ANALYSIS AND APPLICATION OF CURRENT PULSE TECHNIQUES IN ELECTROCHEMICAL KINETICS Thesis for the Degree of ‘Ph. D. MICHIGAN STATE UNIVERSITY PETER H. IDAUM 1969 THESIS This is to certify that the thesis entitled Analysis and Application of Current Pulse Techniques in Electrochemical Kinetics presented by Peter H. Daum has been accepted towards fulfillment of the requirements for _..Eh..D. _ degree in_.Chemi.stry @528 l, Major professor 6' Date , ‘7 0-169 _—, A. 4.1—; L‘ _M._“ n _ Michigan . tare Univers: -y ABSTRACT ANALXSIS AND APPLICATION OF CURRENT PULSE TECHNIQUES IN ELECTROCHEMICAL KINETICS By Peter H. Daum The determinate errors involved in the various methods of analyzing the data of the current impulse and coulostatic techniques when the relaxations are neither charge transfer nor diffusion controlled are discussed as a function of the ratio of the charge and diffusional time constants Tc/Td' The validity of the application of the simple charge transfer assumption is found to be dependent on an accurate knowledge of the capacitance and on accurate measurements of the potential at short times in the decay. The accuracies of a nomographic and a curve fitting technique of correcting the data for the influences of diffusion are discussed. The accuracy of the first is found to be dependent on an accurate know- ledge of the capacitance and short time measurements while that of the second is found to be relatively inde- pendent of Tc/Ia and the time at which the measurements are made. providing that a sufficient portion of the observed decay is charge transfer controlled. Peter H. Daum The current impulse technique is used to study the electrochemical kinetics of the hexacyanoferrate(III)/(II) couple on platinum. Some new instrumentation is developed which compensates the ohmic potential and allows measure- ments to be made at extremely short times. The exchange rate of the reaction is found to be strongly dependent on the oxidation state of the electrode surface. An oxidized electrode surface state and a reduced electrode surface state are experimentally defined. The reduced exchange rate at oxidized electrodes could be accounted for in terms of a reduction of the "active area” of the elec- trode. The transfer coefficient and activation energy are not affected by surface oxidation. A11 experi- mental evidence points to a simple first order electron transfer reaction in both cases. The results of the investigation are found to be in concordance with those measured by other techniques when surface oxidation effects are taken into account. The determinate errors involved in the measurement of the exchange current with the galvanostatic technique are examined as a function of Tc/Td. It is found that Delahay's reduced equation for the calculation of the exchange current from the extrapolation of the n Kg. té curve to zero time is not valid for the conditions pre- viously reported. A new approach to galvanostatic Peter H. Daum measurements is suggested which is dependent on high current short time observations. Some new instrumenta- tion for the compensation of ohmic potential is des- cribed which allows these measurements to be made. A computer program is developed to analyze the experi- mental data. Some preliminary results are reported on the hexacyanoferrate(III)/(II) couple. ANALYSIS AND APPLICATION OF CURRENT PULSE TECHNIQUES IN ELECTROCHEMICAL KINETICS By 4" It? Ly Peter H? Daum A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1969 ACKNOWLEDGMENTS The author would like to express his appreciation to Professor C. G. Enke for his helpful guidance and encouragement in the course of this work. He would also like to express appreciation to his wife, Mary. whose encouragement, understanding, and typing of the final draft made this work possible. Support from the National Science Foundation and the Lubrizoil Foundation is also gratefully acknowledged. 11 I. II. III. IV. TABLE OF CONTENTS INTRODUCTION TO THE CURRENT IMPULSE TECHNIQUEcso.ooooocooocso... A. Description of the Current Impulse Technique . . . . . . . . . . . . . . . B. History of the Current Impulse Technique . . . . . . . . . . . . . C. Comparison of the Current Impulse Technique With Other Electrochemical Relaxation Techniques . . . . . . . D. Limitations of the Current Impulse Technique . . . . . . . . . . . . . . . . NUMERICAL AND GRAPHICAL ANALYSIS OF THE DATA OF THE COULOSTATIC AND CURRENT IMPULSE TECH- NIQUES O O O O O O O O O O O O O O O O O O O A. Theory . . . . . B. Graphical Analysis . C. Computer Analysis . . D. Nomographic Analysis E. Conclusion . . . . . THE ELECTROCHEMICAL KINETICS OF THE HEXA- CYANOFERRATE(III)/(II) COUPLE ON PLATINUM A. Introduction . . . . . . . B. Experimental . . . . . . . 1. Instrumentation . . . . 2. Cells and Electrodes . 3. Reagents and Solutions C. Experimental Results . . . 1. Reduced Electrodes . . 2. Oxidized Electrodes . . 3. Mechanistic Conclusions A. Comparison of Results . A NEW APPROACH TO GALVANOSTATIC MEASURE- MEN TS O O O O O O O O O O O O O O O O O O O O 111 Page 14 18 18 20 37 #6 IV. A. B. C. D. Introduction . . . . . . . . . . . . . Scope of Reported Research . . . . . . Theory-000.000.000.000 A New Approach t Galvanostatic Measure- mentS................ 1. Experiment . . . . . . . . . . . 2. Results and Conclusions . . . . . REFERENCES O O O O O O O O O O O O O O O O O O APPENDICES A. PROGRAMS FOR NUMERICAL CALCULATIONS ON COULOSTATIC DATA O O O O O O O O O O O A. Equations for Calculation of Over- potential Time Curves . . . . . . B. Curve Fitting . . . . . . . . . . COMPUTER PROGRAM FOR NUMERICAL CALCU- LATIONS ON THE GALVANOSTATIC TECHNIQUE iv Page 100 108 109 115 116 117 121 124 12h 126 137 Table I. II. A-Ic LIST OF TABLES Page values of the Apparent Exchange Current Density for Solutions of Varying Con- centrations of K Fe(CN)6 and K4Fe(CN)6 in 1.00M KCl at 250 o o o o c s c o o c s 77 Comparison of Present Kinetic Results With Results of Previous Investigators . . . . 99 PROGRAM LESSQ' DATA INPUT 0 o o o o o o o o 129 LIST OF FIGURES Figure Page 1. Electrical analog for a current impulse experiment O O O O O O O O O O O O O O O O 2 2. Theoreticgl decay curves with Cd = 2 I 10- F/Om ’ TC/T = 35.0. IO = 0018, 11:1. DO=D§=1O' cm/sec, Co=CR= 10-5 moles/c . . . . . . . . . . . . . . . 22 3. Determinate error in the measurement of I as a function of Tc/T assuming the capa- citance is known exacgly. for various normalized points on the decay . . . . . . 26 4. Determinate error in measurement of the dis- charge capacitance by extrapolating to zero time from the points where t/td = 1.0 and 2.25 as a function of TO/Td . . . . . . . . 29 5. Determinate error in the measurement of the exchange current using the simple charge transfer approximation by taking the slope of the line between the points where t/Td=lcoar1d20250000000000so 31 6. Determinate error in the measurement of the charge capacitance by taking the slope of the line between the points where t/Td=1.0and2.25............ 35 7. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCI. CO :3 0001“. CR = 0.01” o c c c o 38 8. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl. Co = 0.01M, CR = 0.07M . . . . . 39 9. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl, Co = 0.001M. CR = 0.01M . . . . 40 vi Figure Page 10. Experimental and theoretical relaxation curves for the Hg(I)/H8 system in 1M HClOu, C = OO001M O O O O O O O O O O O O O O O O “3 o 11. Block diagram of experimental system for compensation of ohmic drop . . . . . . . . 5H 12. Block diagram of experimental system without ohmic drOp compensation . . . . . . . . . . 59 13. Experimental cell . . . . . . . . . . . . . . 62 1a. Differential capacitance from charge and dis- charge data of the current impulse tech- nique, of pla inum in IN KCl as a function or 108(F8(CN)§-)'.. o c o o c s o c o c o o 68 15. Differential capacitance from charge and dis- charge data of the current impulse tech- nique, of pla inum in 1M KCl, as a function of log(Fe(CN)§') . . . . . . . . . . . . . 70 16. Equivalent circuit of an electrochemical cell. for the case of simple charge trans- fer. Cd = double layer capacitance. R = electrolyte resistance, R = the fara- dfiic resistance. and Zm(t) = the mass trans- portlmpedance.............. 72 1?. Differential capacitance from discharge data of the current impulse technique, of platinum in 1M KCl, as a function of the measured potential 3g. S.C.E.. . . . . . . . 75 18. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl. Co = 0.003M, CR = 0.01M . . . . 79 19. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl, Co = 0.01M. CR = 0.01M . . . . . 80 20. Reaction-order plot for the hexacyanoferrate (III)/(II) couple on platinum in 1H KCI. CO = 0.01“ O O O O O O O O O O O O O O O O 83 21. Reaction-order plot for the hexacyanoferrate (III)/(II) couple on platinum in in KCl, CE = OOOIM O O O O O O O O O O O O O O O O 85 vii Figure Page 22. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in in KCl. 00 = 0.01M. CR = 0.01M. oxidized electrme O O O O O O O O O O O O O O O O O 88 23. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl. Co = 0.01M, CR = 0.01M, oxidized eleCtrOde O O O O O O O O O O O O O O O O O 92 2h. Reaction-order plot for the hexacyanoferrate (III)/(II) couple on platinum in 1M KCl. CR 2 0.01M. oxidized electrode . . . . . . 96 25. Determinate error incurred in measurement of exchange current in the galvanostatic tech- nique by using Equation Iveh . . . . . . . 113 26. Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl. CO = 0003”. CR = 0001” o s c c o 119 viii I. INTRODUCTION TO THE CURRENT IMPULSE TECHNIQUE A. ‘pescription of the Current Impulse Technique The current impulse technique (1) is a transient perturbation method for the study of rapid electro- chemical reactions. The potential of the electro- chemical cell is observed in response to a very brief impulse of constant current of precisely defined dura- tion I, and amplitude it, as shown in Figure 1. Because the pulse is of extremely short duration. the transient impedance of the double layer capacitance Cd is small relative to the faradaic resistance Rf (Rf = RT/nFIO). and the double layer capacitance is charged to some new potential nt=0' before a significant amount of charge is consumed by the faradaic process. If the impulse is applied so that the cell is decoupled from the pulse source at the termination of the impulse, the discharge of the excess charge stored in the double layer can take place only through the faradaic process. In the absence of mass transport processes the potential follows a simple exponential decay law with time :1: nt=oeXp(-t/Rde)o The Current Pulse IL ——"‘lTI‘—— applied to ZmIt ) r" 105 I2 Cd - Double layer capacitance Rf - Faradaic resistance R - Solution resistance Zm(t) - Mass transport impedance Observed Response /(d'q/dt) . It/Cd 17 = 77,,O°*P(’I/Rfcd) Figure 1 Electrical analog for a current impulse experiment exchange current 10 of the reaction can be calculated directly from the slope of the log(n) gs. t curve if we know the capacitance, from the relationship slope = (nF/2.303RT)(Io/Cd). There are two ways of obtaining a measure of the double layer capacitance from a current impulse experiment. The log(n) 13. t curve can be extra- polated to zero time (defined as the time of the termination of the impulse), and the capacitance can then be calculated from the relationship Cd = itr/”t=0° It can also be determined by measuring the slope of the overpotential time curve while charging the double layer with a constant current from the relationship Cd = it/(dn/dt). These capacitances are known as the charge and discharge capacitances respectively. and theoretically should have the same value. When mass transport processes become important. the slope of the log(n) zg, t curve decreases as time proceeds. and it becomes much more difficult to obtain the kinetic parameters from the overpotential time data. Specialized mathematical techniques may be needed to extract the kinetic data from the experimental curves. 3- mammm mwm The principles of the coulostatic technique, which is the historical and logical predecessor to the current impulse technique, were developed independently and simultaneously by Reinmuth (2,3) and Delahay (h-7), though each admitted the priority of the principle of the method to Barker (8,9). The theoretical treatments of the two authors. although from slightly different points of view. lead to essentially the same results. However, Reinmuth's formalism is more elegant and is easier to relate to physical concepts than Delahay's. Both authors considered the problem including mass transport processes. and derived a general equation which they showed could be reduced to two simpler forms when the relaxation was either essentially charge trans- fer controlled or essentially mass transfer controlled. The two authors took alternative experimental approaches to the problem of generating a coulombic impulse in a time negligible with respect to the time constant of an electrochemical reaction. Delahay (10) used a simple device consisting of a small. high quality capacitor which was simply shunted across the cell through a system of relays after being charged with a battery to a known voltage. Thus a charge of accurately known coulombic content q = CV, where C is the capaci- tance of the capacitor, and V the voltage of the battery, was rapidly injected into the system. Reinmuth (2) coupled the cell to a fast rise time pulse generator through a small capacitor. Pseudo- differentiation of the leading edge of the pulse by the R-C combination of the cell resistance and coupling capacitor resulted in the application of a coulombic pulse of a magnitude fixed by the pulse voltage and the capacitance of the coupling capacitor. Both of these experimental techniques gave pulses of very short duration but of a rather undefined form. The major experimental difficulty of these two approaches is the large ohmic drop due to uncompensated solution resistances, which appears throughout the duration of the pulse. This ohmic drop may be many times the magnitude of the relaxation signal. and may 'drive the amplifiers of the measuring system into saturation. Recovery of the amplifiers from this over- drive may be quite slow. This often prevents the observation of any meaningful relaxation data for several microseconds after the start of the experiment. Since the first portion of the relaxation curve is obscured in many cases by relaxation of the amplifiers and by residual IR drop, the capacitances obtained in this method by extrapolation of the relaxation curve to zero time are necessarily somewhat uncertain due to the length of the extrapolation. This inability to make accurate short time measurements also limits the rates of the reactions which can be studied to those in which the half times are much greater than the time interval before reliable measurements can be made. The currentlimpulse technique was conceived by Weir and Enke (i)to minimize some of the experimental difficulties encountered with the coulostatic technique. It is a simple but important modification of the coulo- static technique. Instead of using a charged capacitor to generate the coulostatic