Model fluid in a porous medium: results for a Bethe lattice

TL;DR

This paper models fluid behavior in porous media using a Bethe lattice, revealing how the effective field distribution splits into continuous and discrete parts, with implications for understanding wetting, randomness, and confinement effects.

Contribution

It provides an exact recursive solution for the fluid in a porous medium on the Bethe lattice, distinguishing continuous and discrete spectra of the effective field distribution.

Findings

The effective field distribution has both continuous and discrete components.

The continuous spectrum becomes rougher at lower temperatures.

Numerical solutions are limited in accuracy at low temperatures due to spectrum roughness.

Abstract

We consider a lattice gas with quenched impurities or `quenched-annealed binary mixture' on the Bethe lattice. The quenched part represents a porous matrix in which the (annealed) lattice gas resides. This model features the 3 main factors of fluids in random porous media: wetting, randomness and confinement. The recursive character of the Bethe lattice enables an exact treatment, whose key ingredient is an integral equation yielding the one-particle effective field distribution. Our analysis shows that this distribution consists of two essentially different parts. The first one is a continuous spectrum and corresponds to the macroscopic volume accessible to the fluid, the second is discrete and comes from finite closed cavities in the porous medium. Those closed cavities are in equilibrium with the bulk fluid within the grand canonical ensemble we use, but are inaccessible in real experimental situations. Fortunately, we are able to isolate their contributions. Separation of the discrete spectrum facilitates also the numerical solution of the main equation. The numerical calculations show that the continuous spectrum becomes more and more rough as the temperature decreases, and this limits the accuracy of the solution at low temperatures.