Constitutive theory of thermoelectrics

Thermoelectric materials are characterized by coupled, or interdependent conduction of heat and electricity. When heat is flowing in a thermoelectric material, an electric current arises. The converse effect also happens when a voltage is applied thus generating a heat flow. The static governing equations of a thermoelectric material are given by,

and

where

and

The tensor components , and correspond to the thermal conductivity, Seebeck, and electrical conductivity tensors respectively which require information of the material crystallographic orientations. Within our block-restricted polycrystal approach, this allows for the thermoelectric properties to be evaluated in a computational box with a \textit{real} grain structure by rotating the tensors via,

where is a second rank tensors and is a rotation operator using the internal RotationTensor operation in MOOSE utils. The rotation operator accepts Euler angles in the standard Bunge sequence (). More details of this method can be found in the recent work of Basaula et al. (2022) published recently using FERRET. We aim to expand this capability by introducing material specific grain boundary conductivities that can be parameterized by first-principles calculations.