A/C /nferfase
A Simulator for d
F
Voltammetric Responses ith their 1964 paper (I), Nicholson and Shain launched the modem era of theoretical characterization of cyclic voltam metric responses. Thus began the evolution of cyclic voltammetry (CV) as the electrochemical technique most commonly used (and abused) by electrochemists as well as physical, organic, and inorganic chemists. Our effort to develop a CV simulator was generated by the need for a simple and manageable way to analyze complex electrochemical responses. A cyclic voltammogram is a complicated, time-dependent function of a large number of physical and chemical parameters. Unraveling the coupled dynamics of heterogeneous electron transfer, homogeneous first- and second-order chemical kinetics, and diffusion presents a formidable experimental and computational challenge (2-4). Even computation of the response for a simple electron transfer (A + e G B) requires numerical integration (I) or summation of a series ( 5 ) .Until now there has been no effective compu-
Chemists can use a generalized CV simulator to explore the behavior of virtually any electrochemicalchemical mechanism Manfred Rudolph Friedrich- Schiller Universitat
David P. Reddy Radio Logic Inc.
Stephen W. Feldberg Brookhaven National Laboratory
tational tool for describing the cyclic voltammetric responses for a wide range of mechanisms-at least not without a lot of ad hoc adjustments. Researchers using numerical approaches (6) have been stymied by the “stiffness” of many systems (i.e., the phenomena of interest occur over a large dynamic range of time and/or space). Because of these difficulties,researchers attempting to produce a general CV simulator have had limited success. Speiser has used orthogonal collocation as the basis of EASIEST, which currently is linked to a mainframe computer and is used to access a library of algorithms to compute the cyclic voltammetric responses for specific mechanisms (7).Gosser has devised CVSIM, a simulator based on an explicit finite difference algorithm that is useful for simulating responses for simpler systems but becomes extremely inefficientwhen simulating stiff systems (8).Bieniasz has devised ELSIM, a potpoum of mathematical and numerical techniques that can be used to simulate
Analytical Chemistry, Vol. 66, No. 10, May 15, 1994 589 A
various electrochemical responses but demands a mathematically sophisticated user (9,lO). This computational challenge is effectively met by a generalized CV generator based on Rudolph’s recently developed fast implicit finite difference (FIFD) algorithm (11-14, a refinement of an a p proach described by Newman (15).Rudolph’s CV simulator accepts any userspecified mechanism (any combination of one-electron heterogeneous electron transfers as well as first- and/or secondorder homogeneous chemical reactions), efficiently simulates CV responses for stiff systems, and-perhaps most importantdemands no mathematical or computational skills from the user. Despite the effectiveness of the FIFD algorithm, the easy access to fast, powerful, and inexpensive computers has been essential for the development of a practical simulation tool. The purpose of this A/C Interface is to explain why the FIFD algorithm is so powerful and to describe the capabilities of an operating version of Rudolph’s CV simulator and its applications. Basics of CV
CV is a relatively straightforward technique (16-18). A working electrode (where the reactions of interest occur) is immersed in a solution containing some electroactive analyte, an excess of electrolyte, a reference electrode, and a counter electrode.The applied potential (the potential between the working and reference electrodes) is controlled by a potentiostat that passes the required current between the working and auxiliary electrodes to maintain a desired potential. The common experimentalpractice is to choose a starting potential for the working electrode ESm, where the analyte is electroinactive, and then to change the potential at a constant rate v (= dE/dt) moving from E,, through a potential range where oxidation or reduction of the analyte can occur, to a reversing potential E, where the sweep direction is reversed (-v) and the potential usually returns to Estart.These steps constitute a single cycle, but the process can be repeated any number of times. A wide range of values of lul (19) has been used (0.05-2 x lo6V/s). The interfacial electron transfer for the heterogeneous process A + e s B can be 590 A
described by the Butler-Volmer equation (20)
wheref, is the flux of electrons from the electrode to the solution; k, is the standard rate constant (cm/s); a is the transfer coefficient for A + e B; F is the Faraday constant; R is the gas constant; T is temperature (K); E is the applied potential as measured between the working and reference electrodes; and Eo is the formal potential (volts). Equation 1is written for a one-electron transfer; the simulator can be used only with one-electron transfers. (An nelectron transfer can be described by a sequence of one-electron transfers with appropriatelyselected values of the heterogeneous parameters.)
