JOURNAL OF COLLOID AND INTERFACE SCIENCE 187, 283–295 (1997) ARTICLE NO. CS964671 Calculations of Double-Layer Electrostatic Interactions for the Sphere/Plane Geometry P. WARSZYN ´ SKI AND Z. ADAMCZYK 1 Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek, 30-239 Krako ´w, Poland Received March 11, 1996; accepted October 31, 1996 interactions, numerous attempts to quantify them have been A numerical scheme for solving the nonlinear Poisson–Boltz- described in the literature, including the pioneering work of mann equation for the sphere/sphere and plane/sphere geome- Derjaguin (1) and Verwey and Overbeek (2) known as the tries has been developed. The method is based on an alternating DLVO theory. The force and energy of interactions between direction overrelaxation procedure using the Newton–Raphson plates and spherical particles were determined in the DLVO iteration to solve the nonlinear equation stemming from finite- theory using the simple Guy–Chapman–Stern (3–5) dou- difference discretization. The novelty of the algorithm consists in ble-layer model based on the continuous Poisson–Boltz- using the grid transforming functions that allow a more uniform mann (PB) equation in an uncorrected form. distribution of mesh points in the vicinity of the particle. The Later, many attempts were made to derive more general method was used to perform extensive calculations for opposite surface potentials of the interface and the particle immersed in a equations formulated on the basis of statistical mechanics symmetric electrolyte solution. The electric potential distribution (6–11). This led to complicated nonlinear integro-differen- (within and outside the sphere) was calculated, as well as the tial-difference equations whose solutions became prohibitive force and energy of interaction (from the integral of the force mathematically. More successful were the phenomenologi- over separation distance), for the constant potential, the constant cal theories based on the local thermodynamic balance (12– charge, and the mixed cases. The energy profiles calculated for 15) in which various corrections of the governing PB equa- various ka were compared with the analytical approximations tion were considered, e.g., the dielectric saturation effect derived using the Hogg–Healy–Fuerstenau method and the lin- ( 12, 15, 16 – 18 ) , the polarization effect ( 19 ) , the image and ear superposition approach (LSA). These calculations enabled self-atmosphere effect (19–22), medium compressibility us to conclude that the LSA can be used as a good estimate of (the electrostriction effect), and the discreteness of charge interaction energy for a broad range of ka values at distances (discrete ion) effect (20, 21). For a dilute electrolyte, ion greater than k 01 , i.e., for problems pertinent to colloid particle adsorption.  1997 Academic Press density fluctuations in the diffuse part of the double layer Key Words: colloid interactions; double layers; electrostatic in- pose an additional complicating factor (23). teractions; spherical double layers. On the other hand, at higher electrolyte concentrations, ion activity increases due to the excluded volume effect considered in (12–15, 24). It was shown that this effect, INTRODUCTION although significantly influencing potential distributions, plays a rather minor role in particle interactions except for Electrostatic interactions among charged surfaces in elec- very short separations of the order of 5–10 A ˚ (24). trolyte solutions determine the rate of many dynamic phe- It should be mentioned that many of the above corrections nomena occurring in disperse systems, e.g., aggregation, co- cancel each other in practical situations when other factors agulation, coalescence, flocculation, as well as interactions such as surface charge heterogeneity, roughness, and particu- with boundary surfaces leading to adsorption, adhesion, and larly surface deformations (neglected in the above theories) deposition. A quantitative description of these phenomena become more important. For this reason, in this work we has implications not only for polymer and colloid science, concentrate on the classical form of the PB equation for biophysics, and medicine but also for many modern technol- sphere / plane geometry. ogies involving various separation procedures, e.g., water Despite the considerable significance of this electrostatic and waste water filtration, membrane filtration, protein and problem (and the related problem of two spheres in electro- cell separation, and immobilization of enzymes. lyte solution), no closed-form analytical solutions have yet Due to the large significance of electrostatic double-layer been reported in the literature. The known solutions are of an approximate character usually based on the linearized 1 To whom correspondence should be addressed. form of the PB equation; hence they are applicable for low 283 0021-9797/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.