Phase Coexistence and Common Tangents in Binary Systems: Gibbs Free Energy, Tie-Lines, and Spinodal Decomposition

An in-depth look at phase coexistence in binary systems, focusing on the common tangent construction and the stationarity of Gibbs free energy. The lecture connects thermodynamic criteria to spinodal decomposition and the formation of tie-lines in phase diagrams.

Introduction and Thermodynamic Framework

The lecture examines phase coexistence in a system with two phases and two components A and B that can freely exchange between phases. At equilibrium, the Gibbs free energy is stationary, which for fixed temperature and pressure means the chemical potential of each component is equal across the two phases.

"At equilibrium, chemical potential of A is the same in both phases and the chemical potential of B is the same in both phases." - Instructor

Gibbs Free Energy and Mass Conservation

The total differential of Gibbs free energy is expressed as a sum over components and phases: mu_A1 d n_A1 + mu_B1 d n_B1 + mu_A2 d n_A2 + mu_B2 d n_B2. Conservation of mass provides constraints DN_A1 = -DN_A2 and DN_B1 = -DN_B2, reducing the number of independent variables from four to two. In this form, the coefficients of the independent variables are recognized as the unconstrained variables that determine how G changes with composition in each phase.

"DGPT equals zero requires coefficients to be zero" - Instructor

Partial Molar Properties and Mixing

Each phase has its own solution model and partial molar properties. The chemical potential of each component can be written as mu_A1 = mu_A^0 + Δμ_A^mixing,1 and mu_A2 = mu_A^0 + Δμ_A^mixing,2, with analogous expressions for B. The reference states mu_A^0 and mu_B^0 are fixed; thus the equilibrium conditions reduce to Δμ_A^mixing,1 = Δμ_A^mixing,2 and Δμ_B^mixing,1 = Δμ_B^mixing,2.

"Delta mu of mixing 1 equals delta mu of mixing 2 for each component at equilibrium" - Instructor

Graphical Solution: Common Tangent Construction

Using a two-phase, two-component solution model, the partial molar Gibbs free energy of mixing for each phase can be represented as a curve. A common tangent to both curves defines the coexistence compositions, with the intercepts corresponding to the partial molar properties of mixing. When a common tangent exists, mu_A1 = mu_A2 and mu_B1 = mu_B2, signaling phase equilibrium. If no common tangent can be drawn, two-phase equilibrium is not possible.

"A common tangent on the free energy diagram corresponds to phase coexistence, satisfying mu_A1 = mu_A2 and mu_B1 = mu_B2." - Instructor

Spinodal Systems and Tie-Lines

In spinodal systems, the free energy of mixing leads to spontaneous demixing and the emergence of two coexisting compositions, connected by tie-lines in the phase diagram. The intercepts of the common tangent provide the compositions of the two phases (x_B1 and x_B2), and the corresponding changes in mixing free energy (Δμ_B^mixing and Δμ_A^mixing) define the driving forces for phase separation. The two-phase mixture has a lower overall Gibbs energy than the fully mixed single-phase state, by exactly the amount represented by the tie-line on the diagram.

"In spinodal systems, the common tangent defines the tie-lines and the two coexisting compositions emerge at equilibrium" - Instructor

From Free Energy to Phase Diagrams and Kinetics

As the temperature is varied, the zero of ΔG_mix moves, creating or removing the two-phase region and sweeping out a family of tie-lines. The lecture emphasizes that the common tangent construction provides equilibrium configurations (tie-lines, coexistence compositions) but does not predict kinetics of phase segregation. Kinetics and pattern formation depend on diffusion and other transport properties and will be explored in later discussions.

"The common tangent construction tells us about equilibrium configuration, not the kinetics of segregation" - Instructor