Chemistry in Motion

Chemistry in Motion: Reaction–Diffusion Systems for Micro- and Nanotechnology Bartosz A. Grzybowski

© 2009 John Wiley & Sons Ltd. ISBN: 978-0-470-03043-1

Chemistry in Motion:

Reaction–Diffusion Systems for

Micro- and Nanotechnology

Bartosz A. GrzybowskiNorthwestern University, Evanston, USA

This edition first published 2009

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Library of Congress Cataloging-in-Publication Data

Grzybowski, Bartosz A.

Chemistry in motion : reaction-diffusion systems for micro- and

nanotechnology / Bartosz A. Grzybowski.

p. cm.

Includes bibliographical references and index.

ISBN 978-0-470-03043-1 (cloth : alk. paper)

1. Reaction mechanism. 2. Reaction-diffusion equations.

3. Microtechnology–Mathematics. 4. Nanotechnology–Mathematics. I. Title.

QD502.5.G79 2009

5410.39–dc22 2008044520

A catalogue record for this book is available from the British Library.

ISBN: 978-0-470-03043-1 (HB)

Typeset in 10/12pt Sabon by Thomson Digital, Noida, India.

Printed and bound in Singapore by Fabulous Printers Pte Ltd

To Jolanta, Andrzej and Kristianawith gratitude and love

Contents

Preface xi

List of Boxed Examples xiii

1 Panta Rei: Everything Flows 1

1.1 Historical Perspective 1

1.2 What Lies Ahead? 3

1.3 How Nature Uses RD 4

1.3.1 Animate Systems 5

1.3.2 Inanimate Systems 8

1.4 RD in Science and Technology 9

References 12

2 Basic Ingredients: Diffusion 17

2.1 Diffusion Equation 17

2.2 Solving Diffusion Equations 20

2.2.1 Separation of Variables 20

2.2.2 Laplace Transforms 26

2.3 The Use of Symmetry and Superposition 31

2.4 Cylindrical and Spherical Coordinates 34

2.5 Advanced Topics 38

References 43

3 Chemical Reactions 45

3.1 Reactions and Rates 45

3.2 Chemical Equilibrium 50

3.3 Ionic Reactions and Solubility Products 51

3.4 Autocatalysis, Cooperativity and Feedback 52

3.5 Oscillating Reactions 55

3.6 Reactions in Gels 57

References 59

4 Putting It All Together: Reaction–Diffusion Equations

and the Methods of Solving Them 61

4.1 General Form of Reaction–Diffusion Equations 61

4.2 RD Equations that can be Solved Analytically 62

4.3 Spatial Discretization 66

4.3.1 Finite Difference Methods 66

4.3.2 Finite Element Methods 70

4.4 Temporal Discretization and Integration 80

4.4.1 Case 1: tRxn � tDiff 81

4.4.1.1 Forward Time Centered Space (FTCS)

Differencing 81

4.4.1.2 Backward Time Centered Space (BTCS)

Differencing 81

4.4.1.3 Crank–Nicholson Method 82

4.4.1.4 Alternating Direction Implicit Method

in Two and Three Dimensions 83

4.4.2 Case 2: tRxn � tDiff 83

4.4.2.1 Operator Splitting Method 83

4.4.2.2 Method of Lines 84

4.4.3 Dealing with Precipitation Reactions 86

4.5 Heuristic Rules for Selecting a Numerical Method 87

4.6 Mesoscopic Models 87

References 90

5 Spatial Control of Reaction–Diffusion at Small Scales:

Wet Stamping (WETS) 93

5.1 Choice of Gels 94

5.2 Fabrication 98

Appendix 5A: Practical Guide to Making Agarose Stamps 101

5A.1 PDMS Molding 101

5A.2 Agarose Molding 101

References 102

6 Fabrication by Reaction–Diffusion: Curvilinear

Microstructures for Optics and Fluidics 103

6.1 Microfabrication: The Simple and the Difficult 103

6.2 Fabricating Arrays of Microlenses by RD and WETS 105

6.3 Intermezzo: Some Thoughts on Rational Design 109

6.4 Guiding Microlens Fabrication by Lattice

