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Simplifications in the theory of step experiments at microelectrodes
91
J. Electroanal. Chem., 235 (1987) 97-106 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
SIMPLIFICATIONS MICROELECTRODES
IN THE THEORY
OF STEP
EXPERIMENTS
AT
D.K. COPE Divwon
of MathematicalScrences,
D.E. TALLMAN Department
North Dakota State University,
Fargo, ND 5810.5 (U.S.A.)
l
of Chemistry,
North Dakota State Universrty, Fargo, ND 58105 (U.S.A.)
(Received 19th December 1986)
ABSTRACT Theoretical studies of the chronoamperometry of the reaction 0+ n e- + R often consider (1) the diffusion-limited reaction, (2) the totally irreversible reaction, and (3) the reversible reaction (governed by the Nemst equation), and these can all be considered as special limiting cases of (4) quasi-reversible reactions, electron transfer kinetics governed by the genera1 current-potential characteristic. Cases (1) and (2) simplify to one-component diffusion systems for the concentration of 0; cases (3) and (4) are necessarily two-component diffusion systems involving the concentrations of both 0 and R. We show that. when 0 and R have equal diffusion coefficients, the solution for the reversible reaction (3) can be expressed in terms of the solution for the diffusion-limited reaction (1) and that the solution for the quasi-reversible reaction (4) can be expressed in terms of the solution for the totally irreversible reaction (2). These results hold for arbitrary electrode geometries.
INTRODUCTION
Microelectrodes, which have at least one dimension comparable to the diffusion layer thickness (typically of the order of a few micrometers or smaller), continue to receive much attention in the electrochemical literature [l-12]. This interest is due in part to the enhanced mass transfer at microelectrodes, a result of convergent (or non-linear) diffusion, commonly referred to as the edge effect for planar electrodes [13-H]. For time scales typically used in voltammetry, convergent diffusion at microelectrodes often leads to time independent (steady state) current [16-181 or nearly time independent (quasi steady state) current [11,19,20] and also to higher current densities [9] than generally observed at conventional sized electrodes (having
l
To whom correspondence
0022-0728/87/$03.50
should be addressed.
0 1987 Elsevier Sequoia S.A.
98
dimensions of the order of a millimeter). These properties of microelectrodes make them advantageous for use in analytical detection as well as for the study of heterogeneous electron transfer kinetics [1,9]. The responses of microelectrodes of various geometries to a potential step have been observed experimentally [10,11,20,21] and considered theoretically [11,16-20,221. Even for the simplest case of diffusion-limited current the theoretical treatment of the problem is rather complex due to mixed boundary conditions in the mathematical formulation [23]. Research in our laboratory involving the extension of our recently developed integral equation method [19] to more general electrode boundary conditions has led us to observe simplifications which do not appear to have been previously reported in the electrochemical literature. The main simplifications require only that the oxidized and reduced species have equal diffusion coefficients and are applicable to experiments involving a potential step or sequence of potential steps at an electrode consisting of one or many active regions of arbitrary geometry in a bounded or unbounded cell of arbitrary geometry. We show (1) that the solution for reversible electron transfer can be expressed in terms of the solution for the diffusion-limited case and (2) that the solution for the quasi-reversible case can be expressed in terms of the solution for the totally irreversible case. The approach is illustrated for planar, spherical, and cylindrical electrode geometries. We also show that the solution for R initially present can be expressed in terms of the solution for R initially absent; this result holds for all diffusion coefficients and arbitrary cell and electrode geometries. THEORETICAL
FORMULATION
AND
SOLUTION
This section gives the physical description dimensionless variables, and the results. We consider simple electron transfer O+ne-+R
of the problem,
a reformulation
in
(I)
in a stationary electrolyte with the electrolyzable species 0 and R both soluble. Ion migration is neglected and the diffusion coefficients are assumed independent of concentration. A potential step or sequence of potential steps is applied and the resulting current measured. Physical quantities involved are time T, spatial variables X, Y, Z (Cartesian coordinates), concentrations c,(T, X, Y, Z) and ca( T, X, Y, Z), current I(T), diffusion coefficients Do and D,, and rate constants kf and k,, described by k,=k”
exp(-a(E-E”‘)(nF/RT))
k, = k” exp((1 - a)(E--E”‘)(nF/RT))
(2)
using standard notation. The boundary of the electrochemical cell consists of insulated portions, where no electrical charge passes, and the electrode(s), where the electrochemical reaction takes place. No other assumptions on cell geometry are made; in particular, the cell