pulse, a constant current pulse generator is used. This produces several significant advantages. When the pulse is applied to the cell, the double layer capacitance charges linearly, and from the slope of this charging curve and the magnitude of the applied current, the capacitance may be calculated. This eliminates the unnecessarily long extrapolations to zero time of the coulostatic method to obtain the capa- citance. Second. all ohmic contributions to the meas- ured potential vanish instantaneously upon the cessation of the current pulse, in contrast to the coulostatic case where the charge is injected by a small capacitor. Third, the charge can be injected more rapidly with a constant current generator than with the discharge of a capacitor. This decreases the amount of the injected charge which is consumed by the faradaic process during the time of the injection. Finally, overdrive of the amplifiers of the measuring system can be eliminated by compensating for the IR drop of the solution, since the form of the perturbing impulse is precisely defined. These two closely related techniques have been applied to several experimental systems with varying amounts of success. Delahay and Aramata (10) applied the coulostatic technique to the study of the Zn(II)/Zn(Hg) reaction in 1.0M KCl. Hamelin (11,12) used the tech- nique to investigate the Zn(II)/Zn(Hg) reaction in a number of electrolytes, and the Bi(III)/Bi(Hg) reaction (13) in perchloric acid and nitric acid. Wilson (14) investigated the Zn(II)/Zn(Hg) reaction in several con- centrations of KCl as the supporting electrolyte, Fe(III)/Fe(II) in 0.1M and in 0.3M oxalic acid, and the Cd(II)/Cd(Hg) reaction in 1.0M KCl and in 1.0M KNOB. In all cases the results were in reasonable agreement with the theoretically predicted behavior, and with the results of other investigators. Kooijman (i5) conversely concluded that his measurements of the Hg(I)/Hg couple in 1M perchloric acid were essentially meaningless since adsorption processes were involved and the relaxation time constant therefore included not only the double layer capacitance, but also a pseudoecapacitance which could not be separated from the total capacitance. He also claimed (16) that the reaction was so rapid that mass transport processes dominated the decay and it was therefore difficult to obtain reasonable estimates of the charge transfer parameters. The current impulse technique has been applied to the electrochemical reduction of the Hg(I)/Hg couple in 1.0M perchloric acid by Weir and Enke (i7) and the hexacyanoferrate(III)/(II) system in 1.0M KCl on platinum by Daum and Enke (18). In the first system Weir and Enke found. as Kooijman later found, that the Hg(I)/H8 couple is indeed not only a very rapid reaction but also a very complicated one. Much of the data could not be interpreted unambiguously. and evidence for adsorption and a preceding reaction was obtained. The second system, however, was found to be quite simple with results which agreed well both with theory and with the results of other investigators. This investi- gation will be discussed in detail in Section III of this thesis. Other methods based on the idea of the coulostat have been developed. Wilson (14) described a double pulse method for the determination of the half time of a relaxation. The technique consists of the application of two pulses to the system. The second pulse, of one-half the magnitude of the initial pulse and of opposite sign. is applied at a time such that the potential is returned to its equilibrium value. The time which the second pulse takes to restore the poten- tial to the equilibrium potential is the half-time of the relaxation. Levy (19) developed a small amplitude impulse chain method. The pulses are applied to the system in an evenly spaced train in which successive pulses are of equal magnitude but of opposite sign. The method was used to investigate several systems and good agreement was reported between experiment and theory. Delahay (20) described a large amplitude tech- nique as an alternate analytical method to polarography. In this method an impulse is applied to the system of such a magnitude that it is perturbed to a potential where the reaction is diffusion controlled. This corresponds to the situation in polarography of rapidly perturbing the system from the foot of the wave to the potential where the current is diffusion limited. C. Comparison'gf‘thg Current Impulse Technique EEEEHQEEEE Electrochemical Relaxation Techniques Experimental relaxation methods for the study of electrochemical kinetics can be conveniently divided into two classes, periodic techniques and transient 10 techniques. In the former the electrochemical cell is perturbed with some periodic variation of the potential and the cell current is observed as a function of time. With transient techniques the response of the cell is observed when the system is perturbed from equili- brium with a step function of either the current or the potential. The comparison of the current impulse method will be limited to a comparison with the common transient relaxation techniques. and will further be restricted to those which are limited to small ampli- tudes for which the current-voltage characteristic can be linearized. The first of these techniques is the voltostatic technique (21), in which a voltage step is applied to the cell and the current is observed as a function of time. This method is limited to relatively slow times, and consequently relatively slow processes, because the potential change at the interface is controlled by the time constant of the double layer capacitance and the series resistance of the cell. A closely related technique is the potentiostatic method (22), in which the electrode potential is per- turbed by a fast rise time potentiostatic control system and the current is observed as a function of time. The problem here is essentially the same as that of the voltostatic method. A rapid change of the interface 11 potential requires a large current to charge the double layer. Therefore, the amount of time which is necessary for the potential to reach its controlled value is dependent on the current output and the rise time of the potentiostat. The best experimental systems which have been devised to date require somewhat more than a microsecond to charge the double layer and permit the accurate observation of the cell current. With the galvanostatic technique (23) the cell potential is observed in response to the application of a constant current. The problems of this method can be traced to two sources. The first of these is of course the double layer capacitance. Initially a large portion of the applied current goes to the charging of the double layer capacitance. and very little to the faradaic process. It takes a significant amount of time for the cell potential to reach that required by the applied current, and measurements must be obtained at relatively long times with respect to the start of the experiment, and extrapolated back to zero time to obtain the charge transfer overpotential. The second experimental problem which afflicts the galvanostatic method is the ohmic potential result- ing from the solution resistance of the electrochemical cell. This problem is especially acute when studying fast reactions. In these cases high current densities 12 must be applied to the test electrode in order to obtain measurable values of the charge transfer over- potential. Current densities of this magnitude invari- ably cause an ohmic potential which is many times the magnitude of the charge transfer overpotential. This unwanted voltage must either be compensated experi- mentally, or be accurately measured and subtracted from the signal mathematically. The double pulse galvanostatic method (2#) was developed to avoid the double layer charging problem of the galvanostatic method by pre-charging the double layer_with a high magnitude short duration current pulse. Experimentally the magnitude of the pre-pulse was adjusted so that the overpotential time curve started with a horizontal tangent at the beginning of the second pulse. Thus it was hoped that the current passing through the cell would be entirely faradaic at that instant. The experimental problems associated with obtaining a pre-pulse of precisely the right mag- nitude have recently been reconsidered (25). and it has been concluded that the double pulse method offers no significant advantages over the classical galvano- static method. The experimental problems of the above techniques are due to the fact that the double layer capacitance is in parallel with the electrode reaction. This 13 contributes significantly to the morphology of the potential or current characteristic of these methods at a time scale where the most meaningful information on fast reactions exists. Moreover, in the methods which are current perturbed, the signal of interest must be extracted from a total signal which includes the ohmic potential. With the current impulse technique these experi- mental difficulties are largely circumvented. Initially, when the double layer is charged, the time scales are so fast that essentially no reaction takes place and the two phenomena are not competing. After the charging process is complete, the only path for the discharge of the excess charge stored in the double layer is through the faradaic reaction to ground. No current is diverted from the reaction to charge the double layer; it is already charged, and the path of its discharge is well defined. Moreover, since no significant current flows through the cell during the relaxation, there is no need to correct the observed overpotentials for the effects of ohmic drop. There are other significant advantages in using the current impulse technique. One is the inherent simplicity of the experimental apparatus. All that is required in the case of the coulostatic technique is a battery to charge the capacitor, a system of 1L» relays to apply the pulse to the cell, and a moderately fast oscilloscope to observe the potential. The current impulse technique requires only a slightly more compli- cated high output, fast rise time current pulse gene- rator to perturb the system. Another advantage of the method is its ability to obtain a relatively unambigu- ous estimate of the double layer capacitance under very reactive conditions. These characteristics are especially important in the study of film formation reactions. These reac-‘ tions can be studied as a function of surface coverage, since a known and very small quantity of surface can be applied or removed with each pulse according to its coulombic content and sign. The exchange current and capacitance can be tabulated as a function of the sur- face coverage. From this information a great deal can be learned about the mechanism of film formation in terms of how these variables change as the surface changes. D. ‘Limitations 22,222 Current Impulse Technique The only significant experimental problem of the current impulse method is the overdrive of the amplifier system during the charging process. To minimize the charging time for fast reactions, very high currents and short times are used. These conditions typically 15 cause ohmic drops of 200 to 500 millivolts, driving the oscilloscope amplifier into saturation. Recovery from overdrive may-take up to several microseconds, depending on the oscilloscope and the amount of over- drive. This limits the rate of the reaction which can be studied, as was disCussed previously. The other limitations of the method are of a more theoretical nature. The relaxation curves obtained from this method are featureless monotonic decays, as they are from all relaxation techniques. The experi- mental decays usually fit any one theoretical model as well as any other. Even when large differences are evident in theoretical decays for the various models, the limited accuracy of the experimentally low level signal usually precludes the adoption of any one theor- etical model over any other. The second theoretical problem is the separation of the charge transfer parameters from experimental relaxations which are predominately mass transfer con- trolled. With rapid reactions the rate of conversion of reactant to product is so fast that a deficit of reactant and an excess of product is built up at the surface of the electrode, and a diffusion gradient is set up. Ultimately, if the reaction is fast enough, the rate of the decay of potential becomes limited to the rate at which the reactant can diffuse to the 16 electrode surface. When this happens, the relaxation is diffusion controlled and no kinetic information is available from the decay. This imposes a distinct upper limit on the rate of the reaction which can be studied. In their original papers Delahay (7) and Reinmuth (3) proposed several conditions which must be satisfied in order for the general equation to be reduced to either the charge transfer or the diffusion limiting equation. Unfortunately, most of the electrochemical reactions which have been studied with this technique produce relaxations which are neither purely charge transfer nor purely diffusion controlled, but a combi- nation of the two. Weir and Enke (17) and Daum and Enke (18) have said that satisfactory estimates of the charge transfer parameters can be obtained from decays of this type by obtaining the slope of the log(n) Kg. t curves at times sufficiently short to apply the simple charge transfer assumption. However this assumption has not been sub- jected to a rigorous numerical analysis, and corres- pondingly there has been some concern in the literature about the validity of some of the kinetic parameters which have been obtained by this method (26). There have been several attempts to correct relaxation data for mass transport phenomena (16,27), 17 but all of these techniques require a significant amount of tedious calculation, since the arguments of the general function which includes mass transport become complex in the region of greatest experimental interest. The purpose of Section II of this thesis is to clarify some of the ambiguities involved in obtaining kinetic data from this measurement technique by dis- cussion of some of the determinate errors which are present in the various methods of analyzing the relaxation data as a function of the ratio of the charge and dif- fusional time constants, and to suggest techniques which experimenters can use to minimize these errors. II. NUMERICAL AND GRAPHICAL ANALYSIS OF THE DATA OF THE COULOSTATIC AND CURRENT IMPULSE TECHNIQUES A- 232221 The general equation derived by Reinmuth (3) to describe the decay process is n - nt_0(a+ - B_)-1 [8+exp(83t)erfC(B_€%) - B_exp(83t)erfC(B+t%)](1) where é 8: = Id /2Tc i l/Tc%(Td/4Tc - 1)%' (2) and the charge transfer time constant is re - RTCd/nFIO (3) and the diffusional time constant is t - [RTCd/n2F2(1/CODO + l/CRD§E>]2- I“) Equation 1 reduces to the simple charge transfer equation Td 71: ntaoexp(-t/Tc) (which is the result which could be obtained if diffusion processes were ignored in the derivation) if To »> Td' If-Id >> I c' Equation 1 reduces to n g nt'oexp(t/rd)erfc(t/Td)é. which is the diffusion limiting equation. It is