Extending the range of validity of Fourier's law into the kinetic transport regime via asymptotic solution of the phonon Boltzmann transport equation

We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of a small but finite mean-free path from the asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation. Our approach uses the ratio of the mean-free path to the characteristic system length scale, also known as the Knudsen number, as the expansion parameter to study the effects of boundaries on the breakdown of the Fourier description. We show that, in the bulk, the traditional heat conduction equation using Fourier's law as a constitutive relation is valid at least up to second order in the Knudsen number for steady problems and first order for time-dependent problems. However, this description does not hold within distances on the order of a few mean-free paths from the boundary; this breakdown is a result of kinetic effects that are always present in the boundary vicinity and require solution of a Boltzmann boundary layer problem to be determined. Matching the inner, boundary layer solution to the outer, bulk solution yields boundary conditions for the Fourier description as well as additive corrections in the form of universal kinetic boundary layers; both are found to be proportional to the bulk-solution gradients at the boundary and parametrized by the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Our derivation shows that the traditional no-jump boundary condition for prescribed temperature boundaries and the no-flux boundary condition for diffusely reflecting boundaries are appropriate only to zeroth order in the Knudsen number; at higher order, boundary conditions are of the jump type. We illustrate the utility of the asymptotic solution procedure by demonstrating that it can be used to predict the Kapitza resistance (and temperature jump) associated with an interface between two materials. All results are validated via comparisons with low-variance deviational Monte Carlo simulations.

Figure 1

Temperature profile associated with ${\tau }_{K1,1}=T_{K1,1}/(\partial T_{G0}/\partial x_1){|}_{\eta =0}$, for three relaxation time models. The $x$ axis is scaled by the maximum free path ${\mathrm{\Lambda }}_{\text{max}}$ of each model.

Figure 2

Validation of the first-order asymptotic theory for prescribed temperature boundaries. The solid line denotes the normalized (by the zeroth-order traditional Fourier result) difference between the heat flux predicted by the asymptotic theory and MC simulation results. The dashed line denotes a slope of 2.

Figure 3

Order 0 (dot-dashed line), order 1 (dashed line), and order 2 (solid line) solutions compared to the solution computed by highly resolved Monte Carlo simulation of the problem considered in Sec. 6a, for $\langle \text{Kn}\rangle =0.1$.

Figure 4

Order 1 solution (dashed line) and “infinite order” solution (solid line) compared to the solution computed by a finely resolved Monte Carlo simulation for the problem considered in Sec. 6a, for $\langle \text{Kn}\rangle =0.4$. At this Knudsen number the boundary layer contribution is clearly visible (the solution is no longer a straight line).

Figure 5

Zeroth-order, Monte Carlo, and implicit asymptotic solution for the temperature along the line $x_1=0$ in the two-dimensional example considered in Sec. 6c, for $\langle \text{Kn}\rangle =0.1$. The inset shows a contour plot of the order 0 temperature solution.

Figure 6

Convergence of asymptotic temperature solutions at $(x_1=0,x_2=1)$ in the two-dimensional example considered in Sec. 6c.

Figure 7

Deviational temperature at the center of a square particle after initial heating.

Figure 8

Temperature profile in a 1D system with an Al/Si interface.