Henri Gouin
email: henri.gouin@univ.u-3mrs.fr

Tommaso Ruggeri
email: ruggeri@ciram.unibo.it

Keywords: Fluid mixtures, acceleration waves, interfacial layers PACS: 47.55, 52.35, 68.05, 68.60

Through an Hamiltonian action we write down the system of equations of motions for a mixture of thermocapillary fluids under the assumption that the internal energy is a function not only of the gradient of the densities but also of the gradient of the entropies of each component. A Lagrangian associated with the kinetic energy and the internal energy allows to obtain the equations of momentum for each component and for the barycentric motion of the mixture. We obtain also the balance of energy and we prove that the equations are compatible with the second law of thermodynamics. Though the system is of parabolic type, we prove that there exist two tangential acceleration waves that characterize the interfacial motion. The dependence of the internal energy of the entropy gradients is mandatory for the existence of this kind of waves. The differential system is non-linear but the waves propagate without distortion due to the fact that they are linearly degenerate (exceptional waves).

Introduction

Liquid-vapor and two-phase interfaces are generally represented by a material surface endowed with an energy related to Laplace's surface tension. In fluid mechanics and thermodynamics, the interface appears as a surface separating two media. This surface has its own characteristic behavior and energy properties [START_REF] Levitch | Physicochemical Hydrodynamics[END_REF]. Theoretical and experimental detailed studies show that, 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 in "Structure of liquids[END_REF]. Molecular models such as those used in kinetic theory of gas lead in interfacial layers to laws of state associated with non-convex internal energies, e.g., the van der Waals models [START_REF] Cahn | Free energy of a non-uniform system III[END_REF][START_REF] Rowlinson | Molecular theory of capillarity[END_REF][START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF]. These models appear advantageous as they provide as even more precise verification of Maxwell's rule applied to isothermal phase transition [START_REF] Gouin | Dynamics effects in gradient theory for fluid mixtures[END_REF]. Nonetheless, they present two disadvantages: First, for densities that lie between phase densities, the pressure may become negative. Simple physical experiments can be used, however, to cause traction that leads to these negative pressure values [START_REF] Rocard | Thermodynamique[END_REF][START_REF] Bruhat | Cours de Physique Générale[END_REF]. Second, in the field between bulks, internal energy cannot be represented by a convex surface associated with the variation of densities and entropy. This fact seems to contradict the existence of steady equilibrium state of the matter in this type of region. To overcome these disadvantages, the thermodynamic investigative replaces the non-convex portion corresponding to internal energy with a plane domain. The fluid can no longer be considered as a continuous medium. The interface is represented as a material surface with a null thickness. In this case, the only possible representation of the dynamic behavior of the interface is one of a discontinuous surface, and its essential structure remains unknown.

In the equilibrium state it is possible to eliminate the above disadvantages by appropriately modifying the stress tensor of the capillary layer, which is expressed in an anisotropic form. As a consequence, the energy of the continuous medium must change [START_REF] Cahn | Free energy of a non-uniform system III[END_REF][START_REF] Rowlinson | Molecular theory of capillarity[END_REF][START_REF] Bongiorno | Molecular theory of fluid interfaces[END_REF]. A representation of the energy near the critical point therefore allows the study of interfaces of non-molecular size. This approach is not new and, in fact, dates back to van der Walls [START_REF] Van Der Waals | Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité[END_REF] and Korteweg [START_REF] Korteweg | Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires[END_REF]; it corresponds to what is known as the Landau-Ginzburg theory [START_REF] Hohenberg | Theory of dynamic critical phenomena[END_REF]. The representation proposed in the present study is based on the notion of internal energy which is more convenient to use when the temperature is not uniform. One of the problems that complicates this study of phase transformation dynamics is the apparent contradiction between Korteweg's classical stress theory and the Clausius-Duhem inequality [START_REF] Gurtin | Thermodynamics and the possibility of spatial interaction in elastic materials[END_REF]. Proposal made by Eglit [START_REF] Eglit | A generalization of the model of an ideal compressible fluid[END_REF], Dunn and Serrin [START_REF] Dunn | On the thermomechanics of interstitial working[END_REF], Casal and Gouin [START_REF] Casal | Connection between the energy equation and the motion equation in Korteweg's theory of capillarity[END_REF] and others rectify this apparent anomaly for liquid-vapor interfaces of a pure fluid.

To study capillary layers and bulk phases, the simplest model in continuum mechanics considers an internal energy ε 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 | Free energy of a non-uniform system III[END_REF][START_REF] Van Der Waals | Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité[END_REF]. The second term is approximated by a gradient expansion, typically truncated to the second order. In the simplest version of the theory we have ε = ρ α(ρ, s) + 1 2 m (∇ρ) 2 , where ρ is the matter density, s the specific entropy, α the specific internal energy of the fluid assumed to be homogeneous and m is a coefficient independent of s, ∇ρ and of any higher derivatives. Obviously, the model is simpler than models associated with the renormalization-group theory [START_REF] Domb | The critical point[END_REF]. Nevertheless, it has the advantage of easily extending well-known results for equilibrium cases to the dynamics of interfaces [START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF][START_REF] Truskinovsky | Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium[END_REF]. In such a model, s varied with ρ through the interface in the same way as in the bulk, then s would always be that function of ρ which, at given temperature T satisfied

∂α ∂s (ρ, s) = T. (1)
With this assumption, s = s(ρ) and ε = ε(ρ), so that the points representing single-phase states in the ρ, s, ε space lie on a curve instead of on a surface ε = ε(ρ, s). This was the original assumption of van der Waals which was later justified by Ornstein in 1909 for a system composed of molecules with long-ranged but weak attractive force; this assumption is not exact for more realistic potentials1 . As coexistence curves these are no way peculiar; the only peculiarity is that the single-phase states -in this version of this approximation-have collapsed onto the coexistence curve instead being represented by the points of an extended two-dimensional region of which the coexistence curve is merely a boundary. There is then no proper two-density description of the one-phase states of a one-component system in the lowest order of the mean-field approximation.

There is such a description of the two-states, where not even in mean-field approximation is there any discernible peculiarity; but in practice the potential for the two-density form of the van der Waals theory is then not constructed by the prescription in (1) but by other means. For example Rowlinson and Widom introduce in [START_REF] Rowlinson | Molecular theory of capillarity[END_REF], chapter 9, an energy arising from the mean-field theory and depending on the deviations of the densities s and ρ, say, from their values at the critical point and the gradients of these densities. It is also seen that in c-components systems, c + 1 densities -the densities ρ 1 , ..., ρ c of the c components and the entropy density s-may vary independently through the interface.

Aside from the question of accuracy, there are also qualitative features of some interfaces in physical-chemistry, especially in systems of more than one component, that require two or more independently varying densities for their description. An example is strong positive or negative adsorption of a component i associated with a non-monotonic profile ρ i (z) where z is the spatial variable.

In the one-density theory based on the approximation ∂α ∂s -T = 0 and ∂α ∂ρ i = 0 for all j = i, the resulting one-density model leads inevitably to a monotonic ρ i (z). In a theory based on two or more densities, by contrast, we may have a realistic trajectory with which is associated non-monotonic behavior of one or more of the components if we suppose ρ j (z) to be monotonic. We must also allow the independent variation of at least two densities, entropy included, if we are to account contact angles in three-phase equilibrium : J. Cahn made the remark that we might use a two-(or more-) density van der Waals theory to describe the case of non-spreading in the three phase equilibrium, ( [START_REF] Rowlinson | Molecular theory of capillarity[END_REF], chapter 8). Then, at equilibrium, Rowlinson and Widom pointed out that for single fluids the model must be extended by taking into account not only the strong variations of matter density through the interfacial layer but also the strong variations of entropy. Also in dynamics, for an extended Cahn and Hilliard fluid, the volume internal energy ε is proposed with a gradient expansion depending not only on grad ρ but also on grad s corresponding to a strong heat supply in the change of phases: ε = f (ρ, s, ∇ρ, ∇s). The medium is then called a thermocapillary fluid [START_REF] Casal | Equations of motions of thermocapillary fluids[END_REF][START_REF] Gouin | Adiabatic waves along interfacial layers near the critical point[END_REF].

To extend the model to fluid mixtures corresponding to c-component systems and realistic potentials in molecular theory of fluid interfaces, the internal energy is assumed to be a functional of the different densities of the mixture. In all the cases where strong gradient of densities occurs -for example shocks or capillary layers -the internal energy is chosen as a function of successive derivatives of densities of matter and entropies. To be in accordance with the physical phenomena presented in [START_REF] Rowlinson | Molecular theory of capillarity[END_REF], one will consider the internal energy of a two-component fluid mixture with an internal energy depending also on the gradients of entropy of each component. The internal energy is a Galilean invariant, it does not depend on the reference frame; hence the internal energy depends also on the relative velocity between the two components of the mixture. The conservative motions of thermocapillary fluid mixture are relevant to the so-called second gradient theory [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF] and we obtain a complete set of balance equations for conservative motions; we extend this result to the dissipative case. Our goal is simply to verify the consistency of our model with the second law of thermodynamics. We consider a special case of dissipative thermocapillary mixtures where the introduction of dissipative forces is only done in the framework of the first gradient theory; we deduce the Fick law as a consequence of a friction behavior between the components and from the equations of motion of the components. In such a case, extended thermodynamic principle (as Gibbs identity) provides a set of equations that satisfy the entropy principle, thereby making these irreversible motions compatible with the second law of thermodynamics.

The idea of studying interface motions as localized traveling waves in a multigradient theory is not new and can be traced throughout many problems of condensed matter and phase-transition physics [START_REF] Gouin | Material waves of a fluid in the vicinity of the critical point in Symposium on waves in liquid/gas and liquid/vapor two-phase systems[END_REF]. In Cahn and Hilliard's model [START_REF] Cahn | Free energy of a non-uniform system III[END_REF], the direction of solitary waves was along the gradient of density [START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF][START_REF] Gouin | Material waves of a fluid in the vicinity of the critical point in Symposium on waves in liquid/gas and liquid/vapor two-phase systems[END_REF]. The introduction of the model of thermocapillary fluid mixture provides a better understanding of the behavior of motions in fluid mixture interfaces: it is possible to obtain the previous solitary waves but also a new kind of adiabatic waves may be forecasted. These waves are associated with the spatial second derivatives of entropy and matter densities. For this new kind of adiabatic waves, the direction of propagation is normal to the gradient of densities. In the case of a thick interface, the waves are tangential to the interface and the wave velocities depend on the constitutive equations. Finally we observe that, also if the differential system associated with the wave motions is non-linear, the waves propagate without distortion due to the fact that they are linearly degenerate (exceptional waves) [START_REF] Boillat | Non linear hyperbolic fields and waves in Recent mathematical methods in nonlinear wave propagation[END_REF].

