Potential distribution around

For the problem of potential distribution around a point charge (here the central ion) it is convenient to use the spherical system of coordinates, where in this case all parameters depend only on the distance r from the central ion, regardless of the direction. 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

Debye-Huckel approximation — In calculating the potential distribution around a charge in a solution of a strong -> electrolyte, - Debye and -> Hiickel made the assumption that the electrical energy is small compared to the thermal energy ( zjei (kT), and they solved the -> Poisson-Boltzmann equation V2f = - jT- gc° eexp( y) by expanding the exponential... 

Forx a, Eq. (1.73) reduces to the potential distribution around the planar surface given by Eq. (1.19). This result implies that in the region very near the particle surface, the surface curvature may be neglected so that the surface can be regarded as planar. In the limit of k 0, Eq. (1.72) becomes... 

Potential Distribution Around a Sphere with Arbitrary Potential... 

By using an approximation method similar to the above method and the method of White [6], one can derive an accurate analytic expression for the potential distribution around a spherical particle. Consider a sphere of radius u in a symmetrical electrolyte solution of valence z and bulk concentration n[7]. The spherical Poisson-Boltzmann equation (1.68) in this case becomes... 

We obtain the potential distribution around a sphere of radius a having a surface potential i/ o immersed in a solution of general electrolytes [9]. The Poisson-Boltzmann equation for the electric potential i//(r) is given by Eq. (1.94), which, in terms of/(r), is rewritten as... 

Equation (1.176) gives the general expression for the potential distribution around a cylinder. For the special case of a cylinder in a 1-1 electrolyte, in which case F(y) is given by Eq. (1.134), we obtain Eq. (1.157) with z=l. For other types of electrolytes, one can calculate by using Eq. (1.176) with the help of the corresponding expression for F(y). 

Consider here the asymptotic behavior of the potential distribution around a particle (plate, sphere, or cylinder) at large distances, which will also be used for calculating the electrostatic interaction between two particles. 

We give below a simple method to derive an approximate solution to the hnear-ized Poisson-Boltzmann equation (1.9) for the potential distribution i/ (r) around a nearly spherical spheroidal particle immersed in an electrolyte solution [12]. This method is based on Maxwell s method [13] to derive an approximate solution to the Laplace equation for the potential distribution around a nearly spherical particle. 

Potential Distribution Around a Nonuniformly Charged Surface and Discrete Charge Effects... 

POTENTIAL DISTRIBUTION AROUND A NONUNIFORMLY CHARGED SURFACE... 

We calculate the self-atmosphere potential of an ion near the planar surface. Imagine a planar uncharged plate in contact with a solution of general electrolytes composed of N ionic mobile species of valence z, and bulk concentration (number density) nf (i=l,2,. . . , N) (Fig. 3.10). Let and p, respectively, be the relative permittivities of the electrolyte solution and the plate. Consider the potential distribution around an ion. By symmetry, we use a cylindrical coordinate system r(s, x) with its origin 0 at the plate surface and the x-axis perpendicular to the surface. 

In Chapter 1, we have discussed the potential and charge of hard particles, which colloidal particles play a fundamental role in their interfacial electric phenomena such as electrostatic interaction between them and their motion in an electric field [1 ]. In this chapter, we focus on the case where the particle core is covered by an ion-penetrable surface layer of polyelectrolytes, which we term a surface charge layer (or, simply, a surface layer). Polyelectrolyte-coated particles are often called soft particles [3-16]. It is shown that the Donnan potential plays an important role in determining the potential distribution across a surface charge layer. Soft particles serve as a model for biocolloids such as cells. In such cases, the electrical double layer is formed not only outside but also inside the surface charge layer Figure 4.1 shows schematic representation of ion and potential distributions around a hard surface (Fig. 4.1a) and a soft surface (Fig. 4.1b). 

FIGURE 4.1 Ion and potential distribution around a hard surface (a) and a soft surface (b). When the surface layer is thick, the potential deep inside the surface layer becomes the Donnan potential. 

The potential distribution outside the surface charge layer of a soft particle with surface potential j/g is the same as the potential distribution around a hard particle with a surface potential xj/g. The asymptotic behavior of the potential distribution around a soft particle and that for a hard particle are the same provided they have the same surface potential xj/o- The effective surface potential is an important quantity that determines the asymptotic behaviors of the electrostatic interaction between soft particles (see Chapter 15). 

The Poisson-Boltzmann equation for the potential distribution around a cylindrical particle without recourse to the above two assumptions for the limiting case of completely salt-free suspensions containing only particles and their counterions was solved analytically by Fuoss et al. [1] and Afrey et al. [2]. As for a spherical particle, although the exact analytic solution was not derived, Imai and Oosawa [3,4] smdied the analytic properties of the Poisson-Boltzmann equation for dilute particle suspensions. The Poisson-Boltzmann equation for a salt-free suspension has recently been numerically solved [5-8]. 

In this chapter, we first discuss the case of completely salt-free suspensions of spheres and cylinders. Then, we consider the Poisson-Boltzmann equation for the potential distribution around a spherical colloidal particle in a medium containing its counterions and a small amount of added salts [8]. We also deals with a soft particle in a salt-free medium [9]. 

That is, Teff = 2eff- Th n the potential distribution around the particle except the region very near the particle surface is approximately given by... 

We derive some approximate solutions to Eq. (6.93). It can be shown that as in the case of completely salt-free case, there are three possible regions I, II, and III in the potential distribution around the particle surface for the dilute case. As will be seen later, region I (where y(r) y(R)) is found to be much wider than regions II and III, and in region III the potential is very high so that there are essentially no colons. We may thus approximate Eq. (6.91) as... 

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