H Yamaguchi

M T T Ho

Y Matsuda

T Niimi

Irina Martin Graur

Conductive heat transfer in a gas confined between two concentric spheres: From

Keywords: heat transfer, vacuum, Knudsen number, kinetic model, concentric spheres, thermal accommodation coecient

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

Introduction

The heat transfer through a rareed gas conned between two surfaces with dierent temperatures is a fundamental issue which has been studied for a long time [START_REF] Goodman | Dynamics of Gas-Surface scattering[END_REF][START_REF] Saxena | Thermal Accommodation and Adsorption Coecients of Gases[END_REF]. In parallel to the study of the theoretical aspects of the heat transfer as its dependence on the gas nature, composition and pressure various practical applications were found, such as the principle of operation of the Pirani sensor, which uses the dependence of heat ux on the pressure for the pressure measurements. Despite the long history of investigations of heat transfer between the two surfaces, there are several aspects which remain not very well known until now. One of these problems lies in the interaction of gas molecules with surfaces, which signicantly aects the intensity of the heat transfer between the surfaces. This interaction may be explained in terms of thermal accommodation coecient. The thermal or energy accommodation coecient α is dened as [START_REF] Goodman | Dynamics of Gas-Surface scattering[END_REF][START_REF] Saxena | Thermal Accommodation and Adsorption Coecients of Gases[END_REF],

α = E i -E r E i -E s , ( 1
)
where E is the mean energy of the molecules colliding to a surface; the subscripts i and r correspond to the incident and reected molecules, respectively, the subscript s corresponds to the molecules fully accommodated to the surface.

This coecient is useful in the analysis and the management of heat transfer in micro-and nano-devices, where gas ow should be treated as rareed, even if the operating pressure is an atmospheric pressure, due to their small characteristic dimension. Additionally, the surface-to-volume ratio of the uid for microand nano-devices becomes much larger than that for the conventional devices, therefore, the gas-surface interaction plays an essential role.

In this study, we focused on the heat transfer problem between two concentric spheres. This geometry was employed in a novel measurement system of the thermal accommodation coecient, which characterizes the mean energy transfer through the gas-surface interaction [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF]. This measurement system is based on the low pressure method, rstly introduced by Knudsen, later employed by many researchers [START_REF] Goodman | Dynamics of Gas-Surface scattering[END_REF]. This method employs the particular property of heat transfer at low pressure: the heat transfer between two surfaces maintained at dierent temperatures is proportional to the pressure between them and the thermal accommodation coecient. Therefore, the heat ux through rareed gas conned in concentric spherical shells is measured as a function of pressure, then the thermal accommodation coecient is extracted. The novel measurement system is able to measure the thermal accommodation coecient on non-metal surfaces, which is rarely reported in the literature due to specicity of the measurement methods [START_REF] Saxena | Thermal Accommodation and Adsorption Coecients of Gases[END_REF][START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF], and it is especially important for micro-and nano-devices because of the materials employed.

Recently, the heat transfer between the two concentric spheres system is also simulated [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF] on the basis of the nonlinear S-model kinetic equation [START_REF] Shakhov | Generalization of the Krook kinetic relaxation equation[END_REF]. In the free-molecular, slip and continuum ow regimes, the analytical expressions are provided for arbitrary temperature and spheres' radius ratios. In the transitional regime, the S-model kinetic equation is solved numerically. Then, the limits of the applicability of the obtained analytical expressions and the previous empirical relation are established by confronting the numerical and analytical solutions.

The objective of this study is to evaluate the accuracy of the empirical heat ux expression between the free-molecular and continuum ow regime, and re-vise the expression valid for the whole range of the ow regime. The analytical heat ux expressions in the free-molecular and continuum ow regimes and the numerical data in the transitional ow regime, obtained in Ref. [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF], are used to derive this revised heat ux expression. Then, based on the measurements of the heat ux and the pressure between two spherical shells, provided in Ref.

[3], the values of the thermal accommodation coecients are derived using proposed revised expression of the heat ux. Finally, the numerical simulations are carried out using the S-model kinetic equation with the obtained thermal accommodation coecients, and the measured and simulated heat uxes are compared to validate the revised expression.

Analysis of Experimental Heat Flux Data

The experimental data on the heat ux reported in Ref. [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF] are analyzed in the following. The two concentric spherical shells conguration is chosen as the experimental setup to measure the thermal accommodation coecient on non-metal surfaces. A tiny heater, maintained at temperature T H by an analog electrical circuit, is xed at the center of a spherical vacuum chamber.

The tiny heater has a thin at-plate shape, and the test sample surfaces are attached to the heater. The surface of the vacuum chamber made by Pyrex is maintained at the temperature T C by immersing the chamber in a water bath. The inner radius of the chamber is equal to R C = 49.5mm. The heat ux from the tiny sample to the chamber surface is measured together with the pressure in the chamber. By using the expression relating the heat ux between the surfaces and the pressure, the thermal accommodation coecient is derived.

The conguration of the experimental setup is explained in detail in Ref. [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF].

The heat ux from a heated surface surrounded by a monatomic gas in the the free-molecular ow regime is expressed as,

q F M = α 2 v T p (T H -T C ) , v = 8kT πm , (2)
here T , p and v are the temperature, pressure, and mean molecular speed of the gas, respectively; k is the Boltzmann constant and m is the molecular mass of the gas. It is important to underline that the heat ux is proportional to the pressure of the gas and, therefore, to the gas number density. The heat ux between two surfaces maintained at dierent temperatures in the free-molecular ow regime, Eq. ( 2), is independent of the geometrical conguration of the system, see Refs. [START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF][START_REF] Trott | An experimental assembly for precise measurement of thermal accommodation coecients[END_REF][START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF]. The accommodation coecient α can be obtained by tting heat uxes measured as a function of pressure by Eq. ( 2).

