Applicability of d.c. relaxation techniques to multi-step reactions

ELSEVIER Journal of Electroanalytical Chemistry 421 (1997) 33-44

JOO faOtl. OF

Applicability of d.c. relaxation teclwdques to multi-step reactions

Z. Nagy, N.C. Hung l, K.C. Liddell 2, M. Minkoff, G.K. Leaf Argonne National Laboratory, Dirisions of Materials Science, Chemical Technology, and Mathematics and Computer Science,

Argonne, IL 60439-4837, USA

Received 24 Apri! 1996; revised 9 July 1996

Abstract

The general theory of d.c. relaxation techniques has been developed mainly for single-step electrochemical reactions and for multi-step reactions with a clearly defined rate-determining step and no intermediate accumulation either on the electrode surface or in the solution. A few workers have considered the case of multi-step reactions, but, because of the complexity of these systems, different approximations and simplifications were introduced in every treatment, limiting the general usefulness of the conclusions. Using numerical calculational methods, we have investigated the behavior of two-step (metal deposition/dissolution) reactions for potentiostatic and galvanostatic single- and double-pulse relaxation experiments. We have carded out a large number of numerical simulations using a wide range of variable values. The main purpose was to determine the conditions under which the techniques are applicable for the measurement of the rate constant of the fast and the slow step of the reaction sequence. In particular, two 'critical times' were determined: (i) the time to reach 'steady-state' conditions with ",.he transient techniques and (i i) the time available for the determination of the fast-step kinetics at the beginning of the measuring pulse, we have succeeded in representing these conditions in graphical fomi as a function of pa~meters involving only a few (mostly known) variables. We also found that the appearance of a maximum/minimum in the relaxation curves indicates that only the fast-step kinetics can be determined.

Keyword~: Multi-step reactions; Relaxation techniques; Numerical modeling

1. Introduction

The general theory of d.c. relaxation techniques has been developed mainly for single-step electrochemical reactions and for multi-step reactions with a clearly defined rate-determining step (rds) and no intermediate accumulation either on the electrode surface or in the solution [1-3]. A few workers have considered the case of multi-step reactions, but because of the complexity of the systems, different approximations and simplifications were introduced in every treatment, limiting the general usefulness of the conclusions. Plonski treated the cases of two- and three-step reactions for both galvanostatic and potentiostatic techniques in a series of papers [4-11]. In these treatments, the double-layer charging and the mass-trans-

t Permanent address: Chemistry Department, Wheaton College, Wheaton, IL 60187, USA.

2 Permanent address: Department of Chemical Engineering, Washing- ton State University, Pullman, WA 99164-2710, USA.

port effects were either ignored or treated in approximate fashion to permit derivation of analytical solutions for potential/current and concentration as a function of time. Bachmann and Bertocci [12-14] treated the potentiostatic technique with full consideration of the mass transport effects, but ignored the double-layer charging and the effect of the uncompensated resistance. The treatment of the galvanostatic technique was also limited to the linear current-density/overpotential range [14]. Kopistko and coworkers have treated the double-pulse galvanostatic method for two- and three-step metal deposition/dissolution reactions, using an approximate treatment of the mass transport effect, and also applied their equations in the evaluation of experimental data [15-19]. They also treated the two-step redox reaction for the galvanostatic single- pulse case in the linear current-density/overpotential range [20]. The case of the coulostatic transients was treated by Reller and Kirowa-Eisner in the linear current- density/overpotential range [21].

In this work, we consider the behavior of a two-step (metal deposition/dissolution) reaction for potentiostatic

0022-0728/97/$17.00 Copyright © 1997 Elsevier Science S.A. All rights reserved. PI! S0022-0728(96)04877-2

34 Z. Nagy et al. / Journal of Electroanalytical Chemistry 421 (1997) 33-44

and galvanostatic single- and double-pulse relaxation experiments. Full consideration is given to the mass-transport effects, double-layer charging, uncompensated solution resistance, and the (linear) rise time of the excitation pulses. Some general conclusions given in Section 2.1 are valid for any two-step reaction: redox reaction, gas evolution, and metal deposition/dissolution.

The main purpose of this work was to determine the applicability of the 'classical' data evaluation methods of the d.c. relaxation techniques [1-3] for two-step metal deposition/dissolution reactions. These classical methods generally are based on simplified models of the reactions that permit analytical solution of the differential equations involved. These simplifying assumptions are enumerated in detail in Ref. [3]. Some of the more important assumptions are (i) the validity of the Butler-Volmer kinetic equation, (ii) the assumption that the measurements were carded out under conditions that the Frumkin (double-layer) correction is negligible, and (iii) the assumption of potential independence of the double-layer capacitance. All these limitations apply to our treatment. For a two-step reaction there are further assumptions: (i) that the reaction is in the 'steady-state,' that is, the two steps proceed with the same rate without any changes in intermediate species concentration at the surface, and (ii) only the rate of the slow (rate-determining) step is to be determined. These last two aspects are the main focus of our treatment: under what conditions has the steady-state been achieved, and what are the practical conditions permitting the measurement of the rate of the fast step in the reaction sequence?

The examination of the classical methods is important because they have been, and still are, often used for the examination of multi-step reactions [22-30]. There are more modem data evaluation methods available, usually based on numerical solution of the differential equations and possibly coupled with numerical nonlinear least squares curve fitting methods for the determination of kinetic parameters. For these methods, all the simplifying assumptions can, in principle, be relaxed [3]. For example, the potential dependence of the charge-transfer coefficients (Marcus-type kinetics) can be incorporated in the model [31 ]. Furthermore, the problem of steady state investigated in this treatment is not relevant to these numerical procedures because the changes occurring during the approach to steady state can be included in the model. However, the problem of appropriate condition selection for the determination of fast-step kinetics is relevant even for numerical data evaluations, since one must ensure that the experimental data contain sufficient 'kinetic information' about the fast step. However, these numerical approaches also have some disadvantages: they can require an excessive amount of computer time, and, as the complexity of the model and the number of unknown parameters increases, the statistical significance of the calculated parameters may be in question; these techniques must be combined with good statistical sensitivity analysis for every calculation [3].

