Standard electrode potential, Tafel equation, and the solvationthermodynamicsDmitry V. Matyushov Citation: J. Chem. Phys. 130, 234704 (2009); doi: 10.1063/1.3152847 View online: http://dx.doi.org/10.1063/1.3152847 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v130/i23 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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Standard electrode potential, Tafel equation, and the solvationthermodynamics

Dmitry V. Matyushova�

Center for Biological Physics, Arizona State University, P.O. Box 871504,Tempe, Arizona 85287-1504, USA

�Received 11 April 2009; accepted 20 May 2009; published online 18 June 2009�

Equilibrium in the electronic subsystem across the solution-metal interface is considered to connectthe standard electrode potential to the statistics of localized electronic states in solution. We arguethat a correct derivation of the Nernst equation for the electrode potential requires a carefulseparation of the relevant time scales. An equation for the standard metal potential is derived linkingit to the thermodynamics of solvation. The Anderson–Newns model for electronic delocalizationbetween the solution and the electrode is combined with a bilinear model of solute-solvent couplingintroducing nonlinear solvation into the theory of heterogeneous electron transfer. We therefore arecapable of addressing the question of how nonlinear solvation affects electrochemical observables.The transfer coefficient of electrode kinetics is shown to be equal to the derivative of the free energy,or generalized force, required to shift the unoccupied electronic level in the bulk. The transfercoefficient thus directly quantifies the extent of nonlinear solvation of the redox couple. The currentmodel allows the transfer coefficient to deviate from the value of 0.5 of the linear solvation modelsat zero electrode overpotential. The electrode current curves become asymmetric in respect to thechange in the sign of the electrode overpotential. © 2009 American Institute of Physics.�DOI: 10.1063/1.3152847�

I. INTRODUCTION

The equilibrium between the electronic subsystems of aredox couple in solution with the conduction electrons of ametal electrode requires their corresponding electrochemicalpotentials to be equal: �̄m= �̄s, where “m” and “s” subscriptsrefer to the metal and solution, respectively. The electro-chemical potential1,2 is defined as a sum of the chemicalpotential �m,s and the potential energy of the electron in theelectrostatic field �inner potential �m,s� inside the medium

�̄m,s = �m,s − e�m,s, �1�

where e is the elementary charge. The equality of the twoelectrochemical potentials then connects the chemical poten-tial of the electrons in solution, i.e., electrons localized at thedonor and acceptor electronic levels of the solvated redoxcouple, to the interfacial potential drop and the chemical po-tential �m of the metal conduction electrons,

Eabs = − �s/e = ��m − �s� − �m/e . �2�

This notion establishes the absolute half-cell potential Eabs inTrasatti’s definition.3,4 This convention defines the potentialof a single electrode that does not require specifying a refer-ence electrode. The difference of absolute half potentialsthen measures the cell potential, which in turn is completelydetermined by the chemical potentials of the correspondinghalf reactions in solution.

While these relations are exact within both the thermo-dynamic and statistical-mechanical framework,5 somesubtleties still remain. The absolute potential as defined by

Eq. �2� is not directly measurable and the connection to mea-surable work functions of the metal electrode and solutionrequires the consideration of the surface potential.4,6,7 Re-gardless of these difficulties, the chemical potential �s is afundamental and well-defined property of a solvated redoxcouple, independent of the nature of the metal electrode. It istherefore amenable to at least a theoretical calculation. Suchcalculations are becoming increasingly realistic by employ-ing either analytical techniques or numerical simulations ofensembles determined through force-field potentials. We hereare interested in the latter aspect of the problem. More spe-cifically, we want to provide a theoretical framework for thecalculation of �s from the energetics of localized electronicstates in solution and to connect the solvation thermodynam-ics to observables of electrochemical experiments.

The commonly presented thermodynamic arguments linkthe chemical potential of the electrons in solution �s to thedifference in thermodynamic chemical potentials of the re-duced, �red, and oxidized, �ox, forms of the redox couple��s=�red−�ox. This procedure can be represented by animaginary reaction,

ox�s� + em → red�s� , �3�

in which the electron is brought from the metal to the oxi-dized particle in solution; vacuum is commonly adopted asthe reference state for all energies involved.4

The redox chemical potentials �ox,red are thermodynamicproperties based on ensemble averages over all possible de-grees of freedom in the system. They therefore include,among other contributions, the equilibrium solvation free en-ergies of the solvent relaxing to the solute’s charge distribu-a�Electronic mail: dmitrym@asu.edu.

THE JOURNAL OF CHEMICAL PHYSICS 130, 234704 �2009�

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tion in each redox state. While the thermodynamic cycle inEq. �3� seems quite reasonable and is widely used, one has torecognize that every thermodynamic argument involves anassumption of a certain separation of time scales,8 whichneeds to be carefully articulated, in particular, for problemsrelated to the interaction of slow nuclear with fast electronicdegrees of freedom.9 Because equilibration establishing theelectrode potential occurs in the electronic subsystem only, aconnection between �s and the thermodynamic properties ofsolvated redox species, which incorporate ensemble averagesover the nuclear degrees of freedom, needs to be clearlyestablished.

Below we will derive such relations and show that theoften anticipated direct link between the electrode potentialand the Gibbs energy ��s of the half reaction can be easilyestablished for linear solvation, but does not generally extendto nonlinear response of the solvent. We first present somegeneral statistical-mechanical arguments and then provide afully consistent statistical-mechanical derivation involving aspecific form of the electronic coupling between the redoxcouple and the metal electrode. This derivation is thensupplemented by a nonlinear solvation model with the goalof understanding of how nonlinear solvation is reflected inthe observables commonly reported in the electrochemicalexperiment. We show that the transfer coefficient of the elec-trode kinetics appears as a fundamental parameter quantify-ing the extent of nonlinear solvation of a redox couple in thebulk.