2 Equations of thermocapillary mixtures

Conservative motions of thermocapillary mixtures

To derive the governing equations and boundary conditions in the dissipativefree case, we use the Hamilton principle of least action [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF]. In continuum mechanics, the principle with a system endowed with an infinite number of degrees of freedom was initiated by Lin [START_REF] Lin | A new variational principle for isoenergetic flows[END_REF], Herivel [START_REF] Herivel | The derivation of the equations of motion of an ideal fluid by Hamilton's principle[END_REF], Serrin [START_REF] Serrin | Mathematical principles of classical fluid mechanics in Encyclopedia of Physics VIII[END_REF], Berdichevsky [START_REF] Berdichevsky | Construction of models of continuous media by means of the variational principle[END_REF] and many others; it was proposed by Gouin for fluid mixtures [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]. The main idea is to propose a Lagrangian which yields the behavior of the medium as the difference between a kinetic and a potential energy per unit volume. Then, the variations of the Hamilton action obtained as a linear functional of virtual displacements allow to find the governing equations and boundary conditions. For real media, the irreversibility is introduced both in equations of motion and equation of energy by using a classical approach through the dissipative function, diffusion and heat fluxes.

We study a mixture of two fluids: the motion of a two-fluid continuum can be represented using two surjective mappings (i=1,2)

(t, x) → X i = Λ i (t, x)
where (t, x) belongs to [t 1 , t 2 ] × D t , a set in the time-space occupied by the fluid between times t 1 and t 2 . Variables X i denote the positions of each component of the mixture in reference spaces D 0i . Variations of particle motions are deduced from virtual motions

X i = Ξ i (t, x,κ i ),
where scalars κ i are defined in a neighborhood of zero; they are associated with a two parameter family of virtual motions of the mixture. The real motion corresponds to κ i = 0, the associated virtual displacements are [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]

δ i X i = ∂Ξ i ∂κ i | κ i =0 .
They generalize what is obtained for a single fluid [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF]. To the virtual displacements δ i X i , we associated its image ζ i in the physical space D t occupied by the fluid mixture at time t [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF][START_REF] Gouin | Noether theorem in fluid mechanics[END_REF],

ζ i = - ∂x ∂X i δ i X i .
Conservation of matter for each component requires that

ρ i det ∂x ∂X i = ρ io (X i ) , (2)
where ρ io is the reference volume mass in D 0i and det ∂x ∂X i the Jacobian determinant of the motion of component i. In differentiable cases eqs (2) are equivalent to the equations of balance of matter densities ρ i

∂ρ i ∂t + div(ρ i u i ) = 0, (3)
where u i denotes the velocity vectors of each component i. Now, we assume that the mixture has an entropy for each component [START_REF] Müller | Rational Extended Thermodynamics[END_REF]; for conservative motions, the equations of conservation of specific entropies s i are

∂ρ i s i ∂t + div(ρ i s i u i ) = 0. (4)
Then, relations s i = s io (X i ) define an isentropic motion of the fluid mixture. We deduce the following relations of tensorial quantities [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF][START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF]

δ i u i = d i ζ i dt - ∂u i ∂x ζ i , δ i ρ i = -div (ρ i ζ i ), δ i s i = - ∂s i ∂x ζ i . (5)
where

d i dt = ∂ ∂t + u i .
∇ denotes the material derivative relatively to the component i. We assume that the volume potential energy of the mixture is in the form ε = ǫ(ρ i , s i , ∇ρ i , ∇s i , w), with w = u 1u 2 the relative velocity of the two components of the mixture. This means that the fluid mixture is a function not only of the densities of matter ρ i and specific entropies s i but also of the gradients of ρ i and s i . The fact that ε depends on two entropies is classically adopted in the literature [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF][START_REF] Bedford | Recent advances. Theories of immiscible and structured mixtures[END_REF][START_REF] Müller | Thermodynamics, Interaction of Mechanics and Mathematics Series[END_REF]. Moreover, for a two-velocity medium, there is no coordinate system within the framework of which any motion could be disregarded. So, the standard definition of potential energy leads to its dependence on the relative motion of the components. The dependence of ε with respect to the relative velocity is analog to take into account the added mass effect in heterogeneous two-fluid theory as done by berdichevsky [START_REF] Berdichevsky | Construction of models of continuous media by means of the variational principle[END_REF] and Geurst [START_REF] Geurst | Virtual mass in two-phase bubbly flow[END_REF][START_REF] Geurst | Variational principles and two-fluid hydrodynamics of bubbly liquid/gas mixtures[END_REF]. Let us note we can assume also that ε is depending on (t, x); by this way, we introduce directly the extraneous potential of the body forces.

The potential ε is related with the volume internal energy ̟ of the mixture through the transformation

̟ = ε - ∂ε ∂w w, so that, e = 2 i=1 1 2 ρ i u 2 i + ̟
is the total energy of the system [START_REF] Gavrilyuk | Hyperbolic models of homogeneous two-fluid mixtures[END_REF][START_REF] Gavrilyuk | A variational principle for twofluid models[END_REF].

The equation of motion of component i is given by a variational method associated with a Hamilton action; the vector field x ∈ D t → ζ i is two time continuously differentiable. The Lagrangian of the mixture is

L = 2 i=1 1 2 ρ i u 2 i -ε,
and consequently, the Hamilton action between the times t 1 and t 2 is

I = t 2 t 1 Dt L dxdt.
From the definition of virtual motions, we obtain immediately two variations of the action of Hamilton associated with i = 1, 2,

δ i I = t 2 t 1 Dt 1 2 u 2 i -ǫ ,ρ i δ i ρ i -ǫ ,ρ i,γ δ i ρ i,γ + ρ i K iγ δ i u iγ -ǫ, s i δ i s i -ǫ ,s i,γ δ i s i,γ dxdt ,
where subscript γ corresponds to spatial derivatives associated with gradient terms; as usually summation is made on repeated subscript γ from 1 to 3;

K i = u i + (-1) i 1 ρ i ∂ǫ ∂w
T where index T denotes the transposition. Then, by integration by part we obtain,

δ i I = t 2 t 1 Dt 1 2 u 2 i -ǫ ,ρ i + (ǫ ,ρ i,γ ) ,γ δ i ρ i + ρ i K iγ δ i u iγ -ǫ, s i -(ǫ ,s i,γ ) ,γ δ i s i -(ǫ ,ρ i,γ δ i ρ i ) ,γ -(ǫ ,s i,γ δ i s i ) ,γ dxdt.

Let us denote by

ρ i θ i ≡ ∂ǫ ∂s i and h i ≡ ∂ǫ ∂ρ i , ( 6
)
where ∂ is the variational derivative operator. That is to say,

ρ i θ i = ǫ, s i -(ǫ ,s i,γ ) ,γ ≡ ǫ ,s i -div Ψ i and h i = ǫ ,ρ i -(ǫ ,ρ i,γ ) ,γ ≡ ǫ ,ρ i -div Φ i ,
with,

Ψ i ≡ ∂ǫ ∂∇s i and Φ i ≡ ∂ǫ ∂∇ρ i . Introducing R i = 1 2 u 2 i -h i ,
and taking into account of the expressions for θ i and h i given by eqs (6), we get

δ i I = t 2 t 1 Dt {R i δ i ρ i + ρ i K i • δ i u i -ρ i θ i δ i s i -div (Φ i δ i ρ i + Ψ i δ i s i )} dxdt,
and from relations (5), we obtain

δ i I = t 2 t 1 Dt -R i div (ρ i ζ i ) + ρ i K i • d i ζ i dt - ∂u i ∂x ζ i + ρ i θ i ∂s i ∂x ζ i -div (Φ i δ i ρ i + Ψ i δ i s i ) dxdt.
Consequently,

δ i I = t 2 t 1 Dt ρ i ∂R i ∂x + θ i ∂s i ∂x - d i K T i dt -K T i ∂u i ∂x ζ i + ∂ ∂t (ρ i K T i ζ i ) -div ρ i R i ζ i -ρ i u i K T i ζ i + Φ i δ i ρ i + Ψ i δ i s i dxdt.
The Stokes formula and relation (5) yield

δ i I = t 2 t 1 Dt ρ i ∂R i ∂x + θ i ∂s i ∂x - d i K T i dt -K T i ∂u i ∂x ζ i dxdt (7) + t 2 t 1 ∂Dt g ρ i K T i ζ i -n. ρ i R i ζ i -ρ i u i K T i ζ i -Φ i ∂ρ i ∂x ζ i -Ψ i ∂s i ∂x ζ i -ρ i Φ i div ζ i dσ x dt,
where ∂D t (of mesure dσ x ) is the boundary of D t , n is the unit external normal vector to ∂D t and g is the velocity of ∂D t . If we consider a vector field x ∈ D t → ζ i and its first derivatives vanishing simultaneously on the boundary ∂D t , the Hamilton principle expressed in the form:

∀ ζ i , δ i a = 0 leads to ∀ ζ i , t 2 t 1 Dt ρ i ∂R i ∂x + θ i ∂s i ∂x - d i K T i dt -K T i ∂u i ∂x ζ i dxdt = 0
and consequently,

d i K i dt + ∂u i ∂x T K i = ∇R i + θ i ∇s i . (8)
Let us note that the value of the first member of eq. ( 8) is equal to

d i u i dt + (-1) i d i dt 1 ρ i ∂ǫ ∂w T + ∂u i ∂x T u i + (-1) i ρ i ∂u i ∂x T ∂ǫ ∂w
T and eq. ( 8) yields

ρ i d i u i dt + (-1) i   div u i ∂ǫ ∂w T + d i dt ∂ǫ ∂w T + ∂u i ∂x T ∂ǫ ∂w T   = ρ i θ i ∇ s i -ρ i ∇ h i .
Taking into account of eq. ( 3) of conservation of mass of component i, we deduce

∂ρ i u i ∂t + div(ρ i u i ⊗ u i ) + (-1) i   div u i ∂ǫ ∂w T + ∂ ∂t ∂ǫ ∂w T + ∂ ∂x ∂ǫ ∂w T u i + ∂u i ∂x T ∂ǫ ∂w T   = ρ i θ i ∇ s i -ρ i ∇ h i ,
and due to the fact that div u i ∂ǫ ∂w

T + ∂ ∂x ∂ǫ ∂w T u i = div   ∂ǫ ∂w T ⊗ u i   ,
we get the equations of motion of the two components in the form

∂ρ i u i ∂t + div(ρ i u i ⊗ u i ) + (-1) i   ∂ ∂t ∂ǫ ∂w T + ∂u i ∂x T ∂ǫ ∂w T + div   ∂ǫ ∂w T ⊗ u i     = ρ i θ i ∇ s i -ρ i ∇ h i . (9)
We consider only the isotropic case where the potential energy ε of the mixture can be written in terms of the isotropic invariants

β ij = ∇ρ i •∇ρ j , χ ij = ∇ρ i •∇s j , γ ij = ∇s i •∇s j , (i, j = 1, 2) and ω = 1 2 w 2 . ε = ε(ρ i , s i , β ij , χ ij , γ ij , ω),
Then, the equation of motion of each component of the mixture is

∂ρ i u i ∂t + div(ρ i u i ⊗ u i ) + (10) (-1) i   ∂ ∂t (aw) + a ∂u i ∂x T w + div(a w ⊗ u i )   = ρ i θ i ∇ s i -ρ i ∇ h i ,
where a = ∂ε ∂ω .

