Potential Scan Voltammetry with Finite Diffusion-A Unified Theory H. E. Keller Department of Chemistry, Northeastern University, Boston Mass. 02115
W. H. Reinrnuth Department of Chemistry, Columbia University, N e w Y o r k , N . Y. 10027
Beginning with equations generally accepted as descriptive of electrochemical reactions and mass transport, a solution for potential scan voltammetry is derived in the form of infinite series. The mathematical and physical interpretations of these series are discussed in depth. The discussion is based on the properties of the introduced “linear scan” functions ( L P , J and transfer functions fi). Criteria are developed for determining which L,,, function fits experimental data. Nonplanar electrodes work into the theory quite readily, and derivations of spherical and cylindrical corrections are made. The most significant feature of the theory presented is its ready applicability to any linear complications. Thus it is presented as a “unified approach” to potential scan voltammetry.
IN 1938 MATHESONAND NICHOLS ( I ) published the first account of potential scan voltammetry (PSV). Their primary aim was simply to provide more rapid readout than that available with classical polarography. Although it was soon apparent that many distinctive features set this technique apart from polarography, the mathematical developments required for full utility did not occur for ten years. In 1948, Randles (2) and Sevcik (3) developed the first rigorous mathematical theory of this modification of polarography. PSV has become sufficiently popular that virtually all complications such as deposition, irreversible charge transfer, stirred solutions, thin layer electrodes, nonplanarity, and homogeneous kinetics have been treated (see Reference 4 for bibliography). Here solutions with finite boundaries are treated. The boundary, at a finite distance from the electrode, is a surface beyond which the effects of the electrochemical reaction are assumed to be nil. Two types of boundaries are considered. At one the flux(es) of material is zero; at the other the concentration(s) is assumed constant. These will be referred to as the bounded (B) and nonbounded (NB) cases, respectively. Systems in which the layer between the boundary and the electrode surface is so thin that the concentration gradients can be assumed to be linear will be called infinitesimal bounded (ISB) or infinitesimal nonbounded (ISNB) depending on the boundary type. Delahay (5) has presented equations for stirred solutions with linear diffusion and reversible, irreversible, or quasireversible charge transfer using the ISNB approximation. These equations are readily derived from Fick’s first law. Bowers and Wilson (6, 7 ) applied these results to membrane covered electrodes and confirmed the mathematical analysis.
In a series of papers on stripping voltammetry, DeVries and Van Dalen (8-10) treated the thin film mercury electrode as a bounded layer. Their solution was in the form of an integral equation which was evaluated numerically. The limit of small layer thickness (or slow scan rate) led them to the ISB approximation, and they also checked the limit of large thickness (I) against the results of Randles and Sevcik. Therefore, they solved the ISB-I, B-I, and 1-1 cases where, in denoting the case type, the reactant layer is indicated first. Roe and Toni (11) solved the ISB-ISNB case for stripping from a mercury film into a stirred solution. The ISB-ISB case was solved by Hubbard and Anson (12) in connection with their thin layer cells. In both the latter instances, the solution was readily obtained from Fick’s first law and Faraday’s law. In several of the foregoing cases, workers have verified the theoretical treatment by comparison with experiment. This verification provides a base from which it is possible to justify extension of the theory to include experimental situations which have not been tested in practice. Theoretical treatments generally are reported in one of two ways-for simple cases in closed form, for more complicated cases in the form of numerical solutions to the corresponding integro-differential equations. Mejman (13) [see also Smutek (1411 was probably the first to express current-potential relationships in infinite series form. Reinmuth (15, 16) discussed general methods of obtaining solutions for PSV in the form of an exponential series while Buck (17) showed how another series might be utilized. Nicholson and Shain (18) in their treatment of kinetic and catalytic systems gave series for each of the cases they considered. None of these approaches yielded a generally applicable series solution. Moreover, even in the cases they purported to describe, the series were often divergent, and no practical evaluations in those cases were proposed. This paper presents a new method of obtaining exponential series for PSV. The technique used is an extension of Reinmuth’s work (16) but has the capability of much greater generality. In particular, the method allows systems involving finite diffusion, quasi-reversibility, and linear complications (e.g., first order kinetic or catalytic reactions, linear adsorption isotherms) to be treated. A general equation describing a recursion relationship for computing coefficients in the exponential series is derived. The characteristics of specific systems appear in this equation as functions of the summation -.