2. Theoretical

2.1. Multi-step kinetic equations

The steady-state kinetic equation for a charge transfer reaction with two consecutive steps involving single-electron charge transfers

MZ + e - - - M z- ' (1)

M z-! + e - = M z-2 (2)

was derived by Vetter [32] (ignoring mass-transport effects) as:

: = 2jo,, :o,2

]o.l exp ~-~ rl + ]o,2 exp - ~ ~7

(3)

This equation has seldom been used in its original form because of its complexity. It is probably best used with nonlinear curve-fitting data evaluation for the determination of the exchange current densities and symmetry coe.f- ficients of both reaction steps from steady-state experimental data [33]. A more general, and even more complex, equation was derived by Hurd for n-electron transfer reactions [34].

However, Eq. (3) can be simplified for (Jo.l/Joa) << 1 or (Jo.2/Jo.i) << 1 to Eqs. (4) and (5) respectively:

j = 2 Jo.~ exp RT 71

-- exp[ -- ~1 F ] -if'71} I .

{ ] j = 2jo. 2 exp RT rl

-exp[-(1 +fl2)F~'~ r/I}

(4)

(5)

In these forms, the equations can be used to determine the kinetic constants of the slow reaction step. Equations identical to Eqs. (4) and (5) can also be obtained from the general, multi-step Butler-Volmer equation derived using the rds approximation [35]:

- ~ --R-~-- r/ (6)

where or

ac = -- + p/3c (7)

Z. Nagy et ai. / Journai of Electroanatytical Chemistry 421 (1997) 33-44 35

and

n C a+ ac = - (8)

//

In past work, investigations of kinetics and mechanisms of multi-step metal-deposition/dissolution reactions were very often based on Eq. (6). A few representative applica- tions using a variety of measurement techniques for vari- ous electrode reactions are given in Refs. [22-30].

Vetter [32] has also shown that Eq. (3) can be simplified, for very large overpotentials, to forms permitting the determination of the kinetic parameters of the fast reaction step under some conditions. For Irll >> (RT/F)ln(jo, l/jo,2) using cathodic polarization and for 7/ >> (RT/F)ln(jo,2/jo. l) using anodic polarization these relations are given by Eqs. (9) and (10) respectively: 3

j = -2 jo . ,ex p - - - ~ - - r I for(jo.,/Jo.2 ) > 1 (9)

and

j=2 j ° '2exp RT rl for(]o,2)/Jo,,)> 1 (10)

The limitations of the rds approximation (F_xls. (4)-(6)) and the large-overpotential approximation (F_xls. (9) and (10)) can be quantitatively determined by comparing the current predicted by these equations with the current calculated from Eq. (3). For the case o f /3 ! = /3 2 - 0.5 (generally, for ¢t~ = /3 2), the limitations can be simply, presented with the use of one applicability diagram given in Fig. 1. This figure shows the rds zone on the left-hand side, the large-overpotential zone on the right-hand side, and an intermediate zone where neither of these approximations is applicable. (Examples will be discussed below in connec- tion with Figs. 9 and 10.) Fig. 2 presents a Tafel plot illustrating the three potential ranges and the determination of the slow- and fast-step kinetics from steady-state data 4.

From these evaluations, some interesting generalizations can be made for the applicability of the above kinetic equations under steady-state conditions.

(i) When the first step of the reaction sequence is slow in the direction of the polarization, the kinetics of the slow step can be determined in an overpotential range that has

l o 0

10 "1

10 -2

10 "s QO

~ 10 "4

O

"-" lO-S

10 .0

10 -7

10 "e -5 0 5 10 18

+~F/RT

Fig. i. Applicability diagram of simplified multi-step kinetic equations (for/3 i =/32). Kinetics of the slow and the fast step can be measured (to within 10%) in the shaded zones, using (i) the rds approximation (Eqs. (4) and (5)) and (ii) the large-overpotential approximation (Eqs. (9) and (10)) respectively. The 1% limits are indicated by dashed lines. For Jo.i/Jo.2 use ~F/RT; for Jo.2/Jo.i use -71F/RT (order of steps defined in the cathodic direction).

10 "1 m m

10 .=

? E o 10 -3

>- ! - (,o Z 10 .4 U.I E3

7 .1 ,,- 10-s

U

10 "e

3 It is worth noting that Eqs. (9) and (10) are also valid for reversed exchange current density ratios, for IT/l> >(RT/F) . However, for reversed exchange current density ratios, they can be used only for the determination of the kinetics of the slow step in the reaction sequence, being equivalent to the cathodic part of Eq. (4) and to the anodic part of Eq. (5) respectively.

4 Fig. l Fig. 2 and the following generalizations are valid for any reaction described by Eqs. ( l) and (2): redox reaction, gas evolution, and metal deposition/dissolution. However, the following treatment of the relaxation techniques is limited to metal deposition/dissolution reactions.

-0.3 -0.2 -0.1 0.0 0.1 0.2

OVERPOTENTIAL IV

0.3

Fig. 2. Example of Tafel plot for multi-eleclron charge-transfer reaction. Conditions: jo . l=10 -6 and Jo.2=10-4Acm-2; /31=/32=0.5; and T = 298.15K. The extrapolated exchange current densities are twice the actual values, while the cathodic transfer coefficient is 0.5, the low-overpotential anodic transfer coefficient is 1.5, and the high-overpotential anodic transfer coefficient is 0.5, according to Eqs. (4) and (10).

36 Z. Nagy et ai. / Journal of Electroanalytical Chemistry 421 (1997) 33-44

no high-overpotential limit, while the kinetics of the fast- step cannot be determined at all. This is the situation for (Jo.l/Jo.z) < 1 and cathodic polarization, and for (Jo.2/Jo.i) < 1 and anodic polarization.

(ii) When the first step of the reaction sequence is fast in the direction of the polarization, the kinetics of the slow step can be determined in an overpotential range that has a high-overpotential limit. Neither the fast- nor the slow-step kinetics can be determined with sufficient accuracy at higher overpotentials (in an intermediate overpotential range). Finally, the fast-step kinetics can be determined at even higher overpotentials. This is the situation for (Jo.I/Jo,2) < 1 and anodic polarization, and for (Jo.2/JoA) < 1 and cathodic polarization.

(iii) Consequently, for certain conditions, the rds approximation is applicable in one polarization direction only. The limit of the rds approximations is not far from zero polarization when the exchange current density ratio is not much smaller than one. Therefore, for 1 > (Jo.JJo.~) _>3× 10 -3 only cathodic polarization, and for 1 > (joa/Jo.t)>_ 3 × 10 -3 only cathodic polarization can be used to determine the slow step kinetics with the rds approximation. This may be one of the reasons why experimentally a a + a c is often less than n/1, in contrast with ~ . (8).

