Solubility of poorly soluble species in bulk and nanopores from grand-canonical Wang-Landau sampling

Abstract

Poorly soluble species can have mean populations below one in small systems such as nanopores. In this regime, the grand-potential minimum, which identifies the most probable particle number, differs from the ensemble-averaged solute loading. We use grand-canonical Wang-Landau sampling (GCWLS) to calculate the density of states $Ω(N_\mathrm{A}, N_\mathrm{B})$ for a binary Lennard-Jones mixture of solute B in solvent A, both in bulk and in slit pores. One set of GCWLS simulations gives ensemble averages over a range of chemical potentials, whereas conventional grand-canonical Monte Carlo (GCMC) simulation requires a separate simulation at each state point. Independent GCMC simulations confirm the GCWLS ensemble averages. The solvent number distribution is nearly Gaussian, whereas the dilute solute distribution is skewed and Poisson-like. The most probable solute number therefore lies below the ensemble-averaged loading $\langle N_\mathrm{B}\rangle$ and is zero whenever $\langle N_\mathrm{B}\rangle<1$. Reading the solute loading from the grand-potential minimum underestimates it and can falsely indicate that no solute is present. Defining the oversolubility as $\mathcal{O}_\mathrm{B}=x_\mathrm{B}^{\mathrm{conf}}/x_\mathrm{B}^{\mathrm{bulk}}$, where $x_\mathrm{B}$ is the solute mole fraction, we find $\mathcal{O}_\mathrm{B}\approx12$ in a pore of 3.5 particle diameters width when both wall interactions are repulsive. Adding wall-solute attraction moderately raises it to $\approx13$, whereas wall-solvent attraction lowers it to $\approx3.4$. This comparison shows that the wall-solvent interaction controls most of the enhancement. As the pore widens, $\mathcal{O}_\mathrm{B}$ decreases toward unity. In bulk and in nanopores alike, the solubility of poorly soluble species should be obtained from ensemble averages rather than from grand-potential minima.