The Language of Shape - The Role of Curvature in Condensed Matter, Physics, Chemistry and Biology

THE LANGUAGE OF SHAPE THE ROLE OF CURVATURE IN CONDENSED MATTER: PHYSICS, CHEMISTRY AND BIOLOGY

Cover illustration: Beyond the Euclidean desert: hyperbolic radiolaria skeletons. Adapted from SEM imoge by Roger Heady ond Michoel Ciszewski.

THE LANGUAGE OF SHAPE THE ROLE OF CURVATURE IN CONDENSED MATTER" PHYSICS, CHEMISTRY AND BIOLOGY STEPHEN HYDE DEPARTMENT OF APPLIED MATHEMATICS INSTITUTE OF ADVANCED STUDIES AUSTRALIAN NATIONAL UNIVERSITY CANBERRA, 0200, AUSTRALIA

STEN ANDERSSON SANDVIK RESEARCHINSTITUTE S. LJ~NGGATAN27 38074 L~TTORP, SWEDEN

KARE LARSSON DEPARTMENT OF FOOD TECHNOLOGY LUND UNIVERSITY BOX 124, 22100 LUND, SWEDEN

ZOLTAN BLUM DEPARTMENT OF INORGANIC CHEMISTRY LUND UNIVERSITY BOX 124, 22100 LUND, SWEDEN

TOMAS LANDH DEPARTMENT OF FOOD TECHNOLOGY LUND UNIVERSITY BOX 124, 22100 LUNDI SWEDEN

SVEN

LIDIN

DEPARTMENT OF INORGANIC CHEMISTRY LUND UNIVERSITY BOX 124, 22100 LUND, SWEDEN BARRY

W.

NINHAM

DEPARTMENT OF APPLIED MATHEMATICS INSTITUTE OF ADVANCED STUDIES AUSTRALIAN NATIONAL UNIVERSITY CANBERRA, 0200, AUSTRAUA

1997 ELSEVIER

AMSTERDAM -

LAUSANNE

-

NEW YORK -

OXFORD

- SHANNON

-

TOKYO

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

ISBN: 0 444 81538 4 91997 Elsevier Science B.V. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. This book is printed on acid-free paper.

Printed and bound by Antony Rowe Ltd, Eastboume Transferred to digital printing ZOO5

Acknowledgments This project has gone on too long - far too long to recall all those who deserve our thanks. Nevertheless, we are very grateful to all those scientists who have kindly furnished us with data, figures and other comments. In particular, we thank Profs. Hans-Georg von Schnering (Stuttgart) and Reinhard Nesper (Ziirich), Takeji Hashimoto and Hiro Hasegawa (Kyoto) for substantial support. We have been assisted beyond the call of duty by our secretaries: Diana Wallace (Canberra) and Ingrid Mellqvist (Lund). Fiona Meldrum patiently hunted through the text, tracking down a number of typographical and language errors. Finally, we cannot forget those fresh Clyde River oysters served nightly at the Malua Bay Bowling Club, which - ably assisted by local champagne - fortified us during a week's intense work, where the form of the book was thrashed out.

Canberra, December 22, 1996.

This Page Intentionally Left Blank

o,

VU

Preface

During the latest decade we have worked on periodic surfaces with zero average curvature and their significance in chemical structures, ranging from atomic and molecular arrangements in crystals to complex self-assembled colloidal aggregates. This approach has proved to be fruitful, not only in the determination of complex structures, but also in the understanding of phase behaviour and relations between structure and physical properties. Our aim has been to summarise our own understanding of this growing field, and to provide a complete description of relevant shapes and the forces behind their formation. This book deals with the role of curvature, a neglected dimension, in guiding chemical, biochemical and cellular processes. The curved surfaces that concern us might be those traced out by the head groups of phospholipid molecules that spontaneously self-assemble to form membranes and other building blocks of biology. Or they can be the surfaces of proteins involved in catalysis. They are provided in abundance par excellence by inorganic chemistry. In biology these dynamic entities have a marvellous capacity for self-organisation and selfassembly which is beginning to be understood. They transform from one shape to another under the influence of the forces of nature with an astonishing ease that allows them to manage resources, direct complex sequences of reactions, and arrange for delivery, all on time. Shape determines function, and the energetics of function dictates the optimal structure required. At least that is our thesis. The cognition and recognition of shape and form are one of the earliest tasks presented to the brain. Shape and form are so much a part of our mental processes that we tend to take them for granted. Almost any word in any language that describes objects conjures up an image that involves form. And indeed one of the deepest expressions of our sense of being is representational art. Painting and sculpture deal exclusively with colour, shape and form. Yet despite the vaunted successes of physics and mathematics that underlies m o d e m science, science remains antithetic to art because it reduces diversity to too sterile order through the imposition of Euclidean symmetry. According to conventional texts, forces act between point atoms, spheres, cylinders and planes in a kind of pythagorean and ptolemaic imperative that ignores curvature. There is nowhere an awareness that shape may have a role to play, except to please the eye. Nature ever geometrised, said somebody. True. But it has good reasons. In cell and molecular biology where mechanisms of enzyme action are not understood and attributed to some kind of Maxwell demon, all is specificity, and the lipids of membranes serve to do no more than act as a passive matrix for proteins and as a protection for the procreation of a uni-dimensional, machine like and stolidly boring DNA. There is more to it than that.

viii The thesis of this book is that two circumstances may have contributed to our present situation. The one has to do with the forces acting between chemical assemblies, and the interplay between these forces, set by the environment in which they work, and curvature. The other has to do with the absence of any language describing shapes of physically associated assemblies that are part of the subject of cellular and molecular biology. When shape is taken into account one comes to the realisation that curvature, and forces, set by constraints, are meaningful thermodynamic variables, (derived from classical thermodynamics). The key problem in the reductionist chain is how to build a statistical mechanism that uses a language of shapes. This language draws on topology and differential geometry. What we will attempt to show is that once that language is learnt, the world begins to take on a richer and more colourful unity. Through a consideration of minimal surfaces and other shapes the bewildering chaos of nature makes more sense. We are convinced that a structural description based on curvature is useful in physical and biological sciences, and the numerous examples presented here support that view. Finally, we hope that our speculations on the role of these shapes in chemical reactions and in molecular organisation in living systems will inspire new work in this field.

ix

Table of Contents Chapter The

1 Mathematics

of Curvature

1.1 ................. I n t r o d u c t o r y r e m a r k s ............................................................................................... 1 1.2 ................. C u r v a t u r e ................................................................................................................... 2 1.3 ................ D i f f e r e n t i a l g e o m e t r y of s u r f a c e s ........................................................................... 4 1.4 ................ T h e G a u s s m a p .......................................................................................................... 6 1.5 ................. G e o d e s i c c u r v a t u r e a n d g e o d e s i c s ......................................................... . ............... 7 1.6 ................ T o r s i o n ........................................................................................................................ 8 1.7 ................. T h e G a u s s - B o n n e t t h e o r e m ..................................................................................... 10 1.8 ................. T o p o l o g y .................................................................................................................... 11 1.9 ................. A p r o v i s i o n a l c a t a l o g u e o f s u r f a c e f o r m s .............................................................. 14 1.10 ............... A h i s t o r i c a l p e r s p e c t i v e ............................................................................................ 18 1.11 ............... P e r i o d i c m i n i m a l s u r f a c e s ....................................................................................... 21 1.12 ............... T h e B o n n e t t r a n s f o r m a t i o n : t h e P - s u r f a c e , t h e D - s u r f a c e a n d t h e g y r o i d ........ 27 1.13 .............. P a r a l l e l s u r f a c e s ......................................................................................................... 32 1.14 ............... F u t u r e d i r e c t i o n s ....................................................................................................... 32 Appendix: ..... A c a t a l o g u e of s o m e m i n i m a l s u r f a c e s .................................................................. 33 M a t h e m a t i c a l B i b l i o g r a p h y ............................................................................................................. 40 R e f e r e n c e s ........................................................................................................................................ 41

Chapter 2 T h e Lessons of Chemistry Inorganic C h e m i s t r y - . F r o m t h e d i s c r e t e

43

l a t t i c e of c r y s t a l s y m m e t r y to t h e c o n t i n u o u s ....... ...................... m a n i f o l d s of d i f f e r e n t i a l g e o m e t r y ......................................................................... 43 2.1 ................. T h e b a c k g r o u n d ........................................................................................................ 43 2.2 ................. T h e u n r a v e l l i n g of c o m p l e x s t r u c t u r e s .................................................................. 44 2.3 ................. D e f e c t s ........................................................................................................................ 46 2.4 ................. T h e i n t r i n s i c c u r v a t u r e of s o l i d s ............................................................................. 49 2.5 ................. H y d r o p h o b i c z e o l i t e s a n d a d s o r p t i o n ................................................................... 52 2.6 ................. P h a s e t r a n s i t i o n s , o r d e r a n d d i s o r d e r .................................................................... 55 2.7 ................. Q u a n t i t a t i v e a n a l y s i s of h y p e r b o l i c f r a m e w o r k s : s i l i c a t e d e n s i t i e s .................. 58 2.8 ................. T e t r a h e d r a l f r a m e w o r k s : T h r e e - o r t w o - d i m e n s i o n a l s t r u c t u r e s ? ..................... 63 2.9 ................. Q u a s i c r y s t a l s ............................................................................................................. 66 O r g a n i c C h e m i s t r y : T h e S h a p e of M o l e c u l e s ............................................................................. 73 2.10 ............... T h e h y p e r b o l i c n a t u r e of sp 3 o r b i t a l s ..................................................................... 3'3 2.11 ............... O r g a n i c s c u l p t u r e s : c a r c e r a n d s , c r o w n s , etc ......................................................... 75 2.12 ............... B e y o n d g r a p h i t e : f u U e r e n e s a n d s c h w a r z i t e s ....................................................... 78 Appendix: ..... T h e p r o b l e m o f q u a s i c r y s t a l s .................................................................................. 80 R e f e r e n c e s ........................................................................................................................................ 84

Chapter

3

Molecular

Forces

and

Self-Assembly

87

3.1 ................. T h e b a c k g r o u n d ........................................................................................................ 87 3.2.1 .............. T h e n a t u r e of force, s h a p e a n d s i z e ........................................................................ 88 3.2.2 .............. S e l f - e n e r g y , m o l e c u l a r s i z e a n d s h a p e ................................................................... 89 3.2.3 .............. S e l f - e n e r g y a n d a d s o r p t i o n ..................................................................................... 91 3.2.4 .............. T h e s h a p e of b o n d s ................................................................................................... 94 3.3 ................. T h e b a c k g r o u n d to s u r f a c e f o r c e s ........................................................................... 96 3.4 ................. M o l e c u l a r f o r c e s i n d e t a i l ......................................................................................... 98 3.4.1 .............. v a n d e r W a a l s f o r c e s ................................................................................................. 98 3.4.2 .............. L i f s h i t z f o r c e s ............................................................................................................. 100

3.4.3 .............. D o u b l e - l a y e r forces ................................................................................................... 103 3.5 ................. A g a l l i m a u f r y of forces ............................................................................................. 105 3.5.1 .............. Forces d u e to l i q u i d s t r u c t u r e ................................................................................. 105 3.5.2 .............. S u r f a c e - i n d u c e d l i q u i d s t r u c t u r e ............................................................................ 106 3.5.3 .............. H y d r a t i o n forces in p h o s p h o l i p i d s ........................................................................ 106 3.5.4 .............. Surface d i p o l e c o r r e l a t i o n s ...................................................................................... 107 3.5.5 .............. S e c o n d a r y h y d r a t i o n forces a n d i o n - b i n d i n g ....................................................... 108 3.5.6 .............. R a n g e of t h e d o u b l e - l a y e r force a n d i m p l i c a t i o n s ............................................... 109 3.5.7 .............. H y d r o p h o b i c i n t e r a c t i o n s ........................................................................................ 110 3.5.8 .............. N o n - i o n i c s u r f a c t a n t forces ..................................................................................... 111 3.5.9 .............. Forces of t h e r m o d y n a m i c o r i g i n ............................................................................. 111 3.5.10 ............ T h e H e l f r i c h force ..................................................................................................... 112 3.5.11 ............ Forces of v e r y l o n g r a n g e ......................................................................................... 112 3.5.12 ............ S u m m a r y .................................................................................................................... 113 3.6 ................. S e l f - o r g a n i s a t i o n in s u r f a c t a n t s o l u t i o n s ............................................................... 113 3.6.1 .............. A g g r e g a t e s t r u c t u r e in the E u c l i d e a n d e s e r t ......................................................... 116 3.6.2 .............. C u r v a t u r e as t h e d e t e r m i n a n t of m i c r o s t r u c t u r e ................................................. 117 3.6.3 .............. G e n e s i s of t h e s u r f a c t a n t p a r a m e t e r ....................................................................... 119 3.6.4 .............. T h e t y r a n n y of t h e o r y ............................................................................................... 122 Appendix A:. E v o l u t i o n of c o n c e p t s o n l o n g r a n g e m o l e c u l a r forces r e s p o n s i b l e for ........... ...................... o r g a n i s a t i o n a n d i n t e r a c t i o n s i n c o l l o i d a l s y s t e m s .............................................. 124 Append/x B: .. M o d e m c o n c e p t s of s e l f - a s s e m b l y ......................................................................... 128 Appendix C:.. R e m a r k s o n t h e n a t u r e of the h y d r o p h o b i c i n t e r a c t i o n a n d w a t e r s t r u c t u r e .. 129 R e f e r e n c e s ........................................................................................................................................ 137

Chapter

4

Beyond The

Flatland

Geometric

Forms

due to

Self-Assembly

4.1 ................ I n t r o d u c t i o n : m o l e c u l a r d i m e n s i o n s a n d c u r v a t u r e ............................................ 4.2 ................. T h e local g e o m e t r y of a g g r e g a t e s ........................................................................... 4.3 ............... T h e c o m p o s i t i o n of s u r f a c t a n t m i x t u r e s : t h e g l o b a l c o n s t r a i n t .......................... 4.4 ................. Bilayers i n s u r f a c t a n t - w a t e r m i x t u r e s .................................................................... 4.5 ................. M o n o l a y e r s i n s u r f a c t a n t - w a t e r m i x t u r e s ............................................................. 4.6 ................. G e o m e t r i c a l physics: b e n d i n g e n e r g y .................................................................... 4.7 ................. T h e m e s o p h a s e b e h a v i o u r of s u r f a c t a n t - a n d l i p i d - w a t e r m i x t u r e s ................ 4.8 ................. T h e h y p e r b o l i c r e a l m : c u b i c a n d i n t e r m e d i a t e p h a s e s ........................................ 4.9 ................. M e s o s t r u c t u r e in t e r n a r y s u r f a c t a n t - w a t e r - o i l s y s t e m s : m i c r o e m u l s i o n s ........ 4.10 ............... Block c o p o l y m e r m e l t s : a n i n t r o d u c t i o n ............................................................... 4.11 .............. C o p o l y m e r s e l f - a s s e m b l y ......................................................................................... 4.12 ............... R e l a t i o n b e t w e e n m a t e r i a l p r o p e r t i e s a n d s t r u c t u r e ........................................... 4.13 ............... P r o t e i n a s s e m b l i e s i n bacteria: a m e s h p h a s e ....................................................... 4.14 ............... S e l f - a s s e m b l y of c h i r a l m o l e c u l e s ........................................................................... R e f e r e n c e s ........................................................................................................................................

141 141 143 146 149 154 157 160 163 170 176 177 185 186 187 194

Chapter 5 Lipid Self-Assembly and Function In Biological Systems 199 Self-association of lipids in an aqueous environment ............................................................. 199 5.1.1 .............. I n t r o d u c t i o n ............................................................................................................... 199 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6 5.1.7

.............. .............. .............. .............. .............. ..............

G e n e r a l b e h a v i o u r of l i p i d s in w a t e r ..................................................................... C u b i c p h a s e s .............................................................................................................. C u b i c l i p i d - p r o t e i n - w a t e r p h a s e s ........................................................................... D i s p e r s i o n s of b i c o n t i n u o u s c u b i c p h a s e s : c u b o s o m e s ....................................... L i p o s o m a l d i s p e r s i o n s ............................................................................................. Vesicles a n d m e m b r a n e s ..........................................................................................

200 203 206 207 208 209

xi

5.1.8 ............. C h o l e s t e r i c l i q u i d - c r y s t a l s a n d l o w - d e n s i t y l i p o p r o t e i n s t r u c t u r e s .................. 211 Cell membranes.............................................................................................................................. 213 5.2.1 .............. I n t r o d u c t i o n ............................................................................................................... 213 5.2.2 .............. O n i n t r i n s i c p e r i o d i c b i l a y e r c u r v a t u r e i n m e m b r a n e f u n c t i o n : ...................... A m o d e l m e m b r a n e b i l a y e r p h a s e t r a n s i t i o n i n v o l v i n g p e r i o d i c c u r v a t u r e ... 215 5.2.3 .............. L i p i d c o m p o s i t i o n c o n t r o l i n m e m b r a n e s ............................................................. 215 5.2.4 .............. T h e n e r v e m e m b r a n e , s i g n a l t r a n s m i s s i o n a n d a n a e s t h e s i a .............................. 218 ...................... B i l a y e r c o n f o r m a t i o n d u r i n g t h e a c t i o n p o t e n t i a l ................................................ 218 ...................... A n a e s t h e t i c effects .................................................................................................... 220 5.2.5 .............. A n a e s t h e t i c a g e n t s a n d c a n c e r , i m m u n o s u p p r e s s i o n ......................................... 222 5.2.6 .............. O n t h e m e t a s t a t i c m e c h a n i s m of m a l i g n a n t ceils ................................................ 224 5.2.7 .............. M e m b r a n e s i n m i c r o - o r g a n i s m s a n d a n t i - m i c r o b i a l a g e n t s ............................... 224 5.2.8 .............. P l a n t cell m e m b r a n e s ............................................................................................... 226 5.2.9 .............. T h e C 2D c o n f o r m a t i o n a n d m e m b r a n e f u s i o n ...................................................... 226 5.2.10 ............ M e m b r a n e s e n c a p s u l a t i n g o i l / f a t s a n d b i l i q u i d f o a m s ...................................... 227 5.2.11 ............ S u g a r g r o u p s , r e c e p t o r - l i g a n d b i n d i n g a n d c o o p e r a t i v i t y ................................. 229 5.2.12 ............ A C 2D m e m b r a n e s t r u c t u r e i n a Streptomyces s t r a i n ............................................ 230 R e f e r e n c e s ........................................................................................................................................ 232

Chapter

6

Folding and Function In Proteins and

DNA

237

6.1 ................. O v e r a l l f e a t u r e s of p r o t e i n s t r u c t u r e ...................................................................... 237 6.2 ................. a - h e l i x d o m a i n s ......................................................................................................... 239 6.3 .............. a - h e l i x / ~ - s h e e t d o m a i n s ....................................................................................... 239 6.4 ............... [3-sheet d o m a i n s ........................................................................................................ 241 6.5 ................ M e m b r a n e p r o t e i n s ................................................................................................... 242 6.6 ................. E n z y m a t i c a c t i o n ....................................................................................................... 243 6.7 ................. P r o t e i n f u n c t i o n a n d d i o x i n p o i s o n i n g .................................................................. 2 4 7 6.8 ................. G e o m e t r y i n h o r m o n e - r e c e p t o r i n t e r a c t i o n s ......................................................... 248 6.9 ................. Self / n o n - s e l f r e c o g n i t i o n ....................................................................................... 250 6.10 ............... D N A f o l d i n g .............................................................................................................. 251 6.11 ............... S e l f - a s s e m b l y a n d c r y s t a l l i s a t i o n of p r o t e i n s ....................................................... 253 R e f e r e n c e s ........................................................................................................................................ 256