(8) W. T. DeVries and E. Van Dalen, J . Electround. Chem., 8, (1) . , L. A. Matheson and N. Nichols. Trans. Elecrrochem. SOC..73. 193 (1938). (2) J. E. B. Randles, Trans. Faraday SOC.,44, 327 (1948). (3) A. Sevcik, Coli. Czech. Chem. Commun., 13, 349 (1948). (4) H. E. Keller, Ph.D. thesis, Columbia University, New York, N.Y., 1968. (5) P. Delahay, “New Instrumental Methods in Electrochemistry,” Interscience, New York, N.Y., 1954. (6) R. C . Bowers and A. M. Wilson, J. Amer. Ckem. SOC.,80, 2968 (1958). (7) Zbid., 81, 1840 (1959).
434
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
366 (1964). (9) W. T. DeVries, ibid., 9,448 (1965). (10) W. T. DeVries and E. Van Dalen, ibid., 14,315 (1967). 37, 1503 (1965). (11) D. Roe and J. Toni, ANAL.CHEM., (12) A. T. Hubbard and F. C. Anson, ibid., 38, 58 (1966). (13) N. Mejrnan, 2. Fiz. Chim., 22, 1454 (1948). (14) M. Smutek, Coll. Czech. Chem. Commun., 20, 247 (1955). 33,1793 (1961). (15) W. H. Reinmuth, ANAL.CHEM., (16) Zbid., 34, 1446 (1962). (17) R. P. Buck, ibid., 36,947 (1964). (18) R. Nicholson and I. Shain, ibid., p 706.
variable. These functions are separately derived from equations describing the non-electrochemical features of the system. In general two such functions must be obtained, one for the reactant and one for the product. For systems subject only to linear diffusion there are five functions corresponding to the five boundary types (I, B, NB, ISB, ISNB) and therefore twenty-five combinations. For nine of these combinations (pairs of I, ISB, ISNB), simple functions are derived. These functions can frequently be applied to considerably more complex situations. There are two problems inherent in any general theoretical approach to PSV. The first is that the complexity of the theory itself makes representation of current-potential relationships difficult. Second, in applying theory to experiment, a large number of parameters must be considered. There are generally so many of these parameters that several simplifying assumptions must be made. The theory presented here keeps simplifying assumptions to a minimum and provides a few readily characterized functions applicable to a large variety of cases. However, the method still has two significant limitations. It assumes the initial potential to be far from equilibrium so that the product concentration is virutally zero, and the results are exponential series. THEORY
All of the systems dealt with in this paper are assumed to obey the absolute rate theory expression relating current and potential with surface concentration. i/nFAk,
=
(Coo - ACo) exp (CRO
1
+ ACR)exp
See Appendix I for definition of the symbols. The relationship between concentration and current is taken to be in a form related to that derived in Appendix 11.
AC = i.
sb
i (t
- T)f(T)dT
= i;(S)T(s)
where s is the transform variable. The function, J'(s), is familiar in engineering as the transfer function relating i(t) and AC(r). The equation for potential in a linear scan experiment is
E
=
E; - Ut
(3)
In the following discussion u is always assumed positive; that is only cathodic scans are considered as conversion to negative v is trivial. The experimental system is further assumed to be initially at electrochemical equilibrium, a condition represented by COO/CRO= exp
[z -
Zcjz"j
Li
~ o ( T )exp
(-a,nFL;r/RT)dT -
Coo exp (-nFut/RT)
-
Zcjzaj-l l t f ~ ( rexp ) (-ajnFur/RT)dT
(7)
The next step is to take the limit as t tends toward infinity while holding z constant. If a j is not negative, the result of this operation puts the integrals into the form of Laplace transforms (with argument ainFo/RT). From Equation 6 it can be seen that the above limiting process is equivalent to taking the limit as E?tends toward infinity while holding z constant. Of course this cannot be done experimentally, but for practical purposes it is only necessary that Et be anodic enough that its value has no effect on the experiment. There are experimental situations where this criterion cannot be satisfied, and the theory presented here does not apply. Normally, however, this condition can be fulfilled by initial potentials only a few hundred millivolts anodic of the half-peak or halfwave potential. Exact values are best determined by experiment. Initial potentials which influence the final result are to be avoided because their effect is unlikely to be of any experimental interest and they add an unwanted source of complexity. The result of the limiting process is
(2)
This may be recognized as a convolution integral which, when Laplace transformed, gives the form found in the appendices. Z(S)
Equation 5 is a completely general way of representing the relationship between i and z. Once aj is defined, as it is below, the generality of the expression is reduced. Utility is lost as well when the coefficients, cj, produce a series which is quite difficult to sum and therefore cannot be evaluated completely. This limitation is a theoretical one and could be removed by a good summation method. Substitution of Equations 2 through 6 in Equation 1 yields Equation 7.
