S T Rakotonarivo
email: sandrine.rakotonarivo@univ-amu.com

C Payan

J Moysan

C Hochard

Local damage evaluation of a laminate composite plate using ultrasonic birefringence of shear wave

Keywords: Non-destructive testing, Ultrasonics, Damage induced anisotropy, Laminates, Shear wave birefringence

This paper presents an ultrasonic non-destructive testing method to locally characterize composite material damage through the analysis of the anisotropy behavior variation induced by damage. The approach is based on shear wave contact measurements and analyses shear wave birefringence to estimate variation of the attenuation coefficient and shear moduli ratio between warp and weft directions. The method only requires a single piezo-electric contact sensor and thus, is easy to implement for practical applications. Moreover, unlike other methods, it does not require any reference measurement in a calibrated medium such as water to evaluate attenuation coefficient variation and shear moduli ratio. It is shown that the proposed method allows to locally qualify damage with good repeatability and robustness.

Introduction

Composite materials are widely used in many sectors of activity such as aeronautics, automotive and naval industries, or civil engineering that require the use of both lightweight and mechanically efficient materials. In order to ensure integrity and safety of any composite structure, it is primordial to conduct mate-rial testing throughout the structure life from the manufacturing process to the in service phase. Non-destructive testing techniques based on ultrasounds have shown great potential throughout the last decades to reach this goal. Indeed ultrasonic waves allow to achieve local or global composite structure inspections to evaluate the structure health and determine if it is damaged [START_REF] Markham | Measurement of the elastic constants of fibre composites by ultrasonics[END_REF][START_REF] Scott | A review of non-destructive testing of composite materials[END_REF][START_REF] Harizi | Mechanical damage characterization of glass fiber-reinforced polymer laminates by ultrasonic maps[END_REF].

When a composite structure gets damaged, a loss of stiffness is generally observed with respect to a pristine structure [START_REF] Thollon | A general damage model for woven fabric composite laminates up to first failure[END_REF][START_REF] Thollon | Analyse du comportement à rupture de composites stratifiés constitués de plis tissés sous chargements statique et de fatigue[END_REF]. This loss of stiffness, also called damaged factor, allows to qualify the material damage at the macroscopic scale. In order to get an estimate of this factor, one needs to measure first the elasticity constants of the composite and follows their evolution. One approach to measure elasticity properties of the composite consists in measuring ultrasound velocity of bulk waves on plates immersed in water and for many incident directions [START_REF] Markham | Measurement of the elastic constants of fibre composites by ultrasonics[END_REF][START_REF] Castagnede | Optimal determination of the elastic constants of composite materials from ultrasonic wave-speed measurements[END_REF][START_REF] Rokhlin | Double through-transmission bulk wave method for ultrasonic phase velocity measurement and determination of elastic constants of composite materials[END_REF][START_REF] Aristégui | Optimal recovery of the elasticity tensor of general anisotropic materials from ultrasonic velocity data[END_REF][START_REF] Castellano | Ultrasonic goniometric immersion tests for the characterization of fatigue post-lvi damage induced anisotropy superimposed to the constitutive anisotropy of polymer composites[END_REF]. However, those approaches based on immersed ultrasound measurements are difficult to implement for practical applications.

The Impulse Excitation Technique was recently applied to damaged composite laminates for measuring the residual elastic properties [START_REF] Paolino | Damaged composite laminates: Assessment of residual young's mod-ulus through the impulse excitation technique[END_REF]. As it requires two side access, in situ inspection of structures with the later technique remains challenging. Other methods exploit Lamb waves dispersion curves for characterizing composite plates from laser measurements [START_REF] Wu | On the measurement of anisotropic elastic constants of fiber-reinforced composite plate using ultrasonic bulk wave and laser generated lamb wave[END_REF] or from air-coupled ultrasonic transducers requiring a single plate side access [START_REF] Castaings | Lamb and sh waves generated and detected by air-coupled ultrasonic transducers in composite material plates[END_REF]. An other category of approaches utilizes acoustic emission techniques to monitor composite structure and to detect damage mechanisms by detecting the released energy due to these events in a structure [START_REF] Nikbakht | Delamination evaluation of composite laminates with different interface fiber orientations using acoustic emission features and micro visualization[END_REF][START_REF] Kumar | Characterization of indentation damage resistance of hybrid composite laminates using acoustic emission monitoring[END_REF].

Alternatively, it was shown that acoustic birefringence provides information about the stiffness of composite plates and is sensitive to damage [START_REF] Solodov | Nondestructive evaluation of anisotropy in composite materials via acoustic birefringence[END_REF][START_REF] Fey | Quantification of fatigue state in cfrp using ultrasonic birefringence[END_REF]. The approach is based on the analysis of the anisotropy behavior of orthotropic composite plates. Acoustic birefringence represents difference in the ultrasonic velocities of two orthogonally polarized shear waves propagating through the thickness of a plate and polarized into both principal perpendicular directions to their average value. It is sensitive to anisotropy in the microstructure (texture, fibre alignment) and stress [START_REF] Arora | Effect of texture and stress on acoustic birefringence[END_REF]. Hence, analysis of acoustic birefringence through measurements of shear wave velocity in various directions of polarization provide information about the direction of stress [START_REF] Clark | On the use of acoustic birefringence to determine components of plane stress[END_REF] or the fibre orientation [START_REF] Solodov | Nondestructive evaluation of anisotropy in composite materials via acoustic birefringence[END_REF][START_REF] Varadan | Rotation of elastic shear waves in laminated, structurally chiral composites[END_REF]. In addition to the primary anisotropy of the pristine material, the damage modifies its mechanical properties (loss of stiffness) and therefore its anisotropy according to the encountered sollicitations. The material loss of stiffness modifies the ultrasonic shear wave velocities and thus the shear wave birefringence behavior of the material. Ultrasonic shear wave birefringence is then correlated to the material damage. In particular, it was shown that birefringence has a good potential for characterizing damage on composite laminates [START_REF] Solodov | Nondestructive evaluation of anisotropy in composite materials via acoustic birefringence[END_REF][START_REF] Fey | Quantification of fatigue state in cfrp using ultrasonic birefringence[END_REF]. In this paper, this approach is extended to locally evaluate damage on a laminate plate through estimation of shear moduli ratio and attenuation coefficient variation from ultrasound contact pulse-echo measurements. Using the experimental procedure proposed in [START_REF] Solodov | Nondestructive evaluation of anisotropy in composite materials via acoustic birefringence[END_REF][START_REF] Fey | Quantification of fatigue state in cfrp using ultrasonic birefringence[END_REF][START_REF] Rheinfurth | Ultrasonic birefringence as a measure of mechanically induced fatigue damage in laminated composites[END_REF] for shear wave birefringence analysis, a new method of estimation of the ultrasound parameters is proposed for characterizing damage. In the presented approach, the inversion algorithm exploits the whole frequency content of the signal and applies a global optimization scheme for estimating the shear moduli ratio and the attenuation coefficient variation. Moreover, reference measurements in a calibrated medium is no more needed for estimation of the attenuation parameter as in [START_REF] Rheinfurth | Ultrasonic birefringence as a measure of mechanically induced fatigue damage in laminated composites[END_REF]. Experimental robustness/accuracy of the method is tested by performing many measurements at different day time. The method successfully manages to locally characterize damage on a unbalanced woven composite plate made of fiber glass reinforcement in an epoxy matrix. Interest for woven ply laminates relies on their better resistance to delamination than unidirectional ply and their low sensitivity to transverse rupture [START_REF] Hochard | Optimum design of laminated composite structures[END_REF].

Theoretical basis of the method are presented in section 2. Then, section 3 presents the experimental tests conducted on pristine and damaged samples.

Results are discussed in the following paragraph. Conclusions are presented in section 5.

Damage evaluation from shear wave birefringence analysis

This section presents the shear wave birefringence theory with general formulations for layered composites made of orthotropic plies with any orientation between plies. Then, simplified expressions are derived for a composite made of plies with identical orientation and properties. Further, the method for measuring off plan shear moduli and attenuation variation for characterizing the material is presented in section 2.2.

Theory of shear wave birefringence

The approach is based on the analysis of shear wave phase and amplitude variations versus polarization angle from contact pulse-echo measurements following the experimental set-up depicted in Fig. 1. Fiber-reinforced composites have mechanical properties varying according to manufacturing process. In consequence, due to their anisotropic nature, the ultrasonic shear wave propagates with different velocity and attenuation for each direction of polarization. Ply of a fiber-reinforced layered composite manifest an orthotropic anisotropy with two symmetry axes. Thus shear behavior of each ply is characterized by two shear modulus in planes perpendicular to directions 1 and 2 (axes represented in Fig. 1), as well as two attenuations for each direction.

As shown in Fig. 1, a shear wave transducer, located at the surface of the composite plate, transmits a shear wave into a layered composite of thickness d and at normal incidence with a polarization angle θ in the plane [ 1 2]. In pulse echo mode, this transducer receives a reflected wave u(t) by the plate backwall.

Its Fourier transform U (ω) = 1 √ 2π +∞ -∞ u(t) exp(-iωt)dt is expressed in terms of amplitude U Φ and phase Φ U (ω, G 13 , G 23 , ρ, d, θ, ∆α, ∆θ) = U Φ exp(iωt + iΦ), (1)
where ω is the pulsation, ρ the layered composite density. Parameters G 13 , G 23 represents for each ply, the shear modulus in planes perpendicular to warp and weft directions, respectively. Variables ∆α, ∆θ are also (1xN ) vectors whose elements correspond to the ultrasonic attenuation of each layer and the ply orientation according to first layer, respectively.

90

Following the matrix formalism given in [START_REF] Rheinfurth | Ultrasonic birefringence as a measure of mechanically induced fatigue damage in laminated composites[END_REF], each ply is characterized by a propagator matrix P n ,

P n =   P n|| 0 0 P n⊥ ,   =   e -jdn(ω ρn G n|| +jα n|| ) 0 0 e -jdn(ω ρn G n⊥ +jα n⊥ )   (2)
where ρ n and d n are the density and the thickness of the n th ply, respectively.

For this orthotropic material, note that shear moduli in the warp and weft directions, denoted G n|| and G n⊥ , are related to the elasticity constants c 55

MHz, the attenuation is well approximated by a linear law [START_REF] Kim | Ultrasonic wave propagation in carbon fibrereinforced plastics[END_REF],

α n|| = k n|| f and α n⊥ = k n⊥ f , (3)
where k n|| and k n⊥ are the frequency independent attenuation coefficients in the warp and weft direction. The latter attenuation coefficient will be studied for characterizing the composite material.