Chapter 7 Cytomembranes and Cubic Membrane Systems Revisited

257

7.1 ................. M e m b r a n e o r g a n i s a t i o n ........................................................................................... 257 7.2 ................ R e c o g n i t i o n o f h y p e r b o l i c p e r i o d i c c y t o m e m b r a n e m o r p h o l o g i e s i n e l e c t r o n ...................... m i c r o s c o p i c s e c t i o n s ................................................................................................. 259 7.3 ................ T h e s t r u c t u r e a n d o c c u r r e n c e of c u b i c m e m b r a n e s ............................................ 266 7.4 ................ C u b i c m e m b r a n e s in u n i c e l l u l a r o r g a n i s m s : p r o k a r y o t e s a n d p r o t o z o a ......... 272 7.5 ................ C u b i c m e m b r a n e s i n p l a n t s ..................................................................................... 275 7.6 ................ C u b i c m e m b r a n e s i n f u n g i ...................................................................................... 284 7.7 ................. C u b i c m e m b r a n e s i n m e t a z o a ................................................................................. 286 7.8 ................. R e l a t i o n s h i p s b e t w e e n t u b u l o r e t i c u l a r s t r u c t u r e s , ...................... a n n u l a t e l a m e i l a e , a n d c u b i c m e m b r a n e s ............................................................. 314 7.9 ................. B i o g e n e s i s of c u b i c m e m b r a n e s .............................................................................. 317 7.10 ............... R e l a t i o n s h i p s b e t w e e n c u b i c m e m b r a n e s a n d c u b i c p h a s e s .............................. 321 7.11 ............... F u n c t i o n a l i t i e s of c u b i c m e m b r a n e s ........ ~.............................................................. 323 7.12 ............... C e l l s p a c e o r g a n i s a t i o n a n d t o p o l o g y ................................................................... 324 7.13 .............. S p e c i f i c s t r u c t u r e - f u n c t i o n r e l a t i o n s ...................................................................... 327 A b b r e v i a t i o n s .................................................................................................................................... 330 R e f e r e n c e s ........................................................................................................................................ 331

Chapter

8

Some Miscellaneous Speculations

339

Templating 8.1.1 .............. T e m p l a f i n g a n d c u r v a t u r e : D N A t e m p l a t i n g ....................................................... 339 8.1.2 .............. T e m p l a t i n g b y electric fields: e q u i p o t e n t i a l a n d t a n g e n t i a l field surfaces ....... 339 8.1.3 .............. Diffusion w i t h i n fast-ion c o n d u c t o r s ..................................................................... 340 8.1.4 .............. T h e t e m p l a t i n g of zeolites ....................................................................................... 342 8.1.5 .............. T e m p l a t i n g o r g a n i c m o l e c u l e s : the c a e s i u m effect .............................................. 344 8.1.6 .............. T e m p l a t i n g of the m o r p h o l o g y of a calcite c r y s t a l ............................................... 344 Supra s e l f - a s s e m b l y 8.2.1 .............. Biological s u p e r s t r u c t u r e s b a s e d o n s e l f - a s s e m b l y .............................................. 348 8.2.2 .............. C o l l a g e n a n d p l a n t cell w a l l s .................................................................................. 349 8.2.3 .............. The m o l e c u l a r p a c k i n g in n a t i v e s t a r c h ................................................................. 350 8.2.4 .............. S a d d l e s in the kitchen: b r e a d f r o m w h e a t flour ................................................... 352 8.2.5 .............. M u s c l e c o n t r a c t i o n .................................................................................................... 355 8.3 ................. The origin of life: a role for c u b o s o m e s ? ................................................................ 359 8.4 ................. A final w o r d ............................................................................................................... 362 References ........................................................................................................................................ 363 I n d e x .................................................................................................................................................. 3 6 5

Chapter 1

The Mathematics of Curvature

1.1. Introductory remarks his book deals with shape and form, and especially the role of curvature in the natural sciences. Our search is for a connection between structure and function posed by D'Arcy Thompson in his famous book "On Growth and Form" [1] almost a century ago. Our theme will be that curvature, a neglected dimension, is central. Some of the curved surfaces that will preoccupy us and recur are shown in the Appendix to this Chapter. The reader is invited to pursue them at once. They are not just computer generated art or mathematical abstractions, and will be seen later to be ubiquitous in nature. They represent situations as diverse as:

T

*equipotential surfaces dividing space between the atoms of a crystal *real structures formed spontaneously by the constituent molecules of biological membranes *the shapes of bio-macromolecules, from proteins to starch

Euclidean geometry underlies practically all of science and our intuition has depended on it. The shapes provided: planes, cylinders, spheres, polyhedra, all have constant or even zero curvature. Only in theoretical physics, in subjects like general relativity where the curvature of space-time is essential, has non-Euclidean geometry and especially so-called hyperbolic geometry played any part in the scheme of things. The scientific community has been prepared to leave such matters to physicists alone. It can do so no longer, and the idea of curvature is becoming an essential tool to the understanding of many phenomena. This Chapter is concerned with some of the mathematical tools required to describe special properties of curved surfaces. The tools are to be found in differential geometry, analytical function theory, and topology. General references can be found at the end of the Chapter. The reader uninterested in the mathematics can skip the equations and their development. The ideas we want to focus on will be clear enough in the text. A particular class of saddleshaped (hyperbolic) surfaces called minimal surfaces will be treated with special attention since they are relatively s t r a i g h t f o r w a r d to treat mathematically and do form good approximate representations of actual physical and chemical structures.

2

1.2

Chapter I

Curvature

The concept of curvature was d e v e l o p e d by Isaac N e w t o n in the m i d d l e of the 17th century, as a natural extension to his w o r k on the calculus. At that time, the d e t e r m i n a t i o n of the p e r i m e t e r of p l a n a r c u r v e s a n d the area u n d e r curves w e r e major p r o b l e m s . In particular, N e w t o n ' s n e w analytical tools a l l o w e d h i m to d e t e r m i n e the "quadrature" (area) of a circle. It o c c u r r e d to N e w t o n that the radius of the circle of best fit to an arbitrary planar curve at all points on the curve w a s a useful measure, for w h i c h he coined the term "crookedness"[2]. This is curvature (Fig. 1.1).

Figure 1.1: The curvature of a planar curve at a point (P) is equal to the reciprocal of the radius of the circle of best fit to the curve at P, r. The c u r v a t u r e of a planar curve relates arc length along the curve to changes of tangent vector (Fig. 1.2).

~s

.x

Figure 1.2: Tangents TP and QT at two points, P and Q, on a planar curve.

Curvature

3

The tangents TP and QT in Fig. 1.2 subtend angles ~, p+Sp with the x-axis, so that 8 ~ is the angle between the two tangents. If 5s is the length of the arc PQ along the curve, then 8_E is the average curvature of the planar curve along 8s the arc PQ. The curvature at the point P is defined to be the limit of this expression as Q approaches P, i.e. 8-E. Ss

Figure 1.3: The radii of curvature, rp and rQ at two points, P and Q, on a planar curve. The centres of the circles of best fit to the curve at P and Q lie on opposite sides of the curve and the curvature changes sign at the point of inflection on the curve between these points. The curvature at Q is positive and at P it is negative.

If PQ is the arc of a circle of radius r, the angle ~ , between the tangents at P and Q is equal to the angle subtended at the centre of the circle by the arc PQ, d~ = ~. 1 The curvature is constant at all points of a so that 8s=r5 ~, whence ~ circle, and the radius is equal to the reciprocal of the curvature (Fig. 1.1). If the curve is described in cartesian coordinates by a function y=y(x): r = ds _

dl//

ds dx _ s e c ( I g ) ~--~-" tan(I//) = d)_.[ S O dx dlg dtp dx

SCC2(Vr )d~ _d~" dx

that:

and r -

dx=

d2y

dx2 The curvature, ic, is thus given by the expression:

K'=

dx2

[' ,~,j

(1.1)

4

Chapter I

If the positive value of the root of the denominator is taken, the sign of the curvature will be the same as that of d2y" i.e. positive if the curve lies above dx--5-, the tangent, and negative below it. At a point of inflection (or a straight line), d2~---2 ' is zero and therefore the curvature is zero in these cases. The sign of the dx2 curvature signifies the convex or concave nature of the curve. It is also related to the side of the curve at which the centre of the circle of best fit is located (cf. Fig. 1.3).

1.3

The differential geometry of surfaces

The curvatures of a surface are more complex entities, but can be understood as a generalisation of the c u r v a t u r e of planar curves. Imagine a plane containing a point P on the (smooth) surface, which contains the vector (n) passing through P, normal to the surface (Fig. 1.4).

Figure 1.4: The intersection of a surface with the plane containing the normal vector (n) to the surface at the point P. The intersection of this plane with the surface is clearly a planar curve, whose curvature at P can be evaluated as described above. This curvature is equal to the value of the normal curvature, ~:n, at P in the direction prescribed by the orientation of the plane. N o w let the plane rotate about an axis coincident with the normal vector, n. The n o r m a l curvature will vary periodically, so it must attain m a x i m u m and m i n i m u m values. These values are defined to be the principal curvatures, K1 and Ic2, of the surface at P (Fig. 1.5). The directions

Differential geometry

5

at which these extremes occur are referred to as the principal directions at P. In special cases, all these curves of intersection are of equal curvature (e.g. a point on a sphere), the point is an umbilic, and principal directions cannot be defined. If the n o r m a l curvatures at the umbilic are zero (so that all the intersection curves are straight lines) the surface is locally p l a n a r at that point, which is then called a fiat point. At regular points (excluding umbilics) the principal directions are orthogonal. The principal curvatures can be combined to give two useful measures of the c u r v a t u r e of the surface, the Gaussian curvature (K) and the mean curvature

(H): K = ~q.K'2 and H - rl+~:2 2

(1.2)

(a) Both principal curvatures are of equal sign.

(b) One principal curvature is equal to zero.

(c) Principal curvatures of opposite sign.

Figure 1.5: The extrema of normal curvatures define the

principal curvatures of a surface.

The surfaces in Fig. 1.5 h a v e (a) positive G a u s s i a n c u r v a t u r e , (b) zero Gaussian curvature and (c) negative Gaussian curvature. The Gaussian c u r v a t u r e has the dimensions of inverse area and the m e a n curvature has dimensions of inverse length. The topology of the surface

6

Chapter I

(introduced below) is related to a (dimensionless) measure of the integral geometry of the surface, the integral curvature, which is equal to the areaweighted integral of the Gaussian curvature over the surface, /

K da.

The Gaussian curvature and integral curvature bear a fascinating relation to the normal vectors on the surface, and belong to the realm of intrinsic geometry, i.e. the geometry that can be deducedwithout reference to the space within which the surface is embedded. Some further results on the intrinsic geometry of surfaces will be needed throughout the book. We outline them briefly below.

1.4

The Gauss map

The Gaussian curvature has a n u m b e r of interesting geometrical interpretations. One of the more striking is connected with the Gauss map of a surface, which maps the surface onto the unit sphere. The image of a point P on a surface x under the mapping is a point on the unit sphere. This point is given by the intersection of the unit normal n to the surface at P with a unit sphere centred at P. The Gauss map of the surface x is the collection of all such points on the sphere, generated by sliding the surface through the centre of the (fixed) sphere (Fig. 1.6). If a closed curve on a surface is traversed in the opposite sense on the sphere under the mapping, the surface is saddleshaped, and the Gaussian curvature is negative. Clearly the spherical image under the Gauss map of a highly curved surface patch will be larger than that of less curved patches of the same area, since the divergence in direction spanned by the normal vectors is wider for the highly curved patch. An extreme example is the plane, which is mapped onto a single point, whose location depends on the orientation of the plane. The Gauss map is closely related to the Gaussian curvature of the surface. In fact, the surface area of the Gauss-mapped region on the unit sphere is equal to the integral curvature of the region, J K da. An alternative definition of J~ufface the Gaussian curvature follows from this result. Imagine shrinking the region progressively to an infinitesimal area about a point. In the limit the quotient of the area of the surface element and its spherical image is 1/K. If the Gauss map of a surface comprises only a single point (e.g. the plane) or a curve (e.g. the cylinder), the Gaussian curvature is zero at all points on the surface.

The Gauss map

7

Figure 1.6: The Gauss map of a surface. The normal vectors in the triangular ABC region of the

saddle-shaped surface define a region on the unit sphere, A'B'C', given by the intersection of the unit sphere with the collection of normal vectors (each placed at the centre of a unit sphere) within the ABC region. Notice that for the example illustrated the bounding curve on the surface and on the unit sphere are traversed in opposite senses. This is a necessary feature of saddle-shaped surfaces, with negative Gaussian curvature. The G a u s s i a n curvature, K, is a b e n d i n g invariant. This m e a n s that if we can b e n d a s i m p l y connected surface x into a n o t h e r s i m p l y connected surface y w i t h o u t stretching or tearing, there exists a c o n t i n u o u s t r a n s f o r m a t i o n from x to y that preserves the Gaussian curvature at every point. For example, the plane, w i t h G a u s s i a n c u r v a t u r e , K = 0, is easily rolled into a cylinder for w h i c h also K - 0. On the other h a n d there is no w a y to form a sphere (K c o n s t a n t , b u t strictly p o s i t i v e ) f r o m e i t h e r c y l i n d e r or p l a n e w i t h o u t s t r e t c h i n g , t e a r i n g or gluing. Surfaces t h a t are related b y a c u r v a t u r e p r e s e r v i n g t r a n s f o r m a t i o n (like the p l a n e a n d the c y l i n d e r ) are called

isometric.

1.5

Geodesic curvature and geodesics

A n o t h e r entit3, that we shall need belongs to the realm of intrinsic geometry:

geodesic curvature. Consider a surface x, a point P on x and a curve ~ on x p a s s i n g t h r o u g h P. The c u r v a t u r e vector of ~ at P joins P to the centre of c u r v a t u r e of ~. This c u r v a t u r e vector m a y be d e c o m p o s e d into m u t u a l l y o r t h o g o n a l c o m p o n e n t s . These c o m p o n e n t s are given by projection of the

8

Chapter1

curve ~ onto two orthogonal planes: (i) the tangent plane to x at P and, (ii) the plane containing the normal vector to x at P and the tangent vector of ~ at P. The curvature of the latter projection is the normal curvature, lCn, introduced in section 1.3. The geodesic curvature, ~Cg of ~ at P on x is equal to the curvature of the projection of ~ onto the tangent plane to x at P (Fig. 1.7). If the geodesic curvature is zero, the curvature of ~ is identical to the normal curvature. A curve whose geodesic curvature is zero everywhere is called a geodesic, and it is (locally) the shortest distance between two points on the surface. Along geodesic curves, the normal vectors to the geodesic coincide with the normal vectors to the surfaces. An infinite number of geodesics passes through any point, one for every direction emanating from the point. Geodesics on curved surfaces are rarely straight lines. Geodesics on a cur~ed surface linking two points can be constructed by stretching a string (constrained to lie on the surface) between the p o i n t s - the path taken up by the string will always follow a geodesic.

Figure 1.7: Decomposition of a curve in a surface (left) into orthogonal geodesic and normal curvatures (right).

1.6

Torsion

One further measure of the bending of curves needs mention. A nonplanar space curve exhibits curvature (which is measured by the radius of the circle of best fit to the curve) and torsion.

Torsion

9

The torsion of a curve describes its pitch: a helix exhibits both c o n s t a n t c u r v a t u r e and torsion. Its c u r v a t u r e is m e a s u r e d by its projection in the t a n g e n t plane to the curve - which is a circle for a helix - while its torsion describes the degree of non-planarity of the curve. Thus a curve on a surface (even a geodesic), generally displays both curvature and torsion. The m e a s u r e of torsion is u n a m b i g u o u s for an isolated curve p i c t u r e d in three-dimensional space. H o w e v e r , the torsion of a curve lying in a surface has a more complicated p r o p e r t y related to the g e o m e t r y of the surface. The geodesic torsion (~g) is a further m e a s u r e of the local b e n d i n g of a surface curve c o m p l e m e n t a r y to the n o r m a l and geodesic curvatures. The geodesic torsion at a point on a surface in a certain direction is equal to the torsion of the geodesic on the surface t h r o u g h that point in that direction. This can be stated more formally as follows. A triple of orthogonal vectors can be defined at a n y point on a curve lying in a surface. This triple contains the n o r m a l vector to the surface at that point, n, the tangent vector to the curve at that point, t, and a vector orthogonal to both of those vectors, k n o w n as the geodesic normal vector, u = n x t. The rate of change of n w i t h arc l e n g t h along the curve projected onto t is equal to the normal curvature. The rate of change of n projected onto u is equal to the geodesic torsion, tg. The geodesic torsion thus c o m p l e m e n t s both the geodesic and n o r m a l c u r v a t u r e s in a natural way, although (along with the normal curvature) the geodesic torsion is not a concept of intrinsic geometry. All three m e a s u r e s of b e n d i n g of curves on surfaces can be unified by the Bonnet-Kovalevsky formulae: g~- =

ds

~gU +

ICnn

du =-g~gt+ tgn

ds

da

ds

=

-~nt - tgU

Both the normal curvature and the geodesic torsion of a curve on a surface d e p e n d on the variation of the normal vectors to the surface along the curve, w h i c h implies some unexpected results. For example, a straight line certainly displays neither curvature nor torsion. H o w e v e r , the geodesic torsion of a s t r a i g h t line vanishes only if the line is e m b e d d e d in a surface of z e r o G a u s s i a n curvature. The normal c u r v a t u r e a n d geodesic torsion of a c u r v e on a surface are related to the principal curvatures of the surface (ICl and Ic2) and the angle the curve subtends with a principal direction, r tcn : Ic, cos2(+) + to2 sin2(r

(1.3)

~:s = ( K',- K'2)r

(1.4)

~) sin(~)

cItapter I

10

For a surface characterised by ~c1=-~2, the Gaussian curvature is simply related to the normal curvature and geodesic torsion: K-/('n 2 + ~g2

(1.5)

In this case, the magnitude of the geodesic torsion at a point on a straight line lying in the surface is equal to the m a g n i t u d e of the principal curvatures of the surface at that point.

1.7

The Gauss-Bonnet theorem

The Gauss-Bonnet theorem is a p r o f o u n d theorem of differential geometry, linking global and local geometry. Consider a surface patch R, b o u n d e d by a set of m curves ~i- If the edges ~i meet at exterior angles Oi and they have geodesic curvature ~g(Si) where si labels a point on ~i then the theorem says

/('g(Si) dsi + ~-' boundaD"

R

K da + ~ Oi = 211: i=1

(1.6)

Figure 1.8 illustrates the case for a surface patch consisting of four b o u n d a r y arcs, ~I=AB, ~2=BC, ~3=CD and ~4=DA.

Figure 1.8: Four arcs belonging to a surface. From the Gauss-Bonnet theorem, the integral curvature within the region of the surface bounded by the arcs (ABCD) is determined by the vertex angles (0/) and the geodesic curvature along the arcs AB, BC, CD and DA.

