estimated deection of an ideally processed device. w P V DF : active layer width w P EN : total width δ: measured deection As an other way to characterize the 'dead' width inuence we used COMSOL, a nite element based software. Using equation 2.7 or nite element simulations we are able to estimate δ th , the deection of an ideally processed device. This theoretical deection is the value to be used in equation 2.5 if we want to extract the polymers physical constants.

In order to estimate the inuence of width on deection, cantilevers taken from the same sheet but with 3 dierent widths are submitted to a 100 V bias. The deections are measured and displayed g 2.10 as the three green dots. The 10mm large cantilevers have a displacement 50% higher than 2mm ones. The red and purple curves are the two aforementioned corrections to estimate how an ideal device would deect. This gives an estimation of the substrate impact. However it is hard to pinpoint an exact value for the corrections because the two approaches do not yield the same results. Therefore, we chose to work with only 4mm wide cantilever so there is no need to normalize the deection with respect to their width.

Conclusion 2: Devices geometry

The screen printing technique allows for the processing of dierent geometries in a single run. Using this feature the printed PEN sheets have cantilevers with 14 dierent geometries.

The following study is done on 4mm wide cantilevers and the deections are normalized to that of an equivalent 1cm long cantilever.

Impact of the CTFE content

In this section we will study the impact of the CTFE content on the electromechanical response of the terpolymer.

Electrical evolution

Leakage currents

First, we take a look at the leakage currents in polymers because it can aect our polarization measurements. The leakage currents are the charge owing through the sample and a net loss from an energetic standpoint. It is not an intrinsic response of the material and these charge movements do not participate to the polymer strain. Leakage currents could indirectly bend the cantilever with the heat they generate but we dismissed this phenomenon by monitoring the temperature with an IR cam. The leakage currents depend strongly on the fabrication process and the device itself. The choice of electrodes alone can lead to decades of dierence in the amount of leaked charges [START_REF] Hanna | Characterization of current transport in ferroelectric polymer devices[END_REF].

However, through these important variations, identically processed copolymers and terpolymers will not have the same conductivity. Overall, terpolymers turn out to be much more leaky than copolymers. The distortion of polarization measurement due to leakage can be monitored by equation 2.8.

Q l = 1 2 T /2 0 i l (t)dt (2.8)
Q l : increase in the remanent polarization due to leakage i l : leakage current In equation 2.8 t = T 2 corresponds to the rst point where V=0, i.e the value in the polarization cycle that we read as the remanent polarization. Using equation 2.8 we can assess how much we overestimate the remanent polarization of polymers.

Q l = GU 0 2ω (2.9) G: Device conductance ω: sinusoid pulsation U 0 : sinusoid amplitude
The increase in a copolymer remanent polarization is around 0.01µC.cm -2 at 1 Hz, 100V and with leakage currents taken from g 2.11. In terpolymers the remanent polarization is overestimated by 0.3 µC.cm -2 , the same order of magnitude as the actual value. These overestimations correspond to an error of 0.1% in copolymer and 100% in terpolymer, which is problematic in the latter. At 20 Hz however, this deviation is about 20 times lower and the error represents only 5% of the measured value in terpolymers. For this reason the following polarization cycles are acquired at 20Hz.

Polarization

The polarization cycles of the nine compositions are displayed in g 2.12. The radiant tester allows for dierent types of cycles and we measured here the bipolar (-100 V µm -1 to 100 V µm -1 ) and the unipolar (0 to 100 V µm -1 ) response to a triangle waveform. In gure 2.12, the bipolar cycles are on the left and the unipolar cycles on the right. The rst thing we observe is that instead of a continuous change with the addition of CTFE, there is an abrupt transition. As a visual aid we use a color code for each family: red for the copolymer and the 2% CTFE terpolymer, green for terpolymers with a CTFE content between 4 and 7.5% and blue above. The 3 categories will be hereafter referred to as copolymer, ferroelectric terpolymers (FT) and relaxor terpolymers (RT). It makes things easier as we can focus qualitative studies on one representative of each class (0%, 4% and 8.3% CTFE) instead of the nine compositions.

The copolymer is ferroelectric with a remanent polarization of 7 µC.cm -2 which is expected for this kind of VDF/TrFE ratio (g 2.2). Its unipolar response (red cycle on the right panel) is much smaller and only reaches 1 µC.cm -2 at 100 V µm -1 . Provided that the ferroelectric domains are already aligned adequately, there is no ferroelectric switching during the unipolar cycle.

Therefore the dierence between g 2.12 a. and b. is the contribution of ferroelectric domains and we can monitor its evolution with the CTFE content.

The FT have a remanent polarization value near 2.5 µC.cm -2 and their unipolar response reaches the same value. Compared to copolymers, they have a lower ferroelectric response but an otherwise higher polarization. This trends is accentuated with RT polarization cycles (in blue) which are not ferroelectric anymore but possess an even higher unipolar response.

Permittivity

The polymer permittivity was measured with an impedance meter at 0V and 20Hz. It characterizes the polarization response when submitted to a small electrical signal. The link between the capacitance and the full polarization cycle of polymers will be discussed in the following chapter.

The aim here is to monitor how the amount of CTFE impacts the permittivity in terpolymers.

The results are displayed g 2.13.

Conclusion 3: Impact of CTFE on electrical properties

The impact of CTFE on the electrical response is a reduction of the ferroelectricity, an increase of permittivity and leakage currents. Instead of a continuous transition from copolymer to relaxor (RT), the terpolymers between 4 and 7.5% (FT) are almost identical with an abrupt change of permittivity and ferroelectrictity around 8% CTFE.

Mechanical evolution

The cantilever equations have been detailed above, and the deection is an image of the polymer strain (eq 2.4). In order to observe how strain evolves with composition, we measured the deection of cantilevers with dierent amount of CTFE and displayed the results in g 2.14, with one representative of each polymer type. Figure 2.14 shows the normalized cantilever deection for the three dierent categories of polymers, when a bipolar or a unipolar electrical eld waveform are applied. First, P(VDF-TrFE) exhibits a buttery shape in bipolar regime and a piezoelectric linear response in unipolar regime (Fig. 3a and3d). In the case of the ferroelectric ter-polymer (Fig 2 .14 b and e), the smaller dierence between bipolar and unipolar cycles is consistent with a reduced ferroelectric contribution. For the relaxor ter-polymer (Fig. 3c and3f ), the unipolar and bipolar cycles superimpose exactly, likely due to an absence of ferroelectricity. Beyond a specic electrical eld (around 60 V µm -1 ), the mechanical response exhibits a linear regime with very little hysteresis.

Conclusion 4: Impact of CTFE on the polymers behaviour

The copolymer has a marked buttery shape due to its ferroelectric nature. Below 7.5%

CTFE the buttery still exists in terpolymers although strongly reduced and beyond 7.5%

CTFE the strain can only be negative but retains an hysteretic shape. The changes in strain behaviour are consistent with the change in ferroelectric properties observed above.

Performances as actuator

We want to compare the performances of terpolymers as actuators, but the dierent behaviours make the results dependent on test parameters such as the range of electrical eld. For example if we take a bipolar regime, copolymer has a much larger deection than terpolymers, that cannot reach a positive strain (g 2.14). However, the ferroelectric domain switching comes at the cost of a huge energy consumption that would not appear otherwise. The results of a comparative study in bipolar regime are opposed to that of a study in unipolar regime in terms of performance and energy cost. The chosen settings are the following:

-Unipolar regime: only positive potential is applied -Low frequency: 1 Hz -Established regime: from the 2 nd to the ∞-th cycle

Mechanical performances

Figures of merit

The objective when using an actuator is to generate either a displacement or a force. The force generated by a cantilever is equivalent to the minimum force required to block the deection.

It is expressed as eq 2.10:

F b = 3t s lt s t p Y p 4L S (2.
F b = Y s t 3 s l 4L 3 δ (2.11)

δ: displacement of the cantilever free of any load

The blocking force of a cantilever is proportional to its displacement when it is free of any load. Therefore, a comparative study on the cantilevers deection also communicates their ability to generate a force.

Another way to assess the mechanical performances of a polymer is through its stored elastic energy. In the literature this gure of merit is given for the polymer free of any constraint (equation 2.12) [START_REF] Zhao | Electromechanical properties of electrostrictive poly(vinylidene uoridetriuoroethylene) copolymer[END_REF] [19].

E mp = Y p 2 S 2
(2.12)

E mp : mechanical energy density of a polymer

This quantity is much larger in a terpolymer than in a copolymer and the terpolymer appears more than 10 times better in this regard [20]. However eq 2.12 is not suited in our case, where the polymer has to drag the substrate along with it. The relevant parameter is then the elastic energy of the whole cantilever and we can calculate it using eq 2.13

E m = L 0 Y Iδ ( x) 2 dx (2.13)
E m : mecanical energy of a bent cantilever I: Moment of inertia of the beam Considering small deections, δ(x) is a parabola and δ is expressed as

δ (x) = x 2 L 2 δ (L) (2.14)

δ(L): deection measured with our sensor

This gives a simpler form of eq 2.13

E m = Y s t 3 s l 6L 3 δ 2 (2.15)
This physical value is proportional to δ 2 and once again a simple study on the cantilevers deection will be enough to compare the dierent polymers. We will use eq 2.15 to calculate the electromechanical coupling later on but we can already infer the dierence with the common gure of merit in the literature. Using equations 2.12, 2.5 and 2.15 we get the ratio between the mechanical energy of the free standing polymer and that of the whole device (eq 2. 16).

E polymer E device = 3t p Y s t s Y p (2.

16)

This ratio corresponds to the ability of the active layer to transmit its elastic energy to the rest of the device. As we see it is proportional to the polymer rigidity and the terpolymers are likely to lose their overwhelming superiority once embedded into an actual device.

Conclusion 5: Figures of merits of polymers

Measuring the deection of cantilevers also gives the force they can generate and the elastic energy stored. The energy is that of the entire device and not the usual gure of merit of free standing polymers. As a result this gure of merit is less favourable to terpolymers and their interest for actuation is put into question.

Deection proles

The studied terpolymers FT and RT have respectively 4% and 8.3% CTFE content. Their deection below 20 V µm -1 are compared in g 2. 15.

The copolymer based cantilever (green curve in gure 2.15) deects more than both terpolymers (purple and yellow curves). At 20 V µm -1 the copolymer reaches 1.4µm while the FT is at 0.8µm and the RT deection is only 0.2µm. The dierence between RT and copolymer is even

higher than what we estimated from the electromechanical coecients taken in the literature

(table 2.5).
Attempts at improving the actuation performances by tuning the fabrication process [21] or mixing the RT with other blends [19] do not seem promising for low electrical eld applications. A sevenfold increase of the relaxor terpolymer stress would only allow it to reach the copolymer performances. We increase the electrical eld and display the results in gure 2.16 to see how this comparison evolves. The left pannel displays the complete unipolar deection cycles of the three polymers and the right panel only the forward direction (increasing eld) used hereafter to calculate the cantilever elastic energy.

At high electrical elds the copolymer based cantilever has the lowest deection and even the RT performs slightly better beyond 40 V µm -1 . As it turns out, the increased strain of P(VDF-TrFE-CTFE) relaxor terpolymers is almost perfectly balanced by the decrease in Young's modulus reported in dierent studies [20] [19]. The FT on the other hand diverges from the other polymers with a 6µm dierence at 140 V µm -1 . That is 35 % higher compared to a copolymer and this dierence seems to only improve going at higher electrical elds.

In early studies FT have been quickly dismissed as potential candidates for applications, most likely because they were not as ferroelectric as a copolymer and reached lower strains compared to higher CTFE/CFE concentrations [START_REF] Klein | Inuence of composition on relaxor ferroelectric and electromechanical properties of poly(vinylidene uoride-triuoroethylene-chlorouoroethylene)[END_REF]. As we can see in gure 2.16 they turn out to outperform both copolymer and relaxor terpolymers. A consequence of the improved deection is that FT also possess the highest mechanical energy (g 2.17).

Figure 2.17: Elastic energy stored in the cantilevers as a function of the electrical eld.

Calculated from g 2.16

Figure 2.17 is the elastic energy of cantilevers as a function of the electrical eld and will be of use to get the polymers coupling eciency. It is calculated using the displacement values displayed in g 2.16 and equation 2.15.

To simplify the reading, the gures presented above only display a single polymer in each category. Although polymers of a given family (copo/FT/RT) have the same kind of response, the amplitude they reach can vary. Therefore, a similar study was performed on the nine polymer compositions.

Representing the strain curves on the same gure would only make it unreadable so we chose an arbitrary high electrical eld value of 100 V µm -1 and compared the nine compositions at that point. ). Because of the dierent polymers nature, g 2.18 changes depending on the chosen elctrical eld value. However the trend (dotted line) is similar for any electrical eld above 80 V µm -1 . With this gure, we can narrow down the optimal composition of terpolymer to a CTFE content near 5%.

Conclusion 6: Mechanical performances

A higher deection implies that the cantilevers also possess a higher blocking force and elastic energy. Used at low electrical elds (< 20V µm -1 ) copolymer based cantilevers have the highest deection. At higher electrical elds the ferroelectric terpolymers are better with a CTFE content around 5-6 %.

Electrical energy and coupling eciency

Figure of merit

We now focus on the energy cost of these actuators. The electrical energy stored in a device is written as eq 2.17

.

W e = V (t).i(t)dt (2.17)
W e : electrical energy brought to the cantilever V: electric potential at the electrodes i: current in the circuit An electrical current is the derivative of the charge versus time. Thus eq 2.17 can be reformulated into eq 2. 18. Here we can make a parallel with energy harvesting for which it is interesting to have a high value for 3 and a low one for 2. For its part, actuation needs 2+3 to be as small as possible to obtain a high coupling eciency; these objective are not exact opposites but are quite dierent nonetheless.

W e = Qmax 0 V dQ (2.

Electrical energy and coupling

Applying the principle above to our polymer gives g 2.20.

Figure 2.20: Electrical work of three cantilevers, same as g 2.17

The copolymer requires much less energy than both terpolymers; at 140 V µm -1 the electrical energy stored in the device is 40 µJ around one third the amount stored in the other ones. Now that we have both mechanical and electrical energy we can calculate our second gure of merit, the eective electromechanical coupling coecient. This coecient is adimensional and represents the eciency of the device, it is dened with equation 2.19. Each application requires its own coupling eciency study. There is a strong dependence on the electrical eld but even at 140 V µm -1 the RT coupling eciency is half that of the copolymer. The FT are also less cost ecient than copolymers but better than RT. Overall the coupling coecient is very low, under 0.01% for all polymers. This is due to the nature of our device where only the thin polymer layer receives electrical energy and have to transmit it to the whole device. Based on eq 2.16 the coupling coecient of a free standing copolymer lm would be around 0.8 % at 60 V µm -1 while it would reach 1.5 % for a RT. Applied to an actual device it turns out that the copolymer is by far the most ecient among the dierent compositions.

k 2 = E meca W elec (2.19)

Conclusion 7: Electrical energy and coupling eciency

The electrical work is calculated from the polarization cycles and the terpolymers require several times more energy to reach equivalent deections. The increase in electrical cost is much larger than the increase in mechanical output. This translates into a coupling coecient of terpolymers about half that of the copolymer.

Scope of the comparative study

We evaluated the performances of nine dierent terpolymers processed identically. However, there are countless combinations of compositions and annealing conditions and we want to assess the scope of the results presented above.

