Condensation pressures in small pores: An analytical model based on density functional theory

Abstract

Integral methods are used to derive an analytical expression describing fluid condensation pressures in slit pores bounded by parallel plane walls. To obtain this result, the governing equations of density functional theory (DFT) are integrated across the pore width assuming that fluid densities within adsorbed layers are spatially uniform. The thickness, density, and free energy of these layers are expressed as composite functions constructed from asymptotic limits applicable to small and large pores. By equating the total free energy of the adsorbed layers to that of a liquid-full pore, we arrive at a closed-form expression for the condensation pressure in terms of the pore size, surface tension, and Lennard-Jones parameters of the adsorbent and adsorbate molecules. The resulting equation reduces to the Kelvin equation in the large-pore limit. It further reproduces the condensation pressures computed by means of the full DFT equations for all pore sizes in which phase transitions are abrupt. Finally, in the limit of extremely small pores, for which phase transitions may be smooth and continuous, this simple analytical expression provides a good approximation to the apparent condensation pressure indicated by the steepest portion of the adsorption isotherm computed via DFT.

Kinetic theory of heterogeneous nucleation; effect of nonuniform density in the nuclei

“…The contributions to the free energy listed in Section 2 can be expressed as follows [25,26]: The weighting function W(jr À r 0 j) is selected in the form [28] Wðjr À r 0 jÞ ¼ where r = jr À r 0 j. The contribution to the free energy due to the attraction between the fluid-fluid molecules is calculated in the mean-field approximation Minimizing the total free energy with respect to q(r) provides / ff,y (jx À x 0 j, jh À h 0 j) and W y (jx À x 0 j, jh À h 0 j) are obtained by integrating the potential / ff (jr À r 0 j) and the weighted function W(jr À r 0 j) with respect to the y-direction, from À1 to +1, respectively.…”

Contact angle of a nanodrop on a nanorough solid surface

“…The weighting function W(|r − r′|) in eqn (15) is selected in the form 18 W ðjr À r′jÞ ¼ 3 πσ ff 3 where r = |r − r′|. The third contribution is the excess free energy due to fluid-fluid attractive interaction which is accounted for in the mean-field approximation…”

GCMC Simulations of Gas Adsorption in Carbon Pore Structures

“…In general, the gas-liquid phase transition in pores (capillary condensation) is a very interesting feature, which is observed when condensable gases are confined in small cavities. The phenomenon has been extensively studied in pores of various geometries both experimentally and theoretically (Evans & Tarazona, 1984 ;Ravikovitch et al, 2001 ;Nilson & Griffiths, 1999 ;Neimark et al, 2003). Capillary condensation takes place at pressures below the bulk condensation pressure at a given temperature.…”