The Gauss-Bonnet theorem

11

Choose a triangle traced on a surface, whose three edges are geodesics. From the theorem, we have 3 [ Kda + ~_~ 0i= 2~: R i=1

(1.7)

The external angles, e i, are related to the internal angles, ai, by e i = ~ - a l so that the area of the geodesic triangle is: 3

~_~ a i = ~ + ~ Kda i=1 R

(1.8)

Thus, for a Euclidean triangle (which is located on a surface of zero Gaussian curvature, such as the plane), the s u m of the vertex angles of a triangle is indeed ~. However, if the triangle decorates a surface of negative Gaussian curvature, the s u m of the angles is less than ~, if the integral surface has positive integral curvature, the s u m of the angles exceeds ~. The angle "excess" (i.e. its difference from ~) is thus a measure of the integral curvature within the region b o u n d e d by the geodesic edges.

1.8

Topology

G e o m e t r y , differential or otherwise, deals w i t h the metric relationships of rigid objects. There are s o m e f u n d a m e n t a l aspects of shapes that are preserved if the objects studied consist of stretchable rubber sheets. A rubber sphere m a y be deformed into an ellipsoid, or a long, n a r r o w cylinder with caps, or indeed any globular object (Fig 1.9).

Figure 1.9: A sphere can be stretched and bent (without any rupture or fusion of the surface) into an infinite variety of globular surfaces, all topologically equivalent.

ChapterI

12

Similarly, a torus can be d e f o r m e d into any two-sided surface containing a single handle, or a single hole (Fig 1.10).

Figure 1.10: A donut-shaped torus (with a single hole) can be deformed into any two-sided surface containing a single handle, such as a cup. Surface topology is "rubber sheet" geometry, since only those geometrical characteristics of the surface that are maintained u p o n stretching or squeezing are relevant. The usual geometrical notions of area, length, etc., are excluded from topological analysis. Suppose a surface, x, is facetted, subdividing x into a n u m b e r of faces, edges ( b o u n d i n g the faces) and vertices. Denote the n u m b e r of faces by F, the n u m b e r of their edges by E and the n u m b e r of vertices by V. Descartes, a n d later Euler, discovered that (F-E+VO =2 for all polyhedra (Table 1.1). Table 1.1: Relation between numbers of faces (F), edges (E) and vertices (lO of conventional polyhedra. F

E

V

4 6 8 12 20

6 12 12 30 30

4 8 6 20 12

(F-E+V) 2 2 2 2 2

polyhedron tetrahedron cube octahedron dodecahedron icosahedron

Topology

13

This p r o p e r t y holds because (F-E+V) is a topological characteristic, d e p e n d e n t o n l y o n the t o p o l o g y of the f a c e t t e d surface. Since all p o l y h e d r a are topologically e q u i v a l e n t to the s p h e r e (Fig 1.9), (F-E+V) is c o n s e r v e d . The value of this integer is k n o w n as the Euler-Poincar6 characteristic Z(x): F - E + V = Z (x)

(1.9)

N o w imagine a net lying in a s m o o t h surface. This net also facets the surface into c u r v e d faces, c u r v e d edges a n d vertices. If each n o d e o n the n e t has z edges (so that the net is z-connected), a n d the ring size of each ring in the net is n, Euler's relation for the net is: F = Vz.

E =

(1.!0)

so that -- = n + z 1

Table 1.2: Relation between average ring size, n, connectivity, z, and Euler-Poincar~ characteristic per vertex, x/V, for a range of networks that are regular tessellations of surfaces 9 The nature of the network is set by the value of z/V: cages, planar networks and threedimensional frameworks are characterised by positive, zero and negative z/V respectively.

z

n

3 3 3 4 5

3 4 5 3 3

3 4 6 3 3 9

network

type

3/2 1 1/2 1 1/2

tetrahedron cube dodecahedron octahedron icosahedron

cage " " " "

6 4 3

0 0 0

hexagonal tiling square thing triangular thing

sheet " "

7 8

-1/2 -1

"skew" polyhedra

9

4 9

5 ~

X/V

framework ,I

o

-1 o

This result implies that the ring size a n d connectivity of a n e t w o r k d e t e r m i n e the t o p o l o g y of the surface w h i c h c o n t a i n s t h a t n e t w o r k . This a l l o w s for simple characterisation of cage, sheet a n d f r a m e w o r k nets, distinguishable b y the value of their Euler-Poincar~ characteristic (Table 1.2).

A n o t h e r topological characteristic, the genus, g(x), of a surface, is a m e a s u r e of its connectedness. It is equal to the n u m b e r of holes or handles in the surface a n d simply related to the Euler-Poincar~ characteristic by

14

Chapter I

g(x) = [2- X'(x)]/2

(1.11)

(The equation applies only to 'orientable' surfaces, those with distinct sides. This excludes one-sided surfaces, such as the M6bius strip.) Thus, a sphere has genus zero. A torus (or a sphere with one handle) has genus one, and so on.

Remarkably, the topology is linked to the integral curvature of a surface by the simple equation: K da = 2~X

(1.12)

lurfac~ This means that all surfaces with the same number of handles or holes have the same integral curvature! In other words, no amount of bending or squeezing of a surface can add any net integral Gaussian curvature to that displayed by the simplest topologically equivalent surface. For example, although regions of negative Gaussian curvature can be formed in a sphere by squeezing the surface to produce a region which is saddle-shaped, this contribution to the integral curvature will always be compensated by a corresponding positive integral curvature in other regions. A more general, global version of the Gauss-Bonnet theorem can now be stated: Let x be an oriented surface and R be a bounded region of x. As before, let the boundary of R be the union of m simple curves ~i that do not selfintersect, and let 0i be the external angles at the m vertices. Then we have

E

rg d~i +

i= !

naary

f K da + ~,~m Oi = 2~z(R)

(1.13)

i= I

1.9 A provisional catalogue of surface forms It is obviously impossible to offer a complete catalogue of curves. Similarly, no comprehensive list of surface forms can be drawn up. The language of surface shape is a rich one: some familiar forms like the sphere and the plane, are deeply imbued in our consciousness, while others remain difficult to describe and visualise in terms that are intuitively reasonable to all of us raised on the limited vocabulary afforded by the simpler forms. But some attempt at exhaustive classification is necessary: the wealth of form in natural structures draws on the richness of abstract form, so that if we are to

A catalogue of surfaces

15

understand natural structures, it is necessary to obtain as full an intuition about surface forms as possible. To do this we must draw on non-Euclidean geometries. These different geometries emerge if the "parallel postulate" of Euclid's "Elements" is no longer taken as axiomatic. Following the work of Bolyai, Lobachevsky, Gauss and Riemann early last century it became clear that if the parallel postulate is relaxed, three quite distinct geometric classes exist. The three classes are elliptic (single and double), parabolic (Euclidean) and hyperbolic. Riemann's great contribution to geometry lay with his program of abstracting and relating the notion of form to the concept of differential form. The Riemannian approach classifies a geometry by its functional (pointwise) structure - characterised by the curvatures and metric. The triad of local surface geometries is characterised by Gaussian curvature: elliptic shapes have positive Gaussian curvature, Euclidean shapes have zero Gaussian curvature and hyperbolic shapes have negative Gaussian curvature. The vocabulary of surface forms can be developed by marrying this local geometry to (global) topology. Any surface form can be characterised by its intrinsic shape (its metric and curvature) and its global (extrinsic) embedding in space. Since natural structures are embedded in our particular space, we are concerned in this book with two-dimensional (surface) embeddings in threedimensional Euclidean space. (We leave the cosmologists to debate the nature of the deviations from Euclidean space due to matter. However, it is certain that at the length scales we deal with here, the space we live in can be safely approximated by Euclidean space.) Thus, although the embedding space is always approximately Euclidean, its intrinsic geometry may be locally Euclidean (parabolic) or non-Euclidean, viz. elliptic or hyperbolic. In the vicinity of elliptic points, the surface can be fitted to an ellipsoid, whose radii of curvature are equal to those at that point (Fig. 1.5a). The surface lies entirely to one side of its tangent plane, it is "synclastic" and both curvatures have the same sign. About a parabolic point, the surface resembles a cylinder, of radius equal to the inverse of the single nonzero principal curvature (Fig. 1.5b). Hyperbolic ("anticlastic") points can be fitted to a saddle, which is concave in some directions, fiat in others, and convex in others (Fig. 1.5c). At hyperbolic points, the surface lies both above and below its tangent plane. The most familiar surfaces have constant Gaussian curvature over the surface. In our space this is only possible in elliptic and parabolic cases: the sphere and the cylinder and plane respectively. It is impossible to form hyperbolic surfaces of constant (negative) Gaussian curvature without singularities. In general surfaces contain elliptic, parabolic and hyperbolic regions. The "average geometry" of a surface can be characterised by the average value of its Gaussian curvature, . This is equal to the integral curvature divided by the surface area A:

16

C/u~ter 1

L (K>=

Kda rfac~

(1.14)

m'l'~:e The integral curvature of a surface is linked to the Euler-Poincar~ characteristic of that surface (X) by eq. (1.12). This allows the average geometry of orientable surfaces to be related to the number of holes or handles, characterised by the surface genus, g, and the area of the surface, A, by the relation: 2-2g {K}-- A

(1.15)

Figure 1.11: A random sponge-like surface. (Picture courtesy of Peter Pieruschka.)

Thus, surfaces that are free of holes or handles like the sphere have positive and X; they are elliptic. Surfaces with a single handle or a hole (a donutshaped torus, or - plus or minus a single p o i n t - a cylinder or a plane) are on average parabolic, or two-dimensionally Euclidean (=x=O).All surfaces with more than one hole (or handle), are hyperbolic (negative , X)-

A catalogue of surfaces

17

Within a topological context, most surfaces are hyperbolic, yet our knowledge of this geometric class is less developed than for the other two classes. This book is entirely concerned with these hyperbolic surface geometries. In particular, we focus on the most exotic topological species within the hyperbolic realm, namely surfaces of infinite genus. The most general surface within this class is a boundless sponge structure (Fig. 1.11). The mathematics of such random surfaces is at present poorly developed. We confine our treatment to the more tractable cases of periodic surfaces of unbounded genus.

Figure 1.12: A square mesh surface, which is two-periodic.

Periodic hyperbolic surfaces of infinite genus can be further subdivided into those which exhibit a lattice in one-dimension (one-periodic surfaces), twoperiodic and three-periodic surfaces. These periodic surfaces can be divided into equivalent regions, bounded by a unit cell of space that contains the smallest region of the surface that reproduces the complete surface upon translation of the unit cell alone. For geometric purposes, the surface contained within the unit cell (also called a "lattice fundamental region", to distinguish it from a unit cell of the space group to which the surface belongs) is characteristic of the complete surface. To form connected surfaces of unbounded genus, one-periodic surfaces must have genus per unit cell at least equal to unity, two-periodic surfaces must contain genus two (or higher) surfaces within the unit cell and three-periodic surfaces must have a genus per unit cell greater than or equal to three. One-periodic surfaces are relatively unexplored to date. (The double strands of DNA lie on the simplest one-periodic surface, the helicoid.) Two-periodic surfaces deserve some comment. The most interesting examples of these surfaces can be visualised as confined between two parallel bounding planes, with a regular network of pores joining the two parallel sheets. We call these surfaces "mesh surfaces", due to their characteristic twodimensional porous network, which resembles a mesh. A square mesh surface is shown in Fig. 1.12. The mean curvature of these surfaces can be

18

CA~ter1

constant [3], although it cannot equal zero. The inner and outer volumes on either side of these surfaces are quite different: the exterior volume consists of two half spaces, interconnected via a lattice of tunnels and the interior is a two-dimensional tubular network. Three-periodic hyperbolic surfaces of infinite genus carve space into two intertwined sub-volumes, both resembling three-dimensional arrays of interconnected tubes. They are simple candidates for the interfaces in bicontinuous structures, consisting of two continuous subvolumes [4, 5]. As such they have attracted great interest as models for microstructured complex fluid interfaces, biological membranes, and structures of condensed atomic and molecular systems, to be explored in subsequent Chapters. The simplest three-periodic hyperbolic surfaces are "Infinite Periodic Minimal Surfaces" (IPMS, named by Alan Schoen [6]). For these surfaces, the mean curvature is constant on the surface, and everywhere identically zero. This is a defining characteristic of minimal surfaces. For these structures, the sub-volumes can be geometrically identical. This occurs if the IPMS contains straight lines. Such surfaces have been called '~alance surfaces" by Koch and Fischer [7]. We focus primarily on IPMS in this book. Some further discussion of general properties of minimal surfaces is in order here, since a number of their geometrical and topological properties will be required for later chapters.

1.10 A historical perspective The study of minimal surfaces arose naturally in the development of the calculus of variations. The problem of finding the surface forming the smallest area for a given perimeter was first posed by Lagrange in 1762, in the appendix of a famous paper that established the variational calculus [8]. He showed that a necessary condition for the existence of such a surface is the equation __~

Zx

+

ax 41 + z~ + z,? .

-

=0

(1.16)

I+Z?+

Here Z---Z(x,y) is the equation of any surface bounded by the perimeter; Z x, Zy denote partial derivatives. The partial differential equation for a surface is then: (1 + Z ~ Z x x - 2 ZxZvZxy + (1 + Z 2 ~ y y = O .

(1.17)

Lagrange pointed out that the plane would be a trivial solution to the equation but made no further investigations to see what other possibilities existed.

History of minimal surfaces

19

In 1744 Euler discovered the catenoid, the first non-planar minimal surface. This surface is readily realised by a soap film, spanning coaxial circular b o u n d i n g wires. The film shrinks under the action of its surface tension, forming the minimal surface (Fig. 1.13).

Figure 1.13. (Top): A region of the catenoid formed by a soap film. (Bottom left): Computergraphics image of a portion of a catenoid. (Bottom right): larger view showing "trumpet" ends.

The link between curvature and minimal surfaces was made by Meusnier in 1776 [9] . He proved that eq. (1.16) implies that the mean curvature is zero everywhere on a minimal surface. In his own words: "la surface de moindre ~tendue entre ses limites a cette propri~td, que chaque element a ses deux rayons de Courbure de signe contraire & egaux". This is the defining property of a minimal surface;

20

Chapter I

For a minimal surface, the principal curvatures are equal, but opposite in sign at every point. The Gaussian curvature is then always non-positive, and the mean curvature is zero. In addition to the catenoid, Meusnier also found a further non-trivial solution to eq. (1.16), the helicoid, shown below.

Figure 1.14: The helicoid, one of the earliest discovered examples of minimal surfaces.

The helicoid is the only minimal surface built up entirely of straight lines (a ruled surface) and the catenoid is the only minimal surface of revolution. These surfaces are related through the Bonnet transformation that will be discussed later. The term "minimal" is misleading. (The terminology is however timehonoured and we shall live with it.) In fact, the differential equation (16) is satisfied by any surface that constitutes a local critical point to the area function (i.e. a minimum or a maximum). The minimal area property of minimal surfaces is characteristic only of a finite patch of the surface with prescribed boundary. For the systems that concern us in subsequent chapters, this area property is irrelevant. It is the curvature characteristic of minimal surfaces that is important. On any surface, the principal directions are mutually orthogonal at regular points (recall section 1.3). On minimal surfaces, this is true for asymptotic directions as well. (An asymptotic direction is that along which the normal curvature vanishes.) Orthogonality of the asymptotic directions can be shown

History of minimal surfaces

21

to be a requirement that is equivalent to that of zero mean curvature. Hence the orthogonality property can be used to define minimal surfaces. Gauss' paper of 1827 "Disquisiones generales circa superficies' curvas" [10] marks the birth of differential geometry. Following the advances of Gauss, it became possible to deal with surfaces by their intrinsic geometry, which includes those surface features that can be determined without reference to the external space containing the surface. A central tool introduced by Gauss was the Gauss map, discussed in section 1.4. The Gauss map of a surface is conformal (angle-preserving and representable by a complex analytic function) if and only if the surface is a sphere or a minimal surface. This property is a very useful one, since it allows minimal surfaces to be analysed from their Gauss map. The Gauss map of a surface exhibits singularities at special points on a surface, k n o w n as umbilics. On minimal surfaces, the umbilics are easily recognised: they are the points on the surface where its Gaussian curvature is equal to zero, and the surface is locally planar. The nature of these flat points can vary, leading to distinct classes of singularities in the Gauss map. It turns out that the distribution of these singularities in the Gauss map uniquely determines the (intrinsic) geometry of minimal surfaces.

1.11 Periodic minimal

surfaces

Following the discovery of the helicoid by Meusnier, sixty years elapsed before additional examples of minimal surfaces were given in explicit form. In 1835 Scherk published five more examples of minimal surfaces [11]. Of these, two are periodic. They are often referred to simply as Scherk's first and second surfaces (Fig. 1.15).* Two of the leading mathematicians of the nineteenth century, Riemann and Weierstrass, played crucial roles in the further d e v e l o p m e n t of minimal surface theory. Their most i m p o r t a n t result, for which Weierstrass is accredited, is the triplet of integrals k n o w n as the Weierstrass equations. These e q u a t i o n s offer a useful route to periodic minimal surfaces. A summary of the theory follows. A minimal surface can be represented (locally) by a set of three integrals. They represent the inverse of a m a p p i n g from the minimal surface to a Riemann surface. The mapping is a composite one; first the minimal surface is mapped onto the unit sphere (the Gauss map), then the sphere itself is mapped onto the complex plane by stereographic projection. Under these operations, the minimal surface is transformed into a multi-sheeted covering of the complex plane. Any point on the minimal surface (except fiat points), characterised by cartesian coordinates (x,y,z) is described by the complex n u m b e r co, which

*According to Nitsche [28,29], this terminology is grossly misleading, and has occasioned much debate among those concerned with such matters. Such is life.

22

Chapter I

defines its m a p p e d location in the complex plane. The value of ca is dependent only on the surface orientation of (x,y,z), via the Gauss map (Fig. 1.16).

Figure 1.15: Scherk's first and fifth minimal surfaces, d i ~ o v e r e d in 1831. The first is twoperiodic, the second one-periodic.

The Weierstrass equations allow calculation of the cartesian coordinates ((x,y,z) with respect to an origin (xo,Yo,zo)) of the minimal surface at all points on the surface - except flat points - in terms of a complex analytic function R(60). The Weierstrass equations are:

x = x0 + R e

e ie (1 - 0 2) R(60) do)

Y = Yo + Im

e i8 (1 + 602) R(60) d60

z = zo- Re

e ia (260)R(60)d60

(1.18)

Integration is carried out on an arbitrary path from 60o to 601 in the complex plane, for a fixed value of 0 between 0 and n/2. Any analytic function R(60) can be plugged into the equations, to give a minimal surface. Further, the function R(60) uniquely determines a family of surfaces related by the Bonnet transformation as 0 varies. The converse does not hold; any surface will support an infinity of functions R(60) corresponding to different orientations of the original surface. The essence of the problem in elucidating interesting minimal surfaces lies in the choice of suitable functions, R(60).

Periodic minimal surfaces

23

Figure 1.16: Mapping of a minimal surface from real space to the complex plane. A point P on the surface, whose normal vector at P is n, is transformed to a point P' by the Gauss map, given by the intersection of n (placed at the origin of the unit sphere O) with the sphere. P' is mapped into a point P" on the complex plane (real and imaginary axes o"and i"resp.) by stereographic projection from the north pole of the sphere, N, onto the complex plane, which intersects the sphere in its equator.

