ENTER_HOMOGENIZATION_FUN

Use this with the homogenization model for multiphase simulations. The homogenization model is based on the assumption of local equilibrium at each node point, which yields the local chemical potentials at each node point from which the local chemical potential gradients may be estimated. The chemical potential gradients are the driving forces for diffusion. The local kinetics must also be evaluated by some averaging procedure, the choice of which is determined by this command. The local kinetics is evaluated by considering the product of mobility times u-fraction for each component in each phase and the volume fraction of each phase.

The Diffusion Module (DICTRA) in Graphical Mode uses a Diffusion Calculator instead of the Console Mode commands. Most of the Console Mode functionality is available on the Diffusion Calculator. However, you cannot use both Graphical Mode and Console Mode in the same simulation.

Syntax

enter_homogenization_fun

Prompt

Enter homogenization function # /5/

Enter a digit between 1 and 14 (default is 5). The options corresponding to the numbers are listed below.

Enter a digit between 1 and 14 (default is #5) to assign the homogenization function then follow the prompts. The homogenization functions are:

No. Function name

1

General lower Hashin-Shtrikman bound*

2

General upper Hashin-Shtrikman bound*

3

Hashin-Shtrikman bound with prescribed matrix phase*

4

Hashin-Shtrikman bound with majority phase as matrix phase*

5

Rule of mixtures (upper Wiener bound)

6

Inverse rule of mixtures (lower Wiener bound)

7

Labyrinth factor f with prescribed matrix phase

8

Labyrinth factor f**2 with prescribed matrix phase

9

General lower Hashin-Shtrikman bound with excluded phase(s) *

10

General upper Hashin-Shtrikman bound with excluded phase(s) *

11

Hashin-Shtrikman bound with prescribed matrix phase with excluded phase(s) *

12

Hashin-Shtrikman bound with majority phase as matrix phase with excluded phase(s) *

13

Rule of mixtures (upper Wiener bound) with excluded phase(s)

14

Inverse rule of mixtures (lower Wiener bound) with excluded phase(s)
 

* For the Hashin-Shtrikman bounds, see Hashin, Z. & Shtrikman, S. “A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials”. J. Appl. Phys. 33, 3125–3131 (1962).

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.