Elsevier Sequoia S.A., Lausanne. JEC 02754. Closed-form approximate expressions for the electrochemical response of microelectrodes in the zero range limit.
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J. Electroanal. Chem., 358 (1993) l-20
Elsevier Sequoia S.A., Lausanne
JEC 02754
Closed-form approximate expressions for the electrochemical response of microelectrodes in the zero range limit Anatol M. Brodsky, Sergey F. Burlatsky
l
and William P. Reinhardt
Department of Chemistry and CPAC, University of Washington, Seattle, WA 98195 (USA)
(Received 10 August 1992; in revised form 29 January 1993)
Abstract We show that the electrochemical response of one or more microelectrodes can be described, in the limit of small size, by closed-form solutions of diffusion-kinetic equations, which, under appropriate and well-defined conditions, should give a good approximation to the actual response. The predictions of the analytical solutions for the diffusion-kinetic model are compared with microelectrode array experiments of Bard et al. and show excellent agreement. The analytical results make clear the various scaling properties of the solution, as a function of electrode size and spacings, as well as indicating the range of validity of the diffusion-kinetic model itself.
1. INTRODUCTION
Experimental investigations and applications of ultramicroelectrodes in the last decade [1,2] have given rise to new theoretical questions connected with the effects of their small scale in transport phenomena, electrochemical reaction kinetics and potential distribution. The subsequent introduction of microelectrode arrays of many different possible geometries turns out to be especially important [3,4]. Since each microelectrode in an array can be individually potentiostated, it is possible to determine simultaneously the concentrations of different species, their spatial distributions, and their micro- and macrokinetics laws. Such microelectrode arrays may be characterized by several distinct and different length scales, some of which may be controlled independently in a given experimental situation. Such length scales include the following: individual microelectrode size (and geometry); microelectrode spacing; the total array dimensions; the ratio of diffusion constant to reaction rate constant; the Debye-Hiickel shielding length. The possibility of
l
Permanent address: Institute for Chemical Physics, 4 Vorobjevy Gory, Moscow, Russian Federation.
0022-0728/93/$06.00
0 1993 - Elsevier Sequoia S.A. All rights reserved
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independent variation of these length scales leads to quantitatively and qualitatively different regimes in the current-time dependence of the response of individual microelectrodes in the array. This, in turn, allows extraction of important information about the dynamics of the underlying diffusive and kinetic processes. Traditionally, the macrokinetics problem for microelectrodes has been treated within the framework of continuous-media, classical diffusion models, with the assumption of the charge neutrality hypothesis [51. The analytical expressions for current from regular and random arrays of semispherical electrodes have been discussed in the literature (see refs. 6 and 7, and the references cited therein) with the assumption of the equality of reaction constants of all the electrodes. In the practically important case of microelectrode arrays with unequal potentials, the corresponding partial differential equations or equivalent integral equations of this model have been solved using computer simulations, as, for example in ref. 8, and references cited therein. In this article, we introduce an approximate analytical scheme for the solution of microelectrode problems, which can be applied to non-equivalent electrodes, in the same classical model as discussed in refs. 5 and 8. The proposed approximate scheme, which resembles the so-called zero range approximation in atomic physics [9] and, like that approximation, yields analytically soluble equations, is based on the assumption of the smallness of characteristic electrode sizes in comparison with all other length scales in the model. In such an approximation, it is thus possible to disregard the concentration variations along the surface of an individual electrode, as well as assuming that the electrode scale size is much less than the length scales set by the microelectrode spacing, the total array dimensions or the ratio of the diffusion constant to reaction rate constant. Such an approach was used in electrochemical calculations as early as 1941 by Levich and Frumkin [lo]. Analogous approaches have also been applied to the solution of other diffusionkinetics problems [ 11,121. The availability of closed-form solutions to an approximation to the diffusion kinetics problem allows not only direct comparison of the theory with the experiments and computer simulations in the parameter regime where the continuousmedia classical model is adequate, but also provides the possibility of making specific estimates of limits of the direct computer calculations and of the different effects of the breakdown of the model itself. Such effects will become important in the case of extremely small electrodes, as in the case of nanometer-sized electrodes, as described in the work of Lewis et al. [41. In this case, the electrochemical response can be affected by additional microscopic length parameters, completely neglected in the classical diffusion-kinetic picture: for example, fluctuations, lateral correlation length, the electronic Fermi length and the free flight length in very small electrodes which behave more like those for metallic clusters than for metals proper 1131.We plan to discuss the corresponding problems in future articles. The paper is organized as follows. In Section 2 we formulate the chosen model problem and write down the basic equations. The main mathematical development
