Pressures Inside a Nano-Porous Medium. The Case of a Single Phase Fluid

Abstract

We define the pressure of a porous medium in terms of the grand potential, and compute its value in a nano-confined or nano-porous medium, meaning a medium where thermodynamic equations need be adjusted for smallness. On the nano-scale, the pressure depends in a crucial way on the size and shape of the pores. According to Hill [1], two pressures are needed to characterize this situation; the integral pressure and the differential pressure. Using Hill's formalism for a nano-porous medium, we derive an expression for the difference between the integral and the differential pressures in a spherical phase α of radius R, p α − p α = γ/R. We recover the law of Young-Laplace for the differential pressure difference across the same curved surface. We discuss the definition of a representative volume element for the nano-porous medium and show that the smallest REV is half a unit cell in the direction of the pore in the fcc lattice. We also show, for the first time, how the pressure profile through a nano-porous medium can be defined and computed away from equilibrium.

Gibbs Ensemble Monte Carlo Simulation of Fluids in Confinement: Relation between the Differential and Integral Pressures

“…Based on the work of Galteland et al [11], the slope of the fitted line can also be related to the effective surface tension between the fluid particles and the wall in Box 2, i.e., P − p ∼γ fr effective R . The linear relation between P − p and R −1 shows that the effective surface tension does not depend on the curvature of the wall.…”

“…The pressure of a nanoconfined fluid is one of the most important thermodynamic properties which is needed for an accurate description of the flow rate, diffusion coefficient, and the swelling of the nanoporous material [7][8][9][10]. Various approaches for calculating the pressure of a fluid in a nanopore have been proposed [1,[11][12][13]. The main difficulty of the pressure calculation arises from the ambiguous definition of the pressure tensor inside porous materials due to the presence of curved surfaces and confinement effects [7,14,15].…”

“…Recently, Galteland et al [11] reported a new approach for the calculation of pressure in nanoporous materials using Hill's thermodynamics for small systems [20]. In this approach, two different pressures are needed to account for confinement effects in nanoporous materials: the differential pressure P, and the integral pressurep.…”

“…The differential and integral pressures are different for small systems and become equal in the thermodynamic limit. For a system with volume V, the two pressures are related by [11,20]:…”

“…As shown in Equation (1), the two pressures are different only when the integral pressurep depends on the volume of the system. Galteland et al [11] performed equilibrium and non-equilibrium molecular dynamics simulations of Lennard-Jones (LJ) fluids in a face-centered lattice of spherical grains representing a porous medium. Using Hill's thermodynamics of small systems [20] and the additive property of the grand potential, Galteland et al [11] defines the compressional energy (pV) of the representative elementary volume (REV) in the simulations as follows:…”