Inuence of the processing conditions

There are few studies focused on the impact of annealing duration and temperature on terpolymers mechanical performances. Sigamani & al studied the impact of annealing duration, up to 24 hours with no signicant increase of the terpolymers strain [START_REF] Sigamani | Eect of carbon nanollers on the microstructure and electromechanical properties of electroactive polymers[END_REF]. While this reassuring, the annealing duration of our process is even lower at only 15 min. We veried on a few compositions that for durations this short the terpolymer behaviour is not dramatically modied. The dierence is smaller than the statistical variation from supposedly identical samples (g 2.18). The second annealing parameter is the temperature. Annealing below the Curie temperature or above the melting point is detrimental to the actuation performances [21] so we chose a temperature in between at 115°C. We made a few dierent samples to check if our choice of an identical annealing temperature for all terpolymers could impact the results. Left panel is the RT, there is little to no dierence between the two annealing temperatures.

Regarding the FT, a slightly dierent rst cycle hints that higher annealing temperatures might favor the ferroelectric behaviour. FT could possibly be slightly enhanced by dierent annealing

VDF-TrFE ratio

The main dierence between our nine terpolymers composition is their CTFE content. However, it is likely the VDF/TrFE ratio also plays an important role in the electromechanical behaviour. The left panel is the permittivity of a 65.6/26.7/7.7 terpolymer and shows a frequency dependence typical of relaxor ferroelectrics. The right panel is the terpolymer 80.7/11.6/7.7 and its response is that of a dielectric material, not of a relaxor. This shows how our study only holds for a particular VDF-TrFE ratio around 60/30 in P(VDF-TrFE-CTFE). However, it is easier to perform electrical measurements than mechanical ones and what we showed with this study is that the form to look for is that of a ferroelectric terpolymer.

Conclusion 8: Scope of the comparative study

The results of our comparative study should hold true for dierent processing conditions such as annealing temperature and duration. However the considerations on the terpolymer nature are very dependent on the VDF-TrFE ratio, a dimension we did not investigate.

Conclusion

In this chapter we assessed the capability of P(VDF-TrFE-CTFE) terpolymers to act as the active layer of an actuator. The actuators are cantilevers made using screen printing, which allows making dierent geometries in one run. With dierent lengths and widths it is posible to verify the match between the thin beam theory and our measurements. The length and thickness have the expected inuence on the deection which allows for a normalization of the data. Mostly because of the additional substrate around the edges, cantilever width also impacts the deection. For the study presented here we get around this obstacle by only comparing beams with an identical 4mm width.

We studied nine dierent compositions starting with the copolymer (0% CTFE) up to 9.7 % CTFE content. The addition of CTFE transforms the copolymer into a relaxor but the transition is not gradual and the polymers can be separated into three families. From 2 to 7.5 % the terpolymers are still ferroelectric with a remanent polarization at 2-3 µC.cm -2 . We refer to them as ferroelectric terpolymers or simply FT. Past 7.5% CTFE content there is another change in the electromechanical response and the polymers fully translate into relaxors. Electrically there is no remanent polarization and the permittivity increases from 9 in copolymer to 18 in FT and 28 in RT. Mechanically the buttery strain cycle disappears although the cycle is still hysteretic.

The abrupt changes and the three distinct behaviours allow us to focus on only one composition of each polymers family. We measured the deection and the polarization as a function of electrical eld. Deection is a gure of merit in itself but it also bears information on the blocking force and the elastic energy of the device. The electrical consumption is obtained with the polarization cycle and coupled with the elastic energy it gives our second gure of merit: electromechanical coupling. The main conclusions of the comparative study are displayed in the following table 2.25. For actuation purposes the choice of a polymer depends on the importance given to the stress generated and the energy consumption. In terms of deection and blocking force the ferroelectric terpolymers turns out to be the best at high electrical elds and copolymer has the highest displacement below 40 V µm -1 . Regarding electromechanical coupling eciency, the copolymer coecient is about twice that of terpolymers and that is even without considering the increased leakage currents of terpolymers.

The choice in composition was based on the state of the art in the literature. However, there are a lot of other formulations for the VDF/TrFE/CTFE ratio, not to mention recent materials such as terpolymers blends with even more possible combinations. To avoid the tedious characterization of countless new recipes, it seems important to better understand the mechanisms at the origin of terpolymer performance. In the following chapter we will analyse the polarizationstrain relationship of co-and ter-polymers in an eort to uncover the strain mechanisms at hand.

Polymers under low electrical eld

In this chapter we will try to understand the PVDF-TrFE copolymer and PVDF-TrFE-CTFE terpolymers behaviour through a study of their polarization-strain relationship.

Leakage measurements

In the previous chapter we got rid of the leakage currents contribution by acquiring at higher frequencies. However, it is likely that other contributions to the polarization cycle are also frequency dependent. Ideally, we want polarization measurements to be done at the actuation frequency and thus we need to isolate and remove the contribution of leakage currents.

The conduction mechanisms are not clearly identied in our polymers. In PVDF derivatives, electrons are injected in at the electrodes and can move through the sample with dierent regimes and conduction laws [1]. In a rst attempt to remove the leakage currents from the polarization data, we performed IV measurements using an Agilent 4156C. The measurement principle as well as the results are displayed in g 3.1. The amount of leakage in the sample depends on the measurement parameters and mostly on the soak time. This is likely to be due to slow relaxations processes taking place in the amorphous part of terpolymers [2].

To understand what is the impact of the leakage currents on the measured polarization response we can calculate the cycle of polarization of a purely resistive component using equation good portion of its opening. The blue cycle is a standard polarization measurement acquired using the radiant tester, as described in the previous chapter. The red and purple cycles are the calculated cycles of a pure resistor whose resistance value is taken calculated from the red and purple leakage dataset in g 3.1. The order of magnitude is correct but the uncertainty on the device conductance makes it dicult to make sure that leakage currents alone are responsible for the polarization cycle opening.

3.1. Q(V ) = tmax 0 GV (t)dt

Permittivity and dielectric losses

The polymers are dielectric materials and as such can be polarized when an electrical eld is applied. From a physical point of view this polarization may come from several contributions as represented in g 3.3. >> 0 , D can be approximated by the material polarization P . This equations shows why an ideal dielectric material is the electrical equivalent of a pure capacitance.

However, a material is never ideal and regardless of their speed the physical mechanisms at the origin of the polarization need some time to reach equilibrium. This translates into a polarization response slightly out of phase with the electrical eld and is described by complex permittivity, following equation 3. ). We switched from a triangular waveform in g 3.2 to a sinusoidal waveform for the electrical potential. This does not aect the following considerations but is more suited to the complex formalism and easier to manipulate. is added in series with the ideal capacitance. A parallel resistance the same as the leakage contribution also results in a phase shift but only a series resistor introduces a time constant in the system that can model properly the transient response and the relaxation eects. This equivalent R-C series branch is also suited to model the mechanism frequency evolution, as described in the Debye relaxation theory.

In an actual dielectric several mechanisms can contribute to the polarization response, as presented in gure 3.3. Each one has its own time response and adds up to make the total electrical current, which is the value that we actually measure. Electrically, the physical mechanisms correspond to R-C series branches in parallel with one another. Figure 3.5 shows the polarization cycle and equivalent electrical circuit of a theoretical dielectric with two dierent mechanisms. (orange and blue cycles). What we would measure with our radiant tester is only their sum, i.e the green cycle. From the tilt and the ellipse opening we could extract a capacitance and a resistor equivalent to the whole circuit but not identify the two contributions. This is pretty much the process done in a LCR meter with the measurement of the current gain and phase.

Conclusion 9: Leakage current permittivity and dielectric losses

Leaked charges are free charges owing through the sample while dielectric losses are bound charges lagging behind the electrical eld. We want to remove the leakage currents from the polarization cycles but their contribution to the charge measurement is mixed to the capacitive response. IV measurements give a good idea of their amplitude but the dynamic relaxations in the material render this approach unsuitable for a quantitative analysis.

Making use of the raw data

The polymers studied here are dielectric materials with possibly several contributions to the polarization response. As we discussed in the section above, each one is electrically equivalent to a RC branch in parallel with the rest. Based on our observations detailed hereafter and on the work in [2], a good electrical equivalent of the polymers is the circuit in g 3.6. is the reason why we said in the previous examples that the only accessible parameters are the ellipse tilt and opening (or gain and phase in an impedance-meter). There is a post-measurement processing step in the Radiant tester, consisting in a removal of the rst polarization cycle and a centering of remaining cycle. This is the default conguration of the tester and also a standard of polarization measurements. In ferroelectric materials the charge state at the electrode is not null at t=0, thus the centred cycle is the actual charge value at the electrode. However, in a dielectric material there is no reason to do so and we lose potentially useful information.

1-The transient response is the consequence of polarization lagging behind the electrical eld (R 2 ). Yellow cycle in gure 3.7 2-The oset is only due to the pure losses (R 1 ). Green cycle in gure 3.7 The rst point is pretty straightforward, a pure resistor or capacitance does not have a transient response. To justify the second point we will show that once the transient response is over in an R-C series branch, the positive and negative "remanent polarization" have opposite values. Therefore the higher positive value is only coming from the pure losses.

Q RC (t) = Q 1 (sin(ωt + δ) -sin(δ)e -t τ ) (3.5) Q RC : charge Q 1 : amplitude τ : time constant ω: pulsation δ: phase shift
The above equation describes the temporal evolution of the charge in an R-C series branch.

Calling T the sinusoid period, the "remanent polarization" is positive when the time is an odd multiple of the half period and negative when it is an even multiple. This gives eq 3.6 The remaining polarization is a linear slope, due to the parallel capacitance response. With these four equations, we get an analytical system to identify the dierent components. The system is not linear and a numerical solution is more easily obtained. Applying it to the breadboard circuit of g 3.7 gives the values in table 3 The R 2 -C 2 branch decays rapidly which is consistent with a phenomenon such as ions displacement [4], interfacial polarization [5] or, relaxations in the amorphous phase [2]. Our electromechanical measurements are conducted at 1 or 2 Hz (our maximum actuation frequency)

Q RC+ (n) = Q 1 sin(δ)(1 -e -(2n+1)T 2τ ) Q RC-(n) = Q 1 sin(δ)(-1 -e -nT τ ) ( 3
and the dielectric losses are limited. The leakage currents have an important impact on the measurement but can be easily removed as discussed above. The "low electrical elds" denomination includes electrical elds up to 20V.µm -1 . Above that value, the aforementioned description cannot properly describe the polarization cycle. approach is that it constitutes a simple way to remove the contributions of leakage current directly from the remanent polarization asymmetry.

Transition to high electrical elds

At low electrical elds, both copolymer and terpolymers behave as lossy dielectric materials.

Moving on to higher electrical elds the non-linear behaviour due to their ferroelectric and relaxor nature are expected to come into play.

Copolymer

Aside from the losses, we observed that the slope of the polarization curve matched the capacitance value below 20 V.µm -1 . We check if this holds as the electrical eld is increased from 20 to 80 V.µm -1 . The measurements are done with unipolar cycles in order to avoid the switching of ferroelectric domains and the results are displayed in g 3.10. We also added the theoretical response of an ideal dielectric material with a permittivity of r = 10 corresponding to the polymer value. At 80 V.µm -1 the polarization reaches 1 µC.cm -2 instead The second deviation from the lossy dielectric behaviouris due to the ferroelectric nature of the copolymer. The ferroelectric domains can be switched if the electrical eld is strong enough to overcome their cooperation. If we submit the copolymer to a unipolar signal, the domains may or may not be switched depending on their initial orientation. Using that feature we can get the signature of the ferroelectric domain switching as the dierence between the response to the same solicitation but with dierent starting polarization states (up or down). That is the principle behind the established PUND method available in the Radiant tester software. Figure 3.11 shows the measured cycles and the resulting ferroelectric cycle.

Conclusion 11: Copolymer at high elds

The capacitance measurements cannot describe the unipolar polarization cycles in their

entirety. An hysteretic phenomena also contributes to the polarization response, it accounts for 20% of the total value at 80 V.µm -1 . In bipolar regime, that contribution is superimposed to the polarization variation induced by ferroelectric domains switching. The ferroelectric cycle can easily be isolated with the adapted method.

Terpolymer

As we mentioned before, the leakage current in terpolymers cannot be neglected. We apply the principle discussed previously to remove the contribution of said current. This approach supposes that no ferroelectric contribution is present in the material. The leakage correction applied to a terpolymer cycle is shown g 3.12

The left panel of g 3.12 displays the polarization cycle at 1 and 20Hz of a terpolymer. The right panel shows the corrected 1 Hz cycle assuming an ohmic leakage current, and the 20Hz cycle as an element of comparison. The dierence between the two cycles is due to the frequency dependence of other contributions.

Once the leakage correction is applied we have polarization data only depending on the bound an r of 29, 3.5 times that of the copolymer while at 100 V µm -1 it is only twice as high. Understanding the origin of this electrical eld dependence would require to know the origin of the increased permittivity, which is still debated to this day [7], [8]. Regardless of the origin, what we want to estimate here is how this impacts our polarization measurements, using equation 3.8.

P capa = Emax 0 0 r (E)dE (3.8)
P capa = reversible polarization when the material is submitted to small electrical variations.

Phase transition in the litterature

There is no direct evidence of an electrically induced transition toward a ferroelectric phase in terpolymers but the existence of such a transition was inferred soon after the discovery of their important strain response [10]. The current understanding of the micro-structural arrangement of terpolymer chains is for a good part based on the work of Yang & al. Yang worked toward a description linking the terpolymers polarization response and their chains conformation [7] [4] [11]. The following picture is a summary of the impact of CFE and CTFE in the copolymer chain. In gure 3.16 the case of CTFE is represented in the lower row and diers from that of CFE.

The idea is similar except that a stronger pinning renders the orientation of the CTFE monomers impossible which results in a single hysteresis loop (SHL) with no phase transition. Zhu & al proposed a classication of the dierent dielectric materials based on the degree of cooperation between dipoles (g 3.17).

Figure 3.17: Classication of the categories of dielectrics with dierent levels of cooperation.

Taken from [12] In gure 3.17 lower row, the CFE terpolymers would be the 4th electroactive behaviour starting from the left, and CTFE the 5th. However, the residual polarization cycle we isolated in P(VDF-TrFE-CTFE) reminds of the DHL i.e the 4th schematic representation. Therefore, we propose that the green polarization cycle in gure 3.15 is also the signature of a eld-induced phase transition toward a ferroelectric FE phase. Although Yang & al suggested that such transition does not occur in CTFE, it could simply be less discernible. This DHL behaviour is actually visible in our polarization cycles and has also be reported in another study [11].

It should be noted that the polymers in gure 3.16 were uni-axially stretched, maybe making it more dicult to observe the DHL in P(VDF-TrFE-CTFE). Still, the DHL does not need to be visible to infer the presence of the so-called phase transition. This can be observed with the discrepancy betwee permittivity values calculated from the polarization cycle slope and those measured with an impedance-meter.

If extracted from the polarization cycle in g 3. 16, P(VDF-TrFE-CTFE) permitivity reaches 70 while with impedance measurements we measure it at 30. Therefore the polarization cycle must be enhanced by an additional hysteretic contribution.

We can note that in the case of CFE terpolymers, the two methods give an identical permittivity value, around 50 (g 3.13). As the "eld induced phase transition" is only present at higher electrical elds, it does not impact the polarization value near 0 V V.µm -1 and the explanation on RFE behaviour is consistent with both polarization and impedance measurements.

There are no direct evidence of a eld-induced phase transition in terpolymers and the mechanisms behind the relaxor ferroelectric nature are still debated [8]. Still, this hypothesis describes quite well the polarization cycles of PVDF derivatives and explains the anti-ferroelectric like cycle in gure 3.15. In the following section we will try to see if this explanation is consistent with the strain response of terpolymers.

