Repulsion between charged spheres

Given the case of single spheres, the absence of closed-form solutions for the two-sphere problem comes as no surprise. Moreover, because of curvature, the repulsive force derives from both osmotic pressure and an electric stress. We can use the general relation, equation 2.63, along with the equilibrium condition, to show that the repulsive force can be obtained from an integration over the central plane as 

A variety of approximations provides insight into the qualitative and quantitative behavior. The Derjaguin approximation, for example, is applicable for separations small compared with the radius of the spheres (Derjaguin, 1934). Under such conditions, elements on each sphere interact as parallel plane elements at the same separation the total interaction is a sum over the infinitesimal elements. To proceed formally, we adopt a polar cylindrical coordinate system with its axis joining the centers of the spheres of radius a, separated by the distance h and centered at midpoint. r is the interparticular distance on the midplane and z the distance on the center to center axis. A sphere surface is defined by  

The problem has been reduced to one dimension but further analytical progress requires linearization of the differential equation, i.e. small potentials. The force derived from equation 2.77, 

Extensions of the analytical solution show that the error for constant potential boundary conditions remains small as the gap is diminished. Conversely, the terms neglected in the constant charge calculation grow without bound, showing that this approximation is invalid when the gap is much smaller than the Debye thickness. This problem stems from the radial gradients in the potential neglected in equation 2.80. 

Linear superposition of single sphere potentials also provides a useful approximation for the repulsive force. From the Debye-Hckel solution around a single sphere, equation 2.62, we find 

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Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

A. B. Glendinning and W. B. Russel,/. Colloid Interface Sci., 93,95 (1983). The Electrostatic Repulsion between Charged Spheres from Exact Solutions to the Linearized Poisson-Boltzmann Equation. 

A number of refinements and applications are in the literature. Corrections may be made for discreteness of charge [36] or the excluded volume of the hydrated ions [19, 37]. The effects of surface roughness on the electrical double layer have been treated by several groups [38-41] by means of perturbative expansions and numerical analysis. Several geometries have been treated, including two eccentric spheres such as found in encapsulated proteins or drugs [42], and biconcave disks with elastic membranes to model red blood cells [43]. The double-layer repulsion between two spheres has been a topic of much attention due to its importance in colloidal stability. A new numeri-... 

Fig. 1.1 Schematic plot of a typical double layer repulsion between charged colloidal spheres (top), of the Van der Waals attraction (bottom) and their sum, which is the DLVO interaction potential...

Wlien describing the interactions between two charged flat plates in an electrolyte solution, equation (C2.6.6) cannot be solved analytically, so in the general case a numerical solution will have to be used. Several equations are available, however, to describe the behaviour in a number of limiting cases (see [41] for a detailed discussion). Here we present two limiting cases for the interactions between two charged spheres, surrounded by their counterions and added electrolyte, which will be referred to in further sections. This pair interaction is always repulsive in the theory discussed here. 

In an aqueous system with large particles weU-separated by a distance, s, (D and 5 > t ) the electrostatic repulsion energy between two identical charged spheres may be approximated (1) ... 

Many simple sehemes have been put forward for these repulsion integrals, which are usually written. They are taken to depend on the type of atoms that basis funetion x, and Xj are centred on, and on the distance between the atomic centres. Pariser and Parr made use of the uniformly charged sphere representation illustrated in Figure 8.1. 

This value of kn is actually low by an order of magnitude for dilute suspensions of charged spheres of radius Rg. This is due to the neglect of interchain correlations for c < c in the structure factor used in the derivation of Eqs. (295)-(298). If the repulsive interaction between polyelectrolyte chains dominates, as expected in salt-free solutions, the virial expansion for viscosity may be valid over considerable range of concentrations where the average distance between chains scales as. This virial series may be approxi-... 

The Huggins coefficient kn is of order unity for neutral chains and for polyelectrolyte chains at high salt concentrations. In low salt concentrations, the value of kn is expected to be an order of magnitude larger, due to the strong Coulomb repulsion between two polyelectrolyte chains, as seen in the case of colloidal solutions of charged spheres. While it is in principle possible to calculate the leading virial coefficients in Eq. (332) for different salt concentrations, the essential feature of the concentration dependence of t can be approximated by... 

The first phenomenon, to be discussed in 2.2, concerns the saturation of the force of repulsion between two symmetrically charged bodies (particles) in an electrolyte solution as their charge increases. This effect is a direct consequence of the saturation of the electric field at a finite distance from the surfaces of the bodies and of the field properties at infinity. In the one-dimensional case (for parallel plates) the relevant features follow from a direct computation (see, e.g., [9]). In 2.2 the corresponding effect will be discussed for parallel cylinders and spheres [10]. 

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