## Abstract

In a Knudsen layer with thickness comparable to the mean free path, collisions between heat carriers and solid walls play an important role in nanoscale heat transports. An interesting question is that whether these collisions also induce the slip of heat flow similar to the velocity slip condition of the rarefied gases on solid walls. In this work based on the discrete Boltzmann transport equation, the slip boundary condition of heat flux on solid walls in the Knudsen layer is established. This result is exemplified by the slip boundary condition of heat flux in nanowires, which has been proposed in a phenomenological way.

## 1. Introduction

The Fourier law is fundamental to heat conduction theory, which reads
1.1
where ** r** is the position vector,

*t*is the time variable,

**(**

*q***,**

*r**t*) is the heat flux, ∇

*T*(

**,**

*r**t*) is the temperature gradient and

*k*is the thermal conductivity. However, it is well recognized that the Fourier law is inappropriate to heat transport with characteristic times comparable to the mean free time [1–6] or with sizes comparable to the mean free path of heat carriers [7–12]. Much effort has been devoted to modifications of the Fourier law, which leads to many non-Fourier heat conduction models. Among them, the most famous one is the CV model proposed by Cattaneo and Vernotte [13–15] 1.2 where

*τ*is the relaxation time. The natural extension of this model gives 1.3 which is called the single-phase-lagging heat conduction model. Model (1.3) was further extended to the following dual-phase-lagging heat conduction model by Tzou [1] 1.4 where

*τ*

_{q}and

*τ*

_{T}are the phase lags of the temperature gradient and the heat flux vector, respectively. The first-order Taylor expansion of both terms in equation (1.4) yields 1.5 which is also called the dual-phase-lagging heat conduction model in the literature.

Models (1.2)–(1.5) usually lead to the wave-like heat transport which avoids the infinite heat propagation speed implied by the Fourier law [1,16–20]. However, they cannot reflect the non-local effect of nanoscale heat conductions, for example the size effect, which has been observed experimentally [8–12]. The Guyer–Krumhansl model accommodates both the lagging and non-local effects [21–26], which is expressed as follows:
1.6
where *l* is the mean free path of phonons and *Δ* is the Laplace operator.

Recently, based on the following special case of equation (1.6) 1.7 the heat conduction of nanowires was investigated in [24,25,27,28]. They proposed that the phonon flow exhibits a similar behaviour as rarefied gas flows or microscale flows in microchannels with the rarefied effect. As the velocity on the walls of microchannels satisfies the slip boundary condition, analogously the slip boundary condition of heat flux for the phonon flow in nanowires was assumed in [24,25,27,28]. However, the legitimacy of this slip boundary condition of heat flux on the boundary surface of nanowires needs to be proved. In this work, we attempt to derive the slip boundary condition of heat flux on solid walls in Knudsen layers from the discrete Boltzmann transport equation (BTE).

## 2. Boltzmann transport equation

Consider a system consisting of *N* particles. The distribution function *f*_{pv} is defined as
where ** v** and

**are, respectively, velocity and position vector. From the conservation of the number of particles in a given volume in phase space under the assumption that the particles do not interact with each other, we have [29] 2.1 This is called the Liouville equation which implies that when we follow the particles in a given volume element along a flow line in phase space without particle collisions, the distribution is conserved, that is 2.2 If collisions between particles occur, the distribution will change, thus 2.3 where the term on the right-hand side of this equation reflects the impact of particle collisions,**

*r**τ*

_{c}is the average time elapsed between consecutive collisions, and is called the mean free time,

**is the average displacement of particles between two consecutive collisions, thus its magnitude has the same order as the mean free path of heat carriers. By redefining the distribution function as follows: equation (2.3) becomes 2.4 If**

*l***and**

*l**τ*

_{c}are infinitesimal in comparison with the characteristic time and size of the energy transport phenomenon under consideration, respectively, then expanding

