Electroanalytical Chemistry a Series of Advances Volume 22 Electroanalytical Chemistry

8/2/2019 Electroanalytical Chemistry a Series of Advances Volume 22 Electroanalytical Chemistry

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The Library of Congress Cataloged the First Issue of This Title as Follows:

Electroanalytic chemistry: a series of advances, v. 1

New York, M. Dekker, 1966-

v. 23 cm.

Editors: 19661995 A. J. Bard

1966- A. J. Bard and I. Rubinstein

1. Electromechanical analysisAddresses, essays, lectures

1. Bard, Allen J., ed.

QD115E499 545.3 66-11287

Library of Congress

0-8247-4719-4 (v. 22)

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This series is designed to provide authoritative reviews in the field of mod-

ern electroanalytical chemistry defined in its broadest sense. Coverage is

comprehensive and critical. Enough space is devoted to each chapter ofeach volume so that derivations of fundamental equations, detailed de-

scriptions of apparatus and techniques, and complete discussions of im-portant articles can be provided, so that the chapters may be useful without

repeated reference to the periodical literature. Chapters vary in length and

subject area. Some are reviews of recent developments and applications of

well-established techniques, whereas others contain discussion of the

background and problems in areas still being investigated extensively

and in which many statements may still be tentative. Finally, chapters on

techniques generally outside the scope of electroanalytical chemistry, but

which can be applied fruitfully to electrochemical problems, are included.

Electroanalytical chemists and others are concerned not only with

the application of new and classical techniques to analytical problems, but

also with the fundamental theoretical principles upon which these tech-

niques are based. Electroanalytical techniques are proving useful in such

diverse fields as electro-organic synthesis, fuel cell studies, and radical ion

formation, as well as with such problems as the kinetics and mechanisms of

electrode reactions, and the effects of electrode surface phenomena,

adsorption, and the electrical double layer on electrode reactions.

It is hoped that the series is proving useful to the specialist and non-

specialist alikethat it provides a background and a starting point for

graduate students undertaking research in the areas mentioned, and that it

also proves valuable to practicing analytical chemists interested in learning

about and applying electroanalytical techniques. Furthermore, electro-

chemists and industrialchemists with problems of electrosynthesis, electro-

plating, corrosion, and fuel cells, as well as other chemists wishing to apply

electrochemical techniques to chemical problems, may find useful material

in these volumes. A. J. B.

I. R.

INTRODUCTION TO THE SERIES

iii


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L. DAIKHIN Tel Aviv University, Ramat Aviv, Israel

STEPHEN W. FELDBERG Brookhaven National Laboratory, Upton,

New York, U.S.A.

E. GILEADI Tel Aviv University, Ramat Aviv, Israel

MARSHALL D. NEWTON Brookhaven National Laboratory, Upton,

New York, U.S.A.

JOHN F. SMALLEY Brookhaven National Laboratory, Upton, New

York, U.S.A.

GREG M. SWAIN Michigan State University, East Lansing, Michigan,

U.S.A.

V. TSIONSKY Tel Aviv University, Ramat Aviv, Israel

M. URBAKH Tel Aviv University, Ramat, Israel

v

CONTRIBUTORS TO VOLUME 22


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CONTENTS OF VOLUME 22

Introduction to the Series iii

Contributors to Volume 22 v

Contents of Other Volumes xiii

LOOKING AT THE METAL/SOLUTION INTERFACE

WITH THE ELECTROCHEMICAL QUARTZ-CRYSTAL

MICROBALANCE: THEORY AND EXPERIMENT

V. Tsionsky, L. Daikhin, M. Urbakh, and E. Gileadi

I. Introduction 2

A. Is It Really a Microbalance? 3B. Applications of the Quartz Crystal Microbalance 4

C. The Impedance Spectrum of the EQCM 5

D. Outline of This Chapter 8

II. Theoretical Interpretation of the QCM Response 8

A. Impedance 8

B. The Effect of Thin Surface Films 12

C. The Quartz Crystal Operating in Contact

with a Liquid 16

D. Quartz Crystals with Rough Surfaces 26

III. Electrical Double Layer/Electrostatic Adsorption 33

A. Introduction 33

B. Some Typical Results 34C. The Potential Dependence of the Frequency 36

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IV. Adsorption Studies 43

A. The Adsorption of Organic Substances 43

B. The Adsorption of Inorganic Species 53

V. Metal Deposition 60

A. D eposition on the Same Metal Substrate 60

B. Early Stages of Metal Deposition on a Foreign

Substrate 64

VI. The Influence of Roughness on the Response of the

QCM in Liquids 70

A. The Nonelectrochemical Case 71

B. The Electrochemical Case 76VII. Conclusion 83

VIII. Appendix 86

A. Nonuniform Film on the Surface 86

B. Experimental Remarks 86

References 94

THE INDIRECT LASER-INDUCED TEMPERATURE

JUMP METHOD FOR CHARACTERIZING FAST

INTERFACIAL ELECTRON TRANSFER: CONCEPT,

APPLICATION, AND RESULTS

Stephen W. Feldberg, Marshall D. Newton,and John F. Smalley

I. Introduction 102

A. Why Measure Fast Interfacial Electron Transfer

Rate Constants? And How? 103

B. Background 104C. The Underlying Principles of the ILIT Method

The Short Version 106

D. Definition of Terms 108

II. The Evolution of the ILIT Method for the Study of Fast

Interfacial Electron Transfer Kinetics 108A. The Temperature-Jump Approach for Studies of

Homogeneous Kinetics 108

Contents of Volume 22viii


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B. The Temperature-Jump Approach for Studies ofInterfacial Kinetics 108

III. Relevant Electron Transfer Theory: Marcuss

Description of Heterogeneous Nonadiabatic Electron

Transfer Reactions 112

A. Chidseys Approach 112

B. Temperature Dependence 116

C. How Well Does the Butler-Volmer Expression

Approximate Marcuss Formalism? 118

IV. Analysis of the ILIT Response 120

A. Response of the Open-Circuit Electrode Potential toa Change in the Interfacial Temperature in the

Presence of a Perfectly Reversible Redox CoupleAttached to the Electrode Surface 121

B. The Relaxation of the ILIT Response When the

Rate of Electron Transfer Is Not Infinitely Fast 126

C. When Is the ILIT Response Purely Thermal (i.e.,

Devoid of Kinetic Information)? 126

D. The Shape of the Ideal ILIT Perturbation 130

E. Nonidealities of the Shape of the ILIT Perturbation

and ResponseExtracting the Relaxation Rate

Constant, km 134

F. Correlating km to Meaningful Physical Parameters 137

V. Experimental Implementation of ILIT 143

A. The Cell 143

B. The Working Electrode: Preparation and Thermal

Diffusion Properties 148

C. Preparation of Self-Assembled Monolayers 150

D. The Electronics 151

E. Potential Problems 152

F. Energetic and Timing Considerations for Single and

Multiple Pulse Experiments 156

G. Some Suggested Experimental Protocols 160

VI. A Few Examples of Measurements

of Interfacial Kinetics 161

A. Some Typical Transients 161B. Determining the Value ofko 163

C. Arrhenius Plots and Evaluation ofDH p and DHk 163

Contents of Volume 22 ix


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VII. The Potential of the ILIT Approach 166

VIII. Some Thoughts About Future Experiments 166

IX. Glossary of Terms 170

X. Appendix: One-Dimensional Thermal Diffusion into

Two Different Phases 173

References 175

ELECTRICALLY CONDUCTING DIAMOND

THIN FILMS: ADVANCED ELECTRODE MATERIALS

FOR ELECTROCHEMICAL TECHNOLOGIESGreg M. Swain

I. Introduction 182

II. Diamond Thin Film Deposition, Electrode

Architectures, Substrate Materials, and Electrochemical

Cells 185

III. Electrical Conductivity of Diamond Electrodes 194

IV. Characterization of Microcrystalline and Nanocrystalline

Diamond Thin Film Electrodes 195

V. Basic Electrochemical Properties of Microcrystalline and

Nanocrystalline Diamond Thin Film Electrodes 201

VI. Factors Affecting Electron Transfer at Diamond

Electrodes 212

VII. Surface Modification of Diamond Materials and

Electrodes 216

VIII. Electroanalytical Applications 219

A. Azide Detection 219

B. Trace Metal Ion Analysis 221

C. Nitrite Detection 224

D. NADH Detection 225

E. Uric Acid Detection 225

F. Histamine and Serotonin Detection 226

G. Direct Electron Transfer to Heme Peptide andPeroxidase 227

Contents of Volume 22x


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H. Cytochrome c Analysis 228I. Carbamate Pesticide Detection 228

J. Ferrocene Analysis 229

K. Aliphatic Polyamine Detection 230

IX. Electrosynthesis and Electrolytic Water Purification 238

X. Optically Transparent Electrodes for

Spectroelectrochemistry 239

XI. Advanced Electrocatalyst Support Materials 251

A. Composite Electrode Fabrication and

Characterization 252

B. Oxygen Reduction Reaction 259C. Methanol Oxidation Reaction 264

XII. Conclusions 267

References 268

Author Index 279

Subject Index 295

Contents of Volume 22 xi


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CONTENTS OF OTHER VOLUMES

VOLUME 1

AC Polarograph and Related Techniques: Theory and Practice,

Donald E. SmithApplications of Chronopotentiometry to Problems in Analytical

Chemistry, Donald G. Davis

Photoelectrochemistry and Electroluminescence, Theodore Kuwana

The Electrical Double Layer, Part I: Elements of Double-Layer Theory,

David M. Monhilner

VOLUME 2

Electrochemistry of Aromatic Hydrocarbons and Related Substances,

Michael E. Peover

Stripping Voltammetry, Embrecht Barendrecht

The Anodic Film on Platinum Electrodes, S. GilamanOscillographic Polarography at Controlled Alternating Current,

Michael Heyrovksy and Karel Micka

VOLUME 3

Application of Controlled-Current Coulometry to Reaction Kinetics,

Jiri Janata and Harry B. Mark, Jr.

