Extended electron transfer and the Frumkin correction

www.elsevier.nl/locate/jelechem

Journal of Electroanalytical Chemistry 491 (2000) 103–110

Extended electron transfer and the Frumkin correction

David J. Gavaghan 1, Stephen W. Feldberg *Department of Chemistry, Monash Uni6ersity, Clayton, Victoria 3168, Australia

Received 3 March 2000; received in revised form 11 April 2000; accepted 25 May 2000

Dedicated to Professor E. Gileadi on the occasion of his retirement from the University of Tel Aviv and in recognition of his contribution toelectrochemistry

Abstract

It is now well established that the rate constant for electron transfer between a metal electrode and a given redox moiety (atdistance x from the electrode) decays according to exp[−bx ]; values of b are of the order of 108 cm−1 and depend primarilyupon the nature of the intervening medium. The present work demonstrates how this ‘extended’ electron transfer can modify theFrumkin correction for diffuse layer effects in the measurement of heterogeneous rate constants. We quantify the effect of this‘extended’ electron transfer by comparing the electron transfer rate constants deduced with (k %eff) and without (keff) extendedelectron transfer. Butler–Volmer kinetics and a single electron transfer are assumed. The ratio k %eff/keff is deduced as a functionof b/k, where k is the reciprocal Debye length; (F/RT)fPCA where fPCA is the diffuse layer potential at the plane of closestapproach; and zox−a, where zox is the charge of the oxidized redox species and a is the Butler–Volmer transfer coefficient. Thedeviation of k %eff/keff from unity is greatest when the diffuse layer potential, fx, decays over distances of a few angstroms from theplane of closest approach, as is the case with higher concentrations of supporting electrolyte and/or large values of the diffuselayer potential. When �zox�5 �zse� and �zred�5 �zse� the classical Frumkin expression is applicable as long as

− ln�

1+b

k

n5

F

RTfPCA(zox−a)5 ln

�1+

b

k

nWhen �zox�\ �zse� or �zred�\ �zse� and (F/RT)fPCA(zox−a)B0 redox ions can become a significant component of the diffuse layerand the classical Gouy–Chapman theory is no longer applicable. To avoid this complication and apply the classical Frumkincorrection the condition to be met is

05 (F/RT)fPCA(zox−a)5 ln[1+b/k ]

thus ensuring that the redox species is not disproportionately accumulated in the diffuse layer. We also show that when b/k50.5,fPCAzox]0 and fPCAzred]0 then virtually all ET occurs outside the diffuse layer and there is no Frumkin effect. However, thiscondition is not easily achieved experimentally. © 2000 Elsevier Science B.V. All rights reserved.

Keywords: Butler–Volmer theory; Capacitance; Diffuse layer potential; Extended electron transfer; Frumkin correction; Marcus theory

1. Introduction

In this paper we explore how electron transfer (ET)over a distance, ‘extended’ electron transfer (EET), can

significantly modify Frumkin’s correction for diffuse-layer effects on the measurement of heterogeneous rateconstants [1,2]. The Frumkin correction, in principle,corrects for the effect of the diffuse layer potential onthe concentrations of the redox moieties at their planeof closest approach (PCA).

For some time workers have realized that homoge-neous bimolecular ET between an acceptor and donorcan occur over a distance [3–11] and does not occuronly when the reacting molecules are at the distance ofclosest approach, r0. The probability of ET between anacceptor and donor separated by distance, r, is

* Corresponding author. Present address: Energy Science andTechnology Department, Brookhaven National Laboratory, Upton,NY 11973, USA. Tel.: +1-631-3444480; fax: +1-631-3443137.

E-mail address: [email protected] (S.W. Feldberg).1 On leave from Oxford University Computing Laboratory, Wolf-

son Building, Parks Road, and Nuffeld Department of Anaesthetics,University of Oxford, Radcliffe Infirmary, Oxford, UK.

