Potential Impulse Method, Operational Immittance, and the Study of Electrode Processes Ronald L. Birke Department of Chemistry, Uniuersity of South Florida, Tampa, Fla. 33620
The theory and methodology for studying electrode processes with a potential impulse function is considered. A treatment is given of relationships between time domain and operational domain functions in both the Laplace and Fourier (frequency) domains. The response functions which are generated by the application of a potential impulse to various equivalent circuits for different electrode processes are derived. I n addition a computerized system for on-line calculation of operational impedances and admittances is discussed.
VARIOUSEXPERIMENTAL METHODS have been developed to study electrode processes, and these methods are usually classified by the type of input function applied to an electrochemical cell. In addition, these methods can be classified by whether the perturbing signal is of a small amplitude or large amplitude excursion. Experiments designed with a perturbation of small amplitude excursion offer two advantages: a linearized model of the system can be made, and the potential dependency of the parameters can be minimized. Among the many methods which are based on a perturbing signal of small amplitude are those which utilize a current step, a potential step, a sinusoidal voltage, and a charge impulse. The charge impulse method, known as the coulostatic impulse method, could equally well be called a current impulse method. It has, in fact, been pointed out by Weir and Enke ( I ) that a constant current pulse of short duration is identical t o a charge impulse. More generally it can be stated that an impulse of charge is identical to the integral of an impulse of current. In the formalism of electrical engineering, it has long been the practice to speak of both a current and a potential impulse as an input or excitation function for a system. This paper will deal with a new method for studying electrode processes in which the input function is a potential impulse. The characteristics of the many methods developed for the study of electrode processes are different, and each method may have particular advantages for a given situation. The capabilities of the different methods have been reviewed by Damaskin ( 2 ) . One can by no means state that the ultimate measuring system has been developed, and even after many years of exploration, the study of electrode processes is a subject of active and demanding research. There are several characteristics of experimental methods based on a potential impulse which appear t o have advantages over other methods. For many electrode systems, it is preferable to control the potential of the electrode being studied. With a current impulse, the electrode cannot be potentiostated since the system should be at open circuit after impulse decay. With a potential impulse, it is possible t o control the potential between the working electrode and a reference electrode and still apply an extremely fast rise potential impulse to the system. (I) W. D. Weir and C. G. Enke, J . Phys. Chem., 71, 275 (1967). (2) B. B. Damaskin, “The Principles of Current Methods for the Study of Electrochemical Reaction,” McGraw Hill, New York, N. Y . , 1967.
The use of a single one-shot impulse input leads naturally to the conception of another type of input or excitation function, which is made up of a train of repetitive impulses. If the condition is maintained whereby the response of the first impulse dies away completely before the next impulse of the train, then sampling techniques can be used. The use of sampling detection not only allows the measurement of very fast signals but also allows one to use averaging techniques to improve signal to noise ratio. It should be noted that Reinmuth (3) suggested the use of sampling techniques for a current impulse train, but the state of the art of sampling techniques was not sufficiently developed at the time to be feasible. Of course, a potential step method could be used with a repetitive signal, but such a method would necessitate longer times between the repetitive signals than would a potential impulse method. The basic measurement goal in the study of electrode processes is a determination of all the parameters which describe the interfacial system. These parameters include the differential capacitance of the double layer, adsorbed surface concentrations and their variation with potential and solution concentrations, rate constants for adsorption processes, heterogeneous and homogeneous reactions, and coefficients for mass transfer processes such as diffusion. In addition, the potential dependency of these parameters, if there is any, must be examined. The response of an electrode to an input signal will be a function of these system parameters. This response signal when measured as function of time will be a complicated function of both the system parameters and the input function. A simplifying conception is to view the system parameters as making up a linear equivalent circuit of impedance elements. Thus in a two-electrode system if the impedance of the test electrode is much larger than that of the other electrode, the measurement of the impedance of the system is the impedance of an equivalent circuit which represents the electrode process at the test electrode. Using an ac signal, components of this impedance can be measured directly from the response signal since the response will be in the frequency domain where impedance is defined. An analogous measurement to that of ac impedance can be made using any type of a small signal amplitude perturbation. To make such a measurement with an input signal other than a sinusoid, the response signal must be mathematically operated on to transform it to either the frequency domain or to the Laplace domain where impedance or related functions are in fact defined. We wish to emphasize that a rigorous treatment of an electrode process can be made in the transformed domain in terms of an equivalent network of circuit elements consisting of the system parameters as long as the system is linear. The impedance elements of an electrode which are measured by a transformation of the time domain signal can be called the operational impedance of the electrode. The reciprocal of the impedance function is called the ad(3) W. H. Reinmuth, ANAL.CHEM., 34, 1272, (1962).