Normalizing the matrix propagator in (2) by its first element

P n11 = e -jdn ω ρn G n|| +jα n|| (4)
yields the normalized propagator matrix P n

P n = 1 P n11 P n (5) =    1 0 0 exp -jd n ω ρn G n⊥ - ρn G n|| + j ∆kn 2π    . (6)
The attenuation coefficient difference ∆k n , defined as ∆k n = k n|| -k n⊥ , quantifies the dissymmetry between attenuations in the warp and weft directions.

Then, an interface matrix I n,m allows to account for different ply orientations within a layered composite between plies n and m:

I n,m =   cos(∆θ n,m ) sin(∆θ n,m ) -sin(∆θ n,m ) cos(∆θ n,m )   , (7)
where ∆θ n,m represents the orientation difference between plies n and m.

Under the assumption of perfect contact between plies and rigid boundaries, the (2x2) transfert matrix H accounting for the ultrasonic wave propagation within the composite material in a pulse-echo mode is given by

H = N -1 n=1 P n I n,n+1 P N I N,N P N 2 n=N I n,n-1 P n-1 , for N ≥ 2, ( 8
)
where N is the number of plies constituting the composite material. From the normalized propagator matrix in Eq. ( 6), the normalized transfer matrix H is defined as

H = N -1 n=1 P n I n,n+1 P N I N,N P N 2 n=N I n,n-1 P n-1 , for N ≥ 2. (9) = 1 h H, ( 10
)
where h is the normalizing factor defined as h =

  U || U ⊥   = I θ H I -θ   U in 0   , (11)
where U in is the transmitted shear wave and, U || and U ⊥ are components of the reflected wave with polarization parallel and perpendicular to the receiving transducer, respectively. The interface matrices I θ and I -θ account for difference of orientation between the transducer polarization and the first ply (weft direction). They are defined as Eq. ( 7) where ∆θ n,m = θ or -θ. As the transducer is only sensitive to waves with identical polarization as istelf, the 120 measured signal U (ω) from Eq. ( 1) corresponds to U || ,

U (ω, θ) = U || = I θ H I -θ U in . (12)
Normalizing the measured signal in [START_REF] Castaings | Lamb and sh waves generated and detected by air-coupled ultrasonic transducers in composite material plates[END_REF] by its value at θ = 0 • yields

U (ω, θ) = U (ω, θ) U (ω, 0) = I θ H I -θ H 11 . ( 13
)
where H 11 is the first element of the normalized transfert matrix H when θ = 0 • . Finally amplitude and phase of the reflected signal are computed from

U Φ (ω, θ) = U (ω, θ) , (14a)
Φ(ω, θ) = arg(U (ω, θ)). (14b)
Note that for sake of conciseness, dependance of amplitude and phase, U Φ and Φ, with respect to composite properties are omitted from the notation in above expressions.

In the case of a laminate composite plate with identical elastic properties for each ply, the material is characterized by two out of plane shear moduli, G ⊥ and G || , in the warp and weft directions of the ply and a single variation of attenuation coefficient ∆k. Considering a constant ply angle orientation, ∆θ, implementation of expression ( 13) yields the following simplified expression:

U (ω, θ) = cos 2 ∆θ[cos 2 θ cos 2 (2∆θ) + sin 2 θ(cos 2 ∆θe -j2dωχ -sin 2 ∆θe -jdωχ ) + sin(2∆θ) sin(2θ) 4 (e -j 1 2 dωχ -e -j 3 2 dωχ )], . (15)
where

χ = ρ G ⊥ - ρ G || + j ∆k 2π .
In that configuration, d refers to the composite plate thickness (and not the ply thickness). This formulation will be used as a forward modeling for estimating variations of the shear modulus dissymmetry and the attenuation coefficient in order to characterize damage in the case of an unbalanced woven composite plate. The method is presented in the following section.

Inversion algorithm

This study considers laminate plate made of plies with identical orientation and properties. Thus the material is characterized by a single attenuation variation coefficient ∆k and, by two out of plane shear moduli, in the warp and weft directions, that are denoted G 13 = G || and G 23 = G ⊥ . For the considered laminate, the alignement between ply might be not perfect thus misalignment (= orientation variation) between ply is considered through the average orientation ∆θ. In order to evaluate material damage, the method assesses variations of shear moduli and attenuation dissymmetries between the warp and weft directions. The following parameters are estimated from the shear wave birefringence: the variation of the attenuation coefficient difference ∆k and shear moduli in warp and weft directions in order to quantify the shear modulus ratio variation defined as R g = G23 G13 . To characterize the composite material, the inversion algorithm estimates the two shear moduli, the attenuation coefficient difference and the average ply misalignment by minimizing the following cost function

F (G 13 , G 23 , ∆k, ∆θ) = ω f ω0 2π 0 dω dθ |U -U Obs | 2 ω f ω0 2π 0 dω dθ |U Obs | 2 . ( 16
)
The received displacement U refers to the modeled normalized displacement as defined in Eqs. [START_REF] Nikbakht | Delamination evaluation of composite laminates with different interface fiber orientations using acoustic emission features and micro visualization[END_REF] or [START_REF] Solodov | Nondestructive evaluation of anisotropy in composite materials via acoustic birefringence[END_REF] according to formalism provided in section 2.

Signal U Obs refers to experimental reconstructed reflected displacement from measured phase and amplitude signal. The global optimization method "direct search" (patternsearch function from Matlab) is applied on the cost function [START_REF] Fey | Quantification of fatigue state in cfrp using ultrasonic birefringence[END_REF] to find the averaged misalignement ∆θ, the shear moduli ratio R g from estimation of G 13 and G 23 and the attenuation coefficient variation ∆k. The two latter estimated parameters, R g and ∆k, are then used for characterizing the material state. Note that the averaged misalignment is estimated in order to get a better fit between the data and the model but is not used for assessing it in the present paper. thickness at different locations on the pristine and damaged specimen is represented in Fig. 2.

The damaged sample was obtained from tensile tests with discharge until fracture. Damage behavior study of this composite sample is well detailed in [START_REF] Thollon | A general damage model for woven fabric composite laminates up to first failure[END_REF]. Elastic damage is observed in the warp direction, and elastoplastic damage behavior in the weft direction. Although they are various and complex, main damage processes are fiber-matrix debonding, matrix cracking on the scale of the fibres and small transverse cracks occurring in the material. A loss of stiffness was observed in the sample with a non linear evolution of the damage in the weft direction starting from a loading threshold and reaching a limit corresponding to saturation of the ply with small cracks. Measured damages in the weft and warp directions were around 0.48 and 0.15 just before fracture, respectively. These values were measured on the tensile diagram after the unloading and refer to the loss of stiffness in the Young modulus for each direction

d = 1 -E/E 0
, where E is the damaged material stiffness and E 0 the initial stiffness (undamaged material).

The ultrasound experimental set-up and processing are now presented. The method allows to estimate shear moduli ratio from off plan coefficients as well as attenuation coefficient difference between warp and weft direction. 16) is performed to evaluate the attenuation coefficient variation and the shear moduli ratio through estimation of G 13 and G 23 . The misalignment ∆θ is also estimated with average values ranging from -0.3 • to 1.8 • but is not used for damage characterization.

Results

Birefringence measurement at 1.5 MHz in terms of phase and amplitude are plotted in Fig. 5 for the pristine sample and the damaged sample at 3 different locations (referenced as P 1 , P 2 and P 3 to the sample rupture and thus to the most damaged region. Fig. 5 shows that the trend of both amplitude and phase differs completely between the pristine and the damage sample. The relative amplitude decay in the 90 • polarization with respect to the position to the rupture is noticeably sensitive to damage.

The opposite trend is observed with the phase indicating that the 90 • shear wave takes longer to propagate when getting closer to the rupture, which is characteristic of a stiffness loss. These results are highlighted in Figs. 6a and6b plotting the variations of the attenuation coefficient and the shear moduli ratio measured at 3 different positions on the pristine and damaged samples. Indeed, the latter parameters are estimated from experimentally measured phase and amplitude (Fig. 5) following the inversion scheme presented in section 2.2. The experiment is reproduced 10 times at 10 different days to test reproducibility and robustness of the method (see error bar in Fig. 6). Results shows that both parameters (∆k and R g ) are insensitive to measurement position for the pristine plate. Moreover ratio R g is higher for the damaged sample indicating a stronger dissymmetry (unbalance) between the warp and weft shear moduli.

No trend is observed on R g versus the measurement position on the damaged sample whereas attenuation coefficient decays with positions at larger distance to the rupture indicating that attenuation anisotropy gets stronger when getting closer to the rupture.

DISCUSSIONS

For this fiber-reinforced composite plate subjected to tensile stress in the weft direction (0 • ), damage appears first under the form of micro-cracks along the interface between glass fiber in the warp direction (90 • ) and the epoxy.

As there are 83% of fiber in the warp direction (compared to 17% in the weft direction) and due to the direction of sollicitation (tensile stress in the weft direction), shear stiffness is then more affected along the weft direction than in the warp direction. Consequently, the shear modulus in the warp direction, G 23 , does not vary significantly, whereas a loss of shear rigidity is observed in the weft direction, that is a decay of shear modulus G 13 in the weft direction.

In overall, this manifests as an increase of the shear moduli ratio R g from about 1.1 on the pristine sample to 1.16 -1.2 for the damage one as seen in Fig. 6b.

Also, as the wave encounters more micro-cracks in the damaged medium, the wave is more attenuated in the warp direction. This results in accentuating the difference of the attenuation coefficient ∆k between warp and weft directions.

Results also shows that attenuation coefficient ∆k varies with position and decays when getting further from the rupture. In Fig. 6a, decay of the attenuation coefficient from 75 Np/m/MHz to -0.9 Np/m/MHz supports the fact that the wave is less attenuated when moving away from the rupture. Larger density of micro-cracks or larger size of micro-cracks can contribute to higher attenuation of the ultrasonic wave. Thus, this result suggests that attenuation is sensitive enough to locally distinct different damage state of the material when moving away from the rupture, probably due to larger density or larger size of microcracks. Ultrasonic attenuation is more sensitive to damage than shear stiffness.

This trend is in agreement with results from reference [START_REF] Rheinfurth | Ultrasonic birefringence as a measure of mechanically induced fatigue damage in laminated composites[END_REF] (see Fig. 13 of that reference) where attenuation varies by 75% as opposed to 20 % for shear stiffness after 5000 cycles of fatigue test on a composite laminate with respect to their initial value.

To summarize, ultrasonic birefringence method is a local method as opposed to the impulse excitation technique that is a global method. The birefringence approach allows to qualify laminate plate damage at a macroscopic scale through estimations of attenuation coefficient variation and the shear moduli ratio. It is complementary to other ultrasound inspection techniques such as acoustic emission. In acoustic emission methods, the major source of acoustic data is the matrix cracking and a minimum sound level from matrix cracking is required.

