Aymen Jallouli

Najib Kacem
email: najib.kacem@univ-fcomte.fr

Gilles Bourbon

Patrice Le Moal

Joseph Lardies

Experimental characterization of

Keywords: electrostatic resonator, geometric imperfections, experimental characterization, nonlinear dynamics, bifurcation topology PACS 05.45.-a, 07.07.Mp, 46.40.Ff

Introduction

Nano and microelectromechanical systems (NEMS/MEMS) are commonly made from microbeams and microplates [START_REF] Rhoads | Nonlinear dynamics and its applications in micro-and nanoresonators[END_REF]. These type of resonators are used in several applications such as logic devices [START_REF] Hafiz | Microelectromechanical reprogrammable logic device[END_REF], microswitches [START_REF] Jia | Pull-in instability of geometrically nonlinear micro-switches under electrostatic and casimir forces[END_REF][START_REF] Duffy | Mems microswitches for reconfigurable microwave circuitry[END_REF], microphone [START_REF] Mallik | Development and performance analysis of a low-cost mems microphone-based hearing aid with three different audio amplifiers[END_REF][START_REF] Ganji | Slotted capacitive microphone with sputtered aluminum diaphragm and photoresist sacrificial layer[END_REF][START_REF] Williams | An aln mems piezoelectric microphone for aeroacoustic applications[END_REF], mass and gas sensors [START_REF] Sakhaee-Pour | Applications of single-layered graphene sheets as mass sensors and atomistic dust detectors[END_REF][START_REF] Comini | Solid state gas sensing[END_REF][START_REF] Walter | Electrostatic actuation to counterbalance the manufacturing defects in a mems mass detection sensor using mode localization[END_REF][START_REF] Rabenimanana | Mass sensor using mode localization in two weakly coupled mems cantilevers with different lengths: Design and experimental model validation[END_REF][START_REF] Rabenimanana | Functionalization of electrostatic nonlinearities to overcome mode aliasing limitations in the sensitivity of mass microsensors based on energy localization[END_REF] and biological and microfluidic applications [START_REF] Braun | Micromechanical mass sensors for biomolecular detection in a physiological environment[END_REF][START_REF] Alneamy | Electrostatic arch micro-tweezers[END_REF]. However, these micro devices are subject to some imperfections due to a residual stress gradient caused by the microfabrication process [START_REF] Younis | MEMS linear and nonlinear statics and dynamics[END_REF][START_REF] Chu | A study of residual stress distribution through the thickness of p/sup+/silicon films (thermal oxidation effects)[END_REF].

Despite the fact that the geometric imperfections can be very small compared to the size of the micro-system; it has a major effect on its dynamic and static responses. Initial imperfections have been the focus of researchers in the last few years [START_REF] Wang | Nonlinear coupled vibration of electrically actuated arch with flexible supports[END_REF][START_REF] Alcheikh | Highly sensitive and wide-range resonant pressure sensor based on the veering phenomenon[END_REF][START_REF] Alneamy | Electrostatic arch micro-tweezers[END_REF].

The majorities of the researches were focusing on the dynamic and static responses of beams with initial deflection. For cantilevers, the residual stress changes the shape of the beam to a curved shape [START_REF] Ouakad | Simple and accurate analytical solution to the post-buckling response of an electrostatically actuated mems curled cantilever[END_REF]. This initial deflection has a major effect on the pull-in threshold depending on the bending direction, which can cause the failure of the microstructure when the two electrodes comes into contact [START_REF] Younis | MEMS linear and nonlinear statics and dynamics[END_REF][START_REF] Younis | Analytical expressions for the electrostatically actuated curled beam problem[END_REF]. Several studies have been also conducted on microbeams with clamped-clamped boundary condition thanks to its rich dynamic and static behavior (snap-through, bistablity, pull-in ).

Younis et al [START_REF] Younis | Nonlinear dynamics of mems arches under harmonic electrostatic actuation[END_REF][START_REF] Younis | MEMS linear and nonlinear statics and dynamics[END_REF][START_REF] Younis | Analytical expressions for the electrostatically actuated curled beam problem[END_REF] and Ouakad [START_REF] Ouakad | The dynamic behavior of mems arch resonators actuated electrically[END_REF] presented the effect of the initial deflection on the nonlinear dynamic and static behavior of a clamped-clamped beam such as softening and hardening behavior, dynamic snap through and dynamic pull-in. From experimental point of view, curved polysilicon microbeam can be used as bandpass filters and low-powered switches. The effects of axial force resulting by the residual stress or temperature variation have been investigated by Wang et al [START_REF] Wang | Three-to-one internal resonance in mems arch resonators[END_REF]. The governing equation of motion has been discretized using the Galerkin method, the obtained equations have been solved using the method of multiple scale to investigate the dynamic response of the beam in the presence of three-to-one internal resonance by obtaining the frequency response, force response time history phase plane portrait, Poincare section and Fourier spectrum.

The nonlinear behavior of a micro-system has been also studied for the case of a microplate with initial deflection. Saghir et al. [START_REF] Saghir | Initially curved microplates under electrostatic actuation: theory and experiment[END_REF][START_REF] Saghir | An experimental and theoretical investigation of the mechanical behavior of multilayer initially curved microplates under electrostatic actuation[END_REF][START_REF] Saghir | An investigation of the mechanical behavior of initially curved microplates under electrostatic actuation[END_REF] developed the von Kármán plate equation that introduces the electrostatic and geometric nonlinearities. The Galerkin method has been used to discretize the obtained partial differential equations. The numerical model, supported with experimental tests, have been used to study the static and dynamic behaviors of the microplate. They showed the effect of the DC voltage on the static deflection and the natural frequency. Under a harmonic AC voltage, the microplate can have a hardening behavior, due to the effect of the midplane stretching nonlinearity, and a softening behavior due to the effect of the electrostatic nonlinearity. The effects of initial deflection on the natural frequencies have been studied numerically. The increase of the initial deflection leads to an increase of the first natural frequency until it reaches a maximum before it starts decreasing. However the higher modes showed a steady increase.

Other studies [START_REF] Celep | Free flexural vibration of initially imperfect thin plates with large elastic amplitudes[END_REF][START_REF] Celep | Shear and rotatory inertia effects on the large amplitude vibration of the initially imperfect plates[END_REF] investigated the effects of initial deflection of a rectangular plate under different boundaries conditions by presenting an approximate solution for the free flexural vibration.

For the case of circular microplate, Medina et al [START_REF] Medina | Bistable behavior of electrostatically actuated initially curved micro plate[END_REF][START_REF] Medina | Modeling strategies of electrostatically actuated initially curved bistable micro plates[END_REF][START_REF] Medina | Bistability criterion for electrostatically actuated initially curved micro plates[END_REF] developed a reduced order model (ROM) of an initially curved circular plate actuated with electrostatic force. Their studies showed that due to the electrostatic nonlinearity this kind of MEMS structure can have a snap-through. To transform the ROM to a more compact one, they used Berger's approximation for which a single mode can be used to predict the bistable behavior for the case of low thickness to gap ratios microplate.

