Interaction at Constant Surface Charge Density

Here i/ o is the actual surface potential of particle i (i.e., the surface potential at the final stage) and may differ at different points on the particle surfaces. 

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The first term is the Verwey—Overbeek expression for the interactions at constant surface charge density and the second accounts for the variation of the surface charge with the separation distance. 

The form of the Helmholtz free energy F depends on the type of the origin of surface charges on the interacting particles. The following two types of interaction, that is (i) interaction at constant surface charge density and (ii) interaction at constants surface potential are most frequently considered. We denote the free energy F for the constant surface potential case by F and that for the constant surface... 

ARBITRARY POTENTIAL INTERACTION AT CONSTANT SURFACE CHARGE DENSITY... 

The interaction energy between spheres at constant surface potential involves only the function G (0 (which depends only on the sphere radius a,), while the interaction at constant surface charge density is characterized by the function H (i) (which depends on both sphere radius u, and relative permittivity Ep,). The interaction energy in the mixed case involves both G (i) and // (/) ... 

The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions). It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used. 

FIGURE 9.2 Potential distribution i/ (x) across two interacting plates 1 and 2 at constant surface charge density a calculated with Eqs. (9.31)-(9.33) for Kh—1 (upper) and 2 (lower). The dotted lines, which stand for unperturbed potentials, correspond to the case of infinite separation (k/i = oo). 

For the interaction between two hard spheres 1 and 2 having radii ai and a2 for the case where sphere 1 is maintained at constant surface potential J/i and sphere 2 at constant surface charge density [Pg.330]

It can be shown that the interaction energy V°(R) per unit length between cylinder 1 (of relative permittivity pi) and cylinder 2 (of relative permittivity p2) at constant surface charge density is obtained by the interchange G (0 Hn i) with the result that... 

Equations (15.49) and (15.50), respectively, agrees with the expression for the electrostatic interaction energy between two parallel hard plates at constant surface charge density and that for two hard spheres at constant surface charge density [4] (Eqs. (10.54) and (10.55)). 

The van der Waals part is of electromagnetic origin and is, therefore, virtually instantaneous De(vdW) 0. However, for the electrostatic part this is not so obvious. Verwey and Overbeek [1] discussed the difference between interaction at constant (smface) charge (density), (t°, and at constant (surface) potential, °. Both cases lead to repulsion, although for different reasons. When cr° remains fixed upon reduction of h, the potential increases. The work to be done is of a purely electrostatic nature. However, when il/° remains constant, the charge should diminish upon approach. Now work has... 

The potential energy V"(h) of the double-layer interaction per unit area between two parallel plates 1 and 2 with constant surface charge density a at separation h is given by... 

We calculate the potential energy of the double-layer interaction per unit area between two parallel dissimilar plates 1 and 2 at separation h carrying constant surface charge densities cti and <12, as shown in Fig. 10.1[8]. Equation (9.116) for the interaction energy F (h) is generalized to cover the interaction between two parallel dissimilar plates as... 

FIGURE 11.4 Scaled double-layer interaction energy = K/64nkT)V h) per unit area between two parallel similar plates as a function of scaled separation Kh at the scaled unperturbed surface potential >>o = 1. 2, and 5 calculated with Eq. (11.14) (dotted lines) in comparison with the exact results under constant surface potential (curves 1) and constant surface charge density (curves 2). From Ref. [5]. 

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