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The behavior of microdisk and microring electrodes
225
J. Electroanal. Chem., 263 (1989) 225-236 Elsevier Sequoia S.A., Lausanne - Printed
in The Netherlands
The behavior of microdisk and microring electrodes Prediction of the chronoamperometric response of microdisks and of the steady state for CE and EC catalytic reactions by application of Neumann’s integral theorem M. Fleischmann, Derek Pletcher and Guy Denuault Department
of Chemistry,
The University, Southampton,
Hunts. SO9 SNH (Great Britain)
John Daschbach and Stanley Pons * Department (Received
of Chemistry, 8 August
University of Utah, Salt Lake City, UT 84112 (lJ.S.A)
1988; in revised form 25 November
1988)
ABSTRACT It is shown that the chronoamperometric response at microdisk electrodes can be derived from Neumann’s integral theorem of two parameters. The form of the transients can be predicted for a wide range of boundary conditions and this is illustrated by the relaxation behavior of irreversible reactions in addition to that of the widely investigated example of zero surface concentration of the reactant. Corrections to the transitions due to the tertiary current distribution are derived for relaxation experiments and it is shown that the methods developed can also be applied to linear sweep voltammetry. The steady state behavior of CE and EC catalytic reactions is formally similar to the chronoamperometry of irreversible electrode reactions and it is shown that complete descriptions can be obtained for the voltammetry of such systems.
INTRODUCTION
We have shown recently that the steady state behavior of simple reactions at microdisk electrodes can be predicted [1,2] by the application of Neumann’s integral theorem [3]. The concentration distribution for a reactant (or product) in such a reaction is governed by &!?+!?ac+&!L(J lb+’
r
ar
* To whom correspondence
0022-0728/89/%03.50
0)
a.z2
should be addressed.
Q 1989 Elsevier Sequoia
S.A.
226
and we obtain
c(r, z) =
ImadajaexP(;ar) 0
~JO(c&(~p)p
0
dp
(2)
Equation (2) makes a formal linkage between the distribution of sources, Q(p), (or sinks) at the radial position p and the concentration throughout space. We have shown that the use of the simple form for the distribution of sinks
-Q(e)
=
a
ist8){ ca + c* cos( 8) + c2 cos(28)
+ . . . + c, cos( ne)}
(3)
where p=a
sin(f3)
(4)
leads to the recovery of known results [4,5] (derived by the application of the discontinuous integrals of Bessel functions) for constant surface concentrations provided we assume c, = c2 = c3 = . - . = c, = 0
(5)
and co =
+qc-c”)
and of constant, uniform, surface flux provided we assume c0 = c2 = cj = . 1. = c, = 0 and ci = Qu It has also been shown that the reexpression of eqn. (3) in the form
-Q(e)
= a
c;siej (do +
d, c0s(e) + -.
is particularly useful in discussing conditions such as O
D
+d,
c0sye))
the application
of more general
z=O
boundary
(10)
The use of eqn. (9) in eqn. (2) leads to the simple result
c(r)
= -
(11)
(at z = 0) where I is the gamma function and *Fl denotes the appropriate hypergeometric function. Application of eqn. (11) at (n + 1) radial positions in any particular boundary condition (such as eqn. 10) then allows the evaluation of the
221
unknown coefficients in eqn. (9) thereby giving a complete solution for any particular problem. The results obtained by the application of Neumann’s integral theorem have been related also to results derived by an extension [6] of the application of the discontinuous integrals of Bessel functions [4,5]. We have shown also that the use of Neumann’s integral allows the evaluation of the effects of the distribution of potential in the solution (the tertiary current distribution) and of the non-linearities in the concentration terms of the electrode reactions [1,2]. In this paper we extend the approach to the discussion of the chronoamperometric response of microdisk electrodes, comment on the effects of the distribution of potential in the solution, and outline the analysis of linear sweep voltammetric experiments. In view of the formal similarity of the derived equations to those obtained for the steady state behavior of reactions following the CE and EC mechanisms, we also derive the relevant results for these reaction pathways. CHRONOAMPEROMETRY
We consider first of all the behavior of a simple diffusion controlled where mass transfer of the reactant is controlled by
reaction
(12) subject to the initial condition c=P
,
O-crcco,
zb-0,
t=O
and the boundary conditions c=O, D$=O,
Oa,
z=O,
t>O
z = 0,
t>o
Laplace transformation
of eqn. (12) subject to eqn. (13) gives
