## Abstract

The effective thermal conductivity of nanocomposites constituted by nanoparticles and homogeneous host media is discussed from the point of view of extended irreversible thermodynamics. This formalism is particularly well adapted to the description of small length scales. As illustrations, dispersion of Si nanoparticles in Ge (respectively, SiO_{2} in epoxy resin) homogeneous matrices is investigated, the nanoparticles are assumed to be spherical with a wide dispersion. Four specific problems are studied: the dependence of the effective thermal conductivity on the volume fraction of particles, the type of phonon scattering at the interface particle–matrix, the radius of the nanoparticles and the temperature.

## 1. Introduction

Nanofluids and nanocomposites have considerably matured during the last decades, both from the fundamental and the applied points of view. They have been used in a wide variety of applications linked to their potential to develop significant modifications of thermal heat transfer properties [1–4]. The change in thermal conductivity has also been exploited to obtain enhancements in the figure of merit *Z* of thermoelectric materials [5] which behaves as the inverse of the heat conductivity. Many nanostructured materials have overcome the limit *ZT*=1, for instance *ZT*=2.4 in Bi_{2}Te_{3} or Sb_{2}Te_{3} [6] and a *ZT*=2.6 in layered SnSe crystals [7].

In this work, we focus on the discussion on how the presence of nanoparticles fundamentally modifies the effective thermal conductivity of nanocomposites. The subject is of interest as confirmed by the numerous publications during the last decade. The change in the effective heat conductivity is, among others, linked to the nature and properties of the host matrix and the nanoparticles, and more particularly to the volume fraction of nanoparticles, their dimension, the nature of particle–matrix interface and the temperature. At the theoretical level, one of the main problems that occur is related to the choice of the analytical expression of the effective heat conductivity of the nanocomposite in terms of the heat conductivities of the matrix and the embedded particles. The problem can be approached by solving directly the Boltzmann equation for a phonon gas but it raises many difficulties from a mathematical point of view and more particularly with regard to the nature of the boundary conditions. Another option is to use *ad hoc* analytical expressions, obtained by correlating several data (e.g. [8,9]); the shortcomings of such formulations is their limited applicability and their lack of physical background. Several works are based on Fourier's heat conduction law (e.g. [10]; see also [9,11] for an overview), which is not applicable when the dimensions of the system are comparable or smaller than the mean free path of the heat carriers [12,13]. Here, our objective is to go beyond Fourier's law and to avoid solving Boltzmann's transport equation. This is achieved by constructing a phenomenological approach axed on one of the latest developments of non-equilibrium thermodynamics, namely extended irreversible thermodynamics (EIT), e.g. [14,15]. This formalism has proved to be particularly well suited to describe systems at short time and small length scales and has been applied in a previous work for the description of a transient temperature profile through a nano-film. It is the use of EIT which provides a new and original approach to the problem.

The working hypotheses of the present theoretical study are the following:

— the nanoparticles take the form of rigid homogeneous non-porous spheres;

— the spheres are distributed randomly in the matrix;

— the matrix element is homogeneous;

— nanoparticles aggregations are not taken into account; and

— the material parameters used in the calculations are those of the Debye model [16,17].

One may find in the literature several mathematical expressions of the effective heat conductivity of nanosystems (e.g. [9,11]). In this work, we will use the following relation that finds its origin in an analogous derivation for the electrical conductivity of rigid particles in a fluid by Maxwell [17], and improved later on by Bruggeman [18]: accordingly, the effective heat conductivity will be given by
*φ* stands for the volume fraction of the dispersed particles, λ_{m} and λ_{p} for the heat conductivities of the matrix and the particles, respectively, *α* is a dimensionless parameter accounting for the interactions at the particle–matrix interface [19] and given by
*r*_{p} is the radius of the spherical particles, *R* is a measure of the interfacial boundary resistance and *Rλ*_{m} is the so-called Kapitza radius. If *R*=0, whence *α*=0, the interface is called a perfect interface. Relation (1.2) leads to satisfactory predictions in the case of diffuse scattering for which Chen [16] establishes the result
*v*_{i}(*i*=*m*,*p*) designating the volumetric specific heats and phonon group velocities, respectively.