1
(E; - E o ' )
(4)
Z C , ~ ~ - ~ ~ R ( ~ , ~(8)F ~ / R T ) The? functions are relatively easy to calculate even for complex processes, while their inverse transforms are ordinarily extremely difficult to obtain. One of the more propitious results of this approach is that the inverse transforms need not be known. The coefficients, cj, are calculated by recursion and substituted into Equation 5 to yield an expression for the current. In order to obtain the coefficients the exponents, a,, must be defined. The simplest definition that fits Equation 8 is one of the form aj
=j
+ k(l
- a);
j , k integers such t h a t j
+ k(l
- a) 3 0
(9)
With the substitution, the series for the current, Equation 5, becomes m
m
We represent the current as a series of the form i
=
Z cjzaj
(5)
where z is defined by the following expression z = exp [-,(vi nF RT
+ E"'
- E;)]
a double sum. The series in Equation 10 is not necessarily or even usually convergent, and no general methods exist for summing divergent double sums. Fortunately the series can be simplified to a single sum by several expedients. A reversible system is one in which the sum on the left of Equation 8 ANALYTICAL CHEMISTRY, VOL. 44, NO, 3, MARCH 1972
435
can be neglected, and an irreversible system is one in which the second sum on the right can be neglected. Both of these simplifications yield a single series; in the former case aj = j and in the latter u j = aj. A more sophisticated method of arriving at a single sum is available when a is a rational number, say m/n. This assumption places no restrictions on the values that a may have since a n irrational number may be approximated to give any degree of accuracy by a rational number. In fact, the usual representation of experimental values of a is a decimal fraction which is always a rational number. With this assumption a valid representation of a, is simply aj = jjn. These quasi-reversible series have several peculiarities which limit their usefulness in computation. First, they are usually divergent. Second, the magnitudes of the coefficients do not vary uniformly but rather in a roughly cyclical fashion with period n. Several periods of the cycle are required in the summing process. Third, when ks d R T / n F v D is even slightly less than unity, the coefficients diverge extremely rapidly. The net result is that when the process is not “close to reversible,” the summing process requires a precision beyond the capacity of the computer. RESULTS
The equations defining f for uncomplicated diffusion are derived in Appendix 11. For a plane electrode ,-
where hypf signifies coth for a bounded system and tanh for a nonbounded system, li is the layer thickness, and the other terms are defined in Appendix I. Of particular interest are the limiting cases where 1 + 0 and I + rn . There are three such limiting cases because the latter limit yields the same expression for nonbounded and bounded systems : 1. Infinitesimal nonbounded (ISNB), f ( s ) = I/nFAD. 2. Semi-infinite ( ~ ) , j ( s )= l / n F A d s D . 3. Infinitesimal bounded (ISB), f ( s ) = l/nFAls. In these cases f can be written as 70 = a ~ s - * / and ~ f~ = ~ ~ s where - 9 ~ a0 ~and a R are appropriate constants. The variables p and q may have values of 0, 1, or 2. Substituting these expressions in Equation 8 and solving for cy in the reversible charge-transfer case gives a series describing the relationship between i and E. This relationship is equivalent to a function which we shall denote as L,,,, the “linear scan function,” and define as
Table I. Linear Scan Functions
Lo.0 = x / ( l
+x) m
L0,l
dJ? (-X)j
= -
LOA
= 1
- (l/x)exp:(l/x)El(l/x) m
LLO=
-
(-x)j/d(j j=1
=
-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
Absolutely convergent Conditionally convergent
dJ’(-X)j j-1 m
Divergent G,o= xe-* m b.1 =
-
d j / ( j - I ) ! (-x)j
Absolutely convergent
j=1
maticians (19). All of the series encountered here fall into the class of “summable series,” and can be summed to meaningful limits by appropriate methods. One such method due to Euler has already been discussed in the chemical literature (13). Most of the divergent series developed here cannot be summed by Euler’s method, and a more powerful method is desired. Although many methods are available, those which are classified as linear, such as Euler’s, are severely limited because they involve summing large numbers to obtain small results. For extreme divergence they do not work at all. Some nonlinear methods are also available and were used here with great success. The Z i transformation of Shanks (20) which we used is readily applicable to computer calculation and gives excellent results for the most strongly divergent series encountered. Although much is said about summability and methods of summation, little attention has been paid to the practical use of divergent series. The propriety of using them to calculate results is not immediately obvious. A heuristic justification for the procedure is that if a physically meaningful model is used in calculations, the results of the calculations must have significance. However, much more convincing are the results themselves. Several divergent series were checked against their closed form equivalents before the method was applied. One series was particularly convincing. It is m
(- l ) j f l j ! x ’
j=l
436
- l)!