where s is the variable of the Laplace transformation. The solution of eqn. (16) can therefore be expressed in terms of Neumann’s integral of two parameters (p, s) using Q(p, S) as p)p
dp (17)
By analogy with the expansion (9) used to describe the distribution steady state, we write
-Q
a
--&) (d,(s)
of sinks in the
+ c&(s)co@) + . * * +&(s) COW))
228
where all the coefficients di are now functions of s. Substitution (4) in eqn. (17) gives at z = 0 c(r,
s) = $
- “/ D 0
Q) J,(cur)cr
of eqns. (18) and
dcu
[a2+s/D]“2
X[“td,(s)J,(aa
sin(t9)) sign
cosi(8)
de
j=O O" 4("')J((j+l)/2) [
for a distribution of sinks a(@)
(~444
(19)
a2 + s,D]1/2(aa)((j-1)/2)
over the surface where
ej = 2(cj-‘)/2)r(( j + 1)/2)
(20) The coefficients c?~(s) can then be obtained by substituting eqns. (18) and (19) into the Laplace transform of the boundary condition appropriate to the particular form of the experiment. For example, for the simple condition F=O,
O
z=O
(21)
(cf. eqn. 14) we solve the set of equations
n dT,(s)ej
c-
j=O
m J0(~~k)Jc(j+1)/2)
J o
Da
(444
[a2 +s,D]1/2(cya)((j-1)/2)
coo =
(22)
s
at (n + 1) radial positions. We therefore obtain the unknown coefficients as functions of the parameter s. Substitution of these coefficients into eqn. (18) and integration over the radius of the disk then gives the Laplace transform of the total rate of reaction, R, rdr
%=2nD
@‘_
= 27i-aJ$o[‘2Zj(s)
cod(e)
r2)t/2
sin(@) de
n cJj(S) =277aC
-
(23)
j=O(i+l)
Numerical inversion of eqn. (23) gives the total rate of reaction in the t-domain. An alternative procedure is to invert eqn. (22) to the relevant convolution integrals
= 1
(24)
229
In the numerical integration over say 1 intervals of A7 we then have to solve the set of eqns. (24) for the values dj in the Ith interval. This forward integration is simplified by the fact that we know that the system follows the Cottrell equation at short times [7] for which at sufficiently large s
d,(s) = 5
(25)
and do=d2=d3=
(26)
. . . =d,=O
OTHER BOUNDARY CONDITIONS; SWEEP VOLTAMMETRY
THE TERTIARY
CURRENT
DISTRIBUTION;
LINEAR
We have pointed out elsewhere that the application of the boundary condition (14) (or, more generally, of a constant surface concentration boundary condition) is unrealistic as the flux would have to become infinite at the edges of the disk [1,2,4,8] (see also ref. 6). However, the flux cannot become infinite for two reasons: firstly because the rate constant governing the electrochemical reaction must itself be finite and, secondly, because an infinite rate of reaction would also require an infinite overpotential at the edges of the disk. The current distribution must therefore be more uniform than that predicted using the boundary condition (14) under most experimental conditions (the tertiary current distribution) as we have shown in the previous paper. As the application of Neumann’s integral theorem provides a link between the assumed form of the flux distribution, eqn. (18), and the concentration distribution, eqn. (19), it is straightforward to assess the effects of boundary conditions more complex than eqn. (14). We restrict attention here to irreversible reactions which are described by eqn. (10). Substitution of cqn. (19) into the Laplace transform of eqn. (10) gives
[,,s)+,,.,(~~‘*+~2~s)(~) D(y; “’)= _1r2)l,2 +...+JJq_jLy”] (u2
= a
,d,& (Jo(s) + c&(s)cos(8) + . . * +d,(s) COW)) (27)
The dependence of the coefficients gj on s can therefore be obtained as for the case of zero surface concentration governed by eqn. (22). The distribution of the flux and
230
concentration and the total rate of reaction can then be derived for the Laplace plane using eqns. (18), (19), and (23). Numerical inversion gives the dependencies in the t-plane. It therefore becomes possible to assess the effects of changes in k on the behavior of the system and the relaxation of the rate of reaction under potentiostatic conditions can be evaluated. The application of Neumann’s integral theorem also allows the evaluation of the potential distribution in the solution, as in the case of the discussion of the steady state [1,2]. However, in contrast to that case, the effects of this potential distribution on the chronoamperometric transient can only be evaluated in limiting cases. The potential distribution in the presence of excess support electrolyte is governed in the Laplace space by Ka26 ar2
I Kdrp+Ka26 -0 r ar az2 ’
&=O,
r=co,
z=w
(28)
while the distribution of sources (for a cathodic reaction) over the disk is given by -
K$ = -zFij(e)
= a -Ire)(d,(s)
+ d,(s) cos(8) + . *. +&(s) cos”(8))
(29) We therefore obtain the potential at the surface of the disk, z = 0, as lwjo”‘*a(8, -zFa = -/,
m J,(ar)
K
=
$
s)J,(ar)J,,(aa daJ$c[‘*aj(,r)JO(a.