Whereas ultrasonic birefringence allows to measure effects of micro-cracks with any sound level on the shear wave propagation. As acoustic emission, birefringence could be well suitable for continuous monitoring of composite laminates by integrating in situ sensors into the structure as performed with film sensors [START_REF] Ji-Hun | Characterization of low-velocity impact-induced damages in carbon/epoxy composite laminates using a poly(vinylidene fluoride-trifluoroethylene) film sensor[END_REF].

CONCLUSIONS

This paper presented a method to locally characterize damage of laminate composite plate from analysis of ultrasonic birefringence of shear wave. The approach simultaneously evaluates two parameters, the attenuation coefficient difference and the shear moduli ratio between warp and weft direction, to qualitatively assess the composite structure state without requiring any reference measurement in a calibrated medium such as water. The method is based on shear wave contact measurement in pulse echo mode and only requires a single piezo-electric contact sensor and thus, is easy to implement for practical applications. The approach was tested on layered unbalanced woven composite specimens, damaged and pristine ones made of glass fiber reinforcements in an epoxy matrix. It successfully shows that change of shear wave birefringence induced by damage is well characterized by shear moduli ratio and attenuation coefficient variation. In particular, attenuation coefficient variation allows to locally assess the material state at a lateral scale of few cms and is more sensitive to damage than ratio shear moduli. In spite of difficulty induced by ultrasound contact measurement (pressure applied on the transducer can strongly affects measured amplitude and thus changes value of attenuation), the method is reproductible and robust: it was performed 10 times on 10 different days with human operator and results remained stable.

ACKNOLEDGEMENTS

This works was supported by a CNRS PEPS funding. The authors would like to thank Anthony Salomon and Bruno Giraud for performing the experiments.

Figure 1 :

 1 Figure 1: Scheme of the ultrasonic wave measurement set-up in a pulse-echo mode. The transducer emits into the composite sample a shear wave polarized along the direction represented by the line on the transducer (polarization angle θ with respect to axe 1). The reflected wave by the backwall is measured by the same transducer at the same polarization angle θ.

N - 1 n=1

 1 P n11 P N 11 P N 11 2 n=N P (n-1)11 . For a polarization angle θ of the transmitted shear wave, components of the reflected shear wave in a pulse-echo mode are expressed as,

3 . 3 . 1 .

 331 Local damage evaluation on a layered unbalanced woven ply composite plate Description of the layered unbalanced woven ply laminates Shear wave birefringence measurements are performed on pristine and damaged layered composite plates. Each composite plate is made of 8 unbalanced woven plies all positioned with the same orientation (meaning that in theory, ∆θ = 0 • ). The material used in this study is a glass reinforced epoxy: the reinforcement is a glass five-harness satin in which 83% of the fibers ran in the warp direction and 17% in the weft direction. The unbalanced woven ply shows different behavior in the warp and weft directions due to the larger number of fibres in the warp direction. Density of each specimen is ρ =1819 kg/m 3 , and

Figure 2 :

 2 Figure 2: Picture of the (a) pristine and (b) damaged unbalanced composite woven specimen with respective location of the measurement point. The basis is such that shear modulus G 13 and G 23 are modulus in weft direction (1) and warp direction (2), respectively.

3. 2 .

 2 Experimental set-up and pre-processing Ultrasound measurements are performed at different locations on the specimen as shown in Fig. 2 with a 2.25 MHz piezo-electric shear wave transducer (Panametric V156) in a pulse-echo mode. The polarization angle evolves between 0 • and 360 • with a stepsize of 15 • (see Fig. 3). Generation and reception of impulse signals are monitored with a square wave pulser/receiver (model 5077PR from Sofranel Olympus). And time series are acquired using a Lecroy WS422 oscilloscope. For each set of acquisitions, time series are averaged 30 times at each positions. In order to test robustness and reproducibility of the method, this set of measurements are performed ten times at ten different days for each position and polarization angle. The averaged time series measured on the pristine sample are illustrated in Fig. 4(a) versus the polarization angle. Windowing is applied to the time series to extract the echo from the backwall (see Fig. 4a with echo between the two dashed lines) and to compute its Fourier transform. Phases and modules at each polarization (Figs. 4a, 4b and 5) are extracted from the frequency domain signal (0,5 -2 MHz) and normalized according to their values at polarization angle θ = 0 • . The latter experimental modules and phases, U Φ,obs and Φ obs are used to reconstruct the displacement U Obs = U Φ,obs exp(iωt + iΦ obs ). Then minimization of the cost function in (

Figure 3 :

 3 Figure 3: Experimental set-up of the ultrasonic wave measurement. Polarization of the shear wave transmitted by the transducer is represented by a line on the transducer in the zoom picture.

Figure 4 :

 4 Figure 4: (a) Measured time series (absolute amplitude in mV) versus polarisation angle on the pristine composite sample at position P 1 as shown in Fig. 2. Windowing is applied to extract the backwall echo between the two dashed line and to compute its Fourier transform. (b) Normalized module (no unit) of the first echo in the frequency domain; the amplitude is normalized by amplitude of the module at θ = 0 • . (c)Phase shift (in radian) of the first echo in frequency domain with respect to the phase at polarization angle θ = 0 • .

Figure 5 :Figure 6 :

 56 Figure 5: Birefringence measurement at 1.5 MHz for the pristine sample at position P 2 and the damaged sample at positions P 1 , P 2 and P 3 . Averaging is performed over 10 acquisitions. (a) Module of the backwall echo over 10 acquisitions ; the amplitude is normalized by amplitude of the module at θ = 0 • . (b) Phase of the backwall echo over 10 acquisitions; the phase is normalized by the phase at θ = 0 • .

  ). Position P 3 refers to the closest position

	Oscilloscope	Pulser/Receiver
		Zoom
	Composite
	plate	Transducer
	Experimental set-up	Protractor

and c 44 through following relations c 44 = G n|| and c 55 = G n⊥[START_REF] Rokhlin | Physical ultrasonics of composites[END_REF]. Due to the anisotropic nature of the material, there are also different attenuations in the warp and weft directions, denoted α n|| and α n⊥ . In the frequency band 1-5


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Introduction

In the frame of microtechnologies, aluminium oxide (Al2O3) has been investigated for several applications: as a "high-k" material for MOS gate oxide [START_REF] Wallace | High-k dielectric materials for microelectronics[END_REF][START_REF] Kittl | high-k dielectrics for future generation memory devices[END_REF], as an ion-sensitive and passivation layer for FET-based microsensors [START_REF] Knopfmacher | Nernst limit in dual-gated Si-nanowire FET sensors[END_REF][START_REF] Chen | Al2O3/silicon nanoISFET with near ideal Nernstian response[END_REF], as a passivation layer for OLED and solar cell devices [START_REF] Dingemans | Status and prospects of Al2O3-based surface passivation for silicon solar cells[END_REF][START_REF] Maindron | Investigation of Al2O3 barrier film properties made by atomic layer deposition onto fluorescent tris-(8-hydroxyquinoline) aluminium molecular films[END_REF], as a luminescent material for dosimeters [START_REF] Yukihara | Optically stimulated luminescence dosimetry in medicine[END_REF],... As a consequence, many processes based on physical and/or chemical deposition techniques were developed in order to integrate alumina thin films with optimized properties into the corresponding fabrication processes: electron beam evaporation [START_REF] Proost | Evolution of the growth stress stiffness and microstructure of alumina thin films during vapour deposition[END_REF][START_REF] Nayar | Structural, optical and mechanical properties of amorphous and crystalline alumina thin films[END_REF], pulsed laser deposition [START_REF] Hibert | Properties of aluminum oxide thin films deposited by pulsed laser deposition and plasma enhanced chemical vapour deposition[END_REF], sputtering [START_REF] Kelly | Control of the structure and properties of aluminum oxide coating by pulsed magnetron sputtering[END_REF], chemical vapour deposition [START_REF] Ruppi | Deposition, microstructure and properties of texture-controlled CVD -Al2O3 coatings[END_REF], metal-organic chemical vapour deposition [START_REF] Shao | Growth and characterization of Al2O3 gate dielectric films by low-pressure metalorganic chemical vapor deposition[END_REF], plasma-enhanced chemical vapour deposition [START_REF] Johnson | Physical and electrical properties of nanocrystalline prepared by remote plasma enhanced chemical vapour deposition[END_REF],... Among them, atomic layer deposition (ALD) was thoroughly studied for the conformal deposition of Al2O3 thin films [START_REF] Leskelä | Atomic layer deposition (ALD): from precursors to films structures[END_REF][START_REF] Sneh | Thin film atomic layer deposition equipment for semiconductor processing[END_REF][START_REF] Puurunen | Surface chemistry of atomic layer deposition: a case study for the trimethylaluminum/water process[END_REF], emphasizing the use of tri-methyl-aluminium Al(CH3)3 (also called TMA) as an aluminium source and water (H2O) as an oxidant source [START_REF] Ott | Al2O3 thin film growth on Si (100) using binary sequence chemistry[END_REF][START_REF] Yun | Dependence of atomic layerdeposited Al2O3 films characteristics on growth temperature and Al precursors of Al(CH3)3 and AlCl3[END_REF][START_REF] Matero | Effect of water dose on the atomic layer deposition rate of oxide thin films[END_REF][START_REF] Elam | Viscous flow reactor with quartz microbalance for thin film growth by atomic layer deposition[END_REF][START_REF] Barbos | Characterization of Al2O3 thin films prepared by thermal ALD[END_REF][START_REF] Borylo | Structure and properties of Al2O3 thin films deposited by ALD process[END_REF][START_REF] Ha | Influence of oxidant source on the property of atomic layer deposited Al2O3 on hydrogen-terminated Si substrate[END_REF][START_REF] Van Hemmen | Plasma and thermal ALD of Al2O3 in a commercial 200 mm ALD reactor[END_REF]. Nevertheless, other sources such aluminium trichloride (AlCl3) as well as dioxygen (O2) and ozone (O3), were also developed in parallel using plasma-enhanced atomic layer deposition (PE-ALD) in order to reduce the impurity content in the grown Al2O3 film and improve their dielectric properties [START_REF] Ha | Influence of oxidant source on the property of atomic layer deposited Al2O3 on hydrogen-terminated Si substrate[END_REF][START_REF] Van Hemmen | Plasma and thermal ALD of Al2O3 in a commercial 200 mm ALD reactor[END_REF][START_REF] Rai | Influence of surface temperature on the mechanism of atomic layer deposition of aluminum oxide using an oxygen plasma and ozone[END_REF][START_REF] Potts | Room-temperature ALD of metal oxide thin films by energy-enhanced ALD[END_REF][START_REF] Aarik | Atomic layer deposition of high-quality Al2O3 and Al-doped TiO2 thin films from hydrogen-free precursors[END_REF].

