3374
CHARLES A. JOHNSON AND SIDNEY BARNARTT
Constant-Potential Reactions at Cylindrical Electrodes by Charles A. Johnson and Sidney Barnartt E. C. B a i n Laboratory for Fundamental Research, United States Steel Corporation, Monroeville, Pennsylvania (Received A p r i l 7, 1969)
16146
A new approximate analytic equation is developed for the current-time behavior of a cylindrical electrode at con-
stant potential, where both charge transfer and mass transfer control the electrode reaction rate. The equation is restricted to first-order charge-transfer mechanisms in which electrical work is done only during the rate-determining step. This approximate equation for a cylinder of radius a has the same form as that for a sphere of radius 8a/3. It is compared with exact numerical solutions for cylindrical electrodes and exhibits good agreement over the entire reaction time of interest. I t provides an accurate method of determining charge-transfer currents from experimentalcurrent-time curves obtained with cylindricalelectrodes. The theory of electrode kinetics a t constant potential, under conditions where diffusional mass transfer and the charge-transfer mechanism both control the reaction rate, is well developed for planar and spherical electrodes. I n the case of planar electrodes, the exact solution of the boundary-value problem has been derived in closed form for first-order charge-transfer mechanisms under the restriction that all of the electrical work involved occurs in the rate-determining step.’ For higher-order mechanisms an approximate closedform solution has been derived for planar electrodes;2 this gives satisfactory agreement with the true currenttime curves over almost the entire time during which there is appreciable control of the reaction rate by the charge-transfer p r o ~ e s s . ~I n~ ~the case of spherical electrodes the exact solution of the boundary-value problem has been given in closed form for first-order charge-transfer mechanisms ;4--8 no theory for higher order mechanisms has appeared. I n the case of cylindrical electrodes it is not possible to obtain the full current-time behavior in terms of known functions.’ We have previously given an approximate treatmen t’ of first-order reactions which is, unfortunately, valid only for short reaction times (i-e., only for about the first 15% of the mass-transport-controlled decrease in the current). I n the present paper we develop an approximate closed-form solution for first-order reactions at cylindrical electrodes that closely reproduces the true current-time curves over longer reaction times (the time span over which the current decreases to half the initial value). Thz new approximate equation is remarkably good over the entire reaction time of interest and over the range of experimental conditions normally encountered. It provides a method of determining charge-transfer currents from experimental current-time curves which is both simple and accurate.
Development of the New Approximate Current-Time Equation Our approach to the problem may be described as T h e Journal of Physical Chemistry
having two stages. In the first stage it is shown that, for moderately short reaction times at least, the currenttime behavior for a cylindrical electrode of radius a is closely similar to that for a spherical electrode of radius 8a/3. This result is established by examination of the Laplace transforms of the current for both cases. It leads to a closed-form approximate solution for a cylindrical electrode which is just the known solution for a spherical electrode. This, unfortunately, is still too cumbersome for application to the reduction of experimental data. In the second stage, therefore, we make a further approximation in that this solution is replaced by a simpler, approximate form which has been shown previously8 to yield good results when applied to spherical geometry. The latter simpler form is the approximation for cylindrical electrodes that we shall test. Consider an electrode reaction of the form Oe+ + ne- = R(”-n)+ (1) for which the net anodic current density i at any time 2 is given by
i/io=
(c~cRO)
exp[(l
- p)nev] (CO!CO~)
exp [-@ne7 I
(2)
Here overpotential 7 represents the potential step applied to the electrode, which is initially at the reversible potential; io is the exchange current density, p the transfer coefficient, and E = F/RT where F is the Faraday constant, R the gas constant, and T the absolute tem(1) H. Gerischer and W. Vielstich, 2. Phys. Chem. (Frankfurt) 3, 16 (1955). (2) 5. Barnartt and C. A. Johnson, Trans. Faraday Soc., 63, 431 (1967). (3) 9. Barnartt and C. A. Johnson, J . Phys. Chem., 71,4430 (1967). (4) I. Shah, K. J. Martin, and J. W. Ross, ibid., 65,259 (1961). (5) J. R. Delmastro and D. E. Smith, J . Electroanal. Chem., 9, 192 (1965). (6) J. R. Delmastro andD. E. Smith, Anal. Chem., 38,169 (1966). (7) C. A. Johnson and S. Barnartt, J . Phys. Chem., 71,1637 (1967). ( 8 ) S. Barnartt and C. A. Johnson, Trans. Faraday SOC.,65, 1091 (1969).