sin(@))
sin(8)
cos’(8)
sin(@) sin(e))
de da
cosi(8)
k(ij(s)ejJ;(r/a) j=O
where ej and h(r/a) ej = 2”j-‘)‘2’r((j
de
(30)
are given as in the steady state case [l] by
+ 1)/2)
(31)
and *l/2
I;(i)=2((j+l),2)r((j+2),2)02F1(~'
-f’ ” 5)
Modification of eqn. (10) to take into account the distribution solution gives
D(ac(i;“)
=k
exp( @(:;)F)cS=
k exp( “(:i’F)(cm
(32) of potential in the
- c(r,
t))
(33)
where k is the value of the rate constant of the electrode reaction in the absence of any ohmic potential drops in the solution and c(r, t) is the inverse of the second term in eqn. (19). A general Laplace transform of eqn. (33) can evidently be
231
obtained only for sufficiently small perturbations of the potential and concentration for which
(34) giving &z= ( aZ 1
a
co;(e) (d,(s)
+
za~~~mu
+ d,(s) cos(8) + * * * +d,(s) cos”(B))
=
cdj(s)ejf(r/u)
(35)
j=O
Special solutions using eqn. (33) include approximated by linear functions in time
those cases where the di can be
d,=g,+h,(t)
(36)
A further interesting case related to eqn. (36) is that of linear sweep voltammetry. We again restrict attention to the case of an irreversible reaction and, neglecting the effects of the tertiary current distribution, write eqn. (10) as D(ac~~‘))=kexp(~)(cm-c(r,t))=kexp(~l)(ca-c(r,f))
(37)
where k is the rate constant at t = 0. Laplace transformation
&cc
(aZ 1
a
,‘,(e)(d,(s)
of eqn. (37) gives
+ d,(s) cos(lq + * * * +dn (4 ~W))
(38) an equation system which can be solved in the t-domain where a
cos(e) (d,(t)
+ d,(f) cos(e)+ * ** +U)
= kc” e&4
-
X iej/‘/“dj(f-7) j=lJ
O
0
-“W)
kc" exp( pt) D1,2Tl,2
exp(-Dcu27) p2
J,(ar)~(j+l)/2)("a) (aa)(ww2)
d(aa)
dr
232
=
kc” expW x
kc* exp( pt) D1,2,1,2a
exp( -M(t ke./‘I , o om~jb) (t
j=O
CE AND EC CATALYTIC
-
- 7)) &(ar)J,,j+1)/2)(aa)
d7
(aa)((j-1)/2)
7y2
REACTIONS
d(aa)
(39)
IN THE STEADY STATE
We first consider the behavior of the CE reaction ASB k,
N
B=k in the steady state with the reactant A present in excess concentration. Then rf denotes the concentration of B, the diffusion of this species is governed by D ac
D~+p~+D~+k;c,-k,e3D~+T,
+De+k
1
-k
2
c=O
3Z2
(40) Equation (40) is subject to the boundary condition c=cm=kl/k2,
r=m,
z=m,
allt
(41)
and again we consider initially the simplest condition at the electrode surface c=O,
O
z=O,
all t
z=O,
allt
(42)
and with Dz
ac
=O,
r>a,;
The similarity of eqn. (40) in standard form
to eqn. (16) is apparent and we obtain the solution for the concentration m
c(r,
z, k2) = 2 2
X
exp (-
[ a2 + k,/D]l”z)
+ ‘/ Do
J,,(ar)cx [ a2
+
k,)Jo(wb $o”Q(~,
k,/D]
dar
1’2
dp
(45)
By analogy to the distribution of sinks for the chronoamperometric assume that for the CE reaction we can write
-ew = a -&~ (4,@,) + 4(k,)
of B as
co@) + . . . +4(b)
transient we
cosn@‘>)
(46)
233
where the dj are now functions of k,. Then at t = 0 we obtain c(r,
k2)
=
2 _ &J$odj(k2)ejjm 4(ar)4(j+1)/2)(“a) d(aa) 0
[a2 + k,/D]
Then, if the boundary condition (42) applies, the (n + 1) coefficients derived by solving the set of equations
O
[ a2
+
k2/D]
‘I*(
aa)((j-1)/2)
(47)
“‘( aa)((i-1)/2)
=
dj(k2)
can be
(48)
k,
J=o
at (n + 1) radial positions. A working curve of the kinetically limited flux can then be constructed from R(k,)
= 2aax
n &d@d
j=O
(I+
(49)
‘)
Mass transfer of the reactant species F in the EC (catalytic) reaction scheme E+re-SF
(iii)
F+Gk;‘E+H
(iv)
is described by
in the presence of excess of the substrate G. Provided the reaction (iii) can be driven sufficiently hard in the forward direction, eqn. (50) will be subject to the boundary conditions C=C*
)
O
c=o 9 r= 00,
allt
z=o,
(51)
allt
z=m,
(52)
where cm is the concentration of species E in the bulk of the solution. With a distribution of sources over the disk having the form (46), we therefore obtain the concentration (c(r,
[a’+
k2/D]1’2z) Jo(
z, k2) = k/“““” 0
[a* +
k,/D]
da
“*
and at z = 0 (54) The d,(k,) are evaluated by solving this set of equations and the working curve of the kinetically limited flux can again be constructed using eqn. (49).