These limitations and generalization will be used re- peatedly in the following treatment of the relaxation techniques.

The more complex case of three consecutive reactions involving single-electron charge transfers has also been studied. Losev and Gorodetskii [36,37] have shown that different steps can control the reaction at different overpotentials, depending on the exchange current-density ratios and the direction of the polarization. They also found that the first step in the reaction sequence will always control the reaction at extremely large overpotentials, independent of exchange current-density ratios. This occurs because the second and third steps of the reaction speed up sufficiently (due to an increase of intermediate concentrations) to become non-rate-determining.

2.2. Relaxation techniques

In the galvanostatic and potentiostatic relaxation techniques, the electrode system is perturbed from its equilib- rium state by a nonrepetitive current (or potential) excitation signal, and the potential (or current) relaxes with time [1-3]. In the following treatment, we consider single- and double-pulse techniques with instantaneously rising pulses, as an ideal case, and with linearly rising and falling pulses, as an approximation of physically real pulses, as shown in Fig. 3. In addition to the charge-transfer kinetics (Eq. (6)), we also take into account the diffusional transport of the reacting species to and from the electrode surface, the

/ \ II__ ! I . . . . i

t = O t| t z t 3 t = O t 3

c d

/ t = 0 t I

L _

t=O

Fig. 3. Perturbing signal shapes. (a), (b): double pulse; (c), (d): single pulse.

charging of the double-layer capacitance, and the effect of the uncompensated IR drop. Several simplifying assumptions are used in the model; these are the same assumptions as are traditionally used in the treatment of transient techniques, and they are enumerated in Ref. [3]. The steady-state condition is, of course, not fulfilled during the relaxation, but the system approaches steady state at sufficiently long times.

We consider a two-step (metal deposition/dissolution) reaction described by Eqs. (1) and (2), with z - 2, taking place on the same metal. The potential-current-time relation of the relaxation process and the change of the concentration of M+, and M 2÷ as a function of time can be obtained, at least in principle, by solving a set of partial differential equations for the diffusion, coupled with an ordinary differential equation for the overpotential.

The diffusion equations are

13c)( x , t ) a2C~(x,t) 13t = D j 0X 2 (11)

where c~ stands for the concentration of M 2+, and c e stands for the concentration of M +. The following initial and boundary conditions apply:

c~(x,t) =cj for x>_0, t = 0 (12)

cSi ( x, t) ~ cj for x ~ ~, t > O (13)

Oc~ ( x , t ) Jl( t) = ~ for x = 0 , t>_0 (14)

i) x FD i

ac~(x,t) J2(t) - j , ( t ) --- - for x = 0 , t > 0 (15)

ax FD 2

The boundary conditions of Eqs. (14) and (15) express the equivalence of the faradaic current density arid the diffusion flux. The faradaic current density of the two steps is,

Z. Nagy et al. / Journal of Electroanalytical Chemistry 421 ( i 997) 33-44 37

in turn, related to the overpotential and the concentration of the reacting species at the electrode surface [38], that is

(c~(O,t)[(l-13,)Fexp 7(t) c~(O,t) Jr(t) =Jo.I c2 RT C 1

( [ ( I - ~ 2 ) F ] c[(O't) .~( t ) = A.2 exp RT 11(t) c2

x e x p - n ( t ) (17)

where the overpotential is the sum of the charge-transfer (activation) and mass-transport (concentration) overpotential terms.

The ordinary differential equation for the overpotential is

dT/(t) j(t) = j r ( t ) +Co, dt (18)

where

jf( t) =jl( t ) + j 2 ( t ) (19)

indicating that the total current is the sum of the faradaic current and the double-layer charging current. Finally, the overall overpotential invariably includes an additional term due to the uncompensated IR drop

E(t) = rl( t) + j( t) g s (20)

These equations cannot be solved analytically for the case considered in this paper, and we had to resort to numerical solution using procedures described below.

3. Calculationai procedures

Details of the numerical calculational methods are given in Appendix A. We have carried out a large number of numerical simulations using a wide range of values for the variables. The variables used in these simulations and their ranges are given in Table 1. The simulations provided (i) the current/potential relaxation with time, (ii) the change of partial current densities (jl and J2) and the distribution of the overpotential into activation and concentration overpotential terms (using two distribution schemes [39]) as a function of time, and (iii) the change of concentrations (c l and c 2) as a function of time and distance from the electrode. Wherever possible, we have used analytical solutions to confirm the numerical calculations for limiting cases. Bachmann and Bertocci [12] derived analytical solutions for the potentiostatic single-step case with instantaneously rising pulses and a double-layer capacitance of zero, and we have derived analytical solutions for the galvanostatic single-pulse technique using the lineafized current-density/overpotential approximation (see Ap- pendix B and Ref. [20]).

Table 1

Ranges for variable values

Variable Units Range

c s m o l c m - a 1 0 - 2 _ 1 0 - 6

c 2 mol cm - 3 10- 3 _ 10- 9

c l / c 2 10-107 Dt cm 2 s - i 10 -4_10 , -6

D 2 cra 2 s - I 1 0 - 4 - 1 0 -6

~slow 0.25-0.75 flfast 0.25-0 .75 jj Acre - 2 1 0 - 4 _ l

T/ V 0.050-0.30O

Jo.slow A c r e - 2 1 0 - 7 _ 1 0 - !

Jo.fast A c r e - 2 10- 7_ I 0 - !

Jo.fast/Jo.slow 10-- 10 6 Col Fcm -2 0_i0-3

R s II cm 2 0- I

We have not considered in this work the data-evaluation procedures for the relaxation techniques. That is, the sepa- ration of the activation overpotential from other types of overpotential and the procedures needed to extract the kinetic parameters from the data. Neither have we considered the question of kinetics vs. diffusion control of the experimental data. There are numerous reviews [1-3] cov- eting these subjects for single-step reactions, and those techniques should also be suitable for two-step reactions as long as the measurements are carded out under conditions ensuring the validity of the equations used (see Fig. l), and as long as the measuring time is within appropriate limits, as discussed in this paper. We also assumed that the measurements are carried out under conditions eliminating, or at least minimizing, electrocrystallization contributions to the overpotentiai; i.e. that the surface morphology provides sufficient defects sites to avoid nucleation and surface diffusion overpotentials, and that the pulses are short enough to avoid dendritic-type growth. Furthermore, we assumed that the system is in a thermodynamic equilib- rium before the application of a measuring pulse. There- fore, even if only M 2+ is present originally, M + will be generated during equilibration.

