The Question of Instrumental Artifact in Linear Sweep Voltammetry with Positive Feedback Ohmic Drop Compensation Eugene E. Wells, Jr. Power Sources Division, Electronic Components Laboratory, USAECOM, Fort Monrnouth, N . J . The question of artifact introduced by the severe demands of positive feedback on the instrumentation has been treated for the particular case of linear sweep voltammetry. It was possible to formulate experimentally applicable theoretical criteria for rejecting voltammograms which suffer from either or both of the maladies which define the upper limit of scan rate for this technique: uncompensated solution resistance and bandpass related instrumental artifact. The effect of the faradaic reaction has been specifically included in the treatment.
A GROWING BODY of literature attests to the popularity of positive feedback techniques as a means of conquering the ohmic drop problem in potentiostatic studies ( I , 2 and references therein). Several attempts to predict, using the criteria for system stability, the amount of resistance which can be “removed” by positive feedback have appeared, and the severe demand of the technique on instrumental bandpass has been duly noted ( I , 2). Yet there has been little interpretation of how these problems relate to the user of positive feedback compensation and his goal of obtaining a cell response which is uncontaminated by the masking effect of ohmic resistance, but which is a true measure of the remaining electrochemical par ameter s. Zero effective (apparent) solution resistance is virtually impossible to achieve with existing apparatus and normally studied solutions, because if sufficient positive feedback is applied, the system becomes unstable and oscillates (1). Thus it is important to determine the effective resistance of each experimental solution at the maximum achievable degree of feedback, for this quantity determines the current which may be passed in a potentiostatic experiment before ohmic drop becomes a detectable artifact. Part of the present work, therefore, describes a convenient procedure for resistance measurements. A second possible source of artifact, instrumental bandpass, determines how “instantaneously” the apparatus can correct an applied voltage signal for ohmic drop. Consider how the perfect compensator would function in response to the first trickle of current through the cell, itself the result of a changing applied voltage (Figure 1). A voltage signal proportional to the instantaneous current through the cell would be generated by resistor Ri,amplified by the compensating operational amplifier, At, and mixed with the signal from the voltage generator at the potentiostat input. The signal which now appears at the potentiostat control points would not be the instantaneous signal generator voltage, but a signal increased by such an amount that it becomes identical to the (desired) generator voltage in the interphase region, after dropping through the resistance of the solution. The electrochemical system would respond with a current which is dependent upon the kinetics of the (1) A. A. Pilla, R. B. Roe, and C. C. Herrmann, J. Electrochem. SOC.,116, 1105 (1969). (2) E. R. Brown, D. E. Smith, and G. L. Booman, ANAL.CHEM., 40,141 1 (1968).
electrochemical process, thus providing the beginning of another cycle. If it is now realized that every real potentiostat has noninfinite frequency response and therefore a propagation delay, it should be clear that for some (large) rate of change of the generator voltage, the correcting signal from the compensating amplifier will be significantly late in arriving at the mixing point. The exact speed at which this behavior becomes detectable as artifact in the observed current would depend on the rate of change of current with voltage (a function of the kinetics of the electrode process) and the sensitivity of the electroanalytical technique to deviations from the assumed voltage input. The theoretical development of this paper is pointed toward linear sweep voltammetry (.?), principally because of the almost (4) total neglect of the technique in the literature of ohmic drop compensation. However, most of the results are independent of the voltage function, and the approach is applicable to other voltammetric techniques. The significant feature is that the faradaic reaction has been considered in determining bandpass behavior. THEORY
The present treatment allows for instrumental nonideality by incorporating the potentiostat gain function in terms of its associated time constant. The compensating operational amplifier used in this work has several orders of magnitude greater frequency response than the potentiostat and is assumed to possess an ideal (Le., frequency independent) gain function. In the mathematical derivations which follow, extensive use has been made of Laplace transformation ( 5 ) as a means of obtaining simple algebraic relationships. Thus, the variable parameters appear in terms of the (complex) Laplace variable, s, which has the dimensions of frequency. The theory is most conveniently formulated through the use of an analog electrical circuit to describe the electrochemical portion of the system. An adequate description of working electrode behavior for most cases of interest is provided using the circuit of Figure 2, in which the solution resistance terminates in the double layer capacitance, which is itself shunted by a faradaic impedance (6 and references therein). In terms of the Laplace variable the impedance function, ZE, for such a circuit would be written as follows ( I ) :
in which Z&) is the faradaic impedance function. The specific form of Z F would depend on the type, complexity, (3) R. S . Nicholson and I. Shain, ANAL.CHEM., 36,706 (1964). (4) E. R. Brown, T. G. McCord, D. E. Smith, andD. D. DeFord, ibid., 38, 1119 (1966). (5) R. V. Churchill, “Operational Mathematics,” McGraw-Hill New York, N. Y . , 1958. ( 6 ) A. A. Pilla and G. J. DiMasi, Aduan. Chem. Ser., 90, 1969.
ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971
87
Figure 2. General equivalent circuit representation of the working electrode Re is the solution resistance; Cd, the electrode capacitance; and ZF, a general faradaic impedance
i (s) =
Figure 1. Instrumentation for positive feedback ohmic drop compensation Z Erepresents the working electrode impedance, AI and A Zthe potentiostat and feedback amplifier, respectively, Ri the adjustment for degree of feedback and Ro,the potentiostat output impedance
and kinetics of the particular faradaic reaction (6), but for reasons which will become apparent momentarily, ZF need not be specified in detail if we restrict ourselves to the goals stated in the introduction. Note that the component values need not be independent of potential over the several hundred millivolt voltage excursions common in linear sweep voltammetry. Thus, unless otherwise stated, c d and ZF have voltage as a functional argument, even though the symbology has been omitted for brevity. Solution resistance is assumed to be independent of electrode voltage. Equation 1 implies that a separation of current flow between the faradaic and capacitative branches is possible, at
least in principle, for by Ohm's law: (2) (3)
The ratio of Equations 2 and 3 provides the following convenient working form: k(s) 3
~c
rU=
cdszF(s)
=
(Re
(Re
c d
+ Ri - RiK2) Cd = R,Cd
+
(6)
(7) (8)
Equation 9 predicts the current response to any perturbing voltage function V(s)for a cell of the type of Figure 2 in the circuit of Figure 1 . If ZF in Equation 4 is expressed in terms of k(s) and the result substituted into Equation 9, the following expression is obtained: i(s) =
[K5] x [1
(4)
+ Ro + Ri)
re = r / ( K 1) the current may be expressed as follows:
1 ~
where k is a function of potential as are Cd and ZF. This result will be used later. The procedure by which the system transfer functions may he combined to provide a meaningful description of the current-voltage behavior of the circuit of Figure l has been outlined in some detail ( I ) and will not he repeated here. Thus, the following equation is the result of straightforward procedures and is a general form of an equation already derived and tested for the same instrumentation using the particular case of an RC analog cell ( I ) . 88
In Equation 5 , K is the potentiostat gain, V(s) the signal generator voltage function, z the potentiostat time constant, and K2 the (constant) gain of the feedback operational amplifier. Other parameters are as indicated in Figure 1. The equation is applicable to the system whether stable o r oscillatory, and if Z , were known, could be used to theoretically predict the critical damping point following the procedure that has been used successfully for RC cells (1). For faradaic cells, the procedure involves considerable difficulty and many assumptions, and for this work a nonoscillating system will be assumed, leaving to the experimentalist the task of assuring that this is indeed the case. Upon substituting Equation 1 into 5 , simplifying, and making the substitutions:
[l
+ k(s)l v(s) cds
+ k ( s ) ] r o r d+
The goal of this analysis was to determine under what conditions a compensated voltammogram is truly a measure of the cell response. The answer hinges upon finding conditions under which instrumental parameters become insignificant in Equation 10. In mathematical terms frequency derived (bandpass) artifacts will be absent when:
ANALYTICAL CHEMISTRY, VOL. 43, NO. 1, JANUARY 1971
ASSUMP E 1
( 1 1)
unity for potentiostats with high gain, Equation 13 reduces to:
where ASSUMP =
Upon substituting the voltage function for a linear sweep :
V(s) = Since the time constants are easily measured in a manner to be described below, further progress depends upon finding k(s) and s, which are functions of the presumably unknown electrochemical mechanism, scan rate, voltage, etc. Unfortunately, it is not convenient to derive these values theoretically because of the nonequilibrium conditions which prevail in linear scan voltammetry. It is possible, however, to measure these parameters, after the fact, from the observed voltammogram. In so doing, the range of s which need be considered is limited to those frequencies actually present t o significant degree. Furthermore k(s), which is potentially troublesome because the degree of s is unknown (see Equation 4), is bounded and finite and has observable potential dependence. When evaluated in this manner, Equation 12 becomes a rejection criterion for artifact contaminated data. Examination of Equation 12 suggests that large values of s are a worst case, as might be expected. Thus, of the frequencies required t o synthesize the voltammogram t o any given point, it is the maximum significant contributor which should be used to determine if the data is valid. Procedures for finding and using maximum s will be presented in the discussion section. As for k(s), the worst case value depends upon the absolute magnitude of the time constants. Both extremes should therefore be tested. It should also be noted that Equation 12 is a sluggish function of k(s) in any case, and that the significance of this parameter in determining the accuracy of data is usually not as great as that of the frequency. For the rerraining derivations, it is convenient t o adopt the fundamental assumption of equivalent circuit impedance analysis that the circuit components are independent of potential (6). In the absence of faradaic reactions, the assumption is reasonable over even moderately large potential excursions, as is evidenced by the nominally flat background traces frequently encountered in linear sweep voltammetry. The assumption of RC cell behavior with potential independent components makes possible the convenient use of Equation 9 for direct analysis of the cell response, a n approach which leads to a n extremely convenient method of determining solution resistance and capacitance. From the equivalent circuit point of view, choosing a potential region where faradaic processes are absent corresponds t o selecting conditions under which ZF is infinite. After taking the indicated limit in Equation 9, we have:
Assume at this point that T C
+
Kfl
TeTeS