II. ELECTRONIC CHEMICAL POTENTIALIN SOLUTION

We start with the standard statistical-mechanical argu-ments by considering a solution of Nox particle in the ox stateand Nred particle in the red state. The red state is occupied bya single electron and the ox state is empty. Since the chemi-cal potential of the localized electronic states �s character-izes the redox system only, we do not need to consider theelectrode electrons at this stage. We just note that the stan-dard setup of an electrochemical experiment assumes that oxand red states are present in access in solution and chargingthe metal electrode to the potential equalizing the electro-chemical potentials of the two systems does not change ei-ther Nox or Nred, which can therefore be kept constant.

The energies of the localized electron in the ox and redstates of a molecule immersed in a fluctuating nuclear envi-ronment can be written as follows:

�ox,red�q� = �ox,red + ���q� . �4�

Here, q denotes the manifold of the nuclear degrees of free-dom, �ox,red are the average energies of the electron in thetwo states, and �� is the fluctuation of the electronic levelcaused by thermal motions of the nuclear degrees of free-dom. The grand-canonical free energy of this two-componentfermionic system then becomes10

�� = − Nox�ln�1 + e���̄s−�ox�q����q

− Nred�ln�1 + e���̄s−�red�q����q, �5�

where �=1 / �kBT� is the inverse temperature. The angular

brackets �¯ �q in Eq. �5� indicate an ensemble average overthe nuclear configurations. The average of the free energyover the distribution of nuclear coordinates is based on theidea of quenched disorder.11 It cannot be introduced bypurely thermodynamic reasoning and requires a certain sepa-ration of time scales, which we discuss first.

The position of a localized electronic state in a polarcondensed solvent is regulated by a number of degrees offreedom with a hierarchy of the corresponding time scales.First, since we are considering a two-state system with twoenergy levels, the characteristic time of transitions betweenthem �0= /�� is determined by their average energy gap��=�ox−�red. This is the electronic time scale of the solute.The solvent is characterized by at least two principle timescales, the time scale of bound electrons �e, and a character-istic timescale of the nuclear motions �n.9 Solvents com-monly used in electrochemical measurements do not absorblight at the frequency �0

−1 at which the charge-transfer tran-sition can be studied by means of spectroscopy. One thencommonly assumes �e�0. With this separation of timescales, the instantaneous energies of the donor and acceptorelectronic levels at a given �frozen� nuclear configuration ofthe system and a frozen population of the localized states �oxor red� are obtained by tracing out the electronic degrees offreedom of the solvent from the entire density matrix of thesystem �i=exp�−�Hi�,

e−��i�q� = Trel,s�e−�Hi� . �6�

Here, the trace �Trel,s� is over the electronic degrees offreedom of the solvent. The partial free energies �i�q��i=ox, red� can then be used in the standard golden rule per-turbation equations to calculate the probabilities of nonadia-batic transitions.9

The next step in the hierarchy of time scales is to recog-nize that the electronic time of the solute �0 is much shorterthan the nuclear time �n. The electronic equilibrium betweenthe two redox states will be established at any nuclear con-figuration of the system. Therefore, the electronic free energy�and not the partition function� needs to be averaged overdifferent nuclear realizations �Eq. �5��, as is normally donefor quenched disorder encountered in glasses.11 We note herein passing that the time of establishing the equilibrium po-tential of the electrode, which is typically much longer thanboth �n and �0, is irrelevant here since we are discussing thechemical potential of the localized electrons in solution.

The chemical potential �s is found by equating−��� /��s�T to the number of electrons in the system, i.e.,Nred. This condition results in the equation

lnNox

Nred= ln

�1 − nred�q

�nox�q, �7�

where

nox,red = �1 + e���ox,red�q�−�̄s��−1 �8�

are the Fermi–Dirac populations of the localized states insolution. If ox and red states are sufficiently split by solva-tion so that ���ox− �̄s��1 and ���̄s−�red��1, one can getthe Nernst equation in the form

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�̄s = �̄s0 − �kBT/2�ln�Nox/Nred� . �9�

Here, the standard electrochemical potential �̄s0 is given by

the mean of the average energies of the ox and red states plusa logarithmic correction

�̄s0 =

�ox + �red

2+

kBT

2ln

z+

z−, �10�

where

z = �e ����q. �11�

The parameters z are fugacities of the hole �“+”� andelectron �“�”� solvations. It should be stressed that theseparameters refer to solvation of electronic states only and notto the entire solute. They are not known a priori as theydepend on the statistics of the electronic energy fluctuations.When the statistics of �� is Gaussian, with the response func-tion independent of the charge, the positive and negativedeviations from the average occur with equal probabilitiesand z=z is related to the trapping �reorganization� energy �s

as z=exp���s�. The logarithmic term in Eq. �10� will vanishin this limit. Finally, we note that for typical conditions ofsolvation in polar media ��=2�s�2 eV and the conditionfor the splitting between the ox and red energies exceedingthe thermal energy, used in deriving Eq. �9�, are well justi-fied.