Conclusion 13: Phase transition in the literature

The DHL (double hysteresis loop) has been observed in P(DVF-TrFE-CFE) and ascribed to a eld-induced phase transition. It is very likely that the same thing occurs in P(DVF-TrFE-CTFE). Having removed leakage and capacitive contributions from the polarization cycle we even isolated the electrical signature of this supposed transition.

Polarization at higher temperatures

We break down the polarization cycle of our terpolymer as the sum of leakage currents, (lossy) capacitance and phase transition cycle. To strengthen this approach we look at how polarization evolves with temperature and check whether this evolution is consistent with our description. dierence between the two cycles.

On the left panel of g 3.19 we measured the polarization cycle of a device at 22°C and 50°C.

We chose this temperature and not a higher one because the dielectric losses increase dramatically with the temperature and interfere with our measurements. The measurements are made at 20Hz on this particular dataset thus we do not apply a leakage correction and these are raw data. As it turns out, the dierence between the two polarization cycles results in a clean signature, much like the anti-ferroelectric cycle in gure 3.15. This result supports the aforementioned hypothesis: polarization cycles of PVDF-TrFE-CTFE at ambient temperature are the sum of a capacitive response and an hysteretic cycle attributable to eld induced phase transition. Above the Curie transition, this transition is not present any longer.

Conclusion 14: Polarization at higher temperatures

By comparing the polarization cycle below and above the Curie temperature we isolate once again the polarization signature of phase transition.

Strain

We now have a good understanding of the meaning of a polarization cycle and will try to correlate it with the strain response of the polymers.

In terpolymers

State of the art of strain in terpolymers

We start with a quick overview of the current understanding of the strain origin in terpolymers.

In this state of the art, the term "terpolymers" refers indistinguishably to our polymers (PVDF-TrFE-CTFE) or to PVDF-TrFE-CFE. This is because the litterature is rather scarce on the subject and these polymers are likely to share similar strain mechanisms.

Terpolymers are said to be electrostrictive materials [5], [13] but saying of a material that it is electrostrictive does not explain the underlying physical mechanisms. It is merely a statement that follows the phenomenological law:

S = QP 2 (3.9)
S: strain Q: electrostrictive coecient

P: Polarization

The squared term in equation 3.9 implies that an electrostrictive material can only be strained in one direction. Ceramics typically have a positive Q 33 coecient while PVDF and its derivatives have a negative Q 33 (the subscript has been introduced in chapter 2). There are several physical eects that can induce strain in a material submitted to an electrical eld. For an overview of electrostrictive mechanisms, one can look at the work of Li & al [14]. In terpolymers there are three likely candidates contributing to electrostriction.

1-Electrically induced phase transition 2-Intrinsic electrostriction 3-Maxwell stress

The phase transition would induce a strain in the material because of the lattice dierence between the two conformations. After the discovery of the relaxor properties of terpolymers, it was proposed that such a phase transition existed and explained the large strain in terpolymers [15] [16]. Following studies showed that this contribution could not explain the majority of the observed strain [17] although this idea can still be found in a recent work [18]. The intrinsic electrostriction relates to how the displacement of atoms in the lattice can translate into a macroscopic strain [14]. Maxwell strain is due to the coulombian attraction between two surfaces with opposite charges. The attraction between the electrodes meets the mechanical resistance of the material and the induced strain is then obtained as eq 3.10 [19].

This eect is completely disregarded in copolymers but terpolymers are less rigid and have a higher permittivity. Sometimes Maxwell strain is discarded entirely, sometimes it supposedly accounts for part of the response [20] and in [9] it is regarded as the sole mechanism behind the

Q 3 ≈ 18 m 4 C -2 .
This value is to be compared with the electrostrive coecient we measure in the following section at

Q 1 ≈ 23 m 4 C -2 . Q 1 characterizes
the longitudinal strain and in both co-and terpolymers the thickness strain is larger than the longitudinal one:

Q 3 > Q 1 .
Although we cannot conclude precisely because of this dierence as well as incertitudes on Y and ν, the Maxwell strain is likely to contribute signicantly to the total strain.

Conclusion 15: State of the art in terpolymers

The knowledge on the origin of strain in terpolymers is limited. Three phenomena are likely to contribute: Field induced phase transition in the crystalline phase, Maxwell strain and intrinsic strain.

Impact of phase transition on the strain in PVDF-TrFE-CTFE Both panels display the cantilever deection but the left panel x-axis is the electrical eld while the right panel x-axis is the squared polarization. As we can see, the deection and therefore the strain are perfectly described by an electrostrictive law. We can extract a value for the coecient Q = 23 m 4 C -2 using the formalism detailed in the previous chapter 2.10. We reiterate this measurement but under a higher electrical eld. The results are displayed in g 3.21 evolves quadratically with the polarization but the coecient Q increases by 50% from 24 to 37 m 4 .C -2 on a limited range between 2 and 7 µC.cm -2 . In the previous section we proposed that the phase transition impacted the electrical response of terpolymers. We singled out its contribution to polarization and located it on an electrical eld range between 40 and 60 V.µm -1 . This seems to coincide with the rupture in the Q -P 2 evolution of gure 3.21. We take a closer look at this correlation by putting in front of one another g 3.21 and g 3.15 in gure 3.22.

Figure 3.22 displays data from the same measurement: electrical eld, polarization (raw and phase transition contribution), and displacement. Each letter (A,B,C) is given as a visual aid and corresponds to a unique state of the material in time. This way we can highlight the following result: the increase in electrostrictive coecient (A 1 -B 1 ) happens precisely where we placed phase transition with the anti-ferroelectric like polarization cycle (A 2 -B 2 ). As a consequence we can ascribe the increase in strain between 40 and 60 V.µm -1 to the dierence in lattice parameters between two phases involved in the transition. It should be noted that with this phenomenological description we quantify the relative contribution of each mechanism in average (macroscopic measurements). The lattice dierence due to phase transition (E) happens only in the crystalline phase. We do not now how much of it, if any at all, is transmitted to the amorphous phase. On the contrary, mechanisms such as Maxwell stress are likely to strain mostly the amorphous phase, because of its lower rigidity compared to the crystalline phase.

Strain at high temperatures

Just like we did with the polarization response (g 3.18 and 3.19), we can check if the phenomenological description presented above is consistent with the behaviour at higher temperatures. We put the cantilevers above a hotplate and measured the deection as a function of the electrical eld. The temperature in gure 3.25 is that of the hotplate and not of the sample itself. This is consistent with the idea of a eld induced transition, hindered by temperature and responsible for the hysteresis. To further check the sturdiness of our description, we measure the polarization and take a look at the polarization-strain relationship. In gure 3.26 we show the polarization alongside the cantilever deection at 80°C and 0.8Hz.

In gure 3.26 the left panel shows the polarization cycle of the terpolymer at 80°C. At that temperatures the dielectric losses are much more important and conduction mechanisms are exacerbated and potentially non-linear [21], hence the yellow cycle. Instead of trying to t the polarization with complex laws we assume that the lossless cycle is the one represented in green.

This cycle is the dielectric polarization calculated from C(V,T=80°) measurements (annex) and using equation 3.14. Without phase transition that would be the only contribution left as per the decomposition shown in gure 3.15. We apply equation 3.9 to the green cycle and compare it to the cantilever deection. The results are displayed on the right panel and are consistent with the description presented so far. Above the Curie temperature, there is no phase transition and we are able to t the strain with dielectric measurements alone.

Conclusion 17: Strain at high temperatures

The strain measurements at high temperature are consistent with our phenomenological description of strain mechanisms in terpolymers. The eld-induced phase transition is most likely hindered by the increasing temperature.

Copolymer and ferroelectric terpolymers

We proposed an explanation for the impact of eld-induced phase transition in terpolymers but the majority of the electrostrictive mechanisms are still not explained (box B in g 3.24). Beside a few considerations on Maxwell strain, there are very few attempts to ll this void in the literature.

Given that copolymers have been studied more extensively, it seems appropriate to start from the strain mechanism in copolymers and see how they evolves with the addition of CTFE in the chain.

State of the art of strain in copolymers

The origin of piezoelectricity in PVDF and PVDF-TrFE is supposed to be electrostriction biased by the remanent polarization [22]. Using equation of electrostriction 3.9 in presence of remanent polarization gives equation 3. Away from the coercive eld, polarization is formulated as the sum of a constant term due to the alignement of the ferroelectric domains (P r , g 3.11) and of the electrical eld induced polarization (P diel , g 3.10). This expression is injected into the strain equation and its devel- opment gives equation 3.11. The rst term (QP 2 r ) is a constant and the quadratic term (QP 2 diel ) is negligible at rst order before the linear one (2QP r P diel ). Therefore, the polymer response is piezoelectric but only as a consequence of electrostriction.

Regarding the physical origin of electrostriction, until 2015 there were two main hypothesis to explain it. The rst one is the so called dimensional model [23], [24] and the other one is an instrinsic eect in the crystalline phase. In 2015, Katsouras & al [25] observed through synchrotron measurements that the lattice parameter change in crystallites corresponds to the macroscopic strain. With this experiment they discarded at once the dimensional model and showed that strain came from the displacement of atoms inside the crystalline lattice. Thanks to these recent observations the understanding of strain in PVDF and PVDF-TrFE is pretty much complete.

1-Piezoelectricity is only a consequence of an electrostrictive behaviour.

2-

The electrostriction itself comes from the displacement of atoms inside the crystalline phase.

However there is a problem with statement 1 above: It fails to describe the strain cycle in bipolar regime. This was rst evidenced rst by Yuki & al [26] and recently addressed by Katsouras & al [25] in an attempt to round o the discussion. They proposed that this discrepancy is due to an additional piezoelectric mechanism adding to the PVDF strain, giving equation 3.12.

S = Q.P 2 + d coupling p.E (3.12)
d coupling : piezoelectric coecient p: parity function as explicited in g 3.27

This hypothesis was based on the observation that adding a linear term in equation 3.11 allowed for a complete description of the bipolar strain cycle, as shown in 3.27. As a physical explanation of its origin, Katsouras & al proposed that it comes from the amorphous phase stressing the crystalline phase under the application of an electrical eld.

Conclusion 18: State of the art of strain in copolymer

Most of the strain in PVDF and its copolymer PVDF-TrFE would come from electrostriction in the crystalline phase. Away from the coercive eld, the induced polarization is biased by a constant ferroelectric polarization and this results in an apparent piezolectric behaviour.

Electrostriction fails to account for the whole copolymer strain and an additional piezoelectric eect is needed to fully describe the bipolar strain cycle.

Copolymer strain measurements

The aim of the following study is to propose a dierent interpretation of the strain in copolymer, in particular of the missing 'piezoelectric' eect.

As we did with terpolymers we start at low electrical elds and confront our measurements to the theoretical framework. At low electrical eld we can approximate P diel with E (g 3.10).

This yields an analytical expression for the piezoelectric coecient d introduced in chapter 1. (to the best of our knowledge) of similar measurements [5] in the literature.

With the copolymer polarized in the positive state we increased the electrical eld while keeping it positive. That way we can observe possible non-linearities without the inuence of ferroelectric switching. We extract the piezoelectric coecients d and g following eq 3.

Conclusion 19: Copolymer strain measurements

The hypothesis of electrostriction biased by the ferroelectric polarization describes perfectly the behavior of the copolymer and ferroelectric terpolymers even at high electrical eld.

Origin of buttery shape in copolymer

The electrostrictive formalism describes correctly the strain response of copolymers. That description does not function when the electrical solicitation is bipolar. In that regime, the switching of ferroelectric domains is at the origin of a buttery shaped strain cycle, as displayed in gure 3.30.

The blue curve is the deection of a copolymer based cantilever measured at 1 Hz. At the same time we acquired the copolymer polarization and used it to calculate the strain according to equation 3.9. We normalized the data so that the model t the measurement away from the coercive eld. Normalizing is just a matter of xing a value for the coecient Q and gure 3.30 highlights how electrostriction fails to describes the copolymer strain around the coercive eld.

The polarization used in the electrostriction equation is the charge surface density measured at the electrode (provided that D ≈ P ). Away from the coercive eld, all the ferroelectric domains amplitude does not imply that the dipoles disappeared but that they are not aligned with the electrical eld. That is to say an empty circle can be interpreted as an in-plane polarization vector, orthogonal to the electrical eld.

In gure 3.31 the left and right columns correspond to two dierent types of domain switching, although they both result in the same polarization measured experimentally. Applying the electrostriction equation (S = QP 2 ) directly to the measured polarization is valid if the material follows an evolution such as the one represented in the left column. When all the domains are in an identical state, the average polarization describe correctly the polarization inside any ferroelectric domain. Applied to copolymer this approach gives the modelled strain in g 3.30 hinting that this description is not adapted to copolymer. The calculated polarization value is plummeting around the coercive eld, as P 2 = 0 implies no strain of the ferroelectric domains.

The right column is another possible mechanism where ferroelectric domains switch one after the other. This representation is a simplication but it has recently been observed to be much closer to the actual evolution of PVDF-TrFE [27]. At the coercive eld (P=0) the dipoles are oriented with or against the electrical eld and therefore the overall strain is not necessarilly null. The strain-polarization relationship around the coercive eld cannot be calculated with an an average polarization value that do not reect the internal inhomogenities in the material.

Isolated domain hypothesis

We want to see how the coexistence of opposite orientations can impact the strain response. As a start we will consider that the ferroelectric domains inside the copolymer are not inuenced by the rest of the material. A crystalline domain only 'sees' the external electrical eld we apply, and not the eld generated by dipoles in other domains or in the amorphous matrix. The polarization of such isolated ferroelectric domains depends on their orientation and we write it as equation 3.

15

P + = P r + apparent E P -= -P r + apparent E (3.15)
P + : Polarization of an up domain P -: Polarization of a down domain P r : remanent polarization of a ferroelectric domain apparent : apparent permittivity In equation 3.15, the P r term is the remanent polarization, a positive constant. This value is the one we would measure in a copolymer monocrystal and we take it as 16 µC.cm -2 . This is mostly a guess [28] based on the macroscopic value of P r and the copolymer 50% crystallinity but an error on P r is not troublesome as it merely impacts the extracted value of electrostrictive coecient Q. It does not change the qualitative considerations hereafter provided that 2P r >> apparent E is still respected.

CHAPTER 3. ORIGIN OF STRAIN IN P(VDF-TRFE) AND P(VDF-TRFE-CTFE)

For a given orientation, up or down, the inuence of electrical eld is the term apparent . We saw previously (gure 3.10) that at high electrical eld the permittivity does not account totally for the unipolar polarization and an hysteretic contribution is also present. We neglect the hysteresis and approximate the unipolar polarization eld by a linear law. Therefore apparent has no physical meaning and is just a way of simplifying the equations hereafter.

With the isolated domains polarization in hand we use an electrostrictive law to obtain their strain, (equation 3.16).

s + = Q(P r + apparent E) 2 s -= Q(P r -apparent E) 2 (3.16)
s + : strain of an up domain s -: strain of a down domain To estimate the total strain as the sum of the local strains, we also need the relative amount of each orientations inside the material. This information can be obtained from the ferroelectric cycle similar to the one in g 3.11. We call f + the ratio of ferroelectric domains in the up state and f -the ratio of domains in the down state. In this simplied approach, the sum of both orientation accounts for all the ferroelectric domains inside the copolymer which is equivalent to equation 3.17.

f + + f -= 1 (3.17)
f + : ratio of positively oriented domains f -: ratio of negatively oriented domains In gure 3.31, at P = -0.5P r we have f + = 1 4 and f -= 3 4 . More generally, functions f + and f -relate to the ferroelectric cycle through equation 3.18.

