Alexey Ph Polikarpov
email: alexey.polikarpov@gmail.com

Irina Martin Graur

Felix Sharipov
email: sharipov@fisica.ufpr.br

Sublimation and deposition in gaseous mixtures

Introduction

The sublimation and deposition processes are very important in many different natural phenomena and industrial fields. To develop efficient systems of weather and avalanche forecasting, the information about the rate of the atmospheric water vapor deposition onto snow surface and of the snow sublimation is needed [1,[START_REF] Vionnet | Simulation of wind-induced snow transport and sublimation in alpine terrain using a fully coupled snowpack/atmosphere model[END_REF]. The exploration of particularities of the solar system and especially of the evolution in time of the climatical conditions on the planets requires the development of the model which allows to estimate the rate of mass loss from ice in a planet atmosphere [START_REF] Dundas | Modeling sublimation of ice exposed by new impacts in the martian mid-latitudes[END_REF]. The solar controlled sublimation of ice has been accepted as the key process governing the cometary activity [START_REF] Crifo | Improved gas-kinetic treatment of cometary water sublimation and recondensation: application to coment P/Halley[END_REF]. The key for the explanation of most phenomena associated with the activity of comets is a proper understanding of the processes taking place within a thin layer around the nucleus/coma interface [START_REF] Skorov | A model of heat and mass transfer in a porous cometary nucleus based on a kinetic treatment of mass flow[END_REF][START_REF] Skorov | Direct statistical simulation of the near-surface layers of the cometary atmosphere. I. A spherical nucleus[END_REF]. Heat and mass transfer at sublimation of solid particles is important in the development of new technologies of chemical vapor deposition [START_REF] Igumenov | Heat and mass transfer at sublimation of a single chromium (iii) and zirconium (iv) beta-diketonate particle in the inert gas mixture[END_REF][START_REF] Tung | Vapor sublimation and deposition to build porous particles and composites[END_REF]. For intensifying and optimizing the operation modes for the low-temperature drying of pharmaceutical and food products the knowledge about the sublimation-deposition phenomena are also indispensable [START_REF] Xiang | Investigation of freeze-drying tional[END_REF][START_REF] Oddone | Influence of controlled ice nucleation on the freeze-drying of pharmaceutical products: the secondary drying step[END_REF].

The simulations of the sublimation and deposition phenomena are essentially restricted to the continuum modeling. However, the evaporation and condensation phenomena, analogous to the sublimation and deposition, have been intensively simulated using the gas kinetic theory. For single-component systems consisting of a pure vapor and its condensed phase, many successful results have been obtained and the summary of them can be found in Ref. [START_REF] Sone | Molecular Gas Dynamics[END_REF].

The case of a gas mixture of a vapor and a non-condensable gas was also actively studied in the series of papers. Several kinetic models were applied to mimic the Boltzmann collision operator in case of evaporation and condensation in mixtures, for instance, the Hamel model [START_REF] Hamel | Kinetic model for binary gas mixture[END_REF] in Ref. [START_REF] Onishi | The spherical-droplet problem of evaporation and condensation in a vapour-gas mixture[END_REF], the Carzó-Santos-Brey model [START_REF] Garzó | A kinetic model for a multicomponent gas[END_REF] in Refs. [START_REF] Sone | Kinetic theory analysis of gas flows condensing on a plane condensed phase: case of a mixture of a vapor and a noncondensable gas[END_REF], and the new kinetic model proposed in Ref. [START_REF] Kosuge | Model Boltzmann equation for gas mixtures: Construction and numerical comparison[END_REF].

The full Boltzmann collision operator was used for the same purpose in Ref. [START_REF] Aoki | Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas[END_REF]. Four main geometries were considered: two parallel condensed surfaces [START_REF] Aoki | Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of a noncondensable gas[END_REF], half-space problem (one surface) [START_REF] Taguchi | Vapor flows along a plane condensed phase with weak condensation in the presence of a noncondensable gas[END_REF], two coaxial cylinders [START_REF] Taguchi | Vapor flows in the continuum limit in the presence of a small amount of noncondensable gas[END_REF][START_REF] Yoshida | Cilindrical Couette flow of a vapor-gas mixture: Ghost effect and bifurcation in the continuum limite[END_REF], and evaporation from a sphere into surrounding mixture [START_REF] Onishi | The spherical-droplet problem of evaporation and condensation in a vapour-gas mixture[END_REF][START_REF] Chernyak | The kinetic theory of droplet evaporation[END_REF]. The particularities of the continuum limit approximation and specific phenomena appearing there, like ghost effect, were also largely discussed in Refs. [START_REF] Aoki | Vapor flows with evaporation and condensation in the continuum limit: effect of a trace of noncondensable gas[END_REF][START_REF] Taguchi | Vapor flows in the continuum limit in the presence of a small amount of noncondensable gas[END_REF][START_REF] Yoshida | Cilindrical Couette flow of a vapor-gas mixture: Ghost effect and bifurcation in the continuum limite[END_REF]. However, in most of these work the hypothetical molecular masses and hard sphere potential with hypothetical diameters were used.

Like evaporation and condensation, the sublimation and deposition phenomena can be also described on the basis of the kinetic theory. In the present paper, we develop a methodology based on the numerical solution of the Boltzmann type kinetic equation to simulate the sublimation-deposition processes.

In practice, the sublimation and depositions happen in a gas mixture, where the presence of some amount of gas, which can neither sublimate nor deposit, could affect the whole process. Therefore, the mixture of Helium with Argon was chosen for the specific temperature range, where Helium is being only in a gaseous state while Argon can sublimate and deposit. An experimentally obtained dependency of the saturation pressure on the temperature [START_REF] Ferreira | The sublimation of argon, krypton, and xenon[END_REF] is used for the chosen temperature range. In order to investigate the influence of the intermolecular potential on mass and heat transfer through the mixture in question, three potentials, namely, the Hard Sphere (HS), Lennard-Jones (LJ), and ab initio (AI) [START_REF] Sharipov | Transport coefficients of argon and its mixtures with helium and neon at low density based ab initio potentials[END_REF], are used to describe the intermolecular collisions. The corresponding numerical results are analysed and compared between them.

Problem statement

Let us consider a mixture of the monatomic gases, Helium and Argon, confined between two parallel infinite solid layers of Argon separated by a distance H. One component of this mixture, Helium, referred in the following with subscript 1, is completely reflected from the solid surface. The other component of the mixture, Argon, subscript 2, represents the gaseous phase of the solid surfaces so that it can sublimate or deposit (de-sublimate).

The lower solid surface is maintained at the temperature T L = T 0 -∆T /2, and located at y = -H/2, while the upper one is kept at the temperature T U = T 0 + ∆T /2 and located at y = H/2, where T 0 is the equilibrium temperature, ∆T is the temperature difference between the surfaces, which is small in comparison with the equilibrium temperature, i.e., ∆T T 0 . The equilibrium molar fraction of Helium, C 0 , is defined as:

C 0 = n 01 n 0 , n 0 = n 01 + n 02 , (1)
where n 0α is the equilibrium number density of species α (α = 1, 2). The equilibrium number density of the mixture, n 0 , is related to the equilibrium pressure, p 0 , through equation of state

p 0 = n 0 k B T 0 , where k B is the Boltzmann constant.
The gas rarefaction is characterized by the rarefaction parameter defined as

δ = H , = µv 0 p 0 , (2)
where is the equivalent free path [START_REF] Sharipov | Rarefied Gas Dynamics. Fundamentals for Research and Practice[END_REF], µ is the mixture viscosity [START_REF] Sharipov | Transport coefficients of argon and its mixtures with helium and neon at low density based ab initio potentials[END_REF] at the equilibrium temperature T 0 , v 0 is the characteristic molecular speed of the mixture defined as:

v 0 = 2k B T 0 m , m = C 0 m 1 + (1 -C 0 ) m 2 , (3)
m α is the molecular mass of the species α.

Since the second species of the mixture can sublimate, its pressures are different at the two surfaces. Let p L2 and p U 2 be pressures of the saturated vapor of Argon at the temperatures T L and T U , respectively. Thus, a temperature difference ∆T leads to some specific pressure difference ∆p 2 = p U 2 -p L2 . However, the pressure ∆p 2 and temperature ∆T differences can be considered independent of each other when solving the problem. Let us introduce two independent thermodynamic forces related to the pressure and temperature differences as

X P = ∆p 2 p 02 , X T = ∆T T 0 , (4)
where p 02 = (1 -C 0 ) p 0 is the partial equilibrium pressure of Argon. Since

|X P | 1, |X T | 1, (5)
the thermodynamic forces can be used as small parameters to linearize the kinetic equation.

Our aim is to calculate the mass Ṁ and energy Ė flow rates from the upper surface to lower one as function of the temperature T 0 , the molar fraction C 0 , the rarefaction parameter δ, and the differences ∆p 2 , ∆T . The problem is solved on the level of the velocity distribution functions f α (y, v α ) of the two species, where v α is the molecular velocity vector of species α, see e.g. [START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF]. The flow rates of our interest are defined via the distribution functions as

Ṁ = -m 2 v 2y f 2 (y, v α )dv 2 , (6)
Ė = - 2 α=1 1 2 m α v 2 α v αy f α (y, v α )dv α , (7)
so that the quantities Ṁ and Ė are positive. Note that these quantities are independent of the coordinate y because of the mass and energy conservation laws.

Kinetic equation

The distribution functions f α (y, v α ) needed to calculate the mass and energy flow rates are obtained from the Boltzmann type kinetic equation, which can be linearized using the assumption (5), i.e.

f α (y, v α ) = f M α (1 + h P α X P + h T α X T ), α = 1, 2, (8)
where h P α and h T α are the perturbation functions. The equilibrium distribution function, f M α , reads

f M α = n 0α m α 2πk B T 0 3/2 exp - m α v 2 α 2k B T 0 . (9)
The perturbation functions of species α obey the two coupled linearized Boltzmann type kinetic equations per species [START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF] ∂h

(i) α ∂t + v yα ∂h (i) α ∂y = 2 β=1
Qαβ h (i)

β , i = P, T, α = 1, 2, (10)
where Qαβ h

(i)
β is the linearized collision operator between the species α and β. A numerical solution of the Boltzmann equation itself represents a hard computational task, see e.g. Refs. [START_REF] Sharipov | Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number[END_REF], [START_REF] Dodulad | Knudsen pumps modeling with Lennard-Jones and ab initio intermolecular potentials[END_REF], [START_REF] Su | Accurate and efficient computation of the Boltzmann equation for Couette flow: Influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient[END_REF]. At the same time, the model kinetic equations provide reliable solution with modest computational effort [START_REF] Sharipov | Benchmark problems for mixtures of rarefied gases. I. Couette flow[END_REF][START_REF] Strapasson | Ab initio simulation of heat transfer through a mixture of rarefied gases[END_REF][START_REF] Ho | Comparative study of the Boltzmann and McCormack equations for Couette and Fourier flows of binary gaseous mixtures[END_REF].