Nonaqueous Solvents for Electrochemical Use, Charles K. Mann

Use of the Radioactive-Tracer Method for the Investigation of the

Electric Double-Layer Structure, N. A. Balashova and

V. E. Kazarinov

Digital Simulation: A General Method for Solving ElectrochemicalDiffusion-Kinetic Problems, Stephen W. Feldberg

xiii


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VOLUME 4

Sine Wave Methods in the Study of Electrode Processes, Margaretha

Sluyters-Rehbach and Jan H. Sluyters

The Theory and Practice of Electrochemistry with Thin Layer Cells,

A. T. Hubbard and F. C. Anson

Application of Controlled Potential Coulometry to the Study of

Electrode Reactions, Allen J. Bard and

K. S. V. Santhanam

VOLUME 5

Hydrated Electrons and Electrochemistry, Geraldine A. Kenney and

David C. Walker

The Fundamentals of Metal Deposition, J. A. Harrison and H. R. Thirsk

Chemical Reactions in Polarography, Rolando Guidelli

VOLUME 6

Electrochemistry of Biological Compounds, A. L. Underwood and

Robert W. BurnettElectrode Processes in Solid Electrolyte Systems, Douglas O. Raleigh

The Fundamental Principles of Current Distribution and Mass Transport

in Electrochemical Cells, John Newman

VOLUME 7

Spectroelectrochemistry at Optically Transparent Electrodes; I. Electrodes

Under Semi-infinite Diffusion Conditions, Theodore Kuwana and

Nicholas Winograd

Organometallic Electrochemistry, Michael D. Morris

Faradaic Rectification Method and Its Applications in the Study ofElectrode Processes, H. P. Agarwal

Contents of Other Volumesxiv


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VOLUME 8

Techniques, Apparatus, and Analytical Applications of Controlled-

Potential Coulometry, Jackson E. Harrar

Streaming Maxima in Polarography, Henry H. Bauer

Solute Behavior in Solvents and Melts, A Study by Use of Transfer

Activity Coefficients, Denise Bauer and Mylene Breant

VOLUME 9

Chemisorption at Electrodes: Hydrogen and Oxygen on Noble Metals

and their Alloys, Ronald Woods

Pulse Radiolysis and Polarography: Electrode Reactions of Short-lived

Free Radicals, Armin Henglein

VOLUME 10

Techniques of Electrogenerated Chemiluminescence, Larry R. Faulkner

and Allen J. Bard

Electron Spin Resonance and Electrochemistry, Ted M. McKinney

VOLUME 11

Charge Transfer Processes at Semiconductor Electrodes,

R. Memming

Methods for Electroanalysis In Vivo, Jir Koryta, Miroslav Brezina,

Jir Prada c , and Jarmila Prada cova

Polarography and Related Electroanalytical Techniques in

Pharmacy and Pharmacology, G. J. Patriarche,

M. Chateau-Gosselin, J. L. Vandenbalck,

and Petr Zuman

Polarography of Antibiotics and Antibacterial Agents,Howard Siegerman

Contents of Other Volumes xv


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VOLUME 12

Flow Electrolysis with Extended-Surface Electrodes, Roman E. Sioda

and Kenneth B. Keating

Voltammetric Methods for the Study of AdsorbedSpecies, Etienne Laviron

Coulostatic Pulse Techniques, Herman P. van Leeuwen

VOLUME 13

Spectroelectrochemistry at Optically Transparent Electrodes,II. Electrodes Under Thin-Layer and Semi-infinite Diffusion

Conditions and Indirect Coulometric Iterations, William H.

Heineman, Fred M. Hawkridge, and Henry N. Blount

Polynomial Approximation Techniques for Differential Equations in

Electrochemical Problems, Stanley Pons

Chemically Modified Electrodes, Royce W. Murray

VOLUME 14

Precision in Linear Sweep and Cyclic Voltammetry, Vernon D. Parker

Conformational Change and Isomerization Associated with Electrode

Reactions, Dennis H. Evans and Kathleen M. OConnellSquare-Wave Voltammetry, Janet Osteryoung and John J. ODea

Infrared Vibrational Spectroscopy of the Electron-Solution Interface,

John K. Foley, Carol Korzeniewski, John L. Dashbach, and

Stanley Pons

VOLUME 15

Electrochemistry of Liquid-Liquid Interfaces, H. H. J. Girault and

D. J. Schiffrin

Ellipsometry: Principles and Recent Applications in Electrochemistry,

Shimson Gottesfeld

Voltammetry at Ultramicroelectrodes, R. Mark Wightman andDavid O. Wipf

Contents of Other Volumesxvi


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VOLUME 16

Voltammetry Following Nonelectrolytic Preconcentration, Joseph Wang

Hydrodynamic Voltammetry in Continous-Flow Analysis, Hari

Gunasingham and Bernard Fleet

Electrochemical Aspects of Low-Dimensional Molecular Solids, Michael

D. Ward

VOLUME 17

Applications of the Quartz Crystal Microbalance to Electrochemistry,

Daniel A. Buttry

Optical Second Harmonic Generation as an In Situ Probe of

Electrochemical Interfaces, Geraldine L. Richmond

New Developments in Electrochemical Mass Spectroscopy,

Barbara Bittins-Cattaneo, Eduardo Cattaneo, Peter Ko nigshoven,

and Wolf Vielstich

Carbon Electrodes: Structural Effects on Electron Transfer Kinetics,

Richard L. McCreery

VOLUME 18

Electrochemistry in Micelles, Microemulsions, and Related

Microheterogeneous Fluids, James F. Rusling

Mechanism of Charge Transport in Polymer-Modified Electrodes,

Gyo rgy Inzelt

Scanning Electrochemical Microscopy, Allen J. Bard, Fu-Ren F. Fan,

and Michael V. Mirkin

VOLUME 19

Numerical Simulation of Electroanalytical Experiments: Recent Advances

in Methodology, Bernd Speiser

Electrochemistry of Organized Monolayers of Thiols and RelatedMolecules on Electrodes, Harry O. Finklea

Contents of Other Volumes xvii


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LOOKING AT THE METAL/SOLUTION INTERFACEWITH THE ELECTROCHEMICAL QUARTZ CRYSTAL

MICROBALANCE: THEORY AND EXPERIMENT

V. Tsionsky, L. Daikhin, M. Urbakh, and E. Gileadi

School of Chemistry

Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University

Ramat Aviv, Israel

I. INTRODUCTION 2

A. Is It Really a Microbalance? 3

B. Applications of the Quartz Crystal Microbalance 4

C. The Impedance Spectrum of the EQCM 5

D. Outline of This Chapter 8

II. THEORETICAL INTERPRETATION OF THE QCM

RESPONSE 8

A. Impedance 8

B. The Effect of Thin Surface Films 12

C. The Quartz Crystal Operating in Contact

with a Liquid 16D. Quartz Crystals with Rough Surfaces 26

III. ELECTRICAL DOUBLE LAYER/ELECTROSTATIC

ADSORPTION 33

A. Introduction 33

B. Some Typical Results 34C. The Potential Dependence of the Frequency 36

IV. ADSORPTION STUDIES 43

A. The Adsorption of Organic Substances 43

B. The Adsorption of Inorganic Species 53

V. METAL DEPOSITION 60

A. Deposition on the Same Metal Substrate 60

B. Early Stages of Metal Deposition on aForeign Substrate 64

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VI. THE INFLUENCE OF ROUGHNESS ON THERESPONSE OF THE QCM IN LIQUIDS 70

A. The Nonelectrochemical Case 71

B. The Electrochemical Case 76

VII. CONCLUSION 83

VIII. APPENDIX 86

A. Nonuniform Film on the Surface 86

B. Experimental Remarks 86

References 94

I. INTRODUCTION

The literature concerning the quartz crystal microbalance (QCM) and its

electrochemical analogue, the electrochemical crystal microbalance

(EQCM) is wide and diverse. Many reviews are available in the literature,

discussing the fundamental properties of this device and its numerous

applications, including its use in electrochemistry [15]. In this chapter we

concentrate on electrochemical applications, specifically in studies of

submonolayer phenomena and the interaction of the vibrating crystal with

the electrolyte in contact with it.