Our task is to determine the values of the quantities λ_{m}, λ_{p} and *α* that enter in expression (1.1) of the effective heat conductivity λ_{eff} in terms of the volume fraction, nanoparticles radius, degree of specularity of the interface particle–matrix and temperature. The heat conductivity λ_{m} as well as the coefficient *α* will not raise many problems as they will generally be obtained from experimental data and/or well-established models; the determination of the quantity λ_{p} is a source of difficulty because of the small dimensions of the particles and demands a special treatment; it represents the essential motivation of the present analysis based on EIT [14,15]. As shown in the appendix, wherein we briefly recall the ingredients of this formalism, the expression for λ_{p} is given by
*Kn* is the Knudsen number, defined as *Kn*=*l*/*r*_{p} with *l* the mean free path of the phonons. It is checked that for *Kn* indicating a linear dependence with respect to the radius *r*_{p} which is the observed asymptotic behaviour in nanostructures.

In the first part of §2, we study the influence on the effective heat conductivity of the volume fraction of particles concomitantly with their dimension and the nature of the interface between particles and matrix. Two illustrations are considered: uniform dispersion of Si particles in Ge and SiO_{2} particles embedded in epoxy resin; the results are compared with other models and experimental data. The second part of §2 is devoted to the study of the variations of the effective heat conductivity with the temperature. Final comments and comparison with two other models are found in §3.

## 2. Modelling effective heat conductivity in nanocomposites

Our objective is to determine the dependence of the effective heat conductivity λ_{eff} of nanocomposites on the volume fraction of the nanoparticles, their size, the nature of the interface (either diffusive or specular) and the temperature.

### (a) Volume fraction, size and interface dependence

We first consider the simplified case of a fixed temperature, say room temperature. To determine the volume fraction and size dependencies of λ_{eff}, we need the expressions of the heat conductivities λ_{m} and λ_{p} of the matrix and the particles, respectively. Determining λ_{m} will not raise many problems, indeed it is sufficient to use the classical expression
*c*^{v}_{m} is the volumetric heat capacity, *v*_{m} the speed of sound and *l*_{m} the mean free path expressed by the empirical Matthiesen rule
*et al.* [25]. Note that the maximum volume fraction *φ* is the one corresponding to the maximum packing of hard spherical particles (i.e. *φ*_{max}=*π*/√18<1).

To take into account the nature of the collisions at the interface matrix–particle, we have, following Dames & Chen [26], replaced the particle radius *r*_{p} by the quantity
*s* standing for the probability of specular diffusion of the phonons on the particle–matrix interface; *s*=0 is characteristic of diffusive collisions, while *s*=1 denotes pure specular interactions.

Let us now determine the expression of λ_{p} which will be different from that of λ_{m} as we must take into account the size dependence of the thermal conductivity. In the following, we will use the result (1.4) provided by EIT and write λ_{p} in the form

The above model will be illustrated by two examples, namely Silicium (Si) particles dispersed in Germanium (Ge), and Silica (SiO_{2}) particles embedded in an epoxy resin. The latter example has been selected because it allows to compare with experimental data; moreover, the effective heat conductivity is seen to increase with the volume fraction instead of decrease as observed for Si–Ge. The system Si–Ge has been the subject of much attention during the last years as attested by the works of Wang & Mingo [27] and Kim *et al*. [28].

First, calculations have been performed for the couple Si–Ge with three different values for the radius of the Si nanoparticles (*r*_{p}=5, 25, 100 nm) and for *s*=0, 0.2, 1. The material parameters [16] used in the calculations are given in table 1. The corresponding results are reported in figure 1, in this figure and in the following ones, the volume fraction is limited to the value *φ*=*π*/√18, which corresponds to the maximum packing of hard spherical spheres, as discussed earlier.

Although expression (1.3) of the thermal boundary resistance coefficient *R* was derived in the diffuse limit [16], we have checked its validity by taking a non-zero *s*-value (*s*=0.2), and even the extreme case of a pure specular surface *s*=1, by comparing our results with those of [21], who include specular scattering in their developments. We observe good agreement indicating that approximation (1.3) used for *R* is valid outside its strict range of applicability. Our results agree also with Monte Carlo simulations as performed by Jeng *et al*. [29]. Figure 1 indicates that whatever the values of the radius, λ_{eff} remains practically constant up to values of *φ* close to 0.01 after which, it gradually decreases. This is true for small *s*-values but not for *s*=1 for which a steep increase takes place especially at high *φ*-values. The same behaviour was noted by Behrang *et al*. [21] but the correspondence is only qualitative because it must be kept in mind that the applicability of our model is restricted to small *s*-values. It should also be noted that our numerical values are slightly larger than those of Behrang *et al*. [21] and Minnich & Chen [19], principally for large radii (around 100 nm). This is not surprising as these authors use different values for the material parameters, based on the dispersion rather than on Debye's model. When we repeat our calculations with the dispersion approximation, the differences between our description and those of Minnich-Chen and Behrang *et al*. become minute.

