Theory of the voltammetric response of electrodes of submicron

Anal. Chem. 1891, 65,3343-3353

Theory of the Voltammetric Response of Electrodes of Submicron Dimensions. Violation of Electroneutrality in the Presence of Excess Supporting Electrolyte Christopher P.Smith Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455

Henry 5. White* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112

The steady-state voltammetric behavior of spherical electrodes of nanometer dimensions ( I and 10 nm) is computed by numerical solution of the Nernst-Planck and Poisson equations. The results demonstrate that the reversible oxidation or reduction of an electroactive species, assumed to be present in solution at millimolar concentrations, is strongly affected by the electric field from the electrode surface, even in the presence of a large excess quantity of supportingelectrolyte (0.5 M). The assumptions of electroneutrality or diffusion-controlled transport are shown to be generally inappropriate for analyzing the voltammetric response (wave shape and limiting current) of electrodes of dimensions less than -0.1 pm. Transport-limited currents corresponding to the oxidation or reduction of a charged electroactive species are significantly enhanced or depressed relative to values calculated by assuming electroneutrality. When a neutral reactant is oxidized or reduced, the limiting current is unaffected by the electric field, but the voltammetric wave no longer has a classical Nernstian shape and is shifted in potential with respect to a reference electrode. Depending on the relative values of the formal redox potential (EO')and the potential of zero charge (PZC),the nonclassical wave shape may lead to situations where a transport-limited electrochemical reaction appears to be limited by the rate of heterogeneous electron transfer. Conversely, it is possible for a kinetically limited oxidation or reduction to appear as a reversible reaction from an analysis of the voltammetric wave shape. The results are discussed in terms of a recent report of unusually large heterogeneous rate constants obtained by steady-state voltammetry at electrodes of nanometer dimensions. I. INTRODUCTION Electrochemical methods involving the diffusion or migration of a soluble redox-active species are frequently based on the implicit assumption that charge neutrality is maintained throughout the solution, except in the region adjacent to the electrode that defines the electrical double layer. The assumption of electroneutralityis warranted in most instances and is a useful simplifcation of the mathematical descriptions of the interrelationship between faradaic current, electrode 0003-2700/83/03663343$04.00/0

potential, and time. Indeed, the electroneutrality approximation haa had such a long history of successful application in electrochemistrythat the validity of the approximation is seldom questioned. In dealing with transport-based electrochemical phenomena, the eledroneutrality approximation can be interpreted as a statement that the electric field resulting from the flow of current through a resistive solution is not large enough to induce a separation of positive and negative ions. As shown by Levich, this assumption is generally valid in solutions containing a relative excess of supporting electrolyte.' In addition, in electrochemical studies, the electroneutrality approximation also invokes the assumption that the overall mass-transport rate of ions and molecules is unaffected by the presence of the electrical double layer at the electrode/ solution interface. This assumption is also generally valid, since the mass-transport resistance of the thin doublalayer region is generally orders of magnitude smaller than that associated with the depletion boundary layer (the latter being d e f i e d by the concentration gradients of the electroactive species).2 The recent developmentof electroanalyticalmethodab a d on electrodes of micron and submicron dimensions (i.e,, microelectrodes)has led to a number of interesting situations where the use of the electroneutralityapproximation appears to be questionable. The first of these applications is the measurement of the steady-state voltammetric response in solutions containing very little supporting electrolyte."'@ (1)Levich, V. G. Physicochemical Hydrodynamics; Prentice-Hall, Inc.: Englewood Cliffs, 1962. (2) Norton,J. D.; Whita, H. 5.;Feldberg, S. W . J. Phys. Chem. 1990, 94,6772. (3) Oldham, K. B. J. Electroanal. Chem. 1988,260, 1. (4) Caeeidy, J.; Khoo, S. B.; Pone, S.; Fleiechmann, M. J . Phyr. Chem. 1986,89, 3933. (5) Dibble, T.; Bandyopadhyay, S.; Ghoroghchian, J.; Smith, J. J.; Sarfarazi, F.; Fleischmann, M.; Pone, S. J. Phys. Chem. 1986,90,5276. (6) Ciszkomka, M.; Stojek, 2.;Ostaryoung, J. Anal. Chem. 1990,62, 349. (7) Bond, A. M.; Fleiachmann, M.; Robinson,J. J. Electoanal. Chem. Interfacial Electrochem. 1984, 168, 299.

(8)Bond, A. M.;Lay, P. A. J. Electoanal. Chem. Interfacial Electrochem. 1986,199,285. (9) Amatore, C.;Deakin,M. R.;Wightman,R.M.J.Electoana1. Chem. Interfacial Electrochem. 1987,220,49. (10) Pena,M. J.; Fleischmann, M.; Garrard, N. J. Electoanal. Chem. Interfacial Electrochem. 1987,220, 31. (11) Amatore, C.; Foeset, B.; Bartelt, J.; Deakin, M. R.; Wightman, R. M. J. Ekctoanal. Chem. Interfacial Ekctrochem. 1988,266,256. (12) Bond,A. M.;Fleischmann,M.;Robhn, J. J . Electoanal. Chem. Interfacial Electrochem. 1984, 172, 11. (13) -u, G. N.; Rueling, 3. F.; Bond, A. M.J . Ekctoanal. Chem. Interfacud Ekctrochem. 1990,292, 187. (14) Bruckenetein, S. Anal. Chem. 1987,69, 2098. (16) Pendley, B. D.; Abruna, H. D.; Norton,J. D.; Benson, W.E.;White, H. S . Anal. Chem. 1991,63,2766. @ 1803 American Ctmmlcal Sockdy

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ANALYTICAL CHEMISTRY, VOL. 65, NO. 23, DECEMBER 1, 1993

Oldham has demonstrated that the small size of a microelectrode results in a convergent flux of ions to the electrode which reduces the ohmic potential loss associated with current However, the electric fields within the depletion layer, calculated under the assumption of electroneutrality, are sufficiently large to question whether charge neutrality is in fact actually maintained. The second application of microelectrodes that casts doubt on the electroneutrality approximation is the measurement of ultrafast electron rate constants using electrodes of nanometer dimensions. For electrodes of such small dimensions, the thickness of the double layer, in the presence or absence of excess supporting electrolyte, approaches that of the depletion layer, leading to the expectation that transport of charged electroactive species will be affected by the field from the electrode.2 Under these conditions, it appears questionable to employ the electroneutrality approximation in analyzing the voltammetric response to obtain kinetic information. In this paper, we reexamine the use of the electroneutrality approximation in describing the voltammetric response of spherical microelectrodes. In section 11, we briefly review the fundamental equations that describe the distribution of ions and electric potential in an electrochemical experiment. In section 111, several approximations that have been previously employed to calculate the voltammetric response of microelectrodes are discussed. In section IV, the electroneutrality approximation, as applied to spherical microelectrodes, is tested for self-consistency. The results show that this approximation, and the theories which are based upon it, are invalid when the radius of the electrode is less than 0.1 pm and the ratio of the concentration of the supporting electrolyte to that of the electroactive species is less than -10. In section V, we present the results of an exact calculation of the voltammetric response that is based on the simultaneous solution of the Nernst-Planck and Poisson equations. The results from these calculations and those from section VI demonstrate that the assumption of electroneutrality is inappropriate for electrodes of submicron dimensions, regardless of the electrolyte concentration. Finally, in section VII, we discuss some limitations of existing theories that are based on the assumption of electroneurality. We specifically show that the misuse of these theories can readily lead to large systematic errors in measurements of heterogeneous electron-transfer rate constants.

