bockris - modern aspects of electrochemistry no 20

MODERN ASPECTS OFELECTROCHEMISTRY

No. 20

Edited by

J. O'M. BOCKRISDepartment of Chemistry

Texas A&M UniversityCollege Station, Texas

RALPH E. WHITEDepartment of Chemical Engineering

Texas A&M UniversityCollege Station, Texas

and

B. E. CONWAYDepartment of Chemistry

University of OttawaOttawa, Ontario, Canada

PLENUM PRESS • NEW YORK AND LONDON

LIST OF CONTRIBUTORS

HECTOR D. ABRUNADepartment of ChemistryCornell UniversityIthaca, New York 14853

H. P. AGARWALDepartment of ChemistryM.A. College of TechnologyBhopal, India

ALEKSANDAR DESPfcFaculty of Technology

and MetallurgyUniversity of BelgradeBelgrade, Yugoslavia

JERRY GOODISMANDepartment of ChemistrySyracuse UniversitySyracuse, New York 13244

A. M. KUZNETSOVA. N. Frumkin Institute for ElectrochemisAcademy of Sciences of the USSRMoscow V-71, USSR

VITALY P. PARKHUTIKDepartment of MicroelectronicsMinsk Radioengineering InstituteMinsk, USSR

ISAO TANIGUCHIDepartment of Applied ChemistryFaculty of EngineeringKumamoto UniversityKurokamiKumamoto 860, Japan

A Continuation Order Plan is available for this series. A continuation order will bringdelivery of each new volume immediately upon publication. Volumes are billed onlyupon actual shipment. For further information please contact the publisher.

Contents

Chapter 1

THEORIES FOR THE METAL IN THEMETAL-ELECTROLYTE INTERFACE

Jerry Goodisman

I. Introduction 11. The Metal in the Interface 12. Metal-Electrolyte Interaction 6

II. Separation of Metal and Electrolyte Properties 91. Electrostatics 92. Experimental Results 14

III. Metal Structure 201. Electrons in Metals 202. Band Structure 253. Ion-Electron Interactions 304. Screening 33

IV. Metal Surfaces 391. Density Functionals 392. Self-Consistent Theories 433. Screening 46

V. Metal Electrons in the Interface 541. Metal Nonideality 542. Density-Functional Theories 573. Metal-Solvent Distance 684. Models for Metal and Electrolyte 725. General Dielectric Formalism 83

vii

viii Contents

VI. Conclusions 88References 90

Chapter 2

RECENT ADVANCES IN THE THEORY OFCHARGE TRANSFER

A. M. Kuznetsov

I. Introduction 95II. Interaction of Reactants with the Medium 95

1. Adiabatic and Diabatic Approaches: A ReferenceModel 96

2. A New Approach to the Interaction of theElectron with the Polarization of the Medium inNonadiabatic Reactions 101

III. Nonadiabatic Electron Transfer Reactions 1041. Energy of Activation or Free Energy of

Activation? 1042. Effects of Diagonal and Off-Diagonal Dynamic

Disorder in Reactions Involving Transfer ofWeakly Bound Electrons (A ConfigurationalModel) 110

3. Feynman Path Integral Approach 1174. Lability Principle in Chemical Kinetics 1195. Effect of Modulation of the Electron Density on

the Inner-Sphere Activation 122IV. Elementary Act of the Process of Proton Transfer . . . 127

1. Physical Mechanism of the Elementary Act and aBasic Model 127

2. Distance-Dependent Tunneling in theBorn-Oppenheimer Approximation 130

3. Hydrogen Ion Discharge at Metal Electrodes 1344. Charge Variation Model 137

V. Processes Involving Transfer of Atoms and AtomicGroups 142

Contents ix

1. Fluctuational Preparation of the Barrier and RolePlayed by the Excited Vibrational States in theBorn-Oppenheimer Approximation 142

2. Role of Inertia Effects in the Sub-Barrier Transferof Heavy Particles 147

3. Nonadiabaticity Effects in Processes InvolvingTransfer of Atoms and Atomic Groups 151

4. Ligand Substitutions in Alkyl Halides 155VI. Dynamic and Stochastic Approaches to the

Description of the Processes of Charge Transfer 1581. Dynamic and Fluctuational Subsystems 1592. Transition Probability and Master Equations 1603. Frequency Factor in the Transition Probability... . 1614. Effect of Relaxation on the Probability of the

Adiabatic Transition: A Dynamic Approach in theClassical Limit 163

5. Stochastic Equations 1696. Effect of Dissipation on Tunneling 172

VII. Conclusion 173References 173

Chapter 3

RECENT DEVELOPMENTS IN FARADAICRECTIFICATION STUDIES

H. P. Agarwal

I. Introduction 177II. Theoretical Aspects 178

1. Single-Electron Charge Transfer Reactions 1792. Two-Electron Charge Transfer Reactions 1823. Three-Electron Charge Transfer Reactions 1844. Zero-Point Method 185

III. Instrumentation and Results 1901. Faradaic Rectification Studies at Metal

Ion/Metal(s) Interfaces 190

x Contents

2. Faradaic Rectification Studies at RedoxCouple/Inert Metal(s) Interfaces 204

IV. Faradaic Rectification Polarography and ItsApplications 219

1. Studies Using Inorganic Ions as Depolarizers 2192. Studies Using Organic Compounds as Depolarizers 240

V. Other Applications 246VI. Conclusions 247Appendix A 250Appendix B 254Notation 259References 260

Chapter 4

X RAYS AS PROBES OFELECTROCHEMICAL INTERFACES

Hector D. Abrufia

I. Introduction 265II. X Rays and Their Generation 267

III. Synchrotron Radiation and Its Origin 269IV. Introduction to EXAFS and X-Ray Absorption

Spectroscopy 273V. Theory of EXAFS 277

1. Amplitude Term 2782. Oscillatory Term 2803. Data Analysis 281

VI. Surface EXAFS and Polarization Studies 286VII. Experimental Aspects 287

1. Synchrotron Sources 2872. Detection 288

VIII. EXAFS Studies of Electrochemical Systems 2911. Oxide Films 2922. Monolayers 2983. Adsorption 3034. Spectroelectrochemistry 305

Contents xi

IX. X-Ray Standing Waves 3101. Introduction 3102. Experimental Aspects 3153. X-Ray Standing Wave Studies at Electrochemical

Interfaces 316X. X-Ray Diffraction 320

XI. Conclusions and Future Directions 321References 322

Chapter 5

ELECTROCHEMICAL ANDPHOTOELECTROCHEMICAL REDUCTION

OF CARBON DIOXIDE

Isao Taniguchi

I. Introduction 327II. Electrochemical Reduction of Carbon Dioxide 328

1. Reduction of Carbon Dioxide at Metal Electrodes 3282. Mechanisms of Electrochemical Reduction of

Carbon Dioxide 3363. Pathways for Carbon Dioxide Reduction 3434. Reduction of Carbon Dioxide at Semiconductor

Electrodes in the Dark 344III. Photoelectrochemical Reduction of Carbon Dioxide . . 349

1. Reduction of Carbon Dioxide at Illuminatedp-Type Semiconductor Electrodes 349

2. Photoassisted Reduction of Carbon Dioxide withSuspensions of Semiconductor Powders 363

IV. Catalysts for Carbon Dioxide Reduction 3671. Metal Complexes of N-Macrocycles 3682. Iron-Sulfur Clusters : 3743. Re, Rh, and Ru Complexes 3754. Other Catalysts 380

V. Miscellaneous Studies 3831. Photochemical Reduction of Carbon Dioxide 3832. Reduction of Carbon Monoxide 388

xii Contents

3. Thermal Reactions for Carbon Dioxide Reduction 3894. Carbon Dioxide Fixation Using Reactions with

Other Compounds 389VI. Summary and Future Perspectives 390References 394

Chapter 6

ELECTROCHEMISTRY OF ALUMINUM INAQUEOUS SOLUTIONS AND PHYSICS OF

ITS ANODIC OXIDEAleksandar Despic and Vitaly Parkhutik

I. Introduction 401II. Overview of the System 403

III. Kinetics of Aluminum Anodization 4081. General Considerations 4082. Open-Circuit Phenomena 4213. Kinetics of Barrier-Film Formation 4234. Formation of Porous Oxides 4295. Active Dissolution of Aluminum 433

IV. Structure and Morphology of Anodic AluminumOxides 447

1. Methods of Determining Composition andStructure 447

2. Chemical Composition of Anodic AluminumOxides 450

3. Crystal Structure of Anodic Aluminas 4574. Hydration of Growing and Aging Anodic

Aluminum Oxides 4605. Morphology of Porous Anodic Aluminum Oxides 464

V. Electrophysical Properties of Anodic AluminumOxide Films 467

1. Space Charge Effects 4672. Electronic Conduction 4703. Electret Effects 4774. Electric Breakdown of Anodic Alumina Films 480

Contents xiii

5. Transient and Aging Phenomena in AnodicAlumina Films 482

6. Electro- and Photoluminescence 484VI. Trends in Application of Anodic Alumina Films in

Technology 4871. Electrolytic Capacitors 4882. Substrates for Hybrid Integrated Circuits 4893. Interconnection Metallization for Multilevel LSI... 4914. Gate Insulators for MOSFETs 4915. Magnetic Recording Applications 4926. Photolithography Masks 4927. Plasma Anodization of Aluminum 492

References 493

Index 505

Theories for the Metal in theMetal-Electrolyte Interface

Jerry GoodismanDepartment of Chemistry, Syracuse University, Syracuse, New York 13244

1. INTRODUCTION

1. The Metal in the Interface

Since the electrochemical interface is usually the interface betweena metal and an electrolyte, all properties of the interface may beexpected to involve contributions of the metal and of the electrolyte.However, most theories of the electrochemical interface are theoriesof the electrolyte phase, with no reference to the contributions ofthe metal. Here, we discuss more recent theoretical work whichattempts to redress this inequity. As we shall see, it is not, in general,possible to separate, experimentally, the metal contribution fromthe electrolyte contribution.

It is instructive to see how this comes about for the potentialof a simple electrochemical cell. We consider the potential of zerocharge for the cell1

Hg|Na|Na\S|Hg'

The solution phase S is supposed to contain Na+. Here, the Na|Na+

electrode, which is reversible, is the reference electrode. This cellis not actually realizable, but its potential of zero charge can becalculated from measurements on other cells.

2 Jerry Goodisman

The cell potential is the difference in electrical potential (innerpotential) between the right-hand and left-hand mercury terminals:

The third potential difference may be expressed as a difference ofchemical potentials by invoking the equality of the electrochemicalpotentials of the electron in the two metallic phases in contact. Thefirst potential difference may be written as a difference in outerpotentials plus a difference in surface potentials of the free surfacesof S and Hg, corresponding to a path from the inside of Hg to theinside of S passing through their free surfaces and through vacuumoutside; it may also be written as a difference of surface potentialsplus an "ionic" or free-charge contribution, corresponding to apath passing through the interface between Hg and S. At zeroelectrode charge, the free-charge contribution vanishes, and A"g isjust a difference in surface potentials, so that the potential of zerocharge is

Epzo = *HE(S) - *s(Hg) - ASN> + F" V r a - M?g) (1)

Here, /YHg(s) is the surface potential of mercury in contact with the

solution phase S. The difference of metal and solution surfacepotentials, *Hg(s) - #s(Hg), is the dipolar potential difference, oftenwritten2 as g^(dip) = gM(dip) - gs(dip), with the metal M hereequal to Hg.

The surface potential of mercury in contact with S may bewritten as the surface potential of the free surface of mercury, ^Hg,plus the change due to the presence of S, 5*"8. Since F^Hg - /A"8

is the work function of mercury, 4>Hg, we have

£Pzo = ®Hg/F - XS(Ug) + S*?8 - A?V + fiVVF (2)

Most of the quantities appearing in this equation are measurableor calculable. Thus, the potential of zero charge for the cell ismeasured as 2.51 V, and the work function of mercury as 4.51 V,while the chemical potential of the electron in sodium can be reliably

The Metal-Electrolyte Interface 3

calculated as —3.2 eV. Bockris and Khan1 estimated the free-surfacepotential ^ s for aqueous solutions as 0.2 V, and the effect of thesolution on the mercury surface potential, 8x™B, as 0.26 V, to calcu-late the difference of inner potentials A™a</>. Note that 8xss is largerin magnitude than the surface potential of the electrolyte phase /^

s,but it is generally believed that the latter quantity varies more withsurface charge than does the former, so that 8x™ is unimportantwhen capacitances are being considered.

In Eq. (1), -A^a0 + F"V^ a is a property of the Na|Na+

reference electrode. In order to get information not depending onthe reference electrode, one has to study Epzc and <J>M with changingmetal M. For a metal M, the right-hand side of Eq. (2) may berearranged3 to <DM/e + 8X¥ - 8x1 - Ek9 where 8X

SM = *s(Hg) - *S

and Ek is the "electrode potential on the vacuum scale" of thereference electrode,4 which can be determined experimentally.

The quantity (8x™g - 8xHg) is the difference between the effectof mercury on the surface potential of water and the effect of wateron the surface potential of mercury. Experimentally, there is noway to separate2 these two contributions to g^(dip); theoretically,they are discussed and calculated in radically different ways. Thisis already clear in Eq. (1), which involves *Hg(s) - *s(Hg). If onekeeps the same solution but substitutes another metal M for mer-cury, *M(S) will replace # Hg(s), contributing to the change in thecell potential, but, in addition, *S(M) will probably differ from/V

S(Hg), since xS is ascribed to the orientation of solvent dipoles atthe metal surface, which will be different for a different metal. Toseparate 8%™ and 8XM, some assumption or model is required. Forexample, in interpreting plots of adsorption potential versus chainlength for a series of aliphatic alcohols at the water-air and water-mercury interfaces, the difference between the lines may beascribed3 to 8XM, thus measuring one of the two separately.

When polar or polarizable species are adsorbed on a metal,the work function changes. This is partly due to gs(dip) but is alsodue to the change in x™ and other effects.2 As for the interfacialpotential in the electrochemical cell, the contributions of adsorbateand metal cannot be separated. Usually, the latter gets ignored. Itis precisely this term that interests us here.

The basic double-layer model considers the solid as a perfectconductor, so that gM(dip) is charge independent and the potential

4 Jerry Goodisman

drop (metal minus solution) is just g(ion) — gs(dip) plus a constant.Although early theories considered penetration of the electric fieldof the interface into the metal (the work of Rice5 actually ascribedthe entire inner-layer capacitance to the metal, rejecting Stern-typetheories of the compact layer), this idea was subsequently dropped.The history of the subject has been well reviewed by Vorotyntsevand Kornyshev.6 Mott and Watts-Tobin7 criticized ascribing theinner-layer capacitance entirely to the metal and also argued againstusing, unchanged, results for the capacitance of the metal-vacuuminterface in discussion of the metal-solution interface. Althoughthey did not reject a metal contribution to the capacitance, theirwork seems to have been interpreted68 to mean that the metal'scontribution to the reciprocal of the capacitance is unimportant,so that only solution species should be considered in a model forthe interface. Of course, penetration of the electric field into theinterface is important for semiconductors, and, in 1979, its possiblerelevance for semimetals like Bi and Sb was suggested.9

If the metal is ideal, the model for the interface is a model forthe electrolyte phase. The capacitance of the interface correspondsto two capacitances in series, that of the diffuse layer, for whichthe Gouy-Chapman theory is considered adequate, and that of thecompact layer. The simplest model for the latter is a layer containingN dipoles per unit area, each dipole having an average normalcomponent pN (pN depends on the field). Then gs(dip) = NpN/e,where s is the dielectric permittivity in the dipole layer. If s isconsidered to arise from electrons, the high-frequency dielectricconstant of the solvent would be used for it, but other values maybe appropriate. Writing the compact-layer capacitance per unit areaat the point of zero charge as Cc = ec/47rd (ideal parallel-platecapacitor), one can derive10 values for ec from experimental Cc

values for various metals. These values increase for different metalsalong with the (measured) potential drop g across the compactlayer. If g measures orientation of solvent dipoles, increased gshould be associated with decreased ec\ the contradiction has beenadduced11 as evidence for a missing metal contribution in the model.

In their classic treatment of the compact-layer capacitances,MacDonald and Barlow12 affirmed that the thickness of the spacecharge or penetration region in the metallic electrode is so smallfor a good conductor that its effect may be neglected. Their theory


of the compact layer, together with the Gouy-Chapman theory forthe diffuse layer, successfully fitted most capacitance data forHg/water and Hg/methanol interfaces with nonadsorbing elec-trolytes (except in the anodic region). Interaction of the solventdipoles with the mercury was treated by summing multiple idealimage interactions, other forces involved in physisorption beingneglected.

Models for the compact layer of the metal-electrolyte interfacehave become more and more elaborate, providing better and betterrepresentations of observed electrocapillary data for differentmetals, solvents, and temperatures, but almost always leaving themetal itself out of consideration, except for consideration of imageinteractions of the solvent dipoles. For reviews of these models,see Parsons,13 Reeves,14 Fawcett et al,15 and Guidelli,16 who givesdetailed discussion of the mathematical as well as the physicalassumptions used.

Fawcett et al15 presented a four-state model, involvingmonomers and clusters, for water in the inner layer of the mercury-aqueous solution interface with no specific adsorption. By treatingdipole-dipole interactions on a molecular level, they avoidedintroduction of an effective dielectric constant. The contribution tothe dipole-dipole interaction of dipole images in the metal wasincluded, along with non-nearest-neighbor interactions, by usingan effective coordination number. Values for seven parameters,including this effective coordination number, were chosen to givethe best agreement with experimental capacitances at various sur-face charges and temperatures. Recently, Guidelli17 has developeda statistical mechanical treatment for a monolayer of hydrogen-bonded molecules, interacting with a surface and with each otherand orienting in an electric field. Capacitance as a function ofcharge was calculated and compared to compact-layer capacitancesderived from experiment by removing the diffuse-layer contribution.The model accounted qualitatively for some of the observedfeatures, without considering any contribution of the metal itself.

It is perhaps unnecessary to note that the fact that a model,with a suitable choice of parameters, can account for experimentaldata does not prove that the model is correct or that no physicaleffects of importance have been left out. Yeager,18 in his review ofnontraditional approaches to the study of the metal-electrolyte

6 Jerry Goodisman

interface, stated that compact-layer models that do not adequatelytake into account the structure of the electrode surface and theinteraction of water in the compact layer with water outside willnot yield an understanding of the interface. Further, he noted thelack of molecular specificity of traditional electrochemical measure-ments. Among methods which can distinguish between molecularspecies in the interface are ellipsometry, Raman spectroscopies,Mossbauer spectroscopy, and electron spin resonance; perhapsthese can be of help in disentangling metal and electrolyte contribu-tions to interfacial properties.

The importance of the structure of the metal was also sug-gested19 for the capacitances at the point of zero charge of metal-molten salt interfaces. As compared with metal-electrolyte inter-faces, the striking difference is that the capacitances of the latterdecrease with increasing temperature and those of the formerincrease sharply with temperature. (It is perhaps just as strikingthat the sizes of the capacitances for the two systems are quitesimilar, although the natures of the systems are so different.)According to the Gouy-Chapman theory, the capacitance at thepoint of zero charge is proportional to T~l/2\ improved statisticalmechanical theories such as the mean spherical approximation20

give a weaker decrease with T. It is suggested by March and Tosi20

that the increase in capacitance with temperature is connected withincreased penetration of the metal surface by the ions of the elec-trolyte, this penetration increasing with increase in temperature.

2. Metal-Electrolyte Interaction

One might expect, as a first approximation, that at the same cellpotential relative to the potential of zero charge, one would havethe same surface charge in the interface, regardless of the metal,but there are many exceptions to this,21 ascribed to the effect ofthe metal on the orientation of solvent molecules. Similarly, theincrease of the capacitance for the Ga/aqueous solution interfacewith decreasing negative surface charge is ascribed to specificadsorption of water, with the negative end of the water moleculetoward the metal. This chemisorption is likely to be stronger forGa than for Hg.


The nonelectrostatic forces between metal and solution havesometimes been subsumed22 into a "natural field," tending to orientthe solvent dipoles, whose effect is represented by an energy contri-bution -pAE cos 0 where A£ is the natural field strength, p theorienting dipole moment, and 6 the angle between the dipole andthe direction normal to the interface; this term, independent of thesurface charge, is to be added to the energy of a dipole in anexternal field (due to surface charge), -pEx cos 0. Experimentalevidence916 indeed indicates that, at the point of zero charge, watermolecules in the metal-electrolyte interface are preferentially orien-ted with their negative (oxygen) ends toward the metal. This givesa negative contribution to the metal-solution potential differenceA^V. AS the temperature is increased, this contribution shoulddecrease in size, which explains the fact that A^V increases withtemperature. If this preferential orientaiton is due to nonelectricalforces of adsorption, it is reasonable that a slightly negative surfacecharge on the metal is necessary to give zero net orientation ofdipoles. Below we shall mention another mechanism, due directlyto penetration by the metal's conduction electrons of the solventlayer, which produces the same preferential orientation of dipolesat zero surface charge.

In fact, the orientation of water at the potential of zero chargeis expected to depend approximately linearly on the electronegativ-ity of the metal.9 This orientation (see below) may be deduced fromanalysis of the variation of the potential drop across the interfacewith surface charge for different metals and electrolytes. Suchanalysis leads to the establishment of a hydrophilicity scale of themetals ("solvophilicity" for nonaqueous solvents) which expressesthe relative strengths of metal-solvent interaction, as well as therelative reactivities of the different metals to oxygen.23

In addition to the nonelectrostatic adsorptive force, there isan image force between a dipole and a metal, which will be presentwhenever charged or dipolar particles in a medium of one dielectricconstant are near a region of another dielectric constant. If themetal is treated as an ideal conductor, the image-force contributionto the energy of a dipole in the electrolyte is proportional to p2/z3,where z is the distance of the dipole from the plane boundary ofthe metal (considered ideal, with no surface structure), and to1 + cos2 0. This ideal term is, of course, the same for all metals. If

8 Jerry Goodisman

the metal structure is considered more realistically, the image-typeinteractions will differ for different metals.

Specific adsorption of ions (probably anions) of the electrolytephase on the metal also should depend on the metal. Assuming aLangmuir-type equilibrium, one has22 for ions of charge qv andsolution concentration c,

Here K is a constant, Te and Tf are the number of empty and filledsites per unit area on the metal surface, (f>t is the adsorptionpotential, and if/A is the electrostatic potential of the empty site; <f>t

depends on surface charge. The sum Fo = Te + Tf, total number ofsites per unit area, depends on the metal, as does <£,.

The above effects are more familiar than direct contributionsof the metal's components to the properties of the interface. In thischapter, we are primarily interested in the latter; these contributeto xMiS)- The two quantities *M(S) and *S(M) (or 8%™ and 8XM) areeasily distinguished theoretically, as the contributions to the poten-tial difference of polarizable components of the metal and solutionphases, but apparently cannot be measured individually withoutadducing the results of calculations or theoretical arguments. Amodel for the interface which ignores one of these contributionsto A (/> may, suitably parameterized, account for experimentaldata, but this does not prove that the neglected contribution is notimportant in reality. Of course, the tradition has been to neglectthe metal's contribution to properties of the interface. Recently,however, it has been possible to use modern theories of the structureof metals and metal surfaces to calculate, or, at least, estimatereliably, ^M(s) and 8x™ (as well as discuss 8XM9 which enters sometheories of the interface). It is this work, and its implications forour understanding of the electrochemical double layer, that wediscuss in this chapter.

We shall discuss first the theoretical separation of metal andsolution properties and then turn to the modern theories of metalstructure, particularly as they apply to the surface. Then we shallconsider the calculation of quantities relevant to the metal in theinterface and theories of the metal-solution interface which takethe metal, as well as the solution, into account.


II. SEPARATION OF METAL ANDELECTROLYTE PROPERTIES

1. Electrostatics

For an interface between two phases with no common charged orpolarizable components, contributions of the two phases to certainproperties are easily distinguished theoretically. The charge densityand electrical polarization at any point in the interface are eachsums of two contributions which can be assigned to the two phases.The overall electroneutrality of a planar interface may be written

£ (? •*) * - £ (? "*)di+£ (f -*)* - °where n, is the density of species i, a function of z, and qx is thecharge of species i. Of the two sums in the second member, one isover charged components of phase A and the other over chargedcomponents of phase B. The surface charge density of phase A (incharges per unit area) is

= I q. \ n,(z) dz (3)

and is equal and opposite to the surface charge density of phase B,

(4)<7B = I q, [ n,(z) dzi J —oo

The charge density at any point is X, ^g,, the sum being over speciesof both phases. Note that no "geographical" separation of com-ponents of the phases is required, anticipating that, in the realsystem, it may not be possible to divide the system such that allthe components of phase A lie on one side of a geometrical surfaceand all the components of phase B on the other. It is necessaryonly to identify each component as belonging to one phase or theother. In a geographical separation, one would write

and q = J (gn

10 Jerry Goodisman

Because of the overall electroneutrality, qA will equal -qB for anychoice of X.

All nf are determined, in principle, by the equations of statis-tical mechanics, since they are one-particle distributions.4 As such,each nl depends on the interactions between particles of species iand all other particles in the system, whether belonging to the metalor to the other phase. If there is a geographical separation ofparticles of species i from, say, particles of species k (as when iand k belong to different phases), the interaction between particlesk and a particle of species i near the surface may be averaged overpositions of particles k, i.e., no correlation is assumed between theparticles of the two species, so that the particles k become a sourceof external field for particles i. For a particle i far from the surface,the interaction is probably unimportant (unless it is a long-rangeelectrostatic interaction).

For electrochemistry, of course, the most important propertiesof the nt involve the electric field and potential in the system. Ingeneral, to find the electromagnetic field in a medium, one has tosolve the basic equation

V • E = 4TT(P + Pext) (5)

for the statistically averaged electric field E. Here, pext(r) is thecharge density of external field sources and p(r) is the averageddensity of induced charge due to polarization of the medium.24 Thecharge density p is £ ntqt9 the sum extending over species of particlesbelonging to both phases; there is no separation in this generalformalism. One requires, in addition to Eq. (5), a phenomenologicalrelationship between p and the electrical potential </>, where E =—V(f>. Alternatively, one can deal with the polarization field P(r),where p = -V • P, and assume a dependence of P on E: P, like p,involves contributions of particles from both phases. The relationbetween p and <f> is often taken as linear and local, so that p atany point is proportional to the electrostatic potential at that point(although the constant of proportionality may vary from point topoint). However, it may be necessary to go to a nonlinear relation-ship, to take into consideration spatial correlations between chargedensity fluctuations. Then p(r) is given, in the linear approximation,by Jdr' a(r, r')</>(r')- If <Mr) is more slowly varying than the polariza-bility function a(r, r'), one recovers the local limit.


The Poisson equation in electrostatic units for planar symmetrymay be written

dD/dz = 4irp{z) = 4TT £ nl(z)ql

where D = E + 4TTP, p is the free-charge density, and the polariz-ation P = Xi n«(z)Mi(z)> with /A, the average z-component of thedipole moment of a molecule of species i, which may depend onz. Since the z-component of the electric field E is -dV/dz,

-d2V/dz2 = 4TT I nl(z)ql - AirdP/dzi

Integrating twice to get the potential drop across the interface, wehave

V(oo) - V(-oo) = -4TT dz dz'iz nxqx - dP/dz'JJ - c o J —oo \ i /

= 4TT | <fc(zE GII,+ !/*,».) (6)J -co \ i I /

We have used the electroneutrality of bulk phases [X ^i«i(±0°) = 0]and the fact that there is no electrical polarization in bulk phases.

The two terms in the potential drop are the dipole momentsof free charges and the permanent dipole moments. Each may bedivided into contributions of components of the two phases.Equation (6) may be rewritten in terms of the surface chargedensities of Eqs. (3) and (4):

V(oo) - V(-oo) = 47r(qAzA + qBzB) + 4TT | dz(£ /*,*, + £ ^n

Here, the centroids of the charge distributions of phases A and Bhave been introduced:

The term 4rr(qAzA+ qBzB) is a free-charge term, resembling thatof an ideal capacitor of surface charge density qA = -qB andinterplanar spacing zA — zB.

12 Jerry Goodisman

In another notation, and supposing the metal appears forz -> -oo and the electrolyte for z -> oo, the potential difference acrossthe interface may be written as

A V = V(-oo) - V(oo) = A£V

= g^(ion) - gs(dip) + gM(dip) (7)

where g^(ion) is the free-charge contribution, vanishing for qM =qs = 0, and the other terms are dipolar contributions due to com-ponents of solution and metal. In the free-charge term, some fixedposition z° in, say, phase M may be substituted for zM. Then thedifference

47rqM(zM - z°) = 4TT [ dz (z - z°) £ *,(z)J —oo i

is assigned to the dipolar term for metal M.It is usually assumed that the components of a metal are ions

(with tightly bound charge) and electrons, so that there are nopolarizable species in the metal phase. The contribution of themetal to the potential drop across the interface is then

iAt the point of zero charge, the metal is neutral:

M

dz I q,n,(z) = 0J:but this does not make A VM zero if the positive and negative species(ions and electrons) are distributed differently in space. One knowsthat the electrons spill over from a metal surface, corresponding toa surface dipole or double layer with its negative and toward theoutside of the metal. This is responsible for the surface potentialof a metal, which is several volts in size, and makes an importantcontribution to the work function. Although the presence of anelectrolyte phase on the outside of the metal will certainly have aneffect on the distribution of the metal's electrons, one should notexpect the large surface dipole and large potential difference todisappear when the metal is in an interface.


It may be noted that for a polycrystalline surface of a solidmetal electrode, there is a complication in defining the potential ofzero charge because one does not have zero charge density on allcrystal faces simultaneously. The work function is different fordifferent crystal faces but, if the grains are in contact, theelectrochemical potential of the electron has a single value. Thus,the Volta or outer potential is different for different faces, sodifferent surface charge densities are present. An average workfunction can be defined simply, but the definition of an averageEpzc is not as easy.2 Similarly, the surface structure of a real solidmetal makes the electronic structure more complicated than for aliquid metal or for a single-crystal face. Certainly, the use of densityprofiles depending only on distance perpendicular to the interfaceis an oversimplification.

The reciprocal of the capacitance, C"1, is the change of poten-tial drop with the surface charge qM. According to Eq. (7), C"1 isa sum of three contributions, each an inverse capacitance, corre-sponding to three capacitances in series. Like the charge densityand A V, C"1 may be divided into metal and solution contributions.However, if the electronic distribution of the metal does not changerelative to the metal's ionic distribution as the interface is charged,A VM will be independent of qM, and the metal will not contributeto the capacitance of the interface. This is the case for an idealmetal, in which, furthermore, there is no penetration of electricfield so that A VM = 0. Of course, the metal may contributeindirectly, since the interaction between the metal and the solutioncomponents may affect the ability of the latter to reorient as theelectric field is changed.

Differentiating Eq. (7), the inverse capacitance is

= [C(ion)]-1 - [C^dip)]-1 + [CM(dip)]~1 (8)In the notation of Eqs. (1) and (2), A V = *M(S) - *S(M), so that

E = A V + O H g / F - / 8 - £ f e

= <DHg/F + 8x£g - gs(dip) + g^(ion) - Ek

with Ek the contribution of the reversible reference electrode,A^V - F"V^ a . Differentiating,

dE/dqM = d8X¥/dqM - dgs(dip)/dqM + dg?(ion)/dqM

14 Jerry Goodisman

Then 1/C = 1/CM - 1/CS + 1/Cion, the terms arising from themetal electronic polarizability, the reorientation of solvent dipoles,and the electric field. C s is sometimes written as Cdip.

The electric field or ionic term corresponds to an ideal parallel-plate capacitor, with potential drop g^(ion) = qMd/47re. It includesa contribution from the polarizability of the electrolyte, since thedielectric constant is included in the expression. The distance dbetween the layers of charge is often taken to be from the outerHelmholtz plane (distance of closest approach of ions in solutionto the metal in the absence of specific adsorption) to the positionof the image charge in the metal; a model for the metal is requiredto define this position properly. The capacitance per unit area ofthe ideal capacitor is a constant, e/Aird^ often written as Kion. Thecontribution to 1/C is 1/Kxon\ this term is much less important inthe sum (larger capacitance) than the other two contributions.2

2. Experimental Results

According to Eq. (2), the work function for a metal M is relatedto the potential of zero charge for the M/aqueous solution interfaceby

where the constant Ek involves quantities relevant to the referenceelectrode only. The relation between the potential of zero chargeand the work function has been discussed in detail by Trasitti,10'23'25

for nonaqueous as well as aqueous solvents. If experimental valuesfor 3>M are plotted versus Epzc for the same reference electrode, thepoints for different metals fall mostly along two lines.10'25 The slopeof the line for the sp metals differs noticeably from unity, indicatingthat ~xs(M) + fys1 is not independent of M. The point for Ga isfar off this line; it is in fact close to the line for the d metals, whichhas a slope close to unity. This means that - ^ S(M) + &x™ ls aboutthe same for all d metals. At a pzc corresponding to Hg, the linefor the d metals is higher than the Hg work function by about0.25 V. Thus, for a d metal and an sp metal having the same workfunction, Epzc is significantly higher for the latter, i.e., the quantity

g (M) j g m o r e n e g a t i v e for ^ e d metal.


An interpretation of these facts is that the orientation of waterdipoles is essentially complete on all d metals, with the negativeends of the dipoles pointed toward the metal, giving ^S(M) the samepositive value for all M (and that 8%™ is the same as well). Theslope of the line for the sp metals implies that -/YsM) + fy™ becomesmore negative in the order {Hg Sn Bi Pb In Cd}, which could reflectincreasing ordering or strength of adsorption of water on the metal.Thus, one is disregarding the difference between 8x™ from onemetal to another.3 The direct effect of the metal's components,through #Ys\ cannot be separated experimentally from the indirectone, adsorption or orientation forces on the solvent dipoles by themetal; only by theoretical reasoning or calculations can one judgewhich is dominant.3'26 Thus, one can measure Sx^o ~ 8XH2°

a s

-0.25 V and estimate from adsorption data and a reasonableassumption26 that 8x"f =* -0.06 V so that S*Sfo =*-0.31 V.[Gratifyingly (see below), various theories for the metal in theinterface have found values close to this one for this quantity.]From a number of experiments, it is suggested that xHl° is between0.08 and 0.13 V, which means that gH2°(dip) is between 0.02 and0.07 V on a mercury electrode at the point of zero charge (oxygenend of the water molecules toward the metal).

For transition or d metals, the value of the quantity X =SYs1 ~ gs(dip) or 8x™ - *S(M) at the point of zero charge is largeand negative and about the same (—0.67 ± 0.07 V) for all the metalsfor which data are available.26 At the same time, the heat offormation of the metal oxide, a measure of the strength of thephysisorption of water on the metal in the interface, varies from300 to 900 kJ/mol. For sp metals, the heat of formation of the oxideis 300 kJ/mol or less, while the value of X for Hg is estimated as+0.06 V. The interpretation of the results for the d metals is thatthe interaction between metal and solvent is strong enough in allcases to completely orient the water molecules, giving gs(dip) itsmaximum value, while 8x™ does not change much. It is less likely,though possible, that 8x™ and gs(dip) both vary from metal tometal, but in a compensating way, so as to keep their differenceconstant. A strong metal dependence of 8%™ would require areconsideration of the hydrophilicity scale for metals.3

For the sd metals (Au, Ag, and Cu) the situation is morecomplicated. Higher X is associated with lower heat of oxide

16 Jerry Goodisman

formation. A number of explanations for this can be given,26 includ-ing values of 8%™ exceeding gs(dip) on structured surfaces, andprotruding d orbitals at the positions of strong interaction withwater. The data on the effect of crystal face on X, which would beuseful to resolve some ambiguities, have been questioned.26

The experimental data bearing on the question of the effect ofdifferent metals and different crystal orientations on the propertiesof the metal-electrolyte interface have been discussed by Hamelinet al.27 The results of capacitance measurements for seven sp metals(Ag, Au, Cu, Zn, Pb, Sn, and Bi) in aqueous electrolytes arereviewed. The potential of zero charge is derived from the maximumof the capacitance. Subtracting the diffuse-layer capacitance, onederives the inner-layer capacitance, which, when plotted againstsurface charge, shows a maximum close to qM = 0. This maximum,which is almost independent of crystal orientation, is explained interms of the reorientation of water molecules adjacent to the metalsurface. Interaction of different faces of metal with water, ions, andorganic molecules inside the outer Helmholtz plane are discussed,as well as adsorption.

Since the surface dipole potential of a metal ^M is due tospillover of electrons from the metal toward the outside (electrolytephase in the interface), the positive end of the dipole always pointstoward the interior of the metal. The electron density must alsoshow a lateral smoothing as compared to the atomically distributedpositive charge density. This decreases the surface dipole potentialbecause the ion cores protrude into the electronic tail. The rougherthe crystal face, the smaller is the magnitude of %M- Because ofthis, the electronic work function of a metal, which is e\M — fi™(/JL™ = chemical potential of electrons in the metal), differs fordifferent crystal faces. The rougher faces have lower values of ^M

and hence lower work functions, whereas the faces with highernumber densities of ions (which have smoother positive chargedensities in the directions parallel to the interface) have higherwork functions.28

For the metal in the interface, the surface potential of theelectrolyte phase is nearly the same for all crystal orientations.29

Therefore, referring to Eq. (2), the potential of zero charge varieswith the surface potential or the work funtion and is larger for themost densely packed faces. Correspondingly, atomic irregularities


lower the potential of zero charge. Trasatti gives a table (Table 2in Ref. 26) which shows the experimental results. There is also aparallelism between the potential of zero charge and the calculatedsurface energy, which is larger for more densely packed faces.Because the difference of pzc's should be the same as the differenceof electronic work functions in going from one crystal orientationto another, the equilibrium or Nernst potential of an electrochemicalreaction, when there is no adsorption, is independent of crystalorientation.28

It may be noted that the statement made above—that thesurface potential in the electrolyte phase does not depend on theorientation of the crystal face—is necessarily an assumption, as isthe neglect of 8x™. It is another example of separation of metaland electrolyte contributions to a property of the interface, whichcan only be done theoretically. In fact, a recent article29 has dis-cussed the influence of the atomic structure of the metal surfacefor solid metals on the water dipoles of the compact layer. Differentcrystal faces can allow different degrees of interpenetration ofspecies of the electrolyte and the metal surface layer. Nonunifor-mities in the directions parallel to the surface may be reflected inthe results of capacitance measurements, as well as opticalmeasurements.

The capacitances of electrode-solution interfaces vary with themetal, for sp metals, in the same order as their attraction for oxygen(the "hydrophilicity," measured, for example, by the heat of forma-tion of the metal oxide). This is true for nonaqueous solvents aswell as for water, although a change in the solvent changes the sizeof the capacitance. There are few experimental results for capacit-ances of interfaces involving d metals. One interpretation26 is that,for those metals for which the water molecules are strongly orientedby their interaction with the surface, the effect of the electric fieldis to give a large modification of their orientation. One may alsoreason that if the molecules interact strongly with the surface, alarge electric field (large change in the surface charge) is requiredto change their orientation, which gives the solvent contribution tothe potential drop across the interface.

Calculations, which we shall discuss later, show3032 that thedirect contribution of the metal to the interfacial capacitance (fromthe conduction electrons) increases with the electron density, in

18 Jerry Goodisman

the same way as the experimental capacitance. Indeed, the measuredcapacitances for different metal-aqueous solution interfaces can be"calculated" by combining calculated capacitances for the metalwith the same value for the solution contribution in all cases. Themodel used in these calculations seems more reasonable for liquidmetals than for polycrystalline solid metals. Furthermore, thesecalculations (see below) treat the electrolyte phase very crudelyand probably overestimate the importance of the metal in theinterface.26 This is shown by consideration of the changes in capacit-ances when the solvent is changed, some of which are easilyexplained in terms of the properties of the solvent molecules.

Kalyuzhnaya et al33 argued that the reason for the highercapacitances of In and In-Ga alloy as compared to Ga and Hg inthe same solution was due to the different electron density distribu-tions in the surface layer, rather than to adsorption of ions. Inacetonitrile, for which the separation of the plates of the ideal ioniccapacitor is larger than for water, the difference in capacitances issmaller. As we will see later, the penetration depth of the electricfield into the metal, and hence the effective dimension of thecapacitor corresponding to g(ion), should depend on the metal'selectron density. However, from analysis of capacitances of variousinterfaces with different metals, it can be inferred9 that the capacit-ance associated with g™(ion) is independent of metal, so thatg^(ion) itself is independent of metal. Capacitance curves fordifferent metals in the same solution often coincide at large negativecharge densities,34 so that a large effect of metal on the capacitanceis unlikely.

From capacitance-potential curves for various aqueous elec-trolytes on the (110) crystal face of silver, Valette35 derived theadsorption for F~, CIO4, PF^, and BF4. The order of adsorptionstrengths was inverse to that found for mercury, and a possibleexplanation is in terms of the alteration of the potential of zerocharge on the surface defects where the adsorption takes place.Furthermore, the inner-layer capacitance at negative potentials,assumed due to oriented water molecules, was found to have avalue almost twice as high as what is found for other metals; this,too, was explained by a model of atomic-scale surface topography.If this is important, one cannot necessarily use capacitances at thepoint of zero charge to classify the strengths of metal-solvent


interactions. Further, a correct model of the interface mustnecessarily describe the components of both the metal and theelectrolyte phase.

Information about the metal in the electrochemical interfacecan also be obtained36 from measurements on metal surfaces. Thus,information on the potential drop across the metal-vacuum inter-face is obtained from measurements of the work function (see nextsection), which are normally done in vacuum. To simulate thesolution side of the interface, one can adsorb H2O and ions, andstill make measurements in high vacuum. There are problems withmeasuring surface charge, however, and with controlling the poten-tial drop. The latter must be done by adsorbing electropositive orelectronegative species. For example, adsorption of alkalis producesa positively charged adsorbate and a negative charge on the metal,thus behaving like a cathodic potential and decreasing AV. Thisleads to a decrease in the work function of several volts, whenseveral monolayers are adsorbed.

The change in work function as water was adsorbed on aparticular crystal face of Ag or Cu was measured36; the workfunction decreases with increasing water adsorption. From theinitial slope of the curves of work function versus amount adsorbed,one can deduce a dipole moment of 0.9 D for H2O on Cu(110) and0.6 D for H2O on Ag(110), the negative (oxygen) end toward themetal. The work function became independent of water adsorptionafter about two monolayers. Adding other high-vacuum techniquesto work function measurements, one can find out a lot about theway water is adsorbed on these surfaces. However, one still cannotexperimentally distinguish the dipole orientation contribution tothe change in surface potential from the effects of charge transferand intermolecular interactions. If the adsorbed water has an effecton the conduction-electron tail, the resulting contribution to thechange in surface potential can likewise not be detected andseparated out.

From measurements of the rate of change in work functionwith adsorption of Cs on Ag, one can infer a surface dipole of8.3 D. Assuming the Cs is completely ionized, this gives the distancebetween the positive charge of Cs+ and its image charge in themetal as d = 1.73 A. From this, one can calculate the differentialcapacitance (Cdiff = dqM/dkV) of the metal surface according to

20 Jerry Good ism an

the ideal-capacitor formula, as l/Aird = 4.6 x 106 cm/cm2 or5.1 /xF/cm2. Similarly, from the decrease in work function with Bradsorption one can infer a dipole moment of 0.4 D. If Br is assumedto ionize completely, this means a distance d of 0.08 A and adifferential capacitance of 110/xF/cm2.

III. METAL STRUCTURE

1. Electrons in Metals

We begin with a presentation of the ideas of the electronic structureof metals. A liquid or solid metal of course consists of positivelycharged nuclei and electrons. However, since most of the electronsare tightly bound to individual nuclei, one can treat a system ofpositive ions or ion cores (nuclei plus "core" electrons) and freeelectrons, bound to the metal as a whole. In a simple metal, theelectrons of the latter type, which are treated explicitly, are theconduction electrons, whose parentage is the valence electrons ofthe metal atoms; all others are considered as part of the cores. Insome metals, such as the transition elements, the distinction betweencore and conduction electrons is not as sharp.

If there are N metal atoms, each of which can easily lose zelectrons to form an ion Mz+, one would like to find the energyand wave function of a system of zN electrons interacting with Nions and with each other. The total energy is the electronic energyplus the interionic repulsion; it depends on the positions of theions, taken as fixed. From knowledge of the total energy as afunction of ionic configuration, one could calculate the equilibriumarrangement of the ions (crystal structure for a solid metal) andthe force constants associated with displacement of ions from theirequilibrium positions (crystal vibrations). The electron densitywhich enters the equations of electrostatics is obtained by averagingover ionic configurations of low energy.

Since the equilibrium ionic configuration is determined by theenergy of the conduction electrons, it is not surprising that ions ata metal surface, which are in a different electronic environmentfrom ions in the bulk and hence experience different forces, arearranged somewhat differently from ions in the bulk metal. For


some metals, including gold and platinum, the two-dimensionalarrangement of the outermost layer of ions differs from that oflayers beneath it, as revealed37 by low-energy electron diffraction(LEED) from the surface. For others, LEED shows that the spacingbetween the outermost layer and the layer beneath it is as much as10% smaller than interlayer spacings in bulk. The falloff of theelectronic charge density from the uniform bulk value, discussedbelow, produces forces on a surface ion which cause its displace-ment to a position for which the electronic forces are compensatedby forces due to other ions. This, in turn, leads to distortion in thestructure of several layers, not only the outermost, so that sub-sequent interlayer spacings also differ from their bulk values. Inbinary alloys, the same mechanism produces segregation (composi-tions differing from bulk composition) in surface and near-surfacelayers and different displacements of ions of the two components.

One can expect that the electron density corresponding to theelectronic state of lowest energy is roughly constant in the interiorof the metal and decreases to zero outside the metal. This meansthat the potential seen by an electron, due to the ion cores and theother electrons, is roughly constant inside the metal, with a valuesignificantly lower than the potential outside. The simplest modelfor electrons in a metal, the Sommerfeld38 model, takes this potentialas -VQ inside and 0 outside. One is then led to consider theone-dimensional Schrodinger equation

(-fi2/2m)(d2il//dx2)+ Vi/t

Vo forx<0

There are acceptable solutions to this equation for all e > - Vo. Bycombining the electron densities for all solutions with e between-V o and some upper limit el9 one gets the electron density p(x),which approaches a constant, the bulk electron density pb, forx -» —oo, and which approaches zero for x -» oo (if Vo is large enoughso that ex < 0). The difference in energy between the highest filledelectronic state and the lowest, ex+ Vo, is called the Fermi energy(although other definitions of the term are sometimes used).

The solution to the Schrodinger equation (9) is a plane wave,e±lkx, for x < 0 and a decreasing exponential, e~Kx, for x > 0.

22 Jerry Goodisman

Substituting into the Schrodinger equation, we find (fi2k2/2m) -Vo = e and —h2K2/2m = e. A suitable combination of elkx ande~lkx must be taken so that the value and slope of the wave functionare continuous at x = 0 with e~Kx. The electron density is obtainedby adding |i/ |2 for all states occupied by electrons. Taking eachstate as either occupied (if e < et) or empty (if e > ex) correspondsto the Fermi distribution (Fermi-Dirac statistics) for absolute tem-perature T = 0. If we suppose that Vo -» oo, e is large and negativeand K large and positive, so that e~Kx — 0 and the continuityconditions on the wave function are replaced by if/ = 0 at x = 0.

For the infinite-barrier problem, then, if/ = 0 for x > 0 andifj = A sin kx for x < 0, where A is a normalization constant. Thenormalization may be determined by considering the metal as aone-dimensional box of length L and later letting L -> oo. Then theacceptable wave functions are

<t> = (2/ L)1/2 sin nirx/L

($ = 0 for x = 0 and for x = L), with the familiar particle-in-a-boxenergies

e + y0 = n2h27T2/2mL2 = n2h2/SmL2

Because of electron spin, each state can hold two electrons.Adding together contributions for all n from 1 to nl —(8mL2//i2)1/2(ei+ Vo)1/2, we should obtain the total number ofelectrons:

Nz = l 2 = 2(SmL2/h2)1/2(£l + V0)1/2

i

The Fermi energy eF — ex + Vo is then

EF = (Nz/L)2(h2/32m)

which is proportional to the square of the one-dimensional electrondensity Nz/ L.

In three dimensions, the normalized eigenfunctions may betaken as

$ = (2/L)3/2(sin nx7rx/L)(sin ny7ry/L)(sin nz7rz/L)

(cubical box of volume L3) and the energies are - Vo + n2h2/SmL2

where n2 = n2x + nl + n2

z. All states with n2h2/SmL2 < ex -f Vo are


occupied with two electrons. The sum over states may be approxi-mated by an integral, since the energies are very closely spaced:

Nz = \ dnx\ dny dnz(2) = 2(4TT/8) * n2 dn (10)

We have gone from Cartesian to spherical coordinates in the lastmember, as a convenience in imposing the limit on the integrations,and carried out the integration over angles, giving the factor of 4TT;the division by 8 is because only positive values of nx, ny, and nz

are to be considered. Now we have

Nz = 7T(ni)3/3 = (ir/3)[SmL2(el + V0)/h

2]3/2

The average electron density is now pb = Nz/ L3, so that 3pb/ir =(SmeF/h2)3/2 or

sF = (h2/Sm)(3Pb/7r)2/3 = (ti2/2m)(37r2pb)2/3 (11)

Usually,39 periodic boundary conditions on the electronic wavefunctions are imposed instead of requiring that they vanish on theboundary; the result of Eq. (11) is unchanged.

In addition to calculating the total number of electrons andaverage electron density pb for a large system, we may calculatethe variation of electron density with distance near the metal surface(electron density profile) using this model by summing over theelectron densities of the occupied wave functions. We suppose thatthe compensating positive charge density due to the ion cores isgiven by

*<•>-{.'pb z<0

The position of the infinite barrier at which the electronic eigenf unc-tions vanish is given by z = zw. The value of zw will be determinedto make the system electrically neutral overall. The wave functionsfor electrons at an infinite barrier are products of imaginaryexponentials (free-particle wave functions) in the x- and y-direc-tions, and sine functions vanishing at zw in the z-direction. Theelectron density «_(z) is obtained by adding together the squaresof the electronic wave functions, starting with zero energy and

24 Jerry Goodisman

including enough to give the density of Eq. (11) for z -» -oo (bulk).Then, for z < zW9

f 3cos[2Mz-%)] 3 sin[2kF(z - zw)]}W_(Z) = pb\ 1 + — 2 - -j — 3 - -3 \ (12)

I AkF(z-zw)2 %kF(z-zwf Jwhere h2k2

F/2m = eF so kF = 3ir2pb. For electrical neutrality, sincep+(z) = pb for z < 0, we have

3 cos[2Mz ~ Zw)] 3 sin[2kF(z - zw)]}4k2

F(z - zwf 8/c3F(z - zwf JPb\ dz\\+

where L -* oo, or

r (3/2fcF)

- (3/4*

fJ-oc

cos x - x 3 sin x)

Jxx^s inx

where x = 2kF(z - zw). This gives

zw = 37r/SkF (13)

to guarantee electroneutrality.Note that the interactions between electrons are not taken into

account in this calculation, which corresponds to a single-particlemodel, each electron interacting with the average field. One couldgo on to calculate the response to an external potential U(j) in thesame approximation. If U were spatially homogeneous, each elec-tronic energy would be changed by the same constant. Thus, thechange in the electron density due to U would be the same as thechange due to a shift in the Fermi level, and the linear response toa spatially homogeneous potential is obtained40 by differentiatingthe electron density with respect to the Fermi level, obtaining8p = (UkF/7r2a0) (1 -s inx/x) .

By integrating the Poisson equation

d2V/dz2 = -

from —00 to +00 we find

f°(-00) - 4TT

V(oo) - V(-oo) - 4TT dz z(p+ - n_)


We can now calculate a surface potential as follows:

V(oo) - V(-oo) = 4TT dz zpb - 4TT I dz zn_(z)J-L J-L

-\ZWzdz-(3/4k2F)\° dx(x + 2kFzw)

Jo J -oo

x [x~2 COS(JC) - x~~3 sin(x)] |

z^/ 3 3 z wkf/7T I

o Ak Rk I

which, on substituting for pb and zw, is

V(oo) - V(-oo) = (/cF/7r)[(37r2/32) - 1] (14)

The surface potential of the metal,

X = V(-oo) - V(oo)

is thus positive. For example, if pb = 0.01 e/(a0)3 and kF =

O.67(ao)~1, x is 0.016 esu = 4.8 V according to this model.

The surface potential comes about because of the spillovertoward the outside of the metal of the conduction electron wavefunctions, which are delocalized over the metal. As we will seebelow, this result follows from more sophisticated models for themetal surface as well. The core electrons, which are localized, havenot been considered explicitly. In fact, the electronic energy levelsin a crystal form bands. That is, for certain ranges of energy thereare continua of energy eigenvalues (bands), while for others, theenergy gaps, there are no allowed electronic energy eigenvalues. Acrystal acts as a metal if there is a partly rilled band, so thatlow-energy excitations of electrons in the band are possible. Thecontinuum of free-particle energy levels we have been discussingis supposed to represent the highest-energy band, with all bandsof lower energy filled.

2. Band Structure

To have free-electron energy levels, the potential in which theelectrons move must be constant (V = — Voh in a crystalline metalthere are ion cores arranged in a regular array or lattice, which


makes the potential in which the electrons move periodic. The effectof such a potential may be understood as diffraction of the electronwaves by the lattice. In a one-dimensional crystal the free-electronwave functions are elkx with energy fi2k2/2m, where k = 2nir/Lwith L the size of the crystal and n an integer. (This correspondsto periodic boundary conditions; box normalization, as above, hassin kx with kL = nir and n non-negative.) The energy is quadraticin k. The Bragg equation for reflection by a lattice of a waveperpendicular to the layers is 2d = mk where d is the lattice spacing,m is an integer, and the electron wavelength is given by the deBroglie relationship

A = h/p = h/kfi = 2ir/k

Thus, the free-electron wave functions will be seriously disturbedwhen d = mrr/k. The corresponding values of /c,

k = m7T/d (15)

give the positions of the breaks in the energy versus k plot, i.e., thegaps in the band structure.

For these values of k the functions e±lkx for k = rmr/d arereplaced by the linear combinations, sin kx and cos kx. It is easyto see that the potential of the ion cores makes the energies of thesetwo waves quite different, since one corresponds to zero probabilitydensity at the positions of the ion cores and the other to a maximumprobability density at these positions. The former will have thehigher energy since the positive ion cores represent an attractivepotential for the electrons. The difference in energies of the twofunctions is the band gap. For a perturbation theory treatment ofthe effect of the ion cores, see Appendix A of Kittel39 or Ziman.41

The result is that the energy as a function of k follows the parabolaE = h2k2/2m except near k = rrnr/d, where there are distortionssuch that the energy for k just above rmr/d differs by a finiteamount (the band gap) from the energy for k just below mir/d.

The origin of allowed bands and band gaps may also be under-stood in terms of the electronic energy levels of neutral atoms whichare allowed to come together so that their charge distributionsoverlap (tight-binding approximation). The overlap of valence-shellatomic orbitals leads to binding and antibinding molecular orbitals


with energies above and below the energies of the atomic orbitals.For N atoms, each contributing one valence orbital, there will beN molecular orbitals of different energies. The spread in energy,since it comes from the overlap, increases as the atoms get closertogether. Such a band of energy levels will result from each atomicorbital energy level in the separated atoms. Thus, for a collectionof lithium atoms, there will be a Is band, a 2s band, 2p bands, etc.The Is band will be lower in energy than the 2s band and the 2sband lower in energy than the 2p band, but if the interatomicdistances decrease enough, nothing will prevent the bands fromoverlapping: the band gaps necessarily decrease as the widths ofthe bands increase. For a given interatomic distance, the width ofthe Is band will be less than that of the 2s band because Is orbitalson different atoms, being more tightly bound, overlap less than 2sorbitals. It is clear that a regular array of atoms, such as is foundin a crystalline solid, is not necessary for the formation of bandsin the tight-binding picture; one expects band structure for liquidmetals as well.

If, in each atom, the atomic orbital giving rise to a band isfilled with two electrons, there will be 2N electrons in the Nmolecular orbitals of the band formed from N atoms. Then theband will be rilled, provided there is no overlap with other bands(e.g., the Is band for Li). These electrons will not respond to anapplied electric field since it will require some minimum energy(the band gap) for them to change their state, so they do notcontribute to conductivity. If there is only one electron per atomin an atom orbital, as for the 2s orbital of Li, the band will behalf-filled (N electrons in the 2N states of the Li 2s band). Anelectric field will induce a current in these bands, since electronsat the Fermi level can gain momentum (change k) by going fromone state of the band to another with little cost in energy. A solidwill be an insulator if all bands are either completely filled orcompletely empty, and if the band gap between the highest filledand lowest empty bands is much greater than kT. Thus solidhydrogen is an insulator. The alkaline earth metals are not, becauseof the overlap in energy of the valence s band and the valence pbands.

The band structure (widths of bands, energy gaps) willobviously depend on the arrangement of the atoms as well as on

28 Jerry Goodisman

their nature. For the one-dimensional crystal lattice, the positionsof the gaps depend on the interionic spacing d according to k =mn/d in the free-electron picture, but the size of the gap, ascalculated by perturbation theory, depends on the ion-core poten-tial. The ranges of k corresponding to allowed energies are calledBrillouin zones. Thus, the first Brillouin zone in the one-dimensionalcrystal extends from k = —rr/d to k = w/d. The second Brillouinzone includes two ranges of k corresponding to the same energies:-2TT/ d < k < -jr/d and ir/d < k < 2TT/d. In three dimensions, aplane wave with energy E = fi2k2/2m has wave function exp(/k • r).The values of the vector k corresponding to the boundaries ofBrillouin zones are found from

2k-g + 277g2 = 0 (16)

where g is a reciprocal lattice vector (g = \/d for a one-dimensionallattice). The orientations of the planes defined by Eq. (16), whichbound the Brillouin zones, and hence the shape of the zones, dependon the crystal structure through the reciprocal lattice vectors.

For instance, the first Brillouin zone of the simple cubic latticeis a cube of edge length 2v/d, where d is the interatomic distance.The volume of the zone in k-space is {2IT/ d)2. According to Eq.(10), the number of states, including electron spin, with k < kx

(kx = WJTT/L) is (7r/3)(/c1L/7r)3. Dividing by the volume in k-space,4TT/C3/3, and by the volume of the crystal, L3, we find that thenumber of states per unit volume in k-space per unit volume ofcrystal is 2/(2TT)3. Thus, the number of states in the first zone ofthe crystal per unit volume of crystal is 2/d3, which correspondsto two states for each atom. The first zone for the body-centeredcubic structure is a rhombic dodecahedron, and the first zone forthe face-centered cubic structure is a truncated octahedron.39 Thereare two states per atom in the first zone in each case. To the extentthat the energy levels are like those for free particles, the constant-energy surfaces in k-space are spheres. This obviously does nothold near zone boundaries. The Fermi surface is the boundary, ink-space, between filled and unfilled levels. Being a constant-energysurface, the Fermi surface will be spherical as long as it is not toonear the zone boundaries.

For a periodic lattice, it can be shown (Bloch theorem) thatthe solutions to the one-electron Schrodinger equation are of the


form

* = uk(r)elkr (17)

where uk(r) is periodic with the same period as the potential of thelattice, so that a translation of r by one of the lattice vectors v justmultiplies iff by elk v. The function uk depends on k. If k is outsidethe first Brillouin zone, one may write the wave function in Eq.(17) as

\\f = uk(r) exp(27ng • r) exp[/(k - lirg) • r]

= «!'(r) elk r

Here u' = u exp(27n'g • r) is, like w, periodic with the period of thelattice, and k' = k — lirg is a reduced wave vector. Repeating thisas necessary, one may reduce k' to a vector in the first Brillouinzone. In this "reduced zone scheme," each wave function is writtenas a periodic function multiplied by elk r with k a vector in the firstzone; the periodic function has to be indexed, say ujk(r), to distin-guish different families of wave functions as well as the k value.The index j could correspond to the atomic orbital if a tight-bindingscheme is used to describe the crystal wave functions.

The conduction band normally corresponds to the valenceelectrons and is the highest-energy band containing electrons. Asmentioned above, for alkali metals this band is an s-band and ishalf-filled, since each atom contributes one valence orbital and oneelectron. The crystal structure is body-centered cubic; with oneelectron per atom, the Fermi surface should be quite spherical. Thedivalent metals, including the alkaline earths, exhibit hexagonalclosest-packing, face-centered and body-centered cubic, and otherstructures. With two electrons in the outermost s orbital of eachatom, one would have a filled band of highest energy in the solid,and insulator behavior, were it not for the mixing of the s- and/7-bands. For beryllium, there is less mixing than for the otherelements, so that the conductivity is small.

The overlap between s- and /7-bands also occurs for the alkalimetals and for the monovalent noble metals copper, silver, andgold, which have face-centered cubic structures. The noble metalsdiffer from the alkalis because of the filled d-shell just below the5-shell in energy; the d-band and the s-band overlap in the solid.

30 Jerry Goodisman

Furthermore, the electrons of the d- shells, through the exclusionprinciple, produce added repulsions between atoms, giving thesemetals a much lower cohesive energy and a much lower compressi-bility than the alkalis. The shape of the Fermi surface is complicated,and it is thought to touch the zone boundary. The transition elementshave partly rilled d-shells, and there is significant overlap betweenthe d-band and the s-band just above it in energy. The state density(number of one-electron states per unit energy range) may be quitea complicated function of energy.

3. Ion-Electron Interactions

The very use of the energy-band description implies that eachelectron is in a one-electron state, whose wave function is thesolution of a one-electron Schrodinger equation like Eq. (9), exceptwith a more complicated potential V. V must represent the effectof the positively charged nuclei, arranged at the points of a latticeor according to some distribution, and of the electrons other thanthe one considered in the Schrodinger equation. The potential Vis then a self-consistent potential of Hartree-Fock theory; it dependson all the electronic wave functions other than the one beingdetermined in the one-electron Schrodinger equation and is deter-mined along with the wave functions as part of an iterative pro-cedure. The electronic contribution to V includes the coulombpotential of the electronic cloud and an operator representingexchange.

For the conduction electrons, it is reasonable to consider thatthe inner-shell electrons are all localized on individual nuclei, inwave functions very much like those they occupy in the free atoms.The potential V should then include the potential due to thepositively charged ions, each consisting of a nucleus plus filledinner shells of electrons, and the self-consistent potential (coulombplus exchange) of the conduction electrons. However, the potentialof an ion core must include the effect of exchange or antisymmetrywith the inner-shell or core electrons, which means that the conduc-tion-band wave functions must be orthogonal to the core-electronwave functions. This is the basis of the orthogonalized-plane-wavemethod, which has been successfully used to calculate band struc-tures for many metals.41


Formally, each orthogonalized-plane-wave basis function maybe written as (I — P)t/jk, where if/k is a plane wave and P is theprojection operator such that Pif/k gives the core-state componentof i//k:

tJ \ <A?A d r

with ifjtJ the wave function of the jth core state on ion t. Theexpansion in orthogonalized plane waves converges rapidly, so thatthe wave function £ ck(\ - P)if/k has contributions only from smallk and <j> = £ ckij/k is a smooth function. Note that </>, called thepseudo wave function, differs from the true wave function (1 - P)(f>only in the core regions. Inserting (1 - P)</> into the Schrodingerequation, one can derive 42 an equation for cf>:

- (h2/2m)V2cf) + W(/> = Ecf> (18)

where W is called a pseudopotential. W includes the true potentialV plus terms involving the projection operator P, making W anonlocal operator. Since the effect of P is to introduce oscillationsinto a smooth wave function, it raises the kinetic energy. Thiscancels off part of the strongly attractive potential V, making W amuch less attractive potential. W may be small enough for its effectto be considered by perturbation theory, which is not the case forV itself.

Although the pseudopotential is, from its definition, a nonlocaloperator, it is often represented approximately as a multiplicativepotential. Parameters in some chosen functional form for this poten-tial are chosen so that calculations of some physical properties,using this potential, give results agreeing with experiment. It isoften the case that many properties can be calculated correctly withthe same potential.43 One of the simplest forms for an atomic modeleffective potential is that of Ashcroft44: r~l6(r - Rc), where theparameter is the "core radius" Rc and 6 is a step-function.

One now has a picture of conduction electrons in the potentialof the ions, which is really a collection of pseudopotentials. Theenergy of the electronic system obviously depends on the positionsof the ions. From the electronic energy as a function of ionicpositions, say Uel(R), one could determine the equilibrium ionicconfiguration (interionic spacing in a crystal or ion density profile

32 Jerry Goodisman

in a liquid). From C/ei(R) one could also calculate the force exertedby the electrons on each ion. It is clear that the effective interionicforce in a metal differs from the force between ions in a vacuum,because a displacement of one ion produces a change in the elec-tronic wave functions and hence a changed potential felt by theother ions. In fact, the energy of the ion-electron system canapproximately (but only approximately) be written45 as a sum ofpair-interaction terms plus a term independent of ion positions.

One can consider an assembly of ions, each the source of apotential and a pseudopotential which act on the electrons, givingrise to the induced electronic charge density pind(R). The interactionof pind(R) with the ions gives the band-structure energy, whichdepends on the arrangement of the ions. This, combined with theMadelung energy of ions in a uniform, negatively charged back-ground, gives the part of the energy of the system which dependson the ionic arrangement. The result can be written, for ahomogeneous metal, as a volume-dependent term plus a structuralterm, |ZO(R, — R,); the effective interionic interaction potential<J> also depends on volume, since it includes the effects of theinteracting electron gas. In an inhomogeneous system, such as oneinvolving a surface, 3> could depend on position as well as on theinterionic vector R, - R,, but for homogeneous metals, the effectiveinterionic interaction <£> depends on interionic distance R only. Itis strongly repulsive at small R, but includes a small attraction aswell, so that there is a region for which the force -d<£>/dR isnegative and a minimum, for which <$> is less than its value atR = oo. There are oscillations in O at large R, whose origin is thedielectric function of the electrons (see below), which expressesthe screening of the coulomb potential by the electron gas.

For simple monovalent metals, the pseudopotential interactionbetween ion cores and electrons is weak, leading to a uniformdensity for the conduction electrons in the interior, as would obtainif there were no point ions, but rather a uniform positive back-ground. The arrangement of ions is determined by the ion-electronand interionic forces, but the former have no effect if the electronsare uniformly distributed. As the interionic forces are mainlycoulombic, it is not surprising that the alkali metals crystallize ina body-centered cubic lattice, which is the lattice with the smallestMadelung energy for a given density.46 Diffraction measurements


of the melts of Na and K indicate little change in local coordinationfrom the solid. Thus, the interionic forces (coulomb and hard-sphere) dominate in determining structure in the liquid. For poly-valent simple metals, it is necessary to take into account the con-centration of conduction electron density around the multiplycharged ions to understand46 the stable ionic arrangements in thecrystal and in the melt.

4. Screening

The effect of screening by the electrons in a metal is to convert abare coulombic interaction q2/r, where r is the distance betweentwo charges q, to an interaction (q2/r) e~r/l, I being the screeninglength. The Fourier transform of q2/r is 47re2/k2, and the Fouriertransform of (q2/r)e~r/l is 47rq2/(k2+ l~2). The ratio of Fouriercomponents, 1 + (k/)~2, is the /c-dependent dielectric function e(k),which describes the screening of a sinusoidally varying field ofwavelength l/k. The value of / for a system of ions screening eachother, as in the Debye-Hiickel model for an ionic liquid,47 is{kT/4irnq2y/2 if n is the number density. The screening by adegenerate Fermi gas of electrons is characterized by a differentscreening length, which may be estimated from the Debye-Hiickelresult by noting that (kT/m)1/2 is the average velocity of a particlein a classical fluid while (some fraction of) fikF/ m is the correspond-ing velocity in the quantum fluid; kF is the Fermi momentum,related [see Eqs. (12) and (13)] to the density by 3ir2n = kF. Thus,for (kT)l/2 in the Debye-Huckel screening length, one shouldsubstitute m~l/2fikF, and q = e, giving a quantum screening lengthm~l/2fikF(47Tne2)~l/2. In terms of the Bohr radius a0 = fi2/me2, thescreening length is

The screening length is of the order of an angstrom in good metals.It is, of course, usual in discussing the electrochemical interface

to use a dielectric constant, which is the ratio of the electric displace-ment to the electric field. By Fourier transforming the dielectricfunction e(k), one would obtain an effective dielectric constant,which would, however, depend on position. In fact,48 the screening

34 Jerry Goodisman

in the interface is nonlocal, and the effective dielectric constant isactually an integral of a dielectric response function. It can beshown47 that it is not always possible to replace the exact nonlocaltheory by a local theory, even with a position-dependent dielectricconstant.

An expression for e(k) in the case of a Fermi gas of freeelectrons can be obtained by considering the effect of an introducedpoint charge potential, small enough so the arguments of perturba-tion theory are valid. In the absence of this potential, the electronicwave functions are plane waves V~1/2exp(ik • r), where V is thevolume of the system, and the electron density is uniform. Thepoint charge potential is screened by the electrons, so that thepotential felt by an electron, 4>, is due to the point charge and tothe other electrons, whose wave functions are distorted from planewaves. The electron density and the potential are related by thePoisson equation,

V2<l> = -47re(nb - n)

where nb is the density of the positive background charge. Anotherequation relating 4> and n is required.

In the Thomas-Fermi model,49 the kinetic energy density ofthe electron gas is written as

and the potential energy density as — en<l>, where n may vary withposition. The energy of the most energetic electron must be thesame everywhere, so

j(3ir2)2/3n2/3e2a0 - e<S> = constant

Far from the introduced point charge, n = nb and 4> = 0. Assuming4> is small, the deviation of n from nb is small, and

e * = &TT2)2/\n2/i - nl/3)e2a0

Combining this with the Poisson equation, we get

V2<D = l2ne2(3TT2r2/3nl/3<P/(e2a0) (19)


The solution corresponding to a point charge q at the origin is

<l> = (q/r)e-"\

with

l2 = {U7r)-l07T2)2/3n-bl/3a0 (20)

Since n,1/3 = (37r2)1/3/kF, we find / = (7rao/4kF)1/2.In the Hartree-Fock or self-consistent field picture, <£ also

enters the Schrodinger equation which determines the electronicwave functions. One thus has to solve the Schrodinger equation

VV* + (k2 - 2m^/ft2)ijjk = 0

for the electronic wave functions. This equation is equivalent tothe integral equation

il;k= V-1/2exp(ik-r)

+ (2m/ ft2) J G(r - r')*(r')iMr') dr' (21)

where G(r - r'), a Green's function, is -exp(/fc|r - r'|)/(47r|r - r'|),because

V^r-r'l"1 = -47r8(r-r')

In the integral of Eq. (21), one can substitute the unperturbed wavefunction V~1/2 exp(/k • r') for if/k(r') since O is a small perturbation.With the resulting expression for \fjk, one calculates the electrondensity as a sum of densities for the occupied orbitals (with |k| <kF). Replacing the sum by an integral and subtracting the unperturb-ed electron density, one finds50 for the change of electron densitythe Lindhard result

J jx(2kF\x - r'|)|r - r'|= - (mk2F/27r3fi3) J jx(2kF\x - r'|)|r - r'|-2*(r') dr' (22)

where jx{x) = (sin x — x cos x)/x2. Since <J> arises from the intro-duced point charge and from 8n,

V2<£ = -47re28(r) + 4<7re28n

36 Jerry Goodisman

Solving the resulting equation by Fourier transformation, oneobtains for the Fourier components

with

2mfcFe2/ , fc2-4/c22mkFe (

f k \

k-2kF

2kF

(23)7rft2fc2 \ 4kFk

A noteworthy feature of e(k) is the singularity at k = 2/cF, thediameter of the Fermi sphere.

Writing this as 1 + (kl)~2, one can extract the screening length/. For long wavelengths (small /c), the second term in parenthesesvanishes and the screening length obeys

I2 = 7rh2/2mkFe2 = 7raJ2kF

which is twice the estimate of Eq. (20). Alternatively, one notesthat the small-k limit corresponds to a slowly varying potential. Asshown above, the introduction of a small point charge Ze into anelectron plasma results in a potential V(r) which obeys

If one assumes V is slowly varying in space, compared to otherfactors in the integral, one can put V(r) for V(r') and take it outof the integral as a constant; what remains is

jx(2kFs)s~2 ds = 4TT(2/CF)~1 (sin u - u cos u)u~4u2 du

= 2ir/kF

Then

V2V = 47rZe28(r) + 4kFV/7ra0

which has the solution V(r) = - (Ze2/r)e~qr with q2 = 4kF/irao\this is clearly a screened coulomb potential.

The change in the electronic charge density according to Eq.(22) is

(2*p|r-r ' |) |r-r 'r2V(r')dr '


If V is localized, say, near the origin, then for locations far fromthe origin, this behaves like jx(2kFr)/r2, which means ascos(2/cFr)/r3. These damped oscillations of frequency 2kF are theFriedel oscillations, which always arise when an electron gas isperturbed; the frequency of oscillation comes from the "kink" inthe dielectric function at 2kF. We see the Friedel oscillations (inplanar rather than in spherical geometry) for the electron gas at ahard wall [Eq. (12) et seq.] and for the electron density at thesurface of a metal.

More generally, one considers an external test-charge density/to(r, t) acting on a plasma (here, the electron density) in thepresence of a uniform charge density of opposite sign which isfixed. The polarization of the plasma is described by a change inthe electron density 8n(r, t) which is assumed to be related linearlyto the potential Vo of the density n0. The most general linear relationis

8n(r, t) = J J dr' dt' K(\r - r'|, t - t') V0(r', t') (24)

The response function K can depend only on |r - r'| for a uniformisotropic system. On Fourier transformation,

8n(Ka>) = x(K<o)Vo(K<o) (25)

where 8n(k, co) and V0(k9 co) are the Fourier transforms of 8n(r, t)and Vo(r, 0- Since Vo is the electrostatic potential of the chargedistribution n0, V0(k, co) = (47re2/k2)n0(k, w) where no(k, co) is theFourier transform of no(r, t). The Poisson equation for the electricaldisplacement is

V • D(r, t) = -4ireno(r, t)

In the presence of the plasma, the electrical displacement D is thesame, but the electric field now obeys

V • E(r, t) = -47re[no(r, t) + 8n(r, t)]

On Fourier transforming these equations, one finds

k-E(k,o>) , , 8n(k,co) 2

1 + 1k-D(k,cu) n0(k,(o) k2

38 Jerry Goodisman

the ratio of k • D to k • E is just the wavelength- and frequency-dependent dielectric constant, e(k, co). The calculation of x anc* eat various levels of approximation, for classical and quantumplasmas, is discussed by March and Tosi.50

Treating the free electrons in a metal as a collection of zero-frequency oscillators gives rise51 to a complex frequency-dependentdielectric constant of 1 - (O2

p/((o2 - iw/r), with wp = (Airne2/m)1/2

the plasma frequency and T a collision time. For metals like Agand Au, and with frequencies co corresponding to visible or ultra-violet light, this simplifies to give a real part

eff=l-co2

p/co2

and an imaginary part e'} = 0. The complex refractive index, whichis the square root of the dielectric constant, determines the reflec-tivity of a solid. This is the basis for UV-visible reflectance spectros-copy. The electroreflectance is the change in reflectivity of themetal-electrolyte interface with electrode potential.

The electroreflectance of a Cu single-crystal surface in I NH2SO4 is discussed by Kolb.51 The gross features are explained interms of a potential-induced change in the metal surface's opticalconstants. For more anodic potentials, the free-electron concentra-tion near the metal surface is lowered compared to the bulk,changing the plasma frequency to [47r(n + &n)2e2/m]1/2 whereAn = CDLAV7ed, with CDL the double-layer capacitance of theinterface, AV the potential drop across the interface, and d thepenetration depth of a static electric field into the metal. To explainthe electroreflectance in more detail, one must consider boundelectrons as well. Surface states must also be taken into account:their energies depend strongly on the applied field. It is assumedthat, because they are localized at the surface, they sample a fractionof the potential drop across the interface. The surface states ofadsorbates at the electrode-solution interface can be studied by insitu time-modulated reflection spectroscopy, which also shows theeffects of modification on the conduction-electron tail.52

The interaction between two ions in a metal is screened by thegas of conduction electrons. Although corrections for exchange andcorrelation are required, the features of the screened interactionare what one would expect from the preceding calculation of the


screened potential of a single charge.53 It is coulombic at smallinterionic separations R, almost completely screened away at largeR, and oscillatory at intermediate R. From such an effectiveinterionic interaction potential, one can calculate the stability ofvarious liquid and crystal structures, interionic distances, forceconstants, etc. Much of the theory requires, as we have seen above,that the potential of an ion be a weak perturbation on the electrongas. The coulombic potential of an ion core would not seem to beweak, but the ion core is not really a point charge, and thus is asource of a pseudopotential as discussed above. The total effectivepotential is indeed weak.

IV. METAL SURFACES

1. Density Functionals

We now consider the electron density at the surface of a metal. Anoften-used approach to surface properties of many-particle systemsis the density-functional approach, which supposes that the freeenergy of such a system can be written as the integral of a free-energydensity. In the simplest such theories, the free-energy density is afunction of the local particle density. By requiring that the chemicalpotential (electrochemical potential, for a system of charged parti-cles) be independent of z, one derives49 an equation for n(z). Thisapproach has been applied to a one-component plasma, with abackground step-function density of compensating charge, andgives results which agree well with the results of computer simula-tion for this problem.

For liquid metals, one has to set up density functionals for theelectrons and for the particles making up the positive background(ion cores). Since the electrons are to be treated quantum mechani-cally, their density functional will not be the same as that used forthe ions. The simplest quantum statistical theories of electrons,such as the Thomas-Fermi and Thomas-Fermi-Dirac theories,write the electronic energy as the integral of an energy density e(n),a function of the local density n. Then, the actual density is foundby minimizing e(n) + vn, where v is the potential energy. Such


calculations produce surface energies and other properties whichare not in good agreement with experiment. Better theories49 includea term depending on the gradient of the particle density. Thus, ifn{z) is the density as a function of position for a planar interface,the free energy per unit surface area would be fi[n(z)] +fi[n(z)> n'(z)]. On general grounds, f2 cannot depend on the signof n'(z).

For a density which is not too rapidly varying, the electronicenergy can be written as the integral over all space of a sum ofthree terms. In addition to the potential energy term, vn9 there is49

a term, s(n), which represents the density of kinetic energy (includ-ing exchange and correlation) of a homogeneous electron fluid ofdensity n, and the inhomogeneity term, proportional to \Vn\2/n.For a system containing a planar surface, n is a function of z. Thedensity profile n(z) is determined19 by minimizing the integral overall space of the energy density, e(n) + c\Vn\2/n + vn, with respectto n{z). For simple liquid metals, an electron density profile canbe calculated by minimizing this energy functional with respectto «(r).

Smith54 used this approach to calculate electron densityprofiles, and from them, surface potentials and work functions, for26 metals. The positive charge density was modeled as a jellium(step-function) and the quantity £„[«] — fiN was made stationaryto variations in parameters in the density profile function n. Here,Ev is the surface energy, JJL is a Lagrange multiplier, and N =J n(z) dz is the number of electrons per unit surface area. Thefunctional for Ev included the interelectronic repulsion, the interac-tion of n(z) with the potential of the positive background, andterms representing kinetic, exchange, and correlation energy. Forthese last three contributions, Smith used

- 0.056 724/3(0.079 + nl/3yl dz

which are the corresponding energy densities of a uniform electrongas of density n. He also included an inhomogeneity term,


| (Vn)2(72n)~' dz. The density profile was taken as

n+-\n+eb\ z < 0

with n+ the positive charge density and b a parameter determinedby variation. Surface barriers in agreement with experiment (errorof about 10%, with barriers from 4 to 30 eV) were obtained.

Schmickler and Henderson55 have improved on Smith's resultsby using the trial density profile

•«-{:<—[1 - Aeax cos(cx + d)], x < 0n+(Be-bx), x > 0

There are four conditions that mut be satisfied: continuity of n(x)and n'(x), charge balance between n(x) and the background, andthe Budd-Vannimenus half-moment condition (see below) relatingthe potential difference between -oo and 0 to the derivative of theenergy per electron in bulk. The two remaining free parameterswere determined by minimizing the same density functional usedby Smith. The results for work functions are closer to those obtainedby Lang and Kohn56 (see below) than Smith's (taking advantageof the Budd-Vannimenus condition, only the potential dropbetween the jellium edge and vacuum at infinity needs to be calcu-lated). The effect of an external field, due to a low charge densityon a sheet at infinite distance from the jellium, was also calculated.The resulting profile of induced surface charge density, as well asthe position of the effective image plane (center of mass of inducedsurface charge), agreed well with that of Lang and Kohn.57

One knows, however, that the simple density-functionaltheories cannot produce an oscillatory density profile. The energyobtained by Schmickler and Henderson55 is, of course, lower thanthat of Smith54 because of the extra parameters, but the oscillationsin the profile found are smaller than the true Friedel oscillations.Further, the density-functional theories often give seriously inexactresults. The problem is in the incorrect treatment of the electronickinetic energy, which is, of course, a major contributor to the totalelectronic energy. The electronic kinetic energy is not a simplefunctional of the electron density like e(n) + c\Vn\2/n, but a

42 Jerry Goodisman

quantum mechanical expectation value over a wave function of thekinetic energy operator - (h2/2m)V2. Lang and Kohn56 developeda theory in which the electronic kinetic energy is treated correctly,as a sum of expectation values over occupied one-electron states(orbitals). These states are determined self-consistently, as in theHartree-Fock theory, but, as we will see, the theory of Lang andKohn includes interelectronic correlation, absent from Hartree-Fock.

Suppose one seeks the electron density in the presence of anexternal potential V(r). The density is given by

n(r) = N ^*^dr ' (27)

where N is the number of electrons, the integration is over allelectronic coordinates but one, and ^ is the (antisymmetric) eigen-function of the N- electron Hamiltonian. This Hamiltonian includesoperators for electronic kinetic energy, interaction of the electronswith V, and interelectronic repulsion. In the theory of Lang andKohn,56'58 one seeks a potential V, different from V, such that thedensity of noninteracting electrons in the presence of V is equalto n, the density of interacting electrons in the presence of V. Inthe absence of interelectronic interaction, the Hamiltonian is a sumof one-electron Hamiltonians, H = £,- h(i), where

h(i)= -(ft2/2m)V?+V'(r,)

The eigenfunctions of H are determinants formed from one-electronfunctions which are eigenfunctions of h, say,

h(i)4>j(i) = ejtj(i) (28)

Then, the electron density is just

the sum being over the eigenfunctions corresponding to occupiedone-electron states. The problem is the determination of V.

It can be shown rigorously59 that for a system of interactingelectrons in a compensating charge background, the energy can bewritten as an electrostatic part plus a functional of the electron


density G[n]. The energy of the system is

E = J V(r)n(r)

+ (e2/2) I n(r)n(r') ] |r - r'p1 dr dr' + G[n]

so that the density could be determined by minimizing E withrespect to n. However, the form of G[n] is not known, and is likelyto be nonlocal and extremely complex. Theories which use simpleapproximate forms for G[n] do not give very good densities. Theexchange-correlation energy functional is defined as

[n] = G[n]-l f it,f(-j J

sxc[n] = G[n]-l f it,f(-fi2/2m)V2ilfjdT (29)j J

the last term being the kinetic energy of the system of noninteractingelectrons which gives the same density as the system of interest.Although the exact form of sxc[n] is not known either, it is arelatively small part of the total energy, and sufficiently accurateapproximations to it can be constructed. The self-consistent theorythen takes the effective potential V'(r,) as </>(r) + 8exc/8n, where 0is the electrostatic potential (due to positive charges and electrons);if there is an external potential, it is also included in V. Thiscorresponds to choosing the {i/ } to minimize E, with G beingexc + the correct kinetic energy in terms of the {(/>,}.

2. Self-Consistent Theories

The scheme for obtaining the electron density is thus as follows.(i) Given an electron density «(r), one can construct the electrostaticpotential and 8exc/8n. (ii) This gives the potential V and theHamiltonian h. (iii) The eigenfunctions of h are found and com-bined into a new electron density, (iv) One seeks self-consistency;i.e., input and output electron densities should be identical. Inprinciple, though not in practice, this involves iteration of steps(i)-(iii). With the self-consistent functions, G[n] is written as

*M + I [ <A,*(-*72m)V2<A, dr = exc[n] + I [ **(* - V')<A, drj J j J

44 Jerry Goodisman

so that, using Eq. (29), the energy of the system is

E = | V(r)*(r)dr-J V'(r)/i(r)

+ (*72) J n(r)n(r')|r-r'rdrdr'

Thus, for a planar interface one determines the one-electron wavefunctions according to

[(-fi2/2m)(d2/dz2) + Veff(z)]<K(z) = ^ f c (z ) (30)where Veff includes the external potential, the coulombic (Hartree)potential of the electrons, and the exchange-correlation potential,8exc/8n. Veff depends on n through this last term. The one-electronwave functions are actually </ (z) e'k s where k' is {kX9 ky) ands = {*, y}9 so the one-electron energy is actually fi2k'2 + ek. Incombining the squared wave functions to form n(z), one integratesover all values of kx and ky such that sk + fi2kf2/2m < fi2k2

F/2m,where hkF is the Fermi momentum.

For a surface problem, the external potential is usually thatof a semi-infinite slab of positive charge, called jellium. In calcula-tions for the solid (crystalline) metal, the stepfunction charge distri-bution must be replaced by a lattice of ions, each carrying apseudopotential for the electrons which has a contribution frominner-shell exchange repulsion as well as from the coulombic attrac-tion. The effect of this replacement on the electronic energy iscalculated by perturbation theory. For liquid metals, one expectsthat there will be an ion density profile different from a stepfunction.60 It is possible to perform the calculation of electronicwave functions using, in the Schrodinger equation, the potential ofa non-step distribution of ion cores, each carrying inner-shell repul-sion as well as coulombic attraction. If more gradually varying(broader) ionic profiles are assumed, the electronic profile alsobroadens out, remaining broader than the ionic one. The surfacedipole potential, which depends on the difference between the ionicand electronic profiles, decreases in size.

Calculations using oscillatory ion profiles have also been per-formed61 and give better surface profiles and work functions forliquid metals. Such oscillatory profiles are expected on the following


argument: The electronic profile provides the potential in whichthe ions are supposed to move (Born-Oppenheimer or adiabaticapproximation); the potential due to an electron density whichdecreases to zero outside the metals acts like a wall, keeping theions inside the metal, and this should produce oscillations. Otherarguments have been given, and oscillatory ion profiles also resultfrom simulations.62 It is always true that the electrons "spill over"toward the outside of the metal, i.e., that the electronic densityprofile is broader than the positive-charge (ionic) backgroundprofile. This, of course, is the origin of the surface dipole potential,which is an important part of the work function.

It should be noted that the ionic profile referred to is an averageprofile, and electronic and ionic profiles should actually be calcu-lated together, in a self-consistent manner, as follows: With theions fixed at some configuration, the electron density is determined,perhaps by solution of the Schrodinger equation by the Lang-Kohntheory.58 This yields, in addition to an electron density profile, theenergy of the ion-electron system. The energy as a function of ionicconfiguration, say, £(R), becomes the potential energy which deter-mines the arrangement of the ions. In principle, each ionic configur-ation is weighted by a Boltzmann factor, exp[—E(R)/kT]. Thenintegration or averaging, with the Boltzmann factors as weights,gives the average distribution, and averaging the electron densitieswith the same weights gives the average electronic profile. Thepreceding models for liquid metals short-circuit this by assumingthat the average electronic profile, calculated in the above way, canbe replaced by the electronic profile calculated for the average ionicconfiguration.

Since the positive background is in the form of ions, it is notreally a continuous density profile depending only on the coordinateperpendicular to the interface. The electron density, though always"smoother" than the positive charge density, is likewise more com-plicated than a one-dimensional profile. However, it is rare to seecalculations for the electron density which take into accountinhomogeneities in the parallel directions, except when adsorbedatoms are being considered.

The atomic distribution of positive charge must be taken intoaccount when considering work functions for different crystal facesof solid metals. The chemical potential /JL™ is a bulk property and

46 Jerry Goodisman

its value is independent of the crystal face through which the leavingelectrons are supposed to pass, so that the difference in workfunctions observed for different crystal faces is due to the changedsurface potential. As a first approximation, one may take it thatthe smoothing of the electron density in directions parallel to thesurface means that the electron density profile is almost the samefor all crystal faces, so that the different surface potentials comeabout because of the way in which the electron density cloud ispenetrated by the ions. Any penetration of the negative chargedensity by positive charges will lower ^M, so that crystal faces forwhich the positive ions are distributed most uniformly, i.e., thefaces of higher ion density, will have higher xM and higher workfunctions. Similarly, defects and other inhomogeneities in the sur-face will reduce \M a nd 4>. Because of the connection betweenwork function and potential of zero charge when the metal is inan electrochemical cell, a similar explanation can be given for thedifferences in Epzc for different crystal faces of solid metals.

3. Screening

We now consider calculations of screening of external charges bythe electrons at the surface of a metal. It may be noted at the outsetthat one cannot simply transfer results for the metal-vacuum surfaceinto the electrochemical interface, although some general ideas willcarry over. Thus, one can expect that the screening length is greaterat the metal surface than in bulk because of the decreased electrondensity. This was used by Kuklin63 in a theory for the contributionof the metal to the capacitance of the electrochemical interface,which has been criticized8 on a number of points, including mathe-matical errors; it leads to serious discrepancies with experimentalresults.

Newns40 calculated the response to an external electric fieldof an idealized metal, plane waves at an infinite barrier, with densityprofile [see Eq. (12)]

p(x) = po(l + 3 cos x'/x'2 - 3 sin x'/x'3), x' = 2kFx

The response was calculated in the Hartree approximation, andonly a linear response was considered. Suppose an external poten-tial U(r) leads to a change in the electron density 8p. This then


produces an additional electrical potential which has an effect onthe electron density. In the Hartree model, one requires a function8p which is self-consistent, so that the total electrical potential, dueto U and to dp, produces the change 8p. In the linear approximation,one calculates only the change in p which is proportional to U.This approach is like that of Eqs. (19)-(23) for bulk metal.

Calculations were performed for an external field due to apoint charge as well as for a uniform external field. In discussingthe results, a screening length was defined as d = V(0)/V'(0), wherethe potential outside the metal is V(0) 4- V'(0)x and the potentialinside the metal, which is screened by the electrons, approacheszero for x -> -oo. Since - V'(0) is the external electric field, propor-tional to the surface charge density of a capacitor, and V(0) is thepotential drop across the metal from 0 to -oo, the screening lengthis the inverse of a capacitance. For a perfect conductor, d = 0,since there is no penetration of electric field into the metal and nopotential drop in the metal. In the Thomas-Fermi theory, d = A"1,where the Thomas-Fermi screening length A is defined by theThomas-Fermi equation [Eq. (19)], V2V = \2V. Schiff64 also per-formed an approximate calculation of the shielding of the field ofan external charge distribution by the conduction electrons at ametal surface and estimated the penetration into the metal of theelectric field; the calculation was done by a variational method,and the Thomas-Fermi approximations were used to describe theelectrons.

As expected, because the Thomas-Fermi theory is valid forhigh densities, d = A"1 is valid only for rs « 1, where 47rrs

3/3 isthe inverse of the electron density. For higher rs (lower /cF, lowerbulk density) Newns' calculations40 show that d is well approxi-mated (to a few percent out to rs = 6) by A ~* + TT/4/CF. Experimentalmeasurement of d for the metal-vacuum interface may be madevia the change of the work function due to a layer of adsorbedions. If these ions have charge q and are at a distance a from themetal surface, the change in work function is AO = 47rnsq(a + d),where ns is the surface concentration of ions. This assumes thatthere is negligible penetration of ions into the tail of the electrondensity, and neglects nonlinear effects. Penetration would reduced. Experiments for Cs+ adsorbed on tungsten give results varyingwidely with crystal face and with the experimenter,40 and their

48 Jerry Goodisman

interpretation is complicated by the band structure of tungsten, butit is concluded that d is about 1.35-1.85 A. This is greatly in excessof any reasonable Thomas-Fermi screening length but is compatiblewith Newns' results40 if rs is 3-4. The large increase in screeninglength relative to Thomas-Fermi theory is related to the low electrondensity near the surface of the metal.

Beck and Celli65 also calculated the linear response of a metalto an external charge distribution by a method equivalent to thatof Newns,40 with the electrons confined to a half-space by an infinitebarrier. It was concluded that, for most electrical properties, theFriedel oscillations, associated with the discontinuity in the linear-response function at 2/cF, are not important. Rudnick66 also calcu-lated the screening of a static charge distribution in the self-consistent Hartree approximation, but with a more transparentformalism than that of Newns. He emphasized the Friedel oscilla-tions in the screening charge density and how they were affectedby the electron gas boundary. For a very dense electron gas, it wasargued that, since the infinite-barrier model does not give a gooddescription of the electron gas near the surface, it should not be avalid model for the response of the electrons in a metal to anexternal charge distribution.

Ying et al61 applied the density-functional formalism to thelinear-response problem. One starts with an inhomogeneous elec-tron gas at T = 0 in an external potential V"oxt(r) and with electrondensity no(r). Assuming a perturbing potential V?xt(r) =Jpfxt(r')|r - r'p1 dr1, one seeks the change n^r) in the electrondensity. Here, the Lang-Kohn formalism58 is used for both theperturbed and the unperturbed system. The equation for the totalelectrostatic potential V(r),

-V(r) + 8G/8n = /x

is linearized with respect to V^xt and nl9 which yields equationsfor nx and Vx. These equations were solved by Ying et al61 usingthe Thomas-Fermi functional for G,

n5/3(r) dr

In this case, the unperturbed solution outside the metal is Vo =400/(6 + x)2, with b = 5.0813.


Heinrichs68 used the Thomas-Fermi model, with severalphenomenological approximations added, to study the response ofthe metal surfaces to external charge distributions. This includesthe image-potential interaction with a static point charge. An adsor-bed ion and its induced screening charge constitute a dipole, whichis involved in the change in the work function of the metal as wellas in other phenomena, which are reviewed by Heinrichs.68 TheThomas-Fermi response is discussed in detail, as well as the solu-tions to the relevant equations. Results are compared to those fromself-consistent calculations (see below).

The Thomas-Fermi-Dirac model, which adds an exchangeenergy density to the kinetic energy density of the Thomas-Fermimodel,49 has also been used to describe the electrons of a metalsurface.65 One can solve the Thomas-Fermi-Dirac equation numeri-cally to obtain the electron density profile for any surface chargedensity, thus obtaining the capacitance of the metal surface as wellas the surface potential. Here, one is not limited to the linear-response regime. Making the metal one plate of a capacitor rep-resenting the interface, one finds that the inverse capacitance ofthe metal increases as qM becomes negative.

Lang and Kohn57 used their self-consistent density-functionaltheory to calculate the screening charges induced in a metal surfaceby a uniform electric field or an external point charge. In the lattercase, one is calculating the image potential. The positive chargesare described by the uniform-background model (step function orjellium). The theory was worked out for the linear-response regime;i.e., one seeks the induced charge profile which is proportional tothe external charge distribution. It may be noted that one can simplyadd an external field to the potential of the Schrodinger equationand solve the self-consistent equations; it is not necessary to use aperturbation-type expansion and limit oneself to the linear response.

Noting the symmetry of the background in the y- and z-directions, the external charge distribution was written57 as a sumof terms of the form

where s = (y, z) and p = (py,pz); p specifies whether one has asheet of charge, a point charge, or some other distribution in theplane x = xY. Since xx is supposed to be well outside values of x

50 Jerry Goodisman

for which the electron density is appreciable, the external potentialis

i//e = (2TT/P) exp(-/?|x - Xi\)

on solving the Poisson equation. The total perturbing potential is

Jwhere 8n is the induced charge density.

Of great interest is the center of gravity of the induced charge,

-f.zo= z8n(z)dz 8n(z)dzJ —oo / J —oo

which is the effective position of the added or subtracted chargein the metal. (For larger charge densities, the center of gravity willchange with charge density. Like the change in the shape of 8n,this will contribute to the dipolar capacitance of the metal surface.)The charge induced in the metal by an external point charge hasits centroid at z0 if the point charge is far enough away. If the endof the metal-ion (step-function) distribution is at z = 0, and theexternal charge (which would be in the electrolyte phase if themetal were in an interface) is at z = Z, the "ideal capacitor" hasa width, not of Z, but of Z - z0 [see Eqs. (6), (7) et seq.]. It turnsout that z0 = l.9a0 for rs = l.5a0 (bulk density of 0.0236ao3) an<3z0 = 1.2a0 for rs = 6a0 (density of 0.000368^01). As the point chargeapproaches the metal, z0 decreases slightly until for some positiond from the jellium edge, zo(d) becomes equal to d and remainsequal to d inside the metal.

Some exact results for the charge distribution induced at ajellium surface by a static applied field were derived by Budd andVannimenus.70 These authors calculated the linear response of ajellium slab of finite thickness to a charged plane parallel to thesurface, either for isolated jellium (for which total charge is conser-ved) or grounded jellium (for which the charge of the jellium isequal and opposite to that of the perturbing charged plane). Theyshowed that the half-moment of the charge distribution induced ingrounded jellium is related to the unperturbed electric field Eo by

f°E0(z) = 4irpbe[zd(-z) - dz'z'n(z'\z)] (31)

J-oo


Here, n{z'\z) is the charge density induced at z' by a charged plane,of unit surface charge density, located at z, and pb is the bulkjellium density. The Hellmann-Feynman theorem was used. Thefull moment of n(z'\z) obeys

r oo /• z Too

dz' z'n(z|z')= dz' z'n(z'|oo) + z dz'n(z'\oo)J-oo JO Jz

where H(Z'|OO) is the response of the jellium (for z < 0) to a chargedplane at infinity.

In general (beyond linear response), the half-moment ofn(z|oo) is given by

icr2 = - I J dzn(z|oo)J = p b \ dzzn(z\oo)

where a is the surface charge density. The half-moment is thusinversely proportional to the jellium density and is quadratic in theapplied surface charge density. This is shown by considering theforce on the plane infinitely far away, (ae)(—lirae), where — 27rcreis the field at the plane. This must be equal and opposite to theforce on the jellium background, -pbe J°.L dz' E'(z'\oo), whereE\z'\oo) is the electric field at z' induced by the charged plane.Since £'-^0asz'-> —oo, one can let L approach oo. Integrating byparts, one finds

f°z'E'f^ - z' dE'/dz' dz'}

J-oo

f°= pbe z'{-47re«(z'|oo)}J —oo

Therefore, \a2 = pb dzzn(z\oo).J-oo

Perhaps of greater interest to us are results derived by thesame authors71 that relate surface and bulk electronic properties ofjellium. Considering two jellium slabs, one extending from - L to-D and the other from D to L, they calculated the force per unitarea exerted by one on the other. According to the Hellmann-Feynman theorem, this is just the sum of the electric fields acting

-277-o-V = -

52 Jerry Goodisman

on each slice of positive background, multiplied by the positivechage density of the slice. Then,

= r P + ( * ) ( -J D

F/A = P+(z)(-d V/dz) dz = -pb[ V(L; D) - V(D; D)]J D

where V{x\ D) is the electrostatic potential at x when the slabs areseparated by 2D. When the separation distance is zero, the forceper unit area can be shown to be simply the pressure inside thejellium. If L is large, the latter can be calculated in uniform bulkjellium, and V(L; 0) - V(0; 0) is just the difference in electrostaticpotential between a point in bulk jellium (z = 0) and a point at thejellium surface (z = L).

Now suppose the energy density of the electrons in bulk jelliumis given by nf(n), so that the density functional / ( n ) is the energyper electron of a uniform electron gas of density n in the positivebackground. (The integral of nf(n), plus inhomogeneity terms, isthe quantity one minimizes to obtain the density profile.) Thepressure in bulk jellium is then n(df/dn), so that

[n(df/dn)]n=Ph = <M-oo) - 0(0) (32)

We have returned to a former notation, denoting the electrostaticpotential by </>, so that </>(0) is the potential at the jellium surfaceand (f>(—oo) is the potential in bulk jellium. The left member of Eq.(32) is, of course, a bulk property and the right member a surfaceproperty. Another proof of Eq. (32) was given by Vannimenus andBudd.72 They also derived a related important theorem,

dEJdPbJ-oc

Here Es is the surface energy and the left side refers to the changeof Es with change in bulk electron density.

