Scanning Electrochemical Microscopy

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Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Current printing (last digit): 10 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITED STATES OF AMERICA

PREFACE

A little more than ten years have elapsed since publication of the first papers describing the fundamentals of scanning electrochemical microscopy (SECM). During this decade, the field of SECM has evolved substantially. The technique has been used in a variety of ways, for example, as an electrochemical tool to study heterogeneous and homogeneous reactions, for high-resolution imaging of the chemical reactivity and topography of various interfaces, and for microfabrication. Quantitative theoretical models have been developed for different modes of the SECM operation. The first commercial SECM instrument was introduced in 1999. The SECM technique is now used by a number of research groups in many different countries. We think the time has come to publish the first monograph, providing comprehensive reviews of different aspects of SECM. The first five chapters of this book contain experimental and theoretical background, which is essential for everyone working in this field: principles of SECM measurements (Chapter 1), instrumentation (Chapter 2), preparation of SECM ultramicroelectrodes (Chapter 3), imaging methodologies (Chapter 4), and theory (Chapter 5). Other chapters are dedicated to specific applications and are self-contained. Although some knowledge of electrochemistry and physical chemistry is assumed, the key ideas are discussed at the level suitable for beginning graduate students. Through the addition of submicrometer-scale spatial resolution, SECM greatly increases the capacity of electrochemical techniques to characterize interfaces and measure local kinetics. In this way, it has proved useful for a broad range of interdisciplinary research. Various applications of SECM are discussed in this book, from studies of biological systems, to sensors, to probing reactions at the liquid/liquid interface. Although we did not intend to present even a brief survey of those diverse areas of research, each chapter iii

iv

Preface

provides sufficient details to allow a specialist to evaluate the applicability of the SECM methods for solving a specific problem. We hope it will be useful to all interested in learning about this technique and applying it. We would like to thank our students, co-workers, and colleagues who have done so much to develop SECM. The future for this technique, which is unique among scanning probe methods in its quantitative rigor and its ability to study with ease samples in liquid environments, continues to be a bright one. Allen J. Bard Michael V. Mirkin

CONTENTS

Preface

iii

Contributors

ix

1

INTRODUCTION AND PRINCIPLES Allen J. Bard I. II. III.

2

Background of Scanning Electrochemical Microscopy Principles of SECM Applications of SECM References

1

1 2 9 15

INSTRUMENTATION David O. Wipf

17

I. II. III. IV. V. VI. VII.

17 18 42 44 44 53 59 66 71

Introduction Overview of the SECM Apparatus Commercial SECM Instrument Implementation Tip Position Modulation Instrumentation Constant-Current Mode Instrumentation Experimental Difficulties in Data Acquisition Accessory Equipment for SECM Appendix; Suppliers References

v

vi 3

Contents THE PREPARATION OF TIPS FOR SCANNING ELECTROCHEMICAL MICROSCOPY Fu-Ren F. Fan and Christophe Demaille I. II. III.

4

5

6

Introduction Preparation Techniques Nondisk Tips and Tip Shape Characterization References

75

75 75 104 107

SECM IMAGING Fu-Ren F. Fan

111

I. II. III. IV. V.

111 111 115 124 139 141

Introduction Principle and Methodology of SEM Imaging Images in Solutions Images in Humid Air Conclusions and Future Projections References

THEORY Michael V. Mirkin

145

I. II. III. IV. V.

Introduction Feedback Mode of SECM Operation Generation/Collection (G/C) Mode of SECM Operation SECM of More Complicated Chemical Systems Numerical Solution of SECM Diffusion Problems Using PDEase2 Program Package List of Symbols References

145 145 165 170

HETEROGENEOUS ELECTRON TRANSFER REACTIONS Kai Borgwarth and Ju¨rgen Heinze

201

I. II. III. IV. V.

201 203 217 234 237 238

Introduction Principles Studies of Heterogeneous Electron Transfer Applications Conclusion and Outlook References

182 193 198

Contents 7

KINETICS OF HOMOGENEOUS REACTIONS COUPLED TO HETEROGENEOUS ELECTRON TRANSFER Patrick R. Unwin I. II. III. IV.

8

9

Introduction ECi Processes EC2i Processes ECE/DISP Processes References

241

241 244 270 283 297

CHARGE-TRANSFER AT THE LIQUID/LIQUID INTERFACE Michael V. Mirkin and Michael Tsionsky

299

I. II. III. IV.

299 301 325 336 339

Introduction Electron Transfer Ion Transfer at the ITIES Processes with Coupled Homogeneous Reactions References

IMAGING MOLECULAR TRANSPORT ACROSS MEMBRANES Bradley D. Bath, Henry S. White, and Erik R. Scott I. II. III. IV.

10

vii

Introduction Principles of Imaging Porous Membranes Applications Future Directions References

343

343 346 365 392 394

POTENTIOMETRIC PROBES Guy Denuault, Ge´za Nagy, and Kla´ra To´th

397

I. II. III. IV.

397 415 417

V. VI.

Introduction Basic Theory Properties and Behavior of Ion-Selective Probes Potentiometric Measurements in Scanning Probe Microscopies Other than SECM Potentiometric Measurements in SECM Conclusion References

422 423 441 442

viii 11

Contents BIOLOGICAL SYSTEMS Benjamin R. Horrocks and Gunther Wittstock I. II. III. IV.

12

521

I. II.

521

Introduction Measurement of Adsorption/Desorption Kinetics and Surface Diffusion Rates Dissolution Kinetics of Ionic Single Crystals Corrosion Studies Conclusions References

MICRO- AND NANOPATTERNING USING THE SCANNING ELECTROCHEMICAL MICROSCOPE Daniel Mandler I. II. III.