11. STATEMENT OF PROBLEM Voltammetric analyses of mass-transfer and kinetic phenomena are derived from fundamental equations describing the distributions and fluxes of the chemical species participating in the electrochemical reaction. In an unstirred solution, the expression for the flux, J i (mol/cm2s),of species i resulting from its electrochemical potential gradient, is given by the Nernst-Planck equation. Ji

= -DiVCi - (ziF/RT)DiCiV4

(1)

In eq 1, Di, C,, and zi are the diffusion coefficient, concentration, and charge of species i. The electric potential is denoted by 4, and the remaining terms have their usual meaning. In electrochemical experiments, the electrolyte solution comprises several species (reactants, products, electrolyte ions) whose concentrations near the electrode surface (16) Norton, J.D.;Benson, W. E.; White, H. S.;Pendley,B. D.;Abruna, H. D. Anal. Chem. 1991, 63, 1909. (17) Norton, J. D.; White, H. S. J . ElectoanaL Chem. Interfacial Electrochem. 1992, 325, 341. (18) Lee, C.; Anson, F. J. Electoanal. Chem. Interfacial Electrochem. 1992.323.381. (19) C&per, J. B.; Bond, A. M.; Oldham, K. B. J. Electroanal. Chem. 1992, 331, 877.

may be affected by the electrode reaction. Thus, for an N-component solution, there are N instances of eq 1 that describe the individual fluxes. In general, only the fluxes of the reactant and product species, evaluated at the electrode surface, contribute to the observed current in a voltammetric experiment. For a simple redox reaction, such as 0 neR, the current can be written in terms of the flux of 0 or R at the electrode surface: I = nFAJBowf = - nFAJsR!. However, the flux expressions for the reactant and product are coupled to those of the inert electrolyte ions through the migration term in eq 1containing the electric field, -V+. Thus, for the general case of an N-component solution, calculation of the fluxes of the electrochemical reactants and products requires a solution to the set of N equations, containing N + 1unknowns (C1,C2,...CN,and 4). An additional relationship between 4 and the set of Ci’s must be provided in order to obtain the voltammetric current. Although rarely employed in rigorous fashion by electrochemists, the fundamental relationship between 4 and the set of Ci’s is the Poisson equation

+

v 2 4 = -p/tt,

(2)

where p is the local net charge density in the solution, t is the static dielectric constant, and tois the permittivity of vacuum. The net charge density for the electrochemical problem is given by

where the summation is over nonelectroactive as well as electroactive species. In principle, eqs 1-3, along with the appropriate mass conservation laws and boundary conditions, provide sufficient information to describe the flux and distribution of every component of the solution. The solution to this set of equations provides the exact description of the voltammetric wave shape. To our knowledge, a description of a voltammetric wave shape based on the exact solution of this set of equations has not been previously described.

111. APPROXIMATE DESCRIPTIONS OF THE VOLTAMMETRIC WAVE SHAPE Electrochemists use mathematical simplifications, based on sound physical reasoning, to avoid the use of Poisson’s equation in solving the above set of differential equations. Since one purpose of this paper is to demonstrate that the most frequently used mathematical simplifications are inappropriate for computing the voltammetric response of microelectrodes, we will briefly review the origin of these approximations and provide a rough guide to the physical situations where they appear reasonable. A. Excess Supporting Electrolyte. The most common practice that simplifies the analysis of the voltammetric experiment is the addition of a large quantity of an inert electrolyte to the solution, such that the inert ion concentration is in excess of the concentrations of all electroactive species. The assumption made in doing such is that small variations in the local charge density, resulting from oxidation or reduction, are buffered by the presence of a large number of inert ions. Under these conditions, only a slight adjustment of the position of the inert ions is necessary to prevent the development of an electric field. Consequently, V4 = 0 and the set of flux equations reduces to J i = -DiVCi (4) for each of the product and reactant species directly involved in the electrochemical reaction (and any species coupled to

$Sa


the electron-transfer step by homogeneous reactions). This assumption forms the basis of all electroanalytical techniques based on diffusional transport. Microelectrode problems based on the assumption of a purely diffusional transport have been the subject of several recent articles.20.z1 An additional assumption in this approximation is that the electrical double layer is sufficiently thin that it has no significanteffect on the transport of ions between the electrode surface and the bulk solution. This assumptionwillbe shown in section V to be generally invalid for electrodes of submicron dimensions. B. Electroneutrality. A second commonly employed approximation is based on the assumption of local electroneutrality throughout the solution. Mathematically, the statement of electroneutrality is obtained by assuming that the divergence of the electric field can be taken equal to zero. The Poisson equation (eq 2) thus simplifies to 0=

czici

(5)

1

It may appear that eq 5 is a restatement of the approximation made in case A, where an excess quantity of inert ions was considered. Indeed, eq 5 must also apply to that situation, since VIP = 0 was assumed. However, the fact that the divergence of the field is assumed to be zero does not necessarily imply that the electric field itself is vanishingly small. Indeed, the electroneutrality approximation is most frequently employed in physical situations where ion transport generatesan electric field large enough to enhanceor suppress the flux of ions. Such situations arise in a number of problems involving ion transport. In the context of microelectrode voltammetry, the electroneutrality approximation is used when the ratio between the concentrationsof inert supporting electrolyte ions and electroactive species, CBupporJCredox, is small. General treatments of this problem are given in ref 11. As in case A, it is almost always assumed in using the electroneutrality approximation that the electrical double layer has no significant effect on the ion fluxes or voltammetric current. C. Unscreened Electrode. A third approximate treatment has been presented by Norton et al.,2 who considered situations in which the solution contains an insufficient number of ions to screen the electrostatic charge of the electrode surface. Under these conditions, the Poisson equation may be simplified to the Laplace equation 024 = 0

(6)

In contrast to the assumptions made for case A (excess supporting electrolyte) or case B (electroneutrality), the treatment of Norton et al. emphasizes the role that the electric double layer plays in controlling the fluxes of ions. The criterion for applying this approximation to a spherical electrode of radius r, is that the product of the electrode radius and the inverse Debye length ( K ) approach zero, i.e., r d 0. Thus, the unscreened electrode exists for situations where the electrode dimension and/or the total solution ion concentration are exceedingly small.

-

D. Equilibrium and Nonequilibrium Double-Layer Structure. Beginning with Levich,’ several researchersz2 have described the effect of the equilibrium double layer on ion transport. The key assumption in these treatments is that the potential distribution in the double layer, calculated (20)h k i , C.G.J. Electoanal. Chem. Interfacial Electrochem. 1990, 296,317. (21)Mirkin, M. V.;Bard, A. J. J. Electoanal. Chem. Interfacial Electrochem. 1992,323,29; 1992,323,1. (22)Delehay, P. Double Layer and Electrode Kinetics; Interscience: New York, 1965; pp 153-167.

at zero-current conditions and in the presence of excess supporting electrolyte (and other special casesz3)is unperturbed by the current flow. Thus, an analytical expression for the double-layer potential distribution (obtained from the Gouy-Chapman model) is used directly to calculate the flux. This type of approximate approach waa applied by Amatore and Lefrou for the case of ultrafast cyclic voltammetry at spherical electrode^.^^ Others have used both numerical methods and asymptotic approximationsof the Nernst-Planck and Poisson equations to investigate the formation of the equilibrium double layer next to planar cation-exchange membranesz5and electrode surfaces.26s21

IV. BREAKDOWN OF ELECTRONEUTRALITY In this section, we estimate the magnitude of the error incurred as a result of employing the electroneutrality approximation rather than the Poisson equation. To estimate this error, the potential distribution obtained by solving the Nernst-Planck equation with the electroneutrality approximation is substituted into Poisson’s equation. Since the resultant potential distribution is based on electroneutrality and is only an approximation of the true distribution, there will be a discrepancy between the left- and right-hand sides of eq 2, i.e., -(ceo/F)V2c$ # CiziCi. Since it was assumed that CiziCi = 0, the degree to which (eco/F)Vzc$ departs from 0 provides an estimate of the quality of the approximation. This is the sort of reasoning used by Levich’ and Newmana to demonstrate the validity of the electroneutrality approximation for macroscopic systems. The following example uses the same reasoning to show the limitations of the electroneutrality approximation (as was also done by Rubenstein and ShtilmanZ5for a problem involving planar geometry). For the single-electron oxidation of a neutral reactant at a spherical electrode of radius r,, the potential distribution is given by (7)

where is the potential difference between the position r in the solution and the bulk of the solution and I/Il is the current normalized by the limiting current. This is the potential distribution derived by Oldham under the assumption of electroneutrality (eq 30 of ref 3) except that we have assumed the equivalence of the reactant and product diffusivities and have used y to represent the ratio of the supportingelectrolyte and reactant concentrations (y CsuppodCredox). The relationship between the current and the electrode potential, E, is given by eq 31 of ref 3 (with slight rearrangement) as