However, it is dicult in a simple apparatus to maintain the free-molecular ow regime (i.e., low pressure environment) owing to leakage. An alternative way is to build a high vacuum experimental setup; however, in this case, the measurement system becomes huge and costly. In addition, it is also not easy to measure very small heat ux at low pressure environment. To realize the measurement in a low-cost simple apparatus, a much higher pressure condition is favorable for the measurement. A more general model for the heat ux through a gas from a heated surface has to be implemented for the accommodation coecient extraction for this condition extended to the higher pressure. For the whole ow regimes, i.e. from the free-molecular to continuum ow regimes, the expression of the heat ux q can be approximated by a simple empirical interpolation of the free-molecular limit heat ux q F M , Eq. (2), and the continuum limit heat ux, q C , as it was done in Refs. [START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF][START_REF] Trott | An experimental assembly for precise measurement of thermal accommodation coecients[END_REF][START_REF] Springer | Heat transfer in rareed gases[END_REF][START_REF] Sherman | A survey of experimental results and methods for the transitional regime of rareed gas dynamics[END_REF], and it is expressed as

1 q = 1 q F M + 1 q C . (3)
For the continuum limit the heat ux q C is described by Fourier's law and it is independent of pressure and dependent upon the ow geometry. By making the size of the internal heated surface relatively small compared to the external surface of the vacuum chamber, the heat transfer problem is approximated by a simple spatially symmetric heat transfer between two concentric spherical shells, even though the shape of this internal heated surface is not a sphere but a at plate, as mentioned above [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF]. Following this model of two spherical shells, we can calculate the radius of the internal virtual sphere from the surface areas equality: the surface of the internal virtual sphere of a radius R H has the same surface area as the real heated surface. From this equality, the radius of virtual internal sphere is estimated as R H = 4.95mm. Thus, the radius ratio of the concentric spherical shells, R = R C /R H , is equal to 10 and it is relatively large. By assuming the concentric spherical shells geometry, the theoretical heat ux at the continuum limit q C in Eq. ( 3) is expressed as

q C = κ (T H -T C ) R C R H R C -R H 1 R 2 H , ( 4
)
where κ is the thermal conductivity of the gas. The temperature inside the spherical vacuum chamber is assumed to be equal to the temperature of the surface of the external spherical shell T C [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF][START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF]. Therefore, in this analysis, the temperature dependence of the thermal conductivity κ along the radial direction is not taken into account, and the thermal conductivity at the temperature of the external spherical shell, T C , is used for the entire region inside the vacuum chamber: κ = κ(T C ). In addition, the pressure is assumed to be constant between the shells.

To minimize an error which can come from the use of the empirical relation, Eq.( 3), the pressure condition is limited below 1.6Pa in the experiment so that the measurement is realized in the near free-molecular regime.

In order to test this new developed experimental setup, rst a platinum sample is used. A platinum foil with a thickness of 10µm (Nilaco) is selected as a sample surface. Five values of the hot sphere surface temperature, T H , are stated in the experiments, see Table 1. The cold sphere temperature, T C , is almost the same for all ve cases and equal to a room temperature. A number is attributed for each hot temperature value to simplify the reference. The averaged accommodation coecients, extracted by the described above procedure from three measurements of the heat ux and pressure for each case, are provided in Table 1. Surface temperatures were dierent for each measurement, and the mean surface temperatures are listed, with any variation from the mean value less than 0.2 K. The relative standard errors of the accommodation coecient did not exceed 1.6%, showing good repeatability of the measurements.

However, it was not so simple to estimate the measurement accuracy of the system [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF], and the number of signicant digits was decided from the size of the above mentioned relative standard error.

From Table 1, the accommodation coecients exceed unity in some cases. This could be coming from above mentioned several assumptions made in the extraction procedure. Therefore, more accurate expression to describe the heat ux is important to improve the existing methodology of the accommodation coecient extraction.

Analytical Solution and Numerical Simulation

In this section, we present the detailed analytical and numerical analyses on the heat ux problem between two concentric spherical shells in all ow regimes following Ref. [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF]. The assumptions used for derivation of all analytical expressions are provided. These analyses allow to estimate the error, which is made, when using the analytical expression for a set of physical parameters, and also it allows the improvement on the accuracy of the accommodation coecient extraction procedure.

The heat transfer between two spherical shells, of the arbitrary radii R H and R C , for the internal and external shells, respectively, is analyzed for various temperatures of the shells' surfaces T H and T C . The detailed description of the developed approaches can be found in Ref. [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF]. We provide here only one part of the results indispensable for the accommodation coecient extraction procedure. To characterize the level of the gas rarefaction, it is convenient to introduce the rarefaction parameter as following

δ 0 = R 0 , = µ 0 v 0 p 0 , v 0 = 2kT 0 m . (5)
Here R 0 is the reference length of the problem, is the equivalent mean free path, p 0 is the reference pressure, p 0 = n av kT 0 , n av is the number density averaged over the physical space n av = 3/(R 3 C -R 3 H ) n(r)r 2 dr, r is the radial coordinate of physical region between the concentric spheres with the origin at their centers, n(r) is the gas number density, which depends on r coordinate only, µ 0 and v 0 are the gas viscosity and the most probable molecular velocity at the reference temperature T 0 , respectively. For convenience, the reference values in Eq. ( 5) are taken as follows

T 0 = T C , R 0 = R C -R H . ( 6
)
The denition of the rarefaction parameter allows us to choose the appropriate modeling as a function of the rarefaction parameter δ 0 value. The cases of δ 0 = 0 and δ 0 → ∞ correspond to the free-molecular ow and continuum limits, respectively. It is to note that the gas rarefaction parameter δ 0 in Eq.