A few examples are treated in detail here, and the advantages and

limitations of the EQCM as a tool for the study of fundamental phenom-

ena at the metal/solution interface are discussed.When the quartz crystal microbalance was first introduced in 1959

[6], it represented a major step forward in our ability to weigh matter. Until

then, routine measurements allowed an accuracy of 0.1 mg, and highly

sensitive measurements could be made with an accuracy limit of 0.3 Ag

under well-controlled experimental conditions, (see Ref. 7). The QCM

extended the sensitivity by two or three orders of magnitude, into the sub-

nanogram regime.

Even used in vacuum or in an inert gas atmosphere at ambient

pressure, the QCM acts as a balance only under certain conditions, as

discussed below. Then the change of mass caused by adsorption or de-

position of a substance from the gas phase can be related directly to the

change of frequency by the simple equation derived by Sauerbrey [6]:

Df Cm Dm 1

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where Cm is a constant, representing the mass sensitivity, which is related toknown properties of quartz and the dimensions of the crystal, andDm is the

added mass density, in units of g/cm2.

A. Is It Really a Microbalance?

Is the quartz crystal microbalance really a microbalance? For one thing, it

should rightly be called a nano-balance, considering that the sensitivity of

modern-day devices is on the order of 12 ng/cm2 and could be pushed

further, if necessary. More importantly, calling it a balance implies that the

Sauerbrey equation applies strictly, namely that the frequency shift is the

sole result of mass loading. It is well known that this is not the case, andthe frequency shift observed could more appropriately be expressed by a

sum of terms of the form

Df Dfm Dfg DfP DfR Dfsl DfT 2where the different terms on the right-hand side (rhs) of this equation

represent the effects of mass loading, viscosity anddensity of the medium in

contact with the vibrating crystal, the hydrostatic pressure, the surface

roughness, the slippage effect, and the temperature, respectively, and the

different contributions can be interdependent. Even this equation does not

tell the whole story, certainly not when the device is immersed in a liquid or

in gas at high pressure. It does not account for solution occluded between

the ridges of a rough surface or in the pores of a porous substrate. The

nature of the interaction between the liquid and the surface, the type ofroughness, and internal stress or strain could all affect the response of the

quartz crystal resonator. These effects become of major importance

particularly when small changes of frequency, associated with submono-

layer phenomena, are considered. Some of these factors will be discussed in

this chapter.

It should be evident from the above arguments that the term quartz

crystal microbalance is a misnomer, which could (and indeed has) lead to

erroneous interpretation of the results obtained by this useful device. It

would be helpful to rename it the quartz crystal sensor (QCS), which

describes what it really doesit is a sensor that responds to its nearest

environment on the nano-scale. However, it may be too late to change the

widely used name. The QCM or its analogue in electrochemistry, the

EQCM, can each act as a nano-balance under specific conditions, but notin general.

Electrochemical Quartz Crystal Microbalance 3


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B. Applications of the Quartz Crystal Microbalance

The most common commercial use of the QCM is as a thickness gauge in

thin-layer technology. When used to monitor the thickness of a metal film

during physical or chemical vapor deposition, it acts very closely as a nano-

balance, providing a real-time measurement of the thickness. Indeed,

devices sold for this purpose are usually calibrated in units of thickness

(having a different scale foreach metal, of course), and claim a sensitivity of

less than 0.1 nm, which implies a sensitivity of less than a monolayer.

The other common application of the QCM is as a nano-sensor

proper, made sensitive to one gas or another by suitable surface treatment.

Selecting the suitable coating on the electrodes of the QCM can determineselectivity and enhance sensitivity. It is not our purpose to discuss sensors

in the present review. It should only be pointed out that any such sensor

would have to be calibrated, since the Sauerbrey equation would not be

expected to apply quantitatively.

1. Applications for Gas-Phase Adsorption

The high sensitivity of the QCM should make it an ideal tool for the study

of adsorption from the gas phase. We note that the number of sites on the

surface of a metal is typically 1.3 1015/cm2, hence a monolayer of a smalladsorbate, occupying a single site, would be about 2.2 nmol/cm2. A

monolayer of water would therefore weigh about 40 ng/cm2, while a

monolayer of pyridine would weigh 3060 ng/cm2, depending on its

orientation on the surface. Comparing these numbers with the sensitivity

of 2 ng/cm2 shows that adsorption isotherms could be measured in the gas

phase employingthe QCM. This hasnot been done properly until relatively

recently, mainly because the device was treated as a microbalance, i.e., it

was assumed that the Sauerbrey equation could be applied, and several

important terms in Eq. (2) were ignored. Obtaining adsorption isotherm

one has to change the pressure over a wide range. Therefore, the changes of

properties of the surrounding gas cannot be ignored. This shortcoming was

overcome by the present authors [8], who developed the supporting gas

method. When this method is employed, the overall pressure is kept

constant by a large excess of an inert gas, and the frequency shift of the

QCM is measured as a function of the partial pressure of the material being

investigated. In this manner all terms in Eq. (2), other thanD

fm, areessentially zero, and the device acts as a true nano-balance. One intriguing

result that was obtained using this method came from a comparison of

the adsorption of benzene and pyridine on a gold surface. It was found

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that a monolayer of pyridine weighs roughly twice as much as a monolayerof benzene. Since the two molecules have almost the same size and

molecular weight, it must be concluded that their configuration in the

adsorbed state is different. Benzene is probably adsorbed flat on the

surface, while pyridine must be adsorbed perpendicular to it, occupying

only half as many sites.

Although the nominal resolution of 2 ng/cm2 should be enough to

study the adsorption isotherm if the monolayer weighs around 3060 ng/

cm2, it is somewhat marginal, and an increase of sensitivity of about one

order of magnitude would be desirable. Part of this enhancement could be

achieved by increasing the roughness factor on the atomic scale, without

influencing the roughness on a scale relevant to the resonance frequency(see Sec. VI).

2. Use of QCM in Liquids

It was not initially obvious that the quartz crystal resonator would operate

in liquids until this was proven experimentally [9,10]. The term associated

with the influence of the viscosity and density of liquid in Eq. (2) can be

written [11] as

Dfg Cg gq 1=2 3Since the product of

ffiffiffiffiffiffigq

pin liquids is about two orders of magnitude

higher than in gases at ambient pressure, the crystal is heavily loaded when

transferred from the gas phase into a liquid.Once the door had been opened to its use in liquids, the potential of

the QCM for interfacial electrochemistry was obvious, and the EQCMbecame popular.

When a QCM is placed in contact with a dilute aqueous solution, the

frequency should shift to lower values by about 0.7 kHz according to Eq.

(3). In practice, a shift of 1.02 kH is observed, depending on the surface

roughness. The effect of roughness is also related indirectly to viscosity and

density, since the hydrodynamic flow regime at the surface is altered as a

result of roughness [1214]. Roughness is a major issue in the interpreta-

tion of the response of the QCM in liquids, and it is discussed in some detailin the following sections.

C. The Impedance Spectrum of the EQCM

In early studies of the QCM and the EQCM, only the resonance frequency

was determined and conclusions were drawn based on the shift of



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frequency. Unfortunately, in many cases this shift was attributed to massloading alone, and it was used to calculate the weight added or removed

from the surface, disregarding other factors that affect the frequency. In

the past decade, more and more laboratories expanded such studies to

include measurements of the impedance spectrum of the crystal [1525].

This provides an additional experimental variable that can obviously yield

further information and a deeper understanding of the structure of the

interface. For instance, a variation in the resonance width provides

unambiguous proof that mechanisms other than mass loading are also

involved.

A series of typical admittance spectra are presented here. In Fig. 1a

we show a simple case of metal deposition (gold on a gold substrate). TheEQCM acts as a true microbalance in this case. The resonance frequency is

shifted to lower values with increasing load, but the shape of the spectrum

remains unaltered.

In Fig. 1b the effect of viscosity on the admittance spectrum is shown.

Here again the resonance frequency is shifted to lower values with

increasing viscosity, but this has nothing to do with mass loading.