14

445 463 504 510 512

PROBING REACTIONS AT SOLID/LIQUID INTERFACES Julie V. Macpherson and Patrick R. Unwin

III. IV. V.

13

Approaches to Imaging Biological and Biochemical Systems Selected Applications Conclusion and Outlook Abbreviations, Acronyms, and Symbols References

445

Patterning by the Direct Mode of the SECM Patterning by the Feedback Mode of the SECM Perspective Approaches References

523 536 573 588 590

593

594 603 623 625

CONCLUSIONS AND PROSPECTS Allen J. Bard

629

I. II. III.

629 634 636 637

Index

Combining SECM with Other Techniques Novel Interfaces Instrumentation Improvements References

639

CONTRIBUTORS

ALLEN J. BARD

The University of Texas at Austin, Austin, Texas

BRADLEY D. BATH

ALZA Corporation, Mountain View, California

KAI BORGWARTH Institute for Physical Chemistry, Albert Ludwig University of Freiburg, Freiburg, Germany CHRISTOPHE DEMAILLE Texas GUY DENUAULT FU-REN F. FAN ¨ RGEN HEINZE JU Germany

The University of Texas at Austin, Austin,

University of Southampton, Southampton, England The University of Texas at Austin, Austin, Texas Albert Ludwig University of Freiburg, Freiburg,

BENJAMIN R. HORROCKS University of Newcastle upon Tyne, Newcastle upon Tyne, United Kingdom JULIE V. MACPHERSON

University of Warwick, Coventry, England

DANIEL MANDLER The Hebrew University of Jerusalem, Jerusalem, Israel MICHAEL V. MIRKIN Flushing, New York

Queens College–City University of New York, ix

x GE´ZA NAGY

Contributors Janus Pannonius University, Pe´cs, Hungary

ERIK R. SCOTT

Medtronic Corporation, Minneapolis, Minnesota

´ RA TO ´ TH Institute of General and Analytical Chemistry, Technical KLA University of Budapest, Budapest, Hungary MICHAEL TSIONSKY PATRICK R. UNWIN HENRY S. WHITE DAVID O. WIPF Mississippi

Gaithersburg, Maryland University of Warwick, Coventry, England

University of Utah, Salt Lake City, Utah Mississippi State University, Mississippi State,

GUNTHER WITTSTOCK Wilhelm-Ostwald-Institute of Physical and Theoretical Chemistry, University of Leipzig, Leipzig, Germany

1 INTRODUCTION AND PRINCIPLES Allen J. Bard The University of Texas at Austin Austin, Texas

I. BACKGROUND OF SCANNING ELECTROCHEMICAL MICROSCOPY This volume is devoted to a complete and up-to-date treatment of scanning electrochemical microscopy (SECM). In this introductory chapter, we cover the historical background of the technique, the basic principles of SECM, and an overview of some of its applications (covered in more depth in later chapters). A number of reviews of this field have also been published (1–6). SECM involves the measurement of the current through an ultramicroelectrode (UME) (an electrode with a radius, a, of the order of a few nm to 25 ␮m) when it is held or moved in a solution in the vicinity of a substrate. Substrates, which can be solid surfaces of different types (e.g., glass, metal, polymer, biological material) or liquids (e.g., mercury, immiscible oil), perturb the electrochemical response of the tip, and this perturbation provides information about the nature and properties of the substrate. The development of SECM depended on previous work on the use of ultramicroelectrodes in electrochemistry and the application of piezoelectric elements to position a tip, as in scanning tunneling microscopy (STM). Certain aspects of SECM behavior also have analogies in electrochemical thinlayer cells and arrays of interdigitated electrodes. The movement of the tip is usually carried out by drivers based on piezoelectric elements, similar to those used in STM, as described in Chapter 2. Typically, inchworm drivers (Burleigh Instruments, Fishers, NY) are used, since they can move larger distances than simple piezoelectric tube scanners. However, where higher resolution is needed, piezoelectric pushers can be added, so that the inchworms provide coarse drives and the pushers nmresolution drives. Generally the direction normal to the substrate is taken as the z direction, while x and y are those in the plane of the substrate. 1

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There are several modes of operation of the SECM. In the tip generation–substrate collection (TG/SC) mode, the tip is used to generate a reactant that is detected at a substrate electrode. For example, the reaction O ⫹ ne → R occurs at the tip, and the reverse reaction occurs at the substrate. This mode of operation is similar to that at the rotating ring-disk electrode (7). Similar behavior is observed for a pair of side-by-side microband electrodes (8,9) and in thin-layer cells (10). In the SECM, TG/SC is usually used in studies of homogeneous chemical reactions, where the reaction of species R as it transits between tip and substrate causes a decrease in the substrate current (see Chapter 7). An alternative mode, where the substrate is the generator and tip the collector (SG/TC mode), can also be employed and is used in studies of reactions at a substrate surface (Chapters 6, 9, 11, and 12). The SG/TC mode was first used to study concentration profiles near an electrode surface without scanning and imaging (11–13). The most frequent mode of operation of the SECM is the feedback mode, where only the tip current is monitored. As discussed in the next section, the tip current is perturbed by the presence of a substrate at close proximity by blockage of the diffusion of solution species to the tip (negative feedback) and by regeneration of O at the substrate (positive feedback). This effect allows investigation of both electrically insulating and conducting surfaces and makes possible imaging of surfaces and the reactions that occur there. This mode of operation with surface imaging was first described, along with the apparatus and theory, in a series of papers in 1989 (14–16).

II. A.

PRINCIPLES OF SECM Ultramicroelectrodes

An understanding of the operation of the SECM and an appreciation of the quantitative aspects of measurements with this instrument depends upon an understanding of electrochemistry at small electrodes. The behavior of ultramicroelectrodes in bulk solution (far from a substrate) has been the subject of a number of reviews (17–21). A simplified experimental setup for an electrochemical experiment is shown in Figure 1. The solution contains a species, O, at a concentration, c, and usually contains supporting electrolyte to decrease the solution resistance and insure that transport of O to the electrode occurs predominantly by diffusion. The electrochemical cell also contains an auxiliary electrode that completes the circuit via the power supply. As the power supply voltage is increased, a reduction reaction, O ⫹ ne → R, occurs at the tip, resulting in a current flow. An oxidation reaction will occur at the auxiliary electrode, but this reaction is usually not of interest in SECM, since this electrode is placed sufficiently far from the UME

Introduction and Principles

FIG. 1

3

Schematic diagram of a cell for ultramicroelectrode voltammetry.

that products formed at the auxiliary electrode do not reach the tip during the experiment. The potential of the tip electrode is monitored against a stable reference electrode, such as a silver/silver chloride electrode. A plot of the current flowing as a function of the potential of the UME is called a voltammogram; a typical one is shown in Figure 2. As shown, an S-shaped

FIG. 2

Typical voltammogram for an ultramicroelectrde.