( 5) is inversely proportional to the Knudsen number.

Free-molecular ow regime

In the free-molecular ow limit (δ 0 → 0), the Boltzmann equation (or model kinetic equations) [START_REF] Cercignani | Mathematical methods in kinetic theory[END_REF] can be solved analytically for arbitrary temperature and radius ratio, because the collision term in its right-hand side can be neglected in this case. We assume here the complete accommodation of the molecules with the surface on the external sphere α C = 1 following the discussion in the experimental analysis [START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF][START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF], and on the internal sphere surface, the Maxwelltype diuse-specular reection with α H = α is assumed. The heat ux at any point between the spheres reads

q F M (r) = α 2 p 0 v (T -1) K F M R H r 2 , K F M = 1 + α 2 √ T -1 -1 1 - (R + 1) √ R 2 -1 R 2 + R + 1 -1 , ( 7
)
where T is the temperature ratio, T = T H /T C . It is interesting to note that the expression of the heat ux depends not only on the temperature ratio T , but also on the radius ratio R.

Transitional regime

Contrarily to the free-molecular ow regime, where the analytical solution of the kinetic equation was obtained, only numerical solution is possible in the transitional regime. Therefore, in this regime the S-model kinetic equation [START_REF] Shakhov | Generalization of the Krook kinetic relaxation equation[END_REF] was solved numerically for various range of the rarefaction parameter, shell's temperature and radius ratios. The details can be found in [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF].

Continuum regime

In the continuum limit, the temperature variation between two concentric spheres may be obtained from the energy balance

∂ ∂r r 2 κ ∂T ∂r = 0. (8)
It is to note that here the hypothesis of zero macroscopic gas velocity is used and only the conduction heat transfer is considered. The Fourier law can be applied to calculate the heat ux q = -κ dT dr .

(

) 9
For the monatomic gases the gas thermal conductivity is related to the gas viscosity as follows

κ = 15 4 k m µ. (10)
In order to dene the dependence of the viscosity on the temperature, the molecular interaction potential must be specied, and we use the inverse power law potential [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] in the following. This model leads to a power law temperature dependence for the viscosity coecient

µ = µ 0 T T 0 ω , ( 11
)
where ω is the viscosity index, which is equal to 0.5 for the Hard Sphere model and 1 for the Maxwell model. In the presented here analysis, the Variable Hard Sphere model (VHS) [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF] is used, where the viscosity index varies with the gas nature; it is equal to 0.66, 0.81 and 0.85, for Helium, Argon and Xenon, respectively. Taking into account the relation between the thermal conductivity and viscosity, Eq. ( 10), the thermal conductivity has similar temperature dependence to the viscosity.

In the continuum limit, the gas temperature in the vicinity of a wall is equal to the wall temperature, so Eqs. ( 8) and ( 9) are solved analytically for the arbitrary temperature and radius ratios. The heat ux distribution takes form:

q C (r) = κ(T C )K C (T H -T C ) R C R H R C -R H 1 r 2 , K C = T ω+1 -1 (ω + 1)(T -1) . ( 12
)
It is worth to note that contrarily to expression (4), the temperature dependence of the heat conductivity is taken into account in this expression. Table 2: Recalculated energy accommodation coecient by using the complete heat transfer expression, Eq. ( 7), with real radius ratio and taking into account the dependence of K F M coecient from the energy accommodation coecient. The relative dierence from the originally obtained accommodation coecients, provided in Table 1, is given in the brackets.

Accuracy Evaluation of Experimental Analysis

In the experimental analysis, several approximations were made to extract the accommodation coecient. However, they are not validated in detail because of a limitation of the experimental measurement in the simplied system.

We will investigate here several approximations made during the experimental analysis to understand in detail the heat ux behavior between two concentric spherical shells.

Free-molecular heat ux: eects of geometrical and physical parameters

The complete analytical expression of the free-molecular heat ux for the case of the arbitrary shell's temperature ratio T and radius ratio R is given by Eq. ( 7).

If the radius of the external sphere is large compared to the internal one (R → ∞), the case of the heat ux between a sphere and the surrounding gas, then the coecient K F M in Eq. ( 7) tends to 1. In this case, Eq.( 7) for r = R H gives the same results as Eq. ( 2) with p = p 0 .

We evaluate now the value of coecient K F M in Eq. ( 7) for the real radius ratio R = 10 and for several sets of measured temperature ratios and accommodation coecients given in Table 1. In the case of small temperature ratio T min = 1.139 with He, the deviation of K F M coecient from unity is of the order of 0.01%; while, in the case of large temperature ratio T max = 1.543 with Xe, this deviation increases up to 0.15%. Therefore, a posteriorly estimation of the coecient K F M 's deviation from unity seems to be relatively small, so expression of Eq.( 2) gives very accurate results for the used in the experimental setup radius ratio R = 10.

However, the K F M coecient depends also on the accommodation coecient, Eq. ( 7), and it might probably aect the tting result. Therefore, the experimentally measured heat uxes are re-analyzed by using the complete analytical expression, Eq. ( 7). From Table 2, it is clearly shown that neither the assumption of the innitely large radius ratio (R → ∞ if R > 10), nor the dependence of the K F M coecient on α does not aect the results. Therefore, the implementation of Eq. ( 2) in the experimental analysis of the free-molecular ow regime is completely justied with maximum accuracy less than 0.2%.