Here, the McCormack model [START_REF] Mccormack | Construction of linearized kinetic models for gaseous mixture and molecular gases[END_REF] is used to describe the collision term. To write down this model, it is convenient to introduce the following dimensionless quantities:

t = t v 0 H , y = y H , c α = m α 2k B T 0 v α , α = 1, 2, (11)
where c α is dimensionless molecular velocity of the species α. In terms of the above introduced dimensionless variables, the linearized Boltzmann type kinetic equations [START_REF] Oddone | Influence of controlled ice nucleation on the freeze-drying of pharmaceutical products: the secondary drying step[END_REF] become

m α m ∂h (i) α ∂t + c αy ∂h (i) α ∂y = δA m α m 2 β=1
Lαβ h (i) , i = P, T, α = 1, 2, (

where

A = C 0 γ 1 + 1 -C 0 γ 2 . ( 13
)
In case of one-dimensional flow along the y-axis, the McCormack collisional term Lαβ h

(i)
β is provided in Appendix A, together with the υ

(n)
αβ functions. The dimensionless moments of the perturbation functions are given as

ν (i) α = 1 π 3/2 e -c 2 α h (i) α dc α , (14)
u (i) 2 = 1 π 3/2 m m α e -c 2 α h (i) 2 c 2y dc 2 , u (i) 1 = 0, (15)
τ (i) α = 1 π 3/2 e -c 2 α h (i) α 2 3 c 2 α -1 dc α , (16)
Π (i) α = 1 π 3/2 e -c 2 α h (i) α c 2 αy - 1 3 c 2 α dc α , (17)
q (i) α = 1 π 3/2 m m α e -c 2 α h (i) α c αy c 2 α - 5 2 dc α . (18)
The parameters γ αβ are proportional to the collision frequency between species α and β and appear in the collision term (A.1) only in the combinations γ 1 = γ 11 + γ 12 and γ 2 = γ 21 + γ 22 , so we need to define only γ 1 and γ 2 . The collision frequencies γ α can be written in the same manner as in the Shakhov kinetic model [START_REF] Shakhov | Generalization of the Krook kinetic relaxation equation[END_REF][START_REF] Sharipov | Gaseous mixture flow through a long tube at arbitrary Knudsen number[END_REF][START_REF] Sharipov | Plane Couette flow of binary gaseous mixture in the whole range of the Knudsen number[END_REF]:

γ α = p 0α µ α , (19)
where p 0α = n 0α k B T 0 is the equilibrium partial pressure and µ α is the partial viscosity given as

µ α = p 0α S β + υ (4) αβ S α S β -υ (4) αβ υ (4) βα , S α = υ (3) αα -υ (4) αα + υ (3)
αβ , and β = α. [START_REF] Yoshida | Cilindrical Couette flow of a vapor-gas mixture: Ghost effect and bifurcation in the continuum limite[END_REF] Other details on the dimensionless presentation of omega integrals (A.6) and

υ (i)
αβ functions (A.2), as well as S α can be found in [START_REF] Ho | Comparative study of the Boltzmann and McCormack equations for Couette and Fourier flows of binary gaseous mixtures[END_REF][START_REF] Polikarpov | Transient heat transfer in a rarefied binary gas mixture confined between parallel plates due to a sudden small change of wall temperatures[END_REF].

Once the dimensionless moments ( 14)-( 18) are known from the solution of Eqs. [START_REF] Hamel | Kinetic model for binary gas mixture[END_REF], the number density and temperature of species α are calculated as

n α = n 0α 1 + ν P α X P + ν T α X T , (21)
T α = T 0 1 + τ P α X P + τ T α X T . (22)
The corresponding quantities of the binary mixture are defined as

n = n 1 + n 2 , (23)
T = C 0 T 1 + (1 -C 0 )T 2 . (24)
The mass Ṁ and energy Ė flow rates can be expressed in terms of the kinetic coefficients Λ ij defined via the dimensionless moments as

Λ Pi = -(1 -C 0 )u (i) 2 , Λ Ti = -C 0 q (i) 1 -(1 -C 0 )q (i) 2 , i = P, T. (25)
Following the formalism of irreversible thermodynamics [START_REF] Sharipov | Onsager-Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. III. Theory and its application for gaseous mixtures[END_REF], the thermodynamic fluxes are represented as

J P = Λ PP X P + Λ PT X T , (26)
J T = Λ TP X P + Λ TT X T . (27)
Then, the mass Ṁ and energy Ė flow rates defined by ( 6) and ( 7) are obtained in terms of the thermodynamics fluxes as

Ṁ = n 0 m 2 v 0 J P , Ė = p 0 v 0 J T + 5 2 J P . ( 28
)
As it was shown in Ref. [START_REF] Sharipov | Onsager-Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction. III. Theory and its application for gaseous mixtures[END_REF], the matrix Λ ij is symmetric, i.e., Λ PT = Λ TP , so that the mass and energy transfer is determined by the three independent coefficients. Moreover, the entropy production σ (never negative) in the system per area unit of the surfaces is expressed in terms of the kinetic coefficients as

σ = n 0 v 0 T 0 2 i,j=1 Λ ij X i X j > 0. (29)
The matrix Λ ij must be positive definite in order to keep the entropy production positive.

Boundary conditions

According to the problem statement, the first species is diffusely scattered from each solid surface so that the boundary conditions at y = 1/2 and c 1y < 0 read

h P 1 = ν P 1w , h T 1 = 1 2 c 2 1 - 5 2 + ν T 1w , (30)
where ν P 1w and ν T 1w are calculated from the impermeability condition

ν (i) 1w = 2 π c1y>0 h (i) 1 c 1y e -c 2 1 dc 1 , i = P, T. (31)
The perturbation function of particles reflected from the lower surface is given by the symmetry condition

h (i) α (y, c 1y ) = -h (i) α (-y, -c 1y ). ( 32
)
The particles of the second species are not reflected, but they are completely absorbed by the surfaces. At the same time, the surfaces emit particles of this species with the Maxwellian distribution function. For instance, the Maxwellian of the upper surface reads

f M 2U = n U m 2 2πk B T U 3/2 exp - m 2 v 2 2 2k B T U , (33)
where n U is the density of saturated vapor at the temperature T U . Thus, the perturbation function of the emitted particle at y = 1/2 and c 2y < 0 is given as

h P 2 = 1 2 , h T 2 = 1 2 c 2 2 - 5 2 . ( 34
)
The perturbation function at y = -1/2 and c 2y > 0 has just the opposite sign according to [START_REF] Ho | Comparative study of the Boltzmann and McCormack equations for Couette and Fourier flows of binary gaseous mixtures[END_REF].

Method of solution

The Discrete Velocity Method (DVM) is used to solve the McCormack kinetic equations [START_REF] Hamel | Kinetic model for binary gas mixture[END_REF]. To reduce computational efforts, the c αz variable is eliminated by introducing the reduced functions of h

(i) α : Φ (i) α = 1 √ π m m α h (i) α exp -c 2 αz dc αz , Ψ (i) α = 1 √ π m m α h (i) α c 2 αz exp -c 2 αz dc αz , (35)
α = 1, 2 and i = P, T .

The DVM consists of splitting the continuum molecular velocity space c x , c y in Eq. ( 12) into discrete velocity sets c xm , c y k , where m, k = ±1, ... ± N c .

These velocities c xm , c y k , are taken to be the roots of the Hermit polynomial of order N c . Then the set of 2N c kinetic equations, corresponding to 2N c values of discrete velocities, is discretized in time and space by finite difference method (FDM). Here N c is taken to be equal to 20. The grid-independence in molecular velocity space is checked by using a finer grid of 50×50 points (N c = 25) resulting the variation of all macroscopic quantities within 1%. The spacial derivatives are approximated by the second-order accurate Total Variation Diminishing (TVD) type scheme as in Ref. [START_REF] Polikarpov | Unsteady rarefied gas flow through a slit[END_REF]. The number of uniformly distributed points in physical space N y is equal to 1000, which insures the numerical error of 1% of the simulations.

The time derivative is approximated by the time-explicit Euler method. The time step ∆t = 0.42, chosen according to the condition by Courant-Friedrichs-Lewy [START_REF] Courant | On the partial difference equations of mathematicalphysics[END_REF], provides the numerical error less than 1%. The stationary solution was considered for our purpose with a relative convergence criterion equal to 10 -7 for all moments of the perturbation functions.

To check the numerical code, some particular situations corresponding to previously published results have been resolved. In case C 0 = 1, the problem is reduced to the heat transfer through a single non-condensable gas reported previously in Ref. [START_REF] Graur | Comparison of different kinetic models for the heat transfer problem[END_REF]. If we assume C 0 = 0, then the single gas sublimation and deposition processes are simulated. The analogous formulation was considered in Refs. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF][START_REF] Polikarpov | Heat and mass transfer in a rarefied gas confined between its two parallel condensed phases[END_REF] in the case of evaporation and condensation. Finally, if we assume that the second species is also non-condensable, then the problem will be the same as that considered in Ref. [START_REF] Sharipov | Heat flux through a binary gaseous mixture over the whole range of the knudsen number[END_REF]. Applying the elaborated numerical codes to these particular situations, we can reproduced the results reported in

T =50 K T =70 K µ 1 × 10
Refs. [START_REF] Graur | Comparison of different kinetic models for the heat transfer problem[END_REF][START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF][START_REF] Polikarpov | Heat and mass transfer in a rarefied gas confined between its two parallel condensed phases[END_REF][START_REF] Sharipov | Heat flux through a binary gaseous mixture over the whole range of the knudsen number[END_REF] within the numerical error.