desirable to make any electrochemical reaction under study conform as closely as possible to the charge transfer limiting equation, since systems 18 19 which fulfill this requirement produce the most mean- ingful information about the electrochemical charge transfer process which is occurring at the interface. ‘ It can be seen from the definition of Te and1 d that the charge transfer limiting equation is favored at high concentrations of electroactive species and low exchange currents. Experimentally a system can be made to conform more closely to the charge transfer limit if the concentrations are increased. Of course, the con- centration cannot be increased infinitely because of solubility limitations and because the function of the supporting electrolyte becomes ill defined at high concentrations. Experimentally, if 10:» rd and Cd is known, the exchange current can be calculated from the experimental data in a number of simple ways. The first of these is to take the half time of the decay, i.e., the time at which the overpotential is half its initial value, and calculate IO from the relationship t% = 0.69315 Tc' The second method is to plot log(n) pg. t and calculate the exchange current from the slope of the resulting straight line. If Td >> the decay is diffusion To, controlled, the half time and the slope are independent of the charge transfer parameters, and no kinetic infor- mation is available. When neither of the above inequalities is satis- fied, the decay is neither charge transfer nor diffusion 20 controlled, but is a combination of both. In this case To: Ta, and the decay is essentially charge transfer controlled at long times. A theoretical relaxation curve with a Tc/Td ratio of 12.5 is shown in Figure 2. The curves are very close at short times, indicating essential charge transfer control, but deviate sub- stantially at long times, indicating the inception of diffusion control. B. Graphical Analysis Kinetic information can be obtained from relaxa- tions of this type in a variety of ways. The simplest and most direct method of estimating the reaction rate is to assume the simple charge transfer limiting equation. The application of this assumption to data of this type obviously presents some difficulties, but these difficulties can be minimized by proper attention to the way in which the measurements are made. An important consideration for the accuracy of these measurements is the value of the double layer capacitance. There are a variety of methods available for the experimental determination of the capacitance under non-reactive conditions, lpg., when the electrode is ideally polarized. Many investigators seem to feel that the capacitance for an electrode at a certain potential and in a specified solution should be the 21 Figure 2 2 Theoretical decay curves with Cd = 2 x 10'5 F/cm Tc/Td 3 25.0, I0 s 0.18.13 = 1, D0 = DR = 10-5 cmz/sec, Co = CR z 10'5 moles/cm3. Upper curve calculated assuming general equation, lower curve calculated assum- ing charge transfer limiting equation. 22 ommi m2; .06 N6 O'ta/u ¢.o 0.0 $6 0.. 23 same regardless of whether the electroactive species is present or not. There is no 2 priori reason to believe this is true in all cases. For example, if specific adsorption of the electroactive species occurs, then the capacitance may differ widely from its value with no elect- roactive species. Or if solutions of extremely high concentrations of electroactive species are studied, the capacitance may change because of the addition of these ions to the double layer. Furthermore, in the case of solid electrodes, the value of the capacitance is depen- dent on small additions of oxide or other films to the electrode surface and is particularly sensitive to the adsorption of any organic compounds. It is important, therefore, to make the measurement of the double layer capacitance in the same solution and at the same time that the relaxation measurement is made. The value of the capacitance can be estimated from coulostatic data by the extrapolation of the log(n) 35. t curve to zero time. The current impulse method provides an additional estimate from the charging curve, since the charging process is more or less linear. Both of these measurements can involve some very large determinate errors depending on the time at which the measurements are made, the definition of zero time, and the ratio TC/Td of the charge transfer and diffusional time constants. Initially we will 2N assume that the capacitance is known, and will cal- culate errors resulting only from the neglect of mass transport processes in the calculation of the exchange current. Figure 3 is a graph of the determinate error in the measurement of the exchange current as a function of the ratio R - Tc/Td' The calculations were made by assuming a diffusional time constant and by systema- tically varying the exchange current and time to generate a series of theoretical decay curves. Curve A was calculated from half times from the relationship t% = 0.69315Tc. Curves B and C were calculated from fifth times and tenth times_from the relationships t1/5 = 0.223141c and t 0. 10536Tc respectively. 1/10 = It is obvious that as the time at which the measure- ments are made decreases, for a specific ratio of Tc/‘dt the error decreases. This is intuitively satis- fying, since one would expect that at zero time the electron transfer reaction would be purely charge transfer controlled, no matter what the value of rc/Td. At small ratios of Tc/Td' though, the approximation is not very good even at times which are close to zero relatively. In these cases, some means must be employed to obtain the kinetic parameters other than use of the simple charge transfer approximation. 25 Figure 3 Determinate error in the measurement of 10 as a function of Tc/Id assuming the capacitance is known exactly, for various normalized points on the decay. (A) Half times (B) Fifth times (C) Tenth times. 26 ohm I. ON o¢ 00 cm 00. 80833 °/o 27 Next, the errors in the determination of the capacitance using the extrapolation technique will be examined. As we have pointed out above, the extra- polation technique can involve some very large deter- minate errors depending on a number of factors. One of the interesting effects of using the extrapolated value of the capacitance is that it corrects to an extent the value of the exchange current calculated from the slope. The determinate errors in the estimate of the capacitance caused by mass transport processes are always in a positive direction. The slope of the log(n) pg, t curve is always less than that if the relaxation were pure charge transfer controlled. Hence, the value of nt=0 is smaller than it should be and the ratio Cd = Aq/nt=0 deviates in the positive direction. The values of the slope conversely deviate in the nega- tive direction, so the product is somewhat corrected. Unfortunately, there is not a 1:1 correspondence in the change of the two variables. Figure a shows the determinate error in the measure of the discharge capacitance as a function of log(Tc/Id). The errors in this figure were calculated by taking the slopes of the log(n) 1p. t curves between the points 1.00 and 2.25 of dimensionless time t/Td and extrapolating to zero time to find the intercept. Figure 5 shows the determinate error in the exchange 28 Figure A Determinate error in measurement of the discharge capacitance by extrapolating to zero time from the points where t/Id = 1.0 and 2.25 as a function of TC/Td. 29 IOO O a) O O (D V 80883 °/o 20' GO ID 2.0 LOGIR) 30 Figure 5 Determinate error in the measurement of the exchange current using the simple charge transfer approximation by taking the slope of the line between the points where t/Td a 1.0 and 2025 31 IOO 80883 °/e 0 d’ 20? 0.0 2.0 LOGIR) 32 current resulting from using the extrapolated value of the capacitance and the slope for the same values of the dimensionless time corresponding to Figure 4. The errors in the capacitance become larger as R decreases, as expected, and this corrects to some extent the value of the exchange current which is cal- culated from the same points. The current impulse technique provides an alter- nate way of measuring the capacitance. Since the form of the perturbation is a square pulse, the double layer capacitance charges linearly. From the slope of this charging curve and the magnitude of the applied current, the charge capacitance can be calculated. This measurement is also subject to some rather large errors, the origin of which become obvious if a simple model of an electrochemical cell is considered. That is, a model where the double layer capacitance is in parallel with the faradaic resistance Rf, and these are both in series with the solution resistance RS. Phenomenologically, the presence of the faradaic reaction in parallel with the double layer capacitance consumes some of the current which is being applied to the cell, and the slope of the charging curve becomes progressively less as time proceeds. These effects become more pronounced as the faradaic resistance becomes smaller. Thus, in order for a slope measurement 33 to be valid, an insignificant amount of the charge must have been consumed by the faradaic process up to the time of the slope measurement. The errors in the charge capacitance can be formu- lated in terms of the same variables as the errors of the discharge capacitance by invoking the methodology of the galvanostatic technique (23). It will be shown in Section IV that the traditional equations derived by Delahay (28) for the galvanostatic technique can be defined in terms of To and Td' It is sufficient for present purposes to state that this can be done and the reader is directed to Section IVAC for the details. Figure 6 shows the error in the charge capaci- tance as a function of log(Tc/Td). The errors were calculated by generating a set of dimensionless over- potential time curves with the computer, and deter- mining the slope at the same dimensionless times t/Td that were used to calculate the errors in the Figures # and 5. A direct comparison of the errors in Figures 0 and 6 can now be made, and it can be seen that the errors in the charge capacitance are larger than the errors in the discharge capacitance, for all ratios of Tc/Td' This is not surprising because in the case of the charge capacitance the system is being constantly driven by the applied current and faradaic and mass transport processes will show up sooner in time. 34 Figure 6 Determinate error in the measurement of the charge capacitance by taking the slope of the line between the points where t/1d = 1.0 and 2.25. 35 O..I 0.0 .5904 ON 4 l ON O¢ OO 00 00. ON. OV— CO. 80883 % 36 The computation of the exchange current with charge capacitance gives slightly less error than the calculations with the discharge capacitance, since the capacitance values differ in the positive direction more than do the discharge values. The tabulation of the errors in the exchange current seem pointless, though, since a good measure of the capacitance as well as the exchange current is generally desired. In view of the preceding analysis, the appli- cability'of the simple charge transfer approximation to experimental coulostatic data is dependent on Tc/Td and time. The shorter the time at which the measure- ments are made, the less the error. For equivalent times, the error increases as To/Td decreases. For measurements on systems which have small ratios of Tc/Ta, the measurements must be made at very short time scales if the approximation is to give even a rough estimate of the reaction rate and the capa- citance. The availability of the charge capacitance measurement from the current impulse technique seems to add little to the accuracy of the capacitance measurement, since this measurement is always less accurate than the discharge capacitance measurement. 37 C. Computer Analysis A better approach to the problem of obtaining kinetic parameters from relaxations of the type 10:: Td is the technique of curve-fitting suggested by Martin (27). This is a computer based technique which involves computing a relaxation curve based on initial estimates of the capacitance and exchange current and systema- tically varying these two parameters to make the experi- mental and calculated curves fit to a predetermined degree. Martin's original program was found to have serious limitations in the function which computed the theoretical value of the overpotential for a given set of parameters. A new program has been written and is presented as Appendix A in this thesis. The curve- fitting routine has been changed from the simple Gauss- Newton method to a variation of that method by H. O. Hartly (29). This variation guarantees the convergence of the method and significantly decreases the number of iterations required to obtain convergence. This program has been used extensively in this laboratory on several experimental systems. Figures 7, 8, and 9 indicate the kind of result which was obtained in the analysis of the hexacyanoferrate(III)/(II) couple on platinum. The experimental curves are the unbroken lines, and the crosses indicate the value of 38 5F 4.. 3_. 2.... II- o 1 LL 5 1 l 1 1 i 1 1 CI 0L5» IA) |.51 2J3 ZLES ms ,usec Figure 7 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl, Co = 0.01M, CR = 0.01M. -———' Experimental curve x Theoretical curve calculated from curve fit values of IO and Cd Data 0 — 1 00 x 10'5 moles/cm3 D 8 90 x 10"6 cmZ/sec o— O o: 0 CR = 1.00 x 10"5 moles/cm3 DR = 7.40 x 10"6 cmz/sec area = 0.05376 it = 0.056 amps -r= 1.00 x 10'7 sec Estimate from slope loggn) pp. 3 Curve fit estimate IO = 0.26 amps/cm2 IO = 0.28 amps/cm2 Cd = 18.7 uF/sz Cd = 18.2uF/cm2 39 5, - 4. . 17th' a{ b 2: . | . OO 1 0:5 1 lfO l l1.5 1 2.0 J 2.5 TIME ,usec Figure 8 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1H KCl, Co = 0.01M. CR = 0.07M. Experimental curve x Theoretical curve calculated from curve fit values of Io and Cd Data -5 3 -6 2 Co a 1.00 x 10 moles/cm D0 = 8.90 x 10 cm /sec CR 3 7.00 x 10'5 moles/cm3 DR 2 7.40 x 10'6 CmZ/sec area = 0.05376 cm2 it = 0.089 amps I: 1.00 x 10"7 sec Estimate from slope loggn) pp, 2 Curve fit estimate Cd 3 31.3 uF/em2 cd 30.411F/cm2 40 h) L“ 45 (fl O l 1 k L l 1 I l J O | 2 3 4 5 7 TIME [.LSEC Figure 9 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl, Co = 0.001M, CR 8 0.01". --—- Experimental curve x Theoretical curve calculated from curve fit values of IO and C(1 9332 Co = 1.00 x 10"6 moles/cm3 D0 = 8.90 x 10"6 cmZ/sec OR = 1.00 x 10'”5 moles/cm3 DR = 7.40 x 10'6 cmz/sec ‘area = 0.05376 cm2 1t = 0.0613 amps r = 1.00 x 10'7 sec Estimate 2222.