*
This
computational challenge is met by a CV generator bused on Rudolph’s FIFD algorithm. The experimentally measured r e sponse is the current that passes between the working and auxiliary electrodes. The cyclic voltammogram is a plot of the current i versus the applied potential E (i = -F x electrode area x f,),where the negative sign makes the plots consistent with Equation 1 and our plotting convention.A typical cyclic voltammogram for the simple reversible electron transfer A + e e B is shown in Figure 1.The current minimum and maximum appear when diffusion, rather than Equation 1, begins to dominate control of the rate of electron
Analytical Chemistry, Vol. 66,No. IO, May 15, 1994
transfer. The potential and current values of the first cycle agree well with extant theory (1).Values of k,, which are smaller than the theoretical maximum of lo4 cm/s at 300 K (21,22) (e.g., 1or 0.1 cm/s), generate slightly different values for the position of these peaks at a scan rate of 1 V/s. The shape of the voltammogram is a function of temperature, the experimental parameters (Esta,,E,,,, v ) , the heterogeneous electron transfer rate parameters (k,,Eo,a),the diffusion coefficientsand concentrations of each species, and the electrode geometry. When additional electrochemical and/ or chemical reactions are involved, as in the ecec mechanism A+e=B BeC C+e=D B+C=A+D
(ecec 1) (ecec 2) (ecec 3) (ecec4)
the rates of the heterogeneous and homogeneous reactions will modify the shape of the voltammogram in significant and interesting ways. (Ecec denotes the sequence of reactions: electrochemical, chemical, electrochemical, and chemical.) The computation of all the coupled interactions of spatially inhomogeneously distributed species presents a formidable challenge. Systems such as the ecec, which involve second-order processes, are most effectively tackled by using numerical methods. Systems considerably more complicated than the ecec are often encountered, and a truly general CV simulator will have to depend partly, if not entirely, on a numerical rather than analytical or quasi-analytical approach. CV can easily be used to determine the concentration and thermodynamic properties of a redox couple. Extracting mechanistic and kinetic details of a given electrochemical-chemical system is much more difficult. To accomplish this, the analyst must first postulate an electrochemicalchemical mechanism. This requires intuiting or guessing just what is going on from the shape of an experimental voltammogram and from various diagnostics (e.g., scan rate dependence, peak separations, currents) as well as from other, possibly nonelectrochemical, experimental data. A sound understanding of the rudiments of cyclic voltammetric behavior and the chemistry of a particular system is an ob-
tions and are spatially diffusion layer; e number of efore the comgiven sim-
finite diffinite differtinction is ations) of a lues methods
- At, y to
tion (i.e., t = A changes in c see Figure 2). We k sentation of any d must extend a dis (37) from the electrode cies must arise cannot compute a c beyond x = Ax DAt? Homogeneous reac duce a reaction layer (24),and
simulation demands that the th
curate depiction of the concentration within the reaction layer (k is a first-order or pseudo-first-order rate constant). Effiterest requires that kAt >> 1,which leads to the requirement that >> 5Ax equivalently DAt/Ax2 >> 1) (12,34).Th the use of EFD methods is seriously reby propagational inadequacy even the devastating constraint, DAt/
fi
tion involves the di into volume elem ponentially expandin causes an increase in t volume elements as th cient and accurate tre
is removed by the hopscotch or DuFortFrankel algorithms. IFD methods are not constrained by propagational inadequacy. In an IFD com-
first cycle.
Analytical Chemistry, Vol. 66,No. IO, May 15,1994 591 A
Figure 3. Simulated voltammograms f
e e B as parameters are changed. /s; for curve 1, D, =
592 A Analytical Chemistry, Vol. 66,No. IO,
equation for each species and for ea volume element), and changes vant volume elements are compu is involved, or when the nonchemically coupled species, equa
When species are chemic as they often are in many reac nisms (see ecec 1-4 on p. 590 a singleband matrix of order n, . that must be inverted (n, is the n species and nVeis the number of elements; the “+ 1” appears becaus cies may be coupled by Equation boundary corresponding to the e surface). This singleband reduced to a block tri which can be solved e spite earlier descriptions of this (15),application of the implicit been limited ( 1 5 , 3 M 1 complexity and by c
tial grid, easier access to i puting power, and an effi unraveling the resultant
dle the nonlinear equations that evolve from higher order reactions, such as B + C ;=r A + D. By using the fully implicit Laasonen algorithm (42) coupled with Richtmyer modfication (43), tain an accuracy rivaling that iar Crank-Nicolson algorithm (44) and eliminate the instabilities that can occur with the Crank-Nicolson algorithm when very large values of DAt/Ax2 ar (45).