Zhang [START_REF] Zhang | Large deflection of clamped circular plate and accuracy of its approximate analytical solutions[END_REF] derived the equation of motion of deflected circular plate using the principle of virtual work by taking into account the in-plane tension. A new approximated static analytical solution is derived and it showed a much simpler and more direct framework to study the plate-membrane transition behavior compared to other analytical approaches.

The nonlinear behavior of circular plates was also investigated at the macroscale by Touzé and Thomas [START_REF] Touzé | Asymmetric non-linear forced vibrations of free-edge circular plates. part 1: Theory[END_REF][START_REF] Thomas | Asymmetric non-linear forced vibrations of free-edge circular plates. part ii: experiments[END_REF]. The equations of motion of the circular plate with free edge boundary condition were derived by taking into account the plate imperfection. The analytical model was validated with respect to experimental results obtained by measuring the deflections of the antinode. The nonlinear frequency responses of the vibration amplitude and the phase were determined for different configurations. By estimating the input parameter of the system, the developed model is capable to predict the nonlinear phenomena of the plate.

In our previous works [START_REF] Jallouli | Pull-in instability tuning in imperfect nonlinear circular microplates under electrostatic actuation[END_REF][START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF], a computational model was developed that investigates the nonlinear static and dynamic behaviors of circular (Capacitive Micromachined Ultrasonic Transducers) with initial imperfection by transforming the von Kármán equation into a set of ordinary differential equations using the differential quadratic method. The model showed a good performance in predicting the nonlinear behavior of the circular microplate and its bifurcation topology.

Most of the previous experimental studies in MEMS actuators have been focusing on beams or rectangular shape microplate with initial imperfection. In this paper, an experimental nonlinear characterization of an electrostatic resonator with circular shape and initial deflection is presented. The main objective in this context is to highlight the influence of low initial deflection on static and dynamic behaviors of circular microplates. First the fabrication technique of the microplates was optimized in order to reduce the residual stress and minimize geometric imperfections. For the second part, different experimental techniques are used during the characterization of the microplate. For the static characterization, an optical interferometer type Mirau was used to measure the topography of the microplate for different DC voltages and to present the effect of the initial deflection on the static behavior and the pull-in voltage of the resonator. For the dynamic characterization, a Laser Doppler Vibrometer was used to measure the velocity of the microplate surface while changing the frequency. From these experimental results, we presented the effects of the initial deflection on the linear eigenfrequencies of the resonator and its nonlinear frequency response. Also, The dynamic behavior of the microplate is investigated by looking to its time history and Fourier spectrum.

2 Fabrication of the resonator and identification of its design parameters

Fabrication process

The resonator is fabricated using anodic bonding technique, where the microplate and the gap are defined on different wafers and bounded at low temperature and by applying a voltage difference [START_REF] Bellaredj | Anodic bonding using soi wafer for fabrication of capacitive micromachined ultrasonic transducers[END_REF]. The Figure 1 presents the process flow used in this work to fabricate the microsystem. The fabrication process was executed in MIMENTO clean room according to the following steps: we start with a wafer of glass Borofloat 33 and a Silicon On Insulator wafer (SOI) with a 2.3 µm silicon device layer representing the membrane thickness.

The first step in the process is creating the gap in the glass wafer. A thin layer of chromium and gold is deposited on the surface of the wafer, Figure 1a. The gold layer is used as a hard mask for glass etching. Next, the cavity shape is defined by photolithography patterning of a previously coated resin layer and etching of the gold / chromium layers. The wafer is then etched with a buffered hydrofluoric acid solution (BHF) to create the cavity, Figure 1b. After creating the cavity, the hard mask is removed by putting the wafer into a gold and chromium etch solution.

A thin layer of gold was deposed at the cavity to create the bottom electrode of the resonator using the lift off technique, Figure 1c.The bottom electrode thickness has to be controlled because the corresponding deposited metal reduces the electrostatic gap.

The two wafers are placed in the bonding machine at a very low pressure of 10 -2 Pa and heated to 350 o C since the two wafers have an equal CTE 350 o C which is very convenient for the bonding. Once the temperature of the two wafers is stabilized, a force of 1000 N is applied followed with four steps of voltage (300 V, 500 V, 700 V and 900 V) for 10 min each to limit the peak current [START_REF] Bellaredj | Anodic bonding using soi wafer for fabrication of capacitive micromachined ultrasonic transducers[END_REF]. The two wafers are now bonded and a controlled cooling started, Figure 1d. The controlled cooling is very essential to release the stress created during the bonding.

In the next step, the handle layer is removed using KOH solution. The oxide layer protect the silicon device layer from etching. Thereafter, the buried oxide layer is removed with BHF solution, Figure 1e. Next, a thin aluminum layer is deposited on the remaining silicon device layer. Another key concept to reduce the residual stress is to perform a thin and slow deposition rate of the aluminum electrode (the thickness is 150 nm and the deposition rate 49 nm/min). The aluminum layer has two important roles: first it is used as a hard mask for the etching of the device layer defining membranes and second it is used as a top electrode on the membranes. The aluminum layer is etched after the photolithography patterning of a previously coated resin layer. Finally, the silicon device layer is etched by Reactive Ion Etching using the aluminum hard mask to define membranes and access to the bottom electrode, Figure 1f.

Design parameter identification of the resonator

The microplate, as it is presented in figure 2 is mainly composed of three parts:

the membrane, with radius R and a thickness h, which is the movable part that generates the acoustic waves.

the bottom electrode, with radius R ele , representing the fixed electrode which delimits the action area of the electrostatic forces (excluding edge effects).

the gap, with height d g , which represents the distance between the top and the bottom electrode. The plate is a bilayer Silicon / Aluminum but the stiffness (given by the thickness and the Young modulus) of the aluminum layer can be neglected compared to that of the silicon. As the silicon is a brittle material at room temperature [START_REF] Hirsch | The brittle-ductile transition in silicon[END_REF], the material of the plate can thus be considered elastic and homogeneous. Moreover, the circular structure by its axisymmetric nature allows to suppose the material of the plate as isotropic with an average Young modulus in the plane (100) around 149 GPa between two extremal values 130 GPa and 169 GPa. The microplate is subjected to an axisymmetric initial deflection w 0 (r) that can be approximated as follows:

w 0 (r) = w max 1 + cos πr R 2 , (1)
where w max represents the initial deflection of the membrane at its center. It is important before starting the characterization to measure the dimensions of the micro-device like the diameter, the gap distance, the electrode thickness and the membrane thickness. To obtain these values, we used an optical microscope to measure the lateral dimensions (like the radius of the cavity and the bottom electrode) and a contact profilometer for vertical dimensions (like the thickness of the membrane, the depth of the cavity and the thickness of the bottom electrode).

The design parameters of the micro-system are displayed in Table 1 with the errors reported for the measured dimensions.

.3 ± 0.1 [µm] d g Gap distance 0.6 ± 0.1 [µm] 3 Mathematical Model
We consider a circular microplate excited with an electric voltage

V (t) = V dc + V ac cos(ωt)
where V dc is the DC bias voltage and V ac is the amplitude of the harmonic voltage.