Figure 1.17: (a) The minimal surface spanning four straight edges of equal length, subtending vertex angles of 60". (b) "Monkey saddle", formed by six copies of the saddle shown in (a). Each copy is related to its adjacent one by rotation of 180" about the common straight edge. These engravings are from Schwarz' original paper. The w o r k of W e i e r s t r a s s a n d R i e m a n n on a n a l y t i c f u n c t i o n s a n d s u r f a c e g e o m e t r y p r o v i d e d the setting for the w o r k of S c h w a r z w h o p i o n e e r e d the s t u d y of three-periodic m i n i m a l surfaces (IPMS). Schwarz, a s t u d e n t of

24

Chapter 1

Weierstrass, worked out the Weierstrass representation (eqs. 18) for two of the simplest IPMS, now called the P- and D-surfaces. (The latter is sometimes called the F-surface.) He also described three further three-periodic minimal surfaces, the CLP-(Crossed Layers of Parallels), H-, and T- surfaces. (The last is also called "Gergonnes surface"). These IPMS are illustrated in the Appendix. For the first time, the extraordinary complexity of these surfaces was revealed [12]. For the next half-century, his work was extended by others.

Figure 1.17(c): A single node of tile three-periodlc [)-'~urface. Four tunnels Ion one side ~f the surface) meet at each node, at angle~ of 10~.5 . Image courtes\' of David Anderson. (d): A model of a portion of the D-surface. The surface partitions space into two interpenetrating open labyrinths, each lying on a d i a m o n d lattice.

Schwarz' mastery of complex analysis was certainly responsible for his prizewinning work. (He was awarded a medal from the Prussian Academy for his original essay outlining the analysis and geometry of these surfaces.) However, much of the analytical groundwork had already been laid out by his predecessors. In particular, Riemann had derived representations for a number of minimal surface patches of IPMS, apparently without realising the extraordinary complexity of the complete IPMS. Schwarz established that the extended surface can be generated from a small surface patch by a simple procedure, at least in special cases. In particular, if the surface patch is bounded by straight lines, the patch can be extended beyond the straight edge by rotation about that edge through 180 ~. Thus, for example, the saddle spanning four straight edges (each subtending and angle of 60") (Fig. 1.17(a)) leads to a "monkey saddle" (Fig. 1.17(b)) with three peaks and three valleys (one for each leg of the monkey and one for the tail.) Further rotation of the monkey saddle leads to the formation of a continuous hyperbolic surface that partitions space into two interpenetrating networks of tunnels. Each network consists of nodes connecting four tunnels meeting at tetrahedral angles (109.5 ~ (Fig. 1.17(c)). The nodes are arranged on

Periodic minimal surfaces

25

a cubic lattice, which is equivalent to the diamond network (Fig. 1.17(d)). An intuitive picture of the formation of the surface can be seen as follows. Imagine a network model of the diamond structure (Fig. 1.17(e)), blue lattice), constructed from rubber tubes. Now inflate the network, swelling the hollow tubes. The resulting structure is a curved continuous network, enclosing the tunnels in the diamond network. If the inflation procedure is continued, the surface closes up around a complementary diamond network. The D-surface is the "half-way point" during the procedure.

Figure 1.17: (e) Computer image of a surface produced by partial inflations of a diamond network (blue). The "outside" of the surface wraps around a complementary (red) diamond lattice. Picture courtesy of David Anderson.

The P-surface can be found by inflating a simple cubic network, where the tubes connect nearest neighbours, giving a cubic array of nodes, each linking six tunnels (Fig. 1.18(a,b,c)).

Figure 1.18. Portions of the P-surface. (a): a single "unit cell", (b): four unit cells.

26

Chapter I

Figure 1.18(c): Computer image of 6x6x6 conventional unit cells of the P-surface (courtesy P. Pieruschka).

Other IPMS which are free of self-intersections can likewise be described by their tunnel geometries. In general, minimal surfaces display self-intersections. The most usual cases are surfaces that intersect themselves everywhere, and the "surface" wraps onto itself repeatedly, eventually densely filling the embedding space. We are only interested in translationally (or orientationally) periodic minimal surfaces, which are free of self-intersections (thereby generating a bicontinuous geometry) or periodic surfaces with limited self-intersections. Elucidation of these cases of interest requires judicious choice of the complex function R(r in the Weierstrass equations (1.18). Remarkably, the D- and P-surfaces are related by the Bonnet transformation. They are both described by the same function in the Weierstrass equations, shown by Schwarz to be:

R(ca)= 1/~/(1-14r162 The "Bonnet angle", 0, is equal to 0 for the D-surface and n/2 for the P. The function R(r used by Schwarz for the D- and P- surfaces is simply the inverse of the square root of the product over the images of the fiat points under the map from the minimal surface to the complex plane. Using r i to denote the fiat point images, the representation can be written: 8

= [I

(1.19)

Periodic minimal surfaces

27

This product form for the Weierstrass polynomial is readily generalised, and offers a useful route to the discovery and parametrisation of three- periodic minimal surfaces (IPMS). It turns out that for all "regular surfaces" (which are the topologically simplest IPMS), the distribution and character of the fiat point images (the location and type of the branch points (of R(~o)) in the complex plane) alone suffice to construct the Weierstrass polynomial, and thus the complete IPMS, using the Weierstrass equations. The Gauss map of an IPMS (which is a function of the surface orientation only through the normal vectors) must be periodic, since a translationally periodic surface is necessarily orientationally periodic. (The converse, however, is not true.) Consequently, the Gauss map of IPMS must lead to periodic tilings of the sphere. This principle has been used to construct all the simpler IPMS, and has recently been generalised to allow explicit parametrisation of more complex "irregular" IPMS [13-24]. Some of these examples are illustrated in the Appendix to this chapter.

1.12

The Bonnet transformation: the P-surface, the D-surface and the gyroid.

The variety of curves that can be traced out by a length of string is endless. This freedom of movement does not extend to surfaces. Most surface patches can only be moulded into each other with some accompanying change in the area of the sheet. However, anyone who rolls up a sheet of paper knows that a plane may be deformed into a cylinder, and it may also be rolled into a cone. This can be done without any stretching or squashing of the sheet. Thus all angles and lengths on the surface are preserved under this operation, which is known in mathematical jargon as an isometry. All those properties that belong to the intrinsic geometry of the surface remain unchanged, including the Gaussian curvature. While the plane can be "developed" into a cylinder or a cone, a sphere cannot be isometrically deformed at all; a property called "the rigidity of the sphere". Among those surfaces that are able to be deformed isometrically are minimal surfaces and surfaces of constant mean curvature. In 1853 Ossian Bonnet discovered the isometric transformation between minimal surfaces that bears his name. He wrote: "Chaque surface minima a une conjugu~e qui est d~veloppable sur elle" (every minimal surface has a "conjugate" minimal surface into which it can be developed). The distinct sequence of deformations is characterised mathematically by the variation in the Bonnet angle 0 in eqs. (18). The simplest example of this Bonnet transformation is the bending of the catenoid into the helicoid. (Fig. 1.19).

28

Chapter I

Figure 1.19: The isometric Bonnet transformation acting on a helicoid (top left), produces a catenoid (bottom right). This transformation leaves the intrinsic geometry intact (e.g. Gaussian curvature, lengths and angles defined at corresponding points) and preserves the zero mean curvature (characteristic of minimal surfaces) throughout, although the global embedding of the minimal surfaces changes dramatically. If the process is continued beyond the catenoid, a helicoid of opposite handedness is generated.

The identification of Bonnet-related minimal surfaces is facilitated using the Gauss map of the surface. A minimal surface is characterised by its fiat points, so any two surfaces with the same collection of flat point images (and singularity types) under the Gauss map (within a rigid body reorientation of the surface) must be associated by the Bonnet transformation. Thus, for example, the P- and D- surfaces, which have an infinite number of flat points, display only eight distinct fiat points under the Gauss map due to the eight distinct surface normal vectors at the fiat points. If the origin of each normal vector is placed at the centre of a cube, the endpoints of the vectors point to the eight vertices of a cube in both the P- and D-surfaces. This feature is clearest in the P-surface, shown in Fig. 1.20. Due to their identical distribution of normal vectors at the flat points (and type of fiat points), the P- and D-surfaces are "adjoint" (or, to quote Bonnet, "conjugate"), their associated Bonnet angles differ by 7t/2 (Figs. 1.22). In the 1960's Schoen [6] made the remarkable discovery that a third intersection-free IPMS- the gyroid - is generated during the transformation (at a Bonnet angle of approximately 38~ Generic minimal surfaces formed during this transformation are self-intersecting, and aperiodic.

The Bonnet transformation

29

Figure 1.20: A unit cell of the P-surface, embedded in a cube. The normal vectors to the P-surface at its eight fiat points (one obscured) are indicated by the arrowed vectors. These vectors point towards the eight vertices of the cube.

Figure 1.21: The gyroid surface discovered by Schoen in the 1960's. (Top: view down [111] axis of a larger partion of the surface; bottom: solid model.)

30

Chapter 1

The general features of the Bonnet t r a n s f o r m a t i o n can be seen in the simplest e x a m p l e , n a m e l y the i s o m e t r y b e t w e e n the catenoid a n d the helicoid (Fig. 1.19). U n d e r the action of the transformation, each point on the surface traces an ellipse in space, c e n t r e d at the origin. If the C a r t e s i a n c o o r d i n a t e s of identical points on adjoint surfaces are (x,y,z) and (x",y",z"), the coordinates of an associate surface, characterised by a Bonnet angle of e are: (x',y',z') = (x,y,z) cos(e) + (x",y",z") sin(0)

Figure 1.22: (a) The Bonnet transformation relating the D-surface, the gyroid and the P-surface. A simple saddle of the D-surface (bounded by an alternative set of straight lines to those in Fig. 1.17(a)) is twisted, ultimately forming a unit bounded by planar curves, which is a portion of the P-surface.

The Bonnet transformation

31

The effect of the Bonnet t r a n s f o r m a t i o n on IPMS' is to t r a n s f o r m the lattice of catenoidal channels - characteristic of IPMS - into helicoidal strips, t h r o u g h a screw operation on the w h o l e surface. For example, the channels in b o t h the P- a n d D-surfaces are t r a n s f o r m e d into spiral tunnels in the gyroid. Due to its intermediate Bonnet angle w i t h respect to the P- and D-surfaces, it lacks straight lines (2-fold axes) and m i r r o r planes. The labyrinths on both sides are e n a n t i o m o r p h i c : one l a b y r i n t h is l e f t - h a n d e d a n d the other r i g h t - h a n d e d (Fig. 1.21). The P-surface, the D-surface and the gyroid, are the simplest m e m b e r s of a large family of structures w h o s e m e m b e r s are still being identified. In m a n y w a y s these three surfaces are the m o s t important: they have b e e n identified in a variety of physical systems, from silicates to cells. Their i m p o r t a n c e can be traced to the fact that they are the most h o m o g e n e o u s IPMS, w i t h the s m o o t h e s t and s m a l l e s t variations (suitably scaled) of G a u s s i a n c u r v a t u r e over the surface. This feature is d u e to their relative paucity of fiat points: they h a v e the m i n i m u m n u m b e r of fiat points per unit cell of all IPMS. (In a topological sense they are the least complex IPMS, displaying the m i n i m u m genus per unit cell of all IPMS - three.)

Figure 1.22(b): Bonnet transformation acting on a number of unit cells of the D-surface (top), viewed along a axis. (The tunnels of the D-surface are obscured in this orientation.) Successive images are for Bonnet angles of 10~ 20~ 30~ 38.015~ and 90~ The transformation "locks in" to the gyroid at a Bonnet angle of 38.015~ Continuation of the transformation beyond the gyroid leads to the P-surface, with tunnels along the directions (lower image). The images are all scaled equivalently.

Chapter I

32

1.13

Parallel surfaces

Two surfaces x and y are parallel if they have an identical distribution of normal vectors; i.e. their Gauss maps are indistinguishable. Thus, a family of parallel surfaces can be produced by translating a surface in the direction of its normal vectors by an equal amount everywhere on the surface. If x and y are parallel surfaces separated by a distance c, it can be shown that their Gaussian and mean curvatures are related by: Ky = Kx/(1 + 2H x + C2Kx)

(1.20)

Hy = (H x - cKx)/(1 + 2H x + c2Kx)

Two interesting conclusions can be drawn from these formulae. First, the Gaussian and mean curvatures of surfaces parallel to minimal surfaces related by a Bonnet transformation remain unchanged. Secondly, the Gaussian curvature of a surface parallel to a minimal surface increases with c. This means that the minimal surface has a larger area than related parallel surfaces.

1.14

Future directions

There has been a resurgence of mathematical research into minimal surfaces in recent years. A number of new complete minimal surfaces [25, 26] have been elucidated, which will surely lead to novel classes of IPMS. Developments have made the parametrisation of more complicated periodic minimal surfaces possible (see Appendix), and the beautiful work of Karcher and colleagues at Bonn [14, 27] and Fischer and Koch [7] at Marburg has led to a plethora of new minimal surfaces containing straight lines free of selfintersections. Clearly, many new hyperbolic surfaces of infinite genus remain to be discovered, and the program initiated by Meusnier, Gauss, Riemann and Schwarz is now well underway.

33

Appendix: A catalogue of some minimal surfaces Let us finish this chapter with a catalogue of some minimal surfaces that can be generated using the Weierstrass parametrisation (eqs. (1.18)). We start by imposing some extra conditions on the function R(ta), which defines the geometry of the minimal surfaces via the Weierstrass equations (1.18). We will consider only those surfaces whose Gaussian curvature is everywhere finite and whose fiat points are isolated. Since the orientation of the surface in space is arbitrary, the north pole on the Gauss sphere (which maps out the point at infinity in the complex plane u n d e r stereographic projection) may always be chosen to have nonzero Gaussian curvature. The Gaussian curvature within the Weierstrass representation is given by [28, 29]: K=-41 R(0))1-2(1 +1 r J2). This imposes an asymptotic behaviour of the function R(t0). lim(K) = lim {- 4 IR(t0~ (1 + ]o~)-4} = C ta-.r

where C is some negative, finite number. This implies lim IR(ta~ =lim (1+]a~)2 = to4 tO--r

ta--r

Then we can consider a general form

R(co) = ~ I a (ca- r -a /b

(1.21)

/=1

This representation enables a catalogue of minimal surfaces to be built. The least n u m b e r of flat points a minimal surface may possess is one. An example of such a surface is Enneper's surface, which is asymptotically flat (Fig 1.23). In this case the fiat point is not an isolated point on the surface. However, only a single surface orientation is displayed by the asymptotically flat boundary. The surface is unique in that the Bonnet transformation applied to this surface does not produce distinct surfaces. All the members of the isometric family related to Enneper's surface are equivalent - they differ only in their relative orientation in space. The Weierstrass parametrisation for this surface is given by R ( tO)Enneper = ( to- too)-4

( 1.22)

34

Chapter I

where too is an arbitrary complex constant.

Figure 1.23: Enneper's minimal surface, the simplest minimal surface. (Note that this surface is self-intersecting.)

The next possibility is a Gauss map containing two singularities due to fiat points. Examples of this case are the helicoid and the catenoid (Figs. 1.13 and 1.14). The normals of the fiat points on these surfaces (at the asymptotic ends of the surfaces) are antiparallel, and hence the Weierstrass parametrisation is given by R(tO)hel,cat= (to- too)-2(to+ l/too) -2

(1.23)

Three fiat points alone cannot satisfy eq. 1.21. Proceeding to four fiat points we may create the Scherk surfaces (Fig. 1.15) by distributing these evenly along a great circle on the Gauss sphere. The standard Weierstrass parametrisation of these surfaces is R(tO)Scher k = (tt~ - 1)-1

(1.24)

All these surfaces have one important characteristic in common. The fiat "points" are not located within any finite portion of the surface. Rather, the surfaces become asymptotically fiat (e.g. the trumpet-shaped "ends" in the catenoid). As the number of fiat points increases beyond four, the fiat points are located at fixed identifiable sites and the surface closes up to become periodic in three dimensions. This distinction between one- or two-periodic and three-periodic minimal surfaces is a crucial one, since it implies that the average Gaussian curvature () of one-, and two-periodic minimal surfaces is usually zero, due to the overwhelming contribution from the

Appendix

35

asymptotically flat ends. On the other hand, the average Gaussian curvature of three-periodic minimal surfaces is negative. This property is important, and may be responsible for the frequent occurrence in nature of threeperiodic surfaces as compared with topologically less complex surfaces. (This issue will be discussed further in Chapter 4.) The simplest, non self-intersecting three-periodic minimal surfaces have a genus of three per unit cell. Six such surfaces are known (plus lower symmetry cases for some). They belong to three distinct isometric families. The first family comprises the P- and D- surfaces and the gyroid discussed above. The second group comprises the H-surface and the h-CLP surface (Fig. 1.24). The H-surface was discovered by Schwarz last century, the h-CLP IPMS was first described by Lidin [30]. Computer studies of the isometric family of IPMS related to the h-CLP surface have revealed a further IPMS associate to the h-CLP, and free of selfintersections [18] (analogous to the gyroid in the D-P family).

Figure 1.24(a): The H-surface, discovered by Schwarz.

36

Chapter I

Figure 1.24(b): The h-CLP surface. The third family consists solely of the CLP-surface (Fig. 1.25).

Figure 1.25: The CLP surface All these IPMS have eight distinct normal vectors due to flat points. The relative distribution of these normal vectors determines the IPMS. They are described by Weierstrass parametrisations of the form (eqs. 18): 8

R(o) = l-I (w- a~)-~/2 i=1

Appendix

37

Surfaces of higher genera normally have a much more complicated structure, with the exception of the I-WP surface (genus 4) which has a fairly simple representation, namely: 6

n(co) = FI (co- c~)-2/3 i=1

Figure 1.26: The I-WP surface, unit cell only shown. A small selection of more complex cubic and tetragonal IMPS is s h o w n below. Many other IPMS can be found in the publications of Karcher, Fischer and Koch and others [14-18, 20, 22-24, 31].

Figure 1.27: A conventional unit cell of the Neovius surface.

38

Chapter 1

Figure 1.28. Left: one node of the F-RD surface. Right: The other node of the F-RD surface.

Figure 1.29: A unit ceil of the tetragonal IPMS, the H-T surface. P r o p e r t i e s of t h e s i m p l e r IPMS are t a b u l a t e d o v e r l e a f ( a d a p t e d f r o m [21]).

Appendix

39

Table 1.3: List of the simpler three-periodic minimal surfaces (IPMS), together with their crystallographic symmetries. Those surfaces that carve space into two interpenetrating open labyrinths are m a r k e d with a tick, a cross denotes IPMS that are self-intersecting. In most cases, two space groups are listed for each IPMS, the first is that of the surface assuming both sides are equivalent, the second is the symmetry displayed by the surface assuming inequivalent sides.

