Interaction Energy of Particles in Porous Media: New Deryaguin Type Approximation

Abstract

In transport through porous media (e.g. membrane separations,1,2 chromatography, etc.), the passage of macromolecules and other colloids is greatly influenced by the long-ranged particle-pore wall interactions.1-7 Solute (macromolecule) adsorption,8 partitioning, and rejection,1 as well as solvent flux,1 are governed by a balance of at least three particle-pore surface forces:6,7 (a) the apolar Lifshitz-van der Waals (LW) interactions,3,4,6,7 (b) the polar (P) interaction described variously as hydration pressure, hydrogen bonding, hydrophobic interaction, or acid-base interaction in aqueous systems,6,7 and (c) electrostatic (electrical double layer) interaction.5 Theoretical expressions for the LW interaction energy of an atom or a simple spherical molecule,8 as well as for a spherical particle,3,4 confined to a cylindrical pore, have been obtained by numerical summation of pairwise interactions. However, the rigorous expressions for the other components of the energy have not been obtained for the particle-pore geometry. Thus, in ultrafiltration of colloids and macromolecules, the sphere-pore interactions are often modeled as interactions between a sphere and a flat plate, which is a poor approximation for small (<20 nm) pores.4 There is no direct method to measure or to predict the magnitude of the energy (force) of interaction between a colloidal particle and thewall of anarrow cylindrical pore. Measurements of the interaction and adhesion energies per unit area between two plane parallel half spaces are, however, more facile on the basis of contact angle goniometry,7 force balance analysis, and atomic force microscopy. Expressions for the interaction energy of parallel plates are also known. Over half a century ago, Deryaguin9 showed that the interaction between two spheres can be obtained to a very good approximation from the knowledge of the corresponding interaction energy per unit area of plane parallel plates. The celebrated Deryaguin approximation (DA) and its generalization10 for arbitrary curvature and orientation of particles are now standard textbook lores.11 For the Deryaguin approximation to work well, the most significant contribution to the interaction energy should come from the immediate neighborhood of the points of closest approach between the curved surfaces.9-11 The approximation thereforeworksbest for convex surfaces (e.g., spheres, ellipsoids) which recede away from each other rather rapidly from around the point of closest approach. Clearly, the case of a particle in an enclosed (concave) space (e.g., cylindrical pore) does not conform to this picture, as the intersurfacedistance increasesmore slowly away from the point of closest approach, especially when the particle gets progressively closer to the pore center. The entire particle surface may thus contribute significantly to the interaction energy. This note has the following twin objectives: (a) to assess the accuracy of the generalized Deryaguin approximation (DA) for spherical particles in cylindrical poresand (b) todevelopanaccurate new surface element integration (SEI) approach for evaluating the particle-pore interactions from the information on the interaction energy of flat plates. Figure 1 shows the schematic of a spherical particle in a long cylindrical pore. The generalized Deryaguin approximation gives the energy of interaction, ED as10,11