99
may be bounded or unbounded, the electrode region may consist of one or many separate regions, and these regions may have arbitrary shape. A unified discussion applying equally to bounded and unbounded cells requires a precise definition of bulk concentration. In an unbounded cell, the bulk concentrations cg and cg are the concentrations infinitely far from the electrode. In a bounded cell, we define cz and cg as the average concentrations over the cell just before the (new) potential step is applied. If an experiment involves a sequence of potential steps applied at times 0 = T, < T, < T, -=c. . . , then we obtain a sequence of diffusion problems for time intervals T k_l G T < Tk_ These problems are identical except for two factors: (a) the initial concentrations of 0 and R change from interval to interval, since the terminal concentrations of one interval are the initial concentrations of the next; (b) a new pair of rate constants k, and k, occurs on each new time interval, a result of the change in potential. We write the initial concentrations in the following way. If the cell is unbounded, the initial concentrations will be written as co=c;;-F,(X,
Y, z),
cacI:+&(X,
Y, Z)
with the condition F,, FR + 0 at large distances from the electrode. That is, F, and FR represent some perturbation around uniform concentration. Concentrations are normally uniform at the application of the very first potential step and F, and FR are both identically zero. At succeeding potential steps F, and FR will be different from zero, possibly representing larger and larger departures from uniform concentration, but the values c; and cg remain the same during all potential steps. If the cell is bounded, the initial concentrations will be written in the same form as above, but the condition is now that F, and FR each have average value zero over the cell. Again, F, and FR will normally be identically zero at the application of the first potential step and will vary at succeeding potential steps. For a bounded cell, the values c: and cg may be different from one time interval to the next. That is, the average values of 0 and R just before Tk may differ from those taken just before Tk_, because there has been some net transfer between 0 and R during the time interval (Tk_l, T,) (of course, the total quantity of 0 and R remains the same under simple electron transfer). The resulting system is described by the two-component diffusion problem (notice that eqn. 3c applies only to unbounded cells):
ace
ac,
,,=DOv2c,,~=D,v’c,
W
Initial conditions: co=c;-F,(X,
Y, Z),
cR=c;;+FR(X,
Y, Z)
(3’4
100
Boundary AS x2+
conditions:
y2+z2-+
+oe,
On all insulated
ac, _-= ac,
w-aN
and
co + c;;
cR+cIf
(3c)
surfaces,
’
(34
On the electrode(s),
ace
DoE+DRx=O ace Doaiv = k,c,
a+
(34
- kg,
(30
Current: I(T)
= nFDoisdA
(3g)
where the integration is with respect to area (A) over all electrodes (E). The notation ac/aN refers to the derivative normal to the surface. To simplify the discussion, introduce dimensionless variables. Let L be some characteristic length associated with the cell, such as the radius or length of an individual electrode. Define: X=Lx,
Y=Ly,
Z=Lz,
fo=Fo/cc:,
c,=c;(p+h) S’=D,/D,,
P=Lk,/D,,
T=(L2/Do)t,
frt=Wcc1;, 8=k,/k,,
c,=c;f(l-g), p=ciVc;r,>
(4)
I=nFD,czLi(t)
The system now has the form Equations:
F-4 Initial conditions: g=fo, BoundaT As x2+$+z2+
(5b)
h=f, conditions:
+co,
g-+0
and
h-0
(5c)
101
On all insulated surfaces,
On the electrode(s),
ag -z+s
*ah an=0
-g=P[(l-gW’(p+h)l Current: = -j-zda
i(t)
where the integration is with respect to area (a) overall electrodes (E). The general behavior of the system is determined by four dimensionless parameters: (1) p, the bulk concentration of R relative to 0. The value is zero if R is initially absent. (2) 6*, a measure of the relative diffusivity of R with respect to 0. The value is usually close to 1. (3) j3, a measure of forward reaction rate. (4) 8, a measure of the back reaction relative to the forward reaction. Equation (5) describes the behavior of the system during a single potential step. If a sequence of potential steps is applied at 0 = t,, < t, < t, < . . . , then on each interval (t,_,, tk) there will be new initial conditions f. and fR at t = t,_ 1 and new kinetic parameters Pk and 13,. The diffusion parameter 6* remains the same through all intervals. In an unbounded cell p remains the same for all intervals, but in a bounded cell the values pk may change from one time interval to the next. An extremely negative potential forces /I to + cc and 8 to 0, corresponding to the diffusion-limited reaction. A totally irreversible reaction corresponds to 8 = 0. If both /3and Be are very large, the reaction is rapid and reversible. These instances of electron transfer kinetics lead to several forms for boundary condition (5f): Case I. Diffusion-limited (5f):
reaction (/3 -+ + cc, 0 -+ 0)
g=l.
Case 2. Totally irreversible reaction (0 + 0) (5f):
$
=P(g-
1).
Case 3. Reversible reaction (/3 + + cc) (5f):
(l-g)-8(p+h)=O.
102
Case 4. Quasi-reversible (5f):
$
reaction,
or general
current-potential
characteristic
PKg-l)+~(P+~)l.