The partial differential equations that describe the dynamic response of the circular microplate, with an initial deflection w 0 (r), are derived from the von Kármán plate theory [START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF].

ρh 2 D ẅ+ h 2 D c ẇ+∇ 4 w = N 0 D 1 r (w ,r + w 0,r ) + (w ,rr + w 0,rr ) + 0 V 2 (t) 2D(d g -w + w 0 ) 2 + 12 h 2 1 r u ,r (w ,r + w 0,r )+u ,rr (w ,r + w 0,r )+u ,r (w ,rr + w 0,rr )+ 1 2r (w ,r ) 2 (w ,r + w 0,r ) +w ,r (w ,r + w 0,r ) 1 r w 0,r + w 0,rr + 1 2 (w ,r ) 2 (w ,rr + w 0,rr )+w ,r w 0,r (w ,rr + w 0,rr ) + w ,rr (w ,r + w 0,r ) 2 + ν r (u ,r (w ,r + w 0,r ) + u (w ,rr + w 0,rr )) , (2)
u ,rr + 1 r u ,r - u r 2 = - 1 -ν 2r (w ,r ) 2 -w ,r w ,rr , (3)
where u(r, t) and w(r, t) represent the in-plane and out of plane displacement of the membrane in the r, θ, z coordinate system.

∇ 4 = ∂ 2 ∂r 2 + 1 r ∂ ∂r 2
the bi-harmonic operator, D = Eh 3 12(1-ν 2 ) the plate flexural rigidity, c the damping coefficient, ( * ) ,r ; ( * ) ,rr first and second derivative with respect to r and ( * ) ( * ) first and second derivative with respect to t.

For an axisymmetric problem and at the center of the microplate, the out of plane displacement w has always a slope equal to zero and the radial displacement u = 0. Considering the peripheral clamping of the circular microplate, the boundary conditions are finally [START_REF] Zhang | Large deflection of clamped circular plate and accuracy of its approximate analytical solutions[END_REF]:

At r = 0 : w ,r = 0 u = 0 At r = R : w ,r = w = 0 u = 0 Eh 1 -ν 2 du dr + 1 2 dw dr 2 + ν r u = 0, (4)
4 Static response

Static characterization setup

For static characterization, the wafer is placed underneath the interferometer, Mirau type, to measure the topography of the microplate, as shown in Figure 3.

The micro-system is forced by a DC voltage using a DC generator equipped with two probes. This technique can provide accurate 3D and 2D scans of the surface of the microplate for different DC voltages. As an advantage of this characterization method, the microplate surface cannot be damaged since the measurements are based on a non-contact scanning using an interferometer. Also, the static displacement of the membrane can be measured while applying a voltage which is not possible with the use of the profilometer. respectively. Figure 4a shows clearly that the plate has a flat surface for V dc = 0 V

. By increasing the DC voltage, the microplate starts to bend downward until it reaches its maximum at V dc = 35 V. At this stage, the increase of the electrostatic force can cause the collapse and thus the destruction of the micro-device by shortcircuit, because no insulation layer can prevent the electrical contact between the microplate and the bottom electrode.

The characterized micro-system in Figure 4b is initially deflected. In fact, the microplate displays a concave shape initial deflection with a maximum w max = 270 nm for V dc = 0 V. When the electrostatic force increases, the microplate starts to change its form to almost flat for V dc = 49 V, and convex beyond this actuation level.

The static responses of the two micro-system, flat and deflected, are depicted in Figure 4c by presenting the total displacement at the microplate center for different DC voltages. The experimental results are compared to the numerical model presented in [START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF], which is based on Differential Quadratic Method (DQM).

The solid and dashed lines, in Figure 4c, represent the stable and the unstable numerical solutions and the stars ( * ) marked in this graph correspond to the pull-in points, for which the solution changes from stable to unstable. The experimental static response shows a good agreement with the numerical results obtained from the reduced order model [START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF]. The maximum error between the numerical and experimental results is less than 5 %.

As the DC voltage increases, the total displacement of the microplate increases until it reaches the pull-in point, where the slope of the curve tends to infinity. It can noticed from Figure 4c that an upward initial deflection increases the pull-in voltage. This can be explained by the increase of the gap distance between the two plates. A small initial imperfection (w max = 270 nm) can shift the pull-in voltage from V dc = 35 V to V dc = 53.5 V. Thus, It is important to take into account the effect of initial deflection in the design and fabrication of electrostatic resonators.

By analyzing the static response, there is no snap-through behavior since the curve has only one stable branch. This is due to the fact that the initial deflection is small compared to the thickness of the microplate [START_REF] Alkharabsheh | Statics and dynamics of mems arches under axial forces[END_REF][START_REF] Ouakad | An electrostatically actuated mems arch band-pass filter[END_REF][START_REF] Krylov | The pull-in behavior of electrostatically actuated bistable microstructures[END_REF].

5 Linear and nonlinear dynamic analysis

Dynamic characterization

The frequency response of the resonator is determined using a vibrometer OFV-534 (P olytec) equipped with a decoder with different precisions (can go up to 1 pm for the displacement and 10 m/s for the velocity) and wide frequency range (up to 24 MHz), shown in Figure 5. The VD02 velocity decoder transforms the optical signal to an electrical one, which will be displayed on the oscilloscope. In order to exhibit nonlinearities (i.e. higher vibration amplitudes), the fluid interactions with the micro-system like ambient air loading and losses including squeeze film effects [START_REF] Jallouli | Modeling and experimental characterization of squeeze film effects in nonlinear capacitive circular microplates[END_REF] was eliminated by placing the wafer inside a vacuum chamber, as shown in The construction of the nonlinear frequency response is performed by exciting the microplate with a DC voltage combined with a periodic chirp signal with an amplitude V ac . The periodic chirp function is a sinusoidal function that changes its frequency through time defined as follows:

V (t) = V dc + V ac sin (2πf 0 (t)t) , where f 0 (t) = f s + f e -f s T pc t, (5)
V dc =0 V V dc =20 V V dc =26 V V dc =30 V V dc =33 V V dc =34.5 V (a)
Vdc=0 V Vdc=30 V Vdc=44 V Vdc=49 V Vdc=51 V Vdc=52 V Vdc=53.5 V (b)

Linear eigenfrequency analysis

The initial deflection does not only affect the static response of the microplate, but it also affects its resonance frequencies. In order to determine the variation of the resonance frequencies with respect to the applied DC voltage, a laser vibrometer is used equipped with vacuum chamber, presented in Figure 5. The two circular microplates that has been tested have the same design parameters presented in Table 1, however, one has a flat surface and the other one has an initial deflection

w max = 270 nm.
The nondimensional eigenfrequencies, as it is described in [START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF], depend mainly on the axial force N 0 , the DC voltage V dc and the initial deflection w 0 . The axial force is chosen by matching the numerical natural frequencies of the microplate, for a low V dc , with the experimental ones. For parameters identification, the initial stress and the plate thickness of the microplate are fitted to match the experimental and numerical results (without exciding the error bar presented in table 1) [START_REF] Ruzziconi | An efficient reduced-order model to investigate the behavior of an imperfect microbeam under axial load and electric excitation[END_REF][START_REF] Bataineh | Dynamics of a clamped-clamped microbeam resonator considering fabrication imperfections[END_REF]. Therefore, the following error function to minimize can be defined as:

e r = 2 i=1 det K -(2πT f exp,i ) 2 M , (6)
where f exp,i is the i th experimental resonance frequency. In our case, two experimental resonance frequencies are used. M and K are the linear mass and stiffness matrix, respectively. T is the nondimensional parameter defined in [START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF]. The principal axis method [START_REF] Brent | Algorithms for minimization without derivatives[END_REF] is used to determine the plate thickness h and the initial stress N 0 that minimize the error function e r .