E = Eo’ + (RT/F)h[(I/Ii)(4y + I/I1)/(47(1- I/Ii))l (8) where Eo (instead of E O ’ ) was the symbol used for the formal potential. To test the accuracy of the electroneutrality assumption used to derive eq 7, the potential distribution is substituted into Poisson’s equation. With ro2Vz4= (rO/r)* d24/d(rdrI2in ~~~

(23)See refs 18 and 22 for additional referencee to approximate treatments of the static equilibrium double layer. (24)Amatore, C.; Lefrou, C. J. Electroanal. Chem. 1990,296,336. (25)Rubenstein, I.; Shtilman, L. J. Chem. Soc., Faraday Trans. 2 1979,75,231.Rubenstein, I.; Segel, L. A. J. Chem. SOC.,Faraday Trans. 2 1979,75,936. Rubenstein, I. J. Chem. Soc., Faraday Trans.2 1981, 77, 1595. (26)Murphy, W.D.;Manzanares, J. A.; Maf6, S.; hiss, H. J. Phys. Chem. 1992,96,W83. Manzanaree, J. A.;Murphy, W. D.; Maf6, S.;Wi, H. J. Phys. Chem. 1992,96,802. (27)Smyrl, W.H.;Newman, J. S. Tram. Faraday SOC.1966,62,207. (28)Newman, J. S.Electrochemical Systems; Prentice-Hall: Englewood Cliffs, NJ, 1972.

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ANALYTICAL CHEMISTRY, VOL. 65, NO. 23. DECEMBER 1, 1993

,

200

4

1

= R E

0

1

,

, \ ,

1

1

.

10

r I ro

Flgure 1. Variationof ro2cmorwith distance from the electrode surface for y = 0,001. ,1 is the error in &,C,as computed by substituting the potential variation (the iR drop predicted by the solution to the Nernst-PLanck equation with the electroneutrality approximation)into Poisson's equation, as described in the text. The system considered here is the one-electron oxidation of a neutral reactant A B+ -te(P' = o). The error is computed at the electrode surface (r = r,), which is held at E = ,Ell2 (which also varies with y). The potential is measured with respect to the PZC.

,

I

i

N_E

E

lo'

p

10"

NW N

1

10-3 I 0-4

100

102

102

Y Flgure 2. Variation of rO2c, (as described in Figure 1) as a function of the ratio of the supporting electrolyte and reactant concentrations (7= CSW~C,&X). 200,

-

maintained in the bulk of the solution. Figure 2 shows the variation of ~,,,,, (calculated at the electrode surface) as a function of y. This figure indicates that the electroneutrality assumption is best when there is an excess of supporting electrolyte. Finally, Figure 3 shows the variation of the quantity (again, computed at the electrode surface) as a function of the electrode potential. This figure shows that the electroneutrality approximation is most appropriate when the current is small. These plots can be used to estimate the error of the electroneutrality assumption for different electrode radii after dividing the y-axis by the square of the electrode radius. For example, if the electrode being used has a radius of 1 pm, then the value read from the y-axis should be divided by lo6 nm2 to obtain the deviation from electroneutrality in units of millimolar. Consider Figure 2, where is plotted as a function of y. For y = 1, ro2Cerror= 10 nm2 mM. If a 10rm-radius electrode is used, ~,,,,, = le7mM. Obviously, a le7 mM deviation from charge neutrality is vanishingly small when the reactant concentrations are on the order of 1mM. On the other hand, if the electrode radius is 1nm, then = 10mM. Thus, the net charge resulting from the separation of charge by the transport-induced field is comparable to or larger than the concentration of the electroactive species. Under these conditions, it is obviouslyinappropriate to assume electroneutrality to solve for the flux. It is important to remember that these graphs only estimate the error in the electroneutrality approximation as it applies to the electric field in the depletion layer that is induced by ion transport (i.e., the iR drop)-the electrical double layer has been totally ignored. As the radius of the electrode becomes smaller than 100 nm, however, the electrical double layer begins to occupy an appreciable fraction of the depletion layer (Figure 4) and will have a significant effect on transport phenomena. Therefore, although ~,,,,, is on the order of 103 mM when y = 100 and ro = 1 nm (Figure 2), it is doubtful that electroneutrality will accurately predict the voltammetric response because the electric field from the electrode spans ~ ~ 1 of 1 the 0 depletion layer (Figure 4). The breakdown of the electroneutrality approximation resulting from overlapping depletion and double layers is addressed in the following section. The above analysis of C,,, can be applied to any oxidation or reduction reaction for which the potential profile is known (Appendix I), but the results are qualitatively the same. Results obtained for reactions which produce a neutral product (e.g., A+ + eA), however, indicate a larger departure from electroneutrality at the limiting current than is shown in Figure 2. The reader should note that the results shown in Figures 1-3 are only approximate; the true departure from electroneutrality (or rather the ramifications of the departure with regard to a voltammogram) can only be

i f \i

io3,

z

Flgure 4. Variation of potential with distance for a surface potential of F@RT = 4 (- 0.1 V) and 1: 1 electrolyte concentrationof 0.5 M for different electrode radii. The fraction of the depletion layer (which is always 10 times larger than the electrode radius, rlr, = 10) occupied by theelectrode double layer decreasesas the electrode size increases. The abscissa is scaled logarithmically.

I

1

,

I

,

,

1

1

e,,,,

c,,,,

01 0.4

'

'

I

'

'

0 E - Eo

'

L

'

'

-1 -0.4

Flguo 3. Variation of ro2Caor(asdescribedIn Figure 1) at the electrode surface as a function of the electrode potential, €(solid line). The error increases with the current, which is also shown (dashed line). The value of y is 0.001.

spherical coordinates, eq 7 is differentiated twice and multiplied by -(r0/r)*(te,4F)to yield,

=

r?Cerror(9)

where 183.81 nm2 mM is the value of RTee,IF evaluated at T = 298 K and e = 78. Note that when (ze,JF)V2d is multiplied by ro2,the resultant expression is dependent only on the dimensionless distance rlr0. For the following discussion we will call the quantity on the right-hand side of eq 9 "ro2Cerror" since it is an estimate of the error in the summation of the electroneutrality approximation, multiplied by the square of the electrode radius. The closer ~,,,,is to 0 (the assumed value of XiziCi), the better the approximation. Figures 1-3 show how C,, varies with respect to the variables r, y, and electrode potential. Figure 1 shows the variation of with respect to distance from the electrode. As expected, the departure from electroneutrality diminishes with increasing distance as it must since electroneutrality is

c,,,,

-

ANALYTICAL CHEMISTRY, VOL. 85,

addressed by solvingfor the fluxes of the electroactive species using Poisson’s equation.