However, the shape of the spectrum is quite different, and the width at

half-height (see below) increases dramatically with increasing viscosity and

density of the liquid. Line 1 and the inset in this figure show the response of

theQCMinH2 at ambient pressure. The product of viscosity and density is

about four orders of magnitude smaller than in any of the liquids.

Correspondingly, the width of the resonance is only about 20 Hz,

compared to about 2.5 kHz in the liquid corresponding to line 2.

Another aspect of the admittance spectrum is shown in Fig. 1c. Here

the same metal deposition was conducted as in Fig. 1a, but the conditions

were purposely chosen to produce a very rough surface (by plating at a

current density close to the mass-transport limited value). The width of the

resonance is increased and the frequency is shifted to lower values with

increasing roughness.

We chose rather extreme cases of viscosity and roughness in Fig. 1b

and 1c, for the purpose of illustration. The corresponding shift in fre-

quency is very high, in the range of 515 kHz, as compared to changes of

frequency of 540 Hz typically observed in the studies of electrosorption,

double layer, upd, and other submonolayer phenomena. The important

conclusion is that even very small changes of viscosity and/or surfaceroughness (produced inadvertently) could lead to a shift of frequency

comparable to that expected for such submonolayer phenomena, and the

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FIG. 1. (a) The real part of the admittance versus frequency: during depositionof gold on a gold-covered EQCM at a current density of 20 AA/cm2. (c) The same

at 500 AA/cm2. (b) The response of the QCM immersed in different media: 1,

hydrogen, 1 atm; 2, dimethyl ether; 3, water; 4 and 5, 40% and 50% aqueoussolutions of sucrose, respectively. (Inset) Admittance for H2, on an expanded scale.

Arrow gq shows the increase of product gq. (From Ref. 24.)



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change of frequency cannot be generally interpreted to be a result of mass

loading alone.

D. Outline of This Chapter

This chapter contains theoretical and experimental sections. In the theo-

retical section we consider different aspects of the behavior of the vibrating

resonator: when it was loaded by additional mass, immersed in viscous

media, has undergone changes in surface roughness, etc. We discuss the

universal perturbation theory of the influence of slightly rough surfaces on

the QCM response and consider the special model for strong roughness,

noting that a general model does not exist for such surfaces. Specialattention was paid to consideration of the influence of slippage on the

QCM at the solid/electrolyte interface.

The QCM is now so widely and extensively used that, in the frame-

work of this chapter, it is not possible to review all the available litera-

ture. Hence we limited ourselves here to a review of the experimental data

and ideas concerning the studies of submonolayer adsorption and inter-

actions taking place at the metal/solution interface. In other words, this

review is restricted to the use of the QCM in fundamental electrochem-

istry. Furthermore, we did not include studies of electrochemical kineticswith the help of the EQCM, which merits a separate review. The problems

of the interpretaion of the EQCM response caused by changes taking

place at the metal/solution interface are obviously of first priority.

We did not present here a full description of the operation of anEQCM. This topic is well described in previous reviews (see Refs. 1,2) and

in many articles published in readily accessible electrochemical journals.

However, a few aspects of the experiments with the EQCM are covered in

the Appendix (Sec. VIII.B).

II. THEORETICAL INTERPRETATION OF THE QCM

RESPONSE

A. Impedance

The shear mode resonator consists of a thin disk of AT-cut quartz crystal

with electrodes coated on both sides. The application of a voltage between

these electrodes results in a shear deformation of the crystal due to itspiezoelectric properties. The crystal can be electrically excited into a

Tsionsky et al.8


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number of resonance modes, each corresponding to a unique standingshear wave across the thickness of the crystal. If a quartz resonator

operates in contact with an outer medium, the oscillating surface interacts

mechanically with the medium and excites motion in it. The mechanical

properties of the medium in contact are reflected in the response of the

resonator.

The geometry of the system consisting of a quartz crystal in contact

with the outer medium is schematically shown in Fig. 2. The z-axis is

plotted perpendicular to the plane of contact - the plane z =0 coinciding

with the unconstrained face of the quartz resonator, and the plane z = dis

its constrained face. The thickness of the quartz crystal is d.

When an ac voltage is applied between the electrodes, the motion ofthe AT cut quartz crystal can be described by a system of two coupled

differential equations, which constitute the wave equation for elastic

displacements, u(z,t) = u(z,x) exp(ixt), and the equations that establish

FIG. 2. Schematic sketch of the quartz crystal resonator in contact with a liquid.The contacting medium is a thin film rigidly attached to the crystal surface from

one side, at z = d. The opposite surface of the crystal (z = 0) is unconstrained. disthe thickness of the quartz crystal.



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the relationship between displacements and the electrostatic poten-tial,u(z,t) =u(z,x) exp(ixt), [26] are

x2qqu z;x c66d2

dz2u z;x 4

e22d2

dz2u z; x e26 d

2

dz2u z; x 5

Here, c66 lq e226=e22 ixgq; qq;lq are the density and shear modu-

lus of quartz,e22,e26 are the dielectric constant, and the piezoelectric stress

coefficient of quartz, gq, is its fictitious viscosity, x = 2p f is the angularfrequency, and f is the frequency. Equations (4) and (5) are solved under

the following boundary conditions:

1. At the plane z =0, the potential equals u0 and the stress is zero.

2. At the plane z = d, the potential equals u0 and the ratio of theshear stress, c66du(z,x)/dz, acting on the contacting medium to

the surface velocity, ixu(d,x), equals Zout.Here Zout is the mechanical impedance of the medium contacting the

quartz surface. Solution of Eqs. (4) and (5) yields the following expression

for the admittance of the quartz resonator [27,28]:

Y ixC0 Z1m 6where C0=e22/d is the static capacitance and Zm is the motional imped-ance:

Zm 1ixC0

/q

K2q 2tan /q=2 1

" # /q

4K2q xC0

Zout=Zq

1 iZout=Zq2tan /q=2

26664

37775 7

and K2q e226=e22c66;/q kqd; Zq kqc66=x , and kq xffiffiffiffiffiffiffiffiffiffiffiffiffi

qq=c66p

is the

wave number of the shear wave in quartz. The first term in Eq. (7) de-

scribes the motional resistance of an unloaded quartz resonator. The sec-

ond term arises from surface loading and includes the properties of the

electrode surfaces and the contacting medium through Zout.In QCM experiments the surface loading is relatively small [27], that

is, |Zout/Zq|


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nant frequency of the quartz-crystal resonator with respect to the resonantfrequency of the unloaded quartz crystal, f0, can be written as [12,29]

D fu Df iG2

if0p

Zout

Zq8

It should be noted that the frequency shift Dfcan be a complex number,

and its imaginary part, G, reflects the width of the resonance. Equation (8)

shows that the complex frequency shift Dfcontains the same information

as the mechanical impedance Zout.

The admittance of the quartz resonator can be presented in terms of

an electrical equivalent circuit [24,15,3036]. The equivalent circuit forthe unloaded quartz crystal consists of a motional branch, which reflects

the vibration of the quartz, and a static capacitance, which is in parallel

with the motional branch. The motional branch includes a resistance,

capacitance, and inductance connected in series. The relationships between

the electrical elements and the mechanical parameters describing the

crystal motion (mass, compliance, and damping coefficient) were consid-

ered in Refs. 3739. When the surface loading is small, |Zout/Zq|


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example, it is not obvious how one would account for a surface roughness

of unspecified nature (slight roughness, strong roughness, or some combi-

nation of both) in terms of an equivalent circuit.

In order to analyze the influence of the different loading mechanisms

on the QCM response, one has to model a dependence of the mechanical

impedance Zout or the complex resonance frequency shift on the chemical

and physical properties of the contacting medium. Various models for the

mechanical contact between the oscillating quartz crystal and the outermedium are discussed below.

B. The Effect of Thin Surface Films

1. Uniform Film Rigidly Attached to the Surface

First we consider the effect of a thin film, rigidly attached to an ideally flat

crystal surface, on the response of the quartz crystal resonator (see Fig. 2).

FIG. 3. The Butterworthvan Dyke equivalent circuit of the loaded quartzcrystal resonator. The parameters R, C, and L describe the behavior of the

unloaded quartz resonator; Zout is the impedance of the contacting medium.

