The Unified Extrapolation Model

Scope. UEM and UEM–Miedema apply only to solution phases (melts and disordered solutions). They are not used for crystalline solids; those require DFT/AIMD.

The problem: too many reasonable ways to extrapolate

Suppose the excess Gibbs energy is known for each binary pair in a ternary system, but not for the ternary itself. A geometric model supplies a rule for combining those binary contributions at a given ternary composition. The classical rules differ in how they project a multicomponent composition back onto the binary edges. Kohler, Toop, and related models, for example, do not weight the same binary contribution in the same way, and their predictions can therefore differ even when they start from identical binary data.

That difference is not a minor technicality. In alloy development, a change in the estimated excess property can move a predicted activity, shift a phase boundary, or change the estimated effect of one solute on another. If the model must be chosen before the calculation, the result inherits the assumptions of that choice, and there is often no clear, system-independent reason to prefer one classical rule over another. The question is therefore not only how to extrapolate, but how to make the extrapolation itself less arbitrary.

One expression, with contribution coefficients

The Unified Extrapolation Model answers that question by placing the classical geometric models inside a single expression. Instead of selecting one model in advance, UEM introduces contribution coefficients for the components. Each coefficient controls how much weight a component, and hence the binary contributions involving it, carries in the multicomponent estimate. When the coefficients take particular values, the unified expression reduces to the behaviour of a familiar geometric model; when they take other values, it describes intermediate or different weighting schemes. The choice of model is thus replaced by a calculation of coefficients.

This unification matters for two reasons. First, it makes the comparison between models explicit: different models are no longer unrelated recipes, but points within one framework. Second, it gives the calculation a physical anchor. The coefficients are not free fitting parameters chosen after seeing the multicomponent result; they are determined from differences between the components themselves, as explained below.

What a property difference means

In UEM, the property difference between two components is a measure of how dissimilar they are with respect to the property that governs the binary interaction being extrapolated. Dissimilar components interact differently from similar ones, and a model that ignores that dissimilarity treats every pair as if it were geometrically equivalent. By contrast, a contribution coefficient that depends on property differences lets the more distinctive component play a larger or smaller role, depending on the system.

The precise definition and the best expression for this difference are themselves research questions, and the published work refines them over time. The important conceptual point is stable across those refinements: the weighting should reflect the components, not merely their mole fractions. Composition still matters, because the amount of each component present must enter any thermodynamic estimate, but composition alone does not determine the weight. The pair of ideas together, composition and component dissimilarity, is what allows one expression to cover cases that previously required different models.

Coupling with the Miedema model and excess-entropy relations

UEM is an extrapolation framework, so it still needs binary input. Where binary thermodynamic data are incomplete, the Miedema model can supply an estimate of the enthalpy of mixing for a binary metallic pair from atomic properties such as electronegativity and electron density at the boundary of the atomic cell. That estimate does not replace a measurement, but it gives a physically motivated starting point, which UEM can then carry into a multicomponent estimate.

A second coupling concerns dilute metallic solutions, where the quantities of interest are often activity interaction coefficients rather than a full Gibbs energy surface. In a dilute solution, a small amount of one solute can noticeably change the activity of another, and that effect is summarised by an interaction coefficient. Excess-entropy relations connect such coefficients to the entropy change associated with mixing, providing another route from binary or pair-level quantities to a multicomponent prediction. Used together, UEM, the Miedema model, and excess-entropy relations support estimates for systems such as dilute iron-based solutions and other alloy solutions, where direct multicomponent experiments are scarce. Applications here are described in general terms, following the published work, and no unpublished numerical results are presented.

Progress in chronological order

The following timeline follows only the published papers listed on the Publications page. Each step is described in general terms, as a development of the model or its use, rather than as a summary of numerical findings.

• 2018: Activity interaction parameters are estimated for iron-sulphur systems containing a third solute, establishing the dilute-solution problem that later work generalises.
• 2019: Geometric thermodynamic models are reconsidered from a new perspective, clarifying why different models give different results and preparing the ground for a unified treatment.
• 2020: A general model is developed for activity interaction parameters of solutes in dilute metallic solutions, extending the dilute-solution treatment beyond a single host system.
• 2020: The definition of the difference in properties between components is examined within UEM, giving the contribution coefficients a clearer physical basis.
• 2020: A new expression for the property difference is proposed, refining how component dissimilarity enters the unified expression.
• 2023: A predicted model is developed for the activity interaction coefficient between solutes in alloy solutions, applying the framework to solute-solute interactions more broadly.
• 2024: A unified extrapolation thermodynamic model for multicomponent solutions based on binary data is presented, bringing the geometric unification and its applications together in a single framework.

Taken together, the sequence moves from a specific dilute-solution application, through a critique of model choice, to a general framework, a refined definition of its central quantity, a broader predictive use, and finally a unified multicomponent model. That is the arc this site follows: a practical problem, a conceptual unification, and a return to practical estimation.