§2222 105$”I.E§- p 22222 pip estimate IO = 0.08 ampS/sz IO = 0.058 amps/cm2 Cd = 24.9 uF/cm2 Cd = 26.3 uF/cm2 41 the overpotential, which was calculated from the final curve fit values of the exchange current and double layer capacitance. It can be seen that the system follows the simple charge transfer model over a rela- tively large concentration range and that the curve fit values do not differ widely from those obtained by using the simple charge transfer assumption. In this case the use of the simple charge transfer approxi- mation is justified and it is not necessary to use the curve-fitting program to obtain estimates of the exchange current and capacitance. The analysis of the Hs(I)/Hg system in this laboratory by Mrs. J. Kudirka proved not to be quite so straightforward. The rate of this system is quite rapid with the ratio of Tc/Td ranging from about 1 to about 0.1 for the concentration ranges which were studied. There was no possibility of using the simple charge transfer assumption with any reasonable accuracy at any but the highest concentrations which were studied. Moreover, it was found at low concentrations that the simple charge transfer model did not apply to the experimental relaxation data. The relaxations decayed faster than the combination of the concentration and the diffusion coefficient mathematically allowed. It was assumed that the only way this could happen was if the concentration at the interface was larger 42 than the bulk concentration. In other words, it could only take place if some sort of adsorption process occurred. The program was modified to include the concentration of the Hg;+ species as one of the vari- ables to be curve fit. Many of the relaxation curves were analyzed and it was found in many cases that the curve fit value of the concentration was higher than the bulk concentration. A typical relaxation of the Hg(I)/Hg system is shown in Figure 10. The unbroken line indicates the experimental curve, and the crosses, the relaxation curve produced by computation from the final values of the fit parameters. The values of the parameters which would have been obtained by using the simple charge transfer assumption are compared to the final curve fit values. While this approach to adsorption is hard to justify in terms of a rigorous mathematical model, the fact that relaxation curves were computed which agreed quite well with the experimental decays seems to indicate that an adsorption process is involved. In view of this analysis of the Hg(I)/Hg system it can be concluded that the value of the rate constant reported by Weir and Enke (17) is in substantial error. The exchange currents in their study were calculated by assuming that the simple charge transfer approxi- mation applied, and no account was taken of adsorption. 17"“! 43 00 on -b ca o I l t 1 l l I I 1 L fl C) C12 C14» (161 C18: If) TIME psec Figure 10 Experimental and theoretical relaxation curves for the Hg(I)/H8 system in 1x salon. co a 0.001n. Experimental curve 0 Theoretical curve calculated from curve fit values of 10, Cd and Co Data 0 1 023 x 10"6 moles/cm3 D 9 1 10"6 2/ o = O o 8 o 1 cm sec area a 0.0249 on2 it = 0.015 amps T: 1.00 x 10'7 sec Estimate from slope 105(0) pg. 3 Curve fit estimate 10 a 0.41 amps/cm2 I0 = 0.68 amps/cm2 Cd :3 “5.6 uF/cmz Cd 2 “3.0 “F/cmz C g 1.605 x 10"6 moles/cm3 44 Exactly how much the rate constant was in error is still in question. It is possible that the surface ‘excess of the Hg§+ ions was so large that it increased the ratio TO/Td to the point where the simple charge transfer approximation was justified. If this was the case, the exchange currents could be substantially correct, but the rate constant would still be in error because the concentration term used in the calculation of the rate constant would not be the bulk concentration, but rather the value of the concentration at the. surface. In all, the curve-fitting technique is capable of obtaining excellent estimates of the exchange current and the capacitance from experimental data. The uncertainty in the estimates increases as the ratio TG/Td decreases, to the limit where the observed portion of the relaxation is completely diffusion controlled, and only a lower estimate of the exchange current can be made. The chief disadvantages of the method are the amount of computer time which is required to analyze a relaxation curve, especially if'Tc/rd is small, and the associated problems which are inherent in the operation of any computer program of this com- plexity. It also is important to note that the pro- gram is always written for a particular reaction mechanism, in this case first order kinetics with no 45 complicating effects other than mass transport. When other effects, such as adsorption, occur, modification of the program to encompass a new model is necessary to obtain correct estimates of the rate parameters. D. Nomographic Analysis An alternate approach to either of the above two methods is the nomographic technique proposed by Kooijman and Sluyters (16). It is essentially a one parameter curve-fitting technique. values of n/nt=o are tabulated for values of the ratio Tc/Id' and the quantity té/rdé. An experimental value of ”/”t=0 is computed, and by knowing the value of the diffusional time constant and the capacitance, an estimate of the current can be calculated. ' The difficulty of application of this technique is somewhere between that of curve-fitting and the simple decay analysis; consequently the accuracy of the results is also somewhere between that of the other two. The major disadvantage of this technique is that it fits only one of the two unknown parameters which control the rate of decay. A capacitance value must be assumed, and this may be very inaccurate, depending on the method and conditions under which it was obtained, as we have shown previously. Tied inextricably to the accuracy of the capacitance value is the point11t=o 46 (Cd = Q/”t=0)' from which the value of nfi1t=o must be calculated. Given reasonable estimates of these parameters, the method is capable of good accuracy; however, with poor estimates the technique is only use- ful for getting rough values of the exchange current and correcting somewhat for the influences of mass trans- port. E. Conclusion The various ways of obtaining the capacitance and kinetic data from the coulostatic and current impulse methods have been evaluated. The best method for obtaining accurate data has been shown to be the two parameter curve-fitting technique. With this tech- nique no prior knowledge of the capacitance or nt=0 is necessary for the accuracy of the measurement. The experimental system need not be maximized for extremely short time measurements as long as sufficient kinetic control is evident. However, the range of the method and the ease by which it calculates the parameters are increased if these measurements are available. The accuracy of the application of the pure charge transfer equation and the nomographic technique have been found to be dependent on accurate short time measurements. For the use of these methods the experi- mental system must be optimized. Within these limitations 1»? these two techniques provide an easy way of estimating the parameters and of determining whether the curve- fitting technique must be applied. III. THE ELECTROCHEMICAL KINETICS or THE HEXAGYANOFERRATEUIIVUI) COUPLE ON PLATINUM Malia-9.22.2122 Previous studies with the current impulse tech- nique have been limited to just one system, the kinetics of the electrochemical reduction of the Hg(I)/Hg system (17). This reaction appeared to have some mechanistic complications, and the interpretation of some of the relaxation data proved to be somewhat anomalous. It was not clear to the investigators, in some cases, whether the complications arose from the method which was used to study the reaction, or whether the diffi- culties were inherent anomalies of the system. It seemed desirable for the development of the technique to show that it is capable of obtaining unambiguous data from a mechanistically simple reaction, and to extend its usefulness to the study of fast electro- chemical reactions at solid electrodes. A system which fulfills these requirements is the hexacyano- ferrate(III)/(II) couple on platinum. It has been the subject of much investigation, and is regarded 48 LI9 by many as the model of a highly reversible electro- chemical reaction without complicating mechanistic effects. The study of this couple with the current impulse technique presents several interesting problems, and gives some new data which is not available from other methods of studying fast electrochemical reactions. First, the current impulse technique gives a relatively unambiguous estimate of the differential double layer capacitance under the actual conditions of the experiment, a measurement which is not readily available from other techniques with a reaction of this rate. This ability is particularly important in this system, because the hexacyanoferrate(II) and hexacyanoferrate(III) anions ,are highly charged, and one might expect that the double layer capacitance of platinum would be a strong function of their concentration and the potential. Second, these two anions are most certainly associated to a differing degree with the potassium ion of the supporting electrolyte (30), and the oxi- dation of the hexacyanoferrate(II) ion or the reduc- tion of the hexacyanoferrate(III) ion probably involves a change in the number of potassium ions associated with the particular anion as it undergoes the electron transfer reaction. It would be interesting to see 50 whether the current impulse technique is capable of giving any information regarding this proposed mechanism. Third, it has been reported in the literature numerous times that the rates of electrochemical reactions on platinum are very dependent on the oxi- dation state of the platinum surface, and more gene- rally, on the existence of any kind of thin film on an electrode surface. The effects of the platinum oxide film on the rate of an electrochemical reaction vary widely. Many investigators (31-33) have noted increased reversa- bility of an electrochemical process after the electrode had been driven anodically to oxygen evolution, when presumably it was coated with an oxide film of some kind. Some of these investigators (32,33) have invoked an oxide bridging mechanism to explain this increase in reversibility. In this theory the electron transfer reaction is aided by an oxide bridge from the electrode surface to the electroactive species in the double layer. Other investigators (31) think that the entire mech- anism of electron transfer reaction is changed and feel there is no need to invoke a bridging mechanism to account for the increased reactivity of the electrode. Both of these explanations appear to be valid in cer- tain instances. 51 Some reactions (34-36), especially those of anions, have been found to be less reversible, and are some- times completely supressed by the presence of an oxide film. The theory usually invoked to explain this pheno- menon is an active site reduction, where the presence of an oxide film reduCes the number of active sites which are available for the transfer of the electron to the electroactive species. Needless to say, the situation concerning oxide films and their effect on the rate of electrochemical reactions is not well defined. The effects depend on how and how much the surface was oxidized, how much it was reduced, and the kind of electroactive species which was being studied. It was decided to add to the information concerning these effects by seeing what effect the presence of an oxide film had on the apparent rate constant of the hexacyanoferrate(III)/(II) couple. B. E erimental 1) Instrumentation One of the major experimental difficulties of the current impulse technique has been the problem of uncompensated ohmic drop during the application of the perturbing impulse. This ohmic drop, which can be many times the magnitude of the relaxation signal, tends to overdrive the amplifier system. Recovery 52 from this overdrive can be quite slow and can often prevent the accurate measurement of the relaxation potential for relatively long times in comparison with the pulse duration. In systems which are marginally charge transfer controlled, this is where the most meaningful kinetic information exists, so it is import- ant to be able to make measurements in as short a time as possible. An additional difficulty is encountered in the measurement of the charge capacitance. This measure- ment is made during the time of the application of the pulse and thus must be extracted from a signal which includes the ohmic drop. The elimination of the ohmic potential from the signal would allow the measurement of the charging slope at higher sensitivities and correspondingly increase the acCuracy of the measurement. A small bridge cell network has been developed in this study to minimize the IR problem, and in so doing to extend the measurement capability of the current impulse technique to shorter time scales. A diagram of the experimental system is shown in Figure 11. The compensating network is a bridge, the requirement for balance being that IA’ the current through arm A times the uncompensated solution resistance (IA x RS), be equal to 13, the current through arm B times the resistance of potentiometer P1. The current proceeds 53 Figure 11 Block diagram of experimental system for compensation of ohmic drop. 54 do} l we .\. o. doew s éoo. I'll-Nil l'llll I'll ll'llll m2< memn. O OIVOOQ m m .o Titian mililm. rllljlll? I rllllH z. OE... OE... Jmo .ZMO MOI—On. mm on. " u __ _ _ ml maOomOJJGOO mom _ _ _ _ _ _ _ _ _ _ _ 55 from the output of the pulse generator and is split at diodes D1 and D2. Because IA = IB the high speed matched diodes D and D2 turn on the two arms of the bridge simul- 1 taneously, and prevent the discharge of the double layer capacitance through the bridge network to ground when the pulse has terminated. The 100 ohm resistors insure the rapid turn-on of the diodes and serve to minimize the variation of the current in the two arms after the pulse has started. The 100 ohm precision resistor in series with the cell also served as a device to measure the magnitude of the current going to charge the double layer. The entire circuit assembly was mounted on a grounded copper plate to isolate the various circuit components. This plate plugged directly onto the out- put of the pulse generator through a BNC connector. The cell was plugged through a multipin connector to the circuit. Potentials were measured between points 0 and D with a sensitive differential amplifier (Tektronix P6046). This amplifier is a miniaturized probe which makes the difference measurement at the signal source, thereby eliminating the problems associated with transmitting the cell signal and compensating signal through rela- tively long distances to the oscilloscope. In 56 combination with a Tektronix 1A5 amplifier and 556 oscilloscope, the probe has a maximum sensitivity of 1 mV/cm, a risetime of 9 nsec, a bandwidth of 40 MHz, and a CMRR of 100011 at 40 MHz. The amplifier plugs into coaxial jacks which are mounted in the copper plate. Two sets of jacks are used, one for the current measurement and the other for the measurement of the relaxation potential. With this system it was possible to reduce the observed IR drop to several tenths of a mV. This ' enhanced the ability to make an accurate charge capa- citance measurement in several ways. Since the IR . drop was compensated, it was possible to increase the sensitivity of the measurement to 1 mV/cm when necessary, _thus increasing the accuracy of the measurement. Furthermore, since the IR drop was the same in both arms of the bridge, the small irregularities in the output of the current pulse generator cancelled in the difference signal. The system also decreased the time at which a potential measurement could be made after the start of the pulse to about 150 nsec, and decreased the time for which an accurate measurement of the relaxation potential could be made to about 100 nsec after the pulse had terminated. This is significantly better than any other system reported in the literature. 