Gas Modeling 111

viii CONTENTS

6.5 Disjoint Features and Microfabrication

of Multilevel Structures 117

6.6 Microfabrication of Microfluidic Devices 121

6.7 Short Summary 124

References 124

7 Multitasking: Micro- and Nanofabrication

with Periodic Precipitation 1277.1 Periodic Precipitation 127

7.2 Phenomenology of Periodic Precipitation 128

7.3 Governing Equations 130

7.4 Microscopic PP Patterns in Two Dimensions 137

7.4.1 Feature Dimensions and Spacing 139

7.4.2 Gel Thickness 140

7.4.3 Degree of Gel Crosslinking 142

7.4.4 Concentration of the Outer and Inner

Electrolytes 142

7.5 Two-Dimensional Patterns for Diffractive Optics 145

7.6 Buckling into the Third Dimension: Periodic

‘Nanowrinkles’ 152

7.7 Toward the Applications of Buckled Surfaces 155

7.8 Parallel Reactions and the Nanoscale 158

References 160

8 Reaction–Diffusion at Interfaces: Structuring Solid Materials 165

8.1 Deposition of Metal Foils at Gel Interfaces 165

8.1.1 RD in the Plating Solution: Film Topography 167

8.1.2 RD in the Gel Substrates: Film Roughness 172

8.2 Cutting into Hard Solids with Soft Gels 178

8.2.1 Etching Equations 178

8.2.1.1 Gold Etching 180

8.2.1.2 Glass and Silicon Etching 181

8.2.2 Structuring Metal Films 181

8.2.3 Microetching Transparent Conductive Oxides,

Semiconductors and Crystals 186

8.2.4 Imprinting Functional Architectures into Glass 189

8.3 The Take-Home Message 192

References 192

9 Micro-chameleons: Reaction–Diffusion for Amplificationand Sensing 195

9.1 Amplification of Material Properties by RD

Micronetworks 197

CONTENTS ix

9.2 Amplifying Macromolecular Changes using

Low-Symmetry Networks 203

9.3 Detecting Molecular Monolayers 205

9.4 Sensing Chemical ‘Food’ 208

9.4.1 Oscillatory Kinetics 211

9.4.2 Diffusive Coupling 212

9.4.3 Wave Emission and Mode Switching 213

9.5 Extensions: New Chemistries, Applications

and Measurements 215

References 222

10 Reaction–Diffusion in Three Dimensions and at the Nanoscale 227

10.1 Fabrication Inside Porous Particles 228

10.1.1 Making Spheres Inside of Cubes 228

10.1.2 Modeling of 3D RD 230

10.1.3 Fabrication Inside of Complex-Shape Particles 235

10.1.4 ‘Remote’ Exchange of the Cores 236

10.1.5 Self-Assembly of Open-Lattice Crystals 238

10.2 Diffusion in Solids: The Kirkendall Effect

and Fabrication of Core–Shell Nanoparticles 240

10.3 Galvanic Replacement and De-Alloying Reactions

at the Nanoscale: Synthesis of Nanocages 248

References 253

11 Epilogue: Challenges and Opportunities for the Future 257

References 263

Appendix A: Nature’s Art 265

Appendix B: Matlab Code for the Minotaur (Example 4.1) 271

Appendix C: Cþþ Code for the Zebra (Example 4.3) 275

Index 283

x CONTENTS

Preface

This book is aimed at all those who are interested in chemical processes at small

scales, especially physical chemists, chemical engineers and material scientists.

The focus of the work is on phenomena in which chemical reactions are coupled

with diffusion – hence Chemistry in Motion. Although reaction–diffusion (RD)

phenomena are essential for the functioning of biological systems, there have

been only a few examples of their application in modern micro- and nanotech-

nology. Part of the problem has been that RD phenomena are hard to bring under

experimental control, especially when the system dimensions are small.