4. Results and discussion

We focused our work on two 'critical times' of multi- step electrode reactions, which have not been treated quantitatively in prior work. These two critical times are: (i) the time to reach 'steady-state' conditions with the transient techniques and (ii) the time available for the determination of the fast-step kinetics at the beginning of the measuring pulse.

The fulfillment of the steady-state condition is important, because the multi-step kinetic equations described in Section 2.1 are valid only under those conditions (that is, when all elementary steps of the reaction sequence proceed at the same rate). This condition is generally not fulfilled

38 Z. Nagy et al. / Journal of Electroanalytical Chemistry 421 (1997) 33-44

at short times during the transient measurements. Although this aspect has not been completely ignored in the past, it has never been treated quantitatively. Losev and coworkers [37,40,41] pointed out that transient and steady-state measurements will give differing results if a considerable build-up of intermediates occurs during the transient, and Plonski [5-11] calculated the approximate time needed to reach steady state for some conditions. We have determined quantitatively the time needed to reach the steady state. For our calculations, the system was considered to be in steady state when the partial currents of the two reaction steps were equal within a maximum deviation of + 2%.

It is easy to demonstrate the necessity to reach steady state before kinetic measurements are made. The simulated galvanostatic transient response of an electrode reaction system is shown in Fig. 4 on four time scales. The expected activation overpotential can be calculated from Eq. (6) as 7 / - -0 .1 V, and our simulation predicted that steady state will be reached in 0.04s. It is quite obvious that measurements taken at shorter times cannot result in the expected overpotential; consider, for example, the sim- plest graphical extrapolation to t = 0. For the sake of clarity, the double-layer charging was ignored in this example; C01 = 0. Therefore, the initial rise of the transient response is not due to double-layer charging but is due only to the change (depletion) of the intermediate species concentration. The changes in partial currents and surface concentrations with time are shown in Fig. 5.

Kopistko and coworkers [1 5-19] have demonstrated

0.00 -- I I I I

*0.02

-0.04

.,J < F-- Z ~ -O.OO

O,. rt,,. I.M > 0 -0.05

"0.10

full scale= 0.001 8 - 0.01 e 0.1 e---1 ]

r I

- 0 . 1 2 L j I I I . . J 0.0 0.2 0.4 O.a 0.8 1.0

TIME Fig. 4. Effect of the measuring time on the activation-controlled (extrapolated to t = 0) overpotential o f galvanostatic transients. Conditions:

J = - 0 " 0 1 4 A c m - = ; Jo,= = 10-a and Jo,2 = I A c m - 2 ; cl = 10 -3 and c2 = 10-6 m ° l c m - 3 ; Di = D2 = 10 -e cm 2 s - i; /3 z =/32 = 0.5; T = 298.15K; Cd~ = OFcm-2 ; and R s -- 0 l ) cm 2.

'o ,,- 14 e

12 ? E

lO >,.. I,- ,m. tn Z i.u 8 1:3 I-.- Z i.,i..i n,,, 6 a.,. ::3 r,j r.,.) t:3 0

4 ,L o 10 "e

C !

'~ c 2 x 10 s

t I

t t

t

: 1

\

0 10 "s 10 -4 10 -3 10 -= 10 "I 10 o

TIME Is

'o 14 --

8

~0 ? E

"6 E

8 Z 0

Z 4 0

Fig. 5. Change of partial currents and surface concentrations with time during a galvanostatic pulse relaxation. Conditions are the same as for Fig. 4.

theoretically and experimentally that the beginning portion of the transient can be controlled, under certain conditions, by the fast step of the reaction sequence, thereby permitting the determination of the fast-step kinetics. This is clearly a different approach, requiring different conditions, from the determination of the fast-step kinetics (using the large-overpotential approximation) from steady-state data (Fig. 2). We have determined quantitatively the critical time limit for the fast-step kinetic control. For our calculations, the fast-step kinetics was considered to control the overall kinetics when the overall current was equal to the partial current of the fast step within a maximum deviation of 2%.

We of the steady

have succeeded in simple, graphical representation two critical times: (i) the time needed to reach state and (ii) the time available for the determina-

tion of the fast-step kinetics at the beginning of the experi- ment. The results are shown for a wide range of conditions and as a function of parameters involving only a few (mostly known) variables.

Fig. 6 presents the most general representation of our results. It is valid for both potentiostatic and galvanostatic single- and double-pulse measurements, and for both rds- and large-overpotential-approximation ranges (as shown in Fig. 1). Some restrictions and assumptions will be enumerated below. The fight-hand shaded zone indicates the conditions under which steady state has been achieved, while the left-hand shaded zone indicates the conditions under which the overall current is controlled by the fast reaction step at the beginning of transient measurements.

Z. Nagy et aL /Journal of Electroanalytical Chemistry 421 (1997) 33-44 39

ld

l o °

10"

¢4

I.IJ

V- lO-S

10 .4

10 "5

10 "e lO' 1o' lO 4 10' 10 e l d 10 e 10 9

(jo.,towllc,ntl~;t=}l x oxpfSI~I )

Fig. 6. General applicability diagram of the relaxation techniques for two-step metal deposition/dissolution reactions. The right-hand shaded zone indicates the conditions under which steady state has been achieved. The left-hand shaded zone indicates the conditions under which the overall current is controlled by the fast reaction step at short times during the transient measurements (for the galvanostatic technique, the left hand side this diagram is limited to the case when the first step is fast in the direction of the polarization). The symbol S is the inverse Tafel slope in the direction of the polarization: for anodic polarization, S= ( 1 - [3slow)F/RT; for cathodic polarization, S=(~slow)F/RT. The symbol 7/represents the "steady-state' value of the overpotential for the right-hand line, and the 'initial' value tbr the left-hand line.

(For the galvanostatic technique, the left hand side of this diagram is limited to the case when the first step is fast in the direction of the polarization.) For this diagram, the plotting parameter contains a number of variables that are not known before the measurement (namely, the exchange current density and the symmetry coefficient of the slow step, and the bulk concentration and diffusion coefficient of the intermediate species); therefore, these must be esti- mated to use the diagram.