234
The use of the boundary conditions (42) and (51) is, however, unrealistic for the same reasons that eqn. (14) is unrealistic for the evaluation of the chronoamperometric transients. In reality, the electron transfer reactions (ii) and (iii) will take place at finite rates and the EC (catalytic) reactions especially will be subject to the effects of the distribution of potential in the solution. It is better therefore to seek to predict the complete form of the polarization plots taking into account all these effects. It is, in fact, possible to achieve such a complete description of the systems since we are considering their behavior in the steady state; we restrict attention here to the case where the electron transfer processes are irreversible so that they can be described by the boundary condition (10). The distribution of potential in the solution follows as outlined above and we obtain at z = 0
(55) Application of the boundary conditions (10) to the reaction mechanism taking into account eqn. (55) gives
[ a2
+
k,/D]
1’2(
step (ii) in the CE
aa)((‘-
1)‘2)
(56) where k is the rate constant for reaction (iii) for zero ohmic potential at the surface of the disk. The non-linear equation system (56) can be solved at (n + 1) radial positions for the (n + 1) required coefficients dj in the same way as has been described for the case of the tertiary current distribution in simple redox reactions at disk electrodes [1,2]. In a similar way the application of eqns. (10) and (55) to the reaction step (iii) of the EC (catalytic) mechanism gives the equation system
co&) (do(b) +d&)
c1
co@) + . . . +4(h)
cosn@‘>)
(57)
235
where k is now the rate constant for reaction (iii) (assumed here to be irreversible) for zero ohmic overpotential in the solution. The coefficients d, derived from eqns. (56) and (57) are naturally also functions of k (i.e. of the standard rate constant, of (Y and of the overpotential) as well as of the solution conductivity K. The polarization curves can therefore be completely described for this model (and indeed other models) using R(k,,
k,
K)
=27Xlx
n dj(k,, k, K)
j=O
(j+l)
DISCUSSION
It can be seen that the application of Neumann’s integral of two parameters (the radial and s-dependence of the surface source/sink over the disk electrode) allows a more comprehensive discussion of the chronoamperometric transients than has been achieved to date using other analytical as well as simulation techniques [4,7,9-161. In particular, since Neumann’s integral leads to a relation between the assumed form of the flw and the concentration at the surface (eqns. 17 and 18) it becomes possible to derive the transient for boundary conditions such as eqn. (10) which are more realistic than the assumption of zero or constant surface concentration, eqn. (14) which has been used to date; for example, eqn. (27) leads to the definition of the potentiostatic relaxation behavior of disk electrodes. We have pointed out elsewhere [4,8,16] that the combined effects of the finite rates of electrode reactions and of the potential distribution in the solution must make the distribution of the flux over the surface much more uniform than that predicted using the boundary condition (14). Indeed, the predictions of a model of uniform surface flux and zero average surface concentration [4,16] have been shown to be in close accord with experimental measurements [7]. While the methodology outlined here allows a systematic exploration of one of the effects, that of the finite rates of electrode reactions, eqn. (lo), the exploration of the effects of changes in the potential in the solution unfortunately remains restricted to the discussion of the relaxation behavior [17] (cf. eqns. 34 and 35); in effect, we reach the limits of analytical techniques. However, it does become possible to develop a complete discussion of linear sweep voltammetry and of cyclic voltammetry [17], as has been outlined for the first case above, provided the effects of the tertiary current distribution are neglected. It has been shown previously that there are no restrictions in the discussion of the steady state behavior of simple electrode processes [1,2]. The discussion presented here shows that this is equally true for electrode processes coupled to reactions in solution such as for the CE and EC (catalytic) mechanisms; a complete description of such systems can be developed. Comparisons of the predictions with those based on other models and methods of analysis [4,18-201 as well as extensions to other reaction schemes will be presented elsewhere [21].
236 GLOSSARY
OF SYMBOLS USED
k
Homogeneous rate constant, various Heterogeneous standard rate constant, cm s-’ Homogeneous rate constants, various types Laplace transform dummy integration variable
k0 k,, 4 7
l
ACKNOWLEDGEMENT
We thank the Office of Naval Research for support of this work. REFERENCES 1 M. Fleischmann, J. Daschbach and S. Ports, J. Electroanal. Chem., 263 (1989) 189. 2 J. Daschbach, S. Pons and M. Fleischmamr, J. Electroanal. Chem., 263 (1989) 225. 3 G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1948. 4 M. Fleischmamr and S. Pons in M. Fleischmann, S. Pons, D. Rolison and P. Schmidt @Is.), Ultramicroelectrodes, Datatech, Morganton, NC, 1987, Ch. 2. 5 M. Fleischmann and S. Pans, J. Electroanal. Chem., 222 (1987) 107. 6 A.M. Bond, K.B. Oldham and C.G. Zoski, J. Electroanal. Chem. 245 (1988) 71. 7 L.J. Li, M. Hawkins, J.W. Pons, J. Daschbach, S. Pons, M. Fleischmann and L.M. Abrantes, J. Electroanal. Chem., 262 (1989) 45. 8 M. Fleischmann and S. Pans, J. Electroanal. Chem., 250 (1988) 257. 9 K. Aoki and J. Osteryoung, J. Electroanal. Chem., 122 (1981) 19. 10 B. Speiser and S. Pons, Can. J. Chem., 60 (1982) 1352. 11 B. Speiser and S. Pons, Can. J. Chem., 60 (1982) 2463. 12 J. Cassidy and S. Pans, Can. J. Chem., 63 (1985) 3577. 13 D. Shoup and A. Szabo, J. Electroanal. Chem., 140 (1982) 237. 14 K. Aoki and J. Osteryoung, J. Electroanal. Chem., 160 (1984) 335. 15 K.B. Oldham, J. Electroanal. Chem., 122 (1981) 1. 16 M. Fleischmann, J. Daschbach and S. Pons, J. Electroanal. Chem., 250 (1988) 269. 17 M. Fleischmann, D. Pletcher, G. Denuault, J. Daschbach and S. Pons, to be published. 18 M. Fleischmann, F. Lasserre, J. Robinson and D. Swan, J. Electroanal. Chem., 177 (1984) 97. 19 M. Fleischmann, F. Lasserre and J. Robinson, J. Electroanal. Chem., 177 (1984) 117. 20 M. Fleischmann and S. Pans, J. Electroanal. Chem., 250 (1988) 277, 285. 21 M. Fleischmann, D. Pletcher, G. Denuault, J. Daschbach and S. Pons, to be published.
* See ref. 1 for symbols not given here.
