Path-Integral Formulation of the Theory of Thermal Boundary Resistance

Abstract

The thermal contact resistance between two dissimilar insulating solids or a solid and liquid helium arises from the partial transmission of phonons at the interface. In the theories of Khalatnikov and Little it is assumed that there is a temperature drop across the interface but that the media on either side are in thermal equilibrium even though there must be a net heat flux in each medium. These theories also predict a finite temperature discontinuity at an interface between identical media which cannot be true in steady-state heat conduction^1,2. We have used the Chambers path-integral method³ of solving the Boltzmann equation to give a self-consistent solution of the transport problem in which this undesirable feature does not arise. In this method the phonon distribution function at a point is calculated by considering the various paths by which phonons may arrive at that point taking account of the probability of being scattered en route. The reflection and transmission of phonons at the boundary gives rise to a non-local thermal conductivity and the temperature distribution must be calculated self-consistently so as to conserve the heat current throughout the system in the steady state. The integral equation for the temperature distribution is difficult to solve even for simple models of the phonon transmission, but we have shown that in one dimension the temperature gradient is constant in both media apart from the discontinuity at the interface. However in general a nonlinear temperature variation near the interface is to be expected⁴.