4

Bard

curve is produced. The current eventually limits to a value that is completely controlled by the rate of mass transfer by diffusion of O from the bulk solution to the electrode surface, where the electrochemical reaction has decreased its concentration to essentially zero. For a conductive disk of radius a in an insulating sheath, this steady-state diffusion-controlled current when the tip is far from a surface is given by: iT,⬁ = 4nFDca

(1)

where D is the diffusion coefficient of species O, and F is the Faraday. The current at electrodes with other shapes, e.g., hemispheres or cones, can be expressed in a similar way, as discussed in Chapter 3, but almost all SECM experiments are carried out with disk-shaped electrodes, because they show the best sensitivity. The current is also relatively independent of the radius of the insulating sheath, rg , often expressed in the SECM literature as RG = rg /a. Moreover, because the flux of O to a small disk by diffusion (⬃Dc/a) is quite large, the current is relatively immune to convective effects like stirring in the solution. The current at a small disk also reaches steady state in a relatively short time (⬃a2/D). For example, a 10 ␮m radius disk will attain steady state in a fraction of a second. These characteristics imply that an ultramicroelectrode used as a scanning tip and moved in a solution can be treated as a steady-state system. Finally, because of the small currents that characterize most experiments with ultramicroelectrode tips, generally pA to nA, resistive drops in the solution during passage of current are generally negligible. B.

Feedback Mode

The general principles of the feedback mode are shown in Figure 3. As shown in Eq. (1), the current, iT,⬁, is measured at the ultramicroelectrode tip when it is far from any surface (A), the subscript, ⬁, implying this long distance. In fact, as we shall see, this distance only has to be a few tip diameters. The current under these conditions is driven by the hemispherical flux of species O from the bulk solution to the tip (Fig. 3A). When the tip is brought near an electrically insulating substrate, like a piece of glass or plastic (Fig. 3C), the substrate blocks some of the diffusion of O to the tip and the current will decrease compared to iT,⬁. The closer the tip gets to the substrate, the smaller iT becomes. At the limit when the distance between tip and substrate, d, approaches zero, iT also approaches zero. This decrease in current with distance is called negative feedback. When the tip is brought near an electrically conductive substrate, like a platinum electrode, while there is still blockage of diffusion of O to the tip by the substrate, there is also the oxidation of the product R back to O. This O generated at the

Introduction and Principles

5

FIG. 3 Basic principles of scanning electrochemical microscopy (SECM): (A) far from the substrate, diffusion leads to a steady-state current, iT,⬁; (B) near a conductive substrate, feedback diffusion leads to iT > iT,⬁; (C) near an insulating substrate, hindered diffusion leads to iT < iT,⬁. (Reprinted with permission from A. J. Bard, G. Denuault, C. Lee, D. Mandler, and D. O. Wipf, Acc. Chem. Res. 23, 357 (1990). Copyright 1990 American Chemical Society.)

6

Bard

substrate diffuses to the tip and causes an increase in the flux of O compared with iT,⬁. Thus with a conductive substrate iT > iT,⬁. In the limit as d approaches zero, the tip will move into a regime where electron tunneling can occur and the tip current will get very large. This increase of current with distance is called positive feedback. A plot of iT versus d, as a tip is moved in the z direction, is called an approach curve. A quantitative description of approach curves can be obtained by solving the diffusion equations for the situation of a disk electrode and a planar substrate (16), as discussed in Chapter 5. Typical approach curves for a conductive substrate (essentially infinite rate of regeneration of O from R) and an insulating substrate (zero rate of regeneration of O) are shown in Figure 4. These curves are given in dimensionless form by plotting IT = iT / iT,⬁ (the tip current normalized by the current far from substrate) versus L = d/a (the tip-substrate separation normalized by the tip radius). Since this plot involves only dimensionless variables, it does not depend upon the concentration or diffusion coefficient of O. From these curves one can readily find d from the measured IT and a knowledge of a. The approach curves for an insulator actually depend upon rg, since the sheath around the conducting portion of the electrode also blocks diffusion, but this effect is not usually important with most practical tips. If the rate constant for electron transfer at the substrate to species O is kb,s , the limiting curves repesent kb,s → 0 (insulator) and kb,s → ⬁ (conductor). The approach curves for intermediate values of kb,s can be found (Chapter 5) (Fig. 5). These are very useful in finding the rate of heterogeneous charge transfer at an interface (see Chapters 6 and 8). C.

Collection-Generation Modes

As discussed above, there are two modes of this type. In the TG/SC mode, the tip is held at a potential where an electrode reaction occurs and the substrate is held at a different potential where a product of the tip reaction will react and thus be collected. In most cases the substrate is considerably larger than the tip, so that the collection efficiency, given by iS /iT (where iS is the substrate current), is essentially 1 (100%) for a stable tip-generated species, R. If R reacts on transit from tip to substrate, iS /iT becomes smaller, and its change with separation, d, allows determination of the rate constant of the homogeneous reaction (Chapter 7). The alternative mode is the substrate generation–tip collection (SG/TC) mode. In this case the tip probes the reactions that are occurring on a substrate. For example, a scan in the z direction can produce the concentration profile, while a scan over the surface can identify hot spots, where reactions occur at a higher rate.

Introduction and Principles

7

FIG. 4

Diffusion-controlled steady-state tip current as a function of tip-substrate separation. (A) Substrate is a conductor; (B) substrate is an insulator. (From Ref. 2.)

A related method involves the use of the tip reaction to perturb a reaction at a surface; an example of this approach is SECM-induced desorption (SECMID) (22). For example, the adsorption/desorption kinetics of protons on a hydrous metal oxide surface can be studied in an unbuffered solution by bringing the tip near the surface and reducing proton (to hydrogen) at the tip. This causes a local change in pH that results in proton desorption from the surface. The tip current can be used to study the kinetics of proton desorption and diffusion on the surface (Chapter 12).

8

Bard

FIG. 5

Approach curves as a function of the heterogeneous reaction rate constant for electron transfer at the substrate, k, IT = iT /iT,⬁. From top to bottom, k (cm/s) is (a) 1, (b) 0.5, (c) 0.1, (d) 0.025, (e) 0.015, (f) 0.01, (g) 0.005, (h) 0.002, (i) 0.0001. Curve (a) is identical to that for mass transfer control and curve (i) for an insulating substrate.

D.

Transient Methods

Most SECM measurements involve steady-state current measurements. This can be a significant advantage in the measurement of kinetics, even for rapid processes, because factors like double-layer charging and adsorption do not contribute to the observed currents. However, one can also carry out transient measurements, recording iT as a function of time. This can be of use in measurements of homogeneous kinetics (Chapter 7) and for systems that are changing with time. It can also be used to determine the diffusion coefficient, D, of a species without knowledge of the solution concentration or number of electrons transferred in the electrode reaction (23). E.

