Hierarchical reference theory of critical fluids in disordered porous media

Abstract

We consider the equilibrium behaviour of fluids imbibed in disordered mesoporous media, including their gas–liquid critical point when present. Our starting points are on the one hand a description of the fluid/solid-matrix system as a quenched-annealed mixture and on the other hand the Hierarchical Reference Theory (HRT) developed by A. Parola and L. Reatto to cope with density fluctuations on all length scales. The formalism combines liquid-state statistical mechanics and the theory of systems in the presence of quenched disorder. A straightforward implementation of the HRT to the quenched-annealed mixture is shown to lead to unsatisfactory results, while indicating that the critical behaviour of the system is in the same universality class as that of the random-field Ising model. After a detour via the field-theoretical renormalization group approach of the latter model, we finally lay out the foundations for a proper HRT of fluids in a disordered porous material.

Keywords:

• renormalization group
• critical phenomena
• fluids
• disordered porous media
• statistical mechanics

Acknowledgements

It is a pleasure to dedicate this article to Professor Luciano Reatto whose work has been strongly influential to our study of critical behaviour in quenched disordered environments.

Notes

Notes

1. Note that the RFIM can also be considered as a two-component quenched-annealed mixture when the random-field distribution depends on a single positive parameter h ₀ measuring the strength of disorder. This is the case for instance of the Gaussian and bimodal distributions. The two spin variables S_i and are then annealed and quenched, respectively. This point of view may be useful when the direct average over disorder yields a complicated nonlinear replicated Hamiltonian, as occurs with a bimodal distribution (see e.g. 69).

2. This comes from the fact that the HRT equation in the OZ/LPA approximation for the bulk fluid with a sharp cutoff reads

and is not the same as that for the quenched-annealed mixture in dimension d + 2, Equation (59).

3. One may indeed think of the quenched matrix configurations as obtained from an equilibrium system with a possibly complicated interaction energy function; contrary to what is sometimes assumed for simplicity, this energy function is not pairwise additive in general and involves irreducible many-body terms, whose description is irrelevant here. Then, the corresponding direct correlation functions as related to the standard (total or Green's) correlation functions through a Legendre transform. , which is a functional of a solid-density field , is only introduced as a convenient trick to keep track of the structural information about the matrix the matrix. In practice, only the static structure factor is needed.

4. As a warm-up, it could be worthwhile to solve the full HRT equations (not only their asymptotic form) with an OZ-like closure (RPA or ORPA) for the quenched-annealed mixture in d = 5 with the proper set of IR regulators. As we have shown, this approximation predicts that the liquid–gas critical behaviour is the same as that of the bulk fluid in d = 3. While such a dimensional-reduction property is not expected to be exact for 24, the error should be small in d = 5. On the other hand, this closure, which incorporates a microscopic description of confinement, randomness and wettability, would allow one to study crossover phenomena determined by the competition between different fixed points (RFIM, bulk, site-diluted Ising model, percolation) and confirm if the liquid–gas critical behaviour in the presence of a disordered matrix is indeed in the universality class of the RFIM.