57 Unfortunately, this experimental system was developed after the largest part of the following study had been completed, and so a different experimental system described below was used for the majority of the reported work. It was not as good as the system ultimately developed, but was quite adequate for the studies which were performed. A block diagram of this system is shown in Figure 12. In this system no effort was made to eliminate the IR problem. It used essentially the same instru- mentation as the preceding system with the exception of the probe and IR compensating network. The per- turbation source was used directly with no intervening diodes. This was possible because in its quiescent state the pulse generator had an impedance to ground which was a minimum of 10 Kohms: this was quite large with respect to the faradaic resistance of the hexa- cyanoferrate(III)/(II) couple. Thus the cell was essentially at open circuit after termination of the pulse, and relaxation took place only through the faradaic reaction. The Tektronix 556 oscilloscope with a 1A5 differential preamplifier was used to measure the potentials. The combination of the main frame and preamplifier had a rise time of less than 11 nsec, a 1 Mohm input resistance, and a maximum sensitivity of 1 mV/cm. The preamplifier had a 58 Figure 12 Block diagram of experimental system without ohmic drop compensation. 59 _ _ _ _ _ _ _ r _- II: I L.. .033. Jun 11......HIIIIIIIIIIIIIHIIIIIH H H H H HIIIII H .I. IIIIIHH H. quHH HIIIIJ ml II-lu mmoomjnzomo 000 02201.55... .J .252. .5022... ._ [IL—3-"- m0k__mwn_xm Eu L’) H... U L £38 a 3058...“. 55:8 II . - I»: e sou IQIBP amino m 352. .z I >—v u me; E L. moomaomom mozmmmhm 4.30 «n. moomkomqw bmwh 63 fitted to a small shielded box which was attached to male and female BNC connectors. The cell assembly was plugged directly onto the output of the pulse gene- rator and a short length of BNC cable was connected from it to the input of the oscilloscope. Nitrogen was led from the purification train to the cell assembly via a small diameter Teflon tube which was immersed directly into the test solution. A second cell of slightly larger dimensions to accommodate immersion in a con- stant temperature bath and attachment to the IR compen- sating network was developed for use in the activation energy studies. Both cells demonstrated excellent high frequency response with minimal distortions due to stray inductances and capacitances. A three electrode configuration was used in both cases, because it minimized the IR drop and gave slightly less noise than a two electrode system. The test electrode was made by melting a small diameter platinum wire with a gas-oxygen torch into a small sphere of approximately 0.05 on2 area. The geometric area of the test electrode was determined by measurement with a Bausch and Lamb microscope fitted with a micrometer eyepiece, and a calibrated micrometer slide. The area was estimated to be correct to within 2%, though no account was taken of surface roughness effects. The reference electrode was a large diameter platinum wire 64 2 area arranged so that it was of approximately 0.5 on less than 0.5 mm from the test electrode. The area of the reference was large enough so that an insignificant amount of polarization occurred at it under the con- ditions of the experiments. The counter-electrode was a cylinder of platinum gauze of approximately 3 cm2 area, arranged concentrically about the referenCe and test electrodes. Prior to each experiment the cell and electrodes were allowed to stand in hot perchloric acid for ten minutes. This served both to oxidize the surface of the electrodes and to oxidize any adsorbed impurities in the cell electrode system. For the reduced elec- trode experiments, the cell was rinsed several times with triple distilled water, filled with 1M KCl, and the electrodes were reduced for several minutes at hydrogen evolution potentials. Care was taken never to expose the electrode to the air when undergoing the changes of solution to minimize the chance that atmo- spheric oxygen would contaminate and oxidize the surface. For the oxidized electrode experiments, the system was used after the usual rinsing with no further treatment. The electrode surfaces produced in this way were found to be stable in their respective states for long periods of time. The system was deaerated with nitrogen for 65 twenty minutes before each run, though the presence of oxygen in solution was not found to affect the results significantly. 3) Reagents 2p; Solutions Solutions were prepared directly by weight from ACS reagent grade chemicals without further purifica- tion. Water was prepared by the redistillation of an alkaline permanganate solution of laboratory distilled water. Nitrogen used to purge solutions of oxygen was dried over calcium chloride, passed through an oven containing copper turnings at 350°C to remove traces of oxygen, passed through traps containing activated char- coal at liquid nitrogen temperatures, presaturated in a 1M KCl solution, and fed to the cell via a glass and Teflon train. C. Egperimental Results 1) Reduced Electrodes The kinetiCs of the hexacyanoferrate(III)/(II) couple were measured at a total of thirteen different concentrations in 1.0M K01. The concentration of either the hexacyanoferrate(III) or the hexacyanoferrate(II) ion was held constant at 0.01M, and the concentration of the other anion of the couple was varied in seven increments from 5 x 10'“M to 7 x 10’2M. The data presented subsequently is the average of at least three 66 separate experiments at each concentration, and in some cases is the average of as many as twenty separate experi- ments. Estimates of the differential capacitance were obtained in two ways; the lag(n) ye. t curves were extrapolated to zero time, to give what is known as the discharge capacitance, and in a separate experiment, a pulse of approximately 0.8 usec duration was applied to the cell and the capacitance value (denoted here- after as the charge capacitance) was calculated directly from the slope of the charging curve. The capacitance data are presented in Figures 14 and 15 as functions of log(Co) and log(CR). It can.be seen that there is marked agreement between the two estimates of the capa- citance at all but the highest concentration, where the measured charge capacitance is significantly higher than the discharge capacitance. This apparent anomaly can be explained by considering a simple electrical model which describes the system (Figure 16). As it was discussed in Section II-B, calculation of the charge capacitance assumes that the charging process is linear with respect to time; this implies that on the time scale of the measurement (approximately one-half micro- second) an insignificant amount of the charge is used by the faradaic process, which is in parallel with the double layer capacitance. At low concentrations this 67 Figure 14 Differential capacitance from charge and discharge data of the current impulse technique, of platinum in 1M KCl as a function of log(Fe(CN)g'). CHARGE CAPACITANCE A 68 3249202 1.800.... o.“ I 0N! QM! or»: _ q _ .. 0.. m .. om Q 0 Q m 0 O 4 .. on 4 3.0. "so .. 0% 323.320 mmmfiama 0 323653 8558 < sz/afi 3 o NVilOVdVO 69 Figure 15 Differential capacitance from charge and discharge data of the current impulse technique, of platinum in in KCl. as a function of log(Fe(CN)g-). 70 .3 3 >335. Amov 00.. O. _ I O.Nl Ogm... 03V... _ _ _ .. o. w .d W. .0. . Q .. ON mu... 0 Q .. Q ® a s w . 1 CM 3. 2 -0700 fl N .. ow M 325.033 35.183 0 m ”.6256qu uom_-m’ BONVlIOVdVO BSHVHOSIG 0.24 0.23 0.32 E vs S.C.E. 0.20 .l6 76 of 1:1 was measured as a function of the total concen- tration of the two ions, from a concentration of 5 x 10‘2h to 3 x 10‘3h of each. It was found that the differential capacitance remained constant over that concentration range within experimental error. This indicates that the differential capacitance is essen- tially independent of the total concentration of the two anions. at least at the potential of an equimolar solution. Exchange currents. shown in Table I. were cal- culated using the discharge capacitance and assuming that the observed overpotential time curves followed the simple exponential decay law dictated by pure charge transfer kinetics. r13‘nt=O(-t/Rfcd)o The use of this equation presupposes that the contributions of mass transport processes to the observed decay are minimal at the time of measurement. There are several criteria which can be applied to test this assumption. which were discussed in detail in Section II-A. One of these is that Tc >> Td' At the experimental concen- trations this inequality was satisfied by a factor of 25 in the most favorable case and a factor of 0.75 in the least favorable case. While it appears that the linear form of the equation is not applicable at the lower concentrations. it should be recognized that this inequality determines the condition for pure 77 TABLE I Values of the Apparent Exchange Current Density for Solutions of Varying Concentrations of K3Fe(CN)6 and K4Fe(CN)6 in 1.00M KCl at 25° 3- 4— o concn Fe(CN) concn Fe(CN) I 6 6 a mole 1...1 mole 1.“l amp/cm— -2 -2 7.00 x 10 1.00 x 10 0.49 3.00 x 10'2 1.00 x 10‘2 0.32 1.00 x 10'"2 1.00 x 10"2 0.23 7.00 x 10"3 1.00 x 10'2 0.19 3.00 x 10'3 1.00 x 10’2 0.11 1.00 x 10"3 1.00 x 10’2 0.079 5.00 x 10"4 1.00 x 10"2 0.050 1.00 x 10"2 7.00 x 10"2 0.62 1.00 x 10’2 3.00 x 10"2 0.38 1.00 x 10"2 7.00 x 10"3 0.20 1.00 x 10‘2 3.00 x 10’3 0.12 1.00 x 10'2 1.00 x 10'3 0.093 -2 -4 1.00 x 10 5.00 x 10 0.052 78 charge transfer control throughout the entire relaxa- tion time. as has been discussed in Section II-B. Even at the low concentrations. though, the decay is charge transfer controlled at short times. Special care was taken at the low concentrations to use initial slopes in the calculation of the exchange current and to obtain good estimates of the capaci- tance, so that the determinate error due to negligence of mass transport processes was minimized. Some of the experimental data was analyzed with the computer program of Appendix A. Some of the results are presented in Section II-C and additional results are presented here in Figures 18 and 19. The values of the curve fit parameters are compared with the values obtained using the simple charge transfer approximation. It can be seen that both approaches give essentially the same results. The average obtained from both methods for a number of data points was found to be essentially the same. Thus, it was felt that the use of the simple charge transfer assumption to calculate the exchange currents was Justified. The calculated and experimental curves of Figures 18 and 19 fit very well, indicating that the electrode reaction is a simple first order electron process. 79 5 L 4 .. 3 .. 2: p l _ O 1 1 1 1 1 I 1 1 1 L O l 2 3 4 5 TIME [1.886 Figure 18 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in in KCl, Co = 0.003M. CR = 0.001M. -- Experimental decay curve x Theoretical decay curve Data 0 00 ‘6 1 / 3 8 0"6 2/ o = 3. x 10 mo es cm D0 = .90 x 1 cm sec cR = 1.00 x 10'5 moles/cm3 DR 2 7.40 x 10"6 cmZ/sec area = 0.05376 cm2 it = 0.049 amps T = 1.00 x 10'"7 sec Initial estimate Curve fit estimate 10 a 0.093 amps/cm2 Io = 0.090 amps/cm2 Cd 3 18.9 UF/sz Cd = 19.3 UF/Omz 171nVV 80 5. a 4» h 3 1- 2’ _ I .- 0 . l 1 J 1 1 1 1 1 C) I 2. 3» ‘4 TIME p.350 Figure 19 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl, Co = 0.01M, CR 2 0.01“. -—-—- Experimental decay curve x Theoretical decay curve p233 Co = 1.00 x 10'5 moles/cm3 D0 = 8.90 x 10"6 cmz/sec CR = 1.00 x 10’5 moles/cm3 DR = 7.00 x 10'6 cmz/sec area = 0.0835 cm2 it = 0.050 amps r = 1.00 x 10"7 sec Initial estimates from slope 92332 §i£_estimate IO = 0.170 amps/cm2 IO = 2.01 x 100'1 amps/cm2 cd a 16.0 uF/cm2 0d = 15.1 uF/cm2 81 The reaction-order plots shown in Figures 20 and 21 were made using the exchange currents in Table I and the measured concentration. The plots are linear over the entire concentration range and give transfer coefficients (a) of 0.54 and 0.46 reSpec- tively. No significance should be attached to the observed differences in the transfer coefficient since these deviations are within experimental error. The heterogeneous rate constant ks (Io = nFAksCRaCol’a) calculated from the average of the exchange currents at a concentration of 0.01M in each ion was found to be 0.24 cm/sec. The exchange current of a 0.01M solution in each ion was measured at ten degrees, thirty degrees, and fifty degrees. Log(IO) was plotted as a function of i/T. and an activation energy of 3.1 I 0.2 kcal/mole was calculated. 2) Oxidized Electrodes The work completed on oxidized electrodes must be considered of a preliminary nature because of some rather persistent discrepancies which were very diffi- cult to explain. The oxidized electrodes which were studied were produced by oxidation of the platinum electrode with hot perchloric acid. Though this oxidation procedure is extreme, it produced an electrode surface 82 Figure 20 Reaction-order plot for the hexacyanoferrate(III)/(II) couple on platinum in 1M KCl, Co = 0.01M. 83 QT .33 \mo_o_2 $550... QNI QM! 04V: . . Q N N 9N awe/Vt“ (°I)90'| 84 Figure 21 Reaction-order plot for the hexacyanoferrate(III)/(II) couple on platinum in IN KCl. CR = 0.01M. 85 00._I .524 30.22 Aoov 00.. 00.N... 00.»! 00¢... - — .. 00.. - 0N.N - 00.N gum/VW (°I ) 90'] 86 state which was fairly reproducible for kinetic measurements. The number of measurements and variety of solu- tions which were studied were not as numerous as those in the reduced electrode experiments, but were suf- ficient to establish several facts. The first of these is that the apparent rate of exchange at perchloric acid oxidized electrodes is drastically reduced from that measured for reduced electrodes at equivalent experi- mental conditions. The second is that the morphology of the experi- mental overpotential time curves was not consistent with the model which describes simple charge transfer processes. With the apparent reduction of the rate of charge transfer of ten. the inequality To >> rd, which determines whether the decay is charge transfer controlled, is satisfied to the extent that the log(n)‘z§. t curve should be virtually linear at the measured times. However this behavior was consistently not observed. The log(n) XE! t curves deviated at long times in ways that suggested the decay was parti-' ally mass transport controlled. These essential facts are illustrated in Figure 22. The experimental curve is designated by crosses and the theoretical decay calculated from the experimentally determined para- meters is shown as the unbroken line. It can be seen 87 Figure 22 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl. CO = 0.01M. CR = 0.01M. oxidized electrode. x Experimental decay -—— Theoretical decay for experimentally determined values of IO and Cd Data -5 3 -6 2 Co = 1.00 x 10 moles/cm D0 = 8.90 x 10 cm /sec CR = 1.00 x 10-5 moles/cm3 DR = 7.40 x 10'6 cmzsec area = 0.0835 cm2 it = 0.368 amps/cm2 1': 1.00 x 10"7 sec Experimentally determined parameters 10 = 0.034 amps/cm2 c 10 uF/cm2 d 88 ommi m2: >5? 