Fabrication

The SECM can also be used as a tool for modification of surfaces. For example, metals or semiconductors can be etched or metals deposited on a surface by passing the tip close to the surface and carrying out an appropriate electrochemical reaction. Two different modes are possible. In the direct

Introduction and Principles

9

mode, the tip acts as the counterelectrode and the desired electrochemical reaction occurs on the substrate. For example, Cu can be etched from a Cu substrate. Spatial resolution is determined by the current density distribution between tip and substrate. In the feedback mode a reactant is generated at the tip which promotes the reaction on the substrate. For example, Cu can be etched by bromine electrogenerated at the tip. In this case resolution is determined by the lateral (x-y) diffusion of reactant as it diffuses from tip to substrate. Details of fabrication using SECM are covered in Chapter 13. III.

APPLICATIONS OF SECM

The chapters that follow illustrate a wide range of applications of SECM that have appeared. Given below is an overview and some examples that might help put the technique in perspective before the detailed treatments. A.

Imaging

By scanning the tip in the x-y plane and measuring current changes (the constant height mode) (or, less frequently, by maintaining a constant current and measuring the changes in d in a constant current mode), one can obtain topographic images of conducting and insulating substrates (Chapter 4). The resolution of such images is governed by the tip radius, a, and d. However, by working in the thin film of water that condenses on a mica surface in humid air, it is possible to obtain higher resolution with a conical tip that is only slightly immersed in the water film. Of particular interest is the use of SECM to perform ‘‘chemical imaging,’’ observing differences in reaction rates at different locations on the surface. This mode is useful in studying biological materials (e.g., enzyme sites) (Chapter 11) and surfaces that have active and passive sites. B.

Ultramicroelectrode Shape Characterization

It is frequently difficult to determine the actual shape of an ultramicroelectrode by examination using an optical or scanning electron microscope. For example, the conducting portion may be slightly recessed inside the glass mantle, or the shape may be that of a cone protruding from the insulator. Electrodes with radii of the order of 1 ␮m or less are particularly difficult to characterize. Simply determining a voltammogram with the tip in bulk solution is usually not useful in this regard, since almost all ultramicroelectrodes will produce a steady-state wave-shaped voltammogram characteristic of roughly hemispherical diffusion. However, by recording an approach curve, iT versus d, one can frequently identify recessed tips (where iT does not increase at small d when the insulator hits the substrate) or tips with

10

Bard

shapes other than disks, which show different approach behavior (Chapter 5). C.

Heterogeneous Kinetics Measurements

As suggested above, by recording an approach curve or voltammogram with the tip close to a substrate, one can study the rates of electron transfer reactions at electrode surfaces (Chapter 6). Because mass transfer rates at the small tip electrodes are high, measurements of fast reactions without interference of mass transfer are possible. As a rule of thumb, one can measure k⬚ values (cm/s) that are of the order of D/d, where D is the diffusion coefficient (cm2/s). For example, k⬚ for ferrocene oxidation at a Pt electrode in acetonitrile solution was measured at a 1 ␮m radius tip at a d of about 0.1 ␮m yielded a value of 3.7 cm/s (24). The use of small tips and small currents decreases any interference from uncompensated resistance effects. D.

Measurements of Homogeneous Kinetics

Rate constants for homogeneous reactions of tip-generated species as they transit between tip and conducting substrate can be determined from steadystate feedback current or TG/SC experiments or by transient measurements (Chapter 7). Generally rate constants can be measured if the lifetime of the species of interest is of the order of the diffusion time between tip and substrate, d 2/2D. Thus first-order reaction rate constants up to about 105 s⫺1 and second-order reaction rate constants up to about 108 M⫺1 s⫺1 are accessible. E.

Biological Systems

There have been a number of applications of SECM to biological systems (Chapter 11). These include imaging of cells, studies of enzymatic reactions, and oxygen evolution on leaf surfaces. SECM has also been applied in investigations of the transport of species through skin (Chapter 9). Because SECM is capable of monitoring a wide range of chemical species with good specificity and high spatial resolution, it should find wide application in studies of living organisms and isolated tissues and cells. F.

Liquid/Liquid Interfaces

There is considerable interest in ion and electron transfer processes at the interface between two immiscible electrolyte solutions (ITIES), e.g., water and 1,2-dichloroethane. SECM can be used to monitor such processes (Chapter 8). It allows one to separate ion transport from electron transfer

Introduction and Principles

11

and is relatively insensitive to the resistance effects often found with more conventional (four-electrode) electrochemical measurements. G.

Membranes and Thin Films

Different types of films on solid surfaces (e.g., polymers, AgBr) and membranes separating solutions have been examined by SECM (Chapters 6 and 9). SECM is a powerful technique for examining transport through membranes, with the ability to scan the surface to locate positions of different permeability. It has also been used with polymer films, e.g., polyelectrolytes or electronically conductive polymers, to probe the counterion (dopant) flux during redox processes. SECM can be particularly useful in probing film thickness as a film is grown on a surface (25). SECM is unique in its ability to probe inside some thin films and study species and electrochemical processes within the films (26,27). For example, the tip current versus z-displacement curve as a conical tip (30 nm radius, 30 nm height) was moved ˚ thick Nafion from a solution of 40 mM NaClO4 into a nominally 2000 A film containing Os(bpy)2⫹ on a glass/ITO substrate (Fig. 6) (26). The tip 3 was held at 0.80 V versus SCE, where Os(bpy)2⫹ is oxidized to the 3⫹ form 3 at a diffusion controlled rate. The different stages of penetration of the tip into the film, from initial contact to tunneling at the ITO can clearly be seen and the film thickness established. Moreover, with the tip at position c, a voltammogram can be recorded (Fig. 7). From such a voltammogram, one can determine the diffusion coefficient of Os(bpy)2⫹ and information about 3 the kinetics and thermodynamics of the reaction occurring in the film. H.

Surface Reactions

Measurements of the rates of surface reactions on insulator surfaces, such as dissolution, adsorption, and surface diffusion, are possible (Chapter 12). For example, proton adsorption on an oxide surface can be studied using the tip to reduce proton and induce a pH increase near the surface (22). Then, by following the tip current with time, information about proton desorption kinetics is obtained. Studies of corrosion reactions are also possible. Indeed, work has been reported where a tip-generated species has initiated localized corrosion and then SECM feedback imaging has been used to study it (28). In these types of studies, the tip is used both to perturb a surface and then to follow changes with time. I.