The specific temperature range was chosen, where Helium can not be in a kg. The gas viscosities of Helium and Argon calculated ab initio in [START_REF] Bich | Ab initio potential energy curve for the helium atom pair and thermophysical properties of the dilute helium gas. II. Thermophysical standard values for low-density helium[END_REF][START_REF] Sharipov | Transport coefficients of argon and its mixtures with helium and neon at low density based ab initio potentials[END_REF] are provided in Table 1 for two reference temperatures.

Intermolecular potentials

The model collision integral (A.1) contains the quantities ν

(n)
αβ expressed in terms of the Ω integrals by Eq. (A.2). In turn, these integrals are determined by the intermolecular potential via the differential cross section so that the solution of the problem considered here is influenced by the potential. To quantify this influence three types of the potential are used in the simulations: the Hard Sphere (HS), Lennard-Jones (LJ), and ab initio (AI) [START_REF] Sharipov | Energy accommodation coefficient extracted from acoustic resonator experiments[END_REF]. The HS potential is the most simple potential because it contains only one fitting parameter, namely, the molecular diameter d i . Dimensionless numerical results for a single gas based on this potential are independent of the gas species and its temperature. That is why the HS potential is widely used in rarefied gas dynamics simulations. In case of binary gaseous mixture, the Ω integrals are calculated analytically using Eq.(A.6). Since the right-hand-side of Eq. ( 12) contains only the Ω integral ratios, to solve it we need to specify only the mass ratio and diameter ratio,

d 2 /d 1 .
The last one can be expressed via the viscosities of single gases as

d 2 d 1 = µ 1 µ 2 1/2 m 2 m 1 1/4 . ( 36
)
The ratio follows from the analytical expression of viscosity for hard sphere gas [START_REF] Sharipov | Rarefied Gas Dynamics. Fundamentals for Research and Practice[END_REF]. The values of d 2 /d 1 based on the ab initio viscosities are presented in Table 1 for two reference temperatures.

The LJ potential given as

V (r) = 4ε d r 12 - d r d (nm)
/k (K) Ref. He-He 0.26410 10.996

[47] Ar-Ar 0.33577 142.94 [START_REF] Patkowski | Argon pair potential at basis set and excitation limits[END_REF] He-Ar 0.31169 29.760 [START_REF] Cacheiro | Coupled cluster calculations of the ground state potential and interaction induced electric properties of the mixed dimers of helium, neon and argon[END_REF] Table 2: Well-depth and zero-point of AI potentials.

-1 of AI potentials with the LJ one is performed in Figure 1, which shows that the AI potentials are close to the LJ one near the well, but their asymptotic behaviors at r → ∞ are different. Such a difference can cause an influence of the potential on macroscopic characteristics at some temperatures.

-0.5 0 1 1.2 1.4 1.6 1.8 2 V/ε r/d LJ He-He Ar -Ar He-Ar
To implement LJ and AI potentials, the Ω integrals have been calculated numerically by the method described in details in Ref. [START_REF] Sharipov | Transport coefficients of argon and its mixtures with helium and neon at low density based ab initio potentials[END_REF]. In case of the LJ, the well depths and zero-points given in Table 2 have been substituted into the expression [START_REF] Polikarpov | Transient heat transfer in a rarefied binary gas mixture confined between parallel plates due to a sudden small change of wall temperatures[END_REF]. The interpolating expressions given in Refs. [START_REF] Cencek | Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potencial and thermophysical properties of helium[END_REF][START_REF] Patkowski | Argon pair potential at basis set and excitation limits[END_REF][START_REF] Cacheiro | Coupled cluster calculations of the ground state potential and interaction induced electric properties of the mixed dimers of helium, neon and argon[END_REF] have been used for the AI potentials. The numerical values of the Ω integrals for both LJ and AI potentials are provided in Table 3, which shows that the discrepancy between the two potentials is about 2% for most of the integrals and it reaches 9% for Ω for C 0 = 0.9, i.e. for low Argon mole fraction. For all these temperature deviation curves, only for temperature variations due to the pressure gradient, τ P 1 curve for Helium, the difference between the two potentials is significant.

Results and discussions

For all other cases both potentials provide very similar results.

The matrix of the kinetic coefficients Λ ij , calculated with HS, AI, and LJ 14), [START_REF] Kosuge | Model Boltzmann equation for gas mixtures: Construction and numerical comparison[END_REF], vs the coordinate y at C 0 = 0.9 and T 0 = 50 K: red lines -δ = 0.1, green linesδ = 1, blue lines δ = 10; solid lines -HS, dashed lines -AI potential. potentials for equilibrium temperature T 0 = 50 K and for three equilibrium mole fractions C 0 = 0.1, 0.5, and 0.9, are provided in Tables 456, respectively.

It has been verified that the Onsager reciprocity relations, Λ PT = Λ TP , are fulfilled within the numerical accuracy so that the two coefficients are presented in one column of Tables 456. Note that the diagonal kinetic coefficients Λ PP and Λ TT are always positive, while the cross kinetic coefficients, Λ PT and Λ TP , can be negative and positive. The values of the cross coefficients are quite smaller than those of the diagonal coefficients that guarantees the matrix Λ ij to be positive definite. The results obtained with AI and LJ potentials are very close to each other for all considered rarefaction parameters and mole fraction ranges. However, the difference of the results based on the AI and HS potentials is quite larger. Analyzing the numerical data on the kinetic coefficients we can observe that the maximum discrepancy between the coefficient Λ PP obtained for the AI and HS potentials reaches 17% at C 0 = 0.9 and δ = 10. This difference decreases by decreasing the mole fraction C 0 . The same analysis for the coefficient Λ TT shows that the maximum discrepancy because of the potential is about 13% and it takes place at C 0 = 0.5 and δ = 20. As expected, the cross coefficients, Λ PT and Λ TP , are much sensitive to the intermolecular potential. The maximal difference in the cross-coefficients, Λ PT , reaches 100% for the same set of parameters, C 0 = 0.5 and δ = 20. In some cases, see e.g. C 0 = 0.5 and δ = 20, the cross coefficients obtained for the AI and HS potentials, have the opposite sign. For all kinetic coefficients, the difference in their values obtained from two potentials decreases with decreasing of the rarefaction parameter.

To study the temperature influence on the kinetic coefficients, similar calculations have been carried out for equilibrium temperature T 0 = 70 K. corresponding coefficients, obtained for the three potentials, are provided in Table 7 -9, for C 0 = 0.1, 0.5 and 0.9, respectively. These data show that the diagonal coefficients are weakly sensitive to the temperature, while the cross coefficients are strongly sensitive to the temperature T 0 for large values of the gas rarefaction.

One of the important characteristics of the sublimation-deposition process is δ

Hard Sphere ab-initio potential Lennard-Jones potential 9: Kinetic coefficients for HS, AI and LJ potentials vs rarefaction parameter δ at C 0 = 0.9 and T 0 = 70K.

Λ PP Λ PT = Λ TP Λ TT Λ PP Λ PT = Λ TP Λ TT Λ PP Λ PT = Λ TP Λ TT 0.
the sublimation-deposition rate (26) and the heat flux ( 27) through the gas-solid interface. To compare the results of the sublimation rate between the potentials considered here for two reference temperatures, T 0 = 50 K and 70 K, the corresponding saturation pressures, p s , are calculated using Eq. (B.1) proposed in Ref. [START_REF] Ferreira | The sublimation of argon, krypton, and xenon[END_REF]. This equation shows that a very small change in the temperature generates a large variation of the saturated vapor pressure. Therefore here, a very small difference of the temperature is considered, namely ∆T /T 0 = 0.004, which leads to a relatively large pressure difference, i.e., ∆p 2 /p 02 = 0.083 and 0.068 for the temperature T 0 equal to 50 K and 70 K, respectively. These values of the pressure differences are still reasonable and the linearized approach can be applied. The sublimation flow rate ( 26) and the heat transfer through interface [START_REF] Sharipov | Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number[END_REF] for different mole fractions, for different values of the rarefaction parameter and for the driving forces equal to X P = 0.083 and X T = 0.004, are provided in Tables 10 and11 for three potentials, HS, LJ, and AI. The results for the temperature 70 K and X P = 0.068 and X T = 0.004 are provided in Tables 12 and13, also for both potentials.

As it can be observed from Table 10 the sublimation flux J P is directed from the hotter surface, y = 0.5, to the colder one, y = -0.5. The absolute value of the sublimation flow rate decreases by increasing the rarefaction parameter.

In addition, this flow rate decreases also with the increasing of the Helium mole fraction, which is the natural trend because Helium does not sublimate or deposit in the considered temperature range. The values of the sublimation flow rate, obtained for the two potentials, are slightly different. Generally, the values of sublimation flow rate, obtained with AI potential are larger than that calculated with HS one, with the maximal difference of 15% at C 0 = 0.9 and δ = 20. This tendency is conserved with the reference temperature increasing up to 70K, see Table 12.

One can see from Table 11 that the heat flux J T has the direction opposite to the sublimation rate, i.e. the heat flows from the colder surface, y = -0.5, the hotter one, y = 0.5, in case of the small mole fraction of Helium C 0 = 0.1.

When the mole fraction increases up to C 0 = 0.5, the heat flux changes the sign from the rarefaction parameter equal to 1. It is worth to note that the direction of the heat flux does not contradict to any thermodynamic law because the total energy flux given by Eq.( 28) is directed from the hotter surface to the colder one.

In general, the absolute value of the heat flux decreases when the rarefaction parameter increases and it has the maximal values for C 0 = 0.5. The difference between the heat flux values obtained from AI and HS potentials is much larger compared to that for the sublimation flow rate, with the maximal difference of 86% for C 0 = 0.1 and δ = 20. The difference of both J P and J T obtained for the AI and LJ potentials is very small. In most of case, this difference does not exceed the numerical error.