The fact that (i) the critical times of the transient experiments are predominantly controlled by the kinetics of the slow step and (ii) the critical times decrease with increasing current or overpotential has been noted by earlier workers [4,15]. These effects occur because (i) the overall reaction rate is controlled by the slow step, and (ii) a finite charge must be transferred to stabilize the intermediate concentration at the steady-state level. The impor- tance of the Cin t Di~t 5 term is due to the fact that diffusion is an additional source/sink (in addition to the reaction steps themselves) of the intermediate at the electrode surface, and this diffusion delays the approach to steady state, as

has already been noted qualitatively [5,14]. In general, the conditions are more favorable for the determination of the kinetics of the first step in the direction of the polarization, independent of whether this is the slow or the fast step. These rationalizations of the effects of system variables on the critical times are also valid for the simplified diagrams presented in Figs. 7 and 8.

The applicability diagrams shown in Figs. 7 and 8 use fewer unknown variables, but they are restricted to the rds and the large-overpotentia! approximations respectively. Fig. 7 is an applicability diagram specific for the rds approximation. This diagram is limited to the case when the slow step of the reaction sequence occurs first in the direction of the polarization, and the left-hand side of the diagram is also limited to the galvanostatic technique. However, the plotting parameter contains only the bulk concentration and diffusion coefficient of the intermediate species, and the current density of the measurement. Fig. 8 is an applicability diagram specific for the large-overpotential approximation. It is limited to the case when the fast step of the reaction sequence occurs first in the direction of the polarization. The plotting parameter still contains un-

lO'

lO °

10 "1

10 -= 0

I,¢1

~ 10-a

10 "4

10 "s

10"e10' 10 4 10 6 10 e l d 10 e 1~

I j I Ic,..~,;~

Fig. 7. Simplified applicability diagram of the relaxation techniques for two-step metal deposition/dissolution reactions. The right-hand shaded zone indicates the conditions under which steady state has been achieved (for both potentiostatic and galvanostatic techniques). The left-hand shaded zone indicates the conditions under which the overall current is controlled by the fast reaction step at short times during the transient measurements for the galvanostatic technique only. Valid only for the rds approximation, ,and the slow step of the reaction sequence must occur first in the direction of the polarization. The symbol j represents the "steady-state' value of the current density.

40 Z. Nagy et aL / Journal of Electroanalytical Chemistry 421 (t997) 33-44

ld

l o 0

10 "1

10 "2 cO

i i i IE ~ 10-3

10 "4

10 "s

10 4 I ~ 10 4 I 0 e 10 e 10 7 I 0 e I 0 e

(li I Ic,,,O°,;,SllJo.,,°,.IJo.,,,,)

Fig. 8. Simplified applicability diagram of the relaxation techniques for tw~o-step metal deposition/dissolution reactions. The right-hand shaded zone indicates the conditions under which steady state has been achieved. The left-hand shaded zone indicates the conditions under which the overall current is controlled by the fast reaction step at short times during the transient measurements. Valid only for the large-overpotentiai approximation, and the fast step of the reaction sequence must occur first in the direction of the polarization. The symbol j represents the 'steady-state' value of the current density for the right-hand line, and the 'initial' value for the left-hand line.

known variables, namely the exchange current density ratio of the elementary steps, and the bulk concentration and diffusion coefficient of the intermediate species.

The diagrams in Figs. 6 -8 are presented for 'instantaneous response' situations, that is, for negligible double- layer capacitance, solution resistance, and pulse rise time (Cd] = R 3 = t I = 0). However, numerous simulations were also carried out to determine the effect of these variables, and the results can be summarized as follows. The diagrams can be used to determine the short-time or the steady-state critical time limits as long as the rise time of the excitation pulse and the rise time of the relaxation signal (RsCdl for the potentiostatic case, and RrCdi for the galvanostatic case) are both shorter than the critical time read from the diagrams. Otherwise, the longest rise time becomes the critical time limit. Every diagram is applicable to both single- and double-pulse techniques. The double-pulse techniques generally involve shorter rise times because of the 'precharging' of the double-layer capacitance.

There is some uncertainty involved in the determination of the critical times with these applicability diagrams (Figs. 6-8) because of the wide range of the values

considered for most variables. We estimate that the error for the critical time limits is about one order of magnitude for extreme conditions, but it is much smaller in most cases. The ranges of variable values for which these applicability diagrams are valid are given in Table 1, with the exception that the smallest Jo.fast/Jo,slow ratio is 10 3 for the critical time limit for determining the fast-step kinetics at short times. In most cases, the values in Table 1 also indicate the maximum ranges investigated in this work; therefore, the diagrams may be also applicable outside these ranges. The potential range investigated covers es- sentially the Tafel range. As a result, these diagrams may not be applicable in the linear current-density/overpotential range.

While these diagrams are useful for indicating to the experimenter the desired measuring conditions (current- potential-time regimes) and the limitations of the relaxation techniques for certain combinations of reaction sys- tcn~ parame.ter~, the experimental data themselves can, under certain conditions, indicate what may or may not be determinable from the data. Specifically, others have observed that unusually shaped relaxation curves can be obtained under certain conditions [4,12,15]. This is always associated with a maximum/minimum in the relaxation curve. We have observed with our simulations that such unusually shaped relaxation curves never occur under con-

0.5

0.4

N, E o 0.3 ,<

>- I-- U) Z 0.2 I.IJ a

I - z LU n- 0.1 r r Z) tO

0.0

0.26 V

0.18 V

0.10 V

-0.1 10 "e 10 "e 10 .4 10 -3 10 "2 10 "1 10 0

T I M E / s

Fig. 9. Examples of potentiostatic relaxation curves. Dashed lines indicate expected activation-controlled current density according to the rds approximation (Eq. (4)) while the dotted lines indicate the same for the large-overpotential approximation (Eq. (10)). Conditions: potential pulse, as indicated; Joj = 10-6 and Joo2 = 10-3Acre-2; cl = 10-4 and c 2 = 1 0 - g m o l c m -3 (for the 0 .26V pulse, c2= 10-8); D l = D 2 = 1 0 - 6 c m 2 s - I ; 131 = 132 = 0.5; T = 298.15K; Cdl = 2X 10 -5 F c m - 2 ; and R s = 0.1 fl cm 2.

