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Redlich-Kister Criterion

A mathematical criterion used to determine the stability of a phase transition based on the properties of the system's potential energy surface.
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The statement of the theorem

Let F(x,T)\mathcal{F}(\boldsymbol{x}, T) be the Helmholtz free energy density of a multi-component system, where x=(x1,x2,…,xN)\boldsymbol{x} = (x_1, x_2, \dots, x_N) are the mole fractions, subject to ∑i=1Nxi=1\sum_{i=1}^N x_i = 1. Assume the free energy density can be expanded in a polynomial form incorporating interaction parameters LijL_{ij}: F(x,T)=∑i=1NxiFi(T)+12∑i=1N∑j=1NxixjΩij(T)+∑i=1N∑j=i+1N∑k=1MLij(k)(xi−xj)k⋅ψk(x)+…\mathcal{F}(\boldsymbol{x}, T) = \sum_{i=1}^N x_i \mathcal{F}_i(T) + \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N x_i x_j \Omega_{ij}(T) + \sum_{i=1}^N \sum_{j=i+1}^N \sum_{k=1}^M L_{ij}^{(k)} (x_i - x_j)^k \cdot \psi_k(\boldsymbol{x}) + \dots. The Redlich-Kister Criterion dictates the stability of the phase transition by analyzing the coefficients Lij(k)L_{ij}^{(k)}. For a system to exhibit a stable phase transition at composition x∗\boldsymbol{x}^*, the coefficients must satisfy the condition that the Hessian matrix of the free energy density, H={∂2F∂xi∂xj}i,j=1N\mathbf{H} = \left\{ \frac{\partial^2 \mathcal{F}}{\partial x_i \partial x_j} \right\}_{i,j=1}^N, must be positive semi-definite (or negative definite, depending on the definition of F\mathcal{F}) in the vicinity of the equilibrium state x∗\boldsymbol{x}^*. Specifically, the stability requires that the coefficients Lij(k)L_{ij}^{(k)} must be constrained such that the second-order mixing term Ωij(T)\Omega_{ij}(T) dominates the higher-order terms for the transition to be continuous (second-order), or that the coefficients lead to a minimum in the Gibbs free energy, ΔG=min⁡x(F(x,T)−∑xiμi0)\Delta G = \min_{\boldsymbol{x}} \left( \mathcal{F}(\boldsymbol{x}, T) - \sum x_i \mu_i^0 \right), ensuring ∂2ΔG∂xi∂xj≥0\frac{\partial^2 \Delta G}{\partial x_i \partial x_j} \ge 0 for all i,ji, j.
Source: Wikipedia