Double-layer problem, Gouy-Chapman theory

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. 

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. 

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... 

From the rigorous treatment of the double-layer problem on the molecular level, it becomes clear that the Gouy-Chapman theory of the interface is equivalent to a mean field solution of a simple primitive model (PM) of electrolytes at the interface (6). To consider the correlation between ions, integral equations that describe the PM are devised and solved in different approximations. An exact solution of the PM of the electrolyte can be obtained from the computer simulations. This solution can be compared with the solutions obtained from different integral equations. For detailed discussion of this topic, refer to the review by Camie and Torrie (6). In many cases, the molecular description of the solvent must be introduced into the theory to explain the complexity of the observed phenomena. The analytical treatment in such cases is very involved, but initial success has already been achieved. Some of the theoretical developments along these lines were reviewed by Blum (7). 

Proceeding now to the problem of the interaction of two parallel flat double layers, we shall base our considerations, as a first approximation, on the same picture as that underlying the Gouy-Chapman theory. Later on we shall consider possible corrections of the theory by taking into account the finite dimensions of the ions in the sense of the Stern-theory. 

In UnderstandingVoltammetry, the problem of the diffuse double layer as conceived in the Gouy-Chapman theory is posed as ... 

The next step in the development of the theory of the double layer is to remember that the ions are subject to thermal fluctuations as well as to the electric field. Gouy in France and Chapman in Britain solved this problem independently and the result is known as the Gouy-Chapman theory. 

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... 

Both the Gouy-Chapman and Debye-Hiickel are continuum theories. They treat the solvent as a continuous medium with a certain dielectric constant, ignoring the molecular nature of the liquid. Also, the ions are not treated as individual point charges but as a continuous charge distribution. For many applications, this is sufficient and the predictions of continuum theory agree with experimental results. Before we finally calculate the free energy of an electric double layer and force between two double layers, we discuss the limitations and problems of the continuum model. At the end of this chapter, electrostatic forces in nonpolar media are described. 

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