Semiconductor Surfaces

SECM has been used to probe heterogeneous electron transfer reaction kinetics on semiconductor electrodes, such as WSe2 (29). In these studies, as

12

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Introduction and Principles

13

in those at the liquid/liquid interface, the use of a separate metal probe electrode is useful in freeing the measured response from resistance effects. It also allows one to examine differences in behavior at different points on a surface. As discussed in Chapter 13 on applications to fabrication, SECM has also been used to etch semiconductor surfaces and study the nature of the etching reactions. J.

Electrochemistry in Small Volumes of Solution

Because of its ability to position an electrode tip with high spatial resolution in three dimensions, SECM can be used to probe electrochemistry in a small volume of liquid (e.g., on a conductive substrate that serves as a counter/ reference electrode). For example, a solution volume of 3–20 ␮L was used to probe the adsorption isotherms on a mineral surface (30). Probing even smaller volumes, e.g., of liquids contained in pores, should be possible. Since electrochemical generation is an ideal method for producing small, controlled amounts of reactants, studies in which one wants to probe chemistry with very limited amounts of sample appear to be a good application. In such studies, means to maintain the sample volume and prevent evaporation, for example, by close control of the humidity or using an overlayer of an immiscible liquid, will be required. K.

Thin Liquid Layers

The SECM has been used to form thin liquid layers and probe electrochemical reactions in them. When the tip is pushed through the interface between


180⬚), the loop will have positive feedback at higher frequencies, leading to oscillation of the piezoelectric device. The loop gain must be low enough to prevent oscillation but still sufficiently large to rapidly move the tip to the desired position. The simple control loop described above is a proportional control because of the linear relationship between the error signal and control voltage. The low loop gain required for stability in proportional control leads to practical problems. Low gain makes the loop respond slowly, slowing scan speeds. Since the error signal is never truly minimized, the tip position also tends to ‘‘hunt’’ for the desired position, wandering above and below the desired minimum in the error signal. A considerable performance improvement is possible by adding integral and differential control elements to the proportional control loop. Applying a correction proportional to the integral the residual error allows a higher proportional loop gain and reduces ‘‘hunting.’’ The differential element applies a correction proportional to the time derivative of the error signal. Thus, large changes in the error signal, produced, for example, from scanning over a sharp edge, can be rapidly corrected. The proportional, integral, and differential elements (i.e., PID) can be made with analog elements, such as operational amplifier circuitry, or by digitizing the signal and using a digital signal processor (DSP) to implement digital equivalents of the PID control (20,43). The PID controller loop must be ‘‘tuned’’ to give optimum performance. The gain of each of the PID elements and the integration time must be adjusted to give good transient response while maintaining stability. In addition, changing the tip diameter or substrate type (conductive or insulating) requires adjusting the tuning parameters. Finding the optimum PID parameters is not trivial and is the subject of much research in control loop opti-

48

Wipf

mization. Commercial STM and AFM systems use analog or digital control loops that can be optimized through software control (43). SECM instrumentation is not yet at that level, and constant-current operation is not offered with the existing commercial SECM instrument. B.

Constant-Current Mode Designs

The tip current can be used to generate an error signal when operating in either the negative or positive feedback mode. Because the tip current has a definite functional dependence on distance, it can easily be incorporated in the feedback loop. Note that the feedback loop must be modified when scanning in the negative or positive feedback mode. (An unfortunate aspect of SECM terminology is the use of positive and negative feedback as descriptions of scanning in the feedback mode over a conducting or insulating surface. Feedback refers, in this context, to the shuttling of the solution mediator between the substrate and tip. One should not assume the use of the word feedback in this context implies use of a constant-current mode— quite the contrary, feedback is usually used to indicate a constant height mode. In addition, negative feedback is used here to indicate a condition of a stable feedback loop. The meaning of ‘‘feedback’’ should be clear from the context.) In negative feedback operation, the tip current must be compared to a reference signal below iT,⬁ and the control loop must move the tip towards the surface when the tip current is larger than the reference current. In positive feedback operation, the tip current is be compared to a reference signal above iT,⬁ and the control loop must move the tip away from the surface when the tip current is larger than the reference current. The similarity of STM and positive feedback SECM imaging allows published designs for STM control loops to be adapted for use in SECM (11,20,44). Using the feedback current in the control loop can lead to tip crashes. Consider the situation where the tip is operating in constant-current positivefeedback mode. If the sample surface has an inactive region in the scan area, a tip crash can occur as the feedback loop interprets the decrease in tip current over the inactive region as an increase in the tip-substrate separation. A crash also occurs when operating in constant-current negative feedback mode if a conducting region is encountered. The addition of some intelligence to the feedback loop will guard against some of these problems. For example, an ‘‘emergency’’ circuit could detect a tip signal above or below iT,⬁ in negative or positive feedback mode, respectively, and shut off the feedback loop during these ‘‘impossible’’ signal levels. Several SECM researchers have suggested alternate methods in providing constant current imaging. They hope to improve the simple approach outlined above by providing an additional source of information about the

Instrumentation

49

tip-substrate distance. The additional information used in all the proven designs is gathered by a tip oscillation. Interestingly, the oscillation range in amplitude from hundreds of micrometers to a few nanometers, indicating the designs are still evolving towards an optimal solution. Three varieties of constant-current mode devices are discussed below. 1. Tip Position Modulation The tip position modulation (TPM) mode of SECM suggests a method for constant-current imaging where the surface type (insulating or conducting) is automatically detected. As described above, TPM uses a small-amplitude vertical modulation of the tip position. A lock-in amplifier detects the modulated tip current, iAC, and generates an output proportional to the rms magnitude and phase of iAC. A low-pass filter removes the ac component of the tip current from the feedback-mode signal, iDC. Comparing iDC to a reference level produces an error signal. The phase of iAC determines the type of surface because of the 180⬚ phase shift of iAC between insulating and conducting surface. For example, a phase range of 30–150⬚ might be allowed for positive feedback signals while a phase range of 210–330⬚ might indicate a negative feedback situation. A dual-window comparator detects the phase signal and sets the loop gain and reference current level as necessary for the type of surface. An important part of the circuit is detection of outof-range or emergency conditions. If the phase signal is not in the proper range or if the tip signal is above or below iT,⬁ in negative or positive feedback mode, respectively, the feedback circuit stops the piezo motion until the phase signal iT,⬁ are both consistent. Using a simple proportional control loop, this circuit was able to image a surface containing a mixture of insulating and conducting phases with a 2 ␮m diameter tip while automatically maintaining the desired tip position without operator input (45). 2. Picking In the ‘‘picking’’ mode, a surface-induced convective current is caused when the tip rapidly approaches from a large tip-substrate separation (8). Borgwarth and coworkers showed that when a tip approaches a surface at a speed of 50 ␮m/s from an initial height of 200 ␮m above that surface, a significant tip current increase occurs as the tip-substrate separation drops below 1 tip radius. Importantly, the increase occurs at both conducting and insulating substrates. In operation, the picking mode instrument repeatedly approaches the surface at high speed until the tip current exceeds a set value. After halting the tip and allowing the tip current to achieve steady state, tip current is measured. The tip is then withdrawn 200 ␮m, and the tip position is incremented in a lateral direction to accomplish a scan. This mode allows better than 2 ␮m vertical accuracy in positioning with 5–25 ␮m diameter

