Bérénice Grec
email: berenice.grec@u-paris.fr

Srboljub Simić
email: ssimic@uns.ac.rs

A T E X template

The paper studies a higher-order diffusion model of Maxwell-Stefan kind. The model is based upon higher-order moment equations of kinetic theory of mixtures, which include viscous dissipation in the model. Governing equations are analyzed in a scaled form, which introduces the proper orders of magnitude of each term. In the socalled diffusive scaling, the Mach and Knudsen numbers are assumed to be of the same small order of magnitude. In the asymptotic limit when the small parameter vanishes, the model exhibits a coupling between the species' partial pressure gradients, which generalizes the classical model. Scaled equations also lead to a higher-order model of diffusion with correction terms in the small parameter. In that case, the viscous tensor is determined by genuine balance laws.

Introduction

Macroscopic description of diffusion usually relies on two well-known models, the Fick and Maxwell-Stefan ones. They are both based upon the mass conservation of species, but differ in the way diffusion mechanism is described, and thus the system closed. In the Fick model, the diffusion fluxes are determined by constitutive relations of phenomenological nature -they are linearly related to the gradients of species' chemical potentials/densities [START_REF] Giovangigli | Multicomponent Flow Modeling[END_REF]. In the Maxwell-Stefan model, diffusion is regarded as a source of momentum exchange between the species, which is balanced by the gradients of partial pressures [START_REF] Krishna | The Maxwell-Stefan approach to mass transfer[END_REF][START_REF] Bothe | On the Maxwell-Stefan approach to multicomponent diffusion[END_REF]. The system is thus closed through a kind of momentum balance equations for the species. While the Fick model fits well into linear thermodynamics of irreversible processes [START_REF] De Groot | Non-equilibrium Thermodynamics[END_REF], the Maxwell-Stefan model fits in the framework of classical cross-diffusion models [START_REF] Hutridurga | Existence and uniqueness analysis of a nonisothermal cross-diffusion system of Maxwell-Stefan type[END_REF][START_REF] Daus | Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type[END_REF].

It is remarkable that both diffusion models mentioned above may be derived starting from a more sophisticated model-the Boltzmann equations of the kinetic theory of mixtures. To derive the Fick model, one has to apply the Chapman-Enskog method, i.e perform the asymptotic expansion of the velocity distribution function and the conservation laws derived from the Boltzmann equations, using the Knudsen number as small parameter [START_REF] Chapman | The Mathematical Theory of Non-uniform Gases. An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases[END_REF][START_REF] Golse | The Boltzmann equation and its hydrodynamic limits[END_REF][START_REF] Briant | Rigorous derivation of the Fick cross-diffusion system from the multi-species Boltzmann equation in the diffusive scaling[END_REF]. This amounts to a hydrodynamic limit of kinetic equations and inherits a physical assumption that the process occurs in the neighborhood of local equilibrium state.

On the other hand, to derive the Maxwell-Stefan model, one has to extend the system of moment equations and include the momentum balances for the species. The Maxwell-Stefan relations are then obtained as an asymptotic limit of the momentum balance laws in so-called diffusive scaling [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF][START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures with general cross sections[END_REF]. The diffusive scaling reflects the assumption that macroscopic velocities in diffusion processes are small compared to molecular velocities, while asymptotic limit amounts to neglecting inertia terms and convective fluxes in species' momentum balance laws.

In either approach, the diffusion is a dissipative process in the sense that it is compatible with the entropy inequality. It contributes to the entropy balance law through an entropy production [START_REF] Krishna | The Maxwell-Stefan approach to mass transfer[END_REF][START_REF] De Groot | Non-equilibrium Thermodynamics[END_REF]. Nevertheless, other dissipative mechanisms could also be included in the analysis, viscous dissipation being the most prominent one. In the framework of irreversible thermodynamics, it is incorporated in the same manner as Fick diffusion -by means of linear constitutive relation for the viscous stress tensor. However, in the case of Maxwell-Stefan model, such a generalization is less straightforward. It is usually included by assumption, and modeled by means of classical constitutive relations for Newtonian fluids [START_REF] Kerkhof | Analysis and extension of the theory of multicomponent fluid diffusion[END_REF][START_REF] Chen | Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system[END_REF].

The aim of this study is to include the viscous dissipation into the Maxwell-Stefan model by a systematic application of the moment method and diffusive scaling at the same time. The starting point is be the system of Boltzmann equations for non-reactive monatomic species. It is used to build up a corresponding system of moment equations, which consists of mass, momentum and momentum flux balance laws for the species, and the energy conservation law for the mixture. In fact, the momentum flux balance laws are the key ingredient which brings viscous dissipation into the model. Since our goal is to generalize the Maxwell-Stefan model of diffusion, all equations are transformed into a scaled (dimensionless) form, which inherits the already mentioned assumption on diffusion processes. The scaling formally ascribes a proper order of magnitude to each term appearing in the moment equations. Even more, it facilitates the systematic derivation of the higher-order model of diffusion, which inherits viscous dissipation, without the use of ad hoc assumptions.

The rest of the paper is organized as follows. Section 2 contains necessary information about kinetic modelling of mixtures, while Section 3 introduces the diffusive scaling into the modelling process. Section 4 is devoted to a derivation of the approximate velocity distribution function in the scaled form, which is needed for the closure of moment equations. Indeed, the usual ansatz used for the classical Maxwell-Stefan model cannot be generalized in a straightforward way. Therefore, the maximum entropy principle is used as a tool for derivation of a proper approximation of the velocity distribution function in Theorem 2. Section 5 contains the main results of the paper. The closed set of scaled higherorder moment equations are given in Theorem 4 and Proposition 5, whereas the asymptotic limit of the higher-order model is given in Theorem 6. We also provide a compatibility condition which restricts the form of the equation of state in Proposition 7. Finally, the higher-order diffusion model is provided in Proposition 8. The paper ends with appropriate conclusions and outlook to possible further studies.

Kinetic modelling of mixtures

In this section, the basics of mixture modelling in kinetic theory will be presented. This will serve as a reference for further study of the scaled equations and the maximum entropy principle.

Boltzmann equations for mixtures

Consider a mixture of gases consisted of S identifiable species. The state of each species is determined by the velocity distribution function

f i (t, x, v) ≡ f i (v), i = 1, . . . , S.
If the external forces are negligible, the behavior of a non-reactive mixture is described by the system of Boltzmann equations:

∂ t f i + v • ∇ x f i = S j=1 Q ij (f i , f j )(v), 1 ≤ i ≤ S, (1)
where for any 1 ≤ i, j ≤ S, Q ij (f i , f j )(v) is the collision operator which determines the rate of change of distribution functions due to particle collisions of species i and j. It has the form:

Q ij (f i , f j )(v) = R 3 S 2 f i (v )f j (v * ) -f i (v)f j (v * ) B ij (v, v * , σ) dσ dv * , (2)
where B ij (v, v * , σ) are the collision cross sections. These cross sections are assumed for simplicity to correspond to Maxwell molecules, meaning that

B ij (v, v * , σ) = b ij (cos θ). (3)
The collision operators can be written under a weak form as follows. For any φ(v) such that the following integrals make sense, we have for any

1 ≤ i, j ≤ S R 3 Q ij (f i , f j )(v)φ(v) dv = R 3 R 3 S 2 b ij (cos θ)f i (v)f j (v * )(φ(v ) -φ(v)) dσ dv * dv. ( 4
)
Choosing φ(v) = 1, we obtain the obvious conservation property

R 3 Q ij (f i , f j )(v) dv = 0. (5)

Moment equations and moments

The moment method is one of the standard methods of analysis of the Boltzmann equations. It is based upon solving a finite number of transfer equations for the moments of the velocity distribution function, the so-called moment equations, instead of solving the Boltzmann equation itself. In other words, the solution of the Boltzmann equations is approximated by solving the corresponding set of moment equations. The moment equations are derived from the Boltzmann equations and actually correspond to their weak formulation. Starting from the Boltzmann equations (1), after multiplication with a test function ψ i (v) and integration over velocity space, one obtains the set of moment equations for any 1

≤ i ≤ S ∂ t R 3 ψ i (v)f i dv+∇ x • R 3 vψ i (v)f i dv = S j=1 R 3 ψ i (v)Q ij (f i , f j )(v) dv. (6)
The moments of the velocity distribution function R 3 ψ i (v)f i dv may correspond to the densities of observable macroscopic quantities, but they could also be tensorial quantities which may or may not have apparent physical meaning.

Choice of the moments/densities which will be taken into account is crucial for the modelling and the accuracy of approximation. In a standard way, physically motivated moments for species i are the partial mass ρ i , momentum ρ i u i k , 1 ≤ k ≤ 3 and energy ρ i E i densities

ρ i (t, x) := R 3 m i f i dv, ρ i (t, x)u i k (t, x) := R 3 m i v k f i dv, ρ i (t, x)E i (t, x) := R 3 1 2 m i |v| 2 f i dv. ( 7
)
For higher-order models, higher-order moments have to be taken as densities.

To this end, for any species i, partial momentum fluxes P i k and their corresponding fluxes P i k n , 1 ≤ k, , n ≤ 3 will be introduced as follows

P i k (t, x) := R 3 m i v k v f i dv, P i k n (t, x) := R 3 m i v k v v n f i dv. ( 8
)
By defining the peculiar velocities c i k := v k -u i k we may observe the following relations:

ρ i E i = 1 2 ρ i |u i | 2 + ρ i ε i , P i k = ρ i u i k u i + p i k , P i k n = ρ i u i k u i u i n + u i k p i n + u i p i nk + u i n p i k + p i k n , (9)
where the partial pressures p i , partial pressure tensors p i k and partial nonconvective fluxes of the momentum fluxes are defined as follows:

3p i (t, x) := R 3 m i |c i | 2 f i dv, p i k (t, x) := R 3 m i c i k c i f i dv, p i k n (t, x) := R 3 m i c i k c i c i n f i dv. ( 10
)
Note that the partial pressures p i are related to the trace of partial pressure tensors p i k by

p i = 1 3 3 k=1 p i kk . (11)
In the case of monatomic gases, we have the following relation between the partial internal energy densities and the partial pressures

3p i = 2ρ i ε i .
For further computations, it will be useful to split the partial pressure tensors into a sum of spherical part, proportional to identity tensor, and deviatoric part p i k :

p i k = p i δ k + p i k , (12)
where δ k is the Kronecker delta. This has the obvious consequence that 3 k=1 p i kk = 0.

Kinetic approach to the Maxwell-Stefan model

Let us give a brief summary of the existing results about the Maxwell-Stefan diffusion model, obtained from the moment equations [START_REF] Daus | Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type[END_REF]. To derive the classical Maxwell-Stefan diffusion model, one has to use moment equations for the mass densities ρ i and for the momentum densities ρ i u i k [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF]. For derivation of the non-isothermal Maxwell-Stefan model, with or without chemical reactions, apart from moment equations for mass and momentum densities, one has to include the moment equations for energy densities ρ i E i [START_REF] Hutridurga | Existence and uniqueness analysis of a nonisothermal cross-diffusion system of Maxwell-Stefan type[END_REF][START_REF] Anwasia | On the formal derivation of the reactive maxwell-stefan equations from the kinetic theory[END_REF][START_REF] Anwasia | On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting[END_REF]. More precisely, in the standard approximation (the so-called asymptotic limit, to be presented in the sequel), it is sufficient to use the energy conservation equation for the whole mixture.

