Gouy-Chapman theory Boltzmann equation

In Chapter 11 (Sections 11.4 and 11.6) we implicitly anticipated that the ion opposite in charge from the wall plays the predominant role in the double layer, the central observation of the Schulze-Hardy rule. This enters the mathematical formalism of the Gouy-Chapman theory in Equation (11.52), in which a Boltzmann factor is used to describe the relative concentration of the ions in the double layer compared to the bulk solution. For those ions that have the same charge as the surface (positive), the exponent in the Boltzmann factor is negative. This reflects the Coulombic repulsion of these ions from the wall. Ions with the same charge as the surface are thus present at lower concentration in the double layer than in the bulk solution. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

This theory of the diffuse layer is satisfactory up to a symmetrical electrolyte concentration of 0.1 mol dm-3, as the Poisson-Boltzmann equation is valid only for dilute solutions. Similarly to the theory of strong electrolytes, the Gouy-Chapman theory of the diffuse layer is more readily applicable to symmetrical rather than unsymmetrical electrolytes. 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Modified Gouy-Chapman theory has been applied to soil particles for many years (Sposito, 1984, Chapter 5). It postulates only one adsorption mechanism -the diffuse-ion swarm - and effectively prescribes surface species activity coefficients through the surface charge-inner potential relationship contained implicitly in the Poisson-Boltzmann equation (Carnie and Torrie, 1984). Closed-form... 

This model is based on the Gouy-Chapman theory (diffuse double-layer theory). The theory states that in the area of the boundary layer between solid and aqueous phase, independently of the surface charge, increased concentrations of cations and anions within a diffuse layer exists because of electrostatic forces. In contrast to the constant-capacitance model, the electrical potential does not change up to a certain distance from the phase boundaries and is not immediately declining in a linear manner (Fig. 14 a). Diffusion counteracts these forces, leading to dilution with increasing distance from the boundary. This relation can be described physically by the Poisson-Boltzmann equation. 

We use the Gouy-Chapman theory for the diffuse layer which is based on the Poisson-Boltzmann (P.B.) equation for the potential distribution. Although the different corrections to the P.B. equation in double-layer theory have been investigated (20, 21, 22, 23), it is difficult to state precisely the range of validity of this equation. In the present problem the P.B. equation seems a reasonable approximation at 0.1M of a 1-1 electrolyte to 50mV for the mean electrostatic potential pd at the ohp (24) this upper limit for pd increases with a decrease in electrolyte concentration. All the values for pd calculated in Tables I-IV are less than 50 mV— most of them are well below. If n is the volume density of each ion type of the 1-1 electrolyte in the substrate, c the dielectric constant of the electrolyte medium, and... 

The Gouy-Chapman theory of the diffuse layer at a charged interfacial region is based on the well-known Poisson-Boltzmann equation ... 

The most widely used theory of the stability of electrostatically stabilized spherical colloids was developed by Deryaguin, Landau, Verwey, and Overbeek (DLVO), based on the Poisson-Boltzmann equation, the model of the diffuse electrical double layer (Gouy-Chapman theory), and the van der Waals attraction [60,61]. One of the key features of this theory is the effective range of the electrical potential around the particles, as shown in Figure 25.7. Charges at the latex particles surface can be either covalently bound or adsorbed, while ionic initiator end groups and ionic comonomers serve as the main sources of covalently attached permanent charges. 

There are several remarks about the Gouy-Chapman theory In spite of the apparent oversimplification the Poisson Boltzmann equation satisfies an overall dynamic equilibrium condition, that fixes the contact density at the electrode surface. This is the contact theorem... 

In the theoretical approaches of Poisson-Boltzmann, modified Gouy-Chapman (MGC), and integral equation theories such as HNC/MSA, concentration or density profiles of counterions and coions are calculated with consideration of the ion-waU and ion-ion in-... 

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. 

This is the important Poisson-Boltzmann (PB) equation and the model used to derive it is usually called the Gouy-Chapman (GC) theory. It is the basic equation for calculating all electrical double-layer problems, for flat surfaces. In deriving it we have, however, assumed that all ions are point charges and that the potentials at each plane x are uniformly smeared out along that plane. These are usually reasonable assumptions. 

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... 

A commonly used model for describing counterion distribution at a charged surface is based on the Gouy-Chapman diffuse double-layer (DDL) theory. This model assumes that the surface can be visualized as a structurally featureless plane with evenly distributed charge, while the counterions are considered point charges in a uniform liquid continuum. In this simplified picture, the equilibrium distribution of counterions is described by the Boltzmann equation ... 

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... 

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. 

Since the first measurements of the electrostatic double-layer force with the AFM not even 10 years ago, the instrument has become a versatile tool to measure surface forces in aqueous electrolyte. Force measurements with the AFM confirmed that with continuum theory based on the Poisson-Boltzmann equation and appKed by Debye, Hiickel, Gouy, and Chapman, the electrostatic double layer can be adequately described for distances larger than 1 to 5 nm. It is valid for all materials investigated so far without exception. It also holds for deformable interfaces such as the air-water interface and the interface between two immiscible liquids. Even the behavior at high surface potentials seems to be described by continuum theory, although some questions still have to be clarified. For close distances, often the hydration force between hydrophilic surfaces influences the interaction. Between hydrophobic surfaces with contact angles above 80°, often the hydrophobic attraction dominates the total force. 

---