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Representation of reversible current for arbitrary electrode geometries and arbitrary potential modulation
J. Elecrroanal. Chem., 303 (1991) 1-15 Elsevier Sequoia S.A., Lausanne
Representation of reversible current for arbitrary electrode geometries and arbitrary potential modulation Davis K. Cope Department
of Mathematics, North Dakota State University, Fargo, ND 58105 (USA)
Dennis E. Tallman * Department
of Chemistry, North Dakota State University, Fargo, ND 58105 (USA)
(Received 18 June 1990; in revised form 12 September 1990)
Abstract For simple electron transfer 0+ n e- + R with equal diffusion coefficients (Do = DR), it is shown that reversible current can be expressed in terms of diffusion-limited current for arbitrary cell/electrode geometries and arbitrary time-dependent potentials. This result provides a better understanding of a formula due to Aoki and co-workers and suggests new experimental approaches which simplify interpretation of data obtained at microelectrodes.
(1) INTRODUCTION
This paper derives a general result for simple electron transfer 0 + n e- + R: if the diffusion coefficients are equal, then the reversible current i,,(t) can be expressed in terms of the diffusion-limited current i,*(t) by the simple formula
(1.1) Here B,(t) - L$(t)p is determined by the time-dependent potential (notation is defined in the next section). This result holds for arbitrary time-dependent potentials and for arbitrary (unbounded) cell and electrode geometries, including arrays and ensembles as long as the same potential is applied across all electrodes. Aoki et al. [l] obtained a result analogous to eqn. (1.1) but with a complicated interpretation. Their derivation, like ours, assumed equal diffusion coefficients and
l
To whom correspondence should be addressed.
0022-0728/91/$03.50
0 1991 - Elsevier Sequoia S.A.
2
allowed arbitrary cell/electrode geometries and arbitrary time-dependent potentials. However, the diffusion-limited current ii*(t) did not appear as such. Instead, a function f(t) occurs which is defined as the inverse Laplace transform of the integral over the electrode of the “inverse kernel” for the Green’s function for the transformed diffusion problem on the given cell geometry (see the Appendix of ref. 1, especially eqns. (Al&A20) and discussion). Examples of f(t) are given for large planar, spherical, cylindrical, and small disk electrodes which show that f(t) is essentially the diffusion-limited current for these special cases. In this paper we give a simple derivation of eqn. (1.1) which leads directly to the general identification of f( t ) with the diffusion-limited current. We have previously shown that, for equal diffusion coefficients, reversible current in response to a potential step can be expressed in terms of diffusion-limited current (and quasi-reversible current can be expressed in terms of totally irreversible current) for arbitrary cell/electrode geometries [2]. The results derived in this paper extend this previous result to arbitrary time-dependent potentials. Diffusion-limited current is typically easier to compute than reversible current arising from a time-dependent potential. Our result provides a direct means of extending calculations of diffusion-limited current for the band [3,4], ring [5,6], and disk [7,8] to the reversible case with time-dependent potential. The result also offers opportunities for experimental exploitation. The reversible current response to any time-dependent potential is the diffusion-limited current (in response to a potential step) convolved with the potential-dependent function as in eqn. (1.1). In the original derivation, Aoki et al. applied the result to square-wave voltammetry [l]. Recent advances in computer-based electrochemical instrumentation employing high speed direct memory access techniques [9] make possible the application of arbitray potential modulations on millisecond and microsecond time scales. Such experimental capability makes possible the design and implementation of potential modulations which optimize experimental measurement of desired electrochemical parameters or which simplify rigorous comparisons of experiment with theory at microelectrodes. The arbitrary potential modulation permitted by our result is therefore a generalization with experimental significance. The practical significance of the restriction to equal diffusion coefficients depends on the reaction. In many cases, simple electron transfer will cause little change in the properties affecting diffusivity. A brief discussion on this topic including quantitative examples has been provided previously [lo]. In addition to the representation for reversible current, we wish to emphasize the general formulation of simple electron transfer given below. It is shown that the physical formulation can always be reduced to dimensionless form by selection of a single characteristic length for the cell/electrode geometry. A dimensionless formulation provides an especially simple description both for calculations and for the derivation of general results, as done here. A particular choice of characteristic length yields area normalized variables, which are especially suitable for comparing behavior of different cell/electrode geometries. For rectilinear diffusion (shielded planar electrode), it is known (see ref. 11 and
3
references therein) that a representation for reversible current can be found for arbitrary potential-time dependence and arbitrary diffusion coefficients. The simple geometry permits this strong result; the representation reduces to our result when the diffusion coefficients are equal. For spherical diffusion, representations for reversible current for arbitrary potential-time dependence have been found but appear always to involve equal diffusion coefficients. For example, Myland and Oldham [lo] obtain a representation for reversible current at a spherical electrode in terms of reversible current at a rectilinear electrode for arbitrary potential-time dependence, but the relation is complicated and equal diffusion coefficients are introduced for simplification. For cylindrical diffusion, the representation obtained here has been used for linear sweep voltammetry and normal and differential pulse voltammetry [12,13], but always with the assumption of equal diffusion coefficients. The second section of this paper gives a general formulation of simple electron transfer for arbitrary cell/electrode geometries and arbitrary time-dependent potential. The third section derives the relation between reversible and diffusion-limited current. The remaining sections illustrate the result for rectilinear, spherical, and cylindrical geometries.