3375
CONSTANT-POTENTIAL REACTIONS AT CYLINDRICAL ELECTRODES perature. The concentrations of reactants R and 0 a t the electrode surface are CRO, coo initially and CR, co at time t. The solution is assumed to contain an excess of supporting electrolyte. We now compare the theoretical current-time behavior when this reaction takes place at a cylindrical electrode of radius a to that for the reaction a t a spherical electrode of radius b. I n both cases it is assumed that the counter electrode is far away from the test electrode and that diffusion is the only masstransfer process. It is instructive to examine the Laplace transforms J ( o y l(s), ) J ( s p h ) (s) of the currents j ( o ~ l ()2 ) ~ j(sBh) ( t ) , defined as
For a spherical electrode of radius b the transform of the current, as obtained from eq All-A15 of ref 7 , may be written J(sph)(S)
=
it=o
x
with
+ (3)
("
4%)'+
which have been evaluated in a previous paper.' For a cylindrical electrode of radius a, the transform of the current is given by eq A30 of ref 7 J(ayl)(S)
= it=o(s
+
X o m o
x
+
IIX2 ("
hR1/sDR
[Ko(adzR>/K1(al/slDR)]
I-'
As s 4 00, eq 9 reduces to
(5)
The difficulty presented by cylindrical geometry is that eq 4 cannot be inverted to yield j(,,,)(t) in terms of known functions. Since we are particularly interested in fairly small values of t , l , Qwhich correspond to large values of the transform parameter s, we examine the behavior of eq 4 as s + 00. Now the asymptotic expansions of KO($) and K I ( ~are ) lo
Thus the first two terms of both the numerator and denominator in the asymptotic expansion of J ( s p h ) (s) are ) when identical with the corresponding terms of J ( o y l(s) b =8~/3. This result leads us to suppose, although without rigorous grounds, that the current-time curve for a cylindrical electrode of radius a will be nearly the same a t short times as the curve for a spherical electrode of radius 8 a / 3 . An exact analytic equation for the latter curve is known, eq A19 of ref 7
i _ With the aid of these expansions we find the asymp-
2* 2-0
- 1+
J(CYl)(S)
-
+ +
1 ( 8 a / 3 ) ( X ~ Xo)
- 8 a t / 3 ) ( d E - 8aE/3)
totic behavior of eq 4 (8-
+
(4)
Here itPo, the current density immediately after application of the potential step, is the charge-transfercontrolled value; DO and DIE are the diffusivities of species 0 and R; KOand K1 are modified Bessel functions of the second kind of orders 0 and 1, respectively; and A 0 = (io/nFDocoO) exp [-/?ne71 XR = (~~/~PDRcRO) exp[(l - /?)ne71
1, ")'
+
t K o ( a ~ 0 ) / ~ 1 ( a1 ~ o )
(8a/3)2t
m)
exp(t2t)
(9) K.B. Oldham and R. A. Osteryoung, J. EZectroanal. C h m . , 11, 397 (1966).
