The Kim-Kim-Suzuki (KKS) model in its current implementation is a two-phase model (with a single order parameter $$$\eta$$$) with the added complexity of introducing Phase-concentrations $$$(c_a, c_b)$$$, i.e. a concentration variable for each component and each phase, in addition to the global concentrations ($$$c$$$).

$$$$c=\left(1-h(\eta)\right)c_a + h(\eta)c_b$$$$

The main advantage this addition yields is the ability to chose the interfacial free energy of the system *independent* of the interfacial width (and thus length scale). KKS models are especially suited for systems with high heat of solution, which in conventional phase field models can lead to unphysically high interfacial free energies due to the miscibility gap contributions along the smooth interface.

Note that while the KKS implementation is for two-*phase* systems it allows for an arbitrary number of *components*, whereas each component is represented by one global concentration variable and two phase concentration variables.

The total free energy $$$F$$$ of the system is given by the Phase-free energies as

$$$$F = \left(1-h(\eta)\right) F_a(c_a) + h(\eta)F_b(c_b) + Wg(\eta)$$$$

The phase free energies are only functions of the respective phase concentrations.

The additional variables require additional constraint equations, which are the *mass conservation* equation (above) and the pointwise equality of the phase chemical potentials

$$$$\frac{\partial f_a}{\partial c_a} = \frac{\partial f_b}{\partial c_b}$$$$

- Derivation of the KKS system residuals and Jacobians
- Comparison with the analytical solution for an equilibrium interface for a simple 2-component example of the KKS model.
- KKS phase-field model with 3 or more components

- Kim et al., Phys. Rev. E, 60 (1999) 7186-7197