(2) GENERAL FORMULATION FOR SIMPLE ELECTRON TRANSFER
This section provides notation and a general formulation for simple electron transfer. Upper case letters are typically used for physical quantities and lower case for dimensionless quantities. Although this paper is concerned only with diffusionlimited and reversible reactions, the formulation will include totally irreversible and quasi-reversible reactions for completeness and future reference. The electrochemical cell is assumed unbounded and of arbitrary shape with one or more electrodes of arbitrary shape and arrangement on the cell boundary. Cartesian coordinates X, Y, Z are used for position within the cell. The interior of the cell is filled with electrolyte containing solvable electrolyzable species 0 and R with initially uniform concentrations C,* and C,* and diffusion coefficients Do and D,, respectively. The species R is allowed to be present for generality. The diffusion coefficients are assumed independent of concentration and migration of 0 and R is assumed negligible. We further assume that the potential across the electrode surface(s) is spatially uniform at each instant in time, a reasonable assumption for non-resistive solutions. This last assumption does not, of course, imply uniform flux or current distribution across the electrode(s). The electrodes are initially poised at a potential such that no reaction occurs (an equilibrium potential if R is present). At time T = 0, the potential is changed and simple electron transfer 0 + n e-+ R takes place with forward and backward kinetic parameters K, and K, respectively. We are concerned with two cases: (1) a potential step to an extreme value where the reaction is diffusion-limited; (2) a time-dependent potential E(T) producing a reversible reaction. An equal potential is applied across all electrodes but the time-dependence is arbitrary. The concentra-
4
tions Co(T, system:
X, Y, 2)
and
C,(T,
X, Y, Z)
then
satisfy
the following
diffusion
~=Dov2Co
-=D,v2C, 3T
ac,
(2.la)
Initial :
c,=c,*
c,=c,*
(2Sb)
At infinity:
co -+ co*
c,+c,*
(2Sc)
ace= 0
m=O
ac,
(2.ld)
ac0
Insulated
boundary:
Electrode(s) Kinetic
-
aN
ace+&a~3%
:
Dow
conditions
at the electrode(s)
Case 1A. Diffusion-limited: Case 1B. Reversible: Case 2A. Totally
=0
yield:
Co = 0
(2.2a)
K, Co - K,CR = 0
irreversible:
Case 2B. Quasi-reversible:
7 Do% -Do%
(2.le)
(2.2b) = K,C,
= K,C,
- K,C,
(2.2c) (2.2d)
Here a/aN is the normal derivative N. v, where N is the unit outer normal to the cell. The above equations assume all electrodes share the same potential; otherwise, separate conditions would be stated at individual electrodes. The current is given by: I(T)
= -n~Do~lW~rO~.rSj~
(2.3)
dA
Notice that the kinetic conditions (2.2a, c) for the diffusion-limited and totally irreversible cases involve only the concentration Co and the system can therefore be reduced to a single equation for Co. The kinetic condition (2.2b) for the reversible case uses only the ratio Kf/Kb, which is related to the potential E(T) by the Nernst equation: KF/Kb=exp(nf{Eo’-E(T)})
(2.4)
where the notation has its customary meaning (formal potential E “‘, f = F/RT for temperature T, etc.). For totally irreversible and quasi-reversible processes, potential dependence of the individual kinetic coefficients is normally modeled by the Butler-Volmer equations [14]: * K,=K” K,=K”
exp(-anf{E(T)-EO’}) exp((l-cu)nf{E(T)-EO’})
Let L be a characteristic radius of a planar disk,
O
(2.5)
length associated with the cell/electrode geometry (e.g. width of a planar band, diameter of a spherical or
cylindrical electrode, etc.). We introduce dimensionless variables by: t = DOT/L2
i(t) = I( T)/nFD&$L
p = c,*/c,”
S2 = D,/D,
4(t)
= W(Kb
+ 4)
e,(t)
=
Kb/(Kt!
+
Kf)
(2.6)
clz=G(P+gR)
co=G(l-80)
x, y, z = X/L,
Y/L,
Z/L
v(t)
= Do/L&
Notice that d,(t) + O=(t) = 1. The functions go and g, correspond to relative changes around the bulk concentrations. Depending on the cell/electrode geometry, additional dimensionless parameters specifying shape and position may occur (e.g. a planar ring electrode requires two lengths to specify its shape; scaling out a characteristic length will still leave one dimensionless parameter necessary to specify the shape). A general dimensionless formulation for simple electron transfer is then:
ago=v=go
Equation: Initial
:
At infinity
:
Electrode(s)
at
go=0
gR
=
gR
+
go
Insulated boundary:
%I,
at
+
0
ago 0
=
(2.7~)
o
aN
(2.7a) (2.7b)
o
GR -=
-= aN
a202gR
O
(2.7d) (2.7e)
:
For a potential applied across all electrodes equally, the kinetic conditions electrodes become: Case 1A. Diffusion-limited: go=1
at the
(2.8)
Case 1B. Reversible: e,(t>gO+e2(t)gR=el(t)-e2(t>p
(2.9)
Case 2A. Totally irreversible: (2.10) Case 2B. Quasi-reversible:
dt)el(r)~+edt>gO +
e2(t)gR=
edt)
-e2(t)p
(2.11)
The dimensionless current is given by:
i(t)=/.,,,,,,,,,,%da
(2.12)
6
Particular cell/electrode geometries may allow many possible choices for a characteristic length L. One choice of particular significance expresses the characteristic length in terms of total electrode area A. This choice may not lead to the simplest formulation, but it does provide a common scaling meaningful across different geometries and is especially suitable for comparison of behavior between geometries. Specifically, we define area normalized variables by the choice (2.13)
L = (A/7$‘* iA( tA) = I( T)/nFDoC$
t, = nDoT/A
( A/r)l'*
(2.14)
The factor ~7is included to simplify the definition for some basic geometries: for the disk, L = (A/a)‘/* = R,, the radius of the disk; for the sphere, L = ( A/r)‘12 = 2R,, the diameter of the sphere. Such area normalized variables have been used in comparing diffusion-limited current at disk, ring, and band electrodes [15], where the definitions are the same as eqn. (2.14) except for a factor of rl’* in defining i,. The definitions of eqn. (2.14) follow directly from the choice of L in eqn. (2.13) and the general resealing introduced in eqn. (2.6). (3) GENERAL
REPRESENTATION
FOR REVERSIBLE
CURRENT
Our main result follows quickly from the general formulation of the last section. Consider the diffusion problem:
ag x=v*g
(3.la)
Initial:
g=o
(3.lb)
At infinity:
g+o
(3.lc)
Insulated boundary:
ag = 0 a~
(3.ld)
Let glA(t, x, Y, 2) and gm(t, x, y, z) be solutions to this problem satisfying the additional boundary condition on the electrode(s): Case 1A:
g,, = 1
Case 1B:
&B
=
e,(t)
(3.2) -
02tt)p
(3.3)
The current in each case is given by (3.4) and we write i,,(t), i,,(t) for the respective currents. Observe that (1) glA(t, x, y, z) = go solves the diffusion-limited case defined by eqns. (2.7) and (2.8);
(2) g&t, x, y, z) = go = g, solves the reversible case defined (2.9) with 6* = 1. Now apply a Laplace transform with respect to t, x, y, z)] =8( s, x, y, z) = j+%g(t,
=qg(t,
by eqns. (2.7) and
x, y, z) dt
0
to eqn. (3.1) to obtain Equation
:
sg=v*g
(3Sa) (3.5b)
boundary:
jj bounded *1 $$ = 0
At infinity: Insulated
(3.5c)
where g,A and jiB are solutions electrode(s): Case 1A:
&A = l/s
Case 1B:
&a = 4<4
satisfying
-
- &(4P
8,WP}&A(S,
&lb)
x, Y, z> =s{e,(s) =
s{
4(4
Equivalently, g&x,
-
42mJ)
x,
-
Y,
on the
that
z>
e2(4P}&,(s,
x,
y,
gta. for the diffusion-limited x, y, z) = go = g, be the Then:
2)
(3.8)
Lb)
by the convolution
Y, 4 = &i’gi&
condition
(3.7) observe
satisfies eqns. (3.5) and (3.7) and is therefore the solution Altogether: let glA(t, x, y, z) = go be the solution current in eqns. (2.7) and (2.8) and, for 6* = 1, let gia(t, solution for the reversible current in eqns. (2.7) and (2.9). &*(s,
boundary
(3.6)
We now derive our main result: g = s{ e,(s)
the additional
theorem, ~9 ~9 Y, 4{ e,(u)
- &(u)P}
du (3.9)
&a(t)
= $1’4,
(t-4(4(4
-GYP)
du
The following sections illustrate this result by showing case in rectilinear, spherical, and cylindrical diffusion.
how it arises as a special
(4) EXAMPLE: RECTILINEAR DIFFUSION
The cell/electrode geometry is a planar electrode of area A and arbitrary shape with shielded edges, i.e. the cell is a cylinder perpendicular to the electrode forming its base. One-dimensional diffusion occurs in the Z-direction. We choose L = ( A/7ry2
(4.1)
8
corresponding Equation
to area normalized
The resulting
system on 0 d z < + cc is:
a*&3 az2
ago -=at
:
variables.