In this case, Ψ i and Φ i can be written

Ψ i = 2 j=1 D ij ∇ρ j + E ij ∇s j , Φ i = 2 j=1 C ij ∇ρ j + D ij ∇s j , with C ij = (1 + δ ij ) ǫ, β ij , D ij = ǫ, χ ij , E ij = (1 + δ ij ) ǫ, γ ij , where δ ij is the Kronecker symbol.
The simplest model is when

C ij = C ji , D ij , E ij = E ji are constant. Then, ε = e(ρ i , s i , w) + 2 i,j=1 1 2 C ij ∇ρ i • ∇ρ j + D ij ∇ρ i • ∇s j + 1 2 E ij ∇s i • ∇s j (11)
where the associated quadratic form with respect to the vectors ∇ρ i and ∇s i is in the form 2 i,j=1

1 2 C ij ∇ρ i • ∇ρ j + D ij ∇ρ i • ∇s j + 1 2 E ij ∇s i • ∇s j .
This quadratic form is assumed positive such as the effect of gradient terms increases the value of the internal energy with respect to a mixture in a homogeneous configuration.

Equation of total momentum and equation of energy for conservative motions of thermocapillary mixtures

We limit first to the conservative case. We notice that the equation of motion of each component is not in divergence form. Nevertheless, by summing eqs [START_REF] Van Der Waals | Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité[END_REF] with respect to i, we obtain the balance equation for the total momentum in a divergence form. In fact, eqs [START_REF] Van Der Waals | Thermodynamique de la capillarité dans l'hypothèse d'une variation continue de densité[END_REF] imply

2 i=1 ∂ρ i u i ∂t + div(ρ i u i ⊗ u i ) -div(a w ⊗ w) = a ∂w ∂x T w + 2 i=1 ρ i θ i ∇ s i -ρ i ∇ h i (12)
In coordinates, the second member of eq. ( 12) is

a w ν w γ,ν + 2 i=1 ǫ ,s i s i,γ -ǫ ,s i,ν ,ν s i,γ -ρ i ǫ ,ρ i ,γ + ρ i ǫ ,ρ i,νν ,γ ,
where ν is summed from 1 to 3. Noting that

ǫ ,γ = 2 i=1 ǫ ,s i s i,γ + ǫ ,s i,ν s i,νγ + ǫ ,ρ i ρ i,γ + ǫ ,ρ i,ν ρ i,νγ + a w ν w γ,ν , (13)
we obtain

a w ν w γ,ν + 2 i=1 ǫ ,s i s i,γ -ǫ ,s i,ν ,ν s i,γ -ρ i ǫ ,ρ i ,γ + ρ i ǫ ,ρ i,νν ,γ = ǫ ,γ - 2 i=1 ǫ ,s i,ν s i,νγ + ǫ ,ρ i ρ i,γ + ǫ ,ρ i,ν ρ i,νγ + ǫ ,s i,ν ,ν s i,γ + ρ i ǫ ,ρ i ,γ -ρ i ǫ ,ρ i,νν ,γ = ǫ, γ + 2 i=1 -ρ i ǫ ,ρ i + ρ i ǫ ,ρ i,ν ,ν ,γ -Φ iν ρ i,γ + Ψ iν s i,γ ,ν ,
and consequently the equation of motion for the total momentum is

∂ρu ∂t + div 2 i=1 (ρ u i ⊗ u i ) -ρa w ⊗ w -σ = 0 ( 14
)
where ρ = ρ 1 + ρ 2 is the total volume mass, ρu = ρ 1 u 1 + ρ 2 u 2 is the total momentum and σ = σ 1 + σ 2 is the total stress tensor such that

σ iνγ = (-P i + ρ i div Φ i ) δ νγ -Φ iν ρ i,γ -Ψ iν s i,γ , with P i = ρ i ǫ ,ρ i - ρ i ǫ ρ .
Let us notice that if ǫ depends also on (t, x) corresponding to an external force potential, an additive term appears as body force in relation [START_REF] Eglit | A generalization of the model of an ideal compressible fluid[END_REF] and in eq. ( 14) the body force appears in the second member. This is not the case in eqs (9) which include the body forces coming from ǫ (depending on (t, x)) in terms h i . For the sake of simplicity, we do not introduce the body force in equation of the total momentum and equation of total energy.

The equation of energy of the total mixture is obtained in the divergence form.

Let us define

M i = ∂ρ i u i ∂t + div(ρ i u i ⊗ u i ) + (-1) i   ∂ ∂t (a w) + a ∂u i ∂x T w + div (aw ⊗ u i )   -ρ i θ i ∇ s i + ρ i ∇ h i , G i = ∂ρ i ∂t + div(ρ i u i ), S = 2 i=1 ∂ρ i s i ∂t + div(ρ i s i u i ) θ i , E = ∂ ∂t 2 i=1 1 2 ρ i u 2 i + ε -a w 2 + div 2 i=1 ρ i K i • u i - 1 2 u 2 i -σ i u i + ε u -U , where U = 2 i=1 d i ρ i dt Φ i + d i s i dt Ψ i corresponds to the interstitial working:
in the same way as for Cahn and Hilliard fluids, an additional term that has the physical dimension of a heat flux must be added to the equation of energy [START_REF] Eglit | A generalization of the model of an ideal compressible fluid[END_REF][START_REF] Dunn | On the thermomechanics of interstitial working[END_REF][START_REF] Casal | Connection between the energy equation and the motion equation in Korteweg's theory of capillarity[END_REF][START_REF] Berdichevsky | Construction of models of continuous media by means of the variational principle[END_REF].

Theorem: For all motions of a thermocapillary fluid mixture, the relation

E -S + 2 i=1 1 2 u 2 i -h i + θ i s i G i -M i • u i ≡ 0 (15)
is identicaly satisfied.

The proof comes from the following algebraic calculation:

M T i u i - 1 2 u 2 i -h i G i + ρ i θ i d i s i dt ≡ ∂ ∂t ( 1 2 ρ i u 2 i ) + div(( 1 2 ρ i u 2 i )u i ) + h i G i +ρ i θ i ∂s i ∂t + ρ i ∂h i ∂x u i + (-1) i   ∂a w ∂t T + a w T ∂u i ∂x + div a u i w T   u i Let us note that ∂ε ∂t + div 2 i=1 ρ i h i u i ≡ a w T ∂w ∂t + 2 i=1 ε ,ρ i ∂ρ i ∂t + div ∂ρ i ∂t ∂ε ∂∇ρ i - ∂ρ i ∂t div ∂ε ∂∇ρ i + ε ,s i ∂s i ∂t + ∂ε ∂∇s i ∇ ∂s i ∂t + h i div (ρ i u i ) + ρ i ∂h i ∂x u i , yields 2 i=1 h i G i ≡ ∂ε ∂t -a w T ∂w ∂t + 2 i=1 div (ρ i h i u i ) -div ∂ρ i ∂t ∂ε ∂∇ρ i -ε ,s i ∂s i ∂t - ∂ε ∂∇s i ∇ ∂s i ∂t -ρ i ∂h i ∂x u i .
Consequently,

2 i=1 M T i u i - 1 2 u 2 i -h i + θ i s i G i + S ≡ ∂ ∂t 2 i=1 1 2 ρ i u 2 i + ε -a w T ∂w ∂t + 2 i=1 div ρ i 1 2 u 2 i + h i u i -div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i + (-1) i   ∂a w ∂t T u i + a w T ∂u i ∂x u i + div a u i w T u i   ≡ ∂ ∂t 2 i=1 1 2 ρ i u 2 i + ε -a w 2 + 2 i=1 div ρ i K T i u i - 1 2 u 2 i + h i u i -div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i
Taking into account of the relations

∂ρ i ∂t Φ i ≡ d i ρ i dt Φ i -Φ i ∂ρ i ∂x u i and ∂s i ∂t Ψ i ≡ d i s i dt Ψ i -Ψ i ∂s i ∂x u i
and the definition of the total stress tensor,

σ ≡ ε - 2 i=1 ρ i ε ,ρ i -ρ i divΦ i Id -Φ i ∂ρ i ∂x -Ψ i ∂s i ∂x ,
where Id is the identity tensor, we deduce immediately the algebraic identity [START_REF] Casal | Connection between the energy equation and the motion equation in Korteweg's theory of capillarity[END_REF].

Corollary: All conservative motions of a thermocapillary mixture satisfy the equation of energy balance

∂ ∂t 2 i=1 1 2 ρ i u 2 i + ε -a w 2 + div 2 i=1 ρ i K i • u i - 1 2 u 2 i -σ i u i + ε u -U = 0. ( 16
)
This result comes from the simultaneity of relations G i = 0, S i = 0 and M i = 0. Let us note that ε -a w 2 corresponds to the volume internal energy ̟ of the mixture.

Dissipative motions of thermocapillary mixtures

The conservative fluid mixture model presented in section 2.1 is relevant to the so-called second gradient theory [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF], [START_REF] Truesdell | Introduction à la mécanique rationnelle des milieux continus[END_REF]. In our form of equations of mass conservation for each component (2), equation of the total momentum of the mixture ( 14), equation of energy ( 16), the diffusion term J = ρ 1 (u 1u) does not directly appear but is deduced respectively from the velocities and densities of the components.

Our aim is to verify the consistency of our model with the second law of thermodynamics. The introduction of dissipative forces is simply done in the framework of the first gradient theory [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF]: the dissipative forces applied to the continuous medium are divided into volume forces f d i and surface forces associated with the Cauchy stress tensor σ d i . Then, the virtual work of dissipative forces δT i applied to the component i is in the form

δT i = f d i • ζ i -tr σ d i ∂ζ i ∂x ,
where δT i is a differential form. For such dissipative motions, no production of masses due to chemical reactions appears.

For the same virtual displacement of two components, ζ = ζ i , (i = 1, 2) , the total virtual work of dissipative forces is

δT = 2 i=1 f d i • ζ -tr σ d i ∂ζ ∂x .
When ζ is a translation, the work δT is equal to zero and consequently,

2 i=1 f d i = 0 or f d 2 = -f d 1 ≡ f d (17)
We specify later the behavior of forces f d i (they will be associated with the diffusion term) and stress tensors σ d i . Taking into account of the dissipative effects, the equations of motion for each component become

∂ρ i u i ∂t + div(ρ i u i ⊗ u i ) + (-1) i   ∂ ∂t (aw) + a ∂u i ∂x T w + div(a w ⊗ u i )   = ρ i θ i ∇ s i -ρ i ∇ h i + div σ d i + f d i (18)
Taking into account of relation [START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF], the equation of the total momentum writes

∂ρu ∂t + div 2 i=1 (ρ u i ⊗ u i ) -ρa w ⊗ w -σ -σ d = 0 ( 19
)
with

σ d = σ d 1 + σ d 2 .
The introduction of the heat flux vector q and the heat supply r comes from classical methods in thermodynamics [START_REF] Müller | Rational Extended Thermodynamics[END_REF][START_REF] Müller | Thermodynamics, Interaction of Mechanics and Mathematics Series[END_REF][START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF]]. If we write,

M d i = M i -div σ d i -f d i , S d = S -r + div q + 2 α=1 f d i .u i -tr (σ d i ∆ i ), E d = E -r + div q - 2 α=1 div σ d i u i , with ∆ i = 1 2   ∂u i ∂x + ∂u i ∂x T 
 represents the velocity deformation tensor of each component, relation [START_REF] Casal | Connection between the energy equation and the motion equation in Korteweg's theory of capillarity[END_REF] writes as

E d -S d - 2 i=1 M d i • u i -(K i • u i -R i -θ i s i ) G i ≡ 0 (20)
which can be considered as the dynamic form of the Gibbs identity.