89 that there is substantial deviation of the experimental curve from the theoretical decay at long times. The third discrepancy is the fact that the measured value of the capacitance was consistently lower for the oxidized electrodes than it was for the reduced electrodes. Though the capacitance values were fairly scattered. the average value for the capacitance of an oxidized electrode at a concentration of 10'2M in both the hexacyanoferrate(III) and hexacyanoferrate(II) ions was 11 uF/cmz. This is about 40% of the measured value of 25 uF/cm2 for reduced electrodes at the same concentration. Most other investigators (38,39) have reported increased values of the capacitance upon Oxidation of the platinum surface. Several approaches to the explanation of these experimental anomalies have been attempted. The most fruitful of these. which is not without its theoretical limitations. is to consider a reduction in the effective area of the electrode due to blockage by the platinum Oxide of the "active sites“ where electron transfer takes place. If the geometric area of the electrode is taken as A. and the fraction of the surface covered by oxide is taken as 4». the electrode area available for electron transfer is A' = A¢. Some of the oxidized electrode decays were mathematically analyzed by invoking this area mechanism 90 using a simple modification of the program in Appendix A. The capacitance and exchange current were changed pro- portionately (thus the Rfcd time constant remained the same but the increased exchange current allowed dif- fusion processes to appear) by decreasing the effective area. The results of one of these computations is shown in Figure 23. When the effective area was reduced from the geometric area of 0.0835 cm2 to 0.0152 cm2 the experimental and theoretical curves fit quite well. The final values of the capacitance and exchange current which were obtained are 55 uF'/cm2 and 0.184 amps/cm2 for the effective area. Thus it appears that estimate of the rate constant obtained at oxidized electrodes by considering the measured exchange current and the geometric area gives a value which is much too small. It is easy to rationalize the effects that plati- num oxide might have on the exchange current by con- sidering the way in which these films were formed. Boiling perchloric acid is a very good oxidizer and it is not hard to imagine multiple oxide layers on various parts of the electrode. The sites on the electrode surface which are active to electron transfer may be the places where oxide films form preferentially. There may be several layers of oxide built up at these sites and thus electron transfer may not take place 91 Figure 23 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl. 00 = 0.01M, C = 0.01M. oxidized electrode. R x Experimental decay, ——— Theoretical decay for experimentally determined values of IO and Cd 0 Theoretical decay calculated by invoking area mechanism _Data Co = 1.00 x 10-5 moles/cm3 D0 = 8.90 x 10"6 cmZ/sec CR = 1.00 x 10'5 moles/cm3 DR = 7.40 x 10'6 cmZ/sec area = 0.0835 cm2 . it = 0.0368 amps T = 1.00 x 10'7 sec Parameters determined by invoking area mechanism I 0.184 amps/cm2 0 Cd 55 uF/cm2 92 ommi m2: 0 m to m _ a q q d _ a o >5? 93 at these places as readily as across other parts of the electrode surface. The sites which are not oxi- dized to this extent may present a lower energy barrier to electron transfer and thus determine the rate of decay. This does not exclude the possibility that several rate processes occur at the surface, one of which is much faster than the other. It is not so easy to explain why the measured value of the capacitance is lower than the value of 50 uF/cm2 reported by other investigators. The ans- wer to this may also lie in the extreme procedure used to make the oxidized surface. Other investigators (38,39) have made their oxidized surfaces with rela- tively mild electrochemical oxidizing techniques. These mild techniques produce a surface which is essentially a monolayer of oxide coating. The multi- -ple layers which were probably formed with the present procedure on select sites of the electrode would account for the low value of the measured capacitance. The capacitance at these sites would be much lower than at other parts of the electrode and would dominate the measured value of the capacitance. It is not clear whether the apparent value of the capacitance obtained by reducing the effective area has any significance other than a mathematical one. It is the capacitance which was mathematically 94 necessary to make the experimental and theoretical curves fit by reducing the area. The similarity between the "effective” area value of 55 uF/cm2 and the value of 50 uF/cm2 obtained by other investigators for oxidized electrodes for total area should not be construed to mean that the two surfaces were the same. The similarity of the two values may have no physical meaning. Figure 24 is a reaction order plot which shows the exchange current per electrode as a function of log(C°), CB the change in exchange current with concentration and = 10'2M. The slope is dependent only on is independent of the area, all other things being constant. The measured value of the transfer coeffici- ent is 0.46. which is substantially the same as that obtained for reduced electrodes. A rate constant of 0.028 was calculated. based on the geometric area and the average exchange current of a number of experi- ments of a 0.01M hexacyanoferrate(II). 0.01M hexa- cyanoferrate(III) solution at 30°C. The activation energy was also determined by obtaining the exchange current of 0.01M hexacyanoferrate(II). 0.01M hexa- cyanoferrate(III) solution at three temperatures and was found to be 3.5 t 0.5 kcal/mole. The activation energy is also independent of the area. depending only on the change of the exchange current with T. This 95 Figure 24 Reaction-order plot for the hexacyanoferrate(III)/(II) couple on platinum in 1M KCl. CR = 0.01M, oxidized electrode. 96 «up: \m woo—2 A000 00... 0.0.. ON... 0...... « « N0 I W“ (01) 90'] 97 also agrees reasonably well with the value obtained with reduced electrodes. The above evidence, while by no means conclusive, seems to suggest that the essential rate process is the same at oxidized and reduced electrodes. However. it is not really possible to postulate the mechanism by which surface oxidation affects the reaction rate without further rate studies on a variety of care- fully controlled surface oxidation states. 3) Mechanistic Conclusions Nothing in the present investigation suggests that the rate of this reaction is affected by anything other than surface oxidation. It appears to be a first order electron transfer reaction with no com- plicating kinetic steps. The linearity of the reaction order plot over the entire concentration range. the independence of the differential capacitance with respect to either the applied current or the potential. and the marked agreement between the charge and dis- charge capacitance at all but the highest concen- trations all support this conclusion. One could propose a mechanism based upon the association con- stants measured by Eaton. George, and Hanania (30) of (KnFe(CN)6)n'3 +‘ K"' + e" : (Kn+1Fe(CN)6)n'3 but if such steps take place, the current impulse technique gives no information to prove or disprove 98 them. The association-dissociation reaction is pro— bably extremely rapid with respect to the electron transfer reaction. 4) Comparison.g§ Results A comparison of the results of the present investigation with those of previous investigators is presented in Table II. It can be seen that the current impulse technique gives a rate constant for reduced electrodes which is about twice as large as that measured by any of the other techniques, while that measured for oxidized electrodes is substantially less than any of the others. The estimates of apparent transfer coefficient agree very well except for the value obtained by Jahn and Vielstich (42) with the .rotating disc electrode. It is concluded that the observed differences in the rate constants are pri- marily due to differences in electrode conditioning. It has been shown that the apparent rate constant is dependent on the amount of surface oxidation. though the measurement of the transfer coefficient is not. The conditioning procedures reported by previous investigators promote varying amounts of surface oxidation as has been verified in this laboratory. so their results are bound to be lower than the ones of the present investigation. 99 TABLE II Comparison of Present Kinetic Results with Results of Previous Investigatorsa Apparent Std. Rate Transfer Const. Investigator(s) Technique Ref. Coeff. cm/sec“1 Randles and Somerton Faradaic impedance 40 -- 0.09 Jordan Hydrodynamic voltammetry 41 -- 0.08 Jahn and Vielstich Rotating disc electrode 42 0.61 0.05 (C) Agarwal Faradaic rectification 43 0.49 -- Wijnen and Smit Cyclic potential step 44 0.55 0.095 (C) Wijnen and Smit Cyclic coulombic step 44 0.50 0.13 (C) Daum and Enke Current impulse reduced electrode -- 0.50 0.24 (avg.) Daum and Enke Current impulse oxidized electrode -- 0.46 0.028 aResults designated (C) were not reported by the respective authors but are calculated from their data. IV. A NEW APPROACH TO GALVANOSTATIC MEASUREMENTS A. Introduction An alternate method for the study of fast electro- chemical processes is the galvanostatic technique. It is subject to many of the same problems as the coulo- static and current impulse techniques and these problems can be formulated in much the same way. since the charac- teristic parameters. To and Td' which characterize these methods are the same. The galvanostatic technique is very simple con- ceptually. A constant current of precise magnitude is applied to an electrochemical cell and the potential is followed as a function of time. The perturbation is limited in magnitude by the requirement of lineariza- tion of the absolute rate equation to a few millivolts anodic or cathodic of the equilibrium potential. Initially. upon application of the current. the test electrode will depart from its equilibrium value towards the charge transfer overpotential ”c' which is the potential which allows the electrochemical reaction to proceed at a rate and in a direction which 100 101 is consistent with the current which is being applied and the concentrations of the reactants in the bulk of the solution. It is prevented from reaching this potential instantaneously by the existence of the double layer capacitance which is in parallel with the fara- daic process. The charging of the capacitance consumes a significant amount of the applied current initially (it = Cddn/dt), so that not all of the applied current is being consumed by the faradaic process. After the initial charging process, the rate of potential change with time decreases drastically, and essentially all of the current goes to the faradaic process. As time proceeds thereafter, the electrode reaction, if it is of sufficient rate, depletes the concentration of reactant and increases the concentration of product at the electrode surface, and causes the appearance of the mass transport or diffusion overpotential and this gradually increases as time proceeds. Berzins and Delahay (23) simultaneously with Lorenz (45) derived Equation 1, which describes the processes which occurred at the electrodes _ RTi 1 .5 1/2 1 1 n - Eff—”EEC”... + 2(n) nFC°(-6:)m + D—R-mn (1) where C0 = C0 = CR' Initially they considered only the processes of diffusion and charge transfer. It was thought that the charging of the double layer took 102 place in less than a usec, and thereafter changes in the morphology of the n I§° t curve due to further charging of the double layer were not important. The equation which they derived shows a square root depen- dence of the potential on time, with the intercept being proportional to the faradaic resistance. The concept of the method was to extrapolate the part of the curve which was diffusion controlled and followed square root time dependence to zero time, where presumably the process was purely charge transfer controlled and thus obtain the charge transfer over- potential from which the exchange current could be calculated. It was determined later that the charging of the double layer affected the morphology of the 0 Kg. t curve at times longer than those originally considered. Berzins and Delahay (28) reconsidered the problem including both charging and mass transport processes, and derived the following equations: 0 - figfiézlexpmzoerucstl/Z) + zscfi—WZ - 1] -";§{exp(72t)erfC(Yt1/2) + 2Y(§91/2 - 1]} (2) - I 1 1 + I 1 1 2 _ nFI 2 ) 8’7 ‘EQF'(co/bo + cR/bR)‘ [4nQF2(CO/DO + CRJbR) R102] ' (3 This equation was then linearized by eliminating the exp(X2)erfc(x) terms by assuming those terms were negligible when t was greater than 50 microseconds. 103 The linearized form, Equation 4, is essentially the same as that originally derived by Delahay, with the exception of the term containing cat which is the correction term for double layer charging. = -RT1* { ( +-jr-)t n v/TTIIF Col/D01)E - 110:1 _ _ nF [IL—1750+ o —7—CR DR)? IO} (4) There are several problems both experimental and theoretical which impair the general usefulness and mar the essential simplicity of the galvanostatic technique. These problems are especially acute when studying very rapid reactions. As kS increases, larger currents must be applied to the test electrode to obtain a measureable charge transfer overvoltage. Measurements must therefore be made at short times to keep the mass transfer overvoltage comparable to the charge transfer overvoltage. Consequently the con- tribution of charging processes and ill-defined mass transport process to the morphology of the 0 3g. t curve become increasingly large, and nonlinearity of the n Z§° t5 curves become evident. An experimental approach to this problem pro- posed by Gerischer and Krause (24) and used by many investigators was the double pulse galvanostatic method. With this method a pre-pulse of constant current of very short duration and high magnitude was applied to the cell to pre-charge the double layer to some 104 potential close to the charge transfer overpotential. In this way all of the current of the second pulse would be consumed by the faradaic process and not by the charging process. Experimentally the magnitude of the pre-pulse was adjusted so that the overpotential time curve started with a horizontal tangent at the beginning of the second pulse; thus the current passing through the cell would be entirely faradaic at that instant. Kooijman and Sluyters (25) have recently consid- ered the experimental difficulties incurred in adjust- ing the magnitude of the pre-pulse so that the initial cell response to the application of the constant current is a horizontal tangent. They have concluded that with present-day equipment the double pulse technique has little to offer over the conventional galvanostatic technique. They conclude that the maximum rate con- stant which can be studied is 0.5 cm/sec. Several mathematical methods have been developed to obtain charge transfer parameters from very fast systems. The first of these attempts was by Birke and Roe (46) in their study of the highly reversable Hg(I)/Hg system. Birke and Roe observed nonlinearity in their 0 3s. t8 plots at short times, and after a detailed examination of the mathematics, concluded that the nonlinearity was due to the neglect of the 105 exp(X2)erfc(x) terms in Delahay's equation. They then proceeded to expand the exp(x2)erfc(x) terms in a Maclaurin series and included the first few terms in the equation which they used. They calculated their exchange currents by an iterative procedure, the first estimate being that of the rugs. tit intercept. Unfortunately, after doing all of this work, they concluded that the curvature was due to other causes and proceeded to do the major portion of the work with the double pulse galvanostatic technique (47). Kooijman and Sluyters (16) proposed an alternate mathematical solution to this problem. They calculated values of the dimensionless overpotential as a function of at% for a variety of values of (8+Y)2/Ew, where a = nzFZ/RTCd(1/CODO% + 1/CRDR%). The exchange cur- rent is calculated assuming the knowledge of t, Co, CR' Do, DR’ Cd and it, computing a value of at% and the dimensionless overpotential, and reading the value Of (8+Y)2/BY. From the value of this ratio the exchange current can be calculated. This can be done for several points to assure the validity of the model which is used, and to obtain an average value of the exchange. 