Interaction Between Two Parallel Dissimilar Plates

Consider two parallel dissimilar plates 1 and 2 having thicknesses d and d2, respectively, separated by a distance h immersed in a liquid containing N ionic species with valence z, and bulk concentration (number density) (/ = 1, 2,. . . , N). We take an x-axis perpendicular to the plates with its origin at the right surface of plate 1, as in Fig. 10.1. We assume that the electric potential j/ x) outside the plates (—oo x —d, 0 x h, and h + d2 x oo) obeys the following one-dimensional planar Poisson-Boltzmann equation  

Biophysical Chemistry of Biointerfaces By Hiroyuki Ohshima Copyright 2010 by John Wiley Sons, Inc. 

ELECTROSTATIC INTERACTION BETWEEN TWO PARALLEL DISSIMILAR PLATES 

FIGURE 10.1 Schematic representation of the potential distribution (soUd line) across two interacting parallel dissimilar plates 1 and 2 at separation h. Dotted line is the unperturbed potential distribution ath = oo. j/oi and i/to2 are the unperturbed surface potentials of plate 1 and 2, respectively. 

The boundary conditions at the plate surface depends on the type of the double--layer interaction between plates 1 and 2. If the surface potentials of the plates 1 and 2 remain constant at i/ oi i//q2 during interaction respectively, then 

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ELECTROSTATIC INTERACTION BETWEEN TWO PARALLEL DISSIMILAR PLATES then P is attractive for all xh and has a minimum PV,... 

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... 

The method for obtaining an analytic expression for the interaction energy at constant surface potential given in Chapter 9 (see Section 9.6) can be applied to the interaction between two parallel dissimilar plates. The results are given below [9]. 

In this chapter, we give exact expressions and various approximate expressions for the force and potential energy of the electrical double-layer interaction between two parallel similar plates. Expressions for the double-layer interaction between two parallel plates are important not only for the interaction between plate-like particles but also for the interaction between two spheres or two cylinders, because the double-interaction between two spheres or two cylinders can be approximately calculated from the corresponding interaction between two parallel plates via Deijaguin s approximation, as shown in Chapter 12. We will discuss the case of two parallel dissimilar plates in Chapter 10. 

In this chapter, we discuss two models for the electrostatic interaction between two parallel dissimilar hard plates, that is, the constant surface charge density model and the surface potential model. We start with the low potential case and then we treat with the case of arbitrary potential. 

FIGURE 14.1 Interaction between two parallel dissimilar hard plates 1 and 2 at separation h. 

Similarly, the interaction energy V(h) between two parallel dissimilar plates 1 and 2 per unit area in a vacuum, which have molecular densities N and N2, London-van der Waals constant Cj and C2, and thicknesses d and 2. respectively, can be obtained from Eq. (4.11)... 

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