Nonlinear Debye-Hiickel

For the sake of simplicity, in what follows it will be considered that the double layer potential is sufficiently small to allow the linearization of the Poisson—Boltzmann equation (the Debye—Hiickel approximation). The extension to the nonlinear cases is (relatively) straightforward however, it will turn out that the differences from the DLVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the distribution of charge inside the double layer is given by... 

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. 

The linearization that leads here to the Debye-Hiickel model is physically consistent in this argument. But the possibility of a model that is unlinearized in this sense is a popular query. More than one response has been offered including the (nonlinear) Poisson-Boltzmann theory and the EXP approximation see (Stell, 1977) also for representative numerical results for the systems discussed here. 

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... 

In this section, we present a novel linearization method for simplifying the nonlinear Poisson-Boltzmann equation to derive an accurate analytic expression for the interaction energy between two parallel similar plates in a symmetrical electrolyte solution [13, 14]. This method is different from the usual linearization method (i.e., the Debye-Hiickel linearization approximation) in that the Poisson-Boltzmann equation in this method is linearized with respect to the deviation of the electric potential from the surface potential so that this approximation is good for small particle separations, while in the usual method, linearization is made with respect to the potential itself so that this approximation is good for low potentials. 

It should be pointed out that the above questions have been the subject of large number of studies, where a number of important results has been obtained. The effects of nonlinear screening in the thermodynamically equilibrium case of charged colloidal suspension were studied in [11-13], It was found that in the presence of the nonlinear effects the effective potential at distances can be described by the linear Debye-Hiickel theory, however, with some effective charge being less than the bare grain charge. 

The Debye-Hiickel expression for p does not predict, even in its nonlinear form, a saturation effect for very large values oizexpjkTas would be desirable. Our equations show that for zetpjkT co, f and d are of the same order and thus in the limit pjzen+ —— 1. We encounter then the effect of saturation. In working with a linear approximation one cannot expect to find a saturation effect. 

Because of nonlinearity, the Poisson-Boltzmann equation (i.e., Eq. 10) can be solved numerically. Using the Debye-Hiickel approximation, an analytical solution of Eq. 10 for the EDL potential can be obtained as... 

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... 

According to a nonlinear Poisson-Boltzmann analysis, the initial radial decay of f from a cylindrical chain segment is much steeper than predicted by a Debye-Hiickel analysis see Figure 3 (28). The steep decay lessens at large distance, and eventually adopts an asymptotic functional form compatible with a Debye-Hiickel approximation. However, to superimpose across this distant region the predictions of the Poisson-Boltzmann analysis onto those formidated... 

Fig. 3. Schematic plots of the decay of the electrostatic potential if(r) near a polyelectrolyte chain of unspecified chain radius. The potential decay predicted by a nonlinear Poisson-Boltzmann analysis can be superimposed onto one predicted by a linearized Debye-Hiickel approximation in the far field region if the surface potential of the Debye-Hiickel is appropriately adjusted. However, in this case there remains strong deviation between the two approaches in the region nearer the chain.

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-Hiickel theory of electrolyte solutions. It is derived from the classical Poisson equation for the electrostatic potential... 

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. ... 

Bulk Model Nonlinearizing the Debye—Hiickel Solution... 

While no exact analytical solution to Eq. [292] is available, approximate nonlinear expressions corresponding to the PGC and NLDH solutions as well as the weak-field Debye-Hiickel solution are given below. 

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] ... 

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