(IO) G.N.Watson, "A Treatise on the Theory of Bessel Functions," 2nd ed, Cambridge University Press, London, 1958. Volume 73, Number 10 October 1969
3376
CHARLES A. JOHNSON AND SIDNEY BARNARTT
This equation reduces to the corresponding result for ! when the electrode radius O is large : planar electrode^',^ a >> ( X R Xo)-’. Also, for the particular case D = DR = DOit reduces to
“
+
-i - -i t 4
1 + exp[X2(1 + 6 ) 2 t ] X 1+ 6 1+6 6
erfciX(1
+ 6)v‘t~
(12)
where 6 = dB/[X(8a/3) J and X is as defined in eq 8. Equation 11is, unfortunately, much too cumbersome to be of use in the analysis of an experimental i-t curve to evaluate the charge-transfer current. This problem has been encountered previously in connection with the analysis of i-t curves for spherical electrodes, and an approximate simplified i-t relation has been developed which is suitable for analysis and yields the chargetransfer current &, with good precision.* Correspondingly, the final step in the development of a useful general i-t relation for cylindrical electrodes is the replacement of eq 11by the approximate, simpler form
i Zt-0
+ 1 +16 ’ exp[h2(1 + 6 ’ ) 2 t ] X
6’
Ii
02
0
1
2
3
4
J!
5
6 ( 2‘1
7
8
9
1
Figure 1. Approximate (broken) and true (solid) current-time curves, test 128: CRO = coo I :5 x 10-6 mol cm-3; DR = 4 X 10-4 Do = 4 x 10-6 om2sec-1; io = 1 mA cm-3; v = 120mV; p = 0.5; a = 0.1814mm.
-
1+6’
erfc[X(l
+ S’)di](13)
+
where 6’ = [(8a/3)(X~ A,)]-’. This equation has the same form as eq 12 and reduces to the latter when
DR
1
Do.
Equation 13, then, is an approximation to the i-t behavior at a cylindrical electrode of radius a, which we have reason to expect may be sufficiently accurate for the purpose of analyzing experimental data. I n the next section the validity of eq 13 is established by comparison with exact solutions of the i-t behavior which were developed by numerical methods. Subsequently we show how eq 13 is used to provide a simple method of determining the charge-transfer current; this method is then illustrated with examples to indicate the degree of precision obtainable. It should be recalled that eq 13 is intended to apply only to that portion of the i-t curve over which the current decreases to half its initial value. It is not a good approximation a t very long reaction times. This is evident from the fact that it predicts lim i(t)= it,o[6‘/(1 t+
m
+ 691
while, as is shown in the Appendix, the true current at a cylindrical electrode approaches zero for long reaction times.
Validity of the Approximate Current-Time Relation To determine how closely eq 13 represents potentiostatic i-t curves a t cylindrical electrodes, we generated exact numerical solutions of the boundary-value problem for specific reaction conditions and electrode The Journal of Physical Chemistry
radii. The Schmidt methodll was used, the boundary conditions on the concentration gradients being dealt with according to a method which is also ascribed to Schmidt.12 Considerable care in selecting step sizes had to be taken to avoid instability and to establish accuracy in the solution. Many numerical solutions were generated and compared with the corresponding approximate curves from eq 13 over the current range 1 > i/it-o > 0.5. This is the range over which the charge-transfer process exerts appreciable control over the reaction rate a t planar electrode^;'^^ it was assumed that the useful current range would be no more extensive in the case of cylindrical electrodes, an assumption both for supported by the fact that i(t)+ 0 as t --t cylindrical electrodes (Appendix) and for planar electrodes. Figure 1 shows a typical comparison of an approximate i-t curve with the corresponding true curve. The approximate curve is seen to be lower than the true one over the time period examined, as was found in all of the comparisons made. The per cent deviation between the two curves, given by d 100[(itrue iapprox)/itrueJ, increases with time a t first but later the two curves come closer together. Eventually the two curves must cross since, as t -P 00, i,,, approaches zero while iapprox approaches a constant value greater than zero. As a single quantitative measure of the agreement between the two curves, we utilized the maximum deviation d,, within the range 1 > i / i + o > 0.5. Table I
’
-
(11) H. S. Carslaw and J. C. Jaeger, “The Conduction of Heat in Solids,” 2nd ed, Section 18.3,Clarendon Press, Oxford, 1959. (12) P. H.Price and M.R. Slack, Brit. J . Appl. Phys., 3 , 379 (1952). The method used in the present paper is Method I1 of this reference.