(4.2a)
Initial :
go=0
ET,=0
(4.2b)
At infinity:
go -+ 0
g,-+O
(4.2~)
agO a (z = 0) : x = a’-@
Electrode The kinetic
conditions
at z = 0 are:
Case 1A. Diffusion-limited: CaselB.
go = 1
(4.3)
~,(t)go+B2(t)g,=8,(t)-B,(t)p
Reversible:
The current
(4.2d)
simplifies
(4.4)
to (4.5)
z=o Applying a Laplace transform with respect to representations valid for both cases: z) =
$&,
z) = 1rss’/2
i;,(s)
(4.6a)
ss’/2
+)
ew(- zs”*/S)
(4.6b)
eqn. (4.6a) with (4.3) gives the diffusion-limited
current:
= ss-“*
Applying
the convolution
go(t70) = -&$t
-
&* /
gR(t, 0) = Substitution
or(t -
theorem
u)+*i(u) u)-l’*i(u)
to eqn. (4.6) at z = 0 gives du
du
into eqn. (4.4) gives an integral
1 l;j71~t(t-u)-“2i,,(u) This equation arbitrary 6*: i;,(s)
(4.2) and (4.5) leads
@) exp(- zs”*)
go(S,
Combining
to t to equations
= 7&‘*9
du=
has convolution
W) /j(t)
(4.8b) equation
for the reversible
current:
;i$$(t$!8 form and can be solved by Laplace
- W)P + @*(t)/a
(4.9) transforms
for
(4.10)
9
When S2 = 1, this simplifies to i;,(s)
= G,(s){
Jr(s)
-
(4.11)
J2WP)
in agreement with our main result, eqn. (3.8). (5) EXAMPLE:
SPHERICAL
DIFFUSION
The cell/electrode geometry is a spherical electrode of radius R, in a cell infinite in all directions. One-dimensional diffusion occurs in the radial direction. We choose L = ( A/‘T)“~
= 2Rs
(5.1)
corresponding to area normalized variables. The resulting system on l/2 d r < + cc is: Equation: ago a2go at=,,z+--
2 ago r ar
ag, -=
Initial :
go = 0
g,=o
(5.2b)
At infinity :
go-+0
&t-,0
(5.2~)
agO ar = a2%
(5.2d)
at
Electrode ( r = l/2) :
s2 -a% + 2 ag, r ar ( k3r2 i
(5.2a)
The kinetic conditions at r = l/2 are: Case 1A. Diffusion-limited: Case 1B. Reversible:
go = 1
(5.3)
8,(t)go+82(t)g,=8,(t)-62(t)p
(5.4)
The current simplifies to i(t)
ago = -77 7 ( 1 r= l/2
(5.5)
Applying a Laplace transform with respect to t to equations (5.2) and (5.5) leads to representations valid for both cases:
&(s, r) = &(s,
r) =
L(s) 47fr(l
+ s’12/2)
exp{ -(r-
r?(s) 47r8’r (1 + s’12/2S)
exp{ -(r-
1/2)s”‘)
1/2)~‘/~/6}
Combining eqn. (5.6a) with (5.3) gives the diffusion-limited i;*(S)
= $(1+
F/2)
(5.6a)
current: (5 *7)
10
Applying the convolution theorem to eqn. (5.6) at Y = l/2 gives go(t,
l/2) = ~~‘Fs{4(r-
gn(t,
l/2) = ~/01F’s{4~z(~-u)}i(tl)
~[&&)I
du
u)}i(u)
(5.8a) du
(5.8b) (5.8~)
= ~ 1 +lV
Substitution into eqn. (5.4) gives an integral equation for the reversible current: ;{[s,(t)F,(4(t-
u)} + @,(t)F,{4a”(t-
= O,(t) -
~)}]~m(~)
f322(l)P
du (5.9)
general solution appears difficult. However, for S2 = 1, the equation simplifies to convolution form:
The
:@{4(t
- U)}jia(U)
du = e,(t)
(5.10)
- fl,(t)p
Solving by Laplace transforms yields i;,(s)
= 277{1+ =&(s){
F/2} e;(s)
{ e,(s) -
- &;(s)p} (5.11)
~2WP)
in agreement with our main result, eqn. (3.8). (6) EXAMPLE:
CYLINDRICAL
DIFFUSION
The cell is the space between two parallel planes with a cylindrical electrode of radius R, and length H, between and perpendicular to the planes. One-dimensional diffusion occurs in the radial direction. We choose L=2R,
YC= Hc/2R,
(6.1)
where yc is a dimensionless parameter for electrode length. This characteristic length gives a simpler formulation than area normalized variables. The resulting system on l/2 < y < + a is: Equation:
ago a2go 1 ago -=a&t -=at ar2 +737 at
s2
-a*&!,+lag, i ar2 r lb
(6.2a) 1
Initial :
go=0
IT,=0
(6.2b)
At infinity :
go + 0
g,-+o
(6.2~)
11
Electrode (r=1/2):
%
=S2%
(6.2d)
The kinetic conditions at r = l/2 are: Case 1A. Diffusion-limited: Case 1B. Reversible:
go = 1
e,(t)g,
(6.3)
+ f12(l)gR = e,(t)
- 6,(t)p
(6.4)
The current simplifies to
ago
( 1
i(t) = -rye
(6.5)
ar
r=1/2
Applying a Laplace transform with respect to t to equations (6.2) and (6.5) leads to representations valid for both cases:
--i^(s) Ko(rP2) go(s, r) = ~ my@“2
&(s,
r) =
(6.6a)
K,‘( P2/2)
-L(S)
K,( rs”2/6)
~Sy,s”‘~
K; ( s”~/~S)
(6.6b)
where K,(z) is the modified Bessel function of order v with K,‘(z) Combining eqn. (6.6a) with (6.3) gives the diffusion-limited current: - rye i;&)
=
s1/2
K; (~l’~/2)
= -K,(z).
(6.7)
K&1/2/2)
Applying the convolution theorem to eqn. (6.6) at r = l/2 gives
go(t, l/2) = gR(tr l/2)
$-Jd&{4(tu)lj(u)
= $-/bFc{482(t
- u)}i(u)
du du
(6.8a) (6.8b)
(6.8c) Substitution into eqn. (6.4) gives an integral equation for the reversible current:
=4(t) - e,(tb
(6.9)
The general solution appears difficult. However, for S2 = 1, the equation simplifies to convolution form:
&-pi{4(t- u>)b(u)
du = e,(t) - e,(tb
(6.10)
12
Solving
by Laplace
transforms
yields
(6.11) in agreement
with out main result, eqn. (3.8).