All the approximations mentioned above assume that the velocity distribution functions are in the form of a Maxwellian, i.e. in some kind of local equilibrium in which dissipative effects (viscous and thermal) are neglected. However, if one takes them into account, higher-order moments and their corresponding transfer equations have to be exploited. Our study of higherorder Maxwell-Stefan models will be limited to the extension which takes into account only viscous dissipation through moment equations for partial momentum fluxes P i k .

Diffusive scaling

In the analysis of particular processes, not all the terms in the governing equations have equal importance, i.e. some of them may be neglected. Such simplifications require insight into the features of the process which is analyzed and appropriate order-of-magnitude estimates. A usual way to reach that goal is to put the governing equations into dimensionless/scaled form. This leads to a reduction of the number of parameters in the model, and yields an estimate of the order of magnitude of particular terms. Diffusion processes in gaseous mixtures possess two important features: (i) they occur in the hydrodynamic setting, in which the characteristic macroscopic length scale is much larger than the mean free path of the particles, and (ii) the characteristic macroscopic velocity is much smaller than the reference molecular velocity, which is of the order of the speed of sound. A scaling of the governing equations which reflects these features is called diffusive scaling.

Boltzmann equations in diffusive scaling

The dimensionless form of the Boltzmann equations ( 1) is obtained by means of scaling variables similarly as in [START_REF] Anwasia | Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model[END_REF]. To this end, the macroscopic time and length scales are introduced, denoted τ and L respectively, as well as the reference kinetic temperature T 0 . They imply two independent velocity scales: u 0 = L τ , which is the speed of macroscopic transport of the gas over distance L in time τ , and

c 0 = 5 3 k B m0 T 0 1/2
, which is the speed of sound in a monatomic gas, with m 0 being the average atomic mass of the mixture and k B the Boltzmann constant. Using τ , L and c 0 as reference time, space and velocity scales, respectively, one may introduce the following dimensionless quantities:

t = t τ , x = x L , v = v c 0 , f i (v) = L 3 c 3 0 N f i (v), Bij (v, v * , σ) = 1 c 0 4πr 2 B ij (v, v * , σ), Qij ( f i , f j )(v) = L 3 c 3 0 N 2 1 c 0 4πr 2 1 c 3 0 Q ij (f i , f j )(v),
where N is the number of gas molecules in a volume L 3 and r is the average radius of the molecules. In such a way, equations (1) acquire the following dimensionless form:

Ma∂ t f i + v • ∇ x f i = 1 Kn S j=1 Qij ( f i , f j )(v), (13)
where

Ma = u 0 c 0 , Kn = mean free path macroscopic length scale = L 3 N × 4πr 2 × 1 L = L 2 4πr 2 N ,
are the Mach and the Knudsen numbers, respectively. In this paper, we focus on the diffusive scaling, which is a particular form of dimensionless Boltzmann equations [START_REF] Chen | Analysis of an incompressible Navier-Stokes-Maxwell-Stefan system[END_REF] in which the Mach and Knudsen numbers are assumed to be of the same small order of magnitude

Ma = Kn = α 1. ( 14
)
With this assumption, we obtain the Boltzmann equations in diffusive scaling:

α∂ t f i + v • ∇ x f i = 1 α S j=1 Q ij (f i , f j )(v), (15)
where we now dropped the hats for readability. The factor 1/α on the righthand side is typical for a hydrodynamic limit of the Boltzmann equations, Higher-order Maxwell-Stefan model of diffusion while multiplication of the time derivative with α implies that the processes are slow.

Moment equations in diffusive scaling

Starting from the Boltzmann equations in the diffusive scaling [START_REF] Anwasia | On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting[END_REF], after multiplication with a test function ψ i (v) and integration over the velocity space, one obtains the (dimensionless) moment equations in diffusive scaling:

α∂ t R 3 ψ i (v)f i dv + ∇ x • R 3 vψ i (v)f i dv = 1 α S j=1 R 3 ψ i (v)Q ij (f i , f j )(v) dv. ( 16
)
The choice of the moments, i.e. of the test functions, determines the state space on which the Boltzmann equations are projected and their solution is approximated. To close the system of moment equations ( 16) one has to approximate the velocity distribution function. Grad [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF] originally used Hermite polynomial expansions as an approximation, but it was shown by Kogan [18] that equivalent results may be obtained by means of a variational approach, the so-called maximum entropy principle.

The maximum entropy principle

The maximum entropy principle (MEP) is a constrained variational formulation which determines the approximate velocity distribution [START_REF] Müller | Extended Thermodynamics[END_REF][START_REF] Dreyer | Maximisation of the entropy in nonequilibrium[END_REF][START_REF] Kremer | An Introduction to the Boltzmann Equation and Transport Processes in Gases[END_REF]. The functional to be maximized is the kinetic entropy H(t, x), defined as:

H(t, x) := S i=1 H i (t, x), H i (t, x) := -k B R 3 f i log(b i f i ) dv, ( 17
)
where b i is a dimensional constant used to make dimensionless the argument of the log function. The constraints are taken to be the moments/macroscopic variables which determine the state space. To derive the Maxwell-Stefan model of diffusion it is sufficient to choose the partial mass, momentum and energy densities as constraints. This implies the approximate velocity distribution functions to be the local Maxwellians. In this study we extend the usual state space to include the pressure tensor in it. Such an extension will provide us with a more detailed insight into dissipation, other than diffusion, which may occur in the system.

In the sequel we shall follow the procedure given in [START_REF] Anwasia | Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model[END_REF] and exploit the diffusive scaling to derive properly scaled velocity distribution functions f i . To that end we introduce the following dimensionless (scaled) variables:

Ĥi = L 3 k B N H i , bi = N L 3 c 3 0 b i , mi = m i m 0 , ρi = L 3 m 0 N ρ i , ûi = u i u 0 , ĉi = c i c 0 , Êi = E i c 2 0 , T = T T 0 , P i k = L 3 m 0 N c 2 0 P i k , pi k = L 3 m 0 N c 2 0 p i k , P i k n = L 3 m 0 N c 3 0 P i k n , (18)
and we again drop the hats in further computations for convenience. With these, equations ( 7) and ( 8) are the same, except for the substitution u i k → αu i k , whereas the scaled kinetic entropy [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF] reads:

H(t, x) = S i=1 H i (t, x), H i (t, x) = - R 3 f i log(b i f i ) dv. ( 19
)
The MEP can now be formulated as a variational problem with constraints in diffusive scaling.

For the completeness of the study we shall firstly recover the local equilibrium velocity distribution functions, and then derive their higher-order approximations which comprise the partial pressure tensors.

The local equilibrium approximation

The MEP for the local equilibrium approximation consists in finding the velocity distribution functions f i (t, x, v) maximizing the kinetic entropy [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF] subject to the following constraints:

ρ i = R 3 m i f i dv, αρ i u i k = R 3 m i v k f i dv, α 2 ρ i |u i | 2 + 2ρ i ε i = R 3 m i |v| 2 f i dv. (20)
In [START_REF] Dreyer | Maximisation of the entropy in nonequilibrium[END_REF] we dropped (t, x) dependence of macroscopic variables.

Theorem 1 The velocity distribution functions which maximize the entropy [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF] with constraints (20) have the following form:

f i (t, x, v) = ρ i m i 3 4πε i 3/2 exp - 3|v -αu i | 2 4ε i . ( 21
)
Proof Although the proof follows the same steps as in [START_REF] Anwasia | Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model[END_REF], we shall briefly repeat them. Let us define the following extended functional

R 3 L v, f i , ∇vf i dv := R 3 S i=1 f i log(b i f i ) + λ i ρ m i f i + 3 k=1 λ i u k m i v k f i + λ i ε m i |v| 2 f i dv,
where λ i ρ (t, x), λ i u k (t, x), λ i ε (t, x) ∈ R are the unknown multipliers. Taking into account that L is independent of ∇vf i , the necessary condition for extremum reduces to ∂L/∂f i = 0, i = 1, . . . , S, yielding S uncoupled equations whose solutions read for any 1

≤ i ≤ S f i = 1 b i exp -1 + m i λ i ρ + m i 3 k=1 λ i u k v k + m i λ i ε |v| 2 .
By inserting these results into the constraints [START_REF] Dreyer | Maximisation of the entropy in nonequilibrium[END_REF] and with rather straightforward computations the following relations are obtained:

λ i ε = 3 4m i ε i , λ i u k 2λ i ε = -αu i k , m i b i exp -1 -λ i ρ exp m i 4λ i ε 3 k=1 λ i u k 2 π m i λ i ε 3/2 = ρ i .
Returning these expressions into the form of f i , one obtains [START_REF] Kremer | An Introduction to the Boltzmann Equation and Transport Processes in Gases[END_REF], which completes the proof.

The higher-order approximation

Higher-order approximations of the velocity distribution functions aim at extending the state space and capturing non-equilibrium effects. To that end, the kinetic entropy [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF] remains the same, whereas the constraints have the following form:

ρ i = R 3 m i f i dv, ( 22
)
αρ i u i k = R 3 m i v k f i dv, (23)
α 2 ρ i u i k u i + p i k = R 3 m i v k v f i dv. ( 24
)
In the sequel it will be more convenient to use the peculiar velocities

c i k = v k -αu i
k and formulate the MEP for the constraints:

ρ i = R 3 m i f i dc, 0 = R 3 m i c i k f i dc, p i k = R 3 m i c i k c i f i dc. ( 25
)
Note that dv = dc holds.

The extended functional for the constraints ( 22)-( 24) is

R 3 S i=1 f i log(b i f i ) + λ i ρ m i f i + 3 k=1 λ i u k m i v k f i + 3 k, =1 λ i P k m i v k v f i   dv. ( 26
)
The same extended functional written for u i k = 0, i = 1, . . . , S, corresponds to MEP for the constraints ( 25):

H = R 3 S i=1 f i log(b i f i ) + λi ρ m i f i + 3 k=1 λi u k m i c i k f i + 3 k, =1 λi p k m i c i k c i f i   dc, (27)
but with different multipliers. However, since ( 26) and ( 27) must have the same value due to Galilean invariance, comparison leads to the following relations between the multipliers:

λ i ρ = λi ρ -α 3 k=1 λi u k u i k + α 2 3 k, =1 λi p k u i k u i , λ i u k = λi u k -2α 3 =1 λi p k u i , λ i P k = λi p k = λi p k . (28)
Therefore, the whole analysis of the higher-order approximation can be based upon the MEP for the constraints [START_REF] Ruggeri | On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single-and multi-temperature models[END_REF], and expressed afterwards in terms of ( 22)-( 24) if needed.

Theorem 2 The velocity distribution functions which maximizes the entropy [START_REF] Grad | On the kinetic theory of rarefied gases[END_REF] with constraints (25) have the following form:

f i (t, x, c) = ρ i m i ρ i 2π 3/2 1 (det p i ) 1/2 exp - ρ i 2 c i T (p i ) -1 c i , (29)
where

p i = p i k 3 k, =1
is the partial pressure tensor, and superscript T stands for the transpose of the vector.

Proof The strategy of the proof is the same as in Theorem 1. Here, it will be based upon the extended functional [START_REF] Bothe | Continuum thermodynamics of chemically reacting fluid mixtures[END_REF] which we shall briefly denote as:

H = R 3 L c, f i , ∇cf i dc.
Since L is independent of ∇cf i , the necessary condition for extremum reduces to ∂L/∂f i = 0, i = 1, . . . , S, implying S uncoupled equations whose solutions are

f i = 1 b i exp   -   1 + m i λi ρ + m i 3 k=1 λi u k c i k + m i 3 k, =1 λi p k c i k c i     . (30)
To determine the multipliers, we have to compute the moments ( 25), but their direct computation using (30) is cumbersome. Therefore, the proof will be completed in two steps.