NO. 23,DECEMBER 1, 1993 8847

0

V. EXACT SOLUTION USING THE POISSON EQUATION Steady-state solutions to the Nernst-Planck and Poisson equations for the voltammetric response a t 1-and 10-nmradii spherical electrodes were obtained using the finite difference method outlined in Appendix 11. Only singleelectron oxidations of reactants with charge z = -3 to +3 are considered below; A’ + B’+’ + e-. The corresponding reductions may be obtained by changing the sign of the current, potential, and Eo’ reported for the oxidation. For example, the voltammetric response for the oxidation A2+ B3+ e- ( E O ’ = +0.1) is equivalent to the voltammetric response for the reduction A” e- + Bs ( E O ’ = -0.1) after the sign of the current and potential is changed. The following parameters and assumptions were chosen to represent typical conditions in an electrochemical experiment: the diffusion coefficients of the reactant and the product are the same; the closest approach of ions and the position of electron transfer (PET) are both located 0.2 nm from the electrode surface; the region between the electrode and the PET contains only a dielectric having a constaut ej which is assumed to be the same asthat in the aqueous solution (ci = c = 78). The electrolyte solution is assumed to initially contain 5 m M of reactant (A) and 0.5 M of 1:l supporting electrolyte, corresponding to a solution containing excess supporting electrolyte; i.e., y = C8upwdC~,,x = 100. The product of the electrochemical reaction (B)is initially absent in the solution. In addition, reactants having a charge z are accompanied by blcounterionsper molecule. All calculations are reported for a temperature of 298 K. Except where noted, the formal reduction potential, EO’, is taken equal to 0 with respect to the potential of zero charge (PZC) of the electrode. In the following discussion, it is important to remember that when the supporting electrolyte is present in solution at a 100-fold excess relative to the electroactive species, the voltammetric response predicted by assuming electroneutrality is essentially diffusion controlled (Le., migration is negligible). Thus, any departure from the usual sigmoidal shape of a reversible voltammogram is the result of migration caused by the field within the double layer, not by the field produced by transport of ions in the depletion layer. In the figures described below, the responses calculatedon the basis of the assumption of electroneutrality (Appendix I) are presented to allow comparisons with the responses predicted by the model based on Poisson’s equation. Oxidation of Neutral and Negatively Charged Reactants. The voltammetric responses corresponding to the oxidation of neutral ( z = 0) and negatively charged reactants ( z = -1, -2, -3) are shown in Figure 5. At a 1-nm electrode, only a neutral reactant gives rise to a response approaching the eledroneurality limit (dashed line). For negatively charged reactants, the voltammetric curve has an essentially sigmoidal shape, but the currents are significantly enhanced as the result of the electrostatic attraction between the negatively charged reactant and the positively charged electrodeat potentials positive of the PZC. Thisenhancement is qualitatively similar to that computed by Norton et based on the assumption of an unscreened electrode (i.e., Vzd = 0). Whereas the assumption of an unscreened electrode leads to the prediction that the transport-limit current increases without bound, due to the increasing attraction between the reactant and the electrode as the electrode potential is made more positive, the use of Poisson’s equation accounts for the screening of the electrode charge as the

+

+

0.25

I

I

0

-0.25

E-Eo Flguro 5. Vdtemmetrlc response for the slngle-ekctron oxldatbn of a negatively charged reactant, Az F= B*’ e- (P’= O), at a l-nm electrode (solid I h ) . The values of z are lndlcatedon the graph. The dashed llne correspondsto the oxkiation of a specks wlth I = -1 at a lO-nm electrode. The current I Is normalized by the dmuskn-llmhed current Id. The parameters used for the calculation are Indicated at the beginning of sectkn V. The solid polnts (0)lndlcate the response predicteduslngtheelectroneutralltyapproxlmatkm(whlch Is essenUally the dlfWon4lmlted response).

+

E - EO’ Flgwo (1. Vdtemmetrlc response for the slngkekctron oxldatbn of a positively charged reactant, Az B*’ e- (P’ = 0), at a l-nm electrode ( d i d Hnes). The values of zare Indlcated on the graph. The dashed llne correspondstothe oxidation of aspecks wlthz = +1 at a 1O-nm electrode. The current I Is normallzed by the dmudofklhnbd current Id. The parameters used for the calculation are Indicated at the beglnnlng of sectkn V. The solid points (0)Indicate the responw predictedusingthe electroneutralltyapproxlmatbn(whlchIs essenUaiy

*

+

the dmuekn-llmlted response).

electrode potential increases. This screening reduces the attraction between the electrode and reactant and leads to a true potential-independent transport-limited current plateau. The enhancement of the transport-limited current ranges from 50% for a reactant with charge -1 to 75% for a reactant with charge -3. When the radius of the electrode is increased to 10 nm, this current enhancement is considerably leas, but still significant, since transport in the depletion layer is influenced by the electrode double layer (Figure 4). The limiting current for the oxidationof a reactant with charge z = -1 is enhanced by 5 % ,while the response for z = -3 (not shown) is enhanced by -8%. Although the shapes of these voltammetricwaves are nearly sigmoidal,they are not identical to the shape of the reversible voltammogram calculated on the basis of the assumption of electroneutrality, a point of importance in kinetic analyses based on the wave shape (section VI). Oxidation of Positively Charged Reactants. The voltammetric responses corresponding to the oxidation of positively charged reactants ( z = +1, +2, +3) are shown in Figure 6. Unlike the response for negatively charged reactauts, there is no limiting current plateau. Instead, the voltammograms are peak shaped as a result of the repulsion between the positively charged reactant and the positively charged electrode at potentials positive of the PZC. The suppression of the current is more pronounced for the more highly charged readants and persists even when the electrode radius is increased to 10 nm. The peak-shaped voltammograms are alsoin qualitative agreementwith calculationsbased on the assumption of an unscreened electrode.2 The maximum is less pronounced, however, since Poisson’s equation

3348


accounts for the screening of the electrode charge, which reduces the repulsion between the electrode and reactant. In summary, the true voltammetric behavior of electrodes of submicron dimensions in the presence of excess supporting electrolyte (y = 100) is intermediate between the behaviors computed on the basis of assumptions of (i) electroneutrality" and (ii) an unscreened electrode.2 The field arising from the electrode surface (ignored under the assumption of electroneutrality) either enhances or suppresses the flux of negatively/positively charged reactants being oxidized at the electrode surface. The enhancement or suppression of the flux is not as pronounced as in the model based on an unscreened electrode since Poisson's equation accounts for the screening of the electrode, which reduces the effect of the electrode double layer.

VI. LIMITATIONS OF APPROXIMATE THEORETICAL TREATMENTS FOR KINETIC MEASUREMENTS The development of microelectrodes has been motivated, in part, by kinetic measurements of fast electron-transfer reactions at metal/liquid interfaces.2934 As noted by Russell et al.,B a decrease in the electrode dimensions is equivalent to an increase in the rate of mass transport of reactants relative to the rate of heterogeneous electron transfer. Thus, electrochemical reactions that are limited by diffusion and/or migration at macroscopic electrodes eventually become limited by the rate of electron transfer as the electrode size is reduced. A simple and illustrative example of how the relative rates of mass transport and electron transfer vary with electrode dimensions is that of redox reaction occurring a t a spherical electrode of radius r, (cm). Assuming that transport occurs solely by diffusion and without interference due to ion migration within the electrical double layer, the steady-state mass-transport coefficients of the electroactive species are -D/ro ( D is the diffusion coefficient, cm2/s)and the heterogeneous electron-transfer rate constant is ko (cm/ s). When the ratio of these values is greater than unity, Dlk'r, > 1, the reaction is limited by the kinetics of the electrontransfer reaction. Conversely, when Dlkoro< 1, the reaction is limited by mass transport of the reactant to the surface. A number of reports have described electrochemical measurements using electrodes of nanometer dimension^.^"^^ In principle, these electrodes can be used for steady-state measurements of electron-transfer rate constants as large as 100cm/s, an order of magnitude larger than can be measured by conventional electroanalytical techniques employing macroscopic electrodes. This capability, as well as several other advantages, (e.g., use in low ionic strength solutions), has led to the development of numerous theories of varying complexities that describe the steady-state voltammetric response of microelectrodes as a function of the mass-transport coefficient and heterogeneous rate constant. An important

-

(29) Russell, A.; Repka, K.; Dibble, T.; Ghoroghchian, J.; Smith, J. J.; Fleischmann, M.; Pitt, C. H.; Pons, S. Anal. Chem. 1986, 58, 2961. (30) Bond, A. M.; Henderson, T. L.; Mann, D. R.; Mann, T. F.; Thormann, W.; Zoski, C. G. Anal. Chem. 1988,60, 1878. (31) Wipf, D. 0.;Kristensen, E. W.; Deaking, M. R.; Wightman, R. M. Anal. Chem. 1988,60,306. (32) Andrieus, C. P.; Hapiot, P.; Saveant, J.-M. J.Phys. Chem. 1988, 92, 5992. (33) Wightman, R. M.; Wipf, D. 0. Acc. Chem. Res. 1990, 23, 64. (34) Bowyer, W. J.; Engleman, E. E.; Evans, D. H. J. Electroanal. Chem. 1989,262,67. (35) Pendley, B. P.; Abruna, H.D. Anal. Chem. 1990,62, 782. (36) Morris, R. B.; Franta, D. J.; White, H. S. J.Phys. Chem. 1987,91, 3559. (37) Seibold, J. D.; Scott, E. R.; White, H. S. J. Electoanol. Chem. Interfacial Electrochem. 1989, 264, 281. (38) Penner, R. M.; Heben, M. J.; Longin, T. L., Lewis, N. S. Science 1990,250, 1118.

