Poisson-Boltzmann equation planar

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]... 

The next step is to determine the electrical charge and potential distribution in this diffuse region. This is done by using relevant electrostatic and statistical mechanical theories. For a charged planar surface, this problem was solved by Gouy (in 1910) and Chapman (in 1913) by solving the Poisson-Boltzmann equation, the so called Gouy-Chapman (G-C) model. 

The geometry. It is clear that the geometry of the system is much simplified in the slab model. Another possibility is to model the protein as a sphere and the stationary phase as a planar surface. For such systems, numerical solutions of the Poisson-Boltzmann equations are required [33]. However, by using the Equation 15.67 in combination with a Derjaguin approximation, it is possible to find an approximate expression for the interaction energy at the point where it has a minimum. The following expression is obtained [31] ... 

The theoretical inconsistencies inherent in the Poisson-Boltzmann equation were shown in Section 11.4 to vanish in the limit of very small potentials. It may also be shown that errors arising from this inconsistency will not be too serious under the conditions that prevail in many colloidal dispersions, even though the potential itself may no longer be small. Accordingly, we return to the Poisson-Boltzmann equation as it applies to a planar interface, Equation (29), to develop the Gouy-Chapman result without the limitations of the Debye-Hiickel approximation. 

Often this equation is referred to as the Poisson-Boltzmann equation. It is a partial differential equation of second order, which in most cases has to be solved numerically. Only for some simple geometries can it be solved analytically. One such geometry is a planar surface. 

For the simple case of a planar, infinitely extended planar surface, the potential cannot change in the y and z direction because of the symmetry and so the differential coefficients with respect to y and z must be zero. We are left with the Poisson-Boltzmann equation which contains only the coordinate normal to the plane x ... 

In the limit of an infinite micellar radius, i.e. a charged planar surface, the salt dependence of Ge is solely due to the entropy factor. A difficult question when applying Eq. (6.13) to the salt dependence of the CMC is if Debye-Hiickel correction factors should be included in the monomer activity. When Ge is obtained from a solution of the Poisson-Boltzmann equation in which the correlations between the mobile ions are neglected, it might be that the use of Debye-Hiickel activity factors give an unbalanced treatment. If the correlations between the mobile ions are not considered in the ionic atmosphere of the micelle they should not be included for the free ions in solution. 

In this Appendix, equations will be derived for the double layer interaction between two charged, planar surfaces in an electrolyte-free system. W e assume that the potential ip(x) obeys the Poisson—Boltzmann equation... 

Let us first calculate the free energy of interactions between two planar, nonundulating interfeces. The potential obeys the Poisson—Boltzmann equation... 

The corresponding modified Poisson-Boltzmann equations for the planar interface located at x=0 are ... 

A hierarchy of approximations now exists for calculating interactions between a charged particle and a charged, planar interface in electrolyte solutions. At moderate surface potentials less than approximately 2(kT/e the linear Poisson-Boltzmann equation provides a good approximation in many circumstances, provided the solution is a 1 1 electrolyte at low to moderate ionic strength. The relative simplicity of the linear equation makes it particularly useful for examining problems that are complicated in other ways, such as interactions involving many particles, interactions with deformable interfaces, and interactions where the detailed structure and properties of the particle (or macromolecule) play an important role. 

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)... 

We solve the planar Poisson-Boltzmann equation (1.20) subject to the boundary conditions ... 

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. 

PLANAR SOFT SURFACE 4.2.1 Poisson-Boltzmann Equation... 

For the inner region (avery small as compared with that in the outer region (bspherical Poisson-Boltzmann equation [6.139] with the following planar Poisson-Boltzmann equation ... 

We assume that the potential J/(x) in the region outside the plates obeys the one-dimensional planar Poisson-Boltzmann equation ... 

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 ... 

We start with the nonlinear planar Poisson-Boltzmann equation (see Eq. (9.74)),... 

Consider two parallel planar dissimilar ion-penetrable membranes 1 and 2 at separation h immersed in a solution containing a symmetrical electrolyte of valence z and bulk concentration n. We take an x-axis as shown in Fig. 13.2 [7-9]. We denote by Ni and Zi, respectively, the density and valence of charged groups in membrane 1 and by N2 and Z2 the corresponding quantities of membrane 2. Without loss of generality we may assume that Zj > 0 and Z2 may be either positive or negative and that Eq. (13.1) holds. The Poisson-Boltzmann equations (13.2)-(13.4) for the potential distribution j/(x) are rewritten in terms of the scaled potential y = zeif/IkT as... 

Consider two parallel planar ion-penetrable membranes 1 and 2, which may not be identical, at separation h in a symmetrical electrolyte solution of valence z and bulk concentration n (Fig. 16.1). We take an x-axis perpendicular to the membranes with its origin at the surface of membrane 1. The electric potential i/ (x) at position X between the membranes (relative to the bulk solution phase, where is set equal to zero) is assumed to be small so that the linearized Poisson-Boltzmann equation can be employed. Membranes 1 and 2, respectively, consist of N and M layers. All the layers are perpendicular to the x-axis. Let the thickness and the density of membrane-fixed charges of the ith layer of membrane j (7=1, 2) be and The linearized Poisson-Boltzmann equation for the /th layer... 

The sign reversal takes place also in the electrophoretic mobility of a non-uniformly charged soft particles, as shown in this section. We treat a large soft particle. The x-axis is taken to be perpendicular to the soft surface with its origin at the front edge of the surface layer (Fig. 21.8). The soft surface consists of the outer layer —d < x < 0) and the inner layer (x < —d), where the inner layer is sufficiently thick so that the inner layer can be considered practically to be infinitely thick. The liquid flow m(x) and equilibrium electric potential i//(x) satisfy the following planar Navier-Stokes equations and the Poisson-Boltzmann equations [39] ... 

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... 

The solution of the Poisson-Boltzmann equation for this case has been known even longer than for the cylinder ( 1 1 ) from it we can calculate the behavior of the mobile ions as their average concentration approaches zero. Let Q be the amount of fixed charge per unit area on the solid planar surface, and let 1 be the so-called Bjerrum length defined by... 

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