Continuum heat ux: eects of geometrical and physical parameters

Let now analyze the analytical expression of the heat transfer in the continuum ow regime. This expression takes part of the empirical tting formula, Table 3: Recalculated energy accommodation coecient by using the empirical heat transfer expression with the continuum heat ux q C , calculated using Eq. ( 12). The relative dierence from the originally obtained accommodation coecients, provided in Table 1, is given in the brackets.

Eq. ( 3), which is used for the extraction of the accommodation coecient in the case when the experimental pressure range is extended to higher pressure.

Even though the heat transfers are measured at relatively low pressure condition around the near free-molecular regime, the heat transfer in the continuum limit could aect the results through Eq. ( 3). Initially, Eq. ( 4) is used with the constant thermal conductivity, calculated at temperature T C due to the large surface-area ratio of the external to the internal spherical shells R 2 with the large radius ratio R [START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF][START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF]. While, the complete analytical expression of the heat transfer in the continuum ow regime, Eq.( 12), takes into account the dependence of the heat transfer coecient on the temperature. It is clear that only coecient K C makes the dierence between two expressions.

In the limit of the small temperature dierence between the spheres' surfaces 12) for r = R H is equivalent to Eq.( 4). In the cases of small temperature ratio T = 1.139, see Table 1, the discrepancy coming from this K C factor becomes 4.5%, 5.6%, 5.9% for He, Ar, and Xe, respectively. However, for the largest temperature ratio between the surfaces and with Xe, it increases up to 22%. Thus, the approximations of the heat transfer in the continuum limit might be a large error source in the experimental analysis.

T H -T C T C or T → 1, K C → 1 and expression (
The experimentally measured heat uxes are re-analyzed by the analytical form, Eq. ( 12), instead of Eq. ( 4). The extracted accommodation coecients are listed in Table 3. The discrepancy from the originally provided value, see Table 1, are also listed in (). It is clearly shown that the discrepancies are much larger for the conditions with large temperature dierences since the temperature distribution between the shells aects the results. The dierence in the gas species is resulting rst from the dierence in the thermal conductivity coecient for dierent gases which is involved in the q C expression, Eq.( 12). Then, for the same value of pressure, the rarefaction parameters δ 0 , Eq. ( 5), and so the ow regimes are dierent for the dierent gases: the lighter Helium is still in free molecular regime (smaller δ 0 ), whilst the heavier Xenon is already in the transitional regime. Although there is a large dierence in the evaluation of the heat transfer in the continuum limit between the analytical form and the original expression used in the experiment, the eect on the accommodation coecient is less than 5%. This should be because the heat transfers were measured in a low pressure environment, less than 1.6 Pa. However, it is recommended to use the analytical form q C (R H ) by Eq. ( 12), instead of Eq. ( 4), since it is more accurate and easy to integrate in the experimental analysis.

Revised expression for the heat ux interpolation

To express the heat ux in the transitional regime, the empirical expression, Eq. ( 3), is useful and known to show quite good results [START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF][START_REF] Yamaguchi | Measurement of thermal accommodation coecients using a simplied system in a concentric sphere shells conguration[END_REF][START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF]. In order to evaluate the accuracy of this empirical expression in detail, the numerical solution of the S-model kinetic equation is evaluated with the Hard Sphere (HS) model for a wide range of parameters; two radius ratios R = 2 and 10, two temperature ratios T = 1.1 and 1.5 and three accommodation coecients α = 0.6, 0.8 and 1.0, i.e. 12 cases in total. Figure 1 shows the results of the numerical solution of the S-model kinetic equation by markers, and the empirical expression, Eq. ( 3), with q F M and q C calculated by Eq. ( 2) and Eq. ( 12), respectively, by dotted lines. It is to note that the dimensionless heat ux q * = q/(p 0 v 0 ) is plotted. From Fig. 1, the empirical expression coincides with the numerical S-model results up to δ 0 ∼ 1. However, they start to deviate by increasing the rarefaction parameter from δ 0 = 10 -1 up to 10 2 , i.e. in the major part of the transitional regime up to the continuum ow regime. The heat ux from the empirical expression starts to decrease at smaller δ 0 compared to the numerical results.

To make a better tting function to reproduce the results of the S-model solutions, the following expression in a dimensionless form is proposed as a modication of the empirical expression, Eq. ( 3):

1 q * = 1 q * F M + 1 ζq * C , ζ = 1 1 -c1 δ0+c2 , (13)
where the dimensionless heat ux in the free-molecular q * F M and continuum q * C limits are expressed in Eqs. (37) and (23) in Ref. [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF]. The factor ζ is introduced to improve the tting quality in the transitional ow regime. The form of Eq. ( 13) is inspired by the expression of the dimensionless heat ux in the slip ow regime q * S , indicated as q r (r) in Eqs. ( 14) and (17) in Ref. [START_REF] Ho | Heat transfer through rareed gas conned between two concentric spheres[END_REF]. In that paper, q * S is obtained by integrating the energy balance equation with the temperature jump boundary conditions. This heat ux in the slip ow regime, q * S , was found to be dierent from q * C by a factor of ζ s = 1/(1 + c/δ 0 ), where c is a constant dened by temperature jump coecients, temperatures and radii of shells. However, this factor is not suitable as a correction term to the empirical expression, because the factor ζ s vanishes in the free-molecular limit (δ 0 → 0), resulting in the nite limit value for 1/(ζ s q * C ), because q * C is proportional to 1/δ 0 . Then, a constant is added to δ 0 in ζ factor to avoid the problem, so now 1/(ζq * C ) vanishes in the free-molecular limit. The correction factor ζ goes to unity in the continuum limit (δ 0 → ∞).