,
The programming of the FIFD method is quite straightforward.Its versatility, robustness, and power more than compensate for sacrificing the simplicity of EFD methods, which are seriously deficient in simulationsof stiff systems. FIFD-based CVsimulutor. As long as a few simple and invariant rules are followed, the FIFD algorithm is extremely
muly any electrochemicalchemical system over a wide dynamic range of parametric values. Thus it is possible to translate a straightforward description of an electrochemical mechanism into the mathematical expressions required to compute the cyclic voltammetric response. For the previously discussed ecec mechanism, for example, the completed mechanism entry window in ogram would be A+e=B B=C C+e=D B+C=A+D With the equal sign we imply that every reaction must go both ways, a manifestation of microscopic reversibility (46).
Using the simulator to explore simple mechanisms is a good way to hone one’s
e various species as they change during cyclic voltammetric perturbation (i.e,, a graphic display of the changing profiles-CV, the movie!), VeriJ512ngsimulation accuracy. How do we know that a given simulated response is correct? One approach is to select conditions for a particular system that devolve to familiar, previously analyzed, and reported examples. However, doing so does not guarantee that simulations of the intermediate conditions are correct. Another approach is to vary the values of the simulation parameters (e.g., At, Ax, and the parameter p, which controls the rate of expansion of the exponential grid). The default values of At, Ax, and p are selecte by the CV simulator to optimize accuracy, stability, and computational efficiencylarger values will produce faster computation with decreasing and eventually unacceptable accuracy, and smaller values will result in slower computation with increasing and eventually a limiting and perhaps unwarranted accuracy. I€reducing the default values of At, Ax, and/or p for any given computation significantly changes the simulated response, either the default values were not properly gramming errors exist. Exploring basic concepts
Nicholson and Shain (1)fully characterized the dependence of the cyclic voltammetric response on the physical and chemical parameters for simple electron transfers and for simple electron transfers led with first-order chemical reacs. Although most CV practitioners are iliar with this work, the simulator offers an invaluable demonstration and The program is used to construct the matrices required to carry out the simula- explication of these and many other phetion; request values for all the cyclic voltam- nomena that shape the cyclic voltametric response. Several examples of COUtric, physical, and chemical param ed heterogeneous and homogeneous actions will be considered. The practitioners’ familiarity with these fundamental ous reactions discussed on p. 596 A and features of cyclic voltammetric behavior allowing some values to be set and automatically computing the others); and offer will enhance their interpretive skills. Response for a simple electron transfer. options for electrode geometry and plotWe will consider the response of the ting convention.The user can simulate simulator as parameters are changed for CV responses, compare simulations with each other and with the experimental data, perform least-squares fit simulations (Figure 3). to other simulations or to experimental creases the peak by a factor of 2, as shown voltammograms with selected parameter in Figure 3a. ehavior is summarized optimization, and observe the concentraAnalytical Chemistry, Vol. 66,No. 10, May 15, 1994 593 A
’
n of the dimensionless cur-
As k,
A change in the ratio D,/D, causes a shift in the potential as illust ure 3b where D,/D, = 1,lO that the height of the reverse peak changes only slightly because of th changing relative positions of E,,, Epeak. The larger the value of !Erevthe higher the reverse for an apparent Eo is
Eoapp = EO
+ (RT/
duces the familiar capacitive uncompensated resistance R, 3d), as does a reduction in the value of k,. The omnipresent uncompensated resistance presents a fundamental diffic evaluating k, from CV responses (4 The combination of capacitance an compensated resistance induces a nonconstant dE/dt at the outer Helmholtz plane, an effect that is not easily removed from the data, even if Cd,is potential independent. The shape of the cyclic voltammo is significantly changed when the thic
Figure 4. Simulated voltammograms for A + e G B and B e C. (a) k, = 1 .O cm/s; T = 298.2K; area (planar electrode) = 1 cm2; Ivl = 1 .O V/s; C,, = 0;R, = 0;K2 = 1000;k2 = 100 s-'; concentration of A = M; € = -0.3 V; Eo = 0.0 V; DA = DB = D, = cm2/s. Inset is cyclic voltammogram. (b) K2 = 1000;k2 = 0,10,lo3,lo6,and 101os-'. Arrow 1 marks minimum of the voltammogram for k2 = 0; arrow 2 marks the minimum of the voltammogram for k2 = 10" s-'. Values of other parameters are k, = M; Eo = 0.0 V; DA = DB = Dc = lo4 cm/s; T = 298.2K; area (planar electrode) = 1 cm2; IvI = 1.0 V/s; C,,= 0;R, = 0; concentration of A = 1 0-5 cm2/s. (c) k, = 1 O4 cm/s (marked by arrow); K2 = 1000;k2 = 1 O'O s-' ; k, = 3.1622,0.31622,0.031622,and 0.0031622 cm/s in order of increasing peak splitting; other parameter values the same as in (a). (d) Simulated voltammogram for five cycles. K2 = 1000;k2 = 1 O3 s-' ; arrow marks reduction peak for first cycle. Other parameters are the s
594 A AnalyticalChemistry, Vol. 66,No. 10, May 15, 1994
ng the con-
10"' s-l
hanism with
(=
of
RTln[K,] /F,
cies B is very small).