=
In Cases 1 and 2 the boundary condition (5f) refers only to the function g. Condition (5e) can then be dropped and the system uncouples, allowing the determination of solutions for g and h separately. Consequently, the current i(t) is determined by solving a one-component system for g. Our results consist of several simple observations. First, let g, h be the solutions to system (5) for arbitrary parameters, p, S2, p, 13. Then: g(t9 x, .Y* z) = (I -@)g,(t,
x, Y, z)
h(t,
x, y, z)
x, Y, z) = (I - Bp)h,(t, i(t)
@a)
= (1 - @)i,(t)
where g, and condition:
h,
are the solutions
to system
(5) with p = 0 and satisfying
initial
In other words, the general solution of system (5) reduces to solving the system for zero bulk concentration of R, a problem depending on the three parameters a2, j3, 8 instead of the original four. This reduction fails when 1 - 8p = 0, that is, when the bulk concentrations are in equilibrium with the applied potential. Of course, if 1 - 8p = 0 and the initial concentrations are uniform (fo = fR = 0), then they simply remain uniform. Second, let g,, h, be the solution to Case 3 (reversible reaction) with S2 = 1 and initial conditions fo =fa =f (that is, equal diffusion coefficients and equal but opposite perturbations about uniform concentration). Then:
g3(t, x, i3(t)
Y,
1-k
z>=h3(1,x3
ep
z) = -yqp,tt,
x9 Y,
z) (74
= - 1 +e
where gi(t, condition:
i -
Yf
1l(t)
x, y, z) is the solution
for Case 1 (diffusion-limited)
satisfying
initial
Third, let g,, h, be the solution to Case 4 (quasi-reversible reaction) with a2 = 1 and initial condition f. = fa =f (that is, equal diffusion coefficients and equal but opposite perturbations about uniform concentration). Then: g‘$(t, x, y,
z) = h4(f, x,
Y,
z>= 1 -
yqg,(B;
ep t,
x9 Y,
z> (84
103
where g2(fi, t, x, y, z) is the solution for Case 2 (totally initial condition and boundary condition:
g2=sf(x, Y,
z) and
irreversible)
satisfying
g=&g2-1)
where B = /3(1 + f?). All results are checked conveniently by substituting them into system (5). If 0 and R have equal diffusion coefficients and the experiment consists of a single potential step applied to initially uniform concentrations, then the simplifications (7) and (8) certainly hold because f. =fR = 0. If the experiment consists of a sequence of potential steps at times 0 = t, < t, < t, < . . . , then f. =fR = 0 initially and, because g, = h, or g, = h, on each time interval (t,_ i, tk), every initial condition continues to satisfy f. = fR. The simplifications (7) and (8) therefore apply at every step. In other words, for simple electron transfer where 0 and R have equal diffusion coefficients and initial concentrations which are equal but opposite perturbations about uniform concentration, then the concentrations continue to satisfy this condition during the application of a sequence of potential steps. For either Case 3 or Case 4, the solution on each time interval corresponding to a potential step is reduced to obtaining a solution for Case 1 or Case 2. Some implications should be mentioned. In discussing electron transfer kinetics, it is sometimes assumed that 0 and R have equal diffusion coefficients in order to simplify the mathematical treatment. The results, (7) and (8) show that such simplification is not an occasional occurrence but holds for potential step experiments in general. Similarly, it is sometimes assumed that R is initially absent. Result (6) shows that this assumption is unnecessary for potential step experiments applied to uniform concentrations since the initial presence of R simply introduces the factor (1 - @). A complete description of current-time curves for a single potential step under Cases 3 and 4 is complicated by the number of parameters: a2, 13in Case 3 and S2, 8, /I in Case 4. Since our results show the calculations simplify when S2 = 1, it may be useful to approach the general study of Cases 3 and 4 by treating the current-time curves for S2 = 1 as reference curves and describing the variation in the curves when a2 # 1. Finally, since the above simplifications are derived by elementary considerations, similar results should hold for electrode reactions involving preceding and/or following homogeneous chemical steps, as long as such steps are first order in 0 and R.
DISCUSSION
This section gives three electrode geometries where the diffusion-limited current has recently been obtained. The corresponding current for a totally reversible reaction with equal diffusion coefficients is then determined by eqn. (7). The leading order term in the short time asymptotic behavior of the current at a planar electrode of arbitrary shape is simply the Cottrell current. Oldham [15,24]
