Helmholtz-Gouy-Chapman layer

Beyond the IHP is a layer of charge bound at the surface by electrostatic forces only. This layer is known as the diffuse layer, or the Gouy-Chapman layer. The innermost plane of the diffuse layer is known as the outer Helmholtz plane (OHP). The relationship between the charge in the diffuse layer, o2, the electrolyte concentration in the bulk of solution, c, and potential at the OHP, 2> can be found from solving the Poisson-Boltzmann equation with appropriate boundary conditions (for 1 1 electrolytes (13))... 

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. 

Stern layers can be introduced at different levels of sophistication. In the simplest case we only consider the finite size effect of the counterions (Fig. 4.5). Due to their size, which in water might include their hydration shell, they cannot get infinitely close to the surface but always remain at a certain distance. This distance <5 between the surface and the centers of these counterions marks the so-called outer Helmholtz plane. It separates the Stern from the Gouy-Chapman layer. For a positively charged surface this is indicated in Fig. 4.5. 

Most solid surfaces in water are charged. Reason Due to the high dielectric permittivity of water, ions are easily dissolved. The resulting electric double layer consist of an inner Stern or Helmholtz layer, which is in close contact with the solid surface, and a diffuse layer, also called the Gouy-Chapman layer. 

The Stern model modifies the Gouy-Chapman model and divides ions present in the solution into two groups a part of the ions is placed near the solid surface, forming the so-called Stern layer (similar to the Helmholtz layer), and the other part having a diffuse distribution (Gouy-Chapman layer). It implies that the surface potential is linear in the Stern layer, and the exponential in the Gouy-Chapman layer. 

Inner Helmholtz Layer Outer Helmholtz Gouy-Chapman (diffuse) Layer... 

The semiconductor occupies the region to the right of the vertical solid line representing the interface (jc = 0). To the left of it, there is the Helmholtz layer formed by ions attracted to the electrode surface, and also by solvent molecules its thickness, L, is the order of the size of an ion. The space-charge region in the solution (the Gouy-Chapman layer) is adjacent to the Helmholtz layer from the electrolyte side. 

Due to the short range of the adsorption interactions, it is possible for one to subdivide the EDL into two main parts (Fig. Ill-12) a dense part, that is closer to the surface (the Stem-Helmholtz layer), within which the adsorption forces are of importance, and a diffuse part (the Gouy - Chapman layer), which is further away from the surface, and within which the adsorption forces are negligible. The major task in EDL theory can be defined as the problem of finding the quantitative distribution of the concentrations of all ions, n present in the system and that of the potential at any point in the solution, cp, as a function of the distance from the surface, x (if confined to a single dimension). 

Analytical models of double layer structures originated roughly a century ago, based on the theoretical work of Helmholtz, Gouy, Chapman, and Stem. In Figure 26, these idealized double-layer models are compared. The Helmholtz model (Fig. 26a) treats the interfacial region as equivalent to a parallel-plate capacitor, with one plate containing the... 

The ensemble Helmholtz layer/Gouy-Chapman layer constitutes the electrochemical double layer. Its thickness is in the order of a few tens of Angstroms. This layer is generally represented by the series combination of two capacitances relative to the diffuse and compact layers, Cdia and Qomp- The capacity of the double layer, Cd, is thus equal to ... 

The compact, or Helmholtz, layer is the closest to the surface, in which the charge distribution and potentials change linearly with distance from the electrode surface. The potentials change exponentially when the more diffused outer Gouy-Chapman layer is formed. 

Fig. 1.2. (a, b) The double layer. Distribution of ions as a function of distance from the electrode behaving as an anode (a) and variation of potential with distance (b) for the model shown. (1) Helmholtz layer, (2) Gouy-Chapman layer, (3) bulk solution... 

Figure 4.1 Model of the electrochemical double layer on a metal electrode. TF Thomas-Fermi layer, iH layer of specific adsorption (inner Helmholtz layer), H Helmholtz layer (outer Helmholtz layer), and GC Gouy-Chapman layer.

Figure 4.2 Potential changes in the different parts of the double layer. A(p potential drop in the space-charge layer, A

Usually the capacitance of the Helmholtz layer and at higher electrolyte concentrations the capacitance of the Gouy-Chapman layer are much larger than the capacitance of the space-charge layer. Therefore, the reciprocal term can be neglected. The space-charge layer is the dominant element and represents the properties of the double layer for semiconductor electrodes... 

Double-layer capacitance Space-charge layer capacitance Capacitance of the Gouy-Chapman layer Capacitance of the Helmholtz layer Madelung constant Sauerbrey constant Distance... 

The Stern Model is a combination of the Helmholtz and Gouy-Chapman models (Figure 3.47). The potential difference between the metal and the solution is comprised of two terms A h. due to the compact Helmholtz layer and A0gc, due to the diffuse Gouy-Chapman layer. 

The potential difference at the semiconductor-electrolyte interface is the sum of three terms Adi rising from the space-charge layer in the semiconductor A( h from the Helmholtz layer and A0qc from the Gouy-Chapman layer (Figure 3.55). 

The liquid phase in an electrochemical experiment typically consists of a solvent containing the dissolved material to be studied and a supporting electrolyte salt to achieve the required conductivity and hence minimise the IR potential drop. With sufficient supporting electrolyte, the electrical double layer (see also Chap. I.l) at the working electrode occupies a distance of about 1 nm from the electrode surface (Fig. II.1.8). Note that the length scale in Fig. II.1.8 is not linear. This layer has been shown to consist of a compact or inner Helmholtz layer and the diffuse not diffusion layer) or Gouy-Chapman layer [41]. The extent to which the diffuse layer extends into the solution phase depends on the concentration of the electrolyte and the double layer may in some cases affect the kinetics of electrochemical processes. Experiments with low concentrations or no added supporting electrolyte can be desirable [42] but, since the double layer becomes more diffuse, they require careful data analysis. Furthermore, the IR drop is extended into the diffusion layer [43] (see also Chap. III.4.5). 

Figure 1-6). Maxwell-Wagner, Helmholtz, Gouy-Chapman, Stem, and Gra-hame theories have been used to describe the interfacial and double layer dynamics [9]. 

Helmholtz plane. It separates the Stern from the Gouy-Chapman layer. For a positively charged surface, this is indicated in Figure 4.5. 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

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... 

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... 

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