In the different studies related to the ALD process [START_REF] Ott | Al2O3 thin film growth on Si (100) using binary sequence chemistry[END_REF][START_REF] Yun | Dependence of atomic layerdeposited Al2O3 films characteristics on growth temperature and Al precursors of Al(CH3)3 and AlCl3[END_REF][START_REF] Matero | Effect of water dose on the atomic layer deposition rate of oxide thin films[END_REF][START_REF] Elam | Viscous flow reactor with quartz microbalance for thin film growth by atomic layer deposition[END_REF][START_REF] Barbos | Characterization of Al2O3 thin films prepared by thermal ALD[END_REF][START_REF] Borylo | Structure and properties of Al2O3 thin films deposited by ALD process[END_REF][START_REF] Ha | Influence of oxidant source on the property of atomic layer deposited Al2O3 on hydrogen-terminated Si substrate[END_REF][START_REF] Van Hemmen | Plasma and thermal ALD of Al2O3 in a commercial 200 mm ALD reactor[END_REF][START_REF] Rai | Influence of surface temperature on the mechanism of atomic layer deposition of aluminum oxide using an oxygen plasma and ozone[END_REF][START_REF] Potts | Room-temperature ALD of metal oxide thin films by energy-enhanced ALD[END_REF][START_REF] Aarik | Atomic layer deposition of high-quality Al2O3 and Al-doped TiO2 thin films from hydrogen-free precursors[END_REF], the presence of a steep interface was systematically assumed between the Al2O3 films and the silicon substrate, which allows simplifying the determination of the ALD growth rate per cycle as well as defining different growth types: linear, substrate-enhanced and substrate-inhibited [START_REF] Puurunen | Island growth as a growth model in atomic layer deposition: a phenomenological model[END_REF]. This "steep interface" assumption seems obvious when considering the atomic layer deposition principles. Nevertheless, since other physical and chemical phenomena occur during the ALD process, it seems crucial to investigate the ALD film layer structure in more detail [START_REF] Petrik | Comparative measurements on atomic layer deposited Al2O3 thin films using ex-situ table top and mapping ellipsometry as well as X-ray and VUV reflectometry[END_REF]. This paper proposes, firstly, to investigate the interfacial properties between the silicon substrate and PE-ALD alumina thin films deposited from Al(CH3)3/O2 precursors, and, secondly, to study the structural, optical, mechanical and dielectric properties of the grown layers, focusing on the impact of a post-growth rapid thermal anneal step (in the 500°C -1100°C temperature range) on their evolution.

Experimental details

Growth experiments were carried out in a FIJI F200 ALD equipment developed by CAMBRIDGE NANOTECH company. Prior to deposition, 100mm (100) silicon substrates were cleaned into diluted hydrofluoric acid (HF -10%) to remove any oxide layer. After this HF cleaning, the native SiO2 layer was still estimated around 0.7 nm by ellipsometry. Then, Al2O3 films were deposited using conventional experimental conditions (deposition temperature: 300°C, base pressure: ~ 70 mTorr) from tri-methyl-aluminium Al(CH3)3 (TMA) and dioxygen (O2) according to a standard routine: TMA injection in gaseous phase (injection duration: 0.025 s), purge (argon: 180 sccm, duration: 6 s), plasma O2 exposure (O2: 20 sccm, plasma power: 300 W), purge (argon: 180 sccm, duration: 6 s). These experimental conditions and durations were chosen in order to saturate the ALD growth rate while preventing any undesirable gaseous mixing [START_REF] Ha | Influence of oxidant source on the property of atomic layer deposited Al2O3 on hydrogen-terminated Si substrate[END_REF][START_REF] Van Hemmen | Plasma and thermal ALD of Al2O3 in a commercial 200 mm ALD reactor[END_REF][START_REF] Potts | Room-temperature ALD of metal oxide thin films by energy-enhanced ALD[END_REF]. Thus, Al2O3 growth occurs according to the binary reaction sequence [START_REF] Langereis | Surface chemistry of plasma-assisted atomic layer deposition of Al2O3 studied by infra-red spectroscopy[END_REF]:

-AlOH* + Al(CH3)3 -----> -Al-O-Al(CH3)2* + CH4 (1)
-AlCH3* + 2O2 -----> -AlOH* + CO2 + H2O (2)
where -AlOH*, -Al-O-Al(CH3)2* and -AlCH3* represent surface species while monofunctional adsorption is assumed for simplification.

Thermal anneals were performed in an AnnealSys AS-Master-2000HT rapid thermal process (RTP) equipment using two different gaseous atmospheres (N2/H2 or O2). Aluminium oxide films were etched using (i) phosphoric and hydrofluoric acids (H3PO4 85% and HF 5%) for wet etching as well as (ii) a Cl2 dry etching gas in an ICP-RIE (Inductively coupled plasma reactive ion etching)

Trikon Omega 201 equipment (ICP power: 500 W, bias power: 25-100 W) [START_REF] Kwon | Dry etching characteristics of Mo and Al2O3 films in O2/Cl2/Ar inductively coupled plasmas[END_REF].

Structural, optical, mechanical and dielectric properties of aluminium oxide thin films were studied using numerous characterisation techniques. Alumina films were studied by spectroscopic ellipsometry using an Uvisel equipment from Horiba Jobin Yvon, in order to estimate deposited thicknesses as well as optical indexes at a 450 nm wavelength. Growth rates per ALD cycle were finally calculated by dividing the deposited thickness by the ALD cycle number. The accuracy of these measurements was estimated to be ± 2%.

Wafer curvature measurements before and after PE-ALD film deposition were also performed by profilometry. Since the deposited films were very thin compared to the silicon substrate (thickness: ~ 500 m), their residual stress was finally calculated thanks to the Stoney's formula [START_REF] Stoney | The tension of metallic films deposited by electrolysis[END_REF][START_REF] Temple-Boyer | Influences of deposition and crystallization kinetics on the properties of silicon films deposited by chemical vapour deposition from silane and disilane[END_REF]. The accuracy of the whole technique was estimated at ± 5%.

Dielectric properties were studied using a mercury probe (MDC company, model 802C)

coupled with an Agilent 4294A impedance analyser. Using a 4.18 × 10 -3 cm 2 mercury contact area, C-V experiments were performed between -10 V and +10 V for a 100 mV sinusoidal signal at 1, 10, 100 and 1000 kHz. Thus, alumina dielectric constant was calculated from the capacitance value in accumulation thanks to the standard planar capacitor equation. The accuracy of the whole technique was estimated at ± 5%.

TEM (Transmission electron microscopy) was performed in a TEM JEOL JEM-ARM200F cold FEG, STEM corrected, equipment installed in the French "Centre de microcaractérisation Raimond Castaing" (Toulouse, France). At 200 kV, its point-to-point resolution is 0.19 nm in TEM mode (0.078 nm in STEM mode). It is coupled with an EDX (Energy Dispersive X-ray)

spectrometer JEOL CENTURIO in order to analyse atomic compositions at the nanoscale (resolution: 2 nm). Before the experiments, the EDX spectrometer was calibrated using a monocrystalline sapphire sample. Then, ~100 nm-thin TEM samples were prepared from Al2O3 films using a FEI Helios 600i FIB (Focused Ion Beam) equipment. GIXRD (Grazing Incidence Xray Diffraction) experiments were also performed using a Bruker D8 Discovery diffractometer (Xray source and associated wavelength: K1 copper, 0.154056 nm) in order to study alumina film crystallinity.

Results and discussion

Mean growth rate per ALD cycle and refractive index

Alumina films characteristics were studied by spectroscopic ellipsometry taking into account the 0.7 nm native silicon oxide layer present after HF clean. Thus, assuming steep interfaces for the Si/SiO2/Al2O3 structure, the deposited thickness, i.e. the mean growth rate per ALD cycle gr, as well as the refractive index at a 450 nm wavelength n450nm were estimated (figure 1). From these first results, the studied TMA/O2 PE-ALD process appears to be characterized by a substrate-enhanced growth [START_REF] Puurunen | Island growth as a growth model in atomic layer deposition: a phenomenological model[END_REF] with an apparent gr value around 0.09 ± 0.01 nm for the highest number of ALD cycles.

Nevertheless, the associated refractive index variations are quite surprising since the n450nm apparent value increases from 1.47 (i.e. similar to silicon dioxide SiO2) at the early stages of the growth, up to 1.67 for the highest number of cycles, i.e. far lower than the value known for bulk monocrystalline alumina or sapphire: n450nm = 1.78 [START_REF] Palik | Handbook of optical constant of solids, volume II[END_REF].

In order to check the steepness of the different interfaces and to have a better analysis of these results, TMA/O2 samples were characterized by TEM coupled with EDX spectrometry. For the thinnest samples (number of cycles: 30 and 60), investigations evidenced a thickness increase of the initial native oxide layer (0.7 nm), accompanied by the formation of an AlxSiyO aluminosilicate overlayer (figures 2a and 2b). On the contrary, for the thickest samples (number of cycles: 300 and 600) no SiO2 layer is observed. Instead, an AlxSiyO/Al2O3 structure is formed (figures 3a and 3b).

The different transition layers (cf. dashed lines in figures 2b and 3b) are determined from the variation of the atomic profiles extracted from the intensity of the different EDX peaks (arbitrary units). In the case of the upper alumina layer in samples with a high number of cycles, it was possible to quantify the EDX spectra and confirm the effective deposition of a stoichiometric Al2O3 alumina layer. The mean values of the different layer thicknesses are given in table 1.

According to these results, it appears that, for ALD cycle number lower than 100 (i.e. for film thickness lower than 10 nm), silicon oxidation occurs during the "TMA injection -plasma O2" routine. Since only native oxide is present on the silicon substrate, this phenomenon is responsible for the growth of the underlayer oxide layer. Moreover, rather than an alumina Al2O3 film, an aluminosilicate AlxSiyO one is simultaneously formed on top, leading to the formation of a SiO2/AlxSiyO structure. On the contrary, for ALD cycle number higher than 200 (i.e. for film thickness higher than 20 nm), the TMA/O2 PE-ALD routine is efficient and allows to effectively deposit alumina atomic layers on top, leading to a final AlxSiyO/Al2O3 structure.

Such experiments clearly show that silicon oxidation and the associated diffusion phenomena interfere with the TMA/O2 PE-ALD process. This scenario was confirmed by studying a 300-cycle PE-ALD films deposited on top of a thermally oxidized silicon substrate (SiO2 thickness: ~10 nm, cf. table 1, bottom row). In this case, considering the TEM/EDX resolution (2 nm), no aluminosilicate layer was evidenced and only alumina was effectively deposited, leading to a final SiO2/Al2O3 structure with steep interfaces (figure 4). Such result is consistent with the interference of silicon oxidation during the first ALD cycles: the thicker the oxide underlayer, the less effective the silicon oxidation phenomenon will be, resulting in a thinner aluminosilicate intermediate layer.

