Near the critical point of a solvent, supercritical solutions develop pronounced molecular clustering—locally dense shells of solvent (and cosolvent) around solute molecules. Hou, Zhang, Han, Mei, Ning and Yang (Phys. Chem. Chem. Phys. 15 (2013) 10654) proposed a thermodynamic principle in which these molecular clusters are treated as an individual thermodynamic phase, so that a supercritical fluid–solid system is described as a three-phase equilibrium (solid, cluster phase, bulk fluid phase) with equality of each component's fugacity among phases; combined with Monte Carlo simulation for cluster composition and the Peng–Robinson equation of state for fugacity coefficients, this treatment reproduced solubilities in the critical region of supercritical CO2 far better than the conventional single-fluid-phase treatment. The paper left open whether the principle generalizes: per its closing discussion, it has not been tested for other systems where molecular clustering is obvious—liquid-phase clustering, polymer solutions, or colloidal systems—and no corresponding calculation methods exist.

Background

A supercritical fluid (SCF) is a fluid held above its critical temperature and pressure, where gas and liquid become indistinguishable; supercritical CO2 is the most widely used SCF solvent. In the critical region, SCF solutions exhibit molecular clustering (also called local density augmentation): the solvent density in the first shell around a solute can greatly exceed the bulk density, as confirmed by spectroscopy, small-angle scattering, and molecular simulation. Conventional phase-equilibrium thermodynamics ignores this structure: the whole fluid is treated as one homogeneous phase, and equations of state (EOS) such as the Peng–Robinson EOS—an empirical cubic EOS relating pressure, volume, and temperature with parameters from critical constants and the acentric factor—are used to compute fugacity coefficients (correction factors linking a mixture component's chemical potential to an ideal-gas reference). For solid solutes in SCFs this treatment systematically fails near the solvent's critical point, where the pressure dependence of solubility is much steeper than predicted, and numerous EOS modifications did not remove the discrepancy.

Hou, Zhang, Han, Mei, Ning and Yang (2013) proposed a different remedy: regard the collection of molecular clusters as an individual thermodynamic phase. The SCF–solid system is then divided into a solid phase, a cluster phase (the transient, locally dense solute–solvent–cosolvent aggregates), and a bulk fluid phase, and at equilibrium the fugacity of every component is equal across the phases, e.g. fiS=fiB=fiCf_{i}^{\mathrm{S}}=f_{i}^{\mathrm{B}}=f_{i}^{\mathrm{C}} for the solute. Because cluster composition cannot be obtained from macroscopic thermodynamics alone, the calculation method couples molecular-level information to the macroscopic equilibrium: Monte Carlo simulation at the given temperature, pressure, and solvent/cosolvent ratio supplies the cluster-phase composition (clusters being identified by a distance/coordination criterion), the Peng–Robinson EOS supplies the fugacity coefficients of both fluid-like phases, and the coupled fugacity-equality conditions are solved iteratively for the overall solute mole fraction and the number of clusters. For benzoic acid, phenanthrene, aspirin, and naphthalene in supercritical CO2 with and without cosolvents (ethyl acetate, acetone, n-pentane), the three-phase calculation matched experimental solubilities in the critical region distinctly better than the conventional two-phase calculation, while far from the critical point the two treatments coincide, as expected when clustering is negligible. The 2013 validation was, however, graphical (no quantitative error metrics were reported), and no independent re-validation of the principle has appeared since.