The electronic chemical potential given by Eq. �10� is ofcourse analogous to the common placement of the Fermilevel of a semiconductor dividing approximately in half itsforbidden gap.12 In addition, when the populations of ox andred states are in equilibrium, Nox /Nred=exp�−����, the stan-dard electrochemical potential �̄s

0 becomes �red. This energyis an analog of the highest occupied Fermi level in the metalat the equilibrium population.

If fluctuations are neglected ��s=0� and one assumesvery close states �ox��red=�, one gets the result obtainedpreviously by Reiss5,7 under the same approximations,

�s = � − kBT ln�Nox/Nred� . �12�

In both cases, the standard chemical potential is given by themean of two energies,12,13 but the coefficient in front of thelogarithm of the population ratio depends on the assumeddistribution of the electronic energy levels.

The arguments leading to Eqs. �10� and �12� are wellestablished and are routinely used in the theory of radiation-less transitions. They are reiterated here to stress that estab-lishing the chemical potential �s of the localized electrons insolution does not rest on thermodynamic arguments aloneand requires a clear recognition of the underlying separationof the relevant time scales. Of course, the same separation oftime scales9 is the basic physics behind the Marcus theory ofelectron transfer,14 which operates in terms of instantaneousdonor and acceptor electronic states shifted by fluctuations ofthe nuclear polarization P. The result of the Gaussian char-acter of these fluctuations, equivalent to the linear responseof the solvent, is the Gaussian formula for the reaction rates,

kox,red � exp�− ���ox,red − �̄m�2/�4�s�� , �13�

where �ox,red, as above, are the average energies of the ox andred states. The equality of the forward and backward electri-cal currents, koxcox=kredcred, results in the condition,

�̄m = �̄s = ��ox + �red�/2 − kBT ln�cox/cred� , �14�

when the activation barrier is greater than kBT and cox,red arethe surface concentrations. In deriving Eq. �14� the definitionof the reorganization energy in terms of the Stokes shift,15

��=�ox−�red=2�s, was used. Equations �9� and �14� areequivalent when cox,red�Nox,red

1/2 .The main result of these fairly simple arguments is that it

is not the equilibrium chemical potentials �free energies� ofthe red and ox states that define the standard chemical po-tential of the localized electrons, but instead the average en-ergies. This derivation in fact supports Parsons’ earlystatement6 that �̄s is not identical to ��s. One then wondershow to reconcile this result with the common and empiricallysupported notion that the standard electrode potential reflectsthe Gibbs energy ��s of a half reaction such that the differ-ence of standard potentials of two half reactions yields theoverall Gibbs energy of a redox reaction. This connection iseasy to draw for Gaussian energy fluctuations correspondingto linear solvation, but is harder to establish for a nonlinearsolvent response.

In order to make our arguments more specific, we willfollow the standard route of the Marcus theory14 and assumethat the only nuclear mode changing the energies of the lo-calized electronic levels is the nuclear dipolar polarization P.The instantaneous electronic energies �i�P� are then vacuumenergies �i

�0� shifted by the solution inner potential �s and bythe interaction of the electric field of the localized electronEe with the polarization P,

�i�P� = �i�0� − e�s − Ee � P . �15�

The asterisk here refers to the tensor contractions �scalarproduct� and integration over the space occupied by the sol-vent.

The average energies �i �i=ox, red� are given by theinteraction of Ee with the polarization Pi in equilibrium withthe electric field Ei in the corresponding oxidation state. Ifthe response of the solvent to the electric field of the solute islinear, Pi is found from the linear response function �, Pi

=��Ei. The average energies are then �i=�i�0�−Ee���Ei

and their mean is �̄= �̄�0�−�E���E, where E= �Eox

+Ered� /2, and we replaced the field of the electron Ee withthe difference of electric fields in the two states �E=Ee

=Ered−Eox. Clearly, the standard electrochemical potential ofthe solution electrons is related to the difference of solvationfree energies in red and ox states ��s,

�̄s0 = �̄�0� − e�s + ��s. �16�

Correspondingly, the potential drop at the metal-solution in-terface at Nox=Nred becomes

��0 = �0 − �s = ��m − �̄�0��/e − ��s/e , �17�

where �0 is the standard electrode potential. A significantconceptual advantage of this result compared to the com-

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monly employed thermodynamic arguments4 is its indepen-dence of the reference level of zero energy, which cancelsout in both the difference between the metal Fermi energy�m and the vacuum chemical potential of the redox pair �̄�0�

and in the difference of solvation free energies.We now turn to a statistical-mechanical derivation of the

standard potential of the electrode by explicitly consideringthe coupling of the localized electrons of the redox couple tothe conduction electrons of the metal. We will calculate thefree energy surfaces of heterogeneous electron transfer andidentify the conditions of equal activation barriers for theforward and backward electron transfer at equilibrium. TheAnderson–Newns model, widely applied in the past to studyheterogeneous electron transfer,16–20 is used for this goal. Wehowever need this particular model only as a means to arriveat a full statistical-mechanical formulation of the problem.Since the model is exactly integrable at a finitetemperature,21 it allows us to obtain general relations for thestandard electrode potential free from theoretical approxima-tions.