J. Electroanal. Chem., 263 (1989) 225-236 Elsevier Sequoia S.A., Lausanne - Printed
in The Netherlands
The behavior of microdisk and microring electrodes Prediction of the chronoamperometric response of microdisks and of the steady state for CE and EC catalytic reactions by application of Neumann’s integral theorem M. Fleischmann, Derek Pletcher and Guy Denuault Department
of Chemistry,
The University, Southampton,
Hunts. SO9 SNH (Great Britain)
John Daschbach and Stanley Pons * Department (Received
of Chemistry, 8 August
University of Utah, Salt Lake City, UT 84112 (lJ.S.A)
1988; in revised form 25 November
1988)
ABSTRACT It is shown that the chronoamperometric response at microdisk electrodes can be derived from Neumann’s integral theorem of two parameters. The form of the transients can be predicted for a wide range of boundary conditions and this is illustrated by the relaxation behavior of irreversible reactions in addition to that of the widely investigated example of zero surface concentration of the reactant. Corrections to the transitions due to the tertiary current distribution are derived for relaxation experiments and it is shown that the methods developed can also be applied to linear sweep voltammetry. The steady state behavior of CE and EC catalytic reactions is formally similar to the chronoamperometry of irreversible electrode reactions and it is shown that complete descriptions can be obtained for the voltammetry of such systems.
INTRODUCTION
We have shown recently that the steady state behavior of simple reactions at microdisk electrodes can be predicted [1,2] by the application of Neumann’s integral theorem [3]. The concentration distribution for a reactant (or product) in such a reaction is governed by &!?+!?ac+&!L(J lb+’
r
ar
* To whom correspondence
0022-0728/89/%03.50
0)
a.z2
should be addressed.
Q 1989 Elsevier Sequoia
S.A.
226
and we obtain
c(r, z) =
ImadajaexP(;ar) 0
~JO(c&(~p)p
0
dp
(2)
Equation (2) makes a formal linkage between the distribution of sources, Q(p), (or sinks) at the radial position p and the concentration throughout space. We have shown that the use of the simple form for the distribution of sinks
-Q(e)
=
a
ist8){ ca + c* cos( 8) + c2 cos(28)
+ . . . + c, cos( ne)}
(3)
where p=a
sin(f3)
(4)
leads to the recovery of known results [4,5] (derived by the application of the discontinuous integrals of Bessel functions) for constant surface concentrations provided we assume c, = c2 = c3 = . - . = c, = 0
(5)
and co =
+qc-c”)
and of constant, uniform, surface flux provided we assume c0 = c2 = cj = . 1. = c, = 0 and ci = Qu It has also been shown that the reexpression of eqn. (3) in the form
-Q(e)
= a
c;siej (do +
d, c0s(e) + -.
is particularly useful in discussing conditions such as O
D
+d,
c0sye))
the application
of more general
z=O
boundary
(10)
The use of eqn. (9) in eqn. (2) leads to the simple result
c(r)
= -
(11)
(at z = 0) where I is the gamma function and *Fl denotes the appropriate hypergeometric function. Application of eqn. (11) at (n + 1) radial positions in any particular boundary condition (such as eqn. 10) then allows the evaluation of the
221
unknown coefficients in eqn. (9) thereby giving a complete solution for any particular problem. The results obtained by the application of Neumann’s integral theorem have been related also to results derived by an extension [6] of the application of the discontinuous integrals of Bessel functions [4,5]. We have shown also that the use of Neumann’s integral allows the evaluation of the effects of the distribution of potential in the solution (the tertiary current distribution) and of the non-linearities in the concentration terms of the electrode reactions [1,2]. In this paper we extend the approach to the discussion of the chronoamperometric response of microdisk electrodes, comment on the effects of the distribution of potential in the solution, and outline the analysis of linear sweep voltammetric experiments. In view of the formal similarity of the derived equations to those obtained for the steady state behavior of reactions following the CE and EC mechanisms, we also derive the relevant results for these reaction pathways. CHRONOAMPEROMETRY
We consider first of all the behavior of a simple diffusion controlled where mass transfer of the reactant is controlled by
reaction
(12) subject to the initial condition c=P
,
O-crcco,
zb-0,
t=O
and the boundary conditions c=O, D$=O,
O
z=O,
t>O
z = 0,
t>o
Laplace transformation
of eqn. (12) subject to eqn. (13) gives
where s is the variable of the Laplace transformation. The solution of eqn. (16) can therefore be expressed in terms of Neumann’s integral of two parameters (p, s) using Q(p, S) as p)p
