Nonlinear Gouy-Chapman solution

Equation [93] gives the Debye-Hiickel potential, based on an apparent surface charge density, which matches the nonlinear Gouy-Chapman solution far from a charged planar surface. The ADH//NLDH potential for a sphere is easily found from Eqs. [188], [187], and [300] ... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

An important feature of the biogeochemistry of trace elements in the rhizosphere is the interaction between plant root surfaces and the ions in the soil solution. These ions may accumulate in the aqueous phases of cell surfaces external to the plasma membranes (PMs). In addition, ions may bind to cell wall (CW) components or to the PM surface with variable strength. In this chapter, we shall describe the distribution of ions among the extracellular phases using electrostatic models (i.e. Gouy-Chapman-Stem and Donnan-plus-binding models) for which parameters are now available. Many plant responses to ions correlate well with computed PM-surface activities, but only poorly with activities in the soil solution. These responses include ion uptake, ion-induced intoxication, and the alleviation of intoxication by other ions. We illustrate our technique for the quantitative resolution of multiple ion effects by inserting cell-surface activities into nonlinear equations. 

Coimterion condensation has detractors (28-34), who point to flaws in the concept s derivation, such as artificial subdivision of the counterions into two populations, inappropriate extrapolation of the Debye-Hiickel approximation to regions of high electrostatic potential, and inconsistent treatment of counterions. The full nonlinear Poisson-Boltzmann equation offers a more rigorous way to interpret electrostatic phenomena in electrolyte solutions, but the physical picture obtained through this equation is different in some ways from the one suggested by condensation (21,34,35). In particular, a Poisson-Boltzmann analysis does not readily identify distinct populations of condensed and free counterions but rather a smoothly varying Gouy-Chapman layer. Nevertheless, Poisson-Boltzmann-based... 

The publication in 1948 of a monograph by Verwey and Overbeek detailing work done by them and others during World War II on the application of the PB equation, and, in particular, the Gouy-Chapman version of it, to the study of colloids has proved to be as important as the initial publications by Gouy and Chapman and Debye and Hiickel. This study laid the foundation for the modern study of colloids and has served as the primary guidepost for most of the work described here. Today, the PB equation, and in particular the Debye-Hiickel (DH) linearized approximation, forms the foundation for modern descriptions of electrolyte and colloid theory. New theories are compared with and often derived from the nonlinear Poisson-Boltzmann equation and in the appropriate limits reduce to the DH result. As has been shown by modern statistical methods, the Debye-Hiickel theory of electrolyte solutions is analogous to the lowest-order harmonic approximation in potential theory. ... 

The electrostatic potential only can be determined relative to a reference point which normally is chosen to be zero at r — oo. However, this equation is still very difficult to solve and an analytical solutions are only available in special cases. Useful solutions occur at low surface potential, where the PB can be linearized (see Debye-Hiickel below). A famous analytical solution was derived by Gouy [12] and Chapman [13] independently (see below) for one flat surface in contact with an infinite salt reservoir. The interaction between two flat and charged surfaces in absence of salt, can also be solved analytically [14]. In other situations the nonlinearized PB equation has to be solved numerically. 

Gouy [29] and Chapman [30] independently proposed an alternative treatment. It is based on an analytical solution of the nonlinear Poisson-Boltzmann equation for the electric potential created by a system of mobile point charges near a charged wall imitating an electrode surface. However, in the original form of this theory, most of its predictions were at variance with experimental data. 

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