3377
CONSTANT-POTENTIAL REACTIONS AT CYLINDRICAL ELECTRODES
Table I : Deviation d,,, Test no.
4
5 9 10 18 19 30 122 128 145
a, mm
mV
0.142 0.473 1,474 0.295 0.267 0.325 0.102 0.208 0.1814 1.211
10 10 10 10 10 10 10 100 120 15
of the Approximate Current-Time Curve from the True Curve, Selected Tests io, mA/cma
’I#
9,
0.5 0.5 0.5 0.5 0.25 0.75 0.5 0.5 0.5 0.5
1 1 0.2 1 1 1 1 1 1 0.2
(cmZ/sec) X I@
(mol/cm*) X 106
DR
Do
CRO
coo
1 1 1 1 1 1 1 4 4 0.4
1 1 4 4 4 4 1
1 1 5 5 5 5 10 5 5 5
1 1 1 1 1 1 1 1 5 1
0.4 0.4 4
--Maximum dB/aA ‘/a 1/10
1 1 1 1 1 1 1 1
1, 880
XR/XO
1.476 1.476 1.181 1,181 1,181 1.181 0.1476 0.9805 10.68 3.586
1.75 1.44 635 26.1 21.2 31.6 15.4 13.0 11.8 362
deviation-i/it-o &ax# %
0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
1.5 0.52 3.7 3.7 3.7 3.7 2.5
5.1 3.1 4.6
d,,,:25%
I
Ll
01005
4%
5%
01
-
I-
0 10
I
20
’R“0
Figure 2. Deviation d,,, between the approximate and true current-time curves for d/Dlax = 1, with D = max(DR, DO).
lists some selected determinations of dmax. I n these and all of the other comparisons made between the approximate and true i-t curves, d,,, occurred at the = 0.500; usually a t still lower lower current limit i/& currents the deviation first increased above d,,, before diminishing again. Figure 2 presents dma, as a function of experimental conditions for a cylinder of radius a = dD/X, where D is the larger of (DR, D O ) , Here quantities ( D R / D o ) and (XR/XO) are used for the axes because our data showed that these two quantities suffice to fix d,, for a given dB/aX. Thus any changes made in 11, io, p, a, CRO, and coo had no discernible effect on d m a x except insofar as they modified DR/Do and XR/XO (compare tests 9, 10, 18, 19 of Table I). The contour lines of constant dmaXshown in Figure 2 were established by over 100 selected determinations of dma, under the condition d D / a X = 1. The following results appear. (a) When D R = D O , d m a x has a minimum value. (b) The symmetry in the deviation plot is such that simultaneous interchange of XR $ XO and DR e D O leaves the value of d,, unchanged. This is readily seen analytically, for such interchange affects neither the approximate i-t curve (eq 13) nor the true i-t curve (see Laplace transform of the true current, eq 4). (c) At any given ratio DR/Do, d,, reaches a peak value under
Figure 3. Deviation dmSxwhen XR
Xo.