Equation (32) is extremely important to the calculation of workfunctions. The work required to remove an electron from the interiorof the metal to vacuum outside is

where fxe, the chemical potential of the electron, is a bulk property


and the surface potential, xM, is a surface property. In the density-functional model,

Me = d[nf(n)]/dn

and

x = (f>{~0®) ~ fii0®)

Equation (32) shows that there is some cancellation between thebulk and surface contributions; combining Eq. (32) with the aboveequations, we have

<I> = —f(n) + e[0(O) — 0(+°o)]

This helps explain58'73 why simple variational calculations can givegood work functions: only the electronic tail, spilling over fromthe positive background, is involved in the surface contribution.

Equation (32) also holds72 for the charged interface, in the form

[n(df/dn)]n=Ph = 0(-oo) - 0(0) - q2/2Pb (33)

where q = qM = -qs is the surface charge density. The last term isthe contribution to the force on the jellium of a sheet of chargedensity qs far outside the metal electrons. The dipolar contributionof the metal to the capacity is [see Eq. (8)] given by 1/CM =d[0(oo) - (f>(-oo)]/dqM. Using Eq. (33) for 0(-oo) - 0(0) and thePoisson equation in the form

- 0(oo) = 477 [ dz' I dz"[n(z") - P+(z")]J oo J oo

= -477 I dz' z'ln(z') - p+(z')]Joo

we have

Thus, only the tail of the electron density (outside the jellium)contributes. The last term above may further be related11 to theposition of the image plane zxm so, at the point of zero charge,

54 Jerry Goodisman

CM = - Wz\m. It should be noted that the Budd-Vannimenustheorems are obeyed by the electron density n(z) which is the exactsolution to the jellium problem, and not necessarily by any approxi-mate (e.g., variational) solution. There are extensions of the Budd-Vannimenus theorems to a bimetallic surface.74

V. METAL ELECTRONS IN THE INTERFACE

1. Metal Nonideality

As we have mentioned, traditional theories of the electrochemicalinterface almost always assumed that the metal is ideal, with asharp boundary, although it was sometimes recognized that watermolecules and other species in the interface feel the electron densitytail. This should affect the distance of approach of the solventmolecules and the behavior of the solvent dipoles in the surface.For example, the penetration of the solvent layer by the tail of theelectron density means that the field at the position of the first layerof solvent dipoles is not zero when the surface charge density iszero. The fact that the metal is not ideal also means that the imageplane for charges of the electrolyte phase is about 1.0 A in frontof the metal surface (position of the outermost ions, or of the jelliumrepresenting them). Although this has little effect75 on the platespacing of an ordinary (macroscopic) capacitor, it is important inchanging the spacing of the ideal capacitor representing the innerlayer of the metal-electrolyte interface.55 This decreased spacingraises the capacitance of the interface above what it would be foran idealized metal. Note that, if there is negligible electron exchangebetween the ions of the electrolyte and the metal, the image interac-tion is the only interaction (electron exchange implies a chemicalinteraction).

Any theory which includes an infinite barrier to metal electronsat the interface will make the reciprocal capacitance too largebecause it makes the effective interplanar spacing of the inner-layercapacitor too large.76 This is why Rice's early5 results (see below)were so incorrect. The fact that the electron tail penetrates a regionof higher dielectric constant further reduces the calculated inversecapacity.30'77


The interface is, from a general point of view, aninhomogeneous dielectric medium. The effects of a dielectric per-mittivity, which need not be local and which varies in space, onthe distribution of charged particles (ions of the electrolyte), wereanalyzed and discussed briefly by Vorotyntsev.78 Simple models forthe system include, in addition to the image-force interaction, apotential representing interaction of ions with the metal electrons.

The image interaction for a nonideal metal was discussed inmore detail by Kornyshev and Vorotyntsev.24 Let q<t>0 (x

r, x) be theelectrostatic potential produced at a point {x\ 0,0} in ahomogeneous medium by a charge q on an ion at {x, 0, 0}, andq<j>(x',x) the potential produced at {x\ 0,0} by this charge if ametal is present for x < 0. The energy of the ion in either case iscalculated by integrating [c<f>(xf, x)] dc from c = 0 to c = q (Guntel-berg charging process), so that the difference in energy, due to themetal, is

AE = \q2 lim[<Mx', x) - </>0(x', x)]

and the image potential is obtained by dividing A£ by q.The classical result for the image potential is -q/4x, indepen-

dent of the metal, but various theories of the metal which assumean infinite potential barrier for the metal electrons give potentialswhich are reduced in size near the metal boundary, so that theinteraction energy is actually finite24 at x = 0. An interpolationformula which reproduces this behavior is

where a is a fitting coefficient and l//cTF is the Thomas-Fermiscreening length. Classically, the image charge for a charge at x = x0

is at x = —x0, while the above formula puts the image charge at— (x0 + a/KT¥). This is interpreted24 to mean that the distributionof induced charge in the metal is that of a disk of thickness KTF~1

and radius x + K^F-The conduction-electron tail is expected to move into the metal

as the surface charge qM becomes more positive, and away fromthe metal as qM becomes more negative. It has been suggested18

that, for large positive qM, the tail could contract enough to deshieldthe inner-shell d orbitals at the surface, which would have a strong

56 Jerry Goodisman

effect on* the interaction of solvent molecules with the metal. Fur-thermore, the variation of the position of the effective image plane,which determines the size of the inner-layer capacitor, will changewith surface charge.55 This variation is not symmetrical: it is easierto extend the tail outward than to push electrons into the metal.Thus, the position of the image plane changes more rapidly withqM as qM becomes more negative.

The movement and distortion of the conduction-electrondensity profile give the contribution of the metal to the interfacialcapacitance. According to the discussion after Eq. 7, the free-charge or ionic contribution to the potential drop across the compactlayer AV is 47rqM(z° — zs), and the contribution of the metal,assuming no dipolar species, is 4irqM{zM - z°), where zM is thecentroid of the metal's charge distribution. Here, z° is fixed but zM

changes with qM. Then, C(ion)"1 in Eq. (8) is simply 47r(z° - zs),while the metal contribution is

f vCM(dip)"1 = 4T7-(Z° - z s ) + 47rgM - ^ j —r • (34)

dq M

j *!»,<*)This contribution involves the positive-ion and electron densityprofiles of the metal, and the former is often assumed not to changewith charging of the interface. In 1983 and 1984, severalworkers3032'79 showed how certain features of the interfacial capac-ity curves should depend on the metal.

In addition to the effect of the nonideality of the metal on theelectrolyte phase, one must consider the influence of the electrolytephase on the metal. This requires a model for the interactionbetween conduction electrons and electrolyte species. Indeed, thisinteraction is what determines the position of electrolyte speciesrelative to the metal in the interface. Some of the work describedbelow is concerned with investigating models for the electrolyte-electron interaction. Although we shall not discuss it, the penetra-tion of water molecules between the atoms of the metal surfacemay be related3 to the different values of the free-charge or ioniccontribution to the inner-layer capacitance found for different crys-tal faces of solid metals. Rough calculations have been done to


show the effect of different atomic packings on the effective thick-ness d of the inner-layer capacitor which enters Kion = e/d.

2. Density-Functional Theories

Attempts to include the electrons of the metal in a description ofthe electrochemical interface go back to Rice5 in 1928. Riceattempted to explain the measured electrocapillary curves for themercury-aqueous solution interface by combining the Gouy-Chapman theory for the ions on the solution side of the interfacewith a quantum mechanical theory (the Thomas-Fermi model) forthe conduction electrons of the metal. The Thomas-Fermi energyfunctional is ^(3TT2)2 / 3 / I5 / 3 + en<j). The energy of the most energeticelectron in a metal must be the same everywhere, which means thatthe electron density n must vary with the potential so that\OiT2)2/3n2/3e2ao - e<f> is constant, or

dc/> = (h2/Sme)(3/7r)2/3 d(n2/3) = A d(n2/3) (35)

(A more general relation between potential and electronic pressurefor a density-functional treatment of a metal-metal interface hasbeen given.74) For two metals, 1 and 2, in contact, equilibrium withrespect to electron transfer requires that the electrochemical poten-tial of the electron be the same in each. Ignoring the contributionof chemical or short-range forces, this means that — ecf) 4- (/*2/8m) x(3n/ TT)2/3 should be the same for both metals. In the Sommerfeldmodel for a metal38 (uniformly distributed electrons confined tothe interior of the metal by a step-function potential), there is nosurface potential, so the difference of outer potentials, which is thecontact potential, is given by

<t>2 ~ <f>i = (h2/8me)[(3n2/7r)2/3 - ( 3 ^ / T T ) 2 7 3 ]

This gives contact potentials of the right magnitude but the wrongsign, because of the neglect of chemical forces and surface poten-tials.

Now, consider a planar metal surface, n depending on thesingle coordinate x. In the interior of the metal, x -» -oo, </> -» 0,and n -> n°°, which is equal to the positive-charge (ion) density. Atthe surface, <f> = (f>0 and n = n0. Poisson's equation is

58 Jerry Goodisman

where e is the dielectric constant. Using Eq. (35) to eliminate <£,we find

A d\n2/3)/dx2 = 47re(n - n°°)/e

Rice5 used this Thomas-Fermi theory for the metal in the interface.This equation is multiplied by 2 d(n)2/3/dx, and then integratedfrom x = 0 to x = -oo. Since dn(x)/dx -» 0 when x -> -oo one gets

-A{[d(n2/3)/dx]0}2 = (167re/5s)[ (O 5 / 3 - n5/3]

- (Siren™/s)[(O2'3 - nj'3] (36)

Now, n20/3 - (n°°)2/3 = cf>0/A and [d(n2/3)/dx]0 = A~\d<t>/dx)0.

The electric field at the metal surface, — (d(/>/dx)0, is equal to4irqM/£9 with qM the surface charge density of the metal.

For small potentials <f>Oi 8n = n0- n°° is small. Then, n5/3 -(n°°)5/3 is approximated as

( O 5 / 3 [ ( l + 8n/n°°)5/3 - 1] - (n^lKdn/n00) +l(8n/n°°)2]

and

<fi0/A = nl" - ( O 2 / 3 = (n°°)2/3[(l + S H / O 2 / 3 - 1]

= (M0O)2/3[f(5n/n0O)-i(8M/nao)2]

For small 8n/n°°, the solution is

8n/n°° = |A" Vo(«°°)~2/3 + | A - 2 < ^ ( O ~ 4 / 3

so that n50/3 - (n°°)5/3 becomes |A"Vow°° + T ^ " 2 ^ O ( « ° ° ) 1 / 3 and Eq.

(36) reads

~A-1(47rqM/s)2 = - (Sire/e)[A~l fan00 + lA^^Kn00)^3]

(37)

Rice showed5 that these approximations are good as long as <£0 isless than about IV. According to Eq. (37), 4irqM/e =(6ne/ eAy/24>o(n°°y/6. This range of potentials thus corresponds toa constant capacity for the metal surface. Since the capacity perunit area of a plane capacitor with interplanar spacing d is e/d,the effective capacitor width for the metal electrons in this model


is (Sir8A/3e)l/2(n°°yl/6. This is just the Thomas-Fermi screeninglength.

The metal surface capacitance is simply

C M = £MKT F /477

where KTF = (67rnoe2/eeF)1/2 is the inverse Thomas-Fermi screen-

ing length, eF is the Fermi energy, and e is the backgrounddielectric constant.8 Using e = 8.9, which would now be consideredas unrealistically large for a metal, Rice was able to explain experi-mental electrocapillary data by coupling the diffuse-layer theoryfor the solution with this model for the metal. He rejected thecompact or Stern layer as part of the explanation: modern theory,of course, generally rejects the contribution of the metal andexplains electrocapillary curves and capacitances by combiningdiffuse-layer theory with a theory for the ions and solvent of thecompact layer.

The background dielectric constant e for the metal arises fromthe polarizability of the ion cores and the contribution of interbandtransitions.11 For mercury and other simple metals, with a largeband gap and relatively unpolarizable ion cores, one expects abackground dielectric constant close to unity. With e = 1 andn°° = 8.17 x 1022 cm"3 (mercury), the capacitance per unit area is

q/cf>0 = eml/2h~1(3n0O/7r)1/6 = 2.10 x 106 cm"1 = 2.33

Since this capacitance is supposed to be in series with that of thesolution and since capacitances of mercury-solution interfaces aremuch larger than 2 fiF/cm2, this number is too low. The Thomas-Fermi theory as well as the neglect of interactions between metalelectrons and the electrolyte are at fault. To reduce the metal'scontribution to the inverse capacitance, a model must include6

penetration of the electron tail of the metal into the solvent region,where the dielectric constant is higher, as the models discussedbelow do.

Another model for the metal in the interface, also employingthe Thomas-Fermi approximation for the electrons, was presentedby Kuklin.63 It posited, in addition, a sharp boundary betweenmetal and solution, and made other assumptions which have beencriticized.6 In spite of its errors, it was one of the first attemptssince Rice5 at a model for the interface which treated the metal as

60 Jerry Goodisman

well as the electrolyte. Related work by Salem80 also contains errors,as discussed by Vorotyntsev and Kornyshev6.

A better density-functional model for the electrons includes,in addition to the Thomas-Fermi kinetic energy density, exchangeand correlation energies, as well as an inhomogeneity term. Sucha model was used by Smith54'81 for the bare metal surface [seeabove, Eq. (26)]. Capacitances for the Thomas-Fermi-Dirac model,which includes only kinetic and exchange energy densities, haverecently been calculated69 for the metal surface, representing theelectrolyte as a charged plane 5 A from the edge of the metal'spositive charge density, represented as a step function. Althoughno interaction between metal electrons and solvent species wasincluded, the results suggest that the metal can make a significantcontribution to the capacitance of the interface.

With the addition of a pseudopotential interaction betweenelectrons and metal ions, the density-functional approach has beenused82 to calculate the effect of the solvent of the electrolyte phaseon the potential difference across the surface of a liquid metal. Thesolvent is modeled as a repulsive barrier or as a region of dielectricconstant greater than unity or both. Assuming no specific adsorp-tion, the metal is supposed to be in contact with a monolayer ofwater, modeled as a region of 3-A thickness (diameter of a watermolecule) in which the dielectric constant is 6 (high-frequencyvalue, appropriate for nonorientable dipoles). Beyond thismonolayer, the dielectric constant is assumed to take on the bulkliquid value of 78, although the calculations showed that the dielec-tric constant outside of the monolayer had only a small effect onthe electronic profile.

The ionic profile of the metal was modeled as a step function,since it was anticipated that it would be much narrower than theelectronic profile, and the distance dx from this step to the beginningof the water monolayer, which reflects the interaction of metal ionsand solvent molecules, was taken as the crystallographic radius ofthe metal ions, Rc. Inside the metal, and out to <i1? the relativedielectric constant was taken as unity. (It may be noted that thesecalculations, and subsequent ones83 which couple this model forthe metal with a model for the interface, take the position of theouter layer of metal ion cores to be on the jellium edge, which isat variance with the usual interpretation in terms of Wigner-Seitz


cells58 wherein the metal centers are half a cell diameter in.) Insome calculations, dx was reduced to a fraction of the crystallo-graphic radius, to model interpenetration of the metal ions andsolvent molecules.

The electron density profile was assumed to have the exponen-tial form

The requirements that n and its slope be continuous at z0 allowthe expression of A and B in terms of a, b9 and z0; a third condition,that the total charge density of the interface be equal to zero, gives

zo=l/a-l/b + qs/pb

The remaining parameters, a and 6, are determined variationallyfor any value of qs by minimizing the electronic surface energy,which is the difference between the total electronic energy of thesystem and the electronic energy of a homogeneous electron gasof density pb extending from z = —oo to z = 0, the position of thepositive-charge background profile. The total electronic energy itselfis infinite for this system.

In the electronic surface energy, the kinetic, exchange, correla-tion, and inhomogeneity energies of the electrons were taken asfunctional of the local electronic density (and, in the case of theinhomogeneity energy, of its gradient). Using the Thomas-Fermiform for the density of electronic kinetic energy, Kkn

5/3 with Kk =3(3TT2)2/3/10, the contribution of electronic kinetic energy to thesurface energy is

Kk\ [n-(z)5/3-pbd(-z)]dzJ - o o

where 6(z) is a step function. The electronic exchange energydensity is -Kan

4/3 with Ka = | (3 /TT) 1 / 3 , the electronic correlationenergy density is - |[0.115n - 0.0311n In (3n/47r)]l/\ and theinhomogeneity energy density is (12n)~1(dn/dz)2. The electrostaticenergy includes the interaction of the electrons with each other andwith the charges of the ion cores (in a step-function distribution).A pseudopotential ion-electron interaction was included by writing

62 Jerry Goodisman

the total ion-electron interaction potential in the Heine-Animaluform84

-Zjr, r>Rm

with given values for the parameters Ao and Rm, and averagingW(r) over the ion distribution. Finally, a direct repulsive interactionbetween the conduction electrons and the cores of the watermolecules was included in the electronic energy, as

Values for the parameter A were chosen without much justification.With this model, electron density profiles were calculated82 for

the bare metal surface and for the metal in the presence of thesolvent. From these profiles, surface potentials were computed bythe integrated form of the Poisson equation. For the seven metalsinvestigated by this model, the effect of the dielectric film is tolower ^M by some tenths of a volt, as shown in Table 1. This is adirect effect of the penetration of the tail of the electron densityinto the region of higher dielectric constant; the change in theelectron density profile is very small. The effect of the repulsivebarrier is also to lower %M by some tenths of a volt, but this loweringis due to the deformation of the electronic profile by the repulsion:the profile is pushed inward toward the metal, which reduces thedifference between electronic and ionic densities and lowers thesize of the surface potential. The lowering, which was found to be

Table 1Change in Metal Surface Potential Due to

Dielectric Film and Barrier

Metal

HgCdInZnPbGaAl

8x™ (film)(V)

-0.24-0.45-0.63-0.67-0.71-1.01-1.47

8x™ (barrier)(V)

-0.21-0.53-0.68-0.68-0.71-0.83-0.92


proportional to the value chosen for A, was greatest for the metalsof highest electron density and smallest Rc. Although high electrondensity and low Rc normally go together, it was established bycalculations82 that the value of Rc, which determines the distanceof the water monolayer from the metal, was the more importantfactor.

It is interesting that the lowering found for Hg and Cd wasclose to the values suggested for these metals from experimentalresults.85 Trasatti3'26 derived a value of -0.31 V for 8%sg near thepoint of zero charge from adsorption measurements of aliphaticalcohols on the mercury-water and mercury-air interfaces. (Hefound no reliable information on adsorption on other metals andnoted that his assumption that the adsorbing polymers of differentchain lengths had a constant orientation on the surface might beunreliable for more hydrophilic metals; as we have emphasized, itis necessary to make some assumption to separate metal and solventcontributions.) This is quite close to the value of -0.24 V obtainedfrom the dielectric film model.

However, consideration, for a series of metals, of the differenceof X = 8%™ ~ gs(dip) from its value for Hg, which can be derivedfrom measurements of potentials of zero charge and work functions,suggests6 that these calculations82 overestimate the importance ofthe metal electrons to the properties of the interface. If X is plottedversus metal electron density, the data follow no trend, as they doif X is plotted versus some quantity measuring the affinity of themetal for oxygen; this shows the importance of adsorption andgs(dip) in X. Trasatti3 plotted the calculated 8\™ versus X(M) -X(Hg). The plot suggests that these calculations overestimate thesize of 8x™ for higher-density metals; for calculated 8%™ to beconsistent with experiment, gs(dip) would have to be less for othermetals than for mercury.

The successes of hydrophilicity scales in correlating much datamean that one should not underestimate the importance of gs(dip).A plot of AHf (heat of formation of metal oxide, a measure ofhydrophilicity) versus X(M) - X(Hg) shows two lines. Preferentialorientation increases with oxygen affinity. Correlations betweenAHJ and X(M) - X(Hg) exist also for solvents other than water,with the rate of increase of X(M) - X(Hg) with A//° being strongerin the sequence acetonitrile < H2O < DMSO, with increasing

64 Jerry Goodisman

orientational polarizability and dielectric constant; the polarizabil-ity of the metal should not be very different for the three solvents.Note that, in the dielectric film model, the solvent dielectric constantis taken as the electronic dielectric constant, which should not varymuch from solvent to solvent.

It may be noted that the dielectric film model seems to givethe correct 8x™ for Hg, and too large a size for other metals: thismeans the electrons stick out too far for the higher-density metals.As will be seen below, an important term, the repulsion due toinner-shell electrons of solvent molecules, is left out in this model.For transition metals, X = 8x™ - gs(dip) is constant while A///changes, suggesting that 8x™ is of minor importance and gs(dip)0

is metal independent. (Note that, for transition metals, ionic radiiare fairly constant, as are electron densities.) This is ascribed3 tothe different nature of the bond to water, which is here strongenough to hold the water in its orientation (A/// is also muchhigher). For sp metals, the surface bond is of an ionic type, whosestrength is a strong function of surface electronegativity. For dmetals, chemisorption prevents much reorientation of molecules byan electric field.

The model for the metal electrons was used30 in a considerationof the capacitance, i.e., the effect of charging of the system on theelectronic distribution. The solution charge density qs was supposedto be located in the plane which bounds the dielectric layer ofwater: in concentrated solutions, Gouy-Chapman theory showsthat the width of the ionic charge distribution becomes small. Thedielectric constant was supposed to be unity from z = 0 to z = dx

(related to the crystallographic radius of the metal ions), equal toex = 6 for z between dx and d2 (d2 - dx = width of a watermonolayer), and equal to e2 = 78 for z beyond z2. For each choiceof gs, the parameters a and b of Eq. (38) were found by minimizingthe surface energy with, of course, the proper value of qM = -qs

imposed. (For a = b, the explicit form of the profile function forqM T* 0 was given by Partenskii and Smorodinskii.86) The electron-water interaction is a 8-function pseudopotential, with a value ofthe multiplying parameter chosen to be "reasonable."

The potential drop across the double layer, AV, was thencalculated in terms of a, b, qs, pb, dx, and d2. All the terms in AVwhich depended on a and b were assigned to x™ or givi(dip) [see


Eq. (7)]; the remainder, which depends explicitly on qs9 was taken

as the ionic or free-charge term:

g^(ion) = -47rqs[(-qs/2Pb) + d, + (d2 - </,)/£,] (39)

The quantity gM(dip) vanishes when a and b become infinite,corresponding to step-function profiles for electrons as well as forions. It is assumed that d1 and d2 are unchanged on charging (noelectrostriction). The reciprocal of the capacitance involves onlytwo terms, since solvent is not considered explicitly:

C 1 = (dA V/dqM) = C(ion)-1 + C(dip)"1

where

C(ion)" 1 = {4<Tr/Ex){d2 - dx) + Airdx + 4>rrqM/pb

(qM = -qs). It may be noted that C(dip) turns out to be negative,because an increase in qM (charge density on the metal) and in A Vis associated with a decrease in the surface dipole of the metalsurface x™, which makes a negative contribution to AV.

With ex = e2 = 1, capacitances were very small compared to

experimental ones, so that Cs(dip) would have to be much smallerthan generally accepted values to get agreement with experiment.With a dielectric present but no 8- function barrier, more reasonableresults were obtained. The addition of the barrier changed 8%M

9

but had little effect on capacitances. Of course, these calculationsare of interest only in comparing the contributions of two metals,or in investigating the importance of modifications of the model,since there is no solvent.

The main result of this work was stated to be that largercapacitances are found for Ga than for Hg because Ga has a smallercrystallographic radius, allowing closer approach of the Helmholtzplanes in the electrolyte to the metal. Thus, capacitances for mercurywere about half those for gallium, the difference being due to thelarger pb and smaller dx for the latter metal. This means that, evenneglecting the contribution of the dipoles of the electrolyte phase,the capacitance of the gallium-solvent interface would be greaterthan that of the mercury-solvent interface, as is observed experi-mentally. It is interesting that, combined with a solvent-dipolecontribution of 25 F/cm2, the capacitances calculated in thismodel for the metal give capacitances for the interface about the

66 Jerry Goodisman

same as those measured for the Hg-solvent and Ga-solvent inter-faces at the point of zero charge.

However, one cannot reproduce the capacitance as a functionof charge by combining this model with a constant solution capacit-ance. The explanation for the different capacitances found fordifferent metals in the interface is usually considered to lie in thedifferent interactions of solvent dipoles with different metals. Frum-kin et a/.34'87 have ascribed the observed larger capacitances for thegallium interface to the larger interaction of the dipoles of thesolvent with the gallium surface. The present work does not invali-date this but mainly points out that a direct effect of the metalought also to be considered.

The effect of decreasing dx from dx — Rc was investigated: thecapacitance of the metal surface is much increased when dx

decreases, as this gives rise to a substantial repulsion due to theelectron-molecule interaction. Apparently, there is then a largebarrier to deformation of the electronic profile by an electrostaticfield, so that AV is less changed by a change in qs, and thecapacitance is increased. The variation of metal capacitance withsurface charge was also shown to be substantial. However, the largedifference between mercury and gallium capacitances remainedover the entire range of qM investigated. The fact that, experi-mentally, capacitances of the metal-solution interface are found tobecome independent of the nature of the metal for qM <-15/AC/cm2 was noted. The explanation for this, it was stated,must come from a model including both solvent and metal.

It is also interesting that if some of the simple models for thebare metal surface are used to calculate the metal's contribution tothe capacitance, a fit to experimental results would require unrea-sonable values for the solution contribution. Thus, the simpleThomas-Fermi result88 of C(dip) = 4TT/ ATF (ATF = Thomas-Fermiscreening length) is greater than (^(experiment)"1, and the same istrue for the improved Thomas-Fermi results of Newns40 and themodel of free electrons at an infinitely repulsive wall [see Eq. (12)].These models are thus considered to be less realistic than the modelof this work.30

It was also noted that for any model including the extensionof the conduction-electron tail into the interface, the electric fieldat the dipoles of the first solvent layer will not be zero for zero


surface charge, because some of the metal's electron density willbe found on the other side of these dipoles from the metal. To havezero electric field acting on the solvent dipoles, one requires aslightly negative value of qM. It is, in fact, found experimentally89

that the maximum entropy of formation of the compact layer occursfor qM equal to about —5 fiC/cm2. Assuming that the mercurymakes a negligible contribution to the charge dependence of theentropy of formation, the entropy must be associated with disorderof the water in the inner layer. A simple picture of water moleculesorienting in a field would associate maximum disorder with zerofield. These calculations emphasize that, at qM = 0, one does nothave zero field at the dipoles, precisely because of the penetrationof the solvent layer by the electron tail. In fact, the value of qM

required to give zero field at the position of the dipoles turns outto be negative, several /i,C/cm2, in agreement with the position ofmaximum entropy. Of course, it is possible for models for theinterface which ignore the metal to obtain a similar result: thefour-state model of Fawcett et al.15 found the maximum entropyof the solvent monolayer (not the configurational entropy) to occurat about the same charge density.

Trasatti3 suggested that the contribution of the metal to thecapacitance was also exaggerated by these calculations. He wrotethe reciprocal of the capacitance of the interface as the sum ofthree terms:

1/C = l /C M -1 /C d i p +1/K i o n

with the free-charge contribution, Kion, usually considered to beindependent of qM, equal to e/d2 (e = dielectric constant of theinner layer). Interfaces involving Hg and Ga(l) exhibit the samecapacitance for large negative qM, although the electronicpolarizabilities for the metal should be quite different. At largenegative qM, if the contribution of the metal is ignored, the experi-mental inner-layer capacitance is just Kion, since the orientation ofdipoles can make little contribution. For Hg, the experimentalinner-layer capacitance for qM < -10 fiC/cm2 is 16.8 /xF/cm2. Ifd2 is taken as 0.31 nm (diameter of a water molecule), e comes outto be 5.9, a reasonable value for the electronic or high-frequencydielectric constant. Thus, it seems that the metal contribution isunimportant.

68 Jerry Goodisman

(Defenders of the importance of the metal may respond bynoting that for qM large and negative, one is trying to push theelectronic tail far out from the metal, which becomes increasinglydifficult because of the repulsive barrier due to the inner shells ofthe solvent molecules. Eventually, making qM more negative willproduce little change in the electronic tail and dxM/dqM, which is1/CM, will be small for all metals.)

It must be noted that there are a number of more or lessarbitrary assumptions made in this work30'31 which need jus-tification, as well as parameters whose values should be calculatedrather than assumed. For instance, the importance of the distancedu taken as equal to Rc, has been mentioned. In principle, thevalue of this distance is a consequence of the forces betweencomponents of the metal and molecules of solvent, and would becalculated in a consistent model of the complete interface. Thiswas pointed out by Yeager,18 who noted that the electron densitytail of the metal determines the distance of closest approach ofsolvent in the interface, as well as the behavior of the solvent dipoleson the surface. Since changing qM will move the electron densitytail in and out, dx should depend on the state of charge of theinterface. In fact, it turns out31 that if dx varies linearly with surfacecharge according to

d, = Rc - (Rc/S0)qM

the dependence of the metal's capacitance on qM is removed.

3. Metal-Solvent Distance

That dx should depend on the state of charge of the electrode maybe argued36 as follows: Charging of the electrode shifts the centerof gravity of the conduction-electron tail but not the center ofgravity of the image charge corresponding to a charged adsorbedspecies. Thus, adsorption of an electropositive species causes theelectron tail to move outward, and adsorption of an electronegativespecies causes it to move inward, toward the metal. The "effectivemedium theory" of adsorption states that the position of an adsor-bate atom relative to the metal surface is determined by the optimumelectronic charge density for the bonding interaction, so that anadsorbed species will move with the electron tail.


The length dx is just the distance between the metal surface(positive background density) and the layer of adsorbed watermolecules. The change of this distance with surface charge of theelectrode was adduced by Kornyshev and Vorotyntsev90 as thecause of the "capacitance hump." Explanation of this maximumin the plot of capacitance as a function of surface charge, occurringfor positive values of qM for many metal-liquid electrolyte inter-faces, has long been a preoccupation of theoretical elec-trochemists91; naturally, past explanations have been in terms ofthe displacements and orientations of solvent species.

Recently, another explanation for the hump, involving thedistortion of the metal's conduction-electron tail, has been pro-posed92 by Schmickler and Henderson. (Although previousworkers30'31 had found a metal contribution to the capacitancewhich increased monotonically with qM, it was suggested that thiswas because the assumption for the electron density profile wastoo simple.) Schmickler and Henderson92 noted that preferentialadsorption of water with the oxygen end toward the metal wouldshift the maximum in the capacitance toward negative charges,since a negative value of qM would be required to give zero netorientation, for which the capacitance is largest. DifferentiatingV(-oo) - V(oo) [see Eq. (6)] with respect to qM, one obtains 1/4TTC

as a sum of three terms, of which the metal contribution is

1/4TTCM = -d\ \ zpM(z) tL J —oo

where pM is the charge density of the metal. One writes pM asPM + #PM, with pM, the density profile for qM = 0, being indepen-dent of qM by definition. Then, invoking the Budd-Vannimenustheorem [Eq. (31) et seq.], one has

c °° ~i /1 / 4 T T C M = - (qM/epb) - d\\ z8pM(z) dz\ dq

The first term tends to make the capacitance greater for increasingqM. For the interface in vacuum, this term is outweighed by theother, so that the calculated capacitance decreases with qM, reflect-ing the fact that it is easier to spill more electrons out into thevacuum than to push them back into the metal againtst the repulsiveforces. In the metal-solution interface, however, it is surmised92

70 Jerry Goodisman

that the short-range repulsion of the solvent molecules makes itharder for the electrons to expand outward, restoring the originalsituation of CM increasing with qM. This will lead to a capacitancemaximum for positive qM.

The dependence of dx on qM is central in a model, proposedby Price and Halley,93 for the metal surface in the double layerwhich is related to that discussed above. The positively chargedion background profile p+(z) is assumed uniform, with a valueequal to the bulk density pb, from z = —oo to z = 0, with theelectronic density profile n(z) more diffuse. In contrast to theprevious model30 which emphasizes penetration by the conductionelectrons of the region of solvent, this model93 supposes that thedensity profile n(z) is zero for z > dl9 where z > dx defines theregion of the electrolyte. Then the potential at dx is given by

x) = cf>0 - 4TT I ' dz I dz'[p+(z) ~ n(z)]J —oo J —oo

where </>0 is the potential inside the metal, for z -» -oo. Integratingby parts and using the fact that the field at dx is given by

£(<*,) = 477 P dz[P+(z) - »(z)]J-oo

and that the field vanishes at z -* -oo, one has

f.0(rf.) = <t>0 - di£(d,) + 4TT Z[P+{Z) - n(z)] dz (40)

J-oo

The last (dipole) term is supposed to be approximated by thetruncated Taylor series

z[p+(z) - nJ-OO

4TT Z[P+{Z) - n(z)]dz = -A^oJ

and dx is written as dx(0) - \fE(d1)9 emphasizing the dependenceof dx on the field E(dx).

Using qM = E(dx)/47r, one thus has

[C(dip)]"1 = - [d<t>(dx)/dq^ = 47rdx - (K/epb)327r2qM

where K is a dimensionless parameter, since (f>(dx) - <f>0 is the


potential drop across the metal. This predicts93 that [C(dip)]"1,which is identified as the compact-layer capacitance by theseauthors, should be linear in qM for small qM, the slope beinginversely proportional to pb and the intercept being related to theparameter dx, which is the separation between the edge of the metal(as defined by the ion cores) and the solution. (Alternatively, onecan say that one is studying the capacitance for qM small enoughto neglect higher terms in a power series for C~x in qM.) It wasalso argued that di(0) should increase with pb. Two formulas forK were presented on the basis of "slide" and "swing" models:K = l/87randK = 1/3 TT. The "slide" model assumed that the profileis displaced without changing its shape and the "swing" modelassumed that the electron density shifts out in such a way that themost distant part of the charge moves the most. Experimental resultsfor five metals, plotted as \/Cc versus p^1, show that neither is wellobeyed. It was suggested93 that band structure effects and surfacestates were playing a role. It was also noted that this model, whichneglects the compact layer of the solvent phase, could not explainthe temperature dependence of the capacitance and thus should beuseful only above 50°C.

Guidelli,16 reviewing work on the capacitance of the metal-electrolyte interface, writes the equation for calculating capacit-ances from models for the inner layer as:

l /C, = d(A"<t>)/dqM = (K^y1 + dg"i°(dip)/dqM

with Kion independent of qM. This assumes that the surface dipoleof the metal is independent of qM, in contradiction to the resultsjust discussed,31'93 which give an important contribution of theelectron tail to the capacitance. However, Guidelli points out thatthe model of Badiali et al31 predicts a capacitance which decreasesas qM becomes more positive, whereas the experimental curve forHg/H2O shows the opposite behavior, while the slide and swingmodels93 do show a capacitance which increases with more positiveqM but incorrectly predict that Cx is temperature independent. Themoral is perhaps that a good model for one part of the interfacecan only show whether or not there is a large contribution of thatpart to properties of the interface but cannot hope to reproducethese properties unless the other part of the interface is also welldescribed.

72 Jerry Goodisman

The interaction between the metal surface and a collection ofwater molecules needs to be considered more carefully.94 The inter-action of conduction electrons of the metal with molecules of theelectrolyte includes an electrostatic interaction with a fixed chargedistribution, which is attractive since the electrostatic potential islargest near the nuclei, an exchange interaction, which is attractiveif it is approximated by an exchange-correlation potential95 and isapparently more important94 then the electrostatic potential, and arepulsive interaction representing the effect of orthogonality of themetal electron wave functions to the closed-shell cores, which maybe represented by a pseudopotential. There is a also a polarizationinteraction, which has been represented by the use of a dielectricconstant different from unity. The exchange-correlation potentialis a local-density approximation, derived from the uniform electrongas result, i.e., d(nexc)/dn, where exc(n) is the exchange-correlationenergy per electron in a uniform electron gas of density n. It hasbeen shown31 that the electrostatic potential (or the charge distribu-tion of the water molecules which gives rise to it) does not needto be known very accurately in the variational calculations basedon density-functional theory: the presence of a layer of dipoles ora double layer of charges at \{dx + d2) (representing the solventmolecules) hardly affects the capacitance calculated for the metal.The representation of the other potentials must be consideredcarefully, as the effects on electron density profiles can be large.Because of their repulsive interaction with the solvent molecules,the electrons of the metal extend out less from the ions when themetal is in the electrochemical interface than for the free metalsurface.

4. Models for Metal and Electrolyte

Several workers have now presented models of the interface as awhole, treating metal and electrolyte phases simultaneously. Inprinciple, one should treat the entire interface, including speciesof both metal and electrolyte phases. Treatments attempting to dothis [see below, around Eq. (46)] have proved too difficult. Fortu-nately, it seems96 that the details of the metal's electronic profiledo not much affect the distribution of solvent species and vice versa.This allows separate solution of the problems for the two phases,


obtaining the distribution of particles of each one in the field dueto the other.

The metal-solvent interactions were put into the model of Priceand Halley93 in a later paper by Halley and co-workers,97 whichalso remedied some of the deficiencies of the original model, suchas the inability to calculate the slope of a plot of (Q)"1 versus qM

and the dependence of the compact-layer capacitance on crystalface. One can show in general [see Eq. (40)] that

f*2= 47rx2q

M - 47rqM x(p+ - n) dxJ -oo

where the position where the metal electron density vanishes andthe solvent-molecule density begins is here denoted by x2. There-fore, the inverse of the compact-layer capacitance Cc, which in thismodel arises from the metal alone, is given by

(Cc)~l = d[<j>0 - ct>(x2)]/dqM = 4TT[X2 - x + qM d(x2 - x)/dqM]

where x is the first moment of the charge induced on the metal:

x= x'8p(x') dx'/ \ 8p(x')dx' (41)J —oo / J -oo

Here 8p is the difference between the charge density at surfacecharge qM and the charge density at qM = 0. Both x2 and x dependonqM.

In Ref. 97, the value of x2 is found by minimizing the surfaceenergy of the interface, written as a sum of five contributions:

(i) The surface energy of the electron layer in the presence ofthe positive background (a semi-infinite jellium slab), calculatedself-consistently according to the Lang-Kohn theory with the local-density approximation.

(ii) The long-range interaction between solvent and metal,including the closed-shell pseudopotential repulsion, acting on theelectrons, and an image-force attraction of the solvent dipoles.Without the image term, no solvent-metal binding was found.

(iii) The electrostatic interaction of the ions of the electrolyte,supposed to be located at x2 + d, with the metal, given by

(iv) The contribution of the metal-ion pseudopotentials, actingon the metal electrons.


(v) The interaction between the solvent molecules and themetal-ion cores. Electron-solvent repulsions come from averaginga pseudopotential repulsion over the distribution of solventmolecules. Since the latter is a step function, the resulting potentialacting on the electrons is also taken as a step function.

Solvent and other contributions to the surface energy that areindependent of x2 need not be considered, since the surface energyis used only to find x2 by minimization. It is assumed that no changein the electronic structure of the metal-ion cores or in their distribu-tion occurs during charging.

The electron density profile is calculated by solving

(-2-V2 + v)cf>n = encf>n (42)

where v includes a Hartree (interelectronic repulsion) term, alocal-density approximation to the exchange-correlation potential,an attraction due to the jellium background, and a short-rangerepulsion from the solvent molecules. Thus, the presence of thesolvent is felt only through the repulsion term, which is writtenV06(x - x2), where 0 is a step function and Vo is a parameter. Sincethis term is considered large enough to exclude electron densityfrom the region x > x2, one can neglect screening of the Hartreeterm by the solvent, which was represented by the dielectric constant6 in the model of Badiali et al82 The procedure is to solve theLang-Kohn equations [see Eq. (30)] self-consistently for fixedvalues of x2 and qM. For each gM, the proper value of x2 is foundby minimizing numerically the surface energy. Then the inversecapacitance is found as 4TT[X2 - x + qM d(x2 - x)/dqM]. To com-pare with experiment, the Gouy-Chapman part of the capacitanceis subtracted from measured capacitances, since it is assumed thatthe compact-layer capacitance arises wholly from the metal.

For the solid metals Cu, Ag, and Au, x2 has been calculated97

for surface charges between —0.003 a.u. and 0.003 a.u. (1 a.u. =5700 fjuC/cm2). The slopes of \/Cc versus qM at qM = 0 are of theright size for Cu and several times too high for Ag and Au. Theinverse capacitances have values of about 10 a.u. and decrease withqM near qM = 0; they continue to decrease for positive qM but leveloff at about 20 a.u. for qM =* -0.002 a.u. For positive qM, near0.002 a.u., the inverse capacitances show maxima and minima as a


function of qM, which are stated by the authors to be real. Theexperimentally observed dependence of capacitance on crystal faceis reproduced.

Values for several parameters appearing in the ionpseudopotential and the solvent-dipole image forces, which appearin the surface energy, need to be chosen. In some cases, such asthe ion-electron pseudopotential, their values could be obtainedfrom other workers. For other parameters, such as the constant inthe image-force solvent-metal interaction, values had to be chosento fit experiment and were not always the expected ones: theconstant for the solvent-metal interaction corresponded to a solventdipole moment three times the actual one for water, suggesting thatother metal-solvent interactions were being "covered" by the func-tional form used. It was also noted97 that a theory including thepolarization of the solvent molecules in the compact layer must beincluded along with a treatment of the metal to explain the tem-perature dependence of the compact-layer capacitance. However,it was emphasized that the contribution of the metal to 1/CC is ofthe same size as the experimental inverse capacitance. Furthermore,the metal contribution gives, at least qualitatively, the correct chargedependence of capacitance as well as the correct trends with crystalface.

In later work by some of these authors,98'99 the model for therepulsive effect of solvent on metal electrons, previously taken asa square barrier starting at z = x2 and with the height a parameterto be chosen, is improved. This allows the use of the correct valuefor the solvent dipole in the image-force attraction of solvent tometal, rather than treating it as a fitting parameter. Two terms usedin the previous calculation are dropped as unimportant: thepseudopotential correction to the jellium potential and the solvent-metal ion-core interaction. The interaction between a metal electronand a solvent molecule is described by an effective potential depend-ing on the distance from the position of the center of mass of asolvent molecule and on the orientation of the solvent molecule.It includes the long-range dipole potential and the electronicpseudopotential calculated by replacing the actual charge distribu-tion of the water molecule by that of the isoelectronic neon atom,94

solving the Lang-Kohn equations to obtain wave functions for allelectrons of a neon atom, and deriving pseudo wave functions [Eq.

76 Jerry Goodisman

(18) above] which resemble the true wave functions far from theatom. From the equations satisfied by the pseudo wave functions,a pseudopotential, depending on the local electron density, isderived, with some approximations. A single parameter enteringthe pseudopotential is chosen so that a calculation using thepseudopotential yields the binding energy of Ne on jellium inagreement with the result of an all-electron calculation. This poten-tial is then averaged over the distribution of solvent moleculepositions and orientations, obtained from statistical mechanicalcalculations100 for a liquid of atoms, each of which is a hard spherecontaining a point dipole, at a hard wall. The solvent distributionis no longer a step function. The position of the hard wall withwhich the hard dipolar spheres are supposed to interact, xW9 replacesx2 as a parameter which must be found for each qM by minimizationof the surface energy.

For calculation of capacitances, another parameter isrequired—the point at which an ideal Gouy-Chapman system,which would have the same potential drop as the actual system,would begin. It is assumed that the difference between this pointand xw is independent of surface charge; for qM = 0, it can beobtained from the hard-dipole distribution. For Cd, this model gaveinverse capacitances which, for the range of qM investigated,differed from the experimental capacitances by only half as muchas did those obtained with the previous model.96 The variation ofCc with qM is correctly reproduced, although the errors in the valuesof Cc are still substantial. It was expected that the improvement inthe calculation of long-range interactions and inclusion of satur-ation effects in the dipolar response of the solvent would helpimprove agreement with experiment.

Kornyshev et al.76 proposed several models of the interface,including both orienting solvent dipoles and polarizable metalelectrons, to calculate the position of the capacitance hump.Although it had been shown32'79101 that this was one of the featuresof the interfacial capacity curves that should depend on the natureof the metal, available calculations did not give the proper positionof the hump. The solvent molecules in the surface layer weremodeled as charged layers, associated with the protons and theoxygen atoms of molecules oriented either toward or away fromthe surface. These layers also carried Harrison-type pseudopoten-


tials84 associated with the oxygen cores and the electrons aroundthe protons, which interact with the metal conduction electrons.Parameters related to the metal-electron profile and to the solvent-molecule layers were chosen by minimization of the surface energy.Capacitance-charge curves were calculated and discussed.However, due to the crude way in which the solvent was represented,the shapes of the capacitance-charge curves were quite incorrect.

Improving the model, Schmickler32 considered a jellium forx < 0, a layer of point dipoles at xx, and a layer of counterioncharges at x2 (outer Helmholtz plane). The inner layer of solventmolecules was represented by a hexagonal lattice of point dipoles,each of magnitude p0, and with lattice constant a. Schmickler used6.12 x 10~30Cmfor/70and3 x KT10mfor a, giving a surface densityof 1019m~2. Each dipole had three possible orientations and socould be represented as a spin-1 system. The interaction of eachdipole with its neighbors was written as a spin coupling, SXS2J, andthe interaction of a dipole with non-nearest neighbors was represen-ted as a mean-field interaction. In the mean-field theory, which wasshown to give results identical to a Monte Carlo method for fieldsbelow 1010 V/cm, the field of the dipoles is just times (po/4'7Teo)(s) Zr rT3, where (s) is the average of the cosine of the angle betweena dipole axis and the field direction, and r, is the distance of dipolei from that being considered (the sum is over all other dipoles).The quantity (s) is given by the Langevin expression, coth x - x~\where x = (po/kT)Eext and the field Eext is that of the dipoles plusthat due to free charges. The mean-field approximation seems towork quite well for fully orientable dipoles at small fields, althoughit is known to give poor results for two- and three-state dipolemodels.

The electrons of the metal were treated as a plasma of densityn(x), described by the one-parameter profile n+(l - \ebx) for x < 0and \n+ e~bx for x > 0. The parameter b was calculated by minimiz-ing the electronic surface energy for a particular value of surfacecharge or dipole field. The interaction energy of the electrons withthe point dipoles of the solvent turned out to be quite small.32 Thesurface energy was written as a density functional, includinggradient terms, and with an external field, but not including ametal-ion pseudopotential. The external field was that of the surfacecharge density plus that generated by the solvent dipoles. The latter

78 Jerry Goodisman

was found by calculating the potential difference across the point-dipole layer. The potential drop across the solvent layer,-Npo(s)/eo, was spread out over the region 0 < x < a to take intoaccount the finite size of solvent molecules. Self-consistent solutionsfor the electron distribution and the dipole field were found. Theparameters in the model included the value of the dipole on asolvent molecule, the lattice parameter for the solvent molecules,and the locations of the inner and outer Helmholtz planes (thesepositions were also assumed to be independent of surface chargedensity). From the electronic charge distribution and calculated(s), the potential drop across the interface, A VM, was calculated32

as a function of surface charge, and in particular for zero surfacecharge.

This model was first used102 to investigate the dependence ofthe potential of zero charge on the metal. Only the bulk-metalelectron density changes from metal to metal. For a metal electrodeconnected to a standard electrode to form an electrochemical cell,the potential of zero charge is given by Eq. (2) or, for a metal M,

£Pzc = ~XM + XmS) ~ XS(M) + F-(4>M - d>ref) (43)

where <£>ref, the work function of the reversible electrode (foremission to solution), is given by FAsef</> — ixrf and is independentof M. The potential drop across the interface, gs*(dip) =^M(S) _ ^s(M) w a s c a i c u i a t e c i . 1 0 2 The quantities * M and <!>M werecalculated for the jellium using the density-functional model in theabsence of external fields, as had been done by Smith,54 who firstsuggested it. The value of <l>ref was taken from experiments: thereference electrode was taken to be H + /H 2 , so 3>ref was taken as-4.5 eV, and * s was taken as -0.13 V.

Calculated work functions, potentials of zero charge, andabsolute values of surface potentials all increase in size with bulkelectron density (the only parameter changing from metal to metal).Calculated values for the potential of zero charge were too low byabout a volt, but when experimental values for <I>M were substitutedfor calculated values, agreement of Epzc with experiment was good,with errors of only several tenths of a volt. Furthermore, the changesfrom metal to metal were generally reproduced, as shown in Table2. In general, the potential of zero charge becomes more negativeas the electron density of the metal increases. Note that there is no


Table 2Potentials of Zero Charge for

M| sol, H+|H2|M'

Metal

HgGaCdInPbSnTIAgAu

£exp(V)

-0.19-0.69-0.75-0.65-0.56-0.38-0.71-0.60

0.19

£Cdlc(V)

-0.26-0.79-0.67-0.80-0.70-0.52-1.06-0.56

0.28

repulsive interaction between metal electrons and the solvent inthis model, so that the effect of the solvent dipoles is to pull theelectrons out of the metal and increase the metal's surface potentialrelative to ^M , its value for the free metal surface.

As noted by Badiali et al.,30'82 there is a positive electric fieldat the position of the dipoles when the interface is at the point ofzero charge because of the spillover of the electron density pastthe dipoles. This produces a small negative dipole field, correspond-ing to preferential orientation of the oxygen end of the watermolecules toward the metal. Because of this dipole field, </>met - </>so]

at the potential of zero charge is smaller than the surface potential* M for the bare metal surface, even though the attractive effect ofthe solvent dipoles on the metal electrons makes the metal contribu-tion to <f)met - </>soi larger in the interface than in the free surface.The solvent dipole terms are important: neglecting them wouldmake Epzo = <&je - 4.63 V [see Eq. (43)], which, for all the metalsexamined but one, is several tenths of a volt higher than theexperimental value.

Capacitance as a function of charge was calculated.79 Thecapacitance curves showed a single hump, near qM = 0, and leveledoff for qM about 10 /iC/cm2 on either side of the potential of zerocharge, due to the dielectric saturation of the dipole system. Thelimiting values of the capacitance increased with increasing electrondensity of the metal. The nonideality of the metal was shown to

80 Jerry Goodisman

increase the capacitance in two ways: because of the penetrationof the electric field into the metal (increase in the effective valueof the width of the capacitor d), and because of the distortion ofthe electronic cloud of the metal by the dipole field (change ofparameters in the electronic profile).

A more realistic model of the interface, which combines thejellium model for the metal with a modern picture of the electrolytesolution, was used by Schmickler and Henderson103 to calculatethe capacitance of the interface. The ions of the electrolyte arecharged hard spheres and the solvent molecules hard spheres withpoint dipoles at their centers. An approximate solution for thedistribution functions of hard-sphere dipoles and ions bounded bya charged hard plane, valid for small surface charge densities, hadpreviously been derived.102 The electrons of the metal are describedby the usual density functional, with three terms representing theirinteraction with the solvent. Two of these are the interaction withthe charge density and polarization (dipole density) of the elec-trolyte; as had been noted by others,31'96 the details of the interactionof the metal electrons with the solution species are unimportant,only the total field, or charge density, being important. The thirdterm is a repulsive potential energy barrier, represented as a stepfunction at z = 0 of height Vb. The value of Vb, which should berelated to the effective barrier for tunneling of metal electronsthrough a water layer, was taken as 3 eV for Hg.

In addition to Vb, the parameters for which values need to bechosen are the sizes of the hard spheres; other parameters are welldefined. Since the distance of closest approach of spheres to themetal is just the jellium edge, the distance of closest approach dx,which was a crucial parameter in other work,31'83 does not enter.(It is noted that, usually, the jellium edge is not considered torepresent the position of the last layer of metal ions. Rather theions are at the center of the Wigner-Seitz cell, or half a latticespacing behind the jellium edge,11'58 which ensures electroneutralityof the Wigner-Seitz cell. Vorotyntsev and Kornyshev6 have alsoargued that the edge of the jellium is one ionic radius beyond thelast plane of ions, giving the example of a lattice consisting of twomonolayers: if this is to be represented by a jellium slab, its thicknessshould be about two ionic diameters, whereas the choice of Badialiet a/.31'83 would make it half this.)


Calculations of the capacitance of the mercury/aqueous elec-trolyte interface near the point of zero charge were performed103

with all hard-sphere diameters taken as 3 A. The results, for variouselectrolyte concentrations, agreed well with measured capacitancesas shown in Table 3. They are a great improvement over what onegets104 when the metal is represented as ideal, i.e., a perfectlyconducting hard wall. The temperature dependence of the compact-layer capacitance was also reproduced by these calculations.

For metals other than mercury, it was assumed that Vb was3.0 eV plus the difference in work functions <£M - 4>Hg. The changesin Vb had little effect on calculated capacitances compared to theeffect of the changed bulk electron density nb. The capacitanceincreases with nb, eventually diverging and becoming negative fornb > 0.025 a.u. This unrealistic behavior may be related to thejellium model or to the form of the profile function used.96 Theincrease of capacitance with nb is not borne out by experiment, thedeviations between theory and experiment being worst for Sn, Pb,Bi, and Sb. This is ascribed103 to the fact that not all the valenceelectrons are free for these metals, so that the correct value to usefor nb should be less than what is calculated by multiplying thenumber of atoms per unit volume by the number of valence electronsper atom.

For solvents other than water, the model predicts, even in theabsence of calculations, that interfacial capacitances in any solventshould increase in the order Hg < In < Ga because of the increasingelectron densities.103 This is, in fact, the case for DMSO andacetonitrile as well as for water. From the model used for the

Table 3Capacitances for Mercury-Solution Interface

Concentration(M)

1(T3

io-2

io-2

1

Calc.

6.013.622.628.7

Capacitance (/x F/cm2)

Expt.

6132126

Ideal

5.19.8

13.815.9

82 Jerry Goodisman

distributions of hard spheres, one can also expect that, for a par-ticular metal, the capacitances should be smaller for DMSO oracetonitrile than for water, because the smaller size of the moleculesof the latter allows a closer metal-solvent approach. A value forthe barrier height Vb of 1.8 eV was used for Hg/DMSO and a valueof 2.25 eV for Hg/acetonitrile, and Vb for other metals was obtainedfrom that for Hg using the calculated jellium work functions.Calculated inner-layer capacitances for these other solvents agreedreasonably well with experiment. In these calculations, the interac-tion of the polarization with the electric field of the electrons spillingpast the first layer of solvent molecules was neglected, as it playedonly a minor role in the water calculations.

The model was also extended11 to single-crystal surfaces ofsilver. Although the calculated inner-layer capacitances varied inthe right way from one face to another, the values were much toolow. The problem was suspected to be due to the importance ofthe d electrons. What is still needed in this model is a bettertreatment of the solvent phase, valid at higher charge density, anda better way of deriving the repulsive potential of the solvent onthe electrons, perhaps by a direct pseudopotential calculation, asdone by Price and Halley.98'99

Another model which combined a model for the solvent witha jellium-type model for the metal electrons was given by Badialiet al83 The metal electrons were supposed to be in the potential ofa jellium background, plus a repulsive pseudopotential averagedover the jellium profile. The solvent was modeled as a collectionof equal-sized hard spheres, charged and dipolar. In this model,the distance of closest approach of ions and molecules to the metalsurface at z = 0 is fixed in terms of the molecular and ionic radii.The effect of the metal on the solution is thus that of an infinitelysmooth, infinitely high barrier, as well as charged surface. Thesolution species are also under the influence of the electronic tailof the metal, represented by an exponential profile.

Previously derived results for the charge-density profile andpolarization profile in this model solution, valid only for smallfields, were used. Although these did not consider the penetrationof the electrons into the solution, the change in the field is small.A Harrison-type pseudopotential84 was used to represent the effectof core electrons of the solution species on the metal electrons.


This was averaged over the total distribution of ionic and dipolarspheres in the solution phase. Parameters in the calculations werechosen to simulate the Hg/DMSO and Ga/DMSO interfaces, sincethe mean-spherical approximation, used for the charge and dipoledistributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily.It was found that the calculated capacitance depended crucially ond, the metal-solution distance. However, the capacitance wasalways greater for Ga than for Hg, partly because of the differentelectron densities on the two metals and partly because d dependson the crystallographic radius. The importance of d is specific tothese models, because the solution is supposed (perhaps incorrectly;see above) to begin at some distance away from the jellium edge.

It was later shown94 that the use of a density-functional theoryled to errors in the calculated surface potentials, although Smith81

had shown that variational calculations within the density-func-tional formalism could give profiles close to those obtained frommore exact calculations. Furthermore, the variational method,which assumed a two-parameter functional form for n_(z) andallowed variation only of these two parameters, could also be acause of error. For the bare surface of Ga, the value of ^M obtainedfrom the Lang-Kohn self consistent theory is 9.48 V, more than avolt below the value obtained from the two-parameter variationalcalculation.94 Furthermore, if one uses a non-step density profilefor the ions, results are also changed significantly.61105 In fact, itseems necessary to use oscillatory ion-density profiles62 to obtainwork functions for the bare metal surface in agreement with experi-ment. Calculations for the capacitance of the metal surface in theinterface which use such profiles have not been performed, partlybecause one would have to make an assumption about how theprofiles change with charging.

5. General Dielectric Formalism

We have already mentioned a more general approach24 to calculat-ing the properties of the interface, which is elegant but apparentlydifficult to realize. The Poisson equation for the statisticallyaveraged electric field E,

V • E = 4TT(P + pext) (44)

84 Jerry Goodisman

where pext is the charge density of the external field sources andp(r) the averaged density of induced charge due to polarization ofthe medium, describes the entire interface and adjoining phases.One requires, in addition, a phenomenological relationship betweenp and the electrical potential <£, where E = -V • </>. Alternatively,one can deal with the polarization field P(r), where p = -V • P,and assume a dependence of P on E. The relation between p and<f> is usually taken as linear and local, so that p at any point isproportional to the electrostatic potential at that point. Kornyshevand Vorotyntsev24 noted that, although nonlinear behavior of thesolvent phase of the interface had generally been considered, nowork had been published in the electrochemical literature on non-linear response of the metal because the metal was considered asmerely a region of constant potential, as in the original Gouy-Chapman theory.106 Although models of the interface generallyassume a local dielectric constant, it may not always be possible48

to replace the exact nonlocal theory by a local one, even if thedielectric constant is allowed to be position dependent. In a non-local theory,

p(r) = J dr'a(r,r'W)

The mathematical methods for treating problems in nonlocal elec-trostatics were discussed by Kornyshev et al107

For a metal in contact with a dielectric, the fields arise fromspecies of both phases. Kornyshev and Vorotyntsev24 speculatedon the effect of the dielectric on the electron density, pointing outthat there should be a long-range interaction with the polarizationof the dielectric, as well as a quantum interaction associated withoverlap between the metal electron's tail and the occupied electronicorbitals of the dielectric. The former tends to spread out the elec-tronic tail as compared to the tail in vacuum, since it screens theinteractions between the electronic charge and the ion cores of themetal, while the latter pushes the electronic tail back toward themetal. Other authors have incorporated these two effects in theirmodels, as discussed above. In general, it is impossible to predicta priori which effect will dominate, but Kornyshev and Vorotyntsevsuggested24 that the spreading will win out if the electronic affinityfor solvent molecules is greater than the work function. In the linear


approximation, both effects may be described in terms of the (non-local) dielectric function of the system, e(z, z', r - r'), where thedielectric function108 relates the electric displacement to the electricfield according to

rdr' e,,(r, r')£,(r')

Thus, the electric displacement at r is linearly related to the electricfield at positions other than r; a local dielectric constant makesD(r) proportional to E(r). If one has planar symmetry, so that onlythe x- components of D and E are nonzero and these depend on xonly,

- JD(x) = j ex(x,x')E(x')dx'

after integrating over z' and y' where ex is an integrated dielectricfunction.

For small deviations from electroneutrality, the charge densityat x is proportional to -</>(x)//cT, where 4> is the difference of theelectrostatic potential from its (constant) value when there is nocharge density (the density of a species of charge z is proportionalto 1 - z<f>kT on linearizing the Boltzmann exponential). Then thePoisson equation [Eq. (44)] becomes the linearized Poisson-Boltzmann equation:

ex(x, xf)[-d<f>(xf)/dxf] rfx'J = K204>(X) (45)

With the proper definitions of ex and K0, this equation is applicableto the metal as well as to the electrolyte in the electrochemicalinterface.24 Kornyshev et al109 used this approach to calculate thecapacitance of the metal-electrolyte interface. In applying Eq. (45)to the electrolyte phase, ex is the dielectric function of the solvent,x' extends from 0 to oo, and x extends from L, the distance ofclosest approach of an ion to the metal (whose surface is at x = 0),to oo, so that K2

0 is replaced by K2O0(X - L). Here K0 is the inverse

Debye length for an electrolyte with dielectric constant of unity,since the dielectric constant is being taken into account on the leftside of Eq. (45). For the metal phase (x < 0) one takes sx as thedielectric function of the metal and limits the integration over x'

86 Jerry Goodisman

to -oo < x < 0. One thus has a sharp-boundary model, with nometal electron density entering the region of solvent.

One has to solve for <£(x) with <£(~°°) = K <f>(°°) = 0, and <f>and D continuous at z = 0. Since the effect of the metal electronsis incorporated into the dielectric function, there are no free chargesto consider in the metal, so that D is constant inside the metal,and the equation becomes

I0

£M(x, x'){d(t)I dx') dx' = constant (x < 0)

with eM the dielectric function of the metal. The equations aresolved109 for small K0, after insertion of a model which giveseM(x, x') in terms of the bulk-metal dielectric function, ebM(x, x')9

with Thomas-Fermi screening for the metal. The electron densityprofile for qM = 0 is approximated as n0 (1 + Aeax/3)~3 with a avariable parameter. In the electrolyte, the basic equation to besolved is

where e is the macroscopic dielectric constant, K l is the Debyelength, and L is the distance of closest approach of electrolyte ionsto the metal. This is solved with a simplified form for e(x, x'). Themeaning of the various parameters entering the model and theireffect on the results of calculations are discussed.109

The capacitance of the system can be written as that of fourcapacitances in series, i.e.,

Cl = C;1 + C s ! + CM1 + CGC (46)

where Cs arises from the dielectric function of the solution phase,CM is the contribution of the metal phase, CGC is the Gouy-Chapman capacitance, and Cs is the capacitance of the solventlayer between z = 0 and the distance of closest approach for ions(Stern layer). The inverse capacitance of the metal then turns outto be 4TTLM, where

i= (2/TT) dk[k2eM(k)]


with eM(fc) the Fourier transform of the bulk-metal dielectric func-tion ebM(x - x'). This characteristic length may be calculated byany of the models for screening in bulk metal, and turns out to beabout half an angstrom. In fact, Kornyshev et al.109 found that CM

was close to the Thomas-Fermi value for mercury when any of anumber of approximations to the dielectric function of mercurywas used. The resulting inverse capacitance is much larger thanexperimental capacitances, so that no positive values for the otherterms in Eq. (46) can give agreement with experiment.

It was concluded, on the basis of these calculations, that thisand related models are incapable of giving capacitances for themercury-aqueous solution interface in agreement with experiment.The explanation was considered most likely to be the assumptionof a sharp boundary between metal and electrolyte phases, whichdoes not permit the electron density tails to penetrate the first layerof solvent, where they would feel a dielectric constant considerablylarger than unity.24 Model calculations, using a single simplifieddielectric function e(x, x') such that e(x, xf) reduces to the dielectricfunctions of solvent or metal far from the interface, showed thatthe penetration of the electron tails into the solvent region indeedreduces the contribution of the electrons to the capacitance. Thisis obvious from the dependence of the capacitance of a planecapacitor on the interplate spacing. The capacitance is then given by

C 1 = CGC + 4TT I dx'\\ dxe-\x,x')-(\/e)6{x'-L)\

(47)

Here 6 is a step function, L is the distance of closest approach ofions to x = 0, e is the static dielectric constant [the k = 0 Fouriercomponent of s(x, x')L and the inverse dielectric function is definedby

r dz e{x", x)e~l(x, x') = 8(x" - x)

Only by smearing of the conduction-electron tail from the metalinto the electrolyte phase can the integral in Eq. (47) be made smallenough to obtain capacitances for the interface in agreement withexperiment.

88 Jerry Goodisman

The calculations were subsequently extended to "moderate"surface charges and electrolyte concentrations.8 The compact-layercapacitance, in this approach, clearly depends on the nature of thesolvent, the nature of the metal electrode, and the interactionbetween solvent and metal. The work8109 describing the electrode-solvent system with the use of nonlocal dielectric functions e(x, x')is reviewed and discussed by Vorotyntsev, Kornyshev, and co-workers.6'77 With several assumptions for e(x, xr), related to theThomas-Fermi model, an explicit expression6 for the compact-layercapacitance could be derived:

Here, /CQ l is the Thomas-Fermi screening length in bulk metal, ande0 is the effective dielectric constant of the region, of thickness 8,in which metal electrons and the polar liquid interpenetrate. Sinceeo> 1, Cc is increased relative to the Thomas-Fermi estimate,kJAir. This increase is thus due to the polarizability of the interlayerpenetrated by the electron tail of the metal, as was already includedin the first model of Badiali et al}2 Various limiting cases of thisequation are discussed.76 For increasingly negative qM, 8 increasesand the solvent parameter s0 becomes more important than themetal parameter k0; this is proposed as a reason why the compact-layer capacitance is observed to become independent of the metalfor large negative qM. A related model for the interface110 is dis-cussed by Vorotyntsev et al. and compared to theirs. Perspectivesfor development of self-consistent theories for metal and electrolyteof the compact layer are also discussed.

VI. CONCLUSIONS

One can say that the metal in the interface has come a long wayin the last ten years, from a featureless conducting medium to acomponent with its own complicated structure and role. Sinceseparation of properties of the interface into metal and electrolytecontributions cannot be done unambiguously by experiment alone,theoretical calculations on the metal surface have been necessaryto establish the importance of the contribution of the metal. It is


clear that the metal itself, aside from its influence on the electrolyte,makes a contribution to the capacitance of the interface and to thechange of the potential drop across the interface as one substitutesone metal for another. Accurate models of the metal in contactwith electrolyte are thus necessary for a correct understanding ofthe electrochemical interface.

Modern theories of electronic structure at a metal surface,which have proved their accuracy for bare metal surfaces, havenow been applied to the calculation of electron density profiles inthe presence of adsorbed species or other external sources ofpotential. The spillover of the negative (electronic) charge densityfrom the positive (ionic) background and the overlap of the formerwith the electrolyte are the crucial effects. Self-consistent calcula-tions, in which the electronic kinetic energy is correctly taken intoaccount, may have to replace the simpler density-functional treat-ments which have been used most often. The situation for liquidmetals, for which the density profile for the positive (ionic) chargedensity is required, is not as satisfactory as for solid metals, forwhich the crystal structure is known.

For the metal in the electrochemical interface, one requires amodel for the interaction between metal and electrolyte species.Most important in such a model are the terms which are responsiblefor establishing the metal-electrolyte distance, so that this distancecan be calculated as a function of surface charge density. The mostimportant such term is the repulsive pseudopotential interaction ofmetal electrons with the cores of solvent species, which affects thedistribution of these electrons and how this distribution reacts tocharging, as well as the metal-electrolyte distance. Although mostcalculations have used parameterized simple functional forms forthis term, it can now be calculated correctly ab initio.

What one requires is a self-consistent picture of the interface,including both metal and electrolyte, so that, for a given surfacecharge, one has distributions of all species of metal and electrolytephases. Unified theories have proved too difficult but, happily, itseems that some decoupling of the two phases is possible, becausethe details of the metal-electrolyte interaction are not so important.Thus, one can calculate the structure of each part of the interfacein the field of the other, so that the distributions of metal speciesare appropriate to the field of the electrolyte species, and vice versa.

90 Jerry Goodisman

Quantum mechanical calculations are appropriate for the electronsin a metal, and, for the electrolyte, modern statistical mechanicaltheories may be used instead of the traditional Gouy-Chapmanplus orienting dipoles description. The potential and electric fieldat any point in the interface can then be calculated, and all measur-able electrical properties can be evaluated for comparison withexperiment.

The calculations involved in a fully consistent model of theinterface are difficult, but, as may be seen in the work discussedabove, quite practicable. It may well be that when a complete model,calculating all terms properly and giving properties in agreementwith experiment, is available, it will appear that some of the sim-plifications and assumptions used in the less complete models ofthe past, such as separation into diffuse and compact layers, willstill be valid. However, it will no longer be possible to neglect thestructure of the metal and reduce it to an uninteresting idealconductor.

REFERENCES

1 J. O'M. Bockris and S. U. M. Khan, Quantum Electrochemistry, Plenum Press,New York, 1979, p. 17.

2 S. Trasatti, in Trends in Interfacial Electrochemistry, Ed. by A. Fernando Silva,Reidel, Dordrecht, 1986, Chap. I.

3 S. Trasatti, J. Electroanal Chem. 150 (1983) 1.4 J. Goodisman, Electrochemistry: Theoretical Foundations, Wiley, New York, 1987,

Chap. 3.5 O . K. Rice, Phys. Rev. 31 (1928) 1051.6 M . A. Vorotyntsev and A. A. Kornyshev, Elektrokhimiya 20 (1984) 3. [Engl.

transl.: Sov. Electrochem. 20 (1984) 1.]7 N. F. Mott and R. J. Watts-Tobin, Electrochim. Acta 4 (1961) 79.8 A. A. Kornyshev and M. A. Vorotyntsev, Can. J. Chem. 59 (1981) 2031.9 S. Trasatti, Modern Aspects of Electrochemistry, No. 13, Ed. by B. E. Conway

and J. O'M. Bockris, Plenum Press, New York, 1979, p. 81.10 S. Trasatti, Advances in Electrochemistry and Electrochemical Engineering, Vol.

10, Ed. by H. Gerischer and C. W. Tobias, Wiley-Interscience, New York, 1976.11 W. Schmickler, in Trends in Interfacial Electrochemistry, Ed. by A. Fernando

Silva, Reidel, Dordrecht, 1986.12 J. R. MacDonald and C. A. Barlow, Jr., /. Chem. Phys. 36 (1962) 3062.13 R. Parsons, in Modern Aspects of Electrochemistry, No. 1, Ed. by J. O'M. Bockris

and B. E. Conway, Plenum Press, New York, 1954, p. 103; R. Parsons, /.Electroanal. Chem. 59 (1975) 229.

14 R. Reeves, in Comprehensive Treatise of Electrochemistry, Vol. 1, Ed. by J. O'M.Bockris, B. E. Conway, and E. Yeager, Plenum Press, New York, 1980, p. 83.


15 W R Fawcett , S Levine, R M d e N o b n g a , and A D McDona ld , J ElectroanalChem 111 (1980) 163

16 R Guidelh, in Trends in Interfaaal Electrochemistry, Ed by A Fernando Silva,Reidel, Dordrecht , 1986, p 387

17 R Guidelh , J Electroanal Chem 123 (1981) 59, 197 (1986) 7718 E Yeager, Surf Sci 101 (1980) 119 N H March and M P Tosi, Coulomb Liquids, Academic , London, 1984, C h a p 820 N H March and M P Tosi, Ref 19, Section 8 321 A N Frumkin, B Damaskin , N Gngoryev , and I Bagotskaya, Electrochim Acta

19 (1974) 6922 M J Sparnaay, The Electrical Double Layer, Pergamon Press, Oxford, 1972,

Sections 3 2 and 3 323 S Trasatti , Colloids and Surfaces 1 (1980) 1732 4 A A Kornyshev and M A Vorotyntsev, Surf Sci 101 (1980) 2325 S Trasatti, in Comprehensive Treatise of Electrochemistry, Vol I, Ed by J O'M

Bockns , B E Conway, and E Yeager, Plenum Press, New York, 198026 S Trasatti, in Trends in Interfacial Electrochemistry, Ed by A Fernando Silva,

Reidel, Dordrecht , 1986, C h a p II27 A Hamelin, T Vitanov, E Sevastyanov, and A Popov, / Electroanal Chem

145 (1983) 22528 A Hamelin, in Trends in Interfacial Electrochemistry, Ed by A Fernando Silva,

Reidel, Dordrecht , 19862 9 S H Liu, J Electroanal Chem 150 (1983) 3053 0 J -P Badiah, M -L Rosinberg, and J Good i sman , / Electroanal Chem 143

(1983) 7331 J P Badiah, M L Rosinberg, and J G o o d i s m a n , / Electronal Chem 150(1983)

2532 W Schmickler, / Electroanal Chem 150 (1983) 1933 A M Kalyuzhnaya, N B Gngoryev, and I A Bagotskaya, Elektrokhimya 10

(1974) 1717 [Engl transl Sov Electrochem 10 (1974) 1628]34 A N Frumkin, N B Pohnovskaya, N Gngoryev , and I A Bagotskaya, Elec-

trochim Acta 10 (1965) 79335 G Valette, / Electroanal Chem 122 (1981) 28536 E M Stuve, K Bange, and J K Sass, in Trends in Interfacial Electrochemistry,

Ed by A Fernando Silva, Reidel, Dordrecht , p 25537 J R N o o n a n and H L Davis, Science 234 (1986) 31038 A Sommerfeld, Naturwissenschaften 15 (1927) 8263 9 C Kittel, Elementary Solid State Physics, Wiley, New York, 1962, C h a p 54 0 D M Newns , Phys Rev B 1 (1970) 33044 1 J M Ziman, Principles of the Theory of Solids, Cambr idge University Press,

Cambridge, 1964, C h a p 3 24 2 W A Harr ison, Solid State Theory, Dover Publicat ions, New York, 1979, C h a p

II43 M L C o h e n and V Heine , Solid State Phys 1A (1970)44 N W Ashcroft , Phys Lett 23 (1966) 4845 N W Ashcroft, in Interaction Potentials and Simulation of Lattice Defects, Ed

by P C Geh len , J R Beeler, and R I Jaffee, P l enum Press, N e w York, 197246 N H M a r c h and M P Tosi , Ref 19, Sections 2 1 a n d 3 147 J G o o d i s m a n , Ref 4, C h a p 4 B48 M A Vorotyntsev and A A Kornyshev , Electrokhimya 15 (1979) 660 [Engl

transl Sov Electrochem 15(1979)560]

92 Jerry Goodisman

49 J. Goodisman, Diatomic Interaction Potential Theory, Vol. I, Academic, NewYork, 1973, Chap. III.F.

50 N. H. March and M. P. Tosi, Ref. 19, Chap. 4.51 D. M. Kolb, in Trends in Interfacial Electrochemistry, Ed. by A. Fernando Silva,

Reidel, Dordrecht , 1986.52 J. D . Mcln tyre , Surf. Sci. 37 (1973) 658.53 Ref. 19, Section 6.1.54 J. R. Smith, Phys. Rev. 181 (1969) 522.55 W. Schmickler and D. Henderson , Phys. Rev. B 30 (1984) 3081.56 N . D. Lang and W. Kohn , Phys. Rev. B 1 (1970) 4555; N . D. Lang, Solid State

Commun.l (1969) 1047.57 N . D. Lang and W. Kohn , Phys. Rev. B 7 (1973) 3541.58 N. D. Lang, in Theory of the Inhomogeneous Electron Gas, Ed. by S. Lundqvist

and N . H. March , Plenum Press, New York, 1981.59 P. Hohenberg and W. Kohn , Phys. Rev. B 136 (1964) 864.60 J. Good i sman and M.-L. Rosinberg, J. Phys. C 16 (1983) 1143.61 J. Good i sman , Phys. Rev. B 32 (1985) 4835.62 M. P. D'Evelyn and S. A. Rice, / Chem. Phys. 78 (1983) 5081.63 R. N . Kuklin, Elektrokhimiya 14 (1978) 381.64 L. I. Schiff, Phys. Rev. B 1 (1970) 4649.65 D . E. Beck and V. Celli, Phys. Rev. B 2 (1970) 2955.66 J. Rudnick, Phys. Rev. B 5 (1972) 2863.67 S. C. Ying, J. R. Smith, and W. Kohn , /. Vac. Sci. Technol 9 (1972) 575.68 J. Heinr ichs , Phys. Rev. B 8 (1973) 1346.69 J. Good i sman , /. Chem. Phys. 86 (1987) 882.70 H. F. Budd and J. Vannimenus , Phys. Rev. B 12 (1975) 509.71 H. F. Budd and J. Vannimenus , Phys. Rev. Lett. 31 (1973) 1218, 1430.72 J. Vannimenus and H. F. Budd, Solid State Commun. 15 (1974) 1739.73 G. D. M a h a n and W. L. Schaich, Phys. Rev. B 12 (1975) 5585.74 J. Heinrichs and N . Kumar , Phys. Rev. B 12 (1975) 802.75 A. K. Theophi lou and A. Modinos , Phys. Rev. B 6 (1972) 801.76 A. A. Kornyshev, M. B. Partenskii , and W. Schmickler, Z. Naturforsch. A 39

(1984) 1122.77 M. A. Vorotyntsev, V. Yu. Isotov, A. A. Kornyshev, and W. Schmickler, Elektro-

khimiya 19 (1983) 295. [Engl . transl.: Sov. Electrochem. 19 (1983) 260.]78 M. A. Vorotyntsev, Elektrokhimiya 14 (1978) 911, 913. [Engl . transl.: Sov. Elec-

trochem. 14 (1978) 781, 783.]79 W. Schmickler, /. Electroanal. Chem. 149 (1983) 15.80 R. R. Salem, Zh. Fiz. Khim. 54 (1980) 212.81 J. R. Smith, in Interactions on Metal Surfaces, Ed. by R. Gomer , Springer, Berlin,

1975, C h a p . 1.82 J .-P. Badial i , M.-L. Rosinberg , and J. G o o d i s m a n , /. Electroanal. Chem. 130

(1981) 31 .83 J . -P. Badia l i , M.-L. Ros inberg , F . Vericat , a n d L. Blum, J. Electroanal. Chem.

158 (1983) 253.84 W. Harrison, Pseudopotentials in the Theory of Metals, Benjamin, New York,

1966, p . 57.85 J. O'M. Bockris and M. A. Habib, /. Electronal. Chem. 68 (1976) 367; S. Trasatti,

J. Electroanal Chem. 33 (1971) 351.86 M . B. Par tenski i a n d Ya. G. Smorod insk i i , Fiz. Tverd. Tela 16 (1974) 644. [Engl .

t ransl . : Sov. Phys. —Solid State 16 (1974) 423.]


87 A. N . Frumkin , B. Damaskin , N. Grigoryev, and I. Bagotskaya, Electrochim. Acta19 (1974) 75; B. B. Damaskin , /. Electroanal. Chem. 75 (1977) 359.

88 R. N . Kuklin, Elektrokhimiya 13 (1977) 1182, 1796.89 J. A. Harr ison, J. E. B. Randies , and D. J. Schiffrin, /. Electroanal Chem. 48

(1973) 359.90 A. A. Kornyshev and M. A. Vorotyntsev, J. Electroanal Chem. 167 (1984) 1.91 M. A. Habib and J. O'M. Bockris, in Comprehensive Treatise of Electrochemistry,

Vol. I, Ed. by J. O ' M . Bockris , B. E. C o n w a y , a n d E. Yeager , P lenum Press,N e w York, 1980, C h a p . 4.

92 W. Schmickler and D. H e n d e r s o n , /. Electroanal. Chem. 176 (1984) 383.93 D . Price and J. W. Halley, /. Electroanal Chem. 150 (1983) 347.94 J. G o o d i s m a n , Theor. Chim. Acta 68 (1985) 197.95 E. P. Wigner , Phys. Rev. 46 (1934) 1002.96 D . H e n d e r s o n , in Trends in Interfacial Electrochemistry, Ed . by A. F e r n a n d o Silva,

Reidel , Dordrech t , 1986.97 J. W. Halley, B. Johnson , D . Price, and M. Schwalm, Phys. Rev. B 31 (1985) 7695.98 J. W. Halley and D. Price, prepr in t , 1986.99 D . Price, Thesis , Universi ty of Minneso ta , Augus t 1986.

100 S. L. Carnie and D. Y. Chan, /. Chem. Phys. 73 (1980) 2949.101 W. Schmickler, J. Electroanal Chem. 157 (1984) 1.102 W. Schmickler , Chem. Phys. Lett. 99 (1983) 135.103 W. Schmickler a n d D . H e n d e r s o n , J. Chem. Phys. 80 (1984) 3381.104 L. Blum a n d D . H e n d e r s o n , J. Chem. Phys. 74 (1981) 1902.105 J. Goodisman, /. Chem. Phys. 82 (1985) 560.106 G. G o u y , /. Phys. (Paris) 9 (1910) 457; D . L. C h a p m a n , Phil Mag. 25 (1913) 1475.107 A. A. Kornyshev, A. I. Rubinshtein, and M. A. Vorotyntsev, /. Phys. CU (1978)

3307.108 K. L. Kl iewer a n d R. Fuchs , Adv. Chem. Phys. 27 (1974) 356.109 A. A. Kornyshev , W. Schmickler , a n d M. A. Vorotyntsev , Phys. Rev. B 25 (1982)

5244.110 V. A. Ki ryanov , E lec t rokhimiya 17 (1981) 286.

Recent Advances in theTheory of Charge Transfer

A. M. Kuznetsov

A. N. Frumkin Institute for Electrochemistry, Academy of Sciences of the USSR,Moscow V-71, USSR

I. INTRODUCTION

Development of the quantum mechanical theory of charge transferprocesses in polar media began more than 20 years ago. The theoryled to a rather profound understanding of the physical mechanismsof elementary chemical processes in solutions. At present, it is agood tool for semiquantitative and, in some cases, quantitativedescription of chemical reactions in solids and solutions. Interestin these problems remains strong, and many new results have beenobtained in recent years which have led to the development of newareas in the theory. The aim of this paper is to describe the mostimportant results of the fundamental character of the resultsobtained during approximately the past nine years. For earlier work,we refer the reader to several review articles.14

II. INTERACTION OF REACTANTS WITHTHE MEDIUM

At present, it is understood that the change in the configuration ofthe medium molecules due to thermal or quantum fluctuations plays

95

96 A. M. Kuznetsov

a major role in charge transfer processes. This may be clearly seenby considering the electron transfer reaction between the simpleions AZl and B*2 located at some fixed distance R from each other.According to the Franck-Condon principle, electron transfer ispossible if the electron energies in the donor sA and in the acceptoreB are approximately equal to each other. In the reaction underconsideration, the matching of the electron energy levels can occuronly due to the interaction of the electron with the medium mole-cules. Thus, the activation of the reactants in this case is directlyeffected by the fluctuations of the medium molecules. If the reactantshave complex intramolecular structure, the intramolecular vibra-tions may also have some role in the activation process. However,the fluctuations in the medium play an essential role in this caseas well since the change in the energy of the intramolecular vibra-tions is due to their interaction with the vibrations of the mediummolecules.

To show more clearly the difference between this new approachand that used earlier, we will briefly summarize the model whichwas widely used for the calculation of the probability of the elemen-tary act of charge transfer processes in polar media.

1. Adiabatic and Diabatic Approaches: A Reference Model

There are two basic approaches to the calculation of the probabilityof the electron transition. One of them, called the adiabaticapproach, consists in that the electron states </>a of the total Hamil-tonian of the system, H, are chosen as the zeroth-order electronstates in the Born-Oppenheimer approximation. These states andthe corresponding energies ea depend on the nuclear coordinatesQk. In the simplest case, the potential energy surface Ua(Qk)corresponding to the ground electron state <f>a has two minima atthe points QkOl and QkOf. The regions near QkOl and QkOf correspondto the reactants and reaction products, respectively. They are separ-ated by a potential barrier with the top at the point Q* defined asthe saddle point on the surface Ua(Qk) (Fig. 1).

The potential energy surface (PES) Up(Qk) for the excitedelectron state <f>p has its minimum near the point Q* (Fig. 1). Inthe classical limit, the electron transition may be treated as acontinuous motion of the system on the lower PES, Ua, from the

Recent Advances in Theory of Charge Transfer 97

V

Figure 1. Adiabatic potential energysurfaces. Q

point QkOl to Qfc0/. During this motion, the adiabatic redistributionof the electron density from the donor site to the acceptor site takesplace. However, in the region near the top of the potential barrier,the transition of the system to the upper PES Up is possible dueto the effects of nonadiabaticity. In this case, the electron remainsin the donor ion.

If the probability for the system to jump to the upper PES issmall, the reaction is an adiabatic one. The advantage of theadiabatic approach consists in the fact that its application does notlead to difficulties of fundamental character, e.g., to those relatedto the detailed balance principle. The activation factor is determinedhere by the energy (or, to be more precise, by the free energy)corresponding to the top of the potential barrier, and the trans-mission coefficient, K, characterizing the probability of the rear-rangement of the electron state is determined by the minimumseparation AJB of the lower and upper PES. The quantity A£ isthe same for the forward and reverse transitions.

However, if the PES are multidimensional, as is the case forreactions in the condensed phase, the adiabatic approach is in-convenient for practical calculations, especially for nonadiabaticreactions.

Another approach widely used for nonadiabatic reactions isthe diabatic one. The channel Hamiltonians Hf and H} determiningthe zeroth-order Born-Oppenheimer electron states of the donor<f>A and acceptor </>B and the perturbations Vt and Vf leading to theforward and reverse electron transitions, respectively, are separated

98

from the total Hamiltonian in this approach:

H = HN + Het + V?d = HN + H} +

A. M. Kuznetsov

(1)

where HN is the Hamiltonian of the nuclei.The diabatic potential energy surfaces Ut and Uf correspond-

ing to the zeroth-order electron states </>, and <j>f cross along ahypersurface S^ (Fig. 2). In the diabatic approach, the electrontransition is usually treated as follows. Moving along the nucleardegrees of freedom Qk on the initial PES, Ul9 at constant initialdistribution of the charge, the system reaches the point of minimumenergy Q* on the intersection surface 5^. In this configuration,referred to as the transitional configuration, the change of theelectron state from </>, to <j)f is possible. This corresponds to electrontransfer from the donor to the acceptor and to transition of thesystem from the potential energy surface U, to the potential energysurface Uf followed by the relaxation of the nuclear configurationto the equilibrium final position.

The activation energy for the nonadiabatic reaction, £"ad, isdetermined by the point of minimum energy on the intersectionsurface of PES Ul and Uf9 and the transmission coefficient K isdetermined by the electron resonance integral

(2)

Figure 2 Diabatic potential energy surfaces.


where V?d is the off-diagonal part of the operator for the interactionof the electron with the acceptor: V?d - V% = VeB - Vd

eB.The electron resonance integral for the reverse transition has

the form

•Jyf= 0Av2d<£Bd3x (3)

where V0/ is the off-diagonal part of the operator for the interactionA -of the electron with the donor: V0/ = V°e

dA = KA - Vd

e

For the adiabatic reactions, the activation energy fj"ad mustbe reduced by the quantity 8Ea, determined by the resonancesplitting of the potential energy surfaces, AE, and by the slopes ofUl and Uf at the point (?*, and K = 1.

In calculating the transition probability for the nonadiabaticreactions, it is sufficient to use the lowest order of quantummechanical perturbation theory in the operator V?d. For theadiabatic reactions, we must perform the summation of the wholeseries of the perturbation theory.5 (It is insufficient to retain onlythe first term of the series that appeared in the quantum mechanicalperturbation theory.) Correct calculations in both adiabatic anddiabatic approaches lead to the same results, which is evidence ofthe equivalence of the two approaches.

The diabatic approach will be mainly used below, since it ismore convenient for nonadiabatic reactions. However, in SectionVI adiabatic reactions will be also considered using the adiabaticapproach.

A harmonic approximation has usually been used for thedescription of the nuclear potential energy,

UN=$Zha>kQ2

k (4)k

where the Qk are dimensionless normal coordinates of the harmonicoscillators characterized by the frequencies cok.

For reactants having complex intramolecular structure, somecoordinates Qk describe the intramolecular degrees of freedom. Forsolutions in which the motion of the molecules is not described bysmall vibrations, the coordinates Qk describe the effective oscillatorscorresponding to collective excitations in the medium. Summationrules have been derived which enable us to relate the characteristicsof the effective oscillators with the dielectric properties of then\edium.5

100 A. M. Kuznetsov

The interaction of the electrons and other charged particleswith the vibrations of the nuclei in the reference model was usuallyconsidered to be linear in the coordinates Qk:

vzQ = - I ykQk (5)k

It is convenient to split this interaction into two parts: the interactionwith the intramolecular vibrations and with the nearest mediummolecules, Vzv, and the interaction with the rest of the solvent, VzP,which may be described in terms of the inertial polarization perunit volume P(r):

V* = Vzv + VzP (6)

The interaction VzP may be written in the form

= - J P(r)E*VzP = - J P(r)E;(r) d3r (7)

where E"(r) is the electric field in vacuum due to the charge ez.Since the interaction of the electron with the medium polariz-

ation is strong, in the reference model it was usually included inthe zeroth-order Hamiltonians determining the Born-Oppenheimerelectron states:

(Te + VeA + Vt + VeP + VdeB + V»d)cf>A = eA</>A

(Te + VeB + V* + VeP + VdeA + V*d)<f>B :

where Te is the kinetic energy of the electron and the superscriptd denotes the diagonal part of the operator, which does not leadto the electron transition but leads to a distortion of the electronstate.

Note that in the reference model all the interactions of theelectron with the medium polarization VeP are included in Eqs. (8)determining the electron states. The dependence of <j>A and <f>B onthe polarization and intramolecular vibrations was entirelyneglected in most calculations of the transition probability [theapproximation of constant electron density (ACED)]. This approxi-mation, together with Eqs. (4)-(7), resulted in the parabolic shapeof the diabatic PES Ul and Uf. The latter differed only by the shift


of equilibrium coordinates QkOi and Qko/ a n d by the energies ofthe effective oscillators at the minima, /, and //.

The result for the transition probability in the classical limithad the form

W = (coeff/27r)K exp(-tf*/kr) (9)

where

H* = (Er + A/)2/4£r (10)

Here A/ = Jf — Jt and Er is the total reorganization energy of thevibrational subsystem:

Er = \1 t>cok(Qk0i - Qkoff = I Erk (11)k k

Equation (9) was obtained using the assumption that the vibra-tional subsystem is in the state of thermal equilibrium correspondingto the initial electron state. The expression for the effectivefrequency <weff has the form5

(12)

2. A New Approach to the Interaction of the Electron with thePolarization of the Medium in Nonadiabatic Reactions

As noted above, in the reference model the dependence of theelectron wave functions <f>A and <£B on the nuclear coordinates wasentirely neglected and the wave functions </>A and <f)B for the isolatedions or the wave functions calculated for corresponding equilibriumnuclear configurations QkOi and Qkof according to Eqs. (8) wereusually used in the calculations.

Recent analysis has shown that this approximation is, in gen-eral, insufficient.6 This is due to the long-range character of theinteraction of the electron with the medium polarization. Thezeroth-order states determined from Eqs. (8) taking into accountthe total interaction of the electron with the total inertial polariz-ation of the medium VeP may not describe the states of the electronlocalized in the donor or in the acceptor sites. Since the polarizationvaries due to thermal fluctuations, at certain configurations of the

102 A. M. Kuznetsov

polarization field the potential energy of the electron may involveseveral potential wells localized in different points. Therefore, thesolutions of Eqs. (8), taking into account all the interactions of theelectron with the inertial polarization, may give states which, ingeneral, do not correspond to the electron localization in the donoror in the acceptor.

It was suggested that the zeroth-order electron states be calcu-lated using equations similar to Eqs. (8) at initial equilibrium valuesof the polarization P0l.

7 However, it may be seen that if the acceptoris an anion, even in the initial equilibrium configuration the equili-brium polarization of the medium near the acceptor may create apotential well for the electron.

To solve this problem, it was suggested6 that the zeroth-orderelectron states be found from the equations

(Te + VeA + V% + VeA

P + V*B + Vld + V^)cf>A = eAcf>A

(Te + VeB + Vl + VfP + VdeA + Vtd + Vep)<t>* £

Unlike Eqs. (8), the first of Eqs. (13) involves only part of theinteraction of the electron with the medium polarization V^P, which,together with VeA, creates the potential well for the electron nearthe donor A. As for the interaction with the polarization VfP, which,together with VeB, creates the potential well for the electron nearthe acceptor, the first of Eqs. (13) involves only the diagonal partof this interaction, Vfp, leading to a distortion of the state c/>A

without a change in the electron localization. The state <f>B isdetermined in a similar way.

The perturbation operators must be modified in accordancewith the new definition of the zeroth-order electron states:

V?d = ( K B ) 0 d + (VfP)0d + (VfJoa

= K B - viz + vfP - veB

Pd + V " - vfv

d

(14)V0/ = ( V c A ) 0 d + (V e

AP ) 0 d + (Ve

Au)Od

= veA - v?A + v*p - v% + v

The method of separating the interaction VeP into V*P andVfP is discussed in the next section.


This new approach enables us to consider all the physicaleffects due to the interaction of the electron with the mediumpolarization and local vibrations and to take them into account inthe calculation of the transition probability. These physical effectsare as follows:

1. Effect of diagonal dynamic disorder (DDD). Fluctuationsof the polarization and the local vibrations produce the variationof the positions of the electron energy levels eA(Q) and sB(Q) tomeet the requirements of the Franck-Condon principle.

2. Effect of off-diagonal dynamic disorder (off-DDD). Theinteraction of the electron with the fluctuations of the polarizationand local vibrations near the other center leads to new termsV?P - V*$9 Vl- Vld and V*P - V*P

d9 V* - V*d in the perturba-

tion operators V°d and Yf [see Eqs. (14)]. A part of these interac-tions corresponding to the equilibrium values of the polarizationPOl and P0/ results in the renormalization of the electron interactionswith ions A and B, due to their partial screening by the dielectricmedium. However, at arbitrary values of the polarization P, thereis another part of these interactions which is due to the fluctuatingelectric fields. This part of the interaction depends on the nuclearcoordinates and may exceed the renormalized interactions of theelectron with the donor and the acceptor. The interaction of theelectron with these fluctuations plays an important role in processesinvolving solvated, trapped, and weakly bound electrons.

3. Effect of diagonal-off-diagonal dynamic disorder (D-off-DDD). The polarization fluctuations and the local vibrations giverise to variation of the electron densities in the donor and theacceptor, i.e., they lead to a modulation of the electron wavefunctions </>A and <f>B. This leads to a modulation of the overlappingof the electron clouds of the donor and the acceptor and hence toa different transmission coefficient from that calculated in theapproximation of constant electron density (ACED). This modula-tion may change the path of transition on the potential energysurfaces.

4. Additional effect of diagonal dynamic disorder. The vari-ations of the electron densities near the centers A and B due topolarization fluctuations and local vibrations lead to changes in theinteraction of the electron with the medium and, hence, to changesin the shape of the potential energy surfaces Ut and Uf as compared

104 A. M. Kuznetsov

to that in the reference model. As a result, the activation energywill be different from that calculated in the ACED.

III. NONADIABATIC ELECTRON TRANSFER REACTIONS

The above effects manifest themselves both in electron transferreactions and in reactions involving the transfer of heavy particles.However, before discussing these effects in electron transfer re-actions, we will consider some problems arising in the referencemodel.

1. Energy of Activation or Free Energy of Activation?

The expression in Eq. (10) for the exponent in Eq. (9) is quitesimilar to that for the activation free energy in electron transferreactions derived by Marcus using the methods of nonequilibriumclassical thermodynamics8:

Fa = (Er + AF)2/4£r (15)

where AF is the free energy of the transition.Instead of the quantity given by Eq. (15), the quantity given

by Eq. (10) was treated as the activation energy of the process inthe earlier papers on the quantum mechanical theory of electrontransfer reactions. This difference between the results of the quan-tum mechanical theory of radiationless transitions and thoseobtained by the methods of nonequilibrium thermodynamics hasalso been noted in Ref. 9. The results of the quantum mechanicaltheory were obtained in the harmonic oscillator model, and Eqs.(9) and (10) are valid only if the vibrations of the oscillators areclassical and their frequencies are unchanged in the course of theelectron transition (i.e., wl

k = (o[). It might seem that, in this case,the energy of the transition and the free energy of the transitionare equal to each other. However, we have to remember that forthe solvent, the oscillators are the effective ones and the parametersof the system Hamiltonian related to the dielectric properties ofthe medium depend on the temperature. Therefore, the problem ofthe relationship between the results obtained by the two methodsmentioned above deserves to be discussed.


It was shown in Ref. 5 that the quantity A/ involved in Eq.(10) includes, in addition to the difference of the electron energies,quantities of the type

Ep _ ET = -(//87r)(J/27r)3 J d3k [\Vf(k)\2 - |D,(k)|2](2/7r)

fJo

2x da) Ime(k9(o)/(o\e(k,(o)\Jo

where D,(k) and Dy(k) are the Fourier components of the electro-static inductions due to the initial and final charge distributions inthe medium.

One may easily see that the quantities Eaq represent the gen-eralization of the expressions for the electrostatic contribution tothe solvation free energy for the case of spatial dispersion of thedielectric function e(k, w). Thus, it has been shown in Ref. 5 thatthe quantity A/ in Eqs. (9) and (10) for the transition probabilityrepresents the free energy of the transition. A similar result wasobtained later in Ref. 10.

The problem of the physical meaning of the quantity Hx andof the reorganization energy of the medium Es has been analyzedin Ref. 11. Following Ref. 11, we write the expression for thetransition probability per unit time in the form3

W = (1/ihkT) exp(Fl0/kT) I dd Tr[ VlPl(l - 0) VtPf(0)] (16)J c —IOO

where p,(l - 0) = exp[-/3(l - 6)Ht] and pf{0) = exp[-p$Hf],with /3 = I/AT, are the statistical operators (the density matrices)of the initial and final states, and Fl0 is the free energy of the initialstate.

In the Condon approximation, Eq. (16) is transformed to theform11

W = (l/ifikT)\Vl\2 I d0fm(0) (17)

fm(6) = exP[-j3F(0)] • exp(-)30Es - pOAF) (18)

106 A. M. Kuznetsov

where AF is the free energy of the transition, j8 = 1/kT,

Es = -\ \ d3rd3rV'Z ABS(r)

x AF£(r')D^(r, r'; co = 0) a, )8 = x, y, z (19)

exp[-j8F(0)]

Q " J d3r8Pt(r, r)AEv(r)J

= exp(i8FIo)TrJexp(-j8Hr)rTexp| j dr

x J</3rSP,(r,r)AEu(r)]} (20)where AEu(r) = E^(r) - EJ'(r) is the difference of the electric fieldsin vacuum due to the reaction products and the reactants, TT is theoperator of time ordering, 5P,(r, T) = P(r, T) - POl(r, T) is theoperator in the Heisenberg representation describing the deviationof the inertial polarization from its initial equilibrium value P0,(r),and Dap(r9 r'; a)) is the Fourier component of the retarded Green'sfunction:

D*,(r, r'; t) = -id(t)([Pa(r, t), Pfi(i>9 0)]) (21)

In the limit where the vibrational spectrum of the inertialpolarization lies in the classical region (co < kT/h)9 the calculationof the expression in Eq. (20) and of the integral over 6 in Eq. (17)gives11

W=(\Vl\2/fi(kTEs/TT)1/2)exp(-p\F(0*)\) (22)

where2 (23)

(24)

From a comparison of Eqs. (9) and (22) we see that //* =|F(0*)|. To elucidate the physical meaning of the exponent in Eq.(22), we consider first the case when 0* = 1 (barrierless reaction).In this case Eq. (20) determines the change of the free energy ofthe system F(l) when it is polarized by the electric field AEV =E/ - E" (only the free energy related to the inertial polarization isconsidered). It may be easily seen that the absolute value of F(l)is equal to the energy of the reorganization of the medium Es (>0).


This equality determines the physical meaning of Es as the changein the free energy of the system due to a change in the inertialpolarization by the quantity AP(r) = P0/(r) - Poi(r)-

Es=\\ J3rAEu(r)AP(r) (25)

For 0* T* 1 we note that in the classical limit Eq. (20) may bewritten in the form

exp[-j3F(0)] = (rt exp j J dt J d3r5P,(r, t)[0AEv(r)]^ (26)

Thus, in this case F(0*) represents the change in the freeenergy of the medium due to the application of the electric field0*AEu(r). The absolute value of F(0*) is equal to the energy ofthe reorganization of the medium when the inertial polarization ischanged by the value APe*(r) = 0*AP(r):

(r) (27)

Using Eq. (27), we can write the transition probability in theform

W=(\Vi\2/ti(kTEs/Try/2)exp[-Es(e*)/kT] (28)

Equation (28) gives the physical meaning of the activationfactor //* as the free energy required for the system to reach thetransitional configuration.

