Helmholtz model

The inner layer (closest to the electrode), known as the inner Helmholtz plane (IHP), contains solvent molecules and specifically adsorbed ions (which are not hilly solvated). It is defined by the locus of points for the specifically adsorbed ions. The next layer, the outer Helmholtz plane (OHP), reflects the imaginary plane passing through the center of solvated ions at then closest approach to the surface. The solvated ions are nonspecifically adsorbed and are attracted to the surface by long-range coulombic forces. Both Helmholtz layers represent the compact layer. Such a compact layer of charges is strongly held by the electrode and can survive even when the electrode is pulled out of the solution. The Helmholtz model does not take into account the thermal motion of ions, which loosens them from the compact layer. 

Whatever the most acceptable model may be and as we need only a rough estimate of the amount of ions discharged, we start from the Helmholtz model of a simple parallel-plate capacitor, whose potential difference is... 

Thus, in terms of the Helmholtz model equation (2.7) becomes ... 

Thus, we now have a reasonable model of the interface in terms of the classical Helmholtz model that can explain the parabolic dependence of y on the applied potential. The various plots predicted by equation (2.18) are shown in Figures 2.5(a) to (c). The variation in the surface tension of the mercury electrode with the applied potential should obey equation (2.18). Obtaining the slope of this curve at each potential V (i.e. differentiating equation (2.18)), gives the charge on the electrode, [Pg.49]

As a result of the above considerations, the Helmholtz model of the interface now shows two planes of interest (see Figure 2.8). The inner Helmholtz plane (IHP) has the solvent molecules and specifically adsorbed ions (usually anions) the outer Helmholtz plane (OHP), the solvated ions, both cations and anions. It can be seen from Figure 2.8 that the dielectric in the capacitor space now comprises two sorts of water that specifically adsorbed at the electrode surface and that lying between the two Helmholtz planes. Continuing the analogy with capacitance, these two forms of water act as the dielectric in two capacitors connected in series. 

Figure 2Jt The double layer Helmholtz model of the electrode/clectrolytc interface.

Figure 2.9, it can be seen that the interfacial capacitance does show a dependence on concentration, particularly at low concentrations. In addition, whilst there is some evidence of the expected step function away from the pzc, the capacitance is not independent of V. Finally, and most destructive, the Helmholtz model most certainly cannot explain the pronounced minimum in the plot at the pzc at low concentration. The first consequence of Figure 2.9 is that it is no longer correct to consider that differentiating the y vs. V plot twice with respect to V gives the absolute double layer capacitance CH where CH is independent of concentration and potential, and only depends on the radius of the solvated and/or unsolvated ion. This implies that the dy/dK (i.e. straight lines joined at the pzc. Thus, in practice, the experimentally obtained capacitance is (ddifferential capacitance. (The value quoted above of 0.05-0,5 Fm 2 for the double-layer was in terms of differential capacitance.) A particular value of (di M/d V) is obtained, and is valid, only at a particular electrolyte concentration and potential. This admits the experimentally observed dependence of the double layer capacity on V and concentration. All subsequent calculations thus use differential capacitances specific to a particular concentration and potential. 

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

This leads to the Helmholtz model of the interface (Fig. 10.5) when the other contacting phase is a metal. The Helmholtz model of the interface predicts that the value of the double layer capacity (Q,) will be given by ... 

The Helmholtz model of the metal/electrolyte interface seems to be appropriate for such interfaces as Au/Na-/S-Al203. For example the interfacial potential across this interface can be varied by 8 V without appreciable continuous current flowing and over this potential range the measured value of Qi changes by only 20% (Fig. 10.6) (Armstrong, Burnham and Willis, 1976). In addition we note that ... 

Fig. 10.5 Helmholtz model of the interface between a metal and an electrolyte. The metal is shown with a negative charge (excess of electrons) which is balanced by an excess of mobile cations, the centres of which are one atomic radius from the surface.

One complication which may be present, when the Helmholtz model is in other respects appropriate, is that of specific adsorption. If one of the mobile species is to some extent chemically bound rather than being simply electrostatically bound to the metal electrode, Cji may show a dependence on the dc bias potential. Indeed this is the normal method of inferring specific adsorption. Another possibility in this case is that dl exhibits different high frequency and low frequency limits because at high frequencies the specific adsorption being an activated process is too slow to follow changes in interface potential. A further complication which is often present in real systems is the presence of an oxide layer on the surface of the metal electrode. Such an oxide layer can generate a potential... 

The appropriate model of the interface for e.g. Pt/LiCFaSOa-PEO depends on the concentration of charged species in the PEO. When the salt PEO ratio is less than 1 10 the Debye length will also be less than the size of a mobile charge. In this case again the Helmholtz model of the double layer will be appropriate. 

Of course, when the volume concentration of mobile charges is sufficiently high that the Debye length is comparable with the ionic radius of the mobile ion(s), a combination of the Helmholtz and Gouy-Chapman models is required. This is achieved by assuming that the measured Cdi value is a series combination of that due to the Gouy-Chapman model (Cgc) and that due to the Helmholtz model (Ch), i.e. 

Figure 4.4. a) Helmholtz model of a double layer qy, excess charge density on metal excess charge density in solution, on HP b) linear variation of potential in the double layer with distance from the electrode. 

The Helmholtz model was found not to be able to give a satisfactory analysis of measured data. Later, another theory of the diffuse double layer was proposed by Gouy and Chapman. The interfacial region for a system with charged lipid, R-Na+, with NaCl, is shown in Figure 4.10. 

As in the case of the Helmholtz model, the plane AA will be negative due to the adsorbed R-species. Therefore, the Na+ and CL ions will be distributed nonuni-formly due to electrostatic forces. The concentrations of the ions near the surface can be given by the Boltzmann distribution, at distance x from the plane AA, as... 

Figure 2.12 Simple capacitor model of electrode-solution interface as a charged double layer (original Helmholtz model). Negatively charged surface. Positively charged ions are attracted to the surface, forming an electrically neutral interphase.

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