m
d . 1
m
If the? functions for a system are of the form as - P I 2 @ = 0, 1, 2), then its current potential relationship can be described in terms of an L,,* function. For example, when reversible charge transfer is replaced by irreversible charge-transfer, the f R function is a constant, Le., q = 0, and an L,,o function results whenfo is in one of the above special forms. There are, in fact, many other systems that are described by the linear scan functions. Because of their ubiquitousness, it is important to characterize the L , , , functions as thoroughly as possible. The series form of the function tends to discourage numerical evaluaticn because its value is of interest for cases in which it is strongly divergent, Fortunately a theory of divergent series and their summability has been developed by mathe-
Divergent
j=l
j=1
the series expansion for Lo,2. The closed form solution is 1
where El(x) =
- (11‘4 exp U/.+E1(1/.4
Lm‘Ti
-
dt (21). Since both the series and the
closed form expression arise from the same equations, they should be equivalent. Results calculated from the series by (19) G . H. Hardy, “Divergent Series,” Oxford University Press, London, 1949. (20) D. Shanks, J . Math. Phys., 34, l(1955). (21) “Handbook of Mathematical Functions,” M. Abramowitz and I. A. Stegun, Ed., US. Government Printing Office, Washington, D.C., 1963.
Table 11. Characteristics of the Linear Scan Functions L0,o L0,l Lo.2 L o L1.1 L1.2 b . 0 b , 1 L.2 Max = 1.oooO 1 .oooo 1 .m 0.4958 0.4463 0.4157" 0.3679 0.2970 0.25oO at In x = ... ... ... 0.780 1.102 1.413 0.007 -0.055 O.Oo0 112 Max = 0.5OOO 0.5000 0.5000 0.2479 0.2231 0.2078 0.1839 0.1485 0.1250 atlnx = 0.000 0.239 0.494 -1.077 -1.094 -1.044 -1.462 -1.642 -1.763 Maximum calculated from the best fit of a linear combination of functions of the form aez b , ax/@ x ) ~ ,ax/@ ~, x)~. +
+
+
Table 111. Inverse Linear Scan Functions X
0.0
0.05 0.1 0.15 0.2
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1.0
L1,l * W
-2.603 -2.279 -2.017 -1.794 -1.597 -1.418 -1.251 -1.093 -0.942 -0.794 -0.646 -0.498 -0.345 -0.183 -0.006 +O. 197 +O ,458 +1.102
L1,o i
m
-2.520 -2.203 -1.950 -1.735 -1.548 -1.379 -1.223 -1.077 -0.938 -0.803 -0.671 -0.539 -0.405 -0.265 -0.115 $0.055 +O .268 +O. 780
Ll,Z * W
-2.625 -2.293 -2.023 -1.791 -1.583 -1.393 -1.215 -1.044 -0.878 -0.713 -0.548 -0.380 -0.205 -0.018 $0.187 +O .427 +O. 739 fl ,378
In [Lp,p-l(x)I L0,l L0.0 -m
-m
-2.155 -1.671 -1.300 -0.990 -0.715 -0.462 -0.223 -0.009 +O. 239 +O ,470 +O. 707 +O. 956 +1.224 +1.520 +1.858 +2.268 +2.770
O1 transformation were accurate to five decimal places, the precision to which the computations were made. The nine linear scan functions listed in Table I are shown in Figure 1. A tabulation of the peak heights, peak, and halfpeak positions is given in Table 11. It is evident from Figure 1 that the functions fall into three distinct categories depending on the value of p . Whenp is zero, L,,, is always a sigmoid curve which rises to a plateau where LO,, = 1. The inflection point is near In x = 0 in each case. The Lz,, curves nearly have the form of the derivatives of the LO,, curves. They are all peak shaped with the peaks near In x = 0. The LI,, curves lie in between the two extremes since they are seen to be peak shaped with a less steep downward slope than the Lz,, curves. Thus it is possible, knowing that a curve represents an L,,, function, to determine the value ofp visually. While the value of p determines the overall curve shape, the value of q also has an influence, although smaller. When q is zero, the curve has a higher peak or inflection and tends toward its limiting value more rapidly. A q value of two produces a poorly defined peak or inflection. The nine curves all merge at small x since the first term in every L, , series is the same. Generally the linear scan function appears in the currentpotential relationship in the form
where A, B, and C are constants determined by experimental parameters, i is the current, and E is potential. While A can be determined experimentally with relative ease by measurement of peak or plateau height and B can be found from halfpeak or half-wave potential, determination of C is not generally straightforward. It is related to peak width. In addition, situations can arise in which it is not apparent that a current-potential curve is represented by an L,,,function.