In the case when the vibrational spectrum of the system spreadsout in the quantum region and the vibrational frequencies of thereaction complex are unchanged in the course of the transition, thefollowing approximate formula can be obtained instead of Eqs. (9)and (10)3'12:

W = (kT/fi)K exp J - ( £ r + &F)2/4ErkT - £ {EJ ft^)

x [cosh(/3ftet>//2) - 1 -\{ph(Oi/2)2ysmh(pfi(Oi/2)

(29)

108 A. M. Kuznetsov

where

Er = Ef + Z £w( W2fcr)/sinh(W2fcr) (30)i

^c=IUk (3Dm

The symmetry factor has the form

a = 0* = \ + AF / 2£? + 1 ErtffKOi/smhiphcoJl) (32)

The summation over / in Eqs. (30) and (32) means summationover all nonclassical discrete local intramolecular vibrations andintegration (using an appropriate weighting function) over thefrequencies describing the spectrum of the polarization fluctuationsbeyond the classical region which satisfy the condition

pfKOi(l-2a)/2« 1 (33)

The quantity Ef is the energy of the reorganization of all theclassical degrees of freedom of the local vibrations and of theclassical part of the medium polarization, and crc is the tunnelingfactor for quantum degrees of freedom (/3fta>m » 1) which do notsatisfy the condition given by Eq. (33).

An expression of the type in Eq. (29) has been rederivedrecently in Ref. 13 for outer-sphere electron transfer reactions withunchanged intramolecular structure of the complexes where essen-tially the following expression for the effective outer-spherereorganization energy Ers was used:

i:Ers = (l/S7T2)(fi/kT) I dco[Im e(<o)/\e(a>)\2]

I (Df - Df)2 d3r (34)

This formula is easily obtained from Eq. (30) if we use the summa-tion rules relating the parameters of the effective oscillators withthe dielectric properties of the medium.5

The physical reason for the appearance of the free energies inthe formulas for the transition probability consists in the fact thatthe reactive vibrational modes interact with the nonreactive modes.


Averaging over the states of the nonreactive modes leads to theappearance of free energies. This problem was recently analyzedindependently in Refs. 14 and 15.

Following Ref. 14, in the classical limit in the Condon approxi-mation we transform Eq. (16) to the form

W = y | V |2 exp(Fi0/kT) J dqx dq2 • • • dQx

x exp{-[U(ql9 q29...9 Qu Qi) + Si(ql9 q2,.. .)

x 8[ei(ql9 q29...) - ef(ql9 ql9...)] (35)

where U is the potential energy of the nuclear subsystems, ql9

q2y.. .are the coordinates of the reactive modes directly interactingwith the electron, Qx, Q2,... are the coordinates of the nonreactivemodes, and et and ef are the electron energies in the initial andfinal states, respectively.

The integration over the coordinates of the nonreactive modes<?i, <?2,... gives

W = Y | Vi-|2 e x p ( F i 0 / / c T ) J dqx d q 2 - -

, q2,...) + Bi(ql9 ql9.. .)

x 8[ei(qu q2, • • •) ~ ef(ql9 q2,...)] (36)

where F(ql9 q2,...) is the configurational part of the free energyof the system as a function of the coordinates of the reactive modes.

Introducing the diabatic free energy surfaces of the initial andfinal states,

q29...) (iu q2, ) i(lu qi* )

)

the expression for W may be transformed to the form

W = y iVtfexpOuo/kT) | dqx dq2- - •

xexp[-Ui(quq29...)/kT\8[Ui(ql9q29...)-Uf(ql9q29...)]

(38)

Thus, to calculate the transition probability for the non-adiabatic reaction it is sufficient to know the diabatic free energy

110 A. M. Kuznetsov

surfaces of the system, representing the configurational parts of thefree energy of the system as a function of the coordinates of thereactive modes at fixed states of the quantum particles.

2. Effects of Diagonal and Off-Diagonal Dynamic Disorder inReactions Involving Transfer of Weakly Bound Electrons

(A Configurational Model)

The effects of the modulation of electron density by local vibrationsand polarization fluctuations are most pronounced for reactionsinvolving transfer of weakly bound electrons. These effects wereinvestigated in Ref. 16 for the transfer of weakly bound electronsfrom a donor AZl to an acceptor B*2 in a polar medium.

The total wave function of the system was presented in the form

<A = <£A(*; q) I Cln{t)xl

n(q) + <M*; q) I Cfm(t)xi(q) (39)

n m

where \ l a nd xf a r e the wave functions of the vibrational subsystem.The Schrodinger equation for the vector C\ having the

components

X^ X I ^nXn I? ^ X

was written in matrix form as follows:

ihdCx/dt = HCx (41)

where

l 1 ^

\H = (1 - S T - " " ,, „ ' (42)

S — (0A| ^B)^ i — ^BA ~ SHAA; Vf — HAB — SHBB (43)

The elements of the matrix H involve the electron matrixelements of the Hamiltonian of the system. The general expression


for the transition probability in Eq. (16) is transformed to the form

W - ^ A2 J dq\ Vt(x\ q)\2nA(x*9 q)nB(x*9 q) J dS J ds

i0) exp(-/30AF)

Pf(q + q-^e} (44)

where the approximate formula for the electron matrix element

Vi = ] <f>%(x;q)Vi(x,q)<f>B(x',q)d3x

- A VJ(x*, q)4>A(x*; q)<f>B(x*; q) (45)

was used and the electron densities

nA(x; q) = \4>A(X; q)\\ nB(x; q) = \d>B(x; q)\2 (46)

are introduced.The usual Condon approximation (CA) is obtained from Eq.

(44) if we assume that the dependence of the electron factors on qis weaker than that of the other factors and use the approximation«A = |<£A(*; <7O«)|2 and nB = \<f>B(x; qof)\

2 in calculating the densitymatrices of heavy particles p® and p/i

WCA = A2| V,(x*, q*t • "A(X*; q*)nB(x*; q*)

"-^FTr[p°( 1 - d)p°f{d)] (47)

where q* is the point for which the integrand in Eq. (47) ismaximum.

Equation (47) shows that in the Condon approximation theprobabilities of forward and reverse transitions satisfy the detailedbalance principle since the point q* corresponds to the intersectionof the potential energy surfaces (and free energy surfaces) whereHAA = HBB. Therefore, at the point q* we have

Vt(q*) = Vf(q*) (48)

This result was obtained in Ref. 17 and was rederived in a numberof subsequent papers.616'18'19

112 A. M. Kuznetsov

If the dependence of nA and nB on q is taken into account inthe calculation of the statistical operators for heavy particles, weobtain the improved Condon approximation (ICA) which differsfrom Eq. (17) only by the change of p°t and p° to p, and pf,respectively. In the classical limit for p, and pf, the expression forthe transition probability takes the form

<*,q)

(49)

Using Eq. (49) one may consider all the cases of the Condonapproximation (CA, ICA), and in some models go beyond theCondon approximation (BCA).

(i) Condon Approximation

It has been shown16 that in the Condon approximation thevalue of the polarization in the transitional configuration P* isequal to

P*(r) = (1 - 0*)POl + 0*PO/ = P£ + PS

PA(r) = (c/4*r)[DA(r) + (1 - 0*)D(A)(r; P0l)] (50)

P*(r) = (c/4ir)[DB(r) + 0* • D(eB)(r; Po/)]

where 0* is the symmetry factor determined as the saddle point inthe integral over 6 in Eq. (44).

The interactions V^P and V*P involved in Eqs. (13) determiningthe zeroth-order electron states in the transitional configurationrepresent the interactions of the electron with the polarizations P*and P | , respectively. At long transfer distances the perturbationleading to the electron transfer has the form

_ z2e2 0*ce2

' ~ e5|x* - R| |x* - R|

The interaction leading to the reverse transition has a similarform. The first term on the right-hand side of Eq. (51) describes


Figure 3. The dependence of theinverse radius of the localization ofthe electron density (y = A/Ao =(r/ro)~

l) near the initial trap(donor of the electron) on the sym-metry factor 0*: (1) P/S = -0.5;(2) P/S = 0; (3) P/S = 0.5. P =ze2/es; S = 5ce2/l6.

the interaction of the electron with the ion in the static dielectric.The second term describes the interaction with the fluctuation ofthe polarization near the acceptor site corresponding to the transi-tional configuration.

Equations (50) and (51) show that for 0 < 0* < 1 the potentialwell for the electron near the donor site is more shallow than thatin the initial equilibrium configuration. This leads to the fact thatthe radius of the electron density distribution in the transitionalconfiguration is greater than in the initial equilibrium one (Fig. 3).A similar situation exists for the electron density distribution nearthe acceptor site. This leads to an increased transmission coefficientas compared to that calculated in the approximation of constantelectron density (ACED).

(ii) Improved Condon Approximation

To take into account the additional effect of diagonal dynamicdisorder in the improved Condon approximation it was suggestedin Ref. 16 that fluctuations of the polarization of the type

P(r) = (C/4TT)[DA + DB + £D<A)(r; POl) + r jD^r ; Po/)] (52)

where £ and 17 are the independent variables, be considered.The free energy surfaces of the initial and final states, £/,(£, 77)

and Uf(£, 17) were calculated taking into account the dependenceof the electron energies eA and eB on f and 77. The latter were

114 A. M. Kuznetsov

calculated using a direct variational method similar to that used inthe polaron theory.20

The transitional configuration (£*, 77*) was found as the sol-ution of two coupled equations for £ and 77:

, V) = UfU, 77)(53)

In the limit when nA and nB are independent of the mediumpolarization, £* and 77* are related to each other by the equation£* = l - 7 7 * = l - 0 * [see Eqs. (50)]. However, in general, they areindependent quantities. Table 1 shows the results for the symmetrictransition.16 In this case we have £* = 77*.

Table 1 shows that the smaller the binding energy of theelectron, determined by the quantity P/ S9 the stronger are the effectsof the modulation of the electron density by the polarization fluctu-ations on the value of £*, on the radius of the electron densitydistribution, and on the activation free energy. However, since athigh binding energy values the electron is more localized, evenrelatively small changes in the radius of the electron density producea greater effect on the transition probability than at small valuesof the binding energy. The effect of off-diagonal dynamic disorderis characterized by the quantity nAnB/n°An^ and an additional effectof diagonal dynamic disorder is characterized by the quantity

Table 1Kinetic Parameters0 for the Symmetric Transition in the

Improved Condon Approximation

PIS

00.512

-0.1

y

0.6130.7250.7860.8500.547

r0.3760.4180.4390.4590.359

0.8110.8640.8930.9250.795

2\pR

"A«B/«A"B

4.705.215.546.055.11

= 4

W/Wo

56.9976.9293.39

117.2958.34

2\pR

« A « B / " > ° B

22.1127.1130.6936.6026.09

= 8

W/Wo

268.0400.5517.4713.4298.0

aThe meaning of the parameters is as follows: W/ Wo = (WA^B/^A^B)exp[-(Fa - F°a)/kT], \p = 5mce21M>h2, y = A/Ao, P = ze2/es, S = 5ce2/\6,c = 0.5.


Fa/ F°a. Table 1 shows that these quantities vary in the oppositedirection with the variation of the binding energy.

(iii) Beyond the Condon Approximation

Equations (49) and (52) enable us to go beyond the Condonapproximation. The deviations from the Condon approximationare due to the fact that for long-distance electron transfer, theoverlapping of the electron wave functions is exponentially small(as small as exp(-aR) where R is the transfer distance) and evensmall changes in the behavior of the decreasing tail of the electronwave function produce large changes in the values of the electronresonance integral (i.e., of the quantity nAnB). This leads to a strongdependence of nAnB on the polarization which must be taken intoaccount in the calculation of the transitional configuration (£*, 77*):

kTd \n(nAnB)/d( - dUJd£ = 0; !/,(£ V) = Uf((9 V) (54)

The results for the symmetric system are given in Table 2. Acomparison of Tables 1 and 2 shows that the dependence of nAnB

on £ and rj influences the position of the transitional configurationand this effect increases with increase in the transfer distance. Thephysical reason for the change of the path of the transition in thiscase is that the system prefers to shift from the saddle point to the

Table 2Kinetic Parameters for the Symmetric Transition TakingAccount of the Modulation of the Zeroth-Order Electron

Densities (Beyond the Condon Approximation)

P/S

0

1

2

2kpR

14

148

148

r

0.5990.541

0.7830.7740.761

0.8490.8460.840

r0.3600.301

0.4330.4140.388

0.4550.4450.427

FJF°a «

0.8120.825

0.8930.8950.902

0.9250.9260.929

«P(-^?)

9.5608.20

12.9012.3110.50

14.6714.3612.90

1.4936.27

1.546.10

45.78

1.576.35

46.52

W/Wo

14.2851.46

19.9175.10

480.8

23.0891.10

600.5

116 A. M. Kuznetsov

Figure 4. Path of the transition on thepotential energy surfaces in the Con-don approximation and beyond theCondon approximation.

point of higher free energy to increase the electron resonanceintegral (Fig. 4).

The configurational model was used for the calculation of theelementary act in the reactions of solvated electrons21 and in theelectrochemical generation of solvated electrons.22 The results forthe activation free energy of the process of electrochemical gener-ation of solvated electrons as a function of the reaction free energy

Table 3Dependence of the Kinetic Parameters for the Reaction ofElectrochemical Generation of Solvated Electrons on the

Free Energy of the Transition

A F / £ r

-0.5-0.4-0.3-0.2-0.1

0+0.1+0.2+0.3+0.4+0.5

P/S

y

00.3680.4830.5630.6270.6800.7260.7670.8040.8380.870

= 0

FJF°a

00.2950.5070.6530.7560.8280.8800.9170.9440.9640.977

P/S =

y

0.6380.6780.7140.7460.7750.8020.8270.8510.8730.8940.914

1

FJF°a

0.4870.6290.7310.8040.8580.8970.9270.9480.9650.9760.985


are given in Table 3. Table 3 shows that the allowance for thedistortion of the electron wave functions by the polarization fluctu-ations in ICA leads to essential differences between the activationfree energy Fa and the quantity F°a calculated in the approximationof constant electron density. This difference increases on goingfrom endothermic (AF > 0) to exothermic (AF < 0) processes,which considerably affects the shape of the current versus potentialdependence.

3. Feynman Path Integral Approach

Polarization fluctuations of a certain type were considered in theconfiguration model presented above. In principle, fluctuations ofa more complicated form may be considered in the same way. Amore general approach was suggested in Refs. 23 and 24, whereEq. (16) for the transition probability has been written in a mixedrepresentation using the Feynman path integrals for the nuclearsubsystem and the functional integrals over the electron wavefunctions of the initial and final states i/>,(x, t) and ^f{x, t) for theelectron:

i0) I dO I.,-»rvr-,u; , — , dx dx'dq dq'

f *J I / /

x Vf(xf, q*)il/f(x, O)t^(x', O)i//y(x, O)(/J*(X', 0)

x expf-iS.O,, q(t)\ qy q\ Tt] + SJOJ, ^(r); qf, f', 7}]}j

(55)

where

7 = 77(1-0) ; Tf=T/6

A'1 = Dif/a expl - 5a I, a = i, /

s = < * l * l * > (56)

>\ >\ Ak /A

C'e _i_ Qeq \ Qqa — &a ' ^ a ' * a

118 A. M. Kuznetsov

where Sa is the total action of the system in the state a (a = i, / ) ,Se

a and SI are the actions of the electron and nuclear subsystems,respectively, and Se* is the contribution to the action from theinteraction of the electrons with the nuclei.

The calculation of the integrals in Eq. (55) in the classical limitin the improved Condon approximation (for the nuclear subsystem)using the saddle point method leads to two coupled equations forthe electron wave functions of the donor and the acceptor in thetransitional configuration:

dr' DfB(x, T') + DA(x) + Dt(x)]} <Mr, r)

= EA(T)<I>A(T, T ) (57)

Dl(x) + DB(x)]}<£B(r, T)

= EB(r)(f>B(r, T) (58)

where Hl0 and H{ are the electron Hamiltonians of the donor and

the acceptor when the interaction of the electron with the mediumis omitted, T, = 1 - 0*, rf = 0*, DA and DB are the electrostaticinductions due to the ions AZl and BZ2, and the superscript d denotesthe diagonal parts of the operators.

A direct variational method was used in Refs. 23 and 24 to gobeyond the Condon approximation. Functions of the type

<t>P = (A3P/TT)1/2 exp(-Ap|r - R|); p = i, / (59)

were used as the probe electron wave functions. Here, the \p arevariable parameters.


The calculation for the symmetric system gave23'24

y = A/Ao = (1 + IP/So - 1//Z)/2(1 + P/So) (60)

with S0 = 5ce2/169 Ao = Ap(l + P/So), \p = 5mce2/16fi2, andP = ze2/es, and

FJF°a = (3 + 4P/S0 + 1//I2)/4(1 + P/So) (61)

where F°a = (5/32)ce2\0 and /I = pF°J2R\0.Equation (60) shows that the quantity A characterizing the

decrease of the electron wave function in Eq. (59) decreases withthe increase of the transfer distance R. The activation free energyincreases with R9 and the state of the electron in the transitionalconfiguration becomes less localized. Equations (60) and (61) takeinto account both the dependence of the electron matrix elementon the medium polarization and the distortion of the shape of thediabatic free energy surfaces U, and Uf. For strongly bound elec-trons (P/So » 1), the latter effect is small and the result is similarto that obtained in Ref. 25. The comparison of the results obtainedby this method in IC A with the results of the configurational modelshowed that they do not differ greatly from each other. This meansthat the fluctuation of the polarization corresponding to the transi-tional configuration is close to that described by Eq. (52). Effectsdue to deviation from the Born-Oppenheimer approximation inthe sub-barrier region are also possible in nonadiabatic long-dist-ance electron transfer processes. However, such effects are moreimportant in processes involving transfer of heavy particles, whichare considered in Section V.

4. Lability Principle in Chemical Kinetics

Based on the results obtained in the investigation of the effects ofmodulation of the electron density by the nuclear vibrations, alability principle in chemical kinetics and catalysis (electrocatalysis)has been formulated in Ref. 26. This principle is formulated asfollows: the greater the lability of the electron, transferable atomsor atomic groups with respect to the action of external fields, localvibrations, or fluctuations of the medium polarization, the higher,as a rule, is the transition probability, all other conditions beingunchanged. Note that the concept "lability" is more general than

120 A. M. Kuznetsov

the concept "polarizability" which is often used in the discussionof the reactivity of atoms and molecules.27'28 The change of theelectron density and the shift of the atoms in the reactants mayoccur, in general, without any significant change in their dipolemoments. For example, in the models of electron transfer con-sidered above, almost spherically symmetric vibrations of the elec-tron density in the donor and in the acceptor occurred in theactivation process. The lability of the particles manifests itself invarious physical phenomena (polarizability, the dependence of thedipole moment of the optical transition on the nuclear vibrations,anharmonicity of the intramolecular vibrations, the formation ofcharge transfer complexes, etc.).

The lability of the particles has two aspects. One of them isrelated to the interaction of the particles with the other degrees offreedom of the reacting system. The more labile the particle, thegreater is the change of its state due to a change in the configurationof the particles and molecules interacting with it. As a rule, thislability essentially facilitates the transition of the system to thetransitional configuration corresponding to the optimum configur-ation of particles participating in the elementary act of the process.

The other aspect of the lability is the fact that the more labileparticles are usually less localized. The examples considered inSections 111(2) and 111(3) show that weakly bound electrons whosewave function decreases more slowly with increasing distance fromthe center of the electron localization are more labile. In the caseof heavy particles, they are more labile the smaller their forceconstants and hence the smaller their vibrational frequencies. There-fore, the amplitude of vibrations of more labile particles is usuallygreater than that of less labile particles. The smaller degree oflocalization of more labile particles also facilitates the transitioneven in the absence of their interaction with other degrees offreedom due to both the increase of the overlapping of the wavefunctions and the decrease of the energy (or free energy) requiredto reach the transitional configuration in the classical case.

The lability of the transferable electrons or atoms influencesall the factors in the expression for the reaction rate constant,

k - exp[-u,(R*)/kT] • K • exp(-FJkT) (62)

where K is the transmission coefficient, Fa is the activation free


energy for the elementary act of the reaction, and the first factordescribes the probability for the reactants to be separated by thedistance R*.

For nonadiabatic reactions, the higher lability of the transfer-able electrons or atoms leads to the following effects:

1. Smaller values of the activation free energy, Fa9 due to thedistortion of the shape of the free energy surfaces.

2. Larger values of the transmission coefficient, K, due toimproved overlapping of the wave functions of quantum particles(electrons, protons, etc.).

3. A smaller repulsion between the reactants due to theirmutual "polarization."

The lability principle is valid also for adiabatic reactions. Foradiabatic reactions, the higher lability of the transferable particlesleads to the following effects:

1. Smaller values of the activation free energy due to (i) thedistortion of the shape of the free energy surfaces and (ii) theincrease of the resonance splitting, A£, of the potential free energiesfor the classical subsystem due to the increased overlapping of thewave functions of the quantum particles.

2. Smaller values of the energy of the repulsion between thereactants.

Note that the lability principle is formulated first of all fortransferable electrons and atoms. An increase in their lability leadsas a rule to an increase in the overlapping of the wave functions.For atoms the latter means a decrease in the Franck-Condon barrier.

As for the other atomic and molecular species (both reactantsand solvent) which play the role of the "effective medium" for thetransition, the influence of their lability on the reaction rate is notalways unambiguous. The higher lability of the "medium" particlesusually leads to the increase of the Franck-Condon barrier andthus the increase of the reorganization energy. However, the repul-sion of the reactants decreases at the same time. One of the manifes-tations of the lability of the "medium" is the dielectric polarizabilityof the solvent, characterized by the dielectric constant, es. In thesimplest case the dependence of the reorganization energy on ss isdetermined by the factor Es — e2(l/e0 - \/es). Thus, at large valuesof es the reorganization energy depends rather weakly on es. Atthe same time the repulsion energy «,(#*) — zxz2/ es decreases with

122 A. M. Kuznetsov

increasing es and this effect may be stronger than the effect due toa weak increase of Es (i.e., due to the increase of the Franck-Condonbarrier). For processes in the abnormal region, both effects act inthe same direction and the increase of the lability of the "medium"particles leads to the increase of the reaction rate constant.

For reactions in which the approach of the reactants to eachother is not very important (e.g., for the transfer between two centerslocated at a fixed distance in a rigid structure) an increase in thelability of the medium particles leads to a decrease in the rate ofthe transition.

In Sections 111(1) and 111(2) the lability principle has beenillustrated for processes involving the transfer of weakly boundelectrons, including the reactions of solvated and trapped electronsand F-centers and processes of electrochemical generation of sol-vated electrons. In Sections IV and V, it will be illustrated also byatom transfer reactions and, in particular, by reactions involvingadsorbed atoms.

5. Effect of Modulation of the Electron Density on theInner-Sphere Activation

Since new experimental data concerning the change in structure ofcomplex ions in the course of the electron transition have recentlyappeared,29 interest in the estimation of the inner-sphere contribu-tion to the transition probability has increased again. In Refs. 29-31this problem was considered assuming classical behavior of theligands in the course of the electron transfer. It was concluded29

that the outer-sphere contribution to the activation free energy isdominant for fast reactions in the systems Fe(phen)2+/3+,Ru(bpy)2+/3+, and Co(bpy)+/2+. For slow reactions (e.g.,Cr(H2O)6+/3+, Co(en)2+/3+) the inner-sphere contribution isdominant. The conclusion about the dominant contribution of theinner-sphere reorganization has also been made in Refs. 30 and 31.However, we should keep in mind that in electron transfer reactionsbetween complex ions, quantum effects may take place.32"34 Estima-tions show that although the quantum effects are not very large,they can amount to an order of magnitude.32'33

The effects of modulation of the electron density by theintramolecular vibrations on the process of inner-sphere activation


and on the transmission coefficient of the nonadiabatic reactionmay be illustrated using a simple model. Let us consider electrontransfer from the complex ion AL to an acceptor B located at somefixed distance R in the model of a linear complex where it isassumed that the ligand L is located between atoms A and B (Fig.5). If the energy of the unoccupied acceptor orbital of the ligandL (EL) lies considerably higher than the energy of the orbitaloccupied by the electron in the atom A (sA), the electron will bemainly localized on this atom. However, if the energies eA and eL

depend on the interatomic distance Q in the complex AL, theintramolecular vibrations can produce redistribution of the electrondensity.

If the size of the complex is rather small and the intramolecularvibrations along the coordinate Q may be described in the harmonicapproximation, the free energy surfaces of the initial and final statesmay be written in the form

i = i l tuok(qk - qkOi)2 + 1

2k

- <?o)2(63)

where Qo is the equilibrium length of the chemical bond A—L ofthe complex AL in the oxidized form, and, et(Q) is the energy ofthe electron in the complex AL, which, with certain approximations,may be written in the form

- Qo)]2 + 4 V2}1/2) (64)

Here, the dependence of eA and eL on Q is assumed to be linear,

eA=eA-yA(Q-Q0)(65a)

Q)

Figure 5. Linear reaction complex forelectron transfer reactions in the systemAL/B.

124 A. M. Kuznetsov

and

7+ = 7A + yL; r - = 7L - 7A (65b)

The equilibrium value of the coordinate Q for the complex inthe reduced form Qr is determined by the equation

dUJdQ = 0 (66)

In principle, the free energy surface Ul may have two minima. Werestrict ourselves to the case when there is only one minimum(Fig. 6), and Qr < Qo, i.e., the length of the chemical bond A—Lin the reduced form is shorter than that in the oxidized form. Then,if 7L > 7A > 0> the free energy surface Ut has the form shown inFig. 6. Figure 6 and Eq. (65a) show that an increase in the lengthof the chemical bond A—L leads to a change in the localizationof the electron energy levels eA and eL. For yL> yA the quantityCL ~ £A decreases with an increase in Q — Qo and, therefore, theelectron density on the ligand increases and that on the atom Adecreases. This redistribution of the electron density leads to twoeffects:

1. The height of the potential barrier separating the initial andfinal states of the nuclear subsystem decreases and, hence, theFranck-Condon factor increases (Fig. 6). In the classical limit, thisresults in a decrease of the activation free energy.

CL Q

Figure 6. Diabatic potential energy surfaces forelectron transfer reactions in the system AL/B.


2. The electron resonance integral Vt increases. In this modelVi has the form

V, = J frV^d'x

= CA«?)J <f>x(x)V^{x)d3x

+ CL«?) j 0L(x; (?) V^B(x) d3x (67)

where (/>A(X) a n d <£L(*)

a r e the atomic orbitals for the electrons,and CA(Q) and CL(Q) are the coefficients characterizing the contri-bution of the atomic orbitals <f>A and </>L to the molecular orbital</>i = 0 A L -

Equation (67) shows that in addition to a direct overlappingof the electron wave functions of the donor A and the acceptor B,the electron resonance integral involves a term related to the over-lapping of the wave functions of the acceptor B and the ligand L.In the initial equilibrium configuration the contribution of theatomic orbital of the ligand to the wave function </>AL is small( C L « CA); however, the overlap integral (</>A|<£B) is exponentiallysmall as compared to the overlap integral (<£L|<£B) SO that in theinitial equilibrium configuration the contribution of the atomicorbital of the ligand to the resonance integral can be considerable.

When the system approaches the transitional configuration twoeffects take place: (1) the coefficient CL increases and at Q = Q*it may be of the order of, or even greater than, the coefficient CA

(CL a: CA), and (2) the distance between the ligand L and atom Bdecreases with increasing Q at a fixed distance R between atomsA and B, leading to an exponential increase of the overlap integral

<0L|0B>.To make the estimations we use hydrogen-like Is functions

for </>A, <£L, and (j>B and assume that the orbital exponents areapproximately equal to each other, i.e., aA — aL — aB = a, then weobtain

Vi BACA(Q*) exp(-aR) + £LCL«?*) exp[-a(R - <?*)] (68)

where BA and BL are constants and the coefficients CA(Q) and

126 A. M. Kuznetsov

CL{Q) are determined by the equations

CA«?) = [|(1 - [sA(Q) - sL(Q)){[sA(Q) - eL(Q)f

+ 4V2}-' / 2)]1 / 2

C L ( Q ) = [5 (1 [ « ? ) ( Q ) ] { [ ( Q ) ( Q ) f

+ 4 V V 1 / 2 ) ] I / 2

In the classical limit, the transitional configuration qf, Q* inthe Condon approximation is determined by the equations

qt = (1 - 6*)qk0, + S*qkof

hil(Q - Qo)

(70)

e,(Q) ~ I fuok(qM,

where 9* is the symmetry factor.Equations (68)-(70) and Fig. 6 show that, according to the

lability principle, the greater the change of the electron wave func-tion of the donor complex AL due to intramolecular vibrations,the higher is the value of the transmission coefficient K (whichdepends on the electron resonance integral V,). The higher also isthe value of the activation factor since, in the classical limit, thevalue of the activation free energy Fa is smaller than the quantityF°a calculated neglecting the effect of the modulation of the electronwave function of the donor <f>A (see Fig. 6).

A simple model was considered above. A more refined theorytaking into account the modulation of the electron wave functionof the complex AL by fluctuations of the medium polarization isgiven in Ref. 35.


IV. ELEMENTARY ACT OF THE PROCESS OFPROTON TRANSFER

The development of the theory of the processes of proton transferhas taken more than 50 years and the description of earlierapproaches may be found in review articles cited previously.1"5

Some points of earlier models continue to be of interest. However,methods have been developed in recent years which enable us totake into account a number of new physical effects playing certainroles in these processes.

1. Physical Mechanism of the Elementary Actand a Basic Model

To formulate the basic model, we consider the transfer of a protonfrom a donor AHZl+1 to an acceptor B*2 in the bulk of the solution.For reactions in the condensed phase, at any fixed distance Rbetween the reactants, the transition probability per unit time W(R)may be introduced. Therefore, we will consider first the transitionof the proton at a fixed distance R and then we will discuss thedependence of the transition probability on the distance betweenthe reactants.

Unlike the simplest outer-sphere electron transfer reactionswhere the electrons are the only quantum subsystem and only twotypes of transitions are possible (adiabatic and nonadiabatic ones),the situation for proton transfer reactions is more complicated.Three types of transitions may be considered here5:

1. Entirely nonadiabatic transitions, in which the electronscannot adiabatically follow the change in the positions of the protonand the medium molecules.

2. Partially adiabatic transitions, in which the electrons followadiabatically the motion of the nuclei but the state of the protoncannot adiabatically follow the change in the state of the mediumpolarization.

3. Entirely adiabatic transitions, in which both the electronsand the proton adiabatically follow the change in the configurationof the medium molecules.

The transition probability for entirely nonadiabatic transitionsmay be calculated in the framework of the diabatic approach

128 A. M. Kuznetsov

considering the diabatic potential or free energy surfaces36 (Fig. 7).The resonance splitting of these surfaces, 2 Ve9 is small and Eq. (16)is applicable where p, and pf now represent the density matricesdescribing the proton vibrations and the fluctuations of the mediumpolarization in the initial (the proton is in the donor molecule) andfinal (the proton is in the acceptor molecule) states.

For partially adiabatic reactions, the resonance splitting of thediabatic potential energy surfaces, 2Ve, is large (Fig. 7) and onlylower potential energy surfaces corresponding to the ground stateelectrons need be considered. Since the proton is a quantum particleand the probability of its tunneling from one potential well to theother is small, the zeroth-order states describing the localization ofthe proton in each potential well, <f)l

m(r) and <f>fn(r), may be defined.

This means that instead of the lower adiabatic potential energysurface, we consider two new diabatic surfaces, Up(r) and Up

f(r),and some new perturbation, V, leading to the proton transfer. Sincethe probability of a proton transition between weakly excited vibra-tional states is small, Eq. (16) may be also used here. However, themeaning of the quantities involved in Eq. (16) should be changedin accordance with the new definition of the zeroth-order statesand of the perturbation operator. This method of calculation forpartially adiabatic reactions was developed in Ref. 5. A similarapproach was subsequently used in Ref. 37.

Figure 7. Potential energy surfaces for the protonat the transitional configuration for the mediummolecules.


Note that since the profile of the lower adiabatic potentialenergy surface for the proton depends on the coordinates of themedium molecules, the zeroth-order states and the diabatic poten-tial energy surfaces depend also on the coordinates of the mediummolecules. The double adiabatic approximation is essentially usedhere: the electrons adiabatically follow the motion of all nuclei,while the proton zeroth-order states adiabatically follow the changeof the positions of the medium molecules.

The physical mechanism of entirely nonadiabatic and partiallyadiabatic transitions is as follows. Due to the fluctuation of themedium polarization, the matching of the zeroth-order energies ofthe quantum subsystem (electrons and proton) of the initial andfinal states occurs. In this transitional configuration, {q*}, the sub-barrier transition of the proton from the initial potential well tothe final one takes place followed by the relaxation of the polariz-ation to the final equilibrium configuration.

The activation free energy of the transition between two fixedvibrational states of the proton in each potential well is determinedby the free energy of the fluctuation of the polarization to thetransitional configuration corresponding to matching of protonenergies. Possible activation of the chemical bond A—H by meansof the excitation of the proton to various vibrational levels is alsotaken into account in the theory since calculations using equationsof the type of Eq. (16) take into account transitions between anyvibrational energy levels of the proton. It is assumed that themolecule AH is in thermal equilibrium with the medium so that thedistribution over vibrational states of the proton is described bythe Gibbs formula.

From the discussion of this basic model in the literature,38'39

we note two points:1. Due to strong interaction of the reactants with the medium,

the influence of the latter may not be reduced only to the wideningof the vibrational levels of the proton in the molecules AH andBH. The theory takes into account the Franck-Condon factordetermined by the reorganization of the medium during the courseof the reaction.

2. The calculations show that for high, narrow barriers thetransitions between unexcited vibrational states of the proton givethe main contribution to the transition probability. This result is

130 A. M. Kuznetsov

not explained simply by low occupation of the excited vibrationalstates. The reason is that the probabilities of the transitions involvingthe excited states, taking due account of their occupancies, makea small contribution to the total transition probability as comparedto the transition probability between the unexcited states.

The height of the potential barrier decreases with the decreaseof the transfer distance. Therefore, the contribution of the transi-tions between excited vibrational states increases and so does thetransition probability. However, short-range repulsion between thereactants increases with a decrease of R, and the reaction occursat an optimum distance R* which is determined by the competitionof these two factors. In principle, we may imagine the situationwhen the optimum distance R* corresponds to the absence of apotential barrier for the proton. However, we should keep in mindthat the transitions between certain excited states may becomeentirely adiabatic at short distances.40'41 In this case, the furtherincrease of the transition probability with the decrease ofR becomesquite weak, and it cannot compensate for the increased repulsionbetween the reactants, so that even for the adiabatic transition, theoptimum distance R may correspond to sub-barrier proton transfer.

2. Distance-Dependent Tunneling in theBorn-Oppenheimer Approximation

First calculations of the optimum distance between the reactants,#*, taking into account the dependence of the probability of protontransfer between the unexcited vibrational energy levels on thetransfer distance have been performed in Ref. 42 assuming classicalcharacter of the reactant motion. Effects of this type were consideredalso in Ref. 43 in another model. It was shown that R* dependson the temperature and this dependence leads to a distortion ofthe Arrhenius temperature dependence of the transition probability.

A general method for the calculation of the transition probabil-ity in the harmonic approximation developed in Ref. 44 enabledus to take into account, in a rigorous way, both the dependence ofthe tunneling of the quantum particles on the coordinates of otherdegrees of freedom of the system and the effects of the inertia andnonadiabaticity of the tunneling particle, taking into account themixing of the normal coordinates of the system in the initial and


final states. The calculation has been generalized to the case of anonparabolic shape of the potential energy describing the relativemotion of the centers of mass of the reactants, taking into accountthe changes during the course of the transition in the coordinates,describing the motion of the center of mass and the intramolecularvibrations.5 For the limiting case of quantum particles (protons),the equations for the transitional configuration have been obtainedfor both classical and quantum motion of the reactants as a whole.It was also taken into account that the change in the probabilityof the tunneling of the quantum particle is due not only to thechange in the distance between the centers of mass of the reactingmolecules but also to a possible dependence of the equilibriumlength of the chemical bond on the distance between the reactants.5

In Refs. 45 and 46 the dependence of the probability of the tunnelingon the transfer distance was taken into account in the calculationof the hydrogen isotope effect.

The dependence of the proton resonance integral J for theunexcited vibrational states on the vibrations of the crystal latticewas taken into account recently in Ref. 47 for proton transferreactions in solids. The dependence of/ on the nuclear coordinateswas chosen phenomenologically as an exponential Gaussianfunction.

Below we will use Eq. (16), which, in certain models in theBorn-Oppenheimer approximation, enables us to take into accountboth the dependence of the proton tunneling between fixed vibra-tional states on the coordinates of other nuclei and the contributionto the transition probability arising from the excited vibrationalstates of the proton. Taking into account that the proton is theeasiest nucleus and that proton transfer reactions occur oftenbetween heavy donor and acceptor molecules we will not considerhere the effects of the inertia, nonadiabaticity, and mixing of thenormal coordinates. These effects will be considered in Section Vin the discussion of the processes of the transfer of heavier atoms.

First, we shall consider the model where the intermolecularvibrations A—B and intramolecular vibrations of the proton in themolecules AHZ'+1 and BHZ2+1 may be described in the harmonicapproximation.48 In this case, using the Born-Oppenheimerapproximation to separate the motion of the proton from the motionof the other atoms for the symmetric transition, Eq. (16) may be

132 A. M. Kuznetsov

transformed to the form48

W = (pV2/ih) j del dQdQ' \ drdr' g(6) exp[-H(0)] (71)

where g(6) is a nonexponential function of 0, and

H(0) = kl[(Qk + Q'k- 2QkQ)2 tanh [pfitok(l - 6)/2]

+ (Qk + Q'k- 2QL)2 tanh [pha>k0/2]

+ (Qk ~ Q'k? cotanh [j8*ait(l - d)/2]

+ (Qk ~ Q'k)2 cotanh [ph<ok6/2]]

+ ii[r ~ rOi(Q) + r' - ro,(Q')]2 tanh

+ [r - ro/(Q) + r' - ro/(Q')]2tanh

+ [r- rol(Q) -r'+ ro,«?')]2 cotanh p

+ [r - %((?) - r' + ro/((?')]2 cotanh [0ftnp0/2]}

(72)

Here Qk = {qk, R} is the set of dimensionless normal coordinatesdescribing the medium polarization (qk) and intermolecular vibra-tions A—B (R), and r is the proton coordinate. The vibrationalfrequencies are assumed to be unchanged during the course of thetransition. The vibrational frequency of the proton and its initialand final equilibrium positions may depend on the coordinates Q.The dependence of fip on Q can, in general, lead to a deviationfrom the harmonic approximation. However, for large tunnelingbarriers, a small change in the frequency Clp strongly affects theprobability of tunneling of the particle. The effect of this factor onthe other part of the vibrational subsystem is weaker.

The region near (?—<?' and r — r' gives the major contributionto the integrals. For the symmetric transition, the saddle point isequal to 6* = \. It is convenient to choose the origin of the coordin-


ate system in such a way that the equality rOl = -rof = r0 is fulfilled.After integration over r and 0, the transition probability may bewritten as

• - - J Y\dqkdRB(qk,R)

expj - I [(Qk - Q'k0)2 + (Qk- <?{0)

2]I k

x tanh(/3 W 4 ) - 2r20(Qk) tanh[jBftnp(Qk)/4]} (73)

where B(qk, R) is a slowly varying function.Equation (73) involves various limiting cases. In some situ-

ations the integrand in Eq. (73) has a sharp maximum at a pointQt. Then W may be approximately written in the form

W - const. • V2 exp{-£ [(q*k - qk0i)2

+ (q* ~ quo/)2] tanh{ph<ok/4)

- [(/?* - R0i)2 + (R* - Ro/)

2] tanh(jB*nk/4)

(74)

where (?* = { *, /?*} is determined by the set of equations

[«?« ~ <?»<») + (Qn ~ Qnof)] tanh(j8*wII/4)

+ 7~-{2rS(QB) tanh[j8ftftJ,(On)/4]} = 0 (75)

where wM = {wk,ftn}.If the behavior of the medium atoms and the relative motion

of the reactants are classical (pfi(on « kT), we have

- - J UdqkdRB(qk,R)

x cxp{-[U,(qk, R) + Uf(qk,

- 2r20(qk, R) tanh[/3tiap(qk,

x8(U,-U,) (76)

134 A. M. Kuznetsov

where U,(qk9 R) and Uf(qk, R) are the free energy surfacesdescribing the motion of the medium atoms and the reactants inthe initial and final states, respectively.

In this form, Eq. (76) is also valid in the case when Clp dependsstrongly on qk and R and when the free energy surfaces Ut(qk, R)and Uf(qk, R) are nonparabolic. In particular, if r0 and Clp areindependent of qk and the repulsion potential for the reactants isthe same in the initial and final states, i.e., URl(R) = URf(R) =U(R), we obtain from Eq. (76)

^ V2exp(-Es/4kT) dRB(R)

x exp{-pU(R) - 2r20(R) tanh[/3fcnp(tf )/4]} (77)

Equation (77) shows that if pfittp(R*)/4 » 1 at an optimumdistance R* between the reactants, proton transfer occurs by meansof tunneling between the unexcited states. However, the distanceof the proton jump, 2ro(i?*), is not equal to the distance betweenthe points of minima of the potential wells of the proton in theequilibrium nuclear configuration. This case is a generalization ofthe results obtained in an earlier model by Dogonadze, Kuznetsov,and Levich36 (DKL model).

If at the optimum distance R *, the frequency Clp is considerablysmaller than at large distances (/31iClp(R*)/4 — 1), then along withthe shift of the proton equilibrium position when the system goesto the transitional configuration, the contribution of transitionsbetween the excited vibrational states of the proton increases andthe proton transition may occur from the levels located near thetop of the potential barrier. This case corresponds to the Kreevoytype of transition.49 In the limit /3h£lp(R*)/4 « 1 we have the caseof the overbarrier proton transition. However, the formulas of thenonadiabatic theory become inapplicable in this case and thereaction is an adiabatic one.

3. Hydrogen Ion Discharge at Metal Electrodes

The basic model presented above is applicable to hydrogen iondischarge reactions at metals. A characteristic feature of these


processes is that one of the reactants is the metal electrode and thereaction rate depends exponentially on the overpotential with aconstant value of the symmetry factor a = \ in a wide region ofpotential variation (the Tafel law). The results of the quantummechanical model suggested by Dogonadze, Kuznetsov, and Levichfor this reaction36 remain of interest. According to this model,the discharge of the ion H3O

+ located at some fixed distancefrom the electrode occurs in the following way.3'36 In the initialstate, the proton vibrates in the H3O

+ ion in various vibrationalstates according to the thermal distribution. In the initial equili-brium configuration of the medium molecules, the vibrationalenergy levels of the proton are not equal to those for the protonin the adsorbed state at the electrode. A classical fluctuation of themolecular surroundings leads to matching of a given pair of protonenergy levels. In this configuration, a quantum (sub-barrier) protontransition from the vibrational level in the H3O

+ ion to a correspond-ing vibrational level of the adsorbed state occurs. A change in thestate of the electrons in the metal takes place when the proton goesunder the barrier in the region of values of its coordinate near thepoint of intersection of the potential energy curves for the protonin the initial (proton in the H3O

+ ion) and final (adsorbed hydrogenatom) states. This change in the electron state results in a redistribu-tion of the electron density and the formation of a chemical bondbetween the proton and the metal. We may conventionally say thatelectron transfer from the metal to the Me—H chemical bondoccurs.

Thus, for a transition between any two vibrational levels ofthe proton, the fluctuation of the molecular surrounding providesthe activation. For each such transition, the motion along theproton coordinate is of quantum (sub-barrier) character. Possibleintramolecular activation of the H—O chemical bond is taken intoaccount in the theory by means of the summation of the probabilitiesof transitions between all the excited vibrational states of the protonwith a weighting function corresponding to the thermal distribu-tion.3'36 Incorporation in the theory of the contribution of the excitedstates enabled us in particular to improve the agreement betweenthe theory and experiment with respect to the independence of thesymmetry factor of the potential in a wide region of 8(p.50 A similarapproach has been used recently in Refs. 51 and 52, where the

136 A. M. Kuznetsov

dependence of the probability of the electron tunneling to the H3O+

ion on the electrode potential was also taken into account.With respect to the electrochemical reactions of hydrogen ions

where the transfer of two particles (electron and proton) occurs, itis of interest to discuss the problem of whether the electron transferand the process of breaking or formation of the chemical bond aresimultaneous or sequential.53 Since the characteristic times of themotions of light and heavy particles are rather different, we maydetermine along the coordinates of which particles the motion ofthe system occurs at a given part of the complex trajectory in thehyperspace leading from the initial equilibrium configuration tothe final one. For the transition of the proton between unexcitedvibrational states, this trajectory, in a crude approximation, is asfollows: (1) classical motion along the coordinates of the mediummolecules qk from qkOl to q*, (2) sub-barrier motion of the protonat a fixed value of qk from rQl to r*, (3) sub-barrier motion of theelectron from xOl to xof at fixed values of qk and r, (4) sub-barriermotion of the proton from r* to r0/, and (5) classical motion alongthe medium coordinates qk from qk to qkof.

Thus, the change of the electron state ends earlier thanthe change of the proton state. However, since the changeof the localization in space of the electron itself says nothingabout the rearrangement of the chemical bonds, this does not meanthat the electron transfer step occurs first and then the breaking orformation of a chemical bond takes place. The electron transferand the rearrangement of the chemical bond may be a unified step.The character of the process depends on the potential energy surfaceof the system after the change of the electron state.

If prior to the electron transition the potential energy surfacealong the proton coordinate r had a minimum corresponding to astable chemical bond, various situations are possible after thechange of the electron state due to the electron transition:

1. The new potential energy surface has no minima. This meansthat the electron transfer leads to cleavage of the chemical bond.The possibility of the formation of a new chemical bond depends,in this case, on the location of the other PES (see below).

2. The new potential energy surface has a minimum corre-sponding to localization of the proton near another atom ormolecule (or near the surface of a solid). This means that the


electron transfer results in the breaking of one chemical bond andthe formation of another one, both processes proceeding in a singlestep.

3. The new surface has a slightly shifted minimum. In thiscase, the result depends on the location of the other PES. If thenew PES intersects another PES corresponding to the localizationof the proton near another molecule (or a solid), a dynamic or afluctuation-relaxation transition to this PES is possible, leading tothe formation of a new chemical bond. However, the rate of thewhole process depends on the characteristics of the chemical bondformed only if the latter transition is the rate-determining one. Ifit is fast, the rate of formation of the new chemical bond will beindependent of its characteristics and will be determined only bythe characteristics of the original molecule.

Further development of the basic model and the detailedanalysis of the dependence of the symmetry factor on the potentialand the temperature54 have shown that there are additional factorswhich can affect the elementary act of this reaction. These investiga-tions led to the formulation of the charge variation model (CVM)55

which will be discussed in the next section.

4. Charge Variation Model

In the basic model presented above, it was assumed that the hydro-gen atom in the adsorbed state is neutral and weakly influences thestate of the medium molecules. In this model the free energy surfacesof the solvent, determining the activation free energy of the transi-tion between two fixed vibrational states of the proton, m and nwere of parabolic shape,

' 3 rUmi = (ITT/C) [ P2(r) d3r - J P(r)E^(r) d3

(78)

[/n/ = (2TT/C) P2(r) c/3r + const.

where P(r) is the inertial polarization of the medium and EH3O+ is

the electric field due to the H3O+ ion. This resulted in the linear

dependence of the symmetry factor for this local transition, amn,

138 A. M. Kuznetsov

on the free energy of the transition, AFnm,

amn=12[l + AFnm/Es] (79)

and only by allowing for transitions between various excited statesof the proton was this dependence made weaker.

A model taking into account the variation of the charge of theadsorbed hydrogen in the activation-deactivation process has beensuggested in Ref. 55. It may be called the charge variation model(CVM). It is known from the theory of chemisorption that thecharge of an adsorbed atom depends on the position of the electronenergy level in the adsorbed atom.56 Due to the interaction of theelectron with the medium, its energy level in the adatom varies.Therefore, with the fluctuations of the molecular surroundings, thecharge of the adsorbed atom varies and hence so does its interactionwith the medium. The latter becomes nonlinear in the coordinatesof the medium species. This leads to a distortion of the shape ofthe free energy surfaces of the solvent at fixed vibrational states ofthe proton, m and n, as compared to that for the basic model andleads to a change of the dependence of the activation free energyof the local transition on AFnm.

If we describe the state of the medium molecules by oneeffective configurational coordinate q, the free energy surfaces ofthe solvent Uim and Ufn have the form55

q0l) im(80)

Ufn(q) = \hcoq2 + ef(q) + VHs(q) + Efn +j{

where VHs(q) is the energy of the interaction of the proton withthe solvent, and Ef

n is the energy of the nth vibrational state of theadsorbed atom H.

The electron energy ef(q) in the Anderson model56 hasthe form55

ef(q) - (2A/7r){[(*a - E F ) / A ] t a n ^ A / ^ - eF)]

- 1 + 7r[(ea - eF)/A]0(eF - ea)} - Un2/4 (81)

where ea(q) a n d A are, respectively, the energy and the width ofthe electron level in the adatom, eF is the Fermi energy in the metal,and U is the repulsion energy of the electrons in the adatom.


The number of electrons in the adatom, n(q), is determinedby the equation

n = (2/77){tan-1[(EF - e°a - Un/2)/A] + TT/2} (82)

where e°a is the zeroth-order electron energy level in the adatom,and ea is related to e°a by the relationship

ea = e°a + lUn(q) (83)

From Eqs. (80) and (81) we obtain for the activation freeenergy Fa of the local transition m -» n55

FJEr = {y[q0 + Ox + b tan(mc/2)] - I}2 (84)

where the excess number of electrons in the adatom, x = n — 1 isrelated to the local free energy of the transition, AFnm, by theequation

2y[ Ox + q0 + 6(1 - x) tan(7rx/2) - Ux2/2] = 1 - &FnJEr (85)

and the following dimensionless quantities are introduced:

y = (fi(o/2Er)1/2', 0 = U/2(2Erfi(o)l/2; b = k/(2Erfia>yn

q0 = [E0 " eF + U/2]/(2Ertico)1/2- Er = (eg)2/2fico (86)

where e0 is the electron energy in the isolated atom, and g is thecoupling constant for the interaction of the electron and the protonwith the medium.

The local symmetry factor amn is given by the equation

amn = dFJdbFnm = {1 - y[b tan(7rx/2) + q0 + Ox]}

x (1 -jc-2tan(7TJc/2)/7r[l + tan2(7rx/2)]

x {1 +2L/[1 +tan2(7rx/2)]/7r6})"1 (87)

Calculations with the use of Eqs. (84)-(87) show that amn canremain equal to \ over a wide range of the AFnm values for certainvalues of the physical parameters (Fig. 8). The results depend ratherweakly on the value of b/ 0 = 2A/ U for b/J values between 0.001and 0.1 and are quite sensitive to the values of q0 and 0. If we

140 A. M. Kuznetsov

1

s—1Ti

, - —j

0.5

Figure 8. Dependence of the sym-metry factor a on the free energy ofthe transition for the reaction of hydro-gen ion discharge on a metal electrode.

neglect the terms in b9 Eqs. (84), (85), and (87) may be simplifiedas follows:

FJEr = [yUx-\f

amn = (1 - 2yUx)/2(l - x) = (1 - Ux/2Er)/2(l - x)

2y( Ox - Ox112) = -AFnm/Er (88)

where the value 1/(27) is chosen for q0.For small x-values, we obtain from Eq. (88)

/Er] (89)

Equation (89) shows that the allowance for the variation ofthe charge of the adsorbed atom in the activation-deactivationprocess in the Anderson model leads to the appearance of a newparameter 2Er/ U in the theory. If U — 2Er, the dependence of amn

on AFnm becomes very weak as compared to that for the basicmodel [see Eq. (79)]. In the first papers on chemisorption theory,a U value of ~13eV was usually accepted for the process ofhydrogen adsorption on tungsten. However, a more refined theorygave values of =s6eV.57 For the adsorption of hydrogen fromsolution we may expect even smaller values for this quantity dueto screening by the dielectric medium.

It should be noted that a weaker dependence of the symmetryfactor on the free energy of the transition was obtained without


considering contributions from the excited vibrational states of theproton. The latter make this dependence even weaker. The physicalmeaning of the result obtained is that the charge of the adsorbedatom varies in the activation-deactivation process, leading to thedistortion of the shape of the free energy surfaces (FES) of thesolvent. When the coordinate q varies from qof to g*, the depen-dence of Ufn on q deviates from the parabolic shape assumed inthe basic model. This deviation influences both the value of theactivation free energy, which is smaller here than in the basic model,and its dependence on the free energy of the transition.

Note that the results obtained are in accordance with the labilityprinciple. The smaller U is, the more labile are the electrons in theadatom and the stronger is the distortion of the shape of the freeenergy surfaces, leading to a decrease of the activation free energyand to an increase of the transition probability.

The modulation of the charge of the adsorbed atom by thevibrations of heavy particles leads to a number of additional effects.In particular, it changes the electron and vibrational wave functionsand the electrostatic energy of the adatom. These effects may alsoinfluence the transition probability and its dependence on theelectrode potential.

The CVM provides, to some extent, a theoretical explanationfor the constancy of the symmetry factor over a wide range ofpotential variation. However, the problem of the sharp change ofthe symmetry factor a between normal and barrierless regions58

still remains unsolved in the framework of the existing models. Itwas suggested in Ref. 54 that such a sharp change of a may bedue to a local structural transition near the reaction zone due to aphase transition in the near-electrode layer. Transitions of this typeusually occur in a narrow range of values of external parameters(pressure, temperature, electric field, etc.). Together with the CVM,these transitions may, in principle, explain the experimentallyobserved dependence of a on the overpotential.

The effects of charge variation may also play a certain role inother processes involving adsorbed atoms, in particular, in electrontransfer processes.59 The physical nature of these effects is to someextent similar to that of the effects of polarization of the electronplasma of the metal by vibrations in a polar medium consideredin Ref. 60.

142 A. M. Kuznetsov

V. PROCESSES INVOLVING TRANSFER OF ATOMSAND ATOMIC GROUPS

Reactions involving transfer of atoms and atomic groups representa more complicated theoretical problem since they are often par-tially or entirely adiabatic and, in addition, a number of effectswhich are not very important in electron transfer reactions must beconsidered. These effects are:

1. fluctuational preparation of the potential barrier fortunneling

2. inertia3. nonadiabaticity of the motion4. deviations from the Condon approximation5. interrelation of the motions of the atoms and mixing of the

normal coordinates6. anharmonicity of the vibrations.Furthermore, there are some effects related to the interaction

of the reactants with the medium. We shall first consider the effectsof the fluctuational preparation of the potential barrier in non-adiabatic reactions.

1. Fluctuational Preparation of the Barrier and RolePlayed by the Excited Vibrational States in the

Born-Oppenheimer Approximation

The effects of transfer of atoms by tunneling may play an essentialrole in a number of phenomena involving the transfer of atoms andatomic groups in the condensed phase. One may expect that theseeffects may exist not only in the proton transfer reactions consideredabove but also in such processes as the diffusion of hydrogen atomsand other light ions (e.g., Li+) in liquids, tunnel inversion andisomerization in some molecules, quantum diffusion of defects andlight atoms in the electrode at cathodic incorporation of the ions,ion transfer across the liquid/solid interface, and low-temperaturechemical reactions.

A model for the diffusion of light ions in structured liquidshas been suggested recently by Schmidt.61 The elementary act ofdiffusion is considered in this model to be the transfer of the ionfrom one cage formed by solvent molecules to a neighboring one.


A formalism similar to that used for partially adiabatic protontransfer reactions was applied in the calculation of the transitionprobability. This model of the diffusion jump is similar to the modelof the diffusion of light defects in solids which was first consideredin Ref. 62.

The probability of the tunneling of a heavy particle from onecage of the condensed medium to another depends on the shapeof the potential barrier formed by other atoms, which, to certaindegrees, can block the transfer of a given particle. This effect issimilar to the modulation of the electron resonance integral byfluctuations of the medium molecules considered above (see alsoRef. 63). The tunneling of the particle between unexcited vibrationalstates has been considered in Ref. 62, and the dependence of theresonance integral /Oo on the coordinates of the symmetric vibra-tional modes of the medium, qS9 was taken into account. This wasdone essentially by averaging the transition probability with thequantum distribution function for the coordinates of the symmetricmodes.62 It will be shown below that this procedure is valid onlyin the high-temperature limit [see also Eqs. (76) and (77)]. Thisproblem was considered also in Ref. 64, where only the coherenttransitions were taken into account. Therefore, the result obtainedin Ref. 64 in the high-temperature limit differs from that obtainedin Ref. 62. A more general method has been developed in Ref. 65.It takes into account the deviations from the Born-Oppenheimerapproximation which will be discussed later.

Below we will restrict ourselves to the Born-Oppenheimerapproximation and, unlike Refs. 62, 64, and 65, we will take intoaccount the contribution from the excited vibrational states of thetunneling particle and consider the role played by the transversequantum vibrations of the tunneling particle itself in the preparationof the potential barrier.48

(i) Role of Quantum Fluctuations of the Tunneling Particle

First, consider the symmetric transition of a particle betweenunexcited vibrational states assuming classical behavior of themedium atoms which form the microstructure near the tunnelingparticle and determine its potential energy. The states of the systemcorresponding to the localization of the particle in the initial and

144 A. M. Kuznetsov

final potential wells are characterized by different shifts of theequilibrium positions of the medium atoms. The shifts of the equili-brium positions of the medium atoms lead to shifts of the energylevels of the tunneling particle ("polaron" effect). However, inaddition to the polaron effect, the tunneling at equilibrium configur-ations of the atoms may be inhibited due to the blocking effect ofthe medium atoms. In this case, the symmetric vibrational modesof the medium atoms play an essential role by leading to the shiftof the blocking atoms and hence to a decrease of the barrier fortunneling. However, if a face-to-face blocking (or a blocking closeto this type) takes place such that the blocking atom is located ona line connecting the equilibrium positions of tunneling particlesor if the blocking atoms are not located in symmetric positions withrespect to this line, the barrier may be considerably decreased dueto the quantum fluctuations of the tunneling particle in a directionperpendicular to the tunneling direction. We will consider this effectfor the case of face-to-face blocking.48

In this case, we may consider that the resonance integral Vdepends on p = |qs - qp|, where qs is the shift of the blocking atomin the direction perpendicular to the tunneling axes, qa, and qp isthe vector determining the position of the tunneling particle in thesymmetry plane which is perpendicular to the axes qa. Assumingthat the motion of the particle along the coordinate qa can beadiabatically separated from that along the coordinate qp, in theharmonic approximation for the medium atoms we obtain for thetransition probability

W=W0O(V2)/V20O (90)

where Voo is the resonance integral calculated at the equilibriumposition of the blocking atom,

Woo = Vlo(7T/h2kTEr)1/2cxp(-Er/4kT) (91)

where Er is the reorganization energy for the antisymmetric modesqa, and

2

(V2) = J d2qsf(qs)\ J d2qp (92)

where iAo( p) is the wave function of the ground state for thetransverse vibrations of the tunneling particle, and f(qs) is thedistribution function for the symmetric modes in the classical limit.


In the harmonic approximation for the transverse vibrationsof the tunneling particle, we obtain from Eq. (92)

PP'A!

x exp[-(A2 + A2)(p2 + p'2)/A2A2] (93)

where Io is the Bessel function of imaginary argument, and

- 2 .

2

A;2 =

A72 = [-T1

Various limiting cases can be obtained from Eq. (93).48 If thedependence of V on p is an exponential one:

V(p) = Voexp[-£(p)] (95)

where B(p) » 1 and B(p) decreases with an increase of p, then at2A? » A« we obtain48

(V2) - V2(p*) exp[-Mcu2p*72/cT] (96)

where p* is determined by the equation

2 = O (97)

Thus, in this limit the activation energy Ea in the transitionprobability involves, in addition to the quantity Er/4, the termM(o2p*2/2 related to the shift of the blocking atom with respect tothe axis qa. This result coincides with that obtained by Flynn andStoneham62 in the classical limit.

In the opposite limit (Ap/\/2 » AJ we have

(V2) ~ V2(p*) exp(-2mpIV*7/i) (98)

In this case the preparation of the barrier is performed mainly bythe quantum fluctuations of the tunneling particle in the transversedirection. Note that the width of the distribution here is 1/V2 ofthat in the distribution function for the coordinates qp. This is dueto the fact that in this case the fluctuations of the particle are ofquantum character and a coherent averaging of the resonance

146 A. M. Kuznetsov

integral over the wave function of the ground state occurs. Theequation for the optimum configuration q* in this case has the form

dB/dp + 2p*/Ap = 0 (99)

(II) Role Played by the Excited States of the Tunneling Particle andQuantum Effects for the Vibrations of the Medium Atoms

At not very low temperatures, the excited vibrational states ofthe tunneling particle make some contribution to the transitionprobability. Furthermore, at high enough frequencies of the vibra-tions of the medium atoms, quantum effects may be important forthe medium. In the harmonic approximation for the tunnelingparticle we obtain, in a similar way as in the case of proton transfer,the expression for the transition probability in the Condonapproximation48

x f 11 dq8k dqpB(q) exp(-{[r - ro(q, 0)]2 + [r + ro(q, 0)]2}

J k

x tanh|j3fia(g, 0)/4])<f>s(qsk) e x p i - 2 £ q\ tanh(j8ftn£/4) 1

(100)

where q = {qsk, qp}, and r is the tunneling coordinate. The origin

of the coordinate system is chosen in such a way that qk"0 = ~qkfo =

qlo, qsko = qsko = 0, and rOl = - r 0 / = ro(q9 qak).

The value gak = 0 in ro(q9 ql) and il(q, ql) corresponds to the

transitional configuration for the antisymmetric modes. The func-tion <f>s(q

sk) for the symmetric modes

<l>s{q'k) = Z~x exp|^-2X (4D2tanh(/3fi<4/4)] (101)

does not coincide, in general, with the distribution function overthe coordinates qk

s,

fs(qi) = Zlo e x P [ - £ (^t)2tanh(i8fta>i/2)J (102)i/2)J


The functions <t>s(ql) and/9(g£) coincide with each other onlyin the classical limit {fi(ok « kT).

After the integration over r, we obtain from Eq. (100)48

W - Vfexpl-

\X\dqlJ k

l dqp B(q) exp[-2r;fo, 0) tanh(phn(q9 0)/4)

- 21 (qsk)

2 tanh(j8fto>'k/4) - 2 1 gj tanh(/3*n£/4) 1

(103)

Thus, the allowance for the dependence of the resonanceintegral on qs

k may not be reduced in general to averaging thetransition probability over the distribution function in Eq. (102).The function (f>s(qk) plays the role of the distribution function forthe coordinates qk in the case of the symmetric transition. In theclassical limit, the results of Flynn and Stoneham62 can be obtainedfrom Eq. (103), and in the low-temperature limit, the result ofKagan and Klinger64 can be obtained.

2. Role of Inertia Effects in the Sub-Barrier Transferof Heavy Particles

In the calculation of the transfer by tunneling of light atomic species,it is usually assumed that the potential energy of the particle u(r)depends on the distance r — R between the particle and the atomto which it is bound. The dynamic interaction between the motionof the atom creating the potential u(r - R) and the motion of thetunneling particle has usually been neglected. This approximationis good if the mass of the particle is considerably smaller than themass of the atom, as is the case for proton transfer in a potentialfield of heavy atoms. However, if the masses of the species arecomparable, effects of nonadiabaticity and inertia can takeplace.65'66 In this section we shall consider the inertia effect in amodel similar to that used in Ref. 66. However, we will take intoaccount some additional effects67 which were not included in theearlier theory.

148 A. M. Kuznetsov

Let us consider the transition of a particle of mass m (forbrevity, we shall call it a "proton") in a two-minimum potentialu(r — R) formed by an atom of mass Mo in a polar medium. Thetotal Hamiltonian of the system taking account of the interaction,VpL, of the tunneling particle with the vibrations of the mediumatoms (phonons) has the form

H = P2/2M0 + M0co20R

2/2 + p2/2m + u(r - R) + VpL + HL (104)

where HL is the Hamiltonian of the medium atoms, P and p arethe momentum operators of the atom and the proton, respectively,and (o0 is the frequency of vibrations of the atom (col = fc/M0,where k is the force constant).

The Hamiltonian in Eq. (104) may describe both the processof tunnel inversion or isomerization of a molecule and the inertiaeffects arising from the symmetric vibrations of the reaction complexAH--B in the cage of the solvent or solid matrix (Fig. 9). In thelatter case, the coordinate and the frequency of the symmetricvibration correspond to R and <oQ.

Introducing the coordinates of the center of mass and of therelative motion of the proton and the atom

x = r- R, X = (M0R + rm)/M9 M = M0+m(105)

P = (Mop - mP)/M, P = P + p, JJL = mMJM

we may transform the Hamiltonian in Eq. (104) to the form66

H = P2/2M + Mco2X2/2 - no)20xX + p2/2^i + u(x)

+ (fjL2/2M0)co20x

2 - I ykqkx + HL (106)

k

with

o)2 = w20M0/M

where, for the sake of simplicity, the interaction is chosen to belinear in the relative coordinate x which corresponds to the inter-action of the medium with the dipolar momentum created by the

O oM Q Figure 9. Linear reaction complex for

the atom transfer reaction.


shift of the proton with respect to the atom. This approximation isnot of fundamental importance and is chosen only for the sake ofsimplicity.

Equation (106) shows that the interaction of the proton withthe motion of the center of mass, described by the terms propor-tional to /JL, is formally of the same form as the interaction with themedium atoms, and the first three terms in the Hamiltonian in Eq.(106) are equivalent to addition of one more degree of freedom tothe vibrational subsystem. Thus, this problem does not differ fromthat for the process of tunnel transfer of the particles stimulatedby the vibrations which were discussed in Section IV. So we mayuse directly the expressions obtained previously with substitutionof the appropriate parameters.

First, we will consider the symmetric transition and will assumethat proton transfer occurs between unexcited vibrational states,the other part of the vibrational subsystem being described byclassical mechanics. Then we obtain67

W = V2[TT/ h2kT(El: + ET)']'12 exp[-(ErL + ET)/4kT] (107)

where E\: is the reorganization energy of the medium, and ET isthe reorganization energy of the degree of freedom describing themotion of the center of mass:

ET = W<o2(X0l - X0f)2 (108)

where

XOl = (/Lt/AfoX*),; xof = (fi/M0)(x)f (109)

Here (x)t and (x)f denote the mean values of the relative coordinatex over the states of the proton in the first and second potentialwells, respectively. Equation (107) shows that the inertia effectslead to a decrease of the activation factor in the transition probabil-ity due to an increase of the reorganization energy. The greater themass, m of the tunneling particle and the frequency of the vibrationsof the atom, o)0, the greater is this effect. The above result corre-sponds to the conclusion drawn in Ref. 66.

However, the inertia leads to one more effect which wasnot considered in the earlier theory. Equation (106) shows thatthe inertia results in the appearance of an additional termin the potential energy of the proton, which leads to a change of

150 A. M. Kuznetsov

the proton resonance integral.67 To estimate this effect, we willconsider a model potential for u(x) of the type

x - x 0 ) 2 , x > 0

x + x 0 ) 2 , X < 0

Using the wave functions of the harmonic oscillator in eachpotential well of the proton, we can estimate the total effect of theinertia on the transition probability in the high-temperatureapproximation for the medium67:

W/ Wo -

- (ti2/MM0)(Mco20x

20/2)/kT] (111)

At 4kThCl > (ti(o0)2, the first term in the exponent is greater

than the absolute value of the second term and the inertia effectleads to an increase of the transition probability.

If the motion of the center of mass is of quantum character(low temperatures), the inertia effect leads only to the renormaliz-ation of the resonance integral,

V2= V20Qxp[(2mn/fi)(\ - M0n

3/Mfc)x20- Ec

rm/fico] (112)

and at (M/M0)3/2(Q,/(o0) > 1, this leads also to an increase of the

transition probability.At arbitrary temperatures, taking into account the contribution

from the excited vibrational states of the tunneling particle, wehave approximately,67

= Woexp -W= Woexp -2(fiii/h)xltanh(pm/4)

- 2(Mco/ fi)(fji2/M20)x

20tanh(p1iG>/4)\ (113)


where

x0 = x0n2/n2

- 2(mn/h)x20tanh(pfin/4) (114)

Thus, the inertia of the tunneling particle leads to two oppositeeffects: a decrease of the transition probability due to the reorganiz-ation along the coordinate of the center of mass and an increaseof the transition probability due to the increase of the Franck-Condon factor of the tunneling particle. Unlike the result in Ref.66, it is found in Ref. 67 that for ordinary relationships betweenthe physical parameters, the inertia leads to an increase of thetransition probability.

3. Nonadiabaticity Effects in Processes Involving Transferof Atoms and Atomic Groups

The effects of deviations from the Born-Oppenheimer approxima-tion (BOA) due to the interaction of the electron in the sub-barrierregion with the local vibrations of the donor or the acceptor wereconsidered for electron transfer processes in Ref. 68. It was shownthat these effects are of importance for long-distance electron trans-fer since in this case the time when the electron is in the sub-barrierregion may be long as compared to the period of the local vibra-tion.68 A similar approach has been used in Ref. 65 to treat non-adiabatic effects in the sub-barrier region in atom transfer processes.However, nonadiabatic effects in the classically attainable regionmay also be of importance in atom transfer processes. In theharmonic approximation, when these effects are taken into accountexactly, they manifest themselves in the noncoincidence of the

152 A. M. Kuznetsov

normal coordinates with the coordinates of individual particles.44

However, in reactions involving the breaking of chemical bonds,the anharmonicity of the vibrations of atoms may be of importance.A model has been suggested in Ref. 69 which enables us to calculatethe probability of partially nonadiabatic transitions taking accountof the effects of nonadiabaticity both in the sub-barrier region andin the classically attainable one.

According to Ref. 69, we consider potential energy surfacesof the type

u, = Mw2R2/2+ m n V / 2 - V,(x, K) + s l *«*(* - <fc>.)2 (115)

uf = Mco2(R - Ro)2/2 + m()2(x - xo)2/2 - Vf(x, R)

2 (116)

where Vt and Vf describe the interaction of the motion of thetunneling particle (x) with an atom (R), the motion of the latterbeing considered to be classical, and the qk are the dimensionlesscoordinates of the other atoms of the reactants and the mediumwhich do not interact directly with the motion along thecoordinate x.

It is assumed that the interactions Vt and Vf are linear in x:

Vi(x, R) = xft(R) = mn2xx%R) (117)

Vf(x, R) = (x- Xo)ff(R) = mtl2(x - xo)[x%R) - x0] (118)

The original expression in Eq. (116) for the transitionprobability is transformed to the form

W - A \ dd \ dxdR Fq(0)Pi(x, R, x,R9l- 6)Pf(x, R, x, R; 6)

(119)

where Fq(6) is the known expression for the generating functionfor the vibrational subsystem {q}.

The expressions for the density matrices p, and pf are writtenthrough the Feynman path integrals over x(t) and (O-70 The pathintegral over x(t) can be calculated exactly for the arbitrary formof the functions fi(R) and f/(R). The path integrals over R(t)cannot, in general, be calculated exactly. However, taking into


account that the major contribution to the integral over R in Eq.(119) comes from a small region in the vicinity of a point R*,approximate expressions may be used for the functions fi(R) and

?(i?*) + bty (120)

ff(R) = mn2[x°f(R*) - x0] + bfy (121)

where y = R - R*.Then the actions Sy[y(t)](y = i , / ) in the path integrals are

quadratic functions of y(t) and the path integrals over y(t) can becalculated using standard methods,70 viz., by introducing new vari-ables y(t) = z(t) + s(t) where z(t) is the trajectory minimizing theaction. The insertion of the density matrices p, and pf calculatedin this way in Eq. (119) leads to Gaussian integrals over x and R,which can be easily calculated. The remaining integral over 0 iscalculated using the saddle point method. The final expression forW is rather cumbersome but it takes into account, in a rigorousmanner, the interaction of the tunneling particle with the motionalong the other degrees of freedom (R).69

Below we will present the result for the symmetric transition[0* = i, R * = R/2, bi = bf=b> 0, x%R*) = JCO - x%R*)] withdue account of the effects in the lowest order in the quantity b69:

W = Aoexp(-Fa/kT-a) (122)

where

Fa = FBa° + 8Ea; (T = <rB0+ Sa (123)

FBa° = F°a - mn2[x°(#*)]72; F°a = (Er + ER)/4 (124)

ER = Mco2R20/2 (125)

aB0 = <ro+ {m£l/2h){[x°f{R*) - x°(R*)]2 - x20}

= <T0 + (mQ/2h){[x0 - 2x?(R*)]2 - x20};

a0 = mnx20/2h (126)

154 A. M. Kuznetsov

The quantities F°a and a0 describe the activation free energyand the tunneling factor in the absence of interaction between themotion of the tunneling particle and that along the coordinate R(i.e., at b = 0). The quantities F®° and aB0 are the activation freeenergy and the tunneling factor in the Born-Oppenheimer approxi-mation (i.e., in the approximation when the motion along thex-coordinate in the initial and final states is adiabatically separatedfrom the motion along the R-coordinate). The terms 8Ea and 8agive the corrections for the effects of nonadiabaticity:

8Ea = (co2/[l2)b&x • RJA (127)

8a = -(a)2/£l2)bkx' RJfitl

with

Ax = x°f(R*) - *?(#*) (128)

Equations (124)-(126) show that even in the BOA the inter-action between the motion of the tunneling particle and the motionalong the coordinate R affects significantly the kinetic parameters,leading to a decrease of the activation free energy and of thetunneling factor a. This effect is in accordance with the labilityprinciple.26 In this case, the lability of the transferable particle isdetermined by the value of the shift of its equilibrium positionx°(R) with the change of the coordinate R. The greater this shift,the smaller is the tunneling distance for the particle in the transitionconfiguration (i.e., at R = R*) and the smaller is a. Note that werefer here to the change of the equilibrium lengths of the chemicalbonds and valence angles with the change of the nuclear configur-ation of the molecule rather than to the trivial decrease of thetransfer distance due to reactants approaching each other. Forexample, the change may be in the H—C—H valence angles inligand substitution reactions of alkyl halides, or it may be theelongation of the chemical bond of the proton with the change inconfiguration of the reactants in intramolecular proton transferreactions. The lability of the transferable particle leads to a decreaseof its energy by the value ml)2[jc?(JR*)]2/2. The activation energyis decreased by the same quantity.

In the process considered above, the corrections due to non-adiabaticity effects lead to an increase of Fa and to a decrease ofa. However, the first effect is greater than the second, and the joint


effect of the nonadiabaticity is a decrease of the transitionprobability.

4. Ligand Substitutions in Alkyl Halides

Ligand substitutions in alkyl halides,

H H\ /

Y- + H - C - X -» Y - C - H + X"/ \

H Hrepresent an example which enables us to illustrate the effects ofinterrelation of the motions of individual atoms and some additionaleffects due to interactions with the medium polarization.

These reactions were considered in Ref. 71 in the model of thelinear complex, and the motion of all the atoms was described withthe use of classical mechanics. However, the frequencies of theintramolecular vibrations are rather high (500-1000 cm"1). There-fore, a model has been suggested72 which enables us to take intoaccount possible quantum effects. In this model,72 it is assumedthat the reactants are in a "cage" formed by the solvent molecules,and the interaction potentials between the atoms Y, C, and X andthe nearest medium molecules are replaced by the effective onesVYC and Vcx, depending only on the distances rCY and rc x ,respectively. Thus, a linear complex Y- • O • X is considered whosecenter of mass is assumed to be fixed. In the initial state the motionof the ion Y~ relative to C—X is a low-frequency one (co ~40-100 cm"1). In the final state, the motion of the ion X" relativeto CY is a low-frequency one. The reaction is assumed to benonadiabatic since it leads to a considerable redistribution of theelectron density. The effective charge is transferred from Y to X.The linear complex Y- • -C- • -X is characterized by the set of normalvibrations. In this model three normal vibrations are considered:one antisymmetric vibration (qH) describing the motion of theprotons in the CH3 group, and two normal vibrations (qx and q2)describing the relative motions of the atoms Y, C, and X. It isimportant that the dimensionless normal coordinates qx and q2 inthe initial and final states are different (the effect of mixing of the

156 A. M. Kuznetsov

normal coordinates):

q\ == [a)\mcm^/fi{mc + mx)]1 /2rcx

q\ - [col2rnY(mc + mx)/fi(mc + mx + mY)]1/2

x ( r Y C + "*x r c x )

q{— [co{mx(mc + mY)/fi(mc + mx + 1/2

r c xmc mY

rYC

(129)

(130)

q{ —

o>{.where co\, w { »The normal vibrations q\ and q{ are related to the shifts of

the ions Y~ and X". The low-frequency part of the inertial polariz-ation of the medium, £k((ok « co\, co{), cannot follow these shifts.The high-frequency part of the inertial polarization, £/(&>/ »w[, o){), adiabatically follows the shifts of the ions Y~ and X", andthe equilibrium coordinates of the effective oscillators describingthis part of the polarization depend on the normal coordinates ofthe corresponding normal vibrations, viz. ^/0,(^i), ^0/(^2)-

The calculation is performed using Eq. (16) and the modelpotential energy surfaces

U, =

- qH0,)2

(131)

Uf =

- qH0f)2 + l

2tiw{(q{ - q{0/)

72The resulting expression for the transition probability has the form

W = (V?/ft) cxp[-(Aq°H)2/2](27rkT/\d2H(0)/d02\e,y/2g(e*)

x exp[-H(6*)/kT] (132)


where g(0*) is a constant, 0* is determined by the equationdH(0)/dO = 0, and

H($) = 0 • AF + Fouter(0) + Finner(0) (133)

Neglecting the quantum tail of the inertial polarization of themedium, we can write the outer-sphere contribution to H(6) in theform

FouteT(0) = 8(l-0)Es (134)

where

+ 1 1 *a>,[^/o.(9l*) - W(«2*)] 2 (135)

The quantity A^H m Eq. (132) is the shift of the equilibriumpositions of the protons (at fixed transitional configurations of theother nuclei).

The contribution to H(0) from the reorganization of thecomplex has the form72

FmneT(e)/kT = 4>n(Wio)2 + <MA<?{0)2 + 2(/.12A<7{0A<?{0 (136)

where

011 = (61/Q)(CxC2

2axb2xxy2+ Cla2b2x2y2+

022 = (b2/ Q)(Cxaxbxxxyx + CxC\a2bxx2yx

012 = -(CxC2bxb2/Q)(axxx - a2x2)(yxy2)l/2

Q = bxb2yxy2 + Cxaxbxxxyx + CxC\axb2xxy2

H- CxC\a2bxx2yx + C1a2^2^2j;2 + ^^2X^2 (137)

with

ak = tanh[xfc(l - 6)]; bk = tenh{yke);

xk = hcolk/2kT; yk = hco{/2kT

Cx = mxmY/(mx + mc)(mY + m c);

C2 = [(mx + mY + mc)mc/mxmY]l/2

The shifts of the normal coordinates A^{0 and A^{o are deter-mined from Eqs. (130) using the equilibrium values rScx and $

158 A. M. Kuznetsov

The expressions obtained take into account the mixing of the normalcoordinates and possible quantum effects due to the reorganizationof the complex and the different character of the interactions ofvarious normal vibrations with the medium polarization. Estima-tions using Eqs. (132)-(135) for various reaction pairs in which F,Cl, Br, and I serve as the ligands Y and X gave satisfactoryagreement with experiment except in the case of the systemCH3C1/Br.72

VI. DYNAMIC AND STOCHASTIC APPROACHES TOTHE DESCRIPTION OF THE PROCESSES OF

CHARGE TRANSFER

It is known that the interaction of the reactants with the mediumplays an important role in the processes occurring in the condensedphase. This interaction may be separated into two parts: (1) theinteraction with the degrees of freedom of the medium which,together with the intramolecular degrees of freedom, representthe reactive modes of the system, and (2) the interactionbetween the reactive and nonreactive modes. The latter play therole of the thermal bath. The interaction with the thermal bathleads to the relaxation of the energy in the reaction system. Further-more, as a result of this interaction, the motion along the reactivemodes is a complicated function of time and, on average, hasstochastic character.

Recently, much attention has been paid to the investigation ofthe role of this interaction in relation to the calculations for adiabaticreactions. For steady-state nonadiabatic reactions where the initialthermal equilibrium is not disturbed by the reaction, the couplingconstants describing the interaction with the thermal bath do notenter explicitly into the expressions for the transition probabilities.The role of the thermal bath in this case is reduced to that theactivation factor is determined by the free energy in the transitionalconfiguration, and for the calculation of the transition probabilities,it is sufficient to know the free energy surfaces of the system asfunctions of the coordinates of the reactive modes.

Two different approaches are used at present in the theory ofthe processes of charge transfer in polar media. One of them is


based on the dynamic description of the polar medium andintramolecular vibrations. The other one originates from the worksof Kramers73 and is based on the stochastic description of themedium and some local vibrations.

1. Dynamic and Fluctuational Subsystems

The problem of the relaxation of the energy in various subsystemsis of great importance in the discussion of the kinetics of a transition.Strictly speaking, motion with constant energy is possible only inthe isolated system. The relaxation of the energy occurs in anysubsystem which is a part of a large system. It is clear qualitativelythat if the characteristic time of the energy relaxation rE of asubsystem is long as compared to the characteristic time of thetransition re, such a subsystem may be considered as a dynamicone. For systems for which the opposite inequality is valid (rE «re), the exchange of energy between both parts of the system willtake place, and the motion in such a subsystem will be, to someextent, a fluctuational one.

In the condensed phase for the system in which the interactionwith the nonreactive modes is of short-range character, rE increaseswith an increase in the size of the subsystem. However, the largerthe subsystem, the more difficult it is to describe it in the frameworkof the dynamical approach. Thus, the separation of the whole systeminto the dynamic and fluctuational subsystems is not always un-ambiguous and depends on the possibilities of dynamic or stochasticdescriptions of various subsystems. For example, at low tem-peratures, where the harmonic approximation is a good one, thewhole subsystem may be described, in principle, in a dynamic way.At high temperatures, high-frequency (quantum) and some low-frequency intramolecular vibrations with long relaxation times mayalso be described in this way. The behavior of individual degreesof freedom describing the collective state of the solvent is of astochastic (fluctuational) character.

In the stochastic approach, the Markovian random process isusually used for the description of the solvent, and it is assumedthat the velocity relaxation is much faster than the coordinaterelaxation.74 Such a description is applicable at long time intervalswhich considerably exceed the characteristic times of the electron

160 A. M. Kuznetsov

motion. In general, it is inapplicable in the nonadiabaticity regionswhere considerable rearrangement of the electron wave functionoccurs.

The pair correlation function of the velocities and the paircorrelation functions of some time derivatives of the velocity aresometimes taken into account.75 However, the validity of thisdescription in the nonadiabaticity regions also has to be proved.The dynamic description or the description using the differentiablerandom process is more rigorous in this region.76

2. Transition Probability and Master Equations

In general, the equations for the density operator should be solvedto describe the kinetics of the process. However, if the nondiagonalmatrix elements of the density operator (with respect to electronstates) do not play an essential role (or if they may be expressedthrough the diagonal matrix elements), the problem is reduced tothe solution of the master equations for the diagonal matrix ele-ments. Equations of two types may be considered. One of them isthe equation for the reduced density matrix which is obtained afterthe calculation of the trace over the states of the nuclear subsystem.We will consider the other type of equation, which describes thechange with time of the densities of the probability to find thesystem in a given electron state as a function of the coordinates ofheavy particles Pt(R, q, <?, s9...) and Pf(R, q, Q, s,.. .).74-77-80

Let us assume that all the nuclear subsystems may be separatedinto several subsystems (R, q, Q, s,...) characterized by differenttimes of motion, for example, low-frequency vibrations of thepolarization or the density of the medium (q), intramolecular vibra-tions, etc. Let (r) be the fastest classical subsystem, for which theconcept of the transition probability per unit time Wlf(q, Q, s) atfixed values of the coordinates of slower subsystems (q, Q, s) maybe introduced.

Then the master equations for the densities of the probabilitiesto find the system in the state i [P,(<7, Q, s)] or in the state /[Pf(q, Q, s)] at given values of the coordinates q, Q, s have the form

= L,P, - WlfP, + WflPf

dPf/dt = LfPf - WfiPf + WtfP,


where Lt and Lf are the operators describing the evolution of thesubsystems (q, Q, s) in the initial and final electron states.

We may also introduce the transition probability per unit timeat fixed values of the coordinates of slower subsystems, Wlf(q, Q)and WfXq, (?)> and consider the master equations for the corre-sponding probability densities RXq, Q) and Rf(q, Q), etc.

The form of the operators of evolution involved in theseequations depends on the way in which they are described. Thesolution of the master equations enables us, in principle, to findthe average rate of transition for both small and large values of thetransition probabilities Wlf, Wlf and Wfl, Wfl.

3. Frequency Factor in the Transition Probability

The value of the frequency factor (the pre-exponential factor), A,in the expression for the average transition probability

w = Aexp(-EJkT) (139)

is usually of interest when one considers the problem of the roleof relaxation and friction in the kinetics of the transition.

For entirely nonadiabatic transitions, the transition prob-abilities are so small that the reaction does not disturb the equili-brium distribution in the nuclear subsystems (q, Q, s), and thecalculation of the mean transition probability is reduced to averag-ing the corresponding local transition probability over the equili-brium distribution of the coordinates q, Q, s:

Wnad ^(Wyd(q,Q,s))q^s (140)

In this case, A is independent of the frequency characteristics ofthe nuclear subsystems.

As for the adiabatic transitions, various situations are possiblewhen the system involves several subsystems (e.g., r, s, and q).

(i) Adiabatic transition in the subsystem (r). If at a fixed valueof the coordinates q and s, the transition is the adiabatic one, thefrequency factor in the transition probability Wlf(q9 s) has the form3

Kd = ^ff/27T (141)

where (orc^ is determined by the dynamical behavior of subsystem

(r) and depends on the profiles of the corresponding potential

162 A. M. Kuznetsov

energy surfaces. In particular, in the model of harmonic vibrations,Wgff has the form3

E^ (142)k k

where E^ is the energy of the reorganization of the /cth degree offreedom in subsystem (r).

The frequency factor As for the transition probability Wlf(q)in subsystem (s) depends on the values of the probabilities Wlf(q, s)and Wfl(q, s). If the latter are sufficiently small, then we have

Wlf(q) = (Wlf(q,s))s (143)

and in this case, the frequency factor As is still determined mainlyby the frequency characteristics of subsystem (r).

If the probabilities Wlf(q, s) and Wfl(q, s) are large, thefrequency factor As depends on the frequency characteristics ofsubsystem (s)

As ~ a>r£/2ir (144)

The expression for the frequency factor A in the mean transi-tion probability depends on the values of Wlf(q) and Wfl(q). Ifthey are sufficiently small so that the conditions

ww**) < uare fulfilled78 [where q* is the value of the coordinate at which thetransition is most effective, and rq is the relaxation time in thesubsystem (#)], the total transition probability may be obtained byaveraging

(145)

If Wlf(q) is described by Eq. (143), the frequency factor A isstill determined by the frequency characteristics of the fastest sub-system (r). If Eq. (144) holds, A depends on the frequency charac-teristics of the subsystems (r, 5). Finally, if the probabilities Wlf(q)and WfXq) are sufficiently large, the frequency factor A dependson the frequency (or relaxational) characteristics of subsystem (q).

(ii) Nonadiabatic transition in subsystem (r), adiabatic transi-tion in subsystem (s). In this case, the result is independent of thefrequency characteristics of subsystem (r). The frequency factor inthe probability Wlf{q) is determined by the frequency characteristicsof subsystem (5), and the frequency factor in the mean transition


probability depends on the values of the probabilities Wlf(q) andWfi(q) (see above).

(iii) Nonadiabatic transition in the subsystems (r, s). In thiscase, the frequency factor in the total rate of the adiabatic transition,Aad , is determined by the frequency (or relaxational) characteristicsof subsystem (q).

Thus, if the system involves several nuclear subsystems, thefrequency factor is not necessarily determined by the relaxationalcharacteristics of the slowest subsystem. Under some conditions,it may be determined by the frequency characteristics of fastersubsystems (including the dynamic ones).

4. Effect of Relaxation on the Probability of the AdiabaticTransition: A Dynamic Approach in the Classical Limit

A dynamic description of the effect of relaxation on the probabilityof the adiabatic transition may be performed using various methods,e.g., a Feynman path integral approach similar to that presentedin Section III (see also Refs. 81-84). Here we shall present theresults for a simple model obtained by another method.85

Let us consider the one-dimensional motion of a classicalparticle (^-oscillator) in a potential well of the type

{\fuoq2 q<qo/2

U(q) = j \fi(o(q - q0)2 qo> q> qo/2,

q > qotaking account of its interaction with the set of classical oscillators(Q-oscillators) modeling the thermal bath.

This model enables us to investigate the character of the motionof the system in the course of the transition in various limits andto analyze under what conditions the stochastic description isapplicable.

In the second quantization representation, the HamiltonianHL describing the motion of the reactive ^-oscillator in the leftpotential well has the form

(146)

164 A. M. Kuznetsov

where a+ and a are the creation and annihilation operators for thereactive g-oscillator, b^ and bm are the creation and annihilationoperators for the oscillators of the thermal bath (Q-oscillators),and the last term describes the interaction VqQ of the reactiveoscillator with the oscillators of the thermal bath.

We introduce the eigenstates and eigenvalues for the creationand annihilation operators (coherent states86):

a\a) — a\a)', (ct\a+ = a*(a|(147)

bm\(3m) = / U / U ; (Pm\b+= p*m(f3m\

The real and imaginary parts of the eigenvalues a(t) andare related to the expected values of the coordinates and momentaof the corresponding oscillators. In particular,

Rea(f) = q(t)/y/2 (148)

It is known that if the system described by the Hamiltonianin Eq. (146) has been in a coherent state a(0) = a and /3m(0) = pm

at the initial moment of time, then its state will also be a coherentone at subsequent moments of time, and the dependence of theeigenvalues on time will be determined by the coupled equations

da{t)/dt = -icoa(t) - i I \mpm(t)m

(149)dpm(t)/dt = -inm/3m(t) - i\ma(t)

with the initial conditions

a(0) = a; pm(0) = (3m (150)

We will find the probability P(t) for the system to pass thepoint g* = qo/2 up to the moment of time t. This probability givesthe upper estimate for the transition probability since, in principle,there are trajectories for which the system goes back to the leftpotential well after crossing the top of the potential barrier.However, if the contribution of these trajectories is small, as is thecase for not too strong an interaction with the thermal bath at largenarrow barriers, P(t) is close to the exact value of the transitionprobability.

The probability P(t) may be found by integration of thedistribution function $(a, {/3m}) over all possible initial values a


and pm provided the condition

(151)

is fulfilled. Thus, we have for P(t)S5

})Ud2ad2pm (152)m

where the integration is performed over the real and imaginaryparts of the initial eigenvalues a and /3m, which corresponds tointegration over all possible initial values of the coordinates andmomenta of the reactive ^-oscillator and the oscillators of thethermal bath (Q-oscillators).

The transition probability per unit time is determined as

w = dP(t)/dt (153)

To calculate the probability P(t), we must know the form ofthe distribution function <f>(a, {/3W}).

(i) The Transition from the Equilibrium State

First, we shall consider the case when, at the initial momentof time, the distribution of the states in the thermal bath and forthe reactive oscillator is an equilibrium one, i.e.,

(«, i /U) = Z-'o n exp{-[*ftm|/3m|2 + h<o\a\2

m

(154)

where ZqQ is the partition function of the whole system.In this case, to calculate the integral in Eq. (152), we have to

take into account that in addition to the condition given by Eq.(151), there is also a restrictive condition for the initial eigenvalues.

Rea<g*/V2 (155)

which means that at time t = 0, the particle is located in the leftpotential well (L). The exact solution for a(t) in the region (L)may be written formally for an arbitrary type of frequency spectrumof the thermal bath. We will accept the frequently used approxima-tion in which the spectrum of the oscillators of the thermal bath

166 A. M. Kuznetsov

near the frequency CD is dense enough so that the solution for a(t)has the form

a(t) ^ a exp(-id)t) + £ Am0m[exp(-i*>Om

-aJ (156)

where

d) = (o + 8(o - iT (157)

The damping constant T and the frequency shift 8w areexpressed through the coupling constants Am for the interaction ofthe oscillator with the thermal bath and through the frequencycharacteristics of the latter.86 The frequency shift will be neglectedin what follows for the sake of simplicity.

At short times (t < w"1), the condition in Eq. (151) takes theform

Re a + cot Im a + 1I Aw Im /3m > ^*/V2 (158)m

It may be easily seen that the major contribution to the integralover Re a and Im a comes from a small region near the point(Re a = q*/y/2', Im a =0). Therefore, the term VqQ in the exponentin Eq. (154) describing the interaction of the reactive oscillator withthe thermal bath may be considered to be approximately equal, inthis region, to VqQ(Re a, Im a, /3m) - VqQ{q*/^, 0, pm). Then,calculating the integral in Eq. (152) using the distribution functionin Eq. (154), we obtain

= Z*QU \ d2ym(com J

xexp{-[hwq*2/2+hnm\ym\2

+ Vq,Q(q*/V2,0,ym)VkT} (159)

Taking into account that the major contribution to the integralsover Re a and Im a in ZqQ comes from a small region near thepoint Re a = Im a = 0, ZqQ may be written in the form

Z^Q = 7r(kT/fio)) exp[-F,(0)//cr] (160)


where F,(0) = F;(Re a = 0; Im a = 0) is the free energy of thesystem in the initial equilibrium configuration for the reactiveoscillator (q = 0, dq/dt = 0).

With the aid of Eq. (160), the transition probability P{t) maybe finally written in the form

P{t) = (ft>cfrf/27r) Qxp(-FJkT) = wt (161)where

Fa = F(Re a = q*/^/2> Im a = 0) - F;(0)

"eff="2 + LA2m«vnm

1 6 2 )

m

The quantity F(Re a = q*/V2; Im a = 0) is the free energy ofthe system in the transitional configuration, i.e., at the value of thecoordinate of the reactive oscillator q* = qo/2.

Thus, in this limit, P(t) increases linearly with t and the conceptof the transition probability per unit time w may be introduced.The calculation for long times leads to w decreasing with an increaseof r.

(ii) The Transition from the Nonequilibrium State

The above method enables us to calculate the transition proba-bility at various initial nonequilibrium conditions. As an example,we will consider the transition from the state in which the initialvalues of the coordinate and velocity of the reactive oscillator areequal to zero.85 In this case, the normalized distribution functionhas the form

<£(«, {/U) = 5<2)(«) II exp(-|/3m|7<«m»/77</im> (163)m

where 8(2\a) = 5(Re a)8(lm a), and (nm) and (n) (which appearsbelow in eq. (169)) are the average occupation numbers for theoscillators:

(n) = [exp(fto/fcT) - I]"1 - kT/fico(164)

(nm) = [exp(mjkT) - I]"1 - kT/mm.

For P(q91) we obtain from Eq. (152)85

P(q, t) = [ n (d2yj7r) exp(-|7m|2) (165)J m

where ym =

168 A. M. Kuznetsov

For a(t) in the region of the left potential well, (L), in thiscase we have

a(t) = I i{nm))mymvm{t) = I \m({njy/2ym

x [exp(-fof - Tt) - cxp(-iamt)]/(co -nm- iT) (166)

Integration in Eq. (165) is restricted by the condition of thetype in Eq. (151). The rotation of the coordinate system and sub-sequent integration transform the expression for P(q, t) to the form

p(q,t)=\ exp(-s2)ds/^ (167)J\/a(t)

where

f V / 2

= (V2/q)h(nm)\vm(t)\2\ (168)

The last expression may be transformed to the form86

a(t) = (V2/q){{n)[l - exp(-2n)]}1/2

Differentiating Eq. (167) with respect to q and t we obtain

dP/dt = D[d2P/dq2 + (hw/kT)qP] (169)

where

D = TkT/fico (170)

The solution of this equation gives the transition probability

co = (2ry1(Ea/47TkT)1/2 exp(-EJkT) (171)

where

T~1 = 2r; Ea = h<o(q*)2/2 (172)

Thus unlike the previous case where the transition probabilityper unit time exists at some small time and is determined by thefrequency characteristics of the reactive oscillator, here the conceptof the transition probability per unit time exists only at somesufficiently long time. Note two more differences between the for-mulas (161)-(162) and (171)-(172). In the first case the frequencyfactor coeff in the transition probability (i.e., preexponential factor)is determined mainly by the frequency of the reactive oscillator co.In the second case it depends on the inverse relaxation time r"1 = 2Fdetermined by the interaction of the reactive oscillator with thethermal bath.


The activation factor in the first case is determined by the freeenergy of the system in the transitional configuration Fa, whereasin the second case it involves the energy of the reactive oscillatorU(q*) = {\/2)fi(i)q*2 in the transitional configuration. The contrastdue to the fact that in the first case the transition probability isdetermined by the equilibrium probability of finding the system inthe transitional configuration, whereas in the second case the pro-cess is essentially a nonequilibrium one, and a Newtonian motionof the reactive oscillator in the field of external random forces inthe potential U(q) from the point q = 0 to the point q* takes place.The result in Eqs. (171) and (172) corresponds to that obtainedfrom Kramers' theory73 in the case of small friction (F -» 0) butdiffers from the latter in the initial conditions.

5. Stochastic Equations

In recent times, a great deal of interest has been devoted to thedescription of the reactions in condensed media using a stochasticapproach such as that of Kramers.73 As has been noted by Frauenfel-der and Wolynes,87 publications in this area are voluminous andit is impossible even to give a complete list of references. The paperscited in the reference iist

74>75'77'78>81'82>88-101 are only a small part ofwhat exists at present in the literature. Additional references mayalso be found in Ref. 87 and in papers cited therein. Langevinequations for the reactive modes or Focker-Planck equations forthe distribution functions for the coordinates and velocities of thereactive modes are usually the starting points when the classicaldescription of the nuclear motion is used. An approximation isoften used in which it is assumed that the relaxation of the velocitiesoccurs much faster than that of the coordinates. In this case, theequations for the distribution functions are reduced to equationsof the diffusion type. For electron transfer reactions, these equationshave the form77'78

&\dP; 1 d (dUt

= ^ i A w xi_dt Tqldq2 kTdq\dq VJ

170 A. M. Kuznetsov

where Pt and Pf are the densities of the probabilities to find thesystem in the initial and final electron states, respectively, at a givenvalue of the relaxational coordinate q, <olf and cofl are the transitionprobabilities per unit time between these electron states at a givenvalue of q, rq is the relaxation time, A2 = X(0), with K(t) =(q(t)q(O))i = A2 exp(-\t\/ rq) the correlation function of the randomprocess q{t), and Ut(q) and Uf(q) are the initial and final potentialenergies, respectively, as functions of the reactive coordinate q.

Using equations of this type, the expressions for the averagetransition probability at an arbitrary value of the electron resonanceintegral V were obtained.77 For the symmetric transition, Wlf hasthe form

¥I / 1 27rV2exp(- EJkT)if ft (1 + 27rV2Tq/2fiEa)(47rErkTy/2

For the nonadiabatic transition,

2(2irV\/hEa)« 1 (175)

Eq. (174) gives the well-known expression for the transition proba-bility [see Eqs. (9) and (10)]. If the condition opposite to Eq. (175)holds, the transition probability for the adiabatic process takes theform

~1/2exp(-Ea/kT) (176)

Thus, in this case the pre-exponential factor (the frequencyfactor) in the transition probability depends on the relaxation timeTq. Various refinements of this simple model were made in manypapers by considering the change of the intramolecular structureof the reactants,79'80 cases of several relaxation times,88 etc.