(7) IMPLICATIONS
AND CONCLUSIONS
The results of this paper (given in eqn. 3.8 for the Laplace domain and in eqn. 3.9 for the time domain) demonstrate that the current for a reversible electron transfer . reactron z,a at any arbitrarily shaped electrode or collection of electrodes is a convolution of two functions, one function (i,,, the diffusion-limited current in response to a potential step) being dependent on electrode/cell geometry but independent of potential [15], the other function (6’,(t) - tY,(t)p) being dependent on the type of potential modulation but independent of electrode/cell geometry. The original work of Aoki et al. [l] and subsequent papers describing extensions of this earlier work [16-181 did not make this explicit connection between the diffusion-limited current (i,,) and the reversible current (i,,). Necessarily, the reversible current i,, will be a function of both the form of the potential modulation and the electrode/cell geometry. Certain difference techniques may, under a restricted set of conditions, yield voltammograms with shapes rather insensitive to electrode geometry. Such was found to be the case by Aoki et al. for square wave voltammetry, although even in this case the magnitude of the difference current was found to be a function of electrode geometry [l]. The significance of this result is its potential usefulness in the calculation of reversible behavior at planar microelectrodes where direct determination of i,, is complicated by the mixed geometric boundary conditions at such geometries [3,5]. We have previously demonstrated the efficiency and high accuracy with which files of ilA(t) vs. t or 6,(s) vs. s can be generated using the integral equation method for planar electrode geometries including band [3,4], ring [S], and disk [t]. With these files in hand, one needs only the appropriate expression for e,(t) (or e,(s)) to compute ire (recall that B,(t) = 1 - B,(t)). For example, for a linear potential sweep experiment at a planar microelectrode (any geometry) initiated at a potential E,, sweeping negatively at a rate V, in a solution initially containing only the oxidized form of the redox couple (C $ = 0, hence, p = 0), we have 8, =
1 +a
1 exp(-bt)
(7.1)
where a = exp[ nf( E, - E o ‘)] (a dimensionless starting potential) and b = nfVL*/D, (a dimensionless sweep rate). There are now two approaches by which one can obtain i,,. One approach is to solve the convolution integral (eqn. 3.9) numerically using a file of computed i,,(t)
13
values and the function 8,(t) (e.g. eqn. 7.1). This approach appears to be quite general, but does require a rather dense set of i,,(t) vs. r values. One advantage of the integral equation method is that as dense a set of such values as required can be generated from the Laplace domain coefficients with insignificant expenditures of computation time [S]. The second approach involves using the Laplace domain equation (3.8) which requires that we obtain &r(s)-values. For the example given in eqn. (7.1), e,(s) can be computed readily for appropriate values of s (for example, by expansion of eqn. 7.1 as an exponential series and inversion terms-wise), and &.r(,s) can thus be evaluated at selected s-values from eqn. (3.8). Indeed, it appears possible to compute l:,(s) for any potential modulation of electrochemical interest. A potential difficulty is the Laplace inversion of &a(s) to obtain ilB(t). We have shown that numerical inversion is possible for the potential step technique [3,4,5,19], and a similar (or possibly even the same) approach may be applicable to other forms of potential modulation. However, numerical inversion using ;a( s)-values located only on the real line (no complex values are obtained by this approach) is an ill-posed problem and extension of a method successful for one class of transforms to a broader class may require further development of the method. One way to implement the second (Laplace) approach which should permit use of present numerical inversion methods is to design e,(t) such that e,(s) is a rational function. In effect, we would be designing the experiment (potential modulation function) so as to simplify the theoretical computation of the response. This approach leads to rather unusual (yet entirely suitable) potential modulation functions, for example log-trig functions [20], but should facilitate comparisons of theoretical and experimental responses. Such experiments are not only possible with modern computer-based instrumentation [9], but exploit more fully the power of such instrumentation than do more traditional experiments. Work in each of the above areas is in progress and results will be reported in due course. ACKNOWLEDGEMENT
This research was supported by the National Science Foundation No. CHE-8711595.
through Grant
LIST OF SYMBOLS
Capitals (physical variables) A electrode area (m’) local concentrations of electroactive species 0, R (mol mP3) Co(T, X K Z), C,(T, X K Z) bulk concentrations of electroactive species 0, R (mol me3) c,*, C,* diffusion coefficients of electroactive species 0, R (m2 s-‘) Do, D,
14
electrode potential (V) formal potential (V) Faraday constant (96,485 C mol-‘) length (height) of cylindrical electrode (m) faradaic current (A) forward, backward kinetic coefficients for the electron transfer reaction (m s-t) standard rate constant for the electron transfer reaction (m
EC 1
E 0, F
HC I( ) K,, K, K0
SC’)
Ko( 1, K,‘( ) L
Z( > N 0 R
Rc RD Rs T
x,
K
z
modified Bessel function of order zero and derivative characteristic length associated with the cell/electrode
geome-
try (m) Laplace transform unit outer normal to the cell the oxidized form of the electroactive species the reduced form of the electroactive species (or the molar gas constant, 8.314 J mol-’ K-‘) radius of cylindrical electrode (m) radius of planar disk electrode (m) radius of spherical electrode (m) time (s) (or temperature in the definition f = F/RT) Cartesian coordinates (m)
Lower case (normalized or dimensionless variables) normalized Faraday constant, F/RT (V-‘) normalized variation about bulk concentration of species 0, R &I(& x, y> z), (eqn. 2.6) gR(t, XT Y, z) the value of go which solves the diffusion-limited case (case glA(f* x> Y? z)
f
g,*(t,
x7 Y, z)
i(t) IA
QA *IB
n r S
t tA
x, y, z
LA) the value of go = g, (for a2 = 1) which solves the reversible case (case 1B) normalized current (eqn. 2.6) area normalized current (eqn. 2.14) normalized current for the diffusion-limited case (case 1A) normalized current for the reversible case (case 1B) number of electrons transferred in the electrode reaction radial coordinate in spherical or cylindrical coordinate systems Laplace variable normalized time (eqn. 2.6) area normalized time (eqn. 2.14) normalized Cartesian coordinates (eqn. 2.6)
Greek symbols (dimensionless variables) transfer coefficient for the electrode reaction a dimensionless cylindrical electrode length (eqn. 6.1) YC
15
s2 77(t) 4( )9 4(
)
P
Note:
diffusion coefficient ratio, D,/D, (eqn. 2.6) normalized diffusive-kinetic parameter (eqn. 2.6) normalized kinetic coefficients (eqn. 2.6) normalized bulk concentration of the electroactive (eqn. 2.6)
Laplace transforms eqn. 3.5.