50

Wipf

tips over surfaces of unknown type. Better accuracy is found when the surface is known to be insulating or conducting. The approach from large separations minimizes the chance of tip-substrate crashes during lateral motion, and the picking mode has an advantage in imaging high-relief (rough) samples. The picking mode does not require a conventional feedback loop as described above and can be readily implemented using equipment normally present in a standard SECM setup. A potential disadvantage to this mode is that sufficient vertical accuracy might be unavailable when using micrometer or smaller tips. In addition, the method can only be used with the SECM feedback mode since addition of a redox mediator is required to provide the tip-positioning signal. 3. Shear Mode The shear mode was developed for use with near-field scanning optical microscopy (NSOM) (47–49). In NSOM imaging the probe is scanned at a distance of about 25 nm from a surface. A number of designs have been implemented to provide the desired subnanometer positioning accuracy in NSOM, but the shear force method has emerged as the most popular. In the shear force method, a ‘‘dither’’ piezoelectric element is used to vibrate the NSOM probe tip into a mechanical resonance at a frequency from 10 to 50 kHz. The undamped probe oscillates with a 10–50 nm amplitude parallel to the surface. When the probe tip is less than 25 nm from the surface, an interaction force, commonly called the shear force, damps the oscillation amplitude and changes the resonance frequency of the probe. The basis for a control loop is the change in tip-vibration amplitude. The tip-vibration amplitude is monitored at the undamped probe resonance frequency with a lock-in amplifier. The control-loop error signal is the deviation of the tipvibration amplitude from a reference vibration amplitude. The success of the shear mode in NSOM has led to several instruments incorporating variants of the shear mode for SECM. The differences in NSOM and SECM designs primarily arise from the method of sensing the vibration amplitude of the probe and the difference between NSOM and SECM probes. The active participation of the probe in shear mode requires new SECM probe designs. It is likely that only lightweight and fragile electrodes with small tip diameters will be suitable for use with shear mode, i.e., ‘‘disposable’’ probes. These probes can be made using metal-filled pulled-glass capillaries and wax- or polymer-coated etched metal (or carbon) fibers. Metal sputtered optical fiber probes can be used in a hybrid electrochemical and optical scanning instrument (50). An instrument employing an optical detection of the probe vibration amplitude was reported by Schuhmann and coworkers (37). A piezoelectric element is used to induce a probe vibration while a focused laser illuminates

Instrumentation

51

the probe electrode near its end. By detecting the rate at which a diffraction line oscillates between the two segments of a split photodiode, the probe vibration frequency is determined. The probe remains stationary to maintain the laser and photodiode alignment, while the substrate is scanned to make images. The probe for this initial experiment was a 1.5 cm long section of a pulled glass capillary containing a 25 ␮m diameter Pt disk electrode. The long pulled section is necessary to produce the flexibility required for oscillation. Several resonance peaks of the electrode were noted, and the lowest frequency (about 2 kHz) was used in imaging. Modulation amplitudes in this experiment are about 1–2 ␮m, which is significantly larger than commonly employed in NSOM. The large amplitudes produce a proportionately larger shear mode distance so that the sample probe separation was in the micrometer range when shear mode damping was observed. The authors also note that the large amplitude tip oscillations increased the steady-state tip current. Use of optical methods in NSOM is often objectionable because of the interference of stray light on the image signal (a moot point in SECM), difficulty with liquid operation, and difficulty with miniaturization. Karraı¨ and Grober developed a nonoptical method, which employs a miniature tuning fork as the vibration amplitude detector (51,52). Figure 8A is an illus-

FIG. 8 Tip mounts used for implementing shear-force feedback in SECM: (A) the tuning fork mount; (B) proposed piezo tube mount.

52

Wipf

tration of the essential components. The tuning fork is an inexpensive piezoelectric device used in quartz clocks. The dither piezoelectric element is rigidly mounted to one of the tuning fork prongs. The probe is glued to the other prong. Excitation of the fork with the dither piezo produces a vibrating motion of the probe monitored by detecting the induced piezoelectric potential produced between the contact pads of the tuning fork. As in other designs, the resonance frequency changes as the tip approaches to within the shear force boundary. An important advantage of the tuning fork is that the probe need not be flexible since the tuning fork provides the motion to drive the probe oscillation. Smyrl and coworkers adapted the tuning-fork based feedback loop and x, y, z positioning found in a commercial NSOM instrument (the Topometrix Aurora) to produce SECM images (53). In their initial report, an etched tungsten probe insulated with polymer was used as the SECM probe. SECM and AFM imaging gave similar topographic profiles, (the AFM exhibiting higher lateral resolution), suggesting that the tuning fork method worked properly. The exact tip surface distance was not determined but was estimated to be less than 50 nm during SECM imaging with distance feedback. An alternate method for piezoelectric detection is illustrated in Figure 8B. Barenz and coworkers use a four-segment piezoelectric tube to provide the dither and detection elements for NSOM (54). The dither element is one segment firmly attached to the mount. Electrical excitation of this segment produces bending motion in the piezoelectric tube and, hence, in the probe. The other three segments are connected in parallel and used to detect the vibration amplitude by monitoring the induced piezoelectricity. Two advantages recommend this design for possible use with SECM probes. The first is the relatively low Q factor of the device. The Q (quality) factor is the sharpness of the resonance peak and is calculated by dividing the peak frequency by the peak width at ⫺3 dB. High Q factors lead to less stable and slower scanning because sharp resonance peaks make it more difficult to stay at the desired frequency. Dropping completely off the resonance peak leads to loss of feedback control and a crash. In addition, high Q resonances tend to ring, which requires longer settling times. A second advantage is an innovative mounting method. A small spring holds the probe in place, but the probe is rigidly coupled to the piezoelectric element by a polyisobutylene layer. This viscous fluid has a high Young’s modulus at frequencies above 10 kHz, which produces the required rigid coupling at operating frequencies but permits rapid replacement of the probe. The probe length protruding from the piezoelectric element can also be adjusted easily, maintaining a nearly constant resonance frequency from probe to probe.

