Gouy-Chapman-Stern theory

By allowing for surface saturation, the Stern theory overcomes the objection to the Gouy-Chapman theory of excessive surface concentrations. In so doing, however, it trades off one set of difficulties for another. In the Gouy-Chapman theory the functional dependence of 0 on x involves only the parameters k and 0O- The former is known and the latter may be... 

Fortunately, there is a relatively simple semi-empirical extension of the Gouy-Chapman theory, which accounts for most experimental observations. This extension was proposed by Stern.7... 

For higher concentrations, and far from the point of zero charge, the Gouy-Chapman theory predicts interfacial capacities that are much higher than the experimental values. To explain these findings, Stern combined... 

Conversely, according to the description of the electrical double layer based on the Stern-Gouy-Chapman (S-G-C) version of the theory [24], counter ions cannot get closer to the surface than a certain distance (plane of closest approach of counter ions). Chemically adsorbed ions are located at the inner Helmholtz plane (IHP), while non-chemically adsorbed ions are located in the outer Helmholtz plane (OHP) at a distance x from the surface. The potential difference between this plane and the bulk solution is 1 ohp- In this version of the theory, Pqhp replaces P in all equations. Two regions are discernible in the double layer the compact area between the charged surface and the OHP in which the potential decays linearly and the diffuse layer in which the potential decay is almost exponential due to screening effects. 

In the Stern-Gouy-Chapman (SGC) theory the double layer is divided into a Stern layer, adjacent to the surface with a thickness dj and a diffuse (GC) layer of point charges. The diffuse layer starts at the Stern plane at distance d] from the surface. In the most simple case the Stern layer is free of, charges. The presence of a Stern layer has considerable consequences for the potential distribution across the Stern layer the potential drops linearly from the surface potential V s to the potential at the Stern plane, V>d- Often is considerably lower than especially in the case of specific adsorption (s.a.). 

The Gouy-Chapman theory did not prove entirely satisfactory, and in 1924 a considerable advance was made by the Germ an-American physicist Otto Stern, whose model is shown in Figure 11.18c. Stern combined the fixed double-layer model of Helmholtz with the diffuse double-layer model of Gouy and Chapman, As shown in the figure, there is a fixed layer at the surface, as well as a diffuse layer. On the whole this treatment has proved to be satisfactory, but for certain kinds of investigations it has been found necessary to develop more elaborate models. 

Theoretical concepts of specific (per unit true electrode surface area) EDL capacitance are based on the known classical theories of EDL developed by Helmholtz, Stern, Gouy-Chapman, Grahame, and so on. One of the directions of modem smdies of EDL is elucidation of ratios between different surface layer characteristics. These include specific capacitances in zero charge points, electronic work functions of metals, their liophilicity, zero charge potentials. Correlations are established between many of these characteristics in a number of metals and solvents. At the same time, there are significant deviations from main trends. Zero charge points were first determined for different carbon materials in the works of Frumkin et al. 

FIGURE 6. The arrangement of water molecules and counterions near to a negatively charged membrane surface according to the Stern model. Within the Stern layer of polarized water molecules the electric potential falls linearly, and for distances further than this the potential profile follows that predicted by the Gouy-Chapman theory of electrical double layers. For ascites cells the potential drop between the surface potential and the zeta potential has been determined to be around... 

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. 

The exponential nature of the accumulation of oppositely charged ions means that Gouy-Chapman theory predicts local concentrations which are much larger than that in bulk if the potential difference applied between the electrode and the solution greatly exceeds RT/F. In practice, the Stern layer of adsorbed solvent molecules and the specific adsorption of ions mediates the potential perceived by the diffuse component of the double layer to a great extent. 

The electrical potential, ij/, at the interface between the micellar core and the surrounding water may be estimated by the Gouy-Chapman theory of the electrical double layer. In the classical theory, a uniform continuous interfacial surface charge is assumed, which is neutralized by a diffuse ionic layer of charges in the aqueous solution. In a detailed model of the Stern layer proposed by Stigter [35-37], this theory is refined to allow for the size and high concentration of the charge carriers at the micelle surface. 

The charge at the diffuse layer plane is calculated from Gouy-Chapman-Stern-Grahame theory, which for a symmetrical monovalent electrolyte of concentration Cg is given by... 

As the electrode surface will, in general, be electrically charged, there will be a surplus of ionic charge with opposite sign in the electrolyte phase in a layer of a certain thickness. The distribution of jons in the electrical double layer so formed is usually described by the Gouy— Chapman—Stern theory [20], which essentially considers the electrostatic interaction between the smeared-out charge on the surface and the positive and negative ions (non-specific adsorption). An extension to this theory is necessary when ions have a more specific interaction with the electrode, i.e. when there is specific adsorption of ions. 

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. 

Parsons and Zobel plot — In several theories for the electric - double layer in the absence of specific adsorption, the interfacial -> capacity C per unit area can formally be decomposed into two capacities in series, one of which is the Gouy-Chapman (- Gouy, - Chapman) capacity CGC 1/C = 1 /CH + 1 /CGC. The capacity Ch is assumed to be independent of the electrolyte concentrations, and has been called the inner-layer, the - Helmholtz, or Stern layer capacity by various authors. In the early work by Stern, Ch was attributed to an inner solvent layer on the electrode surface, into which the ions cannot penetrate more recent theories account for an extended boundary region. In a Parsons and Zobel plot, Ch is determined by plotting experimental values for 1/C vs. 1/Cgc- Specific adsorption results in significant deviations from a straight line, which invalidates this procedure. 

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