the condition XR 3si XO. Such peak values are plotted in Figure 3 for dD/aX = 1or l / 3 . These results show that, provided O / a X I 1, the maximum deviation of the approximate equation (13) from the true one, in the range 0.5 < i/it-,, < 1, will not exceed 5.2%. The quantity O / a X may be used as a measure of the effect of electrode curvature on the i-t curve; the effect disappears as O / a X --+ 0, for then the approximate and the true i-t curves for cylindrical electrodes each approach the planar-electrode curve. No determinations of d m a x were made for (1/ZT/aX) > 1 since X is then sufficiently small, and the time to reach d,, becomes sufficiently long, that deviations >5% need not be encountered in practice. Analysis of Experimental Current-Time Curves The form of the approximate current-time relation (eq 13) for cylindrical electrodes a t constant potential permits us to utilize the “difference-ratio” method of analysis which was developed for spherical electrodes.* The difference-ratio method is outlined below, and it is applied to illustrative examples of i-t curves at cylindrical electrodes to demonstrate that it does permit evaluation of the charge-transfer current with good accuracy. I n these examples we utilized the exact numerical i-t relation as the “experimental” data to be analyzed. Volume 75, Number 10 October 1969
3378
CHARLES A. JOHNSONAND SIDNEY BARNARTT
Table I1 : Example Analysis of Current-Time Curves a t a Cylindrical Electrode by the Difference-Ratio Method t , sec
X, sea-1’2
P
0.06 0.08 0.12 0.2 0.4 0.8 1.2 1.6
Test 30, 1.193 1.222 1.266 1.340 1.453 1.614 1,708 1 763
3.613 3.571 3.503 3.401 3.234 3.030 2.895 2.795
DR =
DO, d,
Mean 95% confidence limits True value 0.1 0.2 0.4 0.8 1.0 1.2 1.4
To apply the difference-ratio method in analyzing an experimental i-t curve, we select an arbitrary time t and make use of the three current measurements i(t),i(4t), and i(9t). The difference ratio, defined as
[i(t) - i(4t)]/[i(41)
- i(9t)l
permits us to calculate the quantity X ( l in eq 13 through the relation
bo
0.55735, bl = -0.25133,
b2
=
+ 6’) appearing
-0.56485
which was developeds to span the range 1.2 < p < 1.9. Next we evaluate X and 6‘; the procedure used is dependent upon whether or not the diffusivities are known.srla When DO and D R are known, X is obtained from the identities
= ( c o ~ D o / c R ~ DeRx)p ( n 4
(17) [In the special case of equal diffusivities this procedure is greatly simplified, since X ( l 6) = X 3dD/8a.] When DOand DR are unknown, we may obtain 6’ from any two of the three measured currents, e.g. XR/XO
+
i(t)
i(4t)
0.333 0.334 0.328 0.328 0.311 0.307 0.294 0.271 0.313 f0.018 0.310
Test 122, D R = l0D0, d, = 5.1% 1.227 0.337 (0.301) 1.312 0.331 (0.299) 1.425 0.324 (0.297) 1.564 0.310 (0.290) 1.615 0.305 (0.287) 1.656 0.299 (0.284) 1.695 0.295 (0.281) Mean 0.314 (0.291) 95% confidence limits 1 0 . 0 1 4 (0.007) True value 0.303
6.197 5.964 5.667 5.304 5.173 5.062 4.965
p
= 2.5%
+
+ C[X(l + 6 ’ ) 4 ] + C[2h(l + 6 ’ ) 4 ]
6’ = 6’
(18)
3.94 3.95 3.95 3.96 3.94 3.88 3.89 3.81 3.91 2~0.04 3.917 6.95 (6.86) 6.96 (6.85) 6.98 (6.84) 6.96 (6.82) 6.94 (6.80) 6.91 (6.79) 6.90 (6.77) 6.94 (6.82) h 0 . 0 2 5 (0.029) 6.859
are determined. To evaluate it,o, we substitute the measured current density a t time t, as well as X and 6’, into eq 13. Table I1 lists two examples of this analysis. The first example, test 30 of Table I, is one involving equal diffusivities. The derived value of itE0 exhibits little dependence on the value of t selected (column 1). In the second example, test 122, wherein DR = 10D0, the deviation of the approximate from true i-t curve is as large a deviation as one might expect to meet in actual experiments. For this case Table I1 lists values of X and i,,o calculated on the basis of known DO and DR,as well as values (in parentheses) of the same quantities calculated on the basis of unknown DO and DR. Both methods of calculation yield ite0with good accuracy; the individual values of remain remarkably close to the mean value. The individual values of A, however, exhibjt greater percentage variations. Test 145 of Table I in which DR/Do = 0.1 (not listed in Table 11),maybe mentioned as a third example : mean = 120 =k 3.0 pA cm-2; true value derived value 118.4 pA cm-2. These examples illustrate that the difference-ratio method, based upon eq 13, may be used with confidence for analyzing experimental potentiostatic i-t curves a t cylindrical electrodes. Inherently it permits evaluation of the charge-transfer current with good accuracy (i2% if the experimental data are sufficiently accurate). I n practice, the usual experimental errors involved in
+
where C[z] = exp(z2) erfc(x). With X(l 6’) and either X or 6’ evaluated as just described, both X and 6‘ The Journal of Physical Chemistry
(13) S. Barnartt and C. A. Johnson, J . Electroanal. Chem., in press.
3379
CONSTANT-POTENTIAL REACTIONS AT CYLINDRICAL ELECTRODES measuring i ( t ) ,i(4t), and i(9t) are expected to contribute greater uncertainties to the analysis than will the deviation of the approximate equation (13) from the true The above analysis yields the charge-transfer current corresponding to a given overpotential from a single potentiostatic i-t curve. From such curves at other values of 7 we obtain the quantities /3 and io, characteristic of the reaction mechanism, following well-known procedures.
Conclusions The current-time equation (13)) developed to describe the general behavior of cylindrical electrodes at constant potential, is a good approximation to the true behavior over the useful current range 1 > i/i+o > 0.5. I n conjunction with the difference-ratio method of analysis, eq 13 permits evaluation of the chargetransfer current from a given experimental i-t curve with good accuracy ( A2% if the experimental data are sufficiently accurate).
Appendix Concentration Changes and Current at Cylindrical Electrodes As t --+ m . In a system having cylindrical symmetry, the diffusion equations for species 0 and R
=
0)
=
CEO
(A2)
+CEO
(A3)
and boundary conditions
r + : co(r, t) +COO,
c R ( ~ t) ,
lim [CO(%t ) ] =
COO
t-L m
lim
t+
[ C R ( U , 1)
J
=
CRO
+ it=o/[nFDo(XR + X O ) ]
(-45)
- it,o/[nFDR(hR -I- Xo)]
(A6)
m
Substitution of these values into eq 2 of the text then yields lim [i(t)] = 0
(A71
t-c m
The ultimate current a t a cylindrical electrode is thus zero as it is for a planar electrode. However, the ultimate concentrations at a cylinder differ from those at a plane, the latter being7
Coo
+ it,o/nFXd&,
CRO
- it=o/nFXd&
(A8)
That these concentrations differ with electrode geometry is not inconsistent with zero current at each electrode, for zero current requires only that the ultimate concentrations obey the thermodynamic relation
are to be solved under the initial conditions
r 2 a: c ~ ( Yt, = 0) = coo, C R ( Y , t
the Laplace transforms of the concentrations co (a, t ) , cR(a, t ) and of the current density i(t)at the electrode surface have been given in a previous paper,7 in which it was pointed out that the Laplace transforms cannot be inverted in terms of known functions. True values of the ultimate concentrations a t the electrode surface and of the ultimate current density can, however, be established by examining the behavior of the Laplace transforms of the concentrations for small values of the Laplace transform parameter s. When this is done, we obtain
where i ( t ) is given by eq 2 of the text. Expressions for
This requirement is met for each geometry; note that the activity coefficient ratio fo/foo = f R / f R o = 1 in the present problem (excess supporting electrolyte), It should be pointed out that the ultimate concentrations a t cylindrical and planar electrodes become identical in the particular case of equal diffusivities.
Volume 79, Number IO
October 1969