It may seem strange that the thermal conductivity of the composite Si–Ge is smaller than that of the pure matrix Ge when the volume fraction of particles in increased. Indeed, since bulk Si has a larger thermal conductivity than bulk Ge, one should expect that composite conductivity will be higher. The reduction of the conductivity finds its interpretation in the small dimension of the particles. Indeed, relation (1.4) tells us that the thermal conductivity of nanospheres of radii comparable or smaller than the mean free path of heat carriers may be considerably less than that of the bulk, hence a decrease of the effective heat conductivity of the composite. Moreover, the smaller is the radius, the smaller the thermal conductivity of the nanoparticles. Similar results are also observed in Si_{x}Ge_{1−x} alloys [27].

A further check of the validity of the model is provided by calculating the effective heat conductivity of a different material, namely SiO_{2} particles embedded in epoxy resin for which experimental data are available [30]. As shown in figure 2*a*, the effective heat conductivity λ_{eff} is slightly growing linearly with particle volume fraction *φ* up to *φ*=0.1 followed by a steep increase. A zoom of the results in the region 0<*φ*<0.1 (figure 2*b*) exhibits the quasi-linear growth of λ_{eff} and the good accord with the experimental data. In contrast with Si–Ge, the effective heat conductivity of the SiO_{2}–epoxy composite is increasing with the nanoparticle density. Our analysis indicates that the boundary properties and the particles dimension play a decisive role in the decrease or increase of thermal conductivity. A possible interpretation of the observed behaviours may be found in the value of the dimensionless *α* (=*Rλ*_{m}/*r*_{p}) parameter which is much smaller in the case of SiO_{2}–epoxy than for Si–Ge of the order 50–200 depending on the values of *r*_{p} and *s*. For *α*>1, λ_{eff} is decreasing, while for *α*<1, λ_{eff} is increasing. This result reflects the relative importance of the dimension *r*_{p} of the particles with respect to the Kapitza radius *Rλ*_{m}. For a given value of the thermal resistance *R*, the less is the radius of the particles, the less is the thermal conductivity as exhibited by figures 1*a*–*c*. The value of the α-parameter is of importance within the perspective of practical applications: constituents with small α-values should be selected when significant enhancement of the thermal conductivity is aimed at while large values should be preferred when a reduction of phonon transport is the objective.

### (b) Temperature dependence

The temperature dependence of the heat conductivity will appear implicitly through the frequency *ω* dependence of the various quantities appearing in the general expression
_{eff} as given by relation (1.1) requires therefore the knowledge of *j*=*m*,*p* and *l*_{m,coll}(*T*) in terms of *ω* and *T*, recall that in the expression of the particle mean free path, only the bulk free path is needed. The limit of integration, *ω*_{D}, is the Debye frequency cut-off: *ω*_{D}=5.14×10^{−13} s^{−1} for Ge and 9.12×10^{−13} s^{−1} for Si. In agreement with [31–33], we assume that the group velocity *v* is independent on *T* and *ω*, while the heat capacity and the bulk mean free path are given, respectively, by
*B*_{j} and *θ*_{j} are constant quantities obtained by fitting experimental data measured by Glassbrenner & Slack [34]. We are now in possession of all the elements needed to evaluate the effective heat conductivity of the nanocomposite as expressed by relation (1.1). To be explicit, the heat conductivity λ_{m} of the matrix element is directly derived from (2.6) with the mean free path in the matrix *l*_{m} (*T*, *ω*) given by _{p} will be written as
_{eff} of the composite Si–Ge are reported in figure 3*a* through 3*c* as a function of the volume fraction *φ* of Si particles, temperature *T* and the specularity coefficient *s*. It is shown that λ_{eff} decreases with the temperature whatever the radius of the nanoparticles and the *s* coefficient; at high temperature (*T*=500 K) and large *r*_{p}-values (around 100 nm), the heat conductivity remains practically constant. By comparing the curves for *s*=0 and *s*=0.2, one observes no drastic changes. A more detectable modification is observed for *s*=0.5 and especially for *a*_{p}=100 nm for which an increase of λ_{eff} is observable at *φ*-values of larger than 0.5.