Z. Nagy et aL /Journal of Electroanalytical Chemistry 421 (1997) 33--44 41

ditions when the rds approximation is valid (see Fig. 1). That is, a maximum/minimum never occurs when the first step is slow in the direction of the polarization, and a maximum/minimum will occur only above a certain overpotential when the first step of the reaction sequence is fast in the direction of the polarization. This behavior is caused by the depletion or accumulation of the intermediate species in the vicinity of the electrode surface and the consequent changes in the partial currents of the two elementary steps. We have observed a maximum/minimum only in the intermediate zone and the large-overpotential-approximation zone of Fig. 1. These observations are in agreement with previously reported qualitative predictions [4,12,15]. Consequently, it can be concluded that the appearance of a maximum/minimum on the relaxation curves indicates that the data do not contain sufficient information to permit the determination of the kinetics of the slow step, but the kinetics of the fast step may be determinable if the overpotential is large enough.

Some examples of unusually shaped relaxation curves are given in Figs. 9 and 10 for potentiostatic and galvanostatic techniques respectively. The lowest-overpotential case in both figures falls into the rds-approximation zone in Fig. 1, the middle-overpotential case falls into the intermediate zone, while the highest-overpotential ease falls into the

large-overpotential-approximation zone. A clearly visible maximum/minimum is observable only in the large-overpotential approximation zone, while the beginning of the maximum/minimum formation can be observed in the intermediate zone. It is also interesting to compare the overpotentials with those predicted by the rds approximation (Eq, (6)), indicated by the dashed lines at the left-hand edge of the figures, and with those predicted by the large-overpotential approximation (Eqs. (9) and (10)), indicated by the dotted lines. The lowest overpotentials agree with the rds approximation, the highest with the large- overpotential approximation, while the intermediate overpotential is different from either approximation.

Our observations are limited to the potentiostatic and galvanostatic relaxation techniques, but the general princi- ples are valid for all relaxation techniques, and it may be worth re-examining them in light of the obvious, but sometimes overlooked, restriction that 'steady-state' ex- pressions can be utilized only for data taken at times exceeding a 'critical time.' The extension of this investiga- tion to reactions with more than two elementary steps is also desirable, but may be too complex.

Acknowledgements

0.35

0.30

0.25

>

..I , 4 :0 .20

I,u

o. o.15 ,J

O

O.lO

0.05

0.5 A c m "2

0.05 A c m "2

0.000S A ©m "2

0.00 10 .0 10 -6 10 4 10 "a 10 .2 10 "1 10 °

T I M E / s

Fig. 10. Examples of galvanostatie relaxation curves. Dashed lines indicate expected activation-controlled overpotential according to the rds approximation (Eq. (4)) while the dotted lines indicate the same for the large-overpotential approximation (Eq. (10)). Conditions: current density pulse, as indicated; jo.l=10 -6 and jo.2 = 10-3Acre-2; el= 10-3molcm-3; c 2 = 10-gmolcm-a, 10-Smolcm-3, and l0 -7 molcrn -3 for 0.0005Acm -2, 0.05Acre -2, and 0.5 Acre -2 pulses respectively; Di= D 2 = 10-6cm2s-~; /3~ =/32 =0.5; T= 298.15K; Cdl = 2XI0-SFcm-2; and Rs=01"lcm 2.

This work was sponsored by the Division of Materials Sciences, Office of Basic Energy Sciences, US Department of Energy, under Contract No. W-31-109-ENG-38. We acknowledge the help of D. Lioulias, R.E. Hawkins, and M.A. Johnson with some of the calculations. One of us (KCL) is also indebted to the Argonne National Labora- tory, Division of Educational Programs, for partial support during her tenure as a Faculty Research Participant.

Appendix A. Numerical calculational methods

In prior work, we used file DISPL2 software package [42] to numerically solve similar differential equation systems for electrochemical relaxation techniques [31,43,44]. Here, we briefly describe the use of this package and the extension that was done for the present work. As described in Ref. [43], the DISPL2 package provides a numerical approximation of a system of partial differential equations by use of Galerkin's procedure with a B-spline basis set. That is, the concentrations are approximated by piece-wise polyno- mials, and the coefficients of the basis functions produce a system of initial value ordinary differential equations, which are solved by use of a multi-step solver. The overpotentiai equation (whether used in potentiostatic or galvanostatic modeling) is treated simultaneously with this system of ordinary differential equations. Since this solver uses an implicit method, a nonlinear system of algebraic equations needs to be solved at each time step. These

42 Z. Nagy et al. /Journal of Electroanalytical Chemistry 421 (1997) 33-44

equations are solved by means of a Newton method, and thus, at the core of the procedure is the need to approximate the Jaeobian matrix for this nonlinear system. The Jacobian involves the derivatives of the current density- overpotential relation with respect to the surface concentrations. This contribution to the Jacobian was approximated in Ref. [43] by providing derivatives with respect to the concentrations, but not restricted to the surface.

However, in the potentiostatic case we experienced occasional failures in the numerical simulation, and the formulation was revised in Ref. [44] to use a time-differen- tiated form of the boundary condition so as to allow the Jacobian to treat exactly the derivatives with respect to the surface concentrations. In practice, we have found both versions of the potentiostatic simulation formulation to be useful. The method was extended in Ref. [31 ] to allow the formulation to handle a potential dependence of the anodic and cathodic transfer coefficients. This extension uses an expression for these terms and their derivative with respect to the overpotential or the current (depending upon whether the overpotential or current is the measured variable in the mathematical model). In this way an exact Jacobian can be provided. This aids both the efficiency of the computation and the range of parameters that can be treated success- fully.

In the present work, the computational model was extended in several ways. (i) The representation of faradaic current was modified to include the two-step model. The necessary derivatives for the Jacobian calculation were modified to treat this case. In fact, the consistency of the single-step and two-step models allows us to treat both cases in a unified package. (ii) We changed our treatment of the double-pulse techniques. For numerical accuracy, we modified our multi-step solver so as to integrate exactly to t I, t 2, and t 3 rather than use a fixed time step and interpolate to the output time. (iii) We now treat the 'degenerate' cases of t I = 0 (instantaneous rise time) and/or t 2 = t 3 (instantaneous drop time). (iv) We extended the simulation to treat a double-layer capacitance of zero. In this case, the overpotential equation becomes an algebraic equation rather than a differential equation. Treating the equation as an algebraic one is more accurate than solving the differential equation for an arbitrarily small double-layer capacitance. (v) We also extended the model to treat an uncompensated solution resistance of zero.