*f*(

**+**

*r***,**

*l***+**

*v**d*

**,**

*v**t*+

*τ*

_{c}) at the point (

**,**

*r***,**

*v**t*) to the first order and taking the limit by letting

*τ*

_{c}tend to zero, we obtain 2.5 This is the BTE in the differential equation form. If there are no external forces acting on the system, equations (2.4) and (2.5) reduce to 2.6 and 2.7 The nonlinear Boltzmann collision term (∂

*f*/∂

*t*)

_{coll}makes equations (2.6) and (2.7) difficult to solve. In order to simplify these equations, the following Bhatnagar–Gross–Krook (BGK) model [30] is employed 2.8 2.9 where

*f*and

*f*

_{0}are, respectively, the non-equilibrium and equilibrium distribution function and

*τ*is the relaxation time. In the absence of any fields or temperature and electrochemical potential gradients, Boltzmann equation (2.9) reduces to The solution of the above equation is , which shows that the distribution function relaxes to the equilibrium on the scale of

*τ*. Therefore, the relaxation time

*τ*is longer than the mean free time

*τ*

_{c}.

Note that when the characteristic time of the transport process under consideration is large enough in comparison with the mean free time *τ*_{c} so that we can take the mean free time or the mean free path as an infinitesimal quantity, then equation (2.8) reduces to equation (2.9) by taking limit of short time. However, for micro/nanoscale heat conduction or ultrafast heating processes where the mean free time or mean free path is compatible or larger than the characteristic time or length, respectively, we cannot obtain equation (2.9) by taking limit of short time in equation (2.8). In such cases, we have to use equation (2.8) to model the heat transport processes. Equation (2.8) will be called the discrete BTE in the following discussion.

## 3. Slip boundary condition of heat flux on a solid wall

Equation (2.8) can be rewritten into the following form:
3.1
Although we cannot take limit of short time in equation (3.1) for nanoscale heat conduction with large Knudsen number, the mean value theorem is valid. Hence, there exist two constants *s* (0≤*s*≤1) and *τ*′_{c}(0≤*τ*′_{c}≤*τ*_{c}) such that
3.2
Application of the relation ** v**=

*l*/

*τ*

_{c}on equation (3.2) yields 3.3 In the following, we only consider the steady cases, thus equation (3.3) reduces to 3.4 By introducing the kinetic energy

*ε*(

**) for particles, equation (3.4) becomes [18] 3.5 Through the distribution function, the heat flux vector is expressed as [18] 3.6 where**

*v**D*(

*ε*) is the density of states. Multiplying

*εD*(

*ε*)

**on both sides of equation (3.5) yields 3.7 In deriving this equation, the relation has been used. Next, we decompose the distribution function into two parts 3.8 where**

*v**f*

_{1}represents the non-equilibrium part. Substituting equation (3.8) into (3.7) gives 3.9 In view of equation (3.7), we have 3.10 This equation allows us to split the heat flux into two parts 3.11 with 3.12 and 3.13 For

*q*_{d}(

**), applying the relation ∇**

*r**f*

_{0}=d

*f*

_{0}/d

*T*∇

*T*yields 3.14 where is the thermal conductivity. By neglecting the non-local effect in equation (3.14), it reduces to the classical Fourier law. Note that from equation (3.11), one can see that the heat flux includes two parts

*q*

_{d}(

**) and**

*r**q*

_{w}(

**).**

*r**q*

_{d}(

**) refers to the diffusive energy transport and is designated by the classical Fourier law.**

*r**q*

_{w}(

**) originates from the ballistic energy transport. For the conventional heat conductions with small Knudsen number, the ballistic energy transport has little contribution to heat flow,**

*r**q*

_{w}(

**) can be neglected, that is,**

*r**q*(

**)=**

*r**q*

_{d}(

**). For nanosystems with the Knudsen number comparable to the system’s characteristic length, for example Knudsen layers, the ballistic energy transport plays an important role in energy transport processes; the contribution of**

*r**q*

_{w}(

**) to the heat flux is not negligible. In the following, we focus on**

*r**q*

_{w}(

**) arising in the nanoscale heat conductions.**

*r*Consider a nanoscale system, for example a nanowire. Its surface is *S*, and the outward unit vector normal to *S* is denoted as ** n**. Then, there exist two other unit vectors