3
is presented in Section 3 and Appendix A, which can be skipped on the first reading. Specific expressions describing the simplest realistic electrochemical microelectrode devices are given in Section 4. A comparison of the theory with recent experiments is initiated in Section 5: to check the applicability of the theory in a case where good results might be expected, and indeed are found, we have chosen the results of Bard et al. [14], which are easily amenable to a quantitative description. Small discrepancies with computer simulations are attributed to the failure of the discretization of singular (in the limit of small dimensions) expressions for the electrochemical response. Speculative remarks are made concerning the interpretation of the results of Lewis et al. [4] for nanometer-sized electrodes, where the traditional models begin to break down. 2. DESCRIPTION
OF THE MODEL
In a typical microelectrode array, microelectrodes are placed on impenetrable inert plates. We will discuss two general cases, applicable to this half-space problem. In the first case (Fig. l), a finite number of microelectrodes have approximately the same characteristic sizes a in all directions, with the height less than or of the order of the width, and characteristic nearest neighbor separation R. In the second case (Fig. 2), one dimension L of the electrodes is much larger than the other two dimensions, i.e. L z=a,and the electrodes in the array are assumed to be parallel. The solution of the corresponding diffusion reaction problems in half-space with the zero flux boundary condition is equivalent to a symmetrical solution in total space (which will be dealt with in the article) up to a multiplier of l/2 in the resulting expressions for the current.
A
0 L
0 v
Fig. 1. Array of a finite number of microelectrodes with equal dimensions of order a in all directions. The heights of the microelectrodes are of the order of or less than the widths.
_._
Fig. 2. Array of microelectrodes
with one dimension L much bigger than others (L 38 a).
As the simplest example of an electrode process, we will consider reversible oxidation-reduction (n-electron transfer) electrode reactions, i.e. A,+E&
-Ei+A,
(1)
where A, and A, are a redox couple, E, represents microelectrode number i (i= l,..., N) at potential E,, k, and k_i are the corresponding direct and reverse surface reaction rate constants (per unit electrode surface) at electrode i. The constants k, and k_, are exponentially dependent (assuming Tafel law behavior) on the potential drop at the microelectrode, which in experimental situations can depend on both i and time. We will restrict ourselves here to the simplest realistic case, where we follow the electrochemical response following switching on a potential difference at the time t = 0. The generalization to more complex time dependence as well as to more complex reactions than reaction (1) presents no difficulties, in principle, although the details will be more complex. We will suppose that transport of the ions A,,, (the subscript notation R,O will often be used to symbolize either the reduced or oxidized form of species A, as may be appropriate) obeys, in the mean field approximation, the diffusion equation =D
at
R,O
v 2pR,0(F,
t,
(2)
where v 2 is the three-dimensional laplacian and PR,o(?, t) and D,,, are the local concentrations and the effective diffusion coefficients (the latter of which will be
5
taken as being concentration independent. The experimental derivation of precise values of reaction constants in real experiments must take into account the fact and that effective diffusion coefficients Da,, can, in fact, be concentration potential dependent) for A, and A, respectively. Reactions in the bulk are neglected. Equation (2) gives an adequate description in the case when it is possible to assume electroneutrality [5]. Such an assumption is reasonable when all characteristic dimensions are greater than the Debye length, characteristic times are bigger than a’/D, and/or for high concentrations of supporting electrolyte. Equation (2) is supplemented with the following standard boundary and initial conditions: -Da( z V )Pa - kiPa + k-iPo IrESi=O -Do( n V ) PO + kipR - k-ipo
i=l
,.**, N
IrESi=O
(3)
PR,O( 7, t) I r=o = PiRn,g
is the equilibrium concentration of A,,, for t Q 0. The boundary where p& conditions in eqn. (31 are taken on electrode surfaces Si (i = 1,. . . , N) and ?I is the direction normal to these surfaces. The present description of the migration reaction problem thus does not take into account convection. This is reasonable in many cases for microelectrodes arrays, as opposed to the case for macroelectrodes. It is convenient to solve eqns. (2) and (3) with the help of Laplace transforms, such that P&&
PI
= /07
r, t) -P&,]
PR,O(
ew(-pt)
dt
(4)
In terms of these transforms, eqns. (2) and (3) may be rewritten as PP;,o(C
P) =D,,,
V “P;,&
P) in
D,(~V)p,+ki
D,(nV)p,-ki
pr;(~,p)+~
p~(r,
(5a)
1[ 1I -k-i
p) + ~
+k_i
in
p~(~,P)‘~
po’(f,
in
=O
Ii ‘i’s,
P) + ~
(5b)
=O FiESi
The initial conditions at t = 0 for the inverse Laplace transform p) will be satisfied automatically.