dp (17)
By analogy with the expansion (9) used to describe the distribution steady state, we write
-Q
a
--&) (d,(s)
of sinks in the
+ c&(s)co@) + . * * +&(s) COW))
228
where all the coefficients di are now functions of s. Substitution (4) in eqn. (17) gives at z = 0 c(r,
s) = $
- “/ D 0
Q) J,(cur)cr
of eqns. (18) and
dcu
[a2+s/D]“2
X[“td,(s)J,(aa
sin(t9)) sign
cosi(8)
de
j=O O" 4("')J((j+l)/2) [
for a distribution of sinks a(@)
(~444
(19)
a2 + s,D]1/2(aa)((j-1)/2)
over the surface where
ej = 2(cj-‘)/2)r(( j + 1)/2)
(20) The coefficients c?~(s) can then be obtained by substituting eqns. (18) and (19) into the Laplace transform of the boundary condition appropriate to the particular form of the experiment. For example, for the simple condition F=O,
O
z=O
(21)
(cf. eqn. 14) we solve the set of equations
n dT,(s)ej
c-
j=O
m J0(~~k)Jc(j+1)/2)
J o
Da
(444
[a2 +s,D]1/2(cya)((j-1)/2)
coo =
(22)
s
at (n + 1) radial positions. We therefore obtain the unknown coefficients as functions of the parameter s. Substitution of these coefficients into eqn. (18) and integration over the radius of the disk then gives the Laplace transform of the total rate of reaction, R, rdr
%=2nD
@‘_
= 27i-aJ$o[‘2Zj(s)
cod(e)
r2)t/2
sin(@) de
n cJj(S) =277aC
-
(23)
j=O(i+l)
Numerical inversion of eqn. (23) gives the total rate of reaction in the t-domain. An alternative procedure is to invert eqn. (22) to the relevant convolution integrals
= 1
(24)
229
In the numerical integration over say 1 intervals of A7 we then have to solve the set of eqns. (24) for the values dj in the Ith interval. This forward integration is simplified by the fact that we know that the system follows the Cottrell equation at short times [7] for which at sufficiently large s
d,(s) = 5
(25)
and do=d2=d3=
(26)
. . . =d,=O
OTHER BOUNDARY CONDITIONS; SWEEP VOLTAMMETRY
THE TERTIARY
CURRENT
DISTRIBUTION;
LINEAR
We have pointed out elsewhere that the application of the boundary condition (14) (or, more generally, of a constant surface concentration boundary condition) is unrealistic as the flux would have to become infinite at the edges of the disk [1,2,4,8] (see also ref. 6). However, the flux cannot become infinite for two reasons: firstly because the rate constant governing the electrochemical reaction must itself be finite and, secondly, because an infinite rate of reaction would also require an infinite overpotential at the edges of the disk. The current distribution must therefore be more uniform than that predicted using the boundary condition (14) under most experimental conditions (the tertiary current distribution) as we have shown in the previous paper. As the application of Neumann’s integral theorem provides a link between the assumed form of the flux distribution, eqn. (18), and the concentration distribution, eqn. (19), it is straightforward to assess the effects of boundary conditions more complex than eqn. (14). We restrict attention here to irreversible reactions which are described by eqn. (10). Substitution of cqn. (19) into the Laplace transform of eqn. (10) gives
[,,s)+,,.,(~~‘*+~2~s)(~) D(y; “’)= _1r2)l,2 +...+JJq_jLy”] (u2
= a
,d,& (Jo(s) + c&(s)cos(8) + . . * +d,(s) COW)) (27)
The dependence of the coefficients gj on s can therefore be obtained as for the case of zero surface concentration governed by eqn. (22). The distribution of the flux and
230
concentration and the total rate of reaction can then be derived for the Laplace plane using eqns. (18), (19), and (23). Numerical inversion gives the dependencies in the t-plane. It therefore becomes possible to assess the effects of changes in k on the behavior of the system and the relaxation of the rate of reaction under potentiostatic conditions can be evaluated. The application of Neumann’s integral theorem also allows the evaluation of the potential distribution in the solution, as in the case of the discussion of the steady state [1,2]. However, in contrast to that case, the effects of this potential distribution on the chronoamperometric transient can only be evaluated in limiting cases. The potential distribution in the presence of excess support electrolyte is governed in the Laplace space by Ka26 ar2
I Kdrp+Ka26 -0 r ar az2 ’
&=O,
r=co,
z=w
(28)
while the distribution of sources (for a cathodic reaction) over the disk is given by -
K$ = -zFij(e)
= a -Ire)(d,(s)
+ d,(s) cos(8) + . *. +&(s) cos”(8))
(29) We therefore obtain the potential at the surface of the disk, z = 0, as lwjo”‘*a(8, -zFa = -/,
m J,(ar)
K
=
$
s)J,(ar)J,,(aa daJ$c[‘*aj(,r)JO(a.