0

-n 1

-0.5

-1 0.2

0 -0.2 E - Eo' Flgure 7. Voltammetric response for the slngle-electron oxldatbn of a neutral reactant, A B+ i-e-, at a 1-nm electrode (mild Ilnes) for different values of P' as Indicated on the graph. The current I Is normalized by the dlffwion-limited current I.. The parameters used for the calculation are Indicated at the beginning of section V. The sdldpokRs(0)lndlcatetheresponsepredlctedwingtheelectroneubaHty approximation (which 1s essentially the dlffuslon-limited response).

assumption made in these theories is that the electrolyte solution remains electrically neutral. This approximation is necessary to obtain analytical expressions describing the I-V response of reversible, quasi-reversible, and irreversible reactions. Penner et al. have reported the application of hemispherical Pt and Pt/Ir electrodes with radii as small as 1.1 nm for measuring the rate constants for several redox couples.38These authors used the theory of Oldham and %ski39 for extracting electron-transfer rates from the shape of the voltammograms. This theory assumes that transport occurs only by diffusion, which we have shown is an invalid assumption when using electrodes of submicron dimensions. It is thus interesting to consider the magnitude of the error that might be introduced in a kinetic analysis when such an approximate theory is employed. We willuse the data of Penner et al.= to illwtrate the limitations of theories based on the assumption of electroneutrality. We consider three redox systems that correspond to the examples for which Penner et al. reported complete voltammetric curves. Oxidation of Ferrocene. Penner et al. presented a voltammogram for the oxidation of ferrocene (Fc) at a 1.6nm Pt electrode. The half-wave potential, E l p , of this voltammogram was shifted by 16 mV positive of EO', an effect that was attributed to the finite rate of electron transfer. Although they used this value to calculate ko for the Fc/Fc+ couple, it appears likely that a t least some of this shift is due to double-layer effects. Figure 7 shows the voltammetric response for the oxidation of a neutral reactant as a function of E O ' . If Eo' of the redox couple is positive of the PZC, the voltammetric curve is shifted toward more positive potentials due to the electrostatic repulsion of electrogenerated product (e.g., Fc+) from the electrode surface. This shift could mistakenly be attributed to kinetic limitations which tend to shift the curve in the same direction. In fact, if this theoretical curve is analyzedMaccording to the approximate theory of Oldham and %ski,38 it appears that rJz0ID = 1.28 and a = 0.67, suggesting that the electron-transfer kinetics are measurable-even though the reaction is reversible. Using ro = 1.6 nm and D = 2 X 10-5 cm2/s,the heterogeneous rate constant calculate from this apparent value of ko is -160 cm/s which is within error of the value of ko = 220 f 120cm/s reported by Penner et al. Thus, an alternative interpretation of the experimental results is that the shift in Ell2 is due solely to double-layer effects and that the heterogeneous (39) Oldham, K. B.; Zoeki, C. G. J. Electoanal. Chem. Interfacial Electrochem. 1988, 256, 11. (40) Nonlinear regreesion was applied t a the curve, assuming equivalence of the reactant and product diffusivities in the model described in ref 39 (which is based on the electroneutrality assumption).


E - E' Flgm 8. Vdtemmetrlc response for the singleelectron oxldatlon, AIB e-, at a l-nm electrode (sdtd #ne@) for different values of €" as indicated on the graph. The cwemt Z Is notmallzed by the dmuekn Umned current 1.5. The parameters used for the calcutetkm are Indicatedat the beginningof sectknV. The sdid points (0)lndlcate theresponse predicteduslngthe elecironeutralttyapproxlmatbn(whlch is essentielly the d m u s i o n - l M response).

* +

electron kinetic rate constant of the ferrocene redox system is too large to measure even with a nanometer-size electrode. If E"' is negative of the PZC, then Ell2 at a 1-nm-radius electrode is predicted to shift -7 mV negative of E"' due to the electrostatic repulsion of the positively charged product from the electro$e surface. This shift toward cathodic potentials cannot be explained on the basis of slow heterogeneous kinetics, since a voltammetric wave for an oxidation reaction would never be shifted negative of the reversible wave as a result of slow electron transfer. However, it appears reasonable to suggest that the shift to negative potentials resulting from electrostatic forces may cancel out part or all of the normal positive shift due to electron-transfer kinetic limitati0ns;thatis,amemuredshift (e.g.,the 16mV measured by Penner et al.) is due in part to the electrostatic effect (-7 mV) and in part to the slow electron transfer. The consequence of these offsetting effects is that the value of k" reported by Penner et al. may be too large. An approximate correction of the measured shiftin E112can be made by a d d i g the I-mV electrostatic shift to the observed shift in Ell2. Such ashift analyzedaccordingto the theory of Oldham and Zo~ki,3~ suggests that the true value is -30 9% lower than the reported value of 220 f 120 cm/s. If E"' were even more negative of the PZC than the assumed value of -0.1 V, the true value of k" would be even smaller. A key point of this discussion is that the PZC must be known in order to obtain an order of magnitude estimate of ko when electrodesof small dimensionsare used. In principle, a measurement of the PZC for the nanometer-size Pt and Pt/Ir electrodes used by Penner et al. would allow for the correction of their rate constant data. Reduction of Ru(NHs)'+. As shown in Figure 8, the reduction of a positively charged reactant, e.g., Ru(NH3)3+, ispredided to give rise to a sigmoidallyshaped voltammogram with a sie;nificantly enhanced limiting current. (As noted above, the shape of the voltammetric response for the reduction of a positively charged species is identical to that for the oxidation of a negatively charged species if the signs of the current and potential axes and EO'are reversed. Thus, the voltammetric behavior expected for a reduction is also given by Figure 8. Although Figure 8 applies to a reactant with charge -1, the behavior is similarfor more highly charged species as seen in Figure 5.) In this case, the voltammetric Ell2 is predicted to be shifted negative of E"', regardless of whether Eo' is positive or negative of the PZC. Similar to the case considered above for the ferrocene oxidation, this shift could be readily mistaken as an indication of kinetic limitations. In addition, if the enhanced limiting current is used to determine the hemispherical electrode radius using the relationship obtainedfrom the assumption of eledroneutrality the apparent electrode radius, rapp, (Id =

3849

would be larger than the actual electrode radius by a factor of 50-100 9% ,depending on the exact charge of the reactant. Penner et al. reported a El/%shift of 37 mV for the voltammetric reduction of Ru(NH3P+ at a 1.1-nm-radius electrode. The voltammogram they preeented has an appearance similar to the curve in Figure 8 corresponding to Eo' = -0.1 V w PZC (suggesting that the E"' value for the reduction is positive of the PZC). As noted above, our resulta indicate that the reversible voltammetric response for the reduction of a positive species will always be shifted in a directionthat would give rise to an apparent kinetic limitation. The magnitude of this shift, from Figure 8, is between 20 and 50 mV for E"' values between 0.1 and -0.1V VB PZC. Thus, we conclude that the value of k" (79 f 44 cm/s) obtained by Penner et al. underestimates the true value. The magnitude of the error cannot be estimated in this case without a precise value of the PZC. Penner et al. ale0 reported that there was no shift in E112 for the reduction of methylviologen (MV2+)using a 2.2-nm electrode.98 This result is qualitatively consistent with the present model only if E"' for the W + / M V +couple is significantly negative of the PZC. Oxidation of FeZ+.On the basis of the results presented above in Figure 6, it is highly unlikely that the voltammetric response for an oxidation of a positively charged readant at a nanometer-size electrode could be mistaken as being kinetically limited since the wave shape is peaked (even at an electrode with a 10-nm radius). However, because the heterogeneous rate constant for the Fe3+/Fe2+couple is small (0.018 cm/s), it was not necessary for Penner et al. to employ electrodes of submicron dimensions to observe kinetic limitations in the voltammetric wave shape.98 The smallest electrodethat these authors used to measure the rate constant of the Fe3+/Fe2+couple had a radius of 1.3 pm, sufficiently large to justify the use of the theory of Oldham and Zoski.39