Tsionsky et al.12


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For a homogeneous thin film with a thickness smaller than thewavelength of the shear oscillations, the shift of the resonance frequency

can be expressed in terms of the change in surface mass density of the film,

Dmf (in g/cm2). This was given by Sauerbrey [6] as

Df 2f2

0 Dmf

lqqq 1=2 9

This equation coincides with Eq. (1) with Cm 2f20 = lqqq 1=2

. Equation

(9) can be derived by supplementing the wave equation [Eq. (4)] with the

Newtonian equation of motion for the surface film:

Dmfx2uf x lq

d

dzu z; w at z d 10

where uf(x) is the displacement of the film. Here the shear stress,lqdu(z,x)/dz, plays the rule of the external force acting on the film. Hereand everywhere below we use in Eq. (4) lq instead ofc66, neglecting small

values xgq and e226=e22 . Solving Eqs. (4) and (10) under the standard

boundary condition, namely that (1) at the unrestricted surface, z = 0, the

shear stress equals zero and (2) at the surface z = d the quartz surface

displacement is equal to the surface film displacement, one obtains the

Sauerbrey equation. From Eq. (10) one can see that the frequency shift is

determined by the inertial force of the film acting on the quartz surface.

Equation (9) shows that the addition of mass rigidly attached to the

surface of the quartz-crystal resonator leads to a decrease of the resonant

frequency, but it does not influence the width of the resonance.

2. Nonuniform Film Rigidly Attached to the Surface

A natural question arises whether a nonhomogeneous mass distribution

can lead to an additional shift of frequency and/or to a broadening of theresonance, compared to the result given by the Sauerbrey equation?

In order to answer this question, we consider here the effect of

nonuniform mass loading on the response of the QCM. Lateral displace-

ment of the nonuniform film can be described by the equation of motion:

DmfRx2ufR lqd

dzuz;R for z d 11

where uf(R) and u(z,R) represent the displacements of the surface film and

quartz crystal andDmf(R) is the surface film density, as a functionof lateral



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coordinates, R. This equation plays the role of a boundary condition forthe wave equation describing the shear-mode oscillations in the quartz

crystal. Equations (4) and (11), with the standard boundary condition, lead

to the following equation for determination of the resonant frequency (see

Appendix):

lqk tankd x2Dmf x4Dm2lZ

dK

2p2gK

lqpKtanpKd x2Dmf12

Here, k = x(qq/lq)1/2

is the wave number of the shear wave in the quartz,p(K) = k2 K2, and K is the two-dimensional tangential wave vector.Considering mechanical response of the quartz crystal we use k instead

of kq [see text following Eqs. (4) and (5)] because xgq and e226=e22 are

small compared to lq. Also, Dml is the root mean square deviation of the

mass density from the average value Dmf, and g(K) is the correlation

function, which describes a nonuniform mass distribution along the

surface. Equation (12) is a general form of the Sauerbrey equation,

applicable for the case of an inhomogeneous surface film. For uniform

mass distribution (corresponds to Dml= 0), it yields the Sauerbrey

equation in its usual form.

Assuming that the correlation function g(K) has a Gaussian form,

with a lateral correlation length, l, Eq. (12) can be solved analytically for

two limiting cases, kl>>1 and kl1,

splitting of the resonant frequency occurs, and the frequency shift can be

estimate as

Df 2f20ffiffiffiffiffiffiffiffiffiffi

qqlqp DmfFDml

13In contrast to the case of uniform mass loading, Dm

1f R 0, two

values of the resonance frequency appear. This effect can be simulated by

a simple equivalent circuit consisting of two Butterworthvan Dyke [33

35] circuits in series with the inductances corresponding to the two dif-

ferent values of the surface mass densities,D

mf D

mlandD

mf D

ml. Dueto overlap of these two resonance states, splitting can manifest itself as a

broadening of the resonance, which will have an effective width of the

Tsionsky et al.14


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order of 2f20Dml=p lqqq 1=2

. For the 6 MHz quartz resonator thisbroadening effect becomes important when the correlation length l is

larger than 0.02 cm.

In thesecond limiting case, kl


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film yields the following expressions for the changes of the frequency, Df,and the width of the resonance, G:

Df 2f20Dmaffiffiffiffiffiffiffiffiffiffiqqlq

p v2

v2 2pf0Dma 2" #

15

G 4f20Dmaffiffiffiffiffiffiffiffiffiffiqqlq

p 2pf0Dmavv2 2pf0Dma 2

" #16

Note that

G

Df 4pf0

Dma

v 17

Thus, the interfacial friction can be evaluated from measurement ofG and

Df. This procedure has been applied to a number of systems in which weak

physical adsorption occurs, such as the adsorption of Xe, Kr, N2 on Au

and of H2O and C6H12 on Ag [4852]. In all above cases, slippage was

observed and the ratio of the coefficient of sliding friction to the mass

density was of the order v/Dma = (108 109)s1. As an example, the

frictional stress acting on the monolayer Xe film sliding on a Ag (111)

surface at a velocity v =1 nm / s, F=vv, equals about 10 N/m2 [54]. It is

much smaller than typical shear stresses involved in sliding of a steel block

on a steel surface under boundary lubrication condition. The shear stress in

the latter case is of the order c108 N/m2 [53].

In a recent paper [55] the dependence of the slip time, ss, on the

amplitude of the crystal surface oscillations, A, and the surface coverages

was investigated. The results refer to the absorption of krypton atoms on

gold at 85jK. The slip time is related to the interfacial friction coefficient, v,

as ss = Dma /v. It was found that there is a step-like transition between a

low-coverage region, where slippage exists at the solid/film interface, and a

high-coverage region where the film is locked to the surface. The transition

occurs at different coverages depending on the amplitude, A. Independent

of coverage, the film is attached rigidly to the surface for AV 0.18 nm and

slides for A > 0.4 nm. In the region of sliding at small coverages, the values

of the slip time are in the interval 210 nsec, for 0.18 nm < A < 0.4 nm.

C. The Quartz Crystal Operating in Contact with a Liquid

1. General Considerations

When a quartz crystal resonator operates in contact with a liquid, the shear

motion of the surface generates motion in the liquid near the interface. The

Tsionsky et al.16


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velocity field, v(r,x) related to this motion in a semi-infinite Newtonianliquid is described by the linearized Navier-Stokes equation:

ixqvr;x jPr;x gDvr;x 18where P(r,x), g, and q are pressure, viscosity, and density of the liquid,

respectively. Under the conditions of the QCM experiments, where the

shear velocities are much smaller than the sound velocity in the liquid, the

displacement of the crystal does not generate compressional waves and a

liquid can be considered as an incompressible one. If the surface is

sufficiently smooth, the quartz oscillations generate plane-parallel laminar

flow in the liquid, as shown in Fig. 4. The velocity field obtained as the

solution of Eq. (18) for a flat surface has the form

vxz vq0xexp1 iz=d 19where vq0(x) is the velocity of the liquid at the surface and d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g=x0q

p.

Equation (19) represents a damped shear wave radiating into the liquid

from the surface of the oscillating resonator. d is the velocity decay length

of this shear wave, which lies between 250 and 177 nm, for dilute aqueous

solutions at room temperature, for crystals having a fundamental frequen-

cy in the range of 510 MHz. Damping of the shear wave has a number of

important consequences. First, it ensures that the quartz crystal can

operate in liquids, the losses in the liquid being limited by the finite depth

of penetration. Second, a small portion of the liquid is coupled to the

crystal motion and a frequency decrease is observed. Third, the viscousnature of motion gives rise to energy losses, which are sensed by the

resonator, both as a decrease in frequency and as anincrease in the width of

the resonance.

2. The Nonslip Boundary Condition

The response of the QCM at the solid/liquid interface can be found by

matching the stress and the velocity fields in the media in contact. It is

usually assumed that the relative velocity at the boundary between the

liquid and the solid is zero. This corresponds to the nonslip boundary

condition. Strong experimental evidence supports this assumption on the

macroscopic scales [56,57]. In this case the frequency shift, Dfl, and the

width of the resonance, Gl, can be written as follows [10,11]:

Dfl f

3=20

ffiffiffiffiffiffiqg

pffiffiffiffiffiffiffiffiffiffiffiffipqqlq

p 20



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Gl 2f

3=20


pffiffiffiffiffiffiffiffiffiffiffiffipqqlq

p 21

Equations (20) and (21) show that the generation of a damped

laminar flow in the liquid causes a decrease in the resonance frequency

and an increase in the resonance width, which are both proportional toffiffiffiffiffiffiqgp . In contrast to the case of the mass loading, where Dfis proportionalto f0

2, the liquid induced response of the QCM is proportional to f03/2.

FIG. 4. The system geometry and the velocity distribution. Curves 1 and 2represent the velocity distributions at the liquid/adsorbate interface without and

with slippage, respectively. Curve 3 is the velocity distribution in the quartz. The

thickness of various layers is not drawn to scale.