III. FREE ENERGY SURFACES OF HETEROGENEOUSELECTRON TRANSFER

The traditional theoretical description of heterogeneouselectron transfer employs the procedure outlined above in amore rigorous fashion. Specifically, the solvent-induced shiftof the electronic level of the ox or red particle is defined asthe reaction coordinate X=−Ee�P. The total Hamiltonian ofthe solution/metal system is then a sum of the electronicHamiltonian He�X� and the solution Hamiltonian Hs�P�. Theformer is given in the �one-electron22� Anderson–Newnsform,

He�X� = ���0� − e�s − X�c†c + �k

�Vkc†ak + h . c .�

+ �k

�kak†ak, �18�

where the sum runs over the wave vectors of the metal con-duction electrons with the energies �k and Vk is the electroniccoupling between the metal and localized solution electrons.Further, ak

† and c† are the fermionic electron creation opera-tors in the metal and solution, respectively. We also adoptedin Eq. �18� the common approximation that the red and oxstates have the same energy ��0� in vacuum such that theiraverage energies in solution differ only by their correspond-ing solvation energies.

The solution Hamiltonian Hs�P� has two components.The first is the interaction of the electric field E0 produced byall molecular charges of the bound electrons and nuclei ofthe solute, except for the electron exchanged with the metal,with polarization P. The second is the Hamiltonian of theliquid dipolar polarization in the absence of the solute, thebath Hamiltonian HB�P�: Hs�P�=HB�P�−E0�P. We distin-guish here between the field E0 of the “bare” solute, withoutelectron added, and the field Eox of the oxidized solute in thebulk to account for possible charge image effects. To help thereader, the various electric fields used in the paper are sum-marized in Table I.

Once this separation of the total Hamiltonian into thesolvent and electronic components is established, one candefine the reversible work �free energy� required to producea fixed position of the electronic localized state

e−���X� = TrP���X − Ee � P�e−�Hs�P�� , �19�

where the trace TrP is taken over the solvent polarization P.In addition, there will be an electronic component to thesystem free energy obtained by tracing the electronic degreesof freedom in the grand-canonical ensemble,

e−��e�X� = Trel�e���̄mN−He�X��� , �20�

where N=c†c+�kak†ak.

The resulting free energy F�X� along the reaction coor-dinate X is the sum of the solution and electronic compo-nents. The latter can be represented as an integral of theelectronic population of the localized state n��� at a givenvalue of the electronic energy �. The final result is21

F�X� = ��X� + ��X�

n���d� , �21�

where ��X�=��0�−e�s−X− �̄m.For the Anderson–Newns model, the population of the

localized state is given by the following formula:21,23

n��� =1

2+

1

�Im����

2�− i

��

2�� , �22�

where ��x�=���x� /��x� and ��x� is the gamma function.The parameter � in Eq. �22� is an effective broadening of theFermi distribution composed of a temperature term, arisingfrom thermal excitation of the conduction electrons, and anadditional broadening originating from the metal-solutioncoupling:

� = �kBT + �, � = ��F�VF�2. �23�

Here, �F is the density of the electronic states at the metalFermi level and VF is Vk at k=kF. For specific calculations,one needs to keep in mind that the overlap of the solutionelectrons with different bands in metal can vary and, in par-ticular, for d electrons, simple proportionality of � to �F

might be misleading.24

The population n��� in Eq. �22� becomes the Fermi dis-tribution when the metal is electronically decoupled from thesolution �Vk=0� and transforms into the function discussedby Anderson25 in the limit ���1,

TABLE I. Electric fields of the solute used in the paper.

Eox,red Electric fields of the reactants in their ox and red state atan infinite distance from the electrode �bulk solution�.

Ee Electric field of the electron localized at the solute inproximity to the electrode.

E0 Electric field of the solute in ox state in proximity to theelectrode.

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n��� = �−1 arccot��/�� . �24�

This latter form of the electronic population in solution �with� replacing �� is commonly used in theories of adiabaticheterogeneous electron transfer.17,18,26 These models thusoperate in the T=0 limit22 of the full solution21 of theAnderson–Newns model.

The use of Eq. �22� in Eq. �21� results in the free energysurface of heterogeneous electron transfer in the followingform:

F�X� = ��X� + ��X�/2 + �−1 Re ln����

2�− i

���X�2�

� .

�25�

In the limit of ���− i���2�, this function transforms intothe commonly used expression for the adiabatic freeenergies,18

F�X� = ��X� +��X�

�arccot���X�/��

+�

2�ln�����2 + ����X��2� . �26�

Depending on the relative magnitudes of the energy gap��X� and the electronic coupling �, the free energy functionsin Eqs. �25� and �26� have either one or two minima. In thelatter case, when � is not too large, they are separated by anactivation barrier given as �Fox,red

act =F�X†�−F�Xox,red�. Here,X† is the position of the maximum and Xox,red are the posi-tions of two minima corresponding to two redox states. Allthese extrema of the free energy surface can be calculatedfrom the equation,

���X�� = n�, �27�

where n�=n���X��� is the electronic population at the coor-dinate X�. The position of the free energy maximum is af-fected by the electrode overpotential �=�m−�0 through theenergy gap as follows:

��X� = X0† − X + e� , �28�

where

X0† = �0

�0� + e��0 − �s� − �m. �29�

The derivatives of the activation barriers with respect tothe overpotential give the corresponding transfer coeffi-cients; for the cathodic reaction one gets �ox

=�F�X†� /��e��−�F�Xox� /��e��. For barriers large comparedto kBT, the minimum position Xox is not strongly affected bythe electrode overpotential and taking the derivative of thebarrier height is sufficient. It then follows from Eq. �21� thatthe transfer coefficient is equal to the electronic populationof the localized state in the transition-state configuration n†

=n���X†��, as first established by Hush,27

�ox��� = n†��� . �30�

One can also easily notice that F�X����X�−��X� and�F�Xred� /��e���−1 when ��X��0 and ���X����. Fromthese relations, one gets a connection between ox and redtransfer coefficients, �red=1−�ox. This relation leads to the

Butler–Volmer law2 when used to calculate the electrode cur-rents.