δ J P × 10 2 (HS) J P × 10 2 (AI) J P × 10 2 (LJ) C 0 = 0.1 0.5 0.9 0.1 0.5 0.9 0.1 0.5 0.9 27) vs. mole fraction C 0 and rarefaction parameter δ at T 0 = 70 K assuming X T = 0.004 and X P = 0.068.

the number densities obtained by Eq. ( 21). An interesting behavior of the temperature profile can be seen on Figs. 5 and6, where the negative temperature gradient is observed: the gas temperature near colder surface becomes higher compared to the temperature near the hotter one. The analogous temperature behaviors were obtained numerically in [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF][START_REF] Cercignani | The paradox of the inverted temperature profiles between an evaporating and a condensing surface[END_REF], where the evaporation and condensation were simulated for a single gas. This inverted temperature profile is explained by the significant contribution of the term τ P 2 X P into the total temperature T containing T 2 calculated by Eq. [START_REF] Aoki | Vapor flows with evaporation and condensation in the continuum limit: effect of a trace of noncondensable gas[END_REF]. For larger value of the mole fraction, i.e. at C 0 = 0.9, the contribution of this term is significantly smaller than that at C 0 = 0.1. As a result, the temperature gradient in the gap becomes positive at the large mole fraction. In all cases, the temperature jump is observed near the surfaces which is larger for lower mole fraction of Helium, C 0 = 0.1. This jump increases by increasing the rarefaction parameter In contrast to the cross kinetic coefficients, the local pressure, temperature and mole fraction are not so sensitive to the potential of intermolecular interaction.

Conclusion

The Using the numerical data on the kinetic coefficients, the sublimation rate and the heat flux through the mixture have been calculated assuming the relative temperature difference equal to 0.004. The relative pressure difference of Argon has been calculated using the experimental relation provided in Ref. [START_REF] Ferreira | The sublimation of argon, krypton, and xenon[END_REF]. It has been found that the absolute value of the sublimation rate decreases by increasing the rarefaction parameter and it decreases considerably by increasing of the non-condensable gas (Helium) equilibrium concentration. The heat flux through the gap changes its direction depending on the Helium mole fraction:

for its small value, C 0 = 0.1, the heat flows from the cold surface to the hot one, while it flows in the opposite direction at C 0 = 0.9. The total energy flux always directed from the hot plate to the cold one.

Interesting effect, of the negative temperature gradient, where the gas temperature near the colder surface becomes larger than that near the hotter surface is observed for C 0 = 0.1 and 0.5. This phenomenon called inverted temperature profile was found previously in the numerical modeling of the evaporation and condensation phenomena, see Refs. [START_REF] Aoki | Gas flows caused by evaporation and condensation on two parallel condensed phases and the negative temperature gradient: numerical analysis by using a nonlinear kinetic equation[END_REF][START_REF] Cercignani | The paradox of the inverted temperature profiles between an evaporating and a condensing surface[END_REF][START_REF] Polikarpov | Heat and mass transfer in a rarefied gas confined between its two parallel condensed phases[END_REF].

Finally, the original approach has been developed to simulate the sublimation and deposition phenomena in the gap between two solid surfaces filled by a mixture of condensable and non-condensable gases. These results could be used to model gas flows at cryogenic temperatures.

Appendix A. Collision term

Expression of the collisional term of Eq. ( 12):

Lαβ h (i) = -γ αβ h (i) α +γ αβ ν (i) α + 2 m α m γ αβ u (i) α -υ (1) αβ u (i) α -u (i) β - υ (2) αβ 2 q (i) α - m α m β q (i) β c αy + γ αβ τ (i) α - 2m α m α + m β τ (i) α -τ (i) β υ (1) αβ c 2 α - 3 2 +2 γ αβ -υ (3)
αβ Π (i) α + υ

αβ q (i) α + υ where E is the dimensionless energy of colliding particles

E = m αβ |v α -v β | 2 2k B T . (A.5)
The dependence of the differential cross section σ on the energy E and deflection angle χ is determined by the potential of interatomic potential. In case of ab initio and Lennard-Jones potentials, the quantity σ(E, χ) is calculated numerically using the quantum theory to interatomic collision [START_REF] Sharipov | Transport coefficients of argon and its mixtures with helium and neon at low density based ab initio potentials[END_REF][START_REF] Sharipov | Transport coefficients of helium-neon mixtures at low density computed from Ab initio potentials[END_REF]. Then, the Ω

  solid state while Argon can sublimate and deposit. Two values of the reference temperature, T 0 = 50 K and 70 K, are chosen based on the sublimation curve of Argon provided in [23]. The following values of molecular masses are used in numerical simulations: m 1 = 6.6465 × 10 -27 kg and m 2 = 66.335 × 10 -27

Figure 1 :

 1 Figure 1: Comparison of ab initio potentials with the Lennard-Jones one.

Figures 2 -

 2 Figures 2 -4 present the profiles of dimensionless quantities, ν (i) α and τ (i) α , for both species, obtained with HS (solid line) and AI (dashed line) potentials for the mole fraction equal to 0.1, 0.5 and 0.9, respectively. For each value of mole fraction three values of the rarefaction parameter are provided, δ = 0.1, 1 and 10. Note, all profiles are symmetric relatively the middle point of the gap that is a consequence of the linearization of the kinetic equation. The dependence of the density deviation of Helium is linear in the most of cases, while the density deviation of Argon represents a non-linear behavior with respect to the y coordinate near the solid surface. The density deviations of both Helium and Argon are slightly sensitive to the intermolecular potential. Considering the temperature deviation curves, τ (i) α , a very pronounced nonlinear behavior is observed for Helium, τ P 1 curve, for all values of the mole fraction. The temperature deviation of Argon becomes also slightly non-linear

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Deviations of density, ν i α , and temperature, τ i α , α = 1, 2, i = P, T , defined by Eqs. (14),[START_REF] Kosuge | Model Boltzmann equation for gas mixtures: Construction and numerical comparison[END_REF], vs the coordinate y at C 0 = 0.1 and T 0 = 50 K: red lines -δ = 0.1, green linesδ = 1, blue lines δ = 10; solid lines -HS, dashed lines -AI potential.

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: Profiles of pressure p, temperature T , and mole fraction of mixture calculated for the two potentials: solid line -HS, dashed line -AI; red lines -δ = 0.1, green line -δ = 1, blue line -δ = 1; at C 0 = 0.1, T 0 = 50 K, X P = 0.083, and X T = 0.004

  sublimation-deposition process in the Helium-Argon mixture is studied numerically on the basis of the McCormack kinetic equation. The matrix of kinetic coefficients has been calculated for a large range of the rarefaction parameter lying from 0.1 to 20, for three values of the Helium mole fraction C 0 = 0.1, 0.5, 0.9, and for two reference temperatures, 50 K and 70 K. In order study the influence of intermolecular potential on macroscopic characteristics, three different potentials, namely, Hard Sphere, Lennard-Jones, and ab initio, have been implemented. It was observed that the cross kinetic coefficients, based on the Hard Sphere potential, differ significantly from those obtained for the ab initio potential. The discrepancy between coefficients obtained for these two potentials reach 100%. The diagonal kinetic coefficients are less sensitive to the potential. In this case, the potential influence is about 17%. At the same time, the local characteristics, such as pressure, temperature and mole fraction, are weakly sensitive to the intermolecular potential. All kinetic coefficients based on the Lennard-Jones potential just slightly differ from the corresponding coefficients based on the ab initio potentail. Thus, the Lannrd-Jones potential with parameters calculated ab initio provides reliable results as well as the ab initio potential itself.

1

  m α m β m α + m β (A.3)is the reduced mass of the binary mixture. The Ω integrals are defined via the differential cross section σ as Ω -cos i χ σ(E, χ) sin χE j+1 e -E dχdE, (A.4)

Table 1 :

 1 Viscosities of Helium, µ 1 , and Argon, µ 2 , and the ratio of their molecular diameters, d 2 /d 1 , for the reference temperatures.

	6 [Pa•s][45]	6.0842	7.5682
	µ 2 × 10 6 [Pa•s][24]	4.31945	5.79674
	d 2 /d 1	2.109	2.031

Table 3 :

 3 Values of Ω-integrals based on LJ and AI potential vs temperature T .

Table 4 :

 4 Kinetic coefficients for HS, AI and LJ potentials vs rarefaction parameter δ at C 0 = 0.1 and T 0 = 50K.

	The

Table 5 :

 5 Kinetic coefficients for HS, AI and LJ potentials vs rarefaction parameter δ at C 0 = 0.5 and T 0 = 50K.

	δ	Λ PP	Hard Sphere Λ PT = Λ TP	Λ TT	Λ PP	ab-initio potential Λ PT = Λ TP	Λ TT	Lennard-Jones potential Λ PP Λ PT = Λ TP Λ TT
	0.1 0.0116	-0.0050	0.6920 0.0117	-0.0054	0.6926 0.0117	-0.0054	0.6930
	1	0.0088	-0.0006	0.5218 0.0092	-0.0025	0.5267 0.0092	-0.0025	0.5270
	2	0.0071	0.0011	0.4240 0.0077	-0.0013	0.4300 0.0077	-0.0014	0.4303
	5	0.0047	0.0023	0.2795 0.0054	-0.0012	0.2851 0.0054	-0.0002	0.2855
	10 0.0030	0.0022	0.1807 0.0036	0.0003	0.1847 0.0016	0.0002	0.1850
	20 0.0018	0.0016	0.1063 0.0021	0.0003	0.1087 0.0022	0.0003	0.1089

Table 6 :

 6 Kinetic coefficients for HS, AI and LJ potentials vs rarefaction parameter δ at C 0 = 0.9 and T 0 = 50K.

Table 7 :

 7 Kinetic coefficients for HS, AI and LJ potentials vs rarefaction parameter δ at C 0 = 0.1 and T 0 = 70K.

	1 0.2367	-0.1113	0.6737 0.2367	-0.1115	0.6756 0.2367	-0.1115	0.6758
	1	0.2188	-0.0749	0.4884 0.2193	-0.0765	0.4994 0.2194	-0.0765	0.5003
	2	0.2112	-0.0572	0.3878 0.2123	-0.0595	0.4021 0.2124	-0.0595	0.4033
	5	0.1999	-0.0325	0.2461 0.2023	-0.0360	0.2612 0.2026	-0.0360	0.2625
	10 0.1886	-0.0167	0.1545 0.1929	-0.0208	0.1671 0.1934	-0.0209	0.1682
	20 0.1721	-0.0059	0.0889 0.1793	-0.0100	0.0975 0.1803	-0.0101	0.0983
	δ	Λ PP	Hard Sphere Λ PT = Λ TP	Λ TT	Λ PP	ab-initio potential Λ PT = Λ TP	Λ TT	Lennard-Jones potential Λ PP Λ PT = Λ TP Λ TT
	0.1 0.1015	-0.0468	0.8502 0.1016	-0.0474	0.8550 0.1016	-0.0474	0.8554
	1	0.0887	-0.0245	0.6268 0.0899	-0.0288	0.6525 0.0900	-0.0288	0.6544
	2	0.0812	-0.0135	0.5009 0.0833	-0.0198	0.5340 0.0835	-0.0198	0.5363
	5	0.0673	-0.0000	0.3219 0.0710	-0.0082	0.3562 0.0714	-0.0082	0.3588
	10 0.0534	0.0060	0.2053 0.0580	-0.0018	0.2327 0.0586	-0.0018	0.2349
	20 0.0381	0.0074	0.1201 0.0428	0.0014	0.1382 0.0433	0.0014	0.1398