0022-0728/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.PII: S 0022 -0728 (00 )00210 -2

D.J. Ga6aghan, S.W. Feldberg / Journal of Electroanalytical Chemistry 491 (2000) 103–110104

pr=pr 0exp[−b(r−r0)] (1)

Miller (see, e.g. Refs. [5,6] and references therein) hasreported values of b between 1.1×108 and 1.4×108

cm−1 for a variety of non-aqueous glasses; Miller [4]has also measured b=0.9×108 cm−1 for ET betweentrapped electrons and a variety of acceptors in anaqueous glass. Ponce et al. [7] measured b=1.65×108

cm−1 for ET between acceptor and donors in anaqueous glass2. Recently, a number of workers havedemonstrated that Eq. (1) also describes heterogeneousET between an electrode and a redox moiety attachedcovalently to the electrode [13–19] or between an elec-trode coated with any of a variety of self-assembledmonolayers and redox moieties in solution [20,21]. Val-ues of b, generally measured, at room temperature,typically range from �0.4×108 to �1×108 cm−1

and depend primarily upon the nature of the tether (e.g.saturated or unsaturated) attaching the redox moiety tothe electrode or the nature of the monolayer3. It isreasonable to suggest that the appropriate expressiondescribing the probability of electron transfer betweenan electrode and a redox moiety in solution, at distancex from the electrode, will be [25–27]:

px=pPCA exp[−b(x−xPCA)] (2)

where pPCA is the probability for ET at the PCA andxPCA is the distance between the electrode and the PCA.For our present considerations we assume that 0.5×1085b52.0×108 cm−1.

2. The Frumkin correction

The quintessential assumption underlying theFrumkin correction [1,2] is that ET between an elec-trode and a redox species can occur only when theredox species is at the reaction plane, usually defined asthe PCA at xPCA. The concentrations of redox species,cox,PCA and cred,PCA, at xPCA are determined by theextant diffuse layer potential, fPCA, between xPCA andthe bulk solution:

cox,PCA=cox,d exp�

−F

RTfPCAzox

n(3)

and

cred,PCA=cred,d exp�

−F

RTfPCAzred

n=cred,d exp

�−

FRT

fPCA(zox−1)n

(4)

where zox and zred (=zox−1) are the charges on theoxidized and reduced redox species, d is the distancefrom the electrode at which (F/RT)�fd ��14. With theGouy–Chapman–Stern model [2,28–31], the value offPCA is defined by:

fPCA=E−Epzc

1+Cd

Ci

= (E−Epzc)Cdl

Cd

(5)

where E is the applied potential measured between theworking electrode and a reference electrode in the bulksolution, Epzc is the potential of zero charge [2], Cdl isthe overall interfacial capacitance (1/Cdl=1Cdl+1/Ci),Cdl is the integral capacitance of the inner layer and Cd

is the integral capacitance of the diffuse layer (capaci-tance units are F cm−2). Cd can be defined by (equa-tions are from Ref. [2]):

Cd=8RTee0cse sinh

�FfPCA�zse�2RT

nfPCA

(6)

where e is the static dielectric constant of the solvent, e0

is the permittivity of free space, and 9zse is the chargeassociated with the 1:1 supporting electrolyte. Implicitassumptions and restrictions for Eqs. (3)–(6) are1. The basic assumptions of the Gouy–Chapman–

Stern model [2,28–31], notably:� The plane of closest approach is the same for all

species.� The solvent is a continuum with a uniform dielec-

tric constant.� There is no electric field within the metal.� The supporting electrolyte is a zse:−zse

electrolyte.� The contribution of the redox ions to the compo-

sition, structure and potential of the diffuse layeris negligible. This is often the case since theconcentrations of the redox ions are generallymuch smaller than the concentration of the sup-porting electrolyte. Furthermore, if fPCAzox]0and fPCAzred]0, the redox ions will be eitherunaffected (if zox=0 or zred=0) or rejected from

2 b-values have been deduced from measurements of ET ratesbetween donors and acceptors randomly distributed in a glassymatrix — typically at 77 K. The temperature coefficient of b is low[4] so the reported range of b-values can be reasonably applied toroom-temperature experiments. Recently, Read et al. [12] have pro-posed a novel and considerably more direct approach for measuringb values for liquid solvents: the acceptor–donor pair is separated bya rigid C-clamp spacer — thus, in principle, ET proceeds preferen-tially through the solvent rather than through the spacer.

3 There is a plethora of ET data for acceptor–spacer–donor sys-tems in solution at room temperature which also exhibit a similarrange of b values. The spacer is any of a number of rigid organicstructures (see Refs. [22–24]).

4 In the absence of concentration polarization cox,d=cox,bulk andcred,d=cred,bulk. However, when there is concentration polarization,the thickness of the diffusion layer is usually much larger than d. Onthe time-scale of the experiment, all ion concentrations between xPCA

and d are assumed to be in electrostatic equilibrium.

