Thermal conductivity of fibre composites at low temperatures

If one can define a local value of the thermal conductivity, with a volume average K o and a mean square fluctuation 〈 ΔK 2〉, then the effective conductivity is K = K o − α〈ΔK 2〉/K o, where α depends on the orientation of the fibres. For isotropic distributions α = 1 3 . A similar relation can be obtained for the thermal resistivity. Other fibre distributions are also discussed in this paper. When the mean free path of the carriers becomes long compared with the fibre diameter, the relations must be modified, either by regarding the conductivity of the fibre as size dependent or by treating the fibres as scatterers of the carriers in the matrix. In the latter case one must disregard the small scale variations in the conductivity in the estimate of 〈 ΔK 2〉. In the case of phonons, the spectral variation of the mean free path should be considered.