104
has derived Z,(T)
the next higher order term, and the short time current
- nFDocG (A/(
TD~T)~‘~ + p/2)
can be written
as
(94
where A is the area of the electrode, P is the unshielded electrode perimeter, and the electrode is assumed to have only bounded radius of curvature (i.e. no corners). Applying eqn. (7), the current for a reversible reaction with Do = D, is z,(T)-nFDo
k,c;, - k,c; k +k f
(A/(
r~o~)1/2
+ ~12)
b
Heinze [22] has given very accurate numerical results for the diffusion-limited current at the planar disk electrode; see also refs. 17 and 18 for asymptotic work. Taking L as the radius of the disk, Heinze’s result is expressed as Zl(T) = nFD,AcGX(q)/(
DoT)1’2
(loa)
x(q) is tabulated in ref. 22 for 0.002 G n G 5. Applying where n = ( DoT/L2)“2; eqn. (7) the current for a reversible reaction with Do = D, at a planar disk electrode is Z,(T)
=nFD&
k,c;, - k,C;: x(+‘(DoT)“~ k +k f
b
(lob)
For a planar band electrode in the form of a rectangle with dimensions L and W and diffusion across the edges of dimension W with the other two edges blocked, the diffusion-limited current has been obtained in ref. 19; see also ref. 20 for asymptotic and experimental results. The result is expressed as Z,(T)
= nFDoWcdi2(f2)
Ola)
where t, = 4DoT/L2 and i2(t2) is tabulated for 0.01 =Sr, G 100. Applying eqn. (7), the current for a reversible reaction with Do = D, at a planar band electrode is k,c;; - k,c;; z,(T)=nFD,W
k
f
+k
12(t2) b
tllb)
Explicit solutions for the potential step experiment, at least in the form of Laplace transforms, can be obtained for linear, spherical, and cylindrical diffusion for all four cases of boundary conditions. We shall use these solutions to illustrate eqns. (7) and (8) further. The transforms for both concentration and current for linear and spherical diffusion are well known and can be inverted by standard tables [25]. We use the transforms here since they have simpler forms than the exact solutions and illustrate eqns. (7) and (8) just as well. The inversion is more difficult for cylindrical diffusion, see refs. 26-28. For linear diffusion, the cell is semi-infinite with arbitrarily shaped cross section and the electrode occupying the base. Here v *c = 13'c/dZ2 where Z is the distance from the electrode. Taking L to be some characteristic length associated with the electrode, reducing system (3) to a dimensionless form (5), and applying a Laplace
105
transform with respect to r leads to the following of the four cases (s is the transform variable):
i,(s) =
Jg/2
i,(s)
=
s’ypft
f,(s)
=
solutions
for the current
for each
s’/2)
(12)
1 - ep sl/2(1
+
e/s)
(1
F4(s) = s”2(
-
OP)P
p(1 + e/s)
+ S”2)
Notice that Cases 3 and 4 reduce to the expected forms of Cases 1 and 2 when s2 = 1. For spherical diffusion, the cell is the space surrounding a spherical electrode of radius L. Here V 2c = a2c/aR2 + (2/R) i3c/aR. Reducing the problem to the dimensionless form (5) and applying a Laplace transform yields i,(S)
1 + s”2 = ~ s
i,(s)
=
#ql + .s”2) s(p+1+s”2) (I-
i3(S) =
ep)(s
+ &2)(1
(13)
+ 9’2)
+.V)] ~(1 - ep)(s + ~l/~)(i + C2) id(S) = s[(~ + c2)(1 + d/2) + P((S + 2’2) + (e/s)0 + W)] S[(6+S1/2)
+
(e/6)(1
Notice the reduction of Cases 3 and 4 when a2 = 1. For cylindrical diffusion, the cell consists of the space between two planes separated by a distance H and the electrode is a cylinder of radius L perpendicular to the two planes. Here v 2c = a2c/i3R2 + (l/R) ac/i3R. Reducing the problem to the dimensionless form (5) and applying the Laplace transform yields: -Ko’(S”2) G(s)
=
i2(s) =
,l/2Ko(sl/2)
-~K;(s1’2) s”2 [ #fx, (2’2) - s”%,I ( s1’2)] -(I - ep)K,‘(s1’2)K,‘(s”2/6) S1/2[~o(sl/2)~d(s1’2/s) + (e/s)Kd(s”2)K,(s”2/S)]
i3W=
i4w = {-~(1 -
wm
+(e/s)K,‘(s1’2)Ko(s1’2/s))
P)K&“2/S)}
{ F[
/3(K,(s”2)K;(s”2/q
-s~‘~K~(s”~)K~(s~‘~/S)]}-’
(14)
106
Here K,(u) is the modified Bessel function of order 0 [25]. Again, Cases 3 and 4 reduce to the expected forms of Cases 1 and 2 when 6* = 1. These simplifications should prove to be very useful in determining the theoretical response of microelectrodes of various geometries to a potential step or a sequence of potential steps, and we are exploring such applications in further research. ACKNOWLEDGEMENT
We are grateful to the National Science Foundation under grants CHE-85-11697 and CHE-87-11595.