The experimental resonance frequencies, displayed in Figure 7, are obtained by exciting the microplate with a small AC voltage, in order to have a linear response, and different DC voltages. For the numerical results, a mathematical model, developed in [START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF], was used with the physical parameters presented in Table 1. The nonlinear partial differential equations ( 2) and ( 3) were spatially discretized using the Differential Quadrature Method (DQM) to obtain a set of nonlinear differential equations. The nonlinear dynamic behavior of the microplate is studied by discretizing the time variable using the Finite Difference Method (FDM). As it can noticed from Figure 7, the natural resonance frequencies at low V dc of the two resonators are slightly different, this is due to the difference in the residual stress. By increasing the DC voltage, the resonance frequencies decrease due to the increase of the negative electrostatic stiffness which is proportional to V 2 dc . At the pull in voltage, the first resonance frequency drops to zero since the electrostatic stiffness is equal to the mechanical one. Clearly, the numerical model has a good agreement with the experimental data. Figure 7 shows also the influence of the initial deflection on the first resonance frequency. At a constant DC voltage, the initial deflection shifts the resonance frequency of the microplate to higher values. This is due to the increase of the gap distance which reduces the electrostatic force. That means an important DC voltage is needed to bend the microplate.

An other major effect to focus on is the effect of the DC voltage on the dissipated energy of the resonator. Figure 8 shows the measured the quality factor Q of the linear frequency response at different DC voltages using the half-power bandwidth method (Q = f 0 ∆f ). The microplate is actuated with small AC voltage V ac = 0.07 V in order to have a linear frequency response for different DC voltage. For low DC voltage the quality factor for the two resonators is important since they are in vacuum chamber to avoid air losses. By increasing the DC voltage, the quality factor decreases strongly even at low ratios of DC to pull-in voltages.

Losses sources are possibly numerous and discussed in many papers [START_REF] Naeli | Dimensional considerations in achieving large quality factors for resonant silicon cantilevers in air[END_REF] : viscous loss, acoustic radiation, support loss, thermoelastic loss, volume and surface loss, ... .

1 Q tot = 1 Q viscous + 1 Q support + 1 Q thermoelastic + 1 Q surf ace + 1 Q ohmic + .... (7)
Some of them as gap-closing effect [START_REF] Shmulevich | On the quality of quality-factor in gap-closing electrostatic resonators[END_REF] and ohmic losses [START_REF] Sazonova | A tunable carbon nanotube resonator[END_REF] depends on the DC voltage. However, only the ohmic losses from the electrons moving on and off the resonator due to capacitive coupling to a nearby gate can explain the observed important influence of the DC voltage on the evolution of the quality factor, in particular according to an inverse quadratic variation :

Q -1 ohmic = 1 πω R Ω (C V dc ) 2 m ef f , (8)
where ω is the angular frequency of vibration, R Ω is the output resistor C is the gradient of the capacitance and m ef f is the effective mass of the resonator. As a conclusion, the more we approach the pull-in voltage of the resonator the more the quality factor decreases.

Nonlinear frequency analysis

In this section, the nonlinear frequency of a circular microplate is investigated near the primary resonance. In the case of an initially deflected circular microplate, the resonator has two different behavior according to the bias voltage level. For the first case in Table 2, the membrane is excited with a small DC voltage and so the geometric nonlinearities are higher than the electrostatic ones, as shown in Figure 9a. Thus, the frequency response bends to the right displaying a hardening behavior involved by geometric nonlinearities. In the second case reported in Table 2, the micro-system biased by a higher DC voltage shows a softening behavior in Figure 9b. Indeed, the frequency response bends to the left illustrating the predominance of electrostatic nonlinearities over geometric ones [START_REF] Rhoads | Nonlinear dynamics and its applications in micro-and nanoresonators[END_REF]. It can be noticed that experimental and numerical results are in a good agreement. The maximum error between the two results is at the bifurcating points. For example the error at the hardening response 9a is equal 7% however for the softening case is equal to 12%. This is because at high displacement field induced by nonlinear effects other loss sources may occur and the quality factor is more affected than expected. Hardening and softening behaviors are accurately described according to qualitative and quantitative considerations. Fig. 9: Comparison between the experimental and the numerical frequency response of the maximum velocity at the center of the microplate with initial deflection w 0 = 270 nm for: (a) small DC voltage, first case in Table 2. (b) for higher DC voltage, second case in Table 2. The solid line presents the stable solution of the system and the dashed line presents the unstable solution.

The nonlinear frequency resonance of the resonator is investigated with respect to DC and AC voltages. In Figure 10a, the microplate was actuated with V ac = 0.25

V and different DC voltages. For a low DC voltage (V dc = 2.5 V), the microplate vibrates with a low amplitude and the system has a linear behavior. By progressively increasing the DC voltage, the vibration amplitude increases, the resonance peak moves to lower frequency values and the curve bends to the right. As long as the bias voltage stays below a threshold of 20V, Figure 10a displays a hardening behavior involved by geometric nonlinearities. Remarkably, at a critical V dc about 20 V, the resonator frequency response changes form hardening to softening due to the amplification of the electrostatic nonlinear terms.

Unlike the effect of V dc on the resonator frequency response, as shown in Figure 10b and Figure 10c, the variation of the AC voltage has a significant impact on the excitation amplitude and a negligible one on the negative electrostatic stiffness. Consequently, the increase in the V ac leads to an increase in the vibration amplitude and the bistability domain without any remarkable frequency shift. This phenomenon is shown in Figure 10b for V dc = 5 V where the frequency response exhibits a hardening behavior. For this case, the geometric nonlinearities are dominating the electrostatic nonlinearities which explains the hardening behavior of the microplate. However, when the electrostatic nonlinearities dominate the geometric nonlinearities, the microplate exhibits a softening behavior as it is shown in Figure 10c.

In Figure 11, we compare the experimental frequency response of a flat and a deflected microplate excited with the same DC and AC voltages (V dc = 15 V and V ac = 0.25 V). With a flat surface, the microplate has a softening behavior.

Nevertheless, when the microplate has an initial deflection, the behavior of the micro-system changes from softening to hardening. The initial deflection increases the gap distance between the two electrodes which decreases the electrostatic force.