Instrumentation VI.

53 EXPERIMENTAL DIFFICULTIES IN DATA ACQUISITION

Acquiring SECM data requires optimization of three simultaneous factors: signal-to-noise ratio (SNR), experiment duration, and image resolution. Improved SNR is accomplished by isolating external noise sources, designing proper grounds, and employing electrical shielding with Faraday cages (28,55,56). Ultimately, though, increasing SNR requires decreasing the measurement bandwidth, f, since most noise sources are proportional to f 1/2. Measurement bandwidth is decreased by filtering, ensemble (signal) averaging, lock-in amplification, or box-car averaging (38,57). For most SECM experiments, the signal is at sufficiently low frequency to be considered a dc level signal and low-pass filtering will be the primary means of reducing the signal bandwidth. Reducing the experiment duration is desirable. For example, a 100 ⫻ 100 ␮m2 image requires 16 minutes to acquire with 1 ␮m resolution at a scan rate of 10 ␮m/s. That may be too long to resolve chemical or electrochemical changes on a substrate surface. Increasing the probe scan rate will decrease the acquisition time, but this may not be feasible. Decreasing the tip size can increase image resolution. However, difficulties arise in simultaneously improving resolution, experiment duration, and SNR in the SECM experiment. At least one goal must be compromised to improve the others, and improving image resolution will be most challenging. A.

Low-Pass Filtering of I-L Curves

As tip electrodes become smaller, the tip current will decrease, resulting in decreased SNR. Thus, the temptation exists to ‘‘improve’’ the data by filtering. Filtering can increase SNR, and most potentiostats do include a lowpass filter to help reduce noise on the working electrode signal. Some instruments offer only one or two fixed settings, while others offer a variable frequency filter. However, filtering causes distortion of the data, which leads to problems in data interpretation. The effect of low-pass filtering a SECM signal is illustrated for two important SECM experiments shown in Figures 9 and 10. Figure 9 shows theoretical positive feedback IT versus L plots for a 1 and 5 ␮m radius disk-shaped tip over a conducting substrate. Here, IT is iT /iT,⬁, the normalized tip current, and L is d/a, where d is the tip-substrate separation and a is the electrode radius. The tip is initially located at a separation of 0.1 L from the substrate surface in the calculated curves. A filtered version of the theoretical curves is also plotted on the same graph. The filtered response is generated by convolution of the theoretical SECM

54

Wipf

Theoretical IT ⫺ L plots for 1 ␮m (top) and 5 ␮m (bottom) tips over a conducting substrate. The ideal curves are illustrated with calculated curves showing the effect of a 5 Hz low-pass filter on the curve shape as the tip is withdrawn from the surface at a rate of 1 ␮m/s.

FIG. 9

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55

signal with the time response of a single-pole low-pass filter with a 5 Hz cutoff frequency (58). The time scale for the curves is fixed by the scan rate. Note that the simulated time scale starts sufficiently prior to probe motion to avoid a discontinuity at t = 0. The model assumes that the tip is initially located near surface and moves continuously away at t > 0. As the lower curve in Figure 9 illustrates, the effect of filtering is minimal at the 5 ␮m radius tip. For the identical filter and probe scan rate, filtering the 1 ␮m radius tip signal causes significant distortion. The extent of the distortion is clearer when a second undistorted I-L curve with an offset of 0.034 L is overlaid on the filtered response. A good fit to the distorted results is observed for I values up to about 5.4, but the remaining data do not fit the theoretical curve. Low-pass filtering introduces significant distortion I-L curves generated by recording the tip current as the probe is scanned to or from the substrate. A particular danger is noted when I-L curves are used to analyze heterogeneous kinetic parameters. The shape of the filtered curves in Figure 9 could be confused with the effect of heterogeneous kinetics, especially as the curve approaches higher IT values. In addition, the distorted curves can lead to a misunderstanding of the actual separation between tip and substrate. Filtering distortion is distinguished from kinetic perturbation by its dependence on the direction of tip motion. The curve is shifted towards larger or smaller L as the tip moves to smaller or larger L values, respectively, when filtering occurs. Filtering distortion also affects negative feedback curves but with a smaller shift due to the less steep approach curve. The actual distortion observed on the curves will be dependent on the tip radius, the direction and speed of approach, the conductivity of the surface, and the point at which the scan starts. By applying the following rule of thumb, a shift in L of 0.01 or less will occur in I-L curves. af/v > 13

(1)

where a is the electrode radius in ␮m, f is the frequency in s⫺1, and v is the scan rate in ␮m/s. B.

Low-Pass Filtering While Imaging

Use of low-pass filtering while acquiring an image also produces distortion in the observed data. Figure 10 illustrates the effect of a single-pole lowpass filter with a 5 or 15 Hz cutoff frequency on a single line scan across an artificial substrate. The artificial data set consists of three conductive regions 16, 1, and 4 ␮m wide surrounded by insulating substrate (see Fig. 10A). The calculation assumes the tip is scanned at 20 ␮m/s at a tip-substrate distance of 1.0 L. In order to more closely approximate a real image; the

56

Wipf

Instrumentation

57

data are first convoluted with a tip function to simulate the effect of the tip size on image resolution. The tip function is a unit area pulse with a width equal to the tip diameter. The tip resolution effect is plotted as the dotted line in the figure and is most dramatically illustrated in Figure 10A, which models a 5 ␮m radius tip. The additional distortion due to diffusion is not modeled in the figures (58). As predicted by the results of the previous section, the distortion due to low-pass filtering increases as the tip radius decreases. In Figure 10A, the 5 ␮m radius tip signal is not seriously degraded at either the 5 or the 15 Hz filter cutoff. However, the slower filter slightly distorts the data at a 1 ␮m radius tip and seriously distorts the 0.2 ␮m radius tip data, particularly for the 1 ␮m feature (Fig. 10B, C). A 1 ␮m feature should be resolvable with a 0.2 ␮m radius tip, but the filtering process significantly reduces the apparent magnitude of the current. A rule of thumb for filter cutoff frequencies during image acquisition is af/v > 1

(2)

This is a relaxed requirement compared to I-L data acquisition since the distortion due to tip-size function diminishes the effect of low-pass filtering. C.