First, let us prove that λi u k = 0, i = 1, . . . , S, k = 1, 2, 3. To this end, we shall first rewrite (30) as

f i = 1 b i exp -1 + m i λi ρ + m i λiT u c i + m i c iT λi p c i , (31)
where λi u = λi

u k 3 k=1
is the vector of momentum multipliers and λi p = λi

p k 3 k, =1
is the matrix of pressure tensor multipliers. Since λi p is a real symmetric matrix, it can be diagonalized by means of an orthogonal matrix R satisfying R T R = RR T = I. Let us introduce the orthogonal transformation in the space of peculiar velocities

C i = R i c i ⇒ c i = R iT C i . (32)
This implies

c iT λi p c i = C iT R i λi p R iT C i = C iT Λi p C i = 3 k=1 Λi k (C i k ) 2 , ( 33
)
where we introduced the transformed diagonal matrix of multipliers

Λi p = R i λi p R iT = diag Λi k 3 k=1 .
Further, by means of same transformation, we obtain

λiT u c i = λiT u R iT C i = LiT u C i = 3 k=1 Li k C i k , (34)
where we introduced the transformed vector of multipliers

Li u = R i λi u = Li k 3 k=1 . ( 35
)
Taking into account (33) and (34), the velocity distributions (31) may be finally transformed into

f i = 1 b i exp -1 + m i λi ρ + m i 3 k=1 Li k C i k + m i 3 k=1 Λi k (C i k ) 2 = 1 b i exp -1 + m i λi ρ 3 k=1 exp -m i Li k C i k + Λi k (C i k ) 2 . ( 36
)
For the computation of constraints [START_REF] Ruggeri | On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single-and multi-temperature models[END_REF] we have to take into account that dc i = dC i , since Jacobian of the transformation (32

) is |J| = |R iT | = 1.
In what follows, we shall compute the constraints (25) 2 . Applying (32) 2 in component form, c i n = 3 j=1 R i jn C i j , and taking advantage of (36) which facilitates a simple application of Fubini's theorem, after straightforward computation we obtain

0 = - m i b i exp -1 + m i λi ρ π m i 3/2   3 j=1 R i jn Li j 2 Λi j   × 3 k=1 1 Λi k exp m i ( Li k ) 2 4 Λi k .
The only possibility to satisfy these constraints is to impose

3 j=1 R i jn Li j 2 Λi j = 0,
which amounts to a homogeneous system of linear algebraic equations. Since |R iT | = |R i | = 1, the matrix of coefficients is non-singular and there exists only a trivial solution Li

j 2 Λi j = 0 ⇒ Li j = 0. ( 37
)
Using ( 35) and (37), by applying the same regularity arguments we arrive to

Li u = R i λi u = 0 ⇒ λi u = 0, which is equivalent to λi u k = 0.
This result implies a simplified form of the velocity distribution functions (30)

f i = 1 b i exp   -   1 + m i λi ρ + m i 3 k, =1 λi p k c i k c i     . ( 38
)
The second step consists in computation of the constraints (25) 1 and (25) 3 using (38), which leads to

ρ i = m i b i exp -1 -m i λi ρ π m i 3/2 det λi p -1/2 , ( 39
)
p i k = 1 2b i exp -1 -m i λi ρ π m i 3/2 adj λi p k det λi p -3/2 , ( 40
)
where adj (A) denotes the classical adjoint of a matrix A. Using (39) and taking into account that adj λi

p k det λi p -1 = λi p -1 k ,
we can reduce (40) to

p i k = 1 2 ρ i m i λi p -1 k ⇔ p i = 1 2 ρ i m i λi p -1 , (41)
or, equivalently

λi p = 1 2 ρ i m i p i -1 . (42)
Using ( 39) and ( 42) in (38), after some direct computations one easily obtains [START_REF] Hutridurga | Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting[END_REF], which completes the proof.

Remark 1 The result of Theorem 2 presents a generalization of the 10 moments approximation of the velocity distribution function given in [START_REF] Levermore | Moment closure hierarchies for kinetic theories[END_REF] to the case of mixtures. We here provide the computations in the scaled form with a more detailed proof.

Theorem 2 implies a simpler structure of the flux of momentum fluxes P i k n . Indeed, if the velocity distribution function f i has the form (29), since p i k n is given by (10) 3 , a simple parity argument implies that p i k n = 0. We thus have the following corollary.

Corollary 3 For the velocity distribution function determined by the higher-order approximation (29), the scaled total flux of the momentum fluxes reads:

P i k n = α 3 ρ i u i k u i u i n + α u i k p i n + u i p i nk + u i n p i k . ( 43
)

Higher-order Maxwell-Stefan model

In what follows, we shall derive the higher-order Maxwell-Stefan model of diffusion. It will be based upon higher-order moment equations in dimensionless form, keeping up the scaling provided by the general form [START_REF] Anwasia | Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model[END_REF]. They will provide a clear order of magnitude estimate for all the ingredients in moment equations, and thus make further approximations formally consistent.

Our aim is to include viscous dissipation by means of moment equations for the momentum fluxes, and analyze different levels of approximation which may be used to describe diffusion processes.

The moment equations

The set of moment equations which we need for generalization of the Maxwell-Stefan model consists of the mass, momentum, momentum flux and energy balance laws for the species (Theorem 4). In the so-called asymptotic limit, relevant to the derivation of Maxwell-Stefan equations in usual form, we shall need only the energy conservation law for the mixture (Proposition 5), rather than balance laws for each species.

Theorem 4 The Boltzmann equations (15) formally lead to the following set of moment equations, consisting for each species in • the mass conservation equations

α ∂ t ρ i + 3 k=1 ∂x k (ρ i u i k ) = 0, 1 ≤ i ≤ S. (44)
• the momentum balance laws

α 2 ∂ t (ρ i u i ) + 3 k=1 ∂x k (ρ i u i k u i ) + 3 k=1 ∂x k p i k = S j=1 2π b ij L 1 m i + m j ρ i ρ j (u j -u i ), 1 ≤ ≤ 3, 1 ≤ i ≤ S. (45)
• the momentum flux balance laws

∂ t (α 3 ρ i u i k u i + αp i k ) + 3 n=1 ∂x n (α 3 ρ i u i k u i u i n + α(u i k p i n + u i p i nk + u i n p i k )) = S j=1 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j (αρ i u i k u i + 1 α p i k ) + m i αρ i ρ j (u i k u j + u i u j k ) + m j ρ i (αρ j u j k u j + 1 α p j k ) + m j A k (m i + m j ) 2 αρ i ρ j |u i -u j | 2 + 3 α ρ j p i + 3 α ρ i p j , 1 ≤ k, ≤ 3, 1 ≤ i ≤ S. (46)
Moreover, the balance laws (46) imply the following energy balance law for each species

∂ t (α 3 ρ i |u i | 2 + 3αp i ) + 3 n=1 ∂x n (α 3 ρ i |u i | 2 u i n + α(2 3 k=1 u i k p i kn + 3u i n p i )) = S j=1 2π b ij L 1 (m i + m j ) 2 1 α -6m i ρ j p i + 6m j ρ i p j + αρ i ρ j -2m i |u i | 2 + 2(m i -m j )u i • u j + 2m j |u j | 2 , 1 ≤ i ≤ S. ( 47
)
Proof We prove each of the balance laws separately, by taking moments of the Boltzmann equations.

• In a very standard way, mass balances are obtained for each species from ( 16) with ψ i (v) = m i , using the moments of the distribution functions ( 22)-( 23) and the conservation properties of the collision kernel (5) to obtain (44).

• For the momentum balances, the computations are again standard [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF]. We choose ψ i (v) = m i v in ( 16) for any 1 ≤ ≤ 3, and use ( 23)- [START_REF] Truesdell | Rational Thermodynamics[END_REF], which leads to

α∂ t (αρ i u i ) + 3 k=1 ∂x k (α 2 ρ i u i k u i + p i k ) = 1 α S j=1 R 3 m i v Q ij (f i , f j )(v) dv.
Let us compute the term in the right hand side of this last equation. Using the weak form (4) with φ(v) = m i v , we obtain

M ij := R 3 m i v Q ij (f i , f j )(v) dv = m i m j m i + m j R 3 R 3 f i (v)f j (v * )(v * -v ) dv * dv S 2 b ij (cos θ) dσ + m i m j m i + m j R 3 R 3 f i (v)f j (v * )|v -v * | dv * dv S 2 b ij (cos θ)σ dσ, since v -v = m j m i + m j (v * -v ) + |v -v * |σ .
The two integrals in σ are computed using spherical coordinates [START_REF] Boudin | The Maxwell-Stefan diffusion limit for a kinetic model of mixtures[END_REF], and it leads to

M ij = 2π m i + m j b ij L 1 R 3 R 3 m i f i (v)m j f j (v * )(v * -v ) dv * dv = α 2π m i + m j b ij L 1 (ρ i ρ j u j -ρ j ρ i u i ).
The momentum balance laws thus become (45).

• To compute the momentum flux balances we choose 16) for any 1 ≤ k, ≤ 3, using ( 24)-( 43), one obtains:

ψ i (v) = m i v k v . From (
α∂ t (α 2 ρ i u i k u i + p i k ) + 3 n=1 ∂x n (α 3 ρ i u i k u i u i n + α(u i k p i n + u i p i nk + u i n p i k )) = 1 α S j=1 R 3 m i v k v Q ij (f i , f j )(v) dv.
Again, let us focus on the computation fo the source term, using again the weak form [START_REF] De Groot | Non-equilibrium Thermodynamics[END_REF]

with φ(v) = m i v k v Q ij k := R 3 m i v k v Q ij (f i , f j )(v) dv = R 3 R 3 S 2 m i b ij (cos θ)f i (v)f j (v * )(v k v -v k v ) dσ dv * dv = R 3 R 3 2π 0 π 0 m i f i (v)f j (v * )(v k v -v k v )b ij (cos θ) sin θ dθ dϕ dv * dv (48)
where the last equality is the change to spherical coordinates. Using the collision rules, the term v k v becomes

v k v = 1 (m i + m j ) 2 (m i v k + m j v * k + m j |v -v * |σ k )(m i v + m j v * + m j |v -v * |σ ) = 1 (m i + m j ) 2 m 2 i v k v + m i m j (v k v * + v v * k ) + m 2 j v * k v * + m j |v -v * | m i v k σ + m j v * k σ + m i v σ k + m j v * σ k + m 2 j |v -v * | 2 σ k σ (49)
Let us first show that by parity arguments, the terms of the last but one line in the previous expression are all zero when integrated as in (48). Indeed, since all terms involving the velocities do not depend on θ nor ϕ, we have to handle terms of the form ηb ij (η) dη = 0, since the function b ij is even. Now, let us handle the integration of the last term in (49). The only terms depending on σ are of the form

A k := 2π 0 π 0 b ij (cos θ) sin θσ k σ dθ dϕ.
With the same argument as before, the integration in ϕ leads to zero for the terms with k = . It remains to handle the terms A kk . We have, using trigonometry relations and the same change of variables as before

A 11 = π 0 b ij (cos θ) sin 3 θ dθ 2π 0 cos 2 ϕ dϕ = π 0 b ij (cos θ) sin 3 θ dθ 2π 0 1 + cos(2ϕ) 2 dϕ = π 1 -1 b ij (η)(1 -η 2 ) dη = π b ij L 1 -B ij ,
where we defined

B ij := 1 -1 η 2 b ij (η) dη.
With the same reasoning, we can prove that A 22 = π b ij L 1 -B ij . For A 33 , we have, using again the same change of variables

