The roles of bulk and surface thermodynamics in the selective adsorption of a confined azeotropic mixture

Fluid mixtures that exhibit an azeotrope cannot be purified by simple bulk distillation. Consequently, there is strong motivation to understand the behavior of azeotropic mixtures under confinement. We address this problem using a machine-learning-enhanced classical density functional theory applied to a binary Lennard-Jones mixture that exhibits azeotropic phase behavior. As proof-of-principle of a “train once, learn many” strategy, our approach combines a neural functional trained on a single-component repulsive reference system with a mean-field treatment of attractive interactions, derived within the framework of hyperdensity functional theory (hyper-DFT). The theory faithfully describes capillary condensation and results from grand canonical Monte Carlo simulations. Moreover, by taking advantage of a known accurate equation of state, the theory we present well-describes bulk thermodynamics by construction. Exploiting the computational efficiency of hyper-DFT, we systematically evaluate adsorption selectivity across a wide range of compositions, pressures, temperatures, and wall-fluid affinities. In cases where the wall-fluid interaction is the same for both species, we find that the pore becomes completely unselective at the bulk azeotropic composition. Strikingly, this unselective point persists far from liquid-vapor coexistence, including in the supercritical regime. Analysis of the bulk equation of state across a wide range of thermodynamic state points shows that the azeotropic composition coincides with equal partial molar volumes and an extremum in the isothermal compressibility. A complementary thermodynamic analysis demonstrates that unselective adsorption corresponds to an aneotrope (a point of zero relative adsorption) and an extremum in the interfacial free energy. We also find that the two interfaces of the slit pore behave independently down to remarkably small slits.

💡 Research Summary

This paper investigates how bulk and interfacial thermodynamics govern the selective adsorption of an azeotropic binary Lennard‑Jones (LJ) mixture confined in a slit pore. The authors develop a machine‑learning‑enhanced classical density functional theory (ML‑cDFT) that follows a “train‑once, learn‑many” paradigm: a neural network is trained on a single‑component repulsive reference system (the Weeks‑Chandler‑Andersen, WCA, fluid) to learn the one‑body direct correlation functional c^(1). Attractive interactions are then added via a mean‑field (MF) term derived within the hyper‑density functional theory (hyper‑DFT) framework. Because the bulk equation of state (EoS) for the mixture is known accurately (the P‑eTS EoS), the MF term can be constructed directly from the EoS, ensuring that bulk thermodynamics are reproduced by construction while the neural functional handles only the inhomogeneous, confined part of the problem.

The binary mixture consists of species A and B with identical diameters σ but different LJ energy parameters (ε_AA = ε, ε_BB = 0.9 ε, ε_AB = 0.806 ε) and a cutoff rc = 2.5 σ. At temperature kBT/ε = 0.77 the mixture exhibits a positive azeotrope at mole fraction x_B ≈ 0.67 and pressure P_az ≈ 0.0248 ε/σ³. The authors first validate the hyper‑DFT/ML approach against grand‑canonical Monte Carlo (GCMC) simulations for a range of wall–fluid interaction strengths (ε_w) and bulk state points. The agreement is excellent for density profiles, adsorption isotherms, and capillary condensation behavior, confirming that the neural functional captures the essential correlations of the reference system and that the MF term correctly adds the attractive contributions.

The central finding concerns adsorption selectivity when the pore walls interact identically with both components. Across a broad span of bulk compositions, pressures, and temperatures—including deep into the supercritical regime—the selectivity (difference in adsorbed amounts of A and B) vanishes precisely at the bulk azeotropic composition. In other words, the pore becomes “unselective” at x_B = x_az, regardless of how far the state point lies from the liquid–vapor coexistence line. This unselective point persists even when the bulk is far from phase coexistence, indicating that the phenomenon is rooted in bulk thermodynamic properties rather than in capillary phase transitions.

Thermodynamic analysis reveals that the azeotropic composition coincides with two bulk signatures: (i) equal partial molar volumes of the two species (V̄_A = V̄_B) and (ii) an extremum (maximum) in the isothermal compressibility χ_T. Moreover, a complementary surface analysis shows that the same composition corresponds to an “aneotrope” – a point of zero relative adsorption – and to an extremum in the interfacial free energy γ of the fluid–wall interface. Thus, the unselective adsorption can be interpreted as the simultaneous occurrence of bulk and interfacial thermodynamic extremal conditions.

An additional, perhaps surprising, observation is that the two walls of the slit pore behave independently down to remarkably narrow separations (only a few σ). Even when the pore width approaches molecular dimensions, each wall establishes its own equilibrium adsorption layer without significant coupling, implying that nanoconfinement does not necessarily induce cooperative effects for this simple mixture.

The paper concludes by emphasizing the methodological advantages of the hyper‑DFT/ML scheme. By delegating bulk thermodynamics to a known EoS and focusing the neural network on the reference repulsive system, the authors achieve high accuracy with modest training data and computational cost. This “train‑once, learn‑many” strategy is especially powerful for mixtures, where the same repulsive reference can be reused across many compositions and external potentials. The authors suggest that the framework can be extended to more complex fluids (e.g., electrolytes, anisotropic particles) and to realistic porous materials, offering a fast and reliable tool for predicting and controlling selective adsorption in industrially relevant separations.