Here, the coecients c 1 and c 2 are analyzed to satisfy the dimensionless (q * ) S-model results for 12 dierent cases, and it is suggested from ttings by the least-square method by the dimensionless form of heat ux that c 1 = 1.04αT /R and c 2 = 1.97αT /R. The tting results by this revised tting function are plotted by solid lines in Fig. 1. It is clearly shown that the agreement between the S-model solutions and the revised tting function are excellent for the whole range of the rarefaction parameter. It is also important to note that the revised Table 4: To check the eect of the heat ux expression for the transitional regime on the accommodation coecient extraction procedure, the accommodation coecients are extracted from the heat ux of the S-model solutions by using the empirical expression, Eq. ( 3), and the revised tting function, Eq. ( 13).

tting function reproduces very well the numerical solution for wide range of parameters; 1.1 ≤ T ≤ 1.5 and 2 ≤ R ≤ 10.

To validate the tting procedure to obtain the accommodation coecient by the revised function, the accommodation coecient is extracted from the S-model numerical solutions by the same procedure as that used for the experimental data. The extracted value of the accommodation coecient should be the same as that used initially in the numerical simulation, α S , if the extraction procedure is accurate enough. The revised function, Eq. ( 13), and the original empirical expression, Eq. ( 3), are employed as a tting function and the extracted accommodation coecients are compared. The data are tted by the least-square method in the dimensionless forms; q * vs. δ 0 . The extracted accommodation coecients are listed in Table 4. The accommodation coecients, used in the numerical simulations, are denoted as α S in the table. As expected from the discrepancies between the S-model solutions and the empirical expression in Fig. 1, extracted accommodation coecients by the empirical expression can not reproduce the value originally used in the numerical simulation, α S . Compared to the empirical expression, the revised function reproduces the accommodation coecients quite well. Even in the worst case where both R and T are small, the deviation is less than 1%. Therefore, it is not highly accurate to use the empirical expression, Eq. ( 3), for the accommodation coecient extraction procedure in the transitional regime.

The revised tting function, Eq. ( 13), is employed to extract the accommodation coecients from the experimental results. In the accommodation coecient extraction procedure, the experimentally measured heat ux is tted in the dimensional form. This is essential to avoid the eect from the measurement error in a low pressure condition [START_REF] Yamaguchi | Investigation on heat transfer between two coaxial cylinders for measurement of thermal accommodation coecient[END_REF]. In the dimensional form of Eq. ( 13), the dimensional heat uxes q F M and q C instead of the dimensionless q * F M and q * C are expressed by using the dimensional form derived from Eq. ( 7) at the free-molecular ow limit and Eq. ( 12) at the continuum limit. The rarefaction parameter δ = (pR 0 )/(µ 0 v 0 ) using the experimentally measured pressure p is employed instead of δ 0 . Re-calculated accommodation coecients by the revised tting function are listed in Table 5. Compared with the results in Table 3, it is clearly shown that the function to express the heat ux in the transitional regime is as important as the consideration of the temperature distribution in the continuum limit in Section 4.2.

! " (((((((((( ((((((((( a: !;((( (((((((((( ((((((((( a: !"((( (((((((((( ((((((((( (a) (((((((((( ((((((((( a: !;((( (((((((((( ((((((((( a: !"((( (((((((((( ((((((((( (c Table 5: Re-extracted energy accommodation coecient by using the revised tting function, Eq. ( 13), by using the more accurate expressions of the heat transfer in the free-molecular ow regime, Eq. ( 7), and in continuum ow regime, Eq. ( 12). The relative dierence from the originally obtained accommodation coecients, provided in Table 1, is given in the brackets.

! # ! $ ! % ! & ! ' ! ! ! ' !' ' ' '
T = 1.1, R = 2 !" !#$ !# !%$ !% ! $ ! ! ! % !% % % %

Comparison between measured and simulated heat ux

To validate the expressions of the heat ux between the concentric spherical shells, the energy accommodation coecients are re-extracted from the experimental data, and the heat uxes are numerically evaluated using the re-extracted results. The comparison between the measured heat ux as a function of pressure and the numerical results for three gases of He, Ar and Xe for ve cases of the surface temperatures are shown in Fig. 2. The numerical simulations are carried out using the S-model kinetic equation. Two dierent sets of the accommodation coecient are employed; the original values extracted from the measurements listed in Table 1 and the corrected values, listed in Table 5, extracted by using the improvements, explained in the previous sections.

In Fig. 2 (a), (c) and (e), the dimensionless curves of the measured and simulated heat ux, q * = q/(p 0 v 0 ), as the function of the pressure via the rarefaction parameter δ 0 are presented. As it is clear from (a), the measured pressure range is restricted to the near free-molecular ow regime, where the dimensionless heat ux has the constant value, which depends only on the accommodation coecient. However, in (c) and (e), the curves of the heat ux start to decreases, that means the ow is in the transitional, or even in the slip ow regime. From these gures, it is clear that the experimental data scattered around the lines, especially in the near free-molecular ow regime, due to the measurement diculty as it was already explained in Section 4. The corrected accommodation coecients are also compared with those in the literatures [START_REF] Trott | An experimental assembly for precise measurement of thermal accommodation coecients[END_REF][START_REF] Amdur | Thermal accommodation coecients on gascovered tungsten, nickel and platinum[END_REF][START_REF] Thomas | The accommodation coecients of he, ne, a, h2, d2, o2, co2, and hg on platinum as a function of temperature[END_REF] for validation in Fig. 2. Considering the dierence in surface conditions which is not known, the corrected values are in good agreement.