A 33 = π 0 b ij (cos θ) sin θ cos 2 θ dθ 2π 0 dϕ = 2π 1 -1 η 2 b ij (η) dη = 2πB ij .
Therefore, we may summarize:

A k =        π b ij L 1 -B ij , k = = 1, 2; 2πB ij , k = = 3; 0, k = . ( 50
)
It remains to handle the terms involving only the velocities in (49). They all have a common multiplicative factor

π 0 b ij (cos θ) sin θ dθ 2π 0 dϕ = 2π b ij L 1 .
Thus, (48) becomes

Q ij k = 2π b ij L 1 (m i + m j ) 2 R 3 R 3 m i f i (v)f j (v * ) × m 2 i v k v + m i m j (v k v * + v v * k ) + m 2 j v * k v * -(m i + m j ) 2 v k v dv * dv + m 2 j (m i + m j ) 2 R 3 R 3 m i f i (v)f j (v * )(|v| 2 + |v * | 2 -2v • v * )A kk δ k dv * dv
Now, in each term, the variables v and v * can be separated. Combined with the moments ( 22)-( 24) of the distribution functions, it leads to

Q ij k = 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j (α 2 ρ i u i k u i + p i k ) + m i α 2 ρ i ρ j (u i k u j + u i u j k ) + m j ρ i (α 2 ρ j u j k u j + p j k ) + m j (m i + m j ) 2 A k ρ j (α 2 ρ i |u i | 2 + 3 n=1 p i nn ) + ρ i (α 2 ρ j |u j | 2 + 3 n=1 p j nn ) -2α 2 ρ i ρ j u i • u j
This can also be rewritten as

Q ij k = 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j (α 2 ρ i u i k u i + p i k ) + m i α 2 ρ i ρ j (u i k u j + u i u j k ) + m j ρ i (α 2 ρ j u j k u j + p j k ) + m j (m i + m j ) 2 A k α 2 ρ i ρ j |u i -u j | 2 + 3ρ j p i + 3ρ i p j
Finally, this leads to the balance law (46).

• Finally, to derive the energy balance laws we shall take into account the assumption that all the species are monatomic gases, and that relations (10) 1 hold. Thus, energy equations will be derived starting from the balance laws (46), taking into account 3 k=1 p i kk = 3p i , choosing = k and summing (46) over k to obtain:

∂ t (α 4 ρ i |u i | 2 + 3α 2 p i ) + 3 n=1 ∂x n (α 4 ρ i |u i | 2 u i n + α 2 (2 3 k=1 u i k p i kn + 3u i n p i )) = S j=1 3 k=1 Q ij kk = S j=1 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j (α 2 ρ i |u i | 2 + 3p i ) + 2m i α 2 ρ i ρ j u i • u j + m j ρ i (α 2 ρ j |u j | 2 + 3p j ) + m j α 2 ρ i ρ j |u i -u j | 2 + 3ρ j p i + 3ρ i p j , (51)
where we used that

3 k=1 A kk = 2π b ij L 1 .
Let us now compute the term inside the brace in the right-hand side of the previous relation.

-(2m i + m j )ρ j (α 2 ρ i |u i | 2 + 3p i ) + 2m i α 2 ρ i ρ j u i • u j + m j ρ i (α 2 ρ j |u j | 2 + 3p j ) + m j α 2 ρ i ρ j |u i -u j | 2 + 3ρ j p i + 3ρ i p j = α 2 ρ i ρ j -2m i |u i | 2 + 2(m i -m j )u i • u j + 2m j |u j | 2 -6m i ρ j p i + 6m j ρ i p j .
This implies the balance law (47).

The energy conservation law for the mixture is a consequence of the energy balances (47) and the properties of the collision operator. However, to write it in a form more similar to the usual macroscopic equations, one needs to define appropriate mixture variables [START_REF] Truesdell | Rational Thermodynamics[END_REF][START_REF] Ruggeri | On the hyperbolic system of a mixture of Eulerian fluids: a comparison between single-and multi-temperature models[END_REF]. To that end, we define the mass density ρ, momentum density ρu k , internal energy density ρε, pressure tensor p kn and internal energy flux q n , 1 ≤ k, n ≤ 3 of the mixture in dimensionless form:

ρ := S i=1 ρ i , ρu k := S i=1 ρ i u i k , ρε := S i=1 ρ i ε i + α 2 S i=1 1 2 ρ i |u i -u| 2 , p kn := S i=1 p i kn + α 2 S i=1 ρ i (u i k -u k )(u i n -u n ), q n := S i=1 ρ i ε i + α 2 1 2 ρ i |u i -u| 2 (u i n -u n ) + S i=1 3
k=1 p i kn (u i k -u k ). ( 52
)
We can the state the energy conservation law for the mixture.

Proposition 5 The energy balance law (47) also implies a conservation law for the whole mixture

α 3 ∂ t S i=1 1 2 ρ i |u i | 2 + α∂ t 3 2 S i=1 p i + α 3 3 n=1 ∂x n S i=1 1 2 ρ i |u i | 2 u i n + α 3 n=1 ∂x n S i=1 3
k=1 u i k p i kn + 3 2 S i=1 u i n p i = 0. ( 53
)
It can be rewritten under the following form

∂ t α 3 ρ|u| 2 + 2αρε + 3 n=1 ∂x n α 3 ρ|u| 2 + 2αρε un + 2α 3 k=1 p kn u k + 2αqn = 0. ( 54
)
Proof From the computation of the source term Q ij k , we can deduce the following relations using symmetry with respect to i and j

3 k=1 Q ii kk = 0, 3 k=1 Q ij kk + 3 k=1 Q ji kk = 0.
Therefore, summation of (47) over i annihilates the right-hand side and one ends up with the energy conservation law for the mixture (53). To recast the conservation law (54), we use appropriate macroscopic mixture variables (52). Having in mind the relation 3p i = 2ρ i ε i , the energy density of the mixture may be written as:

α 3 S i=1 ρ i |u i | 2 + 3α S i=1 p i = α 3 ρ|u| 2 + 2αρε. (55)
On the other hand, the energy flux of the mixture may be transformed to

α 3 S i=1 ρ i |u i | 2 u i n + α 2 S i=1 3
k=1 u i k p i kn + 3 S i=1 u i n p i = α 3 ρ|u| 2 + 2αρε un + 2α 3 k=1 p kn u k + 2αqn.
In such a way energy conservation law (53) becomes (54).

Asymptotic limit

We formally consider the asymptotic limit α → 0 of the scaled moment equations. In the case of local equilibrium approximation, one obtains the classical Maxwell-Stefan equations or their non-isothermal counterpart. Our aim is to formally analyze the asymptotic limit for the higher-order system of moment equations. First, an isothermal model will be analyzed to underline the difference between the standard Maxwell-Stefan model and the higherorder one. After that, an appropriate non-isothermal extension without heat conduction will be discussed.

Theorem 6 Formally, when the scaling parameter α tends to zero, the macroscopic quantities for each species ρ i , u i k and p i k = p i δ k + p i k defined by ( 22)-( 24) satisfy the following system

           ∂ t ρ i + 3 k=1 ∂x k (ρ i u i k ) = 0, ∂x p i = S j=1 2π b ij L 1 mi+mj ρ i ρ j (u j -u i ), 1 ≤ ≤ 3 p i k = 0, 1 ≤ k = ≤ 3 p i = [M -1 β ] i , 1 ≤ ≤ 3, (56)
where [M -1 β ] i is the i-th component of the product of the S × S matrix M defined by

M ij =            2π b ij L 1 (m i + m j ) 2 m j ρ i , if j = i, 2π b ii L 1 4m 2 i m i ρ i - S j=1 2π b ij L 1 (m i + m j ) 2 (2m i + m j )ρ j , if j = i,
and the vector β defined by

β i = S j=1 π (m i + m j ) 2 ×    b ij L 1 (m j -4m i )ρ j p i + 5m j ρ i p j -3m j B ij ρ j p i + ρ i p j , if = 1, 2 2 b ij L 1 -(2m i + m j )ρ j p i + m j ρ i p j + 3m j B ij ρ j p i + ρ i p j , if = 3.
In this system, the partial pressures p i for each species have to be given as functions of ρ i by an equation of state.

Proof The first relation of ( 56) is obvious, and comes straightforwardly from (44). Let us start with the proof of the last two relations of (56). At order α 0 , the momentum flux balance law (46) gives

S j=1 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j p i k + m j ρ i p j k + 3m j A kk δ k (m i + m j ) 2 ρ j p i + ρ i p j = 0
Using the decomposition of the pressure into its diagonal and traceless part, we obtain

S j=1 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j p i k + m j ρ i p j k +δ k 2π b ij L 1 (m i + m j ) 2 -(2m i +m j )ρ j p i +m j ρ i p j + 3m j A kk (m i + m j ) 2 ρ j p i +ρ i p j = 0 (57)
For fixed k, , observe that these relations, for any 1 ≤ i ≤ S, constitute a linear system for the unknowns p i k . Indeed, if we introduce the vector of pressures for all species P k = (p 1 k , p 2 k , • • • , p S k ) T , equations (57) can be rewritten as

M P k = δ k β .
Let us prove that the matrix M is invertible, by proving that its transpose is diagonally dominant, i.e. that for any 1 ≤ i ≤ S, |M ii | > j =i |M ji |. To this end, let us split the sum over j into the sum for j = i and the term for j = i in M ii to rewrite

M ii = - π b ii L 1 m i ρ i + j =i 2π b ij L 1 (m i + m j ) 2 (2m i + m j )ρ j .
Since all coefficients in the matrix M are non-negative, we can compute

|M ii | - j =i |M ji | = π b ii L 1 m i ρ i + j =i 2π b ij L 1 (m i + m j ) 2 (m i + m j )ρ j ,
which is positive as soon as one ρ i is positive. This proves the invertibility of the matrix M . We thus obtain that for any 1 ≤ i ≤ S and 1

≤ k = ≤ 3, p i k = 0.
Further, the values p i 11 = p i 22 and p i 33 are non-identically zero for all species 1 ≤ i ≤ S, and can be obtained through the inverse of M and β as

P = M -1 β , 1 ≤ ≤ 3. ( 58
)
Thus, the pressure tensor is diagonal, and

p i k = (p i + p i kk )δ k , 1 ≤ i ≤ S, 1 ≤ k, ≤ 3. ( 59
)
We can now obtain the second relation in (56). The formal limit of the momentum balance law (45) at order α 0 gives

3 k=1 ∂x k p i k = S j=1 2π b ij L 1 m i + m j ρ i ρ j (u j -u i )
Using ( 58) and (59) in the previous equation, we obtain

∂x p i + p i = S j=1 2π b ij L 1 m i + m j ρ i ρ j (u j -u i ), 1 ≤ ≤ 3. ( 60
)
Moreover, observe that if we sum over i, the right-hand side cancels, and it leads to

∂x p i + S i=1 p i = 0, 1 ≤ ≤ 3.
Theorem 6 inherits a compatibility condition which restricts the structure of partial pressures p i , which is summarized in the following Proposition. Proposition 7 In order for the system (56) to have a solution, the equation of state giving the pressures p i has to satisfy the following compatibility condition

m i p i ρ i = θ(t, x), 1 ≤ i ≤ S. (61)
This condition is in particular satisfied by the scaled ideal gas law:

p i (t, x) = 5 3 ρ i (t, x) T (t, x) m i , ( 62
)
where T is the common temperature of the species.