Improving Image Resolution with Small Diameter Tips

A number of advantages occur when the tip electrode size is reduced. Lateral resolution during imaging is improved and diffusional transit times for the mediator between the tip and substrate are very short at close tip-substrate separations, which allows rapid tip scan rates and produces larger masstransfer coefficients in heterogeneous and homogeneous rate measurements. However, the faradaic tip signal is proportional to tip size. A 1 nA signal at a 5 ␮m radius tip is reduced to a 10 pA signal at a 50 nm tip. The unfortunate conclusion of both Eqs. (1) and (2) is that decreasing the electrode radius requires a proportionate increase in filter frequency or a decrease in probe scan rate—exactly the opposite of what may be desired. Simply stated, use of a smaller tip effectively shifts the I-L curve or image features to higher frequencies. Since imaging occurs at higher scan rates than collection of


1, and even for RG = 10 it leads to about 2% error, which is present in almost all published SECM simulations. B.

Steady-State Conditions

The formulation of the steady-state SECM problem is significantly simpler (8). It includes a single Laplace equation ⭸2C ⭸2C 1 ⭸C ⫹ =0 2 2 ⫹ ⭸Z ⭸R R ⭸R

(22)

for oxidized (or reduced) form of the mediator with boundary conditions (9) and (11) and one of two pairs of boundary conditions for the tip and substrate surfaces, i.e., either Eqs. (10) and (21a) (diffusion-controlled tip reaction and quasireversible substrate kinetics) or Eqs. (8) and (21b) (diffusion-controlled substrate reaction and finite kinetics tip at the tip). All variables in these

Theory

151

FIG. 3 The effect of ␥ on the chronoamperometric characteristics for the positive feedback process at a tip/substrate separation, L = 0.2. The reduced form initially present in solution is oxidized at the tip and regenerated at the substrate. Normalized tip current is plotted as a function of dimensionless time, ␶ = tDR/a2. (Reprinted with permission from Ref. 6. Copyright 1997 Elsevier Science S.A.)

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equations are now time-independent. The assumption of equal diffusion coefficients is unnecessary under steady state, because the concentrations of O and R at steady state are interrelated as follows cO ⫹ cRDR/DO = c⬚O ⫹ c⬚D R R /DO

(23)

The solution for the finite kinetics at the tip and diffusion-limited substrate reaction is (8) 1 ⫺ ␲JT(R)/4␬⬘ = ␪

冕

0

冕

⬁

1

uJT (u)du

J0(pR)J0 (pu)tanh(pL)dp

(24)

0

where J0 is the Bessel function of the first kind of order zero,

␬⬘ = ␲ak⬚exp[⫺␣f(E ⫺ E⬚⬘)]/(4DO)

(25a)

␪ = 1 ⫹ exp[nf(E ⫺ E⬚⬘)]DO /DR

(25b)

and

However, Eq. (24) has not been utilized in any calculations. Most published computational results represent the long-time limit of the data obtained by solving time-dependent equations. By fitting these results, several analytical approximations have been obtained for different situations including diffusion-controlled, irreversible, and quasireversible reactions at both tip and substrate surfaces. 1. Diffusion-Controlled and Nernstian Steady-State Processes The knowledge of the shape of the IT-L curve for a diffusion-controlled process is critical for both imaging and quantitative kinetic measurements because it allows one to establish the distance scale. The equations describing a diffusion-controlled SECM process represent two limiting cases of the general steady-state problem discussed above, i.e., (1) ‘‘pure positive feedback’’ produced by rapid regeneration of the reactant species at the substrate and (2) ‘‘pure negative feedback’’ observed when no mediator regeneration occurs at the substrate. Both problems were treated numerically by Kwak and Bard (1). The dimensionless current-distance curves were tabulated for both insulating and conductive substrates, assuming the equality of the diffusion coefficients and an infinitely large substrate. Later (9), analytical approximations were obtained for both working curves. Both numerical results in Ref. 1 and analytical expressions in Ref. 9 are suitable only for large values of RG (i.e., RG ⱖ 10). However, tips with much thinner insulating sheathes have often been employed because they allow one to obtain closer tip/substrate separation. A typical RG for a pipet-based tip is ⬃1.1 (7). A more recent treatment of this problem (7) is suitable for 1.1 ⱕ RG ⱕ 10. The diffusion problem in dimensionless form is

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⭸2C ⭸2C 1 ⭸C ⫹ ⫹ =0 ⭸Z2 ⭸R2 R ⭸R

0 ⱕ R, L1 < Z < L

(26)

where L1 = ᐍ1/a shows how far behind the tip surface (Z = 0) the simulation goes (Fig. 4); other dimensionless variables are defined above. The boundary conditions are: 0 ⱕ R < 1, Z = 0 (tip surface): C=0

FIG. 4

(27a)

Geometry of the simulation domain and the parameters defining the diffusion problem for SECM. (Adapted with permission from Ref. 7. Copyright 1998 American Chemical Society.)

154

Mirkin 1 ⱕ R ⱕ RG, Z = 0 and R = RG, L1 < Z < 0 (insulating glass): ⭸C =0 ⭸n

(27b)

where ⭸C/⭸n is the normal derivative on the surface of insulating glass surrounding the conductive core of the tip. 0 ⱕ R ⱕ RMAX, Z = L (substrate); C = 1 (positive feedback)

(28a)

⭸C = 0 (negative feedback) ⭸Z

(28b)

or

RG ⱕ R < RMAX, Z = L1 or R = RMAX, L1 < Z < L (simulation space limits): C=1

(29)

The main difference between Eqs. (26)–(29) and the previous simulation (1) is in problem geometry. The values of L1 = 0 and RMAX = RG were used in Ref. 1, i.e., the concentration of the mediator was assumed to equal its bulk value everywhere beyond the limits of the tip-insulating sheath (R ⱖ RG). However, for RG