Hard plates, Interaction between

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. 

Comparison is made with the results for the two conventional models for hard plates given by Honig and Mul [11]. We see that the values of the interaction energy calculated on the basis of the Donnan potential regulation model lie between those calculated from the conventional interaction models (i.e., the constant surface potential model and the constant surface charge density model) and are close to the results obtained the linear superposition approximation. 

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

FIGURE 14.10 Reduced potential energy V = 167tSrSoU/Kg of the image interaction between a hard plate (plate 1) and a hard sphere (sphere 2) of radius az with 2 = 0 as a function of kH for several values of the reduced radius ku2 of sphere 2. Solid lines = 0 dashed lines Si = ex (plate 1 is a metal). From Ref [14]. 

One can also obtain the interaction energy between a cylinder and a hard plate, both having constant surface potential for the case where the cylinder axis is perpendicular to the plate surface (Fig. 14.12). The result is [14]... 

FIGURE 14.12 Interaction between a hard plate 1 and an infinitely long charged hard cylinder 2 of radius fl2 at a separation R. H =R 02) is the closest distance between their surfaces. 

Einally, we compare the image interactions between a hard cyhnder and a hard plate with the usual image interaction between a line charge and a plate by taking the limit of ku2 —> 0 for the case where the surface charge density of plate 1 is always zero (i/ oi = 0). In this limit, we have... 

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)). 

Before considering the interaction between two ion-penetrable membranes, we here treat the interaction between two similar ion-impenetrable hard plates 1 and 2 carrying surface charge density cr at separation h in a salt-free medium containing counterions only (Fig. 18.1) [2]. We take an x-axis perpendicular to the plates with its origin on the surface of plate 1. As a result of the symmetry of the system, we need consider only the region 0 < x < h 2. Let the average number density and the valence of counterions be o and z, respectively. Then we have from electroneutrality condition that... 

FIGURE 18.1 Schematic representation of the electrostatic interaction between two parallel identical hard plates separated by a distance h between their surfaces. 

The polymer density profile of ideal chains next to a hard sphere for arbitrary size ratio q was first ealeulated by Taniguchi et al. [125] and later independently by Eisenriegler et al. [126]. Eisenriegler also considered the pair interaction between two colloids for Rg< R [127] and for Rg R [128], as well as the interaction between a sphere and a flat wall due to ideal chains [129]. Depletion of excluded volume polymer chains at a wall and near a sphere was considered by Hanke et al. [130]. One of their results is that the ratio /Rg at a flat plate, which is 1.13 for ideal chains [118, 119], is slightly smaller (1.07) for excluded-volume chains. 

In this chapter we consider the depletion interaction between two flat plates and between two spherical colloidal particles for different depletants (polymers, small colloidal spheres, rods and plates). First of all we focus on the depletion interaction due to a somewhat hypothetical model depletant, the penetrable hard sphere (phs), to mimic a (ideal) polymer molecule. This model, implicitly introduced by Asakura and Oosawa [1] and considered in detail by Vrij [2], is characterized by the fact that the spheres freely overlap each other but act as hard spheres with diameter a when interacting with a wall or a colloidal particle. The thermodynamic properties of a system of hard spheres plus added penetrable hard spheres have been considered by Widom and Rowlinson [3] and provided much of the inspiration for the theory of phase behavior developed in Chap. 3. 

Fig. 2.25 Interaction potential between two hard plates due to small hard spheres (< = 0.1)...

In Chapter 11, we derived the double-layer interaction energy between two parallel plates with arbitrary surface potentials at large separations compared with the Debye length 1/k with the help of the linear superposition approximation. These results, which do not depend on the type of the double-layer interaction, can be applied both to the constant surface potential and to the constant surface charge density cases as well as their mixed case. In addition, the results obtained on the basis of the linear superposition approximation can be applied not only to hard particles but also to soft particles. We now apply Derjaguin s approximation to these results to obtain the sphere-sphere interaction energy, as shown below. 

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