1 The major disadvantage of the technique is the necessity of assuming a capacitance value for the system. The values which are usually used are those obtained 106 by some other technique, generally under nonreactive conditions. These values are not necessarily correct, for there is nolg priori reason to believe that the capacitance value obtained under other experimental conditions will be the same as those of the system under highly reactive conditions for which the above corrections are necessary. In fact there is frequently good reason to believe that the capacitance values may be quite different. With solid electrodes there is no good reason to assume a capacitance value at any time, since those measurements are dependent on the time of day at which they are taken and are rarely reproducible to the required precision. The second problem is that of uncompensated solu- tion resistance. As has been mentioned, with fast reactions it is necessary to use high applied currents in order to obtain a measureable charge transfer over- potential. The uncompensated solution resistance may cause an ohmic drop which may be many times the mag- nitude of the overpotential of the system. This must be either known and subtracted from the total signal, or compensated with an electronic circuit. Any un- certainty in the value of the ohmic drop directly affects the estimate of the charge transfer over- potential. For very rapid reactions the extrapolated value of the charge transfer overpotential is almost 107 always less than a millivolt, so the IR drop must be known to less than a tenth of a millivolt in order to obtain measurements of the exchange current which are accurate to 10%. These effects have been pointed out by many authors and several means have been developed to eli- minate the problem experimentally and mathematically (48-50). Most of the experimental systems have been simple variations of the original bridge circuit pro- posed by Berzins and Delahay (28). The problem has always been to find a differential amplifier with a sensitivity of at least 1 mV/cm, which has a high bandpass (greater than 10 MHz) and an extremely high common mode rejection ratio, so that the overvoltage can be accurately extracted from a signal many times its magnitude. Coupled with the IR problem is the problem of the definition of zero time. For extremely rapid reactions it is necessary to define zero time very accurately, and this is often not possible because of noise and initial mismatch of the IR compensator. Finally, but by no means the least of the prob- lems, are the conditions for which the simplified form of the exact equation derived by Berzins and Delahay (28) is applicable. Reduction of the exact equation requires that the terms containing the exp(x2)erfc(x) be very small with respect to the other terms in the 108 equation. The situation where these terms are neg- ligible is somewhat clouded, since for most reason- able values of the parameters for an electrochemical system these arguments are complex. Berzins and Delahay (28) arrived at a general time condition of t ’2 50 nsec by considering sets of electrochemical parameters where the arguments of these functions are real. Obviously this condition is not of general applicability and later Inman, Bokris and Blomgren (48) emphasized that the condition t >> 50/82 should be used in its place. The entire problem of linearization was recon— sidered in detail by Kooijman and Sluyters (51). They derived a generalized time condition of B 2-2 c > 100 (5+7382Y2 BY] (5) for which Equation 4 holds to 1%. They showed that this reduces to somewhat simpler forms when three general cases were considered: (ear? << ev. t > 100/81 <5a) (EH~)2 >> BY. t > 50(B+Y)2/82Y2 (5b) (m)2 = 2 5Y1 t > S/BY (5c) B. Scepe of Reported Research The mathematical formalism of the galvanostatic technique will be reformulated in terms of the charge transfer time constant and diffusional time constant 109 To and Td’ which are equivalent to those in the coulo- static technique. While this transformation is trivial nmathematically, it allows the problems of the technique “to attain a physical significance which is not immedi- ately evident from the traditional formalism. The conditions for the reduction of Equation 2 ‘to Equation 4 will be reexamined in terms of the ratio Tc/ 3 and té/Tdé, and the consequences of the errors involved in assuming the linear equation are discussed. IFinally an approach will be suggested for the study (of rapid electrochemical reactions and some preliminary experimental data on the hexacyanoferrate(III)/(II) couple will be presented. C. Theory We can define a charge transfer time constant Tc -_- arr/maxO (6) and a diffusional time constant Td .-. [(RTcd/n2F2H1/Conf + 1/CRDR% )1 i just as we did in the coulostatic technique. Thenf3,Y of Equation 3 become B,Y = TC; 1' [(rdé-4)/4Ic]%. (8) Equation 4, which is the reduced form of Equation 2. trans forms to n = (it/Cd)(Td%(t/fl)% + Tc-Td (9) 110 and the intercept of this equation at zero time is no = (it/0d)<.c-.d) (10) Some of the difficulties in obtaining estimates (of the heterogeneous rate constant and exchange currents iof fast reactions can now'be discussed in terms of these variables. The ratio rc/rd can be considered .an.indicator of the measureability of the charge trans- fer parameters of an electrochemical reaction. When this ratio is large, the amount of kinetic information is large, since perturbations due to mass transport are small with relation to the charge transfer process. .As the ratio becomes smaller, the amount of kinetic information becomes smaller because mass transport dominates the ".!§° t morphology. In the galvanostatic technique the amount of information on the charge transfer process with relation to other types of infor- tmation is proportional to the intercept. The galvanostatic technique is limited to small overvoltages because of the requirement of lineari- zation of the absolute rate equation. The technique has also been historically limited by the assumption of Equation 4 for the calculation of the intercept, and hence the charge transfer resistance, and this equation is only valid for long times. The ratio of the intercept at time zero to the overpotential at time t where the reduced equation is 111 applicable is 13%(t/")%/Tc-Td. It can be easily seen that as Id and time increase, the intercept becomes a smaller fraction of the potential at time t and that part of the intercept which relates to charge transfer becomes a smaller fraction of the intercept. It would seem that the ratio could be made more favorable by increasing the applied current and reducing the time at which the measurements are made, so that the signal to noise ratio can be made more favorable. However this has not been done, because Equation 4 is not applicable at short times. Figure 25 shows an analysis of the errors invol- ved in the determination of the exchange current as a function of the ratio rd/rc to indicate the time ranges for which the assumption of the linear equation is valid. The errors were calculated by generating a series of tables of the dimensionless overpotential n' = nCd/let, and dimensionless square root time tg/Td% for ratios of Td/Uh from 0.01 to 10.0 with a modified version of the program in Appendix B. The only region where the error approaches zero is the point where Td/Tc‘= 2, no matter from what time the extrapolation is made. This is in apparent disagreement with the criterion developed by other investigators for the assumption of the validity of the reduced equation. 112 Figure 25 Determinate error incurred in measurement of exchange current in the galvanostatic technique by using Equation IV-4. 10.0 and 9.0 A. From the points where té/ng 22.0 and 20.0 1 3. From the points where t%/Td§ l C. From the points where té/Tdf 45.0 and 40.0 00.. .3363 8.6 113 l l 0? ON ON 0? 80883 96 114 The criterion which other investigators have used is that the reduced Equation 4 agree with the general Equation 2 to within 1 %. However, this is not sufficient guarantee that the intercept, hence the value of the exchange current, will be accurate to 1%. For example, consider Kooijman and Sluyters' (51) second case, and the time condition which they derive, 1:2,, for (B+Y)2/BY or in present notation Tdfirc X> 1, t > so< 8+. )2/82Y2. Consider a system with the following parameters, 6 D = DR = 10'5 cmZ/sec, C = C = 10' moles/cm3, o 0 R Cd 3 2.0 x 10'5 F/cmz, I = 0.4501 amps/cmz, n = 1, 0 T = 3000K. The ratiord/rc for the system will be 10. According to Kooijman and Sluyters (51) the con- dition for the application of Equation 4 is that t = 5.743 x 10'“ sec. Taking the dimensionless time points of té/rdé = 10 and 12, one calculates that the times are 1.149 x 10"3 and 1.65 x 10'3 sec respectively, which certainly satisfy the above requirement. The values of the dimensionless overpotential calculated from the exact equation are 10.4288 and 12.6781 res- pectively. Using the reduced equation, values of 10.3838 and 12.6405 are calculated, and these are within 1%. However, the intercept is equal to -0.32, which is about 3% of the total value of the overpoten- tial and gives an error in the value of the exchange 115 current of about 40 z . Thus the requirement that Equation 4 be within 1% of Equation 2 is not sufficient guarantee that the measurement will be accurate. Little is to be gained from extending the experi- ment to longer times where the two equations agree better. For if the value of the overpotential at the above times is about 5 mV, then the intercept will be about -0.15 mV, which is a very marginal signal. The longer the time, with the limitation that n < 5 mV, the smaller the intercept, and it ultimately becomes immeasureable. D- A _Newwa 322 Galvanostatic Measurements A new approach to galvanostatic measurements is proposed here which makes no assumptions about either the validity of the reduced equation or the value of the double layer capacitance. The experimental approach is of different emphasis than that traditionally used by experimenters in that extremely high currents are applied to the test electrode to make the charge transfer overvoltage a significant portion of the magnitude of the allowed overpotential. This requires that measure- ments be made at very short times to insure the 5 mV limit is not exceeded and to allow an experimental estimate of the value of the double layer capacitance. 116 The data is analyzed by curve fitting the experi- mantal overpotential time curve with a computer pro- gram which calculates the theoretical overvoltage for a given set of experimental parameters. The same curve fitting routine which was used for the current impulse technique is used in the galvanostatic technique to manipulate the experimental data to obtain estimates of the exchange current and capacitance. This experimental approach has been made possible by the availability of some new instrumentation which enables the experimenter to apply relatively high currents to the electrochemical cell and still extract the everpotential from a signal which includes a very large ohmic drop. 1) Experimental Some very preliminary galvanostatic measurements have been made on the hexacyanoferrate(III)/(II) system. This system does not fulfill the requirements necessary really to test the validity of this approach to galvano- static measurements for fast systems. Though the rate constant of 0.28 cm/sec is moderately high, the appli- cability of this method would show its real advantages in systems where the heterogenous rate constant is 1 cm/sec and higher. Measurements were made in systems where the concentration of the hexacyanoferrate(III)/(II) was high in order to insure large exchange currents, 117 but the ratio of Tc/Td was so large that it was quite easy to do the measurements in the conventional way. The instrumentation was essentially the same as that described in Section III-B-i. The combination Tektronix 556 oscilloscope, P6046 probe and PG-32 pulse generator were used without significant modifi- cation. The IR compensating circuit was modified by the addition of a 1000 ohm resistor in series with the output of the pulse generator to limit the currents to values which were consistent with the reaction which was studied. With this experimental system it was possible to make measurements from times as short as 100 nanoseconds after the start of the current pulse and IR compensation was good to at least 0.1 mV. The procedure including chemicals, electrode pre- paration and so forth, was exactly the same as that des- cribed in Section II-B for reduced platinum electrodes. 2) Results and Conclusions The experimental results were quite varied, and correspondingly quite inconclusive. A typical galvano- stat of the hexacyanoferrate(III)/(II) couple is shown in Figure 26, along with the curve fit parameters of the exchange current and capacitance. Results at any one concentration varied by as much as 100% above and below the average value obtained with the current 118 Figure 26 Experimental and theoretical relaxation curves for the hexacyanoferrate(III)/(II) couple in 1M KCl, Co = 0.03M, CR = 0.01M. -—-’ Experimental curve x Theoretical curve calculated from curve fit values of IO and Cd Data Co = 3.00 x 10"5 moles/cm3 D0 = 8.90 x 10"6 cmZ/sec CR = 1.00 x 10"5 moles/cm3 DR = 7.40 x 10'"6 cmZ/sec area = 0.0835 cm2 it = 0.0413 amps/cm2 Curve fit estimate 0.353 amps/cm2 13 .9 UF/cmz I0 C d 119 851 m2: >5? 