**and**

*b***, satisfying**

*e***⋅**

*n***=0,**

*b***⋅**

*b***=0,**

*e***⋅**

*n***=0, and constituting an orthogonal coordinate system. Therefore, on the solid surface**

*e**S*we have 3.15 Let the average magnitude of the velocity components

*v*_{n},

*v*_{b}and

*v*_{e}respectively, along the directions

**,**

*n***and**

*b***, be , we can rewrite equation (3.15) as follows: 3.16 Subsequently, 3.17 By applying equation (3.17), equation (3.13) arrives at 3.18 For the heat transfer inside a nanoscale system enclosed by the surface**

*e**S*, there is a Knudsen layer with thickness approximately equal to the mean free path

*l*. If this solid system has a characteristic size

*L*which is comparable to

*l*, then the particles inside the Knudsen layer usually only collide with the solid surface. Therefore, we can set (

*c*is a constant). If we further assume

*l*be independent of the velocity, equation (3.18) becomes 3.19 As the gradient operator in equation (3.19) acts on the position vector, the symbol of the gradient can be put out of the sign of integration. Thus, we have 3.20 In deriving equation (3.20), the relation has been employed. As the integration in equation (3.20) involves all the directions of the velocity field and the function

*f*(

**+**

*r**s*

**,**

*l**ε*)

*εD*(

*ε*) is an even function with respect to the velocity

**, we obtain Subsequently, equation (3.20) becomes 3.21 Equation (3.6) allows us to rewrite equation (3.21) as follows: 3.22 Neglecting the non-local effect in this equation, we have 3.23 According to equations (3.11), (3.12) and (3.22), the total heat flux**

*v***can be expressed as 3.24 On an adiabatic solid wall,**

*q*

*q*_{d}(

**) vanishes according to the classical heat conduction theory. In such a case, we have 3.25 This is the slip boundary condition of heat flux on the solid wall in a Knudsen layer. Obviously, boundary condition (3.25) is very similar to the slip boundary condition of velocity field in rarefied gas flows [31,32]. This slip boundary condition of heat flux is applicable for nanoscale heat transfers with all kinds of heat carriers as long as the particle nature of heat carriers is valid. In dielectric solids, heat is transferred by phonons. It is worthwhile to note that gases of phonon are different from the gases of real particles [33]. However, whenever the system characteristic length is larger than the phonon wave length, the particle assumption for phonons is valid.**

*r*## 4. Nanowire

Now we consider phonon transport in a nanowire with cylindrical geometry. Suppose its radius *R* is comparable to the mean free path of the phonon. Applying equations (3.23) and (3.25) on the adiabatic wall of the nanowire yields
4.1
Note that boundary condition (4.1) has been introduced in [24,25,27,28] by analogy between the rarefied gas flow and phonon flow. Therefore, our work puts this phenomenological result on a solid theoretical base.

Furthermore, the first-order Taylor expansion of ** q**(

**+**

*r**s*

**) yields 4.2 Setting**

*l**cs*=

*α*, and denoting the unit vector along the vector

**as**

*l*

*e*_{l}, gives 4.3 For the nanowire under consideration, equation (4.3) reduces to 4.4 which is the extension of equation (4.1) and has been presented in [24,25]. The negative sign appears in equation (4.4) is owing to the fact that on the surface of the nanowire,

**points to the opposite direction of**

*l***.**

*r*Sellitto *et al.* [24] have applied the simplified Guyer–Krumhansl model and equation (4.1) to investigate the sized effect of nanoscale heat conductions in a nanowire with cylindrical geometry. They expressed the effective thermal conductivity of the nanowire as follows [24]:
where *Kn* is the Knudsen number. The size effect is reflected by the dependence of the effective thermal conductivity on the Knudsen number. The application of slip boundary condition (4.1) makes the dependence of the effective thermal conductivity on the radius more reasonable for high Knudsen numbers [24].