for supporting
this work
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28
R.M. Wightman, Anal. Chem., 53 (1981) 1125A. M.A. Dayton, J.C. Brown, K.J. Stutts and R.M. Wightman, Anal. Chem., 52 (1980) 946. B. Scharifker and G. Hills, J. Electroanal. Chem., 130 (1981) 81. G. Hills, A.K. Pour and B. Scharifker, Electrochim. Acta, 28 (1983) 891. J.O. Howell and R.M. Wightman, Anal. Chem., 56 (1984) 524. M. Fleischmann, F. Lasserre, J. Robinson and D. Swan, J. Electroanal. Chem., 177 (1984) 97. M. Fleischmann, F. Lasserre and J. Robinson, J. Electroanal. Chem., 177 (1984) 115. R.S. Robinson and R.L. McCreery, J. Electroanal. Chem., 182 (1985) 61. K.R. Wehmeyer, M.R. Deakin and R.M. Wightman, Anal. Chem., 57 (1985) 1913. P.M. Kovach, W.L. Caudill, D.G. Peters and R.M. Wightman, J. Electroanal. Chem., 185 (1985) 285. K. Aoki, K. Honda, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 186 (1985) 79. Y.-T. Kim, D.M. Scarnulis and A.G. Ewmg, Anal. Chem., 58 (1986) 1782. F.G. Soos and P.J. Lingane, J. Phys. Chem., 68 (1964) 3821. J.B. Flanagan and L. Marcoux, J. Phys. Chem., 77 (1973) 1051. K.B. Oldham, J. Electroanal. Chem., 122 (1981) 1. K. Aoki and J. Osteryoung, J. Electroanal. Chem., 122 (1981) 19. D. Shoup and A. Szabo, J. Electroanal. Chem., 140 (1982) 237. K. Aoki and J. Osteryoung, J. Electroanal. Chem., 160 (1984) 335. S. Coen, D.K. Cope and D.E. Tallman, J. Electroanal. Chem., 215 (1986) 29. A. Szabo, D.K. Cope, D.E. Tallman. P.M. Kovach and R.M. Wightman, J. Electroanal. Chem., 217 (1987) 417. K. Aoki and J. Osteryoung, J. Electroanal. Chem.. 125 (1981) 315. J. Heinze, J. Electroanal. Chem., 124 (1981) 73. P.L. Mills and M.P. Dudukovic, SIAM J. Appl. Math., 44 (1984) 1076. J.C. Myland and K.B. Oldham, J. Electroanal. Chem., 147 (1983) 295. M. Abramowitz and I. Stegun (Eds.), Handbook of Mathematical Functions, Mathematics Series 55, National Bureau of Standards, Washington, 1964. J.C. Jaeger, Proc. R. Sot. Edinburgh, A61 (1942) 223. J.C. Jaeger and M. Clarke, Proc. R. Sot. Edinburgh, A61 (1942) 229. K. Aoki, K. Honda, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 186 (1985) 79.
J. Electroanal. Chem., 235 (1987) 97-106 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
SIMPLIFICATIONS MICROELECTRODES
IN THE THEORY
OF STEP
EXPERIMENTS
AT
D.K. COPE Divwon
of MathematicalScrences,
D.E. TALLMAN Department
North Dakota State University,
Fargo, ND 5810.5 (U.S.A.)
l
of Chemistry,
North Dakota State Universrty, Fargo, ND 58105 (U.S.A.)
(Received 19th December 1986)
ABSTRACT Theoretical studies of the chronoamperometry of the reaction 0+ n e- + R often consider (1) the diffusion-limited reaction, (2) the totally irreversible reaction, and (3) the reversible reaction (governed by the Nemst equation), and these can all be considered as special limiting cases of (4) quasi-reversible reactions, electron transfer kinetics governed by the genera1 current-potential characteristic. Cases (1) and (2) simplify to one-component diffusion systems for the concentration of 0; cases (3) and (4) are necessarily two-component diffusion systems involving the concentrations of both 0 and R. We show that. when 0 and R have equal diffusion coefficients, the solution for the reversible reaction (3) can be expressed in terms of the solution for the diffusion-limited reaction (1) and that the solution for the quasi-reversible reaction (4) can be expressed in terms of the solution for the totally irreversible reaction (2). These results hold for arbitrary electrode geometries.
INTRODUCTION
Microelectrodes, which have at least one dimension comparable to the diffusion layer thickness (typically of the order of a few micrometers or smaller), continue to receive much attention in the electrochemical literature [l-12]. This interest is due in part to the enhanced mass transfer at microelectrodes, a result of convergent (or non-linear) diffusion, commonly referred to as the edge effect for planar electrodes [13-H]. For time scales typically used in voltammetry, convergent diffusion at microelectrodes often leads to time independent (steady state) current [16-181 or nearly time independent (quasi steady state) current [11,19,20] and also to higher current densities [9] than generally observed at conventional sized electrodes (having
l
To whom correspondence
0022-0728/87/$03.50
should be addressed.
0 1987 Elsevier Sequoia S.A.