P f erro = f + P r,macro -f -P r,macro (3.18) P f erro : polarization due to domain switching (g 3.11) P r,macro : copolymer remanent polarization

The value P r,macro is the remanent polarization obtained from standard P(E) cycle around 8 µC.cm -2 , as measured in the previous chapter. We do not call it P r not to confuse with the remanent polarization of a monocrystal in equation 3.15. Combining equations 3.17 and 3.18 we can calculate f + and f -. f + and f -can be obtained from the ferroelectric polarization cycle. The last step to get the total strain lies in taking the strain of each orientation and weighting it by the amount of domain in the corresponding state. This gives equation 3.20.

f -(E) = 1 2 (1 - P f erro(E) P r,macro ) f +(E) = 1 2 (1 + P f erro(E) P r,macro ) (3.19) S = f + s + + f -s - (3.20)
In equation 3.20, s and f are obtained from data in a polarization cycle and Q is the only degree of freedom to link strain to polarization.

We confront the model to experiment in gure 3.32.

Figure 3.32: Normalized strain cycle of a copolymer. Blue is the measured cycle, green is the calculated strain with the average polarization and the red cycle is the strain calculated with isolated domains hypothesis

In gure 3.32 we represented the measurements of a cantilever displacement and the strain calculated with the two approaches: average polarization (S = QP 2 in green) and isolated domains (in red). The strain calculated with the new approach does not show the dramatic drop around the coercive eld like the standard description, but the buttery is squeezed and is hardly a proper t of the measurement. The hypothesis we made of ferroelectric domains free of any inuence from the rest of the material is likely to be the issue and building on the formalism presented so far we try a more realistic description.

Combined descriptions

We considered the ferroelectric domains in copolymer PVDF-TrFE to be isolated because it is embedded in an amorphous matrix. However, the other domains might not be far enough to be disregarded as they make up for 50% of the volume. In addition the amorphous phase is a dielectric and as such can be polarized. The base hypothesis we make now is that the inuence from the rest of the copolymer (outside the ferroelectric domain) is proportional to the average polarization. Thus, the polarization inside a ferroelectric domain is somewhere between that of an isolated domain and the average polarization of the material, value that we measure at the electrodes. Mathematically we write it as equation 3.21.

P +(E) = α(P r + apparent E) + (1 -α)P (E) P -(E) = α(-P r + apparent E) + (1 -α)P (E) (3.21)
P ± : Polarization of a ferroelectric domain apparent : apparent permittivity P r : remanent polarization P (E) : average polarization, measured at the electrodes α: quanties the deviation from the isolated domain hypothesis In equation 3.21, α=1 is the isolated domain hypothesis as described previously. α=0 corresponds to an homogeneous polarization inside the copolymer, i.e the standard approach with equation 3.9. Therefore α is an unknown parameter quantifying how much a ferroelectric domain 'sees' the rest of the material. The expression of polarization is the only dierence with the rest of the previous development (eq 3.16 -3.20). With the electrostrictive coecient Q, α brings to two the number of tting parameters to explain the copolymer strain. This is the same number of parameters as in [25] with the additional piezoelectric eect. We have yet to see if this description can explain the strain cycle in copolymer. The strain calculated with equation 3.21 is displayed in g 3.33 alongside the actual measurements.

The red cycle in gure 3.33 was found for α = 0.9 that is to say a relatively modest contribution from the surrounding polymer onto the ferroelectric domain (10%). This is enough to change signicantly the cycle in g 3.32 and the model matches well the measurement. The main dierence is the strain away from the coercive eld. That is most likely due to the linear approximation of polarization we made when we neglected the hysteresis in gure 3.10 and used apparent instead. This approach:

-Has only two physical unknown parameters, similar to the only other explanation proposed so far -Has a dierent interpretation of the bipolar strain that takes into account the actual microstruc- The main implication of this phenomenological description is that electrostriction alone can account for every nonlinear response with regard to the electrical eld, whether it is the evolution of d 31 in unipolar regime or the buttery strain cycle in bipolar regime.

Conclusion

In this chapter we studied the polarization response of polymers. At low electrical elds, below 20 V.µm -1 both co-and ter-polymers behave like dielectric materials with losses and leakage currents. Even then, polarization measurements give more information than measurements performed with an impedance meter as they also display the electrical temporary regime. This allows for the identication and removal of the leakage current contribution.

The electrostrictive formalism functions perfectly on the low electrical eld range. The terpolymers are electrostrictive with a quadratic dependence to the electrical eld. The copolymer is also electrostrictive but due to the large remanent polarization it has a linear dependence to the electrical eld. This behaviour is described as piezoelectric and we can impose a sign to the piezoelectric coecients d by choosing the remanent polarization orientation. Owing to their mixed nature, the ferroelectric terpolymers have a strain behaviour in between that of relaxor ferroelectric and copolymers. Their strain is dominated by the piezoelectric coecient.

Deviations from the dielectric and electrostrictive formalism appear at higher electrical elds.

In copolymers we started by studying the unipolar regime: the d coecient is eld-dependent while g remains constant. This indicates that the electrostrictive mechanisms do not change but the nonlinear behaviour comes from the polarization response.

In bipolar regime the electrostriction equation fails to describe properly the buttery shape of the strain response. We proposed that the issue lies with an incorrect use of the electrostrictive formalism and not an additional contribution to strain. Near the electrical eld, the ferroelectric domains have dierent orientations and the inhomogeneous state of the material cannot be properly described by the average value measured at the electrodes.

In terpolymers, we used capacitance versus electrical eld measurements to remove the reversible contribution to polarization. The remaining cycle, once corrected of the leakage currents, looks like that of an anti-ferroelectric material. After that, the electrostrictive description was tested out by studying the strain versus squared polarization curve. This allowed for the extraction of an electrostrictive coecient Q. It appears that Q is not constant on the entire polarization range and during a short period between 30 and 60 V.µm -1 , it is 50% higher. The 30-60 V.µm -1 electrical eld range corresponds precisely to the polarization increase in the anti-ferroelectric like cycle.

As both nonlinearities in polarization and electrostriction coecient happen on the same electrical eld range, they are likely to have the same origin. Based on these observation as well as the state of the art, we propose a phenomenological description of the strain mechanism in terpolymers.

While the existence of a eld induced phase transition seems likely, there is no evidence of its existence. Furthermore, the contribution to strain was ascribed to the lattice parameters dierence between phases, which is nothing more than a guess. It would require conrmation by measurements of said lattice parameters. The next chapter is focused on XRD measurements of polymers electrically stressed to determine the validity of those assumptions.

Experiments

We focused this study on a 61.4/30.3/8.3 VDF/TrFE/CTFE terpolymer which is a composition representative of the relaxor polymers family. In chapter 2 we saw that the electrical and mechanical properties are nearly identical between 7.8 and 9.7% CTFE. Therefore, we assume the results from this study can be applied to other compositions as well. Regarding the ferroelectric terpolymers with a CTFE content below 7.8%, their properties are a mix between those of copolymers and of relaxor terpolymers. Understanding their behaviour rst requires a proper comprehension of the relaxor terpolymers.

The XRD study was performed on an EMPYREAN diractometer, as shown in picture 4.1, and a PANALYTICAL diractometer for temperature measurements. The electrical bias is applied through contact needles or soldered cables and driven by a Keithley 2600. The diractometer is used in Bragg-Brentano conguration (illustration in annex). The relevant information for the following study is that the diracting planes of a crystalline domain need to be parallel to the surface in order to contribute to the XRD signal.

With Bragg-Brentano optical conguration, if the sample is not thick enough a part of the acquired signal comes from the substrate. To facilitate the data interpretation we need a substrate whose background is as transparent as possible to X-ray diraction. The changes are irreversible and correspond to a reduction of crystallinity due to irradiation. In a standard diractometer the radiation energy is much lower but the exposure is also longer so we needed to assess whether polymers could withstand it. In gure 4.3 right panel a hundred 5 minutes long consecutive scans are displayed. They were performed on a copolymer sample using the standard diractometer. The spectrum shows no evolution during measurements as the sample is not aected by the X-rays.

Throughout the study all samples broke down at some point, especially during the bias measurement at high temperatures. We ruled out the irradiation as the reason behind the sample degradation so this diculty comes from electrical breakdown. The chosen study parameters are a trade o between spectrum resolution and sample life expectancy.

Conclusion 20: Experiments

The terpolymer chosen for this study contains of 8.3% of CTFE. Most scans are performed on a glass substrate and on a small 3 degrees angular range with a Bragg-Brentano setup. This allows for scans as short as 2 minutes, which increases our chances of acquiring a full electrical cycle before the sample breaks down.

In-situ study of P(VDF-TrFE)

State of the art: structure and XRD characterization

The three phases α, β and γ have been introduced in chapter 1, gure 1.2. These terms are quite ambiguous because they can designate either the organization of monomers inside a polymer chain or the arrangement of those chains with respect to one another. The chain organization is identied by the nature of successive conformations (trans or gauche) between VDF monomers.

Combined with the chain packing, it is responsible for the polymer crystalline structure. This dual designation is legitimate in PVDF because each chain conformation is associated to a crystalline symmetry [1]. However it can become confusing when this terminology is applied to PVDF derivatives. As an example, it is not uncommon to see the P(VDF-TrFE-CTFE) paraelectric (PE) phase referred to as α-phase [2] [3]. Yet, the α phase of PVDF is monoclinic and the terpolymer PE phase is orthorombic.

To visualize the impact of the conformations nature (T or G) on the chain organization, one should refer to gure 1.2. The α phase corresponds to a succession of trans (T) and gauche (G) conformations (T GT G). The PVDF β phase is a succession of trans conformations (T T T T ) and the γ phase is a succession of trans conformations regularly interrupted by a gauche one (T 3 GT 3 G).

The ferroelectric properties of a given phase arise from both the molecular conformation (trans or gauche) and the chains packing (crystal structure). The monomers arrangement inside the polymer chains is best investigated with FTIR measurements while the crystalline lattice is observable with XRD. Therefore XRD and FTIR are complementary and the most common techniques for the structural analysis of PVDF. Reproduced from [4] As for the P(VDF-TrFE) copolymer, it depends on the composition but a 75/25 VDF/TrFE copolymer can have either one of two phases: paraelectric (PE) above the Curie temperature or ferroelectric (FE) below that temperature. The FE phase is very similar to PVDF β-phase but the PE phase has no PVDF equivalent. Table 4.1 sums up the organization of the dierent phases, the data are taken from [1] [5] and [6].

Molecular conformation

Crystal class

α T GT G Monoclinic β T T T T Orthorhombic γ T 3 GT 3 G Monoclinic Copolymer (PE) T GT G Orthorhombic Copolymer (FE)
T T T T Orthorhombic Table 4.1: Molecular conformation and crystalline class of PVDF and P(VDF-TrFE phases)

The information we will obtain from XRD measurements are related the crystalline organi-zation. Panel A is taken from [7] and represents the three lattice directions of the copolymer crystal.

The lattice is orthorhombic with a quasi-hexagonal symmetry [8]. The dipolar moments of the CF2 atoms are aligned with the b lattice parameter and the c parameter is directed along the chain axis.

Panel B is a schematic representation to visualize how the chains are positioned with respect to the crystalline domains and the rest of the sample. The squares are crystallites; the polymer chain inside them are parallel to one another and to the surface. The ferroelectric polarization is represented by red arrows and is aligned with the electrical eld direction. This means that the b-axis of the crystal lattice is normal to the sample surface and consequently the directions a and c are parallel to the surface. In this illustration, the direction a is perpendicular to the drawing sheet plane but it could actually be in any direction perpendicular to b. One should keep in mind that it is a simplied representation. In an actual material all physical quantities such as the polarization orientation are statistically distributed around a mean value.

This crystalline organisation gives the XRD spectrum displayed in gure 4.6 [9]. Taken from [9] The left part of g 4.6 is a schematic representation of the copolymer crystalline lattice. The right part displays the corresponding XRD spectra. It highlights the point that will be a recurring issue throughout this study: the most intense diraction peak is the superposition of two dierent peaks, (110) and (200).

In its PE phase, the copolymer lattice has an hexagonal symmetry. The (200) and (110) peaks are superimposed which results in a thin diraction peak measured experimentally. When in FE phase, the lattice has a quasi-hexagonal symmetry. The two distances d 200 and d 110 are slightly dierent and as a result the diraction peak is a bit wider. The two copolymer phases are easily distinguishable from one another, owing to their signicantly dierent unit cell parameters.

The dipolar moments along the b-axis are at the origin of the ferroelectric properties of P(VDF-TrFE). During polarization switching, i.e the remanent polarization changing sign, the dipolar moments make a 180 °rotation. This motion is achieved with the dipoles rotating around the chain (c-axis) but the chains nor the crystallites do not move in the process.

During poling or polarization reversal, the dipoles rotation associated with ferroelectricity occurs by 60 °steps to align at best with the electrical eld.

Copolymer

Results

The sample used for this study was a 75/25 VDF/TrFE copolymer and was in a poled state before measurement. We ramped up the temperature rapidly to 100°C then heated up the sample more slowly with 5°C increments. At each step, an XRD measurement was performed after temperature stabilization. Once at 135°C we kept the temperature constant and applied an electrical eld of 25 V.µm -1 . We acquired the XRD spectrum and repeated the measurement 4 times with 25 V.µm -1 increments, up to 125 V.µm -1 . The dataset with all electrical elds and temperatures can be found in annex but for sake of readability we only plotted some of them in gure 4.8.

In gure 4.8-A, the spectra at four dierent temperatures are represented. The peak at 2θ = 19.8 o is the FE phase and the one at 2θ = 18 o corresponds to the PE phase. As expected from a rst order transition, there are no intermediate states and a crystalline domain is either in FE or PE conformation. At 125°C both phases coexist in the sample. The PE diraction peak at 130°C is slightly more intense than the FE peak at 100°C but also thinner due to the (200)/(110) superposition. In order to verify if the width and intensity variations compensate each other we extracted the peaks area at each temperature.

To calculate the diraction peaks total area we tted the experimental data using pseudo-Voigt functions, the resulting values are displayed in gure 4.8-B. The orange curve is the FE peak area and the blue curve is the PE peak area. The green curve is the sum of the two others and is proportional to the total amount of diracting matter. At 135°C, the total diraction pattern area is down to 75% of its original value. We will discuss this in the following section.

In gure 4.8-C, the copolymer is heated at 135°C and the spectra are acquired at four different electrical eld values. At 0 V.µm -1 the copolymer is in its PE phase and the diraction peak is located near 2θ = 18°. Between 0 and 100 V.µm -1 there are no signs of a FE phase but the diraction peak changes signicantly. The width increases, the intensity decreases and the the surface has decreased of 5%. With the FE phase apparition, the total area increases slightly above the initial value in the unbiased sample.

Discussion

Field-induced phase transition

The rst result we get out of XRD measurements is the presence of a eld induced phase transition. In panel C, the two peaks at 125V.µm -1 indicate the coexistence of two phases. Given the position of the newly formed phase it must be the FE phase reappearing. The presence of such a transition was already evidenced with conformational changes in FTIR measurements [11].

However, this is the rst report of this transition using XRD, i.e a direct observation of changes in the crystalline structure. If these spectra were that of a terpolymer instead of a copolymer we would have had all the evidence we needed to support the phenomenological description of strain presented in chapter 3. In gure 3.24, box E would be the dierence in d-spacing between FE and PE peak and the polar nature of FE phase would explain box D. These are copolymer spectra but this example illustrates what we will be looking for in terpolymers.

