Boltzmann distribution, Gouy-Chapman

The second term in equation (9) is the usual electrostatic term. Here vA is the valency of the unit and e is the elementary charge, and ip(z) is the electrostatic potential. This second term is the well-known contribution accounted for in the classical Poisson-Boltzmann (Gouy -Chapman) equation that describes the electric double layer. The electrostatic potential can be computed from the charge distribution, as explained below. 

The Gouy-Chapman model describes the properties of the diffuse region of the double-layer. This intuitive model assumes that counterions are point charges that obey a Boltzmann distribution, with highest concentration nearest the oppositely charged fiat surface. The polar solvent is assumed to have the same dielectric constant within the diffuse region. The effective surface... 

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. 

Figure 2.13 illustrates what is currently a widely accepted model of the electrode-solution interphase. This model has evolved from simpler models, which first considered the interphase as a simple capacitor (Helmholtz), then as a Boltzmann distribution of ions (Gouy-Chapman). The electrode is covered by a sheath of oriented solvent molecules (water molecules are illustrated). Adsorbed anions or molecules, A, contact the electrode directly and are not fully solvated. The plane that passes through the center of these molecules is called the inner Helmholtz plane (IHP). Such molecules or ions are said to be specifically adsorbed or contact adsorbed. The molecules in the next layer carry their primary (hydration) shell and are separated from the electrode by the monolayer of oriented solvent (water) molecules adsorbed on the electrode. The plane passing through the center of these solvated molecules or ions is referred to as the outer Helmholtz plane (OHP). Beyond the compact layer defined by the OHP is a Boltzmann distribution of ions determined by electrostatic interaction between the ions and the potential at the OHP and the random jostling of ions and... 

In the Gouy-Chapman diffuse layer the concentration-distance profile Is given by the Boltzmann distribution ... 

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 ... 

Despite the difficulties in quantitative treatment, there exist theoretical models based on the classical treatment initiated by Gouy, Chapman, Debye, and Hiickel and later modified by Stem and Cjrahame. As shown in Figure 7.3, a reasonable representation of the potential distribution by the Poisson-Boltzmann equation can be given as... 

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... 

Let us now examine how the potential cp changes within the diffuse part of the EDL, assuming that cp=0 in the bulk of the dispersion medium. The theory describing this part of the EDL was developed by Gouy and Chapman, who compared the energy of the electrostatic interaction of the ions with the energy of their thermal motion, assuming that the concentration of ions in the EDL was consistent with the Boltzmann distribution ... 

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... 

The simplest model for the electrical double layer is the Helmholtz condenser. A distribution of counterions in the bulk phase described by a Boltzmann distribution agree with the Gouy-Chapman theory. On the basis of a Langmuir isotherm Stem (1924) derived a generalisation of the double layer models given by Helmholtz and Gouy. Grahame (1955) extended this model with the possibility of adsorption of hydrated and dehydrated ions. This leads to a built-up of an inner and an outer Helmholtz double layer. Fig. 2.14. shows schematically the model of specific adsorption of ions and dipoles. 

This simple idea of Helmholtz, that a layer of ions from the solution becomes attached to the surface, was modified in 1910 by the French physicist Georges Gouy (1854-1926) and in 1913 by the British chemist David Leonard Chapman (1869-1958). These workers pointed out that the Helmholtz theory is unsatisfactory in neglecting the Boltzmann distribution of the ions. They suggested that on the... 

Figure 9.7 gives a representation of the diffuse double layer. This model is also known as the Gouy-Chapman layer (named after the persons who first developed the model). The underlying picture is that the surface is located at jc = 0 and that the counterions are not only attracted by the surface but are also subject to thermal motion. The former force tends to accumulate all counterions at the distance of closest approach to the surface (as in the molecular condenser), whereas the latter tries to spread all counterions homogeneously in the solution. The co-ions are subjected to similar counteracting tendencies. See Figure 9.1. The resulting countercharge distribution is given by the Boltzmann equation ...

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 simplest model of charge shielding and colloidal stability against aggregation was developed around 1910 independently by L-G Gouy (18.34-1926), a French physicist, and DL Chapman (1869-1958), a British chemist. They combined Poisson s equation of electrostatics with the Boltzmann distribution law. 

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