Table 8 :

 8 Kinetic coefficients for HS, AI and LJ potentials vs rarefaction parameter δ at C 0 = 0.5 and T 0 = 70K.

	δ	Λ PP	Hard Sphere Λ PT = Λ TP	Λ TT	Λ PP	ab-initio potential Λ PT = Λ TP	Λ TT	Lennard-Jones potential Λ PP Λ PT = Λ TP Λ TT
	0.1 0.0116	-0.0050	0.6919 0.0118	-0.0054	0.6926 0.0118	-0.0054	0.6927
	1	0.0088	-0.0007	0.5214 0.0093	-0.0023	0.5255 0.0093	-0.0023	0.5255
	2	0.0072	0.0010	0.4234 0.0078	-0.0011	0.4285 0.0078	-0.0011	0.4285
	5	0.0048	0.0023	0.2790 0.0055	0.0002	0.2836 0.0055	0.0002	0.2835
	10 0.0031	0.0022	0.1802 0.0037	0.0006	0.1835 0.0037	0.0006	0.1835
	20 0.0018	0.0011	0.1063 0.0022	0.0005	0.1079 0.0022	0.0006	0.1079
	Table

Table 10 :

 10 Sublimation flow rate J P defined by[START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF] vs. mole fraction C 0 and rarefaction parameter δ at T 0 = 50 K assuming X T = 0.004 and X P = 0.083.

	0.1	2.108	1.495 0.497 2.108 1.497 0.500 2.108 1.497 0.500
	1	1.961	1.319 0.380 1.965 1.334 0.396 1.966 1.336 0.397
	2	1.810	1.215 0.314 1.908 1.241 0.334 1.910 1.245 0.335
	5	1.809	1.017 0.210 1.828 1.063 0.233 1.832 1.071 0.234
	10	1.714	0.812 0.136 1.749 0.872 0.156 1.757 0.883 0.157
	20	1.571	0.582 0.080 1.631 0.644 0.094 1.644 0.657 0.095
		J T × 10 3 (HS)		J T × 10 3 (AI)	J T × 10 3 (LJ)
	δ	C 0 = 0.1	0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9
	0.1	-6.54	-0.48 2.35 -6.55 -0.52 2.33 -6.55 -0.52 2.33
	1	-4.27	0.48 2.04 -4.36 0.18 1.90 -4.36 0.19 1.90
	2	-3.20	0.89 1.79 -3.35 0.43 1.61 -3.35 0.43 1.61
	5	-1.71	1.29 1.31 -1.99 0.64 1.13 -1.99 0.65 1.13
	10	-0.77	1.33 0.91 -1.12 0.66 0.76 -1.12 0.67 0.76
	20	-0.13	1.11 0.56 -0.51 0.56 0.46 -0.52 0.56 0.46

Table 11 :

 11 Heat flux J T defined by[START_REF] Sharipov | Poiseuille flow and thermal creep based on the Boltzmann equation with the Lennard-Jones potential over a wide range of the Knudsen number[END_REF] vs. mole fraction C 0 and rarefaction parameter δ at T 0 = 50 K assuming X T = 0.004 and X P = 0.083.

	Figures 5 -7 present the profiles of the macroscopic parameters of the mix-
	ture, namely of the temperature T calculated by Eq. (24), pressure p = nk B T
	with density given by Eq. (23), and local mole fraction C = n 1 /(n 1 + n 2 ) with

Table 12 :

 12 Sublimation flow rate J P defined by (26) vs. mole fraction C 0 and rarefaction parameter δ at T 0 = 70 K assuming X T = 0.004 and X P = 0.068.

		J T × 10 3 (HS)		J T × 10 3 (ab-initio) J T × 10 3 (Lennard-Jones)
	δ	C 0 = 0.1 0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9
	0.1	-4.87	0.22 2.43 -4.88 0.19	2.41	-4.88 0.20	2.41
	1	-3.14	0.84 2.04 -3.20 0.65	1.94	-3.20 0.66	1.94
	2	-2.33	1.08 1.76 -2.44 0.79	1.64	-2.43 0.80	1.64
	5	-1.22	1.28 1.27 -1.40 0.87	1.15	-1.40 0.88	1.15
	10	-0.52	1.23 0.87 -0.75 0.81	0.77	-0.75 0.82	0.77
	20	-0.04	0.98 0.53 -0.29 0.65	0.47	-0.29 0.65	0.47

Table 13 :

 13 Heat flux J T defined by (

[START_REF] Polikarpov | Transient heat transfer in a rarefied binary gas mixture confined between parallel plates due to a sudden small change of wall temperatures[END_REF] contains two fitting parameters: well-depth ε and zero-point d. In contrast to HS, this potential takes into consideration the attractive force and provides more physical results. In this case, the Ω integrals (A.4) are calculated numerically for each species and for some specific temperature. However, if the reduced temperature T * = k B T /ε is introduced, the Ω integrals are tabulated in terms of T * and then they can be used for any gaseous species and their mixtures.Thus, the use of T * reduces computational cost to solve the kinetic equation.The AI potentials obtained numerically from the Schrödinger equation are free from any adjustable parameter. To make easier the use of these potentials, their numerical values are presented analytically using a set of interpolating coefficients. For our purpose, we used the potentials reported in Refs.[START_REF] Cencek | Effects of adiabatic, relativistic, and quantum electrodynamics interactions on the pair potencial and thermophysical properties of helium[END_REF][START_REF] Patkowski | Argon pair potential at basis set and excitation limits[END_REF] and[START_REF] Cacheiro | Coupled cluster calculations of the ground state potential and interaction induced electric properties of the mixed dimers of helium, neon and argon[END_REF] for the collisions He-He, Ar-Ar, and He-Ar, respectively. The welldepths and zero-points of these potentials are given in Table2. A comparison

Acknowledgements

The work of A. Polikarpov was financially supported by the Ministry of Science and Higher Education of the Russian Federation, the research project No FEUZ-2020-0057. I. Graur would like to acknowledge the financial support provided by the European Union network program H2020, MIGRATE project under Grant Agreement No.643095. F. Sharipov acknowledges the Brazilian Agency CNPq for the support of his research, grant 304831/2018-2. [1] A. Hachikubo, Numerical modelling of sublimation on snow and comparison with field measurements, Annals of Glaciology 32 (2001) 27-32.

integral is calculated numerically for each value of the temperature. For HS potential, the Ω integral is calculated analytically and reads [START_REF] Ferziger | Mathematical Theory of Transport Processes in Gases[END_REF] Ω (i,j)

Appendix B. Sublimation pressure

The saturation pressure, p s /Pa, as function of the temperature T /K, is calculated using the following equation, [START_REF] Ferreira | The sublimation of argon, krypton, and xenon[END_REF]:


Erika Di Giuseppe
email: erika.digiuseppe@gmail.com

Jordi Girones
email: jordi.girones@mines-paristech.fr

Loan T T Vo

Emilie Gineau
email: gineauemilie@gmail.com

Catherine Lapierre
email: catherine.lapierre85@sfr.fr

Maryse Brancourt-Hulmel

Stéphanie Arnoult-Carrier
email: stephanie.arnoult-carrier@inrae.fr

Patrick Navard
email: patrick.navard@mines-paristech.fr

Polysaccharides and phenolics of miscanthus belowground cell walls and their influence on polyethylene composites

Keywords: Miscanthus, Belowground materials, Polysaccharides, Cell wall phenolics, Composites, Mechanical properties

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The use of plant biomass for manufacturing bio-based polymer composites has attracted a large interest due to its multiple advantages such as a decrease of weight compared to glass fiber-filled composites and a good environmental footprint [START_REF] Gallos | Lignocellulosic fibers: a critical review of the extrusion process for enhancement of the properties of natural fiber composites[END_REF][START_REF] Ramesh | Plant fibre based bio-composites: Sustainable and renewable green materials[END_REF][START_REF] Shah | Developing plant fibre composites for structural applications by optimising composite parameters: a critical review[END_REF][START_REF] Girijappa | Natural fibers as sustainable and renewable resource for development of eco-friendly composites: A comprehensive review[END_REF]. Among the many plants which have been considered for reinforcing polymers is miscanthus, an interesting grass producing annually a large amount of biomass [START_REF] Arnoult | A review on miscanthus biomass production and composition for bioenergy use: genotypic and environmental variability and implications for breeding[END_REF]. One of the advantages of miscanthus is its low nitrogen needs (Zapater et al., 2017) due to the efficient recycling of nitrogen for the next growth period by its rhizomes [START_REF] Strullu | Biomass production and nitrogen accumulation and remobilisation by Miscanthus giganteus as influenced by nitrogen stocks in belowground organs[END_REF]. This low-fertilizer demand promoted studies regarding the mechanisms of such nitrogen accumulation in the rhizomes [START_REF] Li | Plant and soil effects on bacterial communities associated with Miscanthus 3 giganteus rhizosphere and rhizomes[END_REF]; [START_REF] Keymer | Contribution of nitrogen fixation to first year Miscanthus 3 giganteus[END_REF]Dželetović & Glamočlija, 2015). Thus, the use of miscanthus combines a high biomass production with a low environmental impact [START_REF] Zub | Agronomic and physiological performances of different species of Miscanthus, a major energy crop[END_REF]. The preparation of polymer composites is done with a filler constituted by elongated miscanthus stem fragments. They act as reinforcing agents in polymer composites and it was shown that they can effectively produce polymer composites with good mechanical properties, using a wide range of matrices like polypropylene and polylactic acid [START_REF] Bourmaud | Investigations on mechanical properties of poly(propylene) and poly(lactic acid) reinforced by miscanthus fibers[END_REF], poly(3-hydroxybutyrate-co-3-hydroxyvalerate)/poly(butylene adipate-coterephthalate) [START_REF] Nagarajan | Sustainable Green Composites: Value Addition to Agricultural Residues and Perennial Grasses[END_REF], poly(3-hydroxybutyrate-covalerate)/polylactide (Nanda et al., 2013), polyesters [START_REF] Rao | Tensile properties of elephant grass fiber reinforced polyester composites[END_REF], poly(vinyl alcohol) [START_REF] Kirwan | Enhancing properties of dissolution compounded Miscanthus giganteus reinforced polymer composite systems. Part 1. Improving flexural rigidity[END_REF], poly(butylene succinate)/poly(butylene adipate-co-terephthalate) (Muthuraj et al., 2017a), poly(hydroxybutyrate-co-hydroxyvalerate) (Muthuraj et al., 2017b) or starch-based polymers [START_REF] Johnson | Improvement of the impact performance of a starch based biopolymer via the incorporation of Miscanthus giganteus fibres[END_REF], sometimes removing different extractives by chemical treatments [START_REF] Kirwan | Enhancing properties of dissolution compounded Miscanthus giganteus reinforced polymer composite systems. Part 1. Improving flexural rigidity[END_REF]. Nearly no domestication or breeding has been undertaken to identify new plant genotypes provided with traits favorable to the industrial production of polymer composites. To this end, there is a need to study the influence of compositional characteristics which may influence composite properties. In a recent work, we used different miscanthus genotypes to prepare polyethylene or polypropylene composites from stem fragments and by means of a robust protocol for preparing such composites [START_REF] Girones | Miscanthus stem fragment -reinforced polypropylene composites: development of an optimized preparation procedure at small scale and its validation for differentiating genotypes[END_REF]. Recently, we evaluated miscanthus belowground fragments (rhizomes plus roots) for preparing polyethylene matrix composites and found that their low axial ratio and cellulose content led to composite mechanical properties inferior to those from stem-based composites (Chupin et al., 2017a). In this preliminary study, we employed only one miscanthus species (M. x giganteus) without considering the possible correlations between cell wall components and the composite mechanical properties. In the present work, we used the belowground biomass of two contrasted miscanthus species, M. sinensis and M. x giganteus and we made a more detailed study of their cell wall polysaccharides and phenolics in order to address the following questions:

• How do the belowground cell walls of these two species compare in terms of polysaccharides, lignin and cell wall-linked CA and FA? • Do the mechanical properties of polyethylene composites made from these miscanthus belowground materials correlate with some of their compositional traits? Due to the very different physiological roles of below and above matter in plants, we make the hypothesis that the mechanical properties of polymer composites prepared with belowground miscanthus fillers will have different correlations with the biochemical composition as compared with similar composites prepared with plant stem fillers.

Materials and Methods

Materials

The polymer matrix was a low-density polyethylene (PE), LDPE 1965T from Sabic© (melt flow index, MFI of 65 dg/min at 190 °C and 2.16 Kg, density of 919 kg/cm3). A high density maleic anhydride-grafted polyethylene, OREVAC 18507 provided by Arkema, with MFI of 5 g/10 min at 190 °C and 2.16 kg, was used as coupling agent. Two miscanthus species were used: Miscanthus x giganteus from ADAS, United Kingdom (GIG) and Miscanthus x sinensis Goliath from Nursery Chombart, France (GOL), which is a large hybrid obtained from a cross between Miscanthus x sinensis and Miscanthus sacchafloris. Plants were grown in the INRA experimental unit in Estrées-Mons, in the Hautsde-France region of Northern France (49°53N, 3°00E) on three different randomized blocks (B1, B2, B3) with three different nitrogen contents (N0 = 0 kg N/ha, N1 = 80 kg N/ha and N3 = 240 kg N/ha). They were harvested in February 2015, at the end of their 8th year of cultivation. The belowground materials (roots plus rhizome) were collected.

Preparation and characterization of the belowground fragments

The GIG belowground materials were ground by means of a coffee mill (Carrefour home). The GOL belowground materials being too hard to be directly ground in the coffee mill, their initial size was reduced in a Hellweg M50 granulator (Germany) equipped with a 2.5 mm grid. The fragments were ground later in the coffee mill. About 20 g of samples of both genotypes were sieved in a Retsch AS200 Digit shaker (Retsch, Germany) operating at 40 mm amplitude for 5 minutes. Six sieves with open pore size of 1000, 600, 400, 300, 200, and 100 µm were employed. Only the fragment fraction collected in the 300-200 µm sieve was used to prepare the composites. The procedure was repeated several times until the required amount for composite preparation was reached. Figure 1 shows the GIG and GOL belowground materials before grinding (a), after sieving (b), after compounding (c), and finally after injection molding (d).

Composite preparation

An important issue regarding the study of detailed and tiny correlations between structural or chemical characteristics of a filler and properties of composites is to ensure that all parameters linked to the preparation of the filler and to the processing are kept under control and fixed. To reach this goal, a robust protocol for preparing fragments of plants and processing them with a polymer was established and validated [START_REF] Girones | Miscanthus stem fragment -reinforced polypropylene composites: development of an optimized preparation procedure at small scale and its validation for differentiating genotypes[END_REF]. It allows to be sure that any difference seen when comparing different genotypes of a plant species is coming from the characteristics of the filler and not to the variation of parameters such as the size of the filler, their dispersion or distribution or the coupling between filler and matrix. Such protocol was successfully applied for studying plant fragment-based composites (Chupin et al., 2017a;[START_REF] Vo | Correlations between genotype biochemical characteristics and mechanical properties of maize stem -polyethylene composites[END_REF]. The preparation used in this work follows this procedure. The composition of the composite was kept constant at 35 wt% of belowground fragments, 1.75wt% of coupling agent (5 wt% on biomass basis) and 68.25wt% of polymer. Prior to compounding, belowground fragments were overnight dried in an air-circulating oven operating at 70 °C. All composites were prepared in a Haake Rheomix 600 intensive kinetic mixer, equipped with counter-rotating 'roller' rotors. Rotors speed was fixed at 60 rpm and temperature of the mixing chamber was set at 150 °C. Belowground fragments, polymer pellets and coupling agent were introduced into the mixer in three steps (at t = 1 min, 3 min, and 5 min) during the first 5 minutes of the process, and then let be mixed for 4 minutes, for a total duration of 9 minutes. After compounding, miscanthus belowground-based composites were granulated in a blade mill provided with a 5 mm mesh. Test specimens were injectionmolded in a Haake Minijet-II (Thermo Fisher Scientific, Germany) using steel molds complying either ISO-527-2-1BA (for tensile bar) or ISO-179 (for impact bar) specifications. Cylinder and mold temperatures of the injection molding machine were fixed at 150 °C and 40 °C, respectively. Injection and post-injection pressure were respectively set at 400 and 300 bars.

Mechanical characterization

Composite test bars were kept in a conditioned room at 23 °C and 50% humidity for at least 5 days before testing. Tensile tests were carried out in a Zwicki Z2.5 tensile testing machine (ZwickRoell) operating at 0.02 mm/s (1.2 mm/min). Tensile strength, Young's modulus, elongation at maximum strength and elongation at break were measured. A pendulum Ceast 9050 (Instron Company, France) with a 1 J swing arm was employed to perform Charpy Vnotch impact tests. 2 mm indents on impact bars (dimension = 80 x 10 x 4 mm3) were conducted by means of a single tooth Ceast NotchVIS manual notching machine (Instron, France). At least 5 specimens were tested per sample and per mechanical test.

Compositional analysis

The belowground samples ground to about 0.1 mm were subjected to exhaustive extraction with water then ethanol in an ASE350 accelerated solvent extractor (Dionex). The extract-free samples, yields given by weight%, mainly composed of cell walls, were then dried at 45°C for 48 h before the analyses of cell wall components, as follows. No correction was made for the few % of water which may remain in samples. The analyses of non-cellulosic polysaccharides and cellulose were performed by two successive hydrolyses using trifluoroacetic acid and sulphuric acid as previously described (Chupin et al., 2017a). The neutral monosaccharide composition of non-cellulosic polysaccharides was measured using high-performance anion exchange chromatography with pulse amperometric detection, as previously described [START_REF] Harholt | ARABINAN DEFICIENT 1 is a Putative Arabinosyltransferase Involved in Biosynthesis of Pectic Arabinan in Arabidopsis[END_REF][START_REF] Vo | Correlations between genotype biochemical characteristics and mechanical properties of maize stem -polyethylene composites[END_REF]. Lignin content was measured as Klason lignin (KL), using the Klason procedure as previously reported [START_REF] Méchin | Impact of the brown-midrib bm 5 mutation on maize lignins[END_REF]. The determination of ester-linked CA and FA units was performed by mild alkaline hydrolysis, followed by solid phase extraction and HPLC chromatography, as recently described [START_REF] Sibout | Structural redesigning Arabidopsis lignins into alkalisoluble lignins through the expression of pcoumaroyl-CoA:monolignol transferase PMT[END_REF].

Statistical analysis

Pearson correlation matrices were calculated and designed using the corrplot package (0.84 version) of the statistical software R (3.6.2 version).

3-Results and discussion

Cell wall polysaccharides and phenolics from the belowground materials of GOL and GIG miscanthus species

All these analyses were performed from the biological triplicates harvested in winter and from three different randomized blocks with three nitrogen fertilization levels (N=0, 80 and 240 kg/ha). The extract-free samples, mainly composed of cell walls, were recovered with an extraction yields ranging between 78 and 83 %, whatever the N level or the miscanthus species. These yields are about 10% lower than those obtained from winter-harvested stems. This result means that the belowground biomass contains more soluble components than the aboveground one in winter, a phenomenon in agreement with the storage role of rhizomes. The analyses of cell wall polysaccharides and phenolics from these extract-free belowground samples are shown in Table 1 while detailed block data are given in Table1S. 18) 309 ( 9) 372 ( 15) 142 ( 6) 144 ( 6) 24 ( 2) 32 ( 5) 136 ( 12) 186 ( 14)

5 ( 1 In agreement with Chupin et al. (2017a), the cellulose content of winter-harvested miscanthus belowground materials is nearly half that of stems while the level of non-cellulosic polysaccharides is higher. The neutral sugar composition of the non-cellulosic fraction reveals a high content of glucose which suggests that a substantial fraction of the starch reserves survived the water and alcohol extraction steps employed to prepare the extract-free belowground samples. The belowground starch abundance is higher in winter, up to values close to 100 mg per g of dry matter for GOL and GIG species [START_REF] Purdy | Seasonal carbohydrate dynamics and climatic regulation of senescence inn the perennial grass miscanthus[END_REF]. It is therefore very likely that some non-cellulosic glucose originates from belowground starch. The non-cellulosic fraction of the samples comprises a high level of xylose, in agreement with the fact that arabinoxylans are major hemicelluloses of grass secondary cell walls [START_REF] Vogel | Unique aspects of the grass cell walls[END_REF]. The arabinose to xylose ratio (Ara/Xyl) reflects the arabinosylation degree of xylan chains. Whatever the nitrogen level, this Ara/Xyl ratio is higher in the GOL samples than in the GIG ones, which indicates that the two species differ in the structure of their belowground arabinoxylans. In addition, this ratio (ranging from 0.17 to 0.23) is also about twice as high as the one reported for analogous stem samples (0.11 ± 0.01 for GIG and 0.12 ± 0.01 for GOL).