ANALYTICAL CHEMISTRY, VOL. 43, NO. 10, AUGUST 1971
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mittance, and the term immittance introduced by H. W. Bode denotes either impedance or admittance. Historically, it was Wijnen (4) who first suggested the advantage of transforming the experimental response function to the Laplace or frequency domain. However, Wijnen did not consider the operational impedance of an electrode system, and it was Levart and Poirier d’Ange d’Orsay (5) who investigated the methodology of using the experimentally determined operational impedance to determine the parameters of an electrode process. More recently Pilla (6) has given a detailed account of real axis (Laplace) and imaginary axis (Fourier) transformations to generate experimentally obtained operational impedances, and he has discussed the advantages of using these methods for data analysis. He points out that if both input and response functions are transformed and the operational impedance is found, then nonideality in the excitation function does not affect the data analysis. For measurements where analysis is to be performed directly on time domain data, an ideal form of the excitation function such as a step function is usually assumed in making the calculations, and this assumption may not be valid. The utility of the operational immittance approach is that measurements can be extended to very high frequencies where the finite rise time of input functions must be considered. Pilla has emphasized this method for potentiostatic techniques; of course, the method is a general one. The operational impedance found by the transformation of experimental time data is for all practical purposes identical with the impedance measured with ac signals. The fact that the “normal” impedance, which is a function of frequency in the complex plane, can be obtained from the Fourier or Laplace transformation of pulse experiments has been pointed out by Van Leeuwen, Kooijman, SluytersRehbach, and Sluyters (7). These authors, however, avoid the direct transformation of pulse data, and they analyze their experimental data by using trial functions whose Laplace transforms are known. However, direct Fourier or Laplace transformation of input and response time domain signals for an impedance measurement should be possible with presently available electronic equipment, In the following sections of this paper, a general treatment will be given for the relationships between Laplace and frequency domain functions and time domain functions. This treatment is definitely worthwhile because it points out several facts which are of consequence for measuring electrode immittances. Following the general treatment, a discussion will be given of equivalent circuits for electrode processes and of the theory of the single impulse potential input method with comparisons of this method and the current impulse method. Finally, a computer based instrumental setup with sampling detection for making accurate immitance measurements will be described. Relationships between the Time Domain and the Laplace or Frequency Domain. An electrode process can be described in the time domain by a system of differential equations with initial and boundary conditions. If linearized conditions are maintained, these equations can usually be reduced to a set of linear equations with constant coefficients. The
input function, sometimes called the forcing function, and the response function will be variables which appear in the equations. A solution of the entire boundary value problem usually involves the solution of a partial differential equation with appropriate boundary conditions for the concentration which is a function of the distance from the electrode and the time after the application of the input signal. However, it is possible to transform the original functions with a Laplace or Fourier transform to a so-called image space where the time variable is replaced by the Laplace variable s i n a Laplace transformation or by the angular frequency variable w in a Fourier transformation. Mathematically, the Laplace transform of the function f ( t ) is defined by
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where the symbol d: means take the Laplace transform and s is the complex variable, s = u j w . If f ( t ) is a causal function which means that f ( t ) = 0 for t < 0, and
+
Lm
‘f(t)ldt
