Stern-Grahame double layer model

A simplified approach, based on the Stern-Grahame double layer model, gives the following relation for the metal surface charge density,... 

Equation 3.74 is a parameterization of ctm =/ (cp ) based on the Stern-Grahame double layer model. This relation requires as input the potential of zero charge and the Helmholtz capacitance. As mentioned, finding =f(

various surface and bulk oxide species, occurring simultaneously with double layer charging. 

See color insert.) Electric double-layer models at interface of electrode and electrolyte solution. (a) Diffuse layer or Gouy-Chapman model, (b) Helmholtz layer or model the d represents the double-layer thickness, (c) Stern-Grahame layer or model in which the IHP represents the inner Helmholtz plane and the OHP represents the outer Helmholtz plane. 

Figure 7.4. Schematic model of the Electrical Double Layer (EDL) at the metal oxide-aqueous solution interface showing elements of the Gouy-Chapman-Stern-Grahame model, including specifically adsorbed cations and non-specifically adsorbed solvated anions. The zero-plane is defined by the location of surface sites, which may be protonated or deprotonated. The inner Helmholtz plane, or [i-planc, is defined by the centers of specifically adsorbed anions and cations. The outer Helmholtz plane, d-plane, or Stern plane corresponds to the beginning of the diffuse layer of counter-ions and co-ions. Cation size has been exaggerated. Estimates of the dielectric constant of water, e, are indicated for the first and second water layers nearest the interface and for bulk water (modified after [6]).

The solvent molecules form an oriented parallel, producing an electric potential that is added to the surface potential. This layer of solvent molecules can be protruded by the specifically adsorbed ions, or inner-sphere complexed ions. In this model, the solvent molecules together with the specifically adsorbed, inner-sphere complexed ions form the inner Helmholtz layer. Some authors divide the inner Helmholtz layer into two additional layers. For example, Grahame (1950) and Conway et al. (1951) assume that the relative permittivity of water varies along the double layer. In addition, the Stern variable surface charge-variable surface potential model (Bowden et al. 1977, 1980 Barrow et al. 1980, 1981) states that hydrogen and hydroxide ions, specifically adsorbed and inner-sphere... 

Many more-sophisticated models have been put forth to describe electrokinetic phenomena at surfaces. Considerations have included distance of closest approach of counterions, conduction behind the shear plane, specific adsorption of electrolyte ions, variability of permittivity and viscosity in the electrical double layer, discreteness of charge on the surface, surface roughness, surface porosity, and surface-bound water [7], Perhaps the most commonly used model has been the Gouy-Chapman-Stem-Grahame model 8]. This model separates the counterion region into a compact, surface-bound Stern" layer, wherein potential decays linearly, and a diffuse region that obeys the Poisson-Boltzmann relation. 

Fig. 10.14 Schematic diagram of the double layer according to the Gouy-Chapman-Stern-Grahame model. The metal electrode has a net negative charge and solvated monatomic cations define the inner boundary of the diffuse layer at the outer Helmholtz plane (oHp).

Figure 12. Diagram of inner region of the double layer showing outer Helmholtz (OHP) plane with oriented solvent dipoles interacting with electrostatically adsorbed solvated ions [schematic based on Stern-Grahame model (Ref. 95) BDM model (Ref. 60) includes an extra layer of solvent dipoles between the metal surface and OHP of cations].

After 20 years. Stern [23] modified these models by including both a compact and a diffuse layer. At the same time, Grahame [24] divided the Stern layer into two regions (i) an inner Helmholtz plane consisting of a layer of adsorbed ions at the surface of the electrode and (ii) an outer Helmholtz plane (referred to as Gouy plane as well), which is formed by the closest approach of diffuse ions to the electrode surface. From the Grahame model, the capacitance C of the double layer is described by Equation 8.1 as follows ... 

It is evident now why the Helmholtz and Gouy-Chapman models were retained. While each one alone fails completely when compared with experiment, a series combination of the two yields reasonably good agreement. There is room for improvement and refinement of the theory, but we shall not deal with that here. The model of Stern brings theory and experiment close enough for us to believe that it does describe the real situation at the interface. Moreover, later work of Grahame shows that the diffuse-double-layer theory, used in the proper context, yields consistent results and can be considered to be correct, within the limits of the approximations used to derive it. 

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