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

TL;DR

The paper demonstrates a neural density-functional theory within hyperdensity functional theory to study a binary azeotropic Lennard–Jones mixture under confinement. By training a neural functional on a repulsive reference and applying a mean-field attraction consistent with a known bulk equation of state, the approach achieves accurate inhomogeneous predictions and efficient evaluation of pore selectivity across a broad thermodynamic range. A key finding is that the azeotropic composition leaves a robust imprint on confinement: pore selectivity crossover occurs near x_B^(az) across conditions, accompanied by equal partial molar volumes and an extremum in isothermal compressibility; an aneotropic composition x_B^(an) where relative adsorption vanishes aligns with x_B^(az) and shifts predictably with wall affinity. Overall, the work links bulk azeotropy to interfacial adsorption in confinement and introduces a scalable, transferable ML-cDFT framework for complex mixtures with potential impact on separation technologies.

Abstract

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.

Figures (5)

• Figure 1: Effects of confinement on a binary LJ mixture. (a) The pressure-composition phase diagram of the mixture at $k_{\rm B}T/\epsilon = 0.77$. An azeotrope forms at $x_{\rm B}^{\rm{(az)}} \approx 0.67$, as indicated by the vertical dashed line. The circle indicates the state point of the density profiles in (b), where the mixture is confined in a slit, for two different wall--fluid interaction strengths (as indicated by the labels). Good agreement between the theory and the simulation data is observed. (c) Pore selectivities vs $x_{\rm B}$ for different wall--fluid interactions, both for component A (orange to purple) and B (green to blue) at the azeotropic pressure, marked by the horizontal dashed line in (a). Lines serve as a guide to the eye. A reversal in selectivity is observed at $x_{\rm B}^{\text{(az)}}$, as indicated by the vertical dashed line.
• Figure 2: The relevance of the azeotropic composition across a broad range of thermodynamic conditions. (a) $S_{\rm B}$ vs $x_{\rm B}$ with varying wall--fluid interaction strengths at different state points, as indicated by the labels. The top and bottom panels correspond to liquid and vapor, respectively, in the reservoir. In all cases, a crossover in selectivity occurs at $x_{\rm B}\approx x_{\rm B}^{\rm (az)}$. (b) $P$--$T$ phase diagram, with the thermodynamic state points used in (a) marked by the circles. The critical azeotropic endpoint is $k_{\mathrm{B}}T_{\text{CAEP}}/\epsilon\approx0.94$staubachInterfacialPropertiesBinary2022 (star) and the upper critical point is $k_{\mathrm{B}}T_{\text{C}}/\epsilon\approx 1.09$ (white square). The inset shows the region around $k_{\mathrm{B}}T_{\text{CAEP}}/\epsilon$ with the lower critical point.
• Figure 3: Bulk thermodynamic properties vs $x_{\rm B}$ for different $P$ and $T$. From left to right, the temperatures correspond to $T \ll T_{\text{c}}$, $T \lesssim T_{\text{CAEP}}<T_{\rm c}$, $T>T_{\text{c}}$, and $T \gg T_{\text{c}}$. (a) For subcritical temperatures, $\Delta_\text{E} \mu$ collapses into liquid and vapor branches that cross at $x_{\rm B}\approx x_{\rm B}^{\rm (az)}$; this crossing point persists at supercritical temperatures. The insets show schematic representations of the $P$--$x_{\rm B}$ phase diagram at subcritical temperatures. (b) The isothermal compressibility is locally extremum at $x_{\rm B}\approx x_{\rm B}^{\rm (az)}$ at all temperatures. In both (a) and (b), discontinuities representing liquid--vapor phase transitions are shown with dotted lines.
• Figure 4: Coincidence of $S_{\rm B}=1$, vanishing relative adsorption, and extremal interfacial tension. (a) $S_{\rm B}$ vs $x_{\rm B}$ at the four different state points from Fig. 2 (and indicated in the inset). (b) $\Gamma_{\rm A}^{\rm (B)}$ vs $x_{\rm B}$, with results corresponding to low pressures (light and dark grey) multiplied by a factor 20 for clarity. (c) wall--fluid interfacial tension. The vertical dashed line indicates the azeotropic composition. All results corresponds to a slit pore with $\beta\epsilon_{\rm w, A} = \beta\epsilon_{\rm w, B} = 2.0$ and $L = 8\sigma$.
• Figure 5: Effect of wall affinity, $\Delta\epsilon_{\rm w} = \epsilon_{\rm w,B}-\epsilon_{\rm w,A}$, on the pore selectivity. In all cases, $\beta \epsilon_{\rm w, B}=2.0$. (a), (b), and (c) respectively show how $S_{\rm B}$, $\Gamma_{\rm A}^{\rm (B)}$, and $\gamma$ vary with $x_{\rm B}$. In all cases, we observe that $S_{\rm B}=1$, $\Gamma_{\rm A}^{\rm (B)} = 0$, and a minimum in $\gamma$ coincide. The vertical dashed and dotted lines indicate $x_{\rm B}^{\rm (az)}$ and $x_{\rm B}^{\rm (an)}$, respectively. The solid black lines in (c) show fits to a fourth-order polynomial, constrained to be minimum at the aneotropic composition. Panel (d) shows how $x_{\rm B}^{\rm (an)}$ varies with $\beta\Delta\epsilon_{\rm w}$ for different $L$ (as indicated in the legend). (e) $S_{\rm B}(x_{\rm B}^{\rm (az)})$ vs $\beta\Delta\epsilon_{\rm w}$ for different widths of the slit pore. When rescaled according to Eq. \ref{['eqn:S-by-DeltaEps']} all data approximately collapse onto the same master curve, as seen in (f).