hemispherical simu < 1). In this
mate establishment oft
more difficult to sim
for the reactions that
factor of lo6would us there is a need
3 ; nential e pro-
fined for B = C. Th written with B+C plications. When the conversion of C to B rium constant K, for principle, there is no which can poise the all of the material as species however, that the more positive of E the slower the conversion o The simulator will compute the equi rium concentrations at the initial Estartfor any mechanism as well values of the thermodynamic param and analytical concentrations that the has entered. When the equilibrium co centrations differ signiiicantly from th corresponding analytical concentrati the user should be wary; if the simul is correctly mimicking a real chemica system, the starting conditions for th experimental voltammogram may be different from what was expected. The ec mechanism is the arch
and temporal stiffness. When the product
reaction (e.g., B = C) , its concentratio near the electrode is lowered and the
for A + e = B is decreased from lo4 c (mow, Figure 4 ~ to ) 3.1622 cm/s or 0.31622 cm/s (values that normally pro duce reversible behavior at ] V I = 1 V/s), the shape of the voltammogram is significantly distorted. The reason is that species A and B are the active redox couple, and the concentration of species B is very small. The effective heterogeneous reaction rate for this process is k,/K,". The values of kJK in Figure 4c are (for a = 0.5) 316,0.1,0.01, and 0.001 cm/s-essentially the same as the values of k , in Figure 3c. When adjusted for the shift in a p parent Eo, Figures 4c and 3c are virtually We saw that, after repeated cycling for a simple electron transfer, the response approaches a steady state (Figure 1);it doesn't change from one cycle to the next.
Although none of these individual obreversible voltam- servations is particularly newsw ease with which the behavior of s-' the reduction
with k2 = 1000 s-' the reducincreased a bit and shifted
simulator. The slowest of the tions discussed (when k2 quires about 10 s on a 48
more positive and the oxidation peak Analytical Chemistry, Vo1.-66, No. 10, May 15, 1994 595 A
ke nearly 1000 years. Even allowing for enhanced speed of a supercomputer, ait would tax the patience of Methu-
namic paramete the values of E" other reactions. ure 5a, exhibits pea V, corresponding to ior of A + e = Band
or K for that reaction. R from a reaction sche values of the concentrations w tem ultimately reache wever, a TSR can have
the simulator as parame
The detection oft
10" and the resulti
cies B is never signi The eec is a good example o ith a TSR. Once the values for
actions (50).A TSR
Figure 5. Simulated voltammogram for A
lytical Chemistry, Vol. 66, No. IO,
x
,curve 2 in Figure 5a,
+ e G B, B + e e C, and 2B e A + C.
one-electron transfer; the simulated ratio is 2.827. As with the ec mechanism, the shape of the voltammogram for the eec under these conditions is very sensitive to the values of the heterogeneous rate constants k,, and ks,,. The assumption that k,, and kq:2 = lo4 cm/s is unrealistic. Resetting each of them to a smaller and more realistic value, such as 1cm/s or 0.1 cm/s, produces a significant change in the shape of the voltammogram (Figure 5b). This may be one reason a reversible twoelectron transfer is rare. Responses for the conditions in Figures 5a or 5b are insensitive to the kinetics of 2B = A + C if the diffusion coefficients of all the species are identical and if the heterogeneous electron transfers are reversible on the time scale of the experiment (i.e., k , , , , / q E G Z % > 1) (51, 52).“Turning on” the homogeneous kinetics creates noticeable changes in the voltammogram when at least one of two electron transfers is quasi-reversible on the time scale of the experiment. However, there is a dramatic effect on the concentration profiles when & plane (also assumed to be the plane of also be ignored if the plane of closest a p where 6 is the thickness of the diffusion proach for the redox moieties is outside closest approach for the redox moieties) medium, D,,, is the largest diffusion co- and the bulk solution. Electron transfer the diffuse double layer (which could h a p efficient, and,,T is the maximum time pen if the radius of the redox moieties rates are modified because the Q2 potenwere much larger than the ions compristhat will have elapsed since the initial per- tial alters the concentration of any charged electroactive species at the outer ing the supporting electrolyte) and if all Helmholtz plane relative to its bulk conheterogeneous reactions are reversible monly encountered by the electroanalyst centration; in addition, the potential that (i.e., Nemstian behavior obtains). Even working with thin-layer cells (56,57) or with Nernstian behavior, the thickness of drives the heterogeneous elecpon transwith modified electrode systems in which fer kinetics (see Equation 1) between the the diffuse double layer must be much FLD occurring within a thin film on the electrode and the outer Helmboltz plane smaller than the thickness of any operaelectrode couples with SILD in the adjative chemical reaction layer. will be E - Q2 rather than E. cent bulk solution (58,59).The cyclic An ab initio determination of a precise Marcusian heterogeneous kinetics. The voltammetric responses for FLD and relationship between q2 and E is difficult if use of the Marcus (68) rather than the not impossible. AlButler-Volmer formalism (Equation 1) to though the Gouydescribe heterogeneous electron transfer Chapman equation must also be considered. A very nice dis(66) is adequate for cussion of Marcus’ treatment of heterogepredicting the relaneous kinetics is presented in Reference tionship between q2 68. Deviations from pure Butler-Volmer behavior created by Marcusian kinetics and charge, determining the relationand/or the Frumkin effect are, within limship between $2 and its, adequately approximated by the ButE is complicated by ler-Volmer formalism with a potential ambiguities in the dependent a (55).Few attempts have potential of zero been made to solve this dilemma in the charge Epzc(difficult context of the analysis of cyclic voltenough to locate on ammetric behavior (69). single crystal surAdsorption phenomena. The effects ot faces, let alone on a adsorption, which are varied and complex polycrystalline sur(67, 70),have not been included in the I face), capacitance of present version of the simulator. It is not the compact layer particularly difficult to simulate the behavFigure 6. Simulated voltammograms for A + e e B and ior of a given adsorption isotherm (71). (67),adsorption, P + B e Q + A. and surface restrpcAn isotherm must be defined for each adRandom noise = 2 x 1 0-5 A; T = 298.2 K; area (planar electrode) = turing. If, as a first sorbing species, but which of a wide selec1 cm‘; IvI = 1.0V/s; C,,= 0;R, = 0;Eo = 0.0 V; a = 0.5;/g = 1.0cm/ approximation,one tion of isotherms (72) should be used? s;K2 = lo4;k2 = lo9 M-’s-’* , concentration of A = M; concentration of P = 1F3M; DA= D, = Dp = 0, = 1 0-5cm2/s; data density can assume that Q2 = When a redox couple adsorbs, two quite is 1 point every 0.005V; no noise for Eo = 0.002 V, a = 0.5, k, = a + bE + cE2 over the likely different isotherms must be consid0.95 cm/s, K2 = 1.14 x 1 04, k2 = 0.95 x 1 O9 M-’s-’ , which are potential range of ered because the charge on the oxidbeg average values deduced from optimal fit of 10 “noisy” simulations interest, incorporatand reduced moieties are necessarily dif(smooth curve). Jagged curve is a typical noisy simulation.
transfer is perfectly reversible (i.e., k, = =) significantly alters the optimized values of K, and k,. When heterogeneous electron transfers are not Nernstian, a double-layer (Frumkin) effect can profoundly modify the shape of the cyclic voltammogram (55).The present version of Rudolph’s simulator (or any other known simulation package) does not consider those effects; however, efforts to do so are under way.
62/0,,2,,,
I
-
590 A Analytical Chemistry, Vol. 66, No. 70, May 15, 7994
of any of the physical parameters. The experimentalist confronting an unfamiliar cyclic voltammetric response does not have to carry out or even understand the underlying complex computations and can consider mechanisms other than those that have been previously analyzed or that are easily computed. The ability to hypothesize reasonable mechanisms can be enhanced by simulating selected systems whose behavior may be characteristic of specific mechanisms or classes of mechanisms. Appropriate texts (1 vant papers will be invaluable sources of information.