D.J. Ga6aghan, S.W. Feldberg / Journal of Electroanalytical Chemistry 491 (2000) 103–110 105

the diffuse layer. However, if fPCAzoxB0 and�zox�\ �zse� or fPCAzredB0 and �zred�\ �zse�, at leastone of the redox species will be disproportion-ately incorporated into the diffuse layer (see Eqs.(3) and (4)), and the simple Gouy–Chapmantheory, valid for zse:−zse electrolytes is no longerapplicable — limited modifications of the sim-ple Gouy–Chapman have been developed[29,32–34] but there is no analytic descriptionthat is completely general.

2. The redox process is a one-electron transfer:

Oxzox+e− =Redzred

where zred=zox−1. The Butler–Volmer equation isan adequate descriptor of the potential dependenceof heterogeneous ET kinetics when e �E−E°%�5�1/2l where e is the electronic charge, E°% is theformal potential for the redox process and l is thereorganization energy for the reaction [14,35,36]5.The Butler–Volmer equation for a one-electrontransfer written as a function of the redox concen-trations at the PCA (i.e. cox,PCA and cred,PCA) is

j=Fk0�

cox,PCA exp�

−aF

RT(E−fPCA−E°%)

n−cred,PCA exp

�(1−a)

FRT

(E−fPCA−E°%)n�

(7)

where j is the current density and E−fPCA−E°% isthe effective potential driving the ET. CombiningEqs. (3), (4) and (7) gives:

j=Fkeff�

cox,d exp�

−aF

RT(E−E°%)

n−cred,d exp

�(1−a)

FRT

(E−E°%)n�

(8)

where keff is the effective standard rate constant forthe classical Frumkin correction defined by:

keff=k0 exp�

−F

RTfPCA(zox−a)

n(9)

3. The effect of extended electron transfer (EET) onthe Frumkin correction

ET between an electrode and redox species in solu-tion is not limited to the redox species at the PCA. ETcan occur over a significant distance (tens of

angstroms). The value of b (see Eq. (2)) for aqueousand non-aqueous solvents [3–11] has a likely range of0.5×108–2.0×108 cm−1. For the purposes of thepresent paper, we will ignore image charge effects whichcan lower the reorganization energy, l, when the redoxspecies are within a few angstroms of an electrode andthereby effect a larger than expected value of the het-erogeneous rate constant [35,36]. We also assume thatthe effect of the jellium edge is negligible at the dis-tances of interest (xPCA will be several angstroms fromthe metal surface [37]). Our main objective in thepresent work is to deduce conditions under which EETcan significantly modify the interpretation of the diffuselayer effects on ET kinetics.

With Eq. (2), which gives us the probability of elec-tron transfer from an electrode to a redox species atdistance x, and fx, the diffuse layer potential at x, wecan write an effective rate constant, k %eff, which includesEET:

k %eff=pPCA&�

x=x PCA

exp�

−F

RTfx(zox−a)

−b(x−xPCA)n

dx (10)

The normalized potential, (F/RT)fx can be defined as afunction of x and (F/RT)fPCA by [2]:

tanh��zse�Ffx

4RT�

tanh�F �zse�fPCA

4RT�=exp[−k(x−xPCA)] (11)

where k is the Debye reciprocal distance defined by

k=�2csez se

2 F2

ee0RT�1/2

(12)

where cse is the concentration (mol cm−3) of the zse:−zse supporting electrolyte. Rearrangement of Eq. (11)gives:

Ffx

RT=

4�zse � arctanh

�exp[−k(x−xPCA)]×

tanh�F �zse�fPCA

4RT��

(13)

Combining Eqs. (10) and (13) gives:

k %eff=pPCA&�

x PCA

exp�

−4(zox−a)

�zse�×

arctanh�

exp[−k(x−xPCA)]tanh�F �zse�fPCA

4RT��n

×

exp[−b(x−xPCA)]dx (14)

Eq. (14) can be written in a more compact form:

5 When e �E−E°%�\�1/2l deviations from the Butler–Volmertheory require an analysis based on the Marcus theory, see, e.g. Ref.[14]. However, under these same conditions the heterogeneous rateswill have frequently become diffusion controlled.


of no consequence. Then:

k %eff =b

k�1

pPCA

k

&�g=0

exp�

−b

kgn

dg=pPCA

b=k0 (19)

In other words, when b/k�1, fPCAzox]0 andfPCAzred]0 there is no Frumkin effect7.