An approach of this type has an advantage in that it is basedonly on rather general characteristics of the random process describ-ing the motion along the reaction coordinate. However, it shouldbe noted that equations of the type (173) describing a nondifferenti-able random Markovian process are, strictly speaking, only validoutside the nonadiabaticity regions. In the nonadiabaticity regions,


where an essential rearrangement of the electron wave functionstakes place, we have to use the dynamic description or the differenti-able random process since the electron "sees" the dynamic motionof the nuclear subsystem. Stochastic equations are more appropriatefor the description of processes where the effects of the nonadiaba-ticity are unimportant.

We note, however, that one more disadvantage is inherent tothe stochastic description. The stochastic approach assumes theaveraging of all the physical values over a time interval A* whichexceeds considerably the time of "free" motion r(A? » r) (r is thetime during which the motion along the coordinate q may beconsidered as a dynamic motion in the corresponding potentialfield). This means that At is the smallest physical time unit and allthe results have the corresponding uncertainty. In particular, forhigh, narrow potential barriers, the uncertainty in the activationenergy in the stochastic approach may exceed kT.

The influence of the fluctuational motion along the reactioncoordinates on the probability of the electron transition has beenconsidered recently in the framework of the Landau-Zenermethod.102 A Hamiltonian of the form

+ /(|1><2| + |2><1|) (177)

was used where v is a constant velocity, / is the resonance integralleading to transitions between the states |1) and |2), and f(t) is arandom function of time.

It was, however, assumed102 that/(O is a random Gaussianprocess. The expressions for the probability P to find the systemin the state |2) at t -» oo if at t -» -oo it was in the state |1) wasobtained in two limiting cases:

1. The case of slow fluctuations corresponds to the inequalities

Ttr«Tc (178)

where

rtr = max(J/|t>|, D/M); rc = 1/y (179)

and D and y are the characteristics of the random process

(f(t)f(t')) = ID2 exp(-y|f - t'\) (180)

In this case, P is equal to the Landau-Zener probability PLZ.

172 A. M. Kuznetsov

2. The case of fast fluctuations corresponds to the relationships

r t r »T c ; D/y«l (181)

(At)f(t'))~(4D2/y)8(t-tf) (182)

In this case, the expression for P has the form102

7 (183)

The Landau-Zener formula in the limit J2/\v\ -> oo gives PLZ = 1whereas Eq. (183) for the case of fast fluctuations in this limit givesP = 5. At small values of J2/\v\, both formulas give the sameresult, P = 2TTJ2/\V\.

6. Effect of Dissipation on Tunneling

A number of papers are devoted to the effect of dissipation ontunneling.81"83103104 Wolynes81 was one of the first to consider thisproblem using the Feynman path integral approach to calculatethe correlation function of the reactive flux involved in theexpression for the rate constant,

k = (1/pti) lim drj(£(O)£(t + ))/(5£(0))2 (184)'^°° Jo

where £ is the unit stepwise Heaviside function determining theoccupation of one side of the double well. One of the main resultsof the paper81 consists in the fact that the dissipation leads to adecrease of the tunnel effects.

A rather general method of the calculation of the tunnelingtaking account of the dissipation was given in Ref. 82. The casesof rather strong dissipation were considered in Refs. 81 and 82,where it was assumed that a thermodynamical equilibrium in theinitial potential well exists. The case of extremely weak friction hasbeen considered using the equations for the density matrix in Ref.83. A quantum analogue of the Focker-Planck equation for theadiabatic and nonadiabatic processes in condensed media wasobtained in Refs. 105 and 106.


VII. CONCLUSION

The brief review of the newest results in the theory of elementarychemical processes in the condensed phase given in this chaptershows that great progress has been achieved in this field duringrecent years, concerning the description of both the interaction ofelectrons with the polar medium and with the intramolecular vibra-tions and the interaction of the intramolecular vibrations and otherreactive modes with each other and with the dissipative subsystem(thermal bath). The rapid development of the theory of the adiabaticreactions of the transfer of heavy particles with due account of thefluctuational character of the motion of the medium in theframework of both dynamic and stochastic approaches should bementioned. The stochastic approach is described only briefly in thischapter. The number of papers in this field is so great that theirdetailed review would require a separate article.

It should be noted that recent work has not been devoted onlyto the application of the theory to new processes and phenomenabut has also been concerned with the basis of the theory. Therefore,new important results have been obtained also for processes whichhave been under theoretical investigation for many years, in par-ticular, for electron and proton transfer reactions. These resultsopen new directions for further investigations.

REFERENCES

1 R. R. Dogonadze and A. M. Kuznetsov, Prog. Surf. Sci. 6 (1975) 1.2 P. P. Schmidt, Specialist Periodical Report, Electrochemistry, Vol. 5, The Chemical

Society, London, 1975, p. 21.3 R. R. Dogonadze and A. M. Kuznetsov, Itogi Nauki i Tekhniki, Ser. Kinetika i

Katalis, Vol. 5, VINITI, Moscow, 1978, p. 2.4 J. Ulstrup, Charge Transfer Processes in Condensed Media, Springer-Verlag,

Berlin, 1979.5 R. R. Dogonadze and A. M. Kuznetsov, Itogi Nauki i Tekhniki, Ser. Fizicheskaya

Khimiya, Kinetika, Vol. 2, VINITI, Moscow, 1973, p. 3.6 A. M. Kuznetsov, Nouv. J. Chimie 5 (1981) 427.7 Sh. Efrima and M. Bixon, J. Chem. Phys. 64 (1976) 3639.8 R. A. Marcus, /. Phys. Chem. 24 (1956) 966, 979.9 M. Bixon and J. J. Jortner, Faraday Disc. Chem. Soc. 74 (1982) 17.

A. A. Ovchinnikov and V. A. Benderskii, Fizicheskaya Khimiya. SovremennyeProblemy, Ed. by Ya. M. Kolotyrkin, Khimiya, Moscow, 1980, p. 159.

11 A. M. Kuznetsov, Elektrokhimiya 17 (1981) 84.

174 A. M . Kuznetsov

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14 A M Kuznetsov, J Electroanal Chem 159 (1983) 24115 A K Churg , R M Weiss, A Warshel , and T Takano , / Phys Chem 87 (1983)

168316 A M Kuznetsov and J Uls t rup, Phys Status Solidi B 114 (1982) 67317 A M Kuznetsov, Ph D thesis, The Moscow Engineering Physics Insti tute,

Moscow 196418 M D Newton , Faraday Disc Chem Soc 74 (1982) 9519 V A Zasukha and S V Volkov, Teor Eksp Khim 18 (1982) 39220 S I Pekar, Untersuchungen uber die Elektronentheone der Knstalle, Akademie-

Verlag, Berlin, 195421 A M Kuznetsov and J Uls t rup, Faraday Disc Chem Soc 74 (1982) 3122 A M Kuznetsov, Poverkhnost 9 (1982) 11923 A M Kuznetsov, Chem Phys Lett 91 (1982) 3424 A M Kuznetsov, Khim Fiz, 1 (1982) 149625 A I Burshtem, G K Ivanov, and M A Kozhushner , Khim Fiz, 1 (1982) 19526 A M Kuznetsov, Elektrokhimiya 19 (1983) 159627 W P Jenks, Catalysis in Chemistry and Biochemistry, McGraw-Hill, New York,

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198038 J O ' M B o c k n s a n d S U M K h a n , Quantum Electrochemistry, P l enum Press ,

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Processes, Plenum Press, New York, 198459 A M Kuznetsov and J Uls t rup (in prepara t ion)60 I G Medvedev, Elektrokhimiya 15 (1979) 713, 88661 P P Schmidt , J Chem Soc Faraday Trans 2 80 (1984) 157, 18162 C P Flynn and A M Stoneham, Phys Rev B 1 (1970) 396663 H Teichler , Phys Status Sohdi B 104 (1981) 23964 Yu Kagan and M I Khnger , Sov Phys—J Exp Theor Phys 43 (1976) 13265 G K Ivanov and M A Kozhushne r , Khim Fiz 2 (1983) 129966 W K u h n and M Wagner , Phys Rev B 23 (1981) 68567 A M Kuznetsov, Elektrokhimiya 21 (1985) 83668 G K Ivanov and M A Kozhushne r , Fiz Tverd Tela 20 (1978) 969 A M Kuznetsov , Elektrokhimiya 22 (1986) 29170 R P Feynman and A R Hibbs, Quantum Mechanics and Path Integrals,

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Recent Developments in FaradaicRectification Studies

H. P. AgarwalDepartment of Chemistry, M.A. College of Technology, Bhopal, India

I. INTRODUCTION

The mid-twentieth century witnessed the discovery of faradaicrectification by Doss and Agarwal.1"4 In the late 1950s whenOldham,5'32 Vdovin,6 Barker,7 and Rangarajan33 each indepen-dently worked out the theoretical formulations which corroboratedthe results reported earlier,3'4 the effect came into the limelight.After the pioneering work of Barker7'8 and Delahay and co-workers9"14 in the early 1960s, its potential and applicability in thestudy of fast electrode kinetics were recognized. The method findsmention in books on electrode processes15"17 and in reviews onrelaxation methods.1718 Earlier reviews on the subject deal withpreliminary details and development in instrumentation tech-niques.19 ~24 Soon, it made an impact and led to the developmentof other related nonlinear phenomena such as radio frequencypolarography,7'8'25'26second-harmonic polarography,27'30 inter-modulation polarography,9'26 and high-level faradaic rectification26

for the study of fast electrode reactions. The first comprehensivereview by the author on the subject appeared in ElectroanalyticalChemistry, Vol. 7. A later review article describes the work donein the field up to early 1972.40 Since then, no voluminous work hasbeen reported, yet even the little progress made is significant as ithas opened up new frontiers in electroanalytical chemistry and in

177

178 H. P. Agarwal

electrodics in particular. The electrodics can be linked directly toproblems of technological importance.

Relaxation methods for the study of fast electrode processesare recent developments but their origin, except in the case offaradaic rectification, can be traced to older work. The other relaxa-tion methods are subject to errors related directly or indirectly tothe internal resistance of the cell and the double-layer capacity ofthe test electrode. These errors tend to increase as the reactionbecomes more and more reversible. None of these methods issuitable for the accurate determination of rate constants larger than1.0 cm/s. Such errors are eliminated with faradaic rectification,because this method takes advantage of complete linearity of cellresistance and the slight nonlinearity of double-layer capacity. Thepotentialities of the faradaic rectification method for measurementof rate constants of the order of 10 cm/s are well recognized, andit is hoped that by suitably developing the technique for measure-ment at frequencies above 20 MHz, it should be possible to measurerate constants even of the order of 100 cm/s.

The present chapter will cover detailed studies of kineticparameters of several reversible, quasi-reversible, and irreversiblereactions accompanied by either single-electron charge transfer ormultiple-electrons charge transfer. To evaluate the kinetic para-meters for each step of electron charge transfer in any multistepreaction, the suitably developed and modified theory of faradaicrectification will be discussed. The results reported relate to thereactions at redox couple/metal, metal ion/ metal, and metalion/mercury interfaces in the audio and higher frequency ranges.The zero-point method has also been applied to some multiple-electron charge transfer reactions and, wheresoever possible, theseresults have been incorporated. Other related methods and applica-tions will also be treated.

II. THEORETICAL ASPECTS

Rectification effects are due to the asymmetry of a current-voltagecurve of an electrode system. The asymmetry of these curves mayarise from the intrinsic asymmetry of the charge transfer reactionor from the extraneous asymmetry produced by inequalities in mass

Faradaic Rectification Studies 179

transfer rates of oxidant and reductant. Devanathan31 has rightlycategorized the former effect as faradaic rectification and the latteras a redoxokinetic effect. However, the term faradaic rectificationhas generally been used to cover both effects. By using the basicequation of a redox electrode superimposed by an alternatingvoltage, being a function of cos cot, Devanathan31 has derived theexpression for the faradaic impedance, faradaic distortion, andfaradaic rectification using only elementary mathematics. He hasestablished that faradaic rectification results because of faradaicdistortion and that these two phenomena are interdependent.

1. Single-Electron Charge Transfer Reactions

Devanathan31 obtained the equation for the total rectification cur-rent due to mass transfer ( /MTR) as

. . (cos 0 + sin 0)W R = -h\<*<$o + oiaqR) • z 1/2 (1)

where Io is the exchange current density, given by

and

L L _ nFq° nFe°oD\£2a>1/2' qR nFCWf2D¥* *~ RT

and 6 is the phase difference between the alternating current andapplied alternating voltage, co = 2irf (f is the frequency of thealternating current), n is the number of electrons involved in thecharge transfer reaction, 1^ is the amplitude of the current corre-sponding to the fundamental frequencies, k° is the rate constant,and F, R, and T have their usual meanings.

The total rectification current, JFR, considering charge transferand mass transfer is given by

r / 2 2x Z2 , x ( C O S 0 + S m 0 ) / XiFR = I0(ac - aa) — - I0{acqo + aaqR)z 1/2 (2)

180 H. P. Agarwal

Charge transfer Mass transfer

Which can be written as31

JFR = h~ (OI2C-a2

a)

2RT(cos 0 + sin 0)

where Zw is the faradaic impedance31 (noting the convention thatthe cathodic current is taken as positive when VA is negative),Zw = [(-Rr + cro>~1/2)2 + o-2^"1]172, with Rr the reaction resistanceand a the Warburg coefficient. It can be shown that

2RT(cos 6 + sin 6)

= ^ + cot e\/aaC0

oD^2->Z \ l + cot20/\ C

Hence,

aaC«oD%2 - acC\DT\ 1C0

oD%2+C°RDr / l+ cot d+cot20

The rectification ratio is obtained by multiplying JFR by the reactionresistance Rr:

= _ /a c - aa\ _ 1\ 4 / 2(ac + «fl)nFV2

A \ / ( c fl)

^ ^ l 1 ^ 1 + cot 6

R l+cot20 W

where AE^ is the shift in mean equilibrium potential. At higherfrequencies and for moderately fast reactions

l + cotfl_ 1 11 + cot2 6 cot B p


where

_W2C°oCP fc°(C0Dj/2

Substituting ac + aa = 1, for single-electron charge transfer reac-tions, the above expression reduces to that of Delahay et al.11

A-EQO VIF I . / CQDQ

= I ( 2c t — 1) — [et

By assuming Do = DR (as a first approximation for the sake ofsimplicity), the equation can be written as

jJ^I^M^ _, _ o _ _ r o

21/2k°

In any fast multielectron transfer reaction all the electronscannot be transferred in one step but only by a succession ofsingle-electron transfer steps, whereas Eq. (6) was arrived at byDevanathan31 for the simple case in which it is assumed that thesame step is rate determining in both directions, irrespective of thenumber of electrons involved in the reaction.

If a two-electron charge transfer reaction takes place in twoseparate steps, each being accompanied by transfer of a singleelectron, the mathematical expression for the determination ofkinetic parameters becomes more involved and complicated.

Recently, Rangarajan has worked out a comprehensivetheory35'36 of the electrode/electrolyte interface using the partialderivative formalism introduced by Grahame34 and consideringfour phenomenological components: (1) charge separation, (2)adsorption-desorption (3) charge transfer at the electrode, and (4)mass transfer with or without volume sources/sinks. It may bementioned that the Rangarajan treatment is restricted to the fourphenomenological components, and it applies only to certainaspects of the general nonlinear case,37 i.e., regime of a generalizedRandies-Ershler scheme. Rangarajan's expressions35'36 are highlycomplicated, very involved, and difficult to apply in the determina-tion of kinetic parameters of each step of a multiple-electron chargetransfer reaction.

182 H. P. Agarwal

It was therefore thought appropriate to suitably modify anddevelop the faradaic rectification theory for the study of multiple-electron charge transfer reactions.

2. Two-Electron Charge Transfer Reactions

In any two-electron charge transfer reaction, the two steps can berepresented as follows:

(a) M2+ + e~^=±±M+

CO J^IO

O ^ R,k°

(b) M+ + e~^± M

CO s^O

R[ <- R

where C°o is the concentration of a bivalent oxidant M2+, C°Rl isthe concentration of an intermediate species formed intermittently,CR is the concentration of reductant present in solution, and k°x

and k°2 are the rate constants for the two respective reactions.

Physical Picture

When a bivalent metallic ion or a redox couple differing by abivalent charge is in solution, it is difficult to assign any particularboundary of separation between a bivalent ion and a univalent ionformed intermittently. It would be very hypothetical to assignseparate interfacial potentials for M2+ £± M+ and M+ ^± M. Itis difficult to isolate the stage when the bivalent metal ion isconverted into a univalent ion which is subsequently converted intoa final reduced state. The only thing that is understood is that inthe Helmholtz layer, the bimetallic ion is first converted to a uni-valent ion state before being finally reduced at the electrode surface.It would be an ideal situation if one could take into account, forseparate transfer resistances, transfer coefficients and the amplitudeVA corresponding to each step of electron charge transfer in anymultistep electron charge transfer theory. In such a case, a verycomplicated and highly involved mathematical expression wouldbe obtained which may be impossible to solve. For purposes of


simplification, the expression has been worked out under the follow-ing assumptions:

1. The value of a is taken to be the same in both of the stepsof electron charge transfer and its value is assumed to be close to0.5.

2. Do = DR.3. VA, the amplitude of the interfacial potential, is considered

for the overall reaction.If AEoo, is the rectification potential due to the first step of the

reaction and A£'OO|I is the rectification potential contribution dueto the second step, then the total rectified potential should be thesum of the rectified potentials for each individual step, i.e., A£oo, +AEOOu. The combined faradaic rectification change for both thesteps of electron charge transfer can be represented as38

+ LE^) _ [CQQ(l-a)-aC°Rl]21/2/c?

V\ K } 1/2(C0y(C0yi-

[C%(l-a)-aCR]21/2k

Assuming that a practically remains constant in both the steps ofelectron charge transfer and putting AE^ + AEOOll = AE^,

[C°R,(1 - a ) - aC0R]21/2k0

2

In order to simplify the expression, it can be assumed that (CR|)a «•

( C R , ) " " " 1 = (CR|)1/2, a condition that is met when a can be

approximated as 0.5. Equation (11) then becomes

[4RTAEX ,„

(C°o)a

J ^ (12)

184 H. P. Agarwal

As the expression involves three unknowns, k°u k2, and C°Rl,it is necessary to carry out the experiment at three different con-centrations of redox couples. This can be achieved in two ways:

1. By keeping the concentration of the reductant constant whilevarying that of the oxidant.

2. By keeping the concentration of the oxidant constant whilevarying that of the reductant.

The separate mathematical expressions for the above two caseshave been obtained, and they are given in Appendix A. Whicheverexperimental condition is chosen, the corresponding expressionswill be used for the determination of the values of C%, k°u k2.

3. Three-Electron Charge Transfer Reactions

In any three-electron charge transfer reaction, the three steps canbe represented as follows. Considering a trivalent metal ionreduction,

(a) M3+ + e~^=±M2+ (b) M2+ + e~ ^=±M+

k°(c) M+ + e~^=± M

CO s^Q

R, ^ R

where C°o is the concentration of a trivalent ion M3+, C°Ru and C°R.are the concentrations of the two intermediate species, C°R is theconcentration of the final reduced state, and k°u k°2, and k°3 are thethree rate constants.

Assuming that a practically remains constant for all three stepsof electron charge transfer, the expression for the total rectifiedpotential can suitably be represented,39 using Eq. (9) as

J[C°o(l-a)-C0

Rll]21/2/c?

- a ) - a C % ] 2 ^ k ° 2 [ C R l ( l - a ) - a C

CRu)%CRy-«D¥ + *>1/2(C%nC°R)l-°D\l2

(13)


The observed rectified potential AE^ may be taken as equalto the total sum of the rectified potentials of the individual steps,A£oo,, Afioo,,, and A£oom, in Eq. (13). Then rearranging terms,

[4RTAEX ,_ ..~\a>l'2DU2

2./2

— a) — aC°R ]/c?

If C°R is kept constant, then the expression involves five unknowns,k°u /c0., /c0, C°Ru, and C°Rl. Hence, it is necessary to carry outexperiments at five different redox concentration ratios. This hasbeen done by using five oxidant concentrations, C°o, C°o, C°o, C°o ,and CQV , keeping the concentration of the reductant constant,i.e., C°R.

In Appendix B, the mathematical derivation is given for obtain-ing experimentally the values of C°Ru, C°Rl9 k°u /c0,, and k% in thecase of any three-electron charge transfer reaction.

From the derivations in Appendix B, it is evident that thepresent faradaic rectification formulations for multiple-electroncharge transfer not only enable the determination of kinetic para-meters for each step of three-electron charge transfer processes butmay also be extended to charge transfer processes involving a highernumber of electrons. However, the calculations become highlyinvolved and complicated.

4. Zero-Point Method

(i) Single-Electron Charge Transfer Reactions

Kinetic parameters can also be obtained by using the zero-pointmethod as described earlier.40 The advantage of this method is thatthe values of a and A:0 can be deduced independent of thedetermination of values of the double-layer capacitance, electrodeimpedance, and potential difference across the electrode/solution

186 H. P. Agarwal

interface. The expression used is40

^ 1/2 . 2 ~~ //~<0 r\\/2 , ^0 i - » l / 2 \ /^_ *\ X*-^)

where

Pk\C°oD

x<>2

In order to apply the zero-point method, it is necessary toknow the value of the frequency at which the rectification voltagetends to zero. The experimental determination of the zero-pointfrequency has some practical difficulties, because over a smallfrequency range, the rectification signal is indistinguishable. Hence,the zero-point frequencies have been determined by extrapolatingplots of A£oo/ V2

A versus co~1/2 for those redox concentration ratioswhich intercept the abscissa (i.e., when AE^ = 0).

(if) Two-Electron Charge Transfer Reactions

(a) Case I: CR is kept constant and C°o varies

The kinetic parameters can also be obtained using the valuesof a) corresponding to zero-point frequencies (o>, c»l9 and co2, i.e.,the frequencies at the respective concentrations). On substitutingthese values in Eq. (12) and Eqs. (a) and (b) of Appendix A, theexpressions reduce to39

= [C°o(l-a)-aCRl]j^

+ [CRl(l-a)-aCR] J12 (16)\^ R)

— (2a — l)(o a (C Rl)

o, o &?

+ [ a C ° R , ( l - a ) - a C ° R ] — ^ | r ^ (17)


- (2a - l)<ol2/2b1/2(C°Ry/2D%2/2i/2

r s^Otr (i \ « /-iO i 1

= [Co (I - a) - baCRl\ 0\*~' O )

+ [bC°Rl(\ - a) - aC0R] J12 (18)

\C R)

For simplification, the second term on the right-hand side in eachof the above three equations may be taken as approximately equal.On subtracting Eq. (17) from Eq. (16) and Eq. (18) from Eq. (16),one obtains

- (2a - l)(C°Rl)1/2D}£2(a>1/2 - a1/2o,j /2)/21/2

= /c?{[C°o(l-a)-aC0Rl]/(C°or

— [Co(l — a) — aaC0Rl]/(C0o)a} (19)

— ( 2 a — 1 ) ( C R l ) L>o \O) — o a)2 ) / 2

= k°1{[C0o(l-a)-aC0

Riy(C0or

- [CS'Cl - «) - baCRiy(C%Y} (20)

On dividing Eq. (19) by Eq. (20),

- (C°or[C2Kl - a) - aaC%]}/

(C2)tt{(C2>T[C2>(l - a) - aC°Rl]

or

1/2 1/2 1/2) -o)l a

: o T [ w i ; 2 - » l / 2 * 1 / 2 ] ~ " a '

- (CS,)°C?{(1 - «) + aaC°R,(C°o)7

188 H. P. Aganval

On rearranging the terms in the above equation,

[(Co) Co - (Co) Co] ( C ? , T

x [(C^)aC°o - (C°o)aC%']\ /-al\_(C°i)a - a(C°o)a

Substituting (o~1/29 u)\l/2, (O21/2, and all other terms given

earlier, C°Rl can be obtained. The value of /c? can be obtained onsubstituting the value of C°Rl and all other terms in Eq. (19). Onsubstituting the values of C°Rl9 k°u and all other terms in Eq. (16),the value of k°2 can be obtained.

(6) Case II: C°o is kept constant and C°R varies

Let the values of co~1/2 at the zero-point frequencies be co~1/2,coi1/2, and a>21/2 at three oxidant/reductant ratios. Referring toEqs. (f), (g), and (h) in Appendix A and substituting AJSoo = 0, theexpressions obtained are39

- (la - \W/2(C°Ry/2D^22-il2

= [C°o(l - a) - aC0Rl]kV(C0

o)a

+ [C°Rl(l - a) - aC0R]k0

2/(C°R)(1-"> (22)

- ( 2 a - l)a,1/2al/2(C%)l/2D^22-1/2

= [Co(\-a)-aaC\]kV(Co)a

+ [aC°Rl(l - a) - aCR']kV(C°RT-a) (23)

- (2a - \)<o1/2b1/2(CRl)1/2D0

/22-1/2

= [C°o(\-a)-baC\]kV(Co)a

+ [bC°Rl(l - a) - aC°R"]k°2/(CR'T-a) (24)

The first term on the right-hand side in each of the above threeequations can be approximately taken as equal so as to simplify


the expressions. Now, on subtracting Eq. (23) from Eq. (22) andEq. (24) from Eq. (23), we obtain

(2a - l)(C°R,)1/2Di /22-1/2[a.! /2a1/2 - w1/2]

= k°2{[C°Rl(l - a ) - aC0R]/(C°R)(1-">

- [aC°Rl(l - a ) - aC°R ']/(C°') (1-a)} (25)

(2a - \)(C0Ry/2D%22->/2[<ol/2b>/2 - « j / 2 a 1 / 2 ]

= k°2{[CRla(l - a ) - aC°R']/(C°R')(1-"'

- [bC%(l - a) - aC°n/(CR")(1-a)} (26)

On dividing Eq. (25) by Eq. (26) and rearranging,

co\/2aV2 - (ol/2

wl2

/2bl/2 - w\/2am

= (C0R")(1-a)[(C0

R')(1-a)C0Rl(l - a ) - aC°R(C°R')(1-a)

- a(C0R)(1-a)C°Rl(l - a ) + aC^{C°R)(1-a)V

(C R ) ( 1 - a ) [aC R l ( l - a)(C°R")°"") - aCR'{CR"Y

- bC°Rl(l - a)(C°R ') (1-a) + aC°R"(C°R')(1-a)]

(C°R) (1-a) [a,i / 2a1 /2 - w1/2]

(C£)(1-a)C£,(l - a) - oCj(CS|) ( l"a)

- a(C°R)(1-a)C°Rl(l - a) + aC°R'(C0R)(1-"V

a(C^")(1"a)CRl(l - a) - aCR'(CR")it-a)

- 6(C°R')(1-a)C°R,(l - a) + aC°R"(C°R')(1-a)

CR,(1 - a)[(CR ' )°-a ) - a(C°R)(l-a)]

- a[C°R(C0R')(I-a) - ^ ' ( C i ) 0 " ^ ] /

C°R,(1 - a)[(C0R")°-a) - b(C°R')(1-a)]

- a[C5|(CSr)( |-o) - (Ci')(l"a)CSr] (27)

190 H. P. Agarwal

On rearranging the terms in the above equation,

c%, = a{[(c£)«-W V 2 - ^2)/(c^'Yl-a)

x (a>l/2bi/2 - «!/2aI/2)][Ci'(C5r)(1-o) - (C0R')°-a)C0

R"]

C°R')a"a) - CS?(C£)(1-a)]}/(l - a){[(C°R)(1-a)

V ' 2 - «I /2)/(C!?) ( I-a)(^ /261/2 - «J'V / 2)]

The value of k% can be obtained on substituting the value ofC°Rl and all other terms in Eq. (25). On substituting the values ofCRI5 k%, and all other terms in Eq. (22), the value of k\ can beobtained.

III. INSTRUMENTATION AND RESULTS

1. Faradaic Rectification Studies at Metal Ion/Metal(s) Interfaces

(i) Experimental Techniques

The study of metal ion/metal(s) interfaces has been limitedbecause of the excessive adsorption of the reactants and impuritiesat the electrode surface and due to the inseparability of the faradaicand nonfaradaic impedances. For obtaining reproducible resultswith solid electrodes, the important factors to be considered arethe fabrication, the smoothness of the surface (by polishing), andthe pretreatment of the electrodes, the treatment of the solutionwith activated charcoal, the use of an inert atmosphere, and theconstancy of the equilibrium potential for the duration of theexperiment. It is appropriate to deal with some of these detailsfrom a practical point of view.

(a) Fabrication of the electrodes

All the three electrodes (test electrode, counter electrode, andreference electrode) are made from the same smooth, bright, pol-ished metal foil (A.R.). The metal foil is cut in rectangular shapes


with continuity provided by a thin contact strip. The other end ofthe strip is attached to a platinum wire using epoxy silver paste.The other end of the platinum wire is connected to silver wire andthe platinum wire is sealed into a glass tube. The size of the testelectrode is 20 times smaller than that of the reference electrode.The counter electrode and reference electrode are of the same size.

The details of the pretreatment of the electrodes andpurification of the charcoal and the solution have already beendescribed in an earlier publication43 and review.40

(b) Measurement of AC and rectified signals

The cell and the circuit diagram are shown in Fig. 1. The cellconsists of a test electrode, Ej, reference electrode, R, and counterelectrode, E2. The ac potential between the test electrode Ej andthe reference electrode R is measured by connecting them to asensitive ac millivoltmeter through the contact key (by connectingpoint a to point b). The rectified voltage between Ej and R ismeasured across a dc microvoltmeter (sensitivity, 1 /JLW/smallest

Figure 1. Circuit diagram for Faradaic rectification studies at metal ion/metal(s)interface.

192 H. P. Agarwal

division) after filtering the ac through a low-pass filter F when pointa is connected to point c by the key K. After switching off the acthe initial dc potential, if any is again measured subtracted fromthe former to obtain the actual magnitude of the rectified voltage.All measurements are made below 5 mV, with the applied ac varyingat frequencies between 50 Hz and 15 kHz.

Before starting any experiment, the potential of the test elec-trode Ej is measured with reference to a saturated calomel electrodewhich is connected to the experimental cell through a bridge con-taining the same supporting electrolyte solution. Such measure-ments are taken whenever the concentration of the metal ion ischanged. The cell is kept immersed in a thermostated bathmaintained at a known temperature.

(c) Diffusion coefficient

For obtaining the value of the rate constant, it is desirable todetermine the value of the diffusion coefficient of the metal ionsor of one of the reactants (in the case of a redox couple) in thesupporting electrolyte at an appropriate temperature. The value ofthe diffusion coefficient is experimentally determined using aMcBain-Dowson cell and the King-Cathard equation, as describedearlier.40

(II) Reactions Occurring through a Single-Electron Charge Transfer

Ag+ + e~^Ag is the simplest reaction which can be studiedeasily in a potassium nitrate solution. In any metal ions metalreaction, CR » C°o, and the theoretical expression in Eq. (9) reduces

= nF/4RT(2a - 1) - ^

to43

Thus, it is evident that the linear portion of a AI?^/ V2A versus (o~l/2

plot can be used for determining the value of a from its interceptwith the ordinate, while the slope of the plot will give the value ofJ°, provided the diffusion coefficient of the metal ion is known.The plots for two concentrations of Ag+ (1.0 mM and 2.0 mM in1.0 M KNO3) are given in Fig. 2. It can be seen that the two plots,


-0.0040.02 0.04

-0.006

O

X

•0.008

0 06

-0.010

Figure 2. bEJV\ versus a)~l/2 plots for Ag+, 1.0 N KNO3/Ag electrode: (a)1.0 mM, (b) 2.0 mM. (From Ref. 43, courtesy Pergamon Press.)

on extrapolation in the high-frequency region, meet at a point onthe ordinate. The kinetic parameters obtained are a = 0.22, k°a =0.30 x 10"2 cm/s, and I°a = 7.3 mA/cm2.43

(in) Reactions Involving Two-Electron Charge Transfer

Very few references are available on the determination of therate constant for each step of electron charge transfer in the reactionM2+ + 2e~ -> M(s), i.e., M2+ + <?"-> M+, M+ + e~ -> M(s). Earlierstudies are mostly related to two-electron charge transfer reactionseither at M2+/Hg(dme), M2+/metal amalgam, or redox couple/Ptinterfaces. Even in these studies, the kinetic parameters have beendetermined assuming that one of the two steps of the reaction ismuch slower and is in overall control of the rate of reaction in both

Table 1Kinetic Parameters of Cu(II)/Cu and Cd(II)/Cd in Various Supporting Electrolytes"

Reaction

Cu(II)/Cu

Cd(II)/Cd

Supportingelectrolyte

1.0 N KC11.0 AT K2SO4

1.0 N KNO3

1.0 N NaClO4

1.0 N KC11.0 N K2SO4

I.ON NaClO4

1.0 N KI1.0 JV KNO3

Do x 106 (meas.)(cm2/s)

4.854.004.124.46

7.46.25.8

11.36.4

a(Cathodic meas.)

0.510.500.510.51

0.500.510.500.500.52

Eliminating C°R

A:?x 10(cm/s)

0.320.180.300.27

0.79—

0.80

fc^xlO(cm/s)

7.30.780.541.7

0.204.970.140.321.42

Taking

fc? x 102

(cm/s)

12.00.721.70.28

5.71.4

39.032.0

1.1

c% = i

k°2 x 105

(cm/s)

2.33.77.31.2

3.41.41.703.807.2

k° x 105 b

(cm/s)

4.62.96.02.4

22.08.06.43.0

10.0

References

39,42

38,39

3 Temperature = 27°C.b k° is the rate constant obtained by considering two-electron charge transfer to occur in a single step or the slower step to be in overall

control of the rate process.

Table 2Kinetic Parameters of Ni(II)/Ni and Zn(II)/Zn in Various

SupportingReaction electrolyte

Ni(II)/Ni

Zn(II)/Zn

.0 N KC1I .ON KNO3

LON KIl .0NNaClO 4

I.ON K2SO4

.0 N KC1

.0 N K2SO4

.0 N NaClO4

.0 N KNO3


10.48.6

12.46.6

10.0

8.44.84.67.8

a(Cathodic meas.)

0.490.490.480.490.44

0.580.510.610.51

Eliminating C°R

k°x 10(cm/s)

0.220.11——

0.94

0.13—

0.060.67

k°2x 10(cm/s)

0.300.290.150.111.70

0.170.130.11

13.7

Supporting Electrolytesa

Taking

k°x x 102

(cm/s)

2.967.011.327.65

52.5

3.853.332.503.2

L^R — 1

k°2 x 103

(cm/s)

0.490.201.210.243.07

0.250.050.020.09

k° x 104 b

(cm/s)

0.220.942.50.999.1

0.331.140.220.91

References

39,51

39,42

f35

11

a Temperature = 27°C.b k° is the rate constant obtained by considering two-electron charge transfer to occur in a single step or the slower step to be in overall

control of the rate process.

196 H. P. Agarwal

directions. The kinetic parameters reported earlier for two-electroncharge transfer reactions, even those obtained by the faradaicrectification method, are based on this assumption. In discussingeach individual reaction in detail below, the results obtained byapplying the recently developed theory of faradaic rectification formultiple-electron charge transfer reactions will be compared withthose reported in the literature.

Some of the two-electron charge transfer reactions whichhave recently been studied are Cu(II)/Cu(s), Ni(II)/Ni(s),Cd(II)/Cd(s), and Zn(II)/Zn(s). Their kinetic parameters indifferent supporting electrolytes are given in Tables 1 and 2.

(a) Cu(II)/Cu(s)

The Cu(II)/Cu reaction has been extensively studied44"50 usinguncomplexed Cu(II) ion species. Hampson and Latham50 havestudied this reaction in an aqueous nitrate electrolyte medium usingfaradaic impedance and galvanostatic pulse methods at varyingtemperatures. From their studies, it is concluded that the reductionof Cu(II) -» Cu(I) is a slow process and that the second step,Cu(I) -» Cu, is fast. This reaction has been studied earlier in 1 NKNO3 by the faradaic rectification method43 assuming that k? « /c°and the slowest step is in overall control of the rate of reaction inboth directions. The value of a reported is 0.45 and k°a -0.11 x 102cm/s.

Recently, the kinetic parameters for each step of this reactionin different supporting electrolytes have been obtained39'42 byapplying the faradaic rectification theory as extended to multiple-electron charge transfer reactions. The kinetic parameters are listedin Table 1.

The A£oo/ V\ versus (o~l/2 plots in l.OJV KC1 at three differentconcentrations of Cu(II) are given in Fig. 3. All three plots tendto be linear in the high-frequency region, and on extension, theymeet on the ordinate. From the intercept on the ordinate, the valueof a is obtained from Eq. (12), which reduces to


.01 .02 .03

- 2

- 6

- 8

Figure 3. kEJ V2A versus <o~l/2 plots for Cu2+, 1.0 N KCl/Cu.

Concentration of Cu2+: (a) 2.0 mM; (b) 1.0 mM; (c) 0.5 mM.

Knowing the value of a, the value of C°Ri is obtained from Eq. (e)in Appendix A as all other terms corresponding to the three con-centrations of Cu(II) are known. On further substituting the valuesof a and C°R{ in Eq. (c) Appendix A, the value of fc? can be obtained.On substituting the values of a, C°Ri, and fc? in Eq. (a) of AppendixA, only, two unknowns, i.e., fc° and C°R, are left. Hence, using Eq.(a), at any two concentrations of Cu(II), the C°R can be eliminatedand the value of k°2 *

s obtained independently of C°R. It is veryinteresting to note that the value of fc° thus determined is of theorder of 10"1 and that the second step of the reaction is faster thanthe first, as has been observed earlier. It should, however, be notedthat the value of k\ in KC1 is exceptionally high. It is 10 timeshigher than that in a potassium sulfate medium and is about 4 timeshigher than the value obtained in sodium perchlorate. The valueof k°2 varies in different supporting electrolytes in the order Cl~ >CIO4 > SO4~ > NO^, whereas k°x in all supporting electrolytesremains practically the same and is much lower than /c°, particularlyin a KC1 medium. From the values of the kinetic parameters listedin Table 1, it becomes evident why there is so much variance in

198 H. P. Agarwal

the values of kinetic data reported in the literature: they hold onlyfor the specific experimental conditions and methods followed bythe worker.

If the rate constant is obtained by assuming that only one ofthe steps is in overall control of the rate of reaction, then the rateconstant obtained for the slowest reaction is of the order of10~5 cm/s, which is the same order of magnitude as obtained whenCOR=1.

Considering the activity of the finally reduced species in thesecond step of reduction, i.e., M+ -» M(s), as 1.0, the value of /c?is invariably found to be much higher than that of k°2 in all support-ing electrolytes. For the sake of comparison, such values are alsoincluded in Table 1.

(b)

The A£oo/ V2A versus *T1/2 plots for the Ni(II)/Ni(s) reaction

in 1.0 N KI are shown in Fig. 4. The values of a in different

06

—20 -

Figure 4. AEJV2A versus oT1 / 2 plots for Ni2+, 1.0 N KI/Ni.

Concentration of Ni2+: (a) 2.0 mM; (b) 1.0 mM; (c) 0.5 mM.


supporting electrolytes are obtained from the intercepts of the plotson the ordinate. Kinetic parameters for this reaction have beenobtained by the faradaic rectification method under the followingconditions.

1. By using Delahay's equation, assuming that both electroncharge transfers occur in a single step. In this case, the rate constantobtained for the slowest reaction is of the order of 1CT5 cm/s(Table 2).

2. By applying the recently developed theory of faradaic rec-tification as applied to multiple-electron charge transfer reactionsunder the condition that /c? » k\ and C°R = 1. Kinetic parametersare obtained for each step of the electron charge transfer. The valueof /c° reported is of the order of 10~6 to 10~9 cm/s whereas that ofk°x is of the order of 10~3 cm/s in different supporting electrolytes.51

3. By applying the faradaic rectification theory of multiple-electron charge transfer reactions under the condition that C°R = 1.The rate constant for the first step of electron charge transfer isfound to be 100 to 1000 times higher than that for the second step.39

The kinetic parameters determined in different supporting elec-trolytes42 are given in Table 2.

4. By applying the theory of multiple-electron charge transferreactions and eliminating C°R by carrying out the experiment atvarying concentrations of Ni(II). It is interesting to find that thevalues of the rate constant for the second step of charge transferare invariably higher than those for the first step of the reaction inall supporting electrolytes. The kinetic parameters are given in Table2. Considering the /c° values, it can be seen that they vary in differentsupporting electrolytes in the following order:

so2.- > c r > NO3" > i" > cio;

The reaction is almost 10 times faster in sulfate media thanin other anionic media, which may perhaps be due to preferentialadsorption of sulfate anions at the electrode surface. It may bepointed out that the second step of the reaction is generally expectedto be fast because of the speed with which the intermediate speciesformed is discharged at the electrode surface. This finds supportin other studies carried out at different metal ion/metal(s)interfaces.

200 H. P. Agarwal

(c) Cd(II)/Cd(s)

The discharge of Cd(II) at dropping mercury or cadmiumamalgam (dropping or hanging) electrodes has been extensivelystudied using polarographic as well as relaxation techniques.Because of complications arising due to preferential adsorption ofmetal ions at the solid electrode surface, studies at metalion/metal(s) interfaces have generally been avoided. Earlier studieswere mostly confined to the study of the overall rate of reactionrather than determination of kinetic parameters for each step ofelectron charge transfer. More recently, Cd(II)/Cd(s) studies havebeen made in different electrolytes for the first time,38 and thekinetic parameters have been obtained for each step of electroncharge transfer by the faradaic rectification method. As in the caseof the Ni(II)/Ni(s) system, the kinetic parameters for this systemare also reported under the conditions when C°R = 1, when C°R iseliminated, and when both electrons are taken to be transferred ina single step and C°R = 1.

The nature of Af^/V^ versus (o~l/2 plots is similar for allsupporting electrolytes to those shown in Fig. 5 for a KN03 medium.The kinetic parameters are given in Table 1.

As has already been explained, when C°R = 1, the rate constantk°x is 100 times higher than k°2 in all supporting electrolytes and thefe° values are comparable to those obtained from Delahay'sexpression, assuming that the slowest step is in overall control ofthe rate of reaction.38 On determining the kinetic parametersindependent of C°R (i.e., by elimination of C°R)9 it is interesting tonote that k\ is invariably found to be higher than k°x and is in therange of 10~2 to 10"1 cm/s. The value of k\ in different electrolytesvaries in the order SO4" > NO^ > I" > C\~ > CIO;.

(d) Zn(II)/Zn(s)

Gaiser and Heusler53 have shown that the electrode reactionZn2+ + 2e~ -> Zn proceeds in two steps: Zn2+ 4- e~ -» Zn+ andZn+ 4- e~ -> Zn(s). Van Der Pol et al.,54 using ac coupled with thefaradaic rectification polarography method, also concluded thatthis reaction is a multistep reaction. Hurlen and Fischer55 havestudied this reaction in an acid solution of potassium chloride and

Figure 5. ±EJV\ versus a>~l/2 plots for Cd2+, 1.0 N KNO3/Cd. Concentrationof Cd2+: (a) 2.0 mM; (b) 1.0 mM; (c) 0.5 mM.

concluded that of the two consecutive charge transfer steps, theion transfer step Zn(I)/Zn(s) is a fast reaction. In all earlier studies,the exchange current density reported is for the overall reaction.

The kinetic parameters for each of the two steps of this reactionhave been obtained in different supporting electrolytes by thefaradaic rectification method and are given in Table 2. The kE^/ V2

A

versus co~l/2 plots in 1.0 N NaC104 are given in Fig. 6. Similarcurves are obtained in other electrolytes as well. On extending thethree plots, they meet at a point on the ordinate. From the interceptat the ordinate, the value of the transfer coefficient determined indifferent supporting electrolytes varies in the range 0.51 to 0.61.Also, in this system, k°x is about 1000 times higher than k°2 inpractically all the media if CR is taken to be 1.0. The k\ values arecomparable to those obtained from Delahay's expression consider-ing that both electrons are transferred in a single step and with thevalue of C°R taken to be 1.

202 H. P. Agarwal

.01 .02

- 4

- 6 L

Figure 6. kEJ V2A versus a)'l/2 plots for Zn2+, 1.0 N NaClO4/Zn. Concentration

of Zn2+: (a) 2.0 mM; (b) 1.0 mM; (c) 0.5 mM.

When the kinetic parameters are obtained by eliminating C°R9

as has been done in the previous cases, k°2 is always higher thank°l9 in accordance with earlier observations.55 It is interesting tonote that the rate constant in a potassium nitrate medium is veryfast (100 times higher) as compared to that in other media. Thek°2 values are, in general, of the order of 10~2cm/s except inpotassium nitrate, for which the rate constant is 1.37 cm/s. Ingeneral, k°x is also of the order of 10~2 cm/s but lesser in magnitudethan k\.

(iv) Reactions Involving Three-Electron Charge Transfer

Very few references are available relating to the study of theAl(III)/Al(s) reaction. Most of the earlier studies have been madein molten cryolite.56'57 Armalis and Levinskas58 have reported theoverall exchange current density of the reaction under a steadystate of deposition and showed that it is a moderately fast reaction.


This reaction is found to be stable in sodium acetate and aceticacid buffer (pH 4.65), and so it has only been studied in this medium.The faradaic rectification theory becomes highly complicated whenextended to three-electron charge transfer reactions due to theformation of the two intermediate species Al(II) and A1(I). In orderto determine the three rate constants and the two unknown con-centration terms, C°Rl and CRu, corresponding to the two intermedi-ate species formed, it becomes necessary to carry out the experimentat five different concentrations of aluminum ion, each below2.00 mM.

The A£oo/ V\ versus w~1/2 plots at the five different concentra-tions of Al(III) meet on the ordinate at a point (Fig. 7), and fromtheir intercept on the ordinate, the value of a obtained is 0.52.

.01 02 03 .04

- 4

- 7

Figure 7. Af^,/ V2A versus a) 1/2 plots for Al3+ in sodium acetate-acetic acid buffer

(pH 4.65)/Al. Concentration of Al3+: (a) 2.0 mM; (b) 1.5 mM; (c) 1.0 mM; (d)0.75 mM; (e) 0.5 mM.

204 H. P. Agarwal

Having determined the value of a, the value of C°Ru is obtainedfrom Eqs. (o) and (p) of Appendix B. On substituting the value ofa and C°Ru in Eq. (m) of Appendix B, the value of C°R] can beobtained. Knowing these parameters, the value of /c° can be deter-mined from Eq. (j) of Appendix B, taking C°R = 1. To determinethe value of k®, Eq. (b) of Appendix B is used. On substituting thevalues of k°u /c°, C°Rl, C°Ru, and all other terms in Eq. (a) of AppendixB, the value of k° is determined.

The three rate constants thus obtained are /c° = 3.68 x 10"5,k°2 = 2.70 x 10~4, and fc° = 5.36 x 1(T4 cm/s, respectively. All threesteps of the reaction are irreversible. However, on comparing thethree rate constants, they vary in the order given below:

k°3 > k°2 > fc?

2. Faradaic Rectification Studies at Redox Couple/InertMetal(s) Interfaces

Faradaic rectification studies in earlier stages were mostly confinedto the commonly known redox couples and platinum electrodeinterfaces. Results relating to Fe2+, Fe3+ in 1.0 N H2SO4 andFe(CN)r , Fe(CN)|~ in 1.0 N KNO3 have already been includedin an earlier review of the subject.40 Some studies were also carriedout with redox couples involving two-electron charge transfer eitherat a platinum interface or at a dropping mercury electrode, assumingthat both the electrons are transferred in a single step and that theslow reaction is in overall control of the rate process. The kineticparameters obtained have already been included in the earlierreview. Those systems for which results were published after 1972will be discussed in this section. The results for single-electron andtwo-electron charge transfer reactions will be given separately.

(i) Redox Couples Accompanied by Single-Electron Charge Transfer

A few reversible redox couples are known for their stabilityand reproducibility such as Fe(CN)^", Fe(CN)4"; Cr3+, Cr2+; Ti4+,Ti3+; Ce4+; Ce3+; and Cu2+, Cu+. For all of these reactions, studieshave been carried out at varying redox concentration ratios, and


for some of these reactions, the zero-point method has also beenapplied. Each of the individual reactions will be discussed in detail.

(i) Fe(CN)l, Fe(CN)46-/Pt(s)

This reaction is fairly stable in a 1.0 N KN03 medium. Earlierstudies have been made using equimolar concentrations of theoxidant and reductant.59 This reaction has been studied again byvarying the redox concentration ratios.60 The plot of AJBoo/ V2

A versusa)~1/2 is shown in Fig. 8. Although the value of a remains unaltered,the rate constant is found to be of the order of 10"1 cm/s and is

Figure 8. bEJ V2A versus (o~l/2 plots for Fe(CN)^, Fe(CN)J" in 1.0 N KNO3/Pt:

(a) 4:1; (b) 2:1; (c) 1:1; (d) 1:2; (e) 1:4. (From Ref. 60, courtesy Elsevier.)

Table 3Kinetic Parameters of Some Single-Electron Charge Transfer Redox Couples at a Platnium Interface0

Redox couple

Ce4+,Ce3+/Pt

C u 2 \ C u 7 P t

Fe(CN)3',Fe(CN)47Pt

Cr3+,Cr27Pt

Ti4+ ,Ti3+/Pt


1.0 N HNO3

1 N H2SO4

1 N HC1

4.5 N HC1 +1.0 ATsodium citrate

1.0 N KNO3

1.0 N H2SO4

1.0 N HC1

Concentrationof redox couple

(oxid: red)

0.5:2.01.0:2.01.0:1.02.0:1.02.0:0.50.5:2.01.0:2.01.0:1.02.0:1.00.5:2.01.0:2.01.0:1.0

0.5:2.01.0:2.0

2.0:0.52.0:1.01.0:2.00.5:2.01.0:1.0

1.0:5.00.5:5.0

0.5:5.0

Do x 106

(cm2/s)

6.8

1.2

4.6

7.16

12.2

5.16

7.14

a(meas.)

0.51

0.60

0.70

0.495

0.49

0.49

0.47

0.49

k°(cm/s)

0.040.060.050.060.030.040.050.310.160.230.330.76

0.030.06

10.5 x 10"2

12.6 x 10~2

14.2 x 10"2

11.5 x 10~2

6.6 x 10~2

1.98 x 10"3

2.00 x 10"3

5.56 x 10"4

Zero-pointmethod

a k°a

0.51 0.01

0.53 0.04

0.51 0.04

0.51 1.12 x 10"2

References

39

39

39

39

60,67

59

60,67

1 Temperature = 27°C.


almost twice the magnitude of the value reported earlier. The kineticparameters are given in Table 3.

(») Cr'\ Cr2+/Pt(s)

The chromous-chromic reaction is highly unstable because ofthe high instability of the chromous salt. This redox couple is onlystable when the concentration of the chromous salt is 5 to 10 timeshigher than that of the chromic salt. This reaction has been studied60

in 1.0 JV H2SO4, and the plots of AJBoo/ V2A versus w~l/2 are given

in Fig. 9. The value of a is 0.47 and k°a = 2 x 10"3 cm/s. The results(Table 3) are comparable to those obtained by direct currentpolarography.

.005 .010 .015- . 4

UJ

- . 8 '

-1 .0 -

Figure 9. kEJV2A versus (o'l/2 plots for Cr3+, Cr2+, 1.0 M H2SO4/Pt.

Cr3+(mM)/Cr2+(mM): (a) 1.0:5.0; (b) 0.5:5.0. (From Ref. 60, courtesy Elsevier.)

208 H. P. Agarwal

U)'l/2

.01 .02

- . 1 5 -

.25-

- . 3 5

Figure 10. &EJV2A versus a>~l/2 plots for Ti4+, Ti3+, 1.0 N HCl/Pt. Ti4+

(mM)/Ti3+(mM): (a) 0.5:5.0; (b) 0.25:5.0. (From Ref. 60, courtesy Elsevier.)

(C) 774+, Ti3+/Pt(s)

This redox couple has been studied in H2SO4 and tartaric acidat the dropping mercury interface by Delahay et alu They onlyreported the value of a for the reaction. This system is only stablewhen the concentration of Ti3+ is 10 to 20 times higher than thatof Ti4+. The ££«,/ V\ versus <o~1/2 plots for this reaction in 1.0 NHC1 are shown in Fig. 10 and the kinetic parameters60 are given inTable 3. The value of a is 0.49 and k°a = 5.56 x 10~4cm/s. Thereaction appears to be irreversible.

(d) Ce4+, Ce3+/Pt(s)

Galus and Adams61 reported that the rate constant for thisredox couple is of the order of 10~4 cm/s. Some studies have beencarried out for the cerous, eerie redox couple in 1.0 M H2SO4 usinga rotating tungsten electrode62 and also in an HNO3 medium using


a platinum electrode.63 A value of a slightly less than 0.5 wasreported in the latter study. This redox couple has been studied atvarying redox concentration ratios in mineral acids, and the kineticparameters have been obtained by the faradaic rectificationmethod.39'42 Kinetic parameters have also been determined by thezero-point method.120

The A£oo/ V\ versus w~l/2 plots in 1.0 N H2SO4 are shown inFig. 11 and the kinetic parameters are given in Table 3. The valueof a is 0.70 in HCl, 0.60 in H2SO4, and 0.51 in HNO3. The rateconstants in the three acids vary in the order given below:

HCl > H2SO4 > HN03

However, the rate constant determined by the zero-pointmethod is 0.04 cm/s in HCl and H2SO4 media and in HNO3 it is

Figure 11. &EJ V2A versus <o~l/2 plots for Ce4 \ Ce3+, 1.0 N H2SO4/Pt. [Ce4

(mM)/Ce3+(mM)]; (a) 0.5:2.0; (b) 1.0:2.0; (c) 1.0:1.0; (d) 2.0:1.0; (e) 2.0:0.5.

210 H. P. Agarwal

only 0.01 cm/s. This shows that the reaction is much slower inHNO3 than in the other two acids.

(<?) Cu2+9 Cu+/Pt(s)

Very few references are available regarding the study of thisredox couple. Gorbachev and co-workers64'65 reported from polariz-ation curves that in citrate and chloride complex electrolytes, theoxidant and reductant reduce to the metal state.

The AEJV\ versus co~1/2 plots for Cu2+, Cu+ in HC1 andsodium citrate media are shown in Fig. 12 and the kinetic

1.0-

.01 .02 .03 .04 .05 .06

-.2

Figure 12. £±EJ V2A versus a)~l/2 plots for Cu2+,

(mM)/Cu+(mM): (a) 0.5:2.0; (b) 1.0:2.0.Cu+ (buffer)/Pt. Cu2


parameters39'42120 are given shown in Table 3. In 4.5 N HC1, thevalue of a is 0.495 and the rate constant is 0.03 cm/s. On comparingthe latter value with that obtained from the zero-point method, itis found to be 3 times higher.

(II) Redox Reactions Involving Two-Electron Charge Transfer

The earlier studies by the faradaic rectification method of redoxreactions involving two-electron charge transfer were done assum-ing that both electron charge transfers occur in a single step. Thiswas so because the earlier theoretical formulations were only appli-cable to the study of single-electron charge transfer reactions. Someof the redox couples which have been studied66'67 are I2, I~/Pt(s);QH2, Q/Pt(s); Sn4+, Sn2+/Pt(s); and Tl3+, Tl+/Pt(s). Each of thesereactions is discussed separately below.

(a) I2, r/Pt(s)

The mechanism for this reaction has been given as66

l~ - e~ ± Iads; Iads + r^I2 ads + e~

The absorption in the form of an adatom has already been reportedby Tyagai and Kolbasov.68 The second-step charge transfer reactionappears to be rate determining in both directions and is in overallcontrol of the reaction. The AE^/ VA versus co~1/2 plots for thisreaction in 1.0JV KNO3 are presented in Fig. 13. On extrapolatingthe curves in the high-frequency region, they meet at a point onthe ordinate. The value of a is determined from the intercept ofthe ordinate and that of k°a from the slope of the curve. The valueof a is 0.49 and k°a = 0.55 x 10~2 cm/s (Table 4). These parametersare comparable to those obtained in KC1.68

(b) 5n4+, Sn2+/Pt{s)

Very few references are available for the kinetic parameters ofthe Sn(IV)/Sn(II) reaction except for the Sn(II)/Sn(Hg) reactionin chloride and perchlorate media.69'70 The &EOD/V2

A versus <o~1/2

plots for this reaction in 1.0 N HC1 are given in Fig. 14. The transfercoefficient is obtained from the intercept of the extrapolated curves

212 H. P. Agarwal

.095 .010 .015

-2O

Figure 13. &EJV2A versus co~l/2 plots for I2, I", 1.0 N KN03/Pt. I2 (mM)/I"

(mM): (a) 0.5:5.0; (b) 1.0:5. (From Ref. 66, courtesy Pergamon Press.)

at a point on the ordinate. Agarwal and Qureshi,66 in their earlierstudies, have reported values of a = 0.48 and k°a = 2.1 x 10~3 cm/s,assuming that the slower step is in overall control of the rate ofthe reaction. Recently, the kinetic parameters have been determinedfor each step of electron charge transfer involved in the reactionand the values of the two rate constants obtained are fc? =5.47 x 10"2 cm/s and k°2 = 1.15 x KT1 cm/s (Table 4).

(c) 773+, 7

Vetter and Thiemke71 studied Tl(III), Tl(I)/Pt(s) and tried toexplain the mechanism of the reaction from a Tafel plot. The kineticparameters of this reaction have been determined by Toshima etal.72 using a potential step method, by Fasco et al73 using an elec-trode vibration technique, and by Agarwal and Qureshi67'74 using the

Table 4Kinetic Parameters of Some Two-Electron Charge Transfer Reactions at a Platinum Interface

Reaction

Iodine,Iodide/ Pt

Tl 3 + ,Tl7Pt

Sn4 + ,Sn2 + /Pt


1.0 M KI

1.0 M KNO3

1.0 N KNO31.0 N K2SO4

1.0 N NaClO4

1.0 N HC1

Concentrationof redox

couple (mM)(oxid:red)

1.0:10002.0:10000.5:5.01.0:5.0

Do x 105

(cm2/s)(meas.)

1.841.761.21

0.68

Diffusioncoefficient,

Do x 106 (meas.)

r(°c)27

25

a(Cathodic

meas.)

0.520.590.51

0.48

(cm2/s)

6.81

cR >

fc?xl0(cm/s)(meas.)

4.521.261.58

0.55

ac

(Cathodicmeas.)

0.50.50.490.49

k°2x\0(cm/s)(meas.)

5.998.551.76

1.15

(Anodicmeas.)

0.50.50.510.51

Zeromethod;

k°x x 103

(cm/s)(meas.)

1138.146.88

—

(cm/s)(meas.) References

5.10 x lO" 5 66,674.94 x 10~5

5.59 x 10"3

4.44 x 10"3

-pointC R > C O

k\ x 102

(cm/s)(meas.)

68.2 39,522.381.92

—

7

I

I

214 H. P. Agarwal

- 2

- 4

- 6

f

0 0 2 ,010

f-2-12

- 1 4 -

Figure 14. /±EJV2A versus o>~1/2 plots for Sn4+, Sn2+, 1.0 N HCl/Pt.

Sn4+(mM)/Sn2+ (mM): (a) 2.0:5.0; (b) 1.0:5.0; (c) 0.5:5.0. (From Ref. 66, courtesyPergamon Press.)

faradaic rectification method. In all the earlier studies, the kineticparameters of the overall reaction were determined assuming thatthe same step is rate determining in both directions. Values ofa = 0.48 and k°a = 0.12 cm/s in 1.0 AT HC1O4 have been reported.

The AEao/ V\ versus a>~1/2 plots in 1.0 AT NaCiO4 are shownin Fig. 15 and the kinetic parameters obtained from extrapolationof these plots and using the zero-point method are given in Table4. It may be pointed out that when C°R is kept constant and C°o isvaried, the value of C°Rl is obtained from Eq. (e) of Appendix A,that of fc? from Eq. (c) of Appendix A, and that of fc° from Eq.(12). For determining the value of the two rate constants by thezero-point method, the theoretical formulations for multiple-elec-tron charge transfer have suitably been modified, and correspondingexpressions for k°2, fc?, and C°R} have been deduced from Eqs. (16),(19), and (21).


002 004~I T "

008 ,010

"T".012

- 2

- 1 0

Figure 15. AEJ V2A versus io'l/2 plots forTl(III), T1(I) in 1.0 N NaClO4/Pt.

Tl3+(mM)/Tl+(mM): (a) 1.0:2.0; (b) 0.75:2.0; (c) 0.5:2.0.

The two rate constants obtained in different supporting elec-trolytes52 are of the order of 10"1 cm/s, indicating that the reactionis fast. It can also be seen that k°2 is invariably higher than fc? inall media but varies less. Thus, it is evident that in such reactionsone cannot assume either of the two reactions as slow and in overallcontrol of the charge transfer reaction. On comparing the valuesof k°2 in different supporting electrolytes, they are found to vary inthe following order:

SOr > NO3" > CIO4

It is interesting to note that the influence of these supportingelectrolytes on k\ is in a different order, i.e.,

NO^ > CIO4 > SO4"

The rate constants k°x and k® determined by the zero-pointmethod are generally 10"1 to 10~2 times lower than those obtainedby extrapolation techniques. The influence of different electrolytes

216 H. P. Aganval

on both rate constants is as given below:

> SOr > CIO4

It is interesting to observe that in nitrate media both rate constantsare 100 times higher than in the other two media. The high valuesof rate constants in NO^ and SO4" media are due to their preferen-tial adsorption at the electrode/solution interface.7'75"77

(d) QH2, Q/Pt(s)

The most widely studied and stable organic redox couple isquinone-hydroquinone, and this reaction can be studied at varyingpH's up to 7.5. The earlier faradaic rectification studies involvedthe determination of the transfer coefficient at varying pH's usingequimolar concentrations of quinone and hydroquinone. Agarwaland Qureshi67 determined the kinetic parameters of this redoxcouple using varying redox concentrations but assuming that thefirst step of electron charge transfer is in overall control of thereaction. More recently, the kinetic parameters for each step ofelectron charge transfer have been determined for the first time atvarying pH's by the usual extrapolation method as well as by thezero-point method. The ^E^/V2^ versus co~1/2 plots at pH 7.4 arepresented in Fig. 16 and the kinetic parameters are given119 inTable 5.

The studies have been carried out with C°o kept constant andlarger than C°R and vice versa, maintaining a total redox concentra-tion not exceeding 4.0 mM. It is generally observed that when theoxidant/reductant concentration ratio is increased, at any frequencythe rectification potential becomes more positive. Conversely, onincreasing the redox concentration ratio, the rectification potentialtends to become more negative at any frequency. On extrapolatingthe plots at high frequencies (Fig. 16) at all redox concentrationratios, they meet on the ordinate at a point and the value of adetermined at all pH's is 0.475 to 0.480. When C°o > C°R, theAEao/ V2

A versus co~l/2 plots on extrapolation intersect the abscissa,and it is possible to determine the two rate constants by the zero-point method in such cases.

The values of C°Rl are obtained from Eq. (j) of Appendix Aand those of /c° and k°x are obtained from Eqs. (f) and (g) and Eq.


5 -

t '

- 2

- 3

- 5 -

015

Figure 16. ±EJV\ versus w"1/2 plots for Q, QH2 (buffer of pH 7.4)/Pt. Q(mM)/QH2 (mM) = (a) 2:0.5; (b) 2:1.0; (c) 2:1.5; (d) 1.5:2; (e) 1:2; (f) 0.5:2.

(f) of Appendix A, respectively. For obtaining the value of C°Rl bythe zero-point method, the values of zero-point frequencies (whenA£oo = 0) are substituted along with other terms in Eq. (28). Know-ing C°Rl, the value of k°2 is determined from Eq. (25) . Finally, thevalue of k°x is obtained from Eq. (22).

It is interesting to note that at all pH's, when C°o > C°R, thevalue of k°x is invariably greater than that of fc°> t>ut o n the otherhand, when C°R > C°o, k°2 is always found to be higher than fe?.These results119 explain clearly for the first time why the results

Table 5Kinetic Parameters of Quinone, Hydroquinone Redox Couple at a Platinum Interface0

pH

4.67.07.4


0.166.96.0

a(Cathodic meas.)

0.4870.4750.475

C°o>

fc? x 102

(cm/s)

0.1428.6

3.19

s-yQ^ R

k°2 x 102

(cm/s)

0.093.051.85

C° ^

fc?xlO(cm/s)

3.81.262.67

> c°o

k°2xl0(cm/s)

6.592.330.56

Zero-pointmethod; C°o > C°R

(cm/s)

0.475.7

k°2 x 103

(cm/s)

7.42.1

References

39,119

1 Temperature = 27°C.


reported earlier using different redox concentration ratios do nottally. The kinetic parameters obtained by the zero-point methodare generally lower in magnitude than those obtained by theextrapolation method. k° is invariably much less than k°x when

IV. FARADAIC RECTIFICATION POLAROGRAPHY ANDITS APPLICATIONS

The merits of the method in which a modulated high-frequencysignal is superimposed on a varying dc polarizing potential wereinvestigated by Barker,7 Delahay and co-workers,11'13 andAgarwal.40'41 Barker initially reported the technique as low levelfaradaic rectification (LLFR) or radio frequency polarography.Later, Van Der Pol, Sluyters-Rehbach, and Sluyters78'79 extendedthe method and carried out a more systematic study, renaming thetechnique faradaic rectification polarography.78 Agarwal andSaxena80'82 superimposed an unmodulated audio frequency alter-nating current on a polarizing dc potential and recorded the rectifiedcurrent at varying frequencies. On plotting the rectified currentversus dc potential, the polarograms (FR polarograms) obtainedare somewhat similar to ac polarograms. Using the faradaic rec-tification polarographic technique, the kinetics of reduction ofseveral metal ions in different supporting electrolytes and of someorganic depolarizers at varying pH's have been studied so as toexamine the potential of the method.

1. Studies Using Inorganic Ions as Depolarizers

Faradaic rectification polarographic studies have been carried outfor a mixture containing several metal ions together and also forindividual inorganic depolarizers so as to explore the applicabilityand limitations of the method and to determine kinetic parametersfor some of them. For comparison, some of the dc and acpolarograms have also been recorded simultaneously. In the follow-ing, the details of the experimental technique used will be describedand the potentiality of the technique in qualitative and quantitativeanalysis will be examined. The applicability of the method in the

220 H. P. Agarwal

determination of kinetic parameters of individual depolarizers willalso be described.

(i) Circuit and Measurement Technique

The circuit diagram for ac polarography and faradaic rec-tification polarography81 is shown in Fig. 17. The output from anaudio oscillator is made incident through a two-way key on aprecision potentiometer. The negative end of the potentiometer isconnected to the dropping mercury electrode (d.m.e.). The pool isgrounded through a standard precision resistance of 100 H. The acand the rectified current are measured across the precision resist-ance. For measuring ac the key K2 is disconnected and key Kx ispressed. The ac signal is amplified by the transistorized mixerpreamplifier. The amplified ac is measured on a Solartron double-beam oscillograph. For some measurements, an ac multivoltmeterhas also been used.

The rectified dc is measured by pressing key K2 and leavingK open, thus enabling filtration of ac through a low-pass filter.The output of the filter is connected to the dc microvoltmeter soas to measure the rectified voltage. The residual dc potential, ifany, is also measured by switching off the ac using the two-way

Figure 17. Circuit diagram for FR polarography.


key K. This is subtracted from the measured rectified voltage so asto obtain the actual magnitude of the rectified voltage. Throughoutthe measurements, the interfacial potential is kept below 15 mV.

The dropping mercury electrode has the following characteris-tics: m = 0.64 mg/s; t = 8.0 s in distilled water in an open circuit.

All solutions are made in bidistilled water by dissolving therequired quantities of analytical-grade reagents. Pure nitrogen gasis bubbled through the solutions to get rid of dissolved oxygen.

The diffusion coefficient of the depolarizer in any supportingelectrolyte is determined using a McBain-Dowson cell and theKing-Cathard equation.

Faradaic rectification polarographic studies have been carriedout in a mixture82 containing seven metal ions—Pb(II), T1(I),In(III), Cd(II), Ni(II), Zn(II), and Co(II), the concentration ofeach being as low as 1.0 mM in 1.0 M KC1 and the half-wavepotential of some of them differing by less than 20 mV—to examinethe applicability of the method. Simultaneously, the dc steps andac polarograms have also been obtained so as to compare thelimitations of the three methods. The FR polarograms at varyingac frequencies have been recorded. For the purpose of determiningthe kinetic parameters of each individual depolarizer in differentsupporting electrolytes, the FR polarogram of each of thedepolarizers has been obtained. The results are given in Figs. 18-20.

The dc polarogram of the mixture containing the above-mentioned metal ions shows distinct steps corresponding to T1(I),Cd(II), Zn(II), and Co(II) ions only (Fig. 18) but does not provideinformation about the presence of the other three ions, viz., lead,indium, and nickel, in the mixture. Further, the limiting currentcorresponding to each step seems to be due to a combined con-centration effect of lead and thallium ions, cadmium and indiumions, and zinc and nickel ions.

The ac polarograms of the mixture at varying frequencies (Fig.19) shows four ac summit peaks corresponding to reduction ofT1(I), In(III), Cd(II), and Zn(II). The summit peaks for In(III)and Cd(II) are very close and so their ac waves are not very sharp.The first summit peak corresponding to T1(I) appears to be due tothe combined reduction of lead and thallium ions, as is evidentfrom the summit peak height. Hence, ac polarographic analysisonly enables the identification of four metal ions out of seven and

0'2 0*4 0*6 0'8 \'O \'2 1*4 1*6

D. C . P O T E N T I A L I N V 0 LT S

Figure 18. DC polarogram of a mixture containing 1.0 mM concentrations of Pb2+,Tl+, In3+, Cd2+, Ni2+, Zn2+, and Co2+ in 1.0 N KC1.

the determination of their concentrations may also not be veryaccurate because of the overlapping of some waves. The ac polaro-graphic technique only permits resolution of ac waves when thehalf-wave potentials of the constituents differ by 40 mV. Further,the method is strictly applicable only to moderately fast reversiblereactions whose rate constants fall between 10~2 cm/s and1CT1 cm/s. The ac summit peak height does not always vary linearlywith the concentration of the depolarizer.


.-t

5 0 0 ^

0'2 0'4 0"6 0*8 I'O ! '2 !'4

D. C . P O T E N T I AL IN V O L T S

Figure 19. AC polarograms of a mixture containing 1 mM concentrations of Pb2+,Tl+, In3+, Cd2+, Ni2+, Zn2+, and Co2+ in 1.0 N KC1.

The FR polarograms obtained for the mixture containing theseven components (Fig. 20) has the following distinct features incomparison to dc and ac polarograms:

(i) When FR polarographic studies are carried out with asupporting electrolyte alone (in the absence of depolarizer), therectified current is found to be zero at all potentials (Fig. 20) unlikethe ac polarogram for the supporting electrolyte alone, in whichsome appreciable initial amount of ac is invariably present.

(ii) Distinct FR summit peaks corresponding to each metalion are obtained (coincident with the half-wave potential) even

Figure 20. FR polarograms of a mixture containing 1 mM concentrations of Pb24

TT, In3+, Cd2+, Ni2+, Zn2+, and Co2+ in 1.0 N KC1.

when the half-wave potentials of the components present differ by20 mV (as can be seen in case of reduction of nickel and zinc ions).This is a great advantage in the qualitative analysis of mixturescontaining several constituents.

(iii) For those reactions which are very fast, such as the reduc-tion of lead and cadmium ions, the FR summit peaks even appearup to very high frequencies (50 kHz) whereas for very slow reac-


tions, such as the reduction of Co(II), the FR summit peak is onlyconspicuous at 50 Hz. Thus, it is obvious that the frequency depen-dence of FR summit peaks gives an indication of the comparativerates of reduction of the different constituents present in the mixture.

(iv) For some ions, i.e., Pb(II), In(III), Cd(II), and Zn(II),the FR summit peaks are seen in the positive quadrant whereasthose corresponding to the reduction of T1(I), Ni(II), and Co(II)appear in the negative region. This shows that it can be concludedfrom FR polarograms that those ions whose summit peaks appearin the positive region should have transfer coefficients greater than0.5, whereas those whose summit peaks fall in the negative quadrantshould have transfer coefficients less than 0.5. This also explainshow a better resolution of FR summit peaks, even when the half-wave potentials of the constituents differ only by 20 mV, is achieved.

(v) Since the FR summit peak current at any frequency varieslinearly with the concentration of the depolarizer, the quantitativeestimation of each of the constituents should be possible.

(vi) By applying the formulations for the faradaic rectificationunder the condition C0oD^2 = C°RD1^2 (which is met at the half-wave potential), it is possible to determine the value of the transfercoefficient (a) and of the apparent rate constant (k°a) for thereaction.

(vii) FR polarographic studies are applicable to reversiblereactions, quasi-reversible reactions, and irreversible reactions.Thus, FR polarography can be used to study all kinds of reactionsand is not limited to either irreversible reactions, as is the case fordc polarography, or moderately fast reactions, as for ac polarogra-phy. The method has been applied for the determination of kineticparameters of some inorganic depolarizers in different supportingelectrolytes. The results for each depolarizer will be discussedseparately.

(M) Single-Electron Charge Transfer Reactions

(a) Tl(I)/d.m.e.

The FR polarogram of 1.0 mM T1(I) in 1.0 N NaClO4 is shownin Fig. 21. The FR summit peaks at all ac frequencies coincide with

the dc half-wave potential of T1(I) in 1.0 N NaClO4 and the summitpeak is conspicuous even up to 600 Hz, indicating that the reactionshould be moderately fast. The kinetic parameters are obtained 83'84

from the plot of kEJ V\ versus (o~l/2 (Fig. 22). When C^D^2 =C°RD](2 (the conditions met at the half-wave potential), the faradaic

Faradaic Rectification Studies

1*4

227

0-8

0'2

2 0 40 60

Figure 22. AE^/V^ versusT1(I) in 1.0 N NaClO4.

plot for 1.0 mM

rectification theory ' gives the following expressions:

nFRT

nFRT ~2k°a\f)'"

(29)

(30)

where all the notations are as usual. Knowing VA, the ac potentialat the electrode/solution interface, and the constant value ofAEQO/ VA obtained in the high-frequency region (Fig. 22), the valueof a is obtained from Eq. (29). Substituting the values of a, D, thediffusion coefficient of T1(I) in 1.0 AT NaClO4, and the slope corre-sponding to the linear relationship in the low-frequency region(Fig. 22) in Eq. (30), the value of k°a is determined.

Similar FR polarograms and versus co~1/2 plots areobtained for T1(I) in other supporting electrolytes, and the respec-tive values of a and k°a have been obtained. The kinetic parametersare given in Table 6. As the FR polarograms in all the supportingelectrolytes are obtained on the negative side of the abscissa, thetransfer coefficient is expected to be less than 0.5, and it lies between

228 H. P. Agarwal

Table 6Kinetic Parameters of T1(I) and Cu(I) in

Various Supporting Electrolytes"

Reaction

Tl(I)/d.m.e.b

Cu(I)/d.m.e.c

a Temperatureb Refs. 52, 83,c Refs. 83 and


1.0 N KNO3

1.0 N KC10.5 N K2SO4

1.0 N NaClO4

l .ONKBr1.0 N KNO31.0 N KC10.5 N K2SO4

1.0 N NaClO4

Do x 106

(meas.)(cm/s)

11.5711.3216.507.90

4.54.14.84.04.4

= 27 ± 1°C; concentration of theand 84.85.

a(Cathodic

meas.)

0.450.400.4550.43

0.550.540.5350.540.55

depolarizer =

k°a x 102

(cm/s)

3.53.24.02.9

2.21.32.02.02.0

1.0 mM.

k°, x 102

(cm/s)

1.3

—

0.380.81—

0.97

0.40 and 0.46 (in different supporting electrolytes). The rate constantis of the order of 10~2cm/s and varies from 2.9 x 10~2 to 4.0 x10~2cm/s in different supporting electrolytes. It should be notedthat these values are of lower order than those reported by Barker7

using the faradaic rectification method at radio frequencies. Veryfew references are available in the literature for this reaction. Thevalues of the apparent standard rate constant vary in differentsupporting electrolytes in the following order:

s o r > NO3- > c r > CIO4

The values of k°a have been corrected for the double layer ina KC1 medium (as <f>2 data were available only in this medium) byapplying the Frumkin theory. They are given in Table 6.

(b) Cu(I)/d.m.e.

Cu(II) is reduced in KC1 and KBr media in two steps, i.e.,Cu(II) -> Cu(I) in the vicinity of zero potential and Cu(I) ->Cu(metal) at -0.225 and -0.15V, respectively. As the acpolarograms and FR polarograms are recorded in different support-ing electrolytes beyond —0.10 V, the summit peaks obtained in both


cases (Fig. 23) correspond to the reduction of Cu(I) -> Cu(metal).The FR summit peaks are conspicuous up to 600 Hz in all supportingelectrolytes and are obtained in the positive region at the half-wavepotential corresponding to the second electron charge transferreaction. This shows that the value of a should be greater than 0.5and the reaction should be moderately fast.85 The kinetic parametersare obtained in the same way as for the reduction of T1(I) ions andare given in Table 6. The value of a lies between 0.535 and 0.55and the rate constant ranges from 1.3 x 10~2 to 2.2 x 10~2 cm/s indifferent supporting electrolytes. The rate constants obtained areconsistent with those obtained by faradaic impedance and acpolarographic methods.86'87

The double-layer correction has been applied in perchlorate,nitrate, and chloride media and the true rate constant, k°t, varies

0*40 r

0*3 035 0*4D, C, POTENTIAL

0*45

IN VOLTS

Figure 23. FR polarograms of 1.0 mM Cu(II) in 1.0 NKNO3.

230 H. P. Agarwal

in the following order:

cio; > c r > NO3"

It is interesting to note that the rate constant is appreciably lowerin the nitrate medium as compared to that in the other supportingelectrolytes.

(iii) Two-Electron Charge Transfer Reactions

(a) Pb(II)/d.m.e.

The FR summit peak has been obtained at the dc half-wavepotential in the respective supporting electrolytes (Fig. 24). The

-o 500 HZ* I kHz-± 5 kHz- 10kHz

!5kH220kHZ30kHz

0*35 0*40 0*45 0*50

D.C.POTENTIAL IN VOLTS

Figure 24. FR polarograms of 1.0 mM Pb(II) in 1.0 NKC1.


kinetic parameters have been obtained using the usual two equations[Eqs. (29) and (30)]. The value of a obtained88 lies between 0.52and 0.53 and k°a ranges from 0.34 x 10"2 to 0.98 x 10~2 cm/s (Table7). The values of a are comparable to those reported by Agarwal,41

but k°a is somewhat lower than given in the literature.7'41 It shouldbe noted that the value of the rate constant in the KCl medium isfairly high and is not as low as reported by Barker.7 The high valuein the KCl medium is due to excessive adsorption of chloride ion,as has been reported by Sluyters-Rehbach et al.89 The k°a values inKNO3 and KBr are comparable to those reported in the literature.However, they are not as low as that reported by Barker.7

The rate constants in different supporting electrolytes vary inthe following order:

Cl" > CIO; > I" > NO^ > Br"

The k°a values in chloride, bromide, and percholate media havebeen corrected by applying the Frumkin theory for the double-layer

Table 7Kinetic Parameters of Pb2+ and Cd2+ in


Reaction

Pb2+/d.m.e.b

Cd27d.m.e.d


1.0 N KNO31.0 N KCl1.0 N KBr1.0 N KI1.0 N NaClO4

1.0 AT KF1.0 N KNO31.0 TV KCl1.0 N KBr1.0 N KI1.0 N NaClO4

Do x 106

(meas.)(cm2/s)

7.59.08.6

7.2

4.56.47.47.7

11.35.8

a(Cathodic

meas.)

0.530.530.5250.520.525

0.540.530.530.550.540.535

k°a x 102

(cm/s)

0.390.980.340.45c

0.56

0.480.411.020.570.900.27

k°t x 102

(cm/s)

1.080.38—

0.50

0.16—

1.100.70——

a Temperature = 27 ± 1°C; concentration of the depolarizer =1.0 mM.b Refs. 83 and 88.c The value of k°a is obtained assuming that the value of D in a KI medium is the

same as that in a KBr medium. The direct determination of the value of D in aKI medium is not possible as PBI2 is not appreciably soluble.

d Refs. 38 and 83.

232 H. P. Agarwal

correction as <£2 data are available in the literature.90"92 The correc-ted rate constant values vary in the following order:

Cl~ > CIO; > Br"

The k°t is obtained in the chloride medium corresponding tothe complex PbCl+ (other complexes possible are PbCl2, PbCl^,and PbClJT), as it correlates with the corrected values in the othersupporting electrolytes. With the increase of charge on the complexfrom neutral to negative values, the rate constant will be enhanced.

(b) Cd(II)/d.m.e.

The reduction of cadmium ions is a fast reaction and has beenwidely studied. The faradaic rectification method using high-frequency modulated signals has been applied for the determinationof kinetic parameters of this reaction by Barker,7 Van Der Pol etal.,54 and Agarwal using a hanging amalgam drop electrode.41 TheFR polarograms of Cd(II) in 1.0 N KF are shown in Fig. 25. Fromthe polarograms, it can be seen that all the FR summit peaks, evenup to 50 kHz, coincide with the half-wave potential (which is about-0.77 V). The rate constant and the value of a are obtained fromAEoo/ V2

A versus (o~1/2 plots using Eqs. (29) and (30). The value ofa in different supporting electrolytes lies between 0.53 and 0.55and its is not as high in potassium iodide as has been reported inthe literature.7'41 k°a lies between 0.27 x 10~2 and 1.02 x 10"2 cm/s(Table 7). In sodium perchlorate, the reaction is slowest whereasit is very fast in potassium chloride. The effect of different supportingelectrolytes on the apparent rate constant is in the following order:

Cl~ > 1" > Br" > F" > NO^ > CIO4

The values of the rate constants obtained are fairly comparable tothose given in the literature, and the present technique appears tobe quite reliable for determination of kinetics parameters of fastreactions.

The k°a values in KF, KC1, and KBr have been corrected byapplying the Frumkin theory for the double-layer correction as </>2

data were available in the literature for these media. The /c° valuesobtained in Cl" and Br~ media correspond to CdCl+ and CdBr"complexes, respectively, as they correlate93 with corrected values

in the other supporting electrolytes. The k°t values in Cl , Br , andF~ also vary in the above order.

(c) Zn(II)/d.m.e.

The reduction of zinc ions at d.m.e. has widely been studiedand the reaction has been reported to be quasi-reversible.94 VanDer Pol and co-workers54 studied this reaction by the faradaicrectification polarographic technique using high-frequency modu-lated signals. The kinetic parameters have been evaluated by the

234 H. P. Agarwal

present technique using unmodulated audio frequency alternatingcurrent and the results have been compared with those given in theliterature.83

The plot of A£oo/ VA versus w~l/2 for reduction of zinc ion in1.0 N KC1 is shown in Fig. 26. At high frequencies (400 Hz andabove), A£oo/ VA tends to be constant, and the value of a is obtainedfrom Eq. (29). Similarly, the rate constant is determined from theslope of the plot using Eq. (30). The kinetic parameters in differentsupporting electrolytes are given in Table 8. The value of a liesbetween 0.52 and 0.535 and k°a is in the range of 2.2 x 10~2cm/sto 2.8 x 10~2 cm/s. It is interesting to note that the rate constant in

0.4 -

LLJ

uQUJ

ULLI

WITH SUPPO^TIHG CUC-T«K)LYTC ALONC AT

1.00 1.05 1.10d.c . POTENTIAL IN VOLTS

Figure 26. FR polarograms of 2.0 mM Zn(II) in 1.0 TV KC1.


Table 8Kinetic Parameters of Cr(III), In(III), Zn(II), and Co(II) in


Reaction

Cr(III)/d.m.e.fc

In(III)/d.m.e.c

Zn(II)/d.m.e/

(

Co(II)/d.m.e.e


1.0 N KC1L.O N KNO3

1.0 N KBrI.ON K2SO4

1.0 N NaClO4

t.O N KC1

L0NKC1I .ON KNO3I .ON KBr).5 N K2SO4

I .ON NaClO4

l.ON KF

Do x 106

(meas.)(cm2/s)

9.51.17.84.76.4

5.6

8.47.86.74.84.6

a(Cathodic

meas.)

0.470.470.450.470.45

0.52

0.520.520.5250.5350.53

0.48

k°a x 102

(cm/s)

1.40.231.60.541.2

8.0

2.52.82.62.22.4

0.54

kQt x 102

(cm/s)

—

—

1.07

——

—

0.048

a Temperature = 25°C; concentration of the depolarizer = 1 mM.b Refs. 83 and 99.c Refs. 82 and 83.d Refs. 81 and 83.e Refs. 82 and 83.

a sulfate medium is the lowest whereas in bromide and nitratemedia it is a high as reported in the literature.94 The values of thekinetic parameters in different supporting electrolytes follow theorder

KNO3 > KBr > KC1 > NaClO4 > K2SO4

Van Der Pol et al.54 have reported the same order for the valuesof the rate constant for the second electron charge transfer step.

{d) Co(II)/d.m.e.

The dc polarograms of Co(II) in a noncomplexation state havebeen obtained95"98 in 1.0 AT KC1, 1.0 N KF, and 1.0 M sodium

236 H. P. Agarwal

citrate + 0.1M NaOH media. The FR polarograms have beenobtained in the latter two media (Figs. 27 and 28). It may be notedthat, as reported in the literature,96 no dc step is obtained in a 1.0 Msodium citrate 4-0.1 M NaOH medium, except a catalytic wavewhich is due to the fact that its half-wave potential is close to thepotential at which catalytic discharge of hydrogen occurs. In FRpolarography, well-defined polarograms are obtained uninterruptedby catalytic discharge of hydrogen. This will be of great advantagein such studies.

The Co(II) reaction in 1.0 N KF is quasi-reversible as is evidentfrom the FR polarograms. The FR summit peaks are quite distincteven up to 600 Hz (Fig. 28). The kinetic parameters for this reactionhave been obtained82 from the Ai^/V^ versus co~l/2 plot usingEqs. (29) and (30). The value of a is 0.48 and k°a is 5.4 x 10~3 cm/s

D.C.POTENTIAL IN VOLTS1*6 1*65 17 1*75

0 + i r 1 » 4

- 0 8Figure 27. FR polarograms of 1.0 mM Co(II) in1.0 M sodium citrate + 0.1 M NaOH.


D.C.POTENTIAL IN VOLTS1*2 1*3 t'4

01

6 0 0 ^

-0*20 L

Figure 28. FR polarograms of 1.0 mM Co(II) in 1.0 NKF + 0.01% gelatin.

(Table 8). The value of k°a has been corrected for the double layerin KF20 and the value of k°t obtained is 0.48 x 10~3 cm/s.

(it?) Three-Electron Charge Transfer Reactions

(a) Cr{III)/d.m.e.

Generally, two steps are obtained in the dc polarograms forthe reduction of Cr(III) in different supporting electrolytes exceptin a KNO3 medium. It is interesting to note that only one FRsummit peak is obtained in different supporting electrolytes exceptin a 1.0 JV KC1 medium. FR polarograms of Cr(III) in 1.0 N KC1at different frequencies of the ac are shown in Fig. 29. The FRpolarogram shows two FR summit peaks at 0.88 V and 1.45 V.

238 H. P. Agarwal

D. C. Potential (V)0 . 8 0 . 9 1 . 0 1 . 1 1 . 2 1 . 3 1 . 4 1 . 5 1.6Of T

» 2 5 *•50cr»H O C *. 2OOcn

- - 1 0LL1

= - 2 0

QLUE L - 3 0

U^ - 4 0

Figure 29. FR polarograms of 1.0 mM of Cr(III) in1.0 AT KC1.

corresponding to the two steps of the reaction, i.e., Cr(III) -> Cr(II)and Cr(II) -> Cr, respectively. As the height of the FR summit peakfor the first step of the reaction is very small and appears only upto 50 Hz, this reaction is irreversible, and it is not possible todetermine its rate constant. The FR summit peak corresponding tothe second step of the reaction is conspicuous even up to 200 Hz,and thus it has been possible to determine the values of a and k°a

for this reaction in different supporting electrolytes." The value ofa lies between 0.45 and 0.47 and k°a ranges from 0.23 x 10"2 to1.6 x 10~2cm/s (Table 8).

The rate constant is moderately fast in bromide, chloride, andperchlorate media and varies in the following order:

Br" > c r > cior

In nitrate and sulfate media, Cr(III) is not very stable and thereaction Cr(II) -> Cr tends to be irreversible. The rate constantsobtained in these media are tentative because of excessive adsorp-tion of chromous ions in these media. The rate constants have been


corrected for the double layer in 1.0 N KC1 and 1.0 N KNO3 media.

(b) In(III)/d.m.e.

The dc polarogram of In(III) gives only one step and the acpolarogram shows one ac summit peak in 1.0 N KC1 at 0.65 V,indicating that there is only one step in the reduction in(III) -» In.In the FR polarogram in 1.0 N KC1 (Fig. 30), the FR summit peakis obtained at 0.65 V and the peak is conspicuous even up to 1000 Hz.This shows that the reaction should be moderately fast. The valuesof the kinetic parameters obtained82 are k°a = 8.0 x 10~2 cm/s anda = 0.52 (Table 8). It is evident that this reaction is not as fast ashas been reported by Barker,7 who found a rate constant of 1.5 cm/s.The reaction is also not irreversible as has invariably been reported

0*5 06 O'7 0'8

D.C. POTENTIAL IN VOLTS

Figure 30. FR polarograms of 1.0 mM of In(III) in1.0 N KC1.

240 H. P. Agarwal

in the literature.100101 Jain102 has recently shown that this reactionis quasi-reversible in halides, which corroborates the aboveresults.

2. Studies Using Organic Compounds as Depolarizers

Polarographic studies of organic compounds are very complicated.Many of the compounds behave as surfactants, most of them exhibitmultiple-electron charge transfer, and very few are soluble in water.The measurement of the capacitance of the double layer, the cellresistance, and the impedance at the electrode/solution interfacepresents many difficulties. To examine the versatility of the FRpolarographic technique, a few simple water-soluble compoundshave been chosen for the study. The results obtained are somewhatexciting because the FR polarographic studies not only help in theelucidation of the mechanism of the reaction in different stages butalso enable the determination of kinetic parameters for each stepof reduction.

The experimental cell, the measurement technique, and themethod used for purification of the solutions are the same as givenin Section IV (l(i)). The organic substances studied are vanillin,isatin, and 5-nitrobenzimidazole. Each of them will be dealt within detail.

(i) Reduction of Vanillin

The electrolytic reduction of vanillin was studied by Shima.103

He reported that it involves a two-electron charge transfer leadingto the formation of the final product, vanillyl alcohol. Brdicka104

and Zuman105 have independently studied the dc polarography ofthis compound at varying pH's. Suzuki106 has reported that vanillinis irreversibly reduced, as it gives no ac polarogram.

The reduction of vanillin has been studied by the FR polaro-graphic technique.107 The FR summit peaks are conspicuous up to200-250 Hz at all pH's. With an increase in pH, the FR summitpeak potential tends to be more cathodic. The FR wave is invariablyobtained on the negative side of the abscissa, indicating that a isless than 0.5. The FR polarograms at pH 10.5 are shown in Fig. 31.


The reduction of vanillin is accompanied by a two-electroncharge transfer and the mechanism of the reaction can be explainedas follows:

OH DCHOH

OCH3 OCH3

Vanillin

OHOCH3 OH

OCH3

Vanillyl alcohol

The values of a and the rate constant k°a at varying pH's havebeen determined107 and are given in Table 9. The values of a aregenerally found to decrease with increasing pH and lie between0.23 and 0.34. The rate constant decreases with an increase in pHexcept in a neutral medium and it ranges from 3.63 x 10"2 cm/s to2.42 x 10"2cm/s.

(11) Reduction of Isatin

The polarography of isatin is interesting because the reductionof this compound involves an eight-electron charge transfer andthe reduced final product formed is indoline. The dc polarographicstudy of the compound108 shows that in fairly acidic media, atwo-step reduction occurs. With a decrease in acidity of the medium,a one-step reduction occurs. The FR polarography of isatin at pH1.1 gives two FR waves, one appearing in the negative quadrant at0.23 V and the other in the positive quadrant at 0.4 V (Fig. 32). Thefirst wave corresponds to a six-electron charge transfer and the

242 H. P. Agarwal

D. C. POTENTIAL IN VOLTS1.5 1.6 1.7 1.8

«40 Hz

*50 Hz

75 Hz

Hz

*150, 200, 250 Hz

P SUPPORTING

ELECTROLYTE

ALONE

-0.6Figure 31. FR polarograms of 1.0 mM vanillin in a buffer of pH 10.5.

second to a two-electron charge transfer. The following mechanismcan be given for the reduction in acidic medium108109:

H2

/CH(OH)

Step II

Table 9Kinetic Parameters of Some Organic Depolarizers0

Depolarizer6

Vanillin

5-NitrobenzimidazoleIsatin

Diffusion coefficient,D2 x 106

(cm2/s)

8.2

8.71.85

pH of buffer

1.14.97.09.3

10.57.0LI

FR summitPeak potential

(V vs. SCE)

-1.09-1.27-1.55-1.60-1.60-0.67

I ( 6 O -0.23I I ( 2 O -0.42

a(Cathodic

meas.)

0.340.310.250.230.270.04

10.49110.75

K(cm/s)

3.63 x 10~2

3.45 x 10~2

4.31 x 10~2

3.22 x 10~2

2.42 x 10~2

2.6 x 10~2

16.4 x 10~5

II6.5 x 10'5

a Temperature = 27 ± 1°C.b Concentration = 1.0 mM.

Figure 32. FR polarograms of 1.0 tnM of isatin in a buffer ofpH 1.1.

Bhargava109 has reported that no ac polarographic wave isobtained for the first step of the reduction, and for the second stepof reduction, the ac wave is conspicuous only at low audio frequen-cies, indicating that both steps are very slow and irreversible. Thediffusion coefficient of isatin in a buffer of pH 1.1 is reported tobe 1.85 x 10"6cm2/s and the value of n for the second step is 2.


As the first wave appears on the negative side of the abscissa, thevalue of a is 0.49 whereas for the second wave, appearing on thepositive side of the abscissa, the value of a is 0.75. The value ofthe rate constant110'111 for the first step of the reaction is 6.4 x10~5 cm/s whereas for the second step, it is 6.5 x 10"5 cm/s (Table9).

(iii) Reduction of 5-Nitrobenzitnidazole

The dc polarograms reported for 5-nitrobenzimidazole112 areof special interest: In a highly acidic medium, the reduction takesplace in two steps at —0.14 V and —0.68 V, respectively, whereas atpH 4.0 and above, there is only one step. Bhargava109 evaluated

D. C. POTENTIAL IN VOLTS0.5 0.6 0.7

-o 40 Hz

- 50 Hz

- a 100 Hz, 150,200 & 250 Hz

V

FR polarograms of 1.0 mM 5-nitrobenzimidazole in buffer of pH 7.0.

246 H. P. Agarwal

the value of n from the Ilkovik equation as 6 in a buffer of pH 7.5and the value of the diffusion coefficient D, as 8.7 x 10~6 cm2/s.From ac polarographic studies at pH 7.5, Bhargava reported thisreaction to be moderately fast.

The FR polarograms of 5-nitrobenzimidazole in a buffer ofpH 7.0 are shown in Fig. 33. The FR polarograms are obtained onthe negative side of the abscissa, indicating that a should be lessthan 0.5. The FR summit peak is obtained at the half-wave potential-0.65 V and it is conspicuous up to 100 Hz. As the reaction involvesa six-electron charge transfer, it may be assigned the followingmechanism:

O2N5-Nitrobenzimidazole

HOHN

H2N5-Aminobenzimidazole

HN

The kinetic parameters (Table 9) obtained for the reaction110

are a = 0.04 and k°a = 2.6 x 10~2 cm/s. The rate constant is of thesame order as reported by ac polarography.109 The low value ofthe transfer coefficient indicates that the compound is excessivelyadsorbed at the electrode surface.

V. OTHER APPLICATIONS

Except for the recent developments in measurement of electrodekinetics of multiple-electron charge transfer reactions and in


faradaic rectification polarography, very few references are avail-able relating to new applications. Some isolated observations madeare of importance, and they should be suitably exploited. Two ofthem are described below.

1. Growth formation in epitaxial electrodeposition. Recently,Sheshadri113 observed that at small overpotentials caused byfaradaic rectification, growth formation occurs in the epitaxialelectrodeposition of copper on various copper single-crystalplanes.

2. Study of instantaneous corrosion rates. Sathyanarayana andSrinivasan114115 have extended faradaic rectification studies to thedetermination of instantaneous corrosion rates of partially revers-ible metal ion/metal interfaces. They emphasized the practicalutility of the method in the fast monitoring of the corrosion of abattery.

VI. CONCLUSIONS

The faradaic rectification effect is a general phenomenon in that itis exhibited in reversible electrode systems, at semiconductor/elec-trolyte interfaces, and in all kinds of redox couples accompaniedby single-electron charge transfer, two-electron charge transfer, andmultiple-electron charge transfer. In the past, substantial progressin this field could not be achieved mainly for two reasons: Firstly,the lack of perfection and standardization of experimental tech-niques for measurements, particularly at metal ion/metal(s), metalion/d.m.e., redox couple/inert metal(s), and metal ion/amalgramelectrode (hanging or dropping) interfaces at audio and radiofrequencies, and secondly, for want of a suitable theory applicableto multiple-electron charge transfer reactions.

Earlier studies generally involved the evaluation of kineticparameters of reactions which are accompanied by single-electroncharge transfer.116 Some reactions involving two-electron chargetransfer were also studied, assuming either that both electrons aretransferred in a single step or that the slower step in the two-stepreaction is in overall control of the rate process. As described inthis chapter for the first time, the faradaic rectification theory for

248 H. P. Agarwal

multiple-electron charge transfer reactions has been suitablydeveloped and modified, and the kinetic parameters for each stepof several multiple-electron charge transfer reactions have beenevaluated. Wherever possible, the zero-point method has beenapplied for the determination of kinetic parameters and the k°a

values have been corrected for the double layer. One interestingobservation is that in any two-electron charge transfer reaction, ifthe two rate constants (for the two steps) differ by a factor of 100or so, it would not be appropriate to assume that the slower reactionis in overall control of the rate process. In any two-electron chargetransfer reaction at a metal ion/metal(s) interface, the second-steprate constant is invariably higher than the first-step rate constantif parameters are obtained by eliminating C°R, the concentrationof the finally reduced species. However, if C°R is taken to be 1.0,/c? is always greater than k°2. The kinetic parameters thus obtainedby different methods explain the variation in the values reportedin the literature. If three or more electrons are involved in thecharge transfer, the faradaic rectification theory becomes compli-cated. The study of two-electron charge transfer reactions in a redoxcouple reveals that when C°o > C°R, k\ is greater than k\, and whenC°R > C o, fc? is less than k\. This explains why the results of earlierworkers who carried out studies at varying redox concentrationratios did not tally.

The FR polarographic technique recently developed has theunique distinction of enabling qualitative detection and quantitativeestimation (for concentrations as low as 0.1 mM) in mixtures ofseveral components whose half-wave potentials may differ by evenless than 20 mV. This method also helps in simultaneous determina-tions of kinetic parameters of reversible, quasi-reversible, andirreversible reactions. The FR polarographic study of organic com-pounds will be of immense use in elucidating the mechanism andin evaluating the kinetic parameters of several steps involved in thereactions of these compounds. Most of the organic reactions aregenerally accompanied by four- to eight-electron charge transfer,and the FR polarographic technique would be very useful in theanalysis and study of such complicated reactions. The systematicstudy of some organic compounds of industrial importance willafford very valuable information that can be used to optimizeproduction processes.


The method can successfully be used in analyses of impuritiesin metals and alloys, for estimation of minor elements inmonomolecular films of oxide layers of Fe-Cr-Ni alloys, for detec-tion of metal impurities in environmental pollution, for studyingthe depression of high-grade semiconducting materials and foranalysis of the corrosion products of contact junction diodes usedin microelectronic circuits. Much sophistication is desirable on theinstrumental side so as to incorporate an automatic recording deviceto make an FR polarograph suitable for wider applications andcommon use.

