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Molecular correlations and solvation in simple fluids

We study the molecular correlations in a lattice model of a solution of a low-solubility solute, with emphasis on how the thermodynamics is reflected in the correlation functions. The model is treated in Bethe-Guggenheim approximation, which is exact on a Bethe lattice (Cayley tree). The solution properties are obtained in the limit of infinite dilution of the solute. With $h_{11}(r)$, $h_{12}(r)$, and $h_{22}(r)$ the three pair correlation functions as functions of the separation $r$ (subscripts 1 and 2 referring to solvent and solute, respectively), we find for $r \geq 2$ lattice steps that $h_{22}(r)/h_{12}(r) \equiv h_{12}(r)/h_{11}(r)$. This illustrates a general theorem that holds in the asymptotic limit of infinite $r$. The three correlation functions share a common exponential decay length (correlation length), but when the solubility of the solute is low the amplitude of the decay of $h_{22}(r)$ is much greater than that of $h_{12}(r)$, which in turn is much greater than that of $h_{11}(r)$. As a consequence the amplitude of the decay of $h_{22}(r)$ is enormously greater than that of $h_{11}(r)$. The effective solute-solute attraction then remains discernible at distances at which the solvent molecules are essentially no longer correlated, as found in similar circumstances in an earlier model. The second osmotic virial coefficient is large and negative, as expected. We find that the solvent-mediated part $W(r)$ of the potential of mean force between solutes, evaluated at contact, $r=1$, is related in this model to the Gibbs free energy of solvation at fixed pressure, $ΔG_p^*$, by $(Z/2) W(1) + ΔG_p^* \equiv p v_0$, where $Z$ is the coordination number of the lattice, $p$ the pressure, and $v_0$ the volume of the cell associated with each lattice site. A large, positive $ΔG_p^*$ associated with the low solubility is thus reflected in a strong attraction (large negative $W$ at contact), which is the major contributor to the second osmotic virial coefficient. In this model, the low solubility (large positive $ΔG_p^*$) is due partly to an unfavorable enthalpy of solvation and partly to an unfavorable solvation entropy, unlike in the hydrophobic effect, where the enthalpy of solvation itself favors high solubility, but is overweighed by the unfavorable solvation entropy.