Mode Resolved Continuum Mechanics Model of Phonon Scattering From Embedded Cylinders

Abstract

Phonon scattering from media with embedded spherical nanoparticles has been studied extensively over the last decade due to its application to reducing the thermal conductivity of thermoelectric materials. However, similar studies of thermal transport in fiber-embedded media have received little attention. Calculating the thermal conductivity tensor from microscopic principles requires knowledge of the scattering cross section spanning all possible incident elastic wave orientations, polarizations and wavelengths including the transition from Rayleigh to geometric scattering regimes. In this paper, we use continuum mechanics to develop an analytic treatment of elastic wave scattering for an embedded cylinder and show that a classic treatise on the subject contains important errors for oblique angles of incidence, which we correct. We also develop missing equations for the scattering cross section at oblique angles and study the sensitivity of the scattering cross section as a function of elas-todynamic contrast mechanisms. In particular, we find that for oblique angles of incidence, both elastic and density contrast are important mechanisms by which scattering can be controlled, but that their effects can offset one another, similar to the theory of reflection at flat interfaces. The solution developed captures the scattering physics for all possible incident elastic wave orientations, polarizations and wavelengths including the transition from Rayleigh to geometric scattering regimes, so long as the continuum approximation holds. The method thus enables incorporation of coherent scattering models into calculations of the thermal conductivity tensor for media with nanofibers.