Electrostriction

Besides phase transition, a noteworthy change induced by the electrical eld is the diraction peak shift in position. In gure 4.8-A between 125 and 130°, the PE peak moves toward lower angles. According to Bragg's Law (2d.sin(θ) = λ) it corresponds to an increase of the interplanar distances i.e a lattice strain. This strain is nothing more than thermal expansion, noticeable in PE phase but barely present in the FE phase of copolymers [12].

In g 4.8-C, the PE peak shifts in the opposite direction compared to thermal expansion.

This negative strain of the lattice corresponds to electrostriction and is consistent with the sign of P(VDF-TrFE) electrostrictive coecient (Q 33 < 0).

There is another consequence of the electrostrictive eect on the XRD spectra. Between 50 and 100 V.µm -1 the diraction peak amplitude goes from 21.5 to 18 (arbitrary units), a 16% drop.

This variation is compensated by a larger width and the total peak area remains somewhat constant. The PE peak shape variations could be due electrostriction, with the (110) and (200) peaks drifting apart from one another.

When the material is strained, whether it is thermal expansion or electrostriction, both the (200) and (110) diraction peaks are aected. However, the only way for the diraction peak to keep an identical shape throughout the strain modications would be for the two interplanar distances d 110 and d 200 evolve identically. This might be the case with thermal expansion because the lattice expands in all directions. The electrostrictive eect on the other hand contracts the lattice in one direction and expands it in the two others: all directions are not equivalent. This results in a separation of the (200) and (110) diraction peaks and most likely explains the change in intensity and width observed in gure 4.8-C. We drew gure 4.9 to illustrate this idea.

In gure 4.9 we have dierent system axis: that of the sample and that of crystallites. The electrical eld is applied in the direction 3 of the sample system axis. The optical setup is so that we observe the diraction planes parallel to the surface (Φ = 90 o in g 4.7); this is also direction 3 of the sample system axis. Aligning the electrical eld and the measurement direction does not depends on our choice of a particular system. The electromechanical response of a given crystallite depends on its orientation and the electromechanical equations must be applied in the lattice system axis.

To keep a consistent description between macroscopic and microscopic formalism, the bdirection of a lattice will be referred to as 3. The direction along a and c will be respectively 1 and 2. The two crystallites families we observe with XRD measurements at Φ = 90 o are displayed in gure 4.9. The strain of the left crystallite is expressed with equation 4.1.

S 200 = S 1 = M 11 .E 2 This case is pretty straightforward because the electrical eld and the measured strain are in the lattice principal direction a. Moving on to the crystallite on the right in gure 4.9, the strain has a more complex expression because the measurement direction does not coincide with a lattice principal axis.

S 110 = S 1 + S 3 (4.2)

S 1 = E 2 (M 11 + √ 3M 31 ) S 2 = E 2 ( √ 3M 33 + M 13 ) (4.3)
There is little use in identifying the dierent elements of the copolymer electrostrictive tensor because we are not working with a monocrystal. However, this point was discussed in order to highlight two issues. First, comparing equation 4.1 and 4.3 we can see that these two values might very well be dierent. This is possibly the explanation behind the change in PE diraction peak with the electrical eld: the (110) and ( 200) peaks are drifting apart.

The second issue is the diculty to compare, at least quantitatively, the macroscopic strain and the crystalline strain. The crystalline strain is a function of several electromechanical coefcients and even knowing them would not be enough. We only observe two particular sets of crystallites while the total crystalline phase response is the sum of all the dierent orientations, each one strained dierently.

Domains rotation

The last unexplained variations in the XRD measurements g 4.8 are the changes in total peak area, panel B and D. There is a 25% decrease in the total diraction area throughout the heating process of the unbiased sample (B). There are no obvious reasons why the domains size would be aected by the Curie transition. This variation is more likely explained by the changes in crystallites orientation following the loss of ferroelectric order.

This can be understood with the work of Bur & al [10] (4.7-A). The rst poling of a PVDF sample aligns the ferroelectric domains with rotation of either 60°, 120°or 180°toward the electrical eld direction. With the apparition of a preferential direction, the amount of crystallites in Bragg's condition is aected. In gure 4.7-A, this would correspond to the transition from (a) toward (b)+(c). We can see that this results in an increase of the diraction peak intensity at Φ = 90 (our setup). Now, this is an ideal case with only a few possible orientations but it helps understanding how a similar evolution can be expected during a change in the anisotropic distribution of crystallites. In gure 4.8-B, as temperature increases the ferroelectric order is lost and so is the preferential orientation of domains. With the progressive return to an isotropic distribution of crystallites, the peak area decreases. This explanation is consistent with the small increase in g 4.8-D once the FE phase has appeared.

There is a dierence between a study comparing the structure before and after poling such as [10] and direct in-situ measurements. In addition to the eect of ferroelectric orientation, the in-situ measurements are also impacted by the dielectric response of the material. In the previous chapter, we separated these two notions of ferroelectric and dielectric (or capacitive) response.

As the polarization is intrinsically linked to the structural changes inside the polymer, the same dichotomy is in order here. Assuming that the dielectric response is also due to orientation mechanisms, one could wonder if it also impacts the XRD spectra. We can infer the crystallites rotation angle due to capacitive response by correlating dipole orientation and polarization cycle. -2 . Assuming half of that contribution comes from the orientation of dipoles inside the crystal, it yields a 5°rotation in average of the crystallites. This is an overestimation because the crystal permittivity is lower than that of the amorphous phase [13] and we disregarded other physical mechanisms. Therefore, that value of 5°is more of an upper limit for the average dielectric rotation. Even so, it is more than 10 times smaller compared to the 60°s tepwise rotations of ferroelectric domains. We assume dielectric orientation does not disrupt the statistical distribution much, especially compared to ferroelectric orientation.

There are numerous other physical phenomena capable of inuencing a diraction pattern.

This includes the number of defects present in the crystal, the crystalline domains size or the sample crystallinity. Any dependence of said quantities on the electrical eld could complicate the data interpretation. As this experiment is a rst attempt at the in-situ study of polymers we will disregard these other contributions to begin with.

Conclusion 22: Copolymer

We reported direct evidence of the eld-induced PE-FE transition in copolymers near the Curie temperature. The eect of electrostriction was observed in the PE crystalline phase of a copolymer and the variations are consistent with the negative sign of its electrostrictive coecient.

Two other eects of electrical elds on the XRD spectra were also discussed. Firstly, the electrostriction distorts the diraction peak because it aect dierently the ( 200) and ( 110) peaks. Secondly, the annealing of copolymer un-poles it and as a consequence the preferential orientation is lost. The changes in anisotropy aects the peaks intensity and total area.

With a general idea of how electrical eld can impact the polymers spectra, we move on to the terpolymer study. These polarization cycles are that of a CFE terpolymer but we already discussed previously that these considerations were most likely applicable to our polymers as well. Figure 4.12-B dis- and the hypothesized crystal order [16] plays RFE+PE to signify the presence of both PE and RFE phase in the terpolymer at ambient temperature .

The exact nature of those phases is still unclear and this notion of RFE+PE phase was recently rened by Bargain & al [9]. They proposed that the terpolymer phase at ambient temperature is in the middle of a continuous transitions from RFE to PE, as illustrated in gure 4.13. Figure 4.13 is taken from [9] and sums up the phenomenological description Bargain & al proposed to explain the phase transition with temperature. On the left, the chains are below 0°C

, in RFE phase. The red dots are the CTFE pinning units making gauche conformation in an otherwise all-trans (TTTT) chain. As temperature increases, gauche conformations progressively appear in-between the CTFE units in the VDF-TrFE sequence. These conformations are called Based on the current state understanding of the terpolymers structure, three main results can be expected from XRD in-situ measurements.

First, an experimental conrmation that an electrical eld induces structural changes in terpolymers crystalline phase. Although polarization measurements strongly point in that direction, there is no evidence as of yet. Second, a link between structural changes and electromechanical observations from the previous chapter. The idea is to provide a physical explanation for the variation of electrostrictive coecient (g 3.22) and the overall description of strain mechanisms in terpolymers (g 3.24). The last objective is to shed some light on the nature of this (FE)

phase and how it relates to the RFE-PE order. While XRD measurements alone are unlikely to characterize the whole structure it can be a starting point for following studies. 4.14-A displays the diraction peak of a terpolymer from 0 to 16V.µm -1 with is no visible dependence to the electrical eld. In gure 4.14-B the electrical eld ranges from 22 to 66 V.µm -1 . The peak becomes more intense and thinner while the asymmetry switches from left to right. Figure 4.14-C displays XRD patterns for electrical elds up to 166 V.µm -1 , the peak transitions towards high 2θ while maintaining an almost identical shape.

TERPOLYMERS 123

On the whole electrical eld range, the diraction peak can be tted with two pseudo-Voigt functions. Three t examples are displayed in gure 4.14 D-F with the two tting functions called peak 1 and peak 2. The t of all spectra are performed in a single run: the boundary conditions and initial parameters are identical throughout the operation. Below 20 V.µm -1 two peaks at least are required to t the asymmetry (D). Between 24 and 72 V.µm -1 (E), the change in shape is matched with one peak increasing at the expense of the other. At higher electrical elds (F), only one peak with a small tail is needed to t the diraction peak. This peak keeps an identical shape while shifting toward high angles.

To go further than discussing the experimental diraction peak alone, we also monitor the evolution of the tting functions. In copolymers, there were countless possibilities to t the spectra and we did not care for the chosen functions. In terpolymers, because of the much more limited number of solutions, we estimate that these tting functions draw a realistic picture of the structural changes. There are several hypothesis to explain their evolution, visible in gure 4.14 D-F.

First, each tting peak could correspond to a dierent phase and the XRD spectra evolution would then be the direct image of a eld-induced phase transition: one phase grows as the other disappears. This is a possibility but it leaves some important questions unanswered and among them the unexpected width of the high electrical-elds diraction peak. In the state of the art section, we saw that the low temperature phase of terpolymers (RFE) and the FE phase of copolymers both exhibit a broader diraction peak due to their pseudo-hexagonal symmetry.

Yet, we observe the opposite evolution as the electrical eld increases.

The second possibility is that each peak corresponds to the (110) and (200) diraction planes of the same crystalline phase. In that case, the asymmetry switching side observed in gure 4.14 B would correspond to the orientation of crystallites in the electrical eld direction. A (110) plane in Bragg's condition makes a 30°angle with the electrical eld while a (200) diracting plane makes a 90°angle with the eld. An orientation in the electrical eld direction would, among other changes invisible to our optical setup, increase the (110) diracting population and reduce the (200) one.

The third hypothesis is that we have a more complex situation with any number of phases, each one with possibly several peaks which may or may not rotate with the electrical eld. While this is a distinct possibility, short of leaving the interpretation to imagination there is not much to do with this hypothesis and XRD measurements alone.

Conclusion 23: Terpolymer

We observed with XRD measurements that the terpolymer displays signicant structural changes with the electrical eld. The data interpretation is not as simple as it was in copolymer and several mechanisms could explain the spectra variation. The idea of crystallites rotation with the electrical eld allows for an easier interpretation of the data and we choose to continue the study in light of this hypothesis.

Discussion

The rst parameter we extracted from gure 4.14 is the d-spacing evolution versus electrical eld. To correlate the structural changes to macroscopic measurements we regrouped in table 4 The notion of phase transition is debated here but it was the terminology previously used in the previous chapter, hence the notation "phase transition" in table 4.2.

The rst interest of table 4.2 is that it shows the strains order of magnitude. As it turns out the crystalline phase strain reaches values similar to that of the entire sample. This was not a given because the terpolymers have a crystallinity ratio between 30 and 40 %. As the amorphous phase is much softer than the crystalline part, it could have born most of the deformation.

The second noteworthy comparison in table 4.2 is the relative contribution of "phase transition" to the total strain. This value is much higher in the XRD measurements than in the macroscopic acquisitions. This observation tends to show that the nonlinear electrostrictive behavior originates from the crystalline phase, as proposed in chapter 3. However in light of the orientation hypothesis, the strain model from chapter 3 should be slightly modied, as illustrated with 4.16. The dipolar moment is along the b-axis of the lattice and the parameter b is about half that of parameter a. So, the additional strain we associated to a dierence in lattice parameter between two phases is most likely a rotation of the crystallites in the electrical eld direction.

Beside the diraction peak position we also extracted their width, intensity and area, displayed in gure 4.17. we observe with XRD measurements is the structural counterpart of the electromechanical nonlinearities measured macroscopically. The area variations are also consistent with the rotation hypothesis. As discussed with copolymer, the apparition of a preferential orientation can increase the number of crystallites in Bragg's conditions.

However, there is one important discrepancy between XRD and electromechanical measurements: the hysteresis width. In gure 4.17 the cycle width is about 20 V.µm -1 wide, about half that of the macroscopic hysteresis (g 3.15). We ascribe this dierence to the frequency dependence of the domains rotation. To validate this hypothesis, the strain response of a terpolymer was measured at dierent frequencies. The results are displayed in g 4.18. -4 Hz), we nd a value of 20 V.µm -1 . This value is identical to that of our XRD measurements in g 4.17 and explains the dierence between macroscopic and XRD hysteresis.

To transition or not to transition ?

So far, we have seen that the XRD spectra and the strain nonlinearities could be explained by the orientation of crystalline domains. As discussed in the previous section, the rotation associated to the dielectric nature of the material is mild and can be disregarded. So what could be at the origin of this dramatic orientation ?

The notion of eld-induced transition toward a ferroelectric phase becomes once again in order and could explain the sudden rotation. If such a transition happens, the terpolymer would be in a state much like that of an unpoled β-PVDF or copolymer. As the transition takes place, the newly formed ferroelectric domains are oriented in the electrical eld direction and this subsequent rotation is actually what we observe. Unfortunately this means that we have no information on the structural changes associated to this hypothetical transition, as they are drowned in texturation eects.

Conclusion 24: Terpolymer at ambient temperature

The XRD measurements of electrically biased samples provide evidence that the macroscopic nonlinearities in the polarization and strain behaviour of terpolymers originate from their crystalline phase. The increase in electrostrictive coecient is most likely due to the rotation of crystalline domains. The sudden orientation of crystallites could be due to a phase transition but we cannot distinguish an eventual structural change from rotation eects.

At higher temperatures

We reproduced the XRD study at higher temperatures in an attempt to obtain some information on the nature of phase transition. The terpolymer sample was rst heated without any bias, the results are displayed g 4.19-A. As a visual element of comparison, we also added the electrical measurements conducted at 40°C. This time we observe that at 40°C the strain is dierent from the 30°C response and is closer to the 50°measurement. A possible explanation is that the (110) and (200) peaks are closer to one another and the jump from (110) to (200) does not aect the d-spacing much.

Two more pieces of information will help us draw a picture of the structural changes. Firstly, the phase transition seems to appear around the same electrical eld value regardless of the temperature, as highligted by the dotted line in gure 4.20. This means the RFE/PE polar ordering does not inuence much the occurrence of this transition. Secondly, the hysteresis is reduced as the temperature increases which means the newly formed phase is less stable. The RFE/PE order might play a role in the FE phase stability.

Even completed with temperature measurements, this study does not allow us to conclude on the nature of the structural changes inside terpolymers. However, building on the current state of the art we can propose a phenomenological description that is consistent with the XRD data acquired so far. induced by temperature. We drew the dipolar moments in an anti-ferroelectric 2D fashion for sake of simplicity. The actual form of gauche conformations is of little importance here and the relevant information is that the resulting dipolar moment is null. The green dots are pinning domains due to CTFE units. They also form gauche conformations with the neighbouring units and are dierent from the blue arrows because they are not due to temperature but to the presence of a chlorine atom. A succession of VDF-TrFE dipoles between two pinning units will be referred to as a sequence. The PE, RFE/PE and RFE phases were described with gure 4.13 and are represented in g 4.21 upper right corner. Now the question is how does the electrical eld aect this dipole arrangement? In copolymers we saw that applying an electrical eld was equivalent to cooling the material. Applied to terpolymers this would mean pushing the RFE/PE equilibrium toward a RFE phase. However, the notion of CTFE 'unpinning' is necessary because the RFE phase is not ferroelectric and the PE->RFE phase transition cannot explain alone a ferroelectric-like behaviour.

