The electrostatic interaction between two spheres of con-An explicit exact analytic expression for the energy of the elecsiderably different sizes can be approximated as the plate/ trostatic interaction between a platelike particle 1 and a spherical sphere interaction. The electrostatic interaction between a particle 2 of radius a 2 immersed in an electrolyte solution of Deplate and a sphere is thus of particular interest (17-22, 4, bye-Hu Β¨ckel parameter k is derived on the basis of the linearized 8). Interactions of this type are also quite often encountered
Poisson-Boltzmann equation. Both particles may have either conin practical situations, such as collection of colloidal partistant surface potential or constant surface charge density. In the cles at the surface of flat substrates. limit of ka 2 r 0, the interaction between a plate with zero surface Because of mathematical difficulties, however, it is not charge density and a sphere having constant surface charge deneasy to derive an interaction energy expression between a sity becomes identical to the usual image interaction between a point charge and an uncharged plate. The present theory thus plate and a sphere from the expressions obtained in our leads to a generalization of the usual image interaction of a point previous papers (6, 9, 10 1 ) for the interaction energy becharge to cover the case of a charged colloidal particle of finite tween two spheres of radii a 1 and a 2 by taking the limit size. α§ 1998 Academic Press a 2 r Ο±, since the spherical polar coordinate is used for both Key Words: sphere/surface interaction; electrostatic interaction; spheres. In the present paper, the cylindrical coordinate and image interaction.
the spherical coordinate are both used. We also elucidate how the image interaction of a charged colloidal particle with a plate is related to the usual image interaction of a BOTH AT CONSTANT SURFACE POTENTIAL tial energy of electrostatic interactions, one needs to solve the Poisson-Boltzmann equation for the electric potential 2.1. Linearized Poisson-Boltzmann Equation distribution in the system of interacting particles. Recently Consider a hard plate (of semi-infinite thickness) carrying we have shown (3-11) that the linearized Poisson-Boltza constant surface potential c 01 (plate 1) and a charged hard mann equation (the Debye-Hu Β¨ckel equation) can exactly sphere of radius a 2 carrying a constant surface potential c 02 be solved for two interacting charged spherical particles for (sphere 2), separated by a distance H between their surfaces, various types of boundary conditions at the surfaces of interimmersed in an electrolyte solution. We employ both a cylinacting particles. On the basis of the obtained potential distridrical coordinate system (s, f, z) and a spherical polar bution, we have derived explicit exact analytic expressions coordinate system (r, u, f), as shown in Fig. , in which for the interaction energy for these systems without recourse the origin of the cylindrical coordinate system, O 1 , is located to Derjaguin's approximation method (12). In the large ka at the surface of plate 1 and the origin of the spherical polar limit (k Γ
Debye-Hu Β¨ckel parameter and a Γ
particle racoordinate system is located at the center O 2 of sphere 2. dius), these exact expressions are found to agree with the Here z and r, respectively, represent the distance measured corresponding approximate formulas derived by Hogg et al.
from any point P to the surface of plate 1 and to the center (13) for the constant surface potential case, by Wiese and O 2 of sphere 2, s is the distance between the point P and Healy ( ) for the constant surface charge density case, and by Kar et al. (15) for the mixed case. We have also shown that the linearized Poisson-Boltzmann equation can exactly 1 In Eqs. [24] and [25] in Ref. (10), ka 2 (H / a 2 ) in the denominator in the square root should be ka 2 (H / a 1 ).
be solved for two interacting parallel charged cylinders (16). 42