Binary solutions

Binary solutions

extensions/chemistry/solution.mojo models the phases of a binary system, such as the liquid and the solid solutions of an alloy. It gives each component's activity and finds the two-phase regions at a temperature.

This page is a reference. For the rest of the library, see Process chemistry.

A phase model

A PhaseModel is one phase of components 1 and 2. Its molar Gibbs energy at the mole fraction x of component 2 is:

G   = (1 - x) G1 + x G2 + R T ((1 - x) ln(1 - x) + x ln x) + Gex
Gex = x (1 - x) sum_k L_k (1 - 2 x)^k

G1 and G2 are the pure components' energies in this phase. The L_k are the Redlich-Kister parameters. Each is a Linear function of temperature, a + b T. With no L_k, the solution is ideal. With L0 only, it is regular.

Method	Gives
excess(x, T)	Gex, in J/mol
gibbs(x, T)	G, in J/mol
log_gamma(component, x, T)	The natural log of a component's activity coefficient
activity(component, x, T)	The activity, x times the coefficient

A reaction can take an activity as the fraction of a condensed term. A dissolved metal then has a lower driving force than the pure metal.

Phase boundaries

phase_boundaries(phases, T) finds the two-phase regions at a temperature. It samples every phase's Gibbs curve and takes their lower convex hull. A stretch of the hull that spans no sampled point is a TieLine. The tie line joins two phases, or two compositions of one phase across a miscibility gap.

The ends of each tie line are the phase boundaries, to the sampling step. The default is 2000 samples, and MAX_SAMPLES is the most. Scan the temperature to draw a phase diagram.

Checks

The tests compare the hull with closed forms that it does not use:

• A symmetric regular solution with parameter W has a miscibility gap below Tc = W / (2 R). Its edges solve ln(x / (1 - x)) = W (2x - 1) / (R T).
• Two ideal phases form a lens. With k_i = exp((Gs_i - Gl_i) / R T), the solidus is (1 - k1) / (k2 - k1), and the liquidus is k2 times that.
• An activity coefficient equals the partial molar excess energy, from a finite difference of excess.

Where the parameters come from

The parameters of a real alloy come from an assessment, such as a CALPHAD database. Most such databases are licensed, so the module takes the parameters as inputs. Give each parameter's source beside it, as a stated bound gives its source.

Limits

A model has two components. Each phase is one sublattice, with no order. Magnetic and pressure terms are not modeled.