Double layer capacitance Typical values

The Faradaic and capacitive components of the current both increase with the scan rate. The latter increases faster (proportionally to v) than the former (proportionally to y/v), making the extraction of the Faradaic component from the total current less and less precise as the scan rate increases, particularly if the concentration of the molecules under investigation is small. The variations of the capacitive and Faradaic responses are illustrated in Figure 1.7 with typical values of the various parameters. The analysis above assumed implicitly that the double-layer capacitance is independent of the electrode potential. In fact, this is not strictly true. It may, however, be regarded as a good approximation in most cases, especially when care is taken to limit the overall potential variation to values on the order of half-a-volt.10 13... 

When one considers a distance scale much smaller than 1 pm, surface roughness also is an issue to observed electrode behavior. The ratio of the microscopic surface area to the projected electrode area is usually designated the roughness factor, and can vary from 1.0 to 5 or so for typical solid electrodes, or much higher for porous electrodes. Capacitance, surface faradaic reactions, adsorption, and electrode kinetics all depend on microscopic area. For example, double-layer capacitance increases with roughness such that the apparent capacitance (C°bs) is larger than the value for a perfectly flat electrode (Cflat) as shown in Equation 10.1 ... 

A typical example of the ideal polarizable electrode is Hg, which shows a double-layer capacitance of 10 20 pF cm in aqueous electrolyte solutions. Since the double-layer capacitance is dependent on electrode potential V, minimum values of differential double-layer capacitance Ca were adopted, as defined below, where Q is the charge ... 

This is about two orders of magnitude higher than typical values observed for the double-layer capacitance. We can also evaluate the range of potential over which is equal to or larger than C. Substituting = 16 pF/cm (i.e., a typical value of C ) into Eq. 501 we find that the two values of 0 which satisfy this relationship are... 

Graphical methods provide a first step toward interpretation and evaluation of impedance data. An outline of graphical methods is presented in Chapter 16 for simple reactive and blocking circuits. The same concepts are applied here for systems that are more typical of practical applications. The graphical techniques presented in this chapter do not depend on any specific model. The approaches, therefore, can provide a qualitative interpretation. Surprisingly, even in the absence of specific models, values of such physically meaningful parameters as the double-layer capacitance can be obtained from high- or low-frequency asymptotes. 

Double layer capacitance Excess charge in an electrode surface is compensated by a build-up of opposite-charged ions (Helmholtz layer), creating an electrical double layer. This layer is mathematically treated as a parallel plate capacitor. Typical values are on the order of tens of micro farads per cm. ... 

Gasteiger and Mathias assume a thin-film structure of the ionomer of 0.5-2 nm covering the entire solid catalyst surface. Experimental support for this electrode structure comes from double-layer capacitance measurements using cyclic voltammetry and AC impedance techniques. Gasteiger and Mathias observed values that are typical of Pt and carbon interfaces with electrolyte and imply that the entire solid surface was in contact with electrolyte for these electrodes. Under several assumptions regarding structure, diffusion, and reactivity, a minimum permeability was derived for a maximum of 20 mV loss. 

The diagnostic ability of these forms of voltammetery are excellent, but their detection limits are poor, being limited to the 10 to lO M level, thus rendering them inappropriate for incorporation into chemical sensors. This rather poor limit of detection arises from the relative contributions to the total cell current of the faradaic and the capacitative currents. The capacitative current contribution is a linear function of sweep rate, whereas the faradaic current is a function of the square root of the sweep rate (for a reversible process). Thus, increasing the sweep rate in an attempt to produce larger faradaic currents inevitably causes a deterioration in the signal to noise ratio. An approximate expression can be used to evaluate the relative contribution of the capacitative (I c) and the peak faradaic currents (ip). Assuming typical values for the diffusion coefficient (10 cm s ), and the double layer capacitance (20 then... 

This last point, which has been ignored until now, in fact imposes limitations on all transient techniques. Essentially, in addition to the faradaic current flowing in response to a potential perturbation, there is also a current due to the charging of the electrochemical double-layer capacitance (for more details see Chapter 5). In chronoamperometry this manifests itself as a sharp spike in the current at short times, which totally masks the faradaic current. The duration of the double layer charging spike depends upon the cell configuration, but might typically by a few hundred microseconds. Since It=o cannot be measured directly it is necessary to resort to an extrapolation procedure to obtain its value, and whilst direct extrapolation of an /Vs t transient is occasionally possible, a linear extrapolation is always preferable. In order to see how this should be done we must first solve Pick s 2nd Law for a potential step experiment under the conditions of mixed control. The differential equations to be solved are... 

The classical circuit element that gives rise to semicircles is a charge transfer resistance shunted by a double-layer capacitance. Indeed the values of the capacitance deduced from the semicircular features are typical of those for a double-layer capacitance. The barrier to charge transfer can be either a barrier to electron transfer at the electrode/polymer interface or a barrier to ion transfer (perhaps through resolvation) at the polymer/electrolyte interface. We now derive expressions for the charge transfer resistance in either case. 

Here, Rj is the solution resistance between the electrodes, which is typically much smaller than the other components. Ret is the charge transfer resistance, which accounts for the ability of the redox compound to interact with the electrode surface via electron transport. C is the capacitance between the electrode and the charged ions in solution. This capacitance is known as the double-layer capacitance, which exists between any metal placed in an electrolyte solution. W is an element called the Warburg impedance which accounts for the effects of mass-transfer limitations. The Warburg impedance itself has both a real and imaginary component and is frequency dependent. One can calculate the total impedance of this circuit with respect to the values of the components and the frequency as shown in Eqn (4.3). 

The actual value of the double-layer capacitance depends on many variables including electrode type, electrochemical potential, oxide layers, electrode surface heterogeneity, impurity adsorption, media type, temperature, etc. [1, pp. 45-48]. Capacitance of the double layer also largely depends on the intermolecular structure of the analyzed media, such as the dielectric constant (or high-frequency permittivity), concentration and types of conducting species, electron-pair donicity, dipole moment, molecular size, and shape of solvent molecules. Systematic correlation with dielectric constant is lacking and complex, due to ionic interactions in the solution. In ionic aqueous solutions with supporting electrolyte ("supported system") the values of -10-60 pF/ cm are typically experimentally observed for thin double layers and solution permittivity e - 80. The double-layer capacitance values for nonpolar dielec-... 

---