3.2. General beha6ior of Eq. (15)

To evaluate the effect of EET, we will compare thevalue k %eff of computed with Eq. (15) to the value of keff

computed using Eqs. (9) and (15) for classical Frumkincorrection. The ratio, k %eff/keff, is an effective mode ofcomparison:

The computation results are presented by showing k %eff/keff as a function of (F/RT)fPCA(zox−a)/�zox−a �8 fordifferent values of b/k and zox−a. Physically sensibleranges of the independent variables are deduced in thenext section.

3.3. Physically sensible ranges of the independent6ariables

3.3.1. The range for (F/RT)fPCA

For the purposes of the present discussion we assumethat the practical limit for any ion concentration is 10M — undoubtedly well beyond the range of accuracyfor the Gouy–Chapman theory. Thus, for a given cse

and a zse:−zse electrolyte:

FRT

�zsefPCA�5 ln�10 M

cse

n(21)

If we assume that zse=1 and that cse]0.001 M weconclude that (F/RT)�fPCA�59.2 or 10, for conve-nience. For systems where �zox�5 �zse� and �zred�5 �zse� wecan consider values of (F/RT)�fPCA over the full range,i.e.:

−105F

RTfPCA510 (22)

However, when fPCAzoxB0 and �zox�\ �zse� or �zred�\�zse�, recall that accumulation of one or both redox ions

k %eff=pPCA

k

&�g=0

exp�

−4(zox−a)

�zse�×

arctanh�

exp[−g ]tanh�F �zse�fPCA

4RT��n

exp�−b

kgn

dg

(15)

where

g=k(x−xPCA) (16)

For most experimental conditions with typical valuesof b the effective integration limit will be a distancethat is considerably smaller than the thickness of thediffusion layer that may be produced adjacent to theelectrode.

3.1. Limiting beha6ior of Eq. (15)

Before examining the generalized behavior of Eq.(15) it is informative to explore two limiting exampleswhere the integral is easily evaluated:

1. b/k�12. b/k�1 and fPCAzox]0 and fPCAzred]0

In the first example, when b/k�1, i.e. fx (Eq. (13)) isa much weaker function of distance than px (Eq. (2)),the decay of the integrand in Eq. (15) is dominated byexp[− (b/k)g ]; the integral will have become very smallbefore exp[−g ] deviates significantly from its initialvalue of unity (the integration is initiated from g=0).Thus, when b/k�1, Eq. (15) simplifies to

k %eff =b

k�1

pPCA

kexp

�−

FRT

fPCA(zox−a)n

×&�g=0

exp�

−b

kgn

dg=pPCA

bexp

�−

FRT

fPCA(zox−a)n

(17)

The form of Eq. (17) is identical to that of Eq. (9) as itshould be when b/k�1 since virtually all the EToccurs within the region where fx$fPCA; thus

k0=pPCA

b(18)

In the second example, when b/k�1 and fPCAzox]0 and fPCAzred]0, virtually all the ET will occuroutside the diffuse layer and the change inconcentrations6 of the redox species at the PCA will be

k %eff

keff

=

b

k

&�g=0

exp�

−4(zox−a)

�zse�arctanh

�exp[−g ] tanh

�F �zse�fPCA

4RT��n

exp�

−b

kgn

dg

exp�

−F

RTfPCA(zox−a)

n (20)

7 This could also happen if the redox species is very large, i.e. itsPCA lies outside the diffuse layer. Our assumption for the presentanalysis is that the PCA is the same for all species.

8 The term (zox−a)/�zox−a � could alternatively be written assign(zox−a).

6 Since we have specified that fPCAzox]0 and fPCAzred]0 theconcentration of a redox species at the PCA will be the same as orless than its concentration at x=d.


in the diffuse layer will render the simplified Gouy–Chapman expressions invalid. Consequently, when�zox�\ �zse� and �zred�\ �zse� the appropriate limit for (F/RT)�fPCAzox is best expressed by:

05F

RTfPCA

(zox−a)�zox−a �510 (23)

In terms of (possibly) known experimental parameters(see Eq. (5)), Eq. (23) becomes:

05F

RT(E−Epzc)

Cdl

Cd

(zox−a)�zox−a �510 (24)

Fig. 3. Plots of log 10[k %eff/keff] vs. (F/RT)fPCA(zox−a)/�zox−a � forb/k=10; remaining details same as for Fig. 1.

