Thermodynamics of quasi-chemical solutions and critical behaviour

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In Guggenheim’s Quasi-Chemical Theory (QCT), pair correlations directly lower or raise the configurational entropy compared to the Bragg-Williams (mean-field) approximation. This shift dictates both the critical temperature for phase separation (T_c) when Ω > 0 and the order–disorder transition temperature (T_od) when Ω < 0.

Phase separation and miscibility gap (Ω > 0)

In a symmetric binary system characterised by mole fractions x_A = x_B = 0.5, the critical point of macroscopic demixing occurs at the inflection point where the second and third composition derivatives of the molar Gibbs free energy of mixing vanish simultaneously:

Bragg-Williams limit (z → ∞)

In the mean-field limit, spatial correlations are entirely neglected, such that pairs are populated according to a purely random probability distribution (P_AB = 2x_Ax_B). The molar free energy of mixing simplifies to:

Evaluating the second derivative with respect to x_B:

Quasi-chemical derivation

The quasi-chemical treatment rigorously incorporates nearest-neighbour pair correlations into the configurational partition function. The exact molar free energy is parameterised by pair probabilities:

By differentiating with respect to the number of atoms, the chemical potential difference between component species is obtained in terms of the auxiliary ordering factor β:

The spinodal condition corresponds to ∂(μ_B - μ_A)/∂x_B = 0. Evaluating this derivative at the equiatomic composition (x_A = x_B = 0.5), where the auxiliary parameter simplifies to β = exp[Ω/(zRT)], yields:

Substituting the definition β_c = exp[Ω/(zRT_c^QCT)] and solving directly for temperature delivers the quasi-chemical critical temperature:

Order–disorder transition (Ω < 0)

When the interaction parameter is negative (Ω < 0), the energetic driving force favours unlike pairs (A–B) over like pairs (A–A, B–B). At sufficiently low temperatures, the system undergoes an order–disorder transformation, populating distinct interpenetrating sublattices. The long-range order parameter η (0 ≤ η ≤ 1) defines the site occupation probabilities across sublattices α and β:

B2 ordering on body-centred cubic (z = 8, bipartite lattice)

The body-centred cubic (BCC) crystal lattice divides naturally into two equivalent, interpenetrating simple cubic sublattices: the cube corners (α) and the body centres (β). Every nearest neighbour of an atom on the α sublattice resides on the β sublattice. This bipartite topological symmetry ensures that the transformation is a continuous, second-order phase transition. Expanding the quasi-chemical free energy in powers of η near the disordered state (η → 0) shows that the susceptibility diverges at:

Owing to the bipartite symmetry of the BCC lattice, the mathematical form for T_od^QCT is strictly isomorphic to the demixing critical temperature T_c^QCT under the inversion Ω → -Ω.

L1₀ and L1₂ ordering on face-centred cubic (z = 12, frustrated triangular coordination)

Unlike the BCC structure, the face-centred cubic (FCC) lattice decomposes into four interpenetrating simple cubic sublattices. Nearest neighbours form closed equilateral triangles and regular tetrahedra. This geometric topology introduces profound geometric frustration: it is mathematically impossible to arrange pairs on a triangle such that all three bonds are simultaneously A–B pairs.

Consequently, the A1 → L1₀ (AB stoichiometry) and A1 → L1₂ (A₃B stoichiometry) ordering transformations are discontinuous, first-order transitions characterised by a latent heat and coexistence regions.

The simple quasi-chemical pair approximation systematically fails to capture many-body multi-site correlations on frustrated geometries. To obtain quantitatively accurate phase boundaries on FCC lattices, higher-order methods—most notably Kikuchi’s Cluster Variation Method (CVM) using tetrahedral clusters—must be employed. In the mean-field approximation, T_od^BW(L1₀) ≈ |Ω|/(4R), whereas pair and cluster correlations depress this transition temperature by approximately 15% to 20%.

Comparison of critical temperatures

The ratio between the quasi-chemical and Bragg-Williams critical temperatures provides a direct metric of the role of nearest-neighbour spatial correlations as a function of the coordination number z:

As the coordination number increases, each atom interacts with an increasing number of neighbours, diluting the localized statistical impact of individual pairs. In the asymptotic limit z → ∞, the ratio approaches unity, rigorously recovering the classical mean-field Bragg-Williams result: