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A Simulator for
Cyclic Voltammetric Responses
W
ith their 1964 paper (i), Nicholson and Shain launched the modern era of theoretical characterization of cyclic voltammetric 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, homogeneousfirst-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 =± B) requires numerical integration (1) 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 electrochemical:hemical mechanism Manfred Rudolph Friedrich - Schiller Universitàt 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 inefficient when simulating stiff systems (8). Bieniasz has devised ELSIM, a potpourri of mathematical and numerical techniques that can be used to simulate
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various electrochemical responses but demands a mathematically sophisticated user (9,10). This computational challenge is effec tively met by a generalized CV generator based on Rudolph's recently developed fast implicit finite difference (FIFD) algo rithm (11-14), a refinement of an ap proach described by Newman (15). Ru dolph'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 important— demands no mathematical or computa tional skills from the user. Despite the effectiveness of the FIFD algorithm, the easy access to fast, powerful, and inexpen sive 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 pow erful and to describe the capabilities of an operating version of Rudolph's CV simula tor and its applications. Basics of CV CV is a relatively straightforward tech nique (16-18). A working electrode (where the reactions of interest occur) is immersed in a solution containing some electroactive analyte, an excess of electro lyte, a reference electrode, and a counter electrode. The applied potential (the po tential 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 experimental practice is to choose a start ing potential for the working electrode Eslari, where the analyte is electroinactive, and then to change the potential at a con stant rate υ (= dE/dt) moving from £ start through a potential range where oxidation or reduction of the analyte can occur, to a reversing potential £ rev , where the sweep direction is reversed (-v) and the poten tial usually returns to £ start . These steps constitute a single cycle, but the process can be repeated any number of times. A wide range of values of \v\ (19) has been used (0.05-2 xl0 6 V/s). The interfacial electron transfer for the heterogeneous process A + e ΐ± Β can be 590 A
described by the Butler-Volmer equation (20)
h
[A]x=0exp
-a — (E RT
[#Uexp (1 - α ) — (E RT
E°) E°)
(1)
where χ. is the flux of electrons from the electrode to the solution; ks is the stan dard rate constant (cm/s); α is the trans fer coefficient for A + e ?± B; F is the Fara day constant; R is the gas constant; Τ is temperature (Κ); Ε is the applied poten tial as measured between the working and reference electrodes; and E° is the formal potential (volts). Equation 1 is written for a one-electron transfer; the simulator can be used only with one-electron transfers. (An «-electron transfer can be described by a sequence of one-electron transfers with appropriately selected values of the heterogeneous parameters.)
This computational challenge is met by a CV generator based on Rudolph's FIFD algorithm. The experimentally measured re sponse is the current that passes between the working and auxiliary electrodes. The cyclic voltammogram is a plot of the cur rent i versus the applied potential Ε (i = -F χ electrode area χ fe), where the nega tive sign makes the plots consistent with Equation 1 and our plotting convention. A typical cyclic voltammogram for the sim ple reversible electron transfer A + e α 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
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transfer. The potential and current values of the first cycle agree well with extant theory (1). Values of ks, which are smaller than the theoretical maximum of 104 cm/s at 300 Κ (21,22) (e.g., 1 or 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 experimen tal parameters (Eslarl, £ rev , v), the hetero geneous electron transfer rate parameters (fta, E°, a), the diffusion coefficients and 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
(ecec 1)
Β— C C + e =± D B+ C—A+D
(ecec 2) (ecec 3) (ecec4)
the rates of the heterogeneous and homo geneous reactions will modify the shape of the voltammogram in significant and interesting ways. (Ecec denotes the se quence of reactions: electrochemical, chemical, electrochemical, and chemical.) The computation of all the coupled inter actions of spatially inhomogeneously dis tributed species presents a formidable challenge. Systems such as the ecec, which involve second-order processes, are most effectively tackled by using numeri cal methods. Systems considerably more complicated than the ecec are often encoun tered, 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 proper ties of a redox couple. Extracting mecha nistic and kinetic details of a given electro chemical-chemical system is much more difficult. To accomplish this, the analyst must first postulate an electrochemicalchemical mechanism. This requires intuit ing or guessing just what is going on from the shape of an experimental voltammo gram 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-
vious prerequisite. The analyst must then confirm the mechanism. Simulations for a given mechanism with set values for the cyclic voltammetric and physical and chemical parameters must (within some margin of error) match the experimental voltammograms obtained for different scan rates, concentrations, and geome tries. Of course, confirmation of one mechanism does not exclude other mech anisms that may work just as well. A Gauss-Newton algorithm (14) is used to refine initial guesses of the values of se lected parameters and to minimize the residual. The first of these skills is perhaps the most elusive because there is as yet no way to automate user intuition. Using the simulator to explore the behavior of nu merous simple mechanisms is a good way to hone one's intuition. FIFO method Any finite difference treatment of an elec trochemical problem requires the discreti zation of space and time. Space discretiza tion involves the division of the relevant space adjacent to the electrode surface into volume elements (Figure 2). An ex ponentially expanding space grid (23) causes an increase in the thickness of the volume elements as the distance from the electrode's surface increases; allows effi cient and accurate treatment of concentra
tion profiles, which are associated with reaction layers (24) produced by homoge neous chemical reactions and are spatially more compact than the diffusion layer; and significantly diminishes (sometimes by orders of magnitude) the number of volume elements, and therefore the com putation time, required for any given sim ulation. The two fundamental types of finite difference equations are explicit finite dif ference (EFD) and implicit finite differ ence (IFD) (6). A simplistic distinction is that EFD methods are used to compute some properties (e.g., concentrations) of a system at t + Δί using only known values of those properties at time t. IFD methods compute the properties at t + At using val ues of those properties at t + At along with known values at t, and sometimes at t - At, ... t - nAt. The EFD method is easy to understand and program. Unfortunately, despite advances in the electrochemical applications of the EFD methods, notably the hopscotch (25-32) and DuFort-Frankel (33-35) variations, the application of EFD methods to stiff problems is thwarted (even with a super computer) by a fundamental limitation known as "propagational inadequacy" (36). This algorithmic dysfunction can be described as follows: For each iteration (i.e., for each time step Δί) with any EFD algorithm, a perturbation in the concentra
tion can propagate only one additional volume element. Thus, after the first itera tion (i.e., t = Δί), the only computed changes in concentration will be for the first volume element 0 = 1, width = Ax; see Figure 2). We know that proper repre sentation of any diffusional perturbation must extend a distance χ = (37) from the electrode surface. If χ » Ax in the first iteration, serious inaccura cies must arise because an EFD algorithm cannot compute a concentration change beyond χ = Αχ during the first iteration. Why not simply decrease the value of DAt? Homogeneous reactions often pro duce a reaction layer (24), and accurate simulation demands that the thickness of the reaction layer \JDlk (or equivalently \jDAtlkAt) be > 5Ax, which ensures ac curate depiction of the concentration within the reaction layer (k is a first-order or pseudo-first-order rate constant). Effi cient simulation of many problems of in terest requires that kAt » 1, which leads to the requirement that \/DAt » 5Ax (or equivalently DAt/Ax2 » 1) (12,34). Thus the use of EFD methods is seriously re stricted by propagational inadequacy even though the devastating constraint, DAt/ Ax2 < 0.5, of the simplest EFD formalism is removed by the hopscotch or DuFortFrankel algorithms. IFD methods are not constrained by propagational inadequacy. In an IFD com-
Figure 1 . Simulated voltammogram for A + e — B. T= 298.2 K; area (planar electrode) = 1 cm 2 ; \v\ = 1.0 V/s; concen tration of A = 10" 3 M; E° = 0.0 V; ks = 10 4 cm/s; α = 0.5; DA = DB = 10~5 cm 2 /s; ncycie = 5. Arrow 1 marks the position of the reduction peak of the first cycle; arrow 2 marks the position of the oxidation peak of the first cycle.
Figure 2. Spatial grid for finite difference simulation. The value of β, the factor for exponential expansion of the grid, is 0-0.5. When β = 0, a uniform grid in which the width of each volume element is Δχ is produced.
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Figure 3. Simulated voltammograms f or A + e = Β as parameters are changed. (a) kB = 1.0 cm/s; \v\ = 1.0 V/s" (curve 1 ), 0.25 V/s (curve 2), and 0.0625 V/s (curve 3). (b) ^ - 1.0 cm/s, DA = 10"5 cm2/s; for curve 1, DB = 10"5 cm2/s; curve 2, DB = 10"4 cm2/s; curve 3, DB = 10"6 cm2/s. (c) ks = 10" (arrow), 0.1, 0.01, 0.0O1, and 0.0001 cm/s in order of increasing peak splitting, (d) ks = '\ cm/s; Cd, = 5 χ 10"5 farads/cm2. Arrow marks the voltammogram for flu = 0 ohms; increase in the peak splitting with Ru = 10, 20, 50, and 100 ohms, (e)fcs= 10" cm/s; curve 1 (cylindrical electrode): rB = 10"4 cm, h = 2 χ 10"" cm, £° = 0.5 V; curve 2 (spherical electrode): r0 = 10"" cm, £° - 0.6 V; curve 3 (cylindrical electrode): r0 = 10"6cm; /7 = 2x 10"6cm, E° = 1.5 V; curve 4 (spherical electrode): r0 = 10~6cm, £° = 1.6 V (h = 2r0 so that cylinder and sphere have identical areas). Current scale for curves 3 and 4 should be multiplied by 10~2. Values of other parameters not given in (a)-(e) are the same as for Figure 1. 592 A
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putation the concentrations at f = / + Δί are robust and can be used for accurate simu considered; thus, the result is a set of si lation of virtually any electrochemicalmultaneous equations (essentially one chemical system over a wide dynamic equation for each species and for each range of parametric values. Thus it is pos volume element), and changes in all rele sible to translate a straightforward de vant volume elements are computed with scription of an electrochemical mecha each iteration. When only a single species nism into the mathematical expressions is involved, or when there are several required to compute the cyclic voltamnonchemically coupled species, equations metric response. For the previously dis for each species can be represented by a cussed ecec mechanism, for example, the tridiagonal matrix that is easily and effi completed mechanism entry window in ciently unraveled (38). the program would be When species are chemically coupled, A+e=B as they often are in many reaction mecha B=C nisms (see ecec 1-4 on p. 590 A) there is C+e=D a single-band matrix of order ns · (nve + 1) B+C=A+D that must be inverted («s is the number of species and «ve is the number of volume With the equal sign we imply that every elements; the "+ 1" appears because spe reaction must go both ways, a manifesta cies may be coupled by Equation 1 at the tion of microscopic reversibility (46). boundary corresponding to the electrode surface). This single-band matrix can be reduced to a block tridiagonal matrix, which can be solved efficiently (12). De spite earlier descriptions of this approach (15), application of the implicit approach for solving electroanalytical problems has been limited (15,39-41) by programming complexity and by computational de mands. The combination of the exponen tial grid, easier access to incredible com puting power, and an efficient approach to unraveling the resultant matrix has pro duced a general, efficient, and robust method for analyzing electrochemical re sponses. The method is easily extended to han dle the nonlinear equations that evolve from higher order reactions, such as Β + C ;± A + D. By using the fully implicit Laasonen algorithm (42) coupled with the Richtmyer modification (43), one can ob tain an accuracy rivaling that of the famil iar Crank-Nicolson algorithm (44) and eliminate the instabilities that can occur with the Crank-Nicolson algorithm when very large values of Ό AtI Ax1 are required (45). The programming of the FIFD method is quite straightforward. Its versatility, robustness, and power more than com pensate for sacrificing the simplicity of EFD methods, which are seriously defi cient in simulations of stiff systems. FIFD-based CVsimulator. As long as a few simple and invariant rules are fol lowed, the FIFD algorithm is extremely
Using the simulator to explore simple mechanisms is a good way to hone one's intuition.
tion profiles of the various species as they change during cyclic voltammetric pertur bation (i.e., a graphic display of the chang ing profiles—CV, the movie!). Verifying simulation accuracy. How do we know that a given simulated response is correct? One approach is to select con ditions for a particular system that devolve to familiar, previously analyzed, and re ported examples. However, doing so does not guarantee that simulations of the in termediate conditions are correct. An other approach is to vary the values of the simulation parameters (e.g., Δί, Αχ, and the parameter β, which controls the rate of expansion of the exponential grid). The default values of Δί, Ax, and β are selected by the CV simulator to optimize accuracy, stability, and computational efficiency— larger values will produce faster computa tion with decreasing and eventually unac ceptable accuracy, and smaller values will result in slower computation with increas ing and eventually a limiting and perhaps unwarranted accuracy. If reducing the default values of Δί, Αχ, and/or β for any given computation significantly changes the simulated response, either the default values were not properly optimized or pro gramming errors exist.