For the components of the mixture, equations of momenta and equations of masses are in the form

M d i = 0, G i = 0 . (21)
The Gibbs identity [START_REF] Gouin | Adiabatic waves along interfacial layers near the critical point[END_REF] and eqs [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF] imply

S d = E d . If we assume that S d = 0, i.e., 2
i=1 ∂ρ i s i ∂t + div(ρ i s i u i ) θ i + f d i .u i -tr (σ d i ∆ i ) -r + div q = 0, ( 22
)
it is equivalent to write E d = 0, i.e., ∂ ∂t

2 i=1 1 2 ρ i u 2 i + ε -a w 2 + (23) div 2 i=1 ρ i K i • u i - 1 2 u 2 i -σ i -σ d i u i + ε u -U + q -r = 0.
Eq. ( 22) is the equation of entropy and Eq. ( 23) is the equation of energy.

In the conservative case, the system is closed with two different temperatures θ i (i = 1, 2). In the dissipative case we need additional arguments to obtain equations for each entropy s i which could remplace equations (4). A possibility is to consider the case when the exchanges of momentum and energy between the two components are rapid enough to have a common temperature (this is not the case of heterogeneous mixtures where each phase may have different pressures and temperatures [START_REF] Lhuillier | From Molecular mixtures to suspensions of particles[END_REF]). Then, in dissipative case, if we know all the dissipative functions, the governing system is closed. Note also that it could be possible to consider a common temperature and entropy both for conservative and dissipative case. This case is connected with a conservative equation for the common entropy s (see Appendix). Another possibility is to assume that the entropy is transported along the ith component (say, for example i = 1) which was used for quantum fluids by Landau [START_REF] Landau | Fluid Mechanics[END_REF] ∂ρ

1 s ∂t + div (ρ 1 s u 1 ) = 0.
In this case, the independent functions are ρ 1 , s, u 1 , ρ 2 , u 2 , where ρ i are submitted to the constraints (3) and the case of Helium superfluid is a special case of our study corresponding to s 2 = 0 and s = s 1 . Using of this hypothesis is nevertheless doubtful for classical fluids.

Hence, we may suppose a common temperature only for dissipative case. This means that θ = θ 1 = θ 2 (24) This hypothesis closes the system [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continus[END_REF][START_REF] Gouin | Material waves of a fluid in the vicinity of the critical point in Symposium on waves in liquid/gas and liquid/vapor two-phase systems[END_REF]. Now we focus on the governing equations for each components of the mixture:

M d i = 0 (i = 1, 2).
For slow motions, we rewrite these equations in the following form :

M d i ≃ ρ i ∇ h i -ρ i θ ∇ s i -div σ d i -f d i = 0.
If we consider the case when the motion of each component is regular enough, an approximative case is the case of solid displacements for the motion of each component; then, div σ d i ≃ 0 and consequently,

M d i ≃ ρ i ∇ µ i -f d i = 0,
where µ i = h i -θs i is the chemical potential of the component i of the mixture at the temperature θ.

Considering the difference M d 2 -M d 1 and using relation [START_REF] Slemrod | Admissibility criteria for propagating phase boundaries in a van der Waals fluid[END_REF], we obtain,

f d 2 ρ 2 - f d 1 ρ 1 ≡ ρ f d ρ 1 ρ 2 = ∇ µ, (25)
where µ = µ 2 -µ 1 is the chemical potential of the mixture [START_REF] Bowen | Theory of mixtures in Continuum physics III[END_REF][START_REF] Nigmatulin | Fundamentals of mechanics of heterogeneous mixtures[END_REF]. Let us introduce the diffusion flux J,

J ≡ ρ 1 (u 1 -u) = ρ 1 ρ 2 ρ w. (26)
Equation ( 25) implies,

f d • w = ∇ µ • J. ( 27
)
The term ∇ µ • J corresponds to the entropy production due to the diffusion process. Expression [START_REF] Serrin | Mathematical principles of classical fluid mechanics in Encyclopedia of Physics VIII[END_REF] is the connection between the mechanical drag force between components of the mixture and the thermodynamical process of diffusion; consequently, Eq. ( 22) yields

2 i=1 ∂ρ i s i ∂t + div(ρ i s i u i ) θ i -tr (σ d i ∆ i ) -r + div q -∇ µ • J = 0.
We have obtained the equations of balance of masses, equation of total momentum, Eq. ( 25) between the components and equation of total energy by using an energetic method. They are the extension of classical mixture equations to equations of mixtures of fluids involving density gradients.

Second law of thermodynamics

Relation ( 22) may be rewritten in the form

2 i=1 ∂ρ i s i ∂t + div(ρ i s i u i ) +div q θ - r θ = 1 θ 2 i=1 tr (σ d i ∆ i ) +f d •w-∇θ • q θ 2
This last equation yields the entropy production due to the diffusion, viscosity and heat flux processes. Due to θ > 0, if we assume

2 i=1 tr (σ d i ∆ i ) + f d • w -∇θ • q θ ≥ 0, (28)
we get the Clausius-Duhem inequality in the form

2 i=1 ∂ρ i s i ∂t + div(ρ i s i u i ) + div q θ - r θ ≥ 0.
Let us notice that ( 28) is satisfied if tr (σ d i ∆ i ) ≥ 0, f d •w ≥ 0 and ∇θ • q ≤ 0. The following constitutive laws are classically adopted for σ d i , f d and q :

The stress tensor σ d i is a symmetric isotropic tensor function of ∆ i such that 2 i=1 tr (σ d i ∆ i ) 0 (a special case of such a function is σ d i = λ i (tr ∆ i )Id + 2µ i ∆ i with µ i 0 and 3λ i + 2µ i 0).

The heat flux vector satisfies the Fourier law q = -χ ∇θ with χ 0.

The linear approximation known in the literature as the Stokes drag formula is adopted [START_REF] Landau | Fluid Mechanics[END_REF][START_REF] Bowen | Theory of mixtures in Continuum physics III[END_REF]

f d = k w, k 0, with f d 2 = -f d 1 ≡ f d and w = u 1 -u 2 .
Let us note that relations ( 25) and ( 26) together with the drag formula yield the property of the diffusion flux

J = 1 k ∇ µ, (29)
which is the general form of the Fick law [START_REF] De Groot | Non-Equilibrium Thermodynamics[END_REF][START_REF] Landau | Fluid Mechanics[END_REF]. So, the Fick law is not directly a linear phenomenological law as the Fourier law but a direct consequence of equations of motion and the Stokes drag force hypothesis which was previously noticed by Bowen for a different model [START_REF] Bowen | Theory of mixtures in Continuum physics III[END_REF].

Acceleration Waves

Now, we will consider conservative motions of thermocapillary mixtures only.

As it is well known, wave phenomena -in particular discontinuity waves (waves across the front of which some derivatives of the field variables have jumps) -are typical of models that are described through hyperbolic differential systems. A classical example of discontinuity waves in continuum mechanics are the so-called acceleration waves in which, among the other variables, the acceleration jumps across the front while the velocity is continuous [START_REF] Whitham | Linear and nonlinear waves[END_REF].

Dissipative systems and in particular models for diffusive processes have usual differential system with a parabolic structure and discontinuity waves are not admissible. A typical example of non admissibility is the one of Navier-Stokes-Fourier fluids. In this case a possible approach to obtain hyperbolic system is the method of the Extended Thermodynamic theory [START_REF] Müller | Rational Extended Thermodynamics[END_REF], valid also for rarefied gases.

Nevertheless in parabolic systems some discontinuity waves may propagate for particular initial data. The aim of this paper is to prove that for the present model of thermocapillarity fluid binary mixtures there exists the possibility of propagation of two tangential acceleration waves, provided that the internal energy is at least a function of the entropy gradient of one component. For this aim, we first briefly recall some very well known questions about discontinuity waves.

A wave is a discontinuity wave if the wave front with Cartesian equation φ(t, x) = 0 separates the space in two subspaces in which there exists regular solutions of the differential system but across the normal direction of the front some derivatives of the field suffers a jump [START_REF] Boillat | Non linear hyperbolic fields and waves in Recent mathematical methods in nonlinear wave propagation[END_REF].

As usual, we indicate the jump with a square bracket,

[ ] = ( ) φ=0 --( ) φ=0 +
and we introduce the map between (t, x) and (φ, ξ) , where ξ ≡ ξ (t, x) represents the tangential manifold of the wave surface in time-space. Therefore the assumptions for the discontinuity waves are expressed for a generic function f in the form,

                                                 [f ] = 0; ∂ k f ∂ξ γ 1 • • • ∂ξ γ k = 0 ∀k;
where there exists p ≧ 1 such that

∂ j f ∂φ j = 0 for 1 ≤ j ≤ p -1; δ k f ≡ ∂ k f ∂φ k = 0 for k ≥ p. (30)
Taking into account [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF] and the Hadamard lemma [START_REF] Hadamard | Leçons sur la propagation des ondes et les équations de l'hydrodynamique[END_REF], we have

∂ p f ∂x γ 1 • • • ∂x γ p = δ p f n i 1 • • • n ip ; ∂ p f ∂t p = (-λ) p δ p f
where λ and here n ≡ (n i ) are respectively the normal velocity and the unit normal vector to the wave front.

The advantages of the previous symbols are that there exists a chain rule between the field derivative in the differential systems and the corresponding jump relation,

∂ t → -λ δ; ∂ γ → n γ δ . (31)
We apply now this procedure to our differential systems assuming that across the wave front ρ i , s i and its first derivatives are continuous and there are jumps for the second derivative (p = 2) while the velocity is continuous and suffer a jump in the first derivative (p = 1)

[ρ i ] = [s i ] = [δρ i ] = [δs i ] = 0; [δ 2 ρ i ] = 0; [δ 2 s i ] = 0; (32) [u i ] = 0; [δu i ] = 0; i = 1, 2.

Jump conditions:

From the balance of mass of the two components (3), taking into account of eqs [START_REF] Müller | Rational Extended Thermodynamics[END_REF] and the chain rule [START_REF] Gouin | Noether theorem in fluid mechanics[END_REF], we obtain that the normal components of the first derivative of the velocities are continuous,

[δu in ] = 0, (33)
where

u in = n • u i .
If we differentiate the entropy balance law of each component (4), with respect x, we obtain for the discontinuities

-v i δ 2 s i + ∇s i • δu i = 0, (34)
where v i = λ -u in are the relative velocities of the wave front with respect to the fluid components. From the balance of momentum we obtain

-ρ i v i δu i + (-1) i a {v i δ(u 2 -u 1 ) + (δu i • (u 1 -u 2 )) n} -ρ i {∇s i [θ i ] -[∇h i ]} = 0. ( 35
)
From eq. ( 6) we obtain

B i ≡ -ρ i [θ i ] = 2 j=1 D ji δ 2 ρ j + E ji δ 2 s j A i ≡ -[∇h i ] = n 2 j=1 C ij δ 3 ρ j + D ij δ 3 s j . (36)
Then relation [START_REF] Geurst | Virtual mass in two-phase bubbly flow[END_REF] becomes

-ρ i v i δu i + (-1) i a {v i δ(u 2 -u 1 ) + (δu i • (u 1 -u 2 )) n} + B i ∇s i -ρ i A i = 0.
If we multiply by n and we take into account of relation [START_REF] Bedford | Recent advances. Theories of immiscible and structured mixtures[END_REF], we obtain

ρ i A i • n = (-1) i a δu i • (u 1 -u 2 ) + B i ∇s i • n, (37)
then, we get the final jump conditions from the momentum equations

ρ i v i δu i + (-1) i av i δ(u 1 -u 2 ) -B i ∇ t s i = 0, (38)
where

∇ t s i = ∇s i -(∇s i • n)n (39)
denotes the tangential component of the gradient of entropy of each component.

