About the Homogenization Functions

The geometrical interpretation of the Hashin-Shtrikman bounds are concentric spherical shells of each phase. For the general lower Hashin-Shtrikman bound the outermost shell consists of the phase with the most sluggish kinetics and vice versa for the general upper bound. The geometrical interpretation of the Hashin-Shtrikman bounds suggest further varieties of the bounds, viz. Hashin-Shtrikman bound with prescribed matrix phase and Hashin-Shtrikman bound with majority phase as matrix phase, where the outermost shell consist of a prescribed phase or the phase with highest local volume fraction, respectively.

The geometrical interpretation of the Wiener bounds are continuous layers of each phase either parallel with (upper bound) or orthogonal to (lower bound) the direction of diffusion.

The labyrinth factor functions (described below) are available in Console Mode and when using TC-Python. 

The labyrinth factor functions implies that all diffusion takes place in a single continuous matrix phase. The impeding effect on diffusion by phases dispersed in the matrix phase is taken into account by multiplying the flux with either the volume fraction (Labyrinth factor f with prescribed matrix phase), or the volume fraction squared (Labyrinth factor f**2 with prescribed matrix phase), of the matrix phase.

The varieties with excluded phases (described below)  are available in Console Mode and when using TC-Python. 

The varieties with excluded phases are useful in several respects. First, if a phase is modeled as having zero solubility for a component, the mobility of that component in that phase is undefined, which causes a (non-terminal) error. Setting a phase as excluded causes the mobility of all components in that phase to be set to zero. Second, often there are some major matrix solid solution phases and some minor precipitate phases. If the mobilities in the minor precipitate phases are zero the lower Hashin-Shtrikman bound is useless as it produces a kinetic coefficient of zero. However, using General lower Hashin-Shtrikman bound with excluded phase(s) the excluded phases are not considered when evaluating what phase has the most sluggish kinetics.