Exploring basic concepts Nicholson and Shain (1) fully character ized the dependence of the cyclic voltam metric response on the physical and chemical parameters for simple electron transfers and for simple electron transfers coupled with first-order chemical reac tions. Although most CV practitioners are familiar with this work, the simulator of fers an invaluable demonstration and The program is used to construct the explication of these and many other phe matrices required to carry out the simula tion; request values for all the cyclic voltam- nomena that shape the cyclic voltam metric, physical, and chemical parameters metric response. Several examples of cou pled heterogeneous and homogeneous (recognizing the thermodynamic con reactions will be considered. The practi straints and thermodynamically superflu tioners' familiarity with these fundamental ous reactions discussed on p. 596 A and features of cyclic voltammetric behavior allowing some values to be set and auto matically computing the others); and offer will enhance their interpretive skills. options for electrode geometry and plot Response for a simple electron transfer. ting convention. The user can simulate We will consider the response of the CV responses, compare simulations with simulator as parameters are changed for each other and with the experimental a simple electron transfer, A + e = Β data, perform least-squares fit simulations (Figure 3). to other simulations or to experimental Decreasing \v\ by a factor of 4 de voltammograms with selected parameter creases the peak by a factor of 2, as shown optimization, and observe the concentra in Figure 3a. This behavior is summarized Analytical Chemistry, Vol. 66, No. 10, May 15, 1994 593 A
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in the definition of the dimensionless cur rent function (2) χ = i /[area FcKi\/(FD
M/RT)]
(2)
A change in the ratio DA/DB causes a shift in the potential as illustrated in Fig ure 3b where DA/DK = 1,10, and 0.1. Note that the height of the reverse peak changes only slightly because of the changing relative positions of £ rev and £ peak . The larger the value of |£ r e v - £ peak | the higher the reverse peak. The equation for an apparent E° is E
app = E + (RT/2F)\n{DB/DA)
(3)
As ks decreases, peak separation in creases. With α = 0.5, decreasing ks from 104 cm/s (perfectly reversible when \v\ = 1 V/s) to 0.1,0.01,0.001, or 0.0001 cm/s increases the peak separation, as indi cated in Figure 3c. A similar set of voltammograms could be obtained by keeping ks constant, increasing \v\, and normalizing the current by plotting the dimensionless current function versus E. The limiting shift in the cathodic peak potential will be 0.118 V as seen between the voltammograms for ks = 10"3 cm/s and 10~4 cm/s. Including a double-layer capacitance Cdl (farads/cm2) in the simulation pro
duces the familiar capacitive current and uncompensated resistance Ru (ohms), which increases the peak splitting (Figure 3d), as does a reduction in the value of ka. The omnipresent uncompensated resis tance presents a fundamental difficulty in evaluating ks from CV responses (47). The combination of capacitance and un compensated resistance induces a nonconstant dE/dt at the outer Helmholtz plane, an effect that is not easily removed from the data, even if Cdl is potential inde pendent. The shape of the cyclic voltammogram is significantly changed when the thick-
Figure 4. Simulated voltammograms for A + e s± Β and Β — C. (a) As = 1.0 cm/s; Τ = 298.2 Κ; area (planar electrode) = 1 cm 2 ; |v| = 1.0 V/s; Cd, = 0; Ru = 0; K2 = 1000; k2 = 100 s~1; concentration of A = 10" 3 M; E = - 0 . 3 V; E° = 0.0 V; DA = DB= Dc = 10" 5 cm2/s. Inset is cyclic voltammogram. (b) K2 = 1000; k2 = 0, 10, 10 3 , 10 6 , and 10 10 sr 1 . Arrow 1 marks minimum of the voltammogram for fc, = 0; arrow 2 marks the minimum of the voltammogram for k2 = 10'° s _ 1 . Values of other parameters are ka = 10" cm/s; T= 298.2 K; area (planar electrode) = 1 cm 2 ; M = 1.0 V/s; Cdl = 0; Ru = 0; concentration of A = 10~3 M; £ ° = 0.0 V; DA= DB = DC = 10" 5 cm 2 /s. (c) ks - 10 4 cm/s (marked by arrow); K2 = 1000; k2 = 10 10 s _ 1 ; /ς = 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 = 103 s" 1 ; arrow marks reduction peak for first cycle. Other parameters are the same as in (b).
