# 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.