As a matter of fact, it finally appears that the initial SiO2/AlxSiyO bilayer transforms into an aluminosilicate monolayer prior to forming the final AlxSiyO/Al2O3 structure. Such transition is expected to be controlled by the diffusion of oxygen, aluminium and silicon atoms during the PE-ALD process (performed at 300°C), as already shown for hafnium oxide thin films deposited by ALD [START_REF] Cho | Chemical interaction between atomic-layer-deposited HfO2 thin films and the Si substrate[END_REF]. In order to check this assumption, the 600-cycle AlxSiyO/Al2O3 film was annealed for 5 minutes under an O2 gaseous atmosphere at different temperatures (750°C and 900°C). Indeed, TEM/EDX characterizations showed that the aluminosilicate AlxSiyO underlayer broke up in order to participate to the formation of both SiO2 lower layer and Al2O3 upper layer (table 2 and figure 5).

Thus, the thermal diffusion and rearrangement of oxygen, aluminium and silicon atoms into a Si/SiO2/Al2O3 structure with steep interfaces was effectively demonstrated.

In view of these results, the values of film deposited thickness and refractive index, initially determined by spectroscopic ellipsometry, were reconsidered assuming the formation of a SiO2/AlxSiyO/Al2O3 structure in all samples, independently of the number of PE-ALD cycles. Thus, in all cases, the refractive index at a 450 nm wavelength n450nm of the Al2O3 upper layer was estimated around 1.64. In parallel, the mean growth rate per ALD cycle gr was finally estimated around 0.08 ± 0.005 nm whatever the number of ALD cycles.

Thus, it is finally shown that the TMA/O2 PE-ALD process is not characterized by steep interfaces. For low ALD cycle numbers (lower than 100), the "TMA injectionplasma O2" routine is responsible for the silicon substrate oxidation, i.e. for the growth of a SiO2 underlayer, and, subsequently, for the deposition of an AlxSiyO aluminosilicate film. Taking into account this phenomenon, it is clear that no substrate-enhanced growth occurs at the early stages of the growth process, as previously assumed. Indeed, apart for the initial regime, the TMA/O2 PE-ALD process occurs in a standard way, allowing the deposition of stoichiometric alumina Al2O3 with a linear growth mechanism, i.e. with a constant growth rate.

In the following, investigations will focus on the physical/chemical properties of TMA/O2 PE-ALD films obtained for high numbers of ALD cycles (higher than 400). For such high ALD cycle numbers, the influence of the aluminosilicate AlxSiyO lower layer is negligible (figure 1), allowing to consider a global SiO2/Al2O3 structure. According to our analyses, this assumption is related to measurement errors around ± 4% for the alumina film thickness and related parameters (see hereafter).

Thermal annealing influences

Since as-deposited alumina films were characterized by refractive index lower than expected, the influence of thermal annealing was investigated in detail. Thus, different rapid thermal annealing processes (duration: 5 minutes, gaseous atmospheres: N2/H2 and O2) were performed at increasing temperatures, i.e. from 500 to 1100°C, on different 400-cycle alumina samples. In both gaseous ambiences, the refractive index at a 450nm wavelength was found to increase with annealing temperature (figure 6). Starting at 750°C from the as-deposited n450nm value around 1.64, a sigmoidal variation was evidenced to reach a final n450nm value around 1.72. Since such variations were already evidenced for ALD alumina films [START_REF] Wang | The impact of thickness and thermal annealing on refractive index for aluminium oxide thin films deposited by atomic layer deposition[END_REF] and associated to crystallization phenomena for CVD silicon film [START_REF] Temple-Boyer | Influences of deposition and crystallization kinetics on the properties of silicon films deposited by chemical vapour deposition from silane and disilane[END_REF], X-ray diffraction analysis was performed on the different samples.

According to a previous study [START_REF] Jakschik | Crystallization behaviour of thin ALD-Al2O3 films[END_REF], XRD spectra were studied in the [30° -72°] 2 angle range and more precisely around 67.8° (figures 7a and 7b). At this 2 angle value, starting at 800°C, the temperature increase is responsible for the appearance of an intensity peak that was previously related to crystallization phenomena (figures 7b and 7c). Thus, the refractive index n450nm increase with annealing temperature was effectively related to the transition from amorphous to polycrystalline alumina. As a result, the corresponding values were determined:

-PE-ALD amorphous alumina: n450nm ≈ 1.64

-PE-ALD polycrystalline alumina: n450nm ≈ 1.72

The n450nm value obtained for polycrystalline alumina was still lower than 1.78, evidenced for bulk monocrystalline alumina or sapphire [START_REF] Palik | Handbook of optical constant of solids, volume II[END_REF]. As already shown for monocrystalline and polycrystalline silicon [START_REF] Temple-Boyer | Influences of deposition and crystallization kinetics on the properties of silicon films deposited by chemical vapour deposition from silane and disilane[END_REF], this discrepancy might be associated to the presence of amorphous grain boundaries in the polycrystalline Al2O3 films (figure 8).

Residual stress

The residual stress  of the different 900-cycle TMA/O2 PE-ALD films was studied by wafer curvature measurements and calculated using the Stoney equation while considering a SiO2/Al2O3 bilayer. Independently of the gaseous atmosphere (N2/H2 and O2), similar variations were evidenced as a function of the annealing temperature (figure 9).

Initially, the as-deposited alumina films were characterized by a "no-stress" configuration ( ≈ +10 MPa). Then, a first increase was evidenced between 500 and 700°C to reach a +300 MPa tensile stress. This phenomenon could be related to the diffusion of oxygen, aluminium and silicon atoms into the film occurring at such temperature and therefore associated to the rearrangement of the AlxSiyO/Al2O3 structure towards the SiO2/Al2O3 one (see section 3.1 above). Then, a second sharp increase was evidenced between 750 and 800°C to reach very high tensile stress values ( ≈ +2200 MPa). As shown by Krautheim et al. [START_REF] Krautheim | Mechanical stress in ALD-Al2O3 films[END_REF] and as seen previously (see section 3.2 above), this phenomenon should be related to film crystallization and therefore to the transition from amorphous to polycrystalline alumina. However, if a sigmoidal stress variation was expected (as shown for amorphous silicon crystallization [START_REF] Temple-Boyer | Influences of deposition and crystallization kinetics on the properties of silicon films deposited by chemical vapour deposition from silane and disilane[END_REF]), a third phenomenon responsible for compressive stress was shown to interfere for temperature higher than 800°C. Thus, a stress decrease was finally evidenced for the highest temperatures. This last phenomenon is not clearly explained. Since it was obtained for both gaseous atmosphere (N2/H2 and O2), it cannot be due to silicon oxidation phenomena (negligible for stress measurement) but should be related to some phase transition into polycrystalline alumina [START_REF] Krautheim | Mechanical stress in ALD-Al2O3 films[END_REF].

Dielectric constant

The dielectric constant r of different 400-cycle TMA/O2 PE-ALD films was studied by capacitance-voltage (C-V) measurements at different frequencies (1, 10, 100 and 1000 kHz) and analysed as a function of the annealing temperature (gaseous atmosphere: O2). Measured C-V curves are characteristics of a Metal-Insulator-Silicon capacitive structure (results not shown), enabling the determination of the capacitance in accumulation Cacc. Finally, the insulator equivalent dielectric constant r was estimated using the planar capacitor equation while considering a SiO2/Al2O3 bilayer. As shown for the refractive index n450nm (see section 3.2 above), a typical sigmoidal variation was obtained, evidencing a dielectric constant increase from 8.5 to 10.2 with temperature (figure 10). Again, this variation should be related to crystallization phenomena of alumina between 750 and 850°C. As a result, the dielectric constants of amorphous and polycrystalline alumina were determined:

-PE-ALD amorphous alumina: r ≈ 8.5

-PE-ALD polycrystalline alumina: r ≈ 10.2

These  values are in good agreement with standard value obtained for bulk monocrystalline alumina or sapphire [START_REF]Metals, Alloys, Compounds, Ceramics, Polymers, Composites, catalogue 1993/1994[END_REF].

Etch properties

Since the integration of TMA/O2-based Al2O3 PE-ALD films is planned for the realization of microtechnological devices, their chemical and physical etch properties were also studied (table 3).

Concerning wet etching, phosphoric acid H3PO4 was found to be ineffective while hydrofluoric acid HF was only effective for amorphous Al2O3 (etch rate: 40 nm/min). In order to etch polycrystalline Al2O3, a Cl2-based dry etch in an ICP-RIE equipment was implemented. Thus, depending on the ICP power, the etch rate was found to increase linearly up to 90 nm/min for amorphous Al2O3, while it saturates at a maximal value of 40 nm/min for polycrystalline Al2O3. According to our results, a bias power of 50 W is proposed in order to effectively and selectively etch any TMA/O2based Al2O3 PE-ALD films.

Conclusion

In summary, PE-ALD Al2O3 thin films (thickness range: 6 -72 nm) were deposited on silicon substrates using TMA/O2 precursors. TEM/EDX investigations evidenced the formation of different transition layers depending on the number of ALD cycles. For low ALD cycle numbers (lower than 100), it was demonstrated that the TMA/O2 PE-ALD routine is not efficient to deposit alumina atomic layers because of silicon oxidation phenomena. As a result, the initial formation of a SiO2/AlxSiyO transition structure with no steep interface was shown. For high ALD cycle numbers (higher than 200), oxidation phenomena are inhibited due to the higher deposited thickness so that the TMA/O2 PE-ALD process operates normally to form alumina layers. Nevertheless, diffusion and rearrangement of oxygen, aluminium and silicon atoms were found to interfere with the PE-ALD process, leading to the formation of an aluminosilicate AlxSiyO transition layer and the deposition of a final AlxSiyO/Al2O3 stack. In view of these different mechanisms, no "substrateenhanced growth" phenomenon was finally evidenced for the TMA/O2 PE-ALD process. As a result, a constant growth rate (0.08 nm per cycle) and refractive index (1.64 at a 450 nm wavelength) were obtained for the deposited Al2O3 upper layer, independently of the number of ALD cycles.

The influence of post-growth thermal annealing was also investigated. Thanks to XRD analysis, crystallization of Al2O3 films was found to start at a temperature of 750°C. Then, thermally induced atomic rearrangement within the Al2O3 structure was found to be responsible for sigmoidal increases of various parameters with increasing temperature, such as the refractive index at a 450 nm wavelength n450nm, the residual stress  and the dielectric constant r. Thus, the corresponding properties of amorphous and polycrystalline alumina were finally determined. The results obtained in this work allow to have a better understanding of the PE-ALD-Al2O3 film deposition process and related properties. They will be helpful for the integration of thick (> 20 nm) alumina layers in the frame of micro/nano-technological applications.