p&F,
t) of
p&(T,
3. APPROXIMATE
GENERAL
SOLUTION
To obtain the approximate analytical solution of eqns. (5a) and (5b), we will use the relative smallness of microelectrode sizes. We will assume, in particular, that
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the ratio H of the distance between two electrodes R,, = 1Ri- EjIto an electrode with dimensions a (see Figs. 1 and 2) satisfies the condition H=Rij/a
> 1
(6)
For typical microelectrode arrays, the small time transient effects which occur for t < a2/D are, as a rule, unobservable and, for that reason, we will suppose the fulfilment of the inequality t > t, = a2/D,,,
(7)
This does not necessarily mean that all results are valid only in the diffusion-controlled, long time limit. Inequality (7) can be satisfied simultaneously with the condition of a kinetics-controlled regime, such that t =
(l/r)
for d = 3
exp(-XR,O~ for d = 2
Kcl( xn,or)
where r=
IFI
x&o = D
P
(14
R.0
and K,(z) is the modified Bessel function of the third kind [15]. Symmetry arguments indicate that the situation of Fig. 1 corresponds to a dimension d = 3 and that arrays of the type shown in Fig. 2 correspond to d = 2 (provided that all the concentrations are constant along the length L of the cylindrical electrodes and the dependence on one of the spatial coordinates is therefore excluded). Substitution of eqn. (11) into the boundary conditions of eqn. (5b) gives a system of linear equations for the determination of the coefficients cR*O in eqn. (11). Taking into account inequalities (6) and (71, we approximate GR,O(T, p), within the framework of the boundary conditions of eqn. (5b), by the following expressions: 1 -
G,,,(+R;,
p) =
for d = 3
XR.0
I’-“’ _ln Xn,o I r-Ri
=U =R,
(X R,O~) -=c 1 and p < l/t,; exp(
Gn>o(r - ‘0 ‘) =
for d = 2
-Y
2 for IF-R~I
( 13a)
1
for d = 3
-XR,ORij>/Rij
K,(xR,oRij)
and
( 13b)
for d = 2
and i+j, where Rij= IRi-EjI and y=O.5712... is Euler’s constant. Introducing eqns. (ll), (13a) and (lob) into eqn. (5b) gives the following system of linear algebraic equations for coefficients cF*O:
for r=R
-D
R$ a
-ki~G$c~ j
+ k_iCG.?cp = i
-k.e+k_.g ‘P
in
-D o&.+ki~G;c;--k_i~G;c~=k.=k_,~ a i i
‘P
‘P
in
(14)
where G!!.” = EJ
(l/u) ev(
-~n,~
for i=j
(d=3)
for i +j
-XR,ORij)/Rij
-ln(xi,0/4)- Y
(15)
for i=j
G!%O = lJ
(d=2)
for i#j
i KO(XR,ORij)
In eqn. (ll), we have dropped terms of the relative order O((xa)*). Thus, eqn. (11) may now be solved by standard linear algebra techniques. Then, it is possible to calculate currents to the ith electrode (i = 1,. . . , IV), the Laplace transform of which is equal to z;+(p) = +Fn;(D,cp-D,cy)
= +?TZW,c;
(16)
where Zi denotes the full current on the ith electrode for d = 3 and the current density per unit length of electrode for d = 2, and F is Faraday’s constant. In eqn. (13), the size a enters only through coefficients CR and, as a result, ZiU3-d depends on a only through renormalized parameters, such as b,O
=
i =
D,,o/a*
kiu
p’in _ pa3
4. APPLICATION TO SPECIFIC MICROELECTRODE ARRANGEMENT
4.1. Single electrode For the single electrode d = 3 with characteristic plate, eqns. (14)-(16) lead to
dimension a on a supporting
/&ji$ _ i-5;
Z'(p)=?