sin(@))
sin(8)
cos’(8)
sin(@) sin(e))
de da
cosi(8)
k(ij(s)ejJ;(r/a) j=O
where ej and h(r/a) ej = 2”j-‘)‘2’r((j
de
(30)
are given as in the steady state case [l] by
+ 1)/2)
(31)
and *l/2
I;(i)=2((j+l),2)r((j+2),2)02F1(~'
-f’ ” 5)
Modification of eqn. (10) to take into account the distribution solution gives
D(ac(i;“)
=k
exp( @(:;)F)cS=
k exp( “(:i’F)(cm
(32) of potential in the
- c(r,
t))
(33)
where k is the value of the rate constant of the electrode reaction in the absence of any ohmic potential drops in the solution and c(r, t) is the inverse of the second term in eqn. (19). A general Laplace transform of eqn. (33) can evidently be
231
obtained only for sufficiently small perturbations of the potential and concentration for which
(34) giving &z= ( aZ 1
a
co;(e) (d,(s)
+
za~~~mu
+ d,(s) cos(8) + * * * +d,(s) cos”(B))
=
cdj(s)ejf(r/u)
(35)
j=O
Special solutions using eqn. (33) include approximated by linear functions in time
those cases where the di can be
d,=g,+h,(t)
(36)
A further interesting case related to eqn. (36) is that of linear sweep voltammetry. We again restrict attention to the case of an irreversible reaction and, neglecting the effects of the tertiary current distribution, write eqn. (10) as D(ac~~‘))=kexp(~)(cm-c(r,t))=kexp(~l)(ca-c(r,f))
(37)
where k is the rate constant at t = 0. Laplace transformation
&cc
(aZ 1
a
,‘,(e)(d,(s)
of eqn. (37) gives
+ d,(s) cos(lq + * * * +dn (4 ~W))
(38) an equation system which can be solved in the t-domain where a
cos(e) (d,(t)
+ d,(f) cos(e)+ * ** +U)
= kc” e&4
-
X iej/‘/“dj(f-7) j=lJ
O
0
-“W)
kc" exp( pt) D1,2Tl,2
exp(-Dcu27) p2
J,(ar)~(j+l)/2)("a) (aa)(ww2)
d(aa)
dr
232
=
kc” expW x
kc* exp( pt) D1,2,1,2a
exp( -M(t ke./‘I , o om~jb) (t
j=O
CE AND EC CATALYTIC
-
- 7)) &(ar)J,,j+1)/2)(aa)
d7
(aa)((j-1)/2)
7y2
REACTIONS
d(aa)
(39)
IN THE STEADY STATE
We first consider the behavior of the CE reaction ASB k,
N
B=k in the steady state with the reactant A present in excess concentration. Then rf denotes the concentration of B, the diffusion of this species is governed by D ac
D~+p~+D~+k;c,-k,e3D~+T,
+De+k
1
-k
2
c=O
3Z2
(40) Equation (40) is subject to the boundary condition c=cm=kl/k2,
r=m,
z=m,
allt
(41)
and again we consider initially the simplest condition at the electrode surface c=O,
O
z=O,
all t
z=O,
allt
(42)
and with Dz
ac
=O,
r>a,;
The similarity of eqn. (40) in standard form
to eqn. (16) is apparent and we obtain the solution for the concentration m
c(r,
z, k2) = 2 2
X
exp (-
[ a2 + k,/D]l”z)
+ ‘/ Do
J,,(ar)cx [ a2
+
k,)Jo(wb $o”Q(~,
k,/D]
dar
1’2
dp
(45)
By analogy to the distribution of sinks for the chronoamperometric assume that for the CE reaction we can write
-ew = a -&~ (4,@,) + 4(k,)
of B as
co@) + . . . +4(b)
transient we
cosn@‘>)
(46)
233
where the dj are now functions of k,. Then at t = 0 we obtain c(r,
k2)
=
2 _ &J$odj(k2)ejjm 4(ar)4(j+1)/2)(“a) d(aa) 0
[a2 + k,/D]
Then, if the boundary condition (42) applies, the (n + 1) coefficients derived by solving the set of equations
O
[ a2
+
k2/D]
‘I*(
aa)((j-1)/2)
(47)
“‘( aa)((i-1)/2)
=
dj(k2)
can be
(48)
k,
J=o
at (n + 1) radial positions. A working curve of the kinetically limited flux can then be constructed from R(k,)
= 2aax
n &d@d
j=O
(I+
(49)
‘)
Mass transfer of the reactant species F in the EC (catalytic) reaction scheme E+re-SF
(iii)
F+Gk;‘E+H
(iv)
is described by
in the presence of excess of the substrate G. Provided the reaction (iii) can be driven sufficiently hard in the forward direction, eqn. (50) will be subject to the boundary conditions C=C*
)
O
c=o 9 r= 00,
allt
z=o,
(51)
allt
z=m,
(52)
where cm is the concentration of species E in the bulk of the solution. With a distribution of sources over the disk having the form (46), we therefore obtain the concentration (c(r,
[a’+
k2/D]1’2z) Jo(
z, k2) = k/“““” 0
[a* +
k,/D]
da
“*
and at z = 0 (54) The d,(k,) are evaluated by solving this set of equations and the working curve of the kinetically limited flux can again be constructed using eqn. (49).