Tables and figures captions

Table 1: TEM characterisation of the different structures deposited by PE-ALD from TMA and O2

Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 2345678910 Figure 2: TEM/EDX investigation of the 60-cycle TMA/O2 PE-ALD sample

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :

 12345678910 Figure 1: Apparent mean growth rate per ALD cycle and refractive index of Al2O3 films

Table 2 :

 2 TEM characterisation of the 600-cycle TMA/O2 PE-ALD Al2O3 films annealed 5 minutes under an O2 gaseous atmosphere at different temperatures

Table 3 :

 3 Wet and dry etching properties of Al2O3 deposited by PE-ALD from TMA and O2 Figure 1: Apparent mean growth rate per ALD cycle and refractive index of Al2O3 films versus ALD cycle number (parameters were estimated by ellipsometry at a 450nm wavelength)

Table 1 :

 1 TEM characterisation of the different structures deposited by PE-ALD from TMA and O2 (* experimental value lower than the TEM/EDX resolution)

	Number of cycles	SiO2 thickness (nm)	AlxSiyO thickness (nm)	Al2O3 thickness (nm)
	30	4	2.5	0
	60	7	5	0
	300	0 *	6	24
	600	0 *	6	48
	300 (on SiO2)	10	0 *	26

Table 3 :

 3 Wet and dry etching properties of Al2O3 deposited by PE-ALD from TMA and O2

Acknowledgements

Technological realizations were partly supported by the French RENATECH network. Furthermore, we would like to thank Vincent Mortet (Institute of physics, Academy of sciences of the Czech Republic) for helpful discussions.


H Gouin

Prof M Antonio

Greco

THE D'ALEMBERT-LAGRANGE PRINCIPLE FOR GRADIENT THEORIES AND BOUNDARY CONDITIONS

Motions of continuous media presenting singularities are associated with phenomena involving shocks, interfaces or material surfaces. The equations representing evolutions of these media are irregular through geometrical manifolds. A unique continuous medium is conceptually simpler than several media with surfaces of singularity. To avoid the surfaces of discontinuity in the theory, we transform the model by considering a continuous medium taking into account more complete internal energies expressed in gradient developments associated with the variables of state. Nevertheless, resulting equations of motion are of an higher order than those of the classical models: they lead to non-linear models associated with more complex integration processes on the mathematical level as well as on the numerical point of view. In fact, such models allow a precise study of singular zones when they have a non negligible physical thickness. This is typically the case for capillarity phenomena in fluids or mixtures of fluids in which interfacial zones are transition layers between phases or layers between fluids and solid walls. Within the framework of mechanics for continuous media, we propose to deal with the functional point of view considering globally the equations of the media as well as the boundary conditions associated with these equations. For this aim, we revisit the d'Alembert-Lagrange principle of virtual works which is able to consider the expressions of the works of forces applied to a continuous medium as a linear functional value on a space of test functions in the form of virtual displacements. At the end, we analyze examples corresponding to capillary fluids. This analysis brings us to numerical or asymptotic methods avoiding the difficulties due to singularities in simpler -but with singularities-models.

Introduction

A mechanical problem is generally studied through force interactions between masses located in material points: this Newton point of view leads together to the statistical mechanics but also to the continuum mechanics. The statistical mechanics is mostly precise but is in fact too detailed and in many cases huge calculations crop up. The continuum mechanics is an asymptotic notion coming from short range interactions between molecules. It follows a loose of information but a more efficient and directly computable theory. In the simplest case of continuum mechanics, residual information comes through stress tensor like Cauchy tensor [START_REF] Germain | Cours de mécanique des milieux continus[END_REF][START_REF] Truesdell | First course in rational continuum mechanics[END_REF] . The concept of stress tensor is so frequently used that it has become as natural as the notion of force. Nevertheless, tensor of contact couples can be investigated as in Cosserat medium [START_REF] Cosserat | Sur la théorie des corps déformables[END_REF] or configuration forces like in Gurtin approach [START_REF] Gurtin | Configurational forces as basic concepts in continuum physics[END_REF] with edge interactions of Noll and Virga [START_REF] Noll | [END_REF] . Stress tensors and contact forces are interrelated notions 6 . A fundamental point of view in continuum mechanics is: the Newton system for forces is equivalent to the work of forces is the value of a linear functional of displacements. Such a method due to Lagrange is dual of the system of forces due to Newton 7,8 and is not issued from a variational approach; the minimization of the energy coincides with the functional approach in a special variational principle only for some equilibrium cases. The linear functional expressing the work of forces is related to the theory of distributions; a decomposition theorem associated with displacements (as test functions whose supports are C ∞ compact manifolds) uniquely determines a canonical zero order form (separated form) with respect both to the test functions and the transverse derivatives of contact test functions [START_REF] Schwartz | Théorie des distributions[END_REF] . As Newton's principle is useless when we do not have any constitutive equation for the expression of forces, the linear functional method is useless when we do not have any constitutive assumption for the virtual work functional. The choice of the simple material theory associated with the Cauchy stress tensor corresponds with a constitutive assumption on its virtual work functional. It is important to notice that constitutive equations for the free energy χ and constitutive assumption for the virtual work functional may be incompatible 10 : for any virtual displacement ζ of an isothermal medium, the variation -δχ must be equal to the virtual work of internal forces δτ int . The equilibrium state is then obtained by the existence of a solution minimizing the free energy. The equation of motion of a continuous medium is deduced from the d'Alembert-Lagrange principle of virtual works which is an extension of the principle in mechanics of systems with a finite number of degrees of freedom: The motion is such that for any virtual displacement the virtual work of forces is equal to the virtual work of mass accelerations. Let us note: if the virtual work of forces is expressed in classical notations in the form

δτ = D {f . ζ + tr [(-p 1 + 2µ ∇V) .∇ζ]} dv + S T. ζ ds (1)
from the d'Alembert-Lagrange principle, we obtain not only the equations of balance momentum for a viscous fluid in the domain D but also the boundary conditions on the border S of D. We notice that expression (1) is not the Frechet derivative of any functional expression. If the free energy depends on the strain tensor F , then δτ must depend on ∇ζ and leads to the existence of the Cauchy stress tensor. If the free energy depends on the strain tensor F and on the overstrain tensor ∇F, then δτ must depend on ∇ζ and ∇ 2 ζ.

Conjugated (or transposed) mappings being denoted by asterisk, for any vectors a, b, we write a * b for their scalar product (the line vector is multiplied by the column vector) and ab * or a⊗b for their tensor product (the column vector is multiplied by the line vector). The product of a mapping A by a vector a is denoted by A a. Notation b * A means the covector c * defined by the rule c * = (A * b) * . The divergence of a linear transformation A is the covector divA such that, for any constant vector a, (div A) a = div (A a). We introduce a Galilean or fixed system of coordinates (x 1 , x 2 , x 3 ) which is also denoted by x as Euler or spatial variables. If f is a real function of x, ∂f ∂x is the linear form associated with the gradient of f and

∂f ∂x i = ( ∂f ∂x ) i ;
consequently, ( ∂f ∂x ) * = grad f . The identity tensor is denoted by 1. Now, we present the method and its consequences in different cases of gradient theory. As examples, we revisit the case of Laplace theory of capillarity and the case of van der Waals fluids.

Virtual work of continuous medium

The motion of a continuous medium is classically represented by a continuous transformation ϕ of a three-dimensional space into the physical space. In order to describe the transformation analytically, the variables X =(X 1 , X 2 , X 3 ) which single out individual particles correspond to material or Lagrange variables. Then, the transformation representing the motion of a continuous medium is

x = ϕ (X,t) or x i = ϕ i (X 1 , X 2 , X 3 , t) , i = 1, 2, 3
where t denotes the time. At t fixed the transformation possesses an inverse and continuous derivatives up to the second order except at singular sur-faces, curves or points. Then, the diffeomorphism ϕ from the set D 0 of the particles into the physical space D is an element of a functional space ℘ of the positions of the continuous medium considered as a manifold with an infinite number of dimensions. To formulate the d'Alembert-Lagrange principle of virtual works, we introduce the notion of virtual displacements. This is obtained by letting the displacements arise from variations in the paths of the particles. Let a one-parameter family of varied paths or virtual motions denoted by {ϕ η } and possessing continuous derivatives up to the second order and expressed analytically by the transformation x = Φ (X,t; η) with η ∈ O, where O is an open real set containing 0 and such that Φ (X,t; 0) = ϕ (X,t) or ϕ 0 = ϕ (the real motion of the continuous medium is obtained when η = 0). The derivation with respect to η when η = 0 is denoted by δ. Derivation δ is named variation and the virtual displacement is the variation of the position of the medium [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF] . The virtual displacement is a tangent vector to ℘ in ϕ (δϕ ∈ T ϕ (℘)). In the physical space, the virtual displacement δϕ is determined by the variation of each particle: the virtual displacement of the particle x is such that ζ = δx when δX = 0, δη = 1 at η = 0; we associate the field of tangent vectors to D

x ∈ D → ζ = ψ(x) ≡ ∂Φ ∂η | η=0 ∈ T x (D)
where T x (D) is the tangent vector bundle to D at x. The concept of virtual work is purposed in the form:

The virtual work is a linear functional value of the virtual displacement,

δτ =< ℑ, δϕ > (2)
where < . , . > denotes the inner product of ℑ and δϕ; then, ℑ belongs to the cotangent space of ℘ at ϕ (ℑ ∈ T * ϕ (℘)). In Relation (2), the medium in position ϕ is submitted to the covector ℑ denoting all the stresses; in the case of motion, we must add the inertial forces associated with the acceleration quantities to the volume forces. The d'Alembert-Lagrange principle of virtual works is expressed as:

For all virtual displacements, the virtual work is null.

Consequently, representation (2) leads to: 2) is a distribution in a separated form, the d'Alembert-Lagrange principle yields the equations of motions and boundary conditions in the form ℑ = 0 .

