Poisson-Boltzmann equation spherical

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

Interactions between cationic micelles and uni- and divalent anions have been treated quantitatively by solving the Poisson-Boltzmann equation in spherical symmetry and considering both Coulombic and specific attractive forces. Predicted rate-surfactant profiles are similar to those based on the ion-exchange and mass-action models (Section 3), but fit the data better for reactions in solutions containing divalent anions (Bunton, C. A. and Moffatt, J. R. (1985) J. Phys. Chem. 1985, 89, 4166 1986,90, 538). 

The spherically symmetric potential around a charged sphere is described by the Poisson-Boltzmann equation ... 

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),... 

Solve the Poisson-Boltzmann equation for a spherically symmetric double layer surrounding a particle of radius Rs to obtain Equation (38) for the potential distribution in the double layer. Note that the required boundary conditions in this case are at r = Rs, and p - 0 as r -> oo. (Hint Transform p(r) to a new function y(r) = r J/(r) before solving the LPB equation.)... 

We return to the solution of the Poisson-Boltzmann equation for a spherical particle, Equation (19), with B = 0 ... 

A new theory of electrolyte solutions is described. This theory is based on a Debye-Hiickel model and modified to allow for the mutual polarization of ions. From a general solution of the linearized Poisson-Boltzmann equation, an expression is derived for the activity coefficient of a central polarized ion in an ionic atmosphere of non-spherical symmetry that reduces to the Debye-Hiickel limiting laws at infinite dilution. A method for the simultaneous charging of an ion and its ionic cloud is developed to allow for ionic polarization. Comparison of the calculated activity coefficients with experimental values shows that the characteristic shapes of the log y vs. concentration curves are well represented by the theory up to moderately high concentrations. Some consequences in relation to the structure of electrolyte solutions are discussed. 

Similar calculations, based on the same principles, were carried out for spherical particles. Since the Poisson-Boltzmann equation cannot be integrated analytically in spherical symmetry, a numerical integration was performed. The computer-generated numerical tables of reduced potential as a function of reduced distance of Loeb, Wiersema, and... 

For a semiconductor microparticle, by employing a Poisson-Boltzmann equation of spherical symmetry with radius R, a potential drop is obtained as a function of r (distance from the center), as shown in Fig. 5.1(A). 2)... 

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

If the electrical potential is low ( 25 mV), a Poisson-Boltzmann equation can be approximated satisfactorily by a linear expression, which is more readily solvable. Various problems have been solved in the literature. These include, for example, planar [24], spherical [25], spheroidal [26], and arbitrary shaped particles [27]. The result for a linear Poisson-Boltzmann equation will not be discussed here. 

The potential distribution, and hence the extent of the band bending, within the space charge layer of a planar macroscopic electrode may be obtained by solution of the one-dimensional Poisson-Boltzmann equation [95]. However, since the particles may be assumed to have spherical geometry, the Poisson-Boltzmann for a sphere must be solved. This has been done by Albery and Bartlett [131] in a treatment that was recently extended by Liver and Nitzan [125]. For an n-type semiconductor particle of radius r0, the Poisson-Boltzmann equation for the case of spherical symmetry takes the form ... 

W.R. Bowen and A.O. Sharif, Adaptive finite element solution of the non-linear Poisson-Boltzmann equation—a charged spherical particle at various distances from a charged cylindrical pore in a charged planar surface, J. Colloid Interface Sci. 187 (1997)... 

To make headway with the colloidal problem, the Poisson-Boltzmann equation must be solved in spherical coordinates. -> Debye and -> Huckel [iv] introduced the following approximation into the spherical case,... 

Strauss et al. [28] has developed a numerical method for the nonlinear Poisson-Boltzmann equation 4 > 25 mV for this spherical particle in a spherical cell geometry. Figure 11.5 is a plot of the osmotic pressure for a suspension of identical particles with 100 mV surface potential and KU = 3.3. In this figure, the configurational osmotic pressure is also given and is much smaller than that of the osmotic pressure due to the double layer. The osmotic pressure increases with increased volume fraction due to the further overlap of the double layers sur-roimding each particle. 

Consider a spherical particle of radius a in a general electrolyte solution. The electric potential >j/ r) at position r obeys the following spherical Poisson-Boltzmann equation [3] ... 

When the magnimde of the surface potential is arbitrary so that the Debye-Hiickel hnearization cannot be allowed, we have to solve the original nonlinear spherical Poisson-Boltzmann equation (1.68). This equation has not been solved but its approximate analytic solutions have been derived [5-8]. Consider a sphere of radius a with a... 

Loeb et al. tabulated numerical computer solutions to the nonlinear spherical Poisson-Boltzmann equation (1.63). On the basis of their numerical tables, they discovered the following empirical formula for the cT-ij/o relationship ... 

The above method can be extended to the case of general electrolytes composed of N ionic species with valence Zi and bulk concentration (number density) n°° (/ = 1, 2,. . . , AO (in units of m ) [8]. The spherical Poisson-Boltzmann equation (1.68) can be rewritten in terms of the scaled potential y = e jj kT as... 

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... 

So far we have treated uniformly charged planar, spherical, or cylindrical particles. For general cases other than the above examples, it is not easy to solve analytically the Poisson-Boltzmann equation (1.5). In the following, we give an example in which one can derive approximate solutions. 

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. 

Consider a spherical soft particle consisting the particle core of radius a covered by an ion-penetrable layer of polyelectrolytes of thickness d. The outer radius of the particle is thus given hy b = a + d (Fig. 4.4). Within the surface layer, ionized groups of valence Z are distributed at a constant density N. The Poisson-Boltzmann equations (4.1) and (4.2) are replaced by the following spherical Poisson-Boltzmann equations for electric potential r being the distance from the center of the particle ... 

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]. 

For the inner region (avery small as compared with that in the outer region (bPoisson-Boltzmann equation ... 

Consider the interaction energy Vsp(H) between two spheres 1 and 2 of radii Uj and fl2 separated by a distance H between their surfaces (Fig. 12.1). The spherical Poisson-Boltzmann equation for the two interacting spheres has been not solved. If, however, the following conditions are satisfied. 

Numerical solution of the Poisson and Poisson-Boltzmann equations is more complicated since these are three dimensional partial differential equations, which in the latter case can be non-linear. Solutions in planar, cylindrical and spherical geometry, are... 

This series arises naturally, when expressing the Coulomb potential of a charge separated by a distance s from the origin in terms of spherical coordinates. The positive powers result when r < s, while for r > s the potential is described by the negative powers. Similarly the solutions of the linearized Poisson-Boltzmann equation are generated by the analogous expansion of the shielded Coulomb potential exp[fix]/r of a non-centered point charge. Now the expansion for r > s involves the modified spherical Bessel-functions fo (x), while lor r < s the functions are the same as for the unshielded Coulomb potential,... 

Wall, T.F., and Berkowitz, J. Numerical solution to the Poisson - Boltzmann equation for spherical polyelectrolyte molecules. Journal of Chemical Physics, 1957, 26, p. 114-122. 

In particular, the potential distribution within a spherical semiconductor particle could be derived [97] using a linearized Poisson-Boltzmann equation. As discussed in [69], two limiting cases are of special importance for light-induced electron transfer in semiconductor dispersions. For large particles, the total potential drop within the particle is... 

Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

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