Conclusion

The measurement technique, called the low-pressure method, was analyzed by comparing the expressions, usually used by this technique, with the analytical expressions and numerical solution of the S-model kinetic equation. In the free-molecular ow limit, it is conrmed that the original expression coincides with the analytical form, and it is reasonable to assume the radius ratio /01 /0! !01 !0! !%&23. 43565%%%%%%%%%%%%%%%%%%%%%%%%%%%%%/%%%%%%%%$%%%%%%%%7%%%%%%%%#%%%%%%%%1 89:6;<*6=>%%%%%%%%%%%%%%%% % % % % ?@*AB6CD%4A;;64>6B%a% % % % % ?@*AB6CD%A;<E<=3C%a%%%% % % % % (f) Xe, dimensional Figure 2: The heat ux as a function of pressure: comparisons between the experimental and numerical data with the dimensionless and the dimensional forms for the ve cases. The numerical results by using the originally obtained accommodation coecients, listed in Table 1, and by using the corrected values, listed in Table 5, are compared.

  ((((((()*+,-./(((0+123245/(((6277289 a:'! (((

) T = 1 . 1 ,

 11 T = 1.5, R = 10

Figure 1 :

 1 Figure1: The dimensionless heat uxes q * are plotted as a function of the rarefaction parameter δ 0 . The S-model solutions, shown by markers in all ow regimes, are compared with the empirical expression, represented by dotted lines, and the revised tting function, shown by solid lines. For the accommodation coecients α = 1.0, the results are plotted in black (S-model: ), for α = 0.8, they are plotted in red (S-model: ) and for α = 0.6,they are plotted in blue (S-model: ), respectively.

3 .

 3 The experimental data seemed to converge in the region where the lines curved. The comparison between the solid and dotted lines indicates that the corrected accommodation gives better agreement with the experimental results.In Figs.2 (b), (d) and (f ), the dimensional quantities are plotted. This form is used for extracting the accommodation coecient by the least square method. It is clearly shown that the agreement is much better from the original to corrected accommodation coecient for the whole pressure range. Therefore, it is conrmed that the revised procedure suits well for the extraction of the accommodation coecient. It is also important to note that even though the at plate-shaped heater is assumed as a sphere in the experiment, as it was explained in Section 2, the experimental heat ux is well simulated by the numerical solutions of the S-model kinetic equation, indicating the validity of the assumption.

  ############################# ########3########4########5########! 67829:(2;<################ # # # # =>(?@2AB#0?9920<2@#a# # # ## =>(?@2AB#?9:C:;/A#a###

Table 1 :

 1 Experimental temperature conditions and mean energy accommodation coecients based on measurements.

			1	2	3	4	5
		T H (K)	335	364	395	424	453
	He	T C (K)	294	294	294	294	294
		α	0.280 0.292 0.308 0.322 0.338
		T H (K)	335	364	394	424	453
	Ar	T C (K)	293	293	293	293	294
		α	0.850 0.867 0.856 0.864 0.886
		T H (K)	335	364	394	424	453
	Xe	T C (K)	294	294	294	294	294
		α	1.024 1.045 1.066 1.053 1.065

Preprint submitted to ElsevierDecember 12, 2016

& & & & ?@+AB6CD&4A;;64>6B&a& & & & & ?@+AB6CD&A;<E<=3C&a&&&& & & & &

Acknowledgments

This work was partially supported by JSPS KAKAENHI Grant Number 16K14157. This work was also granted access to the HPC resources of Aix-Marseille Université nanced by the project Equip@Meso (ANR-10-EQPX-29-01) of the program Investissements d'Avenir supervised by the Agence Nationale pour la Recherche. The authors would like to thank Mr. Takamasa Imai, Mr. Tadashi Iwai and Mr. Akira Kondo for their support on the measurements.

to be innite when it is about 10. In the continuum limit, it is important to consider the temperature dependence of the thermal conductivity coecient.

In the transitional regime, there is a small discrepancy between the empirical


K D Ahose

S Lejeunes

D Eyheramendy

On the thermal aging of a filled butadiene rubber

In this study we investigate the influence of thermal aging on the mechanical properties of a butadiene rubber filled with carbon black. To emphasize the influence of both the crosslink density and the crosslink lengths, we consider three different materials based on the same formulation. Dynamic and quasistatic characterizations are realized periodically at room temperature to study of the impact of aging on various mechanical characteristic such as the equilibrium hysteresis, the Payne effect, etc. Crosslink density is followed by swelling tests.

INTRODUCTION

The impact of thermal aging phenomena on physical properties such as mechanical stiffness, tensile strength limit, loss angle, shore hardness, has been studied by many authors, see for instance [START_REF] Tomer | Cross-linking assessment after accelerated ageing of ethylene propylene diene monomer rubber[END_REF][START_REF] Shabani | Thermal and radiochemical of neat and ATH filled EPDM : establishment of structure/properties relationships[END_REF][START_REF] Kumar | Ageing of elastomers: a molecular approach based on rheological characterization[END_REF], Kartout 2016, Ben Hassine 2013 and references therein. Obviously, aging phenomena are material dependent and different issues arrise depending on monomers or vulcanizate agents (see [START_REF] Choi | Accelerated thermal aging behaviors of epdm and nbr vulcanizates[END_REF] for a comparison of EPDM and NBR). These phenomena are also strongly coupled with the environment, chemical reactions can occur with air (oxidation), salt water (see for instance [START_REF] Gac | Ageing mechanism and mechanical degradation behaviour of polychloroprene rubber in a marine environment: Comparison of accelerated ageing and long term exposure[END_REF][START_REF] Rabanizada | Experimental investigation of the dynamic mechanical behaviour of chemically aged elastomers[END_REF], etc. Furthermore, mechanical state during thermal aging may also have an influence on aging phenomena (see for instance [START_REF] Ciutacu | Accelerated thermal ageing of ethylene-propylene rubber in pressurized oxygen[END_REF]. The chemo-physical evolutions involved during aging may lead both to the formation of new crosslinks and to the dissociation of existing ones. For sulfur vulcanized system a commonly admitted mechanism is the degradation of poly-sulfur crosslinks into di-sulfur of mono-sulfur ones. Free sulfurs can eventually migrate and form new crosslinks. This phenomenon is of interest for mechanicians as it can occur during high-cycles fatigue in particular when the heat-build is significant or when the thermal environment plays an important role.

