Microscopic first-principles determination of the fluctuon parameters in the heterophase-fluctuation theory of the critical compressibility factor

The van der Waals equation of state predicts a universal critical compressibility factor $Z_c=P_cv_c/RT_c=3/8=0.375$, while measured values for real molecular fluids lie roughly between 0.23 and 0.32 — the best-known quantitative failure of van der Waals' mean-field description of the gas–liquid critical point. Bakai (2020) revived van der Waals' 1910 conjecture that molecular "pseudo associations" remove this discrepancy, modeling the near-critical fluid as a heterophase mixture of mesoscopic liquid-like and gas-like droplets (l- and g-fluctuons) and obtaining $Z_c=3/8-0.193/\bar{k}_{\mathrm{fl},c}$. As published, however, the theory describes rather than predicts: the fluctuon interaction parameter $g_2$ is fixed only by consistency with the van der Waals critical temperature, and the mean number of molecules per fluctuon $\bar{k}_{\mathrm{fl},c}$ is back-fitted from the measured $Z_c$ itself. The open problem is a non-circular microscopic determination of $\bar{k}_{\mathrm{fl},c}$ and $g_2$ from intermolecular physics, so that the relation $Z_c=3/8-0.193/\bar{k}_{\mathrm{fl},c}$ becomes a prediction testable against experiment. No first-principles derivation of these parameters is presently available; the neighboring approaches to the critical compressibility factor all require empirical input.

Background

At the gas–liquid critical point of a fluid (pressure $P_c$, temperature $T_c$, molar volume $v_c$), the dimensionless critical compressibility factor is $Z_c=P_cv_c/(RT_c)$, with $R$ the gas constant. Van der Waals' equation of state $(P+a/v^2)(v-b)=RT$, where $a$ measures the attractive interaction between molecules and $b$ the excluded volume, predicts the universal value $Z_c=3/8=0.375$ for all fluids — a cornerstone of the law of corresponding states, in which fluids with reduced variables $P/P_c$, $v/v_c$, $T/T_c$ behave identically. Experimentally, real fluids violate this prediction: simple dispersion-dominated fluids cluster near 0.27–0.30 (ethylene 0.270, xenon and krypton 0.288, argon 0.291, helium-4 0.308), while strongly associating fluids fall near 0.23 and quantum fluids rise toward 0.32. In his 1910 Nobel lecture, van der Waals conjectured that the discrepancy arises from "pseudo associations" — transient molecular complexes in the dense fluid that remove part of the attraction from the surface pressure — but never completed the modification of his theory. Frenkel's 1939 theory of heterophase fluctuations (liquid-like droplets in the gas and gaseous bubbles in the liquid) supplies a physical picture for such complexes, but only in the dilute limit far from criticality. Bakai (2020) revived the idea near the critical point, where droplet and bubble concentrations are large: the near-critical fluid is modeled as a mixture of mesoscopic species called l-fluctuons (liquid-like droplets) and g-fluctuons (gaseous bubbles), with number fractions $\sigma_l$ and $\sigma_g$ ($\sigma_l+\sigma_g=1$). The fluctuon contribution to the free energy per molecule is an Ising-type effective Hamiltonian, $g_2\sigma_l\sigma_g+T(\sigma_l\ln\sigma_l+\sigma_g\ln\sigma_g)$, where $g_2>0$ quantifies the interaction between fluctuons and the logarithmic terms are the mixing entropy; this Hamiltonian bifurcates at $T_c=g_2/2$, where $\sigma_l=\sigma_g=1/2$. Requiring consistency with the van der Waals critical point fixes $g_2=2T_{c,\mathrm{VdW}}=16a/(27b)$. The fluctuonic pressure then lowers the critical compressibility factor to $Z_c=3/8-0.193/\bar{k}_{\mathrm{fl},c}$, where $\bar{k}_{\mathrm{fl},c}$ is the mean number of molecules per fluctuon at criticality. Inverting this relation against measured values, $\bar{k}_{\mathrm{fl},c}=0.193/(Z_{c,\mathrm{VdW}}-Z_{c,\mathrm{exp}})$, yields numbers between about 1.8 (ethylene) and 3.5 (helium-3), correlating with the de Boer parameter, a dimensionless measure of quantum delocalization of the molecule. The framework connects to the lattice gas — the Ising-model analogue of a fluid introduced by Lee and Yang, in which occupied and empty lattice sites play the role of liquid and gas — through Bakai's lattice-fluctuon model, a mesoscopic model of the gas–liquid transition in which the lattice-gas model appears as the special case of one molecule per cell. It also connects to the Kac–Uhlenbeck–Hemmer model, where the van der Waals equation becomes exact for infinitely weak, infinitely long-ranged attractions, implying $Z_c\to3/8$ in that limit. The wider literature treats $Z_c$ as a nonuniversal quantity: in renormalization-group theory it is a combination of critical amplitudes and metric factors with no universality expectation, and phenomenological accounts relate it to the Zeno line (the locus of states where the compressibility factor equals one, Apfelbaum and Vorob'ev), to the lattice gas through Kulinskii's global isomorphism mapping, or to Pitzer's acentric factor (an empirical measure of molecular non-sphericity from the reduced vapor pressure); Sanchez and Boening found instead that the critical fugacity coefficient is near-universal (about 2/3) across 68 fluids. None of these derives $Z_c$ for real fluids from microscopic interactions.

Problem Statement

Derive, from the microscopic intermolecular physics of a fluid and without using the measured critical compressibility factor (or any measured critical datum that fixes it) as input, the mesoscopic parameters of the heterophase-fluctuation correction to the van der Waals equation of state — namely the mean number of molecules per fluctuon at the critical point $\bar{k}_{\mathrm{fl},c}$ and the fluctuon interaction parameter $g_2$ — so that the relation $Z_c=3/8-0.193/\bar{k}_{\mathrm{fl},c}$ becomes a prediction for real fluids rather than a fit. This requires: (a) a microscopic derivation of $g_2$, or a derivation that justifies or replaces the consistency identification $g_2=16a/(27b)=2T_{c,\mathrm{VdW}}$; and (b) a determination of $\bar{k}_{\mathrm{fl},c}$ for named fluids from their intermolecular interactions, with the predicted $Z_c$ values then tested against experiment. A complete answer states the predicted $\bar{k}_{\mathrm{fl},c}$, $g_2$, and $Z_c$ for named fluids together with their microscopic derivation, and thereby settles whether van der Waals' 1910 conjecture gives a quantitatively correct account of the non-universality of the critical compressibility factor.

References

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