We have represented on figure 4*a*–*c* the effective heat conductivity versus the temperature for three values of the volume fraction *φ*=0.01, 0.2 and 0.5, *s* being fixed equal to zero, while the values of *r*_{p} are the same as previously, namely *r*_{p}=5, 25, 100 nm.

To emphasize the role of the presence of nanoparticles on the composite heat conductivity, we have drawn the curve corresponding to a pure Ge crystal; the reduction of λ_{eff} becomes more important as the size of the Si particles becomes small and the volume fraction large. The decrease of λ_{eff} with temperature may be explained as follows: by increasing the temperature, one causes an increase of the thermal resistance whence a diminution of thermal conductivity. This effect is less pronounced for smaller radii of the particles, because of the increase of the particle–matrix interface. This can be interpreted by saying that phonons will meet more obstacles with, as a consequence, a reduction of heat transport. Heat conductivity is practically insensitive to temperature variations at high volume fractions (*φ*>0.5) and small nanoparticles (*r*_{p}<5 nm). Our results are qualitatively similar to those obtained by Behrang *et al*. [21] with the differences that they restrict themselves to diffusive scattering (*s*=0) with nanoparticles radii from 10 to 50 nm versus 5 to 100 nm in this paper.

## 3. Final comments

Our objective is to study the change in heat conductivity resulting from the dispersion of spherical nanoparticles in homogeneous host media in the framework of EIT [14,15], whose basic concepts are recalled in the appendix. The dependence with respect to several factors as volume fraction, particles radius, nature of the interface and temperature is examined. The originality of the present model is the derivation of the expression of the heat conductivity of the nanoparticles.

The most important result of this paper is embedded in equation (2.5). It makes explicit the dependence of the heat conductivity of the nanoparticles on their size through the Knudsen number

The model predicts numerical values which are of the same order of magnitude as those obtained by other authors as Minnich & Chen [19] and Behrang *et al.* [21] whose predictions match Monte Carlo simulations. The observation that our results are in good agreement with other ones based on different models attests of the validity of the present approach.

The results plotted in figure 1 indicate that the effective heat conductivity λ_{eff} of Si–Ge composite is decreasing with the nanoparticles density and that at a fixed volume fraction, λ_{eff} is decreasing with decreasing radii. Such an effect may be of interest within the perspective of an optimal conversion of heat transport into electric current; indeed, the efficiency of this conversion is measured by means of the so-called figure of merit defined by *Z*=*σ*_{e}*ε*^{2}/λ, with *σ*_{e} the electrical conductivity and *ε* the Seebeck coefficient so that a lowering of the heat conductivity λ will clearly contribute to a better efficiency.

Most works are silent about variation of the effective heat conductivity λ_{eff} with the temperature. This subject is discussed in the second part of §2, where it is shown that increasing the temperature results in a decrease of the effective thermal conductivity, in particular, the variations of λ_{eff} with temperature are seen to be less important for small radii and large volume fractions.

Although our work compares well with that of Behrang *et al*. [21], it is important to underline their differences. First, Behrang *et al*. analysis is not based on non-equilibrium thermodynamics but follows a hybrid route mixing an effective medium approach (EMA) and Boltzmann's theory; in particular, they centre all their developments on the notion of probability of phonon transmission from particles to matrix, not used in our approach. To examine the role of volume fraction, Behrang *et al*. make use of the dispersion model [16,20], but they replace it by Debye's one to examine the temperature effects. Here, for the sake of homogeneity, Debye's model is used throughout the whole work. This is also the reason why we use different values for the material parameters. Another important difference is that Behrang *et al*. assume everywhere that the non-dimensional parameter *α* is zero meaning that they do not take into account the Kapitza resistance between nanoparticles and matrix. To account for specular diffusion, we simply redefine the particulate radius (see relation (2.8)) as proposed by Dames & Chen [26], instead, Behrang *et al*., calculate separately the contributions ^{(d)}_{eff} from the specular and diffusive effects, respectively, and write the heat conductivity of the effective medium as the arithmetic average

It is our purpose to extend our analysis to non-spherical nanoparticles, say ellipsoidal or cylindrical shapes and explore important effects like particle agglomeration, which is not dealt with in the present approach. Extensions to include spatially ordered distributions, such as superlattices [35] and graded materials, will also be the subject of future investigations.