In this study we propose to investigate the consequence of thermal aging on a butadiene rubber reinforced with carbon black fillers and vulcanized with sulfur. To study the influence of the poly-sulfur crosslinks we consider a single rubber formulation cured under different conditions (temperature and time of cure). This lead to different crosslink networks and different crosslink densities. We proceed to mechanical characterizations with quasi-static and dynamic tests on unaged and aged specimens.

EXPERIMENTAL SETUP

Materials

The material is a polybutadiene rubber reinforced with carbon black and vulcanized with sulfur. Due to confidentiality reasons, the formulation of this rubber can not detailed here. The vulcanization system is efficient (sulfur to accelerator ratio is greater than one) and the filler mass fraction is about 45%. This system was cured at three different temperatures: 130 o C, 150 o C and 170 o C. The time of cure for each temperature was previously determined with the help of a rheometer: time of cure was obtained when the measured torque reach 98% of the maximum observed one at each temperature. We obtained the following time of cure: 170 o C -7min, 150 o C -17min, 130 o C -55min. To simplify, in the following each couple time/temperature is refereed as a different material even if the initial composition is the same. Each material can be considered as fully vulcanized and differs from each others from its network structure: for high temperature, we have higher crosslink lengths (poly-sulfurs) and a smaller crosslink density than for smaller temperatures of cure.

Thermal aging

Thermal aging of H2 tensile specimens was done with the help of a oven. Each sample was previously put inside an individual plastic bag in which air was partially removed with a vacuum pump. The objective was to minimize thermo-oxidative phenomena and to limit non homogeneous aging. Some specimens, 2 by material, were periodically removed from the oven for mechanical, and chemical characterizations at room temperature.

Mechanical characterizations

Mechanical characterizations on tensile specimen consisted into the following sequence: a softening phase to eliminate the Mullins effect, relaxations by step (relaxations at different increasing and decreasing amplitudes), cyclic sinusoidal tests with two static amplitudes and three dynamics amplitudes (frequencies of 0.1Hz, 3Hz, 6Hz, 10Hz). These tests were all done at room temperature.

UNAGED BEHAVIOR

Chemical and thermal characterizations

We realized swelling tests on small specimens that were put in xylene. Figure 1, shows the evolution of the relative swelling ratio in mass upon the time spent in solvent. We define the relative swelling ratio in mass from:

q = m s -m 0 m 0 (1)
where m s is the mass of the specimen after swelling and m 0 is the initial mass. Swelling tests are good indicators of the crosslink density of the vulcanized network. The crosslink density could be estimated using the equation proposed by [START_REF] Kraus | Swelling of filler-reinforced vulcanizates[END_REF] for filled rubber, however, we do not had access to the volume fraction of unfilled rubber in the swollen rubber phase. Nevertheless, it can be seen from Kraus or Flory-Rhener works [START_REF] Flory | Statistical mechanics of crosslinked polymer networks ii. swelling[END_REF]) that crosslink density is related to the inverse of the previously defined swelling ratio. We can therefore admit that crosslink density is higher for the material cured at 130 o C -55min than for those cured at 150 o C -17min and 170 o C -7min. We also realized thermal characterizations with a DSC and optical dilatometry with DIC. The glass-transition temperature was measured at -82 o C for each materials and the thermal dilatation behavior was also very close for each materials.

Mechanical behavior

Figures 2 and 3 show some obtained results on unaged samples. It can be seen that, in accordance with the statistical theory of rubber network, the higher is the crosslink density the higher is the stiffness. where p is the hydrostatic pressure. For the equilibrium part, we adopt a Mooney-Rivlin potential defined such as:

ρ 0 ψ eq = sC 10 (I 1 -3) + sC 01 (I 2 -3) (3)
where ρ 0 is the reference density, I 1 , I 2 the two first strain invariants. The material coefficients, C 10 , C 01 are classical Mooney parameters (which are identical for each materials) and s is a parameter that is fixed to 1 for the material 130-55 and that is identified for other materials. The identification of C 10 , C 01 , s 150 , s 170 is done by minimizing the leastsquare distance from the end of relaxation points (extracted from figure 3) to the predicted ones for the three materials at the same time. We obtain the material parameters given at table 1 (see also figure 5).

It can be remarked that the obtained values for s are nearly equal to the inverse of the relative swelling ratio such as we recover the hypothesis of proportionality of the rubber elasticity upon the crosslink density that is made in statistical theories. We can postulate:

s x = q 130 q x (4)
where x stands for the material 150-17 or 170-7. C 10 C 01 s 150 s 170 0.185M P a 0.127M P a 0.948 0.9 q 130 /q 150 q 130 /q 170 0.944 0.898 For the non-equilibrium part, we adopt a Neo-Hooken potential and a Maxwell viscous flow:

                         ρ 0 ψ neq = n i=1 (G i /s)ω i (I ei 1 -3) Ḃei = L Bei + Bei LT - 1 sτ i BD ei Bei ωi = - 1 h i ω i - 3 I 1 ( B) r i , ω i (t = 0) = 1 (5)
where Bei is the i th isochoric left Cauchy Green tensor coming from the i th multiplicative split of the transformation gradient into viscous and elastic part, L is the isochoric tensor of eulerian velocities, ω i is the i th internal variable to account of the Payne effect, and G i , τ i , h i , r i are material parameters (i = 1..n).