Therefore we obtain the algebraic system of 8 equations ( 33), ( 34) and [START_REF] Truesdell | Introduction à la mécanique rationnelle des milieux continus[END_REF] for the 10 scalar unknowns δ 2 ρ i , δ 2 s i and δu i (i = 1, 2). Consequently, we needs two more conditions that are obtained by compatibility conditions coming from boundary conditions associated with the equation of motion for each component. In fact, we notice in Appendix that if ρ i , s i , ∇ρ i , ∇s i are continuous through a surface of weak discontinuities, then div Φ i must be also continuous through the surface

[div Φ i ] = 0.
We notice additively that these two conditions are compatible with the Rankine-Hugoniot conditions associated to the total momentum [START_REF] Dunn | On the thermomechanics of interstitial working[END_REF] and the total energy balance law ( 16): In fact, eq. ( 14) yields

[σ 1 + σ 2 ] = 0, while from eq. ( 16), we get

[σ 1 u 1n + σ 2 u 2n ] = 0.
Then, we obtain the two supplementary equations

[div Φ i ] = 2 j=1 C ij δ 2 ρ j + D ij δ 2 s j = 0 ( 40
)
and the system for the discontinuities becomes an homogeneous closed system of 10 equations for 10 scalar unknowns δ 2 ρ i , δ 2 s i and δu i (i = 1, 2) in the form,

                                         [δu in ] = 0 ρ i v i δu i + (-1) i av i δ(u 1 -u 2 ) -2 j=1 D ji δ 2 ρ j + E ji δ 2 s j ∇ t s i = 0 -v i δ 2 s i + ∇s i • δu i = 0 2 j=1 C ij δ 2 ρ j + D ij δ 2 s j = 0 (41)
We observe that the conditions ( 36), [START_REF] Gavrilyuk | A variational principle for twofluid models[END_REF] are constraints for the jump of the third derivatives of densisties and entropies δ 3 ρ j , δ 3 s j .

Now we consider the weak discontinuities near the equilibrium of the fluid mixture; then

u i = 0.
Consequently v = λ is the velocity of the acceleration wave. Due to the fact that thermocapillary mixtures can be considered as a mathematical model for interfacial layers between two mixture bulks [START_REF] Gouin | Variational theory of mixtures in continuum mechanics[END_REF], the gradients of tensorial quantities ρ i and s i are orthogonal to the interfacial layers and consequently are collinear,

∇s 1 = b ∇s 2 .
Then the second and the third equations of system (41) allow to eliminate u i and to get

c 2 s ((a -ρ 1 ) δ 2 s 1 -a b δ 2 s 2 ) + 2 j=1 D j1 δ 2 ρ j + E j1 δ 2 s j (∇ t s 1 ) 2 = 0 and c 2 s -a b δ 2 s 1 + b 2 (a -ρ 2 ) δ 2 s 2 + 2 j=1 D j2 δ 2 ρ j + E j2 δ 2 s j (∇ t s 1 ) 2 = 0 where c 2 s = v 2 (∇ t s 1 ) 2 .
Therefore we obtain a system of compatibility between the variables δ 2 ρ j and δ 2 s j in the form

                                                       C 11 δ 2 ρ 1 + C 12 δ 2 ρ 2 + D 11 δ 2 s 1 + D 12 δ 2 s 2 = 0 C 12 δ 2 ρ 1 + C 22 δ 2 ρ 2 + D 21 δ 2 s 1 + D 22 δ 2 s 2 = 0 D 11 δ 2 ρ 1 + D 21 δ 2 ρ 2 + (E 11 + (a -ρ 1 ) c 2 s ) δ 2 s 1 + (E 12 -a b c 2 s ) δ 2 s 2 = 0 D 12 δ 2 ρ 1 + D 22 δ 2 ρ 2 + (E 12 -a b c 2 s ) δ 2 s 1 + (E 22 + (a -ρ 2 ) b 2 c 2 s ) δ 2 s 2 = 0 (42)
Let us denote

C = C T = C 11 C 12 C 12 C 22 , D = D 11 D 12 D 21 D 22 , E = E T = E 11 E 12 E 12 E 22 , B = B T = ρ 1 -a ab ab (ρ 2 -a)b 2 δ 2 ρ = δ 2 ρ 1 δ 2 ρ 2 , δ 2 s = δ 2 s 1 δ 2 s 2
and system (42) is writting

     C δ 2 ρ + D δ 2 s = 0 D T δ 2 ρ + (E -c 2 s B) δ 2 s = 0 , which implies A -c 2 s B δ 2 s = 0. ( 43
)
where 43) it is simple to verify the property, Theorem: If EC -D 2 is positive definite and if we consider small diffusion, i.e. a < ρ * with ρ * = ρ 1 ρ 2 ρ 1 + ρ 2 then, A and B are both symmetric and definite positive, all the eigenvalues c 2 s of eq. ( 43) are positive and two discontinuity waves exist.

A = E -D T C -1 D . From eq.(

Exceptional waves:

As in the hyperbolic case, a wave is exceptional or linearly degenerate (see e.g. [START_REF] Boillat | Non linear hyperbolic fields and waves in Recent mathematical methods in nonlinear wave propagation[END_REF]) if δλ ≡ 0.

In this case the wave behavior is similar to the behavior in linear case and we do not get any distortion of the wave or shock formation.

It is simple matter to prove that both the waves fulfill the exceptionality condition. In fact, taking into account of relations ( 31),(39), we have δ∇ t s ≡ 0 and from [START_REF] Nigmatulin | Fundamentals of mechanics of heterogeneous mixtures[END_REF] we obtain that λ is function of the modulo of the tangential gradient of entropy λ ≡ λ(|∇ t s|) and then (44) holds.

Results and discussion:

In this paper we prove that the model of thermocapillary fluid mixtures with dissipation yields a system of equations of motions compatible with the second law of thermodynamics at least in simple dissipative cases. The equation of motion and the equation of energy of the barycentric motion of the mixture are in a divergence form in conservative cases. Consequently, Hamilton's principle applied to fluid dynamics is a direct and systematic method to obtain the equations of conservative motions. This principle extended to each component of a mixture of conservative fluids is able to deduce the same number of balance equations than unknown functions. The method yields a non ambiguous framework for the case of non-conservative mixtures (with viscosity, diffusion and heat transfer). Non additional assumption but constitutive behavior compatible with the second law of thermodynamics is necessary. One obtains the dynamic Gibbs relation and Fick's law as a consequence of governing equations.

We have seen that the dependance of an entropy gradient is necessary for the existence of isentropic waves of acceleration along the interfaces: the fact that the internal energy depends not only on the gradient of matter densities but also on the gradient of entropy, yields a new kind of waves which does not appear in simpler models. They are exceptional waves in the sense of Boillat and Lax [START_REF] Boillat | Non linear hyperbolic fields and waves in Recent mathematical methods in nonlinear wave propagation[END_REF] and they appear only in, at least, systems with two dimensions. These second order waves are of weak energy and consequently they are not easy to show up. Recent experiments in space laboratories in micro-gravity conditions, for carbonic dioxide near its critical point, have showed the possibility of such waves [START_REF] Garrabos | [END_REF]. The experimental evidence of such adiabatic waves with other physical reasons we have presented in the introduction should strengthen the necessity to take into account of the dependence of entropy gradients together with density gradients in the expression of the internal energy for continuum models of capillarity and phase transitions.

5 Appendix:

Compatibility conditions for weak discontinuities

We obtain also the compatibility due to the boundary conditions. For this aim we rewrite the variations of the Hamilton action when the equations of the motion (8) are verified. Then, from relation [START_REF] Rocard | Thermodynamique[END_REF], we obtain,

δ i I = t 2 t 1 ∂Dt g ρ i K T i ζ i -n. ρ i R i ζ i -ρ i u i K T i ζ i -Φ i ∂ρ i ∂x ζ i -Ψ i ∂s i ∂x ζ i -ρ i Φ i div ζ i dσ x dt.
For a vector field x ∈ D t → ζ i vanishing with its first derivatives on the boundary ∂D t , we deduce immediately on a surface of discontinuity Σ t (where ρ i , s i and its first derivatives are continuous and there are jumps for the second derivative) the value of the variation of the Hamilton action,

δ i I = t 2 t 1 Σt g ρ i K T i ζ i -n. ρ i R i ζ i -ρ i u i K T i ζ i -Φ i ∂ρ i ∂x ζ i -Ψ i ∂s i ∂x ζ i -ρ i Φ i div ζ i dσ x dt ≡ - t 2 t 1 Σt n . [ρ i R i ] ζ i dσ x dt.
Due to the fact that δ i I = 0 for a vector field x ∈ D t → ζ i , we obtain [ρ i R i ] = 0, and deduce the compatibility conditions (40) across the wave front,

[div Φ i ] ≡ 2 j=1
C ij δ 2 ρ j + D ij δ 2 s j = 0.

Mixture with an entropy for the total fluid

The equations of balance of matter densities ρ, ρ 1 are in the form ∂ρ ∂t + div(ρu) = 0,

∂ρ 1 ∂t + div(ρ 1 u 1 ) = 0, (45)
where ρ and u denote the density and the velocity of the total mixture, ρ 1 and u 1 denote the density and the velocity of one of the two components (it is equivalent to consider the total density ρ and the concentration c = ρ 1 /ρ betwen the two components of the mixture such that eq. ( 45) is equivalent to ∂ (ρc) /∂t + div(ρcu) = 0). For conservative motions, the equation of conservation of the total specific entropy s is ∂ρs ∂t + div(ρsu) = 0.

We consider a volume potential energy of the mixture in the form ε = ε(ρ, ρ 1 , s, s 1 , ∇ρ, ∇ρ 1: , ∇s, ω).

with ω = 1 2 w 2 , where w = uu 1 . As in the section 2, it is possible to deduce the equations of motions through the Hamilton action. We denote ρ θ = ∂ε ∂s , h = ∂ε ∂ρ , h 1 = ∂ε ∂ρ 1 and a = ∂ε ∂ω .

(In fact, the potential energy of the mixture depends only on the insentropic invariants (∇ρ) 2 , (∇ρ 1 ) 2 , ∇ρ • ∇ρ 1 , (∇s) 2 ). By analogous calculations associated with a Lagrangian of the mixture in the form

L = 1 2 ρu 2 -ε,
and virtual motions such as X = Ξ(t, x,κ) and X 1 = Ξ 1 (t, x,κ 1 ), we obtain as in section 2, the equations of motion .