∀ δϕ ∈ T ϕ (℘), δτ = 0 Theorem: If expression (

Some examples of linear functional of forces

Among all possible choices of linear functional of virtual displacements, we classify the following ones: where F i (i = 1, 2, 3) denote the covariant components of the volume force F (including the inertial force terms) presented as a covector. The equation of the motion is

∀ x ∈ D, F i = 0 ⇔ F = 0 (3) 3.1.2. Model B.0
The medium fills a set D and the surface S is the boundary of D belonging to the medium; with the same notations as in section 3.1.1, the linear functional is in the form

δτ = D F i ζ i dv + S T i ζ i ds (4)
T i are the components of the surface forces (tension) T. From Eq. ( 4), we obtain the equation of motion as in Eq. ( 3) and the boundary condition,

∀ x ∈ S, T i = 0 ⇔ T = 0 , 3.2. Model of first gradient 3.2.1. Model A.1
With the previous notations, the linear functional is in the form

δτ = D F i ζ i -σ j i ζ i ,j dv
where σ j i (i, j = 1, 2, 3) are the components of the stress tensor σ. Stokes formula gets back to the model B .0 in the separated form

δτ = D F i + σ j i,j ζ i dv - S n j σ j i ζ i ds
where n j (j = 1, 2, 3) are the components of a covector which is the annulator of the vectors belonging to the tangent plane at the boundary S. It is not necessary to have a metric in the physical space; nevertheless, for the sake of simplicity it is convenient to use the Euclidian metric; the vector n of components n j (j = 1, 2, 3) represents the external normal to S relatively to D; the covector n ⋆ is associated with the components n j . We deduce the equation of motion

∀ x ∈ D, F i + σ j i,j = 0 ⇔ F + div σ = 0 (5)
and the boundary condition The linear functional is expressed in the form

∀ x ∈ S, n j σ j i = 0 ⇔ n ⋆ σ = 0 3.2.
δτ = D F i ζ i -σ j i ζ i ,j dv + S T i ζ i ds
Stokes formula yields the separated form

δτ = D F i + σ j i,j ζ i dv + S T i -n j σ j i ζ i ds (S.0)
and we deduce the equation of motion in the same form as Eq. ( 5) and the boundary condition

∀ x ∈ S, n j σ j i = T i ⇔ n ⋆ σ = T Model B .1 /
0 is the classical theory for elastic media and fluids in continuum mechanics.

Model B.1

The linear functional is expressed in the form

δτ = D F i ζ i -σ j i ζ i ,j dv + S T i ζ i + γ j i ζ i ,j ds (6)
where the tensor γ of components γ j i is a new term. The boundary of D is a surface S shared in a partition of N parts S p of class C 2 , (p = 1, ..., N ) (Fig. 2). We denote by (R m ) -1 the mean curvature of S; the edge Γ p of S p is the union of the limit edges Γ pq between surfaces S p and S q assumed to be of class C 2 and t is the tangent vector to Γ p oriented by n; n ′ is the unit external normal vector to Γ p in the tangent plane to S p : n ′ = t × n. Let us notice that:

γ j i ζ i ,j = -γ j i,j ζ i + V j ,j (7)
where vector V on surfaces S p we obtain,

V j = γ j i ζ i ;
Sp V j ,j ds = - Sp n j V j R m -V j ,l n l ds + Γp n ′ j V j dℓ (8)
We emphasize with the fact that V j ,l n l corresponds to the normal derivative to S p denoted dV j dn . An integration by parts of the term σ j i ζ i ,j in relation ( 6) and taking account of relations (7-8) implies

δτ = D F 1 i ζ i dv + S T 1 i ζ i ds + S L i dζ i dn ds + N p=1 Γp R pi ζ i dℓ (S.

) with the following definitions

   F 1 i ≡ F i + σ j i,j , L i ≡ n j γ j i T 1 i ≡ T i -n j σ j i - d dn (γ j i ) + 1 R m γ j i -γ j i,j , R pi ≡ n ′ j γ j i
Due to theorem 37 in 9 , the distribution (S.1) has a unique decomposition in displacements and transverse derivatives of displacements on the manifolds associated with D and its boundaries: expression (S.1 ) is in a separated form. Consequently, the equation of motion is

∀ x ∈ D, F 1 i = 0 ⇔ F 1 =
0 and the boundary conditions are

∀ x ∈ S, T 1 i = 0, L i = 0 ⇔ T 1 = 0, L = 0 ∀ x ∈ Γ pq , R pi +R qi = 0 ⇔ R p + R q = 0
Term L is not reducible to a force: its virtual work L i dζ i dn is not the product of a force with the displacement ζ; the term L is an embedding action.

Model of second gradient

3.3.1. Model A.2
The linear functional is in the form

δτ = D F i ζ i -σ j i ζ i ,j + S jk i ζ i ,jk dv
Tensor S with S jk i = S kj i is an overstress tensor. An integration by parts of the last term brings back to the model B .1 ,

δτ = D F i ζ i -σ j i + S jk i,k ζ i ,j dv + S n k S jk i ζ i ,j ds
and the virtual work gets the separated form (S.1 ) with:

             F 1 i = F i + σ j i,j + S jk i,jk
volume force

T 1 i = -n j σ j i + S jk i,k - d dn n k S jk i + 1 R m n k S jk i surface force R pi = n ′ j n k S jk i line force L i = n j n k S jk

Model B.2

The linear functional is in the form

δτ = D F i ζ i -σ j i ζ i ,j + S jk i ζ i ,jk dv+ S T i ζ i + γ j i ζ i ,j + U jk i ζ i ,jk ds
This functional yields two integrations successively on S p and on Γ pq with terms at the points A m . With obvious notations, for the same reasons as in section 3.2.3, the virtual work gets the separated form

δτ = D F 1 i ζ i dv + S T 1 i ζ i + L 1 i dζ i dn + L 2 i d 2 ζ i dn 2 ds + p Γp R pi ζ i + M pi dζ i dn ′ dℓ + m φ mi ζ i Am (S.2)
where ζ i Am (i = 1, 2, 3) are the components of ζ at point A m . The calculations are not expended. They introduce the curvature tensor on S p and the geodesic curvature of Γ pq. . Consequently, F 1 i , T 1 i , R pi , φ mi are associated with volume, surface, line and forces at points; L 1 i , L 2 i , M pi are embedding efforts of order 1 and 2 on S and of order 1 on the edge Γ. The equation of motion and boundary conditions express that these seven tensorial quantities are null on their domains of values D, S, Γ p and A m .

Conclusion

It is possible to extend the previous presentation by means of more complex medium with gradient of order n. The models introduce embedding effects of more important order on surfaces, edges and points. The (A.n) model refers to a (B.n-1) model: the fact that boundary surface S is (or is not) a material surface has now a physical meaning. Consequently, we can resume the previous presentation as follows:

a) The choice of a model corresponds to specify the part G of the algebraic dual T * ϕ (℘) in which the efforts are considered: ℑ ∈ G ⊂ T * ϕ (℘). b) In order to operate with the principle of virtual works and to obtain the mechanical equations in the form ℑ = 0, it is no matter that the part G of the dual is separating

(∀ ℑ ∈ G, < ℑ, δϕ >= 0 ⇒ δϕ = 0), but it is important the part G is separated (ℑ ∈ G, ∀ δϕ ∈ T ϕ (℘), < ℑ, δϕ >= 0 ⇒ ℑ = 0). c) The functionals A.1 , B .1 /0 , A.2 , B .
2 are not separated: if ℑ consists in the data of the fields F, σ, T, it is not possible to conclude that the fields are zero. d) Functionals in A.0 , B .0 , S .1 , S .2 . . . are separated: if the fields S 1 , T 1 , R 1 , L, . . . are continuous then, by using the fundamental lemma of variation calculus, their values must be equal to zero. They are the only functionals we must know for using the principle of virtual works; it is exactly as for a solid: the torque of forces is only known in the equations of motion. e) When the fields are not continuous on surfaces or curves, we have to consider a model of greater order in gradients and to introduce integrals on inner boundaries of the medium.

For conservative medium, the first gradient theory corresponds to the compressible case. The theory of fluid, elastic, viscous and plastic media refers to the model (S .0 ). The Laplace theory of capillarity in fluids refers to the model (S .1 ). To take into account superficial effects acting between solids and fluids, we use the model of fluids endowed with capillarity (S .2 ); the theory interprets the capillarity in a continuous way and contains the Laplace theory of capillarity; for solids, the model corresponds to "elastic materials with couple stresses" indicated by Toupin in 12 .

Example 1: The Laplace theory of capillarity

Liquid-vapor and two-phase interfaces are represented by a material surface endowed with an energy relating to Laplace surface tension. The interface appears as a surface separating two media with its own characteristic behavior and energy properties [START_REF] Levitch | Physicochemical Hydrodynamics[END_REF] (when working far from critical conditions, the capillary layer has a thickness equivalent to a few molecular beams [START_REF] Ono | Molecular theory of surface tension in liquid[END_REF] ). The Laplace theory of capillarity refers to the model B .1 in the form (S .1 ) as following: for a compressible fluid with a capillary effect on the wall boundaries, the free energy is in the form

χ = D ρ α(ρ) dv + N p=1 a p Sp ds
where α(ρ) is the fluid specific energy, ρ is the matter density and coefficients a p are the surface tensions of each surface S p . Surface integrations are associated to the space metric; the virtual work of internal forces is

δτ int = D -p ,i ζ i dv+ N p=1 Sp n i p - a p R m ζ i ds+ N p=1 Γp a p n ′ i ζ i dℓ
where p ≡ ρ 2 α ′ (ρ) is the fluid pressure. The external force (including inertial forces) is the body force ρ f defined in D, the surface force is T defined on S and the line force is R defined on Γ. D'Alembert-Lagrange principle yields the equation of motion and boundary conditions:

∀ x ∈ D, -p ,i + ρ f i = 0 ⇔ -grad p + ρ f = 0, ∀ x ∈ S, p n i + T i -a p n i R m = 0 ⇔ p n + T -a p n R m = 0 ∀ x ∈ Γ pq , a p n ′ pi + a q n ′ qi + R i = 0 ⇔ a p n ′ p + a q n ′ q + R
= 0 Boundary conditions are Laplace equation and Young-Dupré condition.