VII. CONCLUSION The approximation of electroneutrality used in developing models of the voltammetric response of microelectrodeehas been shown to be generallyinvalid for electrodesof submicron dimensions. The breakdown in this approximation resulta from two effeds: (i) charge separation in the depletion layer due to electric fields generated by the flux of ions though the electrolyteand (ii) the comparabledimensions of the depletion layer and electrode double layer. The voltammetric response of 1- and 10-nm-radii spherical electrodes is significantly affected by the presence of the double layer, even when there is an excess of supporting electrolyte (C,,,d = 100Cr~oJ. The shapes of the voltammetric curves deviate from the classical Nernstian response based on either the assumption of electroneutralityor a diffueion-controlledresponse. When the magnitude of the charge of the redox species increases, a peak-shaped voltammogram results (e.g., A2+ As+ or A" As); when the magnitude of the charge decreases, an enhanced limiting current resulk, a neutral reactant has the normal limiting current, but the wave shape is different from that based on the assumption of electroneutrality or diffusional transport. In the presence of excess supporting electrolyte, the shift in Ell2 due to electrostatic effects may be similar to that predicted to occur as a result of a fiiite electron-transfer rate. Thus, care must be taken to avoid the misinterpretation of shifts in El12 or small variations in the wave shape as an indication of electron-transferkinetic limitations. Of course, our analysis does not preclude the use of microelectrodes for determining reaction kinetics, but it does require that the

-

-

3350


wave shape be analyzed using a model other than the existing ones that are based on electroneutrality. It might be argued that the specific adsorption of inert electrolyte ions onto the electrode surface would reduce the electric fields at the electrode surface, thereby decreasing themagnitude of the effects we have presented. Two brief arguments suggest otherwise. First, consider the effect of the field within the depletion layer (which is the cause of the failure of the electroneutrality assumption in dilute ionic solutions). While specific adsorption would reduce the field from the electrode, it would not effect the field in the depletion layer. The latter is generated by the flux of ions through the depletion layer, not from the charge on the electrode surface, and is thus insensitive as to whether the electrode is screened or not. Second, consider the effect of the double layer perturbing the flux of ions to the surface (which is the cause of the failure of the electroneutrality assumption at nanometer-size electrodes). If inert ions were allowed to occupy the region between the electrode and the PET, the effect of the double layer would indeed be diminished since part of the electrode charge would be screened. However, our model has treated inert ions as point charges, which allows an unrealistically large number of ions to approach the PET (except at potentials very close to the PZC). If the finite size of the ions or specific adsorption are considered, the ion density adjacent to the electrode would be considerably less than that predicted by assuming point charges. (The same argument applies to the equilibrium double-layer structure constructed from the Gouy-Chapman model.) Ccnsequently, the electrode charge will be screened to a lesser degree, and the potential distribution will extend further into the solution, overlapping more with the depletion layer. Thus, the effect of the electrical double layer on the voltammetric wave shape, discussed in sections V and VI, has actually been underestimated to a small degree by not accounting for specific adsorption. Of course, there may be special instances (e.g., at electrode potentials very near the PZC and for intermediate electrolyte concentrations) where specificadsorption will have a pronounced effect on the behaviors we have described. In concluding, it must be admitted that there does not appear to be any conclusive electrochemical data that directly supports or refutes the new conclusions that we have reached. In part, this is due to the difficulty in constructing and characterizing electrodes of submicron dimensions that have the assumed ideal geometry. The qualitative consistency of the data of Penner et al. with the results does not demonstrate the validity of our proposal, since the experimental results are also reasonable on the basis of other models that assume electroneutrality. However, the description of the voltammetric wave that we have provided appears to stand on more rigorous physical ground, having employed Poisson's equation to obtain the relationship between the electric charge and potential. From the results we have presented, there appear to be at least two experiments that are readily capable of demonstrating the validity (or invalidity) of our theory without a prior knowledge of the precise electrode dimension or shape. The first is a voltammetric measurement corresponding to the oxidationlreduction of a positivelylnegatively charged reactant. As shown in Figure 6, the theory predicts a steadystate, peak-shaped voltammogram (even in the presence of 0.5 M electrolyte), which would be easily recognized. The second experiment is a side-by-sidecomparison of the limiting currents for the oxidation (or reduction) of a neutral species and the oxidationlreduction of a negatively1positively charged reactant. A significant enhancement of the limiting current for only the charged species is predicted to occur as the electrode radius is decreased below -0.1 pm. Until the

validity of the theory is tested, it appears that the use of microelectrodes for measuring fast kinetic rate constants will be shrouded by large uncertainties.

APPENDIX I. GENERAL POTENTIAL PROFILES AND VOLTAMMOGRAMS DERIVED UNDER THE ELECTRONEURALITY ASSUMPTION In order to calculate the spatial variation of the departure from electroneutrality (Figures 1-3) and to compare computed voltammograms and concentration profiles to those predicted on the basis of electroneutrality, it was necessary to obtain explicit expressions for the concentration and potential profiles. To do this, we used the theory presented by Amatore et al.,ll who assumed electroneutrality to determine the effects of diffusion and migration on an arbitrary electrochemical reaction (AZ+ ne- + Bz--", where z and n may be positive or negative) at a spherical electrode in the presence of a finite quantity of supporting electrolyte. Amatore et al. presented three differential equations which describe the spatial variation of the following dimensionless quantities: reactant concentration, a = C A / C A , b a , product concentration, b = CB/CA,bulk,and potential, $ = F4IRT (for which we will use the symbol a): daldy = q,*(l - anzlo) dbldy = -'Py*(l

+ bnZB/cT)

(1.1)

(1.2)

daldy = n\k,*/o (1.3) where eqs 1.1 and 1.2 are eqs 10 and 11 of ref 11with eq 1.3 (eq 13 of ref 11)substituted into them. Symbols used here are z (reactant charge), Z B = z - n (product charge), \k,* (current normalized by the diffusion-limiting current, I/Id), y = 1- rolr (dimensionless spatial variable with ro being the position of the electron transfer-r, in terms of the nomenclature in Appendix 11),and {E

o - z n a = (72 + F,,) 112

(1.4)

with (1.5) 7 = nu + z B ( l + 2 7

+ 121)

(1.6) Although { does not appear in the presentation of Amatore et al., it is introduced here to allow notational simplicity throughout the following discussion. Also, subscripts of 0 or 1used with variables {and 7denote the value of those variables a t the position of electron transfer (y = 0) or the bulk of the solution (y = 1). Furthermore, the analysis of ref 11 is presented in terms of y*, which is a normalized distance, y, divided by a normalized diffusion layer thickness, 6*, for spherical, cylindrical, and convective systems. Since y* = y for a spherical system, we have simplified our notation by using y instead of y*; replacing y with y* makes the results presented in this Appendix applicable to the other systems as well. It is convenient to start by describing the potential distribution. To do this, eq 1.1 is divided by eq 1.3 to give d@/da= n / ( o- zna) which, after rearranging and changing (41) Slotboom, J. W. Electron. Lett. 1969,5 (26), 677. (42) Mock, M. S. Analysis of MathematicalModekr of Semiconductor Deuices; Boole Press Ltd.: Dublin, Ireland, 1983. (43) Another approach is to integrate aCiat until it approaches 0. For a recent application of this method, see ref 26. (44) Loeb, A. L.; Overbeek, J.Th.G.; Wiersema, P. H. The Electrical Double Layer Around a Spherical Colloid Particles; M.I.T. Press: Cambridge, MA, 1960.