By summing eqs [START_REF] Garrabos | [END_REF]47), we obtain the balance equation for the total momentum in a divergence form

∂ρu ∂t + div (ρ u ⊗ u -ρa w ⊗ w -σ) = 0
where σ is the stress tensor such that σ νγ = (-P + ρ div Φ+ρ 1 div Φ 1 ) δ νγ -Φ ν ρ ,γ -Φ 1ν ρ 1,γ -Ψ ν s ,γ , where P = ǫ -ρǫ ,ρ -ρ 1 ǫ ,ρ 1 . The equation of energy of the total mixture is also obtained in the divergence form ∂ ∂t Ψ and it is possible by a similar calculation than in section 3 to deduce an acceleration wave associated with the entropy gradient.

w

  + div(a w ⊗ u) = ρ θ ∇ s -ρ∇ h and ∂ ∂t (aw) + a ∂u 1 ∂x T w + div(a w ⊗ u 1 ) + ρ 1 ∇ h 1 = 0(47)withρ θ = ρǫ ,s -div Ψ, h = ρǫ ,ρ -div Φ , and h 1 = ρ 1 ǫ ,ρ 1 -div Φ 1 ,

1 2 ρu 2

 2 + ε -a w 2 + div 1 2 ρu 2 -a w.u -σ u -U + ε u = 0.

Main sentences in this paragraph and more precisely the comments on static interfaces are issued from the book of Rowlinson and Widom[START_REF] Rowlinson | Molecular theory of capillarity[END_REF] and its bibliography.

Acknowledgements

This paper was developed during a stay of Tommaso Ruggeri as visiting professor in L.M.M.T. of the University of Aix-Marseille III and a stay of Henri Gouin as visiting professor in C.I.R.A.M. of the University of Bologna with a fellowship of the Italian GNFM-INDAM and was supported in part (T.R.) by MIUR Progetto di interesse Nazionale Problemi Matematici Non Lineari di Propagazione e Stabilità nei Modelli del Continuo Coordinatore T. Ruggeri, by the GNFM-INDAM, and by the Istituto Nazionale di Fisica Nucleare (INFN).

The authors are indebted to the anonymous referees for their valuable criticism during the review process.


Ausrine Bartasyte
email: ausrine.bartasyte@gmail.com

Odette Chaix-Pluchery

Jens Kreisel

José Santiso

Samuel Margueron

Michel Boudard

Carmen Jiménez

Adulfas Abrutis

François Weiss

Carmen Jimenez

Raman Spectroscopy and X-ray Diffraction Studies of Stress Effects in PbTiO 3 Thin Films

Introduction

Lead titanate PbTiO 3 (PTO) is a classical example of a ferroelectric oxide among the perovskite family. Thin films of PbTiO 3 and other related materials offer key advantages for a variety of applications in non-volatile memories, microelectronics, sensors technologies and electromechanical systems [START_REF] Scott | The physics of ferroelectric ceramic thin films for memory applications[END_REF]. Physical properties of ferroelectrics in bulk material and in thin films are substantially different due to stresses, grain size effect, domain structure, defects, etc. [START_REF] Okuyama | Si-monolithic miniature ultrasonic sensor using PbTiO 3 thin film prepared on a Si or SiO 2 cantilever[END_REF][START_REF] Scott | Ferroelectric memories[END_REF]. For instance, film stresses can significantly change mechanical, optical and electrical properties, ferroelectric domain structure and the nature of phase transition, which in turn has a potential influence on the reliability of devices. Physical properties such as dielectric, piezoelectric, and optical characteristics critically depend on the ferroelectric domain configuration [START_REF] Damjanovic | Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics[END_REF][START_REF] Ramesh | Effect of crystallographic orientation on ferroelectric properties of PbZr 0.2 Ti 0.8 0 3 thin films[END_REF] which also depends on the various stresses developed during film fabrication. As a consequence of this, several authors have developed a thermodynamic theory which predicts the domain configuration of epitaxial ferroelectric films as function of lattice mismatch and temperature [START_REF] Pertsev | Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films[END_REF][START_REF] Koukhar | Thermodynamic theory of epitaxial ferroelectric thin films with dense domain structures[END_REF][START_REF] Hu | Effect of interfacial dislocations on ferroelectric phase stability and domain morphology in a thin film -a phase-field model[END_REF][START_REF] Li | Phase-field model of domain structures in ferroelectric thin films[END_REF]. Given the complexity of internal and external stresses, an experimental investigation of residual stress remains useful for predicting and controlling physical properties for ferroelectric film -based devices.

Films are usually subjected to substantial stresses during growth and the subsequent cooling process from deposition temperature to room temperature (RT). At the growth temperature, stresses are mainly misfit stresses (also called epitaxial stresses) which result from lattice mismatch between the film and the substrate; intrinsic stresses related to growth technique are rather negligible in metalorganic chemical vapor deposition (MOCVD). During the cooling process, additional stresses may develop in the film due either to thermal stress arising from different thermal expansion coefficients between the film and the substrate or to a lattice distortion related to the structural phase transition. The main stress relaxation mechanisms in epitaxial ferroelectric films are obtained by the formation of dislocations and domains. Dislocation formation can take place only when the layer thickness exceeds a critical value [START_REF] Speck | Domain configurations due to multiple misfit relaxation mechanisms in epitaxial ferroelectric thin films. I. Theory[END_REF]. Ferroelectric domains form at and below the cubic-totetragonal phase transition and are generally observed in the form of tweed patterns in the low temperature phase [START_REF] Ganpule | Imaging three-dimensional polarization in epitaxial polydomain ferroelectric thin films[END_REF].

In the past, optical fluorescence [START_REF] Ma | Stress Measurement in Single-Crystal and Polycrystalline Ceramics Using Their Optical Fluorescence[END_REF], X-ray diffraction (XRD) [START_REF] Ching-Prado | Raman spectroscopy and X-ray diffraction of PbTiO 3 thin films[END_REF], wafer curvature measurements [START_REF] Li | Raman spectroscopy and X-ray diffraction of PbTiO 3 thin film[END_REF], cantilever beam deflection [START_REF] Voncken | New method for the measurement of stress in thin drying gel layers, produced during the formation of ceramic membranes[END_REF], laser reflectance [START_REF] Sengupta | Origins and evolution of stress development in sol-gel derived thin layers and multideposited coatings of lead titanate[END_REF] and Raman spectroscopy [START_REF] Valim | Evaluating the residual stress in PbTiO 3 thin films prepared by a polymeric chemical method[END_REF][START_REF] Sanjurjo | High-pressure Raman study of zone-center phononos in PbTiO 3[END_REF][START_REF] Cerdeira | Effect of pressure on the zone-center phonons of PbTiO 3 and on the ferroelectric-paraelectric phase transition[END_REF][START_REF] Fu | Thickness dependence of stress in lead titanate thin films deposited on Pt-coated Si[END_REF][START_REF] Li | Raman spectroscopy and X-ray diffraction study of PbTiO 3 thin films prepared by sol-gel technique[END_REF][START_REF] Dobal | Micro-Raman investigation of stress variations in lead titanate films on sapphire[END_REF][START_REF] Sun | Phononmode hardening in epitaxial PbTiO 3 ferroelectric thin films[END_REF][START_REF] Lee | Polarized Raman scattering of epitaxial PbTiO 3 thin film with coexisting c and a domains[END_REF] have been used for stress analysis of ferroelectric materials. Among them, XRD and Raman spectroscopy are the most popular techniques, as they are powerful tools for non-destructive investigation of structure. Raman spectroscopy is complementary to XRD methods as it is a local probe and is sensitive to even subtle changes of symmetry. Moreover, ferroelectricity and thus Raman modes are strongly influenced by mechanical deformation of the sample resulting, for example, from hydrostatic pressure or stress [START_REF] Sanjurjo | High-pressure Raman study of zone-center phononos in PbTiO 3[END_REF][START_REF] Cerdeira | Effect of pressure on the zone-center phonons of PbTiO 3 and on the ferroelectric-paraelectric phase transition[END_REF][START_REF] Zheng | Heteroepitaxially enhanced magnetic anisotropy in BatiO 3 -CoFe 2 O 4 nanostructures[END_REF][START_REF] Kornev | Ferroelectricity of perovskites under Pressure[END_REF][START_REF] Kornev | The nature of ferroelectricity under pressure[END_REF][START_REF] Samara | Ferroelectric/relaxor crossover in compositionally disordered perovksites[END_REF].

Here we report a study of domain structure and residual stress evolution with film thickness and of phase transition in epitaxial PTO films on LaAlO 3 (LAO) by using XRD and Raman scattering. In particular, Raman measurements were performed in all polarization configurations attainable in the backscattering geometry which allows us to obtain information on both a-and cdomains.

II. Theoretical considerations

At room temperature PTO is ferroelectric and adopts a tetragonal crystal symmetry (space group C 1 4v ) with one formula unit in the unit cell [START_REF] Burns | Raman studies of underdamped soft modes in PbTiO 3[END_REF]. At 490 °C, a ferroelectric-to-paraelectric phase transition takes place. It has been reported that this structural phase transition has both displacive-and order-disorder-like properties [START_REF] Sicron | Nature of the ferroelectric phase transition in PbTiO 3[END_REF][START_REF] Fontana | Raman spectrum in PbTiO 3 re-examined: dynamics of the soft phonon and the central peak[END_REF] as is the case for most ferroelectric perovskites. In the cubic high-temperature phase (space group O 1 h ), there are 12 vibrational degrees of freedom at k=0, which belong to the irreducible representations 3T 1u + T 2u , where T 2u is a silent mode since it is neither infrared nor Raman active. The triply degenerated T 1u modes are only infrared active. According to these selection rules, Raman modes should disappear in the paraelectric cubic phase. In the tetragonal phase, each T 1u mode splits into A 1 + E modes, resulting in 3A 1 + 3E modes, whereas the T 2u mode splits into B 1 +E modes. The A 1 and E modes are infrared and Raman active and the corresponding phonons are polarized in the z-direction or in the xy plane, respectively; the B 1 mode is only Raman active. The Raman tensors are presented in (1) for each mode, and the direction of polarization is indicated in parentheses.

Í Í Í Î È ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = b a a z A ) ( 1 Í Í Í Î È ⋅ ⋅ ⋅ ⋅ - ⋅ ⋅ ⋅ = c c B 1 Í Í Í Î È ⋅ ⋅ - ⋅ ⋅ ⋅ - ⋅ ⋅ = e e x E ) ( Í Í Í Î È ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = e e y E ) ( (1)
Due to long-range electrostatic forces, all A 1 and E modes split into transverse (TO) and longitudinal (LO) components. Depending on the phonon propagation direction (x, y or z), only pure TO or LO can be observed in Raman spectra, as presented in Table 1.