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seems to be returning (because the kinet ics of Β = C are now fast enough to pro duce species Β from C for reoxidation at the electrode surface) ; and when k2 = 1010 s"1 the cyclic voltammogram be comes reversible again with an apparent E° that has shifted by about + 0.177 V ( = RT\n[K2] IF, from Reference 1) and corresponds to the reversible redox cou ple A + e ^± C (the concentration of spe cies Β is very small). The positive shift of the peak potential with increasing value of k2 and the ulti < \/(F\v\)/[RT(k2 + k_2)] « 1). In this mate establishment of the new reversibil case μΙ4 = \/DB/(k2 + k_2). For the param ity when k2 is very large depends on the eter values in Figure 4a, \iB = 3 χ 10~4 cm ability to sustain Nernstian behavior at the is predicted. Increasing the value of k2 by electrode surface. When the value of ks a factor of 10b would make μΒ = 3 χ 10"7 ; for A + e = Β is decreased from 10" cm/s thus, there is a need for an exponential (arrow, Figure 4c) to 3.1622 cm/s or grid that can efficiently describe the pro 0.31622 cm/s (values that normally pro files of all the species. duce reversible behavior at \v\ = 1 V/s), A+e= B (eel) the shape of the voltammogram is signifi Β=C (ec 2) cantly distorted. The reason is that spe cies A and Β are the active redox couple, Microscopic reversibility demands that and the concentration of species Β is very forward and reverse rate constants be de small. The effective heterogeneous reac fined for Β = C. The mechanism is often tion rate for this process is kJK2. The written with B->C as a completely irre values of kJK 2 in Figure 4c are (for α = versible process, which has amusing im 0.5) 316, 0.1,0.01, and 0.001 cm/s—essen plications. When the rate constant for the tially the same as the values of £s in Fig conversion of C to Β is zero, the equilib ure 3c. When adjusted for the shift in ap rium constant K2 for Β = C is infinite; in parent E°, Figures 4c and 3c are virtually principle, there is no electrode potential E, identical. which can poise the system with virtually all of the material as species A. It is true, We saw that, after repeated cycling for however, that the more positive the value a simple electron transfer, the response of Ε the slower the conversion of A to C. approaches a steady state (Figure 1); it The simulator will compute the equilib doesn't change from one cycle to the next. rium concentrations at the initial potential That will be true with the ec mechanism £ start for any mechanism as well as any as well, but the steady-state voltammo values of the thermodynamic parameters gram may be invisible when viewed on and analytical concentrations that the user the same current scale as the initial voltam has entered. When the equilibrium con mogram. For k2 = 1000 and other parame centrations differ significantly from the ter values given in Figure 4b, the ap corresponding analytical concentrations proach to steady state is shown in Figure When \v\ = 1 V/s, K2 = 1000, and ks = the user should be wary; if the simulation 4d. The voltammogram will eventually 104 cm/s, increasing k2 from 0 to 1010 s"1 is correctly mimicking a real chemical disappear. produces an interesting and perhaps sur system, the starting conditions for the Although none of these individual ob prising effect (Figure 4b). When k2 = k_2 = experimental voltammogram may be quite 0 we observe a classical, reversible voltam servations is particularly newsworthy, the different from what was expected. ease with which the behavior of the ec (or mogram; when k2 = 10 s_1 the reduction any other) mechanism can be simulated The ec mechanism is the archetype for peak has barely changed, but the oxida and explored is noteworthy. Computa tion peak has almost disappeared because the formation of a reaction layer (24) and tional speed is an essential property of the species Β has been converted to the elecclearly demonstrates the problem of spatial simulator. The slowest of the computa troinactive C; with k2 = 1000 s_1 the reduc and temporal stiffness. When the product tions discussed (when k2 = 1010 s~') re tion peak has increased a bit and shifted of a heterogeneous reaction (e.g., species quires about 10 s on a 486DX2/66 MHz positively, the oxidation peak is gone; Β in A + e = B) is removed by a chemical PC; the same computation performed on when k2 = 106 s"1 the reduction peak is reaction (e.g., Β = C), its concentration the same computer, using a simple EFD more positive and the oxidation peak near the electrode is lowered and the
ness of the diffusion layer is large com pared to the radius r0 of a spherical, cylin drical, or disk electrode or the half-width of a band electrode. When the value [\ZRTD/(F\v\)]/rQ is > ~ 40, a spherical or disk electrode has approached the steady state; however, cylindrical and band electrodes will never attain the steady state (48). The simulator demon strates the behavior of cylindrical elec trodes in Figure 3e. Hemicylindrical and hemispherical simulations can serve as reasonable approximations (49) for band and disk geometries, which are much more difficult to simulate (31,32). Effects of a coupled homogeneous reac tion. We shall consider the response of the simulator as parameters are changed for the reactions that comprise the ec mechanism
characteristic thickness of its concentra tion profile (which is the reaction layer thickness μΗ) is greatly diminished. We can explore the general characteristics of the reaction layer by examining the con centration profiles of species A, B, and C at the switching potential Ervv = -0.4 V for a simulation of an ec mechanism with K2 = 1000 and k2 = 100 s"1 as shown in Figure 4a. The concentration profile of species Β indicates that the thickness of the reaction layer μΒ is much smaller than the thickness of the diffusion layer (i.e.,
The ease with which the behavior of the ec (or any other) mechanism can be simulated and explored is noteworthy.
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trochemical reaction whose thermody namic parameter (E" or Κ) is a function of the values of E" or Κ already specified for other reactions. The value of E" or Κ for the TSR cannot be specified by the user. Once a TSR has been identified, the simu lator must compute the correct value of E" or Κ for that reaction. Removal of a TSR from a reaction scheme will not affect the values of the concentrations when the A+e= Β (eec 1) system ultimately reaches equilibrium; Β+e=C (eec 2) however, a TSR can have a profound ef 2B = A+C (eec 3) fect on the rate at which that equilibrium is achieved. The detection of thermodynamically The eec is a good example of a system superfluous reactions (TSRs) is necessary with a TSR. Once the values for E" for A + for any simulation involving chemical re e = Β and E'.] for Β + e = C are entered, the actions (50). ATSR is a chemical or elec
formalism with an exponential grid, would take nearly 1000 years. Even allowing for the enhanced speed of a supercomputer, the wait would tax the patience of Methu selah, not to mention his research budget. TSRs and the disproportionation (eec) mechanism. We will consider the response of the simulator as parameters are changed for the reactions
value of Ka for 2B = Α Λ C is computed. When E", = 0.2 V and E", = -0.2 V, then K, = 1.73 χ 10"' is determined and the resulting cyclic voltammogram, curve 1 in Figure 5a, exhibits peaks separated by 0.4 V, corresponding to the reversible be havior of A + e = Β and Β + e = C. When £',' = -0.2 V and E", = 0.2 V, then Κλ = 5.78 χ 10'' and the resulting voltam mogram, curve 2 in Figure 5a, has a con siderably sharper single peak than curve 1, which corresponds to a two-electron transfer (A + 2e ^= C) because the concen tration of species Β is never significant. The apparent E" = (£',' + £!!)/2 = 0 V. The peak height is 2 ! / - (= 2.828) times as high as the peak height for the corresponding
Figure 5. Simulated voltammogram for A + e — Β, Β + e = C, and 2B — A + C. (a) For curve 1, E° = 0.2 V, E ° = - 0 . 2 V , K3 = 1.73x 10" 7 ; for curve 2, £? = - 0 . 2 V , E° = 0.2 V, K3 = 5.78 χ 106. Other parameters are 7 = 298.2 K; area (planar electrode) = 1 cm 2 ; \v\ = 1.0 V/s; Cdl = 0; flu = 0; concentration of A = 10~3 M; ksi = 10 4 cm/s; a, = 0.5; ks2 = 104 cm/s; a 2 = 0.5, k3 = k_3 = 0; D A = DB = Dc = 10~s cm 2 /s. (b) For curve 1, /