The principle sits inside a broader landscape of cluster-based thermodynamics that must not be conflated with it. The Quantum Cluster Equilibrium (QCE) model—developed from the late 1990s by Weinhold, Kirchner and co-workers—describes pure liquids and their phase transitions as chemical (reaction) equilibria among a finite set of quantum-chemically computed molecular clusters within a single phase, not as a separate cluster phase with cross-phase fugacity equality. Colloidal physics has its own "equilibrium cluster phases" (Groenewold and Kegel 2004): aggregates of colloidal particles stabilized by competing short-range attraction and long-range repulsion form a distinct mesophase in an effective one-component description—a different sense of "cluster phase" from solute–solvent clustering in a multicomponent mixture. Polymer solutions are known to form equilibrium (and shear-induced) mesoscopic clusters (Wolf 2021), documented from crossover-concentration and rheological arguments but without a fugacity-based cluster-phase equilibrium calculation. Recent work on supercritical fluids themselves—on heterogeneous cluster energetics and nonlinear thermodynamic response (Fan, Ly and Ihme 2024) and on reactions enhanced by inhomogeneous microaggregation (Guo, Wang and Liu 2026)—keeps the home domain active but does not engage the clusters-as-a-phase equilibrium principle, whose only journal citation (Noroozi and Paluch 2017) uses a conventional free-energy/activity-coefficient simulation route instead.

Within this landscape, the question left open by the 2013 paper is precise: does the clusters-as-an-individual-phase principle, established only for SCF–solid equilibria, extend to system classes where clustering is likewise obvious—liquids, polymer solutions, colloidal systems—and can usable calculation methods be built for those classes?

Problem Statement

Determine whether the thermodynamic principle proposed by Hou et al. (2013)—that the molecular clusters in a fluid system constitute an individual thermodynamic phase, so that phase equilibrium is computed with the cluster phase as a distinct phase alongside the bulk fluid phase (and any coexisting solid or liquid phase), with equality of each component's fugacity among the phases and with cluster composition and properties derived from molecular-level information—extends beyond supercritical fluid–solid systems to other systems where molecular clustering is evident, specifically liquid-phase clustering, polymer solutions, and colloidal systems; and develop the corresponding calculation methods. Concretely, extending and testing the principle for such clustering systems involves: (i) formulating the multiphase equilibrium treating the clusters as an individual phase, including a defensible criterion for which molecules belong to the cluster phase and how the phase-rule accounting accommodates a phase whose amount and composition are internal variables; (ii) constructing a workable calculation method that couples molecular-level input (simulation or measured cluster structure/composition) with a macroscopic equilibrium calculation to predict a checkable phase-equilibrium observable, such as solubility, liquid–liquid coexistence, or partitioning; and (iii) testing the principle against independent experimental data in the regime where clustering is significant, in comparison with the conventional homogeneous-fluid-phase treatment. The three named system classes are instances of the clustering systems in question, and the research objective is to carry out this full (i)–(iii) program—formulation, a workable calculation method, and a test against independent data—for at least one of them.

References

  1. Hou M., Zhang J., Han B., Mei Q., Ning H., Yang D. (2013), "The molecular clusters in a supercritical fluid–solid system should be considered as a phase—thermodynamic principle and evidence", Physical Chemistry Chemical Physics 15(26):10654–10658, DOI 10.1039/c3cp44670k, https://doi.org/10.1039/c3cp44670k
  2. Hou M., Zhang X., Han B., Song J., Liu G., Zhang Z., Zhang J. (2008), "New topic of supercritical fluids: Local activity coefficients of supercritical solvent and cosolvent around solute", The Journal of Chemical Physics 128:114101, DOI 10.1063/1.2838181, https://doi.org/10.1063/1.2838181
  3. Noroozi J., Paluch A. S. (2017), "Microscopic Structure and Solubility Predictions of Multifunctional Solids in Supercritical Carbon Dioxide: A Molecular Simulation Study", The Journal of Physical Chemistry B 121(7):1660–1674, DOI 10.1021/acs.jpcb.6b12390, https://doi.org/10.1021/acs.jpcb.6b12390
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  5. Kirchner B., Weinhold F., Friedrich J., Perlt E., Lehmann S. B. C. (2014), "Quantum Cluster Equilibrium", book chapter in Many-Electron Approaches in Physics, Chemistry and Mathematics (Mathematical Physics Studies), pp. 77–96, DOI 10.1007/978-3-319-06379-9_4, https://doi.org/10.1007/978-3-319-06379-9_4
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  7. Wolf B. A. (2021), "Polymer solutions: Equilibrium clusters versus shear clusters", Polymer 212:123149, DOI 10.1016/j.polymer.2020.123149, https://doi.org/10.1016/j.polymer.2020.123149
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