Both the transition-state population and the transfer co-efficient in Eq. �30� depend on the electrode overpotential.This dependence arises from the linear shift of the energygap ��X� in Eq. �28� and the dependence of the maximumposition X†��� on the electrode overpotential. Bearing that inmind, the derivative of �ox��� with respect to � can be foundfrom the equation,

��ox�����e��

= � ��2�/�X2���n/����n/�� + �2�/�X2 �

X†. �31�

The two derivatives in this equation are of the order of�n /�������−1 and �2� /�X2��2�s�−1. For the typical con-ditions of electronic transitions in polar solvents, �s�� andEq. �31� can be simplified to

��ox���/��e�� � �2�/�X2�X†. �32�

In the nonadiabatic Marcus theory of heterogeneous dis-charge, �ox��� is a linear function of the overpotential,

�ox��� =1

2+

e�

2�s, �33�

and ��ox��� /��e��= �2�s�−1.The free energy ��X� defines the minimum reversible

work that needs to be applied to the solvated molecule in oxstate to shift its electronic level by the value X. Since elec-tronic coupling to the electrode does not come to the defini-tion of ��X�, only the statistics of the solvent fluctuations isrequired to determine this function. The minimum reversiblework to shift the level of the red state is obviously ��X�−X. If one allows the electronic coupling to the electrode todecay to zero �redox states in the bulk�, the free energies inEq. �25� at the corresponding equilibrium configurations willbecome F�Xox����Xox� and F�Xred����Xred�+X0

†−Xred.Since ��Xred�−Xred is the free energy of the red state, onegets

X0† = − ��s �34�

and, given Eq. �29�, one again comes to Eq. �17� for thestandard electrode potential. The present derivation thus con-firms the traditional connection between the standard poten-tial and the solvation Gibbs energy, independently of thesolvation model employed for ��X�. The linear response ap-proximation used to arrive at Eq. �17� from Eq. �9� is there-fore not required.

IV. NONLINEAR SOLVATION

We now want to look at how the electrode thermody-namics is modified when linear solvation and the corre-sponding picture of Gaussian solvent fluctuations do not ap-ply any more. One faces hard choices in an attempt to gobeyond the Gaussian distributed energy levels as this is aboutthe only condensed-phase model that allows an exact solu-tion for the charge-transfer energetics. If a fundamentallysignificant departure from the Gaussian picture exists andcan be observed2 the corresponding theoretical analysisneeds to be based on an exact model, free of approximations

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potentially creating artifacts in the theoretical results. Onesuch solution, based on a bilinear solute-solvent coupling, ispossible15 and is used here.

The bilinear coupling model assumes that the energylevel of the localized electronic state is a bilinear function ofthe nuclear polarization,

��P� = ��0� − e�s − Ee � P − �1/2�P � � � P . �35�

The term quadratic in solvent polarization represents polar-izability of the localized state. This term is the main math-ematical difference of the bilinear model from the standardMarcus formulation assuming a linear coupling between thesolute and the solvent polarization P.

The quadratic term makes ��P� a curved function of thesolvent polarization, with a maximum when the two-rankpolarizability tensor � is positively defined �see Fig. 1�.Physically, this property anticipates a limit to the shift of theelectronic energy level achievable through the solvent fluc-tuations. It is clear that linear coupling of the Marcus modelis not sustainable for fluctuations requiring large deviationsfrom equilibrium. One can for instance think of a fluctuationresulting in a complete alignment of liquid’s dipoles alongsolute’s electric field. This fluctuation will obviously producethe maximum shift of the energy level in a dipolar solventand will result in a saturation of ��P�. This saturation will notprobably follow the simple quadratic law of Eq. �35�, whichis only used to model this physical reality within the limitedrealm of mathematics allowing for an exact solution for thefunction ��X�.

The polarizability � might have different physical mean-ings. It can obviously be electronic polarizability of the lo-calized electronic state �and thus can be evaluated as thedifference in reactant polarizabilities in red and ox states�,but one can anticipate a number of scenarios leading to thequadratic polarization term.15 We will refrain here from as-signing any specific physical meaning to the polarizability �.The only restriction on its value should demand the mechani-cal stability of the polarization fluctuations, which impliesthat −�+�−1 is positively defined �see below�.

The bilinear dependence of the localized electronic en-ergy on the nuclear solvent polarization required redefiningthe reaction coordinate, which is now given by the equation,

X = P � Ee + ��/2�P2. �36�

Here, in order to simplify the algebra, we assumed that boththe polarizability � and the solvent response function � areisotropic and are given by the corresponding scalar values.The function ��X� can then be determined by Eq. �19� inwhich the above definition of the reaction coordinate is sub-stituted in the delta function. The result of taking the traceover P is the integral

e−���X�+��ox � −�

�

d�eF��,X�, �37�

where the generating function F�� ,X� is

F��,X� = − i���X + ��s� − ��2�����

� + i�. �38�

In Eq. �38�, �= ����−1 is the parameter quantifying theextent of deviation of the solvation thermodynamics from thelinear response. The solvent reorganization energy �s

= �� /2�Ee�Ee is given in the standard quadratic form and thenonlinear reorganization energy ���� is

���� = ��/2��Ee + E0/�� � �Ee + E0/�� . �39�

The condition of mechanical stability requires that ��1 at��0 or, alternatively, ��0. For simplicity, we will consider��0 throughout below; �→� then represents the linear sol-vation regime when the polarizability � can be neglected.

