Effect of Interfacial Roughness on Phonon Radiative Heat Conduction

The interface between a solid and a liquid or two dissimilar solids poses a resistance to heat transport by phonons, which are quanta of lattice vibrational energy. The classical theory explains that the resistance R arises due to a mismatch in acoustic impedances of the two media and predicts R to vary with temperature T as R ~ T⁻³. Experiments with copper and liquid ³He have shown that first, although RT³ is a constant below 0.1 K, its value is much less than that predicted by the classical theory. Second, at higher temperatures the resistance behaves as R ~ T⁻ⁿ where n > 3. This study explains these observations in the temperature range of 0.06 K−1 K and ³He pressure range from zero atmospheres in liquid state to solid state, by including the effects of surface roughness and phonon attenuation in solids by electrons or dislocations. Roughness measurements have shown that solid surfaces have a fractal structure and are characterized by a fractal dimension D lying between 2 and 3 for a surface. Using the fractal characteristics, the thermal boundary resistance is shown to follow the relation RT³ ~ (1 + ηT^β)⁻¹ where β is a function of the dimension D and η is a scaling constant. The predictions are in excellent agreement with the experimental observations, indicating that, in addition to other surface effects, the fractal surface structure could have a strong influence on thermal boundary resistance.