Mesoscale conductivity and scaling function in aggregates of cubic, trigonal, hexagonal, and tetragonal crystals

The framework of stochastic mechanics is used to obtain scale-dependent bounds on the thermal conductivity of random polycrystals. This is done with the help of a scaling function that enables one to define the mesoscale that separates the effective, macroscopic conductivity from the realization-dependent microscale conductivity. We demonstrate that the scaling function depends upon the single-crystal anisotropy measure (k) and the mesoscale (δ) for aggregates made up of cubic, trigonal, hexagonal, and tetragonal single crystals. The proposed methodology unifies the treatment of a variety of crystals over different length scales. Finally, we develop a methodology to construct a material selection diagram in the (k-δ) space.