M i n i m a l surface /

Space Group

Gamus

Bicontinuous

Adjoint surface D /P

Pn3m - F d 3 m / I m a m - Pm~m

3

~4

rPD

R~.m- R~.m (c'=2c)

3

~/

tD/tP

P 4 2 / n n m - I41/amd /

3

~/

(Intersection-free)

I 4 / m m m - P4/mmm oPa / oDa

Immm - Pmmm / Pnnn - Fddd

3

oDb/oPb

Cmma - Imma / Fmmm - Cmmm

3

"~ ~/

mPD

C 1 2 / m l - C 1 2 / m l (c'=2c)

3

CLP / CLP

P 4 2 / m c m - P42/mmc (v)

3

oCLP'/oCLP

Cmmm - Pmmm / Pccm - Ccx~

3

~/

mCLP

P 1 2 / m l - P 1 2 / m l (c'=2c)

3

q

mPCLP/mDCLP

C 1 2 / m l - P12/m1 /

3

~/

P12/cl

-

C 1 2 / c l (a'=2a,b'=2b)

I-WP / S t e s s m a n

Im3m- Im3m/-

4

~//x

VAL/VAL

Cmma - Cmma (c'=2c)

5

~/

H'/H

- / P63/mmc - Prim2

3

x / ~/

40

ChapterI

Mathematical Bibliography: S. Hildebrandt and A. Tromba, "Mathematics and Optimal Form". Scientific American Library, (1985), New York: W.H. Freeman and Co. This book assumes no mathematical competence on the part of the reader (although the authors are distinguished geometers) and offers a good entree into the field of surface geometry and topology for those who find standard mathematical texts forbidding. Filled with beautiful pictures! D. Hilbert and S. Cohn-Vossen, "Geometry and the Imagination". (1952), New York: Chelsea Publishers. This is a very good general introduction to geometry and topology, accessible to the non-specialist. M. do Carmo, "Differential geometry of curves and surfaces". (1976), Eaglewood Cliffs, N.J.: Prentice-HaU Inc. A. Goetz, "Introduction to Differential Geometry". (1970), Reading, Massachusetts: Addison Wesley Publishing Company. A. Gray, "Modern Differential Geometry of Curves and Surfaces". Studies in Advanced Mathematics, ed. S. Krantz. (1993), Boca Raton, FLA.: CRC Press. T.J. Willmore, "Differential Geometry". (1985), Oxford University Press: Delhi. p. 137. All of these are recommended introductions to differential geometry. M. Spivak, "A Comprehensive Introduction to Differential Geometry". Vol. IV, chapter 9. (1979), Berkeley: Publish or Perish, Inc. Spivak gives a modern technical account of all aspects of differential geometry in five volumes, including a good historical section, covering in some detail the original work of Gauss and Riemann (vol 2). J.C.C. Nitsche, "Vorlesungen fiber Minimalfli~chen". (1975), Berlin: Springer Verlag. J.C.C. Nitsche, "Lectures on Minimal Surfaces". Vol. 1. (1989), Cambridge: Cambridge University Press. The gospel according to Nitsche. Everything you ever wanted to know about the classical theory of minimal surfaces . A picture-filled comprehensive alternative is U. Dierkes, S. Hildebrandt, A. Kfister and O. Wohlrab, "Minimal Surfaces", 2 volumes, (1992), Berlin: Springer Verlag. P.A. Firby and C.F. Gardiner, "Surface Topology". 2nd. edn. Ellis-Horwood series in mathematics and its applications, ed. G.M. Bell. (1991), Chichester: Ellis Horwood Limited. This offers a simple introduction to surface topology.

References

41

References 1. D. Thompson, "On Growth and Form". 2nd ed. (1968), Cambridge: Cambridge University Press. 2. R.S. Westfall, "Never At Rest". University Press.

(1980), Cambridge:

Cambridge

H.B. Lawson, Ann. of Math., (1970). 92: pp. 335-374.

~

L.E. Scriven, Nature, (1976). 263: pp. 123-125.

.

5. L.E. Scriven, in "Micellization, solubilization and microemulsions.", K.L. Mittal, Editor. (1977), Plenum Press: New York. pp. 877-893. A.H. Schoen, Infinite periodic minimal surfaces intersections(1970), N.A.S.A.: Technical Note # D5541.

6.

self-

E. Koch and W. Fischer, Acta Cryst., (1990). A46: pp. 33-40.

.

0

without

J.L. Lagrange, Miscellanea Taurinensia, (1760-1761). 2: pp. 173-195.

J.B.M.C. Meusnier, M~m. Math~m. Phys. Acad. Sci. Paris, pr~s. par. div. Savans, (1785). 10: pp. 477-510.

9.

10. K.F. Gauss, "Disquisitiones generales circa superficies curvas (General investigations of curved surfaces)". (1827), reprinted New York: Raven Press (1965). 11.

H.F. Scherk, Crelles Journal, (1835). 13: pp. 185-208.

12. H.A. Schwarz, in "Gesammelte Mathematische Abhandlungen. (1890), Springer: Berlin. 13.

K. Kenmotsu, Math. Ann., (1979). 245: pp. 89-99.

14.

H. Karcher, Manuscripta Math., (1989). 64: pp. 291-337.

15.

S. Lidin and S.T. Hyde, J. Phys. (France), (1987). 48: pp. 1585-1590.

16.

S. Lidin, J. Phys. France, (1988). 49: pp. 421-427.

17. S. Lidin, S.T. Hyde, and B.W. Ninham, J. Phys. France, (1990). 51" pp. 801-813. 18.

S. Lidin and S. Larsson, J. Chem. Soc. Faraday Trans., (1990). 86.

19.

A. Fogden, Acta Cryst., (1993). A49, pp. 409-421.

20.

A. Fogden and S.T. Hyde, Acta Cryst., (1992). A48: pp. 442-451.

21.

A. Fogden and S.T. Hyde, Acta Cryst., (1992). A48: pp. 575-591.

42

Chapter I

22. A. Fogden, M. Haeberlein, and S. Lidin, J. Phys. I (France), (1993). 3: pp. 2371-2385. 23.

A. Fogden and M. Haeberlein, J. Chem. Soc. Faraday Trans., (1994).

90(2): pp. 263-270. 24.

A. Fogden, Z. Kristallogr., (1994). 209: pp. 22-31.

25. D. Hoffman and W.H. Meeks III, J. Differential Geom., (1985). 21: pp. 109-127. 26.

D. Hoffman, Math. Intelt., (1987). 9(3): pp. 8-21.

27.

H. Karcher, Manuscripta Math., (1988). 62: pp. 83-114.

28. J.C.C. Nitsche, "Vorlesungen Springer Verlag.

iiber Minimalfltichen'. (1975), Berlin:

29. J.C.C. Nitsche, "Lectures on Minimal Cambridge: Cambridge University Press.

Surfaces". Vol. 1. (1989),

30. S. Lidin, Periodiska Minimalytor och Kristallstrukturer(1986), Thesis, Lurid University. 31.

W. Fischer and E. Koch, Acta Cryst., (1989). A45: pp. 726-732.

43

Chapter 2

The Lessons of Chemistry

Inorganic Chemistry: From the discrete lattice of crystal symmetry to the continuous manifolds of differential geometry 2.1

The background

n the beginning there was geometry, shape and form, perfect symmetry represented by the crystals of inorganic chemistry. All definitions are tautological, but if such be needed, inorganic chemistry is the study of compounds formed from the elements, excluding a few exceptions, like carbon. The distinction between inorganic and organic chemistry is artificial, and not altogether harmless. But classification is necessary, indeed unavoidable, since the demarcation of boundaries between disciplines does allow a mastery of detail. That is acceptable until new insights are drawn in to go further. This is the stage that inorganic chemistry has reached and its new insights have much to offer other disciplines.

I

The inorganic compounds embrace familiar terms like metals and alloys, salts and hydrides. In an earlier stage of development of the subject, properties of these solids were studied with a main emphasis on how best to prepare them. Pre-eminent among these properties is structure- where the atoms reside in space. Once that is known, our understanding of reactivity, diffusion, strength, catalytic mechanisms, phase behaviour and interrelationships between compounds is possible. A knowledge of structure gives insights into how compounds form and why they self-assemble, and how to predict the properties of unknown materials. The reduction to order of the complexity of nature is relatively easily accomplished in solid state inorganic chemistry compared with say biology. Many inorganic materials are characterised by the simplest possible building scheme, classical crystals. The atoms are repeated in an ordered array which is termed "isometric". By this we mean that a basic unit, the unit cell, is occupied by atoms in a fixed arrangement, and the unit cell is repeated with simple translation. In the great majority of crystals, the unit cell itself possesses certain symmetries, leaving even fewer degrees of freedom in the system. This is why a one gram crystal of NaC1 containing about 2x1022 atoms can be described using only a single distance! Translation and symmetries of the unit cell, examples of which are reflections and rotations, all belong to the group of operations known as isometries. These preserve all distances and angles of the original motif when creating a copy. The unit cells must fill space and this imposes constraints on their shape, and thereby also on allowed interior symmetries. Naturally all these operations are idealisations. Often we find small deviations from this ideal behaviour in a natural crystal. The properties of a crystalline material are heavily dependent on those deviations, which operate to produce non-ideal behaviour.

44

Chapter 2

Let us start by analysing the plane. There is an infinite number of ways to fill the plane with irregular tiles, but if we restrict ourselves to regular polygons, whose edge lengths and angles are all equal we find that only triangles, squares and hexagons will do the job. This is reflected in the fact that the only rotational symmetries in the plane compatible with translational symmetry are two-, three-, four- and six-fold. Pentagons (with five-fold symmetry) fail, and th~ riddle has bothered man since ancient times. The Moors in Granada created the famous palace of Alhambra, and decorated it with intricate geometrical patterns. (Religious considerations forbad the picturing of humans.) In the Alhambra one may find examples of each of the possible 17 discrete planar groups. However, the Arabs went further and tried to create pentagonal tilings [1]. Pentagonal structures are common in nature, but never exhibit translational symmetry. Earlier the Greeks had discovered something even more surprising. While the pentagon does not tile the plane, it does indeed tile the sphere. Twelve pentagons make up the pentagonal dodecahedron whose vertices lie on the sphere. Similarly, the icosahedron is a triangular tiling of the sphere with fivefold rotational symmetry. All this was known to modern crystallographers, and an article of faith enunciated and firmly enshi-ined in an eleventh commandment that said '"Fhou shalt not have five-fold symmetry". Despite this, for decades X-ray crystallography has revealed five-fold symmetries in the atomic arrangements in alloys, but crystaUographers invariably discarded such samples. What was observed was dismissed as nothing but complicated twin structures. (Twinning means that sub-units of a crystal are assembled by reflection or rotation.) It was only recently that Shechtman and Blecht [2, 3] made the bold claim that some rapidly quenched aluminium-manganese alloys exhibited five-fold symmetry within the untwinned "crystal". X-ray diffraction patterns of these alloys exhibit perfect icosahedral symmetry. The fact that the atomic arrangement was clearly not ordered in the classical sense, but still exhibited a perfectly regular diffraction pattern could be explained as an ordering in higherdimensional space, as for the so-called "incommensurate structures". Ordering in "higher" space means that the positions of the atoms in space cannot be labelled by the three cartesian indices, but require extra labels. What is learnt from this goes beyond the statement that geometry is important. Crystallography is by necessity ruled by geometry and its rules are universally valid. But this fact should not induce a state of mind where we think that we can predict all unknown structures from considering how the old ones are built. We must remain open to new ways of looking at old knowledge. 2.2

The unravelling of complex structures

Over the years a plethora of inorganic compounds has been prepared and their structures determined. Making use of the knowledge so gained, inorganic chemists try to predict new structures and how to prepare them, sometimes

Complex structures

45

successfully. Slowly a pattern has emerged, and this pattern has evolved into a new way of looking at complex structures [4, 5].

Figure 2.1: The structure of the zeolite paulingite, cubic, with a lattice parameter of 35.1 ./~, composed of (different coloured) gismondine units.

As a rule almost all complex structures, even those with horrible stoichiometries (like Nb31077F) can be described using simpler structures. There is a natural way to generalise these structures to polyhedral descriptions. Atomic coordinates are grouped to form polyhedra, which in turn form parts of larger structures. This process reduces increasingly complex structures to simpler forms by regarding them as composed of well known and understood packets. The model structures (packets) are put together using the classic crystallographic operations of translation, reflection and rotation. An excellent example of this technique is the unravelling of the structure of the complex zeolite, paulingite (Fig. 2.1). (Zeolites are open alumino-silicate frameworks, widely used for catalysis, discussed in more detail later in this chapter.) When viewed as a set of atomic coordinates in three-dimensional space, the structure of paulingite is monstrous to behold. Once it is realised that the whole can be decomposed into units of the (geometrically much simpler) zeolite gismondine, the picture becomes clearer. If the gismondine is then described as a composite of parts of the even simpler cristobalite structure, we can get a good understanding of the whole spatial arrangement. The introduction of a hierarchical system of ever increasing complexity makes larger packets of information more comprehensible. Another example is the zeolite N - the most complex of all known zeolite structures [6]. In Fig. 2.2 it is shown how this giant structure is decomposed into two much simpler zeolite structures, those of zeolite ZK5 (blue) and sodalite (yellow).

46

Chapter 2

Figure 2.2: The structure of zeolite N, a cubic framework with a lattice parameter of 36.9,/k. The framework consists of two different, interpenetrating zeolite structures, sodalite (yellow) and ZK5 (blue) structure.

To illustrate further, consider giant alloy structures, such as Cu4Cd 3, NaCd 2 and Rh7Mg44. These lend themselves very well to similar structural decompositions and descriptions as those described above. They may all be dissected into simpler structures, joined by reflections, rotations and translations to give a complete description of the atomic positions in the whole structure. The matrices describing these operations are identical to those used by metallurgists to describe the larger scale structure, characterised by the orientation of the grains (crystallites) in ordinary metals and alloys [7]. When small particles are sintered together to form a continuous material, they are on the way to forming a complex structure. Similarly, a complex alloy structure can be viewed as polycrystalline material of simpler structures.

2.3

Defects

To go further towards understanding the concepts of inorganic chemistry we need to consider the subject of defects in solids. They are a key to the behaviour of many materials. They are of central importance to diffusion, phase transformations and reactivity of solid compounds. Defect structures show up as the local occurrence of a grain boundary or sites of a structure building operation. The formation of a structure using translational defects is easily understood by a planar example, illustrated in Figs. 2.3. Modem high resolution electron microscopy (HREM) reveals how frequently these defects occur, and how they provide a setting and a mechanism for intergrowth between related structures.

Defects

47

Figure 2.3(a): Translation by half a unit step in a square lattice creates a linear defect in a two dimensional lattice between the open and filled circles. A defect of this kind may continue to

grow sideways, leading to a whole family of new structures.

Figure 2.3(b): The transformation from a simple square lattice to a centred square lattice through the propagation of a linear defect. In Figs. 2.4 some typical HREM of alloy structures are compared to a freezefracture electron micrograph of a liquid crystal. In the alloy pictures, planar defects, i n t e r g r o w t h of different structures and small f r a g m e n t s of new structures appearing at phase boundaries are clearly seen. The similarities with the lyotropic liquid crystal image are striking. The latter image contains extensive regions of well-ordered material, the boundaries of which are now known to correspond closely to the P- and D- surfaces, described in Chapter 1. Between these regions we find other structures, in some places confined to an interface, but in others extended regions of a new structure can be seen. Several new structures can be inferred to exist from this picture, and these structures are expected to be simply related to known ones. The energy difference between the different phases must be small to enable coexistence or long term metastability. In alloys structural reorientation is sluggish, while in liquid crystalline phases the changes are more rapidly accomplished.

48

Chapter 2

Figures 2.4(a),(b),(c): Typical defects occurring in different alloy structures (similar to steel).

Figure 2.4(d): EM image of a freeze-fractured lipid bilayer folded into a bicontinuous cubic structure. (For details see Chapter 5.) Note the similarity of defe~ts with those in Figs. 2.4(a),(b),(c) although the length scales are very different. (Image courtesy of T. Gulik.)

Intrinsic curvatureof solids

2.4

49

The i n t r i n s i c c u r v a t u r e o f s o l i d s

W e c o m e n o w to a f u n d a m e n t a l p r o p e r t y of solids, t h a t of intrinsic c u r v a t u r e [8]. C o n s i d e r first a salt like N a C l (Fig. 2. 5). T h e crystal can b e c o n s i d e r e d as a c o l l e c t i o n of p o s i t i v e a n d n e g a t i v e c h a r g e s p l a c e d at t h e o r a n g e a n d p i n k lattice sites i l l u s t r a t e d . T h e e n e r g y of f o r m a t i o n p e r u n i t cell is c a l c u l a t e d to first a p p r o x i m a t i o n b y a d d i t i o n of t h e p a i r - w i s e C o u l o m b p o t e n t i a l e n e r g y of i n t e r a c t i o n b e t w e e n t h e c h a r g e s [9]. If w e a s k w h e r e this p o t e n t i a l is z e r o w e h a v e for this p a r t i c u l a r s y m m e t r y t h e set of i n t e r s e c t i n g p l a n e s i l l u s t r a t e d . The m a t t e r is a p p a r e n t l y d e v o i d of interest. If w e c o n s i d e r i n s t e a d CsCl, a v e r y d i f f e r e n t p i c t u r e e m e r g e s . T h e s u r f a c e s of z e r o p o t e n t i a l a r e i l l u s t r a t e d , a n d d i v i d e s p a c e into i n t e r p e n e t r a t i n g l a b y r i n t h s (Fig. 2.6).

l'i~urc 2.~(left): The NaCI structure. The intersecting planes define the "i~erc~" equipotential

-,urface.,-,, between the Na + (red)and CI- (orange) ions. t:igure 2.(~ (right): The CsCI structure with its zero potential surfaces. The vello,,v Cs + ions lie at the centre of the body-centred cubic unit cell. T h e s e s u r f a c e s of z e r o p o t e n t i a l f o r m e d in d i f f e r e n t salts are v e r y close to p e r i o d i c m i n i m a l s u r f a c e s [ 9 ] , w h o s e m e a n c u r v a t u r e , d e f i n e d as t h e a r i t h m e t i c m e a n of the m a i n c u r v a t u r e s , is e v e r y w h e r e z e r o (see C h a p t e r 1 ) ' . On these m i n i m a l s u r f a c e s the G a u s s i a n c u r v a t u r e is e v e r y w h e r e n e g a t i v e or In our first simple example the electrostatic potential set up by CsCl is almost but not quite a minimal surface [10]. The reason is that the Coulomb electrostatic energy is only a part of the whole electromagnetic field. Two body, three and higher order, non-additive van der Waals interactions contribute to the complete field, distributed within the crystal. This leads one to expect that the condition that the stress tensor of the field is zero, as for soap films, yields the condition for equilibrium of the crystal. Precisely that condition is that for the existence of a minimal surface. Strictly speaking the minimal surface might be defined by the condition that the electromagnetic stress tensor is zero. But in any event, we see in this manner that the occurrence of minimal surfaces, should be a consequence of equilibrium (cf. Chapter 3, 3.2.4). Indeed a statement of equilibrium may well be equivalent to quantum statistical mechanics.