1+
J+ -(P/W*‘*]
k[l-
(B/do)i’*]
+
&
60
+ p’/2( k/Z@// + &_/63R/2)
1 1+ k/d,
+ k/B,
(1 +g/lj,+k,Bo)*
1 (17)
9
where 6
p’E,o = pk,oa3 i=ka-’
f_=
R,O =
(1/a2)DR,0
k-a-’
(18)
It is necessary to stress that eqn. (14) can be used only for p < l/t,, where the assumed approximation to the Green’s function is valid. The inverse Laplace transform of eqn. (14) gives the following expression for the current Z(t):
pFn(/+@_i(;)
Z(t) =
1 +f/6,+f_/L5,
1
1+
f/L5 Rs’* + IS_/fio3/2
(Pt)l’*
[
1 +&5R+f_/Z50
1
(19)
for t > to. In the limit k + 03, t + w, eqn. (19) coincides (up to a coefficient of l/2) with the long time limit for diffusion on a sphere (taking into account that a is a diameter in our case) and for the corresponding limit for diffusion on the disc, if we put a =
(2/7r)d
(20)
where d is the disc diameter. For a single electrode d = 2 with characteristic eqns. (ll)-(13) lead to
width a and infinite length L,
pZ+(p)a TF
,$j;
_ i_bE
zi-(~/fiR){h[~(p/~R)“2]+~}-(r_/60)(1n[f(p/~~)“2] +Y) Pa) for (In[+(p/&)“*]
+Y) 1. In this case, using the same mathematical methods and approximations as for single electrodes, we find, for example, for d = 3, that n(k_,p: 12(t)
=
( l/a2 + k,/D,a
x ( herfc[
- k,p;)
+ k_,/D,a)(l/a=
,D:&=]
+ k,/D,a
+ se+
Here the current Z2 is expected to be proportional
+ k_ ,/D,a)
2(D:$=lj
to k_ Ipg - k,pk.
(28)
13
4.3. Collection efficiency for d = 2, t > L2/D
B- L2 /a2
Important experimental applications of microelectrode arrays are connected with measurements of the so-called efficiency [141. The collector efficiency Gij is determined as the modulus of the ratio of the current on collector electrode i to the current on generator electrode j. The collector electrode is potentiostated in such a way that the collector current vanishes when the generator potential approaches equilibrium. Let us discuss the collector efficiency for a system of two microelectrodes. For d = 2 (i.e. when L* B- to), to perform the inverse Laplace transform it is possible to use here the same approach which has led us to eqn. (22) in the case of a single d = 2 electrode. However, in the experiments [141 which we will discuss in the next section, the experimental characteristic time was not less than L*/D and it is incorrect to use the expressions from the previous paragraph for the description of the experiment. In this case, the reasonable expressions for
l-l litt)
+u=
zj(t)
can be obtained in the following way. In the time interval a*/D
< t < L*/D
the time dependence electrodes. For the time
of qij will follow the expressions we obtained for the infinite
t > L*/D
the quasi-steady state will occur when the fixed part (up to about l/(t’/*)) of the generated particles will be picked up by collector electrodes and the remaining part will go to infinity. It is reasonable to expect that the cross-over time interval A will be relatively narrow and of the order A = R*/D =x L*/D
(29)
where R is the gap width. Correspondingly, to estimate the quasi-stationary current in the steady state, it is possible simply to put in the expressions for IClij(t> for infinite d = 2 electrodes, i.e. t = D/aL*, where (Y= 1 + O(R/L)*>. This means that, to obtain *ij for the quasi-steady state, it is necessary simply to substitute p for D/aL*, where D = D, = D, in the corresponding ratios, with
where ZiT(p) corresponds
to infinite d = 2 electrodes.