ANALYTICAL CHEMISTRY, VOL. 65, NO. 23,DECEMBER 1, 1993 8SS1 variables from a to 9 (with n da = d$ gives d@= (q2 + F,)-"'dq

(1.7)

Integrating eq 1.7 from y = 1 (where CP = 0 with respect to the PZC) to some position y gives (1.8)

where the primes denote the variables of integration. The form of the integral of eq 1.8 for ZB # 0 can be inferred from eqs 26 and 28-30 of ref 11. Expressions for all cases are

* = S W ( ~M(a(n( ) + n/(ln(+S;)]

for

ZB

=O

= ZB h[dq11 for bBl 1 = In[(rl+ n/cll, + s;>l for IZB~2 2 (1.9) Equation 1.9 d e f i e s the variation of potential with the reactant concentration. When eq 1.9 is evaluated at the electrode surface (where a = ao), the total iR drop, *o, is obtained. To d e f i e the relationship between the reactant concentration and position, y, we continue with the rearrangement of eq 19 of ref 11 that appears after eq 23: Y,* dy = da + [nza/(u- nza)] da. After changing variables within the bracketed term from a to 1, this expression is integrated between y = 1 (where a = 1) and y to yield *,*@

- 1) = (a - 1) + (z/n)[f-

(1

- ZB(1 + 27 + bl)*] (1.10)

For a specified value of a, the corresponding value of y can be determined if the value of Y,* is known. An expression for the current, Y,*,corresponding to a reactant surface concentration of 00 can be determined from eq 1.10 by setting y = 0 and a = 00:

-*,* =

(a0

- 1) + (z/n)[(o- (1 - ZB(1 + 27 + bl)*O] (1.11)

Finally, for a given concentration of reactant, a, the corresponding product concentration, b, can be found using the condition of electroneutrality. In ref 11, the dimensionless concentration of inert ions in the electrolyte, Cbert/C~,m, is p = 7 (kl- 2)/2 for the cations, and m = 7 (bl+ 2)/2 for the anions. The inert ions comprise those from the 1:l supporting electrolyte (having a concentration C,,,fi = YChbulL) and the bl counterions of A (if it is charged). Electroneutrality requires that p - m za zgb = 0, which can be rearranged to give the product concentration when ZB # 0. When ZB = 0, the product concentration can be determined from eq 1.2 by integrating between y = 0 and the position y where the concentration is a. For either case the value of b is given as

+

+

+ +

b = [ Z cash(@) - zu + (27 + bl) sinh(@)]/zB for Z B # 0 = Y,*(l -y) for Z B = 0 (1.12)

Expression for dWW. The expression for dWdy2 needed to compute the quality of the electroneutrality approximation, as discussed in section IV, is found by differentiating d@/dy(eq 1.3) d2Wdy2= -(nq,*/u') duldy

(1.13)

where the value of duldy can be found by differentiating an alternate form of u given by eq 13 of ref 11: du/dy

-(1-

2')

daldy - (1 - 28') dbldy

(1.14)

Expressions for daldy and dbldy are given in eqs 1.1 and 1.2. Voltammograms. In general, the voltammetric wave shape derived under the assumption of electroneutralitymust include the variation of the potential resulting from the ion

transport, the iR drop. Although Amatore et al. did not include the variation of the ohmic drop in their analyeis,they did mention how it could be incorporated. By use of the expression for *given in eq 1.9,the ohmic drop can be included by replacing their definition of E = -nF(E - Eo')/RT with a d e f i t i o n that corrects for the ohmic lose, I = -nF(E - @O Eo')/RT, where is the value of ip obtained when eq 1.9 was evaluated with a = ao. The quadratic relationship between the surface concentration of the reactant, ao, and the applied overpotential, t, presented by eq 37 of ref 11, contains a minor typographical error. The following is an equivalent corrected expression: [v2(1- 22)- 2nzBu - n21a,2- [2(1 + 27 + bl)vlao +

+ (1 + bl)21= 0 (1.16) While eq 1.16 is quadratic in both a0 and v ('1 + exp[tl), it 147

is written to emphasize the quadratic dependence in 00. Of the two roots for a0 or v determined from eq 1.15, the appropriate value of 00 must be in the range 0 5 a0 5 1 and v must be positive.

APPENDIX 11. FULL NUMERICAL SOLUTION OF THE NERNST-PLANCK AND POISSON EQUATIONS Here, we describe the method that was used to solve the Nernst-Planck and Poisson equations. Before we attempted a solution, it was necessary to decide upon the domain in which the solution was to be sought. The idealized domain we considered has ita origin a distance p from an electrode surface of radius r, and extend to infinity. T h e distance r,, = r, + p (as measured from the center of the spherical electrode) is both the closest approach of ions and the position of electron transfer (PET). The distance between the electrode and the PET is assumed to be filled with a dieledric medium having a dielectric constant, ci. The potential difference between r, and r,,, E - 40, provides the driving force for the electrochemical reaction A z ~ ne- BZs,where n = Z A - ZB, and Z A and ZB may have any charge. As a point of reference, the usual treatment of redox reactions locates the PET just outside the electrode double layer and so the potential difference is E - da, where ~ $ mis the potential drop in the solution resulting from the passage of current. In the presence of a large excess of supporting electrolyte, 48 = 0. Using this domain, the Nernat-Planck and Poisson equations were made dimensionless by introducing the following variables: the dimensionless distance, y = 1 - rJr; the dimensionless potential, = Ft$/RT; and the dimensionless concentration,Ci* = C J C A , multiplied ~, byexp[zi@],togive the quantity, xi = Cj* exp[zjlp], which is a common variable used in describing transport in semiconductors."~'2 Using these variables, the Nernst-Planck and Poisson equations take the simple form

+

Ji

= -DicA,ba eXp[-Zj*Ir,,?l-

y)' dxi/dy

(11.1)

d'@ -1 cA,bd: -=-(11.2) dy' ( l - ~e+T/Ea )~ I For reference, the ratio ceJZTJFis 183.81 nms mM when T = 298 K and e = 78. The special condition of zero flux (Ji= 0) applies to all ions when no current is flowing; it also applies to the inert ions when a steady state has been reached since they do not transport any of the current. Setting J i = 0 yields the simple requirement that dxildy = 0. Since xi = ci,b&* in the bulk and the derivativeie zero, thisis alsothe value of xi everywhere, which leads to the familiar Boltzmann distribution for the

3352


concentration: Ci = Ci,bdk exp[-zi@]. For the electroactive species, the steady state in the presence of current flow requires that aCi/at = -0.J: = 0.43 In spherical coordinates, VeJi = dJi/dr + Wi/r. Transforming variables from r to y gives V J i = r,,-l(l - Y ) ~dJi/dy + 2r,,-l(l - y)Ji, which yields

The final set of equations that was solved comprised Poisson's equation (with the sum running over all ionic species), eq 11.2, and the right-most term within the parenthesis of eq 11.3 for species A and B; Boltzmann distributions were used for the concentration profiles for the inert ions, Note that when zi = 0, the transport is unaffected by migration and the concentration profile varies linearly with respect toy between the value at the PET d xi,^) and the value in the bulk (Cis). The three boundary conditions used for these equations at the PET (y = 0; r = r,, = r, + p) were as follows. (1)A Nernstian relationship between the concentrations is assumed since the electron transfer is considered to be reversible: CA= CBexp[nF(E-$o-E"')/RT], whereE = $ois the potentialdifference between the electrode and the PET. Multiplying both sides by exp[z~@o], and using n = ZA - ZB, yields XA

= X B exp([nF(E -E"')/RTl)

(11.4)

where E is the electrode potential and Eo' is the formal reduction potential, both measured with respect to the potential of zero charge for the electrode. This Nernstian expression can just as easily be replaced with a kinetic expression. (2) The flux of reactant to the surface was set equal to the flux of product away from the surface: J A= -JB. In terms of the flux expression of eq 11.1, this requires that DA(dxA/dy) = -DB exP[n@oI(dxB/dy)