Recently, Reinmuth117 has emphasized the importance of asecond-order method in the study of the kinetics of more complexreactions. Of the several nonlinear relaxation methods, the poten-tialities and applications of the faradaic rectification method aremany and varied. The study of electrode processes in fused saltelectrolytes (which are usually fast) and the rectification innonaqueous solvents, in supersaturated and supercooled solutionsof redox couples, and in fused organic semiconductors seem to bevery promising. The applicability of the method in fast monitoringof corrosion, more exhaustive studies at different semiconduc-tor/electrolyte interfaces, and investigations of rectification throughpolyelectrolytes and single crystals deserve attention. The study ofthe photovoltaic effect produced by ultraviolet,118 infrared, andlaser radiation on the electrode/solution interface and theirinfluence on nonlinear effects would be very interesting.

Studies on the subject are still in their infancy and a moreactive pursuit and exploration is needed to investigate the variousfields just thrown open. The results presented in this chapter mainlycomprise details of our own studies in the field, as practically noreferences from other workers are available.

ACKNOWLEDGMENT

The author expresses his appreciation and gratitude to all thosewho have significantly contributed to the development of this newfield, bringing it to the forefront as a potential tool for the studyof fast electrode kinetics.

250 H. P. Agarwal

APPENDIX A

Case I: CR Is Kept Constant and C°o Varies

Let the concentration of the reductant be C°R and that of the oxidantbe C°o, C°o, C°o' for the three different redox ratios in any experi-ment, the oxidation potential at any concentration can then bewritten as

RT C%

Co

where Ex, £2, and £3 are the oxidant potentials corresponding tothe three respective concentrations. Again,

KT C% RT C?T- £2 = — In —- and ^ - £3 = — In —5-

nt C o w^ C o

The concentration of the intermediate species after the firstelectron charge transfer at the three redox ratios can be representedas C°R], C%, and C°RI9 respectively. The relationship between themcan be given as

_ „ RT^ C% RT, C%Ex-E2 = — \n —Q1 and JE, - E3 = — In —^

nF CRl nF C Rl

Assuming that C% = aC°Rl and C% = bC%9 Eq. (12) correspondingto the other two redox ratios can be written as

it?

^ (a)


(1 - a) - aC°R] 0 j _ a (b)(Cu)

where A £ ^ and A£J> are the rectification potentials for the othertwo redox concentrations. The second term on the right-hand sidein each of the above equations can be approximately taken as equal.On subtracting Eq. (a) from Eq. (12) and rearranging, the expressionobtained is

- (C°onC°6(l - a) - aaC°R[]} (ro^ro,.a

Similarly, on subtracting Eq. (b) from Eq. (12), one gets

~ 21/2

= [(C°jr{C0o(l-a)-aC°R,}

-{Co) \CO (I - a) - baCR]\\ 0 c

Putting

F Vl

Considering the case when the experiment at three differentconcentrations of redox ratios is being carried out, with C°o constantand the concentration of the reductant varying as C°R, C^', andC°R. The oxidation potential is given by


If the concentration of the intermediate species, after the firstelectron transfer, varies in the three cases as C°Rl, CR', and C%,respectively, then the relationships

_ _ RT C% A _ „ RTt C%E E l n ^ E E l n ^

will hold.Considering that C% = aC%, C% = bC% and putting

a>1/2C%2D\i2/21'2 = j ; C°o(l - a) = fc; aC% = I

[p. 183, Eq. (12)] corresponding to the three redox ratios can bewritten39 as

d'j = (k — l)m + (n — 0LCR)k%/C°R " (f)

ej = {k- al)m + (an - aCR')k°2/cT~a) (g)

fj = (fc - bl)m + (bn - aCR")k°2/CR"(l-a) (h)

where A£^ and &E'^ are the rectification potentials for the respec-tive redox concentrations. The first term on the right-hand side ineach of the above three equations can be approximately taken asequal. Subtracting Eq. (g) from Eq. (f) and Eq. (h) from Eq. (f)and rearranging the terms, two equations are obtained. In theseequations, the right-hand side terms and the left-hand side termsof the first equation are divided by the respective terms of thesecond equation (as has been shown in Case I).

C°R = q\ the final expression obtained is

{dr - a1/2e) n(o - ap) + a(pC°Rf - oC°R)

{d' - bl/2f) n(o - bq) + a(qC°R" - oC°R)

On cross multiplication and transferring C°Rl terms on one side,

a[(d' - bx/2f)(pCR' - oCR) - (df - al/2e)(qC0^ - oC°R)]

(i)

o _(1 - a)[(d' - al/2e)(o - bq) - (df - bl/2f)(o - ap)]

(j)

Substituting the experimentally determined value of a, C°R]

can be obtained as all other terms are known. The value of /c° is

254 H. P. Aganval

determined by substituting the value of C°Rl and a in the equationobtained after subtracting Eq. (g) from Eq. (f). Subsequently, bysubstituting the values of /c°, C°Rl, and a in Eq. (f), the value offc? can be determined.

APPENDIX B

The oxidant potentials at varying concentrations can be written as

p - p RT1 C°o^2 - £o in —Q-

= Eo - In —T

nF CR

• = E o ^ In 7^~nF CR

Now from the above relationships

p * T ,E4 = — InnF C o

^ o


If the concentrations of the intermediate species vary at thefive redox ratios as C°Rn, C%, C°Ru, C

0Rl\, and C°Rn after the first

electron charge transfer and C%, C%, C%, C°R"\ and C°Ri afterthe second electron charge transfer, then the relationships

-~- ^ R IKI CR RT

nr c p,, nr K^ Rt nr

Ji - £3 = — In 7 ^ - = r r : l n T^ = —ln bRTnF C°Rn nF"" C\ nF

~ ^ l n 7 ^ = ~ ^ l n 7= " = ^^1 :nF Ci,, nF Ci, nF

c° I V

ii — £ 5 = In Ti = ln n == ln d

nF C nF C nFwill hold for the concentration ratios relating to the intermediatespecies after the first and second steps of electron charge transfer,as explained earlier.

From the above relationships, it would be appropriate toexpress the concentration of the intermediate species by the follow-ing relationship

Now if AEoo, A£L, A£j^, A£^r, and &E™ are the rectifiedpotentials for five different redox concentrations, and putting

ARTkElJFV\- (2a - 1) - d\

4RTkEll/FV\ - (la - 1) = d";

4RTkE™/FV2A - (2a - 1) = dm ;

ARTbE™/FV2A - (2a - 1) = dIV;

D ^ 1 / 2 / 2 1 / 2 = r ; aC°RlI = 5;

C°Rll(l ~ a) = t; C°o(l-a) = u;

256 H. P. Agarwal

r/r . r°{l~a) - v • L°/r°a . r(0(l~a) - v2/ <- Ru ^ R{ — *2, K3/ ^ Ri ^ R ~ -^3?

then Eq. (14), p. 185, corresponding to each redox ratio can bewritten as,

d'r = (k-s) YJC°o + (t - 1) Y2 + (n - aC°R) Y3 (a)

dlr = (u- as)Y1/a(1~a) • C ^ + (t - l)Y2

+ (an-aC°R)Y3/aa (b)

dur = (v- bs) YJb{x-a) • Ct" + (t~l) Y2

+ (bn-aC°R)Y3/ba (c)

dmr = (co- cs)YJC{x-a) • C°oa + (r - /)Y2

+ (cn-aC°R)F3 /C a (d)

dlvr = (x- ds)YJd{x-a) - C°oa + (/ - /)y2

+ (^ - aC 0 ^ )y 3 / d« (e)

On subtracting each of Eqs. (b), (c), (d), and (e) from Eq. (a)and assuming for the sake of simplification that the coefficients ofthe term k2 are almost equal, the expressions obtained are

{d' - dl)r = {(k - s)/C°o - (M - as)/a^"')C0o}Yi

+ {n-aC°R-(an-aC°R)/aa}Y3 (f)

(</' - dn)r = {(k - s)/C°o -(v- bs)/b(x-a) • Cg'°}y,

+ {n-aCR-(bn-aC°R)/ba}Y3 (g)

(d' - dm)r = {(k - s)/C°o - (a. - cs)/c<l-a) • CS""}^

+ {if-aCi-(cfi-oCi)/c-}y3 (h)

{d' - dlv)r = {(*; - s)/C°o -(x- ds)/d(1-a) • C°oa}Y,

+ {n-aC0R-(dn-aC°R)/da}Y3 (i)

Putting,

{n-aCR-{an-aC0R)/aa} = A;

{n - aC°R - (bn - aCR)/ba) = B;

{n - aCR - {en - aC°R)/c°} = C;{n - aC°R - (dn - aCR)/da} = D


Now multiplying [Eq. (f)] by B and [Eq. (g)] by A and subtracting

r{(d'-dl)B-(d'-dll)A}

- {(k - s)/C°o - (v - bs)/b(1-a)C°oa}A] (j)

Similarly multiplying [Eq. (f)] by C and [Eq. (h)] by A andsubtracting

r{(d' - dl)C - (d' - dlu)A}

= r,[{(fc - s)/C°o - (u - as)/a (1-o) • C°o}C

- {(k - s)/C°o -(w- cs)/cu-a)C0o"}A] (k)

Again multiplying [Eq. (f)] by D and [Eq. (i)] by A and subtracting

r{(d' - dl)D - (d' - dlv)A}

= Y,[{k - s)/C°o - (M - as)/a°-a)C°o}D

- {(k - s)/C°o - (x - ds)/d^a)C0oa}A] (1)

Dividing [Eq. (j)] by [Eq. (k)], Yu i.e., kyc%~°\ can be eliminatedand putting

(d'-dl) = z; {(k-s)/C°o} = E;

{E ~{u - as)/a -CO\ = E,

one gets,

{zB - (df - du)A}/{zC - (df - dm)A}

[E'B - {E - (v - bs)/b{l~a) • COpa}A]

" [E'C -{E-(co- cs)/c{l-a)C^a}A] ( m )

Similarly on dividing [Eq. (j)] by [Eq. (1)],

{zB - (df - du)A}/{zD - (df - dlw)A}

_ [E'B -{E-(v- bs)/b{l-a) • C°ga}A]

[E'D -{E-{x- ds)/d{l~a) • C<£ya}A] W

258 H. P. Agarwal

In [Eqs. (m) and (n)] after cross multiplication and transferringall the terms on one side and equating them to 0, two identicalequations are obtained as

C%(1 - a)3 • F + C%C°Rua(l " " ) * * G

+ C°RlC°Ra(l ~a)2-H+ C°R]CRnCRa2(l - a)I

+ C°Rn - CU3 ' J + C°RV(1 -a)-K=0 (o)

C £ ( l - a)3 • F ' + C ^ C 5 r n a ( l - a ) 2 • G'

+ C%C°Ra(l ~ et)2 • H' + C%C°RuCW(l ~ a)I'

+ C0*H • C°R2 • a3 / ' + C°RV(1 - a) • K' = 0 (p)

where F9 G, //, /, 7, K and F', G', H', /', Jf and X' are thecoefficients of the respective terms in the Eqs. (o) and (p).

The coefficients of C°R{ and C°R in the Eqs. (o) and (p) areequated, i.e.,

F = Ff and K = K'

On simplifying them, the value of C°Ru can be obtained.On substituting the value of C°Ru in Eq. (m), the value of C°Rl

can be calculated. On substituting the values of C°Rn and C°Rl inEq. (j), the value of k°x can be obtained,

{B(d'-dl)r-(d'-du)Ar}CRu1 W

[E'B - A{E - (v -

From Eq. (f), the value of k°3 can be calculated,

k% = {(</' - dl)r - fc?£'C°;, • Ct~a)/C%;a)}IA (r)

Now from Eq. (a), the value of k°2 is determined,

k% = [d'r{C°o(l - «) - aC°R|1}fc?/C°Jct°'

CRll(l - a) - aCRl

(s)

where as given above, the values of the different terms used are

(df - dl)r = {kEJV2A - b,E

(df - du)r = (AFoo/ V2A - £,E

d'r = {4RTAEJ V2AF - (2a - \)W/2D^2/2l/2


A = [C°Rl(l -a)- aC°R - {aC°Rl(\ -a)- aC°R}/aa]

B = [C%(1 - a) - aC°R - {bCR{{\ -a)- aC°R}/ba]

E={C0o(l-«)-aC0

Ru}/C°o

- a) - aaC°Rx)/a{x-a) • C°o]

NOTATION

C°o, C°R, C°Rl, C°Ru bulk concentration of oxidant, bulk con-centration of reductant, concentration ofthe two intermediate species in the elec-trode reaction

Do, DR diffusion coefficients of species O and Rac, aa cathodic transfer coefficient and anodic

transfer coefficient/MTR total rectification current due to mass

transferJo, I°a exchange current density and apparent

exchange current density

qR IJnFC°RD)lW2

z nFVJRT

n number of electrons involved in chargetransfer

VA amplitude of ac voltage at the interface6 phase difference between the alternating

current and applied alternating voltage<o 2irf (f is the frequency of the alternating

current)F Faraday's constantR gas constantT temperatureJFR total rectification current

/ * r z 1

nF Zw (ac + aa)

260 H. P. Agarwal

Z^ faradaic impedanceRr reaction resistancea Warburg coefficientA£oo shift in mean equilibrium potential (rec-

tified potential)

P k°(C°oD^2 + C°RDT)

a transfer coefficient of the forward processAfioo,, A£oon, A£ooin rectification potential due to the first-step,

second-step, and third-step reaction,respectively

C%9 C°Ru concentrations of the intermediate speciesformed intermittently after first andsecond electron charge transfer

fc?j ^2, ^3 r a te constants for the first-step, second-step, and third-step reactions

Ex, E2, E3,E4, E5 oxidation potentials corresponding to oxi-dant concentrations C°o, C°o, C°o, C°o\and C°o , respectively

C%> C°R'U concentrations of intermediate speciescorresponding to C°o, C°o

A£oo, AEJo, A £ ^ rectification potentials corresponding toconcentrations C°o, C°d, and C°o

C°R, C R , C°R concentrations of reductants when C°o iskept constant

(ol9w2 zero-point frequencies corresponding totwo different redox concentration ratios

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by Harry Bloom and Felix Gutman, Plenum Press, New York, 1975, p. 195.38 H. P. Agarwal and P. Jain, Electrochim. Acta 26 (1981) 621.39 P. K. Jain, Faradaic rectification studies of electrode solution interfaces, Ph.D.

thesis, Bhopal University, 1981.40 H. P. Agarwal, Electroanalytical Chemistry, Vol. 7, Ed. by A. J. Bard, Marcel

Dekker, New York, 1974, p . 161.41 H. P. Agarwal, Electrochim. Acta 16 (1971) 1395.42 H. P. Agarwal, Proceedings of DAE Symposium on Interactions at Electrode-

Electrolyte Interfaces, 1982, M- l , p. 179.43 H. P. Agarwal and S. Qureshi , Electrochim. Acta 19 (1974) 349.44 J. O 'M. Bockris and H. Kita, /. Electrochem. Soc. 109 (1968) 2021.45 J. O 'M. Bockris and M. Enyo , Trans. Faraday Soc. 58 (1962) 1287.46 J. Hurlen, Acta Chem. Scand. 15 (1962) 630.47 L. N . Nekrasov and N . P. Berezina, Dokl. Akad. Nauk SSSR 142 (1962) 855.48 O. R. Brown and H. R. Thirsk, Electrochim. Acta 10 (1965) 383.49 I. M. Pearson and G. F. Schrader , Electrochim. Acta 13 (1968) 2021.50 N . A. H a m p s o n and R. J. La tham, Trans. Faraday Soc. 66 (1970) 3131.51 H. P. Agarwal and P. K. Jain , Indian. J. Chem. 16A (1978) 126.52 H. P. Agarwal and P. K. Jain , Electrochim. Acta, 30 (1985) 1243.

53 L. Gaiser and K. E. Heusler, Electrochim. Acta 15 (1970) 161.54 F. Van Der Pol, M. Rehbach-Sluyters, and J. H. Sluyters, J. Electroanal. Chem.

Interfacial Electrochem. 58 (1975) 117.55 T. Hurlen and P. Karl Fischer, J. Electroanal Chem. Interfacial Electrochem. 61

(1975) 165.56 P. P. Sagpet, P. Homsi, V. Plichonet, and J. Badoz-Lambling, Electrochim. Acta

20 (1975) 819.57 J. P. Sagpet, V. Plichonet, and J. Badoz-Lambling, Electrochim. Acta 20 (1975) 825.58 S. Armalis and A. Levinskas, Electrokhimiya 12(1) (1976) 1967.59 H. P. Agarwal, J. Electrochem. Soc. 110 (1963) 237.60 H. P. Agarwal and S. Qureshi, J. Electroanal. Chem. 75 (1977) 697.61 Z. Galus and R. N. Adams, J. Phys. Chem. 67 (1963) 866.62 V. V. Emelyanenko and A. M. Skundin, Elektrokhimiya l l ( a ) (1975) 1335.63 A. A. Baranov, G. A. Simakan, E. A. Erin, V. N. Kosyakov, G. A. Timofev, and

A. G. Rykov, Radiokhimiya 21(1) (1970) 59.64 S. V. Gorbachev and E. I. Martynycheva, Tr. Mosk. Khim.-Tekhnol Inst. 67

(1970) 267.65 S. V. Gorbachev, E. I. Martynycheva, E. P. Agasyan, A. I. Kamenvev, and L. A.

Dunaev, Westn. Mosk. Univ. Khim. 17(3) (1976) 382.66 H. P. Agarwal and S. Qureshi, Electrochim. Acta 21 (1976) 465.67 S. Qureshi, Study of kinetics of electrode reaction. Ph.D. thesis, Bhopal University,

1974.68 V. A. Tyagai and G. Ya. Kolbasov, Sov. Electrochem. 6 (1970) 462.69 A. Kozlowska, P. K. Wrone, and Z. Galus, Bull. Acad. Pol Sci. Ser. Sci. Chim.

22(10) (1974) 917.70 S. P. Bukhman and I. O. Krol, Nauk Kaz. SSR, Ser. Khim. 25(3) (1975) 19.71 K. J. Vetter and G. Thiemke, Z Electrochem. 64 (1960) 805.72 S. Toshima, H. Okaniwa, and M. Nishijima, Chem. Abstr. 66 (1966) 110966.73 G. Fasco, Poraico and Maria, Bui Stint. Tech. Inst. Polit. Timibora, 20(2) (1975)

233.74 H. P. Agarwal and S. Qureshi, Indian J. Chem. 14A (1976) 565.75 A. Frumkin and A. Titenskaja, Zh. Fix. Khim. 31 (1957) 485.76 A. Frumkin and N. Polyanooskaja, Zh. Fiz. Khim. 32 (1958) 257.77 J. E. B. Randies, Disc. Faraday Soc. 1 (1947) 11.78 F. Van Der Pol., M. Sluyters-Rehbach, and J. H. Sluyters, /. Electroanal. Chem.

Interfacial Electrochem. 40 (1972) 209.79 F. Van Der Pol. M. Sluyters-Rehbach, and J. H. Sluyters, J. Electroanal Chem.

Interfacial Electrochem. 41 (1973) 512; 45 (1973) 377.80 H. P. Agarwal and M. Saxena, Proceedings of the International Symposium on

Industrial Electrochemistry (SAEST India), 1976, 13.81 H . P. Agarwa l a n d M. Saxena , Indian J. Chem. 16A (1978) 123.82 H. P. Agarwa l a n d M. Saxena , Indian J. Chem. 16A (1978) 754.83 M. Saxena, Studies on electrode processes using a.c. polarography, faradaic rec-

tification and transitional potential decay techniques, Ph.D. thesis, Bhopal Univer-sity, 1978.

84 H . P. Agarwa l a n d M. Saxena , E x t e n d e d Abstract , 31st Mee t ing of ISE , Venice,Italy, 1980, A l , 145.

85 H. P. Agarwal and M. Saxena , Ex tended Abstract , 29th Meet ing of ISE, Budapes t ,Hungary , 1978.

86 J. E. B. Randies and K. N . Somer ton , Trans. Faraday Soc. 48 (1952) 952.87 T. Kambara and T. Ishio, Rev. Polarog. (Japan) 9 (1961) 30.


88 H . P. Agarwal and M. Saxena , Ex t ended Abstract , 30th Meet ing of ISE, Trond-heim, Norway , (1979) 292.

89 M. Sluyters-Rehbach, B. T immer , and J. H. Sluyters , /. Electroanal. Chem. 15(1967) 151.

90 D . C. G h r a m e and R. Parsons , J. Am. Chem. Soc. 83 (1961) 129.91 J. Lowrence , R. Parsons , and R. Payne , /. Electroanal. Chem. 16 (1968) 193.92 R. Payne , /. Electrochem. Soc. 113 (1966) 999 a n d unpub l i shed data .93 J. K. F r i schmann and A. Timnick , Anal. Chem. 39 (1967) 507.94 N . T a n k a a n d R. T a m a m u s h i , Electrochim. Acta 9 (1964) 963.9 5 1 . M. Kolthoff and J. J. L ingane , Polarography, In terscience, N e w York, 1941,

p . 480.96 J. J. Lingane and H. Kerl inger , Ind. Eng. Chem. (Analedu) 13, (1941) 77.97 L. Meites and T. Meites , Polarographic Techniques, In terscience, New York, 1955,

p . 303.98 P. W. West , J. F. D e a n , a n d E. J. Breda, Collect. Czech. Chem. Commun. 13

(1948) 1.99 H. P. Agarwal and M. Saxena , Trans. SAEST Karaikudi 12 (1977) 258.

100 H. Imai , /. Sci. Hiroshima Univ., Ser A 22 (1958) 191.101 S. Inouye a n d H. Imai , Bull. Chem. Soc. Jpn. 33 (1960) 149.102 D . S. Ja in , J. Electrochem. Soc. India 24 (1975) 189.103 G. Shima, Mem. Coll. Sci. Kyoto Imp. Univ. A l l (1928) 419.104 R. Brdicka, Cas. Cis. Lek 58 (1945) 38.105 P. Zuman, Collect. Czech. Chem. Commun. 15 (1950) 1138.106 M. Suzuki, Mem. Coll. Agr. Kyoto Univ., Chem. Ser. 28 (1951) 67.107 H. P. Agarwal and M. Saxena, in Proceedings of the 2nd International Symposium

on Industrial Oriented Basic Electrochemistry, 1980, p. 1.108 W. C. Sumpter, Y. L. Williams, P. H. Wilken, and B. L. Willoughby, J. Org.

Chem. 14 (1949) 713.109 M. B. Bhargava, A.C. polarography of organic compounds, Ph.D. thesis, Bhopal

University, 1972.110 H. P. Agarwal and M. Saxena, in Proceedings II (Abstracts), J. Heyrovsky

Memorial Congress on Polarography, Prague, Czechoslovakia, 1980.111 H. P. Agarwal, in Proceedings of the International Symposium on Recent Aspects

of Electroanalytical Chemistry and Electrochemical Technology, Panjab University,Chandigarh, Dec. 1982.

112 I. M. Kolthoff and J. J. Lingane, Polarography, Vol. 2, Interscience, New York,1952.

113 B. S. Sheshadri, Surf. Tech. 4(3) (1976) 223.114 S. Sathyanarayana and R. Srinivasan, Br. Corros. J. 12(4) (1977) 217.115 S. Sathyanarayana and R. Srinivasan, Br. Corros. J. 12(4) (1977) 221.116 H. P. Agarwal, Electrochim. Acta 17 (1972) 285.117 W. H. Reinmuth, J. Electroanal. Chem 34 (1972) 297.118 G. C. Barker, A. W. Gardner, and D. C. Sammon, J. Electrochem. Soc. 113(1)

(1966) 1182.119 H. P. Agarwal and P. K. Jain, Extended Abstract, 35th Meeting of ISE, Berkeley,

California, (1984) 936.120 H. P. Agarwal and P. K. Ja in , Bull. Electrochem. 2(2) (1986) 185.

X Rays as Probes of Electrochemical Interfaces

Hector D. AbrunaDepartment of Chemistry, Cornell University, Ithaca, New York 14853

I. INTRODUCTION

The study of the structure of the electrode/electrolyte (or moregenerally the solid/electrolyte) interface1'2 represents a problem ofboth fundamental and practical importance in electrochemistry andmany other interfacial disciplines since its properties greatly affect(and often control) reactivity. Its importance relates to a broadrange of problems including corrosion, catalysis, fuel cells, thepotential and ionic gradients at charged interfaces including col-loids and biological membranes, and many others. This problemhas, until recently, proved very elusive to experimental study dueto the difficulty of applying structure-sensitive techniques to thestudy of surfaces in contact with a condensed phase. Thus, mostof our knowledge of the structure of the electrode/solution interfaceis based on indirect evidence which relies primarily on theoreticalmodels to explain thermodynamic, spectroscopic, and kinetic data.

In recent years,3'4 however, there has been renewed interest inthe study of the electrode/solution interface due in part to thedevelopment of new spectroscopic techniques such as surface-enhanced Raman spectroscopy,5"7 electrochemically modulatedinfrared reflectance spectroscopy and related techniques,8'9 second-harmonic generation,10"12 and others which give information aboutthe identity and orientation of molecular species in the interfacial

265

266 Hector D. Abruna

region. Other techniques such as ellipsometry,13 electroreflect-ance and differential reflectance spectroscopy14have been used tofollow adsorption, film formation, and surface reaction.

All of these techniques, although powerful, do not reveal thestructure and geometric arrangement of atomic species at the inter-face. Thus, in spite of its importance, our knowledge of the structureof the electrode/solution interface at the atomic level is still veryrudimentary.

As mentioned previously, this can be attributed in part to thelack of structure-sensitive techniques that can operate in the pres-ence of a condensed phase. Ultrahigh-vacuum (UHV) surfacespectroscopic techniques such as low-energy electron diffraction(LEED), Auger electron spectroscopy (AES), and others have beenapplied to the study of electrochemical interfaces, and a wealth ofinformation has emerged from these ex situ studies on well-definedelectrode surfaces.15"17 However, the fact that these techniquesrequire the use of UHV precludes their use for in situ studies ofthe electrode/solution interface. In addition, transfer of the elec-trode from the electrolytic medium into UHV introduces the veryserious question of whether the nature of the surface examined exsitu has the same structure as the surface in contact with theelectrolyte and under potential control. Furthermore, any informa-tion on the solution side of the interface is, of necessity, lost.

From the foregoing, it is clear that particle spectroscopies (e.g.,Auger, LEED) are unsuited for in situ studies of the electrode/sol-ution interface. Photons, on the other hand, have very large propa-gation distances, and photons in the X-ray region are suitable probesof the atomic structure of interfacial species. The main difficultywith these measurements has been the low intensities available inconventional X-ray sources. The advent of synchrotron radiation18

has dramatically changed the outlook. As a result, a number ofexperiments based on the use of X rays as probes can now beemployed in the study of electrochemical interfaces. These includeEXAFS (extended X-ray absorption fine structure), XANES (X-rayabsorption near edge structure), X-ray standing waves, and surfacediffraction.

In addition, recent instrumental developments have made itpossible to perform kinetic measurements on the millisecond timescale.

X Rays as Probes of Electrochemical Interfaces 267

Clearly, the application of these techniques to the study ofelectrochemical interfaces will allow a much deeper understandingand correlation of structure/reactivity patterns.

In this chapter, I will try to present an introduction to thesevarious techniques with emphasis on EXAFS and X-ray standingwaves and their application to the study of electrochemical inter-faces. Each technique will be treated from theoretical and experi-mental points of view, and selected examples from the literaturewill be employed to illustrate their application to the study ofelectrochemical interfaces.

II. X RAYS AND THEIR GENERATION

X rays comprise that portion of the electromagnetic spectrum whichlies between ultraviolet and gamma rays. The range of wavelengthsis typically from about 0.01 to 100 A. Because of their very shortwavelengths, X rays are powerful probes of atomic structure.

X rays have been traditionally produced by impinging anelectron beam (at energies from about 20 to 50 keV) onto a targetmaterial such as copper, molybdenum, or tungsten. The suddendeceleration of the electron beam by the target material gives riseto a broad spectrum of emission termed bremsstrahlung. The energyof the emitted X rays is given by

A =(hc)/(eV) = 12,400/ V (1)

where A is in angstroms, and V is the accelerating voltage. Theminimum wavelength of emission is obtained when all of theelectron energy is converted to an X-ray photon. The intensity andwavelength distribution of this bremsstrahlung are both a functionof the accelerating voltage, the current, and the target material.Figure 1 shows typical emission envelopes for a tungsten target atvarious accelerating voltages.

When the accelerating voltage reaches a specific value (depen-dent on the nature of the target material), the electrons from thebeam are capable of knocking out core-level electrons from thetarget material, thus giving rise to core vacancies. These are quicklyfilled by electrons in upper levels and this results in the emissionof X-ray photons of characteristic energies which depend on the


0.4 0.6 0.8 1.0Wavelength A

Figure 1. Wavelength distribution of the radi-ation from an X-ray tube with a tungsten target.The numbers above each curve refer to theaccelerating voltages (in keV) employed.

nature of the target. The energies and intensities of characteristiclines depend on the nature of the core hole generated (e.g., K- orL-shell vacancy) as well as the level from which the electron thatfills the vacancy originates. Figure 2 presents a schematic of someof the more important X-ray emission lines. These so-called charac-teristic lines are much more intense than the bremsstrahlungemission and are superimposed on the latter as very sharp emissions.

N

M

K

(32

p3

K series L series

4,f,7/2

H

^ 4,d.3/2^4 ,p ,3 /2

P. 1/2'4.S.1/2

- 3.d.5/2-3,d,3/2- 3.p,3/2- 3,p.l/2" 3.S.1/2

2.P.3/2

2.p,l/2

2.S.1/2

l.s. 1/2

Figure 2. Partial energy level diagram depicting part of theK- and L-series lines.


CDC

Figure 3. Wavelength distribution of theradiation emitted from a molybdenum targetX-ray tube operated at 35 keV.

0 . 2 0 . 4 0 .6 0 . 8 1.0

Wavelength, A

Figure 3 shows an example of characteristic emissions from a Motarget.

A variant of the typical X-ray tube described above is therotating anode, which is capable of generating much higherintensities. Although rotating anode sources can and have beenused in EXAFS experiments, the intensities are of such magnitudethat data acquisition for extended periods of time is required andtheir application is furthermore limited to bulk samples.

An alternative and the most generally employed source of Xrays for EXAFS experiments is that obtained from synchrotronsources based on electron (or positron) storage rings.

III. SYNCHROTRON RADIATION AND ITS ORIGIN

No single development has influenced the field of EXAFS spectros-copy more than the development of synchrotron radiation sources,particularly those based on electron (or positron) storage rings.These provide a continuum of photon energies at intensities thatcan be from 103 to 106 higher than those obtained with X-ray tubes,


thus dramatically decreasing data acquisition times as well asmaking other experiments feasible. We will consider very brieflythe fundamental aspects of synchrotron radiation.

The most attractive features of synchrotron radiation have beensummarized by Winick19 as:

a. High intensityb. Broad spectral rangec. High polarizationd. Pulsed time structuree. Natural collimationf. Small-source-spot sizeg. StabilityWe know from Maxwell's equations that whenever a charged

particle undergoes acceleration, electromagnetic waves are gener-ated. An electron in a circular orbit experiences an accelerationtoward the center of the orbit and as a result emits radiation in anaxis perpendicular to the motion.

As pointed out by Tomboulian and Hartman,20 one can distin-guish two general cases and these relate to whether or not anelectron is orbiting at relativistic speeds. At nonrelativistic speeds(v « c), the pattern of the emitted radiation resembles a doughnut(Fig. 4A). However, at relativistic speeds (v — c), the radiationpattern is highly peaked (Fig. 4B) and one can think of an orbitingsearchlight as a good approximation. This natural collimation effectgives rise to very high fluxes on small targets.

Since the accelerated electrons are constantly emitting radi-ation, we need to resupply the energy if they are to remain in orbit.This is typically done with high-power RF cavities. The spectraldistribution of synchrotron radiation depends on a number offactors, and two that are particularly important are the electronenergy E [expressed in GeV (109eV)] and the bending radius R(in meters) of the orbit. A parameter that relates these is the so-calledcritical wavelength given by

Ac (in angstroms) = 5.6R/E3

In general, useful fluxes are obtained at wavelengths down toAc/4 although in this region the output decreases precipitously.Figure 5 presents some flux curves for the Cornell High EnergySynchrotron Source (CHESS) operated at various energies. A


CentripetalAcceleration, A

A v«c

B. v=c

RF Cavity

Figure 4. Radiation pattern emitted from orbiting electrons whenthe velocities are much smaller than (A) or comparable to (B)the speed of light. (Adapted from Ref. 20.)

closely related parameter is the critical energy and this representsthe midpoint of the radiated power. That is, half of the radiatedpower is above and below this energy.

The fact that Ac is proportional to the bending radius is usedin so-called insertion devices such as wiggler and undulator mag-nets, t Although a description of these is beyond the scope of thischapter, the basic principle behind these is to make the electronbeam undergo sharp serpentine motions (thereby having a veryshort radius of curvature). The net effect is to increase the flux andthe critical energy (see topmost curve in Fig. 5).

Another very important property of synchrotron radiation isits very high degree of polarization. The radiation is predominantlypolarized with the electric field vector parallel to the acceleration

t For an introductory discussion, see Ref. 21.

272

Q

< 10CD

15

§

O

CO

oXa.

ICT

70 MILLIAMPEREOPERATION

, ,,,,.110" 10" I01

ENERGY(KeV)

Figure 5. Photon flux as a function of energy for the Cornell High Ei-chrotron Source (CHESS) operated at various accelerating voltages. Thecurve is the radiation profile from a 6 pole wiggler magnet. (Figure courtedLaboratory for Nuclear Studies at Cornell University.)

vector. Thus, in the plane of the orbit, the radiation is lOOvopolarized. Elliptical polarization can be obtained by going awayfrom the plane. However, intensities also decrease significantly.

Finally, the pulsed time structure, useful for kinetic studies,arises from the fact that in a storage ring the electrons are orbitingin bunches. The specific energy, the number of bunches, and thecircumference of the storage ring dictate the exact time structure.


A. Pre-edgeB. EdgeC. Near edgeD. EXAFS

EnergyFigure 6. Figure depicting the various regions in an X-rayabsorption spectrum.

IV. INTRODUCTION TO EXAFS AND X-RAYABSORPTION SPECTROSCOPY

Extended X-ray absorption fine structuret (EXAFS) refers to themodulation in the X-ray absorption coefficient beyond an absorp-tion edge. Such modulations can extend up to about 1000 eV beyondthe edge and have a magnitude of typically less than 20% of theedge jump.

In order to gain a basic grasp of the EXAFS phenomenon, itis perhaps better to begin by considering the general featuresobserved in an X-ray absorption spectrum. Analogous to obtainingthe UV-vis spectrum of a molecule, when we perform an X-rayabsorption experiment we measure the absorbance of a sample(typically expressed as an absorption coefficient /JL ) as we vary theincident photon energy (Fig. 6). In general, as we increase the

t There have been numerous reviews of EXAFS over the last ten years. A selectednumber of leading references are given as Refs. 22-30.

274 Hector D. Abrufia

energy there is a monotonic decrease in the absorption coefficient.However, when the incident X-ray energy is enough to photoionizea core-level electron, there is an abrupt increase in the absorptioncoefficient and this is termed an absorption edge. There are absorp-tion edges that correspond to the various atomic shells and sub-shells. For example, a given atom will have one K absorption edge,three L edges, five M edges, etc., with the energies decreasing inthe expected order K> L> M (see Fig. 2).

As we continue to scan to higher energies beyond the edge,we can encounter two different situations depending on whetheror not the species that we are investigating has near neighbors(typically at 5 A or closer). If there are no near neighbors, theabsorption coefficient will again decrease in a monotonic fashionuntil its next absorption edge or that of another element presentin the sample is encountered.

In the presence of one or more near neighbors, there will bemodulations in the absorption coefficient as we scan out to energiesabout 1000 eV beyond the edge. The modulations present at energiesfrom about 40 eV to 1000 eV beyond the edge are termed EXAFS.

The phenomenon of EXAFS has been known since the 1930sthrough the work of Kronig31 who stated that the oscillations aredue to the modification of the final state of the photoelectron bynear neighbors. Although up to the 1960s there was controversy asto whether short- or long-range order was responsible for the effect,it is now universally accepted that it is the presence of short-rangeorder that gives rise to the EXAFS oscillations.

The absorption coefficient is a measure of the probability thata given X-ray photon will be absorbed and, therefore, depends onthe initial and final states of the electron. The initial state is verywell defined as it corresponds to the localized core level. The finalstate is represented by the photoionized electron, which can bevisualized as an outgoing photoelectron wave that originates at thecenter of the absorbing atom and that, for S core levels, has sphericalsymmetry. In the presence of near neighbors, this photoelectronwave can be backscattered (Fig. 7) so that the final state will begiven by the sum of the outgoing and backscattered waves. It isthe interference (recall that we are changing the wavelength of thephotoelectron wave as we scan the energy) between the outgoingand backscattered waves that gives rise to the EXAFS oscillations.


X-ray

Figure 7. Depiction of origin of EXAFS. An X-rayphoton is absorbed by A, resulting in the photoionizationof a core-level electron represented as an outgoing (-» )photoelectron wave which is backscattered ( <- ) by anear neighbor, B.

Thus, in a very simple approximation we can see that thefrequency of the EXAFS oscillations will depend on the distancebetween the absorber and its near neighbors, whereas the amplitudeof the oscillations will depend on the numbers and type of neighborsas well as their distance from the absorber. From an analysis ofthe EXAFS, one should be able to obtain information on near-neighbor distances, numbers, and types. A further advantage ofEXAFS is that it can be applied to all forms of matter—solids,liquids, and gases—and that in the case of solids, single crystalsare not required. In addition, one can focus on the environmentaround a particular element by employing X-ray energies aroundan absorption edge of the element of interest without regard forthe rest of the elements in the sample.

The simple description of EXAFS given above is based on theso-called single-electron, single-scattering formalism.3237 Here it isassumed that for sufficiently high energies of the photoelectrons,one can make the plane wave approximation and, in addition, onlysingle backscattering events will be important. This is the reasonwhy the EXAFS is typically considered for energies higher than40 eV beyond the edge since in this energy region the above approxi-mations hold well.

Hector D.

9250 9500Energy, eV

B 9000 9200 9400 9600Energy, eV

Figure 8. Absorption spectrum for (A) CuSO4 and (B) Cufoil.


In addition to the EXAFS region, Fig. 6 shows that there arealso three other regions: the pre-edge, edge, and near-edge regions.

Below or near the edge, there can be absorption peaks due toexcitations to bound states. In some cases these can be so intenseso as to dominate the edge region. For example, Fig. 8A shows anEXAFS spectrum of CuSO4, and the appearance of a very intenseand sharp transition near the edge (often called "white line") isimmediately apparent. This peak is associated with a Is -» 3d local-ized transition. Because of the nature of the transitions, the pre-edgeregion is rich in information pertaining to the energetic location oforbitals, site symmetry, and electronic configuration.

The position of the edge contains information concerning theeffective charge of the absorbing atom. Thus, its location can becorrelated with the oxidation state of the absorber in a way that isanalogous to XPS measurements. For example in Figs. 8A (coppersulfate) and 8B (copper foil) the edge position for Cu is shifted tolower energies (by about 2 eV), consistent with the change in oxida-tion state from 4-2 to 0. Such shifts can be very diagnostic in theassignment of oxidation states.

Finally, we consider the near-edge region generally calledXANES (X-ray absorption near edge structure). [Note: when usingUV or soft X rays this region is generally called near-edge X-rayabsorption fine structure (NEXAFS).] In this region of the spectrumthe photoelectron wave has very small momentum, and as a result,the plane-wave as well as the single-electron single-scatteringapproximations are no longer valid. Instead, one must consider aspherical photoelectron wave as well as the effects of multiplescattering. Because of this factor, the photoelectron wave cansample much of the environment around the absorber. This regionof the spectrum is thus very rich in structure (see Fig. 6); however,the theoretical modeling is very complex. However, increased atten-tion is being given by theoreticians, and in the not so distant future,we will be able to obtain much information from this region.

V. THEORY OF EXAFS

We will consider the theoretical description of EXAFS based onthe single-scattering short-range order formalism. The EXAFS can


be expressed as the normalized modulation of the absorptioncoefficient as a function of energy:

X(E) = [fjL(E) - no(E)]/fji0(E) (2)

Here /JL(E) is the total absorption coefficient at energy E and /x0

is the smooth atomlike absorption coefficient. In order to be ableto extract structural information from the EXAFS, we need to gofrom the energy to the wave vector form using the formulation:

-E0)/fi (3)

where Eo is defined as the threshold energy (typically close but notnecessarily coincident with the energy at the absorption edge) (videinfra). In wave vector form, the EXAFS can be expressed as:

(4)

Here I have divided the expression into two terms whichcorrespond to an amplitude factor (a) and an oscillatory component(b). I will now consider each of these terms in some detail.

1. Amplitude Term

The amplitude term

- ^ NjSAQFjik) exp-<^2 exp-2r;/A(k) (5)

can be subdivided into two main components: a maximum ampli-tude term and an amplitude reduction factor.

For a given shell, the maximum amplitude is given by theproduct of the number (N) of the j type of scatterer times itsrespective backscattering amplitude, Fj(k). This maximum ampli-tude is then reduced by a series of amplitude reduction factorswhich are considered below.


(i) Many-Body Effects

The Si(k) term takes into account amplitude reduction due tomany-body effects and includes losses in the photoelectron energydue to electron shake-up (excitation of other electrons in the absor-ber) or shake-off (ionization of low-binding-energy electrons in theabsorber) processes.

(ii) Thermal Vibrations and Static Disorder

Photoionization and therefore EXAFS takes place on a timescale that is much shorter than that of atomic motions so theexperiment samples an average configuration of the neighborsaround the absorber. Therefore, we need to consider the effects ofthermal vibration and static disorder, both of which will have theeffect of reducing the EXAFS amplitude. These effects are con-sidered in the so-called Debye-Waller factor which is included as

exp— V (6)

This can be separated into static disorder and thermal vibrationalcomponents:

<72j = O"vib + Crs tat ( 7 )

It is generally assumed that the disorder can be representedby a symmetric Gaussian-type pair distribution function and thatthe thermal vibration will be harmonic in nature.

Experimentally, one can only measure a total o\ However, thetwo contributions can be separated by performing a temperaturedependence study of cr or by having an a priori knowledge of C7vib

from vibrational spectroscopy.Whereas there is little that one can do to overcome the effects

of static disorder, the effects of thermal vibration can be significantlydecreased by performing experiments at low temperatures, and, infact, many solid samples are typically run at liquid nitrogen tem-peratures just to minimize such effects. An example of the effectof thermal vibration can be ascertained in Fig. 8 A, where the EXAFSamplitude decreases precipitously due to the large vibrationalamplitude of the Cu—O bond. In general, failure to consider theeffects of thermal vibration and static disorder can result in large


errors in the determination of coordination numbers (number ofneighbors) and interatomic distances.38"40

(iii) Inelastic Losses

Photoelectrons that experience inelastic losses will not havethe appropriate wave vector to contribute to the interference pro-cess. Such losses are taken into account by an exponential dampingfactor,

exp-2r/*<*> (8)

where r is the interatomic distance and \(k) is the electron meanfree path. This damping term limits the range of photoelectrons inthe energy region of interest, and is in part responsible for theshort-range description of the EXAFS phenomenon.

In general, it is the product of all of the above-mentionedfactors that will give rise to the observed amplitude.

2. Oscillatory Term

The oscillatory part of the EXAFS takes into account the relativephases between the outgoing and backscattered waves and, as aresult, includes the interatomic distance between absorber andscatterer. The phase shifts can be visualized by considering that theoutgoing photoelectron wave will experience the absorbing atom'sphase shift S,(/c) on its outward trajectory. It will then experiencethe near neighbor's phase shift as(k) upon scattering and theabsorbing atom's phase shift once again. There is in addition a 2krterm which represents twice the interatomic distance between absor-ber and scatterer. Thus, the oscillatory part of the EXAFS is givenby

sin[2fcr + 28I.(fc) + as(fc)] (9)

Since the accuracy of the determination of interatomic distancesdepends largely on the appropriate determination of the relativephases, a great deal of attention has been given to this aspect. Theproblem can be simply stated as follows: when the outgoing photo-electron wave is backscattered by a near neighbor, it is the neigh-bor's electron cloud and not its nucleus that is largely responsiblefor the scattering. As a result, the distance obtained will always be


necessarily shorter than the true interatomic distance and thereforeneeds to be corrected. The correction can be achieved by ab initiocalculation of the phases involved or alternatively it can be deter-mined experimentally through the use of model compounds. Amore thorough discussion of phase correction will be given furtheron.

3. Data Analysis

The basic intent behind any EXAFS data analysis is to be able toextract information related to interatomic distances, numbers, andtypes of backscattering neighbors. In order to accomplish this, thereare a number of steps involved in the data analysis, and theseinclude:

i. Background subtraction and normalizationii. Conversion to wave vector form

iii. k weighingiv. Fourier transforming and filteringv. Fitting for phasevi. Fitting for amplitude

(i) Background Subtraction and Normalization

The first step in the analysis is the background subtraction.However, since the smooth or atomlike absorption (that is, theabsorption for an isolated atom) is in general not available, it isgenerally assumed that the smooth part of JJL(E) is a good approxi-mation to fJL0(E).

Background removal routines typically employ polynomialsplines of some order (typically second or third order). These aredefined over a series of intervals with the constraint that the functionand a stipulated number of derivatives be continuous at the intersec-tion between intervals. In addition, the observed EXAFS oscilla-tions need to be normalized to a single-atom value and this isgenerally done by normalizing the data to the edge jump.

{ii) Conversion to Wave Vector Form

In order to extract structural information, we need to convertthe EXAFS expressed in terms of energy to wave vector form. To


do this, however, we need to choose a value for the threshold energyEo. This is important because of its effect on the phase of theEXAFS oscillations, expecially at low k values. The difficulty indetermining Eo arises from the fact that there is no way of identifyingan edge feature with Eo. A procedure proposed by Lee and Beni41

is to treat Eo as an adjustable parameter in the data analysis whosevalue is changed until the observed phase shifts are in good agree-ment with theoretical values. When good model compounds areavailable, the use of a fixed value for Eo works well.42"44 However,in many cases it is difficult to assess a priori whether a given materialis a good model compound.

Figure 9 A depicts data after background subtraction, normaliz-ation, and conversion to wave vector form.

®

I I I l v I I I I I I I

xk "3

Vi i i i i i i i

FF

CF

tude

200

300

C

- ;

•

w

k ( A 1 ) '

Ji

d

FT

)

|

- w MA"1) l fc - r ( A ) ' 9

Figure 9. Data reduction and data analysis in EXAFS spectroscopy. (A) EXAFSspectrum x(k) versus k after background removal. (B) The solid curve is the weightedEXAFS spectrum k3x(k) versus k (after multiplying x(k) by k2). The dashed curverepresents an attempt to fit the data with a two-distance model by the curve-fitting(CF) technique. (C) Fourier transformation (FT) of the weighted EXAFS spectrumin momentum (k) space into the radial distribution function p3(r') versus r' indistance space. The dashed curve is the window function used to filter the majorpeak in Fourier filtering (FF). (D) Fourier-filtered EXAFS spectrum k3x'(k) versusk (solid curve) of the major peak in (C) after back-transforming into k space. Thedashed curve attempts to fit the filtered data with a single-distance model. (FromRef. 25, with permission.)


(iii) k Weighing

Once the data have been transformed to wave vector form,they are usually multiplied by some power of fc, typically, k2 or fc3.Such a factor cancels the 1/fc factor in Eq. (4) as well as the 1/fc2

dependence of the backscattering amplitude at large values of fc.Figure 9B depicts multiplication by fc3.

This step is important in that it prevents the large-amplitudeoscillations (typically present at low fc) from dominating over thesmaller ones (typically at high fc). This is critical since the determi-nation of interatomic distances depends on the frequency and notthe amplitude of the oscillations. Other approaches having the sameeffect have also been employed.37'47

(iv) Fourier Transforming and Filtering

Examination of the EXAFS formulation in wave vector formreveals that it consists of a sum of sinusoids with phase andamplitude. Sayers et al.32 were the first to recognize the fact that aFourier transform of the EXAFS from wave vector space (fc ordirect space) to frequency space (r) yields a function that is qualita-tively similar to a radial distribution function and is given by:

1 ' fcn*(fc) exp(2lkr) dk (10)

Such a function exhibits peaks (Fig. 9C) that correspond to inter-atomic distances but are shifted to smaller values (recall the distancecorrection mentioned above). This finding was a major break-through in the analysis of EXAFS data since it allowed readyvisualization. However, because of the shift to shorter distancesand the effects of truncation, such an approach is generally notemployed for accurate distance determination. This approach,however, allows for the use of Fourier filtering techniques whichmake possible the isolation of individual coordination shells (thedashed line in Fig. 9C represents a Fourier filtering window thatisolates the first coordination shell). After Fourier filtering, the datais back-transformed to k space (Fig. 9D), where it is fitted foramplitude and phase. The basic principle behind the curve-fittinganalysis is to employ a parameterized function that will model the


observed EXAFS and then the various parameters are adjusteduntil the fit is optimized.

(v) Fitting for Phase

Accurate distance determinations depend critically on theaccurate determination of phase shifts. The two general approachesto this problem are theoretical and empirical determination. Thetwo main approaches to the theoretical calculations of phase shiftshave been the Hartree-Fock36'48 (HF) and Hartree-Fock-Slater47'49

(HFS) methods. The first treatment begins with tabulated atomicwave functions, and the HF equation of the atom plus the externalelectron is solved by iteration. In the HFS (or local density func-tional) approach, the atom is replaced by an electron gas of varyingdensity. In general, both of these approaches are too involved forgeneral use. Teo and Lee50 used the theoretical approach of Leeand Beni37 to calculate and tabulate theoretical phase shifts for themajority of elements. Use of these theoretical phase shifts requiresthe use of an adjustable Eo in the data analysis (vide supra).

The second, and more commonly employed, approach is theempirical one based on the use of model compounds and theconcept of phase transferability. This approach consists of employ-ing a compound of known structure and which has the sameabsorber/backscatter combination as that of the material of interest.The EXAFS spectrum of the known compound (typically calledmodel compound) is obtained and the oscillatory part of the EXAFSis fitted to the expression in Eq. (9). Since r is known in this case,the phase shift can be determined. Typically, the phase shift isparameterized as a quadratic expression. Implicit in this treatmentis the applicability of phase transferability, meaning that for a givenabsorber/scatterer combination, the phase shifts can be transferredto any compound with the same absorber/scatterer combinationwithout regard to chemical effects such as ionicity or covalency ofthe bonds involved. This is based on the idea that at sufficientlyhigh kinetic energies for the photoelectron (e.g., about 50 eV abovethreshold), the EXAFS scattering processes are largely dominatedby core electrons and thus the measured phase shifts are insensitiveto chemical effects. Thus, determination of the phase shift for anabsorber/scatterer pair in a system of known r allows for the


determination of the distance in an unknown having the same atompair. This was thoroughly demonstrated by Citrin et al. in a studyof germanium compounds.51

With good-quality data and appropriately determined phaseshifts, distance determination by EXAFS are typically good to±0.01 A and sometimes better in favorable cases.

(vi) Fitting for Amplitude

Amplitude fitting is employed in order to determine the typesand numbers of backscattering atoms around a given absorber. Theproblem can be divided in two parts, namely, the identification ofthe types of backscatterers and the determination of their numbers.In the first case, if we have no clue as to the probable nature ofthe backscatterer, identification is difficult, especially among atomsthat have similar atomic number, e.g., N and O. This is becausethe backscattering amplitudes are not a very strong function ofatomic number. For example, Fig. 10 shows the backscatteringamplitude for various elements as calculated by Teo and Lee.50 It

Figure 10. Backscattering amplitudeas a function of wave vector for C,Si, Ge, Sn, and Pb. (Adapted fromRef. 50.)

1.2

1.0

0.8

0.6

0.4 -

0.2

BackscatteringAmplitude; F(k)

Pb

Sn


is clear that for Si and C the differences are small and so these twoelements would be difficult to differentiate. However, for the caseof a heavy-atom backscatterer, there is typically a resonance in thebackscattering amplitude (this is analogous to the Ramsauer-Townsend effect) so that differentiation between light and heavybackscatters can be readily made.

It should be mentioned that when a peak from a Fouriertransform is filtered and back-transformed to k space, the enveloperepresents the backscattering amplitude for the near neighborinvolved.

If the identity of the backscatterer is known, then the interestis in determining the number of near neighbors. In this case, oneneeds to compare the amplitude of the EXAFS of the material ofinterest (unknown) to that for a compound of known coordinationnumber and structure. However, unlike transferability of phase,which is generally regarded as an excellent approximation, thetransferability of amplitude is not. This is because there are manyfactors that affect the amplitude and, except for the case of modelcompounds of very similar structures, these will not necessarily(and often will not) be the same. As a result, determination ofcoordination numbers (near neighbors) is usually no better than±20%.

VI. SURFACE EXAFS AND POLARIZATION STUDIES

EXAFS is fundamentally a bulk technique due to the high penetra-tion of high-energy X rays. In order to make it surface sensitive,one can take one of two general approaches. In the first case, ifone knows a priori that the specific element of interest is presentonly at the surface, then a conventional EXAFS measurement willnecessarily give surface information. Alternatively, one can employdetection techniques or geometries such that the detected signalarises predominantly from the surface or near-surface regions.5256

These include electron detection and operating at angles ofincidence that are below the critical angle of the particular material.These aspects will be discussed further in the experimental section.In addition, there have been a number of reviews of this matter,with Citrin's56 being the most comprehensive.


For studies on single-crystal surfaces, surface EXAFS offersan additional experimental handle, namely, the polarization depen-dence of the signal. As mentioned previously, synchrotron radiationis highly polarized with the plane of polarization lying in the planeof orbit. Since only those bonds whose interatomic vector lies inthe plane of polarization of the beam will contribute to the observedEXAFS, polarization dependence studies can provide a wealth ofinformation concerning adsorption sites and near-neighborgeometries. This is especially significant since it is very difficult (ifat all feasible) to obtain this type of information from conventionalEXAFS measurements.

For a near-neighbor shell of atoms (Nt) whose interatomicvector with the absorber makes some angle 0, relative to the planeof polarization, one can relate the effective coordination number(Nf) and the true coordination number through57

NT = 3 t cos2 8j (11)j

Polarization-dependent surface EXAFS measurements haveprovided some of the best-defined characterizations of adsorbatestructures.

VII. EXPERIMENTAL ASPECTS

1. Synchrotron Sources

There are a number of experimental factors to be considered in asurface EXAFS experiment. First of all, one needs access to asynchrotron source (for the reasons previously mentioned) withsignificant flux in the hard X-ray region. In the United States, threesuch facilities exist and these are:

a. Cornell High Energy Synchrotron Source (CHESS)b. Stanford Synchrotron Radiation Laboratory (SSRL)c. National Synchrotron Light Source (NSLS) at Brookhaven

National Laboratory[It should be mentioned that another synchrotron source, theAdvanced Photon Source (APS) will be built at the ArgonneNational Laboratory and should be operational in the mid-1990s.]

288 Hector D. Abrana

Double crystal monochromator

Figure 11. Schematic diagram of a transmission EXAFS experiment. /0 and Ixrefer to the incident and transmitted intensities, respectively.

In addition, one needs to pay close attention to detection schemesand the design of specialized equipment. Of these, I will focus ondetection schemes at this time and will defer the discussion ofdesign of systems for electrochemical measurements to later sectionsdealing with specific experiments.

2. Detection

Detection schemes are usually dictated by the concentration of thespecies of interest, the nature of the sample, and the experiment.All of these aspects have been considered in great detail by Lee etal}6 so I will only mention some of the most important aspects. Ingeneral, the measurement of any parameter that can be related tothe absorption coefficient can be employed in a detection scheme.

(i) Transmission

For concentrated or bulk samples a transmission experimentis both the simplest and the most effective. In essence, one measuresthe X-ray intensities incident and transmitted through a thin anduniform film of the material. Careful analysis of signal-to-noiseratio considerations indicates that optimal results are obtained whenthe sample thickness is of the order of 2.5 absorption lengths. Sincein this case a simple Beer's law applies, the data are usually plottedas ln(///0) versus E. The intensities are measured using ionizationchambers in conjunction with high-gain electrometers (see Fig. 11).

(li) Fluorescence

For dilute samples, where absorption of the X-ray beam bythe element of interest would be very low, a transmission geometry


cannot be employed. Instead, fluorescence detection is the methodof choice.58"60 Fluorescence can be employed as a detection modebecause the characteristic X-ray fluorescence intensity depends onthe number of core holes generated, which, in turn, depends onthe absorption coefficient. Fluorescence detection is much moresensitive than transmission because one is measuring the signalover an essentially zero background. Typically, the incident andfluorescent beams impinge and leave the surface at 45° (the X-raybeam and the detector are, of course, at 90°). The detector can beeither an ionization chamber or a solid state detector. The formeris much simpler to implement whereas the latter gives the bestresolution. A filter (to minimize the contributions from elastic andCompton scattering) and soller slits are typically placed in frontof the detector. The filter material is chosen so as to have anabsorption edge that falls between the excitation energy and theenergy of the characteristic X-ray photon from the element ofinterest. Thus, the filter is generally made from the Z — 1 or Z — 2element, where Z represents the atomic number of the element ofinterest. For example, for CuXa detection, a nickel filter isemployed. In this way the characteristic fluorescence is only slightlyattenuated whereas both the elastic and Compton intensities aregreatly reduced. There is the problem, however, that often the Kp

emission from the filter material is energetically very close to theKa emission from the element of interest.

A solid state detector, either Si(Li) (lithium-drifted silicon) orintrinsic germanium, offers the ability to discriminate on the basisof energy. The resolution can be as good as 150 eV, but it degradessomewhat with increasing detector area. The main problem with asolid state detector is that it has a limited count rate (approximately15kcps). Since it "sees" a wide range of photon energies fromwhich one chooses the region of interest, it can take significantamounts of time to obtain adequate statistics. In addition, the costof solid state detectors and associated electronics is much higherthan that of ion chambers.

(iii) Electron Yield

Electron yield—Auger, partial, or total—can also be employedas detection means since again it depends on the generation of core


holes.53'54 Because of the very small mean free paths of electrons,electron yield detection is very well suited for surface EXAFSmeasurements. However, for this very same reason, in situ studiesof electrochemical interfaces are precluded. Details of electron yieldEXAFS have been discussed by a number of authors.52"56

(it?) Reflection

When one has a planar surface, one can take advantage ofX-ray optics to enhance surface sensitivity.61'62 The most importantis specular or mirror reflection, and this is due to the fact that atX-ray energies the index of refraction of matter is slightly less thanone and is given by:

n = 1 - 8 - ip (12)

with

8 = (l/27r)(e2/mc2)(N0p/A)[Z + A/']A2

j8 = A/X/4TT

where e2/ me2 is the classical electron radius, (Nop)/A is the numberof atoms per unit volume, JV0 is Avogadro's number, p is the density,A is the atomic weight, Z is the atomic number, and A is thewavelength. The term [Z + A/'] is the real part of the scatteringfactor (including the so-called dispersion term/') and is essentiallyequal to Z The imaginary part of the index of refraction, /3, isrelated to absorption, where /JL is the linear absorption coefficient.Considering an X-ray beam incident on a smooth surface and Snell'slaw, one obtains that the critical angle for total reflection is given by:

0crit = V26 (13)

8 is of the order of 10~5 and 0crit is typically of the order of a fewmilliradians (recall that 17.4millirad equals 1°). Thus, as long asthe beam is incident below this critical angle, it is totally reflectedand only an evanescent wave penetrates the substrate. This has twovery important consequences. First of all, the penetration depth isof the order of 20 A and thus one can significantly discriminate infavor of a surface-contained material. Compton and elastic scatter-ing are also minimized. In addition, the reflection enhances thelocal intensity by as much as a factor of 4 as well as the effective


"path length." All of these factors combined serve to enhance thesurface sensitivity of the technique and, when combined with solidstate fluorescence detection, allow for the detection of less thanmonolayer amounts of material.60

(v) Dispersive Arrangements

Up to this point, the experimental techniques described werebased on the use of monochromator crystals and following thesignal of interest as the energy of the incident photon was scanned.This conventional mode of operation suffers from the fact that onlya very narrow range of wavelengths is used at a given time, so thatobtaining a full spectrum requires a significant amount of time,thus precluding real-time kinetic studies of all but the slowest ofreactions. The alternative is to employ a dispersive arrangement63"68

where, by the use of focusing optics, a range of energies can bebrought to focus on a spot. The exact energy spread will dependon the specific optical elements employed but a range of 500 to600 eV represents a realistic value. Coupling this with a photodiodearray allows for the simultaneous use of the full range ofwavelengths, and thus a spectrum can be obtained in periods asshort as milliseconds rather than minutes. This is of great sig-nificance because a number of relevant dynamic processes takeplace on this time scale. The application of this approach toelectrochemical studies will be discussed in a later section.

VIII. EXAFS STUDIES OF ELECTROCHEMICAL SYSTEMS

In order to simplify the discussion, the EXAFS studies onelectrochemical systems reported to date will be divided into thecategories listed below:

1. Oxide films2. Monolayers3. Adsorption4. Spectroelectrochemistry

Further distinction will be made as to whether a given study was


conducted in situ or ex situ, where by in situ I will mean that theelectrode is both in contact with an electrolyte solution and underpotential control.

1. Oxide Films

The study of passive films on electrode surfaces is an area of greatfundamental and practical relevance. Despite decades of intensiveinvestigations, there still exists a great deal of controversy as to theexact structural nature of passive films, especially when they areformed in the presence or absence of glass-forming additives suchas chromium.

One of the main sources of controversy is that many of thestructural studies performed have been on dried films and, aspointed out by O'Grady,69 this results in the determination of thestructure of dehydrated films whose structure can be significantlydifferent from that of hydrated ones.

The use of surface EXAFS in the study of passive films rep-resents a natural application of the technique and, in fact, thestudies by Kruger and co-workers7073 on the passive film on ironrepresent the first reported.

In their earliest report, Kruger and co-workers vacuumdeposited iron films onto glass slides and subsequently oxidizedthe films in either nitrite or chromate solution. They obtained theEXAFS spectra for the oxidized films employing a photocathodeionization chamber and compared these with spectra for y-FeO(OH), y-Fe2O3, and Fe3O4. Although these studies were notin situ, they did not require evacuation of the samples and thereforerepresent an intermediate situation between dehydrated film andin situ experiments. The spectra for Fe, Fe3O4, and the nitrite- andchromate-generated passive films are shown in Figs. 12A and 12B.The near-edge region for the nitrite-generated film showed evidenceof an enhancement, similar to that observed for Fe3O4, indicativeof an increase in the density of available final states with p character.Such an enhancement is absent in the chromate-formed films. Theseresults point to a more covalent bonding in the chromate versusthe nitrite passivated films.

Upon Fourier transforming of the data, two peaks correspond-ing to Fe—O and Fe—Fe distances are obtained (Fig. 13). The

=3

o

Figure 12. Absorption spectra of(A) Fe and Fe3O4 and (B) ironfilms after treatment in chromateand nitrite solutions. (From Ref.71, with permission.)

Chromate treated Nitrite treated

-J iEnergy

peaks in the Fourier transform of the chromate-generated film aremuch less well resolved than those for the nitrite films, and this isascribed to the presence of a glassy structure associated with thechromium. From a comparison of the edge jump for Fe and Cr,the authors estimate that the films have about 12% Cr.

These results point to the importance of hydration effects onthe structure of passive films on iron. However, these results wereobtained ex situ and therefore are subject to some uncertainty.

Most recently, these same authors have employed an in situcell (Fig. 14) for carrying out these experiments. Again they studiednitrite- and chromate-passivated films. The results obtained in thiscase are quite different from the ex situ measurements. In addition,


Nitrite passivatedChromate passivated

( 1 1 1

(A)

Figure 13. Fourier transform of the EXAFSfor iron films after treatment in chromateand nitrite solutions. (From Ref. 71, withpermission.)

the spectral features of the in situ measurements for both nitrite-and chromate-passivated films are quite similar and these are mosteasily ascertained from the derivative of the near-edge region (Fig.15).

The Fourier transforms for both films are again quite similar,but as for the ex situ measurements, the chromate-passivated filmsappear to have a more glassy structure. It should be mentionedthat these studies employed a rather limited data range which makesspectral differentiation difficult.

Thin film

IT1 INfr/zxyilK

/Connectionfrom the back

X ^ ^ ^ - Reference^*~—*\ electrode\ \\ ^ J L - — P t counter

*] T electrode

' J/ ^ Teflon outery^/ case

Inlet andoutlet

Figure 14. In situ cell for performing EXAFS studies onpassivated iron films. (From Ref. 72, with permission.)


I K \ I

7100 7120 7140

Energy (eV)

Figure 15. Derivatives of the near-edgeregion of spectra for nitrite- (A) and chro-mate- (B) passivated iron films under in situ _• . . _. .( + ) and ex situ ( • ) conditions. (From Ref.72, with permission.) Energy (eV)

Hoffman and Kordesch74'75 have presented a series of studieson the passive films on iron with particular attention to cell design.They have employed a so-called bag cell that allows for the in situpassivation and/or cathodic protection of the iron films. These weredeposited onto gold films deposited on Melinex.

In addition, they employed a setup where the working electrodeis partially immersed in solution and continuously rotated. In thisway, they could expose the electrode with only a very thin film ofelectrolyte covering the electrode.

Under these conditions, they were able to obtain spectra (Fig.16) of the film as prepared, of a cathodically protected film, andof a film passivated in borate solution at 1.3 V.

From these studies, they concluded that the passive film hadan Fe—O coordination with six near neighbors at a distance of2 ± 0.1 A. Although a higher signal-to-noise ratio is required torefine the structure, the approach followed by these authors appearsmost appropriate since they were able to reduce the deposited filmsto the metallic state and subsequently oxidize them. It would bemost interesting to ascertain how the structure of the passive filmvaries through sequential reduction/passivation cycles.


7.0 7.2 7.4 7.6

Energy in KeV7.8

Figure 16. Fluorescence detected X-ray absorp-tion spectra for a 4-nm iron film in emersioncell. Spectra are for: (A) dry film; (B) cathodi-cally protected film; (C) passivated film; (D)background electrolyte. (From Ref. 74, withpermission.)

Forty and co-workers76 have investigated the passive filmsformed on iron and iron-chromium alloys upon immersion insodium nitrite solution. They have investigated these films in a wetenvironment (which they term in situ) as well as after dehydration.For a FeCr alloy (13% Cr) they find that the structure of the wetfilm is analogous to that of y-FeOOH but with a higher degree ofdisorder, consistent with the Mossbauer results of O'Grady.69 Upondehydration, the structure of the passive films transforms to onethat is closer to that of y-Fe2O3 but with reduced long-range order.These authors also looked at the chromium edge and found thatthe local structure around chromium in the passive films was similarto that of Cr2O3. They concluded that the presence of chromiumin alloys stabilized the y-FeOOH-like layers against dehydration,thus forming a glassy-like structure which enhances the stability ofthe passive film.

Froment and co-workers have employed reflexafs77 (reflectionEXAFS) for studying passive films on iron78 and nickel.79'80 Theexperiment consists of measuring the ratio of the reflected andincident intensities as a function of energy. Although an EXAFSspectrum can be obtained from such a measurement, the processis somewhat involved since the reflectivity is a complex function


of angle of incidence, refractive index, and energy. They reportsome preliminary data for passive films on iron and nickel but donot derive extensive conclusions as their major intent was to demon-strate the applicability of reflexafs. Also concerning reflexafs froma general point of view, Heald and co-workers81'82 have made acareful comparison of reflexafs versus measurements at grazingincidence with fluorescence detection. They conclude that, in gen-eral, the latter offers enhanced sensitivity for studies of monolayers.However, the reflexafs technique can be applied in a dispersivearrangement (that is, with a broad range of energies incident onthe sample simultaneously), allowing for faster data acquisitionand the possibility of performing kinetic studies on the millisecondtime scale (vide infra).

The most extensive study of the nickel oxide electrode is thatof McBreen et a/.,83 who employed an in situ cell in a transmissionmode (see cell in Fig. 17). The study of nickel oxide is complicatedby the numerous species present and their interconversion. McBreen

Nickel Current Collector -

Nickel Oxide Electrode -

Nickel Current Collector

Grafofl

Electrolyte Port(Typical)

- X-Ray Window (Typical)

Figure 17. In situ transmission EXAFS cell for the study of Ni oxide electrodes.(From Ref. 83, with permission.)


Figure 18. Raw EXAFS spectra for: (A) dry Ni(OH)2electrode; (B) a Ni(OH)2 electrode charged once; (C) anelectrode discharged once; (D) an electrode charged twice.(From Ref. 83, with permission.)

et al. found that the as-prepared /3-Ni(OH)2 has the same structurewithin the x-y plane as that determined by X-ray diffraction experi-ments but with a significant degree of disorder along the c-axis.Oxidation to the trivalent state results in contraction of the Ni—Oand Ni—Ni distances along the x-y plane. Re-reduction of thismaterial yields a structure that is similar to that of the freshlyprepared Ni(OH)2. Repeated oxidation-reduction cycles result inan increased disorder which is believed to be responsible for facili-tating the electrochemical oxidation to the trivalent state.

Figure 18 shows some of the spectra reported. These are fullyconsistent with the above statements.

2. Monolayers

The study of electrochemically deposited monolayers poses thestrictest experimental constraints since the signals will be necessarilyvery low. On the other hand, these studies can provide much detailon interfacial structure at electrode surfaces as well as on the effects


of solvent and supporting electrolyte ions. The use of underpotentialdeposition84 allows for the precise control of the coverage of elec-trodeposited layers up to a monolayer. This represents a uniquefamily of systems with which to probe electrochemical interfacialstructure in situ.

We and others have been involved in the study of such systemsincluding Cu/Au(lll),85'86 Ag/Au(lll),87 Pb/Ag(lll),88 andCu/Pt(lll).89 The first three systems involved the use of epitaxiallydeposited metal films on mica as electrodes.90"92 Such depositiongives rise to electrodes with well-defined single-crystalline struc-tures. In the last case a bulk platinum single crystal was employed.Because of the single-crystalline nature of the electrodes, polariz-ation dependence studies could be used to ascertain surfacestructure.

The best-characterized system to date is the underpotentiallydeposited copper on gold. In this case we were able to obtainEXAFS spectra of a deposited monolayer with the polarization ofthe X-ray beam either perpendicular (Fig. 19A) or parallel (Fig.19B) to the plane of the electrode. A number of salient featurescan be pointed out. First of all, the copper atoms appear to belocated at threefold hollow sites (i.e., three gold near neighbors)on the gold (111) surface with copper near neighbors. The Au/Cuand Cu/Cu distances obtained are 2.58 and 2.91 ± 0.03 A, respec-tively. This last number is very similar to the Au/Au distance inthe (111) direction, suggesting a commensurate structure. Mostsurprising, however, was the presence of oxygen as a scatterer ata distance of 2.08 ± 0.02 A. From analysis and fitting of the datawe obtain that the surface copper atoms are bonded to an oxygenfrom either water or sulfate anion from the electrolyte. That theremight be water or sulfate in contact with the copper layer is notsurprising; however, such interactions generally have very largeDebye-Waller factors so that typically no EXAFS oscillations (orheavily damped oscillations) are observed (see the spectrum for[Cu(H2O)6]

2+ in Fig. 8A). The fact that the presence of oxygen(from water or electrolyte) at a very well-defined distance isobserved is indicative of a significant interaction and underscoresthe importance of in situ studies.

A pictorial representation of this system is shown in Fig. 20where the source of oxygen is presented as water. However, it


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30000

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9000 9250 9500 9750

Energy, eV

oO

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Energy, eV

Figure 19. In situ X-ray absorption spectrum for acopper upd monolayer on a gold (111) electrode withthe polarization of the X-ray beam being perpendicular(A) or parallel (B) to the electrode surface.

should be mentioned that from the EXAFS experiment we cannotrule out sulfate anions as the source of oxygen. In fact, experimentsby Kolb and co-workers93 indicate that sulfate may be present sinceat the potential for monolayer deposition the electrode is positiveof the potential of zero charge, EPZC9 so that sulfate would be

X Rays a5 Probes of Electrochemical Interfaces 301

Figure 20. Schematic representation of the struc-ture of a copper upd monolayer on a gold (111)electrode surface. The copper atoms sit at three-fold hollow sites on the gold surface and watermolecules are bonded to the copper atoms.

Au (111) SurfaceCu-Au = 2.58ACu-Cu = 2.92 A

Cu-O = 2.08 A

present to counterbalance the surface charge. We can, in addition,follow the edge features to ascertain changes in oxidation state.Figure 21A shows the edge region for the deposited monolayerwhile Fig. 21B shows the spectrum after stripping of the coppermonolayer. The appearance of the characteristic "white" line (res-onance near the edge) as well as the edge shift to higher energiesare fully consistent with the oxidation state assignments.

Studies of Ag on Au(lll)87 yield very similar results in termsof the structure of the deposited monolayer (i.e., the silver atomsare bonded to three surface gold atoms and are located at three-foldhollow sites forming a commensurate layer) with again stronginteraction by oxygen from water or electrolyte (perchlorate).

Figure 21. Near-edge spectra for (A) cop-per upd monolayer on a gold (111) elec-trode surface and (B) after electrochemi-cally stripping the copper monolayer.

8.96 8.98 9.00Energy, keV

9.20


Melroy and co-workers88 recently reported on the EXAFSspectrum of Pb underpotentially deposited on silver (111). In thiscase, no Pb/Ag scattering was observed and this was ascribed tothe large Debye-Waller factor for the lead as well as to the presenceof an incommensurate layer. However, data analysis as well ascomparison of the edge region of spectra for the underpotentiallydeposited lead, lead foil, lead acetate, and lead oxide indicated thepresence of oxygen from either water or acetate (from electrolyte)as a backscatterer.

They were also able to perform a potential dependence studyat -0.53 and -1.0 V. They found that the Pb—O distance increasesfrom 2.33 ± 0.02 to 2.38 ± 0.02 A upon changing the potential from-0.53 to -1.0 V versus Ag/AgCl (Fig. 22). This is consistent withthe negatively charged electrode repelling a negatively charged orstrongly dipolar adsorbate.

Most recently, we have been able to obtain the in situ surfaceEXAFS spectrum of a half-monolayer of underpotentiallydeposited copper on a bulk Pt(lll) single crystal pretreated withiodine. The spectrum shown in Fig. 23 is a bit noisy (due to limitednumber of scans) but at least five well-defined oscillations can beobserved. Preliminary data analysis indicates that the copperadatoms sit on threefold hollow sites with copper neighbors at2.80 ± 0.03 A. This distance is very close to the Pt—Pt distance inthe (111) direction and indicates the presence of a commensurate

0.010

0.004

0.000

-0.004

-0.010

I|Ai

/

1 i i10

k/A ]

12

Figure 22. Fourier-filtered data for a lead updmonolayer on a silver (111) electrode at twoapplied potentials. Solid curve, -1.0 V; dashedcurve, —0.53 V. (From Ref. 88, with permission.)


CuUPDonPt(lll)/I

Energy (keV)

Figure 23. In situ X-ray absorption spectrumfor half a monolayer of copper underpotentiallydeposited on a bulk Pt (111) electrode pre-treated with iodine.

layer. The fact that such well-defined two-dimensional structuresare present at half a monolayer coverage is a strong indication thatthe electrodeposition occurs by initial clustering rather than byrandom decoration of the surface with subsequent coalescence.

3. Adsorption

We have studied the adsorption of iodine on Pt(lll) electrodes94

from solutions containing iodide with the intent of followingchanges in the coverage as a function of applied potential (that is,constructing an in situ adsorption isotherm). We were furtherencouraged to carry out these experiments by the recent report byHubbard and co-workers95 on the determination of iodine adsorp-tion by following Auger electron intensities.

We find that after appropriate normalization, our data (Fig.24A) agree quite well with the results presented by Hubbard forpotentials negative of +0.40 V (Fig. 24B). A particularly gratifyingresult was the change in coverage—from | to |, in going from -0.3to +0.2 V. An even more interesting and tantalizing finding wasthat at potentials positive of +0.40 V we observe a significant


A • = Pt(lll) 0.05mM Nal/O.IM Na2SO4(all measurements started at -0.20V)

B • = Pt(lll) lOmM KI/lOmM KClO4/lmM K3PO4.(measurements started at potential of interest)

1.6

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0.3

0.2

0.1

-0.8 -0.6 -0.4 -0.2 0.0 +0.2 +0.4 +0.6 +0.8

E vs Ag/AgCl

Figure 24. Iodine on Pt adsorption isotherm obtained in situ by followingthe edge-jump intensity at the iodine K-edge as a function of potential (A)and comparison with data obtained by Hubbard et al. via Auger intensities(B). (Adapted from Ref. 95.)

enhancement of the iodine fluorescence signal, whereas Hubbardreports no significant changes up to about +0.70 V where faradaicoxidation of iodine to iodate takes place. We believe that the localincrease in the iodine/iodide concentration is produced by afaradaic charge flow, followed by association of the oxidationproduct(s) with the adsorbed iodine layer at the given potential. Itappears that the associated layer does not survive the transfer tovacuum employed in the ex situ study and again underscores theimportance of in situ measurements.

X Rays •• Protes of Electrochemical Interfaces 305

4. Spectroelectrochemistry

EXAFS and XANES techniques have been applied in the moretraditional type of spectroelectrochemical experiments where athin-layer cell configuration is employed. Drawing from extensiveexperience with the related UV-vis measurements, Heineman incollaboration with Elder96'97 were the first to report on an in situEXAFS spectroelectrochemistry experiment. Their first cell designemployed gold minigrid electrodes similar to those typicallyemployed in traditional UV-vis experiments. They studied the ferrocyanide/ferri cyanide couple in each oxidation state by monitoringthe absorption about the iron X-edge using fluorescence detection.A typical spectrum and related Fourier transform are presented inFig. 25. From analysis of their data, they were able to determine

Figure 25. EXAFS data for K3[Fe(CN)6]:(A) /c2-weighted EXAFS; (B) Fouriertransform of (A) showing Fe—C and Fe—N peaks; (C) Fourier-filtered back-trans-formation of the Fe—C peak. (From Ref.97, with permission.)


that for Fe(II) there are 7.4 carbon atoms at 1.97 ± 0.01 A, whereasfor Fe(III) the numbers are 6.8 and 1.94 ±0.01 A, respectively.Since coordination number determination is usually no better than20%, the numbers they find are in good agreement with the knownvalue of 6. More interesting is the fact that the observed contractionof the Fe—C bond upon oxidation is contrary to results based oncrystallographic studies. This points to the importance of in situmeasurements since by means of the applied potential one canprevent changes in the oxidation state of the species being studied.

In addition, the determination of metal-ligand bond distancesin solution and their oxidation state dependence is critical to theapplication of electron transfer theories since such changes cancontribute significantly to the energy of activation through theso-called inner-sphere reorganizational energy term.

These authors have also developed a cell98 (Fig. 26) thatemploys reticulated vitreous carbon as a working electrode andthey find that such a design allows for much faster electrolysis.Using such a cell, they have studied the [Ru(NH3)6]3+/2+ couple

h(^

X-rays

Figure 26. EXAFS spectroelectrochemicalcell: (A) front view, (B) top view, (C) sideview, (D) assembly; (a) auxiliary electrodecompartment, (b) working electrode well,(c) reference electrode compartment, (d)X-ray window, (e) inlet port, (f) auxiliaryelectrode lead, (g) RVC working elec-trode, (h) Pt syringe needle inlet andelectrical contact, (i) Pt wire auxiliaryelectrode, (j) Ag/AgCl(3M NaCl) refer-ence electrode. (From Ref. 98, with per-mission.)


and a cobalt(III/II) sepulchrate as well as the Fe—C distance incytochrome c.

Most recently, they have developed" a cell configuration forthe study of modified electrodes that employs, as a working elec-trode, colloidal graphite deposited onto kapton tape (typicallyemployed as a window material). Such an arrangement minimizesattenuation due to the electrolyte solution.

Antonio et al.100 have performed an in situ EXAFS spectro-electrochemical study of heteropolytungstate anions.

We have also performed some in situ EXAFS measurementson chemically modified electrodes.101 Specifically, we have studiedfilms of [Ru(v-bpy)3]

2+ (v-bpy is 4-vinyl-4'-methyl-2,2'-bipyridine)electropolymerized onto a platinum electrode and in contact withan acetonitrile/0.1 M tetrabutylammonium perchlorate (TBAP)solution and under potential control. The experimental setup con-sists of a thin-layer configuration employing a thin (6 /xm) Teflonwindow and grazing incidence so as to take advantage of the totalexternal reflection effects mentioned previously. We have focusedon determining the lower limit of detection as well as trying toascertain any differences in the metal-ligand bond distances forthe electropolymerized films as a function of coverage when com-pared to the parent complex. Figures 27A and 27B show spectrafor electrodes modified with one and five monolayers of the com-plex, respectively, whereas Fig. 27C shows the spectrum for bulk[Ru(bpy)3]

2+. In Fig. 27A one can ascertain that only the mostprominent features of the spectrum of the parent compound (Fig.27C) are present. (It should be mentioned that a monolayer of[Ru(v-bpy)3]

2+ represents about 5.4 x 1013 molecules/cm2, whichis about 5% of a metal monolayer. This is mentioned since it is themetal centers that give rise to the characteristic fluorescenceemployed in the detection.) However, at a coverage of fivemonolayers of complex (Fig. 27B) the spectrum is essentially indis-tinguishable from that of the bulk material. This indicates that thestructure of electroactive polymer films can be obtained at relativelylow coverages and this should have important implications in tryingto identify the structures of reactive intermediates in electrocatalyticreactions at chemically modified electrodes.

Furthermore, one can monitor changes in oxidation state bythe shift in the edge position. For example, Fig. 28 shows that upon


40000

22.0 22.2 22.3 22.4 22.5

Energy (keV)22.8

Figure 27. In situ X-ray absorption spectra around the ruthenium X-edge for anelectrode modified with (A) one and (B) five monolayers of [Ru(v-bpy)3]

2+. (C)Spectrum of bulk [Ru(bpy)3]

2+.

oxidation of the polymer film (at +1.50 V) from Ru(II) to Ru(III),the edge position shifts to higher energy by about 1.5 eV. Thus, onecan determine the oxidation state of the metal inside a polymerfilm on an electrode surface.

Tourillon and co-workers102"110 have also reported on a numberof spectroelectrochemical studies, especially of electrodeposition

X Rays a s P r o b e s o f Electrochemical Interfaces 309

6000

4000

CO• * - »

c13oO

2000

_L I22080 22100 22120 22140

Energy, eV22160 22180

Figure 28. In situ X-ray absorption of the near-edge region for anelectrode modified with a polymeric film of [Ru(v-bpy)3]

2+ andits potential dependence. Applied potentials were +0.7 V (A) and+ 1.5 V(B).

of metals, particularly copper, on electrodes modified with poly(3-methylthiophene). What sets their experiments apart is the use ofa dispersive approach (Fig. 29). In such a setup, focusing optics(employing a bent crystal) are employed so as to have a range ofenergies (as wide as 500 eV) come to a tight focal spot at the sample.The beam then impinges a photodiode array with 1024 pixels sothat all energies are monitored at once. The net result is to sig-nificantly decrease data acquisition times so that spectra can beobtained in times as short as a few milliseconds. Thus, this opensup tremendous possibilities in terms of kinetic and dynamic studies.One of the more impressive results using this approach is shownin Fig. 30, which shows spectra obtained around the copper edgefor a poly(3-methylthiophene) film (on a platinum electrode) dopedwith Cu2+ ions. The potential of the electrode is stepped so as toreduce the Cu2+ ions to Cu1+ and subsequently to Cu0. The spectrashown in the figure were taken at 7-s intervals and the transitionsfrom Cu2+ to Cu1+ and then to metallic copper are clearly evident.

310 Hector D. Abnina

1.5 --

- J 0

Figure 29. Schematic of a dispersive EXAFS setup as well as spectra for: (A)Cu2+; (B) Cu l + ; (C) copper foil. (From Ref. 105, with permission.)

These authors have also employed this technique for the study ofother metallic inclusions into poly(3-methylthiophene) films,including Ir, Au, and Pt. Thus, this type of arrangement could openup new exciting possibilities in terms of kinetic studies.

IX. X-RAY STANDING WAVES

1. Introduction

The X-ray standing wave (XSW) technique represents anextremely sensitive tool for determining the position of impurityatoms within a crystal or adsorbed onto crystal surfaces.111'112 Thistechnique is based on the X-ray standing wave field that arises asa result of the interference between the coherently related incidentand Bragg diffracted beams from a perfect crystal. In the vicinityof a Bragg reflection (Fig. 31), an incident plane wave (with wavevector k0) and a reflected wave (with wave vector kH) interfere togenerate a standing wave with a periodicity equivalent to that ofthe (/J, k, /) diffracting planes. The standing wave develops not onlyin the diffracting crystal, but also extends well beyond its surface.

7.9

eo E V

Figure 30. In situ measurements of the time evolution of the Cu X-edge when a platinum electrodecoated with a polymeric film of poly (methylthiophene) is cathodically polarized in an aqueous solutioncontaining 50 mM CuCl2. (From Ref. 105, with permission.)

i

8.


K

surface

"Crystal'

Figure 31. Depiction of the X-ray standing wave fieldformed by the interference between incident and Braggreflected beams.

Estimates of this coherence length range to values as large as 1000 Afrom the interface.113 The nodal and antinodal planes of the standingwave are parallel to the diffracting planes and the nodal wavelengthcorresponds to the d-spacing of the diffracting planes. As the angleof incidence is advanced through the strong Bragg reflection, therelative phase between the incident and reflected plane waves (ata fixed point) changes by TT. Due to this phase change, the antinodalplanes of the standing wave field move in the —H direction byone-half of a d-spacing, from a position halfway between the (h, /c, /)diffracting planes (low-angle side of the Bragg reflection) to aposition that coincides with them (high-angle side of the Braggreflection). Thus, the standing wave can be made to sample anadsorbate or overlayer at varying positions above the substrateinterface.

For an atomic overlayer which is positioned parallel to thediffracting planes, the nodal and antinodal planes of the standingwave will pass through the atom plane as the angle of incidence isadvanced. Using an incident beam energy at or beyond the absorp-tion edge of the atoms in the overlayer, the fluorescence emissionyield will be modulated in a characteristic fashion (Fig. 32) as thesubstrate is rocked in angle. The phase and amplitude of thismodulation (or so-called coherent position and coherent fraction)are a measure of the mean position (Z) and width V(Z)2 of thedistribution of atoms in the overlayer. The Z-scale is mod d and

X Ray* a s Probes of Electrochemical Interfaces 313

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ANGLE

Figure 32. X-ray field intensities at extended Ge (220) lattice positions (0-9) for aperfectly collimated incident X-ray beam. An atomic adlayer whose center falls onone of these positions would have its characteristic fluorescence intensity modulatedin the same fashion. The dashed curve represents the Bragg reflectivity profile. (FromM. J. Bedzyk, Ph. D thesis, SUNY Albany, 1982.)


points in the direction normal to the diffraction planes. Standingwave measurements of (Z) and V(Z)2 can be accurate to within1% and 2% of the d-spacing, respectively.114 Golovchenko andco-workers115 have applied this technique to the study of surfaceadsorbates, monitoring the angular dependence of the fluorescenceyield of bromine chemisorbed onto a Si(lll) crystal. It should bementioned that these experiments were performed while the crystalwas covered by a thin film of methanol, pointing to the feasibilityof performing experiments at the solid-liquid interface. Bedzykand Materlik114 used the angular dependence of the fluorescenceyield of bromine to relate its position to the Ge(lll) and Ge(333)diffracting planes. In addition, they demonstrated the feasibility ofusing higher-order reflection for determining the thermal vibrationamplitude of the bromine adsorbate.

One of the problems associated with the implementation ofthe standing wave technique is the fact that it requires the use ofperfect or nearly perfect crystals. This presents a problem especiallyfor relatively soft materials such as copper, gold, silver, andplatinum, which are not only very difficult to grow in such highquality, but are also very difficult to maintain in that state. Thus,most experiments have been performed on silicon or germaniumsingle crystals.

An alternative to the use of perfect crystals is the use ofsynthetic layered microstructures (LSMs).116 These devices are pre-pared by the sequential deposition of alternate layers of materials,typically of high- and low-electron-density elements such as W/C,Mo/C, W/Si, and Pt/C, onto a smooth substrate material such assilicon, germanium, or float glass. Recent results have hinted at thepossibility of growing these devices on cleaved mica. The numberof layer pairs is such that the d-spacing of the synthetic multilayerdictates the diffracting properties of the interface. Althoughoriginally developed as X-ray mirrors for regions of the spectrumwhere natural crystals are not effective, these devices have foundwidespread use in X-ray standing wave experiments, and, in fact,numerous applications have appeared.117

The advantages that accrue from the use of LSMs are manifold.Firstly, these devices are physically robust and can be handledwithout undue provision. They can be tailored to contain materialsof interest and, furthermore, one can vary the d- spacing of the


diffracting planes, thus varying the probing distance of the tech-nique. Their use also simplifies experimental design since theangular reflection widths for LSMs are of the order of milliradiansas opposed to microradians for crystals.

2. Experimental Aspects

One of the main difficulties with X-ray standing wave measurementsis that they are experimentally very demanding. Although theexperimental setup is not particularly complex, alignment of thesample relative to the beam is critical. A typical setup is shown inFig. 33 and consists of an incident beam monitor Jo, a sample stage,a reflected beam monitor IR, and a detector at 90° relative to theX-ray beam. Of particular importance in this experiment is theangular resolution of the sample stage since a typical reflectionwidth will be of the order of tens of microradians for a single crystaland a few milliradians for LSMs. In both cases, however, highangular resolution is required if we are to have a well-resolvedreflectivity profile. In addition, when measuring fluorescence froman adsorbate layer, care must be taken to accurately subtract back-ground radiation.

Slit

Si(Li)Sol id-State

Slit ] T r ^ "^ C l ! f / ^ ' detector

Sample

Detector

Hgure 33. Experimental setup for X-ray standing wave measurements on an LSM.


3. X-Ray Standing Wave Studies at Electrochemical Interfaces

Due to the experimental difficulties involved, there have been onlythree reports of XSW measurements at electrochemical interfaces.Materlik and co-workers have studied the underpotential depositionof thallium on single-crystal copper electrodes under both ex situ119

and in situ120 conditions. In addition, they report results from studiesin the absence and presence of small amounts of oxygen.

In the ex situ studies, the thallium layer was electrodepositedand the electrode was subsequently removed from solution andplaced inside a helium-filled box where the XSW experiments werecarried out.

For the in situ studies, an electrochemical cell was designedto hold the nearly perfect copper crystal in contact with a thin layer(20 to 50 fxm) of electrolyte. Figures 34 and 35 show the cellsemployed in the ex situ and in situ experiments, respectively. Inaddition, Fig. 34 shows the voltammetric traces obtained for thedeposition of TI in the presence and absence of oxygen. In the

(a)

-Q5POTENTIAL vs. SCE ( V I

Figure 34. Voltammograms for TI deposition onto a copper single crystal in thepresence (a) and absence (b) of traces of oxygen. Inset: electrochemical cell.(From Ref. 120, with permission.)

X Ray5 as Probes of Electrochemical Interfaces 317

to pump

to Refelect.

Valve

Mylar window

Cu crystal

Counter electrode

Electrolyte in/out

Holder for Cu

Sample mountheight adjustment

Figure 35. Electrochemical cell for in situ X-ray standing wavemeasurements. (From Ref. 120, with permission.)

experiments, both the reflectivity and the TI fluorescence intensitywere monitored simultaneously. Figure 36 shows the results for theex situ study.

From an analysis of their data, Materlik and co-workers wereable to determine that for the ex situ case and in the absence ofoxygen, the thallium atoms are located at twofold sites at a meandistance of 2.67 ± 0.02 A. For the in situ case and again in the

9

W

3y

7) 2

O

1 10

—

T'-

0.13 Ml TI1 E = 15.3 KeV

•^J -»^# Cu( l l l )

/FLUORESCENCE \/ N •

/ \

. REFLECTIVITY """

°-"°l l l 1 1 I I Y^H

—

f-

\>

>

CT

T

W

ft

1

20 40 60 80

REFLECTION ANGLE; 6 - 6B (jirad)

Figure 36. Angular dependence of the Cu( 111) reflectivityand the normalized T1L fluorescence yield. Points rep-resent experimental data; curves are least-squares fits.(From Ref. 119, with permission.)


absence of oxygen, the data are consistent with the thallium atomsbeing at 2.58 ± 0.02 A, but at threefold sites.

In the presence of oxygen, there is a significant contraction ofthe mean distance of the thallium to 2.27 ± 0.04 A. This is ascribedto a surface reconstruction of the copper induced by the adsorbedoxygen which results in an inward shift of the copper surface atomsby about 0.3 A. This is consistent with low-energy ion-scatteringstudies. In general, these studies are of great significance since theydemonstrate the applicability of the X-ray standing wave techniqueto the in situ study of electrochemical interfaces, even employingsingle crystals.

We have performed some experiments on the use of LSMs inthe investigation of electrochemical interfaces.121 The system thatwe have studied involves the adsorption of iodide onto aplatinum/carbon LSM followed by the electrodeposition of a layerof copper. The LSM sample consisted of 15 platinum/carbon layerpairs with each layer having 26 and 30 A of platinum and carbon,respectively, with platinum as the outermost layer. We used9.2-keV radiation from the Cornell High Energy Synchrotron Source(CHESS) to excite L-level and ^-level fluorescence from the iodideand copper, respectively. Initially, the LSM was contacted with a35 mM aqueous solution of sodium iodide for 15 min. It was thenstudied by the X-ray standing wave technique. The characteristiciodine L fluorescence could be detected and its angular dependencewas indicative of the fact that the layer was on top of the platinumsurface layer. A well-developed reflectivity curve (collected simul-taneously) was also obtained. Following this, the LSM was placedin an electrochemical cell and half a monolayer of copper waselectrodeposited. The LSM (now with half a monolayer of copperand a monolayer of iodide) was again analyzed by the X-raystanding wave technique. Since the incident X-ray energy (9.2 keV)was capable of exciting fluorescence from both the copper and theiodide, the fluorescence intensities of both elements (as well as thereflectivity) were obtained simultaneously. The results presented inFig. 37 show the reflectivity curve and the modulation of the iodideand copper fluorescence intensities. The most important feature isthe noticeable phase difference between the iodide and coppermodulation, i.e., the location of the iodide and copper fluorescencemaxima, with the copper maximum being to the right of the iodide


O_JUJ

>•

auiN

o

Z(A)

_9GQc!

• • • I o d i n e LFluorescence

_ Theory :Z-A.Qk —

ooo Cu KFluorescence —

Theory: Z=0 .7AReflectivity

10 U 16ANGLE (mrad)

18

Figure 37. Experimental results and least-squares fits of data (solid lines) for a Pt/CLSM covered with an electrodeposited layer of copper and an adsorbed layer ofiodine. Topmost curve: IL fluorescence; middle curve: copper Ka fluorescence;bottom curve: reflectivity.

maximum. Since the antinodes move inward as the angle increases,the order in which these maxima occur can be unambiguouslyinterpreted as meaning that the copper layer is closer than theiodide layer to the surface of the platinum. Since the iodide hadbeen previously deposited on the platinum, this representsunequivocal evidence of the displacement of the iodide layer bythe electrodeposited copper. Similar findings based on Augerintensities and LEED patterns have been previously reported byHubbard and co-workers.122123 In addition, from an analysis of thecopper fluorescence intensity, we were able to determine that theelectrodeposited layer had a significant degree of coherence (53%)with the underlying substrate (Fig. 37). The experimental difficultiesassociated with the X-ray standing wave technique and the needto prepare single crystals of high perfection have limited its wide-


Double CrystalMonochromator

Monitor

SingleCrystal

Figure 38. Experimental setup for back-reflection X-ray standing wavemeasurements.

spread applicability. Recent studies by Woodruff et al.124 and us125

have revealed that in a backscattering geometry (Fig. 38) the reflec-tion widths of single crystals such as platinum are of the order ofmillradians (rather than microradians). This means that single crys-tals with a broader mosaic spread can be employed for X-raystanding wave measurements. Thus, with the use of this geometryfor single crystals or the use of LSMs, I am certain that many morestudies employing this technique will be carried out.

X. X-RAY DIFFRACTION

In addition to surface EXAFS and X-ray standing waves, X-raydiffraction can be employed in the study of electrochemical inter-faces. Although an extensive treatment of X-ray diffraction tech-niques is beyond the scope of this chapter, some brief statementsare appropriate.

A number of surface diffraction techniques can be employedin the structural study of electrochemical interfaces, depending onthe details of the system under study. For bulk materials or thickfilms (such that the X-ray beam only samples that layer) conven-tional diffraction experiments can be performed and, in fact, anumber of in situ X-ray diffraction studies of this type have beenreported.126 129 In the case of thin films or monolayers, two differenttechniques can be employed and these are the reflection-diffractiontechnique introduced by Marra and Eisenberger and thetechnique based on surface truncation rods.133 In the first case, theincident X-ray beam impinges on the sample at an angle below


the critical angle so that the penetration depth of the X ray is veryshallow (approximately 20-50 A), allowing for the study of verythin films on a substrate. This technique has been employed byFleischmann and co-workers in a variety of investigations includingstudies of iodine adsorbed on graphite,134135 lead underpotentiallydeposited on silver,135'136 Ni(OH)2,

135 and hydrogen and COadsorption on platinum.137 They introduced the variation of per-forming potential modulation experiments (akin to the proceduresemployed in FT-IR studies of electrochemical interfaces) to obtaindifference diffractograms which were then interpreted in terms ofdifferences in structure at the two potentials.

For the case of surface truncation rods, the technique is basedon the detection of diffraction peaks between Bragg peaks. Althoughthis requires careful alignment and some a priori knowledge of thestructure, monolayer sensitivity can be achieved. In fact, Samantet a/.138 have recently performed an in situ surface diffraction studyof lead underpotentially deposited on silver employing this tech-nique along with grazing incidence diffraction. It is clear that thistechnique will also find widespread use in the near future.

XL CONCLUSIONS AND FUTURE DIRECTIONS

The use of X rays is providing a rare glimpse of the in situ structureof electrochemical interfaces, and as these experiments becomemore widespread, a wide range of phenomena will be explored. Iam certain that these studies will provide the basis for a betterunderstanding and control of electrochemical reactivity.

ACKNOWLEDGMENTS

Our work was generously supported by the Materials Science Centerat Cornell University, the National Science Foundation, the Officeof Naval Research, the Army Research Office, and the DowChemical Company. Special thanks to Dr. Michael J. Bedzyk(CHESS) and to Dr. James H. White as well as to Michael Albarelli,Mark Bommarito, Dr. Martin McMillan, and David Acevedo. Thework on the copper and silver underpotentially deposited on gold


was in collaboration with Dr. Owen Melroy and Dr. Joseph Gordon(IBM, San Jose, California) and Professor Lesser Blum (Universityof Puerto Rico).

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and B. L. Henke, AIP Press, New York, 1981.117 T. W. Barbee and W. K. Warburton, Mater. Lett. 3 (1984) 17.118 A. Iida, T. Matsushita, and T. Ishikawa, Jpn. J. Appl. Phys. 24 (1985) L675.| 1 9 G. Materlik, J. Zegenhagen, and W. Uelhoff, Phys. Rev. B 32 (1985) 5502.120 G. Materlik, M. Schmah, J. Zegenhagen, and W. Uelhoff, Ber. Bunsenges. Phys.

Chem. 91 (1987) 292.M. J. Bedzyk, D. Bilde/• Phys. Chem. 90 (1986) 4926.J. L.:260.

121 M. J. Bedzyk, D. Bilderback, J. H. White, H. D. Abruna, and M. G. Bommarito,

122 J. L. Stickney, S. D. Rosasco, and A. T. Hubbard, /. Electrochem. Soc. 131 (1984)

326 Hector D . Abruna

123 J. L. Stickney, S. D. Rosasco, B. C. Schardt, and A. T. Hubbard, J. Phys. Chem.88 (1984) 251.

124 D. P. Woodruff, D. L. Seymour, C. F. McConville, C. E. Riley, M. D. Crapper,N. P. Prince, and R. G. Jones, Phys. Rev. Lett. 58 (1987) 1460.