are designated
by &: 5, 8,, etc. See discussion
species
preceding
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
K. Aoki, K. Tokuda, H. Matsuda and J. Osteryoung, J. Electroanal. Chem., 207 (1986) 25. D.K. Cope and D.E. Tallman, J. Electroanal. Chem., 235 (1987) 97. S. Coen, D.K. Cope and D.E. Tallman, J. Electroanal. Chem., 215 (1986) 29. D.K. Cope, C.H. Scott, U. Kalapathy and D.E. Tallman, J. Electroanal. Chem., 280 (1990) 27. D.K. Cope, C.H. Scott and D.E. Tallman, J. Electroanal. Chem., 285 (1990) 49. U. Kalapathy, D.E. Tallman and D.K. Cope, J. Electroanal. Chem., 285 (1990) 71. J. He&e, J. Electroanal. Chem., 124 (1981) 73. D.K. Cope and D.E. Tallman, J. Electroanal. Chem., 285 (1990) 79. D.E. Tallman, G. Shepard and W.J. MacKellar, J. Electroanal. Chem., 280 (1990) 327. J.C. Myland and K.B. Oldham, J. Electroanal. Chem.. 185 (1985) 29. R.S. Nicholson and I. Sham, Anal. Chem., 36 (1964) 706. K. Aoki, K. Honda, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 182 (1985) 267. S. Sujaritvanichpong, K. Aoki, T. Tokuda and H. Matsuda, J. Electroanal. Chem., 199 (1986) 2 71. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Wiley, New York, 1980, Ch. 3. D.K. Cope and D.E. Tallman, J. Electroanal. Chem., 285 (1990) 85. K. Aoki and J. Osteryoung, J. Electroanal. Chem., 240 (1988) 45. 2. Stojek, J. Electroanal. Chem., 256 (1988) 283. K. Aoki, K. Maeda and J. Osteryoung, J. Electroanal. Chem., 272 (1989) 17. D.K. Cope, SIAM J. Num. Anal., 27 (1990) 1345. D.K. Cope, U. Kalapathy and D.E. Tallman, work in progress.
R
Representation of reversible current for arbitrary electrode geometries and arbitrary potential modulation Davis K. Cope Department
of Mathematics, North Dakota State University, Fargo, ND 58105 (USA)
Dennis E. Tallman * Department
of Chemistry, North Dakota State University, Fargo, ND 58105 (USA)
(Received 18 June 1990; in revised form 12 September 1990)
Abstract For simple electron transfer 0+ n e- + R with equal diffusion coefficients (Do = DR), it is shown that reversible current can be expressed in terms of diffusion-limited current for arbitrary cell/electrode geometries and arbitrary time-dependent potentials. This result provides a better understanding of a formula due to Aoki and co-workers and suggests new experimental approaches which simplify interpretation of data obtained at microelectrodes.
(1) INTRODUCTION
This paper derives a general result for simple electron transfer 0 + n e- + R: if the diffusion coefficients are equal, then the reversible current i,,(t) can be expressed in terms of the diffusion-limited current i,*(t) by the simple formula
(1.1) Here B,(t) - L$(t)p is determined by the time-dependent potential (notation is defined in the next section). This result holds for arbitrary time-dependent potentials and for arbitrary (unbounded) cell and electrode geometries, including arrays and ensembles as long as the same potential is applied across all electrodes. Aoki et al. [l] obtained a result analogous to eqn. (1.1) but with a complicated interpretation. Their derivation, like ours, assumed equal diffusion coefficients and
l
To whom correspondence should be addressed.
0022-0728/91/$03.50
0 1991 - Elsevier Sequoia S.A.
2
allowed arbitrary cell/electrode geometries and arbitrary time-dependent potentials. However, the diffusion-limited current ii*(t) did not appear as such. Instead, a function f(t) occurs which is defined as the inverse Laplace transform of the integral over the electrode of the “inverse kernel” for the Green’s function for the transformed diffusion problem on the given cell geometry (see the Appendix of ref. 1, especially eqns. (Al&A20) and discussion). Examples of f(t) are given for large planar, spherical, cylindrical, and small disk electrodes which show that f(t) is essentially the diffusion-limited current for these special cases. In this paper we give a simple derivation of eqn. (1.1) which leads directly to the general identification of f( t ) with the diffusion-limited current. We have previously shown that, for equal diffusion coefficients, reversible current in response to a potential step can be expressed in terms of diffusion-limited current (and quasi-reversible current can be expressed in terms of totally irreversible current) for arbitrary cell/electrode geometries [2]. The results derived in this paper extend this previous result to arbitrary time-dependent potentials. Diffusion-limited current is typically easier to compute than reversible current arising from a time-dependent potential. Our result provides a direct means of extending calculations of diffusion-limited current for the band [3,4], ring [5,6], and disk [7,8] to the reversible case with time-dependent potential. The result also offers opportunities for experimental exploitation. The reversible current response to any time-dependent potential is the diffusion-limited current (in response to a potential step) convolved with the potential-dependent function as in eqn. (1.1). In the original derivation, Aoki et al. applied the result to square-wave voltammetry [l]. Recent advances in computer-based electrochemical instrumentation employing high speed direct memory access techniques [9] make possible the application of arbitray potential modulations on millisecond and microsecond time scales. Such experimental capability makes possible the design and implementation of potential modulations which optimize experimental measurement of desired electrochemical parameters or which simplify rigorous comparisons of experiment with theory at microelectrodes. The arbitrary potential modulation permitted by our result is therefore a generalization with experimental significance. The practical significance of the restriction to equal diffusion coefficients depends on the reaction. In many cases, simple electron transfer will cause little change in the properties affecting diffusivity. A brief discussion on this topic including quantitative examples has been provided previously [lo]. In addition to the representation for reversible current, we wish to emphasize the general formulation of simple electron transfer given below. It is shown that the physical formulation can always be reduced to dimensionless form by selection of a single characteristic length for the cell/electrode geometry. A dimensionless formulation provides an especially simple description both for calculations and for the derivation of general results, as done here. A particular choice of characteristic length yields area normalized variables, which are especially suitable for comparing behavior of different cell/electrode geometries. For rectilinear diffusion (shielded planar electrode), it is known (see ref. 11 and
3
references therein) that a representation for reversible current can be found for arbitrary potential-time dependence and arbitrary diffusion coefficients. The simple geometry permits this strong result; the representation reduces to our result when the diffusion coefficients are equal. For spherical diffusion, representations for reversible current for arbitrary potential-time dependence have been found but appear always to involve equal diffusion coefficients. For example, Myland and Oldham [lo] obtain a representation for reversible current at a spherical electrode in terms of reversible current at a rectilinear electrode for arbitrary potential-time dependence, but the relation is complicated and equal diffusion coefficients are introduced for simplification. For cylindrical diffusion, the representation obtained here has been used for linear sweep voltammetry and normal and differential pulse voltammetry [12,13], but always with the assumption of equal diffusion coefficients. The second section of this paper gives a general formulation of simple electron transfer for arbitrary cell/electrode geometries and arbitrary time-dependent potential. The third section derives the relation between reversible and diffusion-limited current. The remaining sections illustrate the result for rectilinear, spherical, and cylindrical geometries.
