Helmholtz layer capacitance

The traditional treatment of a double layer at electrode-electrolyte interfaces is based on its separation into two series contributions the compact ( Helmholtz ) layer and the diffusive ( dif ) layer, so that the inverse capacitance is... 

The capacitance CH and the charge transfer resistance Rct representing the Helmholtz layer. 

Double-layer capacitance The capacitance C owing to the two Helmholtz layers at the electrode solution interface. 

Cf are the resistance and capacitance due to the particulate semiconductor film R m and are the resistance and capacitance of the parts of the BLM which remained unaltered by the incorporation of the semiconductor particles Rsc and Csc are the space charge resistance and capacitance at the semiconductor particle-BLM interface and Rss and Css are the resistance and capacitance due to surface-state on the semiconductor particles in the BLM. Electrolytes short circuit the porous semiconductor particles (Rf = Rsol = 1.4 kO) such that their contribution, along with that due to the Helmholtz layer, can be neglected. This allows the simplification of the equivalent circuit to that shown in Fig. 108c. As seen, the working electrode is connected (via ions) to the semiconductor particulate film. 

Consider a simple interfacial region at a mercury/solution interface. The electrolyte is 0.01 M NaF and the charge on the electrode is 10 iC negative to the pzc. The zeta potential is -10 mV on the same scale. What is the capacitance of the Helmholtz layer and that of the diffuse layer Galculate the capacitance of the interfaces. Take the thickness of the double layer as the distance between the center of the mercury atoms and that of hydrated K+in contact with the electrode through its water layer. (Bockris)... 

As discussed previously, the surface states responsible for the reduction peak could be intrinsic surface states or states associated with a surface-attached intermediate in the series of reactions leading to O-evolution. The latter possibility was deemed to be more likely since no change in voltage across the Helmholtz layer (no change in capacitance) was observed when these states are in the oxidized form. 

The creation of a large space-charge density in the semiconductor (and a corresponding increase in capacitance) can cause the bands to become unpinned since changes in applied potential can now occur across the Helmholtz layer. 

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. 

Frequently, e.g. in the case of alkane thiol monolayers, the electrode is modified with a low-dielectric-constant layer. Film formation causes the Helmholtz layer to change from a mixture of ions and solvent with a high dielectric constant to an ion-free, often organic, layer with a low dielectric constant. The interfacial capacitance... 

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... 

In series with Csc are the capacitances associated with the Helmholtz layer and the electrolyte. It is easy to show that these are expected to be much larger under the circumstances that eqn. (36) is valid, and so the quadrature response of the semiconductor may be used to calculate a capacitance, Cobs, that is closely approximated by Csc. From eqn. (39), it is seen that... 

If Nt rises above 1013cm 2, it may be anticipated that the complete emptying of such states will affect the potential drop in the Helmholtz layer. From above, the change in potential is rO = rNtfJNM, where Nm is the maximum possible number of surface states. Writing y = riVt/iVM, we may approximate y from the estimated double-layer capacitance of 10-20/iFcm-2 this gives y (0.08-0.16) x 10"13 Nt V. The rate constants k, and ka will also depend on yft specifically... 

In the non-steady state, changes of stoichiometry in the bulk or at the oxide surface can be detected by comparison of transient total and partial ionic currents [32], Because of the stability of the surface charge at oxide electrodes at a given pH, oxidation of oxide surface cations under applied potential would produce simultaneous injection of protons into the solution or uptake of hydroxide ions by the surface, resulting in ionic transient currents [10]. It has also been observed that, after the applied potential is removed from the oxide electrode, the surface composition equilibrates slowly with the electrolyte, and proton (or hydroxide ion) fluxes across the Helmholtz layer can be detected with the rotating ring disk electrode in the potentiometric-pH mode [47]. This pseudo-capacitive process would also result in a drift of the electrode potential, but its interpretation may be difficult if the relative relaxation of the potential distribution in the oxide space charge and across the Helmholtz double layer is not known [48]. 

Introducing the differential capacitances of the inner and outer Helmholtz layer (see [3.6.26]), assuming these to be constant, 13.6.61] can be written as... 

Figure 16. Schematic repmsentation of an electrochemical double layer at a mctal/elcctrolytc solution interface, (a) The jcllium double layer (with electron spill-over region contacts a layer of (ordered) solvent molecules, chemisorption of a negative ion is also shown, (b) Representation of the double layer capacitance as a series connection of the capacitance corresponding to the double layer of the metal surface, and the capacitance of (he Helmholtz layer al the solution side.

Figure 18. Estimation of the capacitance of the electrochemical double layer Cstiso as a function of the charge density, using a series connection of the capacitance of the double layer of the metal surface [Cm < 0) and the molecular Helmholtz layer (Csoi). It is found that the electrochemical double layer capacitance shows a maximum at around the point of zero charge, where the influence of the metal phase is strongest. This is in qualitative agreement with the experimental results (Fig. 17). The bars on the capacitance axis indicate 2 x 10 F/cm, the bars on the horizontal axis indicate 5 x 10 C/cm. ...

Figure 19. The electronic structure of an n-type semiconductor/electrolyte solution interface under conditions of free electron depletion at the surface. Shown are the conduction and valence band edges as a function of the distance from the surface. The interfacial potential drop is distributed over a region in the solid (depletion region, width 4c) and the molecular Helmholtz layer at the liquid side (not shown). The interfacial capacitance is represented by a series connection of the capacitance of the depletion layer (Csc) and the Helmholtz layer (Csoi).

As with metals, the Helmholtz layer is developed by adsorption of ions or molecules on the semiconductor surface, by oriented dipoles or, especially in the case of oxides, by the formation of surface bonds between the solid surface and species in solution. Recourse to band-edge placement can be sought through differential capacitance measurements on the semiconductor-redox electrolyte interface [29j. 

The surface state capacitance Css is different from the space charge layer capacitance Csc and the Helmholtz layer capacitance Ch in that there is in general no distance associated with the surface state capacity. 

In an idealized case when the effect of the Helmholtz layer can be neglected, i.e., when Ch Csc, there is a negligible amount of surface states, that is. Css Qc. The total capacitance of the semiconductor/electrolyte interface described by Eq. (1.45) becomes C Csc. The interface capacitance as a function of the electrode potential then follows the Mott-Schottky equation ... 

K. Uosaki and H. Kita, Effects of the helmholtz layer capacitance on the potential distribution at semiconductor/electrolyte interface and the linearity of the Mott-Schottky plot, J. Electrochem. Soc. 130, 895, 1983. 

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