The description starts near ambient temperature, in an unbiased terpolymer sample: this is the representation A-1. To make the link with XRD data this temperature will be considered to be 30°C. The polymer is in RFE/PE phase and each sequence has a net dipolar moment. Still the RFE phase is not ferroelectric and thus the sequences are not oriented in the same direction. This is due to the pinning units which allow the sequences to rotate freely. In terms of XRD Applying an electrical eld above 30 V.µm -1 makes the terpolymer go to state A-2. Past that value, the electrical eld starts to weaken the pinning applied by CTFE units. This allows the VDF-TrFE sequences to cooperate with one another, forming the terpolymer FE phase. This conformation holds as long as the pinning domain are weak enough. Once the ferroelectric phase is formed, the local eld seen by the pinning domains is not only the external eld but also the contribution of neighboring, oriented, RFE/PE sequences. So even when the electrical eld is decreased past the 30V.µm -1 mark, the pinning unit are kept in a weakened state by the ordered RFE/PE sequence (FE phase). This is at the origin of the hysteresis observed in terpolymers.

Regarding the XRD spectra, we do not distinguish the phase transition but observe mostly the subsequent rotation of FE domains: the (110) peak increases while the (200) peak shrinks.

Removing the electrical eld and increasing the temperature brings us to a state represented in B-1. The terpolymer is still in an RFE/PE phase but closer to the PE side. The average moment of each sequence is lower than in A-1. In terms of XRD pattern the proximity to PE phase means the (110) and ( 200) diraction peaks are closer to one another.

The electrical eld is increased once again past the 30 V.µm -1 mark and we get to the state B-2. The same thing happens as in A-2: the pinning eect weakens, the PE/RFE sequences cooperate and and a ferroelectric order is achieved in the polymer chain. Once again, this order is lost as the electrical eld returns to 0 and the pinning units regain their strength. However, the average contribution of FE order to the pinning units local eld is lower this time. This is due to the the smaller average moment of RFE/PE sequences and results in a less stable FE phase.

This would explain why phase transition is still present but almost hysteresis free right below the Curie temperature. In terms of XRD patterns, the diraction peak variations are similar to A-2 albeit less hysteretic, due to this less stable phase.

At 50°C, the terpolymer is in its PE state with no average dipolar moment in a VDF-TrFE sequence, as illustrated in C-1. At 30 V.µm -1 (C-2) the "unpinning" still occurs but as the terpolymer has achieved a PE order, there is no average moment that could force an alignment with the electrical eld. The absence of ferroelectric domains means there are no rotations nor variation in the XRD pattern.

In this description we completely disregarded the reversal of thermal gauche conformations (blue arrows to red arrows) i.e the PE->RFE transition. If we make the parallel with copolymer it is likely to happen, at least to some extent. A good way to study that question would be completing the experiment with in-situ FTIR acquisitions. FTIR cannot detect changes in the crystal symmetry but it informs about conformation changes inside the polymer chain. That is pretty much the opposite of XRD and the two techniques are complementary. Conformation changes outside the phase transition range (20-60 V µm -1 ) would be ascribed to thermal conformations i.e PE->RFE transition.

While the changes induced by an electrical eld can be called a phase transition from an electrostatic standpoint, they do not imply much of a structural evolution. We saw in the state of the art section how the transition from RFE to PE was a subtle change in terms of structure.

Yet it was about half the terpolymer conformations changing from gauche to trans. Here it is only one conformation every dozen or so that requires transforming to change the phase from RFE/PE to FE. In terms of structural change this should be barely noticeable.

Conclusion

In the previous chapter we observed non linearities in the electromechanical behaviour of PVDF-TrFE-CTFE, namely polarization hysteresis and increase in electrostrictive coecient Q. Based on the literature, these oddities were ascribed to a eld induced phase transition. The aim of this chapter was to ascertain the link between macroscopic and structural evolutions and to validate the terpolymer strain phenomenological description proposed previously. Then if possible, shed some light on the nature of this transition.

To better understand the impact of electrical eld on XRD patterns we started our study with PVDF-TrFE copolymer, a well known material with a rst order PE-FE transition. We reported direct evidence of the eld-induced phase transition above the Curie temperature.

Besides phase transition, two signicant eects of electrical eld were observed: electrostriction and orientation of crystalline domains. The main diculty in interpret XRD data comes from the nature of the diraction peak, a superposition of the (200) and (110) peaks. Unlike temperature variations and the thermal expansion it induces, the electrical eld aects the (200) and (110) peaks dierently, which can make the data interpretation delicate.

Moving on to the terpolymer, we observed a signicant evolution of the diraction patterns with the electrical eld. From the diraction peak position we calculated the strain in the crystalline phase and it turned out to be of the same order of magnitude as in the amorphous phase.

The crystalline phase also appeared to be more aected by the nonlinearities in electrostriction than what we observed macroscopically. At last, we extracted from XRD data the variations in shape and intensity of the diraction peak. Aside from a width discrepancy we ascribed to frequency eects, the XRD cycles matched the polarization and strain measurements. With these results we successfully tied the macroscopic nonlinearities to structural changes happening in the crystalline phase.

Going further required an interpretation of the functions used to t the diraction peaks. The rst possibility is that each tting peak corresponds to a crystalline phase and the XRD changes are direct evidence of a eld induced phase transition. Another possibility is that the two peaks used for the t correspond to the (110) and (200) diraction planes of a single phase. In that case, the XRD changes are due to the rotation of domains in the electrical eld direction. The rotation hypothesis seems more plausible at this time as it explain some experimental oddities such as the thin diraction peak of the FE phase. Either way, there must be a eld-induced transition toward a ferroelectric phase happening at some point in terpolymers to explain the abrupt changes.

The terpolymer strain mechanism proposed in the previous chapter is validated by the insitu experiments. It requires one minor correction: the variation of electrostrictive coecient we ascribed to a dierence in lattice parameter is more likely due to an orientation of crystallites, with the smaller b-lattice parameter in the thickness direction.

The electrical study was reproduced at higher temperatures, past the Curie point and we proposed a description of the eld-induced phase transition to account at best for the XRD changes. The pinning eect introduced by Yang & al [16] can explain on its own the experimental observations. Ferroelectric phase appears when pinning domains lose their strength, due to the application of an electrical eld. The RFE/PE order seems to govern the stability of the newly formed FE phase. It is possible a PE->RFE phase transition is also present but it would be a mild and second order eect. The notion of eld-induced phase transition in terpolymers is relevant from an electrostatic standpoint but this is mostly due to domains rotation and the structural changes are likely to be very small.

Chapter 5

Discussion 137

The work presented so far can be divided into two main contributions. First, a comparative study on dierent P(VDF-TrFE-CTFE) terpolymers, to assess the impact of CTFE content on actuation performances. Building on this work, the comparison could be extended to other PVDF derivatives. We will present here a rst series of measurements on a closely related polymer: P(VDF-TrFE-CFE).

The second part of our study was focused on understanding the origin of strain in terpolymers.

There is still many work to be done on the subject and in this chapter some ideas and preliminary results for follow up experiments are presented. help setting a proper environment to make this experiment successful, as it was done in [1].

Clamping eect

An interesting possibility brought up by the XRD study is to locate the strain inside the material.

In the previous chapter we did not go further than a qualitative comparison but with enough care it is possible to quantify the strain inside and outside the crystalline phase. To that end two issues must be addressed, the rst one is to properly account for the dierent crystallites orientations in the material. The second issue, which is the one discussed here, is the substrate clamping.

The cantilevers were made with PEN as substrate and the samples used in XRD analysis were printed on glass. Ideally both mechanical and XRD measurements would have been performed

on the same material. However, as mentioned in chapter 4 we did not use PEN for the XRD study because of the very high background signal it generates (g 4.2). If we hope to compare the macroscopic strain to the XRD samples strain we need to assess how the glass clamping impacts our measurements. To that end we conducted two XRD studies, one on a sample with PEKK substrate and another with a sample printed onto glass. PEKK as a substrate does not hinder XRD interpretation and, because it is softer than glass, we should be able to see how the substrate rigidity aects the strain measurements. The results are displayed in 5.2. extracted macroscopically in chapter 3, but as mentioned above the quantitative comparison between Q macro and Q cryst cannot be conducted yet.

It appears that the d-spacing values of both samples dier from 0.03A o regardless of the electrical bias. This means that the substrate clamping aects the lattice dimensions during the fabrication process. However, it is not much of an issue because we are interested in the electrically induced variations (strain) and not in the absolute lattice parameters. As it turns out, the electrostrictive coecients Q are almost identical in both cases. Therefore, the clamping does not aect the strain in the polymer thickness direction.

Macroscopic approach

To tackle the problem on dierent fronts, the understanding of strain can be studied from a macroscopic perspective. We briey discussed in chapter 3 the notion of Maxwell stress and how 5.1. FOLLOW UP WORK ON P(VDF-TRFE-CTFE) 141 it is most likely contributing to strain in terpolymers. Despite its simple analytical expression, the inuence of Maxwell strain is not quantied properly and varies from one study to another.

A possible way to estimate its contribution would be to measure separately the electrostriction coecient, free of the Maxwell eect. This can be done through capacitance measurements of a mechanically stressed sample, using equation 5.1 [2]. As there are no charged electrodes, the Maxwell strain is not present and only the other contributions are extracted.

Q i = ∂( 1) ∂T There is a signicant dependence of the capacitance value to the applied stress. This is not a consequence of changes in the device geometry because a reduced thickness of the active layer would have resulted in an increase of the capacitance, not the other way around. Therefore we ascribe the capacitance variation to a change in permittivity and using equation 5.1, we get an intrinsic coecient Q i ≈ 6m 4 .C -2 . As the 2% CTFE terpolymer is a ferroelectric (cf chapter 2) material, a more meaningful value to characterize it is the piezoelectric coecient. We calculate this intrinsic d i using equation 3.13 which give a value of -80.10 -12 m.V -1 about twice as high as the actual piezoelectric coecient.

Had the intrinsic d i been lower than the one measured from electromechanical measurements, we could have ascribed the dierence to Maxwell strain. Given the values we extracted, it is likely that the Maxwell contribution is negligible in 2% CTFE terpolymer. The higher intrinsic coecient could have a physical signicance or be the result of errors on the approximations we made of stress values. This is a preliminary experiment and a proper setup should apply a stress directly to avoid any uncertainty. We reproduced this experiment on a 8.3% CTFE terpolymer and the results are displayed in gure 5.4. polymer. However, the coecient Q is supposed to be higher and therefore we expected to see the same stress induced change. Although the variations are supposed to be of the same order as in the 2% terpolymer, it is dicult to extract any value because of the measurement instability.

The lack of signicant variations could be interpreted as a dominant contribution of Maxwell strain, but drawing any denitive conclusion would require a properly calibrated setup, as we mentioned before. Regarding the temporal dependence of capacitance measurements, a solution could be to apply a periodic solicitation coupled with a Fourier analysis to isolate the eect of stress.

Other polymers

This study was focused on P(VDF-TrFE-CTFE), its performances as actuator and its electromechanical behavior. However, it is only one PVDF derivatives among numerous others. With the groundwork and characterization protocol already laid out in this study, another direction to pursue could be extending the comparison to dierent polymers.

The rst material coming to mind is P(VDF-TrFE-CFE) whose structure is close to that of our terpolymers. Furthermore the state of the art is very similar to that P(VDF-TrFE-CTFE): -It is also used in actuation devices [3] or patents [4] with little regard to its performances compared to other compositions and polymers.

-Its electrical properties are well documented but the reports on its strain cycles are scarcer, especially for low CFE contents.

-The understanding of strain mechanisms is not fully understood and a conformation change is expected to play a signicant role. The cycle opening in CFE appears at higher elds and seems to be milder than with CTFE. This is also visible in the right panel with the strain hysteresis. An educated guess based on the conclusions from the previous chapter would be that a eld induced phase transition also occurs in CFE terpolymer, but more gradually.

At high electrical elds the CFE terpolymer generates more stress than CTFE. If the conclusions from chapter 2 also apply to CFE, an intermediary composition with 4-5% CFE should perform even better. This is just an exemple but there are countless other PVDF derivatives, blends and composites.

Conclusion

Over the course of this work we shed some lights on the strain mechanisms of P(VDF-TrFECTFE).

There are still many unknowns mechanisms and several paths can be pursued building on this study. First, the XRD acquisition could be continued with dierent Φ values to try and characterize the crystallites orientations. This would allow to check the validity of previous assumptions and help building a model of strain. The same goal can be tackled on a dierent front with an analysis of the intrinsic electrostriction. The equations of thermodynamic links the variation of permittivity versus stress to the electrostrictive coecient M. This variation does not take into account mechanisms such as Maxwell strain and this separation can bring insightful information.

On a dierent topic, the analyzes conducted on P(VDF-TrFE-CTFE) could be reproduced on similar materials, taking advantage of the framework provided here. A good starting point would be P(VDF-TrFE-CFE) a closely related polymer.

General Conclusion

The rst objective of this work was to assess the performances of P(VDF-TrFE-CTFE) terpolymers to act as active layer in thin lm actuators. For this application the two gures of merit selected were the actuators deection and the electromechanical coupling. These two quantities relate directly to the terpolymers stress and polarization response. The terpolymers with an intermediate CTFE ratio (≈ 5%) turn out to be the best performing in terms of stress, 30% superior to copolymer at 100 V.µm -1 . Regardless of the CTFE content, terpolymers coupling eciency is about half that of copolymer. Therefore, the choice of a specic composition depend on the prioritizing order between performance and energy cost.

The necessity to perform such tedious characterizations come from the non-linear relationship between strain and electrical eld (or squared electrical eld). Because of it, P(VDF-TrFE-CTFE) cannot be simply characterized by a set of electromechanical coecients. To evaluate the relevance of electrostrictive formalism applied to PVDF derivatives we studied the link between electrical eld, polarization and strain. This study allowed us to propose a model of strain in terpolymers, accounting for both the changes in polarization and electrostrictive coecient. As for the physical origin of the non-linearities, a likely candidate suggested by the literature was a eld induced phase transition. parler copo

In order to correlate the electromechanical oddities to structural changes, we conducted insitu XRD measurements on electrically biased samples. The electrical eld has dierent eects on the XRD spectra such as domain orientation, lattice strain and phase transition. The rst measurements were performed on copolymer where these eects could be isolated and discussed.

We observed directly the presence of a eld induced transition toward the ferroelectric phase past the Curie temperature.

The XRD interpretation out to be a more complex task in terpolymer were the dierent contributions were entangled. There is an ambiguity on the origin of pattern variations in terpolymer:

whether we observe a conformation change toward a ferroelectric phase or a texturation eect due to the rotation of said phase. Still, the electromechanical changes are visible in the structure evolution, conrming that the crystalline phase is at the origin of macroscopic non-linearities.

There is still much to be done before fully understanding the origin of strain in PVDF-TrFE-CTFE. The possibilities brought by XRD measurements alone were barely exploited and should provide many other insightful results. Building on this study, another path to pursue could be extending the comparison of actuation performances to similar PVDF-TrFE derivatives, such as PVDF-TrFE-CFE.

