Beyond the Poisson-Boltzmann Equation

In this section we present in more detail the key assumptions behind the use of the Poisson-Boltzmann equation and discuss some of the ways in which these assumptions have been relaxed. We also give brief introductions to several of the popular alternative approaches to standard PB theory. 

Outhwaite and coworkers (Outhwaite 1978 Bhuiyan, Outhwaite, and Levine 1979 Outhwaite, Bhuiyan, and Levine 1980 Outhwaite and Bhuiyan 1983) introduced a MPB approach, in which a modified PB equation was proposed in the RPM, always in the infinite plane surface geometry  

FIGURE 3.15 Dimensionless mean electrostatic potential (a) and surface-ion distribution function (b) as predicted by the Gouy-Chapman-Stern (GCS) and modified Poisson-Boltzmann (MPB) theories for a 1 1 electrolyte with a = 0.425 nm and c = 0.197 M. (Outhwaite, Bhuiyan, and Levine, 1980, Theory of the electric double layer using a modified Poisson-Boltzmann equation. Journal of the Chemical Society, Faraday Transactions 2 Molecular and Chemical Physics, 76, 1388-1408. Reproduced by permission of The Royal Society of Chemistry.) 

The MPB theory was further improved, yielding the MPB5 formulation (Outhwaite and Bhuiyan 1983) and an alternative formulation that we will call MPB6 (Outhwaite and Lamperski 2001) both introduce new expressions for the excluded volume Q. These theories have been compared (Bhuiyan and Outhwaite 2004) with DFT results (Boda et al. 2004) and MC results from the study by Boda et al. (2002). The reader is referred to these papers for details here, we will 

The other line of thought is based in integral equation schemes to obtain the ion-wall distribution function rather than the electrostatic potential. They are based on 

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Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

It is beyond our scope to analyse all possible cases. In fact. In many multi-component systems with ions of different valencies the Poisson-Boltzmann equation becomes analytically unsolvable. We shall restrict ourselves therefore to the general equations for charge and capacitance and one illustration. 

There have been considerable efforts to move beyond the simplified Gouy-Chapman description of double layers at the electrode-electrolyte interface, which are based on the solution of the Poisson-Boltzmann equation for point charges. So-called modified Poisson-Boltzmann (MPB) models have been developed to incorporate finite ion size effects into double layer theory [61]. An early attempt to apply such restricted primitive models of the double layer to the ITIES was made by Cui et al. [62], who treated the problem via the MPB4 approach and compared their results with experimental data for the more problematic water-DCE interface. This work allowed for the presence of the compact layer, although the potential drop across this layer was imposed, rather than emerging as a self-consistent result of the theory. The expression used to describe the potential distribution across this layer was... 

In summary, the understanding of electrical polarization at liquid-liquid interfaces has improved much over the past decades, expanding beyond the classical Gouy-Chapman analysis. After many years of extensive capacitance data measurements for different solvent pairs, electrolytes, and so on, the different theoretical approaches all point toward a three-layer model in which the outer layers are classical diffuse layers that can be treated in a first approximation by the Poisson-Boltzmann equation, and a central layer that ions from both side can penetrate, and where solvent fingering may take place, and where the surface roughness is... 

The derivation of the Poisson-Boltzmann (PB) equation and its variants are well-known and available in many textbooks.Over seventy years of hindsight has shown us that, despite the relative simplicity of the approach, most of the physics and chemistry of polyelectrolyte solutions are well described by the Poisson-Boltzmann equation. To go beyond the PB equation, however, is not as simple as one might think. We therefore include a discussion of the assumptions behind the equation and attempts at improving it in the final part of this review. 

It follows from the preceding section that the limiting case xa 1 (double layer thin as compared with radius of curvature) is simple then we can simply apply the flat layer theory, discussed extensively in secs 3.5a-d. Beyond this limit, the appropriate Poisson-Boltzmann equation (with p in (3.5.631 depending on the geometry) has to be solved with the appropriate boundary condition, l.e. dy/dr for r = 0, so in the centre of a sphere or infinitely long cylinder, the field strength is zero because of symmetry. However, at that location y is not necessarily zero, because double layers from the opposite sides may overlap. This Is a new feature as compared with convex double layers around non-interacting particles. 

The concept of capillary waves can be used to explain how the surface roughness increases the interfacial capacity beyond the Verwey-Niessen value. For this purpose, Pedna and Badiali [82] have solved the linear Poisson-Boltzmann equation across the interface between two solutions vyith different dielectric constants and Debye lengths separated by a corrugated surface. A major difficulty is the boundary condition at the rough interface. 

Manning proposed a linear counterion condensation theory to account for the low activity of counterions in polyelectrolyte solutions 14). The basic idea of the theo is that there is a critical charge density on a polymer chain beyond which some counterions will condense to the polymer chain to lower the charge density, otherwise the ener of the system would approach infinite. The concept of this theory has been widely accepted. The shortcoming of the linear countmon condensation is that it predicts that counterion condensation is independent of ionic strength in the solution, which is not in agreement with experimental observations. Counterion condensation can be obtained duectly by solving the nonlinear Poisson-Boltzmann equation. 

Poisson-Boltzmann equation for the electrostatic potential of the system and leads to a more complex expression for the interaction energy and the force ° and is beyond the scope of this chapter. 

This chapter explores the number of mean field constructions for ions whose structure goes beyond the point charge description, the representation used in the standard Poisson-Boltzmann (PB) equation. The structural details omitted within a point charge picture can be related to electrostatic structure of an ion,... 

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