Critical dynamics of conformational exchange in a near-critical solvent: A statistical-mechanical three-mode theory

We develop a three-mode statistical-mechanical theory for a two-conformer solute in a near-critical solvent. A compressible lattice model with vacancies yields the Gaussian free-energy functional for a critical conserved density, an orthogonal noncritical conserved mode, and the local conformer fraction. Their couplings follow from packing and cohesive interactions, rather than being introduced phenomenologically. Kawasaki transport of the two conserved fields and local Glauber-type conversion produce three relaxation branches. At the characteristic critical scale k ∼ ξ−1, full critical scaling gives λ1 ∝ ξ−z, while the regular orthogonal diffusive mode gives λ2 ∝ ξ−2 and the local mode remains λ3≃τx−1; the minimal overdamped Gaussian theory has z = 4. The conformational correlator contains both collective poles and, consequently, develops a divergent integral relaxation time. Wave-number integration gives the general Eulerian tail t−(d−2+η)/z, while tagged diffusion gives t−(d−2+η)/2 under the decoupling approximation. The Gaussian three-dimensional limits are t−1/4 and t−1/2, respectively. The theory is illustrated for ibuprofen in supercritical CO2. At 323.15 K and 13 MPa, the measured population xC+D = 0.52 corresponds to a grouped conformer splitting of −0.215 kJ mol−1 and a local curvature of 10.76 kJ mol−1. Taking an illustrative conformer differential-solvation scale, ΔΔGsolv ≡ Gsolv,2 − Gsolv,1 = −1.0 kJ mol−1, as an explicit sensitivity anchor gives gϕx/a = −0.093. An Arrhenius extrapolation of the carboxylic cis–trans relaxation in neat liquid ibuprofen gives the molecular reference scale τx ≃ 14.4 ns at 323.15 K. Exact diagonalization then resolves a narrow critical contribution, a weak intermediate diffusive contribution, and a broad local conformational component. The numerical application is a semi-quantitative test of a possible critical-locus interpretation and does not fit an absolute nuclear magnetic resonance (NMR) line shape.