In other words, belowground arabinoxylans are markedly more substituted than stem ones.

Together with the lower cellulose content, this higher substitution level might hinder molecular interactions between the chains of cell wall polysaccharides and therefore decrease the mechanical properties of the plant fibers. These Ara substituents of grass arabinoxylans are often acylated at O-5 by FA and a fraction of these FA esters can be ether-linked to lignins [START_REF] Jacquet | New ether-linked ferulic acid-coniferyl alcohol dimers identified in grass straws[END_REF]. In addition to the major Xyl, Glc and Ara sugars, the non-cellulosic fraction of the belowground samples contains a low amount of galactose (Gal) units. These Gal units might be linked to pectin components, even though pectins are minor constituents of grass secondary cell walls [START_REF] Vogel | Unique aspects of the grass cell walls[END_REF].

In addition to polysaccharides, the belowground cell walls contain substantial lignin amounts that are slightly higher for the GOL species and at all nitrogen levels tested. These lignin amounts are close to the ones observed for winter-harvested whole stems of M. x gig. (226.5 ± 0.9 mg/g) or M. sin. (206.2 ± 2.4 mg/g). By their water expelling effect, these high levels of hydrophobic lignin polymers may participate to the hydrogen bond-mediated cohesion of cellulose microfibrils and thereby to their mechanical strength. In addition to feruloylated arabinoxylans, another unique feature of grass cell walls is the occurrence of p-coumaric acid units prominently ester-linked to lignin units [START_REF] Ralph | Hydroxycinnamates in lignification[END_REF]. Whatever the nitrogen level, the CA content was found to be higher in GIG than in GOL (Table 1), a phenomenon already observed for winter-harvested stem internodes from GIG (21.54 ± 1.4 mg/g) and GOL (16.40 ± 0.7 mg/g) species. In addition, this p-coumaroylation level (Table 1) was also found to be lower in belowground samples than in stem ones. It is noteworthy that, by contrast to the lignin amount which does not seem to be affected by the nitrogen level, the highest nitrogen level is associated with the lowest p-coumaric acid content in both samples. Taken together and not unexpectedly, these compositional analyses revealed that the structure of polysaccharides and of lignins are markedly different in the miscanthus belowground and aboveground biomass.

The correlations between all compositional and mechanical parameters are given in Table 2.

Table 2: Correlation coefficients between mechanical properties of composites and biochemical composition of belowground fragments used to prepare composites. TensileS: tensile strength; ElongM: elongation at maximum; ElongB: elongation at break; ImpactS: impact strength; CA: ester-linked p-coumaric acid; FA: ester-linked ferulic acid; hemicell: total amount of non-cellulosic polysaccharides, Cell: cellulose. In blue, negative correlations, in red, positive correlation. The magnitude of the correlation is expressed by the intensity of the color.

Correlations of biochemical composition with mechanical properties of belowground-based polyethylene composites

The complete results of tensile and impact tests performed on composites are reported in Table 2S. The mean values averaged over the three blocks are given in Table 3. On average, the tensile strength is about 10 MPa, the tensile modulus about 550 MPa, the elongation at break about 5% and the impact strength about 3.5 kJ/m2. Considering that for the neat matrix, the tensile strength is about 7.2 MPa and the tensile modulus about 130 MPa, the ratio of the tensile strength of the composite over the one of the neat matrix is about 1.4 while the equivalent ratio as for the modulus is 4.2. This is in the same range of ratio than for maize stem-based PE composites [START_REF] Vo | Correlations between genotype biochemical characteristics and mechanical properties of maize stem -polyethylene composites[END_REF] and sorghum stem-based PE composites [START_REF] Vo | Processing and properties of sorghum stem fragment-polyethylene composites[END_REF] with 30% concentration of biomass ratio, showing that miscanthus belowground materials have a moderate reinforcement effect, as for maize and sorghum stems. By contrast, the reinforcement capability of miscanthus stems is much higher, with tensile strength ratio of about 2.5 and modulus ratio of about 15 (Chupin et al., 2017a). A striking result of this work is the large difference of the values of tensile moduli between the first paper on belowgroundbased PE composites (Chupin et al., 2017a), i.e. about 257 MPa, and this work, i.e. about 550

GIG GOL

MPa, although the matrix is the same PE. The difference of concentration of biomass filler, 35% in this paper and 30% in (Chupin et al., 2017a) cannot explain such a huge difference.

Composites have a third component in addition of PE and biomass, a coupling agent whose role is to increase the bonding of biomass to the filler. In (Chupin et al., 2017a), this coupling agent is a terpolymer having 3% of maleic acid. In the present paper, the coupling agent is a maleic acid-PE with a high content of maleic acid (the value is kept confidential by the manufacturer) specifically designed for coupling lignocellulosic biomass and PE. It is the effect of this coupling agent which is giving such a doubling of the tensile modulus. As shown in Table 2, the correlation directions of the tensile modulus and the tensile strength with all the biochemical parameters are similar. This should be expected since a linear strength-stiffness relationship is often seen with short fiber filled composites [START_REF] Sobczak | Polypropylene composites with natural fibers and wood -General mechanical property profiles[END_REF][START_REF] Vo | Processing and properties of sorghum stem fragment-polyethylene composites[END_REF]. Usually, the higher the elongation at break, the higher the impact strength, as seen for GOL. The reason why there is no correlation for GIG is unknown.

There is no apparent relationship between mechanical properties (tensile strength, tensile modulus, elongation at break, impact strength) and nitrogen inputs. These results agree with what was found by Chupin et al. (2017a) studying GIG.

The tensile strength and modulus are positively correlated with the cellulose content and negatively correlated with non-cellulosic polysaccharides in GIG, in agreement with previous results from composites with GIG belowground materials (Chupin et al, 2017a) or maize stem [START_REF] Chupin | Thermal and dynamic mechanical characterization of miscanthus stem fragments: effects of genotypes, positions along the stem and their relation with biochemical and structural characteristics[END_REF] as well as short fiber composites [START_REF] Charlet | Characteristics of Hermès flax fibres as a function of their location in the stem and properties of the derived unidirectional composites[END_REF] or flax woven fabrics [START_REF] Acera Fernández | Role of flax cell wall components on the microstructure and transverse mechanical behaviour of flax fabrics reinforced epoxy biocomposites[END_REF]. It can be understood considering that cellulose chains are constituting the solid backbone of lignocellulosic tissues due to their excellent intrinsic mechanical properties (Saito et al., 2013). It is unclear why cellulose is negatively correlated with modulus and strength for GOL. Hemicelluloses are negatively correlated with strength and modulus, due to their weaker molecular interactions and low thermal resistance [START_REF] Chupin | Influence of the radial stem composition on the thermal behaviour of miscanthus and sorghum genotypes[END_REF]. The influence of cellulose on impact strength and elongation at break is not correlated for GIG composites. GOL composites show a weak correlation with impact strength. Lignin and p-coumaric acid contents are positively correlated with tensile strength and modulus for both genotypes and with tensile modulus for GOL. This correlation between these two biochemical parameters can be understood since the major fraction of p-coumaric acid is bound to lignins. Similar correlations of the mechanical properties with lignins and pcoumaric acid were found with maize [START_REF] Chupin | Thermal and dynamic mechanical characterization of miscanthus stem fragments: effects of genotypes, positions along the stem and their relation with biochemical and structural characteristics[END_REF]. The role of ferulic acid is not clear. There is a weak positive correlation to impact for GIG and negative correlations with elongation and impact strength. FA is not following the correlations of hemicellulose. The major non-cellulosic sugars of the belowground samples are xylose and glucose. Non-cellulosic glucose shows no correlation with mechanical properties for GOL while there are negative correlations for tensile strength and modulus and positive ones for elongation in the case of GIG. Xylose shows a different picture, with positive correlations to modulus and strength for GIG and inverse negative correlations for GOL. From the above-mentioned correlations and data, it is possible to draw the following conclusions. GOL and GIG-based composites show similarities which are reminiscent of what has been found for plant stems [START_REF] Kaack | Variation in morphology, anatomy and chemistry of stems of Miscanthus genotypes differing in mechanical properties[END_REF][START_REF] Kong | Anatomical and chemical characteristics associated with lodging resistance in wheat[END_REF]Kaak and Schwarz, 2001) or other plant stem composites [START_REF] Chupin | Thermal and dynamic mechanical characterization of miscanthus stem fragments: effects of genotypes, positions along the stem and their relation with biochemical and structural characteristics[END_REF], which is the favorable influence of cellulose and lignin and the unfavorable influence of hemicellulose on modulus and strength. However, GOL-based composites do not follow such a scheme for cellulose. An interesting result, in agreement with results found by [START_REF] Chupin | Thermal and dynamic mechanical characterization of miscanthus stem fragments: effects of genotypes, positions along the stem and their relation with biochemical and structural characteristics[END_REF], is that p-coumaric acid content is highly favorable to obtain composites with good modulus and strength. We may postulate that the composites prepared with GIG belowground materials have better mechanical performances than the ones prepared with GOL because GIG belowground materials have larger cellulose and p-coumaric acid contents and a lower hemicellulose content than GOL. GIG has a larger content of noncellulosic glucose than GOL. Although fragments prepared by grinding below-ground miscanthus materials seem to behave like plant stems when preparing polymer composites, there are differences between the two genotypes which are intriguing, such as for example the unfavorable influence of cellulose content on tensile strength and modulus for GOL or the inverse correlations found for xylose between GOL and GIG. There is no easy explanation for such counter-intuitive results besides speculation that other factors linked to structural organization effects could play some roles.