50

Chaljter 2

zero, and varies continuously over the surface. Immediately we have entered the field of non-Euclidean hyperbolic geometry of Lobachevski, Bolyai and Gauss. The simplest periodic minimal surfaces like the P- and D- surfaces, divide space into two congruent labyrinths. More complicated surfaces (such as the IWP surface) divide space unequally. An analysis of a structure based on interpenetration using these partitioning surfaces has proved very useful in inorganic chemistry. An example is the W3Fe3C structure (cutting steel). The tungsten atoms form an octahedral network, with carbon in some of the octahedra. The iron forms units of stellar tetrangulae, which are polyhedra made up of five tetrahedra, with one central tetrahedron sharing faces with the four others. The stellar units form a three-dimensional network through sharing of corners (Fig. 2.7(a)). It is reasonable to expect that these two entirely different structures would react with each other. Indeed they do. They interpenetrate to form a composite structure, separated by the D-surface (Fig. 2.7(b)). This surface, which is at the iron-tungsten boundary, is the location of maximal collective interaction of electrons. (There is a neat correspondence here with the bicontinuous microemulsion structures to be explained in Chapter 4. These form spontaneously from mixtures of oil, water, and surfactant that sits at the interface between phases. Here tungsten carbide plays the part of water, iron that of oil, and the electrons fill the role of the amphiphilic surfactant, able to interact with both phases.)

Figure 2.7: (a) The W3Fe3C structure (cutting steel). The red balls denote iron atoms. (b) Dsurface separating the Fe structure (red stellar tetrangulae) and the W3C structure (yellow octahedra).

Intrinsic curvature of solids

51

The reason that this structure forms is as follows: units of octahedral W3C crystallise on one side of an interface, that separates solid W3C from Fe. The Fe atoms are drawn into the interstices of the W3C elements, with a driving force dependent on the Gaussian curvature of the interface. This is one way of looking at intrinsic curvature in solids. As will become apparent, there are other ways that link the interactions between atoms, molecules and larger aggregates to local curvature. Many more examples of interpenetration in inorganic chemistry lead to a recognition of the ubiquity of hyperbolic surfaces of infinite genus exemplified by three-periodic minimal surfaces - that demands consideration. In the giant structure of C u 4 C d 3 the Cu atoms are separated from the Cd atoms by a surface that resembles a minimal surface. In diamond, cubic ice and cristobalite, all the atoms are located on one side of the surface and the space on the other side is empty. If ice is subjected to very high pressure, the same structure appears on both sides of a minimal surface (double ice or ice IX), with almost double the density of o r d i n a r y ice (Fig. 2.8). Similarly, d i a m o n d is expected to transform to a d o u b l e - d i a m o n d structure with metallic properties at sufficiently high pressure.

Figure 2.8: Cubic ice, d i a m o n d or cristobalite (yellow) on one side of the D-surface. In ice IX or double ice, a t o m s are on both sides of the D-surface, yellow and red.

The concept of intrinsic curvature is particularly useful when dealing with the intricate and beautiful structures formed by zeolites. Zeolites are commonly used as technical materials. They exhibit many special properties, due to their extraordinary ability to selectively absorb a large range of molecules. The forces that act are weak, physical not chemical, and we characterise them by invoking the idea of intrinsic curvature. Zeolites are built up from a negatively charged three-dimensional network of tetrahedra of (Si,Al)O 2 connected by sharing of the tetrahedral vertices with

52

Chapter 2

interstitial positively charged counterions, often alkali metal ions. The zeolites crystallise from alkaline, aqueous solutions at temperatures around 50-200 oc. From similar solutions but at different (higher) temperatures and pressures, feldspars, quartz and cristobalite form. Examples of zeolites sitting close to or on one side of periodic minimal surfaces are shown in Figs. 2.9(a),(b): the zeolite known as Linde-A on the Psurface and faujasite on the D-surface. Other examples are zeolite N in which the D-surface partitions the ZK5 and sodalite structures, and also paulingite which is described by the P-surface.

Figure 2.9(a): The Linde A zeolite on one side of the P-surface. The water-alkali structure is on the other side. (b): The faujasite structure on one side of the D-surface.

2.5

Hydrophobic zeolites and adsorption

In the ensuing discussion we deal exclusively with hydrophobic (dealuminated) zeolites. In these zeolites nearly all Al 3+ ions have been substituted by Si4+ ions, so that the stoichiometries of these solids are close to those of pure silica, SiO2. The diameters of the pores in the framework are typically of the order of 5~, up to 7.5~ (zeolite Y). (Recently, wider pore hydrophobic "zeolites" have been synthesised. These are discussed later in this chapter; for now we restrict our attention to the "classical" zeolites.) When these compounds absorb molecules, the heat developed is dependent on the pore size. This heat comes mainly from the non-bonding interaction between the silica framework and the absorbed molecules.

Adsorption in zeolites

53

What is it that makes these weak forces strong enough to crack hydrocarbons, breaking the carbon-carbon bonds? It is clear that it must be a collective effect. The (interior) surface of the zeolite imparts a cooperative effect to the adsorbate. Some conclusions can be d r a w n from a simple analysis of adsorption phenomena as a function of the substrate curvature. It is easy to see that adsorption energies are dependent on the curvature of the interface. Consider first adsorption on a planar interface. At low pressures, p, a sub-monolayer, gas-like, and eventually a two-dimensional liquid described by a Langmuir isotherm (or decorations thereof) forms. At higher pressures still (p/ps>0.35, where ps is the saturated vapour pressure) multilayer adsorption isotherms can occur depending on adsorbate, molecular size and adsorbate-substrate interactions. This regime is usually described by the theory of Brunauer-Emmet-Teller (BET). In this domain, l n ( p / p s ) _--l/t, where t is the thickness of the film. The BET theory, central to surface area determination, is a simple model that nonetheless captures the essentials of adsorption behaviour. It assigns a free energy of adsorption to the first layer, and a different free energy to subsequent molecular layers, equal to the latent heat of vaporisation of the bulk adsorbate liquid. (There is a continuum theory, valid asymptotically for thick films at higher pressures, the Lifshitz isotherm for which the form l n ( p / p s ) ___-t/13, takes over, cf. section 3.4.2) The BET theory is deficient, as are all extant molecular theories, in an incorrect treatment of the entropy of the adsorbed film. In fact, when molecular size and interaction pressure with these parameters are taken into account, a unified isotherm, which also predicts step-like isotherms in certain circumstances emerges. If we calculate the van der Waals energy of adsorption of a molecule on to a planar substrate its form is E ___--A/d3, where d is a cut-off distance of the order of the molecular dimensions. This energy is acutely sensitive to exactly where the molecule sits on the surface, which is rough at the atomic scale. If we imagine that the surface is corrugated at the length scale of the surface atoms, an adsorbed molecule, visualised as a sphere, has several options. If it is too large to settle into the corrugations formed by the close packed spheres on the surface its adsorption energy is typical of physisorption. If the surface atomic distances are stretched slightly so that the adsorbate molecule can sit closer, the adsorption energies turn out to be of the order of chemical bond energies. Clearly then, the geometry of the substrate plays a critical role in the adsorption energetics. Adsorption within a zeolite can occur everywhere in the crystal. The alumino-silicate framework is a convoluted curved sheet, everywhere exposed to the exterior of the crystal by way of the channels (cf. Fig. 2.9). The process described above holds for zeolites (although the adsorbed molecules hover in the tunnels, and are not bound to the zeolite), and the local geometry of the continuous alumino-silicate sheet determines the adsorption energy. This geometry varies throughout the sheet continuously, since the Gaussian curvature is not constant. It is certain then that the adsorption

54

Chapter 2

energy of an adsorbed species is dependent on the Gaussian curvature. A simple model calculation makes this clear. Consider a spherical adsorbate species hovering over five substrate atoms at a distance d from the central atom. We can calculate the "adsorption energy" for this system as a function of the Gaussian curvature of the substrate of the substrate. If the substrate atoms form an ellipsoidal surface, a sphere of radius d (Gaussian curvature 1/d2), the energy is equal to -5/d 4 (assuming, for simplicity the interaction energy scales as 1/d4). If the substrate is fiat (zero Gaussian curvature), the adsorption energy is equal to -2/d 4. For a symmetric saddle (zero mean curvature) of Gaussian curvature -1/d 2, the adsorbate experiences an adsorption energy equal t o - 3 / d 4. We see then that the heat of adsorption is maximised for an adsorbate molecule sitting in a half-sphere. However, that local arrangement cannot be globally realised. All elliptical interfaces (positive Gaussian curvature) will close up, rather than form a continuous structure like a zeolite. The simple analysis described above suggests that a saddle (negative Gaussian curvature) leads to greater adsorption power than a plane (Gaussian curvature zero). And as long as the adsorbate will fit, the tighter the saddle, the better. Further, a smooth change in the Gaussian curvature, as for a minimal surface will funnel adsorbates to preferred sites, increasing the effective catchment area for adsorption. Certainly this simple calculation is an idealisation of the actual physics of adsorption; the energy depends on the adsorbate size, shape, etc. Nevertheless, the lesson is clear; the adsorption energy depends on the curvature of the substrate. In the case of zeolite adsorption, where the mean curvature of the substrate is everywhere close to zero, the adsorption energy must depend on the variations of Gaussian curvature throughout the alumino-silicate network. Our example suggests that for minimal surfaces, the higher the magnitude of the (negative) Gaussian curvature, the stronger the adsorption energy (cf. the plane and the saddle). Other effects, such as steric phenomena, may alter this trend. However, in general we expect adsorption to occur firstly in the vicinity of saddle points within the zeolite structure. This argument suggests there should be a correlation between the integral heat of adsorption for various adsorbates and the Gaussian curvature of a substrate. Adsorption data is available for de-aluminated faujasite and silicalite. The integral heats of adsorption for various adsorbates can be compared to the integral Gaussian curvature per unit area of the zeolites [11]. This is an entity easily computed from the Gauss-Bonnet theorem once the surface areas of these surfaces are known (see Chapter 1). These arguments can be quantified, and put on a firm basis by the theory developed in section 3.2.3. The combination of the curvature-funneling effect and the wide distribution of adsorption sites within zeolites - some of which will be optimal for a

Adsorption in zeolites

55

particular adsorbate - explains how zeolites can catalyse reactions like cracking and polymerisation. The high intrinsic heat developed at the optimal sites supplies the driving force for the reactions in question. Zeolites combine the features of a general catalyst/adsorbent with those of a very specific one. This is u n d e r s t a n d a b l e in terms of curvature. The fact that a continuous distribution of different sites of varying Gaussian curvature is available vouches for the generality of the adsorption phenomenon, and the fact that the efficacy is governed by shape warrants the selectivity. Small changes in the structure of these compounds will affect the selectivity to a large extent without affecting the generality. It will be seen later in the book that the same principles that underlie their catalysis and reactive properties are exploited to a great advantage by enzymatic proteins in biology.

2.6

Phase transitions, order and disorder

One measure for the level of lack of understanding in an established field is the volume of papers and weight of the books devoted to it. The subject of phase transitions in solid state chemistry constitutes an immense field by weight. (The phenomenon of melting - even for the simplest solid - has hardly yielded to theoretical onslaughts at all since the formulation of Lindemann's rule 80 years ago. This criterion, obeyed for real or model solids which range from a one-component electron gas subject to repulsive Coulomb interactions in a positive neutralising continuum background, to a hard-sphere solid with an attractive potential, long or short range - is universal. The rule has it that a solid melts whenever the root mean square deviation of a particle moving in the mean field of its neighbours is about 1/10 of the lattice spacing. The reasons, presumably, are twofold: If one imagines the electrons of an atom to be smeared into an exponential cloud of probability density, the Rayleigh criterion for resolvability tells us that it is impossible to distinguish between one atom and its neighbour at this level of excitation of the solid. On the other hand, if we imagine the atoms as hard spheres, this mean square displacement just permits rapid diffusion from one lattice site to another. The failure of statistical mechanical theories has to do with the neglect of the anisometric structure that occurs in any real liquid, and indeed the non-existence of any serious theory of liquids.) We will concentrate our attention on a small number of special cases of solidsolid transitions, without presuming to infer too much in generality. The approach taken is different to the usual view, and the examples which yield to this attack are inaccessible to standard ways of thinking. The utility of our approach will be apparent from the examples chosen. Earlier we have given an example of a transition built on the concept of symmetry operations. Groups of atoms move by translation, rotation or reflection. Transitions that operate along these lines tend to be sluggish and have large activation energies. The driving force is temperature or pressure. At high temperature the coordination number has a tendency to decrease and

56

Chapter 2

at a critical strain, this will trigger a transition. Very often, high pressure will trigger transitions that give rise to increased coordination. The graphite to diamond transition is a classic example. The multitude of modifications of ice is another. These examples are easily understood from packing arguments. The rutile (TiO2) to the high pressure TiO2(II) (a-PbO 2type) transition used for the lubrication of guns is easily understood from electrostatic considerations. This transition, schematically shown below, leaves the oxygen atoms virtually untouched, while the titanium atoms move according to the Fig. 2.10

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

9

~

9

9

9

9

9

9

9

9

Figure 2.10: Schematic view of Ti02(I) --~ Ti02(II) transformation.

When the solid is compressed (e.g. upon firing), the enhanced repulsion distorts the linear arrangement of positive cations. They move sideways, and the structure becomes more elastic. The only coordination feature that changes is the cation-cation packing, which is enhanced. The new structure is relaxed and this is reflected in the cation - cation distance, which is larger in the more compressed form. Perhaps the most technologically important phase transition in solids is the martensitic transition. The classical martensitic transition occurs in steel where the face-centred cubic phase (fcc, austenite) transforms to the bodycentred phase (bcc, martensite). The hardness of the material depends on the amount of martensite formed, and this in turn is a function of initial temperature and cooling rate. The transition is characterised by a drastic and complex diffusionless r e a r r a n g e m e n t of atoms, adiabaticity, and a propagation of structural change at sonic velocity. This is very fast for a transformation that involves the making and breaking of bonds, and it is clear that bond breaking and making is minimised. Few phenomena in chemistry involving actual movement of atoms occur at comparable rates. The transportation kinetics are independent of temperature over a wide range. This phenomenon, which is widespread in metals, is triggered by a cooperative phase transformation associated with the ensuing change in size of the atoms (or equivalently the position of the potential minimum in their

Phase transitions

57

i n t e r a c t i o n p o t e n t i a l ) . It is a c c o m m o d a t e d b y t h e b c c lattice. T h e w h o l e t r a n s f o r m a t i o n o c c u r s c o o p e r a t i v e l y o v e r l a r g e v o l u m e s of the crystal.

Figure 2.1 l(a): Schematic view of the effect of the Bonnet transformation on a crystal. The elliptical trajectories of all atoms result in a bulk region abcd being transformed into the related sheared and rotated regions 0~u and a'~'7'8' for increasing and decreasing Bonnet angle, 0, respectively. The Bonnet angle 0, determines the degree of rotation, @.

Figure 2.1 l(b): Monkey-saddles of the gyroid (left) and the D surface (right). These saddles are related by the Bonnet transformation. Atoms, fixed to the fiat points of the saddles, are transformed from the face-centred cubic to a body-centred cubic array. T h e s t a n d a r d ( p o s t u l a t e d ) m e c h a n i s m for t h e t r a n s i t i o n is t h e B a i n d e f o r m a t i o n , w h i c h i n v o l v e s a 20% c o n t r a c t i o n in o n e d i r e c t i o n a n d a 12% e x p a n s i o n in t w o p e r p e n d i c u l a r directions. This w a s d e s c r i b e d by Bowles a n d

58

Chapter2

Mackenzie as a three-step process, with rotation, reflection and translation operations (see, for example, [12]). However there is another more credible and entirely different mechanism [13]. By viewing the atoms of the (facecentred cubic) austenite as placed on all the fiat points of the D-surface to which indeed they fit, and applying a Bonnet transformation to this surface to deform the D-surface into the gyroid (Chapter 1), we arrive at a body-centred cubic arrangement of flat points, characteristic of the martensite lattice. This mapping of the atoms onto the flat points of the IPMS, which then transform subject to the Bonnet transformation, fixes uniquely the relative lattice spacings of parent and product phases, as well as their relative crystallographic orientations. The ratio of lattice parameters is 1.249 under the Bonnet transformation, close to the measured value, 1.269. The measured orientation relation (known to metallurgists as the Kurdjomov-Sachs plane correspondence) is exactly that expected by the Bonnet transformation. Further, the occurence of twinning during the transformation ('tweed") is also consistent with the Bonnet transformation (since Bonnet angles of +0 gives twinned regions, cf. Figs. 2.11(a)). This model may also explain the habit plane, which invariably defines the orientation of the interface between parent and product phases in the martensite transformation [14]. Our alternative mechanism to that of Bowles et al. would cost very little energy, since the Bonnet transformation is isometric (no bond stretching along the surface) and this single operation combines all three features of the older model in one simultaneous step, involving the entire crystal. All atoms move along ellipses as shown in Fig. 2.11(a),(b).

2.7

Quantitative analysis of hyperbolic frameworks: silicate densities

A central problem in the zeolite business is to synthesise novel zeolites containing wide pores. These materials are sought after for a variety of scientific and technological reasons, ranging from fractionation of proteins to the storage of methane. This program has recently leapt ahead following the announcement of MCM-41 "zeolites" by Mobil Oil, with pore radii from 15100]k [14]. Until then, the widest pore aperture in hydrophobic zeolites realised in zeolite Y (isostructural to faujasite) - was a mere 7.5~ in diameter. Debate is continuing about the structure of the Mobil materials. The MCM series are structurally very different to conventional zeolites: they do not exhibit crystallinity at the atomic length scale (wide-angle X-ray diffraction patterns consist only of diffuse bands), and microscopic images indicate the zeolite walls have a thickness larger than 5/~. (We remark in passing that these zeolites are synthesised using ionic lyotropic liquid crystals as templates. In these cases, the templating mechanism appears to be a steric one and the alumino-silica framework geometry follows that of the hydrophobiclipophilic interface in the liquid crystalline mesophase. The formation of hyperbolic MCM frameworks is possible, using bicontinuous "cubic" mesophases of the liquid crystals, discussed in Chapters 4 and 5).

Tetrahedral frameworks

59

Wide-pore zeolites must have a low "framework density" (FD), equal to the number of tetrahedral "T" atoms (silicon or aluminium) per 1000A 3. Two questions arise naturally in this context. What sets the framework density? What is the minimum FD achievable for a hydrophobic zeolite? These are difficult questions to answer within the standard three-dimensional Euclidean view of these frameworks. However, within a two-dimensional hyperbolic perspective, they are readily answered. The latter view recognises that infinite crystalline frameworks are tesselations of periodic hyperbolic surface, as in Figs. 2.9. To many, this is an excessively complex description. However, this description does yield new insights into the nature of these covalent frameworks. The bulk density of a material depends on the surface density and the Gaussian curvature of the surface. Thus, a hyperbolic surface with large (negative) Gaussian curvature packs a greater area into a given volume than one of low curvature, since the former is more convoluted. The bulk density together with the Gaussian curvature of the framework then allows estimation of the surface density of the framework, as follows. Define the average value of the Gaussian curvature (see section 1.9) by

JJ

K da

t]~'~ __-- u n i t ce]]

(2.1)

A

where A is the area of the surface per unit cell. From the Gauss-Bonnet theorem: A = 2=g ~ bcc martensitic phase transformation in metals and alloys. The success of this model suggests that the hyperbolic picture, intuitive and obvious for zeolites, is also valid for other atomic structures.

66

Chapter2

'

"

"I

"

I

(a)

"

I

50

50

'"

9

it

9

Co)

"'...

I

.

"

I

-. ~

m

~

.:..:.---

~176 o. ~ "el

25

-..... t~

9 .o

,

4 '

i .... i .... i .... , .... i..-.....5 6 7 average surface ring-size

.