(2) GENERAL FORMULATION FOR SIMPLE ELECTRON TRANSFER
This section provides notation and a general formulation for simple electron transfer. Upper case letters are typically used for physical quantities and lower case for dimensionless quantities. Although this paper is concerned only with diffusionlimited and reversible reactions, the formulation will include totally irreversible and quasi-reversible reactions for completeness and future reference. The electrochemical cell is assumed unbounded and of arbitrary shape with one or more electrodes of arbitrary shape and arrangement on the cell boundary. Cartesian coordinates X, Y, Z are used for position within the cell. The interior of the cell is filled with electrolyte containing solvable electrolyzable species 0 and R with initially uniform concentrations C,* and C,* and diffusion coefficients Do and D,, respectively. The species R is allowed to be present for generality. The diffusion coefficients are assumed independent of concentration and migration of 0 and R is assumed negligible. We further assume that the potential across the electrode surface(s) is spatially uniform at each instant in time, a reasonable assumption for non-resistive solutions. This last assumption does not, of course, imply uniform flux or current distribution across the electrode(s). The electrodes are initially poised at a potential such that no reaction occurs (an equilibrium potential if R is present). At time T = 0, the potential is changed and simple electron transfer 0 + n e-+ R takes place with forward and backward kinetic parameters K, and K, respectively. We are concerned with two cases: (1) a potential step to an extreme value where the reaction is diffusion-limited; (2) a time-dependent potential E(T) producing a reversible reaction. An equal potential is applied across all electrodes but the time-dependence is arbitrary. The concentra-
4
tions Co(T, system:
X, Y, 2)
and
C,(T,
X, Y, Z)
then
satisfy
the following
diffusion
~=Dov2Co
-=D,v2C, 3T
ac,
(2.la)
Initial :
c,=c,*
c,=c,*
(2Sb)
At infinity:
co -+ co*
c,+c,*
(2Sc)
ace= 0
m=O
ac,
(2.ld)
ac0
Insulated
boundary:
Electrode(s) Kinetic
-
aN
ace+&a~3%
:
Dow
conditions
at the electrode(s)
Case 1A. Diffusion-limited: Case 1B. Reversible: Case 2A. Totally
=0
yield:
Co = 0
(2.2a)
K, Co - K,CR = 0
irreversible:
Case 2B. Quasi-reversible:
7 Do% -Do%
(2.le)
(2.2b) = K,C,
= K,C,
- K,C,
(2.2c) (2.2d)
Here a/aN is the normal derivative N. v, where N is the unit outer normal to the cell. The above equations assume all electrodes share the same potential; otherwise, separate conditions would be stated at individual electrodes. The current is given by: I(T)
= -n~Do~lW~rO~.rSj~
(2.3)
dA
Notice that the kinetic conditions (2.2a, c) for the diffusion-limited and totally irreversible cases involve only the concentration Co and the system can therefore be reduced to a single equation for Co. The kinetic condition (2.2b) for the reversible case uses only the ratio Kf/Kb, which is related to the potential E(T) by the Nernst equation: KF/Kb=exp(nf{Eo’-E(T)})
(2.4)
where the notation has its customary meaning (formal potential E “‘, f = F/RT for temperature T, etc.). For totally irreversible and quasi-reversible processes, potential dependence of the individual kinetic coefficients is normally modeled by the Butler-Volmer equations [14]: * K,=K” K,=K”
exp(-anf{E(T)-EO’}) exp((l-cu)nf{E(T)-EO’})
Let L be a characteristic radius of a planar disk,
O
(2.5)
length associated with the cell/electrode geometry (e.g. width of a planar band, diameter of a spherical or
cylindrical electrode, etc.). We introduce dimensionless variables by: t = DOT/L2
i(t) = I( T)/nFD&$L
p = c,*/c,”
S2 = D,/D,
4(t)
= W(Kb
+ 4)
e,(t)
=
Kb/(Kt!
+
Kf)
(2.6)
clz=G(P+gR)
co=G(l-80)
x, y, z = X/L,
Y/L,
Z/L
v(t)
= Do/L&
Notice that d,(t) + O=(t) = 1. The functions go and g, correspond to relative changes around the bulk concentrations. Depending on the cell/electrode geometry, additional dimensionless parameters specifying shape and position may occur (e.g. a planar ring electrode requires two lengths to specify its shape; scaling out a characteristic length will still leave one dimensionless parameter necessary to specify the shape). A general dimensionless formulation for simple electron transfer is then:
ago=v=go
Equation: Initial
:
At infinity
:
Electrode(s)
at
go=0
gR
=
gR
+
go
Insulated boundary:
%I,
at
+
0
ago 0
=
(2.7~)
o
aN
(2.7a) (2.7b)
o
GR -=
-= aN
a202gR
O
(2.7d) (2.7e)
:
For a potential applied across all electrodes equally, the kinetic conditions electrodes become: Case 1A. Diffusion-limited: go=1
at the
(2.8)
Case 1B. Reversible: e,(t>gO+e2(t)gR=el(t)-e2(t>p
(2.9)
Case 2A. Totally irreversible: (2.10) Case 2B. Quasi-reversible:
dt)el(r)~+edt>gO +
e2(t)gR=
edt)
-e2(t)p
(2.11)
The dimensionless current is given by:
i(t)=/.,,,,,,,,,,%da
(2.12)
6
Particular cell/electrode geometries may allow many possible choices for a characteristic length L. One choice of particular significance expresses the characteristic length in terms of total electrode area A. This choice may not lead to the simplest formulation, but it does provide a common scaling meaningful across different geometries and is especially suitable for comparison of behavior between geometries. Specifically, we define area normalized variables by the choice (2.13)
L = (A/7$‘* iA( tA) = I( T)/nFDoC$
t, = nDoT/A
( A/r)l'*
(2.14)
The factor ~7is included to simplify the definition for some basic geometries: for the disk, L = (A/a)‘/* = R,, the radius of the disk; for the sphere, L = ( A/r)‘12 = 2R,, the diameter of the sphere. Such area normalized variables have been used in comparing diffusion-limited current at disk, ring, and band electrodes [15], where the definitions are the same as eqn. (2.14) except for a factor of rl’* in defining i,. The definitions of eqn. (2.14) follow directly from the choice of L in eqn. (2.13) and the general resealing introduced in eqn. (2.6). (3) GENERAL
REPRESENTATION
FOR REVERSIBLE
CURRENT
Our main result follows quickly from the general formulation of the last section. Consider the diffusion problem:
ag x=v*g
(3.la)
Initial:
g=o
(3.lb)
At infinity:
g+o
(3.lc)
Insulated boundary:
ag = 0 a~
(3.ld)
Let glA(t, x, Y, 2) and gm(t, x, y, z) be solutions to this problem satisfying the additional boundary condition on the electrode(s): Case 1A:
g,, = 1
Case 1B:
&B
=
e,(t)
(3.2) -
02tt)p
(3.3)
The current in each case is given by (3.4) and we write i,,(t), i,,(t) for the respective currents. Observe that (1) glA(t, x, y, z) = go solves the diffusion-limited case defined by eqns. (2.7) and (2.8);
(2) g&t, x, y, z) = go = g, solves the reversible case defined (2.9) with 6* = 1. Now apply a Laplace transform with respect to t, x, y, z)] =8( s, x, y, z) = j+%g(t,
=qg(t,
by eqns. (2.7) and
x, y, z) dt
0
to eqn. (3.1) to obtain Equation
:
sg=v*g
(3Sa) (3.5b)
boundary:
jj bounded *1 $$ = 0
At infinity: Insulated
(3.5c)
where g,A and jiB are solutions electrode(s): Case 1A:
&A = l/s
Case 1B:
&a = 4<4
satisfying
-
- &(4P
8,WP}&A(S,
&lb)
x, Y, z> =s{e,(s) =
s{
4(4
Equivalently, g&x,
-
42mJ)
x,
-
Y,
on the
that
z>
e2(4P}&,(s,
x,
y,
gta. for the diffusion-limited x, y, z) = go = g, be the Then:
2)
(3.8)
Lb)
by the convolution
Y, 4 = &i’gi&
condition
(3.7) observe
satisfies eqns. (3.5) and (3.7) and is therefore the solution Altogether: let glA(t, x, y, z) = go be the solution current in eqns. (2.7) and (2.8) and, for 6* = 1, let gia(t, solution for the reversible current in eqns. (2.7) and (2.9). &*(s,
boundary
(3.6)
We now derive our main result: g = s{ e,(s)
the additional
theorem, ~9 ~9 Y, 4{ e,(u)
- &(u)P}
du (3.9)
&a(t)
= $1’4,
(t-4(4(4
-GYP)
du
The following sections illustrate this result by showing case in rectilinear, spherical, and cylindrical diffusion.