The contour integral in the complex plane of � in Eq.�37� can be closed in the upper half plane when X�−��s,where the integrand is an analytic function and the integral isidentically zero. The free energy ��X� is infinite in this rangeof reaction coordinates setting up an upper-energy boundaryto the fluctuations of the energy level. Within the bilinearmodel, this boundary cannot be crossed without violating thecondition of thermodynamic stability.15 The free energy re-quired to move the energy level beyond that boundary isinfinite and the probability for this to happen is zero.

At X�−��s, the integration contour is closed in thelower half plane of complex � in Eq. �37� and one has to dealwith the essential singularity appearing at �=−i� in the gen-erating function in Eq. �38�. The integral can be calculatedanalytically and is expressed in terms of the modified Besselfunction.15 When X is sufficiently far from the fluctuationboundary −��s such that 2���3�����X+��s��1, one canuse the asymptotic expansion of the Bessel function to arriveat a simple equation,

��X� = �ox + ����X + ��s� − �������2. �40�

The moments of the reaction coordinate X can be ob-tained directly from the generating function, without inte-grating over � in Eq. �37�, and are given by the followingrelation:

�Xn�ox,red = �− i��−ne−F��,0�� �n

��neF��,0���=0,−i

. �41�

Exact relations for the reaction coordinate cumulants in theox and red states follow from this equation and, in particular,the difference in solvation free energies becomes

ε(P)

P

εmax

FIG. 1. The illustration of the bilinear coupling model reaching the satura-tion limit �max of the electronic energy �solid line� in contrast with the linearcoupling model �dash-dotted line� that anticipates an arbitrary shift of theelectronic level, albeit with a small probability.

234704-6 Dmitry V. Matyushov J. Chem. Phys. 130, 234704 �2009�

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��s = − �−1F�− i,0� = ���s −�

� − 1����� . �42�

Combined with Eq. �17�, this relation gives the standardelectrode potential for the bilinear solvation model.

The condition of the extremum of the free energy surface�Eq. �27�� becomes a simple algebraic equation,

X� =�3����

�� − n��2 − ��s, �43�

which, solved together with Eq. �22�, allows one to calculateboth X� and n�. The states at the minima of the free energysurface are either electronically empty or populated,n�=0,1, at which points Eq. �43� recovers �X�ox,red fromEq. �41�.

Given that the electronic population of the activated stateis equal to the transfer coefficient �Eq. �30��, the activationbarrier for the oxidation reaction can be written in a verysimple form for nonadiabatic reactions when electronic de-localization does not affect the barrier height ���1�,

�Foxact��� = ���� ��ox���

� − �ox��� 2

. �44�

This relation converts to �Foxact���=�ox���2�s in the linear

limit of the Marcus theory ��→��.As is clear from Eq. �27�, nonlinear solvation will gen-

erally produce electronic populations in the activated statedeviating from n�=0.5 predicted at �m=�0 by linear models.In order to study this problem in more detail, we consider ahalf reaction of one-electron reduction in which a positivelycharged acceptor A+ accepts an electron in a cathodic processand becomes a neutral particle A:

A+ + e− → A . �45�

In this case Ee=−E0 and ����=�s�1−�−1�2. Figure 2 showsthe free energy surfaces F�X� for this reaction calculated for�=1.5, when substantial nonlinear solvation effects are ex-

pected, and at �=20 approaching the limit of the linearMarcus theory. In Fig. 2�a�, the vertical dashed line showsthe limit −��s representing the boundary for the energy levelfluctuations. The curves are plotted at different values of thesolute-metal electronic coupling � �Eq. �23��.

There is a significant qualitative difference in the effectof electronic coupling on the linear and nonlinear free-energycurves. While for the linear-model curves the increase in �depresses equally the activation barrier for the reduction andoxidation reactions, the effect of � on the nonlinear-modelcurves is quite asymmetric. This asymmetry implies that anadiabatic reaction ����1�, characterized by nonlinear sol-vation, will have distinctly different equilibrium and station-ary electrode potentials, the latter given by zero total current.This point is illustrated in Fig. 3 where we show the totalelectrode current ln�ia− ic� calculated as the difference of thecathodic and anodic currents ic,a�exp�−��Fox,red

act �. With in-creasing � and decreasing �, the anodic and cathodic curvesbecome increasingly asymmetric and the point of zero cur-rent shifts from the thermodynamic standard potential.

The activation barrier for the cathodic reaction decreaseswhen the overpotential � becomes more negative until reach-ing zero at the limiting value �lim. This value is simplye�lim=−�s in the standard nonadiabatic Marcus formulation,but becomes a function of both the electronic coupling � andthe asymmetry parameter � in the current formulation. Thevalue of the limiting potential is found by simultaneouslysolving two equations, �F�X� /�X=�2F�X� /�X2=0. The cor-responding relations are particularly simple in the case ofadiabatic electron transfer ����1�. The population nlim atthe limiting potential satisfies the equation

�� − nlim�3 = �2�3����/����1 + cot2�nlim�−1. �46�

After solving this algebraic equation, the limiting overpoten-tial is given as

e�lim = X� + ��s + � cot �nlim, �47�

where X� is determined from Eq. �43� with nlim used insteadof n�.

