Interaction Between Two Parallel Similar Plates

Double-Layer Interaction Between Two Parallel Similar Plates 

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. 

Consider two parallel similar plates 1 and 2 of thickness d separated by a distance h immersed in a liquid containing N ionic species with valence zt and bulk concentration (number density) nf i=, 2,. . . , N). Without loss of generality, we may assume that plates 1 and 2 are positively charged. We take an x-axis perpendicular to the plates with its origin at the right surface of plate 1, as in Fig. 9.1. From the symmetry of the system we need consider only the region —oo x h/2. We assume that the electric potential i/ (x) outside the plate (—oo x —d and 0 x hl2) obeys the following one-dimensional planar Poisson-Boltzmann equation  

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

FIGURE 9.1 Schematic representation of the douhle-layer interaction between two parallel plates 1 and 2 separated by h. ij/ is the potential at the midpoint between the plates. 

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DOUBLE-LAYER INTERACTION BETWEEN TWO PARALLEL SIMILAR PLATES... 

Honig and Mul [7] suggested that better approximations can be obtained if the interaction energy is expressed as a series of tanh(zei/ c/fcT) instead of i/ o and derived the interaction energy correct to tsDSi ze j/ JkT). The interaction energy V (h) per unit area correct to tanh zeij/o/kT) for the interaction between two parallel similar plates at constant surface potential separated by a distance in a symmetrical... 

We treat the interaction between two parallel similar plates placed at x = 0 and h (Fig. 9.6) (i.e., the plates are at separation h) for the case where i/tq remains constant during interaction. The boundary conditions are... 

DOUBLE-LAYER INTERACTION BETWEEN TWO PARALLEL SIMILAR PLATES Substituting Eqs. (9.193) and (9.195) into Eq. (9.196), we obtain... 

It is possible to derive the interaction energy between particles with multilayer structures [4,10,11]. Consider the interaction between two parallel similar plates separated by a distance h (Fig. 19.18). Let the thicknesses of the plate core and the... 

H. Ohshima, Colloids Surf. A, 146, 213 (1999). Approximate Expression for the Potential Energy of Double-Layer Interaction between Two Parallel Similar Plates with Constant Surface Potential. 

By substituting Eq. (9.63) into Eq. (9.27), we obtain the following expression P (h) for the interaction force at constant potential per unit area between two parallel similar plates 1 and 2 at separation h ... 

In this section, we present a novel linearization method for simplifying the nonlinear Poisson-Boltzmann equation to derive an accurate analytic expression for the interaction energy between two parallel similar plates in a symmetrical electrolyte solution [13, 14]. This method is different from the usual linearization method (i.e., the Debye-Hiickel linearization approximation) in that the Poisson-Boltzmann equation in this method is linearized with respect to the deviation of the electric potential from the surface potential so that this approximation is good for small particle separations, while in the usual method, linearization is made with respect to the potential itself so that this approximation is good for low potentials. 

This is the required expression for the interaction energy per unit area between two parallel similar plates with constant surface potential. Erom the nature of this linearization, the obtained potential distribution (9.177) is only accurate near the plate surface and thus the interaction energy expression (9.177) is also only accurate for small plate separations. Indeed, at h = 0, Eq. (9.177) gives the following correct expression, regardless of the value of yo,... 

FIGURE 9.7 Reduced potential energy V = K/64nkT)V of the double-layer interaction per unit area between two parallel similar plates with constant surface potential as a function of the reduced distance Kh between the plates for several values of the scaled unperturbed surface potential ya = ze j/olkT. Solid lines are exact values and dotted lines represent approximate results calculated by Eq. (9.177). The exact and approximate results for To = 1 agree with each other within the linewidth. (From Ref. 13.)... 

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]. 

The interaction force between two parallel similar plates per unit area Ppi(h) is given by Eq. (13.96), namely,... 

The interaction energy V h) per unit area between two parallel similar plates of thickness d separated by distance h between their surfaces in a vacuum (Fig. 19.4) can be obtained by integrating Eq. (19.11)... 

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)... 

Figure 11.5 Interaction energy per unit area versus distance between two similar parallel plates coated with grafted polymer in a good solvent. Forthe calculation, we used a segment length /s= 1.2 nm and a chain length of N = 80 segments. At low grafting density...

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