Gouy-Chapman theory, discussed

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. 

A rigorous solution of this problem was attempted, for example, in the hard sphere approximation by D. Henderson, L. Blum, and others. Here the discussion will be limited to the classical Gouy-Chapman theory, describing conditions between the bulk of the solution and the outer Helmholtz plane and considering the ions as point charges and the solvent as a structureless dielectric of permittivity e. The inner electrical potential 0(1) of the bulk of the solution will be taken as zero and the potential in the outer Helmholtz plane will be denoted as 02. The space charge in the diffuse layer is given by the Poisson equation... 

A theoretical model for the adsorption of metals on to clay particles (<0.5 pm) of sodium montmorillonite, has been proposed, and experimental data on the adsorption of nickel and zinc have been discussed in terms of fitting the model and comparison with the Gouy-Chapman theory [10]. In clays, two processes occur. The first is a pH-independent process involving cation exchange in the interlayers and electrostatic interactions. The second is a pH-dependent process involving the formation of surface complexes. The data generally fitted the clay model and were seen as an extension to the Gouy-Chapman model from the surface reactivity to the interior of the hydrated clay particle. 

The preceding discussion was limited to the artificial case of a single ion. When multiple ions are present, in addition to the issues discussed, there is the problem of ion-ion interactions and correlations. The main motivation for such studies is to come close to the realistic situation in which a finite concentration of ions exists near the metal surface that is in equilibrium with ions in the bulk. Another important specific goal is to investigate the applicability of continuum models, such as the Gouy-Chapman theory. " Although this has been the subject of several Monte Carlo... 

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). 

Many more functional characteristics beyond the discussed selection exist. For instance, the influence of the composition of the membrane on activity has been almost completely ignored. This includes important topics such as membrane fluidity, phase transition, thickness (and hydrophobic matching), surface potentials 4 0 (and Gouy-Chapman theory), partitioning, heterogeneity ( rafts ), or swelling and shrinking in response to stress. 

Oil/water interfaces are classified into the ideal-polarized interface and the nonpolarized interface. The interface between a nitrobenzene solution of tetrabutylam-monium tetraphenylborate and an aqueous solution of lithium chloride behaves as an ideal-polarized interface in a certain potential range. Electrocapillary curves of the interface were measured. The results are analyzed using the electrocapillary equation of the ideal-polarized interface and the Gouy-Chapman theory of diffuse double layers. The electric double layer structure consisting of the inner layer and the two diffuse double layers on each side of the interface is discussed. Electrocapillary curves of the nonpolarized oil/water interface are discussed for two cases of a nonpolarized nitrobenzene/water interface. 

This contact theorem, as well as other sum rules that are valid for the charged interface will be discussed in the next section. The density profiles obtained from the Gouy-Chapman theory are monotonous, that is they show no oscillations. Since in this theory the contact theorem and the electroneutrality condition are satisfied, then, p,-(2) is pinned at the origin, and has a fixed integral, so that the density profile cannot deviate too much from the correct result. When the contact theorem is not... 

The complex but rather accurate Gouy-Chapman theory (see Appendix 10.2 for a brief discussion) provides the distribution of potential and the ion concentration as a function of the distance from the surface. The potential decreases exponentially with the distance and also with decreasing Debye length. 

Both, the Gouy-Chapman and Debye-Hiickel are continuum theories. They treat the solvent as a continuous medium with a certain dielectric constant, but they ignore 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. At the end of this chapter we discuss the limitations and problems of the continuum model. 

Stern combined the ideas of Helmholtz and that of a diffuse layer [64], In Stern theory we take a pragmatic, though somewhat artificial, approach and divide the double layer into two parts an inner part, the Stern layer, and an outer part, the Gouy or diffuse layer. Essentially the Stern layer is a layer of ions which is directly adsorbed to the surface and which is immobile. In contrast, the Gouy-Chapman layer consists of mobile ions, which obey Poisson-Boltzmann statistics. The potential at the point where the bound Stern layer ends and the mobile diffuse layer begins is the zeta potential (C potential). The zeta potential will be discussed in detail in Section 5.4. 

Earlier theories by Gouy, Chapman, and Hcrzfeld discussed the double layer as wholly of this diffuse type but Stem points out that these give far too high values for the capacity of the double layer, partly because in them the ions are supposed mathematically to be able to approach indefinitely close to the solid surface, which is impossible physically owing to the size of the ions. Stern s theory gives a complicated expression for the capacity of the double layer, but accounts reasonably well for the experimental values. Though the layer is largely diffuse in many cases, the capacity is usually of the same order as if the layer were of the plane parallel type, because most of the ions are fairly close to the fixed part of the layer. 

