Thermodynamics of Binary Phase Diagrams: Two-Phase Coexistence, Reference-State Changes, and CALPHAD in Silicon–Germanium

In this MIT OpenCourseWare lecture, the instructor expands binary phase diagram analysis from single-solution models to multiple models when pure components differ in structure. Using silicon–germanium as a focal example, the talk explains how two-phase regions arise, how reference-state changes impact Gibbs free energy, and how temperature governs phase stability. The session also covers reading unary phase diagrams, the role of common tangents in two-phase coexistence, and the practical need for computational tools like CALPHAD to map complex diagrams.

Key insights include the use of reference-state changes to evaluate transformations, the necessity of different solution models in intermediate temperature ranges, and the power of common-tangent constructions to locate phase boundaries - MIT OpenCourseWare

Introduction and Motivation

The lecture begins by recapping prior notions of notation, bookkeeping, and solution modeling, then introduces a crucial generalization: when two phases in a binary system have different crystal structures, you cannot rely on a single solution model across the entire phase diagram. Silicon–germanium is used as a worked example to illustrate what happens when pure components reside in different reference states, such as germanium liquid versus silicon solid alpha in a given temperature range. This setup naturally yields two distinct reference-state baselines for Gibbs free energy in different regions, which must be reconciled to analyze phase behavior.

"delta G equals delta H minus T delta S" - MIT OpenCourseWare

"Unary phase diagrams are isobaric slices of the unary phase diagram" - MIT OpenCourseWare

Reference-State Changes: A New Concept

The first novel idea is the reference-state change, a two-step thought experiment you can perform mentally or in a calculation. You start with liquid germanium and conceptually force atoms into the alpha diamond structure, then mix solids in that structure. The first step is a reference-state change, and its Gibbs energy cost is encoded by the difference in Gibbs free energy between the two reference states. The second step uses a standard solution model to describe mixing in the solid alpha phase. The instructor emphasizes that this is a thermodynamic construct, not a literal lab procedure, and that the energy cost to enforce a non-equilibrium reference state is positive.

"I want you to recall that when you force a phase into a state it does not occupy at equilibrium, you pay Gibbs free energy to do so" - MIT OpenCourseWare

"The change in entropy goes down when you go from liquid to solid, and the change in enthalpy is negative as well in a crystallization process" - MIT OpenCourseWare

Gibbs Energy Curves and Temperature Dependence

The discussion then moves to plotting Gibbs free energy versus temperature for each phase of germanium and silicon, highlighting how the alpha solid solution and the liquid phase curves cross at coexistence temperatures. The slope and curvature of these curves reflect the standard unary thermodynamics learned earlier, including the sign and magnitude of delta H and delta S, and the role of heat capacities. The teacher notes that, if heat-capacity differences are negligible, delta H and delta S can be treated as temperature-independent approximations, simplifying the algebra and enabling analytic expressions for the temperature evolution of free energy differences.

"The Gibbs free energy curves must cross where the phases coexist, and their Gibbs energies are equal at the coexistence temperature" - MIT OpenCourseWare

"If the transformation heat capacity difference is negligible, delta H and delta S are approximately temperature independent" - MIT OpenCourseWare

Two-Phase Coexistence: Conditions and Computation

With two solution models in play (alpha solid solution and liquid), the coexistence conditions are written as equal chemical potentials for each component in the two phases. The instructor provides the conceptual setup for silicon and germanium: the left-hand side involves silicon in its solid alpha reference state plus mixing in the solid, while the right-hand side involves silicon in the liquid state plus the corresponding reference-state change. The same logic is applied to germanium, with its reference state in the liquid and the subsequent solid solution contribution. This reduces to two sets of equations that tie together the two phases through the corresponding solution-model contributions and reference-state changes. The overarching message is that all prior unary data—melting points, solution-model parameters, heats of transformation, and Cp(T)—are required inputs for building the binary phase diagram.

"Chemical potentials are equal in two-phase coexistence" - MIT OpenCourseWare

"We are accounting for delta G of reference-state changes, then the mixing in each phase, and finally the equilibrium with the other phase" - MIT OpenCourseWare

"Reading unary phase diagrams is essential because they form the backbone for the binary diagram construction" - MIT OpenCourseWare

From Unary to Binary: Data, Models, and CALPHAD

The lecturer emphasizes that constructing binary phase diagrams relies on unary phase diagrams and solution models for each phase. You need temperature-dependent enthalpy and entropy data, often sourced from databases of phase transformations, along with Cp(T) data to capture temperature dependence. In practice, many binary diagrams are solved with computer software because closed-form analytical solutions quickly become unwieldy as the number of phases and phases’ reference states increases. The course points to CALPHAD and ThermoCalc as tools for evaluating equilibrium conditions, calculating common tangents, and producing lens-shaped two-phase regions. A guest lecture by Greg Olson is mentioned as an example of the data-driven approach to high-performance materials using CALPHAD-type tools.

"If those conditions are met, you can have different types of binary phase diagrams and a variety of lens shapes" - MIT OpenCourseWare

"In the days ahead we will rely more on CALPHAD style tools to compute these diagrams" - MIT OpenCourseWare

Conclusion and Forward Look

The lecture closes by tying together unary phase diagrams, reference-state changes, and multiscale modeling. The goal is to assemble coherent binary phase diagrams using multiple solution models, compute coexistence lines by enforcing chemical-potential equality, and leverage databases for transformation enthalpies, entropies, and Cp(T). The instructor underscores that this is a culminating, data-driven enterprise where theoretical constructs meet practical computation, foreshadowing more software use and data-driven predictions in subsequent classes.

"We use computers to evaluate equilibrium conditions and common tangent points because manual calculation becomes impractical" - MIT OpenCourseWare