Film depositions were carried out in a vertical hot wall pulsed injection MOCVD reactor described elsewhere [START_REF] Sénateur | Procédé et dispositif d'introduction de précurseurs dans une enceinte de dépot chimique en phase vapeur[END_REF][START_REF] Sénateur | Advances In Inorganic Films and Coatings[END_REF][START_REF] Sénateur | Innovative processing of films and nanocrystalline powders[END_REF]. The injector injects micro-doses (a few microliters) of an organic solution containing a dissolved mixture of metalorganic precursors. After flash evaporation of the micro-doses, the resulting vapor mixture is transported by an Ar + O 2 gas flow towards the heated substrate. The injector consists of a computer-driven, high-speed, precision electromagnetic valve operating under a repetitive pulsing regime. The deposition conditions were optimized in order to obtain a pure and high quality PTO perovskite phase (bulk lattice parameters a=3.899 Å and c=4.153 Å) deposited on a LAO substrate (a=3.791 Å, supplied by CrysTec, Berlin, Germany) with the surface cut parallel to a (00l) plane (indices refer to the pseudo-cubic setting) [START_REF] Bartasyte | Ferroelectric PbTiO 3 films grown by pulsed liquid injection metalorganic chemical vapour deposition[END_REF][START_REF] Bartasyte | Ferroelectric PbTiO 3 films grown by pulsed liquid injection MOCVD[END_REF]. Films of different thickness from 30 to 460 nm were deposited at 650 °C. The thicknesses were measured from the film cross-section micrographs obtained by scanning electron microscopy (SEM). XRD attests a high quality, single phase film. More details concerning deposition techniques and standard sample characterization can be found in references [START_REF] Bartasyte | Ferroelectric PbTiO 3 films grown by pulsed liquid injection metalorganic chemical vapour deposition[END_REF][START_REF] Bartasyte | Ferroelectric PbTiO 3 films grown by pulsed liquid injection MOCVD[END_REF]. Texture analysis of epitaxial films was carried out using a Bruker D8 Advance diffractometer (Bruker AXS, Karlsruhe, Germany) with a four-circle and general area detection diffraction system (GADDS) (l CuKa1 =1.54056 Å, Ni filter).

The LAO substrate was used as an internal standard assuming that it is stress-free.

Raman spectra were collected using a Labram spectrometer (Jobin-Yvon/Horiba, Villeneuve d'Asq, France) equipped with a N 2 cooled charge coupled device detector. Experiments were conducted in the micro-Raman mode at room temperature in a backscattering geometry. The 514.5 nm line of an Ar + ion laser was focused on a spot size smaller than 1 mm. The system was calibrated using Si spectra at room temperature. The temperature measurements in the range from room temperature to 600 °C have been carried out by using a commercial Linkam heating stage (Linkam Scientific Instruments Ltd., Survey, UK) placed under the Raman microscope.

IV. Results and discussion

A. XRD results

The q/2q XRD patterns of PTO/LAO films are characteristic of the PTO tetragonal structure consisting only of intense (00l) and very weak (h00) reflections [START_REF] Bartasyte | Investigation of thickness-dependent stress in PbTiO 3 thin films[END_REF]. This indicates a dominant c-domain structure containing a small fraction of a-domains. Two-dimensional reciprocal space mappings near the (103) and (301) PTO peaks are shown in Fig. 1a. The reciprocal space mapping integrates the intensity from different j-angles (± 5°), making it possible to observe together the (301) and (310) reflections, which were found at different j-values. The coexistence of (301) and

(310) reflections indicates that there exists two types of a-domains, a 1 and a 2 : both types have their c-axis parallel to the substrate plane but a 2 domains are rotated by 90 ˚ with respect to a 1 domains.

(301) reflection is split in two spots indicating twinning of a-domains. The (103) spot corresponding to c-domains is elongated which indicates that some twinning also appears in c-domains.

In order to clarify the twin structure, w-c contour maps of c-domain (003) and a-domain (300) reflections were measured (Fig. 1b and1c). The maps reveal twinning in a-and c-domains with fourfold symmetry. Twinning is very well pronounced in a-domains (Fig. 1c), whereas it is less evident in c-domains despite the square shape of the spot (Fig. 1b). The evolution with film thickness, of twin angle of a-and c-domains extracted directly from the splitting of the reflections in the XRD maps, is given in Table 2. One can see that the tilt of c-domains increases whereas the tilt of a-domains slightly decreases with increasing film thickness.

The proportion of c-and a-domains was quantified as the percentage of a-domains (A, %) in the films and was calculated by peak integration of the (102) reflection in j-scans corresponding to the different domains:

A = 2 *(I(102) c =58°+ I(102) c =62°) 2*( I(102) c =58°+ I(102) c =62°) + I(102) c= 27°* 100 (2)
, where I is the peak integrated intensity in j-scans measured at different c angles, corresponding to a-axis (c = 58 ° or c = 62 °) and c-axis (c = 27 °) oriented crystallites (i.e., a-and c-domains). More details on domain structure can be found in Ref. [START_REF] Bartasyte | Ferroelectric PbTiO 3 films grown by pulsed liquid injection metalorganic chemical vapour deposition[END_REF]. An evaluation of the volume fraction of adomains, applying Eq. ( 2), is given in Table 2 as a function of PTO/LAO film thickness.

The evolution of the c lattice parameter in c-domains with film thickness is given in Fig. 2. The lowest value is observed in thinnest films and it gradually increases with thickness without reaching the bulk c value even in the 460 nm thick film. The c lattice parameter in PTO/LAO films is lower than the value reported in the literature for bulk PTO. Thin films are usually considered to be subjected to biaxial stresses in the film plane and we thus consider that the normal direction is stressfree. If the films are subjected to a biaxial compressive in-plane stress, the in-plane values would be strained: the in-plane parameter would be shorter and the c-axis (out-of-plane parameter) larger.

Therefore, the c-parameter decrease in c-domains should arise from biaxial tensile stress in the film plane although a compressive misfit stress is expected. The fact that we do not observe a compressive stress at RT indicates that the films do not retain the strain induced by the mismatch with substrate. A more important source of strain is probably produced by other types of stresses. We will address this apparent contradiction later in the manuscript.

B. Raman spectroscopy results

1-Effect of polarization

Raman spectra were measured in crossed and parallel polarization configuration (VH and VV, respectively) on the film surface or cross section, i.e. the incident and scattered laser beams are normal or parallel to the substrate plane, respectively. The consideration of the crystallographic reference setting of the films in our system (a-and c-domain structure) and the use of the abovediscussed Raman selection rules allows us the assignment of the Raman modes as schematically illustrated in Fig. 3.

It follows that when a spectrum is recorded on the film surface, E(TO) modes observed in VH configuration come from a-domains only, whereas A 1 (TO) modes obtained in VV configuration originate in a-domains and A 1 (LO) in c-domains. The spectrum recorded on the film cross section in VH configuration consists also of E(TO) modes, which come from c-and 50 % a-domains, according to Raman selection rules. VV spectra measured on cross section consist of A 1 (LO) modes related to a-domains and A 1 (TO) modes, which come from c-and 50 % a-domains. Considering that the c-domain fraction is greater than 80 % in films of all thicknesses, we can assume that E(TO) and A 1 (TO) modes come mainly from c-domains when spectra are collected on the film cross section. As seen in Fig. 3, E(LO) modes cannot be observed in our measurement conditions if selection rules are strictly obeyed. In conclusion, modes relative to a-and c-domains can be separated in polarized

Raman spectra recorded on the film cross section as well as on the film surface.

As predicted by the C 4v point group selection rules, only E(TO) modes are observed in the PTO/LAO spectrum recorded in VH configuration on the film surface and cross section (Fig. 4); this attests to a good crystalline quality and epitaxy of the film. The spectrum measured in a parallel polarization configuration (VV) contains a mixture of A 1 (TO), A 1 (LO), E(TO) and E(LO) modes instead of the only allowed A 1 modes. Furthermore, when spectra are recorded on film surface, A 1 (TO) modes coming from both c-and a-domains are observed as seen in Fig. 4. This probably is related to the existence of ferroelectric domains leading to a depolarization of the laser beam. In spectra recorded on film surface, the separation of A 1 (TO) modes coming from both a-and cdomains is not evident; as the volume fraction of a-domains is quite small, c-domain A 1 (TO) modes resulting from depolarization may have higher intensity than a-domains modes. Nevertheless, A 1 (TO) modes coming from different domains can be separated because the intensity of A 1 (TO) modes related to c-domains markedly increases in spectra recorded on the cross section.

2-E(TO) modes and residual stresses

It has been reported in earlier studies [START_REF] Sanjurjo | High-pressure Raman study of zone-center phononos in PbTiO 3[END_REF] that Raman modes are sensitive to pressure; their positions in wavenumber w as a function of pressure can be described by tan empirical law. If only E(TO modes are considered,

w[E(TO)] = w 0 [E(TO)] + (dw[E(TO)] /dP)P (3)
where w 0 [E(TO)] is the mode wavenumber at ambient conditions (1 bar, 25 °C), and dw[ E(TO)] /dP is the empirical pressure coefficient as determined from measurements under hydrostatic pressure [START_REF] Sanjurjo | High-pressure Raman study of zone-center phononos in PbTiO 3[END_REF][START_REF] Cerdeira | Effect of pressure on the zone-center phonons of PbTiO 3 and on the ferroelectric-paraelectric phase transition[END_REF]. However, stresses in thin films correspond to non-hydrostatic conditions, and we thus have to introduce the effect of anisotropy in the phonon deformation potential. Supposing the harmonic approximation and using symmetry arguments, the relation between doubly degenerate E mode frequency and applied stresses can be described as follows [START_REF] Briggs | Piezospectroscopic study of Raman spectrum of cadmium sulfide[END_REF] E(TO )

Dw = E(TO )
a'

( XX s + YY s ) + E(TO) b' ZZ s (4)
where

E(TO ) Dw = E(TO ) w -0E(TO )
w ,

E(TO )

a' and E(TO ) b' are the deformation potential constants expressed in terms of compliance and XX s , YY s and ZZ s are stresses along a-, b-and c-axis of tetragonal PTO cell, respectively. In the case of hydrostatic pressure P, Eq. ( 4) can be written as

E(TO ) Dw = E(TO) (2a' + E(TO) b' )P (5)
Epitaxial PTO films, grown on cubic substrates, are under biaxial stress b s in the substrate plane. Thus, the shift of E(TO) mode due to biaxial stress in c-domains can be expressed as

E(TO ) Dw = E(TO) 2a' b s (6)
and in a-domains as

E(TO ) Dw = E(TO) (a' + E(TO) b' ) b s (7)
Although PTO films are not in hydrostatic conditions, it has been argued in the literature that hydrostatic pressure instabilities offer a guide as how a material will react to an external stress as Raman modes also shift linearly under biaxial stress; only the relation coefficients differ. It is to be noted that the stress values in tensile-strained PTO films are, in the literature, often estimated from the shift of the E(1TO) soft mode using Eq. ( 3) [START_REF] Valim | Evaluating the residual stress in PbTiO 3 thin films prepared by a polymeric chemical method[END_REF][START_REF] Fu | Thickness dependence of stress in lead titanate thin films deposited on Pt-coated Si[END_REF][START_REF] Dobal | Micro-Raman investigation of stress variations in lead titanate films on sapphire[END_REF][START_REF] Lee | Polarized Raman scattering of epitaxial PbTiO 3 thin film with coexisting c and a domains[END_REF]; however, its pressure coefficient is experimentally known only for compression (hydrostatic pressure) ( E(1TO) dw /dP = -5.8 ± 0.2 cm -1

GPa -1 [START_REF] Sanjurjo | High-pressure Raman study of zone-center phononos in PbTiO 3[END_REF]) and the soft mode is anharmonic. For this reason, we performed stress value calculations from the shift of the E(3TO) mode which is a real (harmonic) hard mode in the sense that its pressure coefficient is positive ( E(3TO) dw /dP = +7.1 ± 1 cm -1 GPa -1 [START_REF] Sanjurjo | High-pressure Raman study of zone-center phononos in PbTiO 3[END_REF]). The obtained values were in good agreement with the stress values calculated from the c-lattice parameter obtained by XRD (Table 3), as described in Ref. [START_REF] Bartasyte | Investigation of thickness-dependent stress in PbTiO 3 thin films[END_REF]. The similarity between stress values allows us to consider that Eq. ( 3) can be used for E(3TO) in the case of PTO/LAO films. Therefore, Raman data obtained from hydrostatic pressure experiments can be used in our case to provide an understanding and reasonable estimate of the stress state from E(3TO) hard mode in c-domains in PTO films. From comparison of Eqs. ( 3) and ( 4), it results that E(3TO) b'

<< E(3TO)
a' and then from Eqs. ( 6) and [START_REF] Koukhar | Thermodynamic theory of epitaxial ferroelectric thin films with dense domain structures[END_REF], that the shift of the E(3TO) mode

E(3TO)
Dw in a-domains should be half of that in c-domains for a given biaxial stress value.

