Poisson-Boltzmann equation mathematics

If there are ions in the solution, they will try to change their location according to the electrostatic potential in the system. Their distribution can be described according to Boltzmarm. Including these effects and applying some mathematics leads to the final linearized Poisson-Boltzmann equation (Eq. (43)). 

The Poisson-Boltzmann equation is a modification of the Poisson equation. It has an additional term describing the solvent charge separation and can also be viewed mathematically as a generalization of Debye-Huckel theory. 

Oosawa (1971) developed a simple mathematical model, using an approximate treatment, to describe the distribution of counterions. We shall use it here as it offers a clear qualitative description of the phenomenon, uncluttered by heavy mathematics associated with the Poisson-Boltzmann equation. Oosawa assumed that there were two phases, one occupied by the polyions, and the other external to them. He also assumed that each contained a uniform distribution of counterions. This is an approximation to the situation where distribution is governed by the Poisson distribution (Atkins, 1978). If the proportion of site-bound ions is negligible, the distribution of counterions between these phases is then given by the Boltzmann distribution, which relates the population ratio of two groups of atoms or ions to the energy difference between them. Thus, for monovalent counterions... 

These differences between intrinsic and doped, or impurity, semiconductors complicate the mathematics of the solution of the Poisson-Boltzmann equation, but the picture that emerges remains basically the same A charged cloud, or space charge, and therefore a potential drop, develops inside the semiconductor the space charge contributes to the capacity of the interphase, etc. 

The solution of the linearized Poisson-Boltzmann equation around cylinders also requires numerical methods, although when cylindrical symmetry and the Debye-Hiickel approximation are assumed the equation can be solved. The solution, however, requires advanced mathematical techniques and we will not discuss it here. It is nevertheless useful to note the form of the solution. The potential for symmetrical electrolytes has been given by Dube (1943) and is written in terms of the charge density a as... 

Equation (1.68) is the sought analogue of the Poisson-Boltzmann equation. We know of no mathematical studies of (1.68) in the context of known or anticipated effects of finite ionic size. 

K. Tintarev, Fundamental solution of the Poisson-Boltzmann equation, in Differential Equations and Mathematical Physics, I. W. Knowles and Y. Saito, eds., Lecture Notes in Math. 1285, Springer-Verlag, Berlin, New York, 1987. 

In this chapter, mathematical procedures for the estimation of the electrical interactions between particles covered by an ion-penetrable membrane immersed in a general electrolyte solution is introduced. The treatment is similar to that for rigid particles, except that fixed charges are distributed over a finite volume in space, rather than over a rigid surface. This introduces some complexities. Several approximate methods for the resolution of the Poisson-Boltzmann equation are discussed. The basic thermodynamic properties of an electrical double layer, including Helmholtz free energy, amount of ion adsorption, and entropy are then estimated on the basis of the results obtained, followed by the evaluation of the critical coagulation concentration of counterions and the stability ratio of the system under consideration. 

Without going into much mathematical detail, the capacity of the space charge region is derived by solving the Poisson-Boltzmann equation as... 

The limitations imposed on DDL theory as a molecular model by these four basic assumptions have been discussed frequently and remain the subject of current research.In Secs. 1.4 and 3.4 it is shown that DDL theory provides a useful framework in which to interpret negative adsorption and electrokinetic experiments on soil clay particles. This fact suggests that the several differences between DDL theory and an exact statistical mechanical description of the behavior of ion swarms near soil particle surfaces must compensate one another in some way, at least in certain applications. Evidence supporting this conclusion is considered at the end of the present section, whose principal objective is to trace out the broad implications of Eq. 5.1 as a theory of the interfacial region. The approach taken serves to develop an appreciation of the limitations of DDL theory that emerge from the mathematical structure of the Poisson-Boltzmann equation and from the requirement that its solutions be self-consistent in their physical interpretation. TTie limitations of DDL theory presented in this way lead naturally to the concept of surface complexation. 

The popular Poisson-Boltzmann equation considers the mean electrostatic potential in a continuous dielectric with point charges and is therefore, an approximation of the actual potential. An improved model and mathematical solution resulted in the MPB equation (26). This equation is based on a restricted primitive electrolyte model that considers ions as charged hard spheres with diameter d in a continuous uniform structureless dielectric medium of constant dielectric permittivity s. The sphere representing an ion has the same permittivity e. The model initially was developed for an electrolyte at a hard wall with dielectric permittivity and surface charge density a. The charge is distributed over the surface evenly and continuously. 

This layer, which is termed the diffuse electrical double layer, can be described mathematically by the Poisson-Boltzmann equation. Within this layer, the shear plane of the particles is located. The potential at this distance from the surface is particularly important as it is the experimentally accessible zeta-potential. When two different colloidal particles that are electrically charged at their surfaces with ions of the same sign approach each other, they wiU experience a net repulsion force as a result of the interaction between the ions located at their diffuse layers. If the net interaction potential between the particles is repulsive and larger than the kinetic energy of the collision, they wiU not coagulate. 

There are different mathematical models of the EDL which reflect the possible variety of its inner stmcture at different levels of sophistication (Hunter 1993, pp. 379). All of them refer to the approach of Gouy (1909) und Chapman (1913) which neglects the existence of the Stern layer and solely accounts for a diffuse layer in immediate contact with the surface. The potential distribution l/(r) inside the diffuse layer is defined by the Poisson-Boltzmann-equation (PBE) ... 

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

From the linearized Poisson-Boltzmann equation the work, -(Vyg)V8 7teQe k/l + kuy), due to ion screening is mathematically always less than zero, which restricts the activity coefficient calculated by Eq. (63) to a range from zero to unity. 

Solutions to the Poisson—Boltzmann equation in which the exponential charge distribution around a solute ion is not linearized [15] have shown additional terms, some of which are positive in value, not present in the linear Poisson—Boltzmann equation [28, 29]. From the form of Eq. (62) one can see that whenever the work, q yfy - yfy), of creating the electrostatic screening potential around an ion becomes positive, values in excess of unity are possible for the activity coefficient. Other methods that have been developed to extend the applicable concentration range of the Debye—Hiickel theory include mathematical modifications of the Debye—Hiickel equation [15, 26, 28, 29] and treating solution complexities such as (1) ionic association as proposed by Bjerrum [15,25], and(2) quadrupole and second-order dipole effects estimated by Onsager and Samaras [30], etc. 

Since ionic association is an electrostatic effect for equilibrium properties of electrolyte solutions, it may be included in the Debye-Hiickel type of treatment by explicitly retaining further terms in the expansion of the Poisson-Boltzmann relation eqn. 5.2.8. - A similar calculation was attempted for conductance by Fuoss and Onsager. The mathematical approach and the model employed are similar to those used in their previous calculation, but they keep explicitly the exp (—0 y) term in the new calculation. The equation derived for A is... 

So far, all governing equations are presented. It can be seen that a closed mathematical model of the combined pressure-driven flow and electroosmotic flow in microfluidic channels should include the continuity equarirm, the Stokes equation, the Poisson equation, the Boltzmann distribution, and the Laplace equation. A set of the governing equations for such combined pressure-driven flow and electroosmotic flow can be rewritten in dimensional form as... 

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