14
Using such an approach for two electrodes (1, generator; case of d = 2, we obtain the following expressssion:
2, collector) in the
&4R,*/~~)
12
- [ln(a/4L)
+2J= I x I = D/[(k,+k_,)a]
+y]
lng+, G D/[(k,+k_,)a] k,a/D
- [ln(a/dL)
t > L2/D
-=SC 1
R,,/L
We have dropped the coefficient
+Y]
-=K1
(30)
(Yin this expression, because, for
the additional term In (Y((w= 1) in the numerator is negligible. Equation (24a) can be generalized for the system of one generator and N collectors. For example, for two equivalent collectors (i = 2,3) placed at equal distances and on opposite sides of the generator (i = l), we will have 2m
R,,/L) (31)
W[V2
+
k-,b]
b-W4L)
+
rl
+
koCW2/L)
with k, = k,, k_, = k_,, R,, = R,3,k,a/D -=ZX 1 and t > L2/D It is worth mentioning that the expressions for the current in eqns. (30) and (311 do not include the diffusion coefficient, when D/k,a -K 1. 4.4. Many electrodes The case of many electrodes can be treated (for t > R/D, where R is the distance between nearest electrodes) within the framework of the same approach as used in atomic physics for the derivation of the so-called Fermi pseudopotential [9,18]. In this approach, sums over j for i Zj in eqn. (11) are replaced by integrals over the support surface S, with the mean electrode density c(h), where fi E S in the integrand, rather than the sum over the distinct localized electrodes Ei. The simplest representative result is obtained for the case when it is possible to treat all the electrodes as equivalent, with constant concentration c(@ = c. In this case, we find the following expression for the Laplace transform of the current Zi+(p, fii> in the point 6 on the surfaces:
Zj+(P,fl)
rFn Nu =--De
(=a)
15
where the numerator
kG;(a)
De=l+
and denominator
+ k-G;(a)
Dlt
functions are now
ad-l
Do
+ k-cad-’ D,
/
G,:
(
a-a’)
d;n’
(32b)
where, for G &o,(a> it is necessary to take eqn. (15) for i =j and for Gi(WO(-,R) - nO(-,R)‘) with the distance R between fi and h’ in argument.
5. COMPARISON
WITH EXPERIMENT
5.1. A simple case: diffusion limit
To demonstrate the utility of the above results, we will compare the predictions of the relatively simple theoretical expressions for collector efficiency in eqns. (30) and (31) with experimental data and numerical simulation results described by Bard et al. in ref. 14 for the redox behavior of Ru(NH&+ in H,O. In this work, generator-collection experiments were carried out for a microelectrode device with a parallel array of closely spaced microband electrodes (50 pm X 2.3 pm X 0.1 pm). According to eqn. (231, the length a in this case must be approximately equal to a = w/2 = 1.15 pm
(33)
One of the most important results of ref. 12 was the estimation of the collection efficiency I). Characteristic times in the experiments were larger than or comparable with the diffusion time t D = L2/D over the length of the microelectrode (in ref. 12 for L = 50 pm, D = 7 lop6 cm* s-l, t, z 3.5 s). Therefore, the electrode cannot be considered as having an infinite length and it is necessary to use for a quasi-stationary regime description the eqns. (30) and (31). Figure 3 shows the comparison of the results calculated with the help of eqn. (30) with experimental and digital simulation results [141 for a generator single collector couple. In Fig. 4 a similar comparison is made for calculation with the help of eqn. (31) for the case of one generator and two symmetrical collectors. In both cases, we are in the diffusion regime when D (k2 +k-,)a
e
-[In(&)
+7]
(34)
16 0.8-
gap Rgap for a one-collector, one-generator Fig. 3. Collector efficiency $r,a vs. generator-collector device. Squares are experimental data and circles are numerical simulation data from [16]. Full and dashed lines are theoretical dependences (eqn. (35)) with a = 1 pm and a = 1.2 pm respectively D/ka --SC 1; R,, in eqn. (35) is put equal to R 12 = R,,, + A, where the electrode width is A = 2.3 pm. The ‘best fit’ gives a value of a = 1 pm, in excellent agreement with the value estimated directly from the length scales in the experiment.
and it is thus possible to use the simplified expressions $2.1 =
[wG2/=4
+ Yl
-
+ rl
b(WW
2[WR12/W 9 1+3,2
E
-
[In(a/4L)
(35a)
+ ~1
+ y] - [In(R,,/L)
+ y]
(36a)
for eqns. (30) and (31) respectively. As is seen from Figs. 3 and 4, the general agreement between the experiments and our simple analytical results is reasonably good, even in the interval where H ,< 1 and inequality (6) is not strictly fulfilled; the value of ~2= 1 pm gives the best fit. It is interesting to note that the computer simulation described in ref. 14 gave a worse fit than eqns. (30) and (311, which could be attributed to the presence of an adjustment parameter and to the singular character of the corresponding solution of the diffusion reaction problem. 5.2. The kinetic regime The comparison with experiment described earlier was restricted to the simplest case, where only the geometry of the electrode array was of importance. According
17
w 1+3.2
0.6.
