In
electrochemistry
Electrochemistry is the branch of physical chemistry concerned with the relationship between electrical potential difference, as a measurable and quantitative phenomenon, and identifiable chemical change, with the potential difference as an outco ...
, a Randles circuit is an
equivalent electrical circuit that consists of an active
electrolyte resistance in
series
Series may refer to:
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* Caroline Series (born 1951), English mathematician, daughter of George Series
* George Series (1920–1995), English physicist
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with the parallel combination of the
double-layer capacitance Double-layer capacitance is the important characteristic of the electrical double layer which appears, for example, at the interface between a conductive electrode and an adjacent liquid electrolyte. At this boundary two layers of charge with oppo ...
and an
impedance () of a
faradaic reaction. It is commonly used in
electrochemical impedance spectroscopy
Dielectric spectroscopy (which falls in a subcategory of impedance spectroscopy) measures the dielectric properties of a medium as a function of frequency.Kremer F., Schonhals A., Luck W. Broadband Dielectric Spectroscopy. – Springer-Verlag, 200 ...
(EIS) for interpretation of impedance spectra, often with a
constant phase element (CPE) replacing the double layer capacity.
The Randles equivalent circuit is one of the simplest possible models describing processes at the electrochemical interface. In real electrochemical systems, impedance spectra are usually more complicated and, thus, the Randles circuit may not give appropriate results.
Explanation
Figure 1 shows the equivalent circuit initially proposed by John Edward Brough Randles for modeling of interfacial electrochemical reactions in presence of semi-infinite linear diffusion of electroactive particles to flat electrodes. A simple model for an electrode immersed in an electrolyte is simply the series combination of the ionic resistance, , with the double layer capacitance, . If a faradaic reaction is taking place then that reaction is occurring in parallel with the charging of the double layer – so the charge transfer resistance, , associated with the faradaic reaction is in parallel with . The key assumption is that the rate of the faradaic reaction is controlled by diffusion of the reactants to the electrode surface. The diffusional resistance element (the Warburg impedance, ), is therefore in series with .
In this model, the impedance of a faradaic reaction consists of an active
charge transfer resistance and a specific electrochemical element of diffusion , represented by a
Warburg element
:
where
* is the
Warburg coefficient;
* is an
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
;
* is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
.
Identifying the Warburg element
In a simple situation, the Warburg element manifests itself in EIS spectra by a line with an angle of 45 degrees in the low frequency region. Figure 2 shows an example of EIS spectrum (presented in the
Nyquist plot
In control theory and stability theory, the Nyquist stability criterion or Strecker–Nyquist stability criterion, independently discovered by the German electrical engineer at Siemens in 1930 and the Swedish-American electrical engineer Har ...
) simulated using the following parameters: = 20 Ω, = 25 μF, = 100 Ω, = 300 Ω•s
−0.5.
Values of the charge transfer resistance and Warburg coefficient depend on physico-chemical parameters of a system under investigation. To obtain the Randles circuit parameters, the fitting of the model to the experimental data should be performed using complex nonlinear least-squares procedures available in numerous EIS data fitting computer programs.
References
*{{cite journal, last1=Randles, first1=J. E. B., title=Kinetics of rapid electrode reactions, journal=Discussions of the Faraday Society, volume=1, year=1947, pages=11, issn=0366-9033, doi=10.1039/df9470100011
*A. Lasia. Electrochemical impedance spectroscopy and its applications. In: Modern Aspects of Electrochemistry. Volume 32. Kluwer Academic/Plenum Pub. 1999, Ch.2, p. 143.
Analog circuits