Conclusions

The belowground cell walls of GOL and GIB species show a lot of similarities in terms of polysaccharides, lignin and cell wall-linked p-coumaric and ferulic acids contents. There is no clear influence of the level of nitrogen content. The content and structure of polysaccharides and of lignins are markedly different in the miscanthus belowground and aboveground biomass, which explains why fragments prepared from these two parts of the plant give very different reinforcing capacities when added to a polymer. The cellulose content of the studied belowground miscanthus biomass is roughly half that of miscanthus stems while the non-cellulosic fraction has a high level of xylose. The cell walls contain substantial amounts of lignin that are slightly higher for the GOL species, in the same order of magnitude as the ones observed for winter-harvested whole stems. The lignin pcoumaroylation level, lower than in stem, was found to be higher in GIG belowground materials than in GOL ones, as observed for stem internodes. The arabinose to xylose ratio is also about twice higher than the one observed for analogous stem samples, suggesting that belowground arabinoxylans are more substituted than stem ones. The mechanical properties of the belowground miscanthus-polyethylene composites correlate with several of belowground compositional traits, with similar trends as for miscanthus stems or plant stem-polyethylene composites. Considering that the fragments used to prepare polymer composites obtained after grinding have a low aspect ratio (Chupin et al., 2017a), the fact that the amount of cellulose, the polymer which is imparting good mechanical properties, is lower that when using stems, is one of the major explanations for the low mechanical properties of belowground miscanthus-polyethylene composites compared to stem-based ones. Lignin and p-coumaric acid contents are strongly positively correlated with tensile strength for both genotypes. These hydrophobic lignin polymers may participate to the hydrogen bond-mediated cohesion of cellulose microfibrils and thereby to their mechanical strength. Similar correlations of the mechanical properties with lignins and p-coumaric acid were found when using maize stems to prepare composites [START_REF] Vo | Correlations between genotype biochemical characteristics and mechanical properties of maize stem -polyethylene composites[END_REF]. There is a negative correlation between non-cellulosic molecules contents and the mechanical properties, as observed when using plant stems. GIG composites have better mechanical performances than the ones prepared with GOL due to its larger cellulose and p-coumaric acid contents and its lower hemicellulose content. The two main differences between GOL and GIG relate to the influence of cellulose and xylose contents on tensile strength and modulus. The hypothesis that the mechanical properties of polymer composites prepared with belowground miscanthus fillers will have different correlations with the biochemical composition as compared with the same composites prepared with plant stem fillers turned out to be incorrect. Both above and below ground matters have the same correlations with polysaccharides and phenolics. One interesting finding is the role of p-coumaric and ferulic acids which are respectively positively and negatively correlated to the mechanical properties of plant fragment-based polymer composites (this work for belowground miscanthus; [START_REF] Vo | Correlations between genotype biochemical characteristics and mechanical properties of maize stem -polyethylene composites[END_REF] for maize stems and an unpublished work from the same author group for miscanthus stems). However, a recent paper [START_REF] Chupin | Thermal and dynamic mechanical characterization of miscanthus stem fragments: effects of genotypes, positions along the stem and their relation with biochemical and structural characteristics[END_REF] reports inverse correlations between these two hydroxycinnamic acids and the intrinsic mechanical properties of the miscanthus stem fragments. Considering that the mechanical properties of composites are directly positively correlated to the intrinsic properties of the filler, this discrepancy has no simple explanation. More work is needed to first confirm the fact that p-coumaric and ferulic acids are respectively negatively and positively correlated to the mechanical properties of plant stem fragments of other plants such as sorghum or maize. If confirmed, there will be a need to understand the role of these acids on the mechanical properties of plant fragments, considering also their role during the breaking of stems to produce the small pieces used for reinforcing polymers.

5-Author contributions

6-Acknowledgements

This work was supported by the program Investments for the Future (grant ANR-11-BTBR-0006-BFF) managed by the French National Research Agency. The authors thank Frédéric Legée for the determination of lignin content. 9.5 (0.2) 515 (9) 4.9 (0.5) 3.4 (0.1) mean and std. dev. 9.6 (0.2) 522 ( 7) 5.1 (0.2) 3.3 (0.1)

Figure 1 .

 1 Figure 1. Physical aspect of GIG (left) and GOL (right) belowground materials a) before grinding, b) after sieving (fraction collected with 200-300 µm openings), c) after mixing in the internal mixer, and d) after injection molding.

  E. Di Giuseppe: Investigation, Writing-Original draft preparation; J. Girones: Investigation; L. Vo: Investigation; E. Gineau: Investigation; C. Lapierre: Methodology, Investigation, Writing; Maryse Brancourt-Hulmel: Methodology, Formal analysis; Stéphanie Arnoult-Carrier: Investigation; P. Navard: Conceptualization; Writing, Reviewing and Editing.

Table 1 :

 1 Amounts of A) cell wall polysaccharides and B) cell wall phenolics in extract-free belowground samples from GIG and GOL miscanthus species. The data represent means (and SD) from biological triplicates. The main neutral sugars of non-cellulosic polysaccharides are xylose (Xyl), arabinose (Ara), Glucose (Glc) and Galactose (Gal). Lignin content is expressed as Klason Lignin (KL).

			A) Cell wall polysaccharides in mg/g
	N level	Miscanthus	Cellulose Non-cellulosic	Xyl	Ara	Glc	Gal
	kg/ha	species		polysaccharides
	0	GIG	281 (20)	332 (24)	143 (5)	28 (2)	152 (27)	7 (1)
		GOL	233 (9)	357 (16)	146 (2)	33 (1)	168 (18)	10 (0)
	80	GIG	298 (30)
		GOL	220 (

Table 3 :

 3 Mechanical properties averaged over the three blocks for GIG and GOL belowground-based composites for the three nitrogen (N) levels. Tensile strength and modulus are in MPa, elongation at break in % and impact strength in kJ/m 2 . The data represent means (and SD) from biological triplicates.

		N level	Tensile	Tensile	Elongation	Impact
		kg/ha	strength	modulus	at break	strength
	GIG	0	10.3 (0.2) 593 (25)	5.0 (0.4)	3.5 (0.1)
	GIG	80	10.4 (0.1) 605 (5)	5.0 (0.0)	3.7 (0.0)
	GIG	240	10.3 (0.1) 604 (21)	4.8 (0.2)	3.7 (0.2)
	GOL	0	9.8 (0.1)	531 (7)	5.1 (0.2)	3.3 (0.1)
	GOL	80	9.7 (0.1)	525 (17)	5.4 (0.3)	3.3 (0.2)
	GOL	240	9.6 (0.2)	533 (7)	5.1 (0.2)	3.3 (0.1)

Table 1S .

 1S Amounts of A) cell wall polysaccharides and B) cell wall phenolics in extract-free belowground samples from GIG and GOL miscanthus species. The data correspond to values obtained from three different field blocks B1/B2/B3. The main neutral sugars of noncellulosic-polysaccharides are xylose (Xyl), arabinose (Ara), Glucose (Glc) and Galactose (Gal). Lignin content is expressed as Klason lignin (KL).

				A) Cell wall polysaccharides in mg/g
	N	Miscanthus	Cellulose	Non cellulosic	Xyl	Ara	Glc	Gal
	level	species		polysaccharides
	kg/ha
	0	GIG
		GOL

Table 2S .

 2S Mechanical properties of PE-based composites reinforced with GIG and GOL belowground fragments.

	Rhizome Bloc nr. Nitrogen	Tensile	Tensile	Elongation	Impact
			content	strength	modulus	at break	strength
			kg/ha	(MPa)	(MPa)	(%)	(kJ/m 2 )
		Neat LDPE		7.2 (0.8)	130 (10)	na	44.8 (1.1)
		B1		10.2 (0.1)	565 (8)	5.3 (0.2)	3.4 (0.2)
		B2	0	10.5 (0.09)	614 (4)	4.9 (0.1)	3.6 (0.1)
		B3		10.3 (0.09)	600 (17)	4.6 (0.3)	3.4 (0.1)
		mean and std. dev.	10.3 (0.2)	593 (25)	5.0 (0.4)	3.5 (0.1)
	GIG	B2 B3	80	10.3 (0.08) 10.5 (0.07)	608 (12) 601 (6)	4.9 (0.3) 5.0 (0.2)	3.7 (0.1) 3.7 (0.2)
		mean and std. dev.	10.4 (0.09)	605 (5)	4.97 (0.03) 3.68 (0.01)
		B1		10.3 (0.1)	627 (7)	4.5 (0.2)	3.6 (0.1)
		B2	240	10.2 (0.1)	590 (5)	5.0 (0.2)	3.9 (0.2)
		B3		10.4 (0.1)	594 (6)	4.9 (0.1)	3.7 (0.1)
		mean and std. dev.	10.3 (0.1)	604 (21)	4.8 (0.2)	3.7 (0.2)
		B1		9.9 (0.05)	537 (8)	5.0 (0.3)	3.1 (0.1)
		B2	0	9.7 (0.1)	534 (12)	4.9 (0.3)	3.4 (0.2)
		B3		9.8 (0.1)	523 (4)	5.3 (0.3)	3.3 (0.1)
		mean and std. dev.	9.8 (0.09)	531 (7)	5.1 (0.2)	3.3 (0.1)
		B1		9.8 (0.08)	540 (18)	5.3 (0.1)	3.3 (0.2)
	GOL	B2	80	9.7 (0.04)	507 (6)	5.7 (0.4)	3.6 (0.1)
		B3		9.6 (0.08)	527 (7)	5.1 (0.2)	3.2 (0.2)
		mean and std. dev.	9.7 (0.09)	525 (17)	5.4 (0.3)	3.3 (0.2)
		B1		9.8 (0.2)	534 (12)	4.9 (0.3)	3.1 (0.1)
		B2 B3	240	9.5 (0.1)	521 (5)	5.2 (0.2)	3.3 (0.2)

Supplementary Materials

Polysaccharides and phenolics of miscanthus belowground cell walls and their influence on polyethylene composites E. Di Giuseppe 1 , J.