The use of the Galerkin procedure and an implicit multi-step solver leads to accurate, but often time-consum- ing numerical simulations. As an alternative approach to these simulations, we have developed a finite-difference method in conjunction with consistent extrapolation in both time and space [45,46]. In this approach, we use a sequence of grids where each grid is a set of points in space and time, and we approximate the solution at the center of these grid points. That is, we use a second-order accurate finite-difference method in both space and time.

We first discuss the initial grid specification. As in the Galerkin approach [43], we use the half axis truncated at x.. = 5Dv~ f and use spatial grid points, which are logarith- mically distributed toward the electrode surface. We treat the time axis similarly for long times. For times during the pulses (which physically are short times), we use an exponential grid in time between 0 and t I, also between t l and t 2, and similarly between t 2 and t 3. The time grid is then exponentially distributed from t 3 to tf. Thus, on this space-time grid, we are using a second-order accurate fully-implicit method.

While this approach is comparable with other box- centered methods, such as Feldberg's original method described in Ref. [47], we extend it by introducing additional grids, which are refinements of this basic grid. For example, by introducing an additional grid in which spatial and temporal points are added halfway between the basic grid points, we can obtain simulation results on two grids. While this increases the computational effort, the results of these twodifferent second-order accurate grids can be extrapolated to yield a fourth-order accurate result in both space and time. The significance of this approach is that the basic grid can be relatively crude, but accuracy can still be obtained via extrapolation, which requires less effort than directly using a higher-order finite-difference method on a fine grid. We have developed an initial implementa- tion of the above-described approach for the two-step model with finite capacitance. This approach was used to produce some of the results presented herein. A more complete description of this method and comparison with other methods will appear in Ref. [48].

Appendix B. Analytical solution for the galvanostatic single-pulse case

An analytical solution can be obtained for the galvanostatic single-pulse case for the linear current-density/overpotential range. For this case, the diffusional and double- layer charging effects are fully treated, and the rise and fall of the pulses is assumed to be instantaneous (see Fig. 3).

Replacing the concentrations by reduced concentrations Qj = (c~ - c )c j , taking the Laplace transform of Eq. (11), and using the initial conditions in Eq. (12) yields

8x2~j sO-j=Dj 8x 2 (21)

The solution of Eq. (21) is in the form of

+q,0x,(¢- ,/o,x) To fulfill the boundary conditions in Eq. (13), the

coefficients pj must be zero. The coefficients qj can be obtained using the boundary conditions in Eqs. (14) and

Z. Nagy et aL / Journal of Electroanalytical Chemistry 421 (1997) 33-44 43

(15) in conjunction with the linearized current-voltage relations s as:

+ 2 r t r 3 F

ql = s + V~( r, + r 2 + r3) + r tr 2 R"-T ~ (23)

and

• f s ' ( r 2 -- r3) + r tr 2 -- 2 r l r 3 F (24) q2

s + ~ - ( r t + r 2 + r3) + r , r 2

where

r I - -A, I /F~/-Dlcl (25)

r 2 =.~ .2 /FII~2Cz (26)

r 3 = j o , , / F l / ~ 2 C z (27)

Subsequently, a relation between )'and rl can be calculated using the Laplace transform of the faradaic current densities (from the derivative of Qj and Eqs. (14) and (15)) and the Laplace transform of the capacitive current (Cdl St/). For an instantaneously rising current density step changing from zero to jt at t = 0, the Laplace transform of the current density is j l / s , and a relation for 71 can be obtained through somewhat elaborate but straightforward algebraic manipulations as

Jl s + ut¢~" + u 2 : : ( : : +u,) (28)

where

u l - - r ! + r 2 + r 3 (29)

U 2 = r lr 2 (30)

F u 3 = r l r 2 + (Jo.l +Jo.2) RTCd ' (31)

F u4 = ( rl jo. 2 + 4r3Jo.,) ~ (32)

RTCal

This can be rewritten as

Jt 3 r . X 2

~ = ~dl m=lE $ 3 / 2 ( ~ " __ Xm ) (33)

where

X~ + u tX j + u2

Kj = X2 [ X2 _ Xj( X k + X,) + X k X, ] (34)

s The linear approximations of the Butler-Volmer equations Eqs. (16) and (17) are respectively ('=,, Ic.o_,, c.o,,]) Jl(/) = J o . I - - - - C2 m ~ - t - L c, ,"

and

,]}

and Xj, and X, and X t are the roots of the cubic equation

X 3 + u , X 2 + u 3 X + u 4= 0 (35)

The inverse of Eq. (33) from standard Laplace transform Tables [49] is

j~ 3 rl( t) = ~dt E K I N G ( - X . . t ) (36)

m--i

where

G ( v . t ) = exp(v2t) e r f c ( v ~ ) + 2 v ~ - ~ - 1 (37)

Appendix C. Nomenclature

Cdl Cint

cj

Dim

nj

E F

J Jf J1 Jo.i JO.SIow

Jo, fast

R Rr Rs S t

tf tj x

Z

~ a

O¢ c

flslow 7/ /t

P o "

capacitance of the double layer (F cm-2) bulk concentration of the intermediate species (tool cm -3 ) bulk concentration of the reactant in step j (molcm -3) concentration near the electrode surface of the reactant in step j (mol cm -3) diffusion coefficient of the intermediate species (cm 2 s- t) diffusion coefficient of the reactant in step j (cm 2 s- I) total overpotential including IR drop (V) Faraday constant (C mol- l ) total current density (A cm-2) total faradaic current density (A cm-2) faradaic current density of step j (A cm-2) exchange current density of step j (Acm -2) exchange current density of the slow step (A cm-2) exchange current density of the fast step (Acm -2) number of electrons exchanged in the overall reaction universal gas constant (J K- l mol- t) reaction resistance (dr/ /d J)n = o (l'l cm 2 ) uncompensated solution resistance (fl cm 2) inverse Tafel slope, defined for Fig. 6 (V -i ) time (s) end of r ,se (s) rise and fail times, defined in Fig. 3 (s) distance from the electrode surface (cm) electric charge of the cation with highest oxidation state anodic transfer coefficient of the overall reaction cathodic transfer coefficient of the overall reaction symmetry coefficient of step j symmetry coefficient of the slow step overpotential (V) stoichiometric number of the reaction mechanism number of electrons transferred in the rds number of electrons transferred in the cathodic direction before the rds

44 Z. Nagy et al./ Journal of Eiectroanalytical Chemistry 421 (1997) 33-44

References

[1] P. Delahay, in P. Delahay and C.W. Tobias (Eds.), Advances in Electrochemistry and Electrochemical Engineering, Vol. !, Wiley- Interscience, New York, 1961, Chapter 5.