## 5. Heuristic derivation of the slip boundary condition

In order to understand the physical meaning of the parameters *c* and *α*, we attempt to heuristically re-derive the slip boundary condition of heat flux on a solid surface. Firstly, let us recall the heuristic derivation of the slip boundary condition of the velocity of the rarefied gas flow near a solid wall.

For the rarefied gas flow near a solid wall, there exists a Knudsen layer with the thickness approximately equal to the mean free path *l*. Outside the Knudsen layer is the bulk flow which is also called the external flow. Assume that near the solid wall surface, a half of the gas molecules come from the external flow and the other half from the refection of the gas molecules from the wall surface. Thus, the average velocity of these two parts of molecules gives the macroscopic velocity of the rarefied gas flow near the solid wall. Denote the velocity component of the gas along the tangential direction of the solid wall surface as *u* and suppose *u*=*u*(*y*), where *y* is the component of the position vector along the normal direction. The average velocity of the molecules from the external flow arriving at the wall surface should be the mean magnitude of the velocities attained by the molecules in the last collision, i.e. the velocity at a distance *ξl* from the surface, *ξ* being a numerical coefficient near one. Denote the unknown velocity of the gas near the wall by *u*_{s}, then the average velocity of the molecules from the external flow is
5.1
Assume that the *σ* portion of the reflected molecules is diffusely reflecting and the other (1−*σ*) portion is specularly reflecting, then the average velocity of the half of molecules that are reflected from the surface is (the velocity of the surface is supposed to be zero)
5.2
Then *u*_{s} should be the mean of the above two velocities
Thus, the slip velocity is expressed as follows:
5.3
Although the heuristic derivation of equation (5.3) is not strict, the dependence of *u*_{s} on *σ*, *l* and (∂*u*/∂*y*)_{0} is correct.

Next the similar reasoning is applied to derive the slip boundary condition of heat flux on the solid wall surface. Under the same assumption made above, the average heat flux owing to the heat carriers that come from the external flow arriving at the wall surface should be the mean magnitude of the heat flux attained by the heat carriers in the last collision, i.e. the heat flux at a distance *ξl* from the solid surface. Denote the unknown heat flux of the particle gas near the wall by *q*_{w}, then the average heat flux of the heat carriers from the external flow is
5.4
Under the same assumption that the *σ* portion of the reflected heat carriers is diffusely reflecting and the other (1−*σ*) portion is specularly reflecting, then the average heat flux of the half of heat carriers that are reflected from the surface is (the diffuse heat flux is supposed to be zero, i.e. the adiabatic boundary condition)
5.5
Then, *q*_{w} should be the mean of the above two heat fluxes
From this equation, the slip heat flux near the solid wall is expressed as follows:
5.6
The comparison with equation (3.23) leads to
5.7
Therefore, the parameters *c* and *α* depend on the accommodation coefficient *σ* which is related to the gas type and wall material, the gas–wall interaction potentials, and varieties of surface conditions, such as temperature, absorbents, lattice configurations and surface roughness. From the above discussion, one can see that the slip of heat flux on solid walls expressed in equations (3.23) and (4.1) are owing to the combined action of the incident and specularly reflected heat carriers near the solid wall.

## 6. Concluding remarks

In nanoscale heat conductions with large Knudsen numbers, the ballistic transport may dominate the heat conduction process. In such cases, the boundaries of the heat conduction medium, for example the solid walls of a nanoscale system, have significant impact on the heat transport process. In the earlier studies [24,25,27,28], a slip boundary condition of heat flux along the walls of nanowires was assumed by analogy between the rarefied gas flow and phonon flow. In this work based on the discrete BTE, we derive the expression of the heat flux by taking into account particle collisions with walls in a Knudsen layer. This result allows us to establish the slip boundary condition of heat flux on solid walls in a Knudsen layer. The heat flux slip boundary condition on the walls of nanowires used in [24,25,27,28] is viewed as a special case of our general result.

## Acknowledgements

Two anonymous reviewers’ valuable comments and suggestions on the original manuscript are greatly appreciated.

- Received August 29, 2013.
- Accepted October 9, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.