Fig. 1. Plots of log 10[k %eff/keff] vs. (F/RT)fPCA(zox−a)/�zox−a � forb/k=0.5; curves a–f: zox−a=90.5, 91.5, 92.5, 93.5, 90.5,95.5. Dashed lines are for (F/RT)fPCA(zox−a)/�zox−a �B0, �zox�\1 or �zred�\1.

3.3.2. The range for b/kFrom Eq. (12) we can estimate a minimum value for

k (assuming e=78, zse=1, and cse=10−3 M) and amaximum value for k (assuming e=10, zse=1, andcse=1.0 M9) giving

1×1065k51×108 cm−1 (25)

From Eq. (25) and range for b, 0.5×1085b52×108

cm−1, adduced from experimental results [3–11], weobtain the range of values of b/k that should beexplored:

0.55b

k5200 (26)

3.3.3. The range for zox−a

We assume that a=0.5 and −55zox56; thus:−5.55zox−a55.5.

4. Computations

All numerical calculations and graphics were per-formed using Matlab’s routine quad810, an adaptiveNewton–Cotes 8 panel rule, to perform the numericalintegration. Results agree within three significantfigures with those generated by our own numericalintegration codes written in FORTRAN.

Figs. 1–5 show values of k %eff/keff computed for (F/RT)fPCA(zox−a)/�zox−a � from −10 to 10 (see Eq.(23) and associated discussion) for b/k=0.5 (Fig. 1), 2

Fig. 2. Plots of log 10[k %eff/keff] vs. (F/RT)fPCA(zox−a)/�zox−a � forb/k=2.0; remaining details same as for Fig. 1.

9 A supporting electrolyte concentration \1 M and �zse�\1 arerarely used.

10 The MathWorks Inc., 24 Prime Park Way, Natick, MA 01760,USA (http://www.mathworks.com).


(Fig. 2), 10 (Fig. 3), 50 (Fig. 4) and 200 (Fig. 5); eachfigure comprising of six curves corresponding to zox−a=90.5, 91.5, 92.5, 93.5, 94.5 and 95.5. Val-ues of (F/RT)fPCA(zox−a)/�zox−a � greater than zeroare of primary interest (see Eq. (23) or (24)). However,for the case where zox−a=1/2 or −1/2 and zse=1 theconstraint of Eq. (23) no longer obtains and the com-puted values are valid for the full range specified in Eq.(22). In the unlikely event that zse\1 for the support-ing electrolyte then as long as �zox�5 �zse� and �zred�5 �zse�the valid range of (F/RT)fPCA will be

−10

�zse�5

FRT

fPCA510

�zse�(27)

5. Results and conclusions

Qualitatively the results shown in Figs. 1–5 arestraightforward. The larger the value of b/k the sharperthe fall off in px (Eq. (2)) and the closer the value ofk %eff/keff is to unity — in other words, the system be-havior can be described using the classical Frumkincorrection. Other factors, e.g. increases in fPCA(zox−a), can significantly increase k %eff/keff — an increase in�fPCA� collapses the mean thickness of the diffuse layer(see Eq. (13)), with a result similar to increasing thevalue of k. For the most part we have focused on thebehavior of systems where the condition of Eq. (23) ismet, i.e. where the charges on the redox species are suchthat they are rejected from the diffuse layer. For thesmallest value of b/k (Fig. 1 where b/k=0.5) we see avirtual linear relationship between log 10[k %eff/keff] and(F/RT)fPCA(zox−a)/�zox−a �\0. When b/k�1 thevalue of k %eff/keff is deduced easily from Eqs. (9), (18)and (19) giving:

k %eff

keff

=b

k�1

exp� F

RTfPCA(zox−a)

n(28)

In this limit (b/k�1) the deviation of k %eff/keff fromunity reflects the inappropriateness of invoking theFrumkin effect under these conditions. Note that thelimiting behavior is not observed for (F/RT)fPCA(zox−a)�zox−a �B0 since the concentration of at least one ofthe redox species is increased (relative to its bulk con-centration) at the PCA and the mathematical argu-ments that are the basis of Eq. (19) do not obtain.

