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generalized Hamiltonian 

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where µ$_{i}$ is the chemical potential of the species i . We shall require charge neutrality for ionized species 3 and 4 in the bulk, and assume µ$_{3}$ = µ$_{4}$ . For the short-range interaction energy we make the standard nearest-neighbor (NN) approximation for all the species 

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where ˆ e$_{n}$ is the unit lattice vector in the n -th direction and summation convention is assumed for i, j = 1 , ..., 4. The coupling constants are symmetric, J$_{ij}$ = J$_{ji}$ , and represent the sum of all short-range interactions. The Hamiltonian of the lattice model with the electrostatic interactions disregarded takes the form 

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where ∑ 〈 x , x ′ 〉 is the summation over all pairs of nearest-neighbors belonging to the system. The system volume is denoted by V , ∂V denotes the boundary layer, and the new operators are 

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From close packing (1) it follows that ˆ o$_{1}$ + ˆ o$_{2}$ = ˆ s 2 = 1 − ˆ φ $^{2}$. The new coupling constants in (5) are linear combinations of the coupling constants J$_{ij}$ , and Δ µ and Δ µ$_{c}$ are linear combinations of µ$_{i}$ . 

We further assume that the species 3 and 4 are similar, i.e., interact in a similar way with 

