The Kim-Kim-Suzuki model is a method for modeling multiphase systems. A defining feature of the model is that the free energies are described by individual concentrations in each phase and order parameters. The different phases co-exist on each "pixel" of the domain, and thus the chemical potential in each co-existing phase is equal. Here, we will go through the theory for two phases, labeled α and β. A schematical illustration of the free energies is shown below.

In the following we denote the concentration in the α-phase c_α and the concentration in the β-phase c_β. The total free energy is given by

φ is a field variable that goes from 0 to 1. If it is 0, there is only α phase and if it is 1 there is only β. The function h(φ) is an interpolating function given by

g(φ) is a function that creates a barrier between the two basins of attraction φ = 0 and φ = 1

Since the two phases co-exist, the chemical potential has to be equal in both phases. This is expressed via the constraint

Furthermore, the concentration at a point in space is given by the weighted average of the concentrations in the individual phases

We want to know the time evolution of the total concentration c and the field φ The total concentration is conserved, while the field φ is not a conserved quantity

We therefore need partial derivatives of f with respect to the total concentraion c and φ. Thus, c_α and c_β are functions of the variables c and φ. To evaluate the functional derivatives, the partial derivatives of c_α and c_β with respect to c, when φ and with respect to φ when c is fixed, are needed. When the concentration in the two phases changes by a small amount (c_α→ c_α + Δc_α and c_β→ c_β + Δc_β), the equation relating the the chemical potentials in the two phases can be expanded to first order in the changes

Moreover, from the equation defining the total concentration we find

The partial derivatives of c_α and c_β with respect to c (at fixed φ) can now be found from the ratios Δc_α/Δ c and Δc_β/Δ c. Combining the results yields the following relations

We proceed in a similar manner to find the partial derivatives with respect to φ, when c is fixed. The strategy is to find how the individual concentrations in the two phases when φ changes by a small amount, such that c remains unchanged.

By expanding h to first order in φ and neglecting terms of order &Delta: φΔ c_α, we arrive at

After collecting the results we find the following relations

Interface Profile

In this section we explore the interface profile of the KKS model. A sketch of how the volume fraction φ changes across an interface is shown below

An important feature of the KKS is model is that the concentrations in the to phases does not change across an interface. However, the total concentration c can change because the volume fraction of phase α and β changes. Consequently, the excess energy associated with an interface is independendent of f_α and f_β. The only remaining part of the overall free energy density is

We can now match the interface energy σ of a sharp interface with the energy of a diffuse interface via

where the dependence on φ' has explicitly been included in the integrand. To solve the integral, we need to know the profile φ(x). This is done be seeking a stationary profile (e.g. δσ = 0).

which we recognize as the Euler-Lagrange equation. By substituting d/dφ = dx/dφ d/dx and multiplying both sides with dx/dφ we arrive at

as φ →, 1 g → 0 and φ' → 0. Thus, the integration constant above has to be zero. By replacing φ' in the integral and introducing the substitution dx = dφdx/dφ we finally arrive at

The next (and very important) step is to determine how the width of the diffuse interface depends on the parameters W and γ. In other words, we need to solve the first order differential equation

We skip the details on how to solve the equation, but the solution is

From the solution, we find that the width l of the interface is approximately

Specializing The Model

In this section we introduce concrete expressions for the free energies and finally end up with a closed form system of equations that can be solved numerically. We will use two parabolic functions for the free energies

where f₀ is a constant. By inserting the explicit free energy forms into the general relations above, we end up with the following system of equations

In the upper figure in the right column, the concentration was randomized on the interval [0, 1) and φ was set equal to the concentration.