Geometric view of phase saturation affinity

Phase saturation describes whether a phase is expected to be present in the equilibrium assemblage of a system at a particular set of conditions. Saturated phases appear in the assemblage because they minimize the relevant thermodynamic potential at current conditions (e.g. total Gibbs energy at defined T, P, and bulk composition). Any equilibrium phase assemblage can be described in terms of a set of chemical potentials for the system components. In most geological applications, it is usually simplest to work in terms of elements or oxides. This chemical potential vector describes a plane in composition-energy space where all of the saturated phases (present in the assemblage) are confined to lie in the plane. Metastable phases necessarily lie above this plane, since they are energetically disfavored, and no phases lie below the plane if the phase assemblage is indeed the equilibrium assemblage. The bulk composition of the system is likewise expressed as the weighted average composition of all phases present.

In thermodynamic modeling of evolving systems or ones with large kinetic barriers, it is critical to track not only the saturated phases but also the nearly stable undersaturated phases as well. The degree of metastability (or undersaturation) is described by the saturation affinity:

\[ \mathcal{A} = -(\mu_i - \hat{\mu}_i) \]

where \(\mu_i\) is the chemical potential vector for the phase in question, \(\hat{\mu}_i\) is the external chemical potentials of the system, and the saturation affinity \(\mathcal{A}\) is the vertical offset between them. (Note that this expression holds for all components, all values of \(i\).) The fact that the chemical potential plane of every phase lies parallel to—though possibly vertically offset from—the chemical potential plane of the system reflects that the phase is in metastable exchange equilibrium with the system as a whole. It is for this reason that, at equilibrium, every phase has a single saturation affinity that is component-independent. Additionally, the negative sign convention reflects the fact that under-saturated phases, which lie above the chemical potential plane of the system (with positive excess Gibbs energy) are taken to have negative affinities. Using the standard convention, only reactions with positive affinities are allowed to progress forward and drive the system toward equilibrium.

Imposing system chemical potentials on individual phases

The chemical potential of each phase with respect to its endmembers is the most important quantity in all of chemical thermodynamics, since it is responsible for establishing and maintaining chemical equilibrium. The condition of chemical equilibrium is that all phases in equilibrium must maintain equality of chemical potentials. A typical approach to imposing this condition is to linearly map the endmember chemical potentials for a particular phase onto a common component basis set (like the elements or oxides). Due to the linearity of chemical potentials, the equilibrium phase assemblage linearly (and uniquely) determines the chemical potentials of the system components (e.g. oxides or elements):

\[ \nu_{\phi}^{c} \cdot \mu^{c} = \mu^{end}_{\phi} \]

where \(\mu_{sys}^{c}\) and \(\mu_{\phi}^{end}\) are the chemical potentials of the system components and the endmembers for phase \(\phi\), and \(\nu_{\phi}^{c}\) is the stoichiometry matrix for the endmembers of phase \(\phi\) in terms of system components. If the phase of interest is an omnicomponent phase, then this expression provides a complete set of linear equations (i.e. the stoichiometry matrix is square), and the chemical potentials of the system can be determined from the phase chemical potentials using the inverse stoichiometry matrix (\(\nu_{\phi}^{c} = [\mu_{sys}^{c}]^{-1} \cdot \mu^{end}_{\phi}\)). In the general case, the complete set of phases in the assemblage must be combined to infer the chemical potentials of the system.

Inferring system chemical potentials from observed phase assemblage

Even in cases lacking an omnicomponent phase (e.g. typical subsolidus phase assemblages), the chemical potentials of the system components (e.g. elements) can be determined directly from the observed phases, their compositions, and reliable thermodynamic models for each phase. The stoichiometry matrix and chemical potential vector for the entire system are created by combining phase stoichiometries and chemical potentials for all phases in the assemblage. For the system-wide stoichiometry matrix, we create a block matrix composed of the stoichiometry matrices for each phase stacked vertically, with one row for each endmember of every phase present in the assemblage and one column for each system component:

\[ \nu_{sys} = \left[ \begin{array}{c} \nu_1 \\ {\small \vdots} \\ \nu_\phi \\ {\small \vdots} \\ \nu_{N_\phi} \end{array} \right] \]

where \(\nu_\phi\) is the stoichiometry matrix for phase \(\phi\), and there are \(N_\phi\) phases present in the assemblage. Similarly, we construct a vector of endmember chemical potentials:

\[ \mu^{end}_{sys} = [\mu^{end}_1 {\small \dots} \mu^{end}_\phi {\small \dots} \mu^{end}_{N_\phi}]^T \]

where ordering of endmembers and phases is maintained across the two expressions. At thermodynamic equilibrium, chemical potentials are equal for all phases in the assemblage, and thus the system-wide expression holds:

\[ \nu_{sys} \cdot \mu_{sys}^c = \mu^{end}_{sys} \]

where \(\mu_{sys}^c\) is the system chemical potential expressed in system components (typically oxides or elements). The implied chemical potentials of the system consistent with this phase assemblage is then determined by solving this system of linear equations. For purely equilibrium theoretical modeling, this expression has an exact solution, but typical applications involving experiments must rely upon linear least squares to obtain the optimal solution, averaging over measurement and model uncertainties.

Using subsolidus phase assemblages as a ‘lookup’ for chemical potential constraints

In cases where an omnicomponent phase does not exist, as in most subsolidus phase assemblages, the chemical potentials of the system can be inferred directly from the equilibrium assemblage. The previous discussions of chemical potential inference presumed that all phase models were accurate and fixed, but this general approach gains even greater utility when we consider calibrating the model for one (or more) of the phases in the assemblage. The general idea behind this calibration strategy is that the parameters of one phase model are determined assuming that the models for all other phases in the assemblage are sufficiently accurate to be used as a dynamic ‘lookup’ table for the chemical potentials of the system. Thus the assemblage is split into ‘lookup phases’ (whose model parameters are fixed) and the target calibration phase (whose parameters are modified during the fitting procedure).

Subsequent calibration based on these experimental constraints can be rather straightforward if we are lucky enough to have a set of lookup phases that fully span the component space of the system. This simplified case occurs when the number of independent components for the lookup phase assemblage (given by the rank of the lookup stoichiometry matrix) is equal to that of the full system. The system chemical potentials are thus fully determined by the lookup phase assemblage alone—via linear least squares—providing experimental constraints on the chemical potentials of the calibration phase. A more complex, yet common, situation arises when the lookup phase assemblage does not fully span the composition space of the system, necessitating a significantly more sophisticated approach to extracting usable chemical potential constraints.