120 impulse technique under the same experimental conditions. The morphology of the rigs. t curve was dependent on the position of the test electrode with respect to the counter and reference electrodes, among other things. Results could vary up to 50% on the same experimental solution just by varying the position of the test elec- trode with respect to the reference several millimeters. This kind of result points out the difficulties which experimenters encounter when trying to make kinetic measurements during a perturbation. The problems of noise, cell geometry3shielding, and frequency dispersion are all magnified under these conditions. Although these results are inconclusive, there is no reason, in principle, why the method will not work. The disparity of the above results is undoubtedly due to some experimental problem which will take some time and ingenuity to work out. The method holds great promise for the examination of extremely fast reactions with the galvanostatic method without resorting to the usual compromises in either the experimental or mathe- matical parts of the technique. REFERENCES 6. 7. 8. 10. 11. 12. 13. 14. 15. REFERENCES Weir, W. D. and Enke, 0.0., J. Phys. Chem., 2;, 275 (1967)- Reinmuth, W. H. and Wilson, C. E., Anal. Chem., Reinmuth, W. H., Anal. Chem., 24, 1272 (1962). Delahay, P. and Mohilner, D. M., J. Phys. Chem., fig. 959 (1962). Delahay, P. and Mohilner, D. M., J. Am. Chem. Soc., 303, 4247 (1962). Delahay, P., Anal. Chem., 34, 1161 (i962). Delahay, P., J. Phys. Chem., éé, 2204 (1962). Reinmuth, W. H., and Delahay, P., Anal. Chem., 34, 1344 (1962). Barker, G.C., in "Transactions of the Symposium on Electrode Processes," Philadelphia, 1959, pp. 325-365, E. Yeager, ed., John Wiley & Sons, Inc., New York, 1961. Delahay, P. and Aramata, A., J. Phys. Chem., éé, 2208 (1962). Hamelin, A., Compt. Rend., 252, 1709 (1963). Hamelin, A., Electrochim, Acta, 2, 289, (1964). Hamelin, A., Compt. Rend., 252, 362 (1964). Wilson, C. E., Ph.D. Thesis, Columbia University, New York, N.Y., 1966. Kooijman, D.J., J. Electroanal. Chem., 12, 365 (1968). 121 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30- 31. 32. 33- 122 Kooijman, D. J. and Sluyters, J. H., Electrochim. Act-'30. $2. 1579 (1967)- Weir, W. D. and Enke, C. G., J. Phys. Chem., 2;, 280 (1967). Daum, P. H. and Enke, C. 0., Anal. Chem., 4;, 653 (1969)- Levy, P., Ph.D. Thesis, Columbia University, 1965. Delahay. Pep Anal. Chem., 2Q. 1267 (1962). Vielstich, W. and Delahay, P., J. Am. Chem. Soc., 220 187“ (1957). Gerischer, H. and Vielstich, W., Z. Physik. Chem. (Frankfurt), 2, 16 (1964). Berzins, T. and Delahay, P., J. Chem. Phys., 23, 972 (1955). Gerischer, H. and Krause, M., Z. Physik. Chem. (Frankfurt), 19, 264 (1967). Kooijman, D. J. and Sluyters, J. H., J. Electro- anal. Chem., 13, 152 (1967). Anson, F. C., in "Annual Review of Physical Chemis- try," V01- 19, H. Eyring, ed., Annual Reviews, Inc., Palo Alto, Calif., 1968. Martin, R. H., Ph.D. Thesis, Louisiana State University, New Orleans, La., 1967. Berzins, T. and Delahay, P., J. Am. Chem. Soc., 22. 6448 (1955). Hartley, H. 0., Technometrics, 3, 269 (1961). Eaton, W. A., George, P. and Hanania, G. I. H., J. Phys. Chem., 21, 2016 (1967). Davis, D., Talanta, 3, 335 (1960). Koltoff, I. M. and Nightengale, E. R., Analyt. Chim. Acta., 41, 329 (1957). Anson, F. 0.. J. Am. Chem. 500.. fig. 1554 (1959). 34- 35. 36- 37. 38. 39. 40. 41. 42. 43. an. 45. 46. 47. 48. 49. 50. 51. 123 Hickling, A. and Wilson, W., J. E. Chem. Soc., 25. 984 (1951). Baker, B. and MacNeven, W., J. Am. Chem. Soc., 25. 1476 (1953). Anson, F. C. and Lingane, J. J., J. Am. Chem. Soc., 12. 4901 (1957). Piersma, B. J., Schuldiner, S. and warner, T. E., J. Electrochem. Soc., 11 , 1319 (1966). Laitinen, H. A. and Enke, C. 0., J. Electrochem. Soc., 102, 773 (1960). Feldberg, S. W., Enke, C. 0., and Bricker, C. E., J. Electrochem. Soc., 110, 826 (1963). Randles, J. E. B. and Somerton, K. W., Trans. Faraday Soc., 4§, 937 (1952). Jordan, J., Anal. Chem., 22, 1708 (1955)- Jahn, D. and Vielstich, W., J. Electrochem. Soc., 102, 849 (1962). Agarwal, H. P., J. Electrochem. Soc., 110, 237 (1963). Wijnen, M. D. and Smit, W. M., Rev. Trav. Chem., Lorenz, W., Z. Elektrochem., 8, 912 (1954). Birke, R. L. and Roe, D. H., Anal. Chem.. 22. 450 (1965). Birke, R. L. and Roe, D. H., Anal. Chem., 21, 455 (1965)- Inman, D., Bockris, J. O'M. and Blomgren, E., J. Electroanal. Chem., 2, 506 (1961). Blomgren, E., Inman, D. and Bockris, J. O'M., Rev. Sci. Instr., 22, 11 (1961). Kooijman, D. J. and Sluyters, J. H., Electro- chimica Acta., 3;, 1147 (1966). Kooijman, D. J. and Sluyters, J. H., Electro- chimica Acta., $2, 693 (1967). APPENDICES APPENDIX A PROGRAMS FOR NUMERICAL CALCULATIONS 0N COULOSTATIC DATA A. Equations 22; Calculation 23 Overpotential 2122 Curves The mathematics of the theoretical coulostatic equation (Equation II-i) including mass transport are simple when the quantity Td/uTc is greater than one. The exp(X2)erfc(X) are real, and several well-known asymtotic expansions are available to calculate the values of these functions. When Td/4rc is less than one the arguments of these functions become complex and the calculations become more difficult. In these cases the error function complement can be expressed by Z erfc(-iz) = 1 + 3i f' exp(t2)dt (1) /1T 0 where in general 2 = x 1 1y. The function W(z) = exp(-22)erfc(iz) (2) has also been defined and can be expressed as Z W(z) = exp(-z2) [1 + 5; é; eXP(t2)dq (3) 124 125 and this function can be evaluated on the basis of an infinite series expansion, w (iz)n W(Z) =2 2 r(% + 1) (14') n: where F(r) is the Gamma function of r. The function we really desire is exp(z2)erfc(z) ' (5) and we can obtain this by substituting the quantity (iz) for (z) in the above equations. The resulting equation is then (D W(iz) = exp(z2)erfc(z) = 2 (-1)n(z)E . (6) This equation suffices for small values of the argu- ments x and y. For large values of these arguments, however, it is more convenient to use the following asymptotic expansion: on 2 m exp(z )erfc(z) = 7%2__{1.4- 3:1 (-1) i. .2;.. 2m-1 (7) These equations were used to write a FUNCTION PROGRAM in FORTRAN IV for the calculation of the theoretical value of the overpotential for given values of time, exchange current and double layer capacitance. It is shown as FUNCTION ETA(XT,IO,CAP) in the program LESSQ at the end of this section. The gamma functions which are necessary for the use of Equation 6 were read 126 in as data in the main program and designated common with the FUNCTION ETA. B. Curve Fitting The object of curve fitting is to vary the para- meters of interest, namely Cd and ID in a systematic way so as to generate a theoretical curve which agrees to a predetermined degree with the experimental curve in question. The equation which must be minimized is: n Q(Io,cd) a 1-1 (yi - Eta(x1,Cd,IO))2 (a) where y1 is the experimental overpotential for the 1th experimental time :1, and Eta(x1,Cd,Io) is the theoretical overpotential at time x1 for capacitance Cd and exchange current IO. If we approximate Eta by a multiple Taylor's series expansion of first order terms about the point (10°, cd°), take the partials of Q with respect to Cd and 10 and set them equal to zero, we obtain the equations which must be satisfied in order that the sum of the squares of the deviations be a minimum. n z [y1 - E111 - AIOBEta - AC aEta118Eta = 0 (9) -———1 . 1 1 1 310 93 Cd 310 127 n 3Eta z 0 (10) i-l a 1] W1 o a d d These Equations 9 and 10 are solved for‘AIo and 40d, and the Gauss-Newton method which consists of the suc- cessive application of the formulas 10(k+1) = 10m + “000' Canal) = Cam " Acdmm’ is applied. To guarantee the convergence of the method, a modification of the Gauss Newton method by Hartley (29) is used. With this method I0(k+1) 3 I00:) + vminAIO(k) ‘12) cd(k+1) a Cd(k) + vm1n10d(k) (13) where Vmin is defined by me . it + new) - 0(1))/(Q(1)-2Q(t) + 0(0)) (14) in which n 0 a 2 - ETA . I , 0 ~ 12 1 Q( ) 1.1 W1 (11 00:) d(k)) ( 5) n :3 8': “ETA .1 AI Dc Q(1) 1_1 xi (‘1 0(k) + —20(k) d(k) + 33.1,”)12 (16) n e Q(1) a: 1-1 [31 - ETA(11.IO(k)+ 410(k), Cd(k) + manna (17) This method not only guarantees convergence, but it also reduces the number of iterations required for convergence. 128 The partial derivatives of Eta were approximated by the relationships 3Et Et I DI - Et I - DI 3.1—9: 3 __a..(_o_1__o%fi.ro_._a.(_o__ol (18) 0 mm _ Etagcd i D03) - Eta(C - D031 'S—Cd ‘ 21>cd ‘1 (19) where D10 and DCd were small fractions of the total value of the exchange current and capacitance respec- tively. It is possible to evaluate these derivatives analytically; however, the results are complicated and consume a significant amount of computer time to cal- culate. These equations were incorporated in PROGRAM LESSQ written in FORTRAN IV for a Control Data 3600 computer. This program accepts experimental coulo- static data and computes the value of the exchange current and capacitance which produces the best fit with the experimental decay. Table A-1 is a list of the data input for this program including dimensions which the data must have, and the program symbol. Initial parameters were obtained from the slope and intercept of the first two experimental points. Iteration was terminated when the change in both of the parameters at the end of an iteration was smaller than some predetermined fraction of the variable before iteration. The output of the program includes the 129 TABLE A-I PROGRAM LESSQ: DATA INPUT $13? *Definition Uni ts C(K) Gamma functions none CO Concentration of 3 oxidized species moles/cm CR Concentration of reduced species ‘moles/cm3 D0 Diffusion coefficient 2 of oxidized species cm /sec DR Diffusion coefficient 2 of reduced species cm /sec INC Duration of applied pulse sec T Temperature °K CURB Magnitude of applied current amps AREA Area of electrode cm2 BN Number of electrons trans- ferred in the electro- chemical process none Nx Number of experimental points none NN Run number none X(I) 1th experimental time sec 1(1) Experimental overpotential for 1th experimental time volts 130 final curve fit value of the exchange current and capacitance, the concentrations. area of electrode, applied current, run number and a comparison of the overpotentials calculated with the final curve fit values of exchange current and capacitance to the experimental values. 131 aqu.mos..¢..o=oavie34.xfi.x.¢.ocaavuoamo..mv»..cooavdcomvuqhmm .QCeooum nvam.mua .~.mmca.a..ech .=~.>..amcx zo~m7m:_; cume 1..xoa..naia>.\..m:~»..xea.v .axm «pm..xo«..u.an «aw..\\..maz<..xo.n.man..u azmxawu moz+u+o~vuma4no~ .omm~o.a4«10..mnmzoaexamzcouoxuwss.m~.a . om.o uzazs m....acu.nxm.»x.qau-1_.>..dnsxcuu4<:u m....~ou.~xm.ax.¢»m-.~.>v.dzceum:oc nee..ooo.oxw.»x.<»uo-.>..czu~cuoxm~a ._vxuhx xz.auaam ac cuumcxa ouwzoo ouomm~a nnaa.u>xooc«.acswaiz. .m a_.>u»> aavxupx xrufiuh mm CL... cc acca mac” rC¢H mm 13h cxaa-m;nuozma rm. daze c» on otpzrnanaasqu\.auuuxx avrcquexmimuoza:xa nflo <<\Acz:mdu.avu»¥cm< nnxxup:sa-.a.u»xamm sop wow 0» on 61>.envmux.H.\Ao:o.avuu2:mm aAAAAAaAAAAAAARANe.H0+maeveNe.ON+.aveNe.hN+m .H.V.N..nm..fls.N..mu..H.V.N..Hm..Hv.~..oa..a.s.~..~fi..ae.a..mi..a.a ..~..na..fi..~..fia..r...~..o..fi..~..~..H...N..n..q..K..m..s-..Nsnow “ma>..nv\.fiux «on “om ca ow 1»ixma.md>vuaxmu02:an 5.0Hoveca..nAAoueoyonuVcd>+vuvad>onuVod>emu.41»+auvaa>+.d.ua>axm com mam.¢om.vom .o.m.a>.a. sow .1x.nrm~x.a.\.a<..avuuz:a< ”A.A.A»AA.AA“A..3..anouauvc3c.o~o.fivezy.bmom .a...2..nm..av.x..n~..a.v.3..aw..av.g..oa..H-..x..aa..fl..3..ma..H.H v.3..nao.“v.3..aa..a...3c.o+.«v.3..n+.fluv.3..no.4..3..n+.«u..3vuoa .mnx..mvx.wu2 mom com o» oo dxmxm.amdx.daxuuuz:aq 1.0a......AA.Iou.cx.nu..dx+euv.1x.nu..dx.mo..1x.au..nx+.avuoxmmm «on nan.wom.mon .c.n.ax.u_ meod>uma> mcedxuoax pmuummud> pzm.<«ndx .xvhaomukzm mmm .zqe.m.\arm»ammnovumm .zqu.m.\aru»amm.n.um~a+mccxc~onrmzau macdx.i>..com..u>..mu>c~m neuux..Muveeayodro.0uxc~n .m..1>+m..a)v\d>ouoxbn .m..n»+mc.axv\uxnm(o~ flcov a» ow ~3H c» On H+xux q:«.vofi.mofiaao-x.d_ maH.maa.vo~xaN-ho.fiu.dex.umnva.Annxemvuxva..m:zemvnx.nuflrvo ex.muz.fiv oL nHvDuOIDu .a.x...ax.x..muaavc a.xuxv any =.=uuuxx ooam.soan.HH:Han.m-.dx.ummeama<.i_ ooom.gosn.HH=H.n.mu.d>.uma<.d~ ncsa.oo=~.msomIo.fi..ax.uma<.ai coon.sasn.flaaa.c.m..a>.uma<.d_ “cam.flsa~.==omxfl...ax.umm<.ma m.»amud> «.aamudx .xvhuzmubmm A2.mu.maqzm>+oqra>.ospuv.uu emvm-.fi\.npmxx.mmmrx.pmxx.mmx.mzcwvuu Anomzcu..ofle\.n_>maa..mCfl..umc Anomzcu..oasxxnaxcaa..noavumpmmx m..np>aan.~..maxo“muncmzau oeelyoxeeayemeeuxc.cnna meen>eweoaxe.cmfieneea>eoee1xe.cc|1>ameenxe.oum»>c~m occa>.dx..m.o..a>.n..ox..vmufi e..d>.m..ux..ema.m..i».~..1x..on-s..axumaxc_m Amcmxos..m.\.~»»c~¢..mfieumcezma .monzou..n.\i~axc_m..ma-.umpmzx ~..ma>o_n.m..maxu_mumcmzou n.an>um..n».m.caxa.«~+naad>.v.amx..nm.1>.o..ax..num»>c_m o..d>.dx..Nuv..d>.n..ax..mn+m..d>.m..1x..Hmun..axumhxc~m “hoazau..¢s\.a»u_a..m-..»o .aomzou...e\1axoam..m.u»max m.._>a.m+m..»xw_muhcmzou mccd>onu.a>.u..dx.maaoo>.v¢.dx..mu»>c_u vecd>.1x..n+n..d>coc.ix..oaumceaxuhxcum mace nauv APPENDIX B COMPUTER PROGRAM FOR NUMERICAL CALCULATIONS ON THE GALVANOSTATIC TECHNIQUE The equations used to calculate theoretical zg. t curves for the galvanostatic technique are very similar to those of the coulostatic technique. Equation IVAZ could be expressed in terms of the same series in only a slightly altered form. FUNCTION ETA(X.IO,CAP) written in FORTRAN IV computes the value of the over- potential for a given value of X (time). exchange current (IO). and CAP (double layer capacitance). This function was combined with a slight modi- fication of PROGRAM LESSQ to curve fit experimental data. It was necessary to read in initial estimates of the exchange current and capacitance since there is no easy way to estimate the exchange current from short time measurements. 137 138 mom.vchvom Ac.vlu>~u~ row anx.mvmnn.fiixao«+.aiucz_d4 AAAAAAAAAAAAAA.Aza.am+.auvc2c.omo.—v.3..»~+m .fisvcx..mm..a..w..nn..au..z..Fm..fi..2..oa..wu~.3..nr.ufia....mr+.fl-fi v.{..ma+..V.1..r~..wnve3..o+.fi..3c.mo.anv.z..m+.Ha.3..n..r-..:vu3< amnxe.nv\..u3 Mom mam c» 30 axumucamoxyuoxuuc7_m< “.0wcv..aa.hfinou.ux.muvodx.eu..dx+mu..mxonov.oxoau..d>+.a.uuxuxm mom nom.momnwom ac.wuax.m~ meod>una> m..d>unmx aow.«yz vaueckdmudx axvhwovubam mmm smup>onuqzs.dn<»ma .cuczaalpn,nu>mu»> Hum cmm.cmu.mo¢.r-nx»a.d_ .Acacxcav.1naeavxd.zaVinc moenucxpa “n.2ma.nv\~nfirwrec~.ud .NPmC\.H.+»H;wC\.H.nnaymc armrvrmcmvenuumzmc .Aocspmcm..rUunsma ammorvaoro.cnoe nucmwmm0ro.oumu resonmIOro.ouve ounmmnnoro.onmc rmwonmnmva.oumu emuormmoso.ouac beefi.u.u.m.mq3n.».2m.xe.quac.ro zcz,ou urhyzo.n~ mama v~.m~.7an.aa.nN31H~z.mamxu.amwxm.mxmz:m.ayw2:u.n~naa ummasoc ~dwvau nauxu+axuzamumxm33v An n1.u\axe.aomcv.unawxm ao.auxsn o; 1m.£.m.c.umxmsam c.fi.~xuzsmua>m33m fiadxu+axuzzmuaxmaam an .Yvux.xc.AWN..suwauxm mo.auv (s a: 1m.c.m.csuastau HHQH coon.nocn.flfioF.m.m.,AN.umqo.m_ Armub>IJQVXsazouvm .zaw»>na.x-a:oumN ao>I-DX.XIQ}UHNN .d>.cxix-asoufi~ zaw»>.»umua> a.»umuax nxvhnounpmm acxpdno.»aomu>muk> soc «cow c» oc .«amm.«zy.ememnn.fiv\.oa+.a.upzqdc A...A..an“Aaaa.a~c.«m+.au..No.om+..V.N..~m+m .alveNe.mme.dvexe.nmendache.HN+.H.e~e.3«e.a|vene.hw+.avase.mfie.flua v.~c.nfio.fivoNc.ra+.«u..~..o+.fiv.N..n+.aov.N..m..a.eu..mo.a...avn3a .mo>..nv\.wun now .om c» oc u>dmuenma>~doxmuuzaaa n.0Hc....».anacced>+muvad>ovuead>.nuv.1>.no..c>+ac..d>+.H.un>exu com 140 rzw zuspwo urayoa.xo