98
dimensions of the order of a millimeter). These properties of microelectrodes make them advantageous for use in analytical detection as well as for the study of heterogeneous electron transfer kinetics [1,9]. The responses of microelectrodes of various geometries to a potential step have been observed experimentally [10,11,20,21] and considered theoretically [11,16-20,221. Even for the simplest case of diffusion-limited current the theoretical treatment of the problem is rather complex due to mixed boundary conditions in the mathematical formulation [23]. Research in our laboratory involving the extension of our recently developed integral equation method [19] to more general electrode boundary conditions has led us to observe simplifications which do not appear to have been previously reported in the electrochemical literature. The main simplifications require only that the oxidized and reduced species have equal diffusion coefficients and are applicable to experiments involving a potential step or sequence of potential steps at an electrode consisting of one or many active regions of arbitrary geometry in a bounded or unbounded cell of arbitrary geometry. We show (1) that the solution for reversible electron transfer can be expressed in terms of the solution for the diffusion-limited case and (2) that the solution for the quasi-reversible case can be expressed in terms of the solution for the totally irreversible case. The approach is illustrated for planar, spherical, and cylindrical electrode geometries. We also show that the solution for R initially present can be expressed in terms of the solution for R initially absent; this result holds for all diffusion coefficients and arbitrary cell and electrode geometries. THEORETICAL
FORMULATION
AND
SOLUTION
This section gives the physical description dimensionless variables, and the results. We consider simple electron transfer O+ne-+R
of the problem,
a reformulation
in
(I)
in a stationary electrolyte with the electrolyzable species 0 and R both soluble. Ion migration is neglected and the diffusion coefficients are assumed independent of concentration. A potential step or sequence of potential steps is applied and the resulting current measured. Physical quantities involved are time T, spatial variables X, Y, Z (Cartesian coordinates), concentrations c,(T, X, Y, Z) and ca( T, X, Y, Z), current I(T), diffusion coefficients Do and D,, and rate constants kf and k,, described by k,=k”
exp(-a(E-E”‘)(nF/RT))
k, = k” exp((1 - a)(E--E”‘)(nF/RT))
(2)
using standard notation. The boundary of the electrochemical cell consists of insulated portions, where no electrical charge passes, and the electrode(s), where the electrochemical reaction takes place. No other assumptions on cell geometry are made; in particular, the cell
99
may be bounded or unbounded, the electrode region may consist of one or many separate regions, and these regions may have arbitrary shape. A unified discussion applying equally to bounded and unbounded cells requires a precise definition of bulk concentration. In an unbounded cell, the bulk concentrations cg and cg are the concentrations infinitely far from the electrode. In a bounded cell, we define cz and cg as the average concentrations over the cell just before the (new) potential step is applied. If an experiment involves a sequence of potential steps applied at times 0 = T, < T, < T, -=c. . . , then we obtain a sequence of diffusion problems for time intervals T k_l G T < Tk_ These problems are identical except for two factors: (a) the initial concentrations of 0 and R change from interval to interval, since the terminal concentrations of one interval are the initial concentrations of the next; (b) a new pair of rate constants k, and k, occurs on each new time interval, a result of the change in potential. We write the initial concentrations in the following way. If the cell is unbounded, the initial concentrations will be written as co=c;;-F,(X,
Y, z),
cacI:+&(X,
Y, Z)
with the condition F,, FR + 0 at large distances from the electrode. That is, F, and FR represent some perturbation around uniform concentration. Concentrations are normally uniform at the application of the very first potential step and F, and FR are both identically zero. At succeeding potential steps F, and FR will be different from zero, possibly representing larger and larger departures from uniform concentration, but the values c; and cg remain the same during all potential steps. If the cell is bounded, the initial concentrations will be written in the same form as above, but the condition is now that F, and FR each have average value zero over the cell. Again, F, and FR will normally be identically zero at the application of the first potential step and will vary at succeeding potential steps. For a bounded cell, the values c: and cg may be different from one time interval to the next. That is, the average values of 0 and R just before Tk may differ from those taken just before Tk_, because there has been some net transfer between 0 and R during the time interval (Tk_l, T,) (of course, the total quantity of 0 and R remains the same under simple electron transfer). The resulting system is described by the two-component diffusion problem (notice that eqn. 3c applies only to unbounded cells):
ace
ac,
,,=DOv2c,,~=D,v’c,
W
Initial conditions: co=c;-F,(X,
Y, Z),
cR=c;;+FR(X,
Y, Z)
(3’4
100
Boundary AS x2+
conditions:
y2+z2-+
+oe,
On all insulated
ac, _-= ac,
w-aN
and
co + c;;
cR+cIf
(3c)
surfaces,
’
(34
On the electrode(s),
ace
DoE+DRx=O ace Doaiv = k,c,
a+
(34
- kg,
(30
Current: I(T)
= nFDoisdA
(3g)
where the integration is with respect to area (A) over all electrodes (E). The notation ac/aN refers to the derivative normal to the surface. To simplify the discussion, introduce dimensionless variables. Let L be some characteristic length associated with the cell, such as the radius or length of an individual electrode. Define: X=Lx,
Y=Ly,
Z=Lz,
fo=Fo/cc:,
c,=c;(p+h) S’=D,/D,,
P=Lk,/D,,
T=(L2/Do)t,
frt=Wcc1;, 8=k,/k,,
c,=c;f(l-g), p=ciVc;r,>
(4)
I=nFD,czLi(t)
The system now has the form Equations:
F-4 Initial conditions: g=fo, BoundaT As x2+$+z2+
(5b)
h=f, conditions:
+co,
g-+0
and
h-0
(5c)
101
On all insulated surfaces,
On the electrode(s),
ag -z+s
*ah an=0
-g=P[(l-gW’(p+h)l Current: = -j-zda
i(t)
where the integration is with respect to area (a) overall electrodes (E). The general behavior of the system is determined by four dimensionless parameters: (1) p, the bulk concentration of R relative to 0. The value is zero if R is initially absent. (2) 6*, a measure of the relative diffusivity of R with respect to 0. The value is usually close to 1. (3) j3, a measure of forward reaction rate. (4) 8, a measure of the back reaction relative to the forward reaction. Equation (5) describes the behavior of the system during a single potential step. If a sequence of potential steps is applied at 0 = t,, < t, < t, < . . . , then on each interval (t,_,, tk) there will be new initial conditions f. and fR at t = t,_ 1 and new kinetic parameters Pk and 13,. The diffusion parameter 6* remains the same through all intervals. In an unbounded cell p remains the same for all intervals, but in a bounded cell the values pk may change from one time interval to the next. An extremely negative potential forces /I to + cc and 8 to 0, corresponding to the diffusion-limited reaction. A totally irreversible reaction corresponds to 8 = 0. If both /3and Be are very large, the reaction is rapid and reversible. These instances of electron transfer kinetics lead to several forms for boundary condition (5f): Case I. Diffusion-limited (5f):
reaction (/3 -+ + cc, 0 -+ 0)
g=l.