Indeed, the E(3TO) mode is observed at a lower wavenumber in PTO/LAO films than in a single crystal (506.2 cm -1 ) (Table 3). This observation provides evidence that films are under tensile residual stress. Stress values evaluated in c-domains decreased from 1.3 to 0.5 GPa when the thickness increased from 30 to 460 nm (Table 3) [START_REF] Bartasyte | Investigation of thickness-dependent stress in PbTiO 3 thin films[END_REF]. E(3TO) modes related to a-domains are shifted to lower wavenumbers in comparison with modes coming from c-domains (Table 3); accordingly, a-domains are more stressed than c-domains. The higher stress in a-domains should result from a larger lattice mismatch between the film and the substrate and more probably from the nano-size of a-domains as they form thin sheets in a matrix of c-domains [START_REF] Ganpule | Imaging three-dimensional polarization in epitaxial polydomain ferroelectric thin films[END_REF], which may generate a supplementary stress at c/a domain interfaces. Therefore, stresses in a-domains are probably not biaxial and the stress relation with

E(TO )
Dw should then be described by Eq. ( 4). Thus, residual stress value in a-domains cannot be estimated from the shifts of Raman modes, because the stress symmetry in a-domains is unknown.

Qualitatively, one can see that the difference in residual stress in a-and c-domains is considerable. Thus, residual stress value in c/a/c/a epitaxial PTO films with dominant c-domain structure can be determined only from polarized Raman spectra recorded on film cross section. The case of polycrystalline films is different as Raman selection rules are not respected and residual stresses can be evaluated from spectra collected in common way on the film surface. Only few literature reports can be found on residual stress values in c-domains in epitaxial PTO films [START_REF] Lee | Polarized Raman scattering of epitaxial PbTiO 3 thin film with coexisting c and a domains[END_REF], and calculations are made by using anharmonic E(1TO) soft mode and Eq. ( 3).

PTO films are under in-plane tensile stress whatever the film thickness, which is in agreement with XRD results. Thus, the origin of the tensile residual stress is not only the mismatch between the film and the substrate, because this would lead to a compressive stress. Other possible origins of tensile stress are related to phase transition and to a difference in film and substrate thermal expansion coefficients. Thermal stress is usually rather small; thus the high residual stress values, especially in thin films, cannot be explained if the stress related to the phase transition is assumed to be completely relaxed. As a matter of fact, residual stress measured in films at room temperature is the sum of all stresses developed during the complex process of film preparation, and it cannot be associated to only one origin of stress. At the present state of our work we cannot say if the epitaxial stress is relaxed to zero or if only the other types of stress are dominating the physics of the system.

3-Domain structure and A 1 (TO) modes

Similar to E(TO) modes, A 1 (2TO) and A 1 (3TO) modes are also shifted to lower wavenumbers in comparison with A 1 bulk modes (Fig. 5 and [START_REF] Bartasyte | Investigation of thickness-dependent stress in PbTiO 3 thin films[END_REF]). As mentioned before, both types of A 1 (TO) modes related to c-domains and coming from a-domains are observed in VV spectra recorded on the film surface due to a depolarization effect (labeled a-and c-components in Fig. 5). Since a-domains are more stressed than c-domains, it results that a-components appear at lower wavenumbers in comparison with c-components. The intensity ratio between a-and c-components is correlated to the volume fraction of a-domains as shown in Fig. 5; it follows the same thickness dependence as the acomponent is the more intense feature in the 125 nm thick film spectrum. Intensities of the A 1 (TO) mode coming from a-domains are increased in comparison with those from c-domains resulting from the depolarization effect. In a recent study of Pb(Zr,Ti)O 3 /MgO epitaxial films [START_REF] Osada | Domains distributions in tetragonal Pb(Zr,Ti)O 3 thin films probed by polarized Raman spectroscopy[END_REF], a remarkable correlation was found between the intensity of the A 1 (1TO) mode and the c-domains volume fraction , thus providing a useful probe for domain characterization.

4-Effect of temperature

The evolution with temperature of VH Raman spectra (E(TO) modes) related to a-and cdomains in a 250 nm thick PTO/LAO film is shown in Fig. 6. Up to 400 °C, spectra are rather similar in both a-and c-domains. However, when the temperature is further increased, the spectral signature of a-and c-domains becomes different: On the one hand, Raman modes in a-domains become gradually larger and less intense and the spectrum almost vanishes. On the other hand, the cdomains maintain a fairly well-defined Raman signature up to 600 °C. This observation suggests that the structural instabilities, with respect to temperature, are different for a-and c-domains. This difference may be explained by various factors, such as the different size and/or orientation of a-and c-domains with respect to the substrate, or a change in the domain structure with temperature.

However, further investigations are necessary to understand this behavior in more detail.

In order to determine the temperature at which the phase transition occurs, the temperature evolution of the E(2TO) mode profile was analyzed for a-and c-domains. The evolution in wavenumber of the E(2TO) mode as a function of temperature is reported in Fig. 7. It can be seen that a-and c-domains indeed show similar behavior at low temperature, in agreement with the above qualitative remarks. On the other hand, while the spectrum of a-domains disappears around 480 °C and is no longer allowed to follow the E(2TO) mode evolution, this mode coming from c-domains presents an anomaly at 500 ˚C. This anomaly provides evidence for a structural rearrangement and we note that this temperature coincides with the ferroelectric-to-paraelectric (tetragonal-to-cubic) phase transition in bulk PTO. Although this coincidence suggests that the c-domains become cubic, the presence of a Raman signature above 500 °C is a direct evidence that the c-domains are not strictly cubic at least on a local level. Although we do not know in detail the stress state of our films, it is interesting to see how our results stand within theoretical predictions. We note that a transition from c/a/c/a to a c-phase and to a paraelectric phase is predicted by Koukhar et al. [START_REF] Koukhar | Thermodynamic theory of epitaxial ferroelectric thin films with dense domain structures[END_REF] and Li et al. [START_REF] Li | Phase-field model of domain structures in ferroelectric thin films[END_REF], which is similar to our experimental observation. However, further investigations are needed to clarify where our films are situated in the so-called Pertsev diagrams. Lets us finally note, that various ferroelectric perovskites present (large) Raman bands even within a cubic average structure, and this is understood by the different coherence lengths obtained by the Raman and XRD techniques. If we consider that XRD probes the long range structure while Raman is a local probe, then the Raman signature of an average cubic structure finds its origin in the existence of local polar disorder which averages out to a cubic structure (one well-known example is the so-called 8-site model [START_REF] Comès | The chain structure of BaTiO 3 and KNbO 3[END_REF]). This could also explain our observed persisting Raman signal. However, the fact that the two bands between 200 and 300 cm -1 remain fairly well-defined also leaves the place for an interpretation where the c-domains adopt above 500 °C a new crystal or domain structure. Raman spectroscopy alone, unfortunately, does not allow to conclude in favor of the first or the second scenario, and high temperature XRD is under way to clarify this point.

V. Conclusions

A systematic study of domain structure and residual stress evolution with film thickness and of phase transition in c/a epitaxial PTO/LAO films is reported by using XRD and Raman spectroscopy.

The results obtained by both techniques provide evidence that the films are under tensile residual stress in the film plane although a compressive misfit stress is expected. The a-domains are more stressed than c-domains, with a significant difference in stress values evaluated from the Raman shift of the E(3TO) mode. Residual stress is partially relaxed with increasing film thickness. As expected by the Raman selection rules, residual stress values in epitaxial PTO films with a dominant c-domain structure can be obtained only from polarized Raman spectra coming from c-domains, i.e. from spectra recorded on the film cross section. The large A 1 (TO) modes are split into two components, respectively assigned to a-and c-domains; their intensity ratio was correlated to the volume fraction of a-domains.

The evolution with temperature of Raman spectra of PTO films shows that Raman modes of adomains gradually evolve until the spectrum almost vanishes at 480 °C whereas modes in c-domains remain well defined up to 600 °C despite an anomaly in the spectrum evolution at 500 °C. This suggests a structural rearrangement at a temperature close to the bulk structural phase transition temperature. Let us finally note, neither the phase transition temperature. Let us finally note that neither the phase transition temperature nor the high temperature symmetry of epitaxial c/a PTO films can be determined from Raman spectra recorded on film surface in backscattering geometry, but that a cross section geometry, as in this work, should be used. Tables Table 1: TO-LO assignment of the A 1 and E modes.

Figure captions

Fig. 1 :

 1 Fig.1: (a) X-ray diffraction omega-2 theta mapping near the (103) and (301)/(310) PTO /LAO peaks. Omega-chi contour maps of (b) c-domain (003) reflection, and (c) a-domain (300) reflection.

Fig. 2 :

 2 Fig. 2: Film thickness dependence of the PTO c lattice parameter in c-domains.

Fig. 3 :Fig. 4 :Fig. 5 :

 345 Fig. 3: Raman scattering geometries and polarization configurations used for the Raman mode assignment in a-and c-domain PTO/LAO films.

Fig. 6 :Fig. 7 :

 67 Fig. 6: Temperature dependence of the VH Raman spectra related to a-and c-domains in a 250 nm thick PTO/LAO film.

Fig. 1

 1 Fig. 1

Fig. 2 20Fig. 3

 23 Fig. 2

Fig. 4 21Fig. 5

 45 Fig. 4

Fig. 6 22Fig. 7

 67 Fig. 6

Table 2 :

 2 X-ray characterization of PTO/LAO films of different thicknesses comprised in the range 30-460 nm.

	Film thickness (nm)	460 250 125 65 30
	Twinning angle of a-domains (˚)	2.36 2.4 2.42 -	-
	Twinning angle of c-domains (˚)	0.69 0.50 0.33 0	0
	Volume fraction of a-domains (%) 4.7 14.4 19.4 9.0 -*
	* Undetermined due to very low peak intensities

Table 3 :

 3 E(3TO) phonon wavenumbers and residual stress values estimated from E(3TO) mode wavenumber and XRD data in PTO/LAO films of different thicknesses comprised in the range 30-460 nm.

		w(E(3TO)) (cm -1 )	Residual stress in c-domains (GPa)
	Thickness (nm) a-domains c-domains	E(3TO)	XRD [37]
	460	500.8	503.2	0.45 ± 0.07 0.40 ± 0.06
	250	499.4	502	0.59 ± 0.10 0.58 ± 0.08
	125	498	501.2	0.70 ± 0.11 0.79 ± 0.11
	65	-	499.7	0.88 ± 0.14 1.00 ± 0.14
	30	-	496.9	1.30 ± 0.21 1.02 ± 0.15

Acknowledgments: This work has been supported by the European network of excellence FAME and EGIDE.