Global energy minimization of multi-component phases with internal degrees of freedom

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Miscibility gap detection is a crucial feature in thermodynamic calculation software to accurately calculate the energy of phases containing regions of compositional instability, i.e., spinodals, and is commonly handled through global minimization (GM) of the Gibbs energy. Inside the spinodal region, all points on the energy surface have negative curvature and will be thermodynamically driven towards demixing. The cause of miscibility gaps in non-ideal solutions is the presence of unfavorable interactions between components that overwhelm the entropically-driven ideal mixing contribution to the Gibbs energy.

From a computational perspective miscibility gaps pose a challenge because they mean that the same phase may appear on multiple points on the equilibrium tie hyperplane but at different compositions. In these cases the software must increase the total degrees of freedom by creating multiple composition sets of the same phase, potentially up to the limit specified by the Gibbs phase rule. For multi-component systems the topology of the energy surfaces can become quite complex, since high-dimensional tangent hyperplanes can interact with the energy surface. Moreover, when handling phases with internal degrees of freedom, i.e., sublattices, it is possible for points on the global energy surface (the composition) to be close together while being far apart in their internal coordinates (the constitution). This is referred to as an internal miscibility gap and is the cause of order-disorder phase transitions.

While few authors have discussed fully generalized GM schemes, there has been work in efficient sampling in low dimensions (Emelianenko 2006) and tie hyperplane calculation for the multi-component case (Perevoshchikova 2012). However, there has not previously been detailed discussion of methods for solving the multi-component case with multiple sublattices.

Reliable convergence to the global energy minimum also requires a good choice of starting points, i.e., phase compositions, for the minimization procedure.