Compact double layer

Atomic Polarization Fields Ionic Fields Electric Dipole Fields The Helmholtz Planes Diffuse Double Layer Compact Double Layer Potential Transients Constant Current Constant Potential Faradaic Processes Non-Faradaic Ideal Polarizable... 

The picture of the compact double layer is further complicated by the fact that the assumption that the electrons in the metal are present in a constant concentration which discontinuously decreases to zero at the interface in the direction towards the solution is too gross a simplification. Indeed, Kornyshev, Schmickler, and Vorotyntsev have pointed out that it is necessary to assume that the electron distribution in the metal and its surroundings can be represented by what is called a jellium the positive metal ions represent a fixed layer of positive charges, while the electron plasma spills over the interface into the compact layer, giving rise to a surface dipole. This surface dipole, together with the dipoles of the solvent molecules, produces the total capacity value of the compact double layer. 

Barlow, C. A., and J. R. MacDonald, Theory of discreteness of charge effects in the electrolyte compact double layer, AE, 6, 1 (1967). 

Classical model of the compact double layer at interfaces... 

Figure 5-11 shows a simple model of the compact double layer on metal electrodes. The electrode interface adsorbs water molecules to form the first mono-molecular adsorption layer about 0.2 nm thick next, the second adsorption layer is formed consisting of water molecules and hydrated ions these two layers constitute a compact electric double layer about 0.3 to 0.5 nm thick. Since adsorbed water molecules in the compact layer are partially bound with the electrode interface, the permittivity of the compact layer becomes smaller than that of free water molecules in aqueous solution, being in the range from 5 to 6 compared with 80 of bulk water in the relative scale of dielectric constant. In general, water molecules are adsorbed as monomers on the surface of metals on which the affinity for adsorption of water is great (e.g. d-metals) whereas, water molecules are adsorbed as clusters in addition to monomers on the surface of metals on which the affinity for adsorption of water is relatively small (e.g. sp-metals). 

Fig. 5-11. A simple model of an interfacial compact double layer on metal electrodes H20,j = adsorbed water molecule H20. = water molecule coordinated with ions H20 . = free water molecule ih = hydrated ions. [From Bockiis-Devanathan-Muller, 1963.]...

Fig. 6-12. Interfacial excess charge in an compact double layer com-prising adsorbed water molecules and hydrated ions = potential of the compact layer (HL).

A simple parallel plate condenser model (Fig. 5-12) gives the electric capacity Ch of the compact double layer as shown in Eqn. 5-8 ... 

Chemisorption of anions at the electrode interface involves dehydration of hydrated anions followed by adsorption of dehydrated anions which, then, penetrate into the compact double layer to contact the interface directly, this result is called the contact adsorption or specific adsorption. The plane of the contact adsorption of dehydrated anions is occasionally called the inner Helmholtz plane... 

For the electron transfer of hydrated redox particles (the outer-sphere electron transfer), the electrode acts merely as a source or sink of electrons transferring across the compact double layer so that the nature of the electrode hardly affects the reaction kinetics this lack of influence by the electrode has been observed for the ferric-ferrous redox reaction. On the other hand, the electron transfer of adsorbed redox particles (the inner-sphere electron transfer) is affected by the state of adsorption so that the nature of the electrode exerts a definite influence on the reaction kinetics, as has been observed with the hydrogen electrode reaction where the reaction rate depends on the property of electrode. 

The reaction path from the initial state to the final state of an elementary step is represented by the potential eneigy curves of the initial and final states of a reacting particle as shown in Fig. 7-6, where the reaction coordinate x denotes the position of a reaction particle moving across a compact double layer on the electrode interface. 

In the case of electrode reactions, the activation energy depends on the electrode potential. We now consider an elementary step in which a charged particle (charge number, zi) transfers across the compact double layer on the electrode interface as shown in Fig. 7-7. In the reaction equilibrium, where the electrochemical potentials of reacting particles are equilibrated between the initial state and the final state (Pk o = Pf( i)), the forward activation energy equals the backward activation energy (P , - Pi = P, i- Pr) P , is the electrochemical potential of the reacting particle at the activated state in equilibrium. 