Annex I: Vibration at higher frequencies A major issue faced in our study was the actuation frequency limitation. Figure 1 illustrate that problematic with the comparison between a deection at 1.5 Hz and at 12 Hz.

Figure 1: Deection measurements of a cantilever at 12Hz and 1.5Hz

We observed cleaner modes with the synchrotron beamline interferometer, gure 2 . The sample was glued onto a glass substrate so we could measure the deformation properly up to 50 Hz. Above that value, new modes started to appear. Figure 3 displays the capacitance measurements versus electrical eld of the sample whose polarization cycles at 22°C and 50°C were compared. The dierence between the two cycles was ascribed to the anti-ferroelectric cycle of phase transition but that cycle was somewhat tilted. This is due to the slightly higher dielectric polarisation at 50°C. Figure 4 is displays the capacitance measurements we used to calculate the dielectric polarization in gure 3.26. Wheter above or below the Curie temperature, the dependence of capacitance to the electrical eld remains similar. This optical conguration introduces an angular error as the incident θ value is not constant. This is corrected by the refocalization of the diracted beam right on the detector. For that to function, the detector must be placed symmetrically to the tube with regard to the sample. That means the Φ angle dened in chapter has to be worth 90°and only the diraction planes parallel to the sample surface can be acquired. In order to perform scans at Φ = 0 o as represented in gure 5.1, a dierent conguration has to be used (g 6).

DHL

  Double hysteresis loop EAP Electroactive polymers FE Ferroelectric FT Ferroelectric terpolymer (< 7:8% CTFE) P(VDF-TrFE) Poly(vinylidene uoride-triuoroethylene) P(VDF-TrFE-CFE) Poly(vinylidene uoride-triuoroethylene-chlorouoroethylene) P(VDF-TrFE-CTFE) Poly(vinylidene uoride-triuoroethylene-chlorotriuoroethylene) PE Paraelectric PEDOT:PSS poly(3,4-ethylenedioxythiophene) polystyrene sulfonate PEN Polyethylene naphtalate PVDF Polyvinylidene uoride PZT Lead zirconate titanate ceramic RFE Relaxor Ferroelectric RT Relaxor terpolymer (> 7:8% CTFE) XRD X-ray diraction 3 Acknowledgments First I want to thank my supervisor Emmanuel Defay for his guidance over these past years. It is thanks to our numerous discussions that I have been able to train my critical mind and grasp

Figure 1 . 3 :

 13 Figure 1.3: Experimental process to obtain the dierent phases of PVDF []

3

 3 Figure 1.3 displays the available paths to obtain the dierent phases of PVDF. Among the

Figure 1 .

 1 Figure 1.4 illustrates the categories of electroactive materials and their links to one another.

Figure 1 . 4 :

 14 Figure 1.4: Schematic representation of the electroactive materials categories.

Figure 1 .

 1 Figure 1.5 displays the results to the query 'PZT' and 'PVDF' in the scopus database. It is presented as the number of annual publications between the year 1990 and 2017.

Figure 1 . 5 :

 15 Figure 1.5: Number of publications per year containing the key-word PZT and PVDF. Source: scopus

Figure 1 . 6 :

 16 Figure 1.6: VOSviewer analysis of the publication on PVDF for the years 2016, 2017 and 2018.

Figure 1 . 7 :

 17 Figure 1.7: Number of publications per year containing the keyword PVDF-TrFE. Source:

Figure 1 . 8 :

 18 Figure 1.8: Structural formula of P(VDF-TrFE-CTFE)

Figure 1 . 9 :

 19 Figure 1.9: Left: soles with embedded piezoelectric and piezoresistive sensors. Right: prototype for an electric piano keyboard

Figure 1 . 10 :

 110 Figure 1.10: Left: Structure developed for a PVDF based piezoelectric harvester, taken from [19]. Right: illustration of a simple yet eective way to generate large strains inside the polymer

Figure 1 . 11 :Figure 1 .

 1111 Figure 1.11: Electromechanical cycle required to harvest energy with an elastomer[20]

Figure 1 . 12 :

 112 Figure 1.12: Left: Principle of an actuator. Right: fully printed actuator, with and without electrical bias

  2008, Neese & al discovered the large electrocaloric eect (ECE) of PVDF-TrFE-CFE near room temperature [23]. From an engineering point of view, ECE is the opposite of the pyroelectric eect. It is the ability to generate a change in temperature with the application of an electrical eld. From a physical point of view, ECE and pyroelectricity are not two sides of the same coin and the large ECE of P(VDF-TrFE-CFE) has no pyroelectric counterpart. Figure 1.13 is an example of what the electrocaloric eect of terpolymers can achieve.

Figure 1 . 13 :

 113 Figure 1.13: Demonstration of a functioning electrocaloric cooler using PVDF-TrFE-CFE as active material.Taken from[24]

Figure 2 .

 2 1 is a representation of the two strain directions we are interested in, and the coecients used to characterize those strains.

Figure 2 . 1 :Figure 2 .

 212 Figure 2.1: Representation of the principal strain directions and the corresponding piezoelectric coecients

Fig 2 . 2

 22 displays the polarization of copolymers with dierent compositions, ranging between 80/20 and 50/50 VDF/TrFE. This gure was furnished by Piezoetech, who also provided the polymers studied in this work.

Figure 2 . 2 :

 22 Figure 2.2: Polarization cycles of six dierent grades of copolymer

Figure 2 . 3 :

 23 Figure 2.3: Left: permittivity versus temperature and frequency of a P(VDF-TrFE-CTFE)

Fig 2 . 4

 24 shows graphic representation of a screen printer, taken from[27].

Figure 2 . 4 :

 24 Figure 2.4: Principle of screen printing[27]

Figure 2 . 5 :

 25 Figure 2.5: Picture of a printed PEN sheet with the 16 actuators. White numbers are the dimensions in mm

Figure 2 . 6 :

 26 Figure 2.6: Picture of the characterization bench

Figure 2 . 7 :

 27 Figure 2.7: Deection of cantilevers from the same PEN sheet with dierent lengths

Figure 2 . 8 :

 28 Figure 2.8: Height prole in the width direction of a cantilever end. COMSOL simulation

Fig 2 . 9

 29 is a picture of a device edge taken with an optical microscope.

Figure 2 . 9 :

 29 Figure 2.9: Picture of a cantilever edge after the laser cut. The dark part is the entire stack and the lightest part is consists of PEN alone

Figure 2 . 10 :

 210 Figure 2.10: Deection of cantilevers from the same print with identical lengths but dierent widths. Green dots are the deection measurements. Red and purple dots are the estimated deection of the device without surrounding PEN substrate. Red from equation 2.5, purple from COMSOL

Figure 2 .

 2 11 is the leakage measurement of two devices with the same thickness measured with an impedance meter (agilent 4156C).

Figure 2 . 11 :

 211 Figure 2.11: Leakage measurements of a copolymer and a terpolymer

Figure 2 . 12 :

 212 Figure 2.12: Polarization response of the nine polymer compositions. Fig b: bipolar solicitation.

Fig a: Unipolar solicitation

  Fig a: Unipolar solicitation

Figure 2 . 13 :

 213 Figure 2.13: Permittivity of the terpolymer as a function of the CTFE content

Figure 2 . 14 :

 214 Figure 2.14: Displacement of three polymers, one for each behaviour identied above. a: copolymer; b: ferroelectric terpolymer; c: relaxor terpolymer. The upper row is the bipolar cycle and the lower one the unipolar cycle.

Figure 2 . 15 :

 215 Figure 2.15: Deection at low electrical eld of three cantilevers, one for each type of polymers.

Figure 2 . 16 :

 216 Figure 2.16: Deection at high electrical eld of three cantilevers, one for each type of polymers. Left panel: full cycle. Right panel: forward direction.

Figure 2 . 18 : 14 Fig 2 .

 218142 Figure 2.18: Deection of cantilevers at 100 V µm -1 as a function of the CTFE content. Each dot represents one cantilever and the squares are the 3 cantilevers shown in g 2.14

  18) Q: Charge at the electrodes A visual representation of the right hand term is shown in g 2.19

Figure 2 . 19 :

 219 Figure 2.19: Electrical charge as a function of the potential in a device. The green area is the electrical energy consumption during a cycle

k 2 : 21 Figure 2 . 21 :

 221221 Figure 2.21: Coupling coecient of the three types of polymers

  Fig 2.22 shows the dierence between annealing a 5.5% CTFE terpolymer for 15mn or 1 hour.

Figure 2 . 22 :

 222 Figure 2.22: Normalized deection of a FT annealed for 15 and 60mn

Fig 2 .

 2 23 shows a RT (8.3% CTFE) and a FT (4%) CTFE annealed at 115°C and 125°C for 15 minutes.

Figure 2 . 23 :

 223 Figure 2.23: Normalized deection of a RT (left) and a FT (right) annealed at 115°and 125 °temperatures

Fig 2 .

 2 Fig 2.24 shows the impedance measurements of two terpolymers with the same CTFE amount but dierent VDF-TrFE contents, taken from [15].

Figure 2 . 24 :

 224 Figure 2.24: Permittivity as a function of temperature of two terpolymers. Left: 65.6/26.7/7.7 VDF/TrFE/CTFE. Right: 80.7/11.6/7.7, taken from[15]

Figure 2 . 25 :

 225 Figure 2.25: Ranking of the three categories of polymer. The two gures of merit are the mechanical stress and the electromechanical coupling

Figure 3 . 1 :

 31 Figure 3.1: Left panel: I(V) measurement principle. Right panel: leakage in a terpolymer with dierent setup parameters

  measured at the electrodes G: device conductance V: electrical waveform This equation is based on the approximation that the leakage current is linear with respect to the electrical potential. We can now compare the estimated eect of leakage currents to the actual polarization cycle of our terpolymer (g 3.2).

Figure 3 . 2 :

 32 Figure 3.2: Electrical response of the device when submitted to a 1Hz triangular waveform. The red and purple curves are the calculated cycles of a pure resistor whose value is taken from two dierent soak times in g 3.1.

Fig 3 . 2

 32 Fig 3.2 shows that the leakage currents distort the charge cycle and are responsible for a

Figure 3 . 3 :

 33 Figure 3.3: Contribution of dierent mechanisms to the overall permittivity of a material, taken from [3]

3 .

 3 The phase dierence experimentally measured is called δ elec and is linked to permittivity through equation 3.4 tan(δ elec ) = (3.4) δ elec = phase dierence between polarization and electrical eld To understand how this phase shift aects the polarization measurements we simulated the polarization cycle of a dielectric with = 30 and dierent values for the dielectric losses (g 3.4

Figure 3 . 4 :

 34 Figure 3.4: Calculated polarization cycle at 1Hz of a dielectric material with a permittivity of 30 and various amounts of dielectric losses.

Figure 3 .

 3 Figure 3.4 shows how polarization mechanisms lagging behind the electrical eld impact the polarization measurements. The eect is similar to that of leakage currents, resulting in an elliptical polarization cycle. To model the lag with an equivalent electrical circuit, a resistor

Figure 3 . 5 :

 35 Figure 3.5: Theoretical material with two polarization mechanisms. Left is the equivalent electrical circuit and right the resulting polarization. Only the green cycle is accessible through measurement

Figure 3 . 6 :

 36 Figure 3.6: Equivalent electrical circuit of a polymer at low frequencies and electrical bias

Figure 3 . 7 :

 37 Figure 3.7: Polarization cycle of the electrical circuit in g 3.6 made with discrete components, at 200Hz. Left: Separate measurement of each branch. Center: polarization cycle of the whole circuit. Left: Polarization cycle actually displayed with default tester settings.

. 6 )Q

 6 RC± : remanent polarization n: positive integer, n-th cycle T: period of the sinusoid Equation 3.6 is the expression of the positive and negative "remanent polarization" as a function of the number of cycles. If n is high enough we are in the permanent regime and the two values are opposite. At the frequencies we deal with, the transient regime is over by the end of the rst cycle (n=2). Therefore, the leakage current is the only possible explanation to the polarization oset and the dierence between the positive and negative "remanent polarization" we observe. Now we can remove the leakage currents directly and calculate the mean conductance in the process (equation 3.7). sinusoid pulsation U 0 : sinusoid amplitude ∆Q remn : dierence between remanent polarizations (±) Once leakage contribution is removed, the ellipse opening is only due to the phase oset δ elec giving the rst equation to identify the R 2 C 2 components. The other equation is obtained with the dierence between rst and second cycle, ascribed to the transient regime in the R 2 C 2 branch.

  With a single polarization cycle we successfully identied each component and especially the pure losses present in the dummy circuit. The following question was whether this could be applied to P(VDF-TrFE) and P(VDF-TrFE-CTFE). Figure3.8 is an example of a terpolymer sample polarization cycle at 0.4Hz and 6.3Hz with a maximum electrical eld inferior to 2V.µm -1 .

Figure 3 . 8 :

 38 Figure 3.8: Polarization of a terpolymer at 6.3Hz and at 0.4Hz with the result of the t

Figure 3 . 9 :

 39 Figure 3.9: Evolution of the extracted electrical components as a function of frequency. On the left series and parallel capacitance and on the right series and parallel resistance

Fig 3 .

 3 Fig 3.10 displays the unipolar polarization cycle of a copolymer at dierent electrical elds.

Figure 3 . 10 :

 310 Figure 3.10: Unipolar polarization cycles of a copolymer with dierent electrical elds amplitudes.

Figure 3 . 11 :

 311 Figure 3.11: Left: polarization cycle of a copolymer with its ferroelectric domains oriented positively. Right: polarization signature of the ferroelectric switching obtained from the left gure

Figure 3 .

 3 Figure 3.12: Left: polarization cycle of a terpolymer at 1Hz and 20 Hz. Right: 1Hz cycle after correction, the 20Hz cycle is left as an element of comparison

Figure 3 . 13 :

 313 Figure 3.13: Capacitance as a function of the electrical eld of the dierent types of polymers

Applying equation 3 .

 3 8 to data in g 3.13 gives us what we refer to hereafter as capacitive or dielectric polarization. The associated mechanisms are reversible and the typical mechanisms are shown in gure 3.3. Although there are associated losses we dismiss them as small enough not to interfere with our study. The dielectric polarization of each polymer is represented in gure 3.14.

Figure 3 . 14 :

 314 Figure 3.14: Calculated polarization response of a dielectric with the measured capacitance values

Figure 3 . 16 :

 316 Figure 3.16: Impact of the introduction of CFE/CTFE monomers in the copolymer chain and resulting polarization cycle.Taken from[7]

Fig 3 .

 3 18 displays the permittivity measured without electrical bias between 12 and 75 o C.

Figure 3 . 18 :

 318 Figure 3.18: Evolution of the permittivity at 0V with temperature of a relaxor terpolymer. The peak position is the Curie temperature

Figure 3 . 19 :

 319 Figure 3.19: Left: polarization cycles measured at 22°C and 50°C at 20Hz (raw data). Right:

S 3 :

 3 Strain in the thickness direction : permittivity Y: Young's modulus ν: Poisson ratio strain. If we estimate the physical parameters to be ν = 0.35, Y=160 MPa and = 30 0 we get a value for the electrostrictive coecient:

First, we need

  to evaluate the strain-polarization relationship of terpolymers in the electrical eld range where they were behaving as linear dielectric materials. Fig 3.20 shows the normalized deection of a terpolymer based cantilever at electrical elds below 20 V µm -1 .

Figure 3 . 20 :

 320 Figure 3.20: Normalized deection of a cantilever as a function of electrical eld (left) and of the squared polarization (right)

Figure 3 . 21 :

 321 Figure 3.21: Left: cantilever displacement as a function of the electrical eld. Right: cantilever displacement as a function of the squared polarization

Figure 3 .