Example 2: Fluids endowed with internal capillarity

For interfacial layers, kinetic theory of gas leads to laws of state associated with non-convex internal energies [START_REF] Cahn | [END_REF][START_REF] Rowlinson | Molecular theory of capillarity[END_REF] . This approach dates back to van der Waals 17 , Korteweg 18 , corresponds to the Landau-Ginzburg theory 19 and presents two disadvantages. First, between phases, the pressure may become negative; simple physical experiments can be used to cause traction that leads to these negative pressure values [START_REF] Rocard | Thermodynamique[END_REF] . Second, in the field between bulks, internal energy cannot be represented by a convex surface associated with density and entropy; this fact seems to contradict the existence of equilibrium states; it is possible to eliminate this disadvantage by writing in an anisotropic form the stress tensor of the capillary layer which allows to study interfaces of non-molecular size near a critical point. One of the problems that complicates this study of phase transformation dynamics is the apparent contradiction between Korteweg classical stress theory and the Clausius-Duhem inequality [START_REF] Gurtin | [END_REF] . Proposal made by Eglit 22 , Dunn and Serrin 23 , Casal and Gouin 24 and others rectifies this anomaly for liquid-vapor interfaces. The simplest model in continuum mechanics considers a free energy as the sum of two terms: a first one corresponding to a medium with a uniform composition equal to the local one and a second one associated with the non-uniformity of the fluid [START_REF] Cahn | [END_REF][START_REF] Van Der Waals | [END_REF] . The second term is approximated by a gradient expansion, typically truncated to the second order. The model is simpler than models associated with the renormalization-group theory [START_REF] Domb | The critical point[END_REF] but has the advantage of easily extending well-known results for equilibrium cases to the dynamics of interfaces [START_REF] Slemrod | [END_REF][27][28] . We consider a fluid D in contact with a wall S. Physical experiments prove that the fluid is nonhomogeneous in the neighborhood of S 16 . The internal energy ε is also a function of the entropy. In the case of isothermal motions, the internal energy is replaced by the free energy. In the mechanical case, the entropy and the temperature are not concerned by the virtual displacements of the medium. Consequently, for isentropic or isothermal motions, ε = f (ρ, β) where β = (grad ρ) [START_REF] Truesdell | First course in rational continuum mechanics[END_REF] . The fluid is submitted to external forces represented by a potential Ω as a function of Eulerian variables x. To obtain boundary conditions it is necessary to know the wall effect. An explicit form for the energy of interaction between surfaces and liquids is proposed in 29 . We denote by B the surface density of energy at the wall. The total energy E of the fluid is the sum of three potential energies: E f (bulk energy), E p (external energy) and E S (surface energy).

E f = D ρ ε(ρ, β) dv, E p = D ρ Ω(x) dv, E S = S B ds
δE S = S δB - 2B R m n * + grad * tg B ζ ds + Γ B n ′ * ζ dℓ
The density in the fluid has a limit value ρ S at the wall S and B is assumed to be a function of ρ S only 29 . Then,

δB = B ′ (ρ S ) δρ S = -ρ S B ′ (ρ S ) div ζ,
where div ζ is computed on S 11 . Let us denote G = -ρ S B ′ (ρ S ); Appendix yields

S δB ds = S G div ζ ds = = S G dζ n dn - 2G R m n * + grad * tg G ζ ds + Γ G n ′ * ζ dℓ δE S = S G dζ n dn - 2H R m n * + grad * tg H ζ ds + Γ H n ′ * ζ dℓ with H = B(ρ S ) -ρ S B ′ (ρ S ). Then, δE = D (ρ grad * Ω -div σ) ζ dv - Γ (A -H) n ′ * ζ dℓ + S (G -A) dζ n dn + 2(A -H) R m n * + grad * tg (A -H) + n * σ ζ ds (9)
At equilibrium, δτ ≡ -δE = 0. The fundamental lemma of variation calculus associated with separated form (9) corresponding to (S.2), yields:

ρ grad * Ω -div σ = 0 ( 10
)
This equation is written in the classical form of equation of equilibrium 24 .

It is not the same for the boundary conditions.

Boundary conditions

Case of a rigid wall

We consider a rigid wall; on S, the virtual displacements satisfy the condition n * ζ = 0. Then, at the rigid wall,

∀ x ∈ S → ζ(x) such that n * ζ = 0, S (G -A) dζ n dn + 2(A -H) R m n * + grad * tg (A -H) + n * σ ζ ds = 0
Due to σ = σ * , we deduce the boundary conditions (11-12)

∀ x ∈ S, G -A = 0, (11)
and there exists a Lagrange multiplier x ∈ S → λ(x) ∈ R such that,

∀ x ∈ S, 2(A -H) R m n + grad tg (A -H) + σ n = λ n (12)
The edge Γ of S belongs to the solid wall and consequently on Γ, ζ = η t: the integral on Γ is null and does not yield any additive condition.

Case of an elastic wall

The equilibrium equation ( 10) is unchanged. On S, the condition ( 11) is also unchanged. The only different condition comes from the fact that we do not have anymore the slipping condition for the virtual displacement on S, (n * ζ = 0). Due to the possible deformation of the wall, the virtual

work of stresses on S is δE e = S κ * ζ ds + Γ R * ζ dℓ where κ = σ e n
is the stress (loading) vector associated with stress tensor σ e of the elastic wall and R is the line force due to the elasticity of the line. Relation ( 12) is replaced by

∀ x ∈ S, 2 (A -H) R m n + grad tg (A -H) + σ n + κ = 0
We obtain an additive condition on Γ in the form (H -A) n ′ + R = 0 and due to condition (11),

∀ x ∈ Γ, B n ′ + R = 0 (13) (If Γ is the union of edges Γ p , B n ′ is replaced by p B p n ′ p ).

Analysis of the boundary conditions

Eq. ( 11) yields C dρ dn + B ′ (ρ S ) = 0; the definition of σ implies:

σ n = -P n -C dρ dn grad ρ
Due to the fact that the tangential part of Eq. ( 12) is always verified, the only condition comes from Eq. ( 11); Eq. ( 12) yields the value of the Lagrange multiplier λ and Eq. ( 13) the value of R. For an elastic (nonrigid) wall we obtain,

κ tg = 0 and κ n = P + 2B R m -B ′ (ρ S ) dρ dn (14)
where κ tg and κ n are the tangential and the normal components of κ.

Taking into account of Eq. ( 14) we obtain the stress values at the non rigid elastic wall. The surface energy is 29

: B(ρ S ) = -γ 1 ρ S + γ 2 2 ρ 2
S where γ 1 and γ 2 are two positive constants and the fluid density condition at the wall is

C dρ dn = γ 1 -γ 2 ρ S
If we denote by ρ B = γ 1 /γ 2 the bifurcation fluid density at the wall, due to the fact C is positive constant [START_REF] Rowlinson | Molecular theory of capillarity[END_REF] , we obtain: if ρ S < ρ B , ( or ρ S > ρ B ), dρ dn is positive ( or negative) and we have a lack ( or excess) of fluid density at the wall. Such media allow to study fluid interfaces and interfacial layers between fluids and solids and lead to numerical and asymptotic methods 30 . The extension to the dynamic case is straightforward: Eq. ( 10) yields

ρ Γ * -div σ + ρ grad * Ω = 0
Vector Γ is the acceleration; boundary conditions (11-14) are unchanged.

Acknowledgments

I am grateful to Professor Tommaso Ruggeri for helpful discussions.

Appendix

Let S be a surface in the 3-dimensional space and n its external normal extended locally in the vicinity of S by the expression n(x) = grad d(x), where d is the distance of a point x to S; for any vector field w, we obtain

E S = S0 B det F det (F -1 n, d 1 X, d 2 X)
where S 0 is the image of S in a reference space with Lagrangian coordinates

X

Fig. 1 .

 1 Fig.1. The boundary S of D is represented by a thick curve and its variation by a thin curve. Variation δϕ of family {ϕη} of varied paths belongs to Tϕ(℘), tangent space to (℘) at ϕ.

3. 1 .

 1 Model of zero gradient 3.1.1. Model A.0 The medium fills an open set D of the physical space and the linear functional is in the form δτ = D F i ζ i dv

2 .

 2 Model B.1/0: (Mixed model with first gradient in D and zero gradient on S)

Fig. 2 .

 2 Fig. 2. The set D has a surface boundary S divided in several parts. The edge of S is denoted by Γ which is also divided in several parts with end points Am.

  the same equation of motion and boundary conditions as in case B .1 .

  We have the results (see Appendix):δE f = D (-div σ) ζ dv + grad * tg A + n * σ ζ ds -Γ A n ′ * ζ dℓ with σ = -P 1-C grad ρ⊗grad ρ ≡ -P 1-C ( ∂ρ ∂x ) * ∂ρ ∂x , where C = 2ρ ε ′ β , P = ρ 2 ε ′ ρ -ρ div(C grad ρ), ε ′ ρ (or ε ′ β )denoting the partial derivative of ε with respect to ρ (or β), ζ n = n * ζ; A = Cρ dρ dn where dρ dn = ∂ρ ∂x n and grad tg denotes the tangential part of the gradient relatively to S. δE p = D ρ ∂Ω ∂x ζ dv ≡ D ρ (grad * Ω) ζ dv; and,

6. 1 .

 1 Equation of equilibrium From any arbitrary variation x ∈ D → ζ(x) such that ζ = 0 on S, we get D (ρ grad * Ω -div σ) ζ ds = 0. Then,

( 15 ) 16 )

 1516 [START_REF] Kobayashi | Foundations of differential geometry[END_REF] : rot(n × w) = n div ww div n + ∂n ∂x w -∂w ∂x nFrom n * ∂n ∂x = 0 and div n = -2 R m we deduce on S,n * rot(n × w) = div w + 2 R m n * wn * ∂w ∂x nWe deduce: for any scalar field A and w = A ζ,A div ζ = A dζ n dn -2A R m ζ n -(grad * tg A) ζ + n * rot (An × ζ) (Let us calculate δE f ; D is a material volume, then δE f = D C grad ρ δρ) -div(C grad ρ) δρ -tr C grad ρ grad * ρ ∂ζ ∂xDue to δ ρ = -ρ div ζ (see[START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF] ),δE f = D ∂P ∂x + div(C grad ρ grad * ρ) ζ dv -D div (C ρ grad ρ div ζ + C grad ρ grad * ρ ζ + P ζ) dv = D -(div σ) ζ dv + S (-A div ζ + n * σ ζ)dsFrom Eq. (16), we deduce immediatly:δE f = D (-div σ)ζ dv + S -A dζ n dn + 2A R m n * + grad * tg A + n * σ ζ ds -Γ A n ′ * ζdℓ Let us calculate δE S ; due to E S = S B det (n, d 1 x, d 2 x) where d 1 x and d 2 x are two coordinate lines of S, we get:

  and F is the deformation gradient tensor ∂x ∂X of components ∂x i ∂X j .Then,δE S = S0 δB det F det (F -1 n, d 1 X, d 2 X) + S0 B δ det F det (F -1 n, d 1 X, d 2 X) . with S0 B δ det F det (F -1 n, d 1 X, d 2 X) = S B div ζ det(n, d 1 x, d 2 x) + B det ∂n ∂x ζ, d 1 x, d 2 x -B det ∂ζ ∂x n, d 1 x, d 2 x = S div(B ζ) -(grad * B) ζ -Bn * ∂ζ ∂x n ds Relation (15) yields: div (B ζ) + 2B R m n * ζn * ∂B ζ ∂x n = n * rot (B n × ζ), S0 B δ det F det (F -1 n, d 1 X, d 2 X) = S0 -2B R m n * + grad * B (nn * -1) ζ ds + S n * rot (B n × ζ) dswhere grad * B (nn * -1) belongs to the cotangent plane to S; we obtain δE S = S δB -2B R m n * + grad * tg B ζ ds + Γ B n ′ * ζ dℓ.