125 M. J. Bedzyk and H. D. Abruna , unpub l i shed results.126 S. U. Falk, J. Electrochem. Soc. 107 (1960) 661 .127 A. J. Sa lk ind , C. J. V e n u t o , a n d S. U . Fa lk , J. Electrochem. Soc. I l l (1964) 493 .128 K. M a c h i d a a n d M . E n y o , Chem. Lett. (1986) 1437.129 G. Nazr i a n d R. H. Mul ler , /. Electrochem. Soc. 132 (1985) 1385.130 W. C. M a r r a , P. Eisenberger , a n d A. Y. C h o , J. Appl. Phys. 50 (1979) 6927.131 P. E i senberger a n d W. C. M a r r a , Phys. Rev. Lett. 46 (1981) 1081.132 W. C. M a r r a , P. H. Fuoss , a n d P. Eisenberger , Phys. Rev. Lett. 49 (1982) 1169.1 3 3 1 . K. R o b i n s o n , Phys. Rev. B 33 (1986) 3830.134 M. F le i schmann , P. J. H e n d r a , a n d J. Rob inson , Nature 288 (1980) 152.135 M. F le i schmann , A. Oliver, a n d J. Rob inson , Electrochim. Ada 31 (1986) 899.136 M. F le i schmann , P. Graves , I. Hill , A. Oliver, a n d J. Rob inson , /. Electroanal.

Chem. 150 (1983) 33.137 M. Fleischmann and B. W. Mao, J. Electroanal Chem. 229 (1987) 125.138 M. G. Samant, M. F. Toney, G. L. Borges, L. Blum, and O. R. Melroy, /. Am.

Chem. Soc. J. Phys. Chem. 92 (1988) 220.

Electrochemical and PhotoelectrochemicalReduction of Carbon Dioxide

Isao TaniguchiDepartment of Applied Chemistry, Faculty of Engineering, Kumamoto University,

Kurokami, Kumamoto 860, Japan

I. INTRODUCTION

The reduction of carbon dioxide has been a subject of active interestfor more than a century.1 Especially in recent years, electrochemicaland photoelectrochemical reduction of carbon dioxide has beenextensively studied.2"4 This is because this reaction has severalattractive features. In view of the increasing possibility of unavaila-bility of oil and other fossil fuels in the near future,5'6 alternativefuels have to be produced from abundant resources such as carbondioxide and water. Carbon dioxide reduction is also an importantbranch of Q chemistry. In addition, the effect of recent excessiveproduction of carbon dioxide on the future climate of the Earth isbeing seriously discussed,7 and carbon dioxide reduction to organicraw materials or fuels would help to reduce this type of atmosphericpollution as well. Carbon dioxide reduction can be used as a suitablereaction for energy storage, as is required, for instance, in theconversion of solar to storable chemical energy.8'9 Moreover, formicacid, which is one of the reduction products of carbon dioxide, hasbeen proposed as a convenient means of hydrogen storage.10

From a fundamental viewpoint, carbon dioxide reduction is amodel reaction which can help us to understand better the mechan-ism of natural photosynthesis.11 Development of artificial photosyn-thetic systems, by mimicking functions of green plants, is one of

327

328 Isao Taniguchi

our ultimate goals. Interest has also been devoted to carbon dioxidereduction as a model of the geochemical carbon cycle12 and ofprebiological photosynthesis.1314 Also, since the reduction of car-bon dioxide does not take place easily, the development of effectivecatalysts is required; such research would lead to an insight intothe activation of stable molecules and would yield informationabout reaction pathways for many-electron transfer that provide asaving of energy and high efficiency. Thus, reduction of carbondioxide has been related to fascinating aspects in various fields ofchemistry.

Here, some recent studies of the electrochemical and photo-electrochemical reduction of carbon dioxide as well as some otherrelated subjects will be reviewed and discussed. Attention is focusedespecially on the work done in the last ten years, to avoid duplicationof previous review articles.2'4

II. ELECTROCHEMICAL REDUCTION OFCARBON DIOXIDE

1. Reduction of Carbon Dioxide at Metal Electrodes

Electrochemical reduction of carbon dioxide usually requires alarge overvoltage and competes with hydrogen evolution, resultingin a low power efficiency. Carbon dioxide, however, is actuallyreduced electrochemically at highly negative potentials (ca. -2.0 Vversus SCE or more negative). Thus, the reaction has been carriedout at metal electrodes having a high hydrogen overvoltage, suchas Hg and Pb. The principal product of CO2 reduction in aqueoussolutions was reported to be formic acid (or formate ion), and highcurrent efficiencies, up to near 100% were achieved at an amalga-mated Zn electrode under a high CO2 pressure.15

Udapa et al.16 showed that CO2 was reduced to formic acid ata mercury electrode in a 0.05 M phosphate buffer (pH 6.8) solution.A current efficiency of 81.5% was obtained at a current density of20 mA/cm2 and a cell voltage of 3.5 V. On the other hand, Bewickand Greener17 reported that malate and glycolate were producedat Hg and Pb electrodes, respectively, using aqueous quartenary

Reduction of Carbon Dioxide 329

ammonium salt electrolytes; unfortunately, the results have neverbeen reproduced.

Instead of mercury, Ito et a/.18 examined systematically somesp metals, such as Zn Pb, Sn, In, and Cd (5 JV purity), as the cathodematerials for CO2 reduction (Fig. 1). Using an In electrode at3.9 mA/cm2, the highest current efficiency (92%) for formic acidproduction was obtained in an aqueous Li2CO3 solution; the poten-tials at which CO2 was reduced at an In electrode were ca. 400 mVless negative than those at an Hg electrode.

A little later, Russell et al.19 tried to obtain methanol fromcarbon dioxide by electrolysis. Reduction of carbon dioxide toformate ion took place in a neutral electrolyte at a mercury electrode.On the other hand, formic acid was reduced to methanol either ina perchloric acid solution at a lead electrode or in a buffered formicacid solution at a tin electrode. The largest faradaic efficiency formethanol formation from formic acid was ca. 12%, with poorreproducibility, after passing 1900 C in the perchloric acid solutionat Pb in a very narrow potential region (-0.9 to -1.0 V versus SCE).In the buffered formic acid solution (0.25 M HCOOH + 0.1M

0 5 10 15

CURRENT DENSITY

20

Figure 1. Current efficiencies for reduction ofC O 2 to formic acid in a 0.1 M Li 2 CO 3 solutionat 25 ± 1°C at various electrodes.1 8 (1) In; (2)Sn (previously anodized) ; (3) Sn; (4) Zn; (5)Pb; (6) Cd.

330 Isao Taniguchi

NaHCO3, pH 3.8), good faradaic efficiencies up to 100% wereobtained, but again only in a narrow potential region (-0.68 to-0.72 V versus SCE) with a current density of less than 4 /x A/cm2.No formaldehyde was produced. Furthermore, the reduction offormaldehyde to methanol took place at an Hg electrode at a currentdensity of ca. 10 mA/cm2 with faradaic efficiencies of more than90% in a basic solution (0.1 M HCHO + 0.1 M Na2CO3, pH -11).No dimerization product such as ethylene glycol was observed.Thus, the authors concluded that the direct reduction of CO2 tomethanol by electrolysis was difficult because of the differences inoptimum conditions for electrolysis of CO2, formic acid, and form-aldehyde. Hori et al20 reported that the current efficiencies forformic acid formation in a CO2-saturated 0.5 M NaHCO3 solutionby constant-current electrolysis at 16 mA/cm2 decreased in the orderIn > Sn > Zn > Pb > Cu » Au, and the partial currents for CO2

reduction were linearly dependent on the hydrogen overvoltage ofthe metals21 (Fig. 2). Later, tin and indium electrodes were againexamined by Kapusta and Hackerman,22 and carbon dioxide reduc-tion to formic acid was confirmed to proceed with a high currentefficiency (ca. 90%), although the overall power efficiency was verylow due to the high overpotential of the reaction. Reduction offormic acid was observed only on a tin electrode at low currentdensities.

More recently, Hori et al23 reported that reduction of CO2 bygalvanostatic electrolysis (5 mA/cm2) at Au and Ag electrodes gave

-1.0 -0.5

Hydrogen Overvoltage /V

Figure 2. Partial currents for CO2

reduction at various electrodes duringconstant-current electrolysis at16 mA/cm2 in a 0.5 M NaHCO3 sol-ution as a function of hydrogen over-voltage of the metal used.20 Values21

of hydrogen overvoltage of the metalswere those obtained in acidic solutionsat 1 mA/cm2.


CO as a main product in a 0.5 M KHCO3 solution (Table 1). Ata Cu electrode, CH4 was produced in appreciable amounts (Table1); production of CH4 was favored below 20°C, and at 0°C thefaradaic efficiency reached ca. 65%, while at 40°C the amount ofC2H4 formed increased to ca. 20% in faradaic efficiency withincreasing hydrogen evolution. At Ru electrodes,24 CO2 was reducedto methane, methanol, and CO in a CO2-saturated 0.2 M Na2SO4

solution at ca. 60°C and ca. -0.55 V (versus SCE) with currentdensities less than 0.5 mA/cm2. Also, Mo electrodes were used25

for CO2 reduction at room temperature: Methanol was reported tobe a major product of electrolysis at -0.7 to -0.8 V for 0.2 MNa2SO4 (pH 4.2) and at -0.57 to -0.67 V (versus SCE) for 0.05 MH2SO4 (pH 1.5), although current densities obtained were rathersmall (ca. 0.1 mA/cm2) and prolonged electrolysis showed aremarkable decrease in current efficiency for methanol formation.Interestingly, when Mo electrodes were pretreated by cycling in aCO2-saturated 0.2 M Na2SO4 solution between -1.2 and 0.2 V(versus SCE), faradaic efficiencies of greater than 100% formethanol formation were observed, indicating that some chemicalreaction of CO2 at the electrode surface is involved in the methanolformation. Taniguchi et al26 also confirmed the formation of COat Au and Ag electrodes, hydrocarbons at a Cu electrode, andmethanol at an Mo electrode, but the current efficiencies of theseproducts, especially the hydrocarbons and methanol, were muchsmaller than those previously reported. Current efficiencies of theproducts depended strongly on the surface conditions such asroughness of the electrode and the presence of deposited impuritiesand, therefore, very small amounts of metallic impurities shouldbe removed from the electrolyte; for example, when ca. 100 ppmof Bi3+ was present in the electrolyte, formic acid became a mainproduct (ca. 60% faradaic efficiency) at a Cu electrode in anaqueous CO2-saturated 0.5 M KHCO3 solution at a constant currentof 5 mA/cm2. These results indicate that metal electrodes themselveshave interesting catalytic activities which effect the distribution ofthe reduction products of CO2; this may be caused by the differencein adsorption behavior of CO2, H2, and the intermediates involvedin CO2 reduction pathways.

In aqueous solutions, solubility of the reactant, i.e., CO2, israther low (ca. 38 mmol/liter in water at 25°C and 1 atm of CO2),

Table 1Results of Cathodic Reduction of CO2 at Various Electrodes0'*

Electrode

Cdc

Snc

Pbc

Inc

Znc

Cu"Ag"Au"Nid

Fe*

Electrodepotential

(V versus SHE)

—-------

1.66 ±0.021.40 ±0.041.62 ±0.031.51 ±0.05L56±0.081.39 ±0.021.45 ±0.021.14 ±0.01

-1.39-1.42

Hcocr65.3/67.265.5/79.572.5/88.892.7/97.617.6/85.015.4/16.51.6/4.60.4/1.0

0.32.1

Faradaic efficiency (lower

CO

6.2/11.12.4/4.10.3/0.60.9/2.23.3/63.31.5/3.1

61.4/89.981.2/93.0

0.01.4

(%)

CH4

0.20.1/0.20.1/0.2

0.00.0

37.1/40.00.00.01.20.0

limit/upper limit)

H2

14.9/22.213.4/40.83.8/30.91.6/4.52.2/17.6

32.8/33.010.4/35.36.7/23.2

96.397.5

Total

93/10094/11094/10093/10290/9887/9299/106

100/10598

101

a Ref. 23.b Concentration of KHCO3: 1.0 mol/dm3 for Cu electrode and 0.5 mol/dm3 for other electrodes.c Current density: 5.5 mA/cm2.d Current density: 5.0 mA/cm2.

§-


and thus the expected maximum current density does not exceed10 mA/cm2. When values of 10~5 cm2 s"1 for the diffusion coefficientof CO2 and 10~3 cm for the thickness of the diffusion layer areassumed, the maximum current density for CO2 reduction toformic acid can be estimated to be ca. 7 mA/cm2. One effectiveway to increase the concentration of CO2 in the solution is touse a high CO2 pressure. Ito et al.21 have examined CO2 reductionunder high CO2 pressures (0-20 kg/cm2). In aqueous Li2CO3 solu-tions under CO2 pressures up to 5 kg/cm2, the higher the CO2

pressure, the greater the current density and also the currentefficiency for formic acid production. When CO2 was reduced inaqueous solutions of tetraalkylammonium salts under a CO2 pres-sure of 10 kg/cm2 at In, Sn, Pb, and Pb-Hg electrodes, the mainproduct was formic acid, but small amounts of propionic acid andn-butyric acid and a trace amount of oxalic acid were detected aswell (Table 2).

In organic aprotic solvents, advantages for CO2 reductionwould be expected; hydrogen evolution which competes with CO2

reduction can be suppressed, and the solubility of CO2 is muchhigher28"30 in organic solvents than in aqueous solutions, althoughthe latter point has not been stressed. Studies on CO2 reduction innonaqueous solvents have been carried out both from electro-analytical31"33 and electrosynthetic34"39 viewpoints, but such studiesare still limited.

Von Kaiser and Heitz34 reported the formation of oxalic acidfrom CO2 in propylene carbonate (PC) and acetonitrile using aCr-Ni-Mo (18:10:2)-steel electrode. Addition of small amountsof water gave rise to additional products, such as glycolic, glyoxylic,tertaric, malic, and succinic acids. Use of a platinum electrodefavored CO formation. N,AT-Dimethylformamide (DMF) was alsoused by Tyssee et al35 and Gambino and Silvestri.36 Gressin et al31

reported that oxalate and carbon monoxide have been observedtogether with formate in nonaqueous solvents, and addition of smallamounts of water favored not only the formation of formate ion,but also glycolate production by further reduction of oxalate.Fischer et a/.38 examined oxalic acid production in various organicsolvents. Optimum current efficiencies for oxalic acid production(ca. 90%) were obtained by preparative-scale electrolysis using anundivided cell with a sacrificial Zn anode and a stainless steel

Table 2Products Obtained by Electrochemical Reduction of Carbon Dioxide under High Pressure of 10 kg/cm2 Gage"

Cathode Electrolyte

QuantityCathode of Currentpotential electricity efficiency^

(V) (C) (%)

Concentration of carboxylic acid ( x 10 3 mol/dm3)

Formic Oxalic Malonic Propionicacid acid acid acid

n- Butyricacid Unknown

Pb

Sn

In

0.10MTBABraq

0.10 M TEAPaq

0.18MTEAPa q

-Hg 0.10 M TEAPaq

0.10 M TBABraq

0.18 M TEAPaq

0.10 M TEAPaq

0.10 M TEAPaq

0.10 M TBABraq

Ref. 27.

-1.6-1.6-1.8-2.2-1.7-1.8-1.8-1.9-1.9-2.3

-1.7-2.0

-1.6-1.8-2.0-2.0-1.6-1.8-1.6-2.0

-1.5-2.0-1.7-2.0

1000196436272000200020004213280020002583

20011650

200200

200040002501456620002000

2000200

2000200

34.931.827.445.168.670.347.847.876.068.6

60.686.0

34.749.542.338.448.842.450.530.0

49.065.034.065.2

90162257232355364522347394451

314368

1826

269398316502262155

25434

17634

tc

4.4tt

tt

tt

tt

t

1

1

1.89.73.92.9

1.8

1.5

t,t,t3

1.7

IH

of HCOOH formation.

Table 3Electrochemical Reduction Products of CO2 at Various Cathodes in 0.10 M TEAP/DMSOa

Cathode

In

ZnSnPb

Potential(V)

-1.5-1.7-2.0-2.0-2.0

Currentefficiency**

(%)

1.81.64.88.252.3

Oxalicacid

1.60.80.72.8

110.0

Tartaricacid

0.7

4.2

Concentrations ( x

Malonicacid

1.50.81.23.80.8

Glycolicacid

1.70.87.92.14.2

10~3 M) c

Formicacid

2.83.211.517.220.1

Propionicacid

6.01.04.9

n- Butyricacid

2.2

2.52.3

a Ref. 40.b Total current efficiency of H2C2O4 and HCOOH formation.c When catholytes used are 20 cm3 and quantity of electricity passed is 965 C.

336 Isao Taniguchi

cathode in acetonitrile with tetrabutylammonium perchlorate.Later, the performance of various cathode materials, catholyteformulations, and cell designs were investigated39 for the formationof oxalic acid from CO2 in DMF under both atmospheric andelevated pressures of CO2.

Ito et al40 examined the electrochemical reduction of CO2 indimethylsulfoxide (DMSO) with tetraalkylammonium salts at Pb,In, Zn, and Sn under high CO2 pressures. At a Pb electrode, themain product was oxalic acid with additional products such astartaric, malonic, glycolic, propionic, and ^-butyric acids, while atIn, Zn, and Sn electrodes, the yields of these products were verylow (Table 3), and carbon monoxide was verified to be the mainproduct; even at a Pt electrode, CO was mainly produced innonaqueous solvents such as acetonitrile and DMF.41 Also, theproducts in propylene carbonate42 were oxalic acid at Pb, CO atSn and In, and substantial amounts of oxalic acid, glyoxylic acid,and CO at Zn, indicating again that the reduction products of CO2

depend on the electrode materials used.

2. Mechanisms of Electrochemical Reduction of Carbon Dioxide

Mechanisms of carbon dioxide reduction in both aqueous andnonaqueous solutions have been studied mainly at metal electrodes.

Van Rysselberghe et al.43 reported the fundamentally veryimportant result that carbon dioxide (CO2) molecules, not bicarbon-ate (HCO^) or carbonate (CO2) ions, were the reacting speciesat a mercury electrode. Recently, Hori and Suzuki44 have studiedthe electrolytic reduction of bicarbonate ion at Hg in aqueousNaHCO3 and Na2CO3 mixtures; formate ion was formed, and theauthors concluded that the dissociation of bicarbonate took placeto give the electroactive species, CO2 molecules. The maximumlimiting current, controlled by the dissociation of HCO^ to CO2

(kd = 6.8 x 10~4s~1), was estimated to be ca. 4 mA/cm2.The works of Eyring and co-workers45 were the first to discuss

CO2 reduction on mercury in detail on the basis of the analysis ofpolarization curves obtained in aqueous solutions. The polarizationcurve showed two regions with different Tafel slopes: region I (lowovervoltage with a Tafel slope of ca. 90 mV/ decade) and region II(high overvoltage with a Tafel slope of more than 200 mV/decade).


The reaction orders with respect to CO2 concentration were almostzero and one in regions I and II, respectively. From these results,the following mechanism was proposed:

ds (1)

C O i a d s + H 2 O -> H C O 2 a d s + O H " (2)

H C O 2 a d s + e~ -> HCO2 (3)

The overall reaction was

C O 2 + H 2 O + 2e~ -> H C O 2 + O H " (4)

where Eq. (3) was the rate-determining step (r.d.s.) in region I,while in region II the first step was the r.d.s.

Ito et a/.18 suppor ted the above reaction pathways for variouscathode materials , such as In, Sn, Cd, and Pb, from the similarityin Tafel slopes. Hori and Suzuki4 6 verified the above mechanismin various aqueous solutions on Hg. Russell et al19 also agreedwith the above mechanism. Adsorbed C O J anion radical was foundas an intermediate at a Pb electrode using modula ted specularreflectance spectroscopy.4 7 This intermediate underwent rapidchemical reaction in an aqueous solution; the rate constant forprotonat ion was found to be 5.5 M " 1 s"1, and the coverage of theintermediate was est imated to be very low (0.02).

Kapus ta and Hackerman 2 2 also repor ted that the reactionpathways on Sn and In electrodes were similar to those postulatedon an Hg electrode. In the case of the Sn electrode, two Tafelregions were observed. One was at potentials less negative than- 1 . 4 5 V versus SCE with a slope of 115 m V / d e c a d e , and the otherwas at more negative potentials than - 1 . 4 5 V with a slope of320 ± 20 mV. At the In electrode, a single Tafel line with a slopeof 140 mV was obtained. The reaction order was one with respectto C O 2 concentrat ion over the entire potential range tested, andthe partial current for C O 2 reduct ion in the region of low overvoltagewas independent of the p H of the solution (pH 1-6.5). These resultswere similar to those observed by the Zakharyan et a/.48 Thecoverage of the electrode by adsorbed species was confirmed to beless than 5% by both capaci tance and potential decay measure-ments.2 2 Thus , the observed Tafel slope, ca. 1 2 0 m V / d e c a d e , andthe independence of p H observed for the partial current for C O 2

338 Isao Taniguchi

reduction were interpreted in terms of Eqs. (l)-(3), with the dis-charge of CO2 [Eq. (1)] as the r.d.s., by assuming Langmuir adsorp-tion isotherms for the adsorbed species. Since the product of CO2

reduction was formate ion, using the value of the reversible potentialfor CO2 + H2O + 2e~ -> HCOJ + OH~, -0.76 V vs. SCE at pH 7,the exchange current densities (A/cm2) were estimated to be 5 xKT11 (at Hg), 1 x 10~9 (at Sn), and 1 x 1(T8 (at In).

Results of photoemission experiments also support the reactionmechanism on both In and Sn as being the same as on Hg,49 andindirect chemical reduction of CO2 with adsorbed hydrogen atomsformed by electrochemical reduction of H+ was suggested to beinvolved. An electrochemically modulated IR spectroscopic investi-gation at a Pt electrode also showed that the reduction of CO2

involved a reaction with adsorbed hydrogen.50 The laser photoelec-tronic emission technique has recently been used to elucidate thereduction mechanism of CO2 on mercury.51 COJads was the onlyintermediate of the two-electron reduction of CO2 on mercury atpH values greater than 3, and the electrode coverage by the inter-mediate at potentials more negative than -1.8 V (versus SCE) wasreported to be negligible (ca. 10~5). Also, the free energies of theelectrode reactions CO2 (gas) + e~ -» COJ (aq) and CO2 (gas) -fe~ -* COJads were calculated to be 2.05 ± 0.12 and 1.60 eV, respec-tively. Later, Vassiliev et al.52 also examined the mechanism andkinetics of electroreduction of CO2 using various metals with highand moderate hydrogen overvoltage. Electrochemical and photo-emission measurements showed basically the same mechanism asthat described above; the effect of the electrode material wasexplained in terms of the adsorption of CO2 molecules initially andof the anion radicals, COJ, produced as well as the effect ofelectrode potential on adsorption behavior of these species. Fromthese results, the optimal electrodes for electroreduction of CO2

were suggested to be those with moderate hydrogen overvoltages;on these electrodes, CO2 reduction took place near the zero-chargepotential, where the maximum adsorption of CO2 on the electrodecan be expected.

As described above, the mechanism for CO2 reduction inaqueous solutions proposed by Eyring and co-workers45 has beenwidely accepted. However, there remains a rather large differencebetween theoretical and observed values for the Tafel slope,45 and


recent measurements on the intermediate and its coverage on theelectrode are not completely consistent with this mechanism. Addi-tional precise experiments and discussion would be required.

In nonaqueous solvents, reduction pathways of CO2 shouldbe different from those in aqueous solutions since different productsare obtained. In dry propylene carbonate, the optical absorptionspectrum observed during reduction of CO2 at a Pb electrodeshowed two strong bands47: The band at 285 nm in PC was attributedto COJ, and the band at longer wavelengths to a second intermedi-ate, but not oxalate. Thus, the following mechanism, involving astep in which COJ is attacked by a CO2 molecule [Eq. (6)], wassuggested:

CO2+e "^COJ(slow) (5)

CO2 4- COJ -> (CO2)J (6)

(CO2) J + e~ -> (CO2 )2 (7)

The first step [Eq. (5)] was postulated to be rate determining becauseof the Tafel slope of 107 mV/ decade and the first-order dependenceof the reduction current on the CO2 concentration. The second-order rate constant of Eq. (6) was estimated to be 7.5 x 103 M"1 s"1.

Lamy et al32 measured the standard potential and kineticparameters for electrochemical reduction of CO2 (1-8 mM) inN,iV-dimethylformamide, from which residual water was deacti-vated using an active alumina suspension, by cyclic voltammetrywith a high scan rate (4400 V/s). They reported a standard redoxpotential for CO2/COJ of -2.21 V versus SCE, a standardheterogeneous rate constant for electron transfer of 6 x 10 3 cm s"1,a transfer coefficient of 0.4, and a dimerization rate constant forCOJ of 107 M~l s"1. In this case, dimerization of COJ was con-sidered, and the values obtained indicate that the r.d.s. in the overallCO2 reduction was the charge transfer to CO2. More recently,Amatore and Saveant53 have discussed the mechanism of CO2

reduction in media of low proton availability, such as DMF, onthe basis of product distribution with the aid of a theory54 relatingthe product distribution to the intrinsic (rates, diffusion coefficients)and operational (concentrations, current densities, thickness ofdiffusion layer) parameters of the system. Three competing reaction

540 Isao Taniguchi

pathways were suggested: (1) Oxalate formation through self-cou-pling of CO J (k = ca. 107 M~l s"1), (2) carbon monoxide formationvia an oxygen-carbon coupling of COJ with CO2 (k = 3.2 x103 M~x s"1), and (3) formate formation through protonation ofCOJ by residual or added water (k = 7.7 x 102 M~l s"1), followedby homogeneous electron transfer from COJ:

(8)

Oxalate formation:

2COJ -> (COJ)2 (9)

Carbon monoxide formation:

O"

COJ + CO2-» C - O - C (10)

o \O" 0"

/ /C-O-C +COJ-> C - O - C +CO2 (11)

/ \ S X0 0 0 O

O"

C-O-C ^CO + COr (12)

S Xo o

Formate formation:

COJ + H20 -* HCOO* + OH" (13)

HCOO* + COJ -> HCOO" + CO2 (14)From the analysis of the experimental results, the authors concludedthat the reaction pathway suggested by Aylmer-Kelly et al47 foroxalate formation via carbon-carbon coupling of COJ with a CO2

molecule was unlikely, but that the dimerization given by Eq. (9)was more probable. This conclusion seems to be in good agreementwith ESR data55 and with the fact that the anion radical producedby oxidation of oxalate ion was unstable in acetonitrile56; if oxalatewere formed by coupling of COJ with a CO2 molecule followed


by further electron transfer, the oxalate anion radical would bemore stable. Moreover, the authors pointed out that the reason whyoxalate formation was higher on lead than on mercury (where COwas the main product) was not due to a specific chemical effect ofthe metals on the course of CO2 reduction, but to a difference inthe current density used for electrolysis. More recent studies ofCO2 reduction in several nonaqueous solvents including DMF andacetonitrile using various metal electrodes41'42 showed that Pb seemsto have a certain catalytic activity for oxalate formation that re-inforces the effect of current density on the distribution of reductionproducts of CO2. Vassiliev et al. have reported57 that in aproticsolvents the r.d.s. was the electron transfer to (CO2) J at Hg, Pb,Sn, In, and Pt electrodes in the first Tafel region, while in the secondTafel region the first electron transfer to an adsorbed CO2 moleculewas the r.d.s. The effect of potential on adsorption of CO2 andanion radicals and on repulsion of negatively charged radicals wasalso suggested. Furthermore, on Pt and Rh, tightly chemisorbedspecies were proposed.

As described above, the mechanism of CO2 reduction is stillunclear. The establishment of unequivocal reaction pathways onthe basis of a better understanding of each elementary step wouldmake it possible to develop an effective means for reduction ofCO2. Investigations employing recently developed spectroscopictechniques, such as FTIR spectroscopy,58 electrochemically modu-lated IR spectroscopy (EMIRS),47 in situ IR reflection spectros-copy,59 photoemission spectroscopy,22'51'52'57 and Raman spectros-copy60 and SERS61 for adsorbed species, together with furtherstudies by techniques used so far, such as flash photolysis,62'63

ESR,64 and electrochemical measurements,65'66 would help to clarifythe reaction intermediates and the reaction mechanism. Recently,EMIRS and FTIR reflection-absorption spectroscopy (FTIRRAS)have been used extensively to study the adsorption and oxidationbehavior of CO,67'68 formic acid,69 and methanol70'71 at variousmetal electrodes. Although the studies done to date are not directlyintended to clarify the mechanism of CO2 reduction, knowledgeabout these species at the electrode surface would be helpful tounderstand the fundamental steps of CO2 reduction at the molecularlevel; these studies showed that dissociative adsorption of thespecies which appear in the reduction pathways of CO2 on the

342 Isao Taniguchi

electrode surface plays an important role in the mechanism of CO2

reduction, and thus in the reduction product distribution.For the reaction of formic acid in aqueous solutions, the

adsorption of formic acid on the electrode was suggested to be therate-determining step at Sn and Pb electrodes.19 The point of zerocharge (pzc) of the electrode was suggested to have a significantimportance, and cadmium was recommended as the cathodematerial for formic acid reduction because of its low value of thepzc. Kapusta and Hackerman22 examined formic acid reduction tomethanol at Sn and In electrodes. The highest current efficiency(ca. 95%) for reduction to methanol was obtained at a tin electrodeat a low current density (ca. 5 /x A/cm2), corresponding to a potentialof -0.95 V versus SCE in a 0.5 M HCOOH + 0.5 M HCOONasolution. However, the formation of an Organometallic complex onthe electrode surface accelerated hydrogen evolution and the cur-rent efficiency of formic acid reduction decreased with time duringelectrolysis. From photoelectrochemical measurements, the authorsconcluded that the rate-determining step in HCOOH reduction atboth Sn and In electrodes was the first electron transfer to theHCOOH molecule, and an HCO# radical was involved as an adsor-bed intermediate.

For reduction of formaldehyde on mercury, the Tafel slopedecreased with an increase in either the formaldehyde concentration(at constant pH) or the pH of the solution (at constant HCHOconcentration).19 The experimental results were in basic agreementwith the previously proposed mechanism,72'73 where formaldehydeis present predominantly in an electroinactive hydrated form,methylene glycol, which undergoes base-catalyzed dehydration togive electroactive formaldehyde. From the Tafel slopes obtained(66-36 mV at pH 6.8-13.0), the authors concluded that it was notthe first electron transfer to the electroactive formaldehyde moleculebut the final step to methanol that was rate determining (the so-called CECE mechanism), although polyoxymethylene glycol pres-ent as an impurity in formaldehyde made the mechanistic studycomplicated and ambiguous. Electrolysis of formaldehyde at a tinelectrode74 was also examined, but a tin complex formed on theelectrode surface made hydrogen evolution more favorable thanHCHO reduction. At a bright Pd cathode,75 electrochemical reduc-tion of bicarbonate, HCO^, was suggested to take place.


3. Pathways for Carbon Dioxide Reduction

If carbon dioxide is reduced directly to give products of interest,the reduction potentials for the half-cell reactions in an aqueoussolution of pH 7 are as follows:

CO2(g) + 2H+ + 2e~ -* HCOOH(aq)

E°=-0.61 (15)

CO2(g) + 4H+ + 4e~ -> HCHO(aq) + H2O

£°=-0 .48 (16)

CO2(g) + 6H+ + 6e~ -* CH3OH(aq) + H2O

£°=-0 .38 (17)

CO2(g) + 8H+ + 8e" -> CH4(g) + 2H2O

E° = -0.24 (18)

CO2(g) + 2H+ + 2e~ -> CO(g) + H2O

E°=-0.52 (19)

2CO2(g) + 2H+ + 2e~ -> H2C2O4(aq)

E° = -0.90 (20)

where the standard redox potentials2 are given in volts versus NHEat pH 7.0, and (g) and (aq) denote the gaseous state and aqueoussolution, respectively.

In aprotic solvents, the reduction of carbon dioxide to formcarbon monoxide and carbonate ion2 is also possible:

2CO2(g) + 2e~ -> CO(g) + CO2," E° = -1.07 (21)

These reactions can be easily combined, if necessary, with ananodic reaction such as oxygen evolution to estimate the thermo-dynamic standard free energy, AG:

2H2O -> O2 4- 4H+ + 4e~ E° = 0.81 (at pH 7) (22)

The values of E° for Eqs. (15)-(20) indicate that if multielectronreductions of CO2 take place, for example, by using suitablecatalysts, the potentials required are much less negative than thatfor single-electron transfer, CO2/COJ, and are also less negative

344 Isao Taniguchi

than the potentials actually required for CO2 reduction (ca. -2.0 Vor more negative). Thus, thermodynamically, but not kinetically,CO2 reduction is comparable in difficulty to hydrogen evolution.For this reason, attempts to find catalysts for multielectron reactionsand to use semiconductor electrodes as multielectron donors havebeen subjects of active interest. Further understanding of theelementary step in CO2 reduction would, hopefully, lead towardsophisticated methods for CO2 reduction.

A calculation of the temperature dependence of the free energyfor the reactions in Eqs. (15)-(18), and hence the electrochemicalpotential, showed that with an increase in temperature, formic acidformation became more unfavorable.4 In the case of formaldehyde,methanol, and methane formation, the calculation indicated a posi-tive shift in the reduction potential, but of very small magnitude:ca. 30 mV for a temperature change from 300 to 500 K, and ca.20 mV from 500 to 1200 K.4

4. Reduction of Carbon Dioxide at SemiconductorElectrodes in the Dark

Semiconductor electrodes seem to be attractive and promisingmaterials for carbon dioxide reduction to highly reduced productssuch as methanol and methane, in contrast to many metal electrodesat which formic acid or CO is the major reduction product. Thispotential utility of semiconductor materials is due to their bandstructure (especially the conduction band level, where multielectrontransfer may be achieved)76 and chemical properties (e.g., CO2 iswell known to adsorb onto metal oxides and/or noble metal-dopedmetal oxides to become more active states77"81)- Recently, severalreports dealing with CO2 reduction at n-type semiconductors inthe dark have appeared, as described below.

Augustynski and co-workers82 showed by cyclic voltammetrythat at an n-TiO2 electrode in a 0.5 M KC1 solution (pH 6), CO2

reduction took place at potentials less negative than those forhydrogen evolution. At both n-TiO2 and ruthenium (1 at. %)-dopedTiO2 electrodes, reduction of carbon dioxide to methanol wasachieved by long-term electrolysis at -0.9 V (versus SCE), althoughno information about the faradaic yield of methanol was given.Inoue et al.83 also reported that CO2 was reduced at a TiO2 electrode


to a mixture of formic acid, formaldehyde, methanol, and methanein acidic solutions.

Recently, however, Tinnemans et a/.84 have questioned theresults of Augustynski and co-workers82 and claimed that the largercurrent obtained in a CO2-saturated solution compared to a N2-saturated solution was not due to CO2 reduction, but rather tohydrogen evolution, with the potentials at which hydrogen evolutionoccurred shifted toward less negative potentials because of thechange in pH in the vicinity of the electrode (Fig. 3). Also, theyreported that long-term electrolysis at -1.0V versus SCE of aCO2-saturated 0.1 M acetate buffer solution at a TiO2/RuO2

(0.5 wt %) cathode gave a mixture of formaldehyde and methanolbut with a current efficiency of at most ca. 1%. Thus, the authorssuggested that CO2 reduction took place by reaction with adsorbedhydrogen generated by the photoassisted decomposition of water.Reactions of formic acid and formaldehyde with adsorbed hydrogenat polycrystalline semiconductor materials were also suggested.85

In commenting on the observations of Tinnemans et a/.,84

Augustynski remarked86 that the importance of the marked affinityof the hydrated TiO2 for CO2 was apparent from anodic peaksobserved on the voltammograms obtained on the reverse sweepafter scanning up to a sufficient negative potential at which CO2

KmA)

0.2

0.0

-0.2

-0.4

f '/W "1 / 1

--- o.— o.

0.0.

5M

1/0

KC1

KC1

KC1

.1M

N2-sat.

C02-sat.

N2/C02-sa

HAc/Ac"

PH

PH

t. pH

6

4

4

•3m£m

.0

.0

.75

-1.0 -0.6 -0.4 -0.2 0

V vs. SCE

Figure 3. Cyclic voltammograms at a TiO2/RuO2 (0.5 wt%)electrode in various solutions.84 Scan rate: 0.01 V/s.

346 Isao Taniguchi

reduction occurred. In addition, the following differences betweenthe results of his group and those of Tinnemans et al. were pointedout: (1) the potential range used by Tinnemans et al.S4 for cyclicvoltammetry was too narrow to find the reduced species of CO2 atthe electrode surface, (2) the acetate and formate buffer solutionsused by Tinnemans et al. introduced confusion in interpreting thei-E curves because the species contained within these buffer solu-tions underwent cathodic reduction at potentials less negative thanthe hydrogen evolution reaction, and (3) the nature of the oxideelectrodes used by these two groups would be different from eachother because of the different preparation procedures employed.

In this connection, cyclic voltammetric measurements on theelectrochemical reduction of CO2 at w-TiO2 and platinized TiO2

film electrodes were reported a little later by Augustynski andco-workers.87 The existence of two electrochemically detectablespecies resulting from CO2 reduction was suggested by anodic peakson the cyclic voltammograms (Fig. 4). Unfortunately, however, no

E (V v s . 0 . 1 N CE) E (V v s . 0 . 1 N CE)

Figure 4. Cyclic voltammograms at Pt-treated TiO2 (a) and TiO2 (b)electrodes in a CO2-saturated 0.5 M KC1 solution at 40°C.87 Scan rate:0.05 V/s.


quantitative analysis of the products of CO2 reduction was givenin this report and the above conflict is still not completely resolved.More recently, Miles et a/.88 have concluded that in aqueous solu-tions containing formic acid, the predominant reaction at TiO2 andother metal electrodes such as Ti, In, Ag, and Pt is the reductionof H3O

+ rather than HCOOH or HCO ~; electrochemical conversionof CO2 into methanol via formic acid is unlikely. To understandbetter the mechanism of CO2 reduction at semiconductor electrodesand to develop effective semiconductor cathodes for CO2 reduction,further experiments including quantitative analysis of products arerequired.

Canfield and Frese89 used M-GaAs for CO2 reduction tomethanol. The faradaic efficiency for methanol formation wasexamined as a function of crystal face, electrolyte, and currentdensity. The highest efficiency (almost 100%) for methanol forma-tion was obtained using the As(lll) face of w-GaAs in a 0.2 Mreagent-grade Na2SO4 solution at a current density of 0.16 to0.2 mA/cm2 (-1.2 to -1.4 V versus SCE). On the other hand, whena solution of ultrapure (99.999%) Na2SO4 prepared with 1.6 x107 il-cm water was used, no methanol was obtained. At the Ga( 111)face of n-GaAs, methanol was produced in both electrolytesdescribed above at lower faradaic efficiencies (14-80%). Also, the(100) and (110) faces of «-GaAs gave low yields of methanol(0-14%). Methanol synthesis was reported to be limited by a surfacechemical step involving adsorbed hydrogen and an unidentifiedintermediate, such as COad, —COH, — CH—OH, —CH2—OH,and —O—CH3, with a chemical rate constant of 6.1 x 10"5 A/cm2

for the rate-determining step. Frese and Canfield90 also observedthe effectiveness of a surface pretreatment of w-GaAs (111) withRu(III) for CO2 reduction to methanol. Later, Sears and Morrison91

indicated that GaAs dissolved (by corrosion) in carbonic acidsolutions reacted with CO2 to form hydroxides of gallium andarsenic plus methanol (or possibly formaldehyde).

Since indium is one of the most effective metals forelectrochemical reduction of CO2, n-TiO2 on which indium hadbeen electrodeposited was examined.92 Enhancement of thefaradaic efficiency of CO2 reduction by one order of magnitude ormore compared to that at undoped n-TiO2 was observed, but theProduct detected was mainly hydrogen with a small amount (<5%

348 Isao Taniguchi

in faradaic efficiency) of formic acid in aqueous tetraethylam-monium perchlorate solutions.

These results suggest that further characterization of the elec-trode surface is required to obtain more reproducible results, but,on the other hand, surface modification would be a very promisingapproach to the reduction of CO2 with high selectivity andefficiency.

Although further experiments are required to establish theutility of semiconductor electrodes for CO2 reduction directly togive highly reduced compounds, the results of gas phase reactionsusing semiconductor materials support the potential reduction ofCO2 to highly reduced products. For example, Hemminger et al93

reported that methane was directly obtained from gaseous waterand CO2 adsorbed on strontium titanate (111) crystals that were incontact with platinum foils by illumination with light of energygreater than the band gap of the semiconductor or by heating to420 K in the dark; surface Ti3+ ions were proposed to act as acatalyst.

In nonaqueous solvents, little has been published to datedealing with CO2 reduction at n-type semiconductors. Tinnemanset al84 suggested oxalic acid formation from CO2 in DMF andDMSO at n-TiO2/RuO2 (0.5 wt%) by cyclic voltammetry. At ann-TiO2 electrode, CO was obtained as the main product,92 with afaradaic efficiency of ca. 80%, by electrolysis at -2.5 V versusAg/Ag+ in CO2-saturated acetonitrile with 0.1 M Et4NClO4, whilewith the introduction of Pb onto an n-TiO2 electrode, oxalate wasalso obtained. However, deposition of small amounts of metalssuch as In, Pt, Rh, Pd, and Ru onto an n-TiO2 electrode did notsignificantly affect the reduction product of CO2; again CO wasformed predominantly.

Thus, although the potential required for polarization wouldbe much larger at n-type semiconductors than at illuminated p-typesemiconductors and despite the fact that not all n-type semiconduc-tors can be used because of corrosion (or reduction) of semiconduc-tor materials themselves, the use of n-type semiconductors toexamine CO2 reduction seems to be indicated because the cathodiccurrent is much larger (the electron is the major carrier for n-typesemiconductors), approaching that of metal electrodes, comparedto the photocurrent obtained at illuminated p-type semiconductors,


so that long-term electrolysis to give detectable amounts of productsrequires less time. Knowledge about the effective surface structurefor CO2 reduction at n-type semiconductors thus obtained wouldalso be applicable for p-type semiconductor electrodes.

III. PHOTOELECTROCHEMICAL REDUCTION OFCARBON DIOXIDE

1. Reduction of Carbon Dioxide at Illuminated p-TypeSemiconductor Electrodes

At illuminated p-type semiconductors, light energy brings about ashift of the applied cathode potential at which CO2 reduction takesplace toward a less negative potential by the photovoltaic effect.94

Thus, light energy can be used to reduce the apparent overpotentialof CO2 reduction.

Halmann reported in 1978 the first example of the reductionof carbon dioxide at a p-GaP electrode in an aqueous solution(0.05 M phosphate buffer, pH 6.8).95 At -1.0 V versus SCE, theinitial photocurrent under CO2 was 6 mA/cm2, decreasing to1 mA/cm2 after 24 h, while the dark current was 0.1 mA/cm2. Incontrast to the electrochemical reduction of CO2 on metal elec-trodes, formic acid, which is a main product at metal electrodes,was further reduced to formaldehyde and methanol at an illumi-nated p-GaP. Analysis of the solution after photoassisted electroly-sis for 18 and 90 h showed that the products were 1.2 x 10~2 and5 x 10~2 M formic acid, 3.2 x 10~4 and 2.8 x 10"4 M formaldehyde,and 1.1 x 10~4 and 8.1xlO"4M methanol, respectively. Themaximum optical conversion efficiency calculated from Eq. (23)for production of formaldehyde and methanol (assuming 100%current efficiency) was 5.6 and 3.6%, respectively, where the biasvoltage against a carbon anode was -0.8 to -0.9 V and 365-nmmonochromatic light was used. In a later publication,4 these valueswere given as ca. 1% or less, where actual current efficiencies weretaken into account [Eq. (24)].

Also, using n-TiO2 as an anode and p-GaP as a cathode in0.1 M lithium carbonate solution, under illumination on both elec-trodes, methanol was produced (3 x 10~3 mol) at a current efficiency

350 Isao Taniguchi

of 60% by electrolysis for 16 h at a constant current of 0.5 mA(2.1 mA/cm2 on the p-GaP cathode). During electrolysis, negativebias to the cathode gradually increased from —0.86 to —1.4 V (versusSCE) to maintain this current. Inoue et al reported a little later83

that formaldehyde and methanol were formed on a />-GaP electrodeat -1.5 V versus SCE under illumination with light of wavelengthshorter than 500 nm in a 0.5 M H2SO4 solution.

To calculate the optical to chemical energy conversionefficiency, Halmann95 used the following equation:

Optical conversion efficiency (%)

= \00Ic[(AH/n)-Vb]/W (23)

where Ic is the current density (mA/cm2), W (mW/cm2) the incidentlight intensity, AH (eV) the heat of combustion ( = 2.962, 2.639,5.315, and 7.259 for hydrogen, formic acid, formaldehyde, andmethanol, respectively), n the number of electrons involved in thereduction of one molecule of reactant to one molecule of product( = 2, 2, 4, and 6 for producing hydrogen, formic acid, formal-dehyde, and methanol, respectively), and Vb (V) the bias voltagebetween the photocathode and the counter electrode.

In a later publication,96 the standard free energy of formationof the products, AG in V, was used instead of AH in Eq. (23) sothat comparisons could be made with the commonly reportedefficiencies of solid state solar cells. For the reduction of carbondioxide to organic compounds, the optical conversion efficiency ofthe system is the sum of the efficiencies for each product. Thus, itcan be given as

Optical conversion efficiency (%)

= 100JcF;[(AG/n) - Vby W (24)

where the values of AG/n for reduction of CO2 to formic acid(liquid), formaldehyde (gas), and methanol (liquid) are 1.48, 1.35,and 1.21 V, respectively.

Another expression which has been used relates the extent ofconversion of the total input energy (both electrical and optical)to chemical energy:

Power conversion efficiency (%)

= 100Jc^(AG/*)/(JcVb + W) (25)


where IcVb represents the electrical energy input. Note that use ofEq. (25) results in much larger values than are calculated using Eq.(23) or (24), and no negative values appear.

Canfield and Frese89 showed that at the As(lll) plane ofp-GaAs and the P(lll) plane of /?-InP, CO2 was reduced tomethanol in aqueous Na2SO4 solutions at -1.2 to -1.4V versusSCE. The highest faradaic efficiency (ca. 80%) was obtained at aphotocurrent density of 60/xA/cm2 (-1.2 to -1.4 V versus SCE)using a p-lnP cathode.

Aurian-Blajeni et al97 examined CO2 reduction at illuminated(600 mW/cm2 was used) /?-GaAs and p-GaP under high CO2 pres-sures, using a specially designed cell (Fig. 5). The products weremainly formic acid with small amounts of formaldehyde andmethanol. The best faradaic efficiency of 80% was obtained under8.5 atm pressure of CO2 using /?-GaP at -1.0 V versus Ag/AgCl ina 0.5 M Na2CO3 solution. The main difficulty reported was theinstability of the electrodes, especially in the case of /?-GaAs.

Since /?-GaAs has a suitable band gap (ca. 1.4 eV) for the solarspectrum, it is an attractive material for solar energy utilization.Zafrir et al.96 examined CO2 reduction at /?-GaAs in the presenceof the vanadium redox couple V(II)/V(III) to overcome the prob-lem of corrosion of the electrode, because /?-Si and p-lnP weresuccessfully stabilized by introduction of the redox couple.98 Sincea V(II)/V(III) chloride solution has a violet to blue color, some ofthe incident light was absorbed by the solution. The highest photo-current at /?-GaAS was obtained in 4 M HC1 solutions havingvanadium ion concentrations of less than 0.1 M. The highest opticalenergy conversion efficiency of 0.21%, calculated from Eq. (24),was obtained in a 4 M HC1 solution containing 0.07 M V(II) at80°C under irradiation with a light flux of 75 mW/cm2. The productsof CO2 reduction observed were formic acid, formaldehyde, andmethanol. In this case, homogeneous catalytic reduction of CO2

with the vanadous ions and also reduction with adsorbed hydrogenatoms formed by photoelectrolysis of water were suggested to beconceivable possibilities, in addition to direct CO2 reduction onP-GaAs.

An interesting result which questions the necessity of metalions for catalysis of CO2 reduction was reported;99 at a polyaniline-coated p-Si electrode, CO2 was effectively reduced to formic acid

352 Isao Taniguchi

Figure 5. Design of a cell for photoassisted electrolysis of CO2 under elevatedpressures.97 (1) Photoelectrode; (2) reference electrode; (3) counter electrode; (4)sampling port with septum; (5) pressure regulator; (6) pressure gauge; (7) O-rings;(8) reaction cell; (9) separator; (10) quartz window; (11) insulated connection; (12)bolts; (13) connections to potentiostat.

and formaldehyde in an aqueous CO2-saturated LiClO4 solution,although the origin of the catalytic activity was unclear.

Taniguchi et al.100 have reported that in the reduction of CO2at/?-GaP in Li2CO3 electrolytes, the current efficiency was enhancedby dissolving 15-crown-5 ether in the electrolyte. The proposedreaction pathway involved the initiation of CO2 reduction by


cathodically deposited Li metal on the GaP surface to give COJ:

Li + CO2 -> Li+ + COO" (26)

where the crown ether facilitated the deposition of Li on the /?-GaPelectrode, and adsorbed crown ether retarded the hydrogen evolu-tion reaction. The reduction of CO2 molecules with lithium metalwas confirmed in propylene carbonate to give COJ, which wasdetected by its absorption spectrum (Amax = 265 nm). In this case,the optical energy conversion efficiency calculated from Eq. (24)was very small (ca. 0.001%) or negative, and even the powerconversion efficiencies [Eq. (25)] were ca. 0.01%.

Recently, results of careful experiments were reported by Itoet al101 They claimed that formic acid, formaldehyde, and methanol,which had been previously reported as photoelectrochemical reduc-tion products of carbon dioxide, were observed also by photolysisof cell materials, such as electrolytes, including 15-crown-5 ether,and epoxy resin, which has often been used as the molding materialof semiconductor electrodes in aqueous solutions. Previouslyreported reduction products were obtained also under nitrogen with(Table 4) and without (Table 5) a /?-GaP photocathode underillumination. These precise experiments under improved condi-tions, where no photolytic products were observed, gave the resultthat the main reduction product of carbon dioxide at a p-GaPphotocathode in aqueous electrolytes was formic acid. Thus, manykinds of products reported in previous papers83'97'100 were suggestedto be due to photolysis of cell materials.

The results of Ito et al.101 indicate that careful experimentsincluding enough blank experiments are necessary in studies ofphotoelectrochemical reduction products of carbon dioxidebecause, unfortunately, the products observed to date are in verylow concentrations. Purification of the carbon dioxide gas itselfshould also be considered, expecially in experiments in which acontinuous flow of CO2 gas is used. Accumulation of organicswhich are present as impurities in CO2 gas is often observed.Purification methods for CO2 gas used are given in somepapers,95"97102 but establishment of a common recommendedmethod would be helpful. Also, it may be advisable to reexamineearlier work on CO2 reduction to exclude meaningless results. Infuture experiments, the use of labeled 13CO2 is to be recommended.

Table 4Photoelectrolytic Products under N2 and CO2 Atmospheres in the Cell with Various Electrolytes and a p-GaP

Photocathode Molded by Epoxy Resin"'*

n (/xmol)

Atmosphere

N2

CO2

Electrolyte

Li2CO3

TEAPTEABrTBABr15-Crown-5

HCOOH

1.51.32.31.96.5

CH3OH + HCHOC

0.040.030.12

tt

(COOH)2

0.1td

0.280.180.25

HOCH2COOH

2.11.34.81.91.5

CH3CHO

0.020.360.90.170.29

C2H5OH

0.020.06

ttt

a Ref. 101.b Photoelectrolysis was performed up to 10 C at -1.2 V versus Ag/AgCl electrode by illuminating with a 300-W xenon lamp alone.c Methanol and formaldehyde were represented as CH3OH + HCHO in Ref. 101 because they were not able to be separated by the

steam chromatographic technique; the amount of product was calculated as an amount of methanol.d t: Trace amount of product.

Table 5Photolytic Products of Various Electrolytes in Quartz Test Tube Illuminated by a 300-W Xenon Lamp Alone

or with the Filter Toshiba UV-37, under N2 Atmosphere"

Filter ofXe lamp

Not used

Used

Time(h)

2528242629

242624

Electrolyte

Li2CO3

TEAPTEABrTBABr15-Crown-5

TEAPTEABr15-Crown-5

HCOOH

nb

nnn

16.7

nn

13.1

CH3OH + HCHO

nnnn

0.55

nnn

n

(COOH)

nnnn

0.22

nn

0.22

(/amol)

2 HOCH2COOH

nnnnn

nnn

CH3CHO

n0.240.280.210.07

nnn

C2H5OH

nnn

0.141.01

nnn

a Ref. 101.b n: No product.

I2,

IO

356 Isao Taniguchi

In photoelectrochemical reduction of carbon dioxide, organicsolvents and their mixtures with water have also been used. Theuse of organic solvents has the advantages103 that (1) competitivehydrogen formation can be suppressed and (2) the increased solu-bility of CO2 in nonaqueous solutions2830 has similar effects to theuse of higher CO2 pressures.

Guruswamy and Bockris104 reported that oxalic acid was quali-tatively detected in a CO2-saturated DMF solution after photoelec-trolysis using an illuminated j?-GaP electrode. Taniguchi et a/.103105

have recently shown that the photocurrent-potential curves at ap~CdTe electrode under monochromatic light (A = 600 nm), in aDMF-0.1 M tetrabutylammonium perchlorate (TBAP) solutioncontaining 5% water, shifted markedly (ca. 0.7 V) toward lessnegative potentials when the bubbling gas was changed from Ar toCO2 (Fig. 6). Controlled-potential electrolysis (-1.2 to -2.4 V ver-sus SCE) under a CO2 atmosphere gave CO in high selectivity. Theilluminated ^-CdTe electrode showed much better performance

-2.0

under CO,

<E

0.0

under Ar—•-/// / / - under CO2 at In

-1.5 -2.0

Potential/V vs SCE

-2.5

Figure 6. Current-potential curves at a p-CdTe electrode in a DMF-0.1 M TBAPsolution containing 5% water under irradiation with monochromatic light of 600 nm,compared with the current-potential curve in a CO2 atmosphere at an In electrode.105

Electrode area: 0.2 cm2 (/?-CdTe) and 1 cm2 (In). Potential scan rate: 0.1 V/s.


than an In electrode, which is the best metal electrode known todate for CO2 reduction in nonaqueous media. In a later publication,various p-type semiconductors were also examined103 in a DMF-0.1 M TBAP solution with 5% water under a CO2 atmosphere. Thephotocurrent (quantum efficiency)-potential curves, the quantumefficiency as a function of wavelength, and the results of photo-assisted controlled-potential electrolysis at various p-type semicon-ductor electrodes are shown in Figs. 7 and 8 and in Table 6. Amongthe p-type semiconductors tested, p-CdTe gave the best results. Thep-Si electrode can be used over a range of wavelengths of the solarspectrum with high quantum efficiencies, and CO2 was reduced toCO with high current efficiency at this electrode, but higher negativepotentials were required for CO2 reduction than at p-CdTe. At bothp-InP and p-GaP electrodes, the current-potential curves showeda more positive onset potential than at /?-CdTe, and good solarenergy utilization was expected from their quantum efficiency-wavelength relationships. However, at p-InP and p-GaP electrodes,

-1.0 -1.5 -2.0

Potential/V vs SCE

-2.5 -3.0

Figure 7. Quantum efficiency versus potential at various p-type semicon-ductors in a DMF-0.1 M TBAP solution containing 5% water under aCO2 atmosphere. Monochromatic light of 600 nm was used for p-Si,p-InP, p-GaAs, and p-CdTe, while light of 400 nm was used for p-GaP.103

Scan rate: 0.1 V/r

358 Isao Taniguchi

p-Si [-2.8 Vvs SC£]p-CdTe [-1.6 V]p-GaP [-2.0 V]p-lnP [-1.8 V]p-GaAs [-1.8 V)

0.0 L900

Figure 8. Quantum efficiency as a function of wavelength (A) for various semi-conductors in a DMF-0.1 M TBAP solution containing 5% water under CO2

atmosphere.103

the faradaic current efficiency for CO2 reduction was low andhydrogen evolution occurred as well. At /?-CdTe, the currentefficiency and product selectivity of the reduction of CO2 to COwere not affected by the water concentration in DMF up to 25%,when 0.1 M TBAP was used as the supporting electrolyte. Thecurrent efficiency was ca. 90% or more when the CO dissolved inthe solutions was taken into account. Various organic solvents canbe used for CO2 reduction to CO at illuminated p-CdTe (Table 7).No remarkable difference in either the current efficiency or theproduct selectivity of CO2 reduction to CO was observed whenvarious tetraalkylammonium salts were used. For CO2 reductionto CO in DMF solutions, the reaction order with respect to CO2

was one and the analysis of the photocurrent-potential curves wasconsistent with the simple expression for the cathodic photocurrentat a semiconductor electrode when the rate-determining step ischarge transfer to CO2 in the presence of surface states.76106107 Onthe basis of these results, the following reaction pathways weresuggested:

(27)

Table 6Photoassisted Controlled-Potential Electrolysis (CPE) in DMF Solutions Containing 5% Water and TBAP

under a CO2 Atmosphere"'*

Run

123456789

Cathodematerials

p-Si(lOO)p-Si(lOO)

p-InP(lOO)p-InP(lOO)p-GaP(lOO)/7-GaP(100)p-GaP(lOO)p-CdTe(lOO)/?-CdTe(100)

Cathodepotential

(V versus SCE)

-2.0-2.0-1.6-1.6-1.6-1.6-2.0-1.6-1.6

Electricitypassed

(C)

4.611.13.66.93.46.25.68.7

16.1

CO

^mol

16.345.9

6.213.95.39.18.9

35.262.5

formedc

Currentefficiency

(%)

68.379.633.238.930.128.330.778.175.0

H2

^tmol

0.61.64.59.36.69.69.00.150.25

formed4

Currentefficiency

(%)

2.52.7

24.126.037.529.931.00.30.3

a Ref. 103.b For p-Si, p-InP, and /7-CdTe, monochromatic light of 600 nm was used, while light of 400 nm was used for p-GaP.c Based on the amount in the gas phase.

Table 7Photoassisted ControUed-Potential Electrolysis (CPE) at p-CdTe at -1.6 V versus SCE in Various

Solvents under Irradiation with Monochromatic Light of 600 nm"

Run

12345678

Solvent*

DMF-5%H2O

DMSO-5%H2O

MeCN-5%H2O

PC-5%H2O

Electricitypassed

(C)

7.920.1

5.718.36.8

21.77.3

18.6

CO

IJLmol

33.085.225.574.119.865.332.576.8

formed6

Currentefficiency

(%)

80.681.886.378.156.258.286.479.9

H2

/xmol

7.620.4

formed'

Currentefficiency

(%)

<0.3<0.3<0.1<0.121.618.1

<0.2<0.2

a Ref. 103.h 0.1 M Bu4NBF4 was used as a supporting electrolyte. DMF = N,N-dimethylformamide, DMSO = dimethylsulfoxide,

MeCN = acetonitrile, PC = propylene carbonate. g1

c Based on the amount in the gas phase. °

f


COJ + CO2 + e--* CO + COi" (28)

and/or

COJ + 2H+ + e" -> CO + H2O (29)

To examine the mechanism of CO2 reduction, it would beuseful to detect intermediates. In this connection, FTIR techniqueswere applied to study an illuminated p-CdTe electrode during thephotoassisted reduction of CO2 in acetonitrile.58 This was the firstreport on IR spectroscopy applied to an illuminated electrode, andadsorbed COJ was detected (Fig. 9). The parallelism between thecoverage of the intermediate and the photocurrent-potential curvewas concluded to be consistent with the mechanism suggested byAmatore and Saveant53 for CO2 reduction to CO [Eqs. (8) and(10)-(12)], when we take into account that COJ was adsorbed onthe electrode.

More recently, Ikeda et a/.108 have examined CO2 reductionin aqueous and nonaqueous solvents using metal-deposited /?-GaPand /?-InP electrodes under illumination. Metal coatings on thesesemiconductor electrodes gave much improved faradaic efficienciesfor CO2 reduction. In an aqueous solution, the products obtainedwere formic acid and CO with hydrogen evolution at Pb-, Zn-, andIn-coated electrodes, while in a nonaqueous PC solution, CO wasobtained with faradaic efficiencies of ca. 90% at In-, Zn-, andAu-coated /?-GaP and p-InP, and a Pb coating on a /?-GaP electrodegave oxalate as the main product with a faradaic efficiency of ca.50% at -1.2 V versus Ag/AgCl.

Bradley et al109 have combined a /?-Si photocathode andhomogeneous catalysts (tetraazamacrocyclic metal complexes,which had been shown to be effective catalysts for CO2 reductionat an Hg electrode110) to reduce the applied cathode potential. Thecatalysts showed111 reversible cyclic voltammetric responses inacetonitrile at illuminated p-S\ electrodes at potentials significantlymore positive (ca. 0.4 V) than those required at a Pt electrode,where the /?-Si used had surface states in high density and Fermilevel pinning112 occurred. Electrolysis of a CO2-saturated solution(acetonitrile-H2O-LiClO4; 1:1:0.1 M) in the presence of 180mM

362 Isao Taniguchi

2150 2100 2050vt c m - 1

2000

Figure 9. FTIR spectra at an illuminated p-CdTeelectrode in CO2-saturated acetonitrile with0.1 M Bu4NBF.58 Spectra recorded at (a) -0.9,(b) - 1 . 1 , (c) -1 .3 , (d) -1 .5, (e) -1.7, (f) -1.9,(g) -2.1 (h) -2 .3 , and (i) -2.5 V versus Ag/AgCl.

of the nickel complex of 5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane, [(Me6[14]aneN4)Ni2+], as the electron-transfer catalyst at -1.0 V versus SCE gave the best results; carbonmonoxide and hydrogen were obtained in a 2:1 molar ratio with


current efficiencies of 95 ± 5%. A 750-W tungsten halogen lampwas used as a light source. In a dry acetonitrile-TBAP-[(Me6[14]aneN4)Ni2+] solution, CO2 reduction was observed atp-Si at -1.3 V versus SCE, and CO and CO3" were the products.Without an electron transfer catalyst, potentials more negative than-1.9V versus SCE were needed for CO2 reduction at p-Si. Theinitial step of the reaction pathway was suggested to be an electrontransfer from the reduced metal complex to CO2. The COJ thusformed would then be protonated (in the case of aqueousacetonitrile solutions) and subsequently a second reduction wouldoccur, resulting in the overall reaction represented by Eq. (19),while in dry acetonitrile Eqs. (10)-(12) were considered. Photoas-sisted electrochemical reduction of CO2 on/?-GaAs(lll) at -0.95 V(versus SHE) was carried out113 in aqueous 0.1 M KC1O4 solution(pH 4.5). Ni2+-cyclam (cyclam = 1,4,8,11-tetraazacy clotetra-decane) catalyzed the reduction of CO2 to CO with a CO/H2 ratioof ca. 2:1. Similar results were also obtained at a p-GaP electrode.Carbera and Abruria114 have shown that at p-Si and polycrystallinethin-film /?-WSe2 semiconductor electrodes on which [Re(CO)3(v-bpy)Cl] (v-bpy = 4-vinyl-4'-methyl-2,2'-bipyridine) was incorpo-rated by electropolymerization, photoelectrocatalytic reduction ofCO2 took place in acetonitrile with 0.1 M TBAP. CO was thepredominant product (>95% current yield). The onset potentialfor CO2 reduction was ca. -0.8 V (/?-Si) and -0.65 V (/?-WSe2)versus a sodium chloride saturated calomel electrode (SSCE), whileat a Pt electrode coated similarly with the catalyst, CO2 reductionbegan to occur at -1.43 V. The turnover numbers of the catalyst atthe photocathodes were >450.

2. Photoassisted Reduction of Carbon Dioxide with Suspensionsof Semiconductor Powders

In experiments investigating conversion of solar energy to chemicalenergy, systems of semiconductor suspensions have also been used.These systems have several advantages and disadvantages:

1. No external energy other than light energy is introduced,and light energy can be stored when uphill reactions take place inthe solutions.

364 Isao Taniguchi

2. Since each semiconductor particle can be considered as amicrophotocell, fast reaction rates can be expected because of theextremely large surface area of the semiconductor on which thereactions take place.

3. Irradiation of the system is much simpler than in the caseof a photoelectrochemical cell using photoelectrodes.

4. No supporting electrolyte is required in the solutions.5. Expensive semiconductor wafers are not required; powders

of the semiconductor suffice. This is convenient because variousmaterials which are not available as electrodes can be used.

6. Furthermore, unique reactions would be expected to occurbecause both oxidation and reduction sites exist close to each other(on the same particle). On the other hand, reverse reactions of thedesired ones easily occur, resulting in low energy conversionefficiency.

7. Another disadvantage of using semiconductor powders isthe difficulty in obtaining kinetic and thermodynamic data.

Inoue et al.83 showed that CO2 was reduced to organic com-pounds such as formic acid, formaldehyde, methanol, and methanein the presence of photosensitive semiconductor powders suspen-ded in water as catalysts. The observed quantum yield of eachproduct was ca. 10~4 to 1(T3. Powders of WO3, TiO2, ZnO, CdS,GaP, and SiC (99.5-99.9999% purity) were used. In each case,1-2 g of semiconductor powder (200-400 mesh) was suspended in100 ml of purified water in a glass cell, into which CO2 gas wasbubbled at a rate of 3 liter/min, and the solution was stirred witha magnetic bar. After 7 h of irradiation using SiC powder as aphotocatalyst, CO2 was reduced to give 1 mM formaldehyde and5.35 mM methanol. Interestingly, the yields of methyl alcohol fromphotocatalytic reduction of CO2 increased as the energy levels ofconduction band of the semiconductor catalysts became more nega-tive with respect to the redox potential of H2CO3/CH3OH, whilein the presence of WO3, of which conduction band level is morepositive than the redox potential of H2CO3/CH3OH, no methanolwas produced.

Halmann and Aurian-Blajeni115 also examined CO2 reductionby irradiation either with sunlight or a high-pressure Hg lamp ofaqueous suspensions of various oxide semiconductors (i.e., TiO2,Fe2O3, WO3, ZnO, and nontronite, an ion-containing clay mineral).


In the case of TiO2 (anatase), a light energy to chemical energyconversion of ca. 0.2% was reported. Moreover, in contrast to theabove observation of Inoue et al.83, the reduction of CO2 wasreported to occur quite well with WO3 as a photocatalyst. In asolution of pH 2.3 with 3.9 g/liter of WO3, with bubbling of CO2

at a rate of 264 ml/min, irradiation with a 70-W high-pressuremercury lamp gave methanol and formaldehyde at rates of 1.17and 0.01 /xmol/h, respectively. Aurian-Blajeni et al.116 comparedthe gas-solid process of CO2 reduction at illuminated semiconduc-tor surfaces with the liquid-solid reaction by illuminating anaqueous suspension of semiconductor powders through which CO2

was bubbled. The latter reaction gave a much higher efficiency thanthe former, and aqueous suspensions of SrTiO3 and WO3 gave ahigh activity for methanol formation (5-7 /xmol/h). Moreover, theauthors measured the band gaps of the semiconductors tested bydiffuse reflectance spectroscopy and showed a poor relationshipbetween the conduction band levels of the semiconductors and theactivity of the semiconductors for the reduction of CO2.

The effect of a rare earth dopant, such as Eu2O3, Sm2O3,Nd2O3, and CeO2, for the large-band-gap semiconductors BaTiO3

and LiNbO3 on the photoassisted reduction of CO2 was examinedusing aqueous suspension systems.117 An enhancement in the yieldof reduction products, formic acid and formaldehyde, was reported,although optical energy conversion efficiencies were 0.01% or less.Strontium titanate powders treated with various transition metaloxides were also examined.118 The predominant reduction productwas formic acid with smaller amounts of formaldehyde andmethanol formed in aqueous solutions. Among the various addi-tives, irridium oxide was the most effective for formic acid formation(the optical to chemical energy conversion efficiency was ca. 0.02%),while for methanol formation, doping with ruthenium oxide wassuitable (the highest optical energy conversion efficiency was0.03%). The lowest doping level, 0.57 mol %, gave the best results.The efficiency of CO2 reduction increased linearly with the quantityof light absorbed in SrTiO3 doped with various additives. Thephotoreduction of CO2 was also examined119 in aqueous sus-pensions of TiO2 powders which had been doped with noble metaloxides and transition elements. RuO2-doped TiO2 showed anincrease in the rate of methanol production, but the optical to

366 Isao Taniguchi

chemical energy conversion efficiency, 100 (heat of combustion ofproducts/incident light flux), was ca. 0.04% at best, and theefficiency declined with prolonged illumination.

The photoassisted reduction of aqueous carbon dioxide in thepresence of inorganic minerals has been examined as a model ofprebiological photosynthesis,120 a potential precursor to the photo-synthetic fixation of CO2 by plants.

The semiconductor suspension systems have serious problems;reduction products of CO2, such as formic acid and methanol, arereoxidized at illuminated semiconductor powders. Also, the re-combination of photogenerated holes and electrons lowers theefficiency of the reactions of holes and/or electrons with speciesin solution.121122 Henglein and Gutierrez123 used semiconductorcolloidal particles instead of powders, because the recombinationof charge carriers generated in illuminated colloidal particles isrelatively slow and thus molecules adsorbed at the colloidal particlesare expected to react efficiently. In the presence of small colloidalparticles of ZnS, formic acid was formed upon illumination of a50 mM aqueous CO2 solution containing 5 mM sulfite; the yieldof formic acid was estimated to be 0.2 molecule per photon adsor-bed. Henglein et al124 also reported that in the presence of analcohol, especially 2-propanol, as a positive hole scavenger, formicacid was produced in ZnS solutions (2 x 10~4 M) containing 50 mMCO2 under irradiation with a Xe lamp (Fig. 10). The quantum yieldfor the formation of HCOOH reached 0.4 molecule/photon, whichcorresponds to a quantum efficiency of 80%, when the 2-propanolconcentration was 1 M. In the absence of alcohol, the formateproduced was effectively oxidized to CO2 at illuminated ZnS. Thereaction scheme is illustrated in Fig. 11. The experimental resultsshowed that 2-propanol reacted with a positive hole via a one-holemechanism to give acetone (80%) and pinacol (20%), whereastwo-electron reduction of CO2 to formate took place in two ways:(1) The COJ radical formed initially was adsorbed too strogly toreact with other radicals, and thus the uptake of the second electronoccurred, and (2) the electrons produced in the colloidal particlescombined with Zn2+ to form Zn atoms at the interface (Zn wasdetected as a product of illumination), and then the Zn atom actedas a two-electron transfer agent toward CO2. By irradiation withan Hg arc lamp of CO2-saturated water (25 ml) containing Pt-doped


Figure 10. Concentration of variousproducts obtained by illumination ofZnS colloidal particles in aqueous sol-ution in the presence of \ M 2-propanol as a function of CO2 con-centration.124

i

o

TiO2 powder (0.1 g), CO2 was reduced to give methane (1and CO (0.25 ^mol) after 24 h, while use of colloidal TiO2 (Ptdoped) increased by ca. 80 times the amounts of products withrespect to a unit weight of TiO2.

125

For CO2 reduction in powder suspension systems, the use ofnonaqueous solvents has not yet been reported.

IV. CATALYSTS FOR CARBON DIOXIDE REDUCTION

Since noncatalyzed carbon dioxide reduction shows a large over-potential and potentials far more negative than -2.0 V versus SCE

ZnS C02

HC02"

(CH3)2C0 + (CH3)2CH0H (80 %)

OH OH

Figure 11. Illustration of the reaction at a ZnS colloidal particlein the presence of CO2 and 2-propanol.124 Two-electron and one-hole mechanism for CO2 reduction and 2-propanol oxidation,respectively, are shown in the figure.

368 Isao Taniguchi

are usually required in preparative-scale electrolysis, a great dealof effort has been devoted to rinding effective catalysts for thisreaction. As was described in the previous section, from a thermody-namic point of view, CO2 reduction can take place at much lessnegative potentials [Eqs. (15)-(20)].

1. Metal Complexes of TV-Macrocycles

The first catalysts reported for the electroreduction of CO2 weremetallophthalocyanines (M-Pc).126 In aqueous solutions oftetraalkylammonium salts, current-potential curves at a cobaltphthalocyanine (Co-Pc)-coated graphite electrode showed a reduc-tion current peak whose height was proportional to the CO2 con-centration and to the square root of the potential sweep rate at agiven CO2 concentration. On electrolysis, oxalic acid and glycolicacid were detected, but formic acid was not. Mn and Pdphthalocyanines were inactive, while Cu and Fe phthalocyanineswere slightly active. At the potentials used for CO2 reduction, M-Pccatalysts would be in their dinegative state, and the occupied dz*orbital of the metal ion in the metallophthalocyanine was suggestedto play an important role in the catalytic activity.

Hiratsuka et al102 used water-soluble tetrasulfonated Co andNi phthalocyanines (M-TSP) as homogeneous catalysts for CO2

reduction to formic acid at an amalgamated platinum electrode.The current-potential and capacitance-potential curves showedthat the reduction potential of CO2 was reduced by ca. 0.2 to 0.4 Vat 1 mA/cm2 in Clark-Lubs buffer solutions in the presence ofcatalysts compared to catalyst-free solutions. The authors suggestedthat a two-step mechanism for CO2 reduction in which a CO2-M-TSP complex was formed at ca. —0.8 V versus SCE, the first reduc-tion wave of M-TSP, and then the reduction of CO2-M-TSP tookplace at ca. -1.2 V versus SCE, the second reduction wave. Recently,metal phthalocyanines deposited on carbon electrodes have beenused127 for electroreduction of CO2 in aqueous solutions. Thecatalytic activity of the catalysts depended on the central metal ionsand the relative order Co2+ > Ni2+ » Fe2+ = Cu2+ > Cr3+, Sn2+ wasobtained. On electrolysis at a potential between -1.2 and -1.4V(versus SCE), formic acid was the product with a current efficiencyof ca. 60% in solutions of pH greater than 5, while at lower pH


values, methanol was also produced but its current efficiency wasless than 5%. Also, carbon electrodes modified by adsorption ofcobalt phthalocyanine, Co-Pc, have been reported128 to be effectivefor CO2 reduction in an aqueous solution of pH 5 to give CO(55-60%) with H2 (35-30%) at a potential ca. 0.3 V more negativethan the thermodynamic CO2/CO redox potential. Oxalate andformate were also detected in the solution, as was previouslyreported,102126 but in only trace amounts, and the major productwas gaseous CO. The turnover numbers of the catalyst exceeded100 s"1, and the dinegative state of Co-Pc was again suggested tobe the active form.

Cobalt porphyrin derivatives were also reported129 to be activefor electrochemical reduction of CO2 to formic acid at an amalga-mated Pt electrode. More recently, Becker et al have reported130

that Ag2+ and Pd2+ metalloporphyrins acted as homogeneouscatalysts for CO2 reduction in dry CH2C12; oxalic acid and H2 (itssource was not clear) were produced, but no CO was detected.

Tetraazamacrocyclic complexes131 of cobalt and nickel werefound110 to be effective in facilitating the reduction of CO2 at —1.3to —1.6 V versus SCE (Table 8). An acetonitrile-water mixture andwater were used as solvents, while in dry dimethylsulfoxide nocatalytic reduction of CO2 took place. Using an Hg electrode, bothCO and H2 were produced, where total current efficiencies weregreater than 90%. The turnover numbers of the catalysts were2-9 rT1. The catalytic activity lasted for more than 24 h and theturnover numbers of the catalysts exceeded 100. A protic sourcewas required to produce both CO and H2, and the authors suggestedthat both products may arise from a common intermediate, whichis most likely a metal hydride. The applied potential for CO2

reduction was further reduced by using illuminated /?-Si in thepresence of the above catalysts.111

Tinnemans et al132 have examined the photo(electro)chemicaland electrochemical reduction of CO2 using some tetraazamac-rocyclic Co(II) and Ni(II) complexes as catalysts. CO and H2

were the products. Pearce and Pletcher133 have investigated themechanism of the reduction of CO2 in acetonitrile-water mixturesby using square planar complexes of nickel and cobalt withmacrocyclic ligands in solution as catalysts. CO was the reductionproduct with no significant amounts of either formic or oxalic acids

Table 8Results of Electrolysis with Various Macrocycles at Hga

AverageElectrode current Turnoverspotential6 efficiency0 Products per h at

Compound (V versus SCE) (%) ratio a 23°Ce Solvent system

CO 93 CO/H2, 1:1 7.8 0.1 M KNO3 in H2O/CH3CN 2:1 (v/v) orH2O only

j:; - : -1.5 90 CO/H2, 1:1 0.1 M KNO3 in H2O/CH3CN 2:1 (v/v) orH2O only

-1.6 98 CO/H2, 2:1 0.1 M LiClO4 in H2O/CH3CN 2:1 (v/v)

f

CN N—i

N N—»

-1.5

Ni

N

-1.3 44 CO 2.1 0.1 M KNO3 in H2O/CH3CN 2:1 (v/v)

a From Ref. 110.b All Controlled-potential electrolysis (CPE) experiments were carried out at the cathodic E1/2 or 0.1 V more negative than the Ej/J

for the M2+/1+ couple in the solvent system used.c Averaged over numerous runs by using the following catalyst concentrations: compounds 1, 2, 3, and 4, 1.2 mM; compound 5, 2.5 mM.d From gas chromatographic data.e Turnovers per hour per mole of catalyst for runs in which the catalyst concentration was 1.4-2A mM. A turnover is defined as 1 equiv

of electrons passed through the electrolysis cell per mole of catalyst. Since the reduction products require two electrons for theirformation, these numbers correspond to twice the moles of product formed per mole of complex per hour.

/ Although catalysis has been observed with this compound in a number of solvent systems, reliable current efficiencies and rates havenot been obtained.

Table 9Electrocatalytic Reduction of CO2 by [Nin(cyclam)]2+ in Water"'5

Run

123456

Electrocatalyst

NiCl2-6H2ONi(cyclam)Cl2

Ni(cyclam)Cl2

Ni(cyclam)Cl2

Ni(cyclam)Cl2

E(V versus N.H.E.)

-1.05-1.05-0.90-0.95-1.00-1.05

Total volume0 ofCO produced (ml)

<0.05<0.05

0.43.6

23.735.6

Turnover frequency'*(h"1); overall

turnover of Ni

—0.3; 1.2

2.9; 10.818; 77.532;116

Average currentefficiency6 (%)

—36829996

Volume of H2

produced/ ml;H2 :CO in

gas produced

0.36; > 101.6; >30

<0 .01 ;<2x 10~2

<0.01; <3 x 10"3

<0 .01 ;<5x 10~4

<0.01; < 3 x 10~4

a Ref. 135.b CO2 (99.995% purity) saturated solutions (75 ml H2O at 25°C; pH ca. 4.1) containing the electrocatalyst (1.7 x 10~4 M) and KNO3

(0.1 M) were placed in a gas-tight electrolysis cell; the working electrode (18 cm2) was mercury (99.99999% purity). The total volumeoccupied by the gases in the electrolysis cell was 86 ml. The gases were analyzed by gas chromatography.

c After 4 h of electrolysis. Md Turnover numbers are calculated from moles of CO produced per mole of electrocatalyst. §e Current efficiency p\p = {2nco x 96 500/C), where nco = moles of CO produced, C = coulombs passed during the run. H


-0.5 E ( y V 5 N H E )

Figure 12. Current-potential cur-ves for Ni(II)-cyclam (1 mM) inan aqueous 0.1 M KC1O4 solution(pH 4.5) under N2 (a) or CO2 (b)at a hanging mercury drop elec-trode.135 Scan rate: 0.1 V/s.

formed, even in the presence of strong proton donors. Thus, themechanism of CO2 reduction in the presence of transition metalsis quite different from that at metal electrodes; the monocation ofthe transition metal reacted with CO2 to give a species which wasfurther reduced at the electrode, where the protonation step wasconstrained to occur at an oxygen rather than a carbon site of theintermediate. Bailey et al.134 used a modified Pt electrode with anickel tetraazaannulene complex electropolymerized film,(Ni[Me4Bro2 [14]tetraeneN4)n, as a catalyst. Electrolysis at -1.85 Vversus SSCE in CO2-saturated acetonitrile-0.1 M tetraethylam-monium perchlorate (TEAP) solutions containing 2 vol % methanolgave formate ion as a product without CO or oxalate.

An extremely selective electrocatalyst for CO2 reduction to COin water has recently been found by Beley et al135 A rather simpleNi complex of 1,4,8,11-tetraazacyclotetradecane, [Ni(II)-cyclam],

Hgure 13. Postulated structure of CO2-Ni-cyclam complex.135

374 Isao Taniguchi

cycles effectively more than 103 times with no significant deactiva-tion to give CO selectively, even in water, by electrolysis at -1.05 V(versus NHE) at an Hg electrode (Table 9 and Fig. 12). Theimportance of the molecular species observed on the electrodesurface was shown. Also, the size of the ligand and the presenceof a secondary amine group have been suggested to be the originof the special properties of the electrocatalyst; the former impartshigh kinetic and thermodynamic stability to the Ni(II) complex,and the latter could favor CO2 fixation by hydrogen bonding(N—H---O) in addition to the carbon-to-Ni(I) binding (Fig. 13).Selective reduction of CO2 to CO in the presence of Ni(II)-cyclamwas again observed at a Pb electrode.136 Also, both Ni(II)-isocyclam(isocyclam = 1,4,7,11-tetraazacyclotetradecane) and cyclams withcentral metals other than Ni, such as Co, Zn, Cu, and Au, showedmuch less catalytic activity; use of Rh(III)-cyclam gave formic acidrather than CO as the main product of CO2 reduction.136 Uniquecatalytic properties of metal cyclams, such as Co- and Ni-cyclam,have also been demonstrated in electrocatalytic reduction ofnitrogen oxyanions.137

2. Iron-Sulfur Clusters

Tetranuclear iron-sulfur clusters of the type [Fe4S4(SR)4]2~,

where R = CH2C6H5 and C6H5, were found138 to catalyze the reduc-tion of CO2 in DMF solutions. Controlled-potential electrolyseswere carried out in a CO2-saturated 0.1 M tetrabutylammoniumtetrafluoroborate (TBAT)-DMF solution at a mercury poolcathode. In the absence of a catalyst, CO2 was substantially reducedonly at potentials more negative than -2.4 V versus SCE, while inthe presence of a cluster, the reduction took place at around -1.7 V;thus, potential shift of ca. 0.7 V was achieved. The products wereanalyzed by means of gas chromatography and isotachophoresis.Without a catalyst, oxalate was the main product, and addition ofsmall amounts of water to the DMF solution favored formateproduction, whereas in the presence of the catalyst, formate wasproduced predominantly even in a dry DMF solution. This resultwas interpreted in terms of indirect reduction of CO2, proceedingby electron transfer from the reduced cluster to CO2 in the bulk


solution and then protonation (tetraalkylammonium ion was con-sidered as a hydrogen source, in part) rather than self-coupling.The catalysts were shown to undergo a two-step reduction to givethe trianion and tetraanion, and the latter was suggested, fromcurrent-potential curves, to be active for CO2 reduction. In addition,[Fe4S4(SR)4]2~ (R = PhCH2 or Bu) and [M2Fe6S8(SEt)9]3~ (M -Mo or W) have recently been reported139 as catalysts for CO2

reduction in DMF. The cubane structure of [Fe4S4(SCH2Ph)4]2~collapsed rapidly during electrolysis at -2.0 V versus SCE underCO2. Addition of an excess amount of PhCH2 SH prevented degra-dation of the cluster, and phenyl acetate was formed (Eq. 30) as areduction product at faradaic efficiencies of 5-15%:

PhCH2S~ 4- CO2 + 2e~ -> PhCH2COO" + S2" (30)

3. Re, Rh, and Ru Complexes

Recently, Hawecker et al.140 have shown Re(bpy)(CO)3Cl (bpy =2,2'-bipyridine) to be an efficient homogeneous catalyst for theelectrochemical reduction of CO2 to CO. The complex wasoriginally found to be a catalyst for CO2 photoreduction by thesame group.141 In a CO2-saturated DMF-water (9:1) solution(60 ml) containing 2 mg of the catalyst and 0.1 M tetraethylam-monium chloride (Et4NCl), electrolysis at -1.25 V (versus NHE)on a glassy carbon electrode gave CO at 98% current efficiency(Table 10). About 300 catalytic cycles without loss of activity wereobserved in 14 h. A similar Re complex, Re(vbpy)(CO)3Cl (vbpy =4-vinyl-4'-methyl-2,2'-bipyridine), also showed142 electrocatalyticactivity for CO2 reduction to give CO in acetonitrile at a Pt electrodecoated with a polymeric film containing the complex; this catalystwas later immobilized on semiconductor photocathodes.114 Nocarbonate was produced at this electrode. The turnover numbersobtained were much larger than those observed with Re(bpy)-(CO)3C1 in solution.140 More recently, a Pt electrode coated witha polypyrrolic film containing Re(bpy)(CO)3Cl was used143 as acatalytic electrode; equal amounts of CO and CO2" were produced,as was the case for Re(bpy)(CO)3Cl in solution.140 The reason forthe difference in the products obtained at Re(vbpy)(CO)3Cl- and

Expt.

Table 10Electroreduction of CO2 Catalyzed by Re(bipy)(CO)3Cl in DMF-H2O Solutions"'6

Medium compositionElectrolysis

time (h)

37

14123451234123.551.535123.5

Volume of COproduced (ml)

6.815.831.60.961.72.42.93.11.63.04.66.02.44.78.2

11.73.67.2

12.01.63.25.5

Coulombsconsumed

55.6129.5259.9

8.314.920.925.928.113.825.940.251.020.540.670.9

100.131.561.6

102.714.328.348.6

Average currentefficiency (%)c

98

92

93

94

94

91

5

6e

DMF-H2O(10%)-NBu4ClO4

DMF-NEt4Cl

DMF-H2O(5%)-NEt4Cl

DMF-H2O(10%)-NEt4Cl

DMF-H2O(10%)-NBu4ClO4

DMF-H2O(20% )-NEt4Cl

a Ref. 140.b CO2 (99.8% purity) saturated solution (60 ml) containing 25 mg of Re(bipy)(CO)3Cl (9.0 x 10~4 M) and 0.1 M supporting electrolyte was placed in a

gas-tight electrolysis cell (three-necked, round-bottomed flask equipped with an oil valve); the working electrode in all the experiments was glassy carbon(ca. 10 cm2). The connection to the working electrode was made with a Pt wire inserted through the lateral part of the flask. The total volume occupiedby the gases in the electrolysis cell was 130 ml. All the solutions were electrolyzed at -1.25 V versus N.H.E. and at ca. 25°C. The gases were analyzedby gas chromatography.

c Averaged over the total duration of the experiment. Since the reduction product requires two electrons for its formation, 2 equivalents of electrons passedthrough the electrolysis cell afford 1 mol of CO.

d Only 7.5 x 1(T5 M Re(bipy)(CO)3Cl was used in this run.ltv this experiment, traces of H2 were detected (about 10 /xl after 3.5 h). After this time, a slight, as yet uncharacterized, precipitate appeared.


Re(bpy)(CO)3Cl-modified electrodes has not yet been explained.However, from the cyclic voltammograms of/ac-Re(bpy)(CO)3Cl(Fig. 14) and from the intermediate complexes formed by electroly-sis in acetonitrile in the presence and absence of CO2, two differentelectrocatalytic pathways (Fig. 15) were suggested144: initial one-electron reduction of the catalyst at ca. -1.5 V versus SCE followedby the reduction of CO2 to give CO and CO3, and initial two-electron reduction of the catalyst at ca. -1.8 V to give CO with noCO\~. The electrochemistry of [Re(CO)3(dmbpy)Cl] (dmbpy =4,4'-dimethyl-2,2'-bipyridine) was investigated145 to obtainmechanistic information on CO2 reduction, and the catalytic reac-

/ ^ co9/ - • *

-1.0 -1 .5 -2 .0

V vs. SCE

Figure 14. Cyclic voltammograms of /<zc-Re(bpy)(CO)3Cl in acetonitrile-0.1 MBu4NPF6 at a Pt electrode.144 Scan rate: 0.2 V/s. The lower voltammograms showthe switching potential characteristics: A and F, reversible one-electron wave; Band D, redox couple due to a dimer of the complex; C, the second metal-basedwave. The upper curves show the effect of CO2 on the voltammogram. See alsoFigure 15.

378 Isao Taniguchi

fac-Re(bpy)(CO)3Cl

-11-[Re(bpy)(C0)3Cir

-2~ Re(bpy)(C0)3 v C02 \ + e -

CQ2 + 2e" Re(bpy)(C0)3C02

CO + LAO]"^ ^ [Re(bpy)(C0)3l" X JCO2

[Re(bpy)(C0)3C02]~

A = an oxide ion acceptor

Figure 15. Postulated reaction pathways for CO2 re-duction in the presence of/ac-Re(bpy)(CO)3Cl.144

tion of such Re complexes was suggested to involve a monocoordi-nated bipyridine intermediate.

Also, it would be worthwhile to investigate catalysts developedfor CO2 reduction in other fields with regard to their possibleapplication to electrochemical and photoelectrochemical reductionof CO2, and vice versa; in fact, catalysts developed for a particularsystem have been applied successfully in various related systemsas described above.

Rhodium and ruthenium complexes have also been studied aseffective catalysts. Rh(diphos)2Cl [diphos = l,2-bis(diphenyl-phosphino)ethane] catalyzed the electroreduction of CO2 inacetonitrile solution.146 Formate was produced at current efficien-cies of ca. 20-40% in dry acetonitrile at ca. -1.5 V (versus Ag wire).It was suggested that acetonitrile itself was the source of the hydro-gen atom and that formation of the hydride HRh(diphos)2 as anactive intermediate was involved. Rh(bpy)3Cl3, which had beenused as a catalyst for the two-electron reduction of NAD+

(nicotinamide adenine dinucleotide) to NADH by Wienkamp andSteckhan,147 has also acted as a catalyst for CO2 reduction inaqueous solutions (0.1 M TEAP) at -1.1 V versus SCE using Hg,Pb, In, graphite, and n-TiO2 electrodes.148 Formate was the main

Table 11Reduction of CO2 at Various Electrodes in H2O-0.1 M Et4NClO4

a

Run Cathode (V versus SCE)

Rhcomplex6

(mM)Q

(C)HCOOH(/xmol)

Current efficiency Volume of gasproduced (ml)

123456789

101112

HgHgPbPbInInIn

GCGC

«-TiO2

n-TiO2

n-TiO2

1.001.01.01.001.01.001.001.0

100180100100300150100300

7300600300

4017

.32694

87070

101848

90510

288

77.41.9

62.918.156.09.0

19.454.5

58.30.4

18.5

20.0 (H2)90.6 (H2)40.0 (H2)86.2 (H2)

72.0 (H2), 2.5 (CO)

103 (H2)

a Ref. 148.b Rh complex 1 mM = 10 fimol in 10 ml electrolyte; Rh complex = Rh(bpy)3Cl3. Possible reaction pathways are: Rh(bpy)3+ + 2e~ ->

Rh(bpy)2f + bpy; Rh(bpy)J + H3O+ ^ [(H-)Rh3+(bpy)2(H2O)]2+; [(H~)Rh3+(bpy)2(H2O)]2+ + CO2 -» [Rh3+(bpy)2(HCOO")

(H2O)]2+ ^ Rh(bpy)^+ 4- H2O + HCOO".

380 Isao Taniguchi

product at current efficiencies of ca. 50-80% (Table 11). Bolingeret al149 have shown that [Rh(bpy)2(O3SCF3)2]

+ and [Ru(trpy).(dppene)Cl]+ [trpy = 2,2',2"-terpyridine; dppene = cis-l,2-bis(di-phenylphosphino)ethylene] work as catalysts in acetonitrile toproduce formate and CO, respectively. The following reactionswere suggested for the Rh complex:

Bu4N+ + CO2 + 2e~ -» HCO2 + Bu3N + CH2=CHEt (31)

and

2Bu4N+ + 2e~ H2 + 2Bu3N + 2CH2=CHEt (32)

while for the Ru complex:

Bu4N+ + 2CO2 + 2e~ -> Bu3N + CH2=CHEt 4- CO + HCO^

(33)

indicating that the reduction product depended on the nature ofthe catalyst.

Tanaka et al150 reported that in the presence of [Ru(bpy)2.CO2](PF6)2, controlled-potential electrolysis of a CO2-saturatedH2O/DMF (9/1 v/v) solution at -1.5 V versus SCE using an Hgpool electrode gave CO and H2 in acidic conditions, and formicacid and CO as well as H2 in alkaline conditions.

From the results described above, CO2 seems to be reducedto CO in the presence of Re complexes in most cases, while RHcomplexes give formic acid and Ru complexes give both formicacid and CO, depending on the conditions.

4. Other Catalysts

The electrochemical reduction of aqueous bicarbonate to formicacid,

HCO3" + 2H+ + 2e~ -> HCO," + H2O (34)

using Pd-impregnated polymer-modified electrodes,151 proceeds atpotentials within 80 mV of the thermodynamic one forHCO^/HCO2, -0.76 V versus SCE; the supported Pd catalyst itself


was previously used152 for reduction of HCO^ with H2 to formHCO2, a nd later a bright Pd cathode was also used.75

Ogura et al153 reduced CO2 to methanol using the so-calledEveritt's salt (K2Fe2+[Fe2+(CN)6])-modified electrode by a some-what complicated but interesting route in the presence of a metalcomplex, such as Fe(II), Co(II), and Ni(II) complexes of 1-nitroso-2-naphthyl-3,6-disulfonic acid, and additional methanol:

LM + CO2 + CH3OH -> LM—O=C=9 (intermediate) (35)

6-HI

CH3

Intermediate + 6H+ + 6e~ -> LM + 2CH3OH + H2O (36)

where LM is the metal complex and the electrons are supplied bythe Everitt's salt (ES); ES is oxidized to become the so-calledPrussian blue (PB, KFe3+[Fe2+(CN)6]), which is again reduced toES electrochemically. Thus, the overall reaction is given as:

CO2 + 6ES + 6H+ -> CH3OH + 6PB + 6K+ + H2O (37)

-6e~

Electrocatalytic reduction of carbon dioxide to Q-C3 hydro-carbons with less than 0.2% electrochemical yield was reported154

at pH 7 in the presence of pyrocatechol, TiCl3, and Na2MoO4 at-1.55 V versus SCE.

As described above, many reports published to date indicatethat metal complexes are promising catalysts for CO2 fixation. Thecatalytic activity is considered basically to be due to a CO2-catalystcomplex formation. Thus, the complexes have to provide a bindingsite for CO2, and this can be realized for some catalysts by losinga ligand on reduction of the catalyst at the electrode. Also, the CO2

molecule is not linear but is rather a bent structure155156 in theactivated state of the CO2-catalyst complexes. Theoretical calcula-tions of CO2-catalyst bonding157 and general ideas about activationof CO2 by metal complexes have been summarized in several recentarticles.158'159

In addition, catalysts for CO2 reduction based on nonmetalliccompounds have also been reported. Taniguchi et al100 reported

382 Isao Taniguchi

that crown ethers such as 15-crown-5 in lithium carbonate solutionsenhanced CO2 reduction, although in this case, the crown etherwas proposed to facilitate the electrodeposition of Li metal on theelectrode, which catalyzed the reduction of CO2. A polyaniline-coated p-Si photocathode was found" to be active for CO2 reduc-tion in a LiClO4 solution. It would also be useful to incorporatecatalysts in a polymer matrix in high density, and modification ofelectrodes with electropolymerized films in which catalysts areincorporated has been widely employed. Recently, ammonium ionwas suggested160 to mediate the photoassisted reduction of CO2 toCO at a /7-CdTe electrode in DMF with small amounts of water.The photocurrent-potential curve shifted toward less negativepotentials when the supporting electrolyte was changed from TBAPto NH4CIO4, accompanied by a change in Tafel slopes. Under aCO2 atmosphere, CO was formed by photoassisted electrolysis ata p-CdTe electrode, but NH3 was the product under an Ar atmos-phere.

The electron transfer between an electrochemically producedperylene dianion and a CO2 molecule was also suggested by cyclicvoltammetry in a DMF solution.161 Later, perylene was used162 inthe photochemical fixation of CO2, as a nonmetal electron carrierto CO2.

Furthermore, a biological catalyst [formate dehydrogenase(FDH)] combined with an illuminated p-InP photocathode was

CB

P-InP

2 MV HCO9H

FDH E -0 .420

,2+

v > 1,35 eV

2 MV2+ C02 + 2H

Figure 16. Scheme for the photoelectrochemicalreduct ion of C O 2 at p-lnP with formate dehy-drogenase ( F D H ) as the catalyst and methylviologen (MV 2 + ) as the electron transfermediator .1 6 3


effective in the reduction of CO2 to give formic acid.163 Photogener-ated electrons in p- InP reduced the enzyme through methyl viologenas a mediator, and then reduction of CO2 took place (Fig. 16), whenphotoassisted electrolysis was carried out at 0.05 V (versus NHE).The turnover numbers of the enzyme exceeded 2 x 104, but loss ofenzyme activity due to denaturation of the protein occurred.

V. MISCELLANEOUS STUDIES

To establish an effective system for CO2 reduction, variousapproaches have to be considered. In this section, miscellaneousstudies of CO2 fixation, other than those involving the usualelectrochemical and photoelectrochemical reduction of CO2, arebriefly reviewed.

1. Photochemical Reduction of Carbon Dioxide

Photochemical fixation of carbon dioxide is a function of greenplants and some bacteria in nature in the form of photosynthesis.All living organisms on the Earth are indebted directly or indirectlyto photosynthesis. Thus, many attempts have been made to simulatethe photosynthetic system and make artificial systems, although todate very little success has been achieved.

Tazuke and Kitamura162 reported the first example of anartificial photosynthetic system based on electron transport sensiti-zation, although the product was not a hydrocarbon, but ratherformic acid. Their system is shown schematically in Fig. 17. In thissystem, the photochemically generated singlet excited state of anaromatic hydrocarbon, such as pyren (Py) or perylene (Pe), was

co -. __> HCOOH and/or

(C00H)2

Figure 17. Schematic representation of an artificial photosyn-thetic system.162

384 Isao Taniguchi

the electron donor (D), and 1,4-dicyanobenzene or 9,10-dicyanoan-thracene was used as an electron-acceptor (A). The electrons accep-ted are successively transferred to CO2 molecules. The speciesdenoted as X in Fig. 17 was not clearly identified but possiblecandidates considered were OH~, HCO^, and/or HCOOH. Usingthis system in an aqueous acetonitrile solution, ca. 1 mmol of formicacid was formed under irradiation with a 300-W high-pressure Hglamp, with a poor quantum yield (< 10~4). The main overall reactionwas given as

CO2 + 2H2O - X HCOOH + H2O2 (38)

Later, an improved system for CO2 photofixation was reported bythe same authors.164 The new system consisted of 6.5 x 10"5 Mtris(2,2'-bipyridine)ruthenium(II), Ru(bpy)3, as the photosensitiveelectron donor, methyl viologen (MV2+, 20 mM) as the electronacceptor, and triethanolamine (TEOA, 0.6 M) as a sacrificial elec-tron donor in a CO2-saturated aqueous solution (Fig. 18). Underirradiation with a 300-W high-pressure Hg lamp with a CuSO4

chemical filter (A > 320 nm), formic acid, which was detected byisotachophoresis, was produced in quantum yields of ca. 0.01%.Recently, however, Kase et al.165 have repeated this experimentusing a 13CO2 tracer and have claimed that the formic acid obtainedwas produced not by CO2 reduction but rather by oxidative cleavageof TEOA.