234
The use of the boundary conditions (42) and (51) is, however, unrealistic for the same reasons that eqn. (14) is unrealistic for the evaluation of the chronoamperometric transients. In reality, the electron transfer reactions (ii) and (iii) will take place at finite rates and the EC (catalytic) reactions especially will be subject to the effects of the distribution of potential in the solution. It is better therefore to seek to predict the complete form of the polarization plots taking into account all these effects. It is, in fact, possible to achieve such a complete description of the systems since we are considering their behavior in the steady state; we restrict attention here to the case where the electron transfer processes are irreversible so that they can be described by the boundary condition (10). The distribution of potential in the solution follows as outlined above and we obtain at z = 0
(55) Application of the boundary conditions (10) to the reaction mechanism taking into account eqn. (55) gives
[ a2
+
k,/D]
1’2(
step (ii) in the CE
aa)((‘-
1)‘2)
(56) where k is the rate constant for reaction (iii) for zero ohmic potential at the surface of the disk. The non-linear equation system (56) can be solved at (n + 1) radial positions for the (n + 1) required coefficients dj in the same way as has been described for the case of the tertiary current distribution in simple redox reactions at disk electrodes [1,2]. In a similar way the application of eqns. (10) and (55) to the reaction step (iii) of the EC (catalytic) mechanism gives the equation system
co&) (do(b) +d&)
c1
co@) + . . . +4(h)
cosn@‘>)
(57)
235
where k is now the rate constant for reaction (iii) (assumed here to be irreversible) for zero ohmic overpotential in the solution. The coefficients d, derived from eqns. (56) and (57) are naturally also functions of k (i.e. of the standard rate constant, of (Y and of the overpotential) as well as of the solution conductivity K. The polarization curves can therefore be completely described for this model (and indeed other models) using R(k,,
k,
K)
=27Xlx
n dj(k,, k, K)
j=O
(j+l)
DISCUSSION
It can be seen that the application of Neumann’s integral of two parameters (the radial and s-dependence of the surface source/sink over the disk electrode) allows a more comprehensive discussion of the chronoamperometric transients than has been achieved to date using other analytical as well as simulation techniques [4,7,9-161. In particular, since Neumann’s integral leads to a relation between the assumed form of the flw and the concentration at the surface (eqns. 17 and 18) it becomes possible to derive the transient for boundary conditions such as eqn. (10) which are more realistic than the assumption of zero or constant surface concentration, eqn. (14) which has been used to date; for example, eqn. (27) leads to the definition of the potentiostatic relaxation behavior of disk electrodes. We have pointed out elsewhere [4,8,16] that the combined effects of the finite rates of electrode reactions and of the potential distribution in the solution must make the distribution of the flux over the surface much more uniform than that predicted using the boundary condition (14). Indeed, the predictions of a model of uniform surface flux and zero average surface concentration [4,16] have been shown to be in close accord with experimental measurements [7]. While the methodology outlined here allows a systematic exploration of one of the effects, that of the finite rates of electrode reactions, eqn. (lo), the exploration of the effects of changes in the potential in the solution unfortunately remains restricted to the discussion of the relaxation behavior [17] (cf. eqns. 34 and 35); in effect, we reach the limits of analytical techniques. However, it does become possible to develop a complete discussion of linear sweep voltammetry and of cyclic voltammetry [17], as has been outlined for the first case above, provided the effects of the tertiary current distribution are neglected. It has been shown previously that there are no restrictions in the discussion of the steady state behavior of simple electrode processes [1,2]. The discussion presented here shows that this is equally true for electrode processes coupled to reactions in solution such as for the CE and EC (catalytic) mechanisms; a complete description of such systems can be developed. Comparisons of the predictions with those based on other models and methods of analysis [4,18-201 as well as extensions to other reaction schemes will be presented elsewhere [21].
236 GLOSSARY
OF SYMBOLS USED
k
Homogeneous rate constant, various Heterogeneous standard rate constant, cm s-’ Homogeneous rate constants, various types Laplace transform dummy integration variable
k0 k,, 4 7
l
ACKNOWLEDGEMENT
We thank the Office of Naval Research for support of this work. REFERENCES 1 M. Fleischmann, J. Daschbach and S. Ports, J. Electroanal. Chem., 263 (1989) 189. 2 J. Daschbach, S. Pons and M. Fleischmamr, J. Electroanal. Chem., 263 (1989) 225. 3 G.N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge University Press, Cambridge, 1948. 4 M. Fleischmamr and S. Pons in M. Fleischmann, S. Pons, D. Rolison and P. Schmidt @Is.), Ultramicroelectrodes, Datatech, Morganton, NC, 1987, Ch. 2. 5 M. Fleischmann and S. Pans, J. Electroanal. Chem., 222 (1987) 107. 6 A.M. Bond, K.B. Oldham and C.G. Zoski, J. Electroanal. Chem. 245 (1988) 71. 7 L.J. Li, M. Hawkins, J.W. Pons, J. Daschbach, S. Pons, M. Fleischmann and L.M. Abrantes, J. Electroanal. Chem., 262 (1989) 45. 8 M. Fleischmann and S. Pans, J. Electroanal. Chem., 250 (1988) 257. 9 K. Aoki and J. Osteryoung, J. Electroanal. Chem., 122 (1981) 19. 10 B. Speiser and S. Pons, Can. J. Chem., 60 (1982) 1352. 11 B. Speiser and S. Pons, Can. J. Chem., 60 (1982) 2463. 12 J. Cassidy and S. Pans, Can. J. Chem., 63 (1985) 3577. 13 D. Shoup and A. Szabo, J. Electroanal. Chem., 140 (1982) 237. 14 K. Aoki and J. Osteryoung, J. Electroanal. Chem., 160 (1984) 335. 15 K.B. Oldham, J. Electroanal. Chem., 122 (1981) 1. 16 M. Fleischmann, J. Daschbach and S. Pons, J. Electroanal. Chem., 250 (1988) 269. 17 M. Fleischmann, D. Pletcher, G. Denuault, J. Daschbach and S. Pons, to be published. 18 M. Fleischmann, F. Lasserre, J. Robinson and D. Swan, J. Electroanal. Chem., 177 (1984) 97. 19 M. Fleischmann, F. Lasserre and J. Robinson, J. Electroanal. Chem., 177 (1984) 117. 20 M. Fleischmann and S. Pans, J. Electroanal. Chem., 250 (1988) 277, 285. 21 M. Fleischmann, D. Pletcher, G. Denuault, J. Daschbach and S. Pons, to be published.
* See ref. 1 for symbols not given here.