Figure 4�a� shows the dependence of �lim on � and � fora cathodic process. Both the increase in the electronic cou-pling and of the solvation nonlinearity shrinks the range ofoverpotentials at which electron transfer occurs as an acti-vated process. Increasing nonlinearity also makes the ca-

-3 -2 -1 0 1X/λ

s

0

0.2

0.4

F(X

)/eV

-1 0 10

0.2

0.4

F(X

)/eV

(a)

(b)

0.20.1

0.01

γ = 1.5

γ = 20

FIG. 2. Free energy surfaces F�X� for the one-electron reaction A++e−

→A at �=1.5 �a� and �=20 �b�. The curves are plotted by using Eq. �25�with different values of � �Eq. �23�� indicated in the plot �in eV�. Thevertical dashed line in �a� indicates the fluctuation boundary −��s; �s

=1 eV.

-0.2 0 0.2eη/λ

s

-4

-2

0

ln|i a

-i c|

γ = 10

γ = 2

FIG. 3. Total electrode current ln�ia− ic� vs overpotential e� /�s for �=10�dash-dotted lines� and �=2 �solid and dashed lines�. The solid lines refer to�=0.2 eV; the dashed and dash-dotted lines denote �=0.1 eV. The curveshave been calculated for a one-electron electrode reaction �Eq. �45�� with�s=1 eV; units of the current are arbitrary.

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thodic and anodic currents asymmetric, as is seen from thedependence of the cathodic transfer coefficient �ox��=0� on� �Fig. 4�b��. With decreasing � �increasing polarizability�,�ox�0� starts to deviate upward from �ox�0�=0.5 predicted bylinear solvation theories. This upward deviation is addition-ally amplified by the solution-electrode electronic coupling.Notice that the Marcus theory allows the transfer coefficientsto deviate from the symmetric case of �ox=�red=0.5 only ata nonzero overpotential,28 while the current theory makes itpossible even at �=0. The corresponding cathodic and an-odic current curves will therefore be asymmetric with respectto the change in the sign of �, as is also seen in Fig. 3.

What is also clear from Fig. 3 is that the lines ln ia,c

versus � �Tafel plots� become increasingly curved with low-ering �. This trend can be quantified by looking at the de-rivative of the transfer coefficient with respect to the elec-trode overpotential �Eq. �32��. In the bilinear model, thisderivative becomes

��ox

��e��=

�� − �ox�3

2�3����. �48�

This function is plotted against � in Fig. 5. Its rise at low �testifies to the increased curvature of the Tafel plot. Thisoutcome is however not universal as it depends on the reac-tion considered. For instance, for the reaction A+e−→A−,one gets ����=�s and the result is just the opposite, the

curvature of the Tafel plot decreases with increasingly non-linear solvation. We also note that we considered positive��1 so far, but the model does not exclude the possibility of��0. It can be the result of the oxidized form being moreelectronically polarizable than the reduced form or someother mechanism. The magnitude of the nonlinearity param-eter � needs to be determined by mapping the model on agiven physical situation. The main outcome is that the modelallows more flexibility in attaining �ox�0��0.5 and in vary-ing curvatures of Tafel plots.

The curvature of the Tafel plot, when recordered,29–31

can be associated with the corresponding observable reorga-nization energy,

�T = �2 � �ox/��e���−1, �49�

which becomes �s in the Marcus model �Eq. �33��. Combin-ing Eqs. �44� and �48� one can represent the activation bar-rier in term of parameters observable from Tafel plots and, inaddition, the nonlinearity parameter �,

�Foxact��� = �T�1 − �ox���/���ox���2. �50�

From this equation, the reorganization energy extractedfrom the Arrhenius slope of the rate constant �A=�T�1−�ox��� /�� should deviate from the Tafel �T. Finklea et al.31

observed �A��T for redox centers attached to alkanethiolmonolayers deposited on gold electrodes. Since � includesthe polarization response function �, the ratio �A /�T is ex-pected to vary with the solvent used.

V. DISCUSSION

The main question addressed in this paper is the relationbetween the standard electrode potential and the solvationthermodynamics of the redox couple in solution. Thestatistical-mechanical derivation presented in Sec. II givesthe standard electrode potential in terms of the mean of theaverage energies of the ox and red states in solution and alogarithmic correction for the asymmetry of the hole versuselectron solvation �Eq. �10��. This asymmetry term vanishesfor symmetric Gaussian fluctuations of the localized elec-tronic energies. In the same approximation, the mean of theaverage energies is related to the difference in solvation freeenergies of red and ox states. Whether the same applies to amore general model of nonlinear solvation is not entirelyclear, but the bilinear model of the solute-solvent couplingproduced the same result. It is probably safe to assume thatthis connection, although probably not exact, is fairly robustand can withstand other scenarios of nonlinear solvation.

The bilinear coupling model was combined with theAnderson–Newns prescription for the electron delocalizationbetween the solution and electrode electronic states. An exactstatistical-mechanical solution was achieved and gave us aninsight at the effect of nonlinear solvation on observablestypically recorded in the electrochemical experiment. Theresulting free energy curves turned out to be asymmetric,with the fluctuation boundary limiting from one side theavailable range of the electronic energies. The correspondingdistribution of the energy levels in solution is schematicallyshown in Fig. 6.

0 5 10 15 20

-0.3

-0.2

-0.1

eηlim

/λs

0 5 10 15 20γ

0.5

0.6

0.7

0.8

α ox(0

)

0.1

0.2

0.3(a)

(b)

FIG. 4. The limiting overpotential �a� and the cathodic transfer coefficient�ox�0� at zero overpotential �b� vs the nonlinearity parameter � at differentvalues of the electronic coupling � indicated in the plot �in eV�; �s=1 eV.