Tsionsky et al.18


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It is interesting to note that for both a surface film rigidly attached tothe resonator and a liquid in contact with the surface of the quartz crystal,

the shift of the resonant frequency can be written in the same form, as

Df f0 qqq

kheff 22

where k x0 ffiffiffiffiffiffiffiffiffiffiffiqq=lqp , q is the bulk density of the medium in contact withthe vibrating surface of the solid, film, or liquid, and heffis the thickness of

the layer that responds to the quartz oscillations. In the case of a film, heffcoincides with the thickness. For a semi-infinite liquid, heff presents a

thickness of liquid involved in the motion, and it should be taken equal to

d/2. The difference in the frequency dependence of the QCM response in

the two cases is a result of the frequency dependency of d. However, in

contrast to the case of pure mass loading, the effect of a liquid results not

only in a frequency shift, but also in a broadening of the resonance.

a. Effect of a Thin Liquid Film at the Interface

The properties (the effective viscosity and density) of the liquid layer in

close vicinity to the interface can differ from their bulk values. There are

various reasons for these phenomena. For example, the properties of a

thin liquid layer confined between solid walls are determined by interac-

tions with the solid walls [58,59]. In electrochemical system the structuring

of a solvent induced by the substrate and a nonuniform ion distribution in

the diffuse double layer can significantly influence the properties of the

solution at the interface. The nonuniform distribution of species, which

influences the properties of the liquid near the electrode, also occurs in the

case of diffusion kinetics. The latter was considered in Ref. 60, where the

ferro/ferri redox system was studied by the EQCM. This was the case

where the velocity decay length (>25 Am) was much less than the thickness

of the diffusion layer (>100 Am), in which the composition of the solution

is different from the bulk composition.

Nonuniform distribution of species results in nonuniform distribu-

tion of the properties of liquid near the vibrating surface of the resonator.

The properties change with distance from the interface, until the values

corresponding to the bulk of solution have been reached. In order to

simplify the description of this nonuniformity on the QCM, it is assumed

that a thin film of liquid, having different values of gfand qf, exists at theinterface [61]. To calculate the effect of this film on the frequency shift, one

has to solve the wave equation for the elastic displacements in the quartz



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crystal [see Eq. (4)] simultaneously with the linearized Navier-Stokesequation for the velocities in the film and in the bulk liquid under standard

nonslip boundary conditions.

The shift of the resonant frequency and the width of the resonance

can be written as

Df f3=20


pffiffiffiffiffiffiffiffiffiffiffiffiplqqq

p 2f20ffiffiffiffiffiffiffiffiffiffi

lqqqp q

1 g

gf

qf q " #

Lf 23

G 2f

3=20 ffiffiffiffiffiffiqgpffiffiffiffiffiffiffiffiffiffiffiffiplqqqp

4f20ffiffiffiffiffiffiffiffiffiffilqqqp q

1

g

gf qf q " # L

2f

d 24

where Lfand qfare the thickness and the density of the film, respectively.

These equations are valid in a particular case, when Lf g the film acts as though it were rigidly attached

to the surface: it causes a shift in frequency equal to that caused by its mass.

3. Slip Boundary Conditions

a. Slippage at Solid/Liquid Interface

Although the nonslip boundary condition has been remarkably successful

in reproducing the characteristics of liquid flow on the macroscopic scale,

its application for a description of liquid dynamics in microscopic liquid

layers is questionable. A number of experimental [6264] and theoretical

[65,66] studies suggest the possibiility of slippage at solid/liquid interfaces.

The boundary condition is controlled by the extent to which the

liquid feels a spatial corrugation in the surface energy of the solid. This

depends on a number of interfacial parameters, including the strength of

the liquid-liquid and liquid-solid interactions, the commensurability of thesubstrate and the liquid densities, characteristic sizes, and also the rough-

ness of the interface. In order to quantify the slippage effect, the slip length,

Tsionsky et al.20


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k, is usually introduced [65,67,68]. The traditional nonslip boundarycondition is replaced by

dvz;xdz

zd

1k

vd;x vq0x 25

where v(z,x) is the velocity in the liquid and vq0(x) is the velocity of thequartz crystal surface. Equation (25) expresses the discontinuity of the

velocity across the interface. For k = 0, Eq. (25) is reduced to the usual

nonslip boundary condition: v(d,x) = vq0 (x). The physical meaning of the

slip length can be clarified by comparing velocity profiles for the nonslip

and slip boundary conditions. These two profiles coincide when the nonslip

boundary condition is imposed at the surface shifted inside the solid on the

distance k with respect to the actual interface.

The slip boundary condition (25) results in the following equations

for the resonant frequency shift and the width of the resonance:

Df f20 qdffiffiffiffiffiffiffiffiffiffiqqlq

p 11 k=d2 k=d2" #

26

G 2f20 qdffiffiffiffiffiffiffiffiffiffi

qqlqp 1 2k=d

1 k=d 2 k=d 2" #

27

Equations (26) and (27) show that the influence of the slippage on the

response of the QCM in liquid is determined by the ratio of the slip length

k to the velocity decay length, d. Even for a small value ofk c 1 nm, the

slippage-induced correction to the frequency shift, Dfsl, will be of the order

of 6.5 Hz for the fundamental frequency of f0 = 5 MHz. This value far

exceeds the resolution of the QCM, but it is difficult to separate it from the

overall QCM signal.

There have been attempts [53] to estimate the slip length at the solid/

liquid interface on the basis of QCM experiments for adsorbed liquid

layers. The slip length can be expressed in terms of the coefficient of sliding

friction, v, at the interface

k gv

28

Using the sliding friction coefficient v = 3 g/cm

2

s, which is obtained for amonolayer of water on Ag [49] and on Au [69], a surprisingly high slip

length of k = 6 104 nm is obtained. Using this value for the interface



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between Au and bulk water, Eq. (26) yields for f0= 5 MHz a value ofDfc7103 Hz, which turns out to be smaller than that observed experimen-tally by a factor of 105. This inconsistency is most likely caused by a

roughness of the electrode surface that reduces the effective slip length.

Another reason could be the difference between friction at the solid/

adsorbed layer and the solid/liquid interfaces. For example, a decrease in

the slip length with increasing film thickness has been observed recently in

QCM studies of Kr films on gold electrodes [55].

Recent molecular dynamics simulations [65,70] demonstrated that

the slip length is determined by the ratio of characteristic energies of liquid-

substrate, els, and liquid-liquid, ell, interactions, k = f(els/ell). The slip

length is negligible for els/ell z 1 and grows with the decrease of theparameter els/ell. The slip length k may be as large as 15 diameters of liquid

molecules for els/ellc 0.5. It should also be noted that, for a given value of

els/ell, the slip length is minimal when substrate and liquid molecules are of

the same size and increases with the increase of incommensurability of the

sizes. For smaller coupling between the liquid and the substrate or

incommensurability of their sizes, the spatial corrugation in the interfacial

energy is weaker and interfacial slip can develop.

The latter conditions are satisfied for partially wetting liquid/solid

interfaces. Wetting is characterized by a contact angle, which can be esti-

mated as [68]

cos h

1

2

qs

q

els

ell 29

where qs and q are the density of the solid and the liquid, respectively.

Thus, the contact angle may be interpreted as a measure of the strength

of interaction between the liquid and the solid, els. One expects a large

value of the slip length for a nonwetting situation (cos(h) ! 1), when elsbecomes much smaller than ell. This conclusion is in agreement with several

experimental observations [62,71] reporting large slip lengths for partially

wetting liquids.

The authors of Refs. 14,72,73 showed that surface treatments

affecting liquid contact angle influence the response of quartz crystal

resonator: resonant frequency changes caused by liquid loading were

consistently smaller for surfaces having large liquid contact angles. These

results were interpreted as arising from the onset of slippage at the solid/liquid interface: the solid-liquid interaction becomes sufficiently weak on a

hydrophobic surface, and shear displacement becomes discontinuous at

Tsionsky et al.22


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the interface. However, this interpretation was called into question by aseries of experiments in which the effect of a hydrophobic monolayer was

examined on devices with various surface roughness [12].

Correlating the wetting properties with the response of the QCM in

contact with liquids seems to be a promising area for future research.

Unfortunately, studies of wetting behavior require ex situ measurements of

the contact angle, which change drastically the properties of the electro-

chemical system at the electrode/adsorbed layer/electrolyte interfaces.

b. Slippage at the Adsorbate/Electrolyte Interface

Slippage is very sensitive to the molecular structure of the interface, as we

have already discussed above. Thus, adsorption can strongly influence thisphenomenon. In order to describe the effect of adsorption, let it be assumed

that the adsorbed layer is rigidly attached to the surface and slippageoccurs at the adsorbate/liquid interface (see Fig. 4). Then the equation of

motion of the adsorbed layer can be written as [74]

ixDmava x lqdu z

dz v va x v1 x at z d 30

where va(x) is the velocity of the adsorbed layer and Dma is its two-

dimensional density, while vl(x) is the velocity of the liquid at the interface.

