Kim-Kim-Suzuki Example: 3 or More Components

Click here to see the input file for this example: kks_example_ternary.i

When additional chemical components are added to the KKS model, a Cahn-Hilliard equation must be added for each additional component. (For $$$n$$$ components, $$$n-1$$$ Cahn-Hilliard equations are required). Each additional Cahn-Hilliard equation requires the kernels:

  • KKSSplitCHCRes
  • CoupledTimeDerivative
  • SplitCHWRes

To enforce the composition and chemical potential constraints, each additional component also requires the kernels

  • KKSPhaseConcentration
  • KKSPhaseChemicalPotential

The Allen-Cahn equation is also modified when additional components are added. The residual becomes

$$$$R=-\frac{dh}{d\eta} \left(F_a-F_b- \sum_{i=1}^{n-1} \frac{dF_a}{dc_{ia}}(c_{ia}-c_{ib})\right) + w\frac{dg}{d\eta}.$$$$

where $$$n$$$ is the number of components. A single KKSACBulkF kernel is needed as in the 2-component case, and an additional KKSACBulkC kernel must added for each additional component.