2 4 6 8 10γ

0

0.2

0.4

0.6

∂(λ sα ox

)/∂(

eη) 1

2

FIG. 5. The derivative of the transfer coefficient with respect to the elec-trode overpotential for the reaction in Eq. �45� �1� and for the reaction A+e−→A− �2�. The horizontal dashed line shows the result of linear solvationtheories ��ox /��e��=0.5 �Eq. �33��.

234704-8 Dmitry V. Matyushov J. Chem. Phys. 130, 234704 �2009�

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The present development is largely driven by our recentobservations of nonlinear electrostatic solvation of large-sizesolutes approaching the typical size of hydrated proteins32 ornanoparticles.33 The main result of those numerical simula-tions is that immersing a large �4–5 solvent diameters� soluteinto a polar solvent dramatically alters the spectrum of elec-trostatic solvent fluctuations from the typical expectations ofGaussian solvation models34 applicable to small solutes.35

The Marcus model assumes that the spectrum of electrostaticsolvent fluctuations is not affected by a small solute, butnumerical simulations suggest that this is true only up to acertain critical solute size. Significant deviations from theGaussian statistics are observed beyond this critical size andthe present development aims to incorporate such effects intotheories of heterogeneous charge transfer at metal electrodes.

We derived �Eqs. �27� and �30�� a general relation be-tween the derivative of the free energy of solvation in thebulk ��X� and the transfer coefficient entering the Tafel lawof electrode kinetics:36

���X†� = �ox. �51�

The transfer coefficient on the right hand side of this equa-tion quantifies the efficiency with which electrical potentialaccelerates the rate of electrode reactions.2 On the otherhand, ��X� on the left hand side of Eq. �51� is the free energyrequired to shift the unoccupied �ox� electronic state in solu-tion by the magnitude X. The relation �red=1−�ox enteringthe Butler–Volmer law of electrode kinetics2 is then a directconsequence of the requirement that the free energy for shift-ing the occupied �red� electronic state is ��X�−X.

From this derivation, the magnitude of the transfer coef-ficient is the free energy derivative, or generalized force, inthe activated state. This result is qualitatively consistent withthe traditional Gurney’s definition of the transfercoefficient37 as a symmetry factor given by the ratio of thechange in the nuclear, �U, to the change in the electronic,�X, energies as a result of thermal fluctuations �ox,red

=�U /�X. While the current formulation gives a more pre-cise definition of the symmetry factor, it also emphasizes itsrelation to the solvation thermodynamics of the solutionelectrons. The value close to 0.5, which is a signature oflinear solvation, is often experimentally observed for smallredox couples. Most deviations from �ox=0.5 reported in the

literature29–31 have been observed at a nonzero overpotential�see Ref. 38 for a recent review�. It is currently not clearwhat the experiment has to say about transfer coefficients oflarge particles, redox proteins in particular.39 We note in thisregard that numerical simulations of small solutes are oftenquite consistent with the linear response,40 and it is only forlarge solutes, significantly exceeding the size of the solventmolecules, that substantial deviations from linear solvationhave been reported.32,33

Although our main concern in this paper has been withthe standard electrode potential, we briefly comment here onthe entropic term in the Nernst equation containing the loga-rithm of the ratio of ox and red concentrations in solution�Eq. �9��. The standard argument in deriving the Nernst equa-tion is to consider the equilibrium between the electrochemi-cal potentials in the half reaction shown in Eq. �3�: �̄ox

+ �̄m= �̄red. The entropic term is then obtained by noting that�̄ox,red is a sum of the standard potential and an ideal, kineticenergy term kBT ln Nox,red.

19 This derivation then results inthe canonical form of the Nernst equation,2

�̄m = �̄s = �red0 − �ox

0 − kBT ln�Nox/Nred� , �52�

missing the factor 1/2 appearing in Eq. �9�.The objection to this approach is that the equilibrium

between the redox couple and the metal is established in theelectronic subsystem only and the translational entropy ofions in solution should not affect the electronic equilibrium.The derivation in Sec. II presents a correct statistical-mechanical approach to the problem,13 as was discussed atlength by Reiss.5,7 Solvation was not, however, a part ofReiss’s derivation and it has been added here. Once solvationsplits the energy gap between the ox and red electronic en-ergy levels to an amount significantly exceeding the thermalenergy, the entropic term gains a factor of 1/2, as is wellestablished in the semiconductor literature.12 We also notethat the Nernst equation was historically derived and testedfor the problem of concentration potential,41 where the use ofthe ideal translational component in the ions’ chemical po-tentials, resulting in Eq. �52�, is fully justified.

ACKNOWLEDGMENTS

This research was supported by the DOE, Chemical Sci-ences Division, Office of Basic Energy Sciences �Grant No.DEFG0207ER15908�.

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metal solution

filled states

empty statesµ

m µs

fluctuation boundary

ε

FIG. 6. Distribution of the electronic energies � of empty ox and filled redstates in solution at �=1.5 and the corresponding levels of the electrochemi-cal potentials of the metal and solution electrons. The dash-dotted lineshows the fluctuation boundary of the localized electronic levels. Electronicstates in solution above this boundary are prohibited by the bilinear model.

234704-9 Standard electrode potential J. Chem. Phys. 130, 234704 �2009�

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234704-10 Dmitry V. Matyushov J. Chem. Phys. 130, 234704 �2009�

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