how it arises as a special
(4) EXAMPLE: RECTILINEAR DIFFUSION
The cell/electrode geometry is a planar electrode of area A and arbitrary shape with shielded edges, i.e. the cell is a cylinder perpendicular to the electrode forming its base. One-dimensional diffusion occurs in the Z-direction. We choose L = ( A/7ry2
(4.1)
8
corresponding Equation
to area normalized
The resulting
system on 0 d z < + cc is:
a*&3 az2
ago -=at
:
variables.
(4.2a)
Initial :
go=0
ET,=0
(4.2b)
At infinity:
go -+ 0
g,-+O
(4.2~)
agO a (z = 0) : x = a’-@
Electrode The kinetic
conditions
at z = 0 are:
Case 1A. Diffusion-limited: CaselB.
go = 1
(4.3)
~,(t)go+B2(t)g,=8,(t)-B,(t)p
Reversible:
The current
(4.2d)
simplifies
(4.4)
to (4.5)
z=o Applying a Laplace transform with respect to representations valid for both cases: z) =
$&,
z) = 1rss’/2
i;,(s)
(4.6a)
ss’/2
+)
ew(- zs”*/S)
(4.6b)
eqn. (4.6a) with (4.3) gives the diffusion-limited
current:
= ss-“*
Applying
the convolution
go(t70) = -&$t
-
&* /
gR(t, 0) = Substitution
or(t -
theorem
u)+*i(u) u)-l’*i(u)
to eqn. (4.6) at z = 0 gives du
du
into eqn. (4.4) gives an integral
1 l;j71~t(t-u)-“2i,,(u) This equation arbitrary 6*: i;,(s)
(4.2) and (4.5) leads
@) exp(- zs”*)
go(S,
Combining
to t to equations
= 7&‘*9
du=
has convolution
W) /j(t)
(4.8b) equation
for the reversible
current:
;i$$(t$!8 form and can be solved by Laplace
- W)P + @*(t)/a
(4.9) transforms
for
(4.10)
9
When S2 = 1, this simplifies to i;,(s)
= G,(s){
Jr(s)
-
(4.11)
J2WP)
in agreement with our main result, eqn. (3.8). (5) EXAMPLE:
SPHERICAL
DIFFUSION
The cell/electrode geometry is a spherical electrode of radius R, in a cell infinite in all directions. One-dimensional diffusion occurs in the radial direction. We choose L = ( A/‘T)“~
= 2Rs
(5.1)
corresponding to area normalized variables. The resulting system on l/2 d r < + cc is: Equation: ago a2go at=,,z+--
2 ago r ar
ag, -=
Initial :
go = 0
g,=o
(5.2b)
At infinity :
go-+0
&t-,0
(5.2~)
agO ar = a2%
(5.2d)
at
Electrode ( r = l/2) :
s2 -a% + 2 ag, r ar ( k3r2 i
(5.2a)
The kinetic conditions at r = l/2 are: Case 1A. Diffusion-limited: Case 1B. Reversible:
go = 1
(5.3)
8,(t)go+82(t)g,=8,(t)-62(t)p
(5.4)
The current simplifies to i(t)
ago = -77 7 ( 1 r= l/2
(5.5)
Applying a Laplace transform with respect to t to equations (5.2) and (5.5) leads to representations valid for both cases:
&(s, r) = &(s,
r) =
L(s) 47fr(l
+ s’12/2)
exp{ -(r-
r?(s) 47r8’r (1 + s’12/2S)
exp{ -(r-
1/2)s”‘)
1/2)~‘/~/6}
Combining eqn. (5.6a) with (5.3) gives the diffusion-limited i;*(S)
= $(1+
F/2)
(5.6a)
current: (5 *7)
10
Applying the convolution theorem to eqn. (5.6) at Y = l/2 gives go(t,
l/2) = ~~‘Fs{4(r-
gn(t,
l/2) = ~/01F’s{4~z(~-u)}i(tl)
~[&&)I
du
u)}i(u)
(5.8a) du
(5.8b) (5.8~)
= ~ 1 +lV
Substitution into eqn. (5.4) gives an integral equation for the reversible current: ;{[s,(t)F,(4(t-
u)} + @,(t)F,{4a”(t-
= O,(t) -
~)}]~m(~)
f322(l)P
du (5.9)
general solution appears difficult. However, for S2 = 1, the equation simplifies to convolution form:
The
:@{4(t
- U)}jia(U)
du = e,(t)
(5.10)
- fl,(t)p
Solving by Laplace transforms yields i;,(s)
= 277{1+ =&(s){
F/2} e;(s)
{ e,(s) -
- &;(s)p} (5.11)
~2WP)
in agreement with our main result, eqn. (3.8). (6) EXAMPLE:
CYLINDRICAL
DIFFUSION
The cell is the space between two parallel planes with a cylindrical electrode of radius R, and length H, between and perpendicular to the planes. One-dimensional diffusion occurs in the radial direction. We choose L=2R,
YC= Hc/2R,
(6.1)
where yc is a dimensionless parameter for electrode length. This characteristic length gives a simpler formulation than area normalized variables. The resulting system on l/2 < y < + a is: Equation:
ago a2go 1 ago -=a&t -=at ar2 +737 at
s2
-a*&!,+lag, i ar2 r lb
(6.2a) 1
Initial :
go=0
IT,=0
(6.2b)
At infinity :
go + 0
g,-+o
(6.2~)
11
Electrode (r=1/2):
%
=S2%
(6.2d)
The kinetic conditions at r = l/2 are: Case 1A. Diffusion-limited: Case 1B. Reversible:
go = 1
e,(t)g,
(6.3)
+ f12(l)gR = e,(t)
- 6,(t)p
(6.4)
The current simplifies to
ago
( 1
i(t) = -rye
(6.5)
ar
r=1/2
Applying a Laplace transform with respect to t to equations (6.2) and (6.5) leads to representations valid for both cases:
--i^(s) Ko(rP2) go(s, r) = ~ my@“2
&(s,
r) =
(6.6a)
K,‘( P2/2)
-L(S)
K,( rs”2/6)
~Sy,s”‘~
K; ( s”~/~S)
(6.6b)
where K,(z) is the modified Bessel function of order v with K,‘(z) Combining eqn. (6.6a) with (6.3) gives the diffusion-limited current: - rye i;&)
=
s1/2
K; (~l’~/2)
= -K,(z).