[2] J. Kuta and E. Yeager, in E. Yeager and A.J. Salkind (Eds.), Techniques of Electrochemistry, Vol. 1, Wiley-lnterscience, New York, 1972, Chapter 3.

[3] Z. Nagy, in R.E. White, J.O'M. Bockris and B.E. Conway (Eds.), Modem Aspects of Electrochemistry, Vol. 21, Plenum, New York, 1990, Chapter 6.

[4] I.H. Plonski, J. Electrochem. Soc., 116 (1969) 944. [5] I.H. Plonski, J. Electrochem. Soc., 116 (1969) 1688. [6] I.H. Pionski, J. Electrochem. Soc., 117 (1970) 1048. [7] I.H. Plonski, Rev. Roum. Phys., 16 (1971) 449. [8] I.H. Plonski, Rev. Roum. Phys., 16 (1971) 467. [9] I.H, Plonski, Rev. Roum. Phys., 16 (1971) 979.

[10] I.H. Plenski, Rev. Roum. Phys., 36 (1991) 203. [11] I.H. Plonski, Ber. Bunsenges. Phys. Chem., 95 (1991) 23. [12] K.J. Bachmann and U. Bertocci, Electrochim. Acta, 15 (1970) 1877. [13] K.J. Bachmann, J. Electrochem. Soc., 118 (1971) 226. [14] K.J. Bachmann, J. Electrochem. Soc., !19 (1972) 1021. [15] O.A. Kopistko and N.V. Grushina, J. Electroanal. Chem., 178

(1984) 21. [16] O.A. Kopistko, N.V. Grushina and D.V. Sokolskii, Dokl. Akad.

Nauk SSSR, 291 (1986) 1157. [17] O.A. Kopistko, J. Electroanal. Chem., 224 (1987) 67. [18] O.A. Kopistko, Electrochim. Acta, 33 (1988) 585. [19] O.A. Kopistko, Elektrokhimiya, 25 (1989) 452. [20] O.A. Kopistko and S.E. Batyrbekova, J. Electroanal. Chem., 336

(1992) 223. [21] H. Relier and E. Kirowa-Eisner, J. Electroanal. Chem., 103 (1979)

335. [22] J.O'M. Bockris and M. Enyo, Trans. Faraday Soc., 58 (1962) ! 187. [23] B. Lovrecek and N. Marincic, Electrochim. Acta, 11 (1966) 237. [24] H.C. Albaya and W.J. Lorenz, Z. Phys. Chem. (Frankfurt am Main),

81 (1972) 294. [25] J.O'M. Bockris, Z. Nagy and A. Damjanovic, J. Electrochem. Soc.,

119 (1972) 285. [26] Q.J.M. Slaiman and WJ. Lorenz, Electrochim. Acta, 19 (1974)791. [27] I.R. Burrows, J.A. Harrison and J. Thompson, J. Electroanal. Chem.,

58 (1975) 241.

[28] A.R. Despic. D. Jovanovic and T. Rakic, Electrochim. Acta, 21 (1976) 63.

[29] J.L. Settle and Z. Nagy, J. Electrochem. Soc., 132 (1985) 1619. [30] Z. Nagy, J.P. Blaudeau, N.C. Hung, L.A. Curtiss and D.J. Zurawski,

J. Electrochem. Soc., 142 (1995) L87. [31] Z. Nagy, G.K. Leaf, M. Minkoff and R.H. Land, Electrochim. Acta,

33 (1988) 1589. [32] K.J. Vetter, Electrochemical Kinetics, Academic Press, New York,

1967, p. 149 and references cited therein. [33] V.V. Gorodetskii, K.A. Mishenina, V.V. Losev, A.N. Grimberg and

G.M. Ostrovskii, Elektrokhimiya, 4 (1968) 46. [34] R.M. Hurd, J. Electrochem. Soc., 109 (1962) 327. [35] J.O'M. Bockris and A.K.N. Reddy, Modem Electrochemistry, Vol.

2, Plenum, New York, 1970, p. 1004. [36] V.V. Loser and V.V. Gorodetskii, Elektrokhimiya, 3 (1967) 1061. [37] V.V. Loser, in B.E. Conway and J.O'M. Bockris (Eds.), Modem

Aspects of Electrochemistry, Vol. 7, Plenum, New York, 1972, Chapter 5.

[38] P. Delahay, Double Layer and Electrode Kinetics, Interscience, New York, 1965, p. 168.

[39] Z. Nagy, Electrochim. Acta, 22 (1977) 191. [40] A.I. Molodov, V,I. Barmashenko and V.V. Loser, Elektrokhimiya, 7

(1971) 18. [41] A.I. Molodov and V.V. Loser, Elektrokhimiya, 7 (1971) 812. [42] G.K. Leaf and M. Minkoff, in R. Vichnevetsky and R. Stepleman

(Eds.), Advances in Computer Methods for Partial Differential Equa- tions V, IMACS, New Brunswick, NJ, 1984, p. 429.

[43] Z. Nagy, R.H. Land, G.K. Leaf and M. Minkoff, J. Eleetrochem. Soe., 132 (1985) 2626.

[44] Z. Nagy, M. Minkoff, G.K. Leaf and R.H. Land, Electrochim. Acta, 32 (1987) 1777.

[45] H.G. Keller, in B. Hubbard (Ed.), Numerical Solution of Partial Differential Equations 2, Academic Press, New York, 1970, p. 193.

[46] H.B. Keller and T. Cebeci, in W. Beiglb~k (Ed.), Lecture Notes in Physics, Proc. Second Int. Conf. on Numerical Methods in Fluid Dynamics, Springer, Berlin, 1971, p. 92.

[47] D. Britz, Digital Simulation in Electrochemi~trq Springer, Berlin, 2rid edn., 1988.

[48] M. Minkoff, G.K. Leaf and Z. Nagy, in preparation. [49] G. Fodor, Laplace Transforms in Engineering, Akademiai Kiado,

Budapest, 1965.

Documents

Applicability of d.c. relaxation techniques to multi-step reactions