Perhaps the most practical application of our analy-sis is that we can specify two zones of behavior: Zone 1behavior, where the classical Frumkin correction can beapplied validly, and Zone 2 behavior where theFrumkin correction need not be applied e6en if there isa significant diffuse layer potential.

5.3.1. Zone 1 beha6iorWhen �zox�5 �zse� and �zred�5 �zse� we can see from

Figs. 1–5 that the classical Frumkin expression is appli-cable (i.e. k %eff/keff is virtual unity) as long as

ln�

1+b

k

n+

FRT

fPCA(zox−a)+ ln�

1+b

k

n(29)

When �zox�\ �zse� or �zred�\ �zse� and (F/RT)fPCA(zox−a)B0 redox ions can become a significant componentof the diffuse layer and the classical Gouy–Chapmantheory is no longer applicable (see discussion followingEq. (6)). To avoid this complication and apply theclassical Frumkin correction the condition to be met is




0+F

RTfPCA(zox−a)+ ln

�1+

b

k

n(30)

We also show that when b/k50.5, fPCAzox]0 andfPCAzred]0 then virtually all ET occurs outside thediffuse layer and there is no Frumkin effect. However,this condition is not easily achieved experimentally. Anadditional complication is that likely small differencesin the xPCA for each species can dramatically modify theanalysis when b/k�1. These differences are less criticalwhen b/k�1.

6. Glossary

Concentration of Ox or Red speciescox,PCA,cred,PCA at xPCA (mol cm−3).cox,d, cred,d Concentration of Ox or Red at xd

(mol cm−3).Concentration of Ox or Red at xcox,x, cred,x

(mol cm−3).Bulk concentration of supportingcse

electrolyte (mol cm−3).Ci, Cd, Cdl Integral capacitance of the inner

layer, diffuse layer, and total doublelayer (see Eqs. (5) and (6)) (F cm−2).

Ci Integral capacitance of the innerlayer (see Eq. (5)) (F cm−2).elementary charge (1.60219×10−19 C)e(C).

E Applied potential (V).E°% Formal potential of redox couple (V).Epzc Potential of zero charge (V).

Faraday’s constant (96485 C mol−1).FCurrent density (see Eqs. (7) and (8))j(A cm−2).Standard rate constant for redox re-k0

action (cm s−1).Effective standard rate constant asso-keff

ciated with the classical Frumkin cor-rection (cm s−1).

k %eff Effective standard rate constant whenEET and diffuse layer effects areconsidered (cm s−1).Outer Helmholtz plane; plane of clos-OHPest approach for supporting elec-trolyte ions.Probability of ET between the elec-pPCA

trode and a redox moiety at distancexPCA.Probability of ET between the elec-px

trode and a redox moiety at distancex.Probability of ET between an accep-pr

tor and donor separated by distancer.

Probability of ET between an accep-pr 0

tor and donor in solution at the dis-tance of closest approach, r0.

r0 Distance of closest approach for anacceptor and donor in solution (cm).Reaction distance for an acceptorrand donor in solution (cm).

PCA Plane of closest approach.Gas constant (=8.3144) (J K−1Rmol−1).Temperature (K).T

xOHP, xPCA, Distance between the electrode andthe OHP or PCA (cm).xd

Charge on Ox or Red species,zox, zred

respectively.Positive value of charge on ions com-zse

prising zse:−zse supporting electrolyte.Transfer coefficient in the Butler–a

Volmer equation (see Eq. (7)).b Decay factor for EET (see Eq. (2))

(cm−1).Normalized distance variable (=g

k(x−xPCA); see Eq. (16)).Distance from electrode at which (F/d

RT)�fx ��1 (cm).Permittivity of free space (=8.8542×e0

10−14 C cm−1 V−1).e Static dielectric constant of the

solvent.Debye reciprocal distance (see Eq.k

(12)) (cm−1).l Reorganization energy (eV).fd Diffuse layer potential at d (V).

Diffuse layer potential at xPCA (V).fPCA

Diffuse layer potential at x (V).fx

Acknowledgements

SWF thanks Robert de Levie and Mary Archer forinsightful and helpful comments, the US Department ofEnergy, Contract no. DE-AC02-98CH10886 for theirsupport, and the Department of Chemistry, MonashUniversity, for their support and hospitality. DJGthanks the Medical Research Council for a CareerDevelopment Fellowship.

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Extended electron transfer and the Frumkin correction