I

'

I

'

I

8 '

0

0

75

75

,

4 '

.

,

I

i

I

5 6 7 average surface ring-size

(d)

Oo

(c)

I

'

|

I

'

I

"

.,..t 9 9

S

~ 9

S

50~

'=

50

,

Q

~

~

E "''.. "~ ,

4

5

"....

I

6

,

"''I

,

7

average surface ring-size

8

25

25

|

4

a

!

5 6 7 average surface ring-size

8

Figure 2.13(a)-(d): Density/ring size data for (a) SiO2 frameworks (some zeolites, clathrasils and dense silicates), (b) H20 frameworks (hydrates and ambient pressure iceI), (c) Ge (clathrates and diamond forms) and (d) Si frameworks (clathrates and diamond). Empty circles denote open frameworks (zeolites, clathrasils/clathrates/hydrates), dots denote dense frameworks (diamond/ice etc.). The full curves are plotted using eq. 2.7 assuming a homogeneity index of 3/4 (open frameworks). The upper dashed curve assumes a value of H=3/4, the lower dashed curve H= 0.65 (dense frameworks). These curves assume no preferred bond angles or lengths, only fixed film density (t~'l). The value of ~ is the average value calculated from eq. 2.7 for all frameworks of a single stoichiometry (12.2-A2 for SIO2, 10.2A 2 for H20, 8.0A 2 for Ge and 7.2/~2 for Si). The dotted curves are calculated from eq. (2.11); they assume preferred bond angles and lengths listed in the text and zero torsion. (Nonzero but constant torsion yields similar curves, displaced upwards.) Two sets of curves are plotted for SiO2 frameworks. The left curves are for a minimttm bond angle of 70~ (all O atoms inside Si rings), the right for 107.5 ~ (half O's inside, half outside Si rings). Curves for vertex angles of 150 ~ (all O's outside) lie beyond the plotted region.

2.9

Quasicrystals

The triacontahedron was discovered by Kepler; Kowalewski [21] t h a t it c a n b e b u i l t f r o m o n e r h o m b o h e d r o n (Fig 2 1 4 )

a n d it w a s s h o w n b y acute and one obtuse

Quasicrystals

67

Figure 2.14: The triacontahedron (left), with the oblate (centre) and prolate (right) rhombohedra. Ten of these rhombohedra form the triacontahedron. The faces of the polyhedra are golden rhombuses, i.e. the quotient between the lengths of the rhombus diagonals is the golden section, ( ~ + 1 ) / 2 . In the language used to describe quasicrystals, the two rhombohedra are called the "oblate" and the "prolate". They have the fascinating property that they fill space, without exhibiting translational symmetry, probably already known by Kepler. Mackay [22] showed that a projection along the five-fold axes gives a pattern equivalent to a Penrose tiling. He also showed, with an optical experiment, that this tiling gives perfectly ordered diffraction patterns, even though the tiling is not an ordered lattice in the classical sense. A diffraction pattern of a quasicrystal of five-fold symmetry, is shown in Fig. 2.15. (This is discussed in more detail in the Appendix.)

9

+

~

9

,

9

9

,

9

.

II "

9

9 . "

9

9

+

.,

9

o

."

9

-;:

9 9

9

9

+

9

9

9

Q

9 Q

9

"

9

9

Q

"

9 9

9

9

9

. 9

. 9

b

Q

9

9

"

,

9

~

o

9

~

9

9

9

9

9

9 ,

9 .

9

'

9

,

9

9

ii

9

. -

9

, 9

9

-

Figure 2.15: Electron diffraction pattern of a quasicrystal.

68

Chapter2

One w a y that this can be pictured in terms of a projection from higher dimensional space is as follows. Take an ordinary square lattice and draw a line across it at 36~ to one of the fundamental lattice directions, and project a part of the lattice onto this line (Fig. 2.16). The projected points form a set of golden Fibonacci sequences. Every point is in principle the start of a new sequence. Were the entire lattice projected onto the line, a dense covering of points w o u l d ensue. Because we only project the part of the lattice that is contained in a window defined by two lines parallel to the line of projection, we get a discrete one-dimensional quasi-lattice (Fig. 2.16). A series of points constructed according to these rules leads to perfect diffraction. We explain that pattern by saying that the one- dimensional structure has translational ordering in a higher dimension (two). A two-dimensional Penrose tiling is a projection from a f o u r - d i m e n s i o n a l translational lattice, and a real quasicrystal is a projection from six-dimensional space.

Figure 2.16: A one-dimensional quasicrystal obtained by projection from a higher-dimensional isometric space. What kind of symmetry is this? It is clearly not translational. (Is it fivefold?) An isometry, or an isometric m a p p i n g , is a transformation that preserves lengths and angles, as in the Bonnet transformation. Certain isometries transform a figure into itself or its mirror image. These are the ordinary symmetry operations that classical crystallography deals with. By combining these we get the discrete groups or transformations allowed for in spaces of different dimensions; 7 groups in one dimension, 17 in two, 230 in three, about 5000 in four and a huge number in six dimensions. Another class of transformations, k n o w n as conformal transformations, preserves angles but not lengths. One example is dilatation, which is repetition by scaling. The golden Fibonnaci sequence is an example of dilatation, as is the pattern of pentagons in Fig. 2.17.

Quasicrystals

69

Figure 2.17- An example of pattern exhibiting dilatational and rotational symmetry based on a regular pentagon. The scaling ratios are golden Fibonacci ratios: 3/2, 5/3, 8/5, 13/8 ..... and the rotations are 36".

Dilatational s y m m e t r y involves expansion of a motif by geometric ratios, while classical (translational) symmetry is a result of arithmetic progression. If space is filled by the packing of the oblate and prolate r h o m b o h e d r a following certain matching rules so that space becomes conformal, a structure is obtained that will give diffraction of icosahedral point symmetry. We assume then that the decoration of the r h o m b o h e d r a with atoms fulfils a Fibonacci sequence. Such a structure has five-fold translation s y m m e t r y in six-dimensional Euclidean space. While the global structure has five-fold s y m m e t r y in three-dimensional space, the local structure will necessarily show deviations. The question of order becomes very complex in systems with dilatational symmetry. Earlier descriptions of order implied isometric repetition. A criterion widely accepted by crystallographers is that a sharp diffraction pattern indicates a well-ordered structure. In that case both Penrose tilings [22] and quasicrystals can be considered to be ordered structures. Conformal s y m m e t r y is very c o m m o n in nature; e.g. we can find it in the nautilus shell and the sunflower. These structures are clearly ordered, even if they do not give sharp diffraction patterns. Here the repetition is nonEuclidean, on a logarithmic spiral (nautilus), or on a torus (sunflower). We are inclined to say that any kind of repetition, conformal or isometric, even in non-Euclidean space, is ordered. However, classification of these more chaotic structures, as for liquids, is less certain. It may be that a liquid can be described as a structure with some of the characteristics of conformal symmetry or perhaps by a representation even more exotic, like a manifold of constant negative curvature. Let us return to quasicrystals and investigate the nature of their intrinsic ordering. It is possible to solve their structure by (almost) classical crystallographic techniques if one a d m i t s structural solutions in sixdimensions. Clearly however this is of little help when we try to grasp the real structure of the crystals. To arrive at the true three-dimensional structure

70

Chapter 2

we have to project the six-dimensional structure down to three dimensions, and the problem is that we do not know what" sort of section should be used to make that projection. Nothing suggests that the section even has to be planar. The problem of elucidating a structure from crystallographic techniques is evidently profoundly difficult.

Figure 2.18: Quasicrystal built by the interpenetration principle. Top (a): The interpenetration shown with only two icosahedra. Bottom (b): Unit containing a larger number of interpenetrating icosahedra. One attempt at full elucidation from six dimensions has been made by Janot et al. [23]. While this attempt is of great elegance, it remains deficient in that the distance between some atoms is too short and partial occupancy must be introduced. The structure can also be described directly using oblate and prolate polyhedra, and combinations of these. Such a structure must be

Quasicrystals

71

constructed from high resolution electron microscopy (HREM) pictures together with the full armoury of intuition and knowledge d r a w n from common structures, and its validity can be checked by comparing diffraction patterns. This has been attempted, starting from HREM images. The proposed structure [24] contains units consisting of an icosahedron surrounded by an i c o s i d o d e c a h e d r o n and this in t u r n is s u r r o u n d e d by a small rhombicosidodecahedron studded with icosahedra: a unit containing in total 174 atoms. These units occupy the corners of the oblate and the prolate rhombohedra. These large units are interconnected by forming parts of wellknown structures like WAll2 and pyrochlore. This would be consistent with a prehistory in the melt of these large units floating around in a loose association. As the temperature drops, these units aggregate to form the quasicrystal. The units are glued together by well-known structures. This model is a member of a series of structures, all built on a simple principle of interpenetration of polyhedra of five-fold point symmetry. The simplest member of the family is shown in Fig. (2.18(a)). Higher members, formed by successive interpenetrations of icosahedra fill space (Fig. 2.18(b)) and a perfectly ordered quasicrystal structure is obtained. There is no need to use oblate or prolate polyhedra; they are generated by a single structure building principle [24]. Whether quasicrystalline structures are limited to alloys remains an open question. It is possible that their occurrence is much more widespread than had been previously thought. Indeed there is evidence for quasicrystallinity in both thermotropic and lyotropic liquid crystals. Diffraction patterns of decagonal symmetry have been recorded in lyotropic liquid crystals [K. FonteU, private communication], (Fig. 2.19), and there is theoretical evidence for the existence of a quasicrystalline structure within the blue phase of cholesterol (Chapters 4, 5). (The decagonal structure has quasisymmetry perpendicular to the tenfold axes, and translation symmetry along them.) Viruses crystallise in icosahedral clusters and the list continues to grow. In addition to five-fold symmetry, it has been shown that eight and ten- fold quasisymmetry is possible.

Figure 2.19: Pseudo-five fold diffraction pattern of a decagonal phase (65~ from the ten-fold axes Cu - K~ radiation, sample to film distance of 200 mm) of a lyotropic liquid crystal (courtesy of the late Krister FonteU). The original image has been traced over to make the image more visible. Compare with Bendersky's study of MnA14[25].

72

~ter 2

The recognition of quasisymmetry as a means of structure building is recent, and has still not made its full impact on the scientific community. In time it will seep through to different branches of chemistry as new examples are identified in different fields. We have already touched on the subject of disorder in solids. One further point deserves comment. Having quasicrystalline materials in mind, it is conceivable that the transition from the liquid to the solid state proceeds via a phase transition exhibiting conformal symmetry. This might be a typical liquid structure, or the structure occurring at the point of solidification. The melt might contain fragments of the solid state, and these would first solidify into a conformal (quasicrystalline) structure if the temperature drops quickly enough. We have already remarked on the observation of a decagonal phase in a liquid crystalline phase. Diffraction data from liquid crystals are often insufficient to determine the symmetry with certainty. It is possible that they could be interpreted in a different way, if the five, eight and tenfold symmetries of the quasicrystalline phases are considered. The expression used for indexing five fold symmetries is Q = 2x/a [(N+zM)/(2(2+z)l 1/2, Q = 4x/~,sinO, (~F5+ 1) / 2 where N is a nonzero integer, M is an integer and the values of M and N are constrained by -0.618N < M < 0.618 N. For the quasicrystal sample of MnA14 type considered by Dubois, a was determined by Janot et al. [23] to be 6.497 A, the lattice parameter of a primitive cubic lattice in six dimensions. It is quite previously dismissed. previously

clear that this possibility permits re-evaluation of some systems, thought to be cubic. This extraordinary possibility cannot easily be Were this to be so, the lessons of inorganic chemistry, a field thought to be relatively complete, would have hardly begun.

73

Organic Chemistry- The Shape of Molecules

2.10

The hyperbolic nature of sp 3 orbitals

We turn now to another area of chemistry where until recently, the notion of shape has been ignored. Organic molecules are constructed from three basic building blocks; sp 3-, sp 2- and sp-hybridised carbons. The latter two hybridisations impart a planar arrangement of substituents around the atom in question while a tetrahedral arrangement will be found around a sp 3 carbon. In geometrical terms, arrays of sp 3 carbons, alone or in combination with other types, will lead to complicated geometrical figures. The sp 3 hybridisation can be described as an idealised saddle configuration and hence an array of such atoms will induce a kind of corrugation in the tessellation, the extent of which is determined by the actual substitution pattern. The resulting surface, necessarily hyperbolic since the ring size normally exceeds four, can be approximated with a portion of a minimal surface. Thus, it should be possible to describe m a n y organic molecules with a suitable minimal surface. Unfortunately, at the present level of m a t h e m a t i c a l dexterity, such a delineation is not readily achievable. Nevertheless some indications as to the minimal surface nature of organic molecules can still be discerned and rationalised. If we start with the simplest possible case, aliphatic hydrocarbons, we can envisage an array of sp 3 carbons, i.e. saddles, joined in some suitable fashion. If we regard each carbon atom in an array as a point belonging to the surface, and consider the fact that a minimal surface is a surface where all such points are saddle points, a minimal surface is formed. For a linear hydrocarbon chain this could be illustrated with a helicoid, a classical analytical minimal surface. This striking resemblance is somewhat stronger if it be recognised that the bonding electrons, rather then the atoms themselves, tessellate the m i n i m a l surface. Bearing that in mind, it is possible to describe conformational movements of hydrocarbon chains in terms of the Bonnet transformation, Fig. 2.20. Not only is it practicable to pinpoint the positions of, and angles between, individual atoms during chain movements, but also an explanation of the concomitant chirality interchange is provided for [26]. The conjecture that bonding electrons rather than atoms should constitute the minimal surface is pleasing, as the notion of bent bonds, a necessity if real molecules are placed on minimal surfaces, is a somewhat ambiguous and a by no means settled issue. If aligned to a minimal surface the "bonds", between atoms in a molecule, could be kept straight, i.e. representing the distances between atoms only, while the actual bonding is along the surface, i.e. curved or bent. This can be interpreted as a situation where the electrons responsible for bonding distend the minimal surface and in so doing secure the over-all geometry. In terms of geometry vs hybridisation one can state that for sp 2 and

74

Chapter 2

sp carbons, substitution geometry and bonding geometry coincide, i.e. the minimal surface in question is a plane. For sp 3 carbons on the other hand substitution geometry is tetrahedral but bonding geometry is saddle shaped, i.e. the bonding electrons are smeared out along a non-planar minimal surface and the appropriate substituents are protruding from the surface.

Figure 2.20: Conformational changes in a C4 fragment of a hydrocarbon chain (large circles denote the C atoms, smaller circles the electrons located between the atoms). These images are produced by fitting bonding electrons to sites on a series of minimal surfaces 0aelicoid-catenoid) related by the Bonnet transformation. The top left image is planar ring fragment, with electrons located around the waist of the catenoid. Successive images show the result of transforming the catenoid by the Bonnet transformation, eventually forming the most extended fragment, with electrons located along the central axis of the helicoid (cf. fig. 1.19). The obvious question to ask w h e n e v e r minimal surfaces are involved is: w h a t is minimised with respect to what? In the case of organic molecules one can apply the same reasoning as for inorganic structures; formation and m a i n t e n a n c e of different c o n t r a p t i o n s with m i n i m a l e x p e n d i t u r e and dissipation of energy. In the construction of molecules where the bonding geometry, be it due to covalent, ionic or van der Waals bonding, is modelled on a minimal surface the required flow of energy is minimised on both enthalpic and entropic grounds. Both actual bond formation and buildingblock re-organisation or relocation is facilitated by the fact that the growing structure tessellates a minimal surface. A feature of curved surfaces is their capacity, loosely speaking, to contain more surface area in a given region than the plane. In situations w h e r e a v a i l a b l e surface area is e v e n t - c o n t r o l l i n g , such as in m o l e c u l a r c o m m u n i c a t i o n , a c u r v e d surface will have the u p p e r edge. Molecules communicate by means of weak forces originating from bonding electron polarisation, either p e r m a n e n t or inducible. The efficacy of the forces is roughly proportional to the n u m b e r of electrons in a molecule and as this in turn is a function of the n u m b e r of atoms, molecular communication is a result of available surface area presented. If the b o n d i n g electrons in a

sp 3 orbitals

75

molecule are situated on a hyperbolic surface their interactive prosperity will capitalise from the area compression mentioned above, i.e. upon each molecular encounter the interactive energy involved is not limited by the actual target area but will be a composite with contributions from the entirety of the molecules. In other words, if the force-field geometries are saddle shaped, molecules are compelled, via the field gradient (including information on both magnitude and direction), to adjust themselves to a "best match" engagement. This is in sharp contrast to a situation with spherical force-field geometry where directional information is essentially useless. Another salient feature of saddle-shaped force fields lies in longrange communication. A solute molecule influences its s u r r o u n d i n g s through the impact of its force field; solvent molecules do not dart around at random but will rather be distributed and oriented according to the forces working on them. Saddle-shaped molecules in solution would be prone to induce saddle-shaped solvation shells, functioning as telltales of low-energy (relatively speaking) interaction routes. In a small molecule it is usually difficult to perceive the saddle surface. This is most certainly due to the conditioning provided by various educational agencies; students are led to believe that the H-C-H angle in, e.g. methane is 109.5 ~ the famous sp 3 angle, illustrated by the hard, pointy geometrical figure of the tetrahedron. Technically speaking it is true; if we replace the atoms with points, connect them with straight lines and measure the angle we will get something like 109.5. The problem is that by doing so we tend to assume that the actual bonding phenomenon is concentrated along those very lines. This delusion is further amplified by the usage of dangerously misleading ball-and-stick models. A methane molecule is not a hard and rigid object, but should be regarded as dense bundle of energy, characterised by smoothness and dynamics. Addition of carbon atoms allows these overall properties to be maintained although some freedom of mobility is lost.

2.11

Organic sculptures: carcerands, crowns, etc.

Organic chemistry is usually presented as a kind of chemical chess; first the chessmen are identified, i.e. the functional groups, second the rules are established, i.e. the possible reactions. Given these precepts the game is ready to begin. Note that this game is confined to single atoms or narrowly restricted regions of a molecule. The rest of the molecule, the part commonly abbreviated as R-, is treated as proud flesh. Interestingly, as the size of organic molecules increases, and structures become more complicated, a striking shift of perspective ensues. As in a chemical glissando the relevance of particular atoms moves out of focus, instead, the shape of the complex structure becomes increasingly germane. Hence the literature is abundant in, e.g. crowns, cavitands, spherands, cryptands, calixarenes, carcerands etc, all with names carrying information on both molecular organisation and chemical capacity. It is noteworthy that in many instances the names given are unconcerned with the identity of the individual building blocks.