In this case, the sign of the overall cubic nonlinearities is positive. Therefore, the frequency response curve bends to the right and exhibits a hardening behavior.

Consequently, the bifurcation topology of the resonator frequency response can be tuned with respect to the DC voltage and the initial deflection. Doing so, the performances of the device can be adjusted in terms of generated acoustic power and frequency bandwidth.

Time history analysis

Another way to characterize the dynamic behavior of a nonlinear resonator is by analyzing the time history of its response [START_REF] Nayfeh | Applied nonlinear dynamics: analytical, computational and experimental methods[END_REF]. In this section, we are interested to the upper stable branch of the nonlinear frequency response of the microplate, In Figure 12a, the excitation frequency is set to be f a = 605 kHz, where the frequency response have one stable solution. As it can noticed from the Fourier spectrum, the response contains components of the first and super-harmonics.

Decreasing the excitation frequency to f b = 598 kHz , which corresponds to a stable response in the bistable region, the Fourier spectrum contains other harmonic peaks around the first and the superharmonics compared to to the frequency f a which explains the quasi-periodic response of the time history response. As shown in Figure 12(c-d), by decreasing more the frequency, the time response becomes periodic and the vibration amplitude increases while the signal becomes more distorted when approaching the bifurcation point. The appearance of superharmonics in the time history of the microplate indicates the presence of quadratic and cubic terms originating from electrostatic and geometric nonlinearities.

Conclusion

The experimental characterization of a circular capacitive transducers and the understanding of their nonlinear behavior were the main focus of this paper.

The micro-system has been characterized with different methods: for the static behavior, an optical interferometer was used to visualize the topography of the upper surface of the microplate for different DC voltages. A comparison between flat and initially deflected resonator was performed showing a good agreement with the theoretical results. Even though this deflection is small, it leads to an important increase of the pull-in voltage due to the increase of the gap distance between the two electrodes.

The dynamic response of the microplate was investigated using Laser Doppler Even though in this work, circular microplates have been tested in vacuum, future work will include the modeling and the experimental analysis of such devices in air and water for practical applications while taking into account the squeeze film effects.

Fig. 1 :

 1 Fig. 1: The process flow of the anodic bonding method. (a) Deposition and patterning of the gold / chromium layer (b) Etching of the cavity in the glass wafer, (d) deposition of the bottom electrode, (d) anodic bonding of the glass wafer with the SOI wafer, (e) etching of the silicon substrate and the buried oxide of the SOI, (f) deposition of the top electrode and dry etching of the membrane.

Fig. 2 :

 2 Fig. 2: (a) A photo of the micro-device after fabrication. (b) A cross section of a circular electrostatic thin microplate.

Fig. 3 :

 3 Fig. 3: (a) Experimental setup for the static response of the resonator. (b) Topography of the membrane surface

Figure 5 .

 5 Figure 5. The electric connections are ensured by two probes placed inside the vacuum chamber. The latter has a transparent window (Plexiglas) enabling the penetration of the laser beam without any distortion. The characterization setup is equipped with two generators in order to excite the resonator with DC and AC voltages combined via a coupling circuit.

Fig. 4 :Fig. 5 :

 45 Fig.4: (a) and (b) Evolution of the static profile for a clamped circular microplate electrostatically excited for the case of flat and deflected microplate (W max = 270 nm), respectively. (c) Comparison of the static displacement between the analytical model described in[START_REF] Jallouli | Investigations of the effects of geometric imperfections on the nonlinear static and dynamic behavior of capacitive micomachined ultrasonic transducers[END_REF] and the experimental results. The solid line presents the stable solution of the system and the dashed line presents the unstable solution.

Fig. 6 :

 6 Fig. 6: Time response of the microplate in the case of (a) forward sweep (from 550 kHz to 620 kHz) and (b) backward sweep (from 620 kHz to 550 kHz) (c) the envelop curves of the time response, where V dc = 20 V, V ac = 0.25 V and T pc = 0.5 s.

Fig. 7 :

 7 Fig. 7: Comparison between the experimental and analytical results of a flat and deflected microplate of the first resonance frequency

Fig. 8 :

 8 Fig.8: The experimental quality factor Q with respect to V dc for the case of a flat and deflected microplate (W max = 270nm) where V ac = 0.07 V.

Fig. 10 :

 10 Fig. 10: Experimental investigation of the effect of the (a) DC voltage (V ac = 0.25 V ) and (b) AC voltage (V dc = 5 V) and (c) AC voltage (V dc = 22.5 V) on the frequency response. The solid and dashed lines represent the vibration amplitude of the microplate by performing a sweep down and a sweep up, respectively.

Fig. 11 :Figure 12 .

 1112 Fig. 11: Experimental frequency response of flat circular microplate and deflected circular microplate with an initial deflection w 0 = 270 nm, excited with the same voltage V dc = 15 V and V ac = 0.25 V

Fig. 12 :

 12 Fig. 12: (e) Nonlinear frequency responce of the microplate for a V dc = 27.5 V and V ac = 1 V (a-d) time response and its FFT for the frequencies f a = 605 kHz, f b = 598 kHz, f c = 544 kHz and f d = 506 kHz

Table 1 :

 1 Physical parameters of the micro-system

	Symbol	Quantity	Dimension
	E	Youngs modulus	149 [GPa]
	ρ	Density	2330 [kg/m 3 ]
	ν	Poisson's ratio	0.27
	R	Radius of the microplate	115 ± 2 [µm]
	R ele	Radius of the bottom electrode	80 ± 2 [µm]
	h	Thickness of the microplate	2

Table 2 :

 2 Excitation parameters used for numerical tests

		V dc	V ac	Q
	Fist case	15 V 0.25 V 700
	Second case 25 V 0.25 V 500

Acknowledgments

This project was supported by the Bourgogne-Franche-Comté region and the EUR EIPHI program (ANR 17-EURE-0002).

Declarations

Funding: This project was funded by the Bourgogne-Franche-Comté region.

Availability of data and material: Raw data were generated using the experimental equipments of FEMTO-ST institute. Derived data supporting the findings of this study are available from the first author Aymen Jallouli on request.

Code availability: The code has been implemented using the commercial software Matlab.

Conflicts of interest/Competing interests:

The authors declare that they have no conflict of interest.


Elie Geagea

Judicaël Jeannoutot

Louise Morgenthaler

Simon Lamare

Alain Rochefort

Frank Palmino

Frédéric Cherioux

Unravelling the growth

Unravelling the growth mechanism of (3,1) graphene nanoribbons on a Cu(111) surface Elie Geagea, a Judicael Jeannoutot, a Louise Morgenthaler, a Simon Lamare, a Alain Rochefort, b Frank Palmino a and Frédéric Chérioux a, *

The growth of graphene nanoribbons has been widely investigated on metal surfaces in ultrahigh vacuum. Here, we re-investigate the growth of graphene nanoribbons obtained by thermal annealing of 9,9'-bianthryl derivatives on a Cu(111) surface by using scanning tunnelling microscopy. On the basis of our results, we propose to complete the reaction mechanism commonly accepted in the literature by adding a hydrogen atom transfer from 2,2'-positions to 10,10'-positions as a key-step in the formation of (3,1)-graphene nanoribbons on a Cu(111) surface.