Proof In equations (57), choosing k = and summing over k cancels the terms involving p i kk , and for any 1 ≤ i ≤ S, it leads to

S j=1 6π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j p i + m j ρ i p j + m j (ρ j p i + ρ i p j ) = S j=1 12π b ij L 1 (m i + m j ) 2 -m i ρ j p i + m j ρ i p j = 0, since 3 k=1 A kk = 2π b ij L 1 . This means that S j=1 b ij L 1 (m i + m j ) 2 m j ρ i p j -m i ρ j p i = 0, i = 1, . . . , S. (63)
These S equations are linearly dependent, since the sum of all these relations is zero. Moreover, we can prove that they imply the compatibility condition (61). Indeed, let

ζ i := m i p i ρ i -m S p S
ρ S . Then, we can replace p i in (63) for 1 ≤ i ≤ S -1, which gives

ρ i S j=1 b ij L 1 (m i + m j ) 2 ρ j ζ j + m S p S ρ S -ρ i ζ i + m S p S ρ S S j=1 b ij L 1 (m i + m j ) 2 ρ j = 0, and for i = S ρ S S j=1 b ij L 1 (m i + m j ) 2 ρ j ζ j + m S p S ρ S -ρ S m S p S ρ S S j=1 b ij L 1 (m i + m j ) 2 ρ j = 0.
This last equation implies that

b ij L 1 (m i + m j ) 2 ρ j ζ j = 0,
which in turns implies in the previous relations, for 1 ≤ i ≤ S -1

ρ i S j=1 b ij L 1 (m i + m j ) 2 ρ j m S p S ρ S -ρ i ζ i S j=1 b ij L 1 (m i + m j ) 2 ρ j +ρ i m S p S ρ S S j=1 b ij L 1 (m i + m j ) 2 ρ j = 0,
which means that for any 1 ≤ i ≤ S -1

ζ i S j=1 b ij L 1 (m i + m j ) 2 ρ j = 0.
This implies that ζ i = 0 for any 1 ≤ i ≤ S -1, which completes the proof.

The main outcome of Theorem 6 (and Proposition 7), describing the higher-order Maxwell-Stefan model in the asymptotic limit, is that momentum exchange between the species is not balanced solely by the gradients of partial pressures. One must take into account the diagonal terms of the traceless part of the pressure tensor, determined by the set of algebraic relations.

In Theorem 6, the system (56) is a closed one when the partial pressures p i are solely determined by the densities, meaning that the quantity θ in (61) is a known constant. This corresponds to the isothermal case. In the nonisothermal case, the temperature varies and is an additional unknown of the system. In order to close the system, we need an additional equation, which is given by the asymptotic limit of the energy conservation for the mixture (53)

∂ t 3 2 S i=1 p i + 3 n=1 ∂ xn S i=1 3
k=1 u i k p i kn + 3 2 S i=1 u i n p i = 0.

Higher-order diffusion model

In the final step of higher-order description of diffusion, a more general model than the one obtained in the asymptotic limit is proposed. To motivate the forthcoming analysis, let us mention that in the mixture models relevant for diffusion, governing equations consist of the balance laws of masses and the balance laws of momenta for the species. They can be derived either in a systematic way [START_REF] Müller | Rational Extended Thermodynamics[END_REF][START_REF] Bothe | Continuum thermodynamics of chemically reacting fluid mixtures[END_REF][START_REF] Ruggeri | Classical and Relativistic Rational Extended Thermodynamics of Gases[END_REF], or in an ad hoc manner [START_REF] Kerkhof | Analysis and extension of the theory of multicomponent fluid diffusion[END_REF]. In the latter case, even viscous dissipation is introduced by assumption. Our analysis will be based upon scaled moment equations given in Theorem 4 and Proposition 5, i.e. mass balances (44), momentum balances (45), momentum flux balances (46) and energy conservation (53). They provide a clear insight into the order of magnitude of all terms needed for the construction of generalized diffusion models. In particular, to build up the model which inherits the inertia terms in the momentum balance equations (45) one has to keep all the O(α 2 ) terms. This corresponds to the models of diffusion proposed in [START_REF] Kerkhof | Analysis and extension of the theory of multicomponent fluid diffusion[END_REF][START_REF] Bothe | Continuum thermodynamics of chemically reacting fluid mixtures[END_REF]. If we want to expand this to higher order models, we have to determine the order of magnitude of terms which will be kept in the governing equations. The model we propose here will be limited to the O(α 2 ) terms in all equations.

Proposition 8 The higher-order diffusion model which inherits up to O(α 2 ) terms consists of the following set of equations:

α∂ t ρ i + α 3 k=1 ∂x k (ρ i u i k ) = 0, (64)
α 2 ∂ t (ρ i u i ) + 3 k=1 ∂x k (ρ i u i k u i ) + 3 k=1 ∂x k p i k = S j=1 2π b ij L 1 m i + m j ρ i ρ j (u j -u i ), (65) α ∂ t p i k + α 3 n=1 ∂x n (u i k p i n + u i p i nk + u i n p i k ) = α S j=1 2π b ij L 1 (m i + m j ) 2 ρ i ρ j -(2m i + m j )u i k u i + m i (u i k u j + u i u j k ) + m j u j k u j + α S j=1 m j A k (m i + m j ) 2 ρ i ρ j |u i -u j | 2 + 1 α S j=1 2π b ij L 1 (m i + m j ) 2 -(2m i + m j )ρ j p i k + m j ρ i p j k + 1 α S j=1
m j A k (m i + m j ) 2 3ρ j p i + 3ρ i p j , (66) It must be emphasized that the model (64)-(67) is independent of the higher-order Maxwell-Stefan model obtained in the asymptotic limit. In particular, O(α -1 ) terms in equation (66) (or in equation ( 46)) vanish in the asymptotic limit, and determine the higher-order corrections to the Maxwell-Stefan model as a solution of the system of algebraic equations (see Theorem 6). However, one cannot discard the same terms in equation (66), since these are now the genuine balance laws, i.e. the rate type equations, which determine the behavior of pressure tensors p i k .

Remark 2 The generalized Maxwell-Stefan model ( 64)-( 67) is non-isothermal by assumption. In the non-isothermal models of previous works [START_REF] Anwasia | Maximum entropy principle approach to a non-isothermal Maxwell-Stefan diffusion model[END_REF][START_REF] Hutridurga | Maxwell-Stefan diffusion asymptotics for gas mixtures in non-isothermal setting[END_REF], all the species bear the same temperature T . Nevertheless, it is a natural question whether the multi-temperature assumption is more appropriate in a generalized setting. Because of the terms of order O(α) in (66), the same reasoning as in Proposition 7 would lead to (61) with a correction of order O(α 2 ). However, since p i ∼ T i , it turns out that O(α 2 ) terms in T i will eventually bring O(α 3 ) contributions to the energy conservation (67), and can thus be neglected in this approximation. As a conclusion, single-temperature assumption is still appropriate in the generalized Maxwell-Stefan model when limited to O(α 2 ) terms.

Conclusion

In this study, the higher-order Maxwell-Stefan diffusion model is derived starting from the kinetic theory of mixtures. It was based upon moment equations for mass, momentum and momentum flux of the species in isothermal case, adjoined with energy conservation law for the mixture in non-isothermal case. All the equations were analyzed in the diffusive scaling. They were closed by the use of an approximate form of the velocity distribution function, obtained by means of maximum entropy principle. The aim was to incorporate the viscous dissipation in the model, in a consistent way, through a proper asymptotic analysis. The higher-order model was derived in two different forms. The first form was obtained in the asymptotic limit. It generalizes the classical Maxwell-Stefan model by including the diagonal terms of deviatoric part of the partial stress tensors, which turn out to be linear functions of partial pressures (Theorem 6). In such a way, the coupling in Maxwell-Stefan relations does not occur only through diffusion fluxes, but also through the gradients of partial pressures.

The second form was obtained when terms of order O(α 2 ) were retained in moment equations, and higher-order terms were neglected (Theorem 8). This results in a set of equations containing complete balance laws of mass and momentum of species, and truncated balance laws of momentum fluxes. On one hand, such a model includes complete partial pressure tensors and determines their behavior through the rate-type equations. On the other hand, it presents a consistent generalization of the diffusion model with respect to the order of magnitude of terms which appear in governing equations.

The higher-order model presented in this study opens several possible paths for further analysis. First, it relies on the moment method applied to the set of Boltzmann equations, and the viscous dissipation is included through the balance laws. It would be of interest to recover the approximation of partial stress tensors for Newtonian fluids. Second, the generalization obtained in the asymptotic limit leads to stronger coupling of equations. It is interesting to see the implications of this coupling through some numerical simulations and to compare the obtained results with a classical Maxwell-Stefan model.

2π 0 π 0 bπ 0 b 1 - 1

 0011 ij (cos θ) sin θσ dθ dϕ. These terms can be computed for = 1, 2, 3, with σ 1 = sin θ cos ϕ, σ 2 = sin θ sin ϕ and σ 3 = cos θ. The terms for = 1, 2 are obviously zero, because of the periodicity of trigonometric functions in the integration in ϕ. For = 3, we use the change of variables η = cos θ to obtain 2π 0 ij (cos θ) sin θ cos θ dθ dϕ = 2π

Proof

  The mass balances (64) and the momentum balances (65) are the exact moment equations (44) and (45), respectively. The momentum flux balances (66) and the energy conservation (67) are obtained from the moment equations (46) and (53), respectively, by neglecting the terms O(α 3 ).

Higher-order Maxwell-Stefan model of diffusion

Acknowledgments. This paper was prepared during the stay of Srboljub Simić at Université Paris Cité thanks to "Guest researchers' faculty programme 2022". The research was also financially supported by the Ministry of Science, Technological Development and Innovation of the Republic of Serbia (Grant No. 451-03-47/2023-01/200125) (S.S.).

Conflict of interest.

On behalf of all authors, the corresponding author states that there is no conflict of interest.


P.-Y Mechin

V Keryvin

J C Grandidier

Effect of the nano-filler content on the compressive strength of continuous carbon fibre / epoxy matrix composites

Keywords: A. Particle-reinforcement, Polymer-matrix composites (PMCs). B. Strength, Mechanical properties. D. Mechanical testing

The axial compressive strength of continuous fibres reinforced composite laminates is a key design parameter. Adding nano-fillers to polymer matrices should increase this property but compressive failure modes may change with increasing the nano-filler content. In this study, the effect of silica 10 nanoparticles content in an epoxy resin on the compressive strength of a composite with high modulus carbon fibres is studied. Significant improvement of the compressive strength is observed with increasing nano-filler content. It is also shown that a single experimental tensile test on ±45 • coupons makes it possible to estimate the compressive strength regardless of the nano-filler content. Indeed, the results of such a test include, on the one hand, a shear failure strain that indicates the type of 15 failure mode in compression, and, on the other hand, a non-linear shear behaviour that is used in a corresponding fibre micro-buckling model.

Introduction 20

Carbon Fibre Reinforced Polymers (CFRP) are widely used in numerous industrial fields (aeronautics, offshore, renewable energies, marine industry) or specific custom applications (leisure, racing cars, racing yachts). Resins play a significant role in the mechanical performances of composites, especially for compressive strength, which is one of the key material properties for engineering design.

One of the underlying compressive failure mechanisms is the interplay between fibres waviness due 25 to manufacturing and curing, and their neighbourhood that contain their displacement [START_REF] Wisnom | The effect of fibre waviness on the relationship between compressive and flexural strengths of Unidirectional composites[END_REF][START_REF] Budiansky | Compressive failure of fibre composites[END_REF]. At some stage, the containment is no longer stiff enough and fibres buckle leading to kinking bands. In this scenario, the matrix plays a crucial role, via its stiffness, and particularly its shear stiffness. The higher the stiffness of the resin is, the higher the compressive strength is. One solution to increase this property is to introduce stiff nano-fillers into the resin system to delay fiber buckling.