Measurements of the surface tension and surface stress of solids are not easy. Some attempts have been made to measure the surface energy, or at least to determine the PZC, of solid electrodes attached to piezoelectric materials (36, 37). More often there is a reliance on studies of differential capacitance (Section 13.4.3) (35, 38). In principle, these measurements could provide all of the information needed to describe the surface charges and relative excesses however, one must first know the PZC. Evaluating it for a solid electrode/electrolyte system is not straightforward, and indeed, as discussed below, the PZC is not uniquely defined for a polycrystalline electrode. The most widely used approach is to evaluate the potential of minimum differential capacitance in a system involving dilute electrolyte. The identification of this potential as the PZC rests on the Gouy-Chapman-Stem theory discussed in Section 13.3,... 

The sum of interfacial potential and distribution potential is termed the iimer or Galvani potential, designated by . It will be discussed below that the observed transmembrane potential difference, Em, is due either to AV, AU, or to AV plus AU, as defined above. It should be remembered that there are two components of (=V + U). They are concerned with only the distribution potential. If compounds, such as phospholipids and interface-active agents, are preferentially adsorbed at the interface, the so-called adsorption potential (V) may also develop. As shown in Fig. 2, both adsorbed fixed charge species and dipoles may contribute to the observed potential. The nature and origin of the adsorption (or interfacial) potential can be discussed in terms of the classical EDL theory of Gouy-Chapman-Stern-Graham (7,15-19]. 

However, the theory neglects the finite size of the ions, and it was Stern who postulated that ions could not approach the electrode beyond a plane of closest approach, thereby introducing in a crude way the ion size (Fig. 5.1c). Although formulated in a complex manner [1], the basis of Stern s model is a combination of the Helmholtz and Gouy-Chapman approaches. It may be noted that Fig. 5.1 also shows the potential and charge distributions resulting from the models. These will be discussed later. 

To date, it has been documented that ILs can be adsorbed onto various electrode surfaces. For example, Nanjundiah et al. found that several ILs used as electrolytes can induce double-layer capacitance phenomena on the surface of an Hg electrode and obtained the respective capacitance values for various ILs. Hyk and Stojek have also studied the IL thin layer on electrode surfaces and suggested that counterions substantially influence the distribution of IL. Kornyshev further discussed IL formations on electrode surfaces, suggesting that IL studies should be based on modern statistical mechanics of dense Coulomb systems or density-functional theory rather than classical electrochemical theories that hinge on a dilute-solution approximation. There are three conventional models that describe the charge distribution of an ion near a charged surface the Helmholtz model, the Gouy-Chapman model, and the Stern model. In the case of ILs, it remains controversial which model can best explain and lit the experimental data. 

In the following sections, different surface complexation models will be introduced. General aspects and specific models will be discussed. The components of surface complexation theory will be presented, as well as some recent developments covering, for example, the use of equations for the diffuse part of the electrical double layer for electrolyte concentrations, for which the traditional Gouy-Chapman equation is not recommended or a generalization of Smit s compartment model [6] for situations in which the traditional models are at a loss. 

Menestrina et al. discuss their interesting results in terms of the Gouy-Chapman electrical double-layer theory and suggest that molluscan hemo-cyanins are a class of channel-forming proteins. 

I am disturbed that following Professor Hammes presentation the language in the discussion has changed. While discussing enzymes we looked at molecule properties closely related to the discussion of small molecules (e.g., atom position and motions). Now in the discussion of complex enzymes, especially in membranes, we have started to use bulk properties (e.g., we talk of phases, dielectric constants, Chapman-Gouy theory, etc.). Is it the view of the discussants that events in membrane-coupled-enzyme systems cannot be described by molecular events because of the complexity of the system resulting from extensive cooperativ-ity within the membrane (e.g., between lipids and proteins) ... 

However, it is interesting to note that the theory of the diffuse double layer was presented independently by Gouy and Chapman (1910) 13 years before the Debye-Hiickel theory of ion ion interactions (1923). The Debye-Hiickel theory was immediately discussed and applied to the diffuse charge around an ion, doubtless owing to the preoccupation of the majority of scientists in the 1920s with bulk properties rather than those at surfaces. 

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