Gouy-Chapman theory for the

Stahlberg has presented models for ion-exchange chromatography combining the Gouy-Chapman theory for the electrical double layer (see Section V-2) with the Langmuir isotherm (. XI-4) [193] and with a specific adsorption model [194]. 

Fig. 10.8 Comparison of the experimental values of with the predictions of the Gouy-Chapman theory for the interface between AsPhiBPh dissolved in PVC and KCl dissolved in water. The concentration of KCl is much greater than that of the AsPhiBPhi so that the double layer contribution from the aqueous phase can be neglected.

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... 

Fig. 13. Potential difference across the inner layer Ao

In Fig. 8 density profiles are presented for several values of charge density a on the wall and for the wall potential h = — and h= Fig. 9 contains the corresponding ionic charge density profiles. For the adsorptive wall potential h < 0) the profiles q z) in Fig. 9(a) and j (z) in Fig. 8(a) are monotonic, as in the Gouy-Chapman theory. For a wall which is neutral relative to the adsorption A = 0 the density profiles are monotonic with a maximum at the wall position. This maximum also appears on the charge... 

The Gouy-Chapman theory for metal-solution interfaces predicts interfacial capacities which are too high for more concentrated electrolyte solutions. It has therefore been amended by introducing an ion-free layer, the so-called Helmholtz layer, in contract with the metal surface. Although the resulting model has been somewhat discredited [30], it has been transferred to liquid-liquid interfaces [31] by postulating a double layer of solvent molecules into which the ions cannot penetrate (see Fig. 17) this is known as the modified Verwey-Niessen model. Since the interfacial capacity of liquid-liquid interfaces is... 

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 Thomas-Fermi model of a metal is similar to the Gouy-Chapman theory for electrolytes. In this model the surface-charge density o is... 

The electrode roughness factor can be determined by using the capacitance measurements and one of the models of the double layer. In the absence of specific adsorption of ions, the inner layer capacitance is independent of the electrolyte concentration, in contrast to the capacitance of the diffuse layer Q, which is concentration dependent. The real surface area can be obtained by measuring the total capacitance C and plotting C against Cj, calculated at pzc from the Gouy-Chapman theory for different electrolyte concentrations. Such plots, called Parsons-Zobel plots, were found to be linear at several charges of the mercury electrode. ... 

According to the Gouy-Chapman theory of the diffuse electric double layer (chapter 3 of [18]), for a uni-univalent electrolyte,... 

This differential equation for the space variation of the potential inside the semiconductor can be easily identified with that for the space variation of the potential inside the electrolyte in the Gouy-Chapman theory of the diffuse layer (Section 6.6.1). The solution can therefore be borrowed from the diffuse-layer theory. One has, from Eq. (6.125),... 

For the evaluation of the non-faradaic component of the response in a more realistic way, different proposals have been made. A useful idea is that corresponding to the interfacial potential distribution proposed in [59] which assumes that the redox center of the molecules can be considered as being distributed homogeneously in a plane, referred to as the plane of electron transfer (PET), located at a finite distance d from the electrode surface. The diffuse capacitance of the solution is modeled by the Gouy-Chapman theory and the dielectric permittivity of the adsorbed layer is considered as constant. Under these conditions, the CV current corresponding to reversible electron transfer reactions can be written as... 

Yi, M., Nymeyer, H., and Zhou, H. X. (2008). Test of the Gouy-Chapman theory for a charged hpid membrane against exphcit-solvent molecular dynamics simulations. Phys. Rev. Lett. 101, 038103-1-4. 

Nature of the Surface Complexes. The constant capacitance model assumes an inner-sphere molecular structure for surface complexes formed in reactions like equation 5a or 7. But this structure does not manifest itself explicitly in the composition dependence of Kc everything molecular is buried in which is an adjustable parameter. This encapsulating characteristic of the model was revealed dramatically by Westall and Hohl (13), who showed that five different surface speciation models, ranging from the Gouy-Chapman theory to the surface complex approach, could fit proton adsorption data on AL O., equally well, despite their mutually contradictory underlying molecular hypotheses [see also Hayes et al. (19)]. They concluded that "... no model will yield an unambiguous description of adsorption. .. . To this conclusion one may add that no model should provide such a description,... 

The application of the Gouy-Chapman theory for describing ion exchange can be illustrated by deriving an exchange isotherm. Consider that the surface excess (in moles per square meter) of an electrostatically sorbed ion i is given... 

In the DL model, assuming Gouy-Chapman theory for a symmetrical electrolyte of charge z and 25°C, the charge density, tr, at some distance away from the surface, is given by... 

In what concerns the tests of the ionic isotherm, they can be carried out only if an analytic expression of is available. That is, the present theory must be necessarily combined with a diffuse layer theory. Therefore, the validity of these isotherms depends to a certain degree on the validity of theory of the diffuse layer adopted for the calculation of For simplicity we used the Gouy-Chapman theory and the well known equation [49] ... 

It may be noticed that Eq. (206) is formally identical with that developed for the Gouy-Chapman theory of the double layer. Substitution of Eq. (189) into (205) gives the expression for the steady-state current on porous electrodes ... 

Stem improved the Gouy-Chapman theory of the DDL by assuming that some ions are tightly retained immediately next to colloid surfaces in a layer of specifically adsorbed or Stem- layer cations. The double layer is diffuse beyond this layer. A satisfactory approximation of the Stem model can be made by assuming that the specifically adsorbed ions quantitatively reduce the surface density of the colloid. The diffuse portion of the double layer then is assumed to develop on a colloid surface of correspondingly reduced charge density. Sample Stem-modification calculations for a series of monovalent cations are shown in Fig. 8.10, Relatively few of the... 

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. 

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). 

Here, C is the inner-layer capacitance, independent of the surface-inactive electrolyte concentration and is the diffuse (Gouy) layer capacitance, expressed according to the Gouy-Chapman theory for a z,z-type electrolyte by... 

---