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Critical dynamics of conformational exchange in a nearcritical 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
$k\sim\xi^{-1}$, full critical scaling gives $\lambda_1\propto\xi^{-z}$,
while the regular orthogonal diffusive mode gives
$\lambda_2\propto\xi^{-2}$ and the local mode remains
$\lambda_3\simeq\tau_x^{-1}$; 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 $t^{-(d-2+\eta)/z}$, while tagged
diffusion gives $t^{-(d-2+\eta)/2}$ 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 CO$_2$. At
$323.15$~K and $13$~MPa, the measured population $x_{C+D}=0.52$ 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,
$\Delta\Delta G_{\mathrm{solv}}\equiv
G_{\mathrm{solv},2}-G_{\mathrm{solv},1}
=-1.0$~kJ\,mol$^{-1}$, as an explicit sensitivity anchor gives
$g_{\phi x}/a=-0.093$. An Arrhenius extrapolation of
the carboxylic cis--trans relaxation in neat liquid ibuprofen gives the
molecular reference scale $\tau_x\simeq14.4$~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 NMR line
shape.