Computational role of compositional-perturbation models
Chemical thermodynamic simulations, especially those involving compositional evolution or model calibration require many repeated calculations of the chemical potentials for each phase, typically dominating simulation computing time. Despite generally modest adjustments in composition from one iteration to the next, the computational burden for each of these calculations is fully realized in every function call, dramatically and unnecessarily increasing overall computing times. We are thus highly motivated to seek out simple local approximations that can achieve dramatically faster runtimes while retaining accuracy over reasonable ranges in composition. In this section, we quantify the accuracy of the ideal and linear solution models, which represent the baseline computational models of chemical thermodyanmics. We also outline the basic framework for a potentially more accurate model, the local regular solution approximation, discussed in detail in the next section.
Ideal and Linear Solution Models
The simplest local approximations for chemical solutions are the physically-motivated ideal solution and the purely mathematical linear solution models. Because the ideal solution is based on the assumption of ideal mixing, it completely ignores any chemical interaction energies and thus generally performs rather poorly for most real-world solutions. In stark contrast, the linear solution model is a purely mathematical representation that has no physical basis, but shows much improved performance as it empirically accounts for (linear) compositional variation in the chemical potentials. Together, these two rather basic yet important models provide the foundation for a much more accurate approximation, the local regular solution, which we will later explore in detail.
The ideal solution model represents a thermodynamic baseline (or null hypothesis) for mixed phases; though it is rarely used directly, it forms the foundation of nearly all more complicated models. The fundamental assumption underlying the ideal solution is that the components of the solution behave as energetically independent entities. The resulting mixture thus has an energy given by the weighted average of the components, modified only by the additional entropy of mixing, yielding the simple molar Gibbs energy expression:
\[ \bar{G}^{ideal} = \sum_i X_i \mu_i^0 + RTm\sum_i X_i \log X_i \]
where \(\mu_i^0\) are the chemical potentials for each pure endmember component, \(X_i\) is the mol-fraction of each component, \(RT\) is the thermal energy scale, and \(m\) is the site multiplicity. The chemical potentials are given by the compositional derivative:
\[ \mu_k = \mu_k^0 + RTm \ln X_k \]
which emphasizes how the chemical potential of the pure endmember component is only modified by ideal entropy mixing. Based on this form, the ideal solution can be transformed into a local perturbation model:
\[ \Delta \mu_k^{ideal} = \mu_k-\mu_k^{ref} \approx RTm \cdot \ln \left(X_k / X_k^{ref}\right) \]
where changes in the chemical potentials are attributed entirely to the expected behavior for the entropy of mixing terms.
Conceptually much simpler is the linear solution model, which is simply a first-order Taylor expansion of the chemical potentials with respect the the molar composition of the phase. Unlike the ideal solution, the linear solution thus requires that both the chemical potentials and their compositional derivatives (the Gibbs curvature matrix) be calculated at the reference composition:
\[ \Delta \mu_k^{linear} = \mu_k-\mu_k^{ref} \approx \sum_l \frac{d \mu_k}{d n_l} \cdot (X_l - X_l^{ref}) \]
where changes in the chemical potentials are assumed to have linear dependence, regardless of the magnitude of the compositional extrapolation. For small perturbations, this assumption is perfectly reasonable, and the linear solution model thus greatly outperforms the ideal solution in this regime. But the lack of logarithmic mixing terms comes to dominate for larger extrapolations, urging us to seek a more accurate model that can capture both empirically observed chemical potential variations and the appropriate form for entropic mixing.
Local regular solution approximation overview
Regular solution models represent the next step in improving accuracy for chemical mixing models, and while they can be quite useful, their simplicity limits their applicability to complex ordered phases. Nevertheless, the ability of the regular solution to capture ideal mixing and roughly represent (in many situations) non-ideal excess interactions makes it attractive for calculations considering local perturbations. We therefore construct the local regular solution approximation, which relies on evaluating the chemical potentials and Gibbs curvature matrix at a reference state. Excess solution properties are extracted from this matrix by subtracting off for expected contributions from an ideal solution. The chemical potential and Gibbs curvature contributions of the regular solution are fortunately linear in the excess mixing parameters \(W_{ij}\), enabling linear least-squares inference of their appropriate values. This approach yields a well-behaved perturbation model whose composition-dependence captures the dominant logarithmic and linear terms. When applied to complex multi-site solution phases, this approximation must be carried out in the augmented dependent-species space (rather than simple independent components) to retain meaningful ideal mixing terms.