 3 Figure 3.22: Left: cantilever displacement as a function of the squared polarization in g 3.12. Right: simultaneoulsy measured polarization cycle and the extracted phase transition signature

Figure 3 . 23 :Figure 3 .

 3233 Figure 3.23: Strain in PVDF-TrFE-CTFE terpolymers at low electrical elds

Conclusion 16 :

 16 Impact of the phase transition on the strain in PVDF-TrFE-CTFE With strain measurements, we observed an increase of the electrostrictive coecient on an intermediate range of electrical elds. This mechanical change matches precisely the polarization increase in the phase transition cycle. Based on theses observations we propose a phenomenological description to explain the role of the eld-induced transition in terpolymers.

Figure 3 . 25 :

 325 Figure 3.25: Deection of a cantilever with 8.5% CTFE at 0.8Hz for three dierent temperatures. The temperature is that of the hotplate below, the cantilever is located a few millimetres above

Figure 3 . 26 :

 326 Figure 3.26: Measurements at T=80°C. Left: polarization cycle, measurement in yellow and calculated from capacitance measurements in green. Right: Cantilever deection, measurement in yellow and the green curve is the green polarization from left panel squared.

11 .S = QP 2 S 2 S 2 dielP

 11222 = Q(P diel + P r ) = QP 2 r + 2QP r P diel + QP diel : Polarization in unipolar regime (g 3.10) P r : remanent polarization

Figure 3 . 27 :

 327 Figure 3.27: Proposed model of strain in PVDF-TrFE-CTFE terpolymers

d = 2QP r 0 r ( 3

 3 .13) d: piezoelectric coecient If the ferroelectric domains have been oriented by a positive bias, the coecient d is negative as Q < 0 in PVDF. And if the copolymer has been negatively polarized d should be positive. We measured the deection of a copolymer and a 4% CTFE ferroelectric terpolymer with the two polarization states. The results are shown in g 3.28.

Figure 3 . 28 :

 328 Figure 3.28: Normalized displacement of two cantilevers: copolymer and 4% CTFE terpolymers. Each one with the two possible orientations of ferroelectric domains

  Figure 3.29: Piezoelectric d 31 and g 31 coecients as a function of the electrical eld exctracted from the same set of measurements

Figure 3 . 30 :

 330 Figure 3.30: Strain cycle in a copolymer and comparison with the electrostrictive law S = QP 2

Figure 3 . 31 :

 331 Figure 3.31: Simplied representation of two dierent switching mechanisms leading to an identical polarization. The circles are ferroelectric domains in the amorphous matrix

Figure 3 . 33 :

 333 Figure 3.33: Normalized copolymer strain and comparison with the model

Bibliography [ 1 ]

 1 A. N. Hanna, U. S. Bhansali, M. A. Khan, and H. N. Alshareef, Characterization of current transport in ferroelectric polymer devices, Organic Electronics, vol. 15, pp. 2228, Jan. 2014. [2] D. Guyomar, R. Belouadah, B. Ducharne, B. Guiard, M.-Q. Le, and K. Yuse, Strain in ferroelectric polymers under low-frequency electric elds: Experiments and modeling, Journal of Intelligent Material Systems & Structures, vol. 25, p. 1323, July 2014.

Figure 4 . 1 :

 41 Figure 4.1: Picture of a 8.3% CTFE terpolymer sample during an in-situ study

Figure 4 . 2 :

 42 Figure 4.2: XRD spectra of an unbiased sample on dierent substrates

Figure 4 . 3 :

 43 Figure 4.3: Left: consecutive acquisitions of an unbiased terpolymer spectrum on the synchrotron beamline. Right: Consecutive measurement on a diractometer

Figure 4 .

 4 4 displays the XRD and FTIR signatures of the α, γ and β PVDF phases.

Figure 4 .

 4 Figure 4.4: a: XRD pattern of the three main phases of PVDF. b: FTIR spectra of the phases.

Figure 4 .

 4 5 is a schematic representation to visualize what this means in terms of polymer structure.

Figure 4 . 5 :

 45 Figure 4.5: A: Representation of the lattice parameters a, b, c and how they relate to the polymer chain orientation, taken from [7]. B: zoom out from A, a representation of the chain arrangement in the amorphous and crystalline phase. The domains are ferroelectric so the lattice parameter b is in the electrical eld direction. Lattice parameter a and c are orthogonal to the electrical eld direction

Figure 4 . 6 :

 46 Figure 4.6: Copolymer PE and FE phases. Left: lattice representation, right: XRD pattern.

Figure 4 . 7 ,

 47 taken from[10] displays a key evidence of this mechanism. This study is of particular interest here because it involves an XRD analyse of the inuence of electrical eld on the PVDF structure.

Figure 4 . 7 :

 47 Figure 4.7: Left: Schematic representation of the crystallites orientation in the tailored β-PVDF sample and the corresponding Φ-scans. A.(a): unpoled, A.(b)+(c): poled [10]. B: principle of a Φ-scan.

Figure 4 . 8 :

 48 Figure 4.8: A: XRD spectra of the copolymer at 0V and four dierent temperatures. B: Surface versus temperature of the two peaks at 2θ = 19.8 and 2θ = 18 in panel A. C: XRD spectrum of the copolymer sample at 135°C and four dierent biases. D: Area versus electrical elds of the two peaks from C.

Figure 4 . 8 -

 48 Figure 4.8-D displays the peak area at each bias value of panel C. Between 0 and 100 V.µm -1

Figure 4 . 9 :

 49 Figure 4.9: Representation of two crystalline domains oriented dierently and contributing to the XRD signal. Left: the (200) plane is in Bragg condition. Right: the (110) plane is in Bragg condition

(4. 1 )

 1 S 200 = strain of the (200) diraction plane S a =strain of the lattice parameter a M 11 =electrostrictive coecient E= electrical eld amplitude

Figure 4 . 10 :

 410 Figure 4.10: Polarization measurement of a copolymer sample poled positively beforehand.

4. 4 P

 4 (VDF-TrFE-CTFE) in-situ study4.4.1 State of the artThe introduction of CTFE monomers in the copolymer chain hinders the crystallization which drops from 50% to 30 % in average. Because of the larger chlorine atom, the inter-chain distance increases and the diraction peak is located at lower 2θ values than for copolymer, as displayed in g 4.11. This crystalline phase is referred to as relaxor ferroelectric (RFE) phase.To describe the impact of CFE and CTFE introduction on the polymer structure, Yang & al introduced the notion of pinning in the polymer chain (g 3.16)[15] [16][17]. The CTFE or CFE units are distributed every few VDF-TrFE repetitions and pin the chain preventing a global cooperation of dipoles. Instead of dipoles cooperating to form ferroelectric domains, the VDF-TrFE sequences would contribute to the permittivity through dipolar orientation. Under the action of an electrical eld, the CFE dipoles would also orient in the electrical eld direction, resulting in a transition toward a ferroelectric phase which would explain the polarization cycles hereafter.

Figure 4 .Figure 4 . 12 :

 4412 Figure 4.11: XDR pattern of a copolymer and a 59.1/31.8/9.1 terpolymer. Taken from[14]

Figure 4 . 13 :

 413 Figure 4.13: Model for the continuous phase transition with temperature in relaxor P(VDF-TrFE-CTFE) [9]

4. 5 Terpolymers 4 . 5 . 1

 5451 Ambient temperature Measurements We acquired the XRD spectra of a terpolymer sample under electrical elds up to 165V.µm -1 at ambient temperature. The raw data are presented in g 4.14 and separated into 3 categories based on the XRD pattern evolution.

Figure 4 . 14 :

 414 Figure 4.14: Raw spectra of a terpolymer under electrical eld. The data are separated into three panels based on their behaviour. Left: 0-17V.µm -1 , middle 23 -71V.µm -1 and right 95 -166V.µm -1 . The lower row displays examples of the t used to extract the peak parameters.

Figure

  Figure 4.14-A displays the diraction peak of a terpolymer from 0 to 16V.µm -1 with is no visible dependence to the electrical eld. In gure 4.14-B the electrical eld ranges from 22 to

Figure 4 .

 4 Figure 4.15: d-spacing calculated from the diraction peaks position.

Figure 4 . 16 :

 416 Figure 4.16: Illustration of the impact of a cell unit orientation on the strain and polarization. Left: before electrical eld, right: after electrical eld

Figure 4 . 17 :

 417 Figure 4.17: Diraction peak parameters versus electrical eld. A:Width, B: Intensity, C: Area

Figure 4 .

 4 17 B displays the diraction peak width, following an evolution opposite to that of intensity. It gets thinner between 20 and 60 V.µm -1 then saturates past that point. The width was an unnecessary degree of freedom for the (110) and (200) peaks as it remains constant throughout the t. Fig 4.17 C displays the total area of each peak versus the electrical eld. With the intensity increasing and the width decreasing, the total area value could have gone either way but it actually increases by 40% on the 20-60 V.µm -1 range. The intensity and width variations conrm the results from the d-spacing extraction: what

Figure 4 . 18 :

 418 Figure 4.18: Left: normalized cantilever deection at dierent frequencies. Right: width of the mechanical cycle versus frequency

Figure 4 . 19 :

 419 Figure 4.19: Left: Diraction peak of an unbiased terpolymer sample at increasing temperatures. Right: electrical cycle at 40°C

Figure 4 . 20 :

 420 Figure 4.20: Electrical study performed at dierent temperatures. A: d-spacing versus electrical eld B: diraction peak width. C:diraction peak intensity

Fig 4 .

 4 Fig 4.20 B displays the diraction peak intensity. Width and intensity are correlated and the same observations as above can be made here. The crystals rotation occurs at 30°C and also at 40°C, albeit less hysteretic this time.

Figure 4 .

 4 21 is based on the models of Yang & al and Bargain & al, presented in the state of the art section. First we need a clear description of the symbols used in the terpolymers representation. The red and blue arrows are VDF-TrFE units, red arrows in the same direction represent trans conformations, resulting in a net dipolar moment. A blue and red arrow represent a gauche conformation

Figure 4 . 21 :

 421 Figure 4.21: Description of structural changes in terpolymers with temperature and electrical eld

5. 1

 1 Follow up work on P(VDF-TrFE-CTFE) 5.1.1 Φ scansIn the previous chapter we proposed that changes in XRD spectra arose from the rotation of crystallites and the (110) diraction peak increasing as the (200) is shrinking. This can be veried experimentally by measuring dierent orientations, as illustrated in gure 5.1.

Figure 5 . 1 :

 51 Figure 5.1: Schematic representation of the expected dierence between a measurement at Φ = 0 o and Φ = 90 o

Figure 5 .

 5 Figure 5.2: d-spacing with data from g 4.14 as a function of squared polarization

(5. 1 )

 1 Q i = intrinsicelectrostrictive coecient permittivity T: Mechanical stress We used equation 5.1 to get the intrinsic electrostrictive coecient of a 2% CTFE terpolymer. The study was performed on a standard cantilever and g 5.3 illustrates the measurement principle along with the results.

Figure 5 . 3 :

 53 Figure 5.3: Capacitance measurements of a 2% CTFE terpolymer as a function of bending

Figure 5 . 4 :

 54 Figure 5.4: Capacitance measurements of a 2% CTFE terpolymer as a function of bending

Figure5. 5

 5 Figure5.5 displays the comparison between 60.4/30.5/9.1% (VDF/TrFE/CFE) and 61.4/30.3/8.3% (VDF/TrFE/CTFE) terpolymers.

Figure 5 . 5 :

 55 Figure 5.5: Comparison between two terpolymers, 60.4/30.5/9.1% (VDF/TrFE/CFE) and 61.4/30.3/8.3% (VDF/TrFE/CTFE). Left: Polarization cycle. Right: Strain cycle

Figure 2 :

 2 Figure 2: Deection measurements at 200 and 500Hz of a glued round capacitance, measured with an interferometer

Figure 3 :

 3 Figure 3: Capacitance measurements of the sample studied in gure 3.19

Figure 4 :

 4 Figure 4: Capacitance measurements of the sample studied in gure 3.26

Figure 5 :

 5 Figure 5: Bragg Brentano measurement principle

Figure 6 :

 6 Figure 6: In plane measurements optical setup

Table 1 .

 1 1: Advantages and drawbacks of the dierent electroactive materials categories.

	.1 some key properties

Table 2 .

 2

	Coecient	Notation	Units
	Permittivity	ij	F.m -1
	Rigidity	c ijkl	P a
	Compliance	s ijkl	P a -1
	Young's modulus	Y	Pa
	Poisson coecient	ν
	Piezoelectric coecients (T,E)

1: Physical quantities of the electroactive material The physical quantities in table 2.1 have subscripts because they are tensors and not scalar values. i and j are Einstein notations and correspond to the dierent directions of space. These quantities are interdependent and are linked to one another by the electromechanical coecients referenced in table 2.2.

Table 2 . 2

 22

: Electromechanical coecients

The constitutive equations of an electroactive materials are the following:

  To give an idea of the typical values for a copolymer we reported a few examples in table 2.3.

		Value	Ref
	d 31 (pm.V -1 )	12.5, 11	[8],[9]
	d 33 (pm.V -1 ) -25, -34, -31, -44 [10], [9], [11], [12]
	Y (GPa)	2.1, 3.3, 1.4	[8],[9], [12]
	Table 2.3: Piezoelectric coecients and Young's modulus of P(VDF-TrFE) from dierent studies
	The d coecients quantify the amount of charges we can get out of a given mechanical
	stress (eq 2.1). For reference, the PZT ceramics commercially available have coecient d 33 >
	300 pm.V -1 [13], ten times superior as the values in table 2.3. Beside their higher d 33 value, PZT
	have a Young's modulus around 60 GPa which is around thirty times more than that of P(VDF-

Table 2 .

 2 5: Estimation of the strain and stress achieved in polymers at 20 V.µm -1 , assuming that equation 2.1 is valid at that electrical eld value.

	.4: Electrostrictive coecient M 31 and Young's modulus of terpolymers found in
		dierent studies.
	Using table 2.3 and 2.4 we can estimate the order of magnitude for the stress and strain of co-
	and ter-polymers at 20 V.µm -1 . Strain is calculated with equation 2.1 and stress is the product
	of strain and Young's modulus (Hooke's law).
		Copolymers	Terpolymers
	Strain	0.02 %	0.1 %
	Stress (Pa)	5.10 5	2.10 5

Table 3 .

 3 .1.

		R p (kΩ) C p (nF) R (kΩ) C (nF)
	Manufacturer	100	10	47	10
	Extracted	98.7	10.1	46.8	9.84

1: Manufacturer and extracted values

Table 4 .

 4 .2 the strain values calculated from XRD measurements and those from chapter 3. 2: Strain values of the terpolymer at 160 V.µm -1 and contribution ascribed to phase transition in chapter 3.

		Macroscopic	XRD
	"Phase transition"	0.3%	-1.4%
	Total strain	2.8%	-3.5%

Chapter 4

In-situ structural study 105 Annex IV: XRD cycles and tting algorithm Figure 7 displays the complete dataset of the temperature study from chapter 4. These are not the raw measurements but the diraction peaks width, intensity, position and area. To obtain these values we used the python script hereafter.

Here are a few remarks on this code. It allow to perform an unlimited number of ts on a single run. The core function with the parameterized tting algorithm comes from the minimize function of the lmt package (Newville & al http://doi.org/10.5281/zenodo.11813). The data must be imported as two list of oat for position and intensity, the function getxrdml is used to transform our xrdml les into oats. Only the function 1 and 3 are parameterized, 2 and 4 get their entry values from them. They account for the ka2 ray of copper.