(6.7)
K&1/2/2)
Applying the convolution theorem to eqn. (6.6) at r = l/2 gives
go(t, l/2) = gR(tr l/2)
$-Jd&{4(tu)lj(u)
= $-/bFc{482(t
- u)}i(u)
du du
(6.8a) (6.8b)
(6.8c) Substitution into eqn. (6.4) gives an integral equation for the reversible current:
=4(t) - e,(tb
(6.9)
The general solution appears difficult. However, for S2 = 1, the equation simplifies to convolution form:
&-pi{4(t- u>)b(u)
du = e,(t) - e,(tb
(6.10)
12
Solving
by Laplace
transforms
yields
(6.11) in agreement
with out main result, eqn. (3.8).
(7) IMPLICATIONS
AND CONCLUSIONS
The results of this paper (given in eqn. 3.8 for the Laplace domain and in eqn. 3.9 for the time domain) demonstrate that the current for a reversible electron transfer . reactron z,a at any arbitrarily shaped electrode or collection of electrodes is a convolution of two functions, one function (i,,, the diffusion-limited current in response to a potential step) being dependent on electrode/cell geometry but independent of potential [15], the other function (6’,(t) - tY,(t)p) being dependent on the type of potential modulation but independent of electrode/cell geometry. The original work of Aoki et al. [l] and subsequent papers describing extensions of this earlier work [16-181 did not make this explicit connection between the diffusion-limited current (i,,) and the reversible current (i,,). Necessarily, the reversible current i,, will be a function of both the form of the potential modulation and the electrode/cell geometry. Certain difference techniques may, under a restricted set of conditions, yield voltammograms with shapes rather insensitive to electrode geometry. Such was found to be the case by Aoki et al. for square wave voltammetry, although even in this case the magnitude of the difference current was found to be a function of electrode geometry [l]. The significance of this result is its potential usefulness in the calculation of reversible behavior at planar microelectrodes where direct determination of i,, is complicated by the mixed geometric boundary conditions at such geometries [3,5]. We have previously demonstrated the efficiency and high accuracy with which files of ilA(t) vs. t or 6,(s) vs. s can be generated using the integral equation method for planar electrode geometries including band [3,4], ring [S], and disk [t]. With these files in hand, one needs only the appropriate expression for e,(t) (or e,(s)) to compute ire (recall that B,(t) = 1 - B,(t)). For example, for a linear potential sweep experiment at a planar microelectrode (any geometry) initiated at a potential E,, sweeping negatively at a rate V, in a solution initially containing only the oxidized form of the redox couple (C $ = 0, hence, p = 0), we have 8, =
1 +a
1 exp(-bt)
(7.1)
where a = exp[ nf( E, - E o ‘)] (a dimensionless starting potential) and b = nfVL*/D, (a dimensionless sweep rate). There are now two approaches by which one can obtain i,,. One approach is to solve the convolution integral (eqn. 3.9) numerically using a file of computed i,,(t)
13
values and the function 8,(t) (e.g. eqn. 7.1). This approach appears to be quite general, but does require a rather dense set of i,,(t) vs. r values. One advantage of the integral equation method is that as dense a set of such values as required can be generated from the Laplace domain coefficients with insignificant expenditures of computation time [S]. The second approach involves using the Laplace domain equation (3.8) which requires that we obtain &r(s)-values. For the example given in eqn. (7.1), e,(s) can be computed readily for appropriate values of s (for example, by expansion of eqn. 7.1 as an exponential series and inversion terms-wise), and &.r(,s) can thus be evaluated at selected s-values from eqn. (3.8). Indeed, it appears possible to compute l:,(s) for any potential modulation of electrochemical interest. A potential difficulty is the Laplace inversion of &a(s) to obtain ilB(t). We have shown that numerical inversion is possible for the potential step technique [3,4,5,19], and a similar (or possibly even the same) approach may be applicable to other forms of potential modulation. However, numerical inversion using ;a( s)-values located only on the real line (no complex values are obtained by this approach) is an ill-posed problem and extension of a method successful for one class of transforms to a broader class may require further development of the method. One way to implement the second (Laplace) approach which should permit use of present numerical inversion methods is to design e,(t) such that e,(s) is a rational function. In effect, we would be designing the experiment (potential modulation function) so as to simplify the theoretical computation of the response. This approach leads to rather unusual (yet entirely suitable) potential modulation functions, for example log-trig functions [20], but should facilitate comparisons of theoretical and experimental responses. Such experiments are not only possible with modern computer-based instrumentation [9], but exploit more fully the power of such instrumentation than do more traditional experiments. Work in each of the above areas is in progress and results will be reported in due course. ACKNOWLEDGEMENT
This research was supported by the National Science Foundation No. CHE-8711595.
through Grant
LIST OF SYMBOLS
Capitals (physical variables) A electrode area (m’) local concentrations of electroactive species 0, R (mol mP3) Co(T, X K Z), C,(T, X K Z) bulk concentrations of electroactive species 0, R (mol me3) c,*, C,* diffusion coefficients of electroactive species 0, R (m2 s-‘) Do, D,
14
electrode potential (V) formal potential (V) Faraday constant (96,485 C mol-‘) length (height) of cylindrical electrode (m) faradaic current (A) forward, backward kinetic coefficients for the electron transfer reaction (m s-t) standard rate constant for the electron transfer reaction (m
EC 1
E 0, F
HC I( ) K,, K, K0
SC’)
Ko( 1, K,‘( ) L
Z( > N 0 R
Rc RD Rs T
x,
K
z
modified Bessel function of order zero and derivative characteristic length associated with the cell/electrode
geome-
try (m) Laplace transform unit outer normal to the cell the oxidized form of the electroactive species the reduced form of the electroactive species (or the molar gas constant, 8.314 J mol-’ K-‘) radius of cylindrical electrode (m) radius of planar disk electrode (m) radius of spherical electrode (m) time (s) (or temperature in the definition f = F/RT) Cartesian coordinates (m)
Lower case (normalized or dimensionless variables) normalized Faraday constant, F/RT (V-‘) normalized variation about bulk concentration of species 0, R &I(& x, y> z), (eqn. 2.6) gR(t, XT Y, z) the value of go which solves the diffusion-limited case (case glA(f* x> Y? z)
f
g,*(t,
x7 Y, z)
i(t) IA
QA *IB
n r S
t tA
x, y, z
LA) the value of go = g, (for a2 = 1) which solves the reversible case (case 1B) normalized current (eqn. 2.6) area normalized current (eqn. 2.14) normalized current for the diffusion-limited case (case 1A) normalized current for the reversible case (case 1B) number of electrons transferred in the electrode reaction radial coordinate in spherical or cylindrical coordinate systems Laplace variable normalized time (eqn. 2.6) area normalized time (eqn. 2.14) normalized Cartesian coordinates (eqn. 2.6)
Greek symbols (dimensionless variables) transfer coefficient for the electrode reaction a dimensionless cylindrical electrode length (eqn. 6.1) YC
15
s2 77(t) 4( )9 4(
)
P
Note:
diffusion coefficient ratio, D,/D, (eqn. 2.6) normalized diffusive-kinetic parameter (eqn. 2.6) normalized kinetic coefficients (eqn. 2.6) normalized bulk concentration of the electroactive (eqn. 2.6)
Laplace transforms eqn. 3.5.
are designated
by &: 5, 8,, etc. See discussion
species
preceding
REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
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R