Incorporating
cols in addition to CV should enhance applicability. A major chall effective organization and c the plethora of data likely to be generated as myriad mechanisms are probed and characterized. We gratefully acknowledge the support of the Division of Chemical Sciences, US.Department of Energy, under contract No. DE-ACO276CH00016. The support of Bioanalytical Systems, Inc., is also greatly appreciated. MR thanks the Deutsche Forschungsgemeinschaft for support. Critical reading of the manuscript as well as many helpful suggestions and comments were contributed by Alan Bond, Justin Harvey, James Anderson, Dennis Evans, and Ralph White. We thank our colleague electrochemists-too numerous to list-whose questions and challenges spurred the development of this simulator and whose comments and suggestions continue to help refine it.
References (1) Nicholson, R S.; Shain, I. Anal. Chem. 1964,36,706. (2) Evans, D. H. Chem. Revs. 1990,90,739,
hat correlate the
recognize some cyclic
voltammetric maBifestations of adsorption. terface can be applied. various pulse metho
The simulator has already shown promise as a tool for teaching the fundamentals of CV in particular and of electrochemistry in general (
ammetric response.
A chemist can use
66,No. 10,
em-
cations;
(56) Kissinger, P. T.; Heinema tory Techniques in Electroa istty; Marcel Dekker: New 106-1 1. (57) Bard, A. J.; Faulkner, L. R. Methods: Fundamentals a John Wiley Rr Sons: New
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(27) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1984,160,l. (28) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1984,160,19. (29) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1984,160,27. (30) Shoup, D.; Szabo, A. J. Electroanal. Chem. 1986,199,437. (31) Deakin, M. R.; Wightman, R. M.; Amatore, c. A. 1.Electroanal. Chem. 1 9 8 6 , 215,49. (32) Michael, A. C.; Wightman, R M. tore, C. A J. Electroanal. Chem. 267,33. (33) DuFort, E. C.; Frankel, S. P. Math, Tables Aids Comput. 1 9 5 3 , 72,135. .J. Electroanal. Chem. (34) (35)
(49) Oldham, K. B.; Zoski, C. G. J. Electroanal. Chem. 1 9 8 8 , 2 5 6 , l l . (50) Luo, W.; Feldberg, S.; Rudolph, M.J. Electroanal. Chem., in press. (51) Pierce, D. T.; Geiger, W. E.J. Am. Chem. SOC.1992,114,6063. (52) Andrieux, C. P.; Savkant, J. M. J. Electroanal. Chem. 1970,28,339. (53) Strojek. J. W.; Kuwana, T.; Feldber S. W. J. Am. Chem. SOC.1 9 6 8 , 9 0 , (54) Feldberg, S. W. J. Am. Chem. SOC. 88,390.
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Saveant, J. M. J. Electroanal. Che 1985,187,205. (59) Andrieux, C . P.; Saveant, anal. Chem. 1984,171 (60) Andrieux, C. P.; Duma Saveant, J. M. J. Electvoa (61) Anson, F. C.; S (62) Bard, A. J.; Fau]
try and Electrochemical Engineering hay, P., Ed.; Interscience: New York
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(66) Bard, A. J.; Faulkner, L.R Electrochem:: Methods: Fundamentals Applicatio
He specializes in medical image processing and format
J. Electroanal. Chem. 1 9 7 3 bach, M.; Mortensen, ectroanal. Chem. 1984,165,61. 42) Laasonen, P. Acta Math. 1949,81,309 yer, R D.; Morton, K. W. Differ Methods for Initial-Value Proble ey & Sons: New York, 1957; pp.
(44) Crank, J.; Nicolson, P. Phil. SOC.1947,43,50. 5) Mocak, J.; Feldberg, S. Chem.. in mess.
__
York. 1982: Vol. 12. DI).53-157. Schulz, C.; Speiser, B.J. Electroanal. em. 1993,354,255. . M . In Electroanalytical * Bard, A. J., Ed.; Marcel Dekrk, 1966;Vol. 1, pp. 241-409. inger, P. T.;Heineman, W. R LaboraTechniq2ces in Electroanalytical ChemMarcel Dekker: New York, 1984; pp. *
Smith, D. E. In Electroanalytical ChemisofAdvances; Bard, A. J., Ed.; ker: New York, 1966;Vol .l,
I
Stephen W Feldberg received his B.A. dein 1958 and 1961 re lied Science, Upton, Feldberg was a postdoctoral fellow ut Br0okhaue.l n m / I a visiting assistant brofes"+ +L,,
s, private communication,
and studies of fast intevacial processes. 600 A
Analytical