The material parameters identification of the non equilibrium part is done only on the data of the material 130-55 (by setting s = 1). We have therefore a viscoelastic model that can account of a variation of the crosslink density, using the relation of eq. (4). A good agreement with the response of the two other materials is observed as can be seen from the figures 6 and 7.

AGED BEHAVIOR

We investigate, in this section, the behavior of aged tensile samples that were put in a oven at 90 o C. No significant variation of mass or volume, of the samples, was observed after aging.

Chemical characterizations

The results of the swelling tests done at different aging time is synthesized with figure 8. We plotted the inverse of the swelling ratio q relative to the initial (unaged) swelling ratio q 0 for each material. It can be seen that the material that has the larger intial mean crosslink length (the larger number of polysulfur crosslink) exhibits the stronger evolution of the crosslink density. After 11 days of aging, the swelling ratio, q, is very close for each material as the initial concentration of sulfur is the same in each material.

Mechanical characterizations

To investigate the results of relaxation tests, we have plotted the stiffness obtained from the end of relaxation tests done at the maximum amplitude. It can be seen from figure 9 that the equilibrium behavior become stiffer. Furthermore, the material 170-7 seems to become more rigid than the two others. We can also notice that the dispersion of the results increases with aging.

For dynamic tests, we calculated the median line and the area of stabilized hysteresis curves. In the following, the term dynamic stiffness stands for the median line. Typical results are shown with figures 10 and 11. As for the equilibrium stiffness, the dynamic stiffness increases with aging while the hysteresis area decreases. The dispersion is stronger than for equilibrium results and seems to increase with aging. Figures 12, 13 and 14 show that the effect of the dynamic amplitude of loading (Payne effect) and the effect of the frequency of loading are not impacted by aging (at least after 15th days of aging): we only see

DISCUSSION

Based on the previous experimental results we can made the following remarks:

• Crosslink density and mean crosslink length.

Under non-oxidative conditions, thermal aging leads to an increase of the crosslink density of a sulfur vulcanized filled rubber and this Figure 14: Hysteresis area on aged samples upon the amplitude of the dynamic amplitude at 3Hz with 50% of elongation preload evolution is related to the initial (and current) mean crosslink length.

• Fillers/rubber network interactions. Aging mechanisms seem to not impact in a significant manner the fillers/network interactions (at least at 90 o C for aging). Amplitude and frequency of loading effects are not modified by aging.

• Rubber network. The crosslink density may not be the only parameter to take into account in a modeling of non-oxidative aging. In this campaign, we show that the aged mechanical behavior can be slightly different between the three materials even if the results of swelling tests are very close after aging. We guess that the aged rubber network is not exactly the same for the three materials after aging, even if the initial formulation is exactly the same.

• Dispersion of experimental results. We saw an increase of the dispersion of experimental results during aging. This effect can have many different origins, this can be due to damage that can occurs during mechanical test. As the limit of chains extensibility is reduced due to aging, damage can occur for smaller load amplitudes. This can also be a consequence of a non homogeneous aging that can have different origins: heterogeneity of temperature in the oven or oxidation (even if we tried to minimize it).

For the modeling part, taking into account of the previous remarks, we will have to introduce at least two supplementary variables that will be related to the mean crosslink length and the crosslink density. We already have introduced a relative crosslink density variable in the unaged behavior (eq. 4). This variable should eventually be complemented with another one.

CONCLUSION

We have investigated the consequence of thermal aging for a sulfur vulcanized filled rubber. From the re-sults of the experimental campaign, we have shown that aging is strongly related to the initial rubber network (crosslink density and mean crosslink length). As already shown by previous authors, the increase of stiffness, the decrease of dissipated energy and the increase of crosslink density are the three main phenomena but the kinetic is dependent on the initial vulcanization system and on the initial crosslink lenghts and crosslink densities.

To investigate further aging mechanisms and their kinetic, we need to do other experimental tests with different temperatures of aging. We will also need to study the impact of a permanent mechanical loads during aging, as done by [START_REF] Johlitz | Thermo-oxidative ageing of elastomers: A modelling approach based on a finite strain theory[END_REF] From the modeling point of view the challenge is to describe the kinetic with few supplementary variables. We also need to have a good knowledge of the initial network structure and swelling tests may not be sufficient for that.

Figure 1 :

 1 Figure 1: Relative mass solvent absorption upon the swelling time for the three materials on virgin samples (not submitted to thermal aging)

Figure 4 :

 4 Figure 4: Stabilized hysteresis in tension on non-aged samples for cyclic tests at 3Hz

Figure 5 :

 5 Figure 5: Identification of the equilibrium contribution with a Mooney-Rivilin potential

Figure 6 :Figure 7 :

 67 Figure 6: Validation of the identified parameters of the nonequilibrium contribution with the material 170-7 (f = 10Hz)

FigureFigure 10 :

 10 Figure 9: Equilibrium stiffness of aged samples

Figure 13 :

 13 Figure 13: Dynamic stifness on aged samples upon the amplitude of the dynamic amplitude at 3Hz with 50% of elongation preload

Table 1 :

 1 Material parameters of the equilibrium part