## Data accessibility

This work contains no experimental data. To obtain further information on the models underlying this paper please contact h.machrafi{at}ulg.ac.be.

## Author' contributions

G.L. performed the theoretical developments with contributions from H.M. This paper was written by G.L. with contributions from H.M and M.G. H.M. performed the calculations. All authors gave final approval for publication.

## Competing interests

We declare we have no competing interests.

## Funding

The work of H.M. is funded by BelSPo.

## Acknowledgements

The authors wish to thank Dr A. Behrang (Ecole Polytechnique de Montréal) and Dr A. Sellitto (University Basilicata, Italy) for providing useful data about the nanomaterials investigated in this work. One of us (G.L) is indebted to the Wallonie-Bruxelles-Quebec 9th CMP (RI 15, biennium 2015–2017) for giving him the opportunity to visit our colleagues of the department of Chemical Engineering at Ecole Polytechnique de Montreal.

## Appendix A. Brief review of extended irreversible thermodynamics

The description of systems at subscales, as nanoparticles and high-frequency processes, requires to go beyond the classical theory of irreversible processes as proposed some decades ago by Onsager [36,37] and Prigogine [38] among others. Indeed, this formalism is based on the local equilibrium hypothesis which limits its range of application to large time and space scales. More recently, some authors have proposed an alternative approach, referred to as EIT, covering a wider class of processes and systems. The principal idea behind EIT is to elevate the dissipative fluxes, as the fluxes of mass, energy and momentum to the status of independent variables at the same level as the classical variable like mass, energy or momentum. As a consequence, the space ** V** of state variables will be formed by the union of the (slow and conserved) classical variables

**and the (fast and non-conserved) flux variables**

*C***so that**

*F***=(**

*V*

*C**U*

**).**

*F*As a case study, let us consider heat conduction in a rigid body at rest, the generalization to more complicated systems as fluids, mixtures, suspensions, polymer solutions, porous media and others have been dealt with in detail in numerous publications and books (e.g. [11,14,15,39]). In the problem of a rigid heat conductor, the only relevant conserved variable is the internal energy *e* (or the temperature *T*), whereas the energy flux (here the heat flux vector ** q**) is the non-conserved flux variable so that the space of state variables is

**=(**

*V**e*,

**). In more complex materials like in nanomaterials, fluxes of higher order should be introduced as shown later on. The corner stone of EIT is to assume the existence of an entropy function**

*q**η*(

**), depending on the whole set**

*V***of variables: here**

*V**η*=

*η*(

*e*,

**), or in terms of time derivatives,**

*q**e*and

*η*are measured per unit volume and a dot stands for the scalar product. The symbol

*d*

_{t}designates the time derivative which is indifferently the material or the partial time derivative as the system is at rest. It is assumed that

*s*is a concave function of the variables to guarantee stability of the equilibrium state and that it obeys a general time-evolution equation of the form

*σ*

^{s}(in short, the entropy production) is positive definite to satisfy the second principle of thermodynamics, the quantity

*J*^{s}is the entropy flux. Let us define the local non-equilibrium temperature by

*T*

^{−1}(

*e*)=∂

*η*/∂

*e*and select the constitutive equation for ∂

*η*/∂

**as given by ∂**

*q**η*/∂

**=−**

*q**γ*(

*T*)

**, where**

*q**γ*(

*T*) is a material coefficient depending generally on

*T*; it is positive definite in order to meet the property that

*s*is maximum at equilibrium, the minus sign in front of

*γ*(

*T*)

**has been introduced for convenience. Under these conditions, expression (A 1), referred to as the Gibbs equation, can be written as**

*q**d*

_{t}

*e*by means of the energy balance which, in absence of heat sources, can be written as

*σ*

^{s}is a bilinear relation in the flux

**and the quantity represented by the two terms between the parentheses that is usually called the thermodynamic force**

*q***. The simplest way to guarantee the positiveness of the entropy production**

*X**σ*

^{s}is to assume a linear flux–force relation of the form

**=**

*q**L*

**, where**

*X**L*is a phenomenological coefficient, this procedure leads to Cattaneo's law [40]

*γL*=

*τ*(relaxation time) and

*L*/

*T*

^{2}=λ (heat conductivity), and wherein

*τ*and λ are proved to be positive quantities [14,15]. Although Cattaneo's relation is useful at short time scales (high frequencies), it is not satisfactory with the purpose to describe heat transport at short length scales, wherein non-localities play a preponderant role.