The first term on the right-hand side of Eq. (30) describes the driving force

acting on the adsorbed layer from the quartz crystal, while the second term

accounts for the friction at the adsorbate/liquid interface.

The velocity fields in the crystal and the liquid are given by thesolutions of the wave equation [Eq. (4)] and the linearized Navier-Stokes

equation [Eq. (18)], respectively. The solution of Eqs. (4), (18), and (30)

with the boundary conditions for shear stresses and velocities leads to the

following equation for the shift of the resonant frequency, Df, and the

change of the width of the resonance, G:

Df 2f20Dma

qqlq1=2 f

3=20 qg1=2

pqqlq1=21

1 a 2a2

" #31

G 2f3=20 qg1=2

pqqlq1=21 2a

1 a2 a2

" #32

Writing Eqs. (31) and (32), we introduced a dimensionless parameter a =

g/ vd = k/d , which is the ratio of the slip length, k = g/v, and the velocity



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decay length in the liquid, d. Equations (31) and (32) include both theinterfacial (adsorption) and the bulk solution contributions to the response

of the QCM, given by Eqs. (20) and (21). The latter remains constant in

adsorption studies and can be subtracted from the overall change given by

Eqs. (31) and (32). As a result, the shift of the resonant frequency and the

change of the width due to adsorption, which are measured experimentally,

are given by the equations:

Df DfluDfm Dfsl 2f20Dma

qqlq1=2 f

3=20 qg1=2pqqlq1=2

aa 11 a2 a2

" #

33G Gl f

3=20 qg1=2pqqlq1=2

4a2

1 a2 a2

" #34

Equation (33) shows that there are two different contributions to the

frequency shift,Dfm andDfsl, which originate from (1) a change of the mass

of the adsorbed layer rigidly coupled to the surface [first term on the rhs of

Eq. (33)], and (2) partial decoupling between the quartz crystal oscillations

and the solution, caused by slippage at the adsorbate/liquid interface

[second term on the rhs of Eq. (33)]. It should be stressed here that, in

contrast to adsorption from the gas phase, electrosorption can result ineither a decrease or an increase of the resonant frequency, depending on its

effect on the mass of the layer rigidly coupled to the surface and on changeof the coefficient of sliding friction, which determines the slip length,

according to Eq. (28).

Consider the effect of adsorption on the parameters Dma and v. The

layer adsorbed at the electrode/electrolyte interface contains two types of

molecules: adsorbate and solvent. In the framework of mean field approx-

imation, the effective interaction between the liquid and the adsorbed layer

can be characterized by the energy elscela&a/&m + ell(1&a/&m), where elais the characteristic energy of the adsorbate/liquid interaction and&m is the

maximum surface excess of the adsorbate. As a result, the slip length at the

adsorbed layerliquid interface can be expressed as

k fela=ell&a=&m 1 &a=&mcfela=ell&a=&m 35showing an increase of k with &a for ela/ell < 1. Equation (35) is the

interpolation formula that describes correctly the behavior of k for small

Tsionsky et al.24


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&a/&m and for &a/&m =1. We note that when the liquid and adsorbatemolecules are of significantly different size, the incommensurability be-

tween the structures of the adsorbed layer and liquid grows with &a, which

may lead to an additional enhancement of the slip length. What is

important here is a relation between scales of corrugations of the poten-

tial energy in the solvent and the adsorbed layer, rather than their physical

sizes of solvent and adsorbed molecules.

The foregoing discussion shows that for ela/ell< 1 the parameter a =

k/d in Eqs. (31) and (32), characterizing the effect of slippage on the

response of the QCM, increases with &a. For instance, for ela/elc 0.5, it

may reach values as high as a c 102 for &a c &m. Correspondingly, the

adsorption-induced slippage leads to a positive frequency shift, whichgrows with &a. This contribution can be larger than the effect of added

weight. As a result, the overall frequency shift due to electrosorption can

be positive and increases with &a [74]. It should be noted that for small

values of the parameter a, the effect of slippage on the resonance frequency

shift is much larger than its effect on the width of the resonance [see Eqs.

(33) and (34)]. Also, slippage will always cause a decrease in the width of

the resonance. Thus, if a positive shift of frequency with adsorption is to be

associated with enhanced slippage, it should also be exhibited as a

reduction of the width of the resonance, although the latter may be hard

to detect experimentally.

Above we discussed the situation where the adsorbed layer is rigidly

attached to the oscillating crystal surface, and there is finite slippage at the

adsorbate/liquid interface. An alternative model based on the assumption

that slippage occurs at the crystal/adsorbed layer interface and nonslip

boundary conditions apply to the adsorbate/liquid interface can also be

considered. For a small slip length, E


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slip length would be equivalent to a decrease of viscosity. Unfortunately,there is at present no suitable theory to describe the effect of the excess

surface charge density (or the corresponding high electrostatic field) on the

viscosity of the electrolyte in the double layer. The derivation of such a

model is complicated by the fact that electro-neutrality does not exist on

the solution side of the interface (except at the potential of zero charge),

although the electrostatic energy is reduced by interaction with the image

charges on the metal side of the interface.

D. Quartz Crystals with Rough Surfaces

1. Quartz Crystals with Rough Surfaces Operating in LiquidsWhen the surface of quartz crystal resonator is rough, the liquid motion

generated by the oscillating surface becomes much more complicated than

for the smooth surface. A variety of additional mechanisms of coupling

between the acoustic waves in the solid and the motion in the liquid can

arise. These may include generation of nonlaminar motion, the conversion

of in-plane surface motion to motion normal to the surface, and trapping

of liquid by cavities and pores. It has been experimentally demonstrated

[12,15,7579] that the roughness-induced response of the QCM includes

both the inertial and viscous contributions. Measurements of the complex

shear mechanical impedance [12] were used to analyze different contribu-

tions to the roughness-induced response of the quartz resonator and to

correlate the experimental results with the surface roughness of the quartz

resonator. Nevertheless, this subject is poorly developed, and the inter-

pretation of experimental results can often be ambiguous.

The dependence of the QCM response on the morphology of the

interface is determined by the relation between the characteristic sizes

of roughness and the length scales of the shear modes in the liquid and

the quartz resonator. The length scales in the liquid (the velocity decay

length, d) and in the crystal (wave length of the shear-mode oscillations,

kq) are defined by the Navier-Stokes equation and by the wave equation

for elastic displacement, respectively. For typical frequencies used in

QCM experiments, f0=510 MHz and the lengths d = (g/pf0q)1/2 and

kq = (lq/qq)1/2f0

1 are of the order 0.1770.25 Am and 0.030.1 cm, re-

spectively.

The surface profile may be specified by a single valued function z =n(R) of the lateral coordinates R that defines a local height of the surface

with respect to a reference plane (z = 0). The latter is chosen so that the

Tsionsky et al.26


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average value ofn(R) will equal zero. Surfaces used in QCM experimentsmay have various scales of roughness. In order to clarify this point, let us

consider the two limiting cases: slight and strong roughness structures,

which are schematically shown in Fig. 5. For the slight roughness (Fig. 5a)

the amplitude of deviation from the reference plane z = 0 is much less

than the lateral characteristic length. In the case of strong roughness (Fig.

5b), the amplitude and period of repetitions are of the same order of

magnitude.

In order to stress the multiscale nature of roughness, the profile

function can be written as the sum of the functions that characterize the

profile of the specific scale i:

nR X

i

niR 36

For the calculation of the response of the QCM, the height-height pair

correlation function is needed [80]. When rough structures having different

FIG. 5. Schematic representation of a slight (a) and a strong (b) roughness. Theprofile of slight roughness is described by the function z =n(R). L is the effective

thickness of the porous film that represents strong roughness. (From Ref. 24.)



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scales do not correlate the total correlation function can be written in theform

< nRVnRVR >X

i

< niRVniRVR > 37

where is the correlation function for the scale iand means averaging over the lateral coordinates.Usually one assumes that the correlation function has a Gaussian form = h2i exp(-|R|2/li2), where hi is theroot mean square height of the roughness and li is the lateral correlation

length, which represents the lateral scale. Thus, the morphology of the

rough surface can be characterized by a set of lengths {hi, li}.

It is impossible at the present time to provide a unified description of

the response of the QCM for nonuniform solid/liquid interfaces with

arbitrary geometrical structure. Below we summarize results obtained

for the limiting cases of slight and strong roughness.

a. Slight Roughness

For slightly rough surfaces, the problem was solved in the framework of

perturbation theory with respect to the parameters |jn(R)|


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integral parameter, the roughness factor, R, which is the ratio between thetrue and the apparent (geometrical) surface area. For slight roughness, the

roughness factor is expressed through the correlation function [81] as

R 1 h2

2

ZdK

2p2 gKK2 41

For the Gaussian random roughness g(K) = pl2 exp(l2K2/4) and Eq. (41)yields R =1+2h2/l2.