Lehn and Ziessel166 have also developed systems for the photo-chemical reduction of CO2. These systems are similar to thoserepresented by Fig. 18. Visible-light irradiation of CO2-saturatedaqueous acetonitrile solutions containing Ru(bpy)3

+ as a photo-sensitizer, cobalt(II) chloride as an electron acceptor, and triethyl-amine as a sacrificial electron donor gave carbon monoxide and

HCOOH

Figure 18. Scheme of an example of animproved photosynthetic system.164 Forother combinations of photosensitizers

TEOA and electron acceptors, see text.


hydrogen simultaneously. Addition of free bipyridine to the solutiondecreased CO generation but increased H2 evolution. When triethyl-amine was replaced by other NR3 compounds, the quantity of gas(CO + H2) produced and the CO/H2 ratio increased in the orderR = Me < Et < Pr. When triethanolamine was used instead oftriethylamine, CO was selectively produced from CO2 in high yield;after irradiation with a 1-kW Xe lamp with a 400-nm cutoff filterfor 22 h, 2.93 ml of CO and 0.12 ml of H2 were obtained (thequantum yields were not reported). In this system, cobalt ion wasan efficient and specific electron mediator for CO2, and the markedeffects of the tertiary amine and of bipyridine were explained interms of the differences in their coordination to the cobalt ion,which would influence the reaction process. Since Co(bpy)J isknown167 to react with bicarbonate to give insoluble [Co(bpy)-(CO)2]2, which decomposes to liberate CO and H2 by acidificationt o p H < l :

[Co(bpy)(CO)2]2 + 6H+ -> 4CO + 2H2 + 2Co2+ + 2bpyH+ (39)

and Co(I) can be photochemically generated using a ruthenium(II)polypyridine complex as a sensitizer, the product (CO) of thephotochemical reduction of CO2 seems to be due to the mediatorused. Using the Rh(bpy)3

+ complex, a similar photochemical systemfor the generation of H2 by reduction of water can be made,168 andby introduction of CO2 into this system, products other than COmay be obtained. Later, Ru(bpy)3Cl2-TEOA-CO2 in DMF wasshown169 to represent a catalytic system for the photoreduction ofCO2 to formic acid. No additional electron acceptor, such as Co(II)ion, was used. DMF was a better solvent than acetonitrile. 13CO2

was used to verify that the reduction of CO2 took place, and thefollowing steps were suggested for the reduction process: photogen-eration of Ru^py)^, ligand photolabilization, hydride formation,insertion of CO2, and release of the formate.

Hawecker et al141 used Re(bpy)(CO)3X (X - Cl, Br) com-plexes as photosensitizers and succeeded in improving markedlythe efficiency of CO formation using a system similar to thatdescribed above, where DMF was used as a solvent and 2,9-dimethyl-l,10-phenanthroline was added, as a ligand for the cobaltion, to a solution containing Ru(bpy)3

+, Co2+, and triethanolamine.

386 Isao Taniguchi

This system produced 8 ml of CO and 19 ml of H2 after irradiationwith a 1-kW Xe lamp for 15 h. Using 13CO2, the CO produced wasverified to come from CO2. Furthermore, Re(L) (CO)3X (L =2,2'-bipyridine or 1,10-phenantroline; X = Cl, Br) complexes werefound to act as both photosensitizers and catalysts for CO2 reductionto CO. Metal carbonyls were considered to be effective because oftheir well-known activity as catalysts for the water-gas shift reac-tion.170 Addition of Cl~ or Br~ stabilized the correspondingRe(bpy)(CO)3X (X = Cl, Br) complexes and resulted in moreefficient production of CO (see runs 2 and 4, 3 and 5, and 8, 9, and10 in Table 12). The Re complexes were also useful as homogeneouscatalysts in the electrochemical reduction of CO2 at a glassy carbonelectrode in aqueous DMF.140 More recently, an improvement ofthis system for the photochemical reduction of CO2 by visible lightwas examined using Ru(bpy)3

+ and Re(CO)3(bpy)Cl as co-catalysts,171 but, unfortunately, the stability of the system was poorfor long-term usage.

The effects of transition metals on the photochemical reductionof CO2 to formaldehyde (0.1 %), formaldehyde to methanol (6-8%),and methanol to methane (ca. 10"5%) were examined172 in aqueoussolutions, but the yields were very low as shown in parentheses foreach reaction.

As a model of photosynthesis in green plants, platinizedchlorophyll a dihydrate polycrystals were used.173 Illumination ofPt-chlorophyll in the presence of CO2 and water gave formic acidby the reaction

2CO2 + 2H2O -> 2HCOOH + O2 (40)

where the products were determined mass spectrometrically.Unfortunately, the photoactivity of the chlorophyll decreased asO2 was produced due to poisoning.

From the viewpoint of a model of prebiotic chemical evolutionand of the primitive atmosphere of the Earth,174175 photosyntheticreactions of CO2 were also examined, and formaldehyde withvarious nitrogen-containing products was obtained.

For other reports dealing with photochemical fixation of CO2,Halmann's review4 is helpful (see also the references cited therein).

This is the present state in the development of chemical systemsfor artificial photosynthesis. For solar energy conversion and

Table 12Generation of CO by Photoreduction of CO2 via Visible Light Irradiation of Solutions

Containing Re(LXCO)3X and CO2 in (HOCH2CH2)3N-DMFa'fc

Expt.

123456789

1011121314 e

15 f

Complex

Re(bipy)(CO)3ClRe(bipy)(CO)3ClRe(bipy)(CO)3ClRe(bipy)(CO)3ClRe(bipy)(CO)3ClRe(bipy)(CO)3ClRe(bipy)(CO)3ClRe(bipy)(CO)3BrRe(bipy)(CO)3BrRe(bipy)(CO)3BrRe(bipy)(CO)3Br

Re(Br-phen)(CO)3BrRe(Br-phen)(CO)3Br

Re(bipy)(CO)3BrRe(bipy)(CO)3Br

Additive0

000

NEt4ClNEt4Cl

NEt4ClO4

NEt4Clg

00

NBu4BrNBu4Br

0NBu4Br

00

Irradiationtime (h)

12424222424223.56

Vol. of COproduced (ml)

6.59.7

16.814.530.06.4

14.07.6

11.412.016.02.73.70.080.04

Turnovernumber**

1116272348102214202128

5.9

——

Q

D

s

a Ref. 141.b Re(bipy)(CO)3 Cl, 8.7 x 10~4 M; Re(bipy)(CO)3Br, 7.9 x 10~4 M; Re(Br-phen)(CO)3Br, 6.6 x 10~4 M; Br-

phen = 5-bromo-l,10-phenanthroline. 30 ml of solution containing Re(L)(CO)3X and 160 ml CO2 (99.8%purity) dissolved in dimethylformamide-(HOCH2CH2)3N(5:1) were irradiated with a 250-W halogen lamp(slide projector) fitted with a 400-nm cutoff filter (Schott GG 420).

c NEt4Cl, 2 x 10~2 M; NBu4Br, 10~2 M.d Obtained by dividing the number of moles of CO produced by the number of moles of ReL(CO)3X.e Experiment carried out without CO2; formal pH of the solution adjusted to 9.5; 1.1 ml of H2 generated./ Same conditions as in experiment 14 but adjusted to 'pH' 8.5; 1.3 ml of H2 generated.g And 25 equiv. of bipy.

388 Isao Taniguchi

storage as well as improvements in efficiency and in product selec-tivity, further efforts are needed.

2. Reduction of Carbon Monoxide

Since carbon monoxide has recently been found as a main productof CO2 reduction in many systems, it would be important to convertCO into further reduced products such as methanol; this is feasiblebecause CO is much more reactive than CO2 and is thus one ofthe starting materials of Q chemistry.176

Uribe et al.177 examined the reduction of CO in liquid NH3-0.1 M KI at -50°C, using various working electrodes such as Pt,Ni, C, and Hg. The reaction of CO with electrogenerated solvatedelectrons produced dimeric species, which precipitated as K2C2O2.Electrochemical reduction of CO in an aqueous solution at porousgas-diffusion and wet-proof electrodes of Co, Ni, and Fe was carriedout,178 and Q to C3 hydrocarbons and ethylene were reported tobe the products.

CO conversion has been investigated for methanol synthesis.The Fischer-Tropsch reaction179 proceeds over catalysts at a syn-thesis gas pressure of near 300 atm and a temperature of near 200°C.Recently, Ogura and Yamasaki180 have reported that CO waselectrochemically reduced selectively to methanol at room tem-perature at atmospheric pressure, using Everritt's salt (ES) in thepresence of a pentacyano iron(II) complex and methanol, as in thecase of CO2 reduction.153 The catalytic reaction of CO with H2 isa well-known thermal gas phase reaction.181"186 Photoelec-trochemical reduction of CO has recently been carried out187 atp-Si in aqueous solution to give formaldehyde with a currentefficiency of less than 5%. An Fe-porphyrin-deposited p-Si elec-trode sometimes showed a positive effect for CO reduction, whileCO reduction was suppressed in the presence of Li+ ions. Yoneyamaet a/.188 have shown that at p-GaP photocathodes coated withheat-treated Fe(II)-TPP (TPP = tetraphenylporphyrin), CO wasreduced to methanol with a current efficiency of ca. 10% in 0.5 MH2SO4 at a constant current density of 0.5 mA/cm2. Using a Cuelectrode, CO was reduced to methanol189 at -1.4 V and 25°C inan aqueous 0.1 M Et4NClO4 solution with a current efficiency ofca. 20% when less than 20 C were passed. Also, in the presence of


1 mM Ni-cyclam, a catalyst135 for CO2 reduction to CO, methanolwas obtained189 by electrolysis of an aqueous CO2-saturated 0.1 MEt4NClO4 solution at —1.5V versus SCE using a Cu electrode,although the current efficiency was rather low (ca. 5%); the mainreduction product of CO2 was CO.

3. Thermal Reactions for Carbon Dioxide Reduction

Hydrogenation of CO and CO2 with H2 to give organic fuels in gasphase thermal reactions is also an attractive subject, and manycatalysts have been developed and used.181"186'190"194 Basic conceptsdeveloped in this field would be applicable to electrochemical andphotoelectrochemical CO2 fixation. For example, as has alreadybeen tried,82 '8487 some noble metals or their oxides can be dopedinto semiconductor electrodes. For hydrogenation reactions of CO2

and CO, transition metal catalysts have commonly been used.Hydrogenation of CO2 to methanol was reported to occur effectivelyover rhenium catalysts such as Re-ZrO2 and Re-Nb2O5, by gasphase thermal reactions under moderate conditions (lOatm and160-220°C) with a selectivity for methanol formation of 50-70% .193

Recently, effective Rh-based catalysts for acetaldehyde formationfrom CO and H2 have been reported.194 Hydrogenation of CO2 inaqueous solution with a Rh hydride complex was also reported.195

Also, knowledge196201 about the adsorption behavior of CO2 oncatalytic metals (such as Rh and Pt) on various metal oxide supportswould be useful to design effective catalysts.

4. Carbon Dioxide Fixation Using Reactionswith Other Compounds

Electrochemical carboxylations of organic molecules such asolefins,202 aromatic hydrocarbons,203 and alkyl halides204"206 in thepresence of CO2 have been examined, as one of the subjects oforganic electrochemistry.207

Attempts have been made to fix CO2 using Organometalliccomplexes in photochemical reactions. A reversible binding of CO2

was achieved with the Cu(I) phenylacetylide-phosphine com-plex,208 which acted as a reversible CO2 carrier at ambient tem-peratures and atmospheric pressure, by CO2 insertion into the

390 Isao Taniguchi

Cu—C bond of the complex. Recently, activation of CO2 as anr^-C metalocarboxylate209 and photoinduced or thermal insertionof CO2 into a metal-hydride bond210 have been reported. Further-more, some metal (Zn2+, Ni2+, and Cd2+) complexes of tetraazacy-cloalkanes have been found to take up CO2 easily in basic alcoholicsolutions, and their structures have also been examined.211 Morerecently facile insertion of CO2 into Rh2(/x-OH)2 to yield a carbon-ate complex of a rather complicated structure has been reported.212

VI. SUMMARY AND FUTURE PERSPECTIVES

In this article, recent developments, up to late 1986, on carbondioxide reduction have been reviewed. These can be summarizedas follows:

1. Reduction of carbon dioxide takes place at various metalelectrodes. The main products are formic acid in aqueous solutionsand oxalate, CO, and formic acid in nonaqueous solutions. Anindium electrode is the most potential saving for CO2 reduction.Due to the difference in optimum conditions between those forCO2 reduction to formic acid and those for formic acid reductionto further reduced products, direct reduction of CO2 in aqueoussolutions without a catalyst to highly reduced products seems tobe difficult at metal electrodes. However, catalytic effects of metalelectrodes themselves have recently become more clear; forexample, on Cu, methane was detected, while on Ag and Au, COwas produced effectively in aqueous solutions. Furthermore, at aMo electrode, methanol was obtained. The power efficiency is,however, still low at any electrode.

2. The reaction mechanism of CO2 reduction is still a subjectof discussion, although, in general, the mechanisms proposed byEyring and co-workers45 and Amatore and Saveant53 have provedacceptable for aqueous and nonaqueous solutions, respectively. Insitu spectroscopic measurement techniques, by which intermediatesand their adsorption behavior can be estimated, will become moreand more important in better understanding each elementary stepof the reaction pathway.

3. If multielectron transfer takes place, the potentials requiredthermodynamically for CO2 reduction are much less negative than


the potential for single-electron transfer to CO2, CO2/COJ. Someeffective catalysts for this purpose have been extensively examined.

4. Semiconductor electrodes are promising for CO2 reductionto highly reduced products, such as methanol and methane, becausethe band structures of semiconductors give rise to a pool of electrons(in the conduction band), which may facilitate multielectron trans-fer. In spite of complications resulting from various conflictingreports, a great deal of progress in the investigation of CO2 reductionon semiconductor electrodes has been made. Unfortunately,however, the usable currents reported are usually very low, evenat n-type semiconductors.

5. The photoelectrochemical reduction of CO2 at illuminatedp-type semiconductor electrodes is also effective for CO2 reductionto highly reduced products. The combination of photocathodeswith catalysts for CO2 reduction leads to a marked decrease in theapparent overpotential. At present, however, light to chemicalenergy conversion efficiencies are still very low, and negative insome cases.

6. The photoassisted reduction of CO2 with suspended semi-conductor powders gives, at present, very low energy efficiencies(at most, ca. 0.01% or less). The use of colloidal semiconductorparticles is more efficient in some cases.

7. Various kinds and types of catalysts for CO2 reduction havebeen developed. Most of them are metal-based complexes. Metalcomplexes of N-macrocyclic compounds are promising. Amongthem, Ni-cyclam is an effective catalyst for selective CO productionfrom CO2, even in water. Re, Ru, and Rh complexes have beeneffectively used in recent years. Also, iron-sulfur clusters are inter-esting as catalysts. Nonmetal catalysts also seem to be possible.The use of polymer-modified electrodes is one of the most interest-ing aspects because polymers may have catalytic functions and alsocatalysts of interest can be incorporated in high density into thepolymer matrix. Fortunately, immobilization of the catalysts onelectrodes sometimes stabilizes their activities, i.e., turnover num-bers increase remarkably. (The review by Murray,213 for example,may be consulted to understand the general features of polymer-modified electrodes.) Although only a few studies focusing oncatalysts for electrochemical and/or photoelectrochemical reduc-tion of CO have been reported, the catalytic reduction of CO would

392 Isao Taniguchi

also lead us to develop multifunctional catalysts with which thedirect reduction of CO2 to highly reduced products can be achieved.

8. Considerable progress has been made on CO2 fixation inphotochemical reduction. The use of Re complexes as photo-sensitizers gave the best results; the reduction product was CO orHCOOH. The catalysts developed in this field are applicable toboth the electrochemical and photoelectrochemical reduction ofCO2. Basic concepts developed in the gas phase reduction of CO2

with H2 can also be used. Furthermore, electrochemical carboxyla-tion of organic molecules such as olefins, aromatic hydrocarbons,and alkyl halides in the presence of CO2 is also an attractive researchsubject. Photoinduced and thermal insertion of CO2 usingOrganometallic complexes has also been extensively examined inrecent years.

9. The following points may be made regarding future poten-tial advances in CO2 fixation:

(i) Precise analysis of CO2 reduction products would be help-ful to understand better CO2 reduction pathways. Products both inthe gas phase and in solution should also be taken into considera-tion. A gas-tight cell is useful.

(ii) To eliminate confusion, much attention should be paid toblank experiments and also to CO2 purification. Photodecomposi-tion of electrolytes as well as cell materials, such as epoxy resins,should be carefully monitored.

(iii) The use of organic solvents is worthwhile because of thehigh solubility of CO2 in these solvents, which has an effect similarto that of high CO2 pressure in aqueous solutions. In fact, muchrecent successful work has been done by using organic solventsand their mixture with water.

10. As mentioned in the text, it is clear that CO2 reduction tovaluable fuels and/or raw chemicals is still at an early stage and,unfortunately, far from having practical application, especially inthe case of solar to storable chemical energy conversion by photo-electrochemical means. Many difficulties remain to be overcome.However, it is also true that there has been much progress in CO2

reduction in recent years.It is difficult to comment about the possibility that CO2 reduc-

tion by various means can be a practical process of use in the future,because of the many difficulties to be overcome, especially the low


energy conversion efficiency. In fact, work in this area has not yetreached the stage for such a discussion. The maximum efficiencyof solar energy conversion at present seems to be, at most, 1 % orless, which is far less than that of green plants in nature, and notheoretical limit of energy conversion efficiency for CO2 reductionusing an ideal system has been established yet. For the time being,the electrochemical reduction of CO2 by using solar cells seems tobe more efficient from the particular viewpoint of solar energyutilization. However, this is another subject. At present, more funda-mental studies are required to design sophisticated systems forartificial photosynthesis in the future. Finally, it should also benoted that when CO2 reduction proceeds successfully, it will besure to have huge benefits. Undoubtedly, a number of new break-throughs will be required to establish a practical system for artificialphotosynthesis and for fuel production. However, the effortsdevoted will also have fruitful influences in various fields ofchemistry.

NOTE ADDED IN PROOF

CO2 reduction is currently one of the most attractive subjects ofinvestigation and new publications have been continuously comingout since the first writing of this manuscript. Some of them areincluded in References 214-222. These recent publications showthat CO2 and CO now show much promise for conversion to highlyreduced products such as CH4, C2H4, and methanol. It is likelythat a new era of CO2 reduction is about to begin.

ACKNOWLEDGMENTS

I wish to express my thanks to Professor J. O'M. Bockris for hisinvitation to write this article. I gratefully acknowledge that thiswork is based on stimulating discussions with Dr. Benedict Aurian-Blajeni and Professor J. O'M. Bockris, who were my co-workers atTexas A & M University several years ago. I also would like tothank Professor Kaname Ito of the Nagoya Institute of Technologyand many other Japanese colleagues as well as Professor Jean-Pierre

394 Isao Taniguchi

Sauvage of CNRS (Strasbourg) for their encouragement and usefuldiscussions about CO2 fixation over the years. Partial financialsupport by the Yazaki Memorial Foundation for some work onCO2 fixation presented in the text is also acknowledged.

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B 34 (1980) 27.173 D . R. Furge , G. D. Fong, and F. K. Fong, / Am. Chem. Soc. 101 (1979) 3694.174 C. E. Folsome, A. Brittain, A. Smith, and S. C h a n g , Nature 294 (1981) 64.175 J. P. P in to , G. R. G l a d s t o n e , a n d Y. L. Yung , Science 210 (1980) 183.176 R. A. She ldon , Chemicals from Synthesis Gas, Reidel , Dordrech t , 1983.177 F. A. Ur ibe , P. R. Sha rp , a n d A. J. Bard, /. Electroanal. Chem. 152 (1983)

173.178 G. A. Kolyagin , V. G. Dani lov , V. L. Korn i enko , I. A. Kedrinski i , and S. P.

G ub in , Elektrokhimiya 19 (1983) 1004; CA 9573 U (1983).179 H. H. Sorch, H. Go lum bic , and R. B. Ander son , The Fischer-Tropsch and Related

Syntheses, Wiley, New York, 1951.180 K. Ogura and S. Yamasaki, private communication.181 T. Iizuka, Y. Tanaka, and K. Tanabe, J. Catal. 76 (1982) 1.182 S. Polizzotti and J. A. Schwarz, /. Catal. 77 (1982) 1.183 H. Miura, M. L. McLanghlin, and R. D. Gonzalez, /. Catal. 79 (1983) 227.184 R. Ki rch , M . Kot te r , a n d L. Ricker t , Ber. Bunsenges. Phys. Chem. 88 (1984) 1054.185 M. A. Vannice and C. Sudhakar, /. Phys. Chem. 88 (1984) 2429.186 K. Kunimori, S. Matsui, and T. Uchijima, Chem. Lett. (1985) 359.187 S. Yamamura, H. Kojima, and W. Kawai, J. Electroanal. Chem. 186 (1985) 309.188 H. Yoneyama, K. Wakamoto, N. Hatanaka, and H. Tamura, Chem. Lett. (1985)

539.I. Taniguchi and Y. Shiraishi, unpublished results.

190 B. A. Sexton and G. A. Somorjai, /. Catal. 46 (1977) 167.191 G. D. Weatherbee and C. H. Bartholomew, /. Catal. 77 (1982) 460.192 H. E. Ferkul, D. J. Stauton, J. D. McCowan, and M. C. Baird, /. Chem. Soc,

Chem. Commun. (1982) 955.193 T. Iizuka, M. Kojima, and K. Tanabe, J. Chem. Soc, Chem. Commun. (1983) 638.194 H. Orita, S. Naito, and K. Tamaru, /. Chem. Soc, Chem. Commun. (1984) 150.195 T. Yoshida, D. L. Thorn, T. Okano, J. A. Ibers, and S. Otsuka, /. Am. Chem.

Soc. 101 (1979) 4212.F. Solymosi, A. Erdohelyi, and T. Bansagi, /. Chem. Soc, Faraday Trans. 1 77(1981) 2465.

197 K. Tanaka and J. M. White, J. Phys. Chem. 86 (1982) 3977.198 A. Amariglio, A. Elbianche, and H. Amariglio, J. Catal. 98 (1986) 355.

400 Isao Taniguchi

199 F. Solymosi and J. Kiss, Surf. Sci. 149 (1985) 17.200 H. A. C . M. Hendr ickx , A. P. J. M. Jongenel i s , a n d B. E. N i e u m e n h u y s , Surf.

Sci. 162 (1985) 269.201 A. Czerwinski , J. Sobokowski , a n d R. Marass i , Anal. Lett. 18 (A14) (1985) 1717;

J. Sobkowski a n d A. Czerwinski , J. Phys. Chem. 89 (1985) 365.202 D . A. Tyssee a n d M. M. Baizer, J. Org. Chem. 39 (1974) 2819, 2823.203 S. Wawzonek and D. Wearr ing, /. Am. Chem. Soc. 81 (1959) 1067.204 M. M. Baizer and J. L. C h r u m a , J. Org. Chem. 37 (1972) 1951.205 J. W. Wagenknech t , /. Electroanal. Chem. 52 (1974) 489.206 S. W a w z o n e k a n d J. M. Shrade l , /. Electrochem. Soc. 126 (1979) 401 .207 M. M. Baizer a n d H. L u n d , Eds . , Organic Electrochemistry, Marcel Dekker , New

York, 1983, C h a p s . 6, 20, a n d 25.208 T. Tsuda , Y. Chujo , and T. Saegusa, J. Chem. Soc, Chem. Commun. (1975) 963.209 T. Forschner , K. Menard , and A. Cutler , / Chem. Soc, Chem. Commun. (1984)

121.210 B. P. Sullivan and T. J. Meyer , /. Chem. Soc, Chem. Commun. (1984) 1244.211 M. Ka to and T. I to, Inorg. Chem. 24 (1985) 504, 509; H. Ito and T. I to, Bull.

Chem. Soc Jpn. 58 (1985) 1755.212 E. G. Lundquis t , K. Folt ing, J. C. Huffman, and K. G. Caul ton , Inorg. Chem.

26 (1987) 205.213 R. W. Murray , in Electroanalytical Chemistry, Vol. 13, Ed. by A. J. Bard, Marcel

Dekker , New York, 1984, p . 191.214 R. L. Cook , R. C. MacDuff, and A. F. Sammelles , J. Electrochem. Soc 134 (1987)

1873; ibid., 134 (1987) 2375.215 S. Ikeda, T. Takagi , and K. I to, Bull. Chem. Soc Jpn. 60 (1987) 2517.216 Y. Hor i , A. Mura ta , R. Takahash i , and S. Suzuki, J. Am. Chem. Soc 109 (1987)

5022.217 K. Ogura and H. Uchida , /. Electroanal Chem. 220 (1987) 333.218 H. Tanabe and K. O h n o , Electrochim. Acta 32 (1987) 1121.219 K. Ogura and I. Yoshida , Electrochim. Acta 32 (1987) 1191; K. Ogura and M.

Fujita, /. Mol Catal. 41 (1987) 303.220 Y. Hor i , A. Mura ta , K. Kikuchi , and S. Suzuki, J. Chem. Soc, Chem. Commun.

(1987) 728.221 H. Ishida, H. Tanaka, K. Tanaka, and T. Tanaka, /. Chem. Soc, Chem. Commun.

(1987) 131.222 S. Daniele, P. Ugo, G. Bontempelli, and M. Fiorani, J. Electroanal Chem. 219

(1987) 259.

Electrochemistry of Aluminum inAqueous Solutions and Physics

of Its Anodic Oxide

Aleksandar DespicFaculty of Technology and Metallurgy, University of Belgrade, Belgrade, Yugoslavia

Vitaly P. ParkhutikDepartment of Microelectronics, Minsk Radioengineering Institute, Minsk, USSR

I. INTRODUCTION

Anodic oxidation of valve metals, particularly, aluminum, hasattracted considerable attention because of its wide application invarious fields of technology. Traditionally, aluminum is "anodized"in order to protect the metal against corrosion, to improve itsabrasion and adsorption properties, etc.1 The more recent andrapidly growing applications of anodic aluminas in electronics aredue to their excellent dielectric properties, perfect planarity, andgood reproducibility in production. Finally, ways have recentlybeen found to use the energy potential of aluminum oxidation forchemical power sources of the metal-air type2'3 and otherelectrochemical applications.

Thus, interest in the electrochemical behavior of aluminum inaqueous solutions and anodic oxides, which, until recently, wasstimulated entirely by attempts to cope with corrosion, has beenenhanced by the wide new areas of application.

A vast body of literature tackles the different aspects of anodicoxidation, including the growth, structure, morphology, and proper-

401

402 A. Despic and V. P. Parkhutik

ties of anodic oxides A number of review articles are available.These are widely cited in a review by Thompson and Wood pub-lished some five years ago.4 The latter can be considered the mostcomprehensive review to date on the mechanism of barrier andporous alumina growth, with special emphasis on chemical compo-sition and morphology. However, some important aspects seem notto have been given adequate attention and much new knowledgehas accumulated since its publication.

Moreover, novel techniques of thin-film analysis (EXAFS,RBS, XPS, etc.) and improved sensitivity of traditional techniques(e.g., IR spectroscopy) have afforded a better understanding ofanodic oxide growth and have even led to a reconsideration ofcommonly accepted concepts.

Also, the increasing application of alumina films in the elec-tronics industry requires that attention be paid to their electrophy-sical properties (dielectric strength, conduction, etc.). However,since the work of Goruk, Young, and Zobel,5 published as longago as 1966 in this same series, no articles reviewing these problemshave appeared. An attempt is made here to emphasize the correla-tion between the electrophysical properties of oxides and the historyof their growth.

Finally, a large number of phenomena connected with activeelectrochemical dissolution of aluminum in the electrolyte, pro-moted by the presence of aggressive anions, are considered todeserve special attention, because understanding of thesephenomena is far from complete, and it is hoped that a review ofthem will stimulate further research.

It is clear that the problem of anodic oxide films is increasinglyof multidisciplinary interest, shared by specialists in different areasof physics and chemistry. It is felt, from reviewing the literature,that these researchers often tend to overlook some aspects whichare somewhat removed from their immediate field. Thus, an other-wise excellent experiment or theory may lose significance becauseof some neglected or ill-defined detail. Hence, it is considered usefulat the outset of this article to try and give an overview of the systemas a whole, summarizing all the factors which contribute to itsextreme complexity. Though some of the points in this overviewmay be considered only too well known by some scientists, it ishoped that they will arouse a new awareness in others.

Electrochemistry of Aluminum 403

II. OVERVIEW OF THE SYSTEM

Formation of aluminum oxide (alumina) upon contact of aluminummetal with pure water occurs because the reaction

2A1 + 3H2O = A12O, + 3H2 (1)

has a free energy of -864.6 kJ/mol under standard conditions.However, the reaction occurs spontaneously ("chemically") onlyuntil a compact layer of the virtually insoluble alumina has formedand separated the points at which hydrogen could evolve, fromfurther supply of water. At that point the reactants in Eq. (1) becomespatially separated. Two interfaces are formed—the metal/oxide(M/O) and the oxide/water (O/S)—and electrical potentialdifferences, A</>M/O = (</>Ai) - (4>o)o and A</>o/s = (</>o)o - (<£s), arebuilt up at both, as shown schematically in Fig. la, due to the

Figure 1. Schematic representation of potential profile and chargedistribution across an anodic oxide film of thickness 8 on aluminum:(a) hypothetical situation in the absence of any current; (b) in thepresence of an anodic current caused by corrosion or by an externalsource. RE, reference electrode to which the potential of aluminum isreferred.


transfer of an excess of aluminum ions and oxygen ions, respec-tively, across the two interfaces, which compensates for the furthertendency for the formation of the oxide.t

These potential differences cannot be directly measured.Nevertheless, the underlying processes at the two interfaces

2A1 + (302-)oxide = A12O3 + 6e~ (2)

and

H2O = 2H+ + (O2)a q (3)

(O2 la q = (02-)oxide (4)

must make significant contributions to the overall free energychange of reaction (1). Hence, the corresponding potential differen-ces must be relatively large.

If there were no metallic impurities at the aluminum surfacepenetrating through the oxide and if the oxide possessed no elec-tronic conductivity and no permeability to water, the measuredpotential difference with respect to a hydrogen electrode in purewater (pH = 7) would correspond to the thermodynamic one of-1.90 V. However, some electronic connection between the outerand the inner interface is always established, allowing reaction (1)to proceed at some rate (corrosion). This changes the situation ofthe potential distribution inasmuch as some electric field must becreated, large enough to provide for the corresponding rate oftransport of the two reacting ions toward each other. Hence, inreality the situation is similar to that shown in Fig. lb, and themetal immersed in water tends to acquire a significantly morepositive rest potential [open-circuit potential (OCP)] than thethermodynamic one.

f The only determinable quantities are the potential difference E = <f>Al - <j>'M, i.e.,the reversible potential with respect to a reference electrode (RE), ER =(0AI)O ~ 0/u (from thermodynamic data), and the cell voltage at a given currentdensity j , E(j) = (<pA])j - <f>'M (which can be measured experimentally). The poten-tial differences at the two interfaces, M/O and O/S, cannot be known exactlybecause of the unknown potential difference </>s — <£M and the volta potentialdifference between the reference electrode metal and aluminum, <f>'M - <f>'M.


The same applies if reaction (1) is forced by anodic currentfrom an external source. The potential must shift further in thepositive direction with respect to the OCP.t

It should be noted that the three parts of the electric field—across the M/O interface, across the bulk of the oxide, and acrossthe O/S interface—are determined by entirely different factors [cf.Section III(l)]. Hence, any imposed change of the potential levelof the metal redistributes itself among the three parts in a mannerwhich cannot be predicted without detailed knowledge of thekinetics of the processes taking place in each of them.

In considering the wide spectrum of phenomena observed inthis system, one must keep in mind the following points, whichmake for an extremely complex situation:

1. Although the anhydrous oxide is the stable reaction productat room temperature, the free energies of dehydration are relativelysmall. Hence, species of different degrees of hydration (or proton-ation) could form during anodic oxidation up to A12O3 • 3H2O [orA1(OH)3]. This is due to the possibility of step wise splitting of water,

+H + (5)

OH~ -> O2~ + H+ (6)

which may be kinetically more favorable than the direct splittingof two protons. Hence, a hydrated alumina should be expected atleast at the O/S interface, [cf. Section IV (4(i))].

2. Anhydrous alumina can have a variety of structures, froman entirely disordered one (truly amorphous), through short-rangeordered amorphous, to highly ordered in a tetrahedral or octahedralarrangement (y,y'-or a-alumina) [cf. Section IV(3)].

t When an external anodic current is applied, it is of interest to know the "faradaicefficiency," -qF, of anodic oxidation, defined as the ratio of the amount of metalactually oxidized to the amount which should be oxidized by the external currentif Faraday's law is strictly applicable with three electrons obtained per atom. Thefact that a part of the metal is oxidized by a corrosion process, without using theexternal current, makes the faradaic efficiency larger than 1. On the other hand,the "material efficiency," r]M, takes into account the quantity of electricity obtainedper quantity of the metal, related again through Faraday's law (of importance forchemical power sources). Obviously, T/M is the reciprocal of rjF. Hence, while thecorrosion process makes r\F > 1, it gives rjM < 1. Other reasons for deviations ofj)F and j}M from unity will be discussed below.


3. The type of conductance exhibited by the oxide and itsvalue are structure sensitive. The oxide is essentially an ionicconductor. One could maintain that it has a relatively high con-centration of low-mobility ionic charge carriers. As far as electronicconductance is concerned although pure alumina is an insulatorwith a band gap of 8 to 9 eV, one has to bear in mind that whenit is produced anodically as a thin film adhering firmly to the metal,an entirely different electronic situation may arise [cf. Section V(2)].

4. The metal virtually always possesses [even in the purestforms known today (99.9999%)] a microheterogeneity of the surfacewith respect to the ease of oxidation or the adsorption affinities forvarious species.

5. Even trace impurities have a profound effect on the open-circuit potential and the rate of corrosion. The electrochemicalbehavior is even more sensitive to alloying with small amounts ofother elements [cf. Section III(5(v))].

6. The three electrons in the outer shell of the metal atoms arenot identical (the electronic configuration is 2s22pl). Hence, theanodic oxidation at the M/O interface is very likely to proceedstepwise, i.e.,

Al -> Al+ + e~ (7)Al+ -> Al3+ + 2e~ (8)

forming the low-valency intermediate [cf. Section III(l(i))]. Thelatter may penetrate through the oxide by some valency transfermechanism and react with water at the O/S interface to formhydrogen [cf. Section III(5(iv))].

7. Aluminum is known to undergo another reaction with water,making the hydride:

Al + |H2O -> A1H3 + |O2 (9)This reaction has a large positive standard free energy change.

However, under certain conditions, it could proceed electrochemi-cally in the cathodic reaction

Al3+ + 3H+ + 6e~ -> AIH3 (10)with formation of an intermediate hydride ion:

Al3+ 4- H+ + 2e~ -> A1H2+ (11)and

A1H2+ + H+ + 2e~ -> AlHj (12)


These reaction products have their domains of stability incertain ranges of potential and pH as shown in the Pourbaix diagramin Fig. 2,6 which may have relevance in cases when open-circuitpotentials are established at highly negative values [cf. Sections111(2) and III(5(v))].

8. At some negative potentials, one could expect implantationof metal atoms obtained by the reduction of, for example, alkalimetal cations.7

9. The oxide is virtually insoluble only in pure water of neutralpH. Its solubility increases sharply in both acid and alkalinesolution, because it undergoes chemical reactions of protonation

H+

A1(OH)2

A1(OH)2+

and aluminate formation

|A12O3

A1(OH)3

A1(OH)2+ H2O

H+-» Al3 H2O

§H2O

OH~

A1(OH)3

A1O2 + 2H2O

(13)

(14)

(15)

(16)

(17)All of the ionic species formed are highly soluble.

This phenomenon enables some aluminum ions to cross theO/S interface and go into the solution. If the efficiency of oxideformation, 7jox, is defined as the ratio of the amount of solid oxideactually formed to the amount which would be formed if noaluminum went into the solution, such a "solubilization" reducesthis efficiency below 1.

Figure 2. The potential-pH (Pourbaix)diagram for aluminum in aqueousmedium, defining regions of thermody-namic stability of the different species.


10. At the O/S interface, for each molecule of alumina formedinside the oxide layer, i.e., three O2~ ions transferred across theO/S interface, six hydrogen ions are formed. Thus, the acidity atthe interface tends to rise to an extent which depends on the rateremoval of these ions by some mechanism. In view of Eqs. (13) to(15), this should lead to oxide dissolution and a further decreaseof r)ox.

11. When the metal is immersed in a solution of a salt of aweak acid (e.g., boric or tartaric), the latter exhibits a bufferingcapacity and thus provides one mechanism for the removal ofhydrogen ions from the interface [cf. Section III(3(iv))].

12. Some anions exhibit a complexing affinity towardaluminum ions. Thus, in such a situation, one should visualize thepresence not only of a series of pure complexes but also of mixedones with oxygen-containing ions, e.g., A1(OH)2C1, A1(OH)C12,A1C13, A1(OH)C1+, and A1C12+. The ionic species are soluble, andthus the interaction with complexing ions may provide an additionalmechanism for solubilization of the oxide.

13. The complexing affinity may cause penetration of anionsinto the oxide by some ion exchange mechanism. The presence ofsuch species inside the oxide may have a profound effect on itsconductivity.

14. Anions may exhibit a tendency toward specific adsorptionat the O/S interface. This may be related in some way to thecomplexing affinity. This effect, occurring at the inner Helmholtzplane of the electrochemical double layer, may significantly changethe charge transfer situation [cf. Section III(5(iii))].

15. Finally, one should note that a significant difference inbehavior could exist between a very thin ( < 1 nm) and a thickoxide layer because of possible interface effects on the bulk oxidein the former case; such effects should be negligible in the latter case.

III. KINETICS OF ALUMINUM ANODIZATION

1. General Considerations

The variety of electrolytes and the wide range of their concentra-tions, temperatures, and anodization regimes provide for a variety


of anodization kinetics, structures, compositions, and properties ofanodic oxides.

Nonporous "barrier" oxides can be grown in neutral (pH 7-8)solutions of borates, citrates, tartrates, phosphates, etc.4'8 They arelimited to a thickness of several hundred nanometers by dielectricbreakdown initiation during growth. Porous oxides are formed inelectrolytes promoting oxide dissolution, i.e., aqueous sulfuric,oxalic, or phosphoric acid solutions, with a thickness of up tohundreds of microns.9 Generally, it is assumed that the structureof porous oxides is a close-packed array of columnar hexagonalcells, each containing a central pore normal to the substrate surfaceand separated from it by a layer of hemispherically shaped barrier-type film as shown in Fig. 3.10 Nonordered, fibrous-like porousoxides are also reported for anodization in chromic acid11 or alka-line baths12 or under pulse anodization.13 Some electrolytes contain-ing "aggressive" anions (halides) cause localized dissolution (pit-ting). In others, depending on the anodization regime, togetherwith barrier or porous oxide formation, pitting,14 uniform oxidedissolution,15 or "burning"16 can be produced as seen in Fig. 4,reproduced from the work of Fukushima et al9 Various types ofcorrosion can also arise.17

It was customary to study these different situations separately.The present state of the art, however, makes it reasonable to attemptconsidering all of these cases in a unified way. In fact, in quite anumber of publications it has been shown that there is no sharpboundary between barrier and porous films. Prolonged anodization

Surface of: fiber

Figure 3. A model of a porous oxide filmformed at 120 V in a phosphoric acid sol-ution, according to Heber.10


1 UNIFORM DISSOLUTION/

UNIFORM

COATING

CONCENTRATION

Figure 4. The temperature-electrolyte concentrationdiagram, defining regions of different anodic dissolutionphenomena, according to Fukushima et al?

of aluminum in barrier-forming electrolytes for 1 to 50 h leads tothe classical porous structure of the oxide.1819 On the other hand,as mentioned above, porous oxide growth comprises, at the initialstage, formation of a barrier oxide. This barrier film is maintainedduring further oxide growth as a semispherical oxide layer at porebottoms.20

It is obvious that some common processes have to take placeduring oxide formation, irrespective of how thick the oxide is orwhich type of electrolyte the metal is immersed in. The two inter-faces and the bulk of the oxide will be considered separately.

(i) M/O Interface

The transfer of an aluminum atom from the metal phase intothe oxide to form an ion should occur by a simultaneous chargetransfer, in much the same way as in all electrochemical metaldissolutions. An electrochemical double layer should be establishedwith aluminum oxide as a solid electrolyte. Since there are no othersolvent molecules present, the likelihood of the presence of a layeranalogous to the Helmholtz layer is small. The electrolyte part ofthe double layer is more likely to be of a Gouy-Chapman type,causing the appearance of a space charge and potential distributionin the oxide according to the Gouy-Chapman equation21

<f>x = cf>oexp(-Kx) (18)

where K is a constant containing the ionic charge and concentrationas well as the dielectric permittivity of the oxide.

Because of the likely high ionic concentration and the smalldielectric constant of the oxide, the diffuse layer thickness is expec-ted to be small, and hence this space charge is limited to a fewnanometers.


The two-step charge transfer [cf. Eqs. (7) and (8)] with forma-tion of a significant amount of monovalent aluminum ion is indi-cated by experimental evidence. As early as 1857, Wholer and Buffdiscovered that aluminum dissolves with a current efficiency largerthan 100% if calculated on the basis of three electrons per atom.22

The anomalous overall valency (between 1 and 3) is likely to resultfrom some monovalent ions going away from the M/O interface,before they are further oxidized electrochemically, and reactingchemically with water further away in the oxide or at the O/Sinterface.23'24 If such a mechanism was operative with activation-controlled kinetics,25 the current-potential relationship should begiven by the Butler-Volmer equation

n yi — \ — -rivi/w/ f i-» T-. — \ rivi/w/ i V^ '7

whereto is the exchange current density for the equilibrium potentialdifference across the interface (Fig. la) and the "overpotential"

A(A0M/O) = (4>Al)j ~ Wo)j ~ [(4>A,)O - (4>'o)o\

is the change of that potential difference needed to pass the currentdensity/ The transfer coefficients could have two sets of values25:

(a) for the case of the first step rate determining

aa = 0.5 and ac = 2.5

and

(b) for the case of the second step [Eq. (8)] rate determining

aa = 2.0 and ac = 1.0

Considerations of the "negative difference effect" [see SectionIII(5(iv))] indicate the second case to be the likely one, but nodirect experimental evidence has been obtained so far.

In any case, experiments reviewed below [see Section 111(3)]indicate that^ is very large, i.e., significant currents can pass withoutmuch polarization. Hence, in such a case, the linearized form ofEq. (19) should be valid, i.e.,

A(A</>M/O) = -r=rj (20)<l/


(ii) O/S Interface

(a) Ion transfer across the interface

The problem of ion transfer across the interface has beentreated in detail by Sato,26'27 Scully,28 and also Valand and Heus-ler,29 following the general theory of Vetter.30 Valand and Heuslerassumed the same type of activation-controlled charge transferkinetics, except that the dominant charge here is that on the O2~ions (or OH~ ions) obtained by splitting water at the interface. Theelectrochemical double layer here is of the usual type for aqueoussystems and the equilibrium p.d. is determined by the main chargetransfer reaction

SDlH2O = OX02~ + 2solH+ (21)

or, alternatively,

SGlH2O = OXOH" + so lH+ (22)

and

s o lOH' = OXOH~ (23)

The species entering the oxide do not, of course, stay as freeentities but are likely to combine into hydrated oxide species ofdifferent charges and degrees of hydration. The equation describingthe kinetics is

(24)

where A(A<£O/S) = (4>o)j ~ (4>o)o is the change of the p.d. from theequilibrium one to the one needed to drive /

One could also expect protons to cross the interface causingprotonation (hydration) of the oxide:

s o lH30+ = OXH+ + SOIH2O (25)

Also, any protons formed in the oxide from OH" ion splittingshould cross the interface in the opposite direction (the Hoar-Yahalom mechanism of field-assisted proton transfer31).


Inasmuch as the protonation of the oxide can be favored bythe direction of the field at the O/S interface (cf. Fig. 1) at equili-brium, the proton current in this direction should decrease exponen-tially with increasing anodic polarization, as the field strength isdecreasing and can even change sign. Conversly, the deprotonationshould be favored, becoming the main mechanism of formation ofthe anhydrous oxide.

However, the Hoar-Yahalom mechanism31 has been ques-tioned by a number of scientists.419'32

Transfer of aluminum ions from the oxide into the solutionwas considered as a statistically independent process, whosekinetics are governed by a rate equation similar to Eq. (24), i.e.(neglecting the return of the ions into the oxide),

= fcJ,'+<i(OHT e x P [ | ^ A(A0O/s)] (26)

One should note that j o ^ - is the oxide-forming current while7*A1

3+ is the dissolution current. Hence, their ratio determines r/ox.Valand and Heusler29 determined jAi3+ experimentally, by

chemical analysis of the solution. Hence, the oxygen ion currentcould also be estimated from the total current and j A l ^ .

The overpotential A( A</>O/S) could not be experimentally deter-mined. However, taking only the first term in Eq. (24) (which is areasonable assumption at any real anodic dissolution currentdensity), one could derive the ratio of the Tafel slopes of the twocurrents as

(d lnjo2-/d lnjM>+)pH = a'Jy (27)

Indeed, in electrolytes containing no "aggressive" anions (asare the halides), over a wide pH range between 0 and 12 (fromsulfuric acid through acetate to phthalate and borate), a doublelogarithmic plot ofj0

2~ versus yAi3+, shown in Fig. 5, yielded straightlines, with slopes of 1.38 ± 0.14.

Similarly, pH dependences of jo2~ andjA13+ (shown in Fig. 6)

give

(d\ogj02-/dpH) = y-afa (28)

and

r - r (29)


log j /Am

Figure 5. Double logarithmic plot of cur-rent density of oxygen ion incorporationinto the oxide, 7J2-, versus aluminum dis-solution current density, j c , at different pHvalues: <J, pH 0.0; A, pH 1.55; O, pH 4.63;V, pH 5.53; T, pH 6.9; • , pH 8.9; D, pH9.85; A, pH 11.0.

Figure 6 indicates a change in the charge transfer mechanismat a pH between 9 and 9.5, corresponding to the pH of zero chargeof aluminum oxide.33'34 Experimental results on the slopes enabledspeculation on the values of transfer coefficients and reaction orders.From that, Valand and Heusler concluded that the most probablemechanism of oxygen ion transfer [reaction (21)] is

(30)

(31)

(32)ad

H2O = adOH" + solH+

OH" = a dO2- + so lH+

adO2 "^ 2 -

with the last step rate determining.

Figure 6. pH dependence of (a) currentdensity (on log scale) of oxygen ion incor-poration into the oxide, at a constant totalcurrent density of 0.1 mA/cm2, and (b) thesteady-state dissolution (aluminum ion)current density of oxide-covered aluminumat 4 V versus SCE.29


As far as the aluminum ion transfer is concerned, the indicatedrate-determining step is

OXA10H2+ -> solA10H2+ at pH < 9 (33)

and

OXA1OH2+ + solOH- -> solAl(OH)X at pH > 9.5 (34)

It is obvious that such an ion transfer must be preceded bysome association of the aluminum ion from the oxide lattice withOH" ion (directly from the solution or adsorbed at the interface)[Eq. (22) or (23)] or by protonation with H+ ions from the solution(Eq. (25)]. Valand and Heusler maintain the first case to be opera-tive. This conclusion must, however, be taken as tentative, andfurther arguments of an experimental nature are warranted.

Additional effects are produced by the presence of "aggressive"anions in the electrolyte. They are treated in Section 111(5).

(b) Chemical dissolution of the oxide

In the above considerations, the O/S interface was taken tobe a clear-cut boundary between the oxide and the electrolyte. Inreality, however, the outer part of the oxide is likely to be hydratedand penetrated by the electrolyte. Hence, the true O/S interface islikely to be withdrawn from the surface to a sufficient depth suchthat some oxide is left without any electric field imposed across it.This is especially true of thick porous oxide layers, but it can occurwith compact layers as well. For example, Hurlen and Haug35 founda duplex film in acetate solution (pH 7-10), composed of a drybarrier-type part and a thicker hydrated part consisting ofA12O3 • |H2O. Although the hydrated part becomes thinner withdecreasing pH and seems to practically vanish at low pH, even athickness of less than a nanometer is sufficient for the surface oxideto stay outside the electrochemical double layer.

In such a case, chemical interaction with the solution can takeplace. This is likely to be primarily the protonation (hydration)reaction leading to formation of some soluble complex ions whichdiffuse toward the bulk of the solution. Hence, this produces amechanism for dissolving and thinning the oxide layer, and therate of this process should be some function of the hydrogen ion


concentration in solution. In such a case, the attainment of a "steadystate" in which the oxide stops growing does not necessarily implyan equilibrium situation with respect to oxygen ion transfer acrossthe O/S interface (yo

2~ = 0), as assumed by Valand and Heusler,but rather the one governed by material balance. Hence, j o

2 should,in such a case, be larger, the higher the rate of chemical dissolution.

(c) Proton buildup effects

The above considerations are based on the assumption thatthe pH at the O/S interface is constant and equal to that in thebulk of the solution. However, in view of the fact that the formationof each molecule of oxide is accompanied by liberation of sixprotons (see Section II, point 10), this need not be so, and this alsoappears to affect the extent to which the oxide layer will growduring anodization (the efficiency of oxide formation, r/ox) and thetype of oxide that will be formed (compact "barrier type" orporous).

The problem of the changes in pH close to and at the O/Sinterface attracted attention primarily in relation to pit growth inlocalized (pitting) corrosion and in attempts to predict the hydrogenion concentration inside the pit. Thus, in 1937, Hoar suggested an"autocatalytic" mechanism of pit propagation,36 the basic reasonbehind which was a drop in pH inside the pit. Indeed, in a numberof investigations,37"42 the pH was found to decrease significantlyinside pits. Galvele43 has attempted to calculate the point, depend-ing on a product of pit depth and current density, at which the pHfalls below that of stability of the insoluble oxide (hydroxide). Hismodel was unrealistic inasmuch as he assumed that the metal isdissolved directly in the electrolyte in the form of a free metal ion(zero current efficiency of oxide formation), the only cause ofdecreased pH being the hydrolysis of this ion. Nevertheless, heshowed that for aluminum ions in solution, the hydrogen ionconcentration could change, for this reason only, by four orders ofmagnitude for a product of current density and diffusion layerthickness ("depth of a pit") of 10~3 A/cm.

Inasmuch as this was denied as the possible cause of pitinitiation (cf. Vetter and Strehblow44), there should be no doubtthat, not only in pits, but wherever anodic dissolution of aluminum


and oxide formation takes place, a significant effect of a pH changecan be expected. Possible events are represented schematically inFig. 7,45 in which consumption of H+ ions by buffering with anionsis also envisaged.

With such a model, the rate of increase in oxide thickness isdetermined by the difference between alumina formation, strictlyfollowing Faraday's law, and its dissolution, the rate of whichshould be some function of hydrogen ion concentration at theinterface, i.e.,

^ = -\^--kc(H+rdt p L3F

(35)

where M and p are the molecular weight and density of the alumina,C represents concentration, and k may be an electrochemical or achemical rate constant, but in view of the above considerations, itis more likely to be the latter.

In any case, the time dependence of the hydrogen ion con-centration at the interface should be obtainable from a mass-balanceequation as

dC(H+)dt

= j ^ - kC(H+)n - kC(A-)C(H+)

(36)

the source being proportional to the current density and the threesinks being (a) the reaction with the oxide, (b) the net association

Figure 7. Schematic representation of eventsat the O/S interface in the presence of abuffering electrolyte (HA/A"). M


with the anion of the electrolyte (with rates of association anddissociation being determined by the rate constants k and k, respec-tively), and (c) molecular (or convective) diffusion into the elec-trolyte represented by the diffusion flux 4>o. The latter is timedependent in a complex way. Hence, Eq. (36) cannot be integratedin a straightforward manner. Nevertheless, different situations canbe discussed:

1. If the buffering capacity of the electrolyte is significant andthe association/dissociation rates are very fast so that the corre-sponding two terms dominate all the others, then

= -£C(A~)C(H+) + kC(HA) = 0 (37)at

i.e., the buffering equilibrium is virtually not disturbed by hydrogenion formation, and C(H+) is equal to that in the bulk of solution:

fcC(HA) C(HA)C ( H } - k C(A-) - K" C(A-) ( 3 8 )

Substituting this in Eq. (35), the rate of growth of the oxide is seento be constant, i.e., the oxide layer thickness increases linearly withtime.

2. If there is no buffering capacity [k -* O and C(HA) -» O]and if the diffusion away is very slow, the concentration of H+ ionsat the interface will grow until the second term in Eq. (36) becomesequal to the first one. Substituting such a condition in Eq. (35),one can see that the rate of growth of the oxide becomes zero, i.e.,the oxide attains a constant thickness. (In fact, some hydrogen ionswill always escape by diffusion and, hence, complete equality ofthe two terms in Eq. (35) can never be attained so that some growthwill have to continue.)

A more detailed discussion of the problem, including someapproximate solutions for the time dependence of oxide growth,is available in Ref. 46.

In reality, the increase of the rate of dissolution due to increas-ing hydrogen ion concentration should increase the overall rateover that of the dissolution via direct transfer of aluminum cationthrough the O/S interface discussed above.


(iii) Transfer of Species through the Oxide

A number of reviews are available on the transfer of speciesthrough the oxide.5'47"49

Ionic conduction studies in solids date back to the work ofGunterschultze and Betz,50 who derived the empirical relationshipbetween the electric field strength, E = [(<K)j•- (4>o)j]/s (cf.Fig. lb), in an oxide and the nonohmic ionic current density, j ,

j = Aexp(BE) (39)

where A and B are constants depending on the temperature.Cabrera and Mott51 have deduced this equation from a modelassuming the influence of an electric field, E, on the barrier heightfor migrating ions at oxide interfaces:

DC0 (zeaE\

= -£-expVWj (40)

where D is a diffusion coefficient, Co is a surface density of movingions, and a is an activation distance of migration. (This equationcannot be directly applied in the case of nonplanar oxide geometry.)

Equations of this type are most often used by experimentaliststo fit their data to theory,5254 as indications of an exponential j(E)dependence are numerous. Vervey55 has used a similar approachto consider the volume-limited processes of ionic migration andobtained the same equation. In order to explain deviations ofexperimental behavior from that predicted by Eq. (40), Vermilyea56

has also taken into consideration the effect of electrostrictionmodifying the activation distance for migrating ions and obtainedthe following equation for the ionic current flow:

DC0 [e x p L = Jwhere a and f$ are constants.

An analogous expression assuming space charge effects andthe double layer structure of the anodic oxide has been obtainedby Goruk et al5 and Bray.57

Dewald58 has introduced a mechanism in which space chargegeneration in the anodic oxide results from the difference betweenactivation energies at the oxide surface and in the bulk. Winkel et


al.59 considered the influence of the amorphous structure of theoxide on ionic migration by assuming the oxide to consist of aseries of potential barriers with Gaussian distributions of activationdistances and heights. The following equation for the ionic fluxresulted:

'L-W'\) ( 4 2 )

where W = (W) + 8w/2kT and a' = (a) - ze8aE/2kT are meanvalues, with 8W and 8a variations of activation heights and distances,respectively. Young and Zobel60 have assumed ions to migrate overramified trajectories and be captured by the oxide region locallycharged by the charge of opposite sign, leading to the followingexpression for the ionic current:

/W-(3E1/2\A kT )j,=./«>exp( ^ ) . (43)

where ji0 is a constant characteristic of the material. Fromhold61

has introduced space charge effects of moving ions disturbing theexponential j(E) law. This approach has been developed furtherin his more recent work.48'49

All the above derivations are based on the assumption of asingle ionic species moving through the oxide. The implicationsof such an approach have been considered most thoroughly byDignam.47 The present state of the art in the field of ionic conductionmodeling needs improvement. The theory should include thefollowing:

(a) Multi-ionic migration. The majority of metals and semi-conductors exhibit both cationic and anionic migration, each ofthem in its turn able to be a multicomponent one.

(b) Amorphous structure of anodic oxides. Since the work ofWinkel et al.,59 there have been attempts to consider ionic conduc-tion in disordered anodic oxides. One such attempt, based onhopping conduction of charged carriers, was made by Parkhutikand Shershulskii62 [cf. Section V(2)].

(c) The heterogeneous chemical structure of anodic oxide filmsacross the surface and perpendicular to it. The effects of a composi-tion varying with depth, causing complications in modeling the


ionic conduction, are currently attracting the attention of research-ers. Fromhold and Fromhold49 have modeled uni- and multipolarionic migration in anodic oxides possessing varying stoichiometryand have shown that moving ions participate in oxidation processesin the bulk of the oxide, thus changing its stoichiometry. Parkhutikand Shershulskii62 have modeled the heterogeneous chemical com-position of the oxide by considering the inhomogeneous spacecharge of heterogeneously incorporated ionic species.

2. Open-Circuit Phenomena

A metal which has been handled in the presence of oxygen fromthe air is always covered by a thin protective layer of oxide.63"65

As soon as it is immersed into an aqueous electrolyte, even if freeof oxygen, a corrosion process starts, imposing after some time asteady-state potential difference between the metal and the solution[open-circuit potential (OCP)]. The corrosion process could berepresented by a Wagner-Traud (Evans) model.66 However, thepartial current of the anodic dissolution of the metal has somewhatunusual characteristics. It remains constant on changing the poten-tial in a positive direction and increases only with an increase inpH. It appears that the dissolution process is controlled by thetransport of OH~ ions in solution.67 (It should be noted, however,that suppression of the dissociation of water [reverse of reaction(30)] would lead to a similar pH effect.) This remains so until acertain potential is reached, which depends on the anion of theelectrolyte. At that potential, a sudden rise in current density occurs[cf. Section 111(5)]. Driving the potential negative from the OCPresults again at a certain point in a sudden increase in the anodicdissolution rate (cathodic pitting corrosion). This resembles thepassive-active transition at passivating interfaces. It is interpreted68

as arising from the hydration of the oxide film due to hydroxyl ionformation accompanying increasing cathodic evolution of hydro-gen. Hence, the Wagner-Traud diagram (in the absence of oxygenin solution) should resemble the one presented in Fig. 8.t

t It is noteworthy that cathodic polarization of aluminum in highly diluted acidscan also cause oxide formation (so-called "cathodization").69 This effect is dueto the fact that proton discharge at the cathodic potentials results in a pH increaseand, hence, coagulation of aluminum hydroxide near the cathode. This causescolloidal A1(OH)3 deposition at the cathode, especially if it is made of aluminum.


OCP

Figure 8. Wagner-Traud (Evans) diagram for aluminum inaqueous solution, in the absence of dissolved oxygen.

Thus, it is interesting to note that high-purity aluminum restsat a potential at which corrosion is at its minimum and is, indeed,relatively very small. It is also largely independent of the anionspresent in the electrolyte.69 This may be attributed to the coulombicrepulsion of anions away from the surface by the negative chargeon the metal. The latter seems not to be completely compensatedin a thin oxide film, as shown schematically in Fig. 9, so that thesolution side of the double layer formed at the O/S interfacecontains excess cations, anions being repelled. The anions couldapproach the O/S interface either at thicker films or at potentialsmore positive than the OCP.

Figure 9. Schematic representation of poten-tial profile and charge distribution in a thinanodic oxide film on aluminum.


When the oxide is formed by anodizing in acid solutions andthe sample is then left to rest at the OCP, some dissolution canoccur. This process has been studied by a numbers of authors,70"75

especially in relation to porous oxides [cf. Section 111(4)]. It wasfound that pore walls are attacked, so that they are widened andtapered to a trumpet-like shape.70'71 Finally, the pore skeletoncollapses and dissolves, at the outer oxide region. The outer regionsof the oxide body dissolve at higher rates than the inner ones.919

The same is true for dissolution of other anodic oxides of valvemetals.76 This thickness dependence is interpreted in terms of adepth-dependent vacancy concentration in the oxide75 or by acidpermeation through cell walls by intercrystalline diffusion,disaggregating the microcrystallites of y-alumina.4

As for the thinning of the barrier film in such a case, it can beunderstood in terms of the effects discussed earlier [cf. SectionIII(l(ii))], as the relaxation of anodic polarization increases therate of proton transfer. Thus, the hydration of the outer regions ofthe film takes place, resulting in double-layer withdrawal andchemical dissolution at the surface.

The open-circuit dissolution is very sensitive to temperature(unlike the field-stimulated dissolution). It was found that a tem-perature rise from 20 to 25°C doubles the rate of dissolution.74 Therate of dissolution is, of course, pH dependent and is virtually nilat higher pH.

In the presence of oxygen in solution, the cathodic reactionbecomes that of oxygen reduction. This can shift the OCP in thepositive direction and bring it to, or close to, the potential of thesharp rise of anodic current ["pitting potential"; cf. Section

3. Kinetics of Barrier-Film Formation

The compact, nonporous anodic alumina film is the most suitablefor fundamental investigations. It is grown by anodization, mostlyunder constant-current (galvanostatic) conditions, in neutralsolutions of borates, tartrates, citrates, and phosphates, all of whichpossess significant buffering capacity and hence do not allowsignificant dissolution of the oxide.


(i) Dependence of Potential and Current on Time

The growth of an anodic alumina film, at a constant current, ischaracterized by a virtually linear increase of the electrode potentialwith time, exemplified by Fig. 10, with a more or less notablecurvature (or an intercept of the extrapolated straight line) at thebeginning of anodization.73 This reflects the constant rate of increaseof the film thickness. Indeed, a linear relationship was foundexperimentally between the potential and the inverse capacitance78

(the latter reflecting the thickness in a model of a parallel-platecapacitor under the assumption of a constant dielectric permit-tivity). This is foreseen by applying Eq. (38) to Eq. (35). It is aconsequence of the need for a constant electric field on the film inorder to transport constant ionic current, as required by Eqs. (39)-(43).

The intercept should reflect the unchanging activation polariz-ation at the two interfaces, as well as some other effects (presenceof a film before anodization, time lag in attainment of the steadystate, etc.). Nevertheless, the fact that it is small or negligibleindicates that charge transfer processes at the interfaces are fastand that the kinetics of the growth are entirely transport controlled.

The linearity leads to another important conclusion: a constantfield for a constant current implies a constant overall conductivitythroughout the film. Since the conductivity is very structure sensi-tive, this implies also that either (i) the film grows homogeneously

" 0 10 20t/min

Figure 10. Experimental records of galvanostatic aluminum anodiz-ation (ja = 5 A/m2) in various electrolytes: (1) adipate, (2) citrate,(3) tartrate, (4) phosphate, (5) oxalate, (6) borate. (a) Anode poten-tial versus time; (b) dissolved aluminum versus time; (c) maximumforming voltage versus electrolyte concentration.20


throughout or (ii) any inhomogeneity expands in proportion to thethickness. (This applies also to the space charge since it affects thefield in the interior of the oxide). It has been shown that, in at leastsome instances,79 the second case is operative [cf. Section IV(2)].

(ii) Current-Field Relationship

The fact is that, on the one hand, a significant field strength,E, is needed to provide significant current. On the other hand, oncein the practical current density range between 0.1 and 10 mA/cm2,a relative insensitivity of the field to the current density is found.In fact, an inverse field of 1.3 to 1.8 nm/V is accepted in the literatureas characteristic of the oxide growth, without mention of the currentdensity used.

Equations (39)-(43) indicate that a Tafel-type relationshipbetween the current and the field should be expected, i.e., from Eq.(40),

23kT la

or, from Eq. (43),

(44)

( 4 5 )

Figure 11 exemplifies experimentally observed dependencesof the current density on the imposed voltage at a constant oxidethickness. It is seen that they fit both equations fairly well. Tafelslopes [Eq. (44)] are in the range of 0.4 V cm"1 dec"1. All other

Figure 11. Steady-state current densityas a function of imposed voltage at aconstant thickness of a barrier-typeoxide, in \ogj versus E and log j versusEl/2 coordinates. (Based on data fromRef. 5.)


constants being known, the values of the slopes in Fig. 11 implyza =0.15 or/3 = 1.1 x 10~19.

In a number of works, a potentiostatic regime has been usedfor the experimental and theoretical study of the anodization ofaluminum and other valve metals.80 Upon the application of aconstant potential step, Va, barrier-forming electrolytes are charac-terized by a sharp increase in the anodic current to a certainmaximum. Both the slope and the maximum are determined by theimpedance of the cell circuit. Subsequently, there is a continuousdecrease in the anodic current, which is due to oxide growth. Thedecay of the anodic current can be described by the expression81

1 1 kE*— = — t (46)ja ln(ja/jao) Jo

where j0 — ja at t = 0, ja0 = DC0/2a, k is a constant, representingoxide volume per migrating ion, and E* is a constant. Within anarrow time interval, Eq. (46) can be approximated by a hyperbolicjat~

l dependence. It then leads to a logarithmic time dependenceof oxide growth52

^ + l ) (47)

where 8' and t' are dimensionless thickness and time, respectively,Eo is an initial electric field, and B is a constant.

Satisfactory agreement of experiments with kinetic laws,described by Eqs. (44) and (45), are observed only for tantalumand niobium, when the current efficiency approaches 100%. Evenfor these metals, certain deviations occur which could be attributedto space charge effects,82 electronic leakage currents,83 or otherfactors. In the case of aluminum, these deviations are relativelylarge, as, even in barrier-forming electrolytes, some oxide dis-solution takes place from the very beginning of voltage supply toan anodized sample.32

(III) Oxide Layer Thickness

The oxide layer thickness can be determined in a number ofways. Direct microscopic observation has been demonstrated byTakahashi et al.79 Results of other methods can be made to agree


Figure 12. Efficiency of oxide formationas a function of anodic current density.79 j /mAcm

with this direct evidence, by assuming certain values of someconstants. Thus, capacitance measurements could render the sameresult if it is assumed that the dielectric constant is 9.8 (a valuehigher than the generally assumed one of 8.4, but explainable whena surface roughness of 1.2 is taken into account). The thicknessdetermined by dissolving the oxide and determining the amount ofaluminum ions analytically would agree with the value obtainedby direct observation, if the density of the oxide is taken to be 2.95and the weight fraction of Al3+ in the oxide, 0.51. Finally, from thevoltage-to-field ratio, one could calculate the same thickness bytaking the inverse field (thickness-to-field ratio) as 1.5 nm/V, whichis in the usually observed range.

(iv) Efficiency of Oxide Formation

The efficiency of oxide formation can be found by comparingthe amount of aluminum determined analytically upon dissolvingthe film to that indicated by Faraday's law. As seen in Fig. 12, theefficiency is found to be less than 1 and increases with increasinganodic current density. On the one hand, this is in line with thefinding of Valand and Heusler29 that the oxygen ion currentincreases more steeply with A(A(/>O/S), and hence with the totalcurrent j , than the aluminum dissolution current (a'a being largerthan y). On the other hand, this is also predicted by Eq. (35),reflecting chemical dissolution. In this instance, it is not possibleto decide which of the two causes is operative.

(v) Ionic Migration in the Oxide Film

The anodic oxide is formed, generally speaking, by migrationof the positive ion (of the oxidized metal) and/or of the negative


ion(s) (oxygen ions, hydroxyl ions, or even anions of the electrolyte)inside the oxide toward the opposite interface under the influenceof the very high electric field created by the externally appliedpotential. The question arises as to what contribution each of theionic species makes to oxide formation, which reflects itself in theposition of the original oxide layer, existing prior to the anodization.The problem amounts to transport number determination and canbe solved by various methods.

Radiotracer techniques involving 18O in the anodization pro-cess are used with subsequent neutron activation analysis84 orSIMS.85 Another method involves implantation of inert ion markersinto the surface layer of the sample prior to anodization andexamination of the position of the markers after the oxide film hasgrown to a certain thickness.86 Assuming immobility of the inertspecies, the ratio of the cation to the anion transport number, t+/ f_,should be equal to the ratio of the outer to the inner layer thickness.Numerous experimental determinations72'87 suggest t+ and f_ to be0.4 and 0.6, respectively.

However, there are indications that these values depend onthe conditions of ionization. Vermilyea88 has interpreted the changefrom compressive to tensile stress, recorded in the oxide, to be dueto the dependence of the transport number of aluminum on theelectric field strength. Brown89 has found this transport number todepend on the electrolyte used in anodization.

A systematic study by Khalil and Leach,90 using a-spec-trometry, has provided values for transport numbers in the valvemetal oxides, which are interesting to compare. These are listed inTable 1.

Takahashi et al,79 in their work on the structure of the barrierlayer [cf. Section IV(2)], have considered phosphate ions, whichare found in the outer layer of the oxide, as immobile markers and,from the position of the boundary between the outer and the innerlayer, deduced the transport number of the cation to vary between0.73 and 0.81 in the current density range between 0.05 and10 mA/cm"2.

A similar method with similar results was used earlier byRandall and Bernard,91 but the objection that the result may be toohigh because of the motion of the phosphate ions was raised byDavies et al92 and Pringle.93 Smaller values independent of the


Table 1Transport Numbers for Various Metals Anodized in

Borate Solutions at 25°Ca

Metal

Ti

Zr

Ta

Al

Current density(mA/cm2)

6506

506

506

50

Oxide thickness(nm)

42.163.3

200179.2160.6133.0110110

Electricfield strength

(V/nm)

0.4270.7420.50.5880.6230.7520.910.91

t+

0.350.390.10.120.280.340.40.49

a Ref. 90.

marker were obtained when inert gas markers were used.92'93

However, even in this case, objections were found in that the methodwas shown to depend on the film existing prior to the anodization94

as well as on the acceleration voltage used to implant the marker.95

The above evidence, however, shows very clearly that in allbarrier film making, both positive and negative ions contribute incomparable proportions to the building of the oxide and hence thatthe oxide grows in both directions leaving the original oxide buriedsomewhere inside, close to the center.

At a certain anodic potential, the compact film breaks downand lets electrons pass through without much resistance, causingoxygen evolution at a high rate. This "dielectric breakdown" isdiscussed in more detail in Section V.

4. Formation of Porous Oxides

(i) Dependence of Potential and Current on Time

Dibasic and tribasic acids, such as sulfuric, oxalic, malonic,and phosphoric acids,96 cause the appearance and development ofa very regular porous structure of the oxide (cf. Fig. 3). Here, thekinetics of galvanostatic anodization are characterized by an initiallinear potential rise, followed, however, at relatively low anodic


potentials, by curving to a maximum value and a decrease to asteady-state value, as exemplified by Fig. 13a.74'97

It should be noted here that the barrier-film-promoting elec-trolytes are also characterized by VA(t) curves similar to those ofthe pore-forming ones, if comparatively small current densities areused (less than 0.5mA/cm2).20

The maximum and steady-state anode potentials depend onthe pH of the electrolyte in a manner shown in Fig. 13b, which wasobtained for anodization in sulfuric acid solutions with the pHadjusted to constant ionic strengths with Na2SO4. This dependencecan be expressed as

VA>0 = A ln(B - C[H+]) (48)

where A, B, and C are empirical constants.93 The same behaviorwas observed by Fukushima et al. for a variety of electrolytes.9

The parameters of the VA(t) function are very sensitive totemperature, as seen in Fig. 13a. As the temperature rises, theposition of the maximum shifts to lower anodization times, withthe value of the potential maximum also decreasing.

In the case of a potentiostatic oxidation with different appliedpotentials, as shown in Fig. 14,98 there is always a dip in the currentdensity corresponding to the potential maximum in galvanostatic

§

" 0 10 20 30 40 50

t / s

Figure 13. (a) Porous oxide growth kinetics foranodization in 15 wt % H2SO4 at ja = 5 mA/m2 andTe = 5°C (1), 15°C (2), 25°C (3), 35°C (4), and 45°C(5).93 (b) Stationary anode potential versus electrolytepH for anodization in H2SO4 solution at ja =4 mA/cm2 (1), 2 mA/cm2 (2), and 1 mA/cm2 (3).97


Figure 14. Anodic current versus time curves forpotentiostatic Al anodization in oxalic (0.2 M) (a),sulfuric (0.5 M) (b), and phosphoric (0.4 M) (c)acid solutions. Ua = 5 V (1), 10 V (2), 15 V (3),20 V (4), 30 V (5), and 40 V (6).

experiments, then a rise to a steady state, with the current densityin some cases significantly larger than the initial current density.

The details of the ja(t) dependence are thoroughly examinedin a number of works,99100 including, most recently, one by Hurlenet al101 His data indicate that aluminum anodization in slightlyacidic electrolytes at low potentials (only slightly higher than thatcorresponding to the active-passive transition at about —1.0 V vs.SCE) reveals a similar current maximum, a sharp decrease, andfurther growth to a steady-state value, but at much higher potentials(Fig. 15).

Such a behavior of the UA(t) or ja(t) functions is consistentwith a fairly well established pore growth mechanism.4 Accordingto this mechanism, the linear potential growth (and current density

Figure 15. Kinetics of potentiostatic Alanodization in ammonium acetate at lowpotentials.35


fall) corresponds to the formation of a planar barrier film. Afterthis film reaches a certain thickness, micropores are nucleated overthe surface, which causes potential saturation and decrease. Thisactually means that, for some reason, microheterogeneity of theoxide comes into play, resulting in easier dissolution of the oxideat some points than at others [implying a varying dissolution rateconstant in Eq. (35) over the surface]. This heterogeneity may bethe same as that causing pitting phenomena under different condi-tions [cf. Section 111(5)].

(w) Steady-State Potential-Current Density Relationship

The steady-state potential (or current density) is related to asteady growth of the porous oxide into the solution, maintaininga constant number of pores and a constant pore radius. This schemeis supported by electron microscopic observations reported by Xuet al102

Typical steady-state voltage-current characteristics in pore-forming electrolytes are shown in Fig. 16. A number of authorshave attempted to interpret these dependences.103104 Ebihara etal.105 used an equation based on a pore model and taking intoaccount a rate-determining transport through the barrier part of

5 10 20 50 100 200 500j a / A r r T 2

5 10 20 50 100 200

j / A m " 2

Figure 16. Relationship between anodiccurrent density and steady-state voltagefor aluminum in (a) sulfuric acid (2 M)(a) and oxalic acid (0.46 M), (b) solutionsat different temperatures.105


the oxide:

W - BE\j (49)

They extracted values of the pre-exponential factor and of W inEq. (49) for two electrolytes (sulfuric and oxalic acids) at differentconcentrations, A being in a reasonable range and W exhibitingremarkable constancy (0.847 and 0.751 for sulfuric acid and oxalicacid, respectively). It should be noted, however, that the modelcontained many adjustable parameters.

(III) Pore Nucleation Phenomena

Two questions that arise are, why does pore formation occuronly after the compact oxide has reached a certain value and, oncethe dissolution starts, why is it not even?

An answer to the first question may be found in noting thatthe electric field in a thin oxide film is different from that in a thickone and that weakening of electrostatic repulsion which preventshydration and withdrawal of the O/S interface from the surface isa prerequisite for chemical dissolution.

One should note that dibasic and tribasic acids still havebuffering capacity, since in the second (or third) dissociation stepthey behave as weak acids. Hence, it takes some time before thehydrogen ion concentration at the surface can increase sufficientlyto start dissolving the oxide. Once this is achieved and a local attackon weaker oxide sites commences, additional buffering by thedissolved aluminum (oxo) ions prevents the further increase inhydrogen ion concentration needed for an equally fast dissolutionof the "harder" oxide. The existence of different-quality oxidesin a hexagonal arrangement, with an amorphous one in themiddle and a crystalline one at the hexagon edges, was found byFranklin.106

5. Active Dissolution of Aluminum

Active anodic dissolution occurs when all the electrochemicallyoxidized aluminum passes into the aqueous phase and the oxidelayer does not grow, i.e., the current efficiency of oxide formation


100-

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.5 1.0E / V vs. SCE

1.5 2.0

Figure 17. Increase in current obtainedon aluminum upon sweeping the poten-tial in the anodic direction in aqueouselectrolytes containing different

falls virtually to zero. This phenomenon appears in the presenceof a number of "aggressive" anions, in particular, halide ions andhydroxyl ion, although the mechanism of action of these two typesof anions appears to be different.

In solutions containing different anions, as seen in Fig. 17, thesudden rise in the anodic current density mentioned earlier [seeSection 111(2)] and characteristic of initiation of active dissolutionoccurs at different potentials. It was shown108 that, at least withhalides, this potential is a linear function of the crystalline radiusof the ion.

(i) Dissolution in the Presence of Halide Ions

If a well-defined compact oxide layer is grown to a certainthickness in a barrier-forming electrolyte (so that the electrodepotential increases to very high values in order to maintain aconstant current), when chloride ions are added, a dramatic decayof the potential results within milliseconds, as shown in Fig. 18.77

20t / s

Figure 18. Decay of oxide formationpotential at a constant current densityof 10mA/cm2 upon addition ofchloride to the electrolyte.


Further oxidation at the same current density takes place at arelatively low constant potential, indicating that the oxide hasstopped growing, i.e., that the efficiency of oxide formation, rjox,has changed from virtually 100% to zero.

As Kaesche pointed out for the example of chloride solutions,on which most of the studies were done,67 at such a particularpotential the current density rises with an infinite slope, i.e., at anycurrent density applied galvanostatically, one and the same dis-solution potential is obtained. Actually, careful determination andsubtraction of the pseudo-ohmic potential drop between the Luggincapillary of the reference electrode and the electrode surfaceresulted in recording a negative slope for the true polarizationcurve, i.e., an increase in the current density led to a somewhatreduced anodic polarization.109 This phenomenon was associatedwith the appearance of localized attack, or "pitting," and hencethe potential plateau at which this occurs is termed the "pittingpotential."110 The pitting potential was found to be independentof pH (-485 ± 10 mV versus SHE for pH 2 to 11) and stirring,down to very low pH values, but was strongly dependent on chlorideconcentration down to 10~3 M. Of the two possible types of pittingfound in corrosion of metals, shown in Fig. 19, aluminum in halidesolutions develops the crystallographic, or "etch," pits.

The constancy of the potential with increasing current densitycould be explained in terms of an automatic adjustment of thenumber of pits while maintaining a constant current per pit. Atpotentials more positive than the pitting potential, Kaesche67 hasfound the total current to increase with time. This complied verywell with a model in which the true current density at the pit(found to be of the order of 300 mA/cm2) and the number of pits,

Figure 19. Schematic drawing of cross sections oftwo types of pits developing in pitting corrosion ofpassive metals: (a) geometric pit; (b) crystallo-graphic or "etch" pit.


counted by optical microscopy, remain constant while the surfacearea of each pit increases as the third power of time. However, thefact that at the constant-potential plateau of active dissolution, theimposed constant current remains constant with time implies adecrease of the true current density at the increasing active surfaceinside the pit or a steady deactivation of some pits (reduction ofthe number of pits), or both. A constant current per pit could becaused either by ohmic control of the process at the pit mouth[which often remains unchanged in diameter while dissolutionoccurs underneath the surface ("undercutting")] or, more likely,by automatic adjustment of the surface film properties (e.g., thick-ness, by the hydrogen ion concentration effect). An interesting factis that the dissolution potential also remains constant when thecurrent density is decreased from any high value (up to 1 A/cm2),even down to the lower limit at which the potential starts deviatingtoward the OCP (below 1 mA/cm2). This implies an automaticdeactivation of dissolution, i.e., either a reduction in the numberof active pits or a decrease in the true current density at their bottom.

Recent studies of the processes of activation and deactivation111

have shown, as seen in Fig. 20, that the time dependences of thepotential, upon the application of current steps, resemble thosecharacteristic of porous film formation and that the differences areof a quantitative nature. The initial part, representing a typicalgalvanostatic charging curve (with the initial jump due to the

0.0

Figure 20. Initial change of potential of aluminumin 2 M NaCl solution upon application of differentconstant anodic current density steps.


pseudo-ohmic resistance and a subsequent charging of the inter-facial capacitance), is not observed in porous film formation simplybecause the voltages involved are an order of magnitude higherthan in the case of films obtained in halide-containing electrolytes.

The capacitance determined from the initial slopes of thecharging curve is about 10 /JL F/ cm2. Taking the dielectric permit-tivity as 9.0, one could calculate that initially (at the OCP) an oxidelayer of the barrier type existed, which was about 0.6 nm thick. ATafelian dependence of the extrapolated initial potential on currentdensity, with slopes of the order of 700-1000 mV/decade, indicatestransport control in the oxide film. The subsequent rise of potentialresembles that of barrier-layer formation. Indeed, the inverse field,calculated as the ratio between the change of oxide film thickness(calculated from Faraday's law) and the change of potential, wasfound to be about 1.3nm/V, which is in the usual range. Themaximum and the subsequent decay to a steady state resemble thebehavior associated with pore nucleation and growth. Hence, onecould conclude that the same inhomogeneity which leads to poreformation results in the localized attack in halide solutions.

The deactivation seems to be as fast as the activation. In thesame recent work,111 a period of time was measured between two

Figure 21. Change of potential ofaluminum in 2 M NaCl solution uponapplication of two constant-current pul-ses (10 mA/cm2) with shortening of thetime interval between them.

500

400

300-

200-

100-

0-

180-

140-

120-

100-

1

ms

250

— —

500

K

1000 ms

V10


pulses of equal current such that the second pulse started at thesame potential at which the previous pulse ended, implying thatno noticeable deactivation occurred. The record of the shorteningof the time between the pulses is seen in Fig. 21. The time until therestoration of the film to the OCP conditions begins is seen to bevery short indeed (1 ms). The kinetics of restoration can be followedfrom the increase of the potential maximum with the increase ofduration of the off period, toward the potential maximum obtainedafter 500 ms, which is similar to that of a nonactivated electroderesting at the OCP.

That the number of sites, or film structure or thickness, dependson the current density is also seen if the current is increased ordecreased stepwise.112 An additional activation peak of potentialis needed at any increase in the current density. Conversely, upondecreasing the current density, an inverted potential peak isobtained, indicating that for some moments the current passes underprevious film conditions.

(II) pH Dependence and Dissolution in Alkaline Solutions

The potential plateau (pitting potential) is said to be insensitiveto pH changes in the medium-pH region.67 This is in line with themodel suggesting the accumulation of hydrogen ions at the O/Sinterface to very high concentrations, when the very small initialconcentration (at pH above 2) becomes irrelevant.

At higher pH (above 8), however, an effect is even noted onthe OCP, shifting to increasingly negative values. According toHurlen and Haug,35 the anodizing of aluminum in acetate bufferedby ammonia exhibits a change in the potentiostatic activation withan increase in pH from 7.2 to 9.9, as shown in Fig. 15. Currenttransients become relatively slow and the current values high, whichindicates an increasing role of alkaline dissolution processes.Similar results have been obtained by others.113

Sysoeva et al.114 made a systematic potentiostatic investigationof anodization in KOH solutions in the concentration range between0.1 and 12.5 M and in the potential range between —1.5 and 0.5 Vvs. SCE. They found a maximum in the aluminum dissolution rateat a KOH concentration of 3-5 M. This is interpreted in terms ofa change in the mechanism of passivation: At low KOH concentra-


j / mAcm

Figure 22. Steady-state polarization curves of aluminumin pure and mixed NaOH + NaCl solutions: D, 4MNaOH; A, 4 M NaOH + 2 M NaCl; O, 1 M NaOH +2 M NaCl; # . 2 M NaCl (pH 1 to 13). Labels on thelines denote measured capacitances of the interface.

tions, aluminum passivation is due to hydrargillite formation,whereas high KOH concentrations cause precipitation of aluminatein the form of K2O-Al2O3-3H2O from the supersaturated solution.It should be noted that the effect of chloride is the same in alkalinesolutions as in neutral ones.115 As seen in Fig. 22, increasing passiva-tion is caused by increasing current density in pure KOH solution.However, when chloride is present, a characteristic free increasein current density is observed at a virtually constant potentialplateau corresponding to that in neutral chloride solutions.

(iii) Mechanism of Active Dissolution of Aluminum

Inasmuch as dissolution at a constant-potential plateau at anycurrent density can be ascribed to continuous activation and deacti-vation of pitting, the latter could be considered a consequence,rather than the cause, of setting conditions of active dissolution. Itonly reflects the very subtle inhomogeneity of the surface, coveringa continuous spectrum of energies of surface sites. Energetic micro-heterogeneity with respect to adsorption centers evenly distributedover the surface is a concept known and used in heterogeneouscatalysis.116 An improved Langmuir adsorption isotherm could be


derived as

6 =[A]

Km Kon

[A] +^ 0 /

(50)

where the surface coverage, 0, depends not only on adsorbateconcentration in solution, [A], but also on adsorption equilibriumconstants ranging continuously between Ko and Km. It is themechanism of active dissolution (at any active site, or, for thatmatter, at the bottom of a pit) that presents a problem.

Three basic mechanisms of pit initiation have been advocatedin the literature (see, e.g., Strehblow117) as applying to pittingprocesses at any passive metal. They are shown schematically inFig. 23.

(a) The penetration mechanism introduced by Hoar et a/.118

assumes fast migration of aggressive anions through the oxide,stimulated by a high electric field, as a rate-determining step in pitinitiation.

[PENETRATION MECHANISM]

Metpl Oxide ( Electrolyte

?.H: °« } (

- - • M e aq ( corrosion )

(film formation)

aq]aqM penetration)

CIO4" aqj

distance

[FILM BREAKING MECHANISM!

passive film (20-100A)

electrolyte withaggressive ions

competitionpassive film —chloride film

(aggressive ions)

[ADSORPTION MECHANISM 1

d 6 / d t = ( V m / z F ) ( j j - j c )

^ M e ^ p x El ^ M e ' ^ j O x ^ El

Figure 23. Models of mechanisms ofpitting initiation at the surface ofpassive metals.


(b) Film cracking as a consequence of mechanical stress pro-duced by adsorbing anions was suggested by Vetter andStrehblow119 as well as by Sato.120

(c) The adsorption mechanism, suggested first byKolotyrkin,121 involves not only adsorption of anions at particularsurface sites, but also their complexing with metal ions from theoxide, producing soluble species. Once such species leave the oxide,it starts thinning locally, with a resulting increase in the electricfield strength, which accelerates the process until the oxide is moreor less completely dissolved.

The fact to be noted here is that halide ions (or any other"aggressive anions") do not possess any buffering capacity and,hence, when the oxide starts being formed, a high hydrogen ionconcentration can be achieved at the interface, leading to a virtualcessation of further oxide growth even in neutral halide solutions.The existence of a hydrogen ion concentration gradient well outsideany pits can be observed experimentally, by the fact that flocculationof A1(OH)3 in a cell is found to occur at a significant distance fromthe electrode, where the pH is sufficiently high for the hydrolysisof aluminum ions to take place. (Quantitative differences betweenthese observations and those in porous film formation may beascribed to different film thicknesses due to different steady-statehydrogen ion concentrations, as a consequence of some bufferingcapacity of dibasic acids.)

While the above effect must play a significant role in the activedissolution under the influence of halide ions, there are reasons tobelieve that some additional effects must be involved. They are:

(a) Active dissolution does not occur until the pitting potentialis reached;

(b) The decay of potential after the short induction period (cf.Fig. 20) is too sudden to be a consequence of establishing a highhydrogen ion concentration; and

(c) The decay of potential upon sudden addition of halidesto barrier-forming buffering electrolytes is equally fast even atrelatively thick films (sustaining over 100 V) and is thus independentof film thickness77; similarly, a sudden increase in the current,associated with the appearance of active dissolution, was observedat a barrier film at the OCP in a barrier-forming electrolyte uponaddition of chloride.122


One can see no way in which the halides could affect thebuffering capacity of these electrolytes. There, indeed, seems to belittle doubt that adsorption of the halides at the oxide surface takesplace and plays a significant role.

The linear dependence of the pitting potential on ionic radiusis likely a reflection of the similarly linear relationship between thelatter and the free energy of formation of aluminum halides.108 Itis reasonable to assume that the energy of adsorption of a halideon the oxide is also related to the latter. Hence, one could postulatethat the potential at which active dissolution takes place is the potentialat which the energy of adsorption overcomes the energy of coulombicrepulsion so that the anions get adsorbed.

It is much less clear how the adsorption leads to such a dramaticchange as a potential decay of several hundred volts, occurringwithin milliseconds. This short time is difficult to associate withfilm thinning, as assumed in the "adsorption mechanism" of pitinitiation. It is not only that the mechanism of dissolution changesso much that the current efficiency falls from virtually 100% tovirtually zero, but also that the resistance of the oxide decreasesby orders of magnitude. The control of the process is, to a greatextent, taken over by the events at the O/S interface, judging fromthe capacitance values measured,115 which approach those typicalof the electrochemical double layer (cf. Fig. 22).

The depth profiling technique used on samples with a barrierfilm before and after the addition of chloride to the buffering borateelectrolyte showed no indication of either chloride penetration orsignificant reduction of the average oxide layer thickness.123 This,of course, does not rule out the possibility of the formation, by anyof the mechanisms suggested above, of pinholes with radii muchsmaller than that of the ion-gun beam, through which the entireactive dissolution could take place, or the possibility that the beammissed pits formed sporadically across the surface. If pinholeswhich are not visible were formed, the dissolution should proceedin them with extremely high true current densities.

Three possible effects of halide adsorption are envisaged:(a) The specifically adsorbed halide ions must affect the elec-

tric field in the double layer at the O/S interface as well as in theoxide as shown in Fig. 24. The center of the negative charges, restingat the inner Helmholtz plane, should accelerate both the transport

Electrochemistry of Alumi 443

Figure 24. Schematic representation of theeffect of anion adsorption on the potentialprofile at the O/S interface, showing thepotential profile before (dashed line) andafter adsorption (solid line).

M DL DL S

of ions through the oxide and the charge transfers across theinterface. While the first effect may be significant in thin oxide films,it must lose importance with increasing film thickness. It is difficultto envisage this effect having a major influence on the oxide whichalready sustains potential differences of the order of 100 V. Hence,this effect cannot explain the major change in the resistance of theoxide layer. On the other hand, if the surface coverage by halideions approaches unity, the water supply to the O/S interface couldbe virtually cut off, so that oxygen ion transfer becomes stronglyinhibited whereas aluminum ion transfer is accelerated, followingan exponential dependence on the field. Thus, the cessation ofoxide growth can be explained in this manner.

One should note that the field in the outer part of the doublelayer should repel OH" ions and attract hydrogen ions. This,however, is not expected to have a major effect on the kinetics (the"Frumkin effect") since the main reactant is likely to be water.

(b) Halides are known to form soluble complexes withaluminum ions. These include neutral ones, such as A1(OH)2C1,A1(OH)C12, and A1C13. Hence, these could pass through the O/Sinterface into the solution without any effect of the electric field inthe double layer. This adds up to the partial current of aluminumtransfer.

(c) The halide ions, once at the oxide surface, can be suckedinto it by the high positive electric field, disrupting the oxidestructure and suddenly increasing the concentration of chargecarriers.

A complete dissolution of the oxide at the rate correspondingto extreme current densities in the pits seems very unlikely since itwould have to involve too much mass transport inside the pinholes.


Deactivation can be understood in terms of the mechanismbased on adsorption of the anions. Although a lower current densitywould need a less positive potential if for example, chloride ionsstayed at the surface, as soon as the potential shifts negative, desorp-tion of chloride should take place, with a corresponding loss ofactivity.

Although a qualitative picture can thus be drawn, the modelmust still be considered as tentative until some quantitative relation-ships are developed and proven experimentally.

(it?) The "Negative Difference Effect9'

When aluminum is anodically dissolved in halide solutions,the rate of hydrogen evolution linearly increases with increasingcurrent density as shown in Fig. 25. This phenomenon is historically,and somewhat misleadingly, termed the "negative differenceeffect"124 (NDE). It is contrary to what one would normally expect,for hydrogen evolution should subside with the potential goingpositive (as indeed is observed in alkaline solutions) or at least stayconstant at a constant-potential plateau.

This phenomenon, however, is not difficult to understand inview of the mechanism of dissolution under such conditions. Sincethe number of active sites increases linearly with current densityand these sites are characterized by a film structure (or thicknessor both) different from that at the OCP, one could expect corre-sponding increases in the corrosion rate. However, as was men-tioned earlier, the active surface area in the pits increases with time,and hence one should expect the corrosion rate to increase corre-spondingly. Therefore, since the effect is not time dependent, one

30-

Al-019% In

100 200 300 400 500

j Q /mAcm-2

Figure 25. Rate of hydrogen evolution(expressed in terms of equivalent currentdensity) as a function of anodic currentdensity for aluminum and an aluminum-0.19% indium alloy immersed in 2 M NaClsolution.


must look for another explanation. This was indeed found inrelating the NDE with the formation of subvalent Al+ ions at theM/O interface. It is likely that the Al+ ions are formed at the M/Ointerface in a concentration which is in some proportion to thecurrent density [if the assumption of the second exchange of twoelectrons as the rate-determining step is accepted; cf. SectionIII(l(i))]. If so, their flux through the oxide layer to the O/Sinterface should retain this proportionality, and so should the rateof their reaction with water, forming hydrogen. As the true currentdensity inside the pit is reduced with time, maintaining a constantcurrent per pit, so should be the rate of hydrogen evolution.

In fact, the NDE can be interpreted along similar lines withoutinvoking Al+ ions: any changes that occur in the oxide layer,automatically reducing the true current density inside the pit, couldcause a corresponding decrease in the corrosion rate, whatever theirorigin. Nevertheless, it is difficult to envisage any other mechanismof hydrogen evolution at the surface of the oxide apart from thatinvolving subvalent ions.

(v) Effect of Alloying Elements and Impurities on ElectrochemicalActivity of Aluminum

A consideration of the electrochemical behavior of the largevariety of aluminum alloys used in practice surpasses by far thescope of this chapter. Nevertheless, we consider it useful to reviewhere the effect of some elements that have a profound effect on thisbehavior.

Reding and Newport125 have pointed out that small amountsof a number of elements (Mg, Ba, Zn, Cd, Hg, Sn, Ga, In) addedto aluminum cause significant shifts of the OCP in chloride solutionsin the negative direction. Despic et al126 have shown, for theexample of low-content (about 0.1%) gallium, indium, and thalliumalloys, that this increased activity is maintained up to a very highanodic current density (up to 1 A/cm2) and that the effect amountsto shifting the entire potential plateau in the negative direction.

The NDE in some alloys was found to be larger than in purealuminum, but in others (with In) to be reduced to very smallvalues, leading to a corrosive loss of the metal of only 0.5% (cf.Fig. 25).


Tuck et al127 have ascribed this effect to the accumulation ofthe alloying element inside the pit, forming a separate metal phasein intimate contact with the base metal. At the surface of such aphase, aluminum coming through it by diffusion could dissolvewithout difficulties imposed by the oxide film. Support for such amodel was found128 (a) in electron microscopic observation andelectron probe detection of a separate metal phase consisting ofthe virtually pure alloying element, and (b) in the fact that sometime is needed, after anodic dissolution starts, to attain increasedactivity. After the usual process of activation, the same as in purealuminum, some time lag is recorded,128 as shown in Fig. 26, before"superactivation" to the negative potential plateau takes place.Similar phenomena are recorded in alkaline solutions, in whichsome elements (Sn) have the additional effect of suppressing other-wise very high hydrogen corrosion rates. A process of superactiva-tion appears to depend on some so far unknown property of thealloying element.

It is interesting to note that, as far as superactivation is concer-ned, a "hierarchy effect," rather than a simple additive or synergiceffect, is found,127 i.e., the "activators" act in the order Sn, Ga, In.Hence, when Sn is present, the superactivation occurs as thoughthe other elements were not present at all.

Many other elements affect the electrochemical activity ofaluminum even at a trace (ppm) level. Thus, copper, zinc, and ironare found to counteract the effect of the activating elements. So

Figure 26. Activation and superactivation of analuminum-gallium (0.2%) alloy at different currentdensities.128


far, there is no basis on which to even speculate about the reasonsfor such effects.

In nonalloyed metal, impurities affect the OCP and the cor-rosion behavior, while they have little effect on the potential plateauof active dissolution.

As shown by Bond et al.,129 the microsegregation of impuritieshas been proven to be a more important factor than their content.Thus, on relatively fast cooling, producing cellular substructure,impurities segregate at the nodes even at a total impurity levelin the ppm range in 99.9993% pure samples. The hydrogen over-voltage, being smaller at the impurity phase than at aluminum,shifts the OCP to a sufficiently positive potential to inducepitting.

A sample of 99.993% Al containing 10-fold higher Fe and Cuconcentrations, but cooled in such a way as to produce noncellularstructure and prevent the segregation of impurities into a separatephase, has maintained the OCP in the same 0.5 M NaCl solu-tion well below the pitting potential and no pitting has beenrecorded.

Similarly, zone-refined aluminum, which has too low animpurity content (in the few hundred ppb range) for developmentof cellular structure, has not been detectably affected by exposureto the pitting solution for periods of up to 260 h.

The fact that impurities do not affect the active dissolution inchloride solutions at current densities larger than 0.01 mA/cm2

shows that the inhomogeneity resulting in a pitting mechanism ofdissolution is unrelated to impurities and is an inherent propertyof the metal.

IV. STRUCTURE AND MORPHOLOGY OF ANODICALUMINUM OXIDES

1. Methods of Determining Composition and Structure

All methods of surface analysis are based on primary particleirradiation of analyzed samples, causing primary flux disturbanceor emission of secondary particles from the surface. Table 2 presentsa classification of the most popular methods of analysis based on


Table 2Excitation-Emission Matrix of Thin-Film Analysis0

Emission

Photons

Electrons

Ions

Photons

IREXAFSNMRESR

XPSSR

LAMMAPID

Excitation

Electrons

EDX

AESELSSEM, TEMLEED

EID

Ions

IAS

SIMSISSRBSNMA

a Abbreviations: AES, Auger electron spectroscopy (Refs. 141-143);EDX, energy-dispersive analysis of X rays (Refs. 135 and 136); ELS,energy loss spectroscopy (Ref. 145); EID, electron-induced iondesorption; ESR, electron spin resonance (Ref. 138); EXAFS, exten-ded X-ray absorption fine structure (Ref. 15); IAS, ion Auger spec-troscopy; IR, infrared spectroscopy (Refs. 133 and 134); ISS, ionsurface scattering (Ref. 150); LAMMA, laser microprobe mass analy-sis; LEED, low-energy electron diffraction (Ref. 147); NMR, nuclearmagnetic resonance (Ref. 137); PID, photoinduced ion desorption;NMA, nuclear microanalysis (Ref. 152); RBS, Rutherford backscat-tering spectroscopy (Refs. 86 and 151); SEM, scanning electronmicroscopy (Ref. 146); SIMS, secondary ion mass spectroscopy(Refs. 148 and 149); SR, synchrotron radiation spectroscopy (Ref.144); TEM, transmission electron spectroscopy (Ref. 146); XPS,X-ray photoelectron spectroscopy (Refs. 139 and 140).

the types of excited and emitted particles. According to Yeager130

and other authors, these methods could be very helpful in studyingsolid-electrolyte interfaces, although they are mostly ex situ tech-niques, and hence the possibility of changes during transfer fromelectrochemical cells to the analyzing apparatus has to be borne inmind.

A comparative analysis of the existing analytical techniques ispresented in a number of works (see, e.g., Refs. 131 and 132). Ascan be seen in Table 3, all problems of anodic alumina analysis


Table 3Comparison of Characteristic Features of Analytic Methods

XPS AES SIMS RBS IR ISS EDX

SensitivitySpeed of analysisLateral resolutionDepth resolutionChemical bonding

informationInformation on

structureDamage abilityDetection of

hydrationComputer controlAvailability of

interpretation

*——*

***

—***

***

**

**

******

*

—*

—***

**

***

*****

—*

****

*

cannot be solved by a single method. If lateral resolution is impor-tant, electron beam methods, such as AES, should be used, aselectron beams can be focused down to several nanometers, toprovide spot sizes lower than characteristic grain sizes at oxidesurfaces. SIMS is preferable for the analysis of oxide hydration, asit offers a unique sensitivity to light elements, particularly hydrogen.In quantitative analysis of the oxide, where the stoichiometry andchemical state of the elements composing the oxide are of interest,the XPS method is to be preferred. To determine the depth profile(composition) of the oxide, the surface analysis method should becombined with a sample sectioning technique. Usually, inert ionsputter profiling methods are employed, and most analyzers areequipped with sputtering facilities. However, ion bombardmentcauses a large number of artifacts, such as preferential sputteringof some components in multicomponent oxides,153 bombardment-stimulated chemical reactions,154 and redistribution of elements dueto knock-on effects. Hence, the results obtained should be checkedby nondestructive analytical methods, such as RBS, and othermethods of depth analysis, such as the chemical sectioningtechnique.155


An important problem in analyzing anodic aluminas is theirporous structure. Ion sputter methods are useless in examininganodic oxides with well-developed porosity, since integral sputter-ing of the entire surface would obviously result in an average,quasi-homogeneous depth composition of the oxide, without resolv-ing the microstructural features (pore center and pore walls).156

Microbeam135 or chemical sectioning19 techniques should be usedin this case.

