Helmholtz capacitance

Figure 2.4 (a) The dependence of the potential as a function of the distance from the electrode surface, taking into account the presence of adsorbed water dipoles, (b) The interface in (a) represented in terms of two capacitors. CD is the dipole capacitance ( = Ef.0/[a - rMj0] and CM is the original Helmholtz capacitance ( - Ee0/r jO). 

Stern used this simplification in his calculations. The simplified model with only one Helmholtz capacitance is commonly referred to as the Stern model (Figure 4b), while the "extended" Stern model (Figure 4a) is designated the triple layer model. 

At high electrolyte concentrations (> 1 mol 1 ) the Helmholtz capacitance CH of the electrode shows values in the order of CH/A=0.5 pF cm-2 [Na7] and is thereby at least one order of magnitude larger than Csc/A. Therefore it can be neglected... 

The electrode/electrolyte interface discussed above exhibits a capacitance whose magnitude depends on the distribution of ions on the solution side of the interface. In relatively concentrated electrolytes, the capacitance of the Helmholtz layer dominates the interfacial capacitance. For most metals, typical Helmholtz capacitances range from 20-60 pF cm-2, and depend substantially on the applied potential, reaching a minimum at the potential of zero charge where there is no excess charge on either side of the interface. 

In general, Csc is orders of magnitude smaller than the Helmholtz capacitance, reflecting the fact that the major apart of any change in electrode potential appears across the space charge region. 

Li and Peter assumed that the space charge capacitance is much smaller than the capacitance of the Helmholtz layer the more generalised theory of Ponomarev and Peter considers the case where the space charge capacitance and Helmholtz capacitances are of comparable magnitude [60]. The attenuated IMPS response is then given by... 

Associated with the double layer at the interface there will be a Helmholtz capacitance whose value may be very roughly estimated from the... 

Jn this case one has a large Helmholtz capacitance on the solu-... 

Figure H. The ratio of the potential drop in the Helmholtz to the total potential change computed as a function of the total potential change. A static dielectric constant of 173 (typical of TiOi) and a Helmholtz capacitance of 10 pF cm were assumed and the doping density was allowed to vary from 10 cm"" (curve 1) to lO " cm" (curve 13).

The Helmholtz capacitance may be described by Ch = e o/d, where e is the local value of the dielectric coefficient and d is the Stem layer thickness. Under conditions where e is independent of surface charge density, we can identify two domains of nearly constant capacitance under varying surface charge densities (i.e., pH). When Cq Ch (at high ionic strength), C == Ch when Cq Ch (lower ionic strength), C Cq. When the surface potential is small, for example, less than 25 mV, then Cq = eeox = 2.3 (25°C) (equation... 

The upper limit to the electron injection rate constants that can be measured by IMPS is determined by Xceii and potentiostat performance. If the space charge capacitance is high, it may be impossible to determine kinj. This is the case, for example, for formic acid oxidation on Ti02, where the IMPS response is dominated by the combination of the space charge and Helmholtz capacitances [84]. It would be interesting to explore the possibilities of using intensity modulated photopotential to overcome this limitation. 

Under conditions of electron accumulation, the interfacial capacitance C of a semiconductor/electrolyte contact tends to that of the Helmholtz layer (see Sect. 2.1.3.1 with Csc > Ch)- The width of the interfacial double layer is reduced to about 0.5 nm hence, it follows the internal surface of a porous electrode. As a result, the overall interfacial capacitance of a nanoporous system can be huge, being determined by the product of the total internal surface area of the system and the Helmholtz-capacitance per unit geometric surface area [148,149]. 

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