Historically, the use of nano-fillers in resin systems was studied as a way to enhance other physical properties (thermal or electrical conductivity,...) [START_REF] Jayasinghe | Effect of networked hybridized nanoparticle reinforcement on the thermal conductivity and mechanical properties of natural rubber composites[END_REF][START_REF] Kúdelčík | The effect of SiO 2 nano-filler on dielectric properties of epoxy resin[END_REF]. Problems linked to the manufacturing of continuous fibre composites with nano-fillers then appeared, such as the difficulty in spreading the nanofillers in the resin or the quality of the fibre/nano-fillers resin interface. Nano-fillers include nanosilica glass beads, clay nano-platelets or carbon nanotubes (CNT). Over the last decade, several approaches were considered by chemists, developing new chemical formulations offering greater rigidity or adding micro/nano-fillers linked to the thermoset network in order to limit its ability to deform.

The enhancement of the mechanical properties of polymers using nano-fillers has been addressed in various studies [START_REF] Quaresimin | Understanding the effect of nano-modifier addition upon the properties of fibre reinforced laminates[END_REF][START_REF] Uddin | Strength of unidirectional glass/epoxy composite with silica nanoparticle-enhanced matrix[END_REF][START_REF] Tsai | Investigating silica nanoparticle effect on dynamic and quasi-static compressive strengths of glass fiber/epoxy nanocomposites[END_REF][START_REF] Jumahat | Effect of Nanofillers on Thermo-Mechanical Properties of Polymers and Composite Laminates[END_REF][START_REF] Jumahat | Fracture mechanisms and failure analysis of carbon fibre/toughened epoxy composites subjected to compressive loading[END_REF][START_REF] Jumahat | Effect of silica nanoparticles on compressive properties of an epoxy polymer[END_REF][START_REF] Jumahat | Compressive properties of nanoclay/epoxy nanocomposites[END_REF][START_REF] Sprenger | Improving mechanical properties of fiber-reinforced composites based on epoxy resins containing industrial surface-modified silica nanoparticles: review and outlook[END_REF]. Jumahat et al. [START_REF] Jumahat | Effect of silica nanoparticles on compressive properties of an epoxy polymer[END_REF] used different nano-fillers contents and types using Epikote 828 and Cycom 977-20 with various nano-filler types (Silica based, CNT and Clay) to quantitatively measure the effect on the mechanical properties. An increase in stiffness from 10 % to 30 %, was found with a nano-filler content varying from 5 to 25 in weight content (wt %).

On CFRP, Tsai and Cheng [START_REF] Tsai | Investigating silica nanoparticle effect on dynamic and quasi-static compressive strengths of glass fiber/epoxy nanocomposites[END_REF] observed an 11 % increase in compressive strength with 30 % increase in nano-silica content. Quaresimin and Varley [START_REF] Quaresimin | Understanding the effect of nano-modifier addition upon the properties of fibre reinforced laminates[END_REF] reported no real increase in compressive strength with different nano fillers (nano-clay, nano-carbon fibre, co-polymers) but a higher resistance to notch sensitivity for nano-clay and nano-fibre (open-hole compression test). Uddin and Sun [START_REF] Uddin | Strength of unidirectional glass/epoxy composite with silica nanoparticle-enhanced matrix[END_REF] observed an increase of ≈ 10 % on the compressive strength with an increase of 8 % of the nano-silica content. These observations were performed on different off-axis laminates coupons, varying from ±5˚to ±15˚. Jumahat [START_REF] Jumahat | Effect of Nanofillers on Thermo-Mechanical Properties of Polymers and Composite Laminates[END_REF] performed in-plane shear tests to get the non-linear behaviour with various nanofillers content (0, 3, 7 and 13 wt %). The in-plane shear modulus increased from 4.3 GPa to 6.4 GPa (+33 %) [START_REF] Astm | D695 -Standard Test Method for Compressive properties of Rigid Plastics[END_REF]. Compressive failure strains were ranging from 1.06 % to 1.3 % with a nano-filler content varying from 0 to 13 wt % with the maximum failure strain for 7 wt %. These tests were suggesting that an optimum nano-filler content would be required to benefit from the maximum compressive strength properties.

Enhancements in compressive strength was also studied for hydrofoils design (racing yacht applications) [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF]. In that study, the effect of resin properties with or without nano-fillers (Gurit products) was compared on two different fibres -high modulus (HR40, Mitsubishi) and intermediate modulus (IM2C, Hexcel) -demonstrating a significant improvement of the compressive strength using the resin with nano-fillers (30-40 % compared to resin without nano-fillers) on IM2C fibres. On the other side, performances were not significantly improved for nano-fillers resins containing HR40. Only a 10 % improvement was observed. Moreover, the standard deviation on the experiments was sufficiently large to lead to equivalent strength design value on both resins with and without nano-fillers to get a sufficient margin of safety. Failure analyses on tests coupons suggested that the failure mechanism under compression was different for HR40 fibres with nano-fillers probably due to a weaker interface between fibre and resin. Besides, estimations of compressive strength suggested that the latter was reduced due to the nano-filler content in the resin.

The following problem arises: what is the contribution of nano-filler content on compressive strength and related damage mechanisms? Could we possibly identify a specific optimum nano-filler content as suggested by Jumahat [START_REF] Jumahat | Effect of Nanofillers on Thermo-Mechanical Properties of Polymers and Composite Laminates[END_REF], sufficiently high enough to stiffen the resin and sufficiently low enough to not modify the failure mechanism [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF]? Using estimations, can we get more insight on these results and propose a targeted nano-filler content delivering optimal performance?

In this aim, it is proposed to characterize in this study the compressive strength of several composites with a single epoxy resin and silica nano-fillers with different contents and the same manufacturing process. The paper is organised as follows. First, the theory describing the prediction of the compressive strength is recalled. Potential physical mechanisms leading to failure are reviewed. Then, the necessary mechanical tests are performed on CFRP involving an epoxy thermoset resin with four different silica nano-filler contents and high modulus carbon fibres. Finally, the experimental and estimated values of compressive strength are compared and discussed in the light of associated failure analysis.

Theory: estimation of compressive strength

Failure under plastic instability

An analytical model for estimating the compressive strength was proposed by Budiansky and Fleck [START_REF] Budiansky | Compressive failure of fibre composites[END_REF]. It requires to know the non-linear shear behaviour of the unidirectional ply (UD) γ UD (shear strain) vs. τ UD (shear stress) assumed to be described by a Ramberg-Osgood (RO)-like behaviour (Eq. ( 1)).

γ UD γ y UD = τ UD τ y UD + 3 7 × τ UD τ y UD n UD (1)
It involves material parameters G UD , n UD , γ y UD and τ y UD . G UD is the shear modulus of the composite. τ y UD is a nominal shear yield stress defined as the shear stress for which a straight line, constructed with a secant modulus reduced to 70 % of the initial shear modulus G UD , intersects the experimental curve. γ y UD is defined by

τ y UD = G UD × γ y UD .
Using the in-plane shear behaviour, the compressive strength, referred to as critical stress generating plastic instability (σ stab UD ), is defined as the maximum stress applied on UD before the micro-buckling of the fibre appears, and is given by Eq. ( 2):

σ stab UD = G UD 1 + n UD 3 7 1 n UD φ 0 /γ y UD n UD -1 n UD -1 n UD (2)
where φ 0 is the initial misalignment of the fibre (fibre waviness).

An additional contribution, referred to as a structural effect [START_REF] Wisnom | On the high compressive strains achieved in bending test on unidirectional carbon-fibre/epoxy[END_REF][START_REF] Grandidier | A fibre direction compressive failure criterion for long fibre laminates at ply scale, including stacking sequence and laminate thickness effects[END_REF], takes place at the laminate scale and involves the thickness of UD ply, the stiffness of the off-axis neighbouring plies and the deformation gradient (e.g. in bending). This mechanism adds a contribution σ stru UD .

σ stru UD = 2 r g f π e b E m E f 1 -ν 2 m V f (1 -V f ) (3)
where r g f is the gyration radius of fibre, E m and ν m are respectively the Young's modulus and the Poisson's ratio of the matrix, E f the longitudinal elastic modulus of the fibre, V f the volume fraction of fibres and e b the characteristic thickness of UD involved in the instability mode (a fraction of the total thickness of UD). For a laminate stacking under bending, e b is 0.4 times the thickness of consecutive UD plies undergoing the higher compressive stress. Details for the calculation of σ stru UD are reported elsewhere [START_REF] Grandidier | A fibre direction compressive failure criterion for long fibre laminates at ply scale, including stacking sequence and laminate thickness effects[END_REF][START_REF] Mechin | An experimental protocol to measure the parameters affecting the compressive strength of CFRP with a fibre micro-buckling failure criterion[END_REF] for sake of clarity. The radius of giration is 1.5 µm for HR40 fibre. The fibre volume fraction is set to 56 % according to the prepreg datasheets. The sum of the two contributions (σ stab UD and σ stru UD ) gives the critical stress that triggers the instability, σ crit UD , as proposed in Ref.

[16] in the form of an engineering criterion, very useful for the fast dimensioning in a design loop.

It is nevertheless grounded on a background of solid numerical simulations describing the possible instabilities [START_REF] Drapier | Structure effect and microbuckling[END_REF][START_REF] Gardin | Homogenized models for the modelling of instability in long fibre media[END_REF]. The initial misalignment angle φ 0 is considered as being constant and set to 2.0˚as previously proposed [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF]. Let us note that this value chosen does not have a significant importance in the context of the comparison of the effect of nano-fillers content on a single fibre type and the same manufacturing process on the compressive strengths of UDs.

Premature shear failure

As related in a previous work [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF], some precautions should be taken on the use of the latter model in the case of premature failure of the fibre/resin interface (or close to). Indeed, Budiansky and Fleck's model assumes that the instability grows up around fibers contained by the plastic behaviour of the resin. The micro-buckling model (see Ref. [START_REF] Budiansky | Compressive failure of fibre composites[END_REF] for details) calculates the maximum allowable shear strain around a candidate fibre for buckling, called critical shear strain at instability γ stab UD , see Eq. ( 4).

γ stab UD γ y UD = 7 3 φ 0 /γ y UD n UD -1 1/n UD + φ 0 /γ y UD n UD -1 (4)
However, the failure can occur due to some material limit which is described using the failure shear strain (γ fail UD ) used as given in Eq. ( 5), and a new stability stress σ stab-mod UD is estimated, from Argon's equilibrium equation [START_REF] Argon | Treatise on Materials Science and Technology[END_REF]:

σ stab-mod UD = τ(γ fail UD ) γ fail UD + φ 0 (5)
The stability stress (σ stab-mod

UD

) is then the compressive strength of UDs, as the premature failure of fibre/resin can not be increased, this time, by the structural effect (σ stru UD ). More details about the use of this specific case are reported in Mechin et al. [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF].

Competition between the two mechanisms

A competition between the two mechanisms may occur. Consequently, a comparison between the computed values of the shear strain at instability (γ stab UD ) and the failure shear strain measured (γ fail UD ) is required to use the relevant strength prediction. Three configurations must be considered (See Fig. 1):

• γ stab UD < γ fail UD (Configuration 1): The shear strain at instability (during compression) remains lower than the failure shear strain. In that case, the instability mode (buckling of fibre) fully develops, extended by the contribution of the structural effect (σ stru UD ):

σ crit UD = σ stab UD + σ stru UD ,
• γ stab UD > γ fail UD (Configuration 2): The shear strain at instability is larger than the failure shear strain measured. In that case, the fibre/resin failure occurs and the instability is triggered prematurely and Argon's equation (Eq. ( 5)) can be used to estimate the strength,

• γ stab UD ≈ γ fail UD (Configuration 3): The shear strain at instability and the failure shear strain are of the same order of magnitude (within experimental uncertainty). In that case, both failure mechanisms can be observed. However, if instability appears, the material limit will be reached immediately.

In the latter case, even if the instability finally occurs at the ply scale, material limit is possibly reached locally. Reaching this limit will lead to shear failure prior to the buckling propagation into the ply. In this context, we use Argon's model [START_REF] Argon | Treatise on Materials Science and Technology[END_REF] since we know the shear failure strain.

Materials and methods

Materials

The study is focused on high modulus HR40 carbon fibres [Mitsubishi] (axial modulus of 373 GPa).

The associated polymer matrix is an epoxy resin system Se84 nano Generation 2 [Gurit]. Using this resin system, four nano-filler (silica nano-particles) contents are used (See Table 1). For confidentiality reasons, the real content will not be given, so that only the normalised content is provided.

Figure 2 provides two fractography images of Se84 nano Generation 1 showing the silica nanoparticles and of Nano Generation 2 M100, where the content is the same for both resins, but the particle size is smaller for the latter. Resin systems have a glass transition temperature (T g ) around 120˚C.

Plates are manufactured from prepregs (raw materials developed by Gurit), where compaction under vacuum is operated every three plies at 3 bars. They are then cured in an autoclave, with a specific heat-up ramp (composite supplier's datasheet) and a pressure of 7 bars. The curing cycle is performed once (one-shot).

Experimental methods

Four-point bending tests were carried out according to ASTM standard [START_REF] Astm | D790 -Standard Test Method for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials[END_REF] with a universal testing machine (Instron 5567, 30 kN load cell) using a distance of 90 mm between the upper rollers (10 mm in diameter), where pure bending occurs, and a distance of 460 mm between the lower rollers.

Mono-axial strain gauges (10 mm in length -Kyowa) were glued on both the compression and the tension sides. Ten samples with the stacking sequence [+45, 0 9 , -45, 0 11 , -45, 0 9 , +45] are tested for each composite material at 5 mm/min. Three were broken with the mould side in compression, three with the vacuum bagging side in compression. The details of the analysis of the flexural tests are reported elsewhere [START_REF] Keryvin | Determination of the longitudinal non linear elastic behaviour and compressive strength of a CFRP ply by bending tests on laminates[END_REF]. Let us note the use of polyethelene pads between the roller and the composite beam surface.

Tensile tests on ±45˚samples are carried out at 2 mm/min (to have the same order of magnitude in terms of strain rate with the four-points bending test)to extract the in-plane shear behaviour of the unidirectional ply (UD). At least, five samples with the dimensions 250 x 25 x 2 mm 3 composed of 8 plies (symmetric layup) were tested for sake of reproducibility at 2 mm/min. A non-linear regression to extract the Ramberg-Osgood (RO) parameters (see. Eq. ( 1)) is made for each sample. Failure shear strains are also measured (γ fail UD ). Besides, fracture surface are observed with a camScan MV2300 (Jeol 7000 series -JSM-700F) scanning electron microscope (SEM). Acceleration voltage is set to 7 kV and image resolution is 1280x1024 pixels.

Results

Representative shear behaviours of the different nano-fillers content resins with HR40 fibres are plotted in Fig. 3. Additional results from Mechin et al. [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF] and Jumahat [START_REF] Jumahat | Effect of Nanofillers on Thermo-Mechanical Properties of Polymers and Composite Laminates[END_REF]] (with HTS40 fibre) involving different materials with nano-fillers were also reported for comparison. In Mechin et al. [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF], Se84 nano Generation 1 was considered, involving specific nano-fillers products. Due to industrial innovation, new products have been considered for the new Se84 nano-fillers Generation 2 system. Therefore, comparison between mechanical performances of Se84 nano Generation 1 and Generation 2 is also proposed.

Table 2 shows the experimental results. The in-plane shear behaviour highlights high stiffnesses for resin with nano-fillers (Se84 nano Generation 1 & Epikote with high nano-fillers content) compared to lower stiffnesses fo resin without nano-fillers (Se84 LV & Epikote with null nano-fillers content).

The variation of the nano-fillers content changes both the stiffness and the strength of the UDs under shear (M50 to M100). The higher the nano-fillers content in resin, the higher the shear elastic stiffness of UD (see Fig. 3). Se84 Nano Generation 1 behaviour is equivalent to M65 resin, except for the failure strain which is larger for M65. As the nano-fillers relative content increases from 50 % to 100 %, the shear elastic stiffness increases (+20 %) (See Table 2). It also increases the hardening exponent n UD , decreases the shear yield strain (γ y UD ) and decreases sharply the failure shear strain values (the shear strength values are significantly less affected). From a failure point of view, the failure shear strain values are slightly higher for all Se84 nano generation 2 configurations compared the Se84 nano generation 1. The scatter in failure shear strains is significantly different depending on the nano-filler content. Standard deviations are of the same order of magnitude for both Se84 nano generation 1 and generation 2.

From the parameters extracted from the RO fit, the compressive strength is estimated classically (see § 2) using Eq. ( 2) and the structural effect. The influence of the non-linear shear parameters on the estimation is rather complex since they appear as numerator or denominator in Eq. ( 2). At first glance, the experimental results and the analytical predictions for the compressive strength do not agree (see Table 2) for the four nano-fillers contents, which confirms the different scenarios depicted in Fig. 1.

Yet, the fractography analysis (see Fig. 4) shows two different mechanisms: combination of fibre buckling (top) and fibre/matrix debonding with possible weak interface (bottom). For all materials, kinking bands are observed, validating the scenario of fibre micro-buckling failure mechanism possibly initiated by the weak fibre/resin interface.

Discussion

With the classical estimation of Budiansky and Fleck [START_REF] Budiansky | Compressive failure of fibre composites[END_REF] combined with the structural effect [START_REF] Grandidier | A fibre direction compressive failure criterion for long fibre laminates at ply scale, including stacking sequence and laminate thickness effects[END_REF], three out of the four nano-fillers content resins (namely M82, M65 and M50) lead to differences between estimations and experimental values. Based on the theory developed in Section 2, an explanation of the failure scenarios is proposed.

Analysis of failure shear strains to identify the mechanisms and compression micro-buckling strength

The failure shear strains (γ fail UD ) and the shear strains at instability (γ stab UD ) are reported in Table 2. Three key points are highlighted with these results and illustrated in Fig. 5 which sums-up the evolution of the failure shear strains with their variability to the shear strain at instability (computed for a specific initial misalignment) according to the nano-fillers content in the resin.

Firstly, for resin M100, the values of failure shear strains (γ fail UD ) are much higher than shear strains at instability (γ stab UD ). Consequently, the plastic micro-buckling propagates in the ply at the mesoscopic scale. The compressive strength prediction is the sum of the micro-buckling contribution and the structural effect, and is in agreement with the experiments (as given in Table 2).

Secondly, for resin M50, the range for failure shear strains is lower than the shear strain at instability.

In that case, the compressive strength prediction is limited by the local limit of the material under shear computed with the critical failure shear strain value (eq. Eq. ( 5)).

Thirdly, for the two remaining resins (M82 & M65), the shear strain at instability is in the range of failure shear strains. The comparison of the model predictions with the experiments allows to propose a scenario. For M65 and M80, compressive strength predictions using the instability model at the ply scale (blue squares) strongly overestimate the strength (see Fig. 5). The propagation of the instability mechanism at the ply scale is not feasible, due to the material limit under shear, reached through all the ply except for the variability. This is confirmed by Argon's prediction (orange), delivering the best estimation comparatively to the micro-buckling model at the mesoscopic scale.

This analysis demonstrates that the interplay between the different failure mechanisms can be estimated for the different fibre/resin couples of the study. Required material properties for the model

(G UD , n UD , γ y UD & γ fail UD
) can be all extracted from a single tension test on ±45˚laminates to get the in-plane shear response. This remarquable conclusion offers to engineers the opportunity to quickly compare compressive strengths of various fibre/resin systems with a reduced amount of experiments, mostly independent of the lamination process.

The remaining parameter (initial misalignment) is considered now. If the compressive strength predictions using the micro-buckling model are particularly overestimated for M50 and M65, it is important to notice that the estimations have been proposed using the arbitrary assumption of an initial misalignment angle set to 2˚. From a global point of view, the measured compressive strength is slightly lower than the one predicted (by ≈ 80 MPa). This comment suggests the following question:

would the micro-buckling model at the ply-scale be able to properly predict the compression strength should the initial misalignment was different? In other words, for which value of the initial misalignment the compressive strength would be equivalent for the estimations and the experiments? This latter point is quantified in § 5.2 to allow concluding on those mechanisms.

The engineering failure criterion therefore helps in illustrating the variability of the compressive strength observed experimentally. According to the variability in failure shear strain, leading to large variations in estimated compressive strength, the two nano-fillers content resins (M65 and M82) should not be considered further for HR40 fibres. The compressive strength would be too significantly variable. On the other side, resin M100 gives significantly higher properties with a minimum variability.

Validation of the explanation proposed by a discussion on the initial misalignment per resin

In the analytical model (criterion), all the parameters were experimentally identified: in-plane shear behaviour, fibre and resin properties, thicknesses and fibre volume fraction. The initial misalignment was chosen from literature review. It remains the single parameter not measured but required for the micro-buckling model estimation (see Eq. ( 2)). It would be actually possible to measure this misalignment using e.g. Yurgartis method [START_REF] Yurgartis | Measurement of small angle fiber misalignments in continuous fiber composites[END_REF]. Our assumption was to suppose that this angle is the same for all specimens since the resin is the same, the manufacturing and curing conditions are also the same. A possible dependence on the nanofiller content is not to be excluded but it is assumed that this dependence could not explain by itself the different failure mechanisms. To validate the explanation suggested (see § 5.1), it is simply assumed that the difference between compressive strength experiments and estimations is due to the remaining unknown parameter: the initial misalignment (φ 0 ). The value for each resin is then estimated considering that the compressive strength prediction is the combination of both limit under shear and micro-buckling with structural effect and matches the mean experimental values. These estimations of φ 0 are reported in Table 3.

For M82 to M50, a larger initial misalignment (than the 2˚baseline value) is required to fit the experiments and compressive strength predictions (considering than a full micro-buckling failure occurs).

According to literature [START_REF] Jumahat | Effect of Nanofillers on Thermo-Mechanical Properties of Polymers and Composite Laminates[END_REF][START_REF] Jumahat | Fracture mechanisms and failure analysis of carbon fibre/toughened epoxy composites subjected to compressive loading[END_REF][START_REF] Jumahat | Compressive properties of nanoclay/epoxy nanocomposites[END_REF], a 2.5˚value (or larger) for the initial misalignment has never been observed for continuous carbon fibres reinforced polymers. Such values should therefore not be considered as consistent.