As we will see shortly, these limitations can be lifted by surprisingly simple

experimental means. The techniques introduced in Chapters 5 to 10 will allow us

to control RD at micro- and nanoscales and to fabricate a variety of small-scale

structures: microlenes, complex microfluidic architectures, optical elements,

chemical sensors and amplifiers, unusual micro- and nanoparticles, and more.

Although these systems are still very primitive compared with the sophisticated

RD schemes found in biology, they illustrate one general thought that underlies

this book – namely that if RD is properly ‘programmed’ it can be a very unique

and practical way of manipulating matter at small scales. The hope is that those

who read this monograph will be able to carry the torch further and construct RD

micro-/nanosystems that gradually approach the complexity and usefulness of

biological RD.

For this to become a reality, however, we must understand RD in quantitative

detail. Since RD phenomena are inherently nonlinear, and the participating

chemicals evolve into final structures via nontrivial and sometimes counter-

intuitive ways, rational design of RD systems requires a fair degree of theoretical

treatment. Recognizing this, we devote Chapters 2 through 4 to a thorough

discussion of the physical basis of RD and the theoretical tools that can be used

to model it.

In these and other chapters, new concepts are derived from the basics and

assume only rudimentary knowledge of chemistry and physics and some

familiarity with differential equations. This does not mean that the things

covered are necessarily easy – not at all! In all cases, however, the material

builds up gradually and multiple examples and literature sources are provided to

illustrate the key concepts.

The book can be used as a text for a one-semester, graduate elective course in

chemical engineering (combining elements of transport and kinetics), and in

materials science or chemistry classes on chemical self-organization, self-

assembly or micro-/nanotechnology. When taught to chemical engineers, Chap-

ters 2 through 4 should be covered in detail. For more practically minded

audiences, it might be reasonable to focus the class on specific self-organization

phenomena and their applications (Chapters 5 to 10), consulting the theory from

Chapters 2 to 4 as needed.

Finally, several acknowledgements are due. The National Science Foundation

generously provided the funding under the CAREER award. I hope the money

was well spent! Sincere thanks go to my graduate students – Kyle Bishop,

Siowling Soh, Paul Wesson, Rafal Klajn, Chris Wilmer and Chris Campbell –

who helped enormously at every stage of the writing process. Last but not least,

the book would have never come into being if not for the constant support, love

and patience of my family – my most fantastic parents and wife to whom I

dedicate this monograph.

Bartosz A. Grzybowski, Evanston, USA

xii PREFACE

List of Boxed Examples

2.1 Unsteady Diffusion in an Infinite Tube 30

2.2 Unsteady Diffusion in a Finite Tube 31

2.3 Is Diffusion Good for Drug Delivery? 37

2.4 Random Walks and Diffusion 42

3.1 More Than Meets the Eye: Nonapparent Reaction Orders 46

3.2 Sequential Reactions 49

4.1 How Diffusion Betrayed the Minotaur 68

4.2 The Origins of the Galerkin Finite Element Scheme 74

4.3 How Reaction–Diffusion Gives Each Zebra Different Stripes 89

6.1 A Closer Look at Gel Wetting 106

6.2 Is Reaction–Diffusion Time-Reversible? 114

6.3 Optimization of Lens Shape Using a Monte Carlo Method 116

7.1 Periodic Precipitation via Spinodal Decomposition 131

7.2 Wave Optics and Periodic Precipitation 144

7.3 Calculating Diffraction Patterns 149

8.1 Stokes–Einstein Equation 176

8.2 RD Microetching for Cell Biology: Imaging

Cytoskeletal Dynamics in ‘Designer’ Cells 184

9.1 Patterning an Excitable BZ Medium with WETS 210

9.2 Calculating Binding Constants from RD Profiles 219

10.1 Transforming Surface Rates into Apparent Bulk Rates 232