-2.197 -1.735 -1.386 -1,099 -0.847 -0.619 -0.406 -0.201 O.Oo0 +o. 201 +O. 406 +0.619 +O. 847 +1.099 +1.386 +1.735 +2.197
Lo4 -03
-2.895 -2.101 -1.591 -1.195 -0.860 -0.561 -0.283 -0.019 +0.239 +0.494 +0.753 $1.020 +1.300 +1.601 +1.933 $2.324 +2.778
L.1
L,O
L ,1
m
f m
f
f m
-2.839 -2.530 -2.284 -2.079 -1.900 -1.741 -1.596 -1.461 -1.334 -1.214 -1.097 -0.982 -0.868 -0.751 -0.630 -0.497 -0.339 +0.007
-2.732 -2.484 -2.275 -2.093 -1.930 -1.780 -1.641 -1.510 -1.384 -1.262 -1.141 -1.019 -0.895 -0.764 -0.619 -0.443 -0.055
-2.887 -2.634 -2.420 -2.232 -2.063 -1.908 -1.763 -1.624 -1.491 -1.360 -1.230 -1.099 -0.962 -0,817 -0.655 -0.455 0.Oo0
In (a)
Figure 1. L,.,(x) us. ln(x) Under these circumstances, a method of determining which, if any, of the L,,, functions applies would be desirable. If Equation 14 is rewritten as B CE = ln(L,,,-~(i/ip)), where i, is the peak or plateau current, and if the function ln{L,,g-l(x)) is known, then C can be computed from the slope of a straight line plot of E us. In{L,,,-l(i/i,)]. In order to allow such a plot to be made In(Lp,q-l(x)] is tabulated in Table 111. In addition to providing a method of computing C (and B), this graph gives a method of deciding which L,,, function (if any) applies to a given experimental situation. The criterion is simply that In { L,,Q-l(i/i,)] us. E be a straight line. If a straight line is not obtained with one of the func-
+
ANALYTICAL CHEMISTRY, VOL. 44,NO. 3,MARCH 1972
437
I
I
I
I
I
I
I
I
po.1
I
I
-3
-2
-I
I
I
0
I
In [LJ~)]
Figure 2. Inverse L,,, functions US. L1.l-l tions, the behavior of the curved line helps predict the next function to attempt. Figure 2 demonstrates this fact with L1,las the initial guess and each of the five In { L p , p - l ( x)func) tions in which p or q equals one plotted against In { L1, l - l ( x ) } . It can be seen that the function expected to represent some instances of post-kinetics, i.e., Ll,0, curves in one direction while in the other case L O ,(representing I pre-kinetics) curves in the opposite direction. Figure 3 shows the same effect with ISB-ISB boundaries. Thus we have evolved a simple criterion for detection and discrimination of kinetics. Other complications resulting in L,,, functions may be dealt with similarly. Discussion of an Application to Simple Diffusion. The treatment in Appendix I1 for planar electrodes corresponds to what we call simple diffusion. In the preceding text, the limiting cases of large and small layer thickness led to the description of simple diffusion systems in terms of L,,, functions. It is our purpose here to show the relationship between the characteristics of the functions and the physical models from which they were derived as well as the dependence of the theoretical expression for current on system parameters. The linear scan function with p = 0 corresponds to a simple diffusion system in which the reactant layer is ISNB. The characteristic feature of the ISNB layer is that the flux of electroactive material, and therefore the current, is proportional to the difference between surface and bulk concentrations of the material. At the beginning of a linear scan experiment, this difference is zero, while at the end the surface concentration is essentially zero. So the current begins at zero and ends at a plateau maximum. The other significant features come from the combination of the a0 and U R constants of the? functionswith their arguments, s = (nFu/RT)ajwhere aj is the exponential term in the current series and is equal to j in reversible systems. In the ISNB case?&) is equal to lo/nFADo. The independence off0 from s reflects the fact that the current in this system does not change with time for any fixed surface and bulk concentrations. An analysis of the relationship between the functions and the current-potential expression derived from them shows that
7
438
e
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
Figure 3. Inverse L,,, functions us. L2,p-1 whenthe maximum current is inversely proportional to ever the system is describable by L,,, functions. The proportionality of the maximum current to D/l in the ISNB case is a result of the nature of the diffusion layer. As 1 becomes smaller, the gradient and hence the current becomes larger. An increase in D produces an increase in the diffusion rate. When p = 2, the L D , gfunction describes the systems in which the reactant layer is ISB. The concentration of material is uniform throughout an ISB layer. Here the rate of change of concentration is the important parameter. The current is proportional to this variable. In a linear scan experiment, a plot of surface concentration us. time usually has a sigmoid shape. The time derivative of the concentration gives the characteristic peak shape of the L 2 , ,functions. Since s = (nFu/RT)j for reversible charge transfer systems, is equal to RT/nZF2Alovin the ISB systems, and hence the maximum current is proportional to lu and independent of the diffusion coefficient. Thus there is no diffusion control. Instead the flux of reacting material is proportional to ldCo/dt, a fact determined by Faraday’s law. Because the surface concentration is Nernst controlled, its rate of change, bCo/dt, is proportional to the scan rate, u. This explains the lu dependence of the maximum current. The p = 1 case, corresponding to semi-infinite diffusion, lies between the other____two cases in mathematical form. Now (TO>,=,= ljnFA d R T l n F D o u which is the geometric mean of the two preceding cases. The maximum current is proportional to indicating the combination of rate of concentration change control before the peak and diffusion control after the peak which is well known to occur in semi-infinite linear diffusion. In all of the reversible charge-transfer systems just discussed ln[~o(s)~a(s)l,,lappears as a term added to the potential in the argument of the linear scan function.