However, predictions using a 2˚value for the initial misalignment deliver results slightly higher than experiments. As a change of the initial misalignment value has no significant effect on the values of shear strain at instability (γ stab UD ), small variations of the initial misalignment can be proposed. The order of shear strain at instability (γ stab UD ) vs. the failure shear strains (γ fail UD ) remains the same. Therefore, the difference observed on the compressive strength regarding the nano-filler content is due to the nano-fillers which modifies the mechanism, not to the initial misalignment. A larger value of the initial misalignment set to 2.2˚offers an excellent agreement between strength predictions and experiments (see Tables 2 and4). The angle of 2.2˚is a compromise that allows to better correlate the experimental measurements with the predictions of the model, knowing that this measurement of the angle has not been realized but that this approximate value is realistic. In addition, each specimen with different nanofiller levels may differ from the others. To refine these estimations, it is essential to better understand these defects to verify that the analytical model is predictive.

Concluding remarks

This work studied the effect of the nano-fillers content in an epoxy resin system combined to a high modulus carbon fibre on the axial compressive strength of CFRP. A thermoset resin system with four different nano-fillers contents was considered. Compressive strengths values were measured by four points bending tests for the UDs. The shear behaviour was characterised using tension tests on ±45˚laminate coupons. Compressive strength values were estimated using two failure criterions dealing with the two competing mechanisms (micro-buckling failure with or without shear failure) and two critical levels of shear strain (estimated shear strain at instability and measured shear failure, respectively).

Two key points are highlighted. Firstly, the nano-fillers content has a significant effect on the compressive strength of UDs as expected. The higher the nano-fillers content, the higher the compressive strength. Secondly, both failure mechanisms are competing and possibly occurring in all the resin/fibre couples considered. When the shear strain at instability is lower than the failure shear strain, the full instability mode develops with the structural effect. At the contrary, in the opposite order for the two shear strains, premature shear failure triggers the instability mode without the structural effect. When both strains have the same values (within experimental uncertainty), the same situation is used giving also scattered estimations in accordance with experiments. The ability to estimate the compressive strength of continuous fibre composites, by selecting the adequate failure criterion based only on a single tension test on ±45˚coupons, has therefore been demonstrated. It

gives the possibility to material scientists to estimate the compressive strength performance for a given nano-fillers content with a straightforward tensile test. It also comforts the use of the microbuckling criterion for mechanical engineers in the fast dimensioning in the design step of composite structures. The approach is based on a very large number of studies on compressive strength, and it appears from our own experience that the simple test (tension on ±45˚coupons) is an interesting indicator that carries the dual information of non-linear behaviour and the strength of the fibre-matrix interface, which are the two mechanisms that compete.

Compressive strength values around 1400 ± 44 MPa were observed for the Nano Gen2 M100 resin, associated to a minimum standard deviation. This result is a significant improvement compared to existing classic low Tg epoxy resins delivering compressive strengths values around 1200 ± 90 MPa [START_REF] Mechin | Limitations on adding nano-fillers to increase the compressive strength of continuous fibre / epoxy matrix composites[END_REF] with HM carbon fibres. In a future work, an objective to continue the search for better compressive strength would be to change the type of high modulus carbon fibre. Indeed, it has been shown that in the case of the highest nano-fillers content, the failure mechanism is plastic micro-buckling (Fig. 5). Consequently, the global failure of the UD is the result of a fibre's failure in bending (in ten-

  γ UD Typical experimental curves Failure distribution for the entire set

Figure 1 .

 1 Figure 1. In-plane shear behaviour of the UD with the three configurations of the failure shear strain and its variability regarding the shear strain at instability: γ stab UD < γ fail UD (1), γ stab UD > γ fail UD (2), γ stab UD ≈ γ fail UD (3).

Figure 2 .Figure 3 .

 23 Figure 2. Fractographic SEM pictures showing the nano-fillers on NanoGen1 (left)and NanoGen2 (M100, right). Both resins have the same nanofillers content but with different sizes.

Figure 4 .

 4 Figure 4. SEM fractography images of the different resins.

Table 1 .

 1 nano-silica weight contents used for the four matrices using the Se84 nano Generation 2 epoxy system.

	sion or compression [21, Fig.1]). Changing to another fibre offering better resistance seems appropri-
	ate. HM63 (Hexcel) or M40J (Toray) are possible candidates to replace HR40 (Mitsubishi).

Table 2 .

 2 Results of the experimental results and estimations (using Eq. (2) and the structural effect) for the four composite materials considered (see the main text for details)

	Se84 nano Gen2	M100 M82 M65 M50
	Nano-filler (wt %)	100	82.4	64.9	50
	φ 0 [˚]	2.0˚2.4˚2.8˚2.8σ
	crit UD [MPa] for φ 0 1417 1245 1149 1108

Table 3 .

 3 Initial misalignment (φ 0 ) required for each resin system to match the experiments and the predictions using the micro-buckling model and structural effect contribution.

	Se84 nano Gen2	M100	M82	M65	M50
	Nano-filler (wt %)	100	82.4	64.9	50
	γ stab UD [ %] (φ 0 = 2.2˚)	1.39	1.93	1.84	2.04
	Failure mechanism	Plastic	Material	Material	Material
		instability	failure	failure	failure
	Strength prediction [MPa] (φ

0 = 2.2˚) 1340 ± 44 1198 ± 121 1234 ± 93 1186 ± 89

Table 4 .

 4 Compressive strength prediction according to each considered mechanism using an updated value for the initial misalignment (φ 0 ) set from the experiments and literature review.

Acknowledgements

The authors are grateful to L. Oudjedi (IRDL -UBS) for experimental support, A. Marchandise (Avel Robotics) for the manufacturing of the coupons and for discussion, and D. Mellier (PPRIME -ENSMA) for providing fractography images. We also thank K. Cadd and T. Chatteley (Gurit company) for providing us with the prepregs and for encouragement.

plotted for both shear strain at instability (γ stab UD -given at 2˚) and failure shear strain (γ fail UD ) with the standard deviation measured. Bottom -Compressive strength plotted regarding the value of failure shear strain vs. stability shear strain. The orange markers are shifted by 1 % for sake of clarity.


A Moussa Tankari

K Medjnoun

M Belaqziz

I Halidou

M Nouiri

K Djessas

Structural, Electrical and Optical Properties of Transparent Conductive Oxides Thin films (TCOs

 Abstract

The aim of this work is to optimize the thickness from 100 nm to 300 nm and doping ration between 1 at% and 2 at% of Al doped zinc oxide thin films deposit on soda lime glass substrate by sputtering RF-magnetron. The realization and characterization of the TCO based on Al doped ZnO nanoparticles to design a high-performance nanostructured optical window intended for photovoltaic applications generally found in the technology of second generation thin-film solar cells based on Cu (In, Ga) Se2 (CIGS) or amorphous silicon require the choice of a purely experimental approach for the optimization of the protocol for synthesizing the Al doped ZnO nanoparticles and an original experimental protocol to develop nanostructured thin films by RF sputtering from the synthesized nanoparticles. Crystal structure, optoelectronic and morphological properties of thin films were investigated by using XRD, SEM, UV-Vis-NIR. The XRD results showed that the films were polycrystalline textured, preferentially oriented along the (002) crystallographic direction normal to the film surface. The SEM images showed that the films has smooth and uniform surface with small ZnO grains. The crystallite size was found to be 27,5 nm. The films show within the visible wavelength region an optical transmittance of more than 90% and an electrical resistivity of 2×10 -4 Ω.cm for the optimum. From optical measurements, the band gap energy value varies from 3.3 to 3.5 eV.

3,2 3,3 3,4 3,5 3,6 3,7 3,8 0,00 51eV 3,2 3,3 3,4 3,5 3,6 3,7 3,8 0

1,50x10 -3 3,00x10 -3 4,50x10 -3 6,00x10 -3 3,53eV h(eV) (h)  (eV.cm 1 ) 2 ZnOAl2%100nm ZnOAl2%200nm ZnOAl2%300nm 3,46eV 3,

 Conclusion

In summary, we have prepared highly transparent Al-doped ZnO thin films on glass substrate using sputtering process at room temperature. Structural, electrical and optical properties were investigated to optimize the thickness and doping ratio for a transparent homogenous oxide films. X-ray diffraction spectra indicated that all samples of Al-doped ZnO films are polycrystalline with a hexagonal wurtzite structure with the maximum crystallite size of about 27.5 nm. The average transmittance of the optimized film is about 90% in the visible range and the band gap energy of about 3.27 eV. From this results, it is possible to conclude that Al-doped ZnO thin films deposited by sputtering process at room temperature with an optimize thickness and doping ratio of 200 nm and 2 at% can be used in optoelectronic devices especially in solar cells .

Optoelectronic measurements for different thicknesses and concentrations show that the best TCO is obtained for a thickness of 200 nm and a concentration of 2 at.% with a low resistivity of around 2.10 -4 Ω.cm and an optical transmission higher than 90% deduced from the optical transmission measurements as shown on table 1. A CIGS solar cell consists of several layers of materials, with Cu(In 1 Ga)Se 2 as the energy absorber layer sandwiched between a substrate, electrical contacts, and a buffer layer. The substrate may be made of glass or made of a polymer material that provides flexibility.

Copper indium gallium selenide (CIGS) thin film solar cells are now in production for energy conversion devices. This cell provides a high absorption coefficient, high temperature stability, and is low-cost compared to other collection materials.

Physical vapor deposition (PVD) processes are deposition processes in which atoms or molecules of a material are vaporized from a solid or liquid source, transported in the form of a vapor through a vacuum or low-pressure gaseous environment, and condense on a substrate. PVD processes can be used to deposit films of elemental, alloy, and compound materials as well as some polymeric materials.

Structural,

  Electrical and Optical Properties of Transparent Conductive Oxides Thin films (TCOs) based on Al doped ZnO nanoparticles for CIGS solar cells A Moussa Tankari, K Medjnoun, Mohamed Belaqziz, I Halidou, M Nouiri, K Djessas To cite this version: A Moussa Tankari, K Medjnoun, Mohamed Belaqziz, I Halidou, M Nouiri, et al.. Structural, Electrical and Optical Properties of Transparent Conductive Oxides Thin films (TCOs) based on Al doped ZnO nanoparticles for CIGS solar cells. 4th International Conference on Nanomaterials, Nanofabrication and Nanocharacterization (NANOMACH), Apr 2023, Fethiye, Turkey, Turkey. hal-04093431



  Electrical properties(Hall effect) The average transmittance is about 90% in the visible range. The Al doped ZnO film showed an optical band gap in the range of 3.3 to 3.5 eV for the studied samples.



  The samples are polycrystalline hexagonal wurtzite structure.  Strange preferential (002) orientation  The film was grown only with the c-axis oriented perpendicular to the surface  The calculated average crystallite sizes of ZnO thin films are of 27.5 nm. Structural and morphological properties  We observe a perfectly columnar and compact structure  The elementary chemical composition studies revealed a rather large quasistoichiometry.  A good uniformity and a high density surface without visible holes or faulty zones on the film surface.  The measured thickness of the films are of the order of some microns. An innovative experimental protocol was developed by coupling two processes:  A green chemistry synthesis process based on aqueous sol-gel avoiding supercritical conditions for the synthesis of nanoparticles for PVD target  A physical process based on RFD PVD sputtering to elaborate ultra-thin nanostructured films Context Table 1: Optoelectronic properties of the thin filmsThis work was supported by the program "Investissements d'Avenir" Labex Solstice managed by the National Research Agency under the code ANR-10-LABX-22-01 CIGS Cell structure



  Target : AZO 1 at.% and AZO 2 at.%  Substrate : glass (1 mm thickness), monocristalline silicon (100) p-type.  Substrate temperature: room temperature  R.f. Power : 60 Watt/cm 2  R.f. Generator : 13,56 MHz  Target-to-substrates distance : 75 mm  Gas : Argon with (99,9999 %) high purity.