Non-local effects are elegantly introduced in the framework of EIT by appealing to a hierarchy of fluxes *Q*^{(1)},*Q*^{(2)},…*Q*^{(N)} with *Q*^{(1)} identified with the heat flux vector ** q**,

*Q*^{(2)}(a tensor of rank two) as the flux of the heat flux,

*Q*^{(3)}as the flux of

*Q*^{(2)}, etc. From the kinetic theory point of view,

*Q*^{(2)},

*Q*^{(3)},…,

*Q*^{(N)}represent the higher moments of the velocity distribution. Written in Cartesian coordinates and designating by

*f*the distribution function, the fluxes are given by,

**=**

*C***−**

*c*

*v*_{m}the relative velocity of phonons with respect to their mean velocity

*v*_{m}. Up to the

*n*th-order moment, the Gibbs equation generalizing expression (A 3) takes the form

*Q*^{(3)}⊗

*Q*^{(2)}stands for

*Q*

_{ijk}

*Q*

_{jk}. We have limited ourselves to the simplest form of the entropy and the entropy flux which are sufficient for the present purpose. The entropy production

*σ*

^{s}which in virtue of (A 2), is given by

*d*

_{t}

*η*and

*J*^{s}from (A 8) and (A 9), respectively, and by eliminating

*d*

_{t}

*e*via the energy balance (A 4), the result is

**,**

*q*

*Q*^{(2)}…

*Q*

^{(N)}. Making use of (A 12) and (A 13), expression (A 11) of the entropy production becomes

*μ*

_{n}≥0(

*n*=1,2,…

*n*) to satisfy the positiveness of the entropy production.

To gain insight about the physical meaning of the various phenomenological coefficients, let us assume absence of non-locality so that the term in ∇⋅*Q*^{(2)} will not appear in (A 12) which reduces to Cattaneo's relation. If in addition, one considers steady situations, the term in *d*_{t}** q** vanishes and one recovers Fourier's law. These observations lead to the following identities:

*μ*

_{1}is related to the heat conductivity λ and

*γ*

_{1}to the relaxation time

*τ*. The identification of the higher order coefficients is not so easy as it demands to compare with higher order evolution equations, but it is expected that the parameters

*μ*

_{n}and

*γ*

_{n}are related to coefficients of thermal conductivity λ

_{n}and relaxation times

*τ*

_{n}of order

*n*, respectively. Moreover, since

*Q*^{(n+1)}is the flux of

*Q*^{(n)}, this implies, by the very definition of a flux, that

*d*

_{t}

*Q*^{(n)}=−∇.

*Q*^{(n+1)}; now, when dividing (A 12) by

*γ*

_{1}and (A 13) by

*γ*

_{n}(

*n*=2,3,…), it follows that

*β*

_{1}/

*γ*

_{1}=−1,

*β*

_{2}/

*γ*

_{2}=−1,… or, more generally,

*γ*

_{n}=−

*β*

_{n}, which reduces considerably the number of undetermined coefficients.

We consider now an infinite number of flux variables and apply the spatial Fourier transform ** k** the wavenumber vector and

**the position vector; this operation leads to the following time-evolution equation of the Fourier transformed heat flux:**

*r**τ*≡

*τ*

_{1}=

*γ*

_{1}/

*μ*

_{1}designates the relaxation time depending generally on

**and λ(**

*k***) the continued fraction for the**

*k***-dependent effective thermal conductivity:**

*k*_{0}is the bulk thermal conductivity independent of the dimension of the system and

*l*

_{n}the mean free path of order

*n*given by

*τ*

_{n}(

*n*>1) corresponding to the higher order fluxes are negligible with respect to

*τ*which is a hypothesis generally well admitted in kinetic theories. We now select the mean free path

*l*

_{n}of order

*n*in terms of

*n*as

*l*identified as the mean free path independent of the order of approximation, this is a natural choice in phonon's kinetic theory as shown by Dreyer & Struchtrup [41]. Under the above conditions, it was shown by Hess [42] that, in the asymptotic limit (

*r*so that it is rather natural to identify the wavenumber

*k*as

*k*=2

*π*/

*r*, expressing (A 19) in terms of the Knudsen number

*Kn*=

*l*/

*r*, one obtains [43] the relation given by (1.4)), namely

- Received March 3, 2015.
- Accepted August 20, 2015.

- © 2015 The Author(s)

Published by the Royal Society. All rights reserved.