2
0
4
8
6
10
&p/pm dependence on generator-collector distance for the case of two Fig. 4. Collector efficiency $,+3,2 symmetrical collectors. Comparison with eqn. (36). AI1 notations are the same as in Fig. 3.
to eqns. (30) and (31), the dependence of I,!Ivs. Rij will change drastically when becomes (owing to potential or size variations) comparable with other terms in corresponding denominators. For example, in opposition to eqn. (35), in the kinetic regime, I,&~ will be proportional to a, such that
D/ka
$2.1 =
(k, + k_,)a
K”(Rd,‘L)
(37a)
for D B (k2
-[l”(G)
(37b)
+Y]
+ k-2)~
and in the cross-over interval, we will have the following dependence 1_
on a:
2(k, + k-duo D
D ‘-
(k_2+k_2)uo
i ln$
2 - [ln(u/4L)
+ r]
1
Pa)
for D
(k2 + k-da
(3gb)
18
where In ao is the root of the equation k fk_ 2
exp(ln a) = -(In
a - ln 4L + Y)
2
Observation of the changes from eqn. (35) to (38b) to (371, with changing a as well as the changes connected with increasing of other parameters in transient regimes, such as D/a2t, D/Lk, DK,/k, etc. can be used for most direct experimental estimations of ki, D and inverse Debye length K~ values. It is necessary to stress here that the functional dependence on parameters in the cross-over intervals of the type in eqn. (38b) has a fairly universal character. 5.3. Microscopic effects
Especially interesting are cases when microscopic effects can become important, as anticipated in the case of the work by Lewis et al. [4] with nanometer-scale electrodes dispersed on a supporting surface. To calculate the reaction rate constants, they first used the diffusion control limit of eqn. (22) for a unit spherical electrode to find out from comparison with experiment the effective value of a/2; they called this the ‘aperture electrochemical radius’. Then they calculated the rate constant by comparison of the experimental data for the kinetic regime with a single electrode of radius a/2. Such an approach is not correct, even in the framework of a simple diffusion model, because, at long times when diffusion layers merge, the current becomes the same as that for one electrode, with sizes of the whole area where the individual electrodes are situated. According to eqn. (27b), the values of the kinetic constants could differ from the true value in eqn. (22) by a factor of
a” (11 2
--&+y
;
The analogous results also could be inferred from equations given in ref. 6. The values of the kinetic constants obtained in ref. 4 were criticized by Baranski [19], who noticed the possible effect of special pecularities in the microelectrode device construction and of contamination effects. The available experimental data are insufficient to determine p; however, its value may be bigger than 10, when N > 10, as is assumed in ref. 4, and (a/R>,, = 1. In the case of nanoelectrodes, there could also be, as was mentioned by the authors [4], the additional substantial effects of a deviation from charge neutrality, the effect of lateral correlations in a water solution structure, and the effect of the specific character of the electronic structure of the electrodes whose dimensions are of the order of the electronic Fermi length and/or mean free electronic path. 6. CONCLUSIONS
We have demonstrated, for several simple examples, the possibilities of an approximate analytical description of the electrochemical response of microelec-
19