Case 2. Totally irreversible reaction (0 + 0) (5f):
$
=P(g-
1).
Case 3. Reversible reaction (/3 + + cc) (5f):
(l-g)-8(p+h)=O.
102
Case 4. Quasi-reversible (5f):
$
reaction,
or general
current-potential
characteristic
PKg-l)+~(P+~)l.
=
In Cases 1 and 2 the boundary condition (5f) refers only to the function g. Condition (5e) can then be dropped and the system uncouples, allowing the determination of solutions for g and h separately. Consequently, the current i(t) is determined by solving a one-component system for g. Our results consist of several simple observations. First, let g, h be the solutions to system (5) for arbitrary parameters, p, S2, p, 13. Then: g(t9 x, .Y* z) = (I -@)g,(t,
x, Y, z)
h(t,
x, y, z)
x, Y, z) = (I - Bp)h,(t, i(t)
@a)
= (1 - @)i,(t)
where g, and condition:
h,
are the solutions
to system
(5) with p = 0 and satisfying
initial
In other words, the general solution of system (5) reduces to solving the system for zero bulk concentration of R, a problem depending on the three parameters a2, j3, 8 instead of the original four. This reduction fails when 1 - 8p = 0, that is, when the bulk concentrations are in equilibrium with the applied potential. Of course, if 1 - 8p = 0 and the initial concentrations are uniform (fo = fR = 0), then they simply remain uniform. Second, let g,, h, be the solution to Case 3 (reversible reaction) with S2 = 1 and initial conditions fo =fa =f (that is, equal diffusion coefficients and equal but opposite perturbations about uniform concentration). Then:
g3(t, x, i3(t)
Y,
1-k
z>=h3(1,x3
ep
z) = -yqp,tt,
x9 Y,
z) (74
= - 1 +e
where gi(t, condition:
i -
Yf
1l(t)
x, y, z) is the solution
for Case 1 (diffusion-limited)
satisfying
initial
Third, let g,, h, be the solution to Case 4 (quasi-reversible reaction) with a2 = 1 and initial condition f. = fa =f (that is, equal diffusion coefficients and equal but opposite perturbations about uniform concentration). Then: g‘$(t, x, y,
z) = h4(f, x,
Y,
z>= 1 -
yqg,(B;
ep t,
x9 Y,
z> (84
103
where g2(fi, t, x, y, z) is the solution for Case 2 (totally initial condition and boundary condition:
g2=sf(x, Y,
z) and
irreversible)
satisfying
g=&g2-1)
where B = /3(1 + f?). All results are checked conveniently by substituting them into system (5). If 0 and R have equal diffusion coefficients and the experiment consists of a single potential step applied to initially uniform concentrations, then the simplifications (7) and (8) certainly hold because f. =fR = 0. If the experiment consists of a sequence of potential steps at times 0 = t, < t, < t, < . . . , then f. =fR = 0 initially and, because g, = h, or g, = h, on each time interval (t,_ i, tk), every initial condition continues to satisfy f. = fR. The simplifications (7) and (8) therefore apply at every step. In other words, for simple electron transfer where 0 and R have equal diffusion coefficients and initial concentrations which are equal but opposite perturbations about uniform concentration, then the concentrations continue to satisfy this condition during the application of a sequence of potential steps. For either Case 3 or Case 4, the solution on each time interval corresponding to a potential step is reduced to obtaining a solution for Case 1 or Case 2. Some implications should be mentioned. In discussing electron transfer kinetics, it is sometimes assumed that 0 and R have equal diffusion coefficients in order to simplify the mathematical treatment. The results, (7) and (8) show that such simplification is not an occasional occurrence but holds for potential step experiments in general. Similarly, it is sometimes assumed that R is initially absent. Result (6) shows that this assumption is unnecessary for potential step experiments applied to uniform concentrations since the initial presence of R simply introduces the factor (1 - @). A complete description of current-time curves for a single potential step under Cases 3 and 4 is complicated by the number of parameters: a2, 13in Case 3 and S2, 8, /I in Case 4. Since our results show the calculations simplify when S2 = 1, it may be useful to approach the general study of Cases 3 and 4 by treating the current-time curves for S2 = 1 as reference curves and describing the variation in the curves when a2 # 1. Finally, since the above simplifications are derived by elementary considerations, similar results should hold for electrode reactions involving preceding and/or following homogeneous chemical steps, as long as such steps are first order in 0 and R.