Graphene nanoribbons (GNRs) are 1D graphene stripes which are widely investigated due to their remarkable electronic properties compared to 2D graphene. 1 To achieve GNRs with well-defined edge configuration and width, and accordingly control these electronic properties, growth was from wisely-chosen molecular building-blocks was optimized thanks to atomic-scale characterizations using scanning probe microscopies under ultra-high vacuum (UHV). 2,3,4,5,6 Among all accessible molecular building blocks for the on-surface synthesis of GNRs, 10,10'-dihalogenated-9,9'-bianthryl derivatives 2 and 9,9'-bianthryl 4,5,6 are the two most commonly used. In the case of 10,10'-dihalogenated-9,9'-bianthryl derivatives, the formation of a large variety of GNRs is based on a thermally-induced Ullmann cross-coupling followed by a cyclodehydrogenation on different kinds of coinage metal surfaces. 2 Non-halogenated 9,9'-bianthryl derivatives lead to chiral (3,1)-GNRs, but only on a copper surfaces. 4,5,6 After few controversies, 7 the currently accepted reactional mechanism for the growth of chiral (3,1)-GNRs on a Cu(111) surface is based on a first thermal annealing leading to a protopolymer thanks to on surface-assisted generation of a 2,2'-diradical species. Then, further heating gives the targeted chiral (3,1)-GNR as result of ring-closure driven by a cyclodehydrogenation (Scheme 1a). Most of the reaction intermediates involved in this mechanism have been observed. 4,5,6 a) b)

Scheme 1: a) Reaction mechanism accepted for the synthesis of chiral (3,1) GNR on a Cu(111) surface by using 9,9'-bianthryl as precursor. The key-step is the surfaceassisted formation of 2,2'-biradical specie which points to the surface. b) Possible chemical pathway for the formation of laterally functionalized (3,1) GNRs based on the reaction mechanism described in a). Dashed-bonds are pointing to the surface while bold bonds are pointing out the surface.

A corollary of this reaction mechanism 6 is the opportunity to synthesis functional (3,1) GNRs by introducing functional groups on the 10,10' positions of the starting building blocks (Scheme 1b). Despite its potential for producing a large variety of GNRs with tailored properties, no implementation of this strategy has been reported so far to our knowledge. In this article, we investigate the growth of laterally functionalized (3,1) chiral GNRs by thermal-induced reaction of 10,10'-di(aryl)-9,9'-bianthryl on a Cu(111) surface in UHV by scanning tunnelling microscopy (STM). We find that the growth of GNRs is not possible when the 10,10' positions of the starting building blocks are substituted by aryl groups. Therefore, we revisit the commonly accepted reaction mechanism accounting for the growth of (3,1) GNR on a Cu(111)-surface to explain this phenomenon. We selected the widely-investigated of 9,9'-bianthryl (BA) as a model compound to secure our experimental procedure for the growth of chiral (3,1) GNRs on Cu(111). We synthesized two other molecules, 10,10'-di(4''-pyridyl)9,9'-bianthryl (PBA) and 10,10'di(4''-cyanophenyl)9,9'-bianthryl (CPBA) respectively, that contain two aryl groups covalently bounded to 10 and 10' positions (Fig. 1 and Fig S1-2 of ESI). We chose cyano phenyl and pyridyl groups as aryl group because they promote the formation of supramolecular on various kinds of surface even at room temperature. 8 The synthesis of molecules and surface preparation are fully described in electronic supplementary information. The molecules were deposited by thermal sublimation under UHV on a Cu( 111 In the case of BA molecules, thermal annealing (673 K) led to the formation of clearly linear nanostructures with lengths up to 17.3 nm (Fig. 2 and Fig. S5 of ESI). A zoom of this STM image (area outlined by blue dashed square in Fig. 2) reveals continuous nanolines, with a uniform apparent height, and clearly seamless. This image is similar to those reported in the literature for chiral (3,1) GNRs. 4,5,6 This experiment confirms that our experimental procedure leads to the formation of (3,1) GNRs on Cu(111).

In the case of PBA and CPBA supramolecular networks, the same thermal annealing (673 K) led to disordered nanostructures (Fig. 3). The STM images show apparently random nanostructures, without any linear feature corresponding to targeted GNRs. We made different thermal annealing, with temperature from 400 to 673 K. However, we never observed linear nanostructures corresponding to functionalized (3,1) GNRs, but only disordered features. This result is very surprising because, according to the reported reaction mechanism, 4,5,6 the presence of aryl substituents at the 10 and 10' positions should be in agreement with the formation of corresponding (3,1) GNRs (Scheme 1b). Moreover, STM images of the starting supramolecular networks suggest that the cyanophenyl or the pyridyl substituents are not pointing to the surface but they are still in the plane of the surface because they are involved in the molecule-molecule interactions (Fig. S3c-d of ESI). As usual, one ring per anthracenyl arm is still pointing to the surface, as observed for BA molecules (Fig. S3b of ESI), indicating that the 2,2'-biradical could be stabilized by the surface if it is (previously) formed.

Therefore, as no (3,1) GNR was observed by using PBA and CPBA as building blocks, we conclude that the surface-induced cleavage of C-H of 2 and 2' positions is not possible by annealing on a Cu(111). Because of the canonical forms of BA governing the reactivity of fused six-membered aromatic rings, [START_REF] March | Advanced Organic Chemistry[END_REF] we assume that the thermally-induced cleavage of C-H bonds is only possible on 10 and 10' positions, even on a Cu(111) surface, to give biradical species (Scheme 2). Then, a hydrogen atom transfer 10 from the 2,2'positions to the 10,10'-positions occurs, leading to the formation of the 2,2'-biradical. To assume this hypothesis, we performed DFT calculations. They indicate that 2,2'-biradical is more stable than 10,10'-biradical by around 290meV on a Cu(111) surface while in the vacuum the 10,10'-biradical is more stable by around 40meV. In addition, the DFT calculations shows that the involved stabilizing surface-molecule interaction originates from the radicals on 2,2'-positions which are pointing to the surface (Fig S6 and S7 of ESI). This hypothesis also justifies why only 2,2'-biradicals are observed by nc-AFM while 10,10'-biradicals are never observed. 6 Finally, the 2,2'-biradical leads to a protopolymer and then, to a (3,1) GNR by a second annealing (Scheme 2). On the contrary, if 10 and 10' positions are substituted by aryl groups, C-H bond on 2,2'-positions are not cleaved by soft annealing (below 500K) to give corresponding radicals even with the assistance of the underlying surface. In this case, a stronger annealing (673 K) is necessary to induce the activation of C-H leading to the growth of very disordered features on the reactive Cu(111) surface due to cyclodehydrogenation and non-directed fusing of aromatic ring. 11

Scheme 2: Completed reaction mechanism for the growth of (3,1) GNR induced by a two-step annealing of BA molecules on a Cu(111) surface. The formation of 10,10'biradical (in red), followed by radical transposition, is added as initial step.

