The interfacial admittance

In this section, the principles outlined above will be applied to derive the expression for the interfacial admittance valid for a simple system as described before. In addition, this treatment will illustrate the use of impedances and admittances in the study of electrochemical kinetics. 

As both the faradaic and the non-faradaic process respond to the same interfacial potential, it is obvious to derive primarily the interfacial admittance 

This solution of the convolution integral is valid if cot 1 practically, cot 50 [28,53]. 

Obviously, the faradaic impedance equals the sum of the two contributions f ct, the charge transfer resistance, and Zw = aco-1/2 (1 — i), the Warburg impedance. Again, the meaning of the parameters Rct and a is still implicit at this stage of the treatment and explicit expressions have to be deduced from an explicit rate equation, e.g. the expressions given in eqns. (51). 

In the more explicit classical theories, the Warburg impedance was 

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The interfacial admittance applied to the study of adsorption parameters... 

Figure 3.40 Equivalent circuit representation of an electrode incorporating an adsorbed redox active species (see text for notation). The components responsible for the ER (electroreflectance) response are shown within the dotted frame, where VF is the interfacial admittance, and the subscript F refer to the current and potential associated with the faradaic or interfacial modulation.

FIG. 10 Faradaic admittance coefficient Yq for the interface between 0.1 M LiCl in water and 0.02 M tetrabutylammonium tetraphenylborate ( ) or tetrapentylammonium tetrakis[3,5-bis(tri-fluoromethyl)phenyl]borate ( ) in o-nitrophenyl octyl ether as a function of the interfacial potential difference A y. (From Ref 73.)... 

The final expression for the operational interfacial admittance, Y(s), is severely complex [17—19] and is thus far not experimentally verified, nor invoked to analyze experimental results. For further details and discussions of simpler limiting cases see ref. 53. 

In this section, the interpretation of interfacial admittance data in the case of an a.c. reversible reaction with adsorption of O is briefly described. The relationships expressing the frequency dependence were derived some time ago [15, 17], but the essential meaning of the parameters involved was fully explained only recently [143], The brief description here is derived from the latter reference. 

As it is inherent to the subject that the charging process and the faradaic process are coupled [17], we consider the total interfacial admittance Yel — + Y i, given by... 

Anticipating the treatment of the faradaic admittance in Sect. 7.4, we derive the a.c. parts of the interfacial concentrations for two cases described in the literature. 

The above analysis shows that in the simple case of one adsorbed intermediate (according to Langmuirian adsorption), various complex plane plots may be obtained, depending on the relative values of the system parameters. These plots are described by various equivalent circuits, which are only the electrical representations of the interfacial phenomena. In fact, there are no real capacitances, inductances, or resistances in the circuit (faradaic process). These parameters originate from the behavior of the kinetic equations and are functions of the rate constants, transfer coefficients, potential, diffusion coefficients, concentrations, etc. In addition, all these parameters are highly nonlinear, that is, they depend on the electrode potential. It seems that the electrical representation of the faradaic impedance, however useful it may sound, is not necessary in the description of the system. The systen may be described in a simpler way directly by the equations describing impedances or admittances (see also Section IV). In... 

The simplest method of taking into account the distribution of pores of different sizes is to use the transmission line ladder network (Fig. 9.4,9.14,9.17, or 9.19) and use different values for the parameters ri, r, and Ci or interfacial impedances z-, and calculate the total admittance by the addition of the admittances of the small pore elements. Such a method was used, for example, by Macdonald et al. [446,447] and Pyun et al. [448]. Although such a model can be used to simulate impedance spectra assuming changes in parameters with the position in the pore, it is difficult to obtain the pore parameters from the experimental spectra. 

Figure 2.1.19. A generalized transmission line where z and y are respectively the series impedance and interfacial admittance per unit length.

An important conclusion from the paper by Brug et al. [1984] is that involvement of a CPE at solid electrodes used for studies of the impedance of Faradaic reactions can severely influence the frequency dispersion of interfacial admittance, leading to large errors in the determined Faradaic rate parameters. However, those authors note that it is feasible to account for the CPE effect correctly and to check the results of impedance analysis with respect to their internal consistency. The latter can be checked by a Kramers-Kronig analysis (cf. Lasia [1999]) which requires, however, detailed frequency-response data. Their approach was supported by experimental impedance studies on proton reduction at single and polycrystalline Au electrodes and on reduction of tris-oxalato-Fe(lII) (Brug et al. [1984]). 

In 2004, Sn et al. reported the voltage-indnced assembly of mercaptosnccinic-acid-stabilized An nanoparticles of abont 1.5 nm diameter at the H2O-DCE interface. Admittance measnrements and qnasi-elastic laser scattering (QELS) were nsed to show that the snrface concentration of the nanoparticle is reversibly controlled by the interfacial polarization. No evidence of irreversible aggregation of the particles at the interface was observed, and the electrocapillary cnrves provide an estimate of the maximnm particle snrface density corresponding to 67% of a sqnare closed-pack arrangement [348]. Similarly, Sn et al. stndied the voltage-indnced assembly and photoreactivity of cadminm selenide (CdSe)... 

Another critical issue is depicted in the inset of Figure 2.9. For most polysaccharides, a commonly used plot of admittance versus frequency does not reveal the appearance of the extra semicircle related to interfacial polarization. It is noteworthy that data treatment proposed in this study allows one to identity and separate these... 

In Section 7.5, we analyze the double layer charge in a solution as a function of the perpendicular distance from the solid surface. No double layer formations are considered in the Maxwell—Wagner theory (Section 3.5.1). However, in wet systems and in particular with a high volume fraction of very small particles, the surface effects from counter-ions and double layers usually dominate. This was shown by Schwan et al. (1962). By dielectric spectroscopy, they determined the dispersion for a suspension of polystyrene particles (Figure 3.10). Classical theories based on polar media and interfacial Maxwell—Wagner theory could not explain such results the measured permittivity decrement was too large. The authors proposed that the results could be explained in terms of surface lateral) admittance. 

The bulk conductance and therefore bulk conductivity of SPE and NCPE thin fUms has been calculated using complex admittance B-G) spectroscopy technique (Chandra et al., 1983). This techniqne is an effective tool for monitoring interfacial phenomena. A typical B-G plot for NCPE film is shown in Figure 4. Using B-G plot, the bulk conductivity (a) has been calculated from the relation... 

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