It should be noted that the roughness factor, R, relevant to the

operation of the EQCM is not the same as the roughness factor commonly

referred to in interfacial electrochemistry, because of the difference in

corresponding length scales. The EQCM roughness factor is mostly

determined by the roughness on the scale of the velocity decay length in

the liquid, d, which assumes values of hundreds of nm, depending on the

frequency of the crystal and the viscosity and density of the liquid. The

interfacial roughness factor is related to charge transfer at the interface

and the double layer structure, and therefore its characteristic scale is

about 1 nm.

The first terms in braces in Eqs. (38) and (39) define the shift and the

broadening of the resonance at the interface between an ideally smooth

crystal and the liquid [11]. The surface roughness leads to an additional

decrease of the resonant frequency and a broadening of the width of the

resonance, expressed by the second terms in this equation.

The particular form of the scaling functions F(l/d) and A(l/d) isdetermined by the morphology of the surface. However, the asymptotic

behavior of these functions for l/d >> 1 and l/d 1 42Fl=d l=db1 b2d=l at l=d > 1 44Al=d l=d2c2 at l=d > 1 the roughness-induced frequencyshift includes a term that does not depend on the viscosity of the liquid,



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the first term in Eq. (42) and Eq. (38). It reflects the effect of the non-uniform pressure distribution, which is developed in the liquid under the

influence of a rough oscillating surface [80]. The corresponding contri-

bution has the form of the Sauerbrey equation. This effect does not exist

for smooth interfaces. The second term in Eq. (42) and Eq. (44) describes

a viscous contribution to the QCM response. Its contribution to Df has

the form of the QCM response at a smooth liquid/solid interface, but

includes an additional factor R that is a roughness factor of the surface.

The latter is a consequence of the fact that for l/d >> 1 the liquid seesthe interface as being locally flat, but with R time its apparent surface

area.

Results obtained in Refs. 80, 81 show that the influence of slightsurface roughness on the frequency shift cannot be explained in terms of

the mass of liquid trapped by surface cavities, as proposed in Refs. 76,

77. This statement can be illustrated by consideration of the sinusoidal

roughness profile. The mass of the liquid trapped by sinusoidal grooves

does not depend on the slope of the roughness, h/l, and is equal to S-h/p,

where Sis the apparent area of the crystal. However, Eq. (38) demonstrates

that the roughness-induced frequency shift increases with increasing slope.

Equation (39) and the asymptotic behavior of the scaling functions

show that in the regions where l/d >> 1 and l/d l2pqf0, the roughness-induced frequency

shift approaches a constant value and the roughness-induced width tends

to zero.

The results obtained make it possible to estimate the effect of

roughness on the response of the QCM if the surface profiles function

n(R) can be found from independent measurements.

b. Strong Roughness

Perturbation theory cannot be applied to describe the effect of the strong

roughness. An approach based on Brinkmans equation has been used

instead to describe the hydrodynamics in the interfacial region [82]. The

flow of a liquid through a nonuniform surface layer has been treated as the

flow of a liquid through a porous medium [8385]. The morphology of

the interfacial layer of thickness, L, has been characterized by a local

permeability, nH, that depends on the effective porosity of the layer, /. Anumber of equations for the permeability have been suggested. For

instance, the empirical Kozeny-Carman equation [83] yields a relationship

Tsionsky et al.30


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between n2H and the effective porosity n2Hfr2/3= 1 / 2 , where r is the

characteristic size of inhomogeneities.

The flow of liquid through the interfacial layer can be described by

the following equation [82]:

ixqvz; x g d2

dz2vz;x gn2H vq0 vz;x 47

where vq0 is the amplitude of the quartz surface velocity and v(z,t) = v(z,x)

exp(ixt) is the velocity of the liquid in the layer. In this equation the effect

of the solid phase on the flow of liquid is given by the resistive force, which

has a Darcy-like form, gn2

H vq0 vz;x . In the case of high effectiveporosity, the resistive force is small and Eq. (47) is reduced to the Navier-Stokes equation, describing the motion of the liquid in contact with a

smooth quartz surface. For a given viscosity, the resistive force increases

with decreasing effective porosity and strongly influences the liquid

motion. At very low effective porosity, all the liquid located in the layer

is trapped by the roughness and moves with a velocity equal to the velocity

of the crystal surface itself.

Brinkmans equation represents a variant of the effective medium

approximation, which does not describe explicitly the generation of non-

laminar liquid motion and conversion of the in-plane surface motion into

the normal-to-interface liquid motion. These effects result in additional

channels of energy dissipation, which are effectively included in the model

by introduction of the Darcy-like resistive force.The liquid-induced frequency shift and the width of the resonance

have the following form [82]:

Df 2f20 q

lqqq1=2Re

&1

q0 L

n2Hq21

1W

1

n2Hq21

2q0q1

coshq1L 1 sinhq1L !'

48

G 4f20 q

lqqq1=2Im

&1

q0 L

n2Hq21

1W

1

n2Hq21

2q0q1

coshq1L 1 sinhq1L !'

49



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where q0= (i2p f0q/g)1/2, q21 = q

20 +n

2H , and W = q1 cosh( q1L)+q0

sinh( q1L). The first terms on the right-hand sides of Eqs. (48) and (49)

describe the response of the QCM for the smooth quartz crystal/liquid

interface [11]. The additional terms present the shift and the width of the

QCM response caused by the interaction of the liquid with a non-uniform

interfacial layer.

When the permeability length scale is the shortest length of the

problem,nH


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non. In this manner, liquid flow at a rough surface has been simulated as aflow at a smooth surface with an effective slip length.

Application of this approach to the QCM problem yields the fol-

lowing equation for the effective slip length:

keff k 1 h0K02

23 4kk01 2kk0

!( ) k0h

20

2

2 3kk01 kk01 2kk0 !

52Equation (52) was derived for a sinusoidal profile of roughness, z(x) =

d+ h0 sin(k0x), with an amplitude h0 and a period of 2p/k0, assuming that

the decay length, d, is the largest characteristic length of the problem,d/k>>1 and dk0 >>1. Beyond these conditions the effective slip length is a

complex function. Equation (52) shows that roughness diminishes the

influence of slippage on the QCM response, namely the effective slip length

becomes smaller than the corresponding length for the smooth interface.

At rough interfaces, the effective slip length decreases with an increase of

the amplitude of the surface corrugation and with a decrease of its period.

It should be noted that an effective slip length is not an intrinsic

property of the surface. Its value depends also on the experimental

configuration, for instance, kefffound for the Poiseuille flow between rough

surfaces [86] differs from the corresponding value obtained for QCM

experiments [Eq. (52)].

So far only a few studies [8689] have been devoted to the effect of

roughness on slippage, and this subject requires additional investigation.

III. ELECTRICAL DOUBLE LAYER/ELECTROSTATIC

ADSORPTION

A. Introduction

We shall restrict our consideration here to the simplest electrochemical

case: the electrical double layer, which is not complicated by charge trans-

fer or by specific adsorption. At first glance it would seem that, if there is no

change in massand nothing happens in the bulk ofthe solution in which the

electrode is immersed, the EQCM response should be zero. However,

essentially all measurements show that in the double-layer region the

frequency of the EQCM depends on potential. The effect is rathersmalla few Hz for crystals with fundamental frequencies of 510 MHz.

In most cases reported in the literature [9096], experiments were

performed employing cyclic voltammetry, with the potential extending to



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the region of oxide formation. All data obtained in this manner exhibitsome degree of hysteresis in the double layer region, which increases with

sweep rate. It also increases when the anodic limit of potential is extended,

i.e., when the potential is swept deeper into the oxide formation region.

There is good reason to believe that this hysteresis is due to a memory

effect of the interface, related to residual adsorbed oxygen or to some

traces of dissolved gold remaining near the surface. In only a few papers

[61,9799] has the response of the EQCM to changes in potential in the

double-layer region been studied under experimental conditions, which

excluded hysteresis. In these studies performed on gold andsilver, thelimits

of potential during cycling were restricted to the double-layer region. Both

metals have sufficiently extensive potential region where only electrostaticadsorption can take place. Moreover, it is well known that the surfaces of

these metals do not undergo any changes in their morphology in the course

of cycling in this restrictedpotential region. Measurements were conducted

in electrolytes that are not specifically adsorbed to a significant extent.

B. Some Typical Results

All attempts to find rigorous quantitative correlation between data

obtained in different laboratories have failed. This is not surprising,considering that the effects are rather small and the surfaces studied have

different histories (technique of producing the EQCM, electrode pretreat-

ment, etc.) and different morphologies. The surfaces employed in EQCM

studies are always polycrystalline, even though they often have a preferredcrystal orientation. Ne

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