Similar types of electric double layer may also be formed at the phase boundary between a solid electrolyte and an aqueous electrolyte solution [7]. They are formed because one electrically-charged component of the solid electrolyte is more readily dissolved, for example the fluoride ion in solid LaFs, leading to excess charge in the solid phase, which, as a result of movement of the holes formed, diffuses into the soUd electrolyte. Another possible way a double layer may be formed is by adsorption of electrically-charged components from solution on the phase boundary, or by reactions of such components with some component of the solid electrolyte. For LaFa this could be the reaction of hydroxyl ions with the trivalent lanthanum ion. Characteristically, for the phase boundary between two immiscible electrolyte solutions, where neither solution contains an amphiphilic ion, the electric double layer consists of two diffuse electric double layers, with no compact double layer at the actual phase boundary, in contrast to the metal electrode/ electrolyte solution boundary [4,8, 35] (see fig. 2.1). Then, for the potential... 

The simplest model of the structure of the metal-solution interphase is the Helmholtz compact double-layer model (1879). According to this model, all the excess charge... 

The compact double layer extends from the electrode to the plane of the fixed charges at a distance x = Xjj from the electrode. The diffuse double layer extends from the distance Xh to the bulk of the solution. This is shown schematically in Figure 4.9. 

The simplest model of the structure of the metal-solution interphase is the Helmholtz compact double-layer model (1879). According to this model, all the excess charge on the solution side of the interphase, qs. is lined up in the same plane at a fixed distance away from the electrode, the Helmholtz plane (Fig. 4.4). This fixed distance xH is determined by the hydration sphere of the ions. It is defined as the plane of the centers of the hydrated ions. All excess charge on the metal, qM, is located at the metal surface. 

In an attempt to rationalize the measured capacitance values, and especially the low value for the basal plane (ca. 3pF/cm2), these authors first concluded that space charge within the electrode is the dominant contribution (rather than the compact double layer with ca. 15-20 pF/cm2, or the diffuse double layer with >100 pF/cm2). They then applied the theory of semiconductor electrodes to confirm this and obtained a good agreement by assuming for SAPG a charge carrier density of 6 x 1018/cm3 and a dielectric constant of 3 for GC, they obtained 13 pF/cm2 with the same dielectric constant and 1019 carriers per cubic centimeter. 

With both the measured capacitance and the diffuse-double-layer capacitance known as functions of E, the capacitance of the compact double layer can be calculated from Eq. 19G as a function of potential. 

How can we judge whether there is agreement between experiment and theory. For one thing, the capacitance of the compact double layer should be independent of concentration. Second, in concentrated solution, the diffuse double layer should have very little effect on the observed capacitance except at, or very close to, E. Thus, repeating these five steps in solutions of different concentrations should yield the same plot of C versus E, and this should coincide with the capacitance measured in concentrated solutions. 

What is the "correct" value of the thickness of the parallel plate capacitor in Eq. 3G. This may be seen by reference to Fig. 6G(a), which shows the surface of a negatively charged electrode covered with a layer of water molecules. The distance of closest approach of a cation is the sum of the diameter of a water molecule (0.27 nm) and its own hydrated radius. For the latter we can use the so-called Stokes radii, calculated from electrolytic conductivity data, which are in the range of 0.2-0.3 nm for most ions. Thus, the thickness of the compact double layer (i.e., the distance of the outer Helmholtz plane from the metal) is 0.47-0.57 nm. 

The solvent also acts as a dielectric medium, which determines the field diji/dx and the energy of Interaction between charges. Now, the dielectric constant e depends on the inherent properties of the molecules (mainly their permanent dipole moment and polarizability) and on the structure of the solvent as a whole. Water is unique in this sense. It is highly associated in the liquid phase and so has a dielectric constant of 78 (at 25 C), which is much higher than that expected from the properties of the individual molecules. When it is adsorbed on the surface of an electrode, inside the compact double layer, the structure of bulk water is destroyed and the molecules are essentially immobilized... 

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