trodes. The proposed description could be also generalized for the description of rough surfaces and for random samples of microelectrodes. It is natural, for example, to infer for established singular behavior, i.e. In I a I for d = 2 and l/ I a I for d = 3, that the corresponding singularity for surfaces with a fractal structure will be of 1a 12-a type, where (Y is the fractal dimension. This also means the inapplicability in the limit a -+ 0 of mean field approximations for the description of random samples of the type used in ref. 7. In future work, we hope to widen the range of applications of the proposed method, including the description of the deviations from the classical diffusion-kinetic model. Most interesting will be the application of the methods described here to the description of a scanning tunnelling microscope operating in an electrolyte solution [20] and to the theory of the scanning electrochemical microscope 1211. ACKNOWLEDGMENTS
We are grateful to Lloyd Burges who initiated this work and to W. Ronald Fawcett for stimulating conversations. We are also grateful for support from the sponsors of the Center for Process Analytical Chemistry, University of Washington, Several comments of the referee have been taken into account in revisions of the manuscript and have led to improvements. We gratefully acknowledge support of the National Science Foundation through Grant CHE 9120206. REFERENCES 1 M. Fleischmann, S. Pons, D. Rolinson and P. Schmidt (Eds.), Ultramicroelectrodes, Datatech Systems, Morganton, NC, 1987. 2 M. Wightman and D. Wipf, in A. Bard (Ed.), Electroanalytical Chemistry, Vol. 15, Marcel Dekker, New York, 1989. 3 A. Bard, J. Craiston, G. KittIesen, T. Shea and M. Wrighton, Anal. Chem., 58 (1986) 2321, and references cited therein. 4 R. Penner, M. Heben, T. Longin and N.S. Lewis, Science, 250 (1991) 1118. R. Penner, M. Heben and N.S. Lewis, Anal. Chem., 61 (19891 1630. 5 V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, New York, 1962. 6 B. Scharifker, J. Electroanal. Chem., 240 (1988) 61. 7 A. Szabo and P. Zwanzig, J. Electroanal. Chem., 314 (1991) 307. 8 M. Mirkin and A. Bard, J. Electroanal. Chem., 323 (1992) 1; 29. 9 I. Demkov and V. Ostrovskii, Zero-Range Potentials and Their Applications in Atomic Physics, Plenum, New York, 1988. 10 V.G. Levich and A.N. Frumkin, J. Fiz. Chem., 15 (19411 789 (in Russian). 11 A. Szabo, J. Phys. Chem., 91 (1987) 3108. 12 S.F. Burlatsky and G.S. Oshanin, J. Stat. Phys. 65 (19911 1095. 13 E.L. Nagaev, Phys. Status Solidi, 167 (1991) 381. 14 A. Bard, J. Crayston, G. Kittlesen, T. Varco Shea and M. Wrighton, Anal. Chem., 58 (19861 2321. 15 I. Gradsteyn and I. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York, 1980. 16 H.S. Carlslaw and J.C. Jaeger, Conduction of Heat in Solids, Oxford University Press, Oxford, 1947, pp. 334-336.
20 17 A. Szabo, D.K. Cope, D.E. Tallman, P.M. Kovach, R.M. Wightman, J. Electroanal. (1987) 417. 18 G. Breit, Phys. Rev., 71 (1947) 215. 19 A. Baranski, J. Electroanal. Chem., 307 (1991) 287. 20 W. Schmickler and D. Henderson, J. Electroanal. Chem., 290 (1990) 283. J. Sass and J. Gimzewski, J. ElectroanaI. Chem., 308 (1991) 333. 21 P. Unwin and A.J. Bard, J. Phys. Chem., 95 (1991) 7814.
Chem., 217
APPENDIX A
In the text, we confronted the problem restoring the time dependence of the current Z(t) for t > t, from the expression for their Laplace transform Z+(p) calculated for p < l/t,. It is known that, in general, such inversion problems are ill-posed without restricting Z+(p) and Z(t) to a specific subclass of functions. In our case, Z+(p) and Z(t) belong to this subclass, because they are connected with a non-singular solution of the diffusion equation. Accordingly, in the d = 2 case, our results for Z+(p) have the form Z+(P) = (l/p)F(ln
P +z l/t,
P)
(Al)
where F (In p> is a smooth (entire) function of In p on and near the real axes for p > l/t,. This conclusion follows from the asymptotic structure of the two-dimensional diffusion equation Green function. In this case, Z(t) will be Z(t)
=F(
-1n t)
t 2-3t,
(fQ)
This result can be confirmed by the asymptotic relation kmF(-In
t) exp(-bt)
dt=/ymF(-u)
exp(-pe’+u)
where u, = --In p is the root of the equation d/dv
dkfoF(-us)i (-p
e” + u) = 0.
(m)