76

Chapter2

Now this leap from particularity to collectivity is not yet another educational deception but rather an appreciation of the impact of shape on chemical phenomena. Organic chemists disclose an impressive imagination when naming new complicated compounds. An imagination obviously inspired by the shapes and properties of the molecules in question. Delectable as it may be that chemists esteem shape as a pertinent constituent, the geometrical interpretation is somewhat erroneous. Chemical symbolism is traditionally based on spherical or partly spherical entities. Accordingly construction of illustrative images is in complicity with tradition. Spheres or objects with spherical geometry can however not be reconciled with saddle geometry. Spherical geometry requires positive Gaussian curvature (K) at each point whereas saddle geometry demands quite the opposite; IC~0. As before the geometry discussed is the interactive geometry, i.e. emanating from the bonding electrons taken as an ensemble. Something of a trend in modem synthetic organic chemistry is what one might somewhat blasphemously call "designer-molecules". In an effort to mimic the performance of biological systems in terms of both specificity and efficacy, chemists have devised synthetic protocols dedicated to display the influence of organisation, i.e. aggregational shape. Relaying on brute force, i.e. strong chemical interactions, rarely accomplish specificity; weak interactions on the other hand are usually inefficient. Nature regularly avoids highly potent reagents but depends rather on the collaborative action of weak yet numerous interactions. To achieve the latter end, some kind of focussing of interactions is imminent. An aggregate with spherical geometry offers excellent opportunities to realise an appropriate organisation but at the same time is seriously hampered when considering range and availability. Maximal interaction focussing will be reached for objects with spherical geometry when the sphere is closed, annulling both interactive range and availability. (Any reader familiar with microemulsions might at this point raise an offended eyebrow but should bear in mind that a micelle, clearly spherical, is essentially non-interactive. The dynamics of microemulsions are catered for by non-spherical intermediates.) Opening up a sphere increases the accessibility and the range, although some focussing is lost, until a hemisphere is formed. At this point the operative range of the system is a cylinder of indefinite extension and with a diameter proportional to the diameter of the hemisphere. Any plausible interactive partner has to enter into that rather narrow cylinder to successfully consummate the interaction. Beyond the hemispherical level, cooperative effects rapidly deteriorate. A superior alternative is an aggregate with saddle geometry, e.g. a minimal surface, where highly efficient cooperativity is combined with, in principle, infinite range- and availability properties. The late 60's and the early 70's saw the advent of the first artificial molecules where these concepts were fully utilised, the crown ethers and their sequels the cryptands. Crown ethers or coronands are hetero-cyclic structures where the hetero-atoms are separated with C2-units; cryptands are bicyclic analogues usually with nitrogens at the bridgeheads. These molecules are used in so called "host-guest" chemistry to confine and solubilise positively charged

Futlerenes

77

metal ions of various kinds, the guests, by means of cooperative interactions via the hetero-atoms and hydrophobic embeddment respectively. The impact of these features is quite impressive; 18-crown-6, a cyclic polyether with six oxygen atoms, can solubilise potassium permanganate in benzene, a solvent in which normally the well-known oxidant is essentially insoluble. In coronand/metal ion complexes, as long as the number of hetero-atoms is low a n d / o r the metal ion in question has a reasonably snug fit, the geometry of the assemblage will be planar, or almost nearly so. An altogether different situation arises when the coronand is larger or when non-optimal metal ions are employed. In those instances planar geometry is abandoned and is replaced with saddle-shaped arrangements. This is the case both when a large coronand is folded around a small metal ion or when a metal ion interacts with more than one coronand molecule. Cryptands, being bicyclic molecules, self-evidently cannot form p l a n a r complexes, hence s a d d l e - s h a p e d arrangements abound. The general trend of forming minimal surfaces abides as molecular complexity increases and is inevitable when it comes to chiral compounds.

Figure 2.21: The carcerand fitted to a unit cell of the P-surface.

78

Chapter2

The concept of coronands and cryptands has been further elaborated throughout the years, resulting in a distinct displacement of the characteristics of both the hosts and the guests. Henceforth hosts are designed to accommodate, e.g. negatively charged ions, neutral organic molecules of differing size, etc. Multi-site hosts, both mixed and isotropic, hosts with integrated reactive groups such as push-pull acid-base systems, hosts bonded to polymeric matrixes, have all become available. The pervading minimal surface nature of these different compounds is especially accentuated in structures providing large surface-carrying units. Spherands(sic!), structures based on cyclic oligomers of p-cresol, upon X-ray analysis disclose explicitly non-planar benzene rings indicating that the so called "enforced cavity" indeed follows the geometry of a minimal surface. Some very recent synthetic achievements are equally suggestive; dodecahedrane, C20H20, a truly spherical molecule yielded to synthetic efforts as late as 1984. Its preparation was regarded as a corner-stone of synthetic organic chemistry- and was preceded by an extensive amount of manpower. On the other hand the synthesis of the so-called carcerand, an equally closed molecule, the structure of which bears a casual resemblance to a sphere, was synthesised without difficulties on an apparently first-attempt basis. This remarkable difference is a reflection of the contrasting geometries involved. The spherical geometry of dodecahedrane is compelling, creating a state of opposition, and indeed congestion, when units preferring saddle geometry are forced together. Attempts to link two hemispherical units in order to complete the dodecahedrane skeleton have been consistently futile. The carcerand structure however is one of saddle geometry. It can be shown that it is an almost perfect fit to the repetitive cell of the surface (Fig. 2.21). Hence in the final synthetic step, when two preformed units are joined, no congestive or other problems are encountered.

2.12 Beyond graphite: fullerenes and schwarzites Organic chemistry is now moving beyond its traditional preoccupation with small molecules. The most spectacular examples of this shift are the fullerenes, which are sp 2 carbon clusters forming giant ball-like molecules: C60, C70 etc [27]. Until their discovery, it was believed that planar graphite consisting exclusively of six-rings - was the sole sp 2 carbon polymer. By Euler's theorem (section 1.8), the formation of carbon shells, whose geometry is elliptic, requires the presence of five-rings in the carbon network. If five-rings are possible, why not seven-rings? The presence of seven-rings in the (three-connected) network must lead to hyperbolic frameworks. Once the average ring-six exceeds six, hyperbolic carbon structures must result, which lie on periodic hyperbolic surfaces, and form three-dimensional extended frameworks. A number of theoretical studies have indicated that these hyperbolic structures, called "schwarzites" in honour of the mathematician Schwarz, should be more stable than the fullerenes [28]. In

79

Beyond graphite

geometric terms, a unique hyperbolic surface can be traced through the threeconnected carbon atoms. Just as for the tetrahedral networks analysed in section 2.8, the curvature of this surface is related to the bonding geometry. Here too, the most stable p o l y m o r p h s are expected to form tessellations of surfaces similar to IPMS [29]. So far, theoretically predicted schwarzites do not display this universality, although a number of predictions give an area per C atom close to that found in graphite and C60 (Fig. 2.22). These theoretical frameworks are the result of c o m p l e x n u m e r i c a l q u a n t u m mechanical calculations. The a p p a r e n t conservation of surface density, irrespective of the curvatures of the surface, is clearly not a direct consequence of standard physics. It will be very interesting to compare the surface densities of actual schwarzites (although they have yet to be prepared in the laboratory) with those of fullerenes and graphite. Given the usefulness of this principle in the study of tetrahedral frameworks, our bet is that they too will lie on the dotted line in Fig. 2.22.

9

I

"

I

"

I

"

lO 8

6

~ _

.~ i

5.5

l

6.0

.~ .

.

I

I

6.5

.

.

~_

9

I

7.0

.

9

7.5

a v e r a g e ring-size

Figure 2.22: Area data calculated as described in Fig. 10 (using the same vertical scale) for a range of hyperbolic "schwarzite" sp 2 carbon networks predicted by various theoreticians. The filled diamonds denote the corresponding areas per vertex for (planar) graphite (n=6) and C60 fullerene (n=5.62). (The latter is calculated from standard crystallographic data assuming a spherical network.)

80

Appendix: The Problem of Quasicrystals The problem posed by the discovery of quasicrystals has occasioned much angst among chemists. Indeed some, like Linus Pauling [30] dispute their existence, and claim that they are an artefact which can easily be accommodated within the framework of conventional theories of space filling structures. The issue remains open. Projections from hyper-Euclidean spaces to three-dimensional and group theory [4, 22, 31], geometric packing arguments [4, 31] are some of the mathematical tools that have been invoked, and some real space models that capture the main features of particular systems have been constructed [24]. The problem is to relate observed diffraction patterns with non standard, supposedly disallowed, crystallographic symmetries, to the atomic distributions that cause them. That problem remains. Because while a physicist living in world made up of equations and group theory has no difficulty in constructing the universe, its scaling laws, and singularities like black holes, as a realisation of a sixteen-dimensional group say, the chemist is more narrowly constrained. A three- dimensional atom has a certain pedestrian reality that does not so easily lend itself to a mapping into six dimensions. The question raised by the quasicrystal debate is much deeper than whether they exist or not. To see this, we recall that the interpretation of diffraction experiments on all known translationally invariant crystals, however complicated, depends ultimately on the existence of the Poisson summation formula. This relation asserts that the Fourier transform of the periodic delta function is itself a periodic delta function, whence the term reciprocal space. Explicitly, the Poisson summation formula is oo

f(x) = ~

~

oo

8(x-m)--- ~

n ~

exp (2~mfix)= 1 + 2 ~

rr~.o

cos(2~mx)

m=l

so that the Fourier transform of a translationally invariant array of atoms represented by the periodic delta function is

f(k) =

dx exp(-2nikx)

8(x-m) ..oo

= ~exp(-27tikx)= ~ -oo

m=-oo

8(k-m)

Appendix: Quasicrystals

81

The determination of crystal structure is 9 immediate, in principle, since any standard diffraction pattern will be related to, e.g., the product of an appropriate combination of three such delta functions (periodic in x,y,z directions), with atomic form factors. Inversion to get the real space atomic positions from the diffraction pattern is then possible via the convolution theorem for Fourier transforms, provided the purely technical problem of the undetermined phase can be solved. Now the Poisson summation formula is at the core of all mathematical analysis [33]. It is equivalent in fact to the calculus, the Jacobi theta function transformations, and to a statement of the Riemann relation connecting the oo

zeta function ~(s)m ~ n!s, Re(s)> I with ~(1-s) so linking up with number n=l

theory itself. (The distribution of zeros of the ~ function in the complex splane is one of the major unsolved problem of analysis.) Fundamental solutions of the time-dependent Schr6dinger equation and the diffusion equation are the theta functions. There has been no basic formula analogous to the Poisson summation formula, characteristic of translational invariance, on which to base an analysis of quasicrystal diffraction patterns. Here successive values of 'reciprocal' space have geometric ratios instead of the arithmetic spacing of the peaked functions observed with ordinary crystalline diffraction. Fig. 2.15 illustrates a two-dimensional section in reciprocal space of a diffraction pattern. The five-fold symmetry is exact, and typically six indices instead of three are required to index each point, with the choice of origin arbitrary, and for assignment of indices, ambiguous. The features of interest are: (1).Along a given five-fold axis, the spacing of the main peaks kn is in the (geometric) ratio kn = 1 where z is the golden ratio (~5 + 1)/2. kn+l (2).Between two such peaks is a sequence of further peaks of lower intensity that all lie in an infinite set of coincident interpenetrating Fibonacci sequences of arbitrary origin. (They can be labelled through a projection from Euclidean two-dimensional space as described in the main text.) (3).Surrounding each five-fold axis are other sequences of lesser intensity that can be connected to their neighbours to form regular self-similar pentagons (icosahedra in three dimensions). (4).Along the five-fold axes the density of points as one approaches the origin becomes infinite. (5).Increasing time of exposure of the recording film results in the appearance of more and more points throughout space in an eventually dense spacefilling array. All points satisfy the same symmetry and self-similarity properties.

82

Chapter2

(6).Through a set of initial points on adjoining five-fold axes a set of 20 intersecting equiangular spirals can be drawn emanating from any chosen origin.

If (1) is taken as the main feature of the structure, the appropriate representation along any axis is [34]: oo

f(k)=~-J ~

8(~_~-m)

(where the factor I kl-j, j=0,1,2 in 1,2,3 dimensions is necessary to preserve self-similarity). This is a very different function from the periodic delta function (arithmetic spacing) that underlies the spectra with which we are most familiar. The Fourier transform of such a function, the "skeleton" of the spectrum, has been investigated [34]. It gives out the equiangular spiral as a fingerprint - the self-similar properties of figures inscribed into such spirals have been a source of much mystery for centuries [35]. The analysis gives more, and the zeros of the Fourier transform give an infinity of other real space points that also satisfy self-similarity and fill up non-occupied space. The intersecting spirals that emerge in two dimensions (or helices in three dimensions) give the Penrose tilings naturally a;td the constraints of self-similarity, scaling, five- fold symmetry and independence of origin also emerge. There is a hint here that new kinds of basis functions other than the usual periodic delta function of ordinary mathematical analysis may provide an extraordinarily rich range of new structures. The emergence of the logarithmic spiral is not surprising, and D'Arcy Thompson [35], James Bernoulli [35], (who was so fascinated by its properties that he had it engraved in his tombstone) and Kepler would have taken it as self-evident. In T h o m p s o n ' s words: "In the growth of the shell, we can conceive of no simpler law than this, namely, that it shall widen and lengthen in the same proportions; and this simplest of laws is that which Nature tends to follow. The shell, like the creature within it, grows in size, but it does not change its shape, and the existence of this constant relativity of growth, or constant similarity of form, is the essence of the equiangular spiral". The maintenance of shape, or the constant change of curvature is indeed of the essence. Quasicrystal structures have been known for a long time to occur in condensed matter and rejected as inexplicable curiosities. They may emerge naturally too in mathematical descriptions of surfaces. (The decagonal variant certainly arises, cf. [36]). It is not an entirely idle speculation to conjecture, e.g. that the principles exploited in construction of quasi-crystals may be precisely those used by nature to build proteins that solve the problem

Appendix: Quasicrystals

83

of self-recognition, the fundamental problem of immunology. Nor, that since the associated infinite incommensurate packings are missing from the usual Hilbert space description of quantum mechanics, that the problem may be related to the difficulties that confront particle physics.

84

Chapter2

Rs163163163 1.

H. Weyl, "Symmetry". (1952), Princeton University Press.

2. D. Shechtman, J.A. Blech, D. Gratias, and J.W. Cahn, Phys. Rev. Lett., (1984). 53: pp. 1951-1953. 3.

D. Shechtman and J.A. Blech, Met. Trans., (1985). A16: pp. 1011-1065.

4. M.V. Jaric, "Introduction to Quasi-Crystals'. Academic Press, (1988 and 1989), New York: Academic Press. 5.

S. Andersson and B.G. Hyde, Z. Kristallogr., (1982). 168: pp. 119-131.

6.

L. F~ilth and S. Andersson, Z. Kristallogr., (1982). 160: pp. 313-316.

7.

Q.B. Yang and S. Andersson, Acta Cryst., (1987). B43: pp. 1-14.

8. S. Andersson, S.T. Hyde, and H.-G. von Schnering, Z. Kristallogr., (1984). 168: pp. 1-17. 9.

R. Nesper and H.G. von Schnering, Z. Kristallogr., (1985). 98: p. 111.

10. I.S. Barnes, S.T. Hyde, and B.W. Ninham, J. Phys. (France), Colloque., (1990). C-7: pp. 19-24. 11. R. Thomasson, S. Lidin and S. Andersson, Angew. Chem. Int. Ed. Engl., (1987)). 26: pp. 1017-1018. 12. J.W. Christian, "The Theory of Transformations in Metals and Alloys.". (1965), New York: Pergamon Press. 13.

S.T. Hyde and S. Andersson, Z. Kristallogr., (1986). 174: pp. 225-236.

14. S.T. Hyde, Infinite periodic minimal surfaces and crystal structures (1986), Ph.D. Thesis, Monash University. 15.

C.T. Kresge, M.E. Leonowicz, W.J. Roth, J.C. Vartuli, and J.S. Beck,

Nature, (1992). 359: p. 710. 16. S.T. Hyde, in "Defects and processes in the solid state. Some examples in earth sciences.", J.N.Boland and J. D. FitzGerald,Editors. (1993), Elsevier: Amsterdam. 17.

S.T. Hyde, Acta Cryst., (1993). AS0: pp. 753-759.

18. W.M. Meier and D.H. Olson, "Atlas of Zeolite Structure Types". 3rd. ed. (1992), London: Butterworth-Heinemann.

References

85

19. S.T. Hyde, Z. Blum, and B.W. Ninham, Acta Cryst., (1993). A49: pp. 596-589. 20. F. Liebau, "Structural Chemistry of Silicates". (1985), Berlin: SpringerVerlag. 21. G. Kowalewski, "Der Keplersche K6rper und andere Bauspiele". Vol. 3. (1938), Leipzig: K.F. Koelers Antiquarium. 2.

A. Mackay, Physica (1982). 114A: pp. 609-613.

23. C. Janot, M. de Boissieu, J.M. Dubois, and J. Pannetier, J. Phys. : Cond. Matter.(1989). 1: pp. 1029-1048. 24. S. Lidin, S. Andersson, J.-O. Bovin, J.-O. Malm, and O. Terasaki, Acta Cryst., (1989). A45: pp. 33-36; M. Jacob, S. Lidin and S. Andersson, Z. Anorg. Allg. Chem. (1993), 619: pp. 1721-1724. 25.

L. Bendersky, J. Phys. (France), (1986). 47(C3): p. 457.

26.

Z. Blum and S.T. Hyde, J. Chem. Res. (S), (1989). (6): pp. 174-175.

27. H.W. Kroto, A.W. Allaf, and S.P. Balm, Chem. Rev., (1991). 91: pp. 1213-1235. 28.

A.L. Mackay and H. Terrones, Nature, (1991). 352: p. 762.

29.

H. Terrones and A.L. Mackay, Carbon, (1992). 30(8): pp. 1251-1260.

30. L. Pauling, in "The Chemical Bond: Structure and Dynamics", A. Zewail, Editor. (1992), Academic Press: New York. 31.

G. Gratias and L. Michel, J. Phys. (France), (1986). C3: p. 7.

32.

R. Penrose, Bull. Inst. Math. Appl., (1974). 10: p. 266.

33. B.W. Ninham, N.E. Frankel, B. Hughes, and L. Glasser, Physica A , (1992). 186: pp. 441-481. 40

B.W. Ninham and S. Lidin, Acta Cryst., (1992). A48: p. 640.

35. D.W. Thompson, "On Growth and Form". 2nd. ed. (1968), Cambridge University Press. 36.

A. Fogden and S.T. Hyde, Acta Cryst., (1992). A48: pp. 442-451.

This Page Intentionally Left Blank

87

Chapter 3 3.1

Molecular Forces and Self-Assembly

The background

hape and form imply the notion of a surface. In biology the shape taken up by an enzyme or DNA is crucial to its catalytic or templating properties. This shape, sometimes a fluctuating structure, is dictated by molecular forces. Intramolecular forces determine shape, and intermolecular forces guide recognition processes. Again, in cell biology, precise recycling patterns occur that require transformations between self assembled aggregates like micelles, vesicles and bilayer membranes. Indeed lipids, the surfactants from which these objects form, provide the most primitive of prebiotic assemblies. These are dynamic, equilibrium structures and even within a given class and order, transmute and reassemble into different structures in response to quite delicate changes in environment- such as temperature, salt and other factors that determine the driving molecular forces. These determine curvature. It is known and understood that lipid membranes provide a matrix for enzymatic catalysis for structural proteins, and for transport, and a protection to cells against the exterior environment.

S

We can make these remarks explicit by considering the mechanism [1, 2] of endocytosis as illustrated in Fig. 3.1. In terms of specific biochemical processes, one focusses on the selectivity of the binding of a ligand to the cell membrane protein, the dissociation of the ligand complex induced by the decrease in pH within the endocytosome, the separation of the ligand and protein in the curl, and the eventual processing of