(11.5)

(3) The electric field is continuous: q(d$/dr)I- = c(d$/dr)l+, where t i is the dielectric constant of the inner region between the electrode and the PET, E is the dielectric constant of the electrolyte solution in the outer region, and the derivatives are evaluated on the inner (subscript "-") and outer (subscript "+") side of the PET. (This relationship between the inner and outer fields was derived from Gauss' law under the simplifying assumption that there is no charge at the PET.) To change variables from r toy in eq 11.5,one must be careful because y = 0 corresponds to the PET, which is at a distance p from the electrode surface. By introducing yi = 1- ro/r as the dimensionless distance in the inner region between the electrode surface and the PET, and changing variables from dr- to dyi (with ro dyi = (1 - y# dr-) and dr+ to dy (with r,, dy = (1 - Y ) dr+), ~ the field continuity relationship becomes ti(rJr,) d$/dyi = t db/dy. Since we assume that there is no charge in the inner region, the right-hand side of Poisson's equation, eq 11.2, is zero, which leads to the requirement that d*@/dy2= 0 (Le., Laplace's equation), the solution of which is linear with respect to yi. With d, = E at yl = 0 and 9 = $0 a t yi = 1 - ro/r,,, the value of d$/dyi is given as (E - $o)r,,/p. Substituting this into the boundary condition (and multiplying by FIR27 gives the relationship between the electrode potential, the potential distribution in the outer region, and the physical parameters of the two regions: FEIRT = 9 0 + (~/).O)(€lti)(d(P/dy!ly=O (11.6) A t the boundary corresponding to the bulk of the electrolyte boundary conditions were XA = 1 (since all concentrations are normalized by the reactant, A, concentration), XB = 0 (assuming that there is no product initially present), and @ = 0 (measured relative the the PZC!. (y = 1) the three

The three equations describing the relationships between potential, reactant, and product were solved using the finitedifference method. The y-domain was divided equally among N nodes, node 1 corresponding to y = 0 and node N corresponding to y = 1, giving a node spacing of A = 1/(N 1). Second-order, centered-finite-difference formulas were used to discretize the three equations at nodes 1through N - 1. The three boundary conditions at y = 0 were discretized in a similar fashion at node 1, and the boundary conditions a t y = 1 were applied to node N. A fictitious node 0 was introduced at y = -A to accommodate the centered-finitedifference formulas for derivatives expressed a t node 1. The resultant algebraic set of 3 N equations in 3 N unknowns (the three quantities XA, XB, and @ at nodes 0 through N - 1) was solved using the Newton-Raphson method. Before continuing, we note that node 0 at y = -A does not corresponding to any physical position in the domain (Le., it is not the electrode surface). Ita sole purpose is to provide a means of computing a centered-finite-difference representation of the derivatives at the node 1. For example, the flux of speciesA at y = 0 is represented in centered-fiite-difference form a s - D ~e x p [ - z ~ @ ( ~ ) ] (XA'")/~A, ~ ~ ( ~ ) -where the numbers in parentheses refer to the nodal position of the quantities. The use of the fictitious node slightly increases the bandwidth of the matrix representation of the equation set if the variables at node 0 are not eliminated. This can be done using the equations written at node 1 (in this case, there are six equations in nine unknowns written at node 1, allowing the three unknowns at node 0 to be eliminated). This elimination is easy if the equations written as node 1 are linear in the quantities to be determined at node 0. In this case, however, the equations are nonlinear so we chose to solve the larger banded-matrix system. Once a solution to the above equations was found, the current density at the PET, Z = F(ZAJA + ZBJB)(which is equivalent to nFJA at the PET where J A = -JB),normalized by the diffusion-limited current density, I d = nFDACA,buLL/r,,, was computed as Z/Z,j = exp [-z~@oI(dXA/dY)l,=o

= eXp[-Z~@o](2(~(~) - XA''')/(~A) (11.7)

The accuracy of the computation was tested by comparing voltammograms computed using 300 and 600 nodal points. Forthe resulta presented in this paper, there wasnosigniiicant difference between the two. While the above equations were solved simultaneously, it is possible to solve them independently if one assumes that the equilibrium double layer is unperturbed by the paesing current. Such an assumption is warranted if the field in the depletion layer is much smaller than the field from the electrode surface (as is the case when there is an excess of supporting electrolyte and when the electrode potential is not too close to the PZC). For example, the potential profiles of Figure 4 are essentially unchanged whether current is flowing or not. Under such an assumption, the electroactive ions are ignored. One may then use tabulated data describing the potential distribution in the electrode double laye+ or simply resolve Poisson's equation (eq 11.2), assuming a Boltzmann distribution for the inert ions. The advantage gained by using this assumption is that the potential distribution for a given potential at r,,need only be determined once. It can then be used to define the field needed to solve the Nernst-Planck equations, which are now only coupled through the boundary conditions at the surface, allowing each flux equation to be solved independently (but still requiring that the boundary conditions for the reactant and product solution be satisfied).

ANALYTICAL CHEMISTRY, VOL. 65, NO. 23, DECEMBER 1, lQQ3 3353

Initial Guess. Whenever one uses the Newton-Rapheon method to solve a set of nonlinear algebraic equations, an initial guess is needed to begin the iterative process toward fiiding an acceptable solution to the equations. In this case, the concentration and potential profiles found by replacing Poisson’s equation with Laplace’s equation were used as an initial guess (which is appropriate when the electrode radius is small). For each voltammogram computed, this initial guess was used only once-at the potential E = E O ‘ . The solution obtained at this potential was used as the initial guess to obtain a solution at a new potential (30 mV away from the previous potential), which in turn was wed as the initial guess at the next potential, etc...., until the desired potential was reached in the positive or negative direction. Since the concentration profiles for ~ 2 =90 are not explicitly given by Norton et al.? they are derived below. Steady-State Concentration and Potential Profiles when V24 = 0. Here we derive the concentration and potential profiies that were used as an initial guess to the Newton-Rapheon procedure described above. Setting the right-hand side of Poisson’s equation (eq II.2) equal to zero, yields the form of Laplace’s equation in sphericalcoordinates, d*@/dy2= 0, the solution of which is a linear in y. If the potential a t y = 0 is 90and the potential in the bulk of the solution is 0, then the solution is 9 = (1- y)90 and the field is constant, given by -d@/dy= 90.Substituting this into the right-hand side of eq 11.3 yields a homogeneous second-order linear differential equation, d2xJdy2+ Z i 9 0 dxildy = 0, having a general solution xi = K1+ Kz exp[-zi9ayl. If the values of xi are xi,o and xi,l at y = 0 and y = 1, respectively, then the values of K Iand KZcan be determined to yield the relationship

between Xi

xi

= Xi,l+

and y

(Xj,o

-~i,~)(exp[-z~@ ex~[-z~9~1)/(1~l-

e x p [ - ~ ~ @(11.8) ~l) where i can be A or B. The values of XA,O and XB,O needed to evaluate eq 11.8 can be determined from the boundary conditions of eqs 11.4 and 11.5. Differentiating eq 11.8 for use in eq 11.5, yields dxildy = -zi*o e x ~ [ - z ~ * a l ( x-~~i,1)/(1,~ ex~[-z~@~l) (11.9) Eliminating the value of XA,O from eqs 11.4 and 11.5 yields

(11.10) - 1); in the limit of Z i 9 0 0, Vi where Vi = -zicPo/(e~p[~i901 = -1. Substituting this value of XB,O into the Nernst relationship of eq 11.4 allows one to calculate the value of XA,O. Finally, substituting dWdy = -90into eq 11.6 gives the relationship between E and @o: +

FE/RT = @ & + (dei)(dro))

(11.11)

ACKNOWLEDGMENT Helpful discussions with Yuming Zhou are gratefully acknowledged. Thiswork is supported by the Office of Naval Research.

RECEIVED for review August 9, 1993. Accepted September 29, 1993.” @

Abstract published in Advance ACS Abstracts, November 1,1993.

Theory of the voltammetric response of electrodes of submicron

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