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We derive the continuum equations and boundary conditions governingphonon-mediated heat transfer in the limit of small but finite mean free pathfrom asymptotic solution of the linearized Boltzmann equation in the relaxationtime approximation. Our approach uses the ratio of the mean free path to thecharacteristic system lengthscale, also known as the Knudsen number, as theexpansion parameter to study the effects of boundaries on the breakdown of theFourier descrition. We show that, in the bulk, the traditional heat conductionequation using Fourier's law as a constitutive relation is valid at least up tosecond order in the Knudsen number for steady problems and first order fortime-dependent problems. However, this description does not hold withindistances on the order of a few mean free paths from the boundary; thisbreakdown is a result of kinetic effects that are always present in theboundary vicinity and require solution of a Boltzmann boundary-layer problem tobe determined. Matching the inner, boundary layer, solution to the outer, bulk,solution yields boundary conditions for the Fourier description as well asadditive corrections in the form of universal kinetic boundary layers; both arefound to be proportional to the bulk-solution gradients at the boundary andparametrized by the material model and the phonon-boundary interaction model(Boltzmann boundary condition). Our derivation shows that the traditionalno-jump boundary condition for prescribed temperature boundaries and no-fluxboundary condition for diffusely reflecting boundaries are appropriate only tozeroth order in the Knudsen number; at higher order, boundary conditions are ofthe jump type. We illustrate the utility of the asymptotic solution procedureby demonstrating that it can be used to predict the Kapitza resistance (andtemperature jump) associated with an interface between two materials.

physical review. b./physical review. b

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