DISCUSSION
This section gives three electrode geometries where the diffusion-limited current has recently been obtained. The corresponding current for a totally reversible reaction with equal diffusion coefficients is then determined by eqn. (7). The leading order term in the short time asymptotic behavior of the current at a planar electrode of arbitrary shape is simply the Cottrell current. Oldham [15,24]
104
has derived Z,(T)
the next higher order term, and the short time current
- nFDocG (A/(
TD~T)~‘~ + p/2)
can be written
as
(94
where A is the area of the electrode, P is the unshielded electrode perimeter, and the electrode is assumed to have only bounded radius of curvature (i.e. no corners). Applying eqn. (7), the current for a reversible reaction with Do = D, is z,(T)-nFDo
k,c;, - k,c; k +k f
(A/(
r~o~)1/2
+ ~12)
b
Heinze [22] has given very accurate numerical results for the diffusion-limited current at the planar disk electrode; see also refs. 17 and 18 for asymptotic work. Taking L as the radius of the disk, Heinze’s result is expressed as Zl(T) = nFD,AcGX(q)/(
DoT)1’2
(loa)
x(q) is tabulated in ref. 22 for 0.002 G n G 5. Applying where n = ( DoT/L2)“2; eqn. (7) the current for a reversible reaction with Do = D, at a planar disk electrode is Z,(T)
=nFD&
k,c;, - k,C;: x(+‘(DoT)“~ k +k f
b
(lob)
For a planar band electrode in the form of a rectangle with dimensions L and W and diffusion across the edges of dimension W with the other two edges blocked, the diffusion-limited current has been obtained in ref. 19; see also ref. 20 for asymptotic and experimental results. The result is expressed as Z,(T)
= nFDoWcdi2(f2)
Ola)
where t, = 4DoT/L2 and i2(t2) is tabulated for 0.01 =Sr, G 100. Applying eqn. (7), the current for a reversible reaction with Do = D, at a planar band electrode is k,c;; - k,c;; z,(T)=nFD,W
k
f
+k
12(t2) b
tllb)
Explicit solutions for the potential step experiment, at least in the form of Laplace transforms, can be obtained for linear, spherical, and cylindrical diffusion for all four cases of boundary conditions. We shall use these solutions to illustrate eqns. (7) and (8) further. The transforms for both concentration and current for linear and spherical diffusion are well known and can be inverted by standard tables [25]. We use the transforms here since they have simpler forms than the exact solutions and illustrate eqns. (7) and (8) just as well. The inversion is more difficult for cylindrical diffusion, see refs. 26-28. For linear diffusion, the cell is semi-infinite with arbitrarily shaped cross section and the electrode occupying the base. Here v *c = 13'c/dZ2 where Z is the distance from the electrode. Taking L to be some characteristic length associated with the electrode, reducing system (3) to a dimensionless form (5), and applying a Laplace
105
transform with respect to r leads to the following of the four cases (s is the transform variable):
i,(s) =
Jg/2
i,(s)
=
s’ypft
f,(s)
=
solutions
for the current
for each
s’/2)
(12)
1 - ep sl/2(1
+
e/s)
(1
F4(s) = s”2(
-
OP)P
p(1 + e/s)
+ S”2)
Notice that Cases 3 and 4 reduce to the expected forms of Cases 1 and 2 when s2 = 1. For spherical diffusion, the cell is the space surrounding a spherical electrode of radius L. Here V 2c = a2c/aR2 + (2/R) i3c/aR. Reducing the problem to the dimensionless form (5) and applying a Laplace transform yields i,(S)
1 + s”2 = ~ s
i,(s)
=
#ql + .s”2) s(p+1+s”2) (I-
i3(S) =
ep)(s
+ &2)(1
(13)
+ 9’2)
+.V)] ~(1 - ep)(s + ~l/~)(i + C2) id(S) = s[(~ + c2)(1 + d/2) + P((S + 2’2) + (e/s)0 + W)] S[(6+S1/2)
+
(e/6)(1
Notice the reduction of Cases 3 and 4 when a2 = 1. For cylindrical diffusion, the cell consists of the space between two planes separated by a distance H and the electrode is a cylinder of radius L perpendicular to the two planes. Here v 2c = a2c/i3R2 + (l/R) ac/i3R. Reducing the problem to the dimensionless form (5) and applying the Laplace transform yields: -Ko’(S”2) G(s)
=
i2(s) =
,l/2Ko(sl/2)
-~K;(s1’2) s”2 [ #fx, (2’2) - s”%,I ( s1’2)] -(I - ep)K,‘(s1’2)K,‘(s”2/6) S1/2[~o(sl/2)~d(s1’2/s) + (e/s)Kd(s”2)K,(s”2/S)]
i3W=
i4w = {-~(1 -
wm
+(e/s)K,‘(s1’2)Ko(s1’2/s))
P)K&“2/S)}
{ F[
/3(K,(s”2)K;(s”2/q
-s~‘~K~(s”~)K~(s~‘~/S)]}-’
(14)
106
Here K,(u) is the modified Bessel function of order 0 [25]. Again, Cases 3 and 4 reduce to the expected forms of Cases 1 and 2 when 6* = 1. These simplifications should prove to be very useful in determining the theoretical response of microelectrodes of various geometries to a potential step or a sequence of potential steps, and we are exploring such applications in further research. ACKNOWLEDGEMENT
We are grateful to the National Science Foundation under grants CHE-85-11697 and CHE-87-11595.
for supporting
this work
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