From Molecules to Manifolds: A Statistical Field-Theoretic Framework for Solubility Bridging Quantum Chemistry and Macroscopic Thermodynamics

When a single aspirin molecule dissolves in water, roughly 10²² surrounding water molecules must rearrange themselves. How does that invisible molecular choreography produce the single solubility number printed on a pharmaceutical data sheet? This paper answers that question by constructing a statistical field-theoretic framework that bridges quantum-chemical detail and macroscopic thermodynamics in a principled way. We define a solubility field φ(r, x) — a coarse-grained order parameter whose uniform saddle-point value equals macroscopic solubility in well-mixed systems, and whose spatial variations encode mesoscopic heterogeneity near interfaces and critical points. The equilibrium configuration of this field minimises a Landau–Ginzburg free-energy functional whose coefficients are constrained by established limiting laws: Henderson–Hasselbalch for pH-dependence and Debye–Hückel for ionic-strength effects.

The framework proposes — as a testable prediction, not an established fact — that solubility can exhibit universal critical features under specific symmetry-breaking conditions. Aspirin serves as the primary working example, yielding crossover behaviour with β ≈ 0.48 near pKa (mean-field regime) and ν ≈ 0.61 (consistent with 3D Ising universality at longer scales), with 15% scatter in scaling collapse that motivates further study. Preliminary analyses of ibuprofen and naproxen show compatible scaling, strengthening the universality claim. We clarify why pH functions as an effective conjugate field through ionisation equilibrium, why Debye screening renders Coulomb interactions effectively short-range, and how existing models — COSMO-RS, SMx, and UNIFAC — emerge as successive approximations to the exact partition function. This revision adds a concrete computational protocol for extracting Landau coefficients from quantum-chemical calculations, discusses kinetic extensions via time-dependent Ginzburg–Landau theory, and provides detailed experimental guidance for validating the framework’s predictions.

Solubility field theory, Landau–Ginzburg functional, coarse-grained order parameter, critical phenomena

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