In conclusion, we have re-investigated the reaction mechanism of growth of (3,1) GNRs on a Cu(111) surface. On the basis of our experimental observations, we propose to complete the reported reaction mechanism by adding an initial step. We state that the first step consists in the C-H bond cleavage of the 10,10' positions, with respect to the canonical forms, followed by a hydrogen atom transfer from the 2,2'-positions to the 10,10'-positions (Scheme 2). Then, the 2,2'-biradical follows the previously accepted reaction mechanism. This new reaction mechanism paves the way to new possibilities for on-surface assisted synthesis of functional nanostructures.

Fig. 1 :

 1 Fig. 1: CPK models of the 9,9'-bianthryl (BA), the 10,10'-di(4''-pyridyl)9,9'-bianthryl (PBA) and the 10,10'-di(4''-cyanophenyl)9,9'-bianthryl (CPBA) molecules respectively.

  ) substrate maintained at room temperature. As revealed in STM images recorded at 110 K (Fig S3-4 of ESI), no isolated molecule was observed but only large extended 2D islands, each one being constituted by the periodic arrangement of several bright protrusions. The repeat unit pattern of each STM image is in accordance with the distance between the edges of the topmost carbon ring of each of the two anthracenyl subunits of molecules (Fig S4 of ESI). Then, we investigated the chemical transformations of these supramolecular networks induced by thermal annealing. Each supramolecular network was exposed to subsequent thermal annealing up to 673 K from 1h to 4h.

Fig. 2 :

 2 Fig. 2: STM images of linear structures formed after subsequent annealing of BA supramolecular network up to 673 K on Cu(111) surface (Vs=-1.4 V, It=10 pA, 30x30nm. On the right, zoomed STM image corresponding to the blue dashed square (4x4 nm²) of linear GNRs with superimposed structural model.

Fig. 3 :

 3 Fig. 3: STM images of disordered features obtained after subsequent annealing of PBA and CPBA supramolecular networks up to 673 K on a Cu(111) surface, respectively. a) PBA (Vs =1.8 V, It =10 pA, 40x40 nm²) and b) CPBA (Vs =-1.2 V, It =10 pA, 70x70 nm²)

The authors acknowledge the financial support from the French National Research Agency through contract OVATION (ANR -19-CE09-0020) and from the Pays de Montbéliard Agglomération. The authors thank Dr S. Benchabane (FEMTO-ST) and Dr J. Coraux (Neel, Grenoble, France) for fruitful discussions.

Author Contributions

L.M. and S.L. synthetised all molecules. E.G., J.J. and F.P. performed STM experiments. E.G., F.P and F.C. wrote the manuscript with inputs from all authors.

Conflicts of interest

There are no conflicts to declare.

Notes and references


Mehdi Zaoui

Alexandre Bourceret

Yves Gaillard

Sylvain Giljean
email: sylvain.giljean@uha.fr

Christophe Rousselot

Marie-Josée Pac

Fabrice Richard

Zaoui Mehdi

Bourceret Alexandre

Giljean Sylvain

Rousselot Christophe

Richard Fabrice

RELATION BETWEEN HARDNESS OF MULTILAYERED (Ti, Al)N BASED COATINGS AND PERIODS OF THEIR STACKING

HAL is

ABSTRACT. This study aims to model, by using a finite element method, the relationship between the hardness and the period Λ of metal/nitride multilayer coatings (Ti0.54Al0.46/Ti0.54Al0.46N)n in order to understand the increase of the hardness at the low periods [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF] and then optimise the multilayer coating architecture to obtain the best mechanical properties.

A 2D axisymmetric finite element model of the Berkovich nanoindentation test was developed. The coating was designed as a stacking of Ti0.54Al0.46 and Ti0.54Al0.46N nanolayers with, in the first hypothesis, equal thickness and perfect interface. The elastoplastic behaviours of the metal and nitride layers were identified by Berkovich nanoindentation experiments and inverse analysis on thick monolayer samples. The indentation curves (P-h) obtained by this model depend on the period Λ of the stacking.

Simulated (P-h) curves were compared with experimental data on 2 µm thick films with different periods Λ ranging from 10 to 50 nm deposited by RF magnetron sputtering using reactive gas pulsing process (RGPP). The model forecasts are very consistent with the experience for the largest period but the model does not reproduce the hardness increase at the lowest periods.

The Λ = 10 nm coating was analysed by electron energy loss spectroscopy (EELS) on a transmission electron microscope. Results show intermixion of the layers with the presence of nitrogen atoms in the metal layer over a few nanometers [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF]. It was concluded that the metal/ceramic interface plays an important role at low periods. The addition in the model of a transition layer in the metal/nitride stacking, with an elastoplastic metal/ceramic medium behaviour, allows to reproduce the nanoindentation experimental curves. The thickness of this transition layer deduced from model updating method is in very good agreement with EELS observations. KEYWORDS: nanoindentation, finite element method, TiAl/TiAlN multilayer films, hardness, indentation modulus.

INTRODUCTION

Titanium and aluminium nitrides (TiAl)N coatings are extensively used in the industry since many years for their hardness, toughness and wear and oxidation resistances [START_REF] Münz | Titanium aluminum nitride films: A new alternative to TiN coatings[END_REF][START_REF] Paldey | Single layer and multilayer wear resistant coatings of (Ti,Al)N: a review[END_REF]. It has been shown that the indentation modulus and the hardness of Ti1-xAlxN coatings depend on the aluminium content and that the optimal properties are obtained when the maximum Al content is substituted for Ti in face centred cubic structure close to x = 0.46 [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF][START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF]. One way to improve the hardness of that kind of coatings consists in depositing multilayer films.

In this study, multilayer coatings are studied and consist in metal/ceramic stacking with alternative deposition of Ti0,54Al0,46 and Ti0,54Al0,46N nanolayers with three different periods of the bilayer metal/ceramic Λ = 50, 16, 10 nm. In order to facilitate the reading of the paper, Ti0,54Al0,46 and Ti0,54Al0,46N are respectively written as TiAl and TiAlN. The main objective of the paper is to study the relationship between hardness of as-design multilayer films and periods of their stacking to understand the elastoplastic phenomenons that are involved in the hardness increasing experimentally observed at the lowest periods [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF].

A 2D axisymmetric model of the Berkovich nanoindentation is developed. Then a method to identify the elastoplastic mechanical properties of the monolayer TiAl and TiAlN is proposed. The results are introduced in a multilayer stacking model that allows to obtain the indentation curves (Ph) as a function of the period Λ. As a first hypothesis the interface between TiAl and TiAlN layers is considered as perfect without thickness. Then a transition layer is added and allows to reproduce the experimental indentation curves.

MATERIALS AND METHODS

TiAl and TiAlN monolayer thin films were deposited onto silicon (100) wafer by RF magnetron sputtering using one sintered titanium/aluminium 66/33 at.% alloy target as described elsewhere [START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF]. (TiAl/TiAlN)n multilayer films with different periods were deposited using Reactive Gas Pulsing Process (RGPP) [START_REF] Martin | Procédé de pulvérisation réactive à signal de commande cyclique et dispositif correspondant[END_REF] with a controlled pulsing flow rate of the nitrogen reactive gas as described elsewhere [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF]. Finally, the samples considered in this paper are summarised in Table 1. Nano-indentation tests were performed using an Ultra Nano-indentation Hardness Tester provided by Anton Paar. Experiments were conducted using a Berkovich tip and the multicycle procedure in load control up to 8 mN [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF][START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF].

TABLE 1. Sample characteristics Sample

A 2D axisymmetric finite element model of the Berkovich nanoindentation test was developed with the ANSYS finite element software as described on Figure 1. An implicit method with non-linear geometric formulation has been used with higher order 2D elements. A deformable indenter, with isotropic linear elastic law, was modelled with a half contact angle of 70,3° and a tip radius to fit the experimental tip defects. A friction coefficient of 0.1 was imposed at the contact between the indenter and the film. The mesh density has been refined in order to the smallest elements are localised in the contact region. The film can be either monolayer TiAl or TiAlN or multilayer (TiAl/TiAlN)n. For the monolayer TiAl and TiAlN, isotropic elastoplastic behaviour laws were considered with linear hardening. Elastic properties were obtained from experimental nanoindentation and plastic properties were estimated using finite element model updating.

The approach used to extract the constitutive parameters of plastic law of TiAl and TiAlN consists in an optimisation problem; the objective of which is to minimise a cost function ω by using an algorithm based on the Levenberg-Marquardt [START_REF] Levenberg | A méthod for the solution of certain non-linear problems in least squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF] method using the MIC2M software [START_REF] Richard | MIC2M Software: Modeling and Identification of the Mechanical Behavior of Materials[END_REF]: [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF] where is the vector of the unknown parameters and ω the cost function i.e., the gap between the experimental data and the numerical results. Then the cost function can be defined as:

(2

)
where T is the number of acquisition times of the load and the maximum value of the experimental indentation load.

RESULTS AND DISCUSSION

3.1

Mechanical properties of TiAl and TiAlN monolayers All the experimental data of the indentation tests performed on the TiAl and TiAlN monolayer films have already been published elsewhere [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF][START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF]. Hardness and indentation modulus values of the metal TiAl are significantly lower than the ones of nitride TiAlN. By postulating the Poisson's ratio as known, the elastic properties of the monolayer films can be deduced from the experimental data. Then, plastic parameters were numerically deduced from the optimisation approach described in method section. Results are presented in Table 2. The comparison of the experimental and model indentation load curves are plotted in Figure 2 and show a good agreement.

3.2

Simulation of (TiAl/TiAlN)n multilayer films with perfect interfaces All the experimental data of the indentation tests performed on the multilayer (TiAl/TiAlN)n films have already been published elsewhere [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF]. Interesting observation is that both the hardness and the elastic modulus of the multilayer films follow a Hall and Petch law as the values linearly increase as a function of . In a first approach, the multilayer film was modelled as a stacking of nanolayers of TiAl and TiAlN with a perfect interface without thickness as showed in figure 3.a. The model was set for each stacking period Λ = 10, 16 and 50 nm. A convergence study was carried out with 1, 2 or 3 elements in the thickness of the nanolayers. Since the relative variation of the load is less than 0.5%, only one element, with a size, was finally used in the thickness (Figure 3.a), in order to minimise the calculation time. Results of the as simulated (P-h) curves are plotted on figure 4. For the largest period Λ = 50 nm, the model is in good agreement with the experience. Furthermore, the gap between the experimental curves and the simulated ones increases when the period decreases to become important for the smallest period. Therefore, the phenomenon that generates an increase in hardness when period decreases is not taken into account in the coating model without interfaces.

3.3

Simulation of (TiAl/TiAlN)n multilayer films with a transition layer In the stacking, between each nanolayer of TiAl and TiAlN, a transition layer was added (figure 3.b.). For this transition layer, an elastoplastic metal/ceramic medium behaviour was considered. Each elastoplastic parameter of the transition layer was assumed to be an average (mixture law 50/50) of the parameters of the TiAl and TiAlN nanolayers. The thickness of this transition layer was optimised by minimising the gap between experimental and simulated data as previously described. Result of the simulate (P-h) curve with a transition layer is plotted on figure 5.a and show a good agreement between experimental and simulated curves for the lowest period. For Λ = 10 nm, the simulated thickness of the transition layer, epTr = 2.12 nm, is close to the one of TiAl and TiAlN layers. This result is in very good agreement with the thickness of rough interfaces observed between TiAl and TiAlN layers specified by Electron Energy-Loss Spectroscopy (Figure 5.b).

CONCLUSION

The elastoplastic properties of Ti0.54Al0.46 metal and Ti0.54Al0.46N ceramic monolayers were identified by comparison of Berkovich nanoindentation experimental data and 2D axisymmetric finite element simulations. The mechanical properties of the monolayers was input in a multilayer model to forecast the mechanical properties of (Ti0.54Al0.46/Ti0.54Al0.46N)n multilayer coatings, with various periods, designed by RF magnetron sputtering using RGPP technique. The multilayer model was considered, in first hypothesis, as a stacking of TiAlN and TiAl nanolayers with equal thickness and perfect interface. That model allows to reproduce the indentation curves of the highest 50 nm period films. To fit the experimental indentation curve at the lowest periods, a transition layer was introduced in the model between each metal/ceramic interface. Using an optimisation process of the model, this transition layer was found to be 2.12 nm thick for the lowest period Λ = 10 nm, which is in good agreement with the results of Electron Energy-Loss Spectroscopy experiments.

FIGURE 1 .

 1 FIGURE 1. Description of the 2D axisymmetric finite element indentation model.

FIGURE 2 .

 2 FIGURE 2. Loading indentation curves (P-h) of (a) TiAl and (b) TiAlN monolayers.

FIGURE 3 .

 3 FIGURE 3. Modelisation of multilayer films, (a) with perfect interfaces (b) with a transition layer.

FIGURE 4 .

 4 FIGURE 4. Comparison of the (P-h) experimental and simulated curves for Λ = 10, 16 and 50 nm with a perfect interface between TiAl and TiAlN nanolayers.

FIGURE 5 .

 5 FIGURE 5. For Λ = 10 nm, (a) Comparison of the (P-h) experimental and simulated curves with a transition layer, (b) EELS results

TABLE 2 .

 2 Elastoplastic parameters of the metal TiAl and the nitride TiAlN.

		Young Modulus	Poisson ratio Yield stress	Linear hardening
		(GPA)		(GPa)	(GPa)
	TiAl	199	0.25	0.88	56.6
	TiAlN	417	0.21	5.70	405.0

ACKNOWLEDGEMENT

This work was performed in the frame of the research project RESEM 2019, managed by the Institut de Recherche Technologique Matériaux Métallurgie Procédés (IRT M2P) and financially supported by the French program Plan d'Investissement d'Avenir (PIA).