dz
Table IV. Diffusion Limited Processes at Planar, Cylindrical, and Spherical Electrodes where
Electrode and boundary 5
Plane
B
coth (1d@)
a
Plane
NB
tanh (1dsp)
= cAO/sF $T/nFA
dz
Cylinder B Cylinder NB 5
coth ( I dsp)- (l/xl) d/o/s 1 - (l/xoxl) [ D / s - I d q s coth (Ida)]
Sphere B Sphere NB
a
1 = x1
1
+
tanh (I (l/xo)
d/slD)
m s tanh(Id@)
- xo.
current ratio of xl/xo where xo is the radius of the sphere and
x1 - xois the layer thickness. Therefore the effect of experimental parameters on the position of the current-potential curve in these systems is of the form ln[~o(s)/~~(s)]j,l = -
nF
2
ln(DRT/nFu12)
(16)
where equal layer thickness and diffusion coefficients have been assumed. The 1-1, ISB-ISB, and ISNB-ISNB systems are those wherep = q. No potential shifts occur on parameter variations in these systems if lo = la and DO = DR. In other (nonsymmetrical) systems, this is not usually true. Instead, potential shifts with the logarithm of the parameters. When charge transfer is irreversiblefR is replaced by l/nFAk,. Thus the current-potential relation reduces to an L , ,a function when reactant mass transport leads to an I, ISB, or ISNB form. Formally then, the irreversible system is equivalent to one in which the product is removed by an ISNB process where D/l of the ISNB system is equivalent to k, of the irreversible one. The correspondence between the two situations is not exact because the argument of the L,,, function in the reversible ISNB case is exp(nF/RT(E”’ - E‘)) while in the irreversible case it is exp(anflRT(Eo‘ - E ) ) . Note that ambiguities could arise if QI is close to unity. Effect of Nonplanar Electrodes. Two types of nonplanarity have been treated in Appendix 11, cylindrical and spherical. The? functions for these cases are shown in Table IV. Their complexity makes characterization of current-potential relations extremely difficult. However, information regarding the range of applicability of planar solutions to nonplanar electrodes would be of great utility since the planar solutions are already known and take on substantially simpler mathematical forms. A current ratio iPLANARIiNONPLANAR, or a n additive term, i P L A N A R - iNONPLANAR, are ways of expressing the change in current due to nonplanarity. These quantities are derived here for a few systems whose current-potential relationships are L, functions, specifically when neither or both boundaries are semi-infinite. With infinitestimal layers, the difference between planar and nonplanar current should be small since the radius of curvature is much greater than layer thickness in usual experimental situations. Straightforward application of the limit (xl - xo)V ’ i i XO. The corrections in all cases of infinitesimal layers vanish as l/xo + 0. The evaluation of the functions in the semi-infinite case gives the following correction terms for
7
7:
For spherical reversible 1-1systems, the current-potential relation can be represented in the form of the sum of the current for a planar electrode and a spherical correction of simple mathematical form (22). This representation can be ob(22) W. H. Reinmuth, J . Amer. Chem. SOC.,79, 6358 (1957). ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
439
0.5,
tained as follows. First, the recursion relation for the coefficients is written.
0.4
I
I
I
I
I
I
-
0.3 -
J
where 6 = (11x0) d D R T l n F u . These coefficients are then placed in the series for i. Careful manipulation yields
{
nF where z = exp - (E"' - E ) } . In irreversible semi-infinite RT systems, the same procedure does not lead to separation of the spherical term (23). The current for a cylindrical electrode in a reversible 1-1 system can be obtained by the same procedure if diffusion coefficients are assumed to be equal.
.-
P
Figure 4. Effect of finite layer thickness. Comparison of bounded and semi-infinitelayers IJ
=
where c is xodnFulDRT. The ratio of modified Hankel functions can be expanded in an asymptotic series for large arguments.
-
m
akxPk k=O
where ao = 1, UI = 1/2, u2 = I/*; see Reference 18, pp 377378. With this substitution Equation 19 becomes, after rearrangement,
2
k=O
akE-k
2
(-l)jj(l-k)/zzj
(22)
j=l
The first term (k = 0) in the expansion is simply I'PLANAR. The second term is one half of the second term for the spherical electrode current. If the third term can be neglected, then the cylindrical correction can be calculated from the wellknown spherical correction given by Equation 20. However, the third term is not necessarily negligible even with an elecm
trode of large radius. It is proportional to E-*
s,'
(L1,1([)/[)d[, an unbounded, monotonically
increasing function of z. Therefore for a given nonzero value of e , a finite maximum value of z exists above which the third term cannot be neglected. Looking at the functions of z appearing in series form in Equation 20, uiz., z/(l
+
+ Z ) ~ , L ~ , Iz/(l ( Z )+ , z), sd [L1,l(D/[ld[
ln(1 z), etc., it is clear for all but the integral that they have about the same magnitude up to z = 1 where they begin to _______
(23) R. D. DeMars and I. Shain, ANAL.CHEM., 81, 2654 (1959). 440
J =
i nFv _ -1 n F A D I W o " RT
sb
[L~,l([)/[ld[ has the same behavior when z is
+
+
less than one and, in fact, lies between z/(l z ) and ln(1 2). This assertion has been checked numerically and found to be correct. Therefore a cylindrical correction term one-half of the corresponding spherical term may be used on the rising portion of a reversible 1-1 curve. Note that again in the irreversible case, the planar and cylindrical contribution cannot be separated by this method. When there is dissolved reactant or product within a spherical or cylindrical electrode, the system corresponds to a situation in which x1 - x o = - r , where r is the radius of the electrode. Two examples are anodic stripping from a mercury drop electrode and electrolysis inside a hollow cylinder electrode. These "electrode contained" systems have the following? functions which are derived by setting x1 = 0 in Table IV. /SPHERICAL
f PLANAR
=
coth(xods/D) -
xo
4%
(23)
I-
fCYLINDRICAL
= ZO(XOV S/ D fPLANAR )-
h(xo d s l D )
(24)
(- l)jj-1/2z5 j=l
which is
E)
diverge rapidly from one another. Thus it is reasonable to assume that
K(x)/Ko(x)
nF RT
-(EO'-
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
Only the bounded cases are given above because the condition x1 = 0 in a nonbounded system would require a point or line
source (or sink) of material at the system center. When xois large, the,? ratios above approximate the inverse of the corresponding semi-infinite ratios. Therefore the spherical correction given in Equation 20 may be used by merely changing the sign of the second term when both the reactant and product are soluble in the electrode. The simplification of large x ois of great value because it is applicable to cylindrical and spherical electrodes of the dimensions normally employed under usual experimental conditions. When x o is small, the behavior of these functions is quite different. This is expressed by the following equations:
APPENDIX I
25
Symbol definitions : = faradaic current = electrode area = initial concentration
i
20
A
of reacting species (oxidized form) C Ro = initial concentration of product species (reduced form) ACR, ACo = difference between initial and instantaneous concentrations = number of electrons transferred per molecule in the electrode process = Faraday = charge-transfer coefficient = gas constant absolute temperature potential formal standard potential of the electrode couple formal rate constant of the heterogeneous charge-transfer reaction at Eo' c oO
15
J IO
05
0 -3
-2
-I
0
2
I
3
P Figure 5. Effect of finite layer thickness. Compariof bounded and infinitesimal bounded layers nF /.L=-(E"'-E) RT -
SPHERICAL
-
CYLINDRICAL
J = -
i
ttFAICo"
nFu RT
time scan rate initial potential (prior to initiation of scan) diffusion coefficient distance normal to the electrode surface Laplace transform variable see Table IV transfer function (see Equation 2a) exponential term in series summed over j (see Equation 5) x1 - x o = layer thickness linear scan functions (see Table I)
30 -~ nFAxO3s2 1 nFAxos
zz _ _
(26)
The cylindricalffunction is of the same form as an ISBTfunction and is inversely proportional to volume. The spherical function shows very unusual behavior. It is inversely proportional to the five thirds power of volume and is not in the same form as previous functions. A complete analysis of the corresponding current functions would involve considerable effort but would be of practical value in only exceptional circumstances. Applicability of the Infinite and Infinitesimal Approximations. Calculations of current-potential curves were made for bounded and nonbounded layers of various thicknesses and the results compared with the simpler approximations of semi-infinite and infinitesimal layers. Curves for the bounded case are shown in Figures 4 and 5. Analysis of the results shows the 0.1 % accuracy point to be 1 2 4.5 d D R T / n F u for infinite diffusion and 1 5 0.2 d D R T / n F u for infinitesimal diffusion. The 5 % points are at 3.0dDRTlnFo and 1.0 X d D R T / n F u , respectively. The effect of scan rate, 0, is particularly to be noted. Increasing scan rate favors the semi-infinite approximation but hurts the infinitesimal approximation. An estimate of the magnitudes involved can be obtained from the size of d D R T / n F u .____ If n = 1, D = cm2sec-', and u = 1 volt sec-', then d D R T / n F u = 2.5 X cm at 25 "C. This is the same order as the thinnest thin layer cells which have been used experimentally. Scan rates two orders of magnitude lower or four orders higher than this typical value are feasible. Note that at the 5 % significance level, a threefold change in layer thickness or a ninefold change in scan rate changes the apparent type from infinitesimal to semi-infinite. At the 0.1% level, the factors are twenty-two and five hundred. These figures indicate that, depending on the precision desired, it is almost always possible to arrange experimental conditions so that one of the limiting simplifications could be employed.
7
7
APPENDIX I1
Simple Diffusion. The system is assumed to obey Fick's second law with constant diffusion coefficients.
where CA is homogeneous concentration, t is time, and x is distance normal to the electrode surface. In nonplanar systems x is zero at the center of the system. The parameter p is the electrode geometry parameter and is zero for planar, one for cylindrical, and two for spherical electrodes. The above equation must be solved with the following boundary conditions : t =
0 , xo 5 x
t
> 0, x
= xo:
t
> 0, x
= xl:
5
x1:
(Bounded system)
bCA ~
-0
dX
(Nonbounded system) CA = CA'
(4a) (4b)
where C A ois the initial bulk concentration. The point x = x o is assumed to be at the electrode surface. In planar systems the zero of the coordinate can be arbitrarily assigned so that those chosen provide the simplest representation. We choose x o equal to zero. In all of the three systems the point x = x1 corresponds to the boundary surface at a uniform disANALYTICAL CHEMISTRY, VOL. 44,
NO. 3, MARCH 1972
441
tance 1 = XI - xofrom the electrode surface. Cathodic currents are defined as being positive, while it is assumed that x1 2 xo so that the plus in Equation 3 refers to the reactant. If x1 < XO,the same derivation applies by changing I to - 1 and =tito Ti. There are two other boundary conditions which give the same results as Equations 4a and 4b for planar systems ( p = 0). They are
The solutions to this equation are of the form (x C D ) ( ’-p)/2Z (l--p)/ZGX
dslD)
where Z(1- - p ) pis a cylinder function of order (1 - p)/2, and i = d-1. General solutions to Equation 5 can be written in terms of appropriate representations of these cylinder functions. p =
-
CA‘
0: CA = S
+
c1
sinh [(x -
XO)
ds/D]
cz cosh [(x
which is equivalent to Equation 4a,and p = 1:
p =
2:
-
CA’ c1 cA= + - sinh [(x - XO) S X C2
- cosh [(x X
+
- XO) dzl
(7)
dz) dz]+ dz]
FA = CaO + Cilo(X dG) + C&dX S
which is equivalent to Equation 4b. If the distance x1 - x oin Equations 4c and 4d is taken as twice that in Equations 4a and 4b, the resulting equations are identical. Formal proof of this assertion is lengthy though uncomplicated and will not be given here. Intuitive justification is somewhat simpler. It should be apparent that if material is removed at x = 0 and added or removed at the same rate at x = 21, there is a point of symmetry at x = I. In the former case (boundary condition 4c), the symmetry is odd and results in constant concentration at x = I (boundary condition 4a). In the latter case (4d), the symmetry is even and results in no net flow across the point x -- 1 (4b). Boundary condition 4c would apply to the physical situation in which the cell consists of two parallel facing electrodes separated by a distance 21 and the electrode reaction is the same at the two electrodes but in the opposite direction. Boundary condition 4d would apply to the same physical arrangement in which the reaction proceeds in the same direction at the two electrodes. The Equations 1 through 4 can be solved readily by Laplace transform techniques. Taking the transform of Equation 1 yields a simple second order differential equation in x that may be recognized as the differential equation for cylinder functions of order (1 - p)/2.
(6)
XO)
(8)
(9)
where Ioand KOare modified Bessel and Hankel functions respectively (24), and c1 and cp are integration constants (different for each case) which can be evaluated by reference to the boundary conditions, Equations 3 and 4. The results of this evaluation for concentration at x = 0 are shown in Table IV. ACKNOWLEDGMENT
We are indebted to the Columbia University Computer Center and the Institute for Space Studies for the generous use of their facilities. Programming assistance provided by Jayne Keller in some parts of this work greatly aided the completion of this paper.
RECEIVED for review April 30,1971. Accepted November 11, 1971. The authors acknowledge the financial support of the Sloan Foundation and the National Science Foundation. (24) G. Korn and T. Korn, “Mathematical Handbook for Scientists and Engineers,” McGraw-Hill, New York, N.Y., 1961, pp 727728.
442
ANALYTICAL CHEMISTRY, VOL. 44, NO. 3, MARCH 1972