2. Chemical Composition of Anodic Aluminum Oxides

The chemical composition of anodic aluminas, with specialemphasis on the depth-dependent incorporation of electrolytespecies (protons, anions, etc.), has been extensively studied.

The major elements composing the oxides are oxygen andaluminum. It has been shown in a number of works that they aredistributed quasi-homogeneously.141143 The stoichiometry ofanodic aluminas, as determined by XPS139157 and RBS,86'151 corre-sponds to nearly perfect A12O3, although there are indications ofboth oxygen158 and aluminum159 deficiency. Deviations from perfectstoichiometry are most significant at the O/S and M/O interfaces.160

The transition layers observed at interfaces increase in thicknessas the oxide grows,143'160 due to the developing roughness of theinterfaces.

Nonstoichiometry of the oxides can be due to a number ofreasons, such as hydration,159 incomplete oxidation,158 and thegeneration of defects at interfaces.157 An important factor affectingthe chemical composition of the oxides is the incorporation ofelectrolyte species into the growing alumina. There have even beensuggestions to use this for impurity doping of oxides and modifyingtheir properties.161 Various kinds of anion distributions andmechanisms of anion incorporation and their influence on oxideproperties have been reported. The problems attracting attentionare:

• What are the kinetics of anion incorporation into the growingoxide and how is this process influenced by anodization conditions?

• Does anion incorporation influence the oxide formation,and what is the mechanism of anion pickup?

Table 4Data on Anion incorporation into Anodic Alumina

Anodizingbath

NH4H2PO4(pH 7)1 M Phosphate,

1 M chromate0.001-0.1 M NaH2PO4

0.4 M phosphate10-20 wt% H3PO4

4% H3PO4, phosphate(pH 5.2-9.3)

0.4 M H3PO4

0.01-5.0 M H3PO4

4% H3PO4

NH4 pentaborate0.5 M H3BO3 + 0.05 M

Na2B4O7 (pH 7.4)0.1-1.0 maleic acid0.2 M H2C2O4

0.2 M H2C2O4

0.5 M H2SO4

0.6 M H2SO4 + 0.4 MMgSO4

H2SO4

56% HNO3HC1 + NH4 tartrateNa2WO3

0.1 M molybdate

Method ofstudy

XPSSIMS

NMASTEM/EDXAESChem. section.

STEM/EDXAESAESRBSXPS

ESR + IRIRAESAESIRXPS

XPSXPSEDX/STEMEDX/EELS/RBS

Chemicalstate of

anion andimpurities

poj-PO2~, Cr+

PhosphorusPhosphorusPhosphorusPO4~

PhosphorusPO^"PO^"B2O3

B2O3

C2O4"C2O4~

c2o5-SO4~S O 4 , SOg"

NO^~, NO 2 , NO2

crTungstenMolybdenum

Meananion

content

1.6 wt%2.6 wt %1-2%

1.6 at. %

0.6 wt %7%

0.72 at. %0.61 at. %

Type ofanion

distribution0

1L

II2

332212

33332

3311

k = LJL

0.70.7,0.16

0.660.60.6

(phosphate)0.7

(H3PO4)

0.33-0.40.33

0.70.91.01.01.0

0.9

0.30.2

Ref.

79162

16310215619

13516316086

159

138165160160166157

167168136102

1 Cf. Fig. 27.


• In what manner are the oxide structure, morphology, andproperties modified by incorporated anions and other electrolytespecies?

Table 4 reviews electrolyte anion distributions in anodicoxides. In the following sections, the items of special interest aresummarized.

(i) Chemical State of Incorporated Electrolyte Constituents

IR and XPS measurements have shown that electrolyte speciesincorporated into the growing oxide are mostly anions of the acidor salt used. There are indications that the molecular form ofincorporated anions may depend on the depth of their location inthe oxide. Fukuda and Fukushima165166 have established, using IRand ESR techniques, that the outermost layer of anodic aluminaformed in oxalic acid contains oxalate anions, C2O4", whereasspecies incorporated deeper than 10 nm are carboxylates, COO".Yaniv et al169 have reported that the molecular form of speciesincorporated in oxides formed in H2SO4 changes from sulfate(SO4) at the surface to elemental sulfur in the body of the oxide.Analogous behavior was observed for aluminum anodized inH2C2O4.

170 This effect is presumably due to field-enhanced reactionsin the oxide during its growth.

(II) Distribution of Electrolyte Species in Oxides

The anion distribution in anodic oxides is usually determinedby ion bombardment or chemical sectioning of alumina sampleswith subsequent analysis by AES or XPS methods, or by the useof the depth-resolving techniques, such as RBS.150 Different typesof concentration profiles are shown in Fig. 27.

Generally speaking, the distribution of anions in alumina filmsformed in neutral electrolytes (presumably borates and phosphates)can be regarded as homogeneous within an "outer" oxide layercomprising about two-fifths to four-fifths of the total thickness ofthe film4 (Fig. 27, type 1 or 2). Relevant data have been obtainedin a number of studies by the Manchester group,86143 as well as byNagayama and Takahashi and co-workers.19'79171


Figure 27. Types of anion concentrationprofiles in anodic oxide layers on aluminum,from the O/S interface (O) inwards to theM/O interface (L). LA: front of anionpenetration.

(D

Another type of anion distribution is observed at anodic oxidesformed in acidic electrolytes. It has been examined by Fukuda165

and can be generally characterized as a single-lump pattern (Fig.27, type 3). This distribution exhibits a maximum shifted inwards,as has been shown in a number of works by various authors160166172

for a variety of acidic electrolytes. Figure 28 illustrates the distribu-tion of sulfur and carbon in thickening oxides formed in H2SO4

and H2C2O4 solutions, as obtained by AES measurements. Sulfate

Figure 28. Distribution of sulfur and carbon inanodic aluminas corresponding to the differentstages of porous structure growth, as determinedby Auger spectroscopy.160

15 20 25 30

5 10 15 20 25 30

Sputtering t i m e / m m


and oxalate species exhibit a maximum concentration that shiftsdeeper into the oxide during its growth. The carbon distribution isbelieved to consist of two components—one a sharply decreasinghydrocarbon contamination and the other due to anion incorpor-ation. The anion distributions of ultrathin (5-7 nm) oxides formedin HNO3 solutions and in HC1 have the same single-lumpshape.167'168

A special case of anodizing in chromic acid is characterizedby the absence of incorporated anions.162'173 It has been shown ina number of studies4 that chromate anions are accumulated at theouter oxide surface and do not penetrate into the oxide body.

Recent data provided by Cocke et al114'175 in an RBS studyof the distribution of heavy anions (tungstates, molybdates,manganates) yield unusual oscillatory anion profiles.

Sokol and co-workers176"178 have studied doping of anodicalumina by rare-earth complex anions. The latter are formed bydissolving rare-earth oxides in citric acid, which exhibits pronoun-ced chelating properties.179 Citrate and polyphosphate complexesof rare earths possess either anionic (at electrolyte pH < 2.5) orcationic (pH > 3) properties,180 and so, by adjusting the electrolytepH to an appropriate value, one may provoke adsorption of rare-earth complexes at the surface of anodized samples. It has beenshown by means of luminescence analysis and SIMS that a widerange of rare earths (Eu, Sc, Er, Y, Nd, La, and Ho) are pickedup by growing alumina films with the concentration of complexanions exponentially decreasing inwards (Fig. 29).176 It has alsobeen found that when added to a phosphoric acid electrolyte, minorcomplex anion impurities diminish the pore growth, presumably

106-

=j iCr-ci

0>

^ 103-

^ N

\ ^s N N

XX)

L / nm

150

Figure 29. SIMS distributions of AlOj ( ),POJ ( ), and Y+ or Nd+ ( ) ions inanodic alumina films formed in H3PO4 containingyttrium (1) and neodymium (2) complex anions.176


by the replacement of phosphate anions at the oxide surface andpreferential incorporation. The mechanism of large complex anionincorporation into growing alumina and its influence on the oxidestructure needs further elucidation.

Very interesting behavior of incorporating anions can beobserved when a multicomponent electrolyte is used for oxideformation. Here, anion antagonism or synergism can be observed,depending on the types of anions used. The antagonism of hydroxylions and acid anions has been observed in a number of cases.Konno et a/.181 have observed, in experiments on anodic aluminadeterioration and hydration, that small amounts of phosphates andchromates inhibit oxide hydration by forming monolayer or two-layer films of adsorbed anions at the oxide surface. Abd-Rabbo etal162 have observed preferential incorporation of phosphate anionsfrom a mixture of phosphates and chromates.

In conclusion, one can say that most anodic oxide films areof a duplex, or even triplex, character, with only the inner portionbeing composed of a pure anhydrous oxide. In the duplex films,the outer layer contains anions and often a degree of hydration.There could exist a third thin oxide layer at the surface, again withsomewhat different properties, which may have a role in the kineticsof oxide growth.

(III) Kinetics of Anion Incorporation into Growing Alumina Films

The rate of anion pickup during the constant-current growthof barrier alumina films is constant but smaller than the rate of filmgrowth, according to the reports of many authors, beginning withthat of Randall and Bernard163 and ending with the recent work ofSkeldon et al86 and Takahashi et al.87 An example of such linearkinetics is presented in Fig. 30. This fact leads to a constant ratiobetween the inner and outer layer thicknesses. An increase in thecurrent density causes some growth of anion content and relativelydeeper penetration.87

Data on anion incorporation into a growing porous oxide wereobtained Fukuda and Fukushima.165166 Their study was the first todemonstrate a correlation between the kinetics of accumulation ofoxalate165 or sulfate166 anions and the change of porous oxidegrowth stages. The results of galvanostatic and potentiostatic


0 20 40 60 80 100 120

Distance from oxide/solution interface, 5/nm

Figure 30. Kinetics of phosphorus incorpor-ation into growing alumina at constant currentdensity (a) and dependence of anion pickup onja value (b).79

anodization regimes are given in Fig. 31. The integral anion contentattains a maximum value somewhat later than the transition tostationary pore growth. Further anodization does not increase theamplitude of anion concentration. Similar results have beenobtained by Parkhutik and co-workers160182 for the oxides formedin sulfuric and oxalic acids (see Fig. 28), whereas anodizing in anH3PO4 bath did not yield such a correlation. Again, the resultsobtained for nitric acid anodization generally resemble those of theformer two cases.172 It seems justified to conclude that, in mostcases, the anion concentration in the growing oxide reaches amaximum value at the moment when intensive pore growth starts.

Figure 31. Correlation of the kinetics of oxidegrowth with kinetics of sulfate incorporation intothe oxide during galvanostatic (a) and potentio-static (b) anodization of Al in H2SO4 solutions.166


(iv) Role of Anions in Anodic Oxide Growth and Their Effect onOxide Properties

The true role of incorporation of anions in the formation ofanodic alumina is being intensively discussed. Baker and Pearson183

have considered the anion effect in modifying the structure of anodicoxides to be due to the coordinative ability of anions to replacealumina tetrahedra in the body of the oxides. Dorsey184185 haspostulated that in porous oxides, anions stabilize the network ofalumina tetrahedra and octahedra.

Thompson, Wood, and co-workers, in a series of papers (cf.,e.g., Ref. 186) have established a correlation between the nature ofanions incorporated into an oxide and the features of the porousoxide formed. They assumed that the electric field applied to theoxide during its growth is inhomogeneous: higher in the inner layerof pure alumina, and lower in the outer part contaminated by anionspecies, as has indeed been observed.79 The outer layer allows theeasy passage of charged particles due to its imperfections. Theimposed anodic voltage is divided between the contaminated andpure alumina regions in proportion to their thicknesses and conduc-tivities. For a given voltage, the larger the thickness of the anion-containing outer layer, the higher is the electric field strength atthe inner layer and the larger is the oxide growth rate. As the ratioof the thickness of the anion-containing layer to that of the pureoxide layer increases in the order chromic acid < phosphoric acid <oxalic acid < sulfuric acid,186 Thompson and co-workers have con-cluded that the rate of oxide growth and dissolution increases inthe same order.

When the results for oxide growth and anionincorporation172160 are compared with the kinetics of space chargeaccumulation in barrier and porous alumina films [see SectionIV(1)], it can be concluded that anion incorporation modifies theelectrostatics of the external oxide interface, thus influencing oxidedissolution and pore formation.172

3. Crystal Structure of Anodic Aluminas

Anodic aluminas are reported in the literature to have both anamorphous and a crystalline structure. The majority of anodic


aluminas exhibit an amorphous structure with a short-range crystal-line order corresponding to octahedral or tetrahedral coordinationof aluminum ions.187"189 According to the existing view,187 amor-phous aluminum oxides should be presented as close-packed arraysof A14O6 molecular units (Fig. 32). Stacked sheets of these units(Fig. 32c) give the appropriate admixture of oxtahedral and tetrahe-dral sites occupied by Al3+.187 The oxide crystallinity is observedfor thick oxides and those possessing composite structure.189 Theextent of crystallinity and the coordination number of aluminumeffect the stability and mechanical properties of oxides and havebeen found to depend strongly on electrolyte and anodizationregimes.

Thus, El-Mashri et al190 have recently studied the Al—O bondlength in thin (50-100 nm) alumina films formed in sodium tartrateand phosphoric acid electrolytes. The average bond length wasestablished to be 0.19 nm (tartrate) and 0.18 nm (phosphate). Ananalysis of these data have yielded the ratio of octahedral (A1O6)to tetrahedral (A1O4) aluminum ion coordination to be 80% : 20%and 30% : 70%, respectively. Popova191 has shown, by transmissionelectron diffraction, a 100% tetrahedral coordination for aluminafilms formed in borates. Oka et al192 have reported A1O6:A1O4

ratios of 30% : 70% and 40% : 60% for films formed in H2SO4 atac and dc regimes, respectively. Parkhutik et al.,193 by measuringthe relative intensities of FT IR peaks of vibrating Al—OA1 bonds,have determined the same ratio in oxides grown in oxalic acidsolution.

Al3+

O 02-

Figure 32. Structure of amorphous alumina showing asingle A14O6 group (a), a sheet of these (b), and a stackof sheets (c).189


As yet, no explanation has been advanced for such specificanion or pH effects.

According to El-Mashri et al.,190 the A1O6:A1O4 ratio deter-mines the hydration capacity of anodic oxides. Tetrahedral sitesare hydrated easily to form a boehmite-like structure, which isknown to be composed of double layers of Al-centered octahedra,weakly linked by water molecules to other layers.184 As the oxideformed in H3PO4 contains about 70% tetrahedral aluminum bonds,its hydration ability should be higher than that of the oxide formedin tartrate solution. However, this has not been found in practice,which is interpreted by El-Mashri et al. as being due to somereduction of A1O4 by incorporated phosphate species.

Besides the amorphous alumina films formed in the majorityof acidic electrolytes (except those formed in chromic acid andexhibiting traces of y-Al2O3

194), there are possibilities of formingoxides with a more or less pronounced crystallinity. These oxidesare formed in alkaline solutions195 and especially in sodiumcarbonate baths.196 According to the data provided by Hiroshi andYoshimura,197 these oxides contain a y- A12O3 phase easily hydratedand converted into a bayerite-like substance.

Specific structural features are observed in the formation ofcomposite oxides. Kobayashi, Shimizu, and their co-workers have,in a series of papers, reported studies of the structure of barrieralumina films, anodically formed on aluminum covered by a thin(5 nm) layer of thermal oxide.198'199 Their experiments have shownthat the thermally oxidized thin layer generally contains y- aluminacrystals of about 0.2 nm size. This layer does not have a pronouncedeffect on ionic transport in the oxide during anodization. Also,islands of y'-alumina are formed around the middle of anodicbarrier oxides. They are nucleated and developed from tiny crystalsof y'-Al2O3 and grow rapidly in the lateral direction underprolonged anodization.198199

The rate of y'-alumina island formation essentially dependson the nature of the electrolyte used. If "outwards migrating" (inthe terms of Xu et al102) anions, such as tungstates and molybdates,are used in the anodization process, y- alumina seed crystals aresurrounded by pure alumina and crystallization occurs easily. Inthe case of "inwards migrating" anions (e.g., citrates, phosphates,tartrates), the oxide material surrounding the y-nuclei is enriched


in the incorporated anions, hindering structural rearrangement atthe amorphous-crystalline boundary. Instead of the thermal treat-ment, samples can be immersed in an anodizing bath at 85°Cfor several minutes to provoke the formation of y- aluminaislands, which are inherited by alumina films under prolongedanodization.200

Recent data reported by Bernard and Florio201 generallyconfirm such a behavior and the appearance of a bilayer oxidestructure, with the outer layer amorphous and the inner one beneathit composed of y-Al2O3.

Composite crystalline-amorphous films are also obtained bycombining the anodization of aluminum with its hydration, as hasbeen shown by Kudo and Alwitt202 and recently confirmed byTakahashi and co-workers192'203'204 as well as by Kobayashi et al205

The anodizing of samples initially covered by hydroxide with apseudoboehmite structure proceeds at higher rates than withoutthe hydrous oxide layer. During anodization, barrier oxide growsunderneath the hydrous oxide layer, consuming its inner part.Incorporation of pseudoboehmite into the anodic oxide leads tothe formation of y-Al2O3 microcrystallites near the middle of thebarrier layer. Under prolonged anodization, these crystal nucleiimpinge upon one another and aggregate to form a band of y-crystalline region, growing rapidly toward the barrier/hydrousoxide interface.

The y- modification of alumina is the only one reported foranodic aluminum oxide. However, thermodynamic data206 and theresults of gravimetric analysis207'208 indicate that a-Al2O3 is alsopossible if the oxide is annealed at temperatures of about 1200°C.

4. Hydration of Growing and Aging Anodic Aluminum Oxides

Two aspects of oxide hydration are generally considered. One ishydration during the growth of the oxide. The other is the interactionwith water of aging oxides immersed therein. This is important forimproving aging stability of oxides and their corrosion resistance.209

(i) Hydration of Growing Oxides

A number of researchers have assumed that oxide growthinvolves inward migration of OH~ groups from the electrolyte and


their reaction with Al3+ ions at the M/O interface.210'211 Thismechanism is in contradiction with that based on transport numberdetermination [cf. Section III(3(v))]. Nevertheless, if there is apenetration of water all the way to the M/O interface, this shouldyield a distribution of OH~ and H+ species in the oxide, and so,in principle, the validity of this mechanism can be verified by directmeasurements of hydroxyl and proton profiles in oxides.

Hydration of growing alumina films was studied by SIMS andXPS methods in the case of barrier oxides159'212'213 and by IRspectroscopy and derivatography207'208 for porous ones. Takahashiand co-workers79159 have interpreted the results of XPS analysisof barrier alumina formed in neutral phosphate solution in termsof oxide hydration. It has been established that the O/Al mole ratiois about 1.7:1 for the outer oxide layer contaminated by acid anions.The excess of oxygen with respect to the 1.5:1 ratio for A12O3 isattributed to the hydration of this layer. It should be noted, how-ever, that XPS cannot be considered a direct method for measuringoxide hydration. The O ls lines corresponding to oxide, A12O3, andhydroxide, A1OOH, practically coincide with one another140 andcannot be resolved separately. Hence, if hydrated, the oxide shouldexhibit anO l s peak with a half-width only slightly larger than thatfor pure A12O3.

213 Besides, Takahashi et al79 have used a chemicalsectioning technique for sample profiling and this procedure itselfmay cause some oxide hydration.

SIMS measurements by Abd Rabbo and co-workers212 haveseemingly presented more direct evidence of hydration of oxidesformed in tartrate solution. Hydrogen was detected throughout anoxide film with a concentration depending on electrolyte pH. Theouter regions of the oxide were found to be more hydrated thanthe inner ones. The results were consistent with the Hoar-Motttheory for barrier oxide growth210 involving OH" movement intothe growing film and movement in the opposite direction of theprotons released at the oxide-aluminum interface. This point ofview is shared by many others.211'213 However, SIMS, combinedwith ion sputter profiling, should not either be considered a directmethod for observing oxide hydration since residual moisture inthe chamber of the analyzer can alter the results obtained.212

To overcome the shortcomings of interpreting the SIMS dataon hydrogen distribution in anodic aluminas, Lanford et al.214 have


used a nuclear reaction technique, based on the measurement ofthe gamma-ray emission output of the reaction of accelerated15N nuclei with H according to the scheme 15N+1H-*12C + 4He + 4.43 MeV. The important feature of this method is thatthe region being analyzed for the presence of hydrogen is notexposed to the atmosphere of the analyzer chamber, and back-ground moisture can be excluded from consideration. Accordingto the data obtained, the proton-enriched portion of 180-nm oxidefilms formed in phosphate, tartrate, and glycol-borate solutionsdoes not exceed 50 nm and the hydrogen content amounts to 0.02-0.17 at. %. Thus, both the penetration and concentration of waterare considerably less than found by Takahashi et al159 Besides, theprotonation of the growing oxides is more than four times lowerthan the anion contamination.159'214 All this suggests that protonsand anions are introduced into oxides by independent processes.Hence, the small extent of oxide hydration opposes the models ofbarrier oxide growth by OH~ inward migration as well as by thedissolution-precipitation mechanism210'211 and rather supports theearlier discussed mechanism, based on aluminum and oxygen ionmovements.

However, this does not preclude the possibility that in a portionof the oxide at least (the outer layer), the OH~ transport mechanismis operative, with the release of protons at the interface betweenthe two oxide layers. Hence, in such a case, some field-assistedproton transfer is likely to take place through the outer layer whilechemical dissolution should be operative at the outer O/S interface.

The hydration of oxides formed in acidic electrolytes, and thuspossessing a porous structure, occurs in the same manner as in thecase of barrier oxides. It is generally recognized that porous oxidesgrown in acidic electrolytes have no bonded water in their bulk.Only chemisorbed OH groups and H2O molecules are detected atthe oxide surfaces. This is established by carefully conducted XPSand IR measurements172'215 as well as by derivatography.208 IRspectroscopy appears not to be very sensitive to oxide protonation,and there has been a good deal of controversy over the interpretationof IR data, concerning the assignment of observed spectral lines.215

As for the porous oxides formed in alkaline solutions, thereis evidence that they are heavily hydrated. Hurlen and Haug35'216

have recently shown that the chemical composition of the nonbarrier


part of an oxide formed in an ammonium acetate buffer at pH 9.9corresponds to A12O3 + ^H2O. These results are supported by theresults of derivatography.113 Belov and Lebedeva217 have estab-lished different degrees of hydration of oxides formed in differentalkaline electrolytes.

(11) Hydration of Aging Oxides

Anodic oxides placed in aqueous media increase their weightby picking up water molecules and hydroxyl ions. The ability ofan oxide to be hydrated during aging at various temperaturesdepends on the conditions of oxide formation. Figure 33218 illus-trates the hydration capacity of porous oxides formed in variouselectrolytes, as well as the capacity of the same oxides to absorbvarious anions during aging at room temperature and at 95°C. Theoxide formed in phosphate solution appears to be the most stableone. This is in good agreement with the data of Konno et a/.181 onoxide hydration in phosphate and chromate solutions. The data ofBelov218 show that the different aging behaviors of oxides formedin different electrolytes are determined by the coordinative ability

1 POROSITY20°C 90min 2 SORPTION OF WATER

3CHROMATESU PHOSPHATES5 SULPHATES6 CHLORATES

95°C 20 mm

0.2-

r o.i-

Figure 33. Hydration capacity of porous oxides formed in various elec-trolytes. The capacity to absorb various anions is also shown.

464 A. Despic and V. P. Parkhntik

Table 5Activation Energy of Hydrationof Oxides Formed in Various

Electrolytes

Electrolyte

Sulfamic acidCrO3

H2C2O4

Na3PO4

H2SO4

Na2BO7

Na2CO3

NaOH

Activation energyof hydration

(eV)

177.9100.545.146.132.122.1

7.72.8-6

of the corresponding anions to form bonds with the oxide surface.According to Belov,218 this ability decreases in the order

H3POJ(HPOj~) > HCrO^ > HSO^SO2,")

> CrO24~ > HCO3-(CC>3~) > MnO;

> F" > OH~(H2O) > Cl~

s" > Br~ > I" > CIO4.

Very useful information concerning the tendency of oxides toundergo hydration in the presence of various anions is presentedby Alwitt and Dyer.209

Various acid anions possess different abilities to competewith OH groups in adsorbing at oxide surfaces, as illustrated byTable 5.219

The apparent large differences in the activation energy ofhydration for oxides formed in acidic and alkaline solutions reflectthe basic differences in the mechanism of oxide growth in thesetwo cases.

5. Morphology of Porous Anodic Aluminum Oxides

Numerous publications have been devoted to the investigation ofthe morphology of porous oxides of aluminum. Pores of virtually


tubular shape with semispherical bottoms have been known to formin a more or less regular way inside alumina cells in hexagonalarrangements such as that shown schematically in Fig. 3. Such aformation is a logical consequence of expanding circles, evenlydistributed over the surface in a (111) type of arrangement (startingfrom active sites), merging after their perimeters hit each other.The question arises as to why the oxide structure changes and theoxide becomes less soluble along the lines of the merger.

Early stages in porous structure development were well docu-mented in the work of Csokan.220 Especially revealing was the workof the Manchester group, conducted with the help of ultramicro-tomy and ion-beam thinning techniques. It enabled the visualizationof pore structure development in anodic aluminas.4146 In recentwork by Nagayama et al10 and Ebihara et al.221 general trends inporous oxide growth have been summarized and their dependenceon the conditions of anodization elucidated. This work as well asearlier investigations reviewed elsewhere222'223 has yielded a largebody of information concerning the geometry, the size, and othermorphological features of oxides formed in a variety of electrolytesand in various regimes. Hence, only a brief summary will be givenhere. Table 6 reviews available information on various aspects ofanodic alumina morphology.

It should be noted that anodization regimes have a major effecton oxide geometry. Palibroda233'234 has summarized the empiricaldependences of the pore diameter, d, the density of pores, n, andthe lifetime of initial barrier growth, r, on the ratio of the anodicvoltage to the limiting voltage, Uamax, as follows:

d -4.986 + 0.709 Ua = 3.64 + 18.89 Ua/Uamax (51)

n = 1.6 x 1012 exp(-4.764£/a/ Uamax) (52)

T = T0(l-l/a/t/amax), where T0 = T for UJ Uamax -> 0 (53)

Ebihara et al224 have put forward a similar dependence of cellsize on the anodic voltage:

14.5 + 2.0 Ua Ua<20V (54)-1.7 + 2.81 Ua Ua>20V (55)

The universal character of this rule is illustrated by Fig. 34.


Table 6Review of Different Aspects of Aluminum Oxide Morphology

Electrolyteused

H2C2O4

0.25-1.0 M H2C2O4-2H2O

0.16 M H2C2O4

180g/liter H2C2O4-2H2O

2-4% H2C2O4

3% H2C2O4

0.5-4.0 M H2SO4

0.6 M H2SO4

H2SO4/H2C2O4

3% Ammonium tartrate0.4 M H3PO4

44 g/liter CrO3

CrO3

0.1 M NaOH, Na2CO3

Na3PO4, NH4OH-NH4F0.3-1.0 M Na2CO3 + NaFHCOOH, CH3COOH +

Na4P2O7

Method ofanalysis

TEM

TEM

Pore fillingTEM

Pore fillingTEMTEMTEM

TEMTEMTEMTEMTEMTEMSEM

SEMSEM

Morphologicalparameters studied

N(Ua,Ce,Te\(Ua9Ce9Te),r(td)9r9

R(Ua,Ce,Te),Ua,ta,Te)9l(Ua)

r, R, N(ta9ja9 Uat TJ,GM(ya, Te)

a

N(Ua)

N, r(Ja, Te)r(td),GM(td)D9l(Ua)AT, r, R{Ua9Te9Ce), {Ua),

N, r{C)hr,R(Ua)r,R(ta,Ua)R,N,GM(ta)GM(Te)GM(Ua)H(ta9 Te)

Kta)H(ta)

Reference

224

225

22670

227221

16710518

228229

11230

231232

' GM: "general morphology" of the oxides registered by TEM-SEM.

Analysis of experimental data shows that the dependence ofthe geometrical parameters of oxides on the temperature and con-centration of electrolyte is different for galvanostatic and potentio-static conditions (Fig. 35).221 It appears that potentiostatic anodiz-ation is limited mainly by processes in the bulk of the oxide andthus is not influenced by temperature (Fig. 35b), whereas thegalvanostatic anodization regime involves oxide dissolution proces-ses at the O/S interface depending both on Tel and Cel.

Characteristic of both dependences is a decrease in the numberof pores with an increase in either the current density or thesteady-state voltage. To date, no clear explanation for thisphenomenon is available.

Electrochemistry of Alwninmn 467

Figure 34. Cell size in porous oxides versusanode potential for different electrolytes: A,oxalic acid; D, phosphoric acid; O, glycolic acid;3 , tartaric acid.224

500-

400-

300-

200-

100-

V. ELECTROPHYSICAL PROPERTIES OF ANODICALUMINUM OXIDE FILMS

1. Space Charge Effects

Every dielectric film, irrespective of the technology used for itsformation, possesses a more or less pronounced space charge. Asignificant space charge is generated in oxide films produced bythe thermal oxidation of materials,235 plasma deposition,236 and

Figure 35. Number of pores versus ja (a) and Ua

(b) at different electrolyte temperatures: D, 10°C;O, 20°C; V, 30°C; A, 40°C.221

CSI

'Eo

3-

2-

1- 1 1


plasma anodization.237 Its value is especially high in the case ofanodic oxidation of metals and semiconductors in electrolytes.238

The incorporated space charge has a negative influence on theparameters of MOS structures with anodic oxide dielectrics239 andcauses a long-term drift of oxide properties.238 It could, however,be useful in preparing electret films, i.e., dielectrics exhibiting anexternal electric field.240 The most important problems in this fieldare:

• the nature of the incorporated space charge.• the distribution of the space charge in an oxide.• the impact of the space charge on the oxide properties.

(i) The Nature of Space Charge in Anodic Aluminum Oxides

Space charge accumulation in anodic alumina is closely relatedto the electrochemical processes taking place at the metal-solutioncontact, as discussed at the beginning of this review (cf. SectionII). This is largely overlooked by physicists considering thesephenomena.

Hence, there is a good deal of controversy about how tounderstand the nature of this space charge. Lobushkin and co-workers,241'242 Zudov and co-workers,243'244 and some others245'246

assume that the space charge is generated in anodic oxides byelectrons injected therein and captured at deep traps. Dyakonovand co-workers247'248 have developed an approach assuming thatthe space charge is associated with lower-valency cationic defectsgenerated in anodic oxides with depth-dependent stoichiometry. Itis assumed by some that the space charge is caused by the decompo-sition of water molecules at the O/S interface, which is closer toelectrochemical reality.

Another approach to the investigation of the nature of thespace charge has been developed in the works of Dewald58 andFromhold and Fromhold49'61 and has been further pursued byParkhutik and co-workers.62'249 The space charge is assumed to begenerated by ionic defects, incorporated and moving in the oxideduring its growth. Such an approach is consistent with theelectrochemical nature and kinetics of oxide growth, the structuralfeatures of oxides, and the specific electric properties of anodicaluminas.62


Parkhutik et al112 have recently shown that there is a directcorrelation between the kinetics of acid anion pickup in growingoxides and the kinetics of space charge accumulation in porousand barrier alumina films. This effect was interpreted in terms ofthe space charge being associated with the electrolyte anions incor-porated into the oxide. This assumption appears to be useful inexplaining the electret properties of anodic alumina films [seeSection V(3)], the asymmetry of the dielectric strength [SectionV(4)] and dc electronic conductivity [Section V(2)], and thetransient behavior of anodic oxides [Section V(5)] and also in thetheoretical modeling of space charge distribution in oxides, as ispresented in the following section.

(11) Distribution of Space Charge

There are a number of papers dedicated to the topic of spacecharge distribution in anodic oxides. Zudova and co-workers,250'251

interpreting thermodepolarization measurements, have claimed thatthe negative space charge centroid is located at the interface separat-ing the outer hydrated layer of the anodic oxide and the inner layerof pure A12O3. At the same time, Morgan et al.252 have assumedthat the body of the oxide has a uniform positive charge, whereasthe oxide boundaries are negatively charged. Berlicki et al253 havepostulated that the space charge is distributed exponentially withthe maximum value located at the oxide boundary.

Parkhutik and Shershulskii249 have modeled the distributionof the space charge of ionic defects inside oxides (sufficiently farfrom the interfaces that the charge distribution near them can beneglected) based on the following assumptions:

(a) The oxide boundaries are permeable to space chargespecies, because of their high solubility in the electrode material.

(b) There is only one type of ionic defect creating the spacecharge.

(c) Migration of ionic defects in the oxide is determined bythe classical high-field mechanism of ionic jumps over a series ofpotential barriers.

The details of modeling the space charge distribution in oxidesare presented elsewhere.249 Figure 36 presents the resulting spacecharge distribution in as-formed (curve 1) and aging oxides (curves


Figure 36. Stored negative space charge (a), coupled electricfield (b), and internal voltage (c) in anodic oxide duringanodization (curve 1) and during isothermal aging at varioustimes (curves 2-7).61

2-7 correspond to different times of aging). The internal electricfield coupled with this space charge is also given in Fig. 36. Thisfield is essentially inhomogeneous and amounts to very high values,sufficient to cause anion redistribution after cutting off the externalelectric field.

One must keep in mind two important points:1. The continuity of the current inside the oxide requires that

the concentration of mobile charge carriers varies with the variationof the field with distance from the interface, so that their productremains constant.

2. The linearity of the change in anodization voltage withchange in oxide thickness [cf. Section III(3(i))] requires that thespace charge distribution inside the oxide remains constant duringoxide growth, i.e., the space charge distribution profile widens inproportion to the thickening of the oxide.

2. Electronic Conduction

Electronic conduction plays a limited role, if any, in anodic oxideformation, since under the anodization conditions and with a high


positive field, any electrons should be pulled into the metal.However, the growing prospects for anodic alumina films beingapplied outside the electrolyte, in electronics, presuppose knowl-edge of their electrophysical properties and, above all, of theelectronic conduction mechanism. Analyzing the literature pub-lished on this subject, one can conclude that the state of the art atpresent is not very different from that reviewed by Goruk et al5

The traditional approach to studying the electronic conduction ofthin film dielectrics, and anodic aluminum oxides in particular, isbased on fitting experimental current-voltage characteristics to oneor two classical mechanisms of electronic conduction, such aselectron tunneling, Schottky or Poole-Frenkel emission, or thespace charge current.254'255 However, the amorphous structure ofanodic oxides, their depth-dependent chemical composition, andthe influence of surface states and other features make the validityof the traditional theoretical assumptions questionable in modelingoxide electronic conductivity.

It appears that the majority of specific features of anodic oxidescan be covered by an approach based on the application of themechanism of hopping electron conductivity through localized sitesin a disordered dielectric. Mott and Twose256 were the first toconsider electron jumps through a random array of impurity centersand to take into account the amorphous structure of anodic oxides.Recent work by Bryksin et al}51 has shown the validity of percola-tion hopping transport in anodic tantalum pentoxide, with thelocalized sites attributed to lower-valency cations, such as Ta3+ andTa4+. Important work in the field of the hopping transport of carriersthrough localized sites (impurity levels, structure defects, smallpolarons, etc.) has been done by Mott and Davis,258 Jonscher andHill,259 Bonch-Bruevitch,260 Shklovskii and Efros,261 Bottger andBryksin,262 Firsov,263 Austin and Mott,264 and Emin.265

However, most of this work has avoided consideration of somecomplicating factors, arising especially from the limited thicknessand real structure of anodic dielectric films. The latter causes thefollowing effects:

(a) Ionic conduction in a dielectric can be of the same orderas or even higher than electronic conduction.266 Hence, the ioniccurrent should be modeled as a modification of hoppingconduction.49'249


(b) Very high electric fields are generated by relatively lowvoltages.

(c) Boundary conditions play an important role as they deter-mine the carrier injection and ejection, image forces, etc.

(d) Carriers injected into the dielectric, localized at deep defectlevels, generate an inhomogeneous and nonstationary space chargeexhibited as an electret effect in the dielectric.267

Both classical268 and more recent269'270 papers have paid little,if any, attention to these complications.

The modeling of hopping conductivity of real amorphousdielectrics of limited thickness, with or without the incorporatedspace charge, has recently been done by Parkhutik andShershulskii.62

The modeling of conductivity of thin disordered dielectricswas based on the following assumptions:

(i) All charged species participating in the conduction processare localized in the dielectric. Localized carriers are adiabaticallyseparated into two subsystems with significantly differing mobilities.One comprises weakly localized carriers, and these carriers partici-pate in hopping conduction. The other group of carriers is con-sidered to be localized at deep centers in the dielectric and possesssuch low mobility that one can neglect their role in the conductionprocess.

(ii) The space charge effects introduced by the subsystem ofstrongly localized carriers are taken into consideration. As themobility of these carriers is very low, the space charge distributionin the dielectric is assumed to be quasi-stationary, and only slowvariations of the space charge are possible under the influence ofvarious aging and relaxation processes.238'271 The space chargedistribution in the dielectric is determined by the nature of theprocess causing its accumulation (oxide formation, UV radiation,272

corona discharge treatments,273 etc.).(iii) The electric field in the dielectric is a result of the

superposition of an external (applied) field and the internal fieldof the space charge, and, hence, it is essentially inhomogeneous.There can be regions in the dielectric where the electric current isdirected against the external electric field.62 So, both the drift anddiffusion modes of hopping transport have to be taken intoconsideration.


(iv) Dielectric interfaces are included by assuming carrierexchange between surface states randomly distributed in energyand the localized states in the bulk of the dielectric by thehopping mechanism or any other process (e.g., Fowler-Nordheimtunneling).

The resulting scheme of localized site distribution in a thin-filmstructure with disordered dielectrics is schematically illustrated inFig. 37 with notations following those introduced by Jonscher andHill.259 The shape of the space charge region, Q (and adiabaticallycoupled with it, the hopping conduction region, W), is chosen tocorrespond to the case of an anodic oxide with an inhomogeneousspace charge [see Section V(l(ii))].

0 L x

Figure 37. Schematic energy diagram of biaseddisordered dielectric. W, energy zone for hop-ping electrons; Q, energy zone for stronglylocalized species forming space charge; N s , sur-face states.61


Using this approach, the hopping transport was modeled as aquasi-Marcovian process. The details of the analytical formulasforming the basis of the modeling and the numerical simulationprocedure are given elsewhere.62 The values of parameters includedin the hopping transport model are listed in Table 7.

The current-voltage characteristics of the anodic oxide, asderived from the model, are given in Fig. 38. The different curvescorrespond to various stages of oxide aging, with the numberingfollowing that in Fig. 36. The inserts in Fig. 38 illustrate linearizationof the j - U curves in Schottky and Poole-Frenkel coordinates. Bothlinearizations, if conducted within the limited range of voltages,are rather satisfactory. This is a good illustration of the difficultiesinvolved in correct identification of the mechanism of anodic oxideconduction. All j - U curves, linearized in logarithmic coordinates,exhibit two changes of slope at increasing external voltage. Suchbehavior, while being well established experimentally,274'275 causesa good deal of controversy in interpretation. A polar assumptionhas been made to the effect that the slope decrease is caused by

Table 7Parameters of the DC Electronic Conduction Model

Based on Hopping Transport"

Symbol

TtLW

Wt

It

we

K

K

Definition

TemperatureDielectric constantThickness of dielectricHopping activation energyHopping distanceActivation energy for carrier injec-

tion from the contact into thedielectric

Activation length for carrierinjection

Activation energy for carrier ejec-tion from the dielectric into thecontact

Activation length for carrierejection

Interface state density

Standardvalue

30010500.310.7

0.8

0.1

1.6

5 x 1016

Variationinterval

20-1200.2-0.80.8-6.00.3-1.0

0.4-1.2

0.01-0.6

0.4-2.0

—

Units

K

nmeVnmeV

nm

eV

nm

m"2

1 Ref. 62.


Figure 38. Calculated j-U curves for dielectricwith negative space charge (solid curves) and refer-ence uncharged one (dashed curve). The insetsillustrate curve linearization in log j-U and logj-Ul/2 coordinates.62

transition from the surface to the volume-limited conductionmode276'277 and by a reverse mode.274'278 Parkhutik and Shershul-skii62 have shown that the j-U slope change is caused by thetransition from bulk-limited hopping conduction to surface-limitedconduction.

The fitting of theoretical curves to experimental data onaluminum278 and tantalum279 anodic oxides is illustrated in Fig. 39.The degree of agreement between theory and experiment is reason-able. The calculated j - U curves for a dielectric with negative spacecharge are essentially asymmetric (Fig. 38). Experimental data showthat such asymmetry is most pronounced in the case of anodicoxides. This polarity has been interpreted in terms of theheterogeneous structure of anodic oxides (p — i — n structureaccording to, for example, Gubanski and Hughes271) with the realnature of such a structure remaining unspecified. Polarizationmeasurements252 show that negative space charge is incorporated


-12-

Figure 39. Comparison of theoretical;- U curveswith experimental data on electrical conductionof aluminum (curve I) and tantalum (curve II)oxides.62

into anodic oxides with a maximum positioned in the vicinity ofan external oxide boundary, which is in line with the electrochemicalconcept of potential profile (cf. Fig. 1). This charge causes asym-metric electric field generation in the manner presented in Fig. 36.The response of such an oxide to an external electric field mustdepend on the polarity of the field. In fact, the modulus of Esc ishigher to the left of the turning point (where E = 0). Therefore,the strength of the external electric field sufficient to neutralize Esc

in the region adjacent to the x = L boundary is lower than thatneutralizing Esc at x = 0, as is schematically illustrated in Fig. 40.The value of the external voltage used for the figure ensures ahigher conductivity of the oxide for positive bias of the x = 0

'/AV

L x

Figure 40. Energy diagrams illustrating thedifference in values of external electric field of bothpolarities able to neutralize the internal electric fieldof asymmetric space charge.62


electrode (the electrode contacting the external oxide boundary)and a lower conductivity for the opposite bias. This kind of polarbehavior is, indeed, exhibited in experiments.271'280 The possibilityarises of influencing the polar behavior and transient phenomenain anodic oxides by manipulating the space charge distributionthrough the judicious variation of experimental factors (electrolytecomposition, anodization regime, etc.).

3. Electret Effects

When an anodic oxide grown on a metal is taken out of the solutionin which anodization was carried out at a few hundred volts, it actsas a charged capacitor, but with the charge distributed inside theoxide (space charge). If one takes an average capacitance of theorder of 1 F/cm2 and a voltage of the order of 100 V across it, onecould expect an integral space charge, Qe, of about 10~4C/cm2 tobe buried inside the oxide. Thus, the oxide is capable of exhibitingan electret effect.267

Since the work of Gunterschultze and Betz,50 who pioneeredthe investigation of space charge effects in anodic oxides, theattention by researchers to this property has constantly been grow-ing. In the original work, freshly obtained anodic oxides wereshort-circuited by a galvanometer, and an exponentially decreasingdischarge current was registered. Later investigations by otherauthors271'272 have confirmed these observations. It has also beenestablished that thermal treatment of oxide samples causesthermally stimulated currents (TSC) directed like the isothermaldischarge current. The similarity to electrets267 has given rise to theterm "anodoelectrets." Another electret property exhibited byanodoelectrets is an external electric field, registered by measuringthe so-called electret (or surface) potential Us of the oxide by anumber of methods (e.g., screened probe, dynamic capacitor). Theessential feature of anodoelectrets is their low thickness (at leastan order of magnitude lower than that of polymer electrets), whichmakes them very promising for application in acoustic wave trans-ducers240 and other important fields.

There have been a number of investigations of the electretbehavior of anodic oxides formed on various valve metals indifferent electrolytes and anodization regimes. The purpose of these


Table 8Electret Parameters of Various Anodoelectrets

Material

Aluminum oxideTantalum oxideNiobium oxideTitanium oxide (solution)Silicon oxide

Qs (C/m2)

2 x 10"4

9.4 x 10"4

1.2 x 10~5

6 x 10~5

2 x 10~7

Tmax C O

9011012010595

studies was to obtain electrets with the highest possible externalelectric field and the best long-term stability.240'251 The resultsobtained by various authors can be briefly summarized as follows:

(i) All the anodic oxides formed on various materials exhibitmore or less pronounced electret properties. Table 8 presents theintegral charges and temperatures corresponding to the TSCmaxima of various anodic oxides.241

(ii) Porous oxides exhibit lower electret parameters than bar-rier ones.

(iii) Electret parameters depend strongly on the electrolyteconcentration, temperature, and other parameters, as is illustratedin Fig. 41. Higher electrolyte concentrations, temperatures (but notexceeding a certain limiting temperature), and current densitiesensure higher values of electret parameters.250'251

(iv) The kinetics of accumulation of electret properties ingrowing oxides with a barrier structure are superlinear in the gal-vanostatic regime and exhibit saturation in the potentiostaticregime.172'242 At the same time, during porous oxide formation, Us

Figure 41. Kinetics of electret potentialdissipation for anodic aluminas formedin (1) 1 wt %H3PO4 , (2) 2 wt % H3PO4,(3) 4wt% H3PO4, (4) 6wt% H3PO4,(5) 0.1 wt% H 3 PO 4 +1.0wt% APB,and (6) 0.1 wt % APB.240


exhibits a sharp maximum corresponding to the moment of com-mencement of pore development. This clearly indicates that theelectret parameters of anodic oxides are coupled with theirchemical, compositional, and structural features.

(v) The electret behavior of anodic oxides is affected by finish-ing treatments of freshly formed oxides: it is higher when samplesare ejected from an anodizing bath, rinsed in water, and dried whilekept at the potential of anodization.

(vi) The electret effect in naturally aging or thermally treatedsamples decreases monotonously, with the rate of decrease depend-ing on the humidity of the atmosphere, temperature, andother factors.241'277 Depolarized anodic oxides can be rechargedby reanodization, UV irradiation, and corona dischargetreatments.272'273

Various mechanisms for electret effect formation in anodicoxides have been proposed. Lobushkin and co-workers241'242

assumed that it is caused by electrons captured at deep trap levelsin oxides. This point of view was supported by Zudov andZudova.244'250 Mikho and Koleboshin272 postulated that the surfacecharge of anodic oxides is caused by dissociation of water moleculesat the oxide-electrolyte interface and absorption of OH~ groups.This mechanism was put forward to explain the restoration of theelectret effect by UV irradiation of depolarized samples. Parkhutikand Shershulskii62 assumed that the electret effect is caused by theaccumulation of incorporated anions into the growing oxide. Theybased their conclusions on measurements of the kinetics of Us

accumulation in anodic oxides and comparative analyses of thekinetics of chemical composition variation of growing oxides.

From the electrochemical point of view, the anodoelectret effectcan be described simply as a residual charge resulting from chargingthe capacitance of the oxide layer in order to create the field neededfor ionic motion in the process of oxide growth. Such a view issupported by the order of magnitude of the charges recorded inanodoelectrets (Table 8). In addition, the anodoelectrets could alsobe considered as poorly defined galvanic cells with solid electrolytes.The M/O interface represents one well-defined electrode, but theinterface between the oxide and the other contact represents anelectrode where the electrochemical process depends largely on theactual environmental conditions, the presence of oxygen from the


air enabling oxygen reduction to become the potential-determiningprocess, and adsorbed water, depending on the humidity of thesurrounding atmosphere, supplying oxygen ions and releasinghydrogen, thus acting as a kind of hydrogen electrode, etc. Thiscould lead to some additional charge.

4. Electric Breakdown of Anodic Alumina Films

Electric breakdown of growing alumina films limits their maximumattainable thickness. It also causes degradation of thin film elec-tronic devices with alumina films. Hence, it is a subject of intensiveresearch and a large number of papers have been published andfurther reviewed. A good deal of controversy exists on variousaspects of alumina oxide breakdown, and a variety of models havebeen proposed in attempts to fit experimental findings.

Generally, the phenomenon of breakdown exhibits the follow-ing features: during galvanostatic anodization of valve metals inbarrier-oxide-forming electrolytes, when the anodic voltage attainsa certain value (the so-called "first-spark voltage" according toIkonopisov284), a single spark appears on the anode surface andfurther growth of Ua is arrested (Fig. 42).

Each breakdown is accompanied by some sound effect and isfollowed by a steady degradation of properties.284 It can also leadto a complete destruction of the oxide with visible fissures andcracks.286 The particular behavior observed depends on a largenumber of factors (electrolyte concentration,287 defect concentra-tion in the oxide,288 etc.). The breakdown of thin-film systems(M-O-M and M-O-S structures) as a rule leads to irreversibledamage of oxide dielectric properties.289

Figure 42. Breakdown voltages for Al anodized in(1) ammonium borate, (2) ammonium adipate, (3)potassium hydrogen phthalate, and (4) ammoniumdihydrogen phosphate.285


There are a number of papers offering explanations of thebreakdown phenomenon. Suggestions have been made that it isdue to the presence of macro- and microdefects in oxides (elec-trolyte-filled fissures, micropores, flaws, etc.).286 Joule heatingeffects were also considered289 as well as volume increase and theresulting increase of internal stresses during anodization,290 elec-trostriction effects,291 or field-assisted ionic migration.292

Ikonopisov284 has conducted a systematic study of breakdownmechanisms in growing anodic oxides. He has enumerated factorssignificantly affecting the breakdown (nature of the anodized metal,electrolyte composition and resistivity) as well as those of lessimportance (current density, surface topography, temperature, etc.).By assuming a mechanism of avalanche multiplication of electronsinjected into the oxide by the Schottky mechanism, Ikonopisov hascorrectly predicted the dependence of Ub on electrolyte resistivityand other breakdown features.

Klein and co-workers have included, in the consideration ofelectronic avalanche breakdown, the effects of charge trapping inthe oxide293'294 and a stochastic nature of the avalanches.295 Accord-ing to these authors, trapped electron charge varies strongly withthe field strength, temperature, oxide thickness, trap density, anddepth distribution. All of this accounts for the encounteredirreproducibility of the experimental results. They have also shownthat breakdown characteristics of anodic alumina films stronglydepend on the polarity of the applied voltage (in thin-film metaloxide/metal systems) and are not influenced by the material usedfor the conducting cover.293 When the polarity of the substrate onwhich the oxide is grown is negative, the breakdown strength istwice as low as it is for positive polarity. This is in line with theobserved asymmetry of dc conductivity [see Section V(2)] andgenerally supports the hypothesis of the influence of negative spacecharge on the properties of anodic oxides.

Further support for this hypothesis was presented by Albellaand co-workers.296"298 They reported evidence that electrolytespecies incorporated into oxides act as a source of avalanchingelectrons. This assumption has yielded the well-known logarithmicdependence of breakdown voltage on electrolyte concentration:

Vb = A - B ln[Anion] (56)


Albella and co-workers have also explained the prebreakdowndeviation of the Ua(t) dependence in galvanostatic anodizationfrom linearity. This is ascribed to increasing participation of elec-trons in the anodic current, resulting in the loss of currentefficiency.296

There are also models assuming the electrostrictive input ofincorporated anions into the breakdown initiation,285'299 ionic driftmodels,300 and many others reviewed elsewhere.283'293 However, themajority of specialists agree that further work is necessary in orderto properly understand the physics of the electric breakdown ingrowing oxide films and that caused by electric stress in thin-filmstructures.

5. Transient and Aging Phenomena in Anodic Alumina Films

Both naturally aging253 and electrically301 or thermally302 stressedanodic oxides exhibit characteristic nonstationary behavior. Onecan distinguish the following transient effects:

1. Thermally stimulated currents in unbiased short-circuitedoxides.

2. Transient currents in biased samples at room temperature.3. Electroforming effects and negative resistivity of oxides.Thermal treatment of short-circuited oxides causes so-called

"thermally stimulated currents" (TSC) due to redistribution ofexcess charge incorporated into the oxide. TSC measurements areimportant for examining the oxide properties (namely, space chargeeffects). Thermal treatment also presents a method for symmetriz-ation of electrophysical properties.274

Figure 43 illustrates the possible current transients duringthermal treatment of Al-anodic Al2O3-Au structures at linearlyincreasing temperature (a) and during isothermal annealing (b).The first case is characterized by a TSC maximum at —400 Kfollowed by a change in current direction and a second maximum(Fig. 43a). In the case of isothermal treatment, 7TSC follows a t~n

dependence, where n is close to unity. These findings are usuallyinterpreted in terms of a release from deep traps of those electronsthat were initially captured there in the process of anodization.There are no clear ideas as to the physical nature of these traps.Parkhutik and Shershulskii249 have postulated that traps are associ-

300 400 500

17 K

Figure 43. Thermodepolarization currents in Al-Al2O3-Au struc-tures244: (a) isochromical annealing of oxide formed up to Ua = 180 V(1) and 100 V (2); (b) isothermal (20°C) resorption current in atmosphere(1) and vacuum (2).

ated with incorporated ionic defects. On the basis of this assump-tion, they carried out a theoretical modeling of TSC. The resultsof the modeling are illustrated in Fig. 44. They are seen to reproducethe experimental behavior (cf. Fig. 43) reasonably well.

Transient effects in naturally aging samples occur with somedelay and are very slow. Nazar and Ahmad274 have observed a slowdecrease of A1-A12O3-A1 capacitance that was attributed toneutralization of Al3+ cations in the vicinity of the internal boundaryand a corresponding increase of the effective oxide thickness.However, the same effect may be due to neutralization of negatively

Figure 44. Resorption currents theoretically modeled for activa-tion energy249: Vt = 0.9eV(l); 1.0 eV (2); 1.1 eV (3). (a) and (b)correspond to Fig. 43.


charged ionic species at the outer oxide interface according to theresults of Section V(l).

A group of scientists have studied current transients in biasedM-O-M structures.271'300 The general behavior of such a systemmay be described by classic theoretical work.268'302 However, thespecific behavior of current transients in anodic oxides made itnecessary to develop a special model for nonsteady current flowapplicable to this case. Aris and Lewis have put forward an assump-tion that current transients in anodic oxides are due to carriertrapping and release in the two systems of localized states (shallowand deep traps) associated with oxygen vacancies and/or incorpor-ated impurities.301 This approach was further supported byothers,271'279 and it generally resembles the oxide band structuretheoretically modeled by Parkhutik and Shershulskii62 (see. Fig. 37).

The negative resistance effect is observed when anodic oxidesare subjected to so-called electroforming (i.e., annealing invacuum).93 Such a treatment removes the special features of theanodic oxides (asymmetry of conduction and electric strength,electret effect, etc.), and the negative resistance effect may beexplained using the general approach developed for amorphousdielectrics.5

6. Electro- and Photoluminescencet

A hundred years ago, Sluginov302 discovered a weak light emissionduring the anodic oxidation of aluminum. This phenomenon (alsofound in other valve metals) has been extensively studied by Gunter-schulze,303 Guminski,304 Anderson,305 Ruzievich,306 van Geel,307

Vermiliyea,308 Smith,309 Ganley and Mooney,310 and many otherinvestigators. Shimizu and Tajima311 explained the effect as an"impact type" electroluminescence (EL) of the oxide films. Electronsinjected into the conduction band of the oxide at the electrolyte-oxide interface are accelerated in high electric fields (106-107 V/cm)and produce nondestructive avalanches, which are quenched atsome distance from the interface as a result of the opposing electricfield created by slowly moving positive charges. Some of the

t This section was written by Vladeta Urosevic, Institute of Physics, University ofBelgrade, Yugoslavia.


avalanche electrons have sufficient kinetic energy to excite lumines-cence centers. EL is always produced in barrier layers (even in thecase of porous alumina), i.e., in the high-field region.

Several types of EL centers have been found:1. For pure Al in inorganic electrolytes which form barrier

oxide films, such as boric acid-borax solution, surface defects("flows") have a dominant role.308 It has been shown312 that in thiscase only, EL vanishes for electropolished samples.

2. For Al doped with 3d or 4/elements (Mn, Cr, Cu, Fe, Zn,Mg, Nd, etc.), sharp and intense emission lines of the dopant aredominant.310

3. For Al in organic electrolytes (for instance, aliphatic acidsand their salts) and in inorganic electrolytes which form porousoxide films (e.g., sulfuric acid), a broad and rather intense emissionmaximum is found,313'314 whose origin is probably connected withincorporated anions.

A representative example of an EL spectrum is shown in Fig.45. The energy levels from which the emission starts are alwaysinside the forbidden band of A12O3.

Photoluminescence (PL) of anodic aluminum oxides was firstinvestigated in films formed in organic acids, the most intense PLbeing in those formed in oxalic acid. Tajima, in his comprehensivereview315 on electro- and photoluminescence in anodic oxide films,concluded that PL centers are carboxylate anions incorporated intothe oxide. On the other hand, Eidel'berg and Tseitina316 proposed

100-

400 500 600

X / n m

Figure 45. EL spectrum of anodic oxide filmformed in an ammonium tartrate solution.


Table 9Positions of Excitation and Emission Maxima of PL

in Anodic Oxide Films Formed in Various Electrolytesby DC Anodization and in Boiling Water

Electrolyte

Oxalic acid

Formic acidPhosphoric acid

Sulfuric acid

Chromic acidSodium carbonateBoiling water (bidistilled)

Excitation maximum(nm)

250310370270340

265,340340320355250

Emission maximum(nm)

420420465

340, 465460

330,460460400460430

1.2-

09-

0.6-

0.3-

0.0

0.6-,

0.3-

0.0

250 300 350

_j i i L__ Figure 46. PI excitation (a) and emission350 ^°° 45° ™ spectrum (b) of anodic oxide film formed

x /nm in a sulfuric acid solution.


that these centers are formed by adsorption of water molecules, orOH radicals, at specific points (defects) in oxide films. The resultsof some recent investigations of PL in anodic oxide films aresummarized in Table 9, and examples of excitation and emissionspectra are given in Fig. 46. Contrary to Tajima's conclusion, it hasbeen found (see Table 9) that inorganic anions and even watermolecules (or OH radicals) can also produce PL centers. Calcula-tions based on ligand field theory and infrared adsorption measure-ments seem to confirm this conclusion.314

If measurements are made in thin oxide films (of thicknessless than 5 nm), at highly polished Al, within a small acceptanceangle (a < 5°), well-defined additional maxima and minima inexcitation (PL) and emission (PL and EL) spectra appear.322 Thisstructure has been explained as a result of interference betweenmonochromatic electromagnetic waves passing directly through theoxide film and EM waves reflected from the Al surface. In a seriesof papers,318"320 this effect has been explored as a means for precisedetermination of anodic oxide film thickness (or growth rate),refractive index, porosity, mean range of electron avalanches, trans-port numbers, etc.

It is to be expected that a more profound investigation of ELand PL can give important information concerning electronic struc-ture of anodic oxide films.

VI. TRENDS IN APPLICATION OF ANODIC ALUMINAFILMS IN TECHNOLOGY

Anodic alumina oxides find steadily growing application in variousspheres of technology. Traditionally, they are most popular in civilindustrial engineering for producing protective and decorative sur-face finish in panels and different objects. These applications arewell reviewed in the literature.321 Anodic alumina is also widelyused in the aircraft and aerospace industry for adhesive bondingof aluminum structures,322"324 composite materials, etc.

Alumina membranes are also produced and used in separationprocesses.325'326

The past decade has been very fruitful for applications ofanodic alumina films in the electronics industry. These applications


have not been presented adequately in the review literature,although they might be of interest to specialists engaged inaluminum anodization either as a science or as a technology.327

Hence, a brief review here of applications of anodic aluminumoxides in electronics is considered worthwhile.

1. Electrolytic Capacitors

Aluminum foil capacitors occupy an important position in circuitapplications due to their unsurpassed volumetric efficiency ofcapacitance and low cost per unit of capacitance.328 Together withtantalum electrolytic capacitors, they are leaders in the electronicdiscrete parts market. Large capacitance is provided by the presenceof extremely thin oxide layers on anodes and cathodes, and highsurface areas of electrodes could be achieved by chemical orelectrochemical tunnel etching of aluminum foils. The capacitanceof etched eluminum can exceed that of unetched metal by as muchas a factor of 50.328

Barrier anodic oxides covering the surface of aluminum etchedfoil are usually formed in borate or phosphate solutions. To improvecapacitor characteristics, high-purity aluminum is desirable with aslow a concentration of impurities as is acceptable in terms of cost.

A significant development in the sphere of aluminum foilcapacitors involves forming the anodic oxide layer after a hot-watertreatment of bare aluminum.329'330 The hydrous oxide layer formedis converted during anodization, in its inner part, into a densebarrier oxide film of high dielectric strength and capacitance.307

Further improvements of aluminum capacitors can be expected inthe direction of increasing breakdown characteristics, long-termstability, and working temperatures.

One way to achieve such improvements is by doping ofaluminum oxide with properly selected impurities. These could beintroduced by implantation into aluminum and subsequent transferinto the oxide during anodization.331 Alternatively, complex anionscontaining impurity atoms could be introduced into the anodizingbath [see Section IV(2)]. The incorporated anions influence thedielectric permittivity, E, of the oxide.176 Hence, one can manipulatethe E value by changing the electrolyte concentration and anodiz-ation regime.91 According to the published data, rare-earth-doped


alumina capacitors exhibit very good dielectric characteristics,approaching those of the bulk rare-earth metal oxides.332

2. Substrates for Hybrid Integrated Circuits

Alumina ceramic substrates were traditionally used for hybridmicroelectronics due to their good dielectric properties.333 At pres-ent, these ceramic substrates are being substituted by metallic platescovered by a dielectric insulating layer so as to improve themechanical strength, the heat transfer ability, and other para-meters.334 This is especially important for high-power hybridintegrated circuits. The problem here is to choose an appropriatemetal-dielectric couple providing for good adhesion, absence ofvarious surface defects in the dielectric (flaws, pores, cracks, etc.),and a good quality of thin films formed at the substrate.

Table 10 presents the parameters of various metallic substrates.It can be seen that the most appropriate are those made of aluminumcovered by anodic oxides.

Usually, an aluminum-magnesium alloy is used for plate for-mation, as it provides good mechanical strength (not less than20 GPa) and uniformity of the porous anodic oxide layer. Optimalalloy composition is 3.2-3.8% Mg, 0.3-0.6% Mn, and 0.5-0.8%Si.335 Oxalic and phosphoric acid solutions are used as electrolytesbecause they render thick (40-60 micron) porous oxide layers. Goodresults are also obtained with a chromic acid electrolyte.336 Theexistence of pores in the oxide is desirable to avoid cracking athigh temperatures. The pores act as a buffer opposing substratedamage. There is a relationship between the allowed damagelesstemperature rise and the porosity of the oxide334:

i )1 1 / 2r r 7 , x-,-1 (,n^[E^-a^] (57)

) J[E^a^]

7r/cr(l + /*!) J

where E is the Young's modulus, i) is the Poisson coefficient, /x isthe surface tension coefficient, a is the linear expansion coefficient,and the indices 1 and 2 correspond to aluminum oxide andaluminum metal, respectively. Ex and /JL1 depend on oxide porosityN and pore radius r as follows:

Ex = {Ex)0{\ ~ r2N); /*, = ( ! - r2N) (58)

i

Table 10Parameters of Various Metallic Substrates"

Substrate material

Al with epoxy resinAl with anodic oxideEnameled steelCovar covered by

dielectricTi with Al anodic oxideSteel with epoxy resin

a Ref. 335.

Priceper unit

area(arb. units)

0.0060.0030.0010.01

0.120.008

Dielectricpermittivityof insulator

47

10-124.6

8-910-12

Maximumworking

temperature(K)

473-523673-723

1073-1273673-773

723-773473-523

Resistivityof dielectric

layer(ft-cm)

106-109

1012-1013

10n-1012

108-1010

1012-1013

106-109

Density(g/cm3)

2.72.86.48.5

4.87.8

Linearexpansioncoefficient

xlO-^K-1)

2416-1810-16

16

2012

Thermalconductivity[W/(m-K)]

4920040

180

291.1

r

1Is-: and V

. P


Hence, Tcr is seen to increase with pore density and pore radius.However, a problem appears at a porous substrate when thin filmsare to be deposited during metallization to form interconnections,thin-film capacitors, etc.335 Sputtered material falls deep into thepores, which affects the planarity of the deposited layer and theelectrical resistivity of the oxide layer underneath.335 To cope withthis effect, the porous oxide should be padded by inorganic (A12O3

and SiO2) or organic (polyimide, negative photoresist) layers.Aluminum plates covered by anodic oxides are also used in

manufacturing magnetic recording disks.336'337

3. Interconnection Metallization for Multilevel LSI

Large-scale integration (LSI) of semiconductor and hybridintegrated circuits requires the use of multilevel interconnectionmetallization. The possible ways of producing multilevel metalliz-ation are reviewed in a number of publications.338'339 Here again,aluminum and its anodic oxide provide a very good planarity ofmetallization patterns, fair electric parameters, low cost, and goodreproducibility.338 There are numerous patents dealing with multi-level aluminum metallization for LSI, differing from one anotherin anodizing procedure, geometry, and other features.

Aluminum metallization in combination with tantalum thinfilms is used for manufacturing thin-film capacitors built into themetallization pattern.340

4. Gate Insulators for MOSFETs

Native oxides, grown thermally or anodically at semiconductorsurfaces, are not suitable for producing gate insulators for metal-oxide-semiconductor field effect transistors (MOSFETs). Oxide-semiconductor interfaces exhibit high densities of surface statesand fixed charge, especially in the case of compound semiconduc-tors.341 Hence, aluminum deposition onto semiconductors and itsfurther anodization to complete oxidation of the metal film isconsidered as a way to improve the situation. There are papersindicating that MOSFETs with anodic aluminum oxide can beformed on silicon substrates342 and A3P5 semiconductors.343 Bestresults are obtained in producing n-channel enhancement-mode


InP transistors for high-speed memory and optoelectronic applica-tions.344 Further work is necessary to investigate the use of anodicalumina in producing high-frequency MOSFETs at other com-pound semiconductors.

5. Magnetic Recording Applications

Polished aluminum-based alloys are used as substrates for produc-ing hard magnetic disks.336 The ferromagnetic layer is usually for-med by sputter deposition onto an aluminum plate covered by athin anodic oxide.337 However, very promising results have beenobtained with electrodeposition of the ferromagnetic metal andalloys in the pores of thick aluminum oxides. The obtained magneticmedia exhibit high coercive force (3200 Oe according to Ref. 345)and good squareness ratios and wear resistance.346 Films with bothvertical and horizontal anisotropy may be formed, and readingdensity is supposed to reach about 15,000 BPI.345

6. Photolithography Masks

The use of aluminum-based masks in photolithography has beenproposed.347 According to the scheme employed, aluminum isdeposited onto a polished glass sheet. The regions of the mask thatshould be light transparent are then converted into porous oxide.As the operation of aluminum anodization exhibits a much bettervertical anisotropy than chemical etching, the masks obtained repro-duce the parameters of standard masks more precisely than thechromium masks usually used.

There are also several proposals to use anodic aluminum oxidesin producing optoelectronic devices. Porous oxides may find useas antireflecting coatings for optical pathways. Anodic aluminafilms doped by Eu and Tb are promising for application in elec-troluminescent cells for TEELs.28

7. Plasma Anodization of Aluminum

The process of plasma anodization takes place when liquid elec-trolyte is replaced by low-temperature oxygen plasma. It was firstreported by Dankov348 and Nazarova,349 but Miles and Smith areconsidered to have pioneered the investigation of the process in


their work of 1963.350 Plasma anodization belongs among the "dry"technological processes and is suitable for other vacuum processesin electronics technology. It provides a high-purity oxidation atmos-phere so that the oxides formed are not contaminated by impuritiesusually occurring in wet anodization. Hence, it is quite promisingfor applications in various spheres of technology and is beingwidely investigated. Still, only a few reviews of this area areavailable.237'351'352 Plasma anodic oxides of aluminum are suitablefor application as dielectrics for MOSFETs,353'354 magnetic memorydisks,355 capacitors,351 etc. However, plasma anodization has severalshortcomings which have to be overcome, including low efficiencyof oxide growth and the small thickness of the oxides formed356

and high fixed-charge density at the oxide-substrate interface.357

Various kinds of discharge (dc glow discharge, high-frequency andmicrowave discharges, arc discharge, corona discharge) have beenused for oxygen plasma excitation and various constructions ofplasma generators have been developed. Hence, the results obtainedby various authors differ very much.237 Besides, the plasma anodiz-ation process is very sensitive to the parameters of the plasma(particle composition of the plasma, energy and concentration ofelectrons and oxygen ions, discharge current, and so on). As aresult, the reproducibility of the results is often poor. To realizethe potential advantages of the process, it is necessary to establishthe proper values of these parameters.

ACKNOWLEDGMENTS

The authors are indebted to Dr. Vladimir Jovic and Mrs. LjiljanaGajic-Krstajic for their help in the preparation of this manuscript.One of the authors (P.V.) wishes to acknowledge the encouragingsupport of Professor Vladimir Labunov and Dr. EugenijaMatweeva.

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Index

Abruna, and electrochemical interphases,265

Absorption spectrawith chromate of passive layers, 293EXAFS, 276

AC polarograms, and faradaic rectifica-tion, 223

AC polarography, and Bahargaba, 245Activation and superactivation, of alumi-

num gallium alloys, 446Activation energy, of hydration layers,

464Active dissolution, of aluminum, 439Adsorbates

effect on work function, 19studied by X-ray standing waves, 314

Adsorption mechanisms, and Kolotyrkin,441

studied by EXAFS, 303Agarwal, discoverer of faradaic rectifica-

tion, 177Aging oxides

and electret formation, 479and hydration, 463

Aging phenomena, in anodic transportfilms, 482

Alloying elements, and electrochemicalactivity of aluminum, 445

Almali and Levinskas, and current densi-ty of reactions under steady-stateconditions, 202

Aluminaanhydrous, 405and doping, 485films, and anion incorporation, 455

Aluminumanodization, 408dissolution, 433, 434, 438, 439dissolution current, and ion incorpora-

tion current, 414and electrochemical activity, affected

by alloying, 445and film breakdown, diagrammed, 440films

and current density relations, 432and electrical conduction, 476and hydration, 461and ionic conduction, 471and kinetics of phosphorus inclu-

sion, 456and oxygen-solution interface, 412and their space charge, 468

oxidesanodic, chemical composition, 450films, 401, 467growing and aging caused by hydra-

tion, 460morphology, 466surface, and the electric field, 405

reaction with water, 406surfaces

and Butler-Volmer equation, 411and impurities thereon, 404

Alvella, and evidence for electrolytespace charges, 481

Amatore and Saveant, and mechanism ofcarbon dioxide reduction in DMF,339

Amorphous structures, and anodic ox-ides, 420

505

506 Index

Amplitude fitting, for EXAFS, 285Amplitude term, in EXAFS, 278Anderson, work with Eyring on carbon

dioxide reduction, 336, 338Angular dependence of copper reflec-

tivity, studied by X rays, 317Anions

aggressive, and the dissolution of alu-minum, 441

concentration, in aluminum oxides,453

distribution, in oxides, work ofFukuda, 453

effects on faradaic rectification, 238incorporation into anodic alumina, 451incorporation into growing alumina

films, 455in oxide films, and their effect on

growth, 457and specific adsorption, 408

Anodic aluminacrystal structure, 457and trends in technology, 487

Anodic dissolution, 442Anodic films, 402Anodic oxides, 402, 403

and EXAFS, 402films, and their breakdown, 480

Anodization, of aluminumkinetics, 431work of Sysoeva, 438

Aprotic solvents, and carbon dioxide re-duction, 333, 343

Artificial photosynthesis, and carbon di-oxide reduction, 383

Aylmer-Kelly et al., and oxalate forma-tion and carbon dioxide reduction,340

Badialiand calculations of potential of zero

charge, 79and capacitance prediction, 71and electron tail, 88and solvent interaction, 74

Bahargaba, and AC polarography, 245

Band structure, in electrodes, 27Bands, at interfaces, 26Barker, and faradaic rectification, 177,

239Barrier, nonporous, diagrammed, 409Barrier film formation, kinetics, 423Basic relationships, and faradaic rec-

tification, 257Beck and Celli, calculation of linear re-

sponse to charge change, 48Bedzyk and Materlik, and angular depen-

dent of fluorescence yield of bro-mine, 14

Beley, model for carbon dioxide reduc-tion, 373

Bernard and Florio, and bilayer oxidestructure, 460

Biological catalyst, and carbon dioxidereduction, 381

Bockris and Habibcalculation of overlap potential, 69calculation of solvent contribution, 66

Bockris and Khan, estimate of surfacepotential, 3

Bockris et al., FTIR spectra involvingcarbon dioxide at illuminated p-CdTe electrodes, 362

Boundary conditions, for electrode sur-faces, 23

Bradley et al., and carbon dioxide reduc-tion at/^-silicon, 361

Breakdownof anodic oxide films, 480mechanisms, and Ikonopisov, 481and sound effects, 480

Brey, expressions for space charge inoxide films, 419

Brillouin zone, in metals, 29Budishka, and faradaic rectification,

240Buffering capacity, of an electrolyte near

an oxide, 418Buildup of protons, in oxides, 416Bunn, and microsegregation of im-

purities, 447Butler-Volmer equation, applied to alu-

minum surface, 411

Index 507

Cadmium, and dropping mercury elec-trode, 232

Capacitancecalculations with solvent contributions,

76and contribution to metal, Kuklin's

work, 46of double layer, McDonald and Bar-

low's treatment, 4of electrochemical interface, 46at interface, 14, 86of mercury-aqueous interface, calcula-

tions of, 81for sp metals, 17total, 14

Capacities, electrolytic, made of alumina,488

Capacity, and dependence on metal, 56Carbon dioxide

cathodic reduction at various elec-trodes, 332

electrochemical reduction, 328fixation, and early history of the earth,

386miscellaneous catalysts, 380photoelectrochemical reduction, 352,

358, 386reduction

aqueous solutions, mechanisms, 337catalysts, 367in DMF by means of light, 355Eyring's mechanism, 336and first electron transfer, 342to formaldehyde, 386to formic acid, efficiency, 329future prospects, 390, 393kinetics and details for various elec-

trodes, 379mechanism, 336, 341at metal electrodes, 328, 329and photoemission experiments, 338on semiconductors, 390thermal, 389

with ruthenium complexes, 375Carbon monoxide, formation of, 388

carbon dioxide reduction to, and Beleyet al, 373

Catalysts, for carbon dioxide reduction,367, 380

Cells, for EXAFS studies, 306Charge transfer

reactionsand kinetic parameters, 212three electrons, 184two electrons, 182

and redox reactions, 211Chemical dissolution, of oxides, 415Chemically modified electrodes, and

EXAFS on metals, 308Chromate-generated films, and Fourier

transfer analysis of EXAFS, 293Chromous and three-electron transfer re-

actions from faradaic rectification,236

Chromous-chromic equilibrium, andfaradaic rectification, 207

Circuits, for faradaic rectification tech-nique, 220

Clusters of iron-sulfur, and carbon diox-ide reduction, 375

Cobaltand faradaic rectification polarograms,

236porphyrins, use in carbon dioxide re-

duction, 369Cocke, work on heavy anion distribution,

454Compact double layer, treatment of by

Guidelli, 5Conduction electrons, density profile, 56Controlled potential electrolysis of carbon

dioxide in DMF, with light as-sistance, 360

Copper upd, appearance studied byEXAFS, 301

Corrosion, and effect on impurities, 406Crown ethers, and reduction of carbon

dioxide, 352Crystal face, effect on surface dipole

potentials, 16Crystal structure, of anodic aluminas, 457Current density

as function of voltage for oxide films,425

508 Index

Current density (cont.)relationships, and formation of alumi-

num films, 432Current relation, and dependence on time

for oxide films, 424Current-voltage curves, for electrical

conduction, 476Cyclic voltammograms

and acetonitrile, 377and reduction of carbon dioxide at

titaniumoxide electrode, 344

Data reduction, and EXAFS, 282Deactivation, in aluminum dissolution,

437Debye-Waller factor, and EXAFS, 299Delahay

contribution to faradaic rectification,177

equations, applied to faradaic rectifica-tion, 199

Density fluctuations, in solution, 10Density function formalism, for a linear

response, 48Depolarization, by inorganic ions, 219Depth profiling, and aluminum oxide

films, 442Despic, and work on anodic oxides, 401Detection, and EXAFS, 288Devanathan, and faradaic rectification,

179, 181Diagrams, for pitting, 435Dielectric constant, for a metal, 59Dielectric film model, for metal-solution

interface, 64Dielectric formalism, 83Diffusion coefficient, and faradaic rec-

tification, 192Dignam, and single ion species going

through oxides, 420Dispersion, and EXAFS studies, 291Dissolution

of aluminum, in presence of halideion, 434

of oxides, 443Distribution of space charge, 469

DMF, as solvent for carbon dioxide re-duction, 339

Dopingand alumina, 485of aluminum, work of Sokol, 454

Doss, discoverer of faradaic rectification,177

Double layer correction, and Faradaicrectification, 229

Dropping mercury electrode, and cad-mium, 232

Ebihara, and cell size as function ofanodic voltage, 465

Efficiency, of oxide film formation, 427Eigenfunctions for surfaces, 22Elastic losses, and EXAFS, 280Electret

effects, 477formation, and aging of oxides, 479parameters, in the space charge region,

478Electric fields

across aluminum oxide surface, 405in the dielectric, for aluminum fields,

472at interfaces, components of, 11

Electrocatalytic reduction of carbon diox-ide, with macrocycles, 372

Electrochemical cell, for in situ X-raystanding waves, 316

Electrochemical interface, studied by Xrays, 315, 316, 318

Electrochemical systems, and EXAFS,291

Electrodes, and measurement of X-raystanding waves, 311

Electrolysis, with various macrocyscleson mercury, 370

Electrolyte species in oxides, 452Electrolytic capacities, made of alumina,

488Electron density, and presence of an ex-

ternal potential, 42Electronic conduction, in alumina film,

470Electron-ion interactions, 30

Index 509

Electron neutralityand charge density at interfaces, 85condition, applied to metal surfaces,

24Electron transfer

and faradaic rectification, 179with pyridine and carbon dioxide re-

duction, 381Electron yield, in EXAFS, 289Electrons

in metal, 20in the interface, 54

Electrophotoluminesence, in aluminafilms, 484

Electrophysical properties, for aluminumfrom space charge, 467

Electroreduction, of carbon dioxide inDMF, effects of various complex-es, 376

Energy band, containing electrons, 29Energy density, in bulk of the electrode,

calculated, 52Er-Mashri et al., and aluminum-oxygen

bond strength, 458Evolution, chemical, and primitive model

of earth, 386EXAFS

adsorption at electrodes, comparisonwith work of Hubbard, 303

and amplitude fitting, 285amplitude term, 278and anodic oxides, 402for copper sulfate, 276data reduction, 281, 282and detection, 288in electrochemical systems, 291and electron yield, 289Hubbard's work, 302and many-body effects, 279for monolayers, 298of nickel oxide, 297origin of radiation, 275and oscillations, 274oscillatory term, 280and oxide films, 292, 299radiation, reflection 290in reflection, 290

EXAFS (cont.)reviews, 272in spectroelectrochemistry, 306in spectroscopy, 282spectrum, for copper on platinum, 302and static order, 279studies of passivated ion, 294for surfaces, 286theory of, 277and thermal vibrations, 279Tourillon's work, 308in transmission, 297

Excitation, for photoemission from al-umina, 486

External field, calculated near an elec-trode, 47

Eyring, mechanism of carbon dioxide re-duction, 336

Faradaic rectificationapplications, 246basic relationships, 254and chromous-chromic relationship,

207circuits for, 220and concentrations, 252conclusions, 247dependence on frequency, 188and Devanathan, 179and double layer correction, 229and electron transfer, 179equations, 258fabrication of electrodes, 190instruments, 190and kinetic parameters, 206, 208and McBain-Dowson cell, 221and nickel-nickelous reaction, 198notations, 259in organic ions, 240parameters of, 195, 228physical picture, 182polarograms, 224, 226, 236and polarography, 219, 248and quinhydrone couple, 216for redox couples, 204references, 260results, 190

510 Index

Faradaic rectification (cont.)and stannous-stannic reactions, 213technique, 220theory, 178, 180, 187, 227and theory of two-electron charge

transfer processes, 186and two-electron transfer reactions,

193with zinc in KC1, 234

Faradaic reduction of isotin, 244Faradaic relaxation methods, 178Fermi energy, 22Fermi gas, considered in electron case,

34Ferrocyanide—ferricyanide, equilibrium

treated by faradaic rectification,205

Field, due to water orientation, 15Film breakdown, for aluminum, diagram-

med, 440Film cracking

and Sato, 441and Vetter and Strehblew, 441

Fluctuations, of density in solution, 10Fluorescence

and EXAFS, 288and reflectivity at electrode, 317

Formalism, for dielectrics, 83Forte and co-workers, and passive films

on iron, 296Fourier transform

application to dielectric constant theo-ry, 33

applied to electrode calculation, 36in chromate-generated film, 293of EXAFS for iron films, 294

Fourier transforming and filtering, ofEXAFS, 283

Free electrons, in copper single crystals,38

Free surface potential, calculation ofBockris and Khan, 3

Friedel oscillations, in metal, 37Froment, and reflected EXAFS, 296Fromholt and Fromholt, modeling of

ionic migration, 421

Frumkin effect, in aluminum deposition,443

FTIR measurements, in carbon dioxidereduction, 341

Fukuda, and anion distribution in oxides,453

Fukuda and Fukushima, on incorporationof anions into alumina, 455

Fukushima, and "burning out" of alumi-num oxide films, 409

Galliumcapacitances, explained, 18

effect of changing double layer, 66compared with mercury for capaci-

tance, 65interface with solutions, 6

Gallium arsenide and photoreduction ofcarbon dioxide, 347

Galus and Adams, and the rate constantfor redox couples, 208

Gamley and Mooney, on photolumines-cence for aluminum, 484

Germanium lattice, and X-ray intensities,313

Golovchenko, and surface adsorbates,314

Gorukand anodic oxides, 402expressions for space charge in oxide

films, 419Gouy-Chapman theory, is it adequate?,

402Graham, and formalism for electrode-

solution interphase, 181Green's function, 35Growth kinetics, of porous oxides, 430Guidelli

capacitance of metal-solution inter-face, 71

treatment of compact double layer, 5Gunterschultze and Betz, and space

charge in oxide layers, 547Guruswamy and Bockris, formation of

oxalic acid in carbon dioxide, 355

Index 511

Hallmanand Aurian-Blajeni, and carbon diox-

ide reduction in powders, 364reduction of carbon dioxide on p-ga\-

lium by means of light, 349review on carbon dioxide fixation, 386

Hartree-Fock and EXAFS, 284Heald and co-workers, and grazing inci-

dents, EXAFS, 297Heavy anion distribution, in aluminum,

work of Cocke, 454Hellmann-Feynmann theorem, applica-

tion of to metal-solution inter-face, 51

Hemminger et al., and methane produc-tion from carbon dioxide, 348

Heusler and Valend, and mechanism ofoxygen ion dissolution current,415

Hiriroshi and Yoshimura, and beta al-umina phase, 459

Hoar-Yashalon, and mechanisms, 413Hoffman, contribution to EXAFS studies

of passive ion, 295Hopping, and aluminum fields, 473Hopping conductivity, and aluminum

films, and modeling, 472Horri, and formation of methane from

carbon dioxide reduction, 330Hubbard

differences to his work obtained byEXAFS, 305

and iodide absorption from Auger, 303work on vacuum systems, 303

Humps, for capacitance, 69Huri and Suzuki

electrolytic reduction of bicarbonate,336

mechanisms for carbon dioxide reduc-tion, 337

Hydration, in growing and aging of alu-minum oxides, 460-463

Hydration capacity, of porous layers,463

Hydrogen evolution, on aluminum films,444

Ikeda, and carbon dioxide reduction innonaqueous solutions, 361

Impuritiesand effect on corrosion, 406and effect on open circuit potential,

406microsegregation, and work of Bunn,

447Indium

and effective mechanism for carbondioxide reduction, 347

and faradaic rectification, 239Indium phosphide, use as photoelectrode

in carbon dioxide reduction, 357Indolin, and faradaic rectification reduc-

tion, 242Induced charge, center of gravity, 50Inorganic ions, as depolarized in faradaic

rectification, 219Integrated circuits, hybrids, 489Interactions, metal-electrolyte, 6Interface

electrochemical, and electron structuretheory, 89

involving gallium, 6metal-electrolyte, 1and presence of electrons, 54

Ion-electron interactions, 30Ionic conduction, of aluminum films,

471Ionic migration

and modeling by Fromholts, 421of oxide film, 427

Iron films, and EXAFS, 294Iron-sulfur clusters, 374Isatin, and its reduction, 241Ito, and photoelectrochemical reduction

of carbon dioxide, 352Ito et al., detailed pathways for carbon

dioxide reduction, 337

Jelliummodel for solvent interactions with

metal, 75properties at the interface, 53and Schmickler contributions, 7

512 Index

Kaesche, and dissolution of aluminum inpresence of halide ions, 434

Kapusta and Hackermanand formic acid reduction, 342and reaction pathways through carbon

dioxide, 337Kenfield and Frese, reduction of carbon

dioxide at semiconductor elec-trodes, 347

Kinetic parametersfor chromous, zinc, and cobalt ions in

various electrolytes, 235for cupric-copper ions, 193in faradaic rectification, 206for lead and cadmium, 231of quinone-hydroquinone, and faradaic

rectification, 218for thallous and copper-thallous

faradaic rectification, 228Kinetics

of aluminum anodization, 408, 431of barrier film formation, 423of electret formation, 478of phosphorus inclusion into aluminum

films, 456King-Cathard equation, and faradaic rec-

tification, 221Klein and co-workers, and electronic av-

alanche mechanism, 481Kolb, work on copper single crystals, 38Kolotyrkin, and film adsorption mecha-

nisms, 441Konno, and anodic alumina deterioration,

455Kornyshev

and capacitance of metal-solution in-terface, 85

and models for the interface, 76review of double layer, 4and Vorotyntsev

and image interactions, 55models at interfaces, 84

Kruger, contributions to EXAFS spectraof passive layers, 292

Kuklin, work on contribution of metal tocapacitance at electrochemical in-terface, 46

Lamy et al., standard potential and ki-netic parameters for electrochemi-cal reduction of carbon dioxide,339

Lang and Kohncalculation of situation near the inter-

face, 41self-consistent density function, 49and self-consistent theory, 83

Lang-Kohn theory, 73Langmuir, equilibrium isotherm, 8LEED, and surface of metals, 21Leed patterns, found by Hubbard, 131Lehn and Ziessel, and systems for pho-

toelectrochemical reduction, 384Linear response, and Beck and Celli cal-

culation, 48Lobushkin, and deep trap levels in ox-

ides, 479

McBreen, and nickel oxide in EXAFS,297

McDonald and Barlow, capacitance ofdouble layer, 4

Macrocyclesand electrolysis on mercury in pres-

ence of carbon dioxide, 371and electrolytic reduction of carbon

dioxide, 368, 372Magnetic recording, with alumina, 492Marra and Asenberger and surface dif-

fraction, 320Masks, photolithographic, made of al-

umina, 492Maxwell's equations, applied to syn-

chrotron radiation, 270Measurements, of X-ray standing waves

on electrodes, 311Mechanism

of active dissolution of aluminum, 439for anodic dissolution of oxides (Hoar

and Yashalon), 413of carbon dioxide reduction, 337

Melroy and co-workers, and EXAFSspectra, 302

Metalscomplexes of N-macrocycles, 368

Index 513

Metals (cont.)and dielectric constant, 59electrodeposited, studied by EXAFS,

308electrons, screening of, 33and electrons therein, 20interface of, 1surface capacitance, 59surface potential, due to dielectric film

on barrier, 62surfaces

and density of state, 39and electronic structure theories, 89

Metal-electrolyte interface, 1Metal-oxygen interface, 410Metal-solvent distance, 68Metal-solvent interaction

contribution of Price and Haley, 73and double layer, 74

Metallic substrates, made of alumina,490

Methane, formation of from carbon diox-ide reduction, 331

Microstructures, 314Model calculations, for the interface, 87Modeling, of hopping conductivity, 472Models

for compact layer, 5at interfaces, as given by Kornyshev

and Vorotyntsev, 84for metal-electrolyte interface, 72

Monolayers, and EXAFS, 298Morphology

of aluminum oxide, 466of anodic aluminum oxides, 447of porous anodic aluminum oxides,

464

O'Grady, contributions to EXAFS spec-tra at passive layers, 292

OH~ transport mechanisms in films andtheir stability, 462

Oldham, early contributions to faradaicrectification, 177

Open circuit phenomena, with aluminumfilms, 421

Organic compounds and depolarization,in faradaic rectification, 240

Orientationof solvent dipoles, 7of water molecules on J-type metals,

15Oscillations, in metal, 32Oscillatory term, in EXAFS, 280Overlap between s and p bands, and

alkaline metals, 29Oxidation states, monitored by EXAFS,

307Oxide films

and current relation as a function oftime, 424

and EXAFS, 292formation, efficiency of, 427and potentiostatic regimes, 426

Oxide layer thickness, 426Oxides

chemical dissolution of, 415and deep trap levels, (Lobushkin), 479solubility in water, 407species transferred through, 419

Oxygen solution, at interface, and pres-ence of buffering electrolyte, foroxide films, 417

Oxygen solution interface, connectedwith aluminum films, 412

National synchrotron light source,(Brookhaven), 287

"Negative difference effect," 444Newns, work on response to electric field

for metal plain, 46Nonaqueous solutions, in carbon dioxide

reduction, 339Nucleation-pore phenomena, 433

Palibroda, and pore diameter in alumi-num oxide films, 465

Parkhutikand anodic oxides, 401and correlation of kinetics of acid

anion pickup and space, 469and space charge of aluminum films,

468

514 Index

Parkhutik and Shershulskai, and distribu-tion of space charge, 469

Passive films on iron (Forte and co-workers), 296

Passive ion, studied by EXAFS, 295Pathways, for carbon dioxide reduction,

343Perturbation theory, applied to electron

transfer, 28pH dependence, of aluminum dissolution,

438Phase, fitting of in EXAFS analysis, 284Photo-assisted reduction

of carbon dioxide in suspensions, 363of powders including rare earth do-

pants, 365Photoelectrocatalysis, in carbon dioxide

reduction on indium phosphide,357

Photoelectrochemical mechanism for re-duction of carbon dioxide, 358

Photoelectrochemical reductionof carbon dioxide and artificial photo-

synthesis, 383of carbon dioxide on several semicon-

ductor electrodes, 349as practiced by Lehn and Ziessel, 384

Photoemission experiments and carbondioxide reduction, 338

Photolithographic masks, made of al-umina, 492

Photolytic products of carbon dioxide re-duction, 355

Photoluminescence, in alumina films,484, 485

Photon emission, from alumina and ex-citation, 486

Photon flux in synchrotron source, 272Photoreduction, of carbon dioxide at p-

silicon, 361Photosynthesis, and carbon dioxide re-

duction, 383, 384Pitting, diagrams, for 435Plain wave, for metal surfaces, 31Plasma, in metals, 37Poisson equation

application to double layers, 11

Poisson equation (cont.)application to metal surfaces, 24

Polarization interactions, at metal-elec-trolyte surface, 72

Polarization studies, and surface EXAFS,286

Polarograms, obtained by faradaic rec-tification, 222, 224

Polarography and faradaic rectification(Barker), 177, 219

Polyanalin-coated p-silicon electrodes,use as photoelectrochemical reac-tions, 350

Popova, and transmission electron dif-fraction, for amorphous alumina,458

Pore-nucleation phenomena, 433Porous oxides

film, diagrammed, 409formation of, 429growth kinetics, 430

Potential-current relation, for oxidefilms, 424

Potential difference at interface, calcula-tion of, 12

Potentialof aluminum, on application of various

current modes, 436as function of component potentials, 2and orientation of solvent dipoles, 7and surface energy, work of Trasatti,

17in terms of other potentials, 14

Potentials, components, for potential ofzero charge, 2

Potentiostatic regimes, for oxide films,426

Pourbaix diagrams, 407Powders, in carbon dioxide reduction,

364Price and Haley

and double layer capacitance calcula-tions, 70

and metal-solvent interactions, 73and pseudopotential calculation, 82

Products, and photoelectrochemical re-duction of carbon dioxide, 354

Index 515

Products obtained in the photoillumina-tion of zinc sulfide, in the pres-ence of carbon dioxide, 367

Profileof charge dissolution at anodic oxide

film, 422of electron density at surface of metal,

60of electron density in metal, 40of oscillating ions, 44

Proton buildup effects, 416Pseudopotential, 31

calculations of Price and Haley, 82electron density profile, 60

Quantum efficiency, as function of wave-length for reduction of carbon di-oxide, 358

Quantum mechanical calculations, formetal-solution interfaces, 89

Quinhydrone couple, and faradaic rec-tification, 216

Rabbo, and measurements on aluminumfilms, 461

Radiation, emitted from orbiting elec-trons, 270

Radiotracer technique, involving 18O,428

Rainmuth, summary of faradaic rectifica-tion, 249

Raman spectroscopy, 265Randall and Bernard, and structure of

barrier layer, 428Rangarajan, contribution to theory of

faradaic rectification, 177, 181Rate constants and faradaic rectification,

201Rate-determining step, in carbon dioxide

reduction, 341Reaction pathways, of carbon dioxide re-

duction, work of Hackerman, 337Recording, magnetic, with alumina, 492Rectification, 183Reding and Newport, work on alloying

on aluminum, 445

Redox couplesand faradaic rectification, 204and rate constant for titanium ions in

redox processes, 208and single charge transfer, 204

Redox processes, and titanium ions, 208Redox reactions

and stannous-stannic, 211and two-electron transfer reactions,

211Reduction

of carbon dioxide, 331, 334, 335of 5-nitrobenzimadosol, its faradaic

rectification, 245Reduction pathway for carbon dioxide

reduction in presence of bi-pyridile, 378

Reflection, of EXAFS radiation, 290Rice's theory, 57, 58Ruthenium complexes, and carbon diox-

ide reduction, 375

Sato, and film cracking, 441Schmickler

and jellium model for the interface,77

and model investigations of zerocharge, 78

Schmickler and Hendersoncalculation of oscillatory density pro-

file, 41a less unrealistic picture of the inter-

face, 80Schrodinger, equation for metal surfaces,

21Screening, of electrons in metals, 33Second harmonic polarography, 177Self-consistent theories, for metal elec-

trodes, 43Semiconductors

and carbon dioxide reduction, 344,348, 359

with suspension systems, problems of,366

Shima, work on faradaic rectification,240

516 Index

SIMSand examination of aluminum oxide

films, 428measurements, on aluminum films by

Rabbo, 461Single electron transfer reactions and po-

larography, 225Sokol, and doping of anodic aluminum,

454Solvent dipole, orientation of, 7Solvent-metal interactions

and double layer, 74and jellium, 75

Solvents, effect on double layers calcu-lated, 81

Sound effects, and breakdown, 480Space charge

in aluminum oxide films (Parkhutik),468

distribution, 469in oxide films, expressions for, 419

Species, transferred through oxide, 419Specific adsorption, of ions, 8Spectra, near edge, for copper upd on

gold, 301Spectroelectrochemistry, 305

and EXAFS, 306Spillover, of electrons outside metals, 45Srinivasan, work on reduction of carbon

dioxide, 347Stability of aluminum films, and amount

of water therein, 462Standing waves

and EXAFS, 312of X rays, 310

Stannous-stannic reactionand faradaic rectification, 213and redox reactions, 211

Structureof barrier layer (Takahashi), 428of metal, importance in capacitance

calculations, 89Suman, work on faradaic rectification,

240Supporting electrolyte, used in faradaic

rectification, 215Surface dipole potential, 16

Surface EXAFS, and polarization studies,286

Surface potential, 2, 25Surfaces, and EXAFS, 286Suspensions of powders, and pho-

toelectrochemical reduction ofcarbon dioxide, 363

Synchroton radiation, 289and electrons, 289origins, 269

Synchrotron source and photon flux, 272,287

Sysoeva, work on potentiostatic anodiza-tion of aluminum, 438

Tafel slopes, and discrepancy with Eyr-ing's mechanism, 338

Takahashi, and concentration of waterand aluminum films, 462

Takahashi et al., and structure of barrierlayer, 428

Taniguchiproduction of carbon dioxide using

crown ethers, 352and reduction of carbon dioxide, 328

Taniguchi et al.remarkable effects of catalysis with

TBAD in carbon dioxide reduc-tion, 356

work on 15-crown-5 ethers for electrontransfer, 381

Tazuke and Kitamura, and artificial pho-tosynthetic systems, 383

Test charge density, in plasma, 37Thallous-thallic reaction and faradaic

rectification, 212Theory

of EXAFS, 277of faradaic rectification, 180

Thermal reduction, of carbon dioxide,389

Thermal vibrations, relevance to EXAFS,279

Thickness, of oxide layers, 426Thomas-Fermi approximation, for elec-

trons in metals, 47

Index 517

Thomas-Fermi model, and model solu-tion interface, 88

Thomas-Fermi statistics, 39Thomas-Fermi-Dirac statistics, 39Three-electron charge transfer

for chromous, 236and faradaic rectification, 202processes, 184

Tinnemans, and reduction of carbon di-oxide, 344, 369

Total capacitance, calculation at the in-terface, 86

Tourillon, and spectroelectrochemicalstudies of EXAFS, 308

Transition metals and photoelectro-chemical reduction of carbon di-oxide, 386

Transmission EX AFS, 297Trasatti

criticism of metal contributions, 67on nonaqueous solutions in double

layer, 14on potential of zero charge and surface

energy, 17Tuck, and recrimination of alloying ele-

ments, effects for aluminum depo-sition, 446

Two-electron charge transfer reactions,182

and faradaic rectification, 186, 193,230

Tyagai and Kolbesov and iodine-iodatereaction, 211

Valend and Heusler, and currents throughaluminum oxide films, 413

Valette, and curves for adsorption, 18Van Rysselberghe et al., and carbon di-

oxide reduction, 336Vanderpol et al., extension of faradaic

rectification technique in polar-ography, 219

Vasilliev et al., mechanism for reductionof carbon dioxide in aprotic sol-vents, 341

Vdovin, early contribution to faradaicrectification, 177

Vermilyee, and interpretation of chargefrom compressive to tensile stress,for oxide films, 428

Vetter and Strehblew, and film cracking,441

Voltammograms, and carbon dioxide re-duction, 346

Vorotyntsev, review of double layer, 4Vorotyntsev and Kornyshev, and jellium

model, 80

Wagner-Traud (Evans) diagram for alu-minum in aqueous solution, 422

Water, reaction with aluminum, 406Wave vector form, in EXAFS, 281Wavelength, distribution and radiation

from X rays, 268Work function

changed by adsorbates, 19changed by water adsorption, 19and potentials of zero charge, 78

X-ray absorption, for copper upd mono-layer on gold, 300

X-ray adsorptionnear edge, 309spectroscopy, 273

X-ray diffraction in electrochemistry, 320X rays

field intensities, on germanium lattice,313

standing waves, 310distribution of energy, 268at electrochemical interfaces, 265,

315, 316, 318experimental arrangement, 315field, pictorial, 312

generation, 267value in electrochemistry, 321

Xu, and gamma alumina islands, 459

Yeager, contributions to electron density,68

Young, and anodic oxides, 402Young and Zobel, and assumption of

migration of ions over trajecto-ries, 420

518 Index

Zafir, and carbon dioxide reduction by Zinc, and dropping mercury electrode,photoelectrochemical means in 233presence of vanadium oxide redox Zinc-zincous reaction, 200couples, 350 Zobel, and anodic oxides, 402

Zekharyan, and coverage of carbon diox- Zudgal, and space charge in aluminumide in an electrode, 337 films, 468

bockris - modern aspects of electrochemistry no 20

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