1 double layer, Stern

Two models of surface hydrolysis reactions and four models of the electrical double layer have been discussed. In this section two examples will be discussed the diprotic surface group model with constant capacitance electric double layer model and the monoprotic surface group model with a Stern double layer model. More details on the derivation of equations used in this section are found elsewhere (3JL). ... 

Distance from Surface Fig. 125. The Stern double layer... 

The method given above for calculating the zeta-potential at a diffuse double layer may be applied to the diffuse portion of the Stern double layer. If charge density on the solid and ai and are the corresponding values on the solution sides of the fixed and diffuse layers, respectively, then the condition of electrical neutrality requires that... 

Figure 3.21. Gouy-Stern double layer model with specific adsorption at the outer Helmholtz plane. The inner layer has a constant capacit mce C. ...

The preliininary problem to be solved will then again be the problem of the distribution of the charge and the nature of the electric potential function for two Stern double layers in interaction. The next problem is the determination of Vp, for this case, as a function of the plate distance. For two diffuse double layers these problems are dealt with in Chapters IV and V.. , . ... 

In many respects we shall be able to make use of the considerations and equations given earlier, as what happens in the interaction of two Stern double layers is primarily an interpenetration of the diffuse Gouy layers of these double layers. As yve need to consider only moderate and large plate distances, the Stern layers do not interfere directly, and will be altered only secondarily by the interaction. The difference from the theory given in Chapter V will be that the electric potential of the Gouy layer, ps, will now he a function of the plate distance. 

In Figure 9.11, the expulsion of co-ions is depicted for a Gouy-Chapman-Stern double layer. Let c+ be the concentration of co-ions at a positively charged surface. In a Gouy-Chapman-Stem double layer, the co-ion expulsion per unit surface area, that is, the negative adsorption of co-ions F+ is given by... 

Figure 2. Three models of the electrochemical interface (a) the Helmholtz fixed (rigid) double layer, 1879 (b) the Gouy-Chapman diffuse double layer 1910-1913 (c)the Stern double layer, 1924, being a combination of the Helmholtz and the Gouy-Chapman concepts.

A similar approach to the boundary condition for the potential at the metal-solution interface has been applied by Biesheuvel et al., in consideration of diffuse charge effects in galvanic cells, desalination by porous electrodes, and transient response of electrochemical cells (Biesheuvel and Bazant, 2010 Biesheuvel et al., 2009 van Soestbergen et al., 2010). However, their treatment neglected the explicit effect of In principle, the PNP model could be modified to incorporate size-dependent and spatially varying dielectric constants in nanopores, as well as ion saturation effects at the interface. However, in a heuristic fashion, such variations could be accounted for in the Helmholtz capacitance of the Stern double layer model. 

Fig. 8. Electrical double layer of a sohd particle and placement of the plane of shear and 2eta potential. = Wall potential, = Stern potential (potential at the plane formed by joining the centers of ions of closest approach to the sohd wall), ] = zeta potential (potential at the shearing surface or plane when the particle and surrounding Hquid move against one another). The particle and surrounding ionic medium satisfy the principle of electroneutrafity.

The behavior of simple and molecular ions at the electrolyte/electrode interface is at the core of many electrochemical processes. The complexity of the interactions demands the introduction of simplifying assumptions. In the classical double layer models due to Helmholtz [120], Gouy and Chapman [121,122], and Stern [123], and in most analytic studies, the molecular nature of the solvent has been neglected altogether, or it has been described in a very approximate way, e.g. as a simple dipolar fluid. Computer simulations... 

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

The physical meaning of the g" (ion) potential depends on the accepted model of ionic double layer. The proposed models correspond to the Gouy Chapman diffuse layer, with or without allowance for the Stern modification and/or the penetration of small counterions above the plane of the ionic heads of the adsorbed large ions [17,18]. The presence of adsorbed Langmuir monolayers may induce very high changes of the surface potential of water. For example. A/" shifts attaining ca. —0.9 (hexadecylamine hydrochloride), and ca. -bl.OV (perfluorodecanoic acid) have been observed [68]. 

Equation (2.33) now defines the double layer in the final model of the structure of the electrolyte near the electrode specifically adsorbed ions and solvent in the IHP, solvated ions forming a plane parallel to the electrode in the OHP and a dilfuse layer of ions having an excess of ions charged opposite to that on the electrode. The excess charge density in the latter region decays exponentially with distance away from the OHP. In addition, the Stern model allows some prediction of the relative importance of the diffuse vs. Helmholtz layers as a function of concentration. Table 2.1 shows... 

Fig. 1 Double layer model for a cathode, (a) Helmholtz model (b) Gouy-Chapman model (c) Stern model. 

In the electrochemical literature one finds the Gouy-Chapman (GC) and Gouy-Chapman-Stern (GCS) approaches as standard models for the electric double layer [9,10]. 

Gouy-Chapman, Stern, and triple layer). Methods which have been used for determining thermodynamic constants from experimental data for surface hydrolysis reactions are examined critically. One method of linear extrapolation of the logarithm of the activity quotient to zero surface charge is shown to bias the values which are obtained for the intrinsic acidity constants of the diprotic surface groups. The advantages of a simple model based on monoprotic surface groups and a Stern model of the electric double layer are discussed. The model is physically plausible, and mathematically consistent with adsorption and surface potential data. 

To complement the models for the surface reactions, a model for the electric double layer is needed. Current models for the electric double layer are based on the work of Stern (21), who viewed the interface as a series of planes or layers, into which species were adsorbed by chemical and electrical forces. A detailed discussion of the application of these models to oxide surfaces is given by Westall and Hohl (2). 

The representation of the data for TiC in terms of the monoprotic surface group model of the oxide surface and the basic Stern model of the electric double layer is shown in Figure 5. It is seen that there is good agreement between the model and the adsorption data furthermore, the computed potential Vq (not shown in the figure) is almost Nernstian, as is observed experimentally. 

It is important to stress that the activity coefficients (and the concentrations) in equation 16.18 refer to the species close to the surface of the electrode, and so can be very different from the values in the bulk solution. This is portrayed in figure 16.6, which displays the Stern model of the double layer [332], One (positive) layer is formed by the charges at the surface of the electrode the other layer, called the outer Helmholtz plane (OHP), is created by the solvated ions with negative charge. Beyond the OHP, the concentration of anions decreases until it reaches the bulk value. Although more sophisticated double-layer models have been proposed [332], it is apparent from figure 16.6 that the local environment of the species that are close to the electrode is distinct from that in the bulk solution. Therefore, the activity coefficients are also different. As these quantities are not... 

Figure 16.6 The Stern model of the double layer. The outer Helmholtz plane (OHP) and the width of the diffusion layer (8) are indicated. The shaded circles represent solvent molecules. The drawing is not to scale The width of the diffusion layer is several orders of magnitude larger than molecular sizes.

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