Journal of Colloid and Interface Science 238, 230–237 (2001) doi:10.1006/jcis.2001.7448, available online at http://www.idealibrary.com on Modeling Disjoining Pressures in Submicrometer Liquid-Filled Cylindrical Geometries Paul Scovazzo 1 and Paul Todd 2 Department of Chemical Engineering, University of Colorado, Campus Box 424, Boulder, Colorado 80309-0424 Received June 6, 2000; accepted January 29, 2001 This work develops models for calculating the disjoining pres- sures of a cylindrical fluid “plug,” specifically in submicro- meter cylindrical pores. This modeling produces closed-form, cylindrical-pore disjoining pressures forLondon/van derWaals and solute/pore-wall adsorption interactions, which are the slit-pore models with the characteristic pore size replaced by the radius and multiplied by 6, resulting in a 48-fold or more increase in mag- nitude. In addition, this work contains a numerical solution for electrostatic interactions. The result of the numerical solution was a 9-fold increase in the modeled disjoining pressure compared to that in the slit-pore model. The cylindrical models may apply to the chemical coating of the interiorwalls of cylindrical pores or to the thermodynamics within droplets after the breakup of a fluid coating a surface. However, the application used as the base case in this paperis the extension of transport and thermodynamic laws for porous media, previously developed with capillary pressure models, to fully saturated porous media with submicrometer-sized pores. As such, the models could apply to mass transport in ultrafiltration, nanofiltration, and reverse-osmosis membranes. C  2001 Academic Press Key Words: disjoining pressures;disjoining pressures in pores; porous media;membrane science. 1.0. INTRODUCTION 1.1. System Definition Interfacial interactions between fluid/fluid and fluid/solid boundaries have effects on transport phenomena (1–4). If the interfacial region is a curved surface between two fluids with a radius of curvature, r, one traditional way of quantifying the interaction force per interfacial area (or the energy per unit vol- ume) is the capillary pressure (see Eq. [1] below)(3). If the fluid is thin (with or without curved surfaces), the interaction force per interfacial area (energy per unit volume) may be quantified as the disjoining pressure (1, 5, 6). Disjoining pressures can be calculated for a film coating a solid surface (fluid bound by dis- similar phases) or for a continuous fluid between two similar phases (solid or another fluid). 1 To whom correspondence should be addressed. 2 Currently at SHOT, Inc., 7200 Highway 150, Greenville, IN 47124. A number of models exist for calculating disjoining pres- sures of fluids in flat geometries (1, 6); however, our work re- quired the calculation of the disjoining pressure of a liquid that is not flat but cylindrical in geometry. In this paper, we de- velop and evaluate three alternative models for calculating the disjoining pressures of a cylindrical fluid “plug” confined at its circumference by another phase. We envision that the mod- els may apply to the chemical coating of the interior walls of cylindrical pores or to thermodynamics within droplets after the breakup of a fluid coating a surface. However, for the purposes of this paper we will focus on applications to mass transfer in membranes with the following system of liquid-filled pores: an ultrafiltration polysulfone membrane in contact with an aque- ous sodium chloride (NaCl) solution. Specifically, the system is a 30-nm cylindrical pore within the ultrafiltration membrane that is completely filled with the sodium chloride solution. We will assume that the Na + ions in the bulk solutions are weak adsorbers onto the polysulfone pore wall, which has a surface charge resulting in an electrostatic surface potential of –18 mV. We have chosen this system for two reasons. First, it describes the base case (a single fluid confined in a cylindrical geome- try by another phase) from which the other more difficult cases of chemically coating a pore wall (7) or the breakup of a film into cylindrical/spherical droplets may be developed. Second, this system allows the use of disjoining pressure concepts for the extension of transport and thermodynamic laws in porous media (Darcy’s law, Kelvin’s law, etc.), previously developed with a capillary pressure model, to fully saturated porous media with submicrometer-sized pores. Nitao and Bear (3) theoreti- cally suggested this extension. 1.2. Background Thermodynamic Considerations Nitao and Bear considered the thermodynamics of transport of aqueous solutions in porous media (3). Their work conceptu- ally extended porous medium transport laws and matrix poten- tials to liquid-filled (fully saturated) submicrometer pores. It did not, however, give models for estimating the necessary thermo- dynamic terms for this extension. This current work proposes models for this extension in cylindrical pores. The following paragraphs summarize the portions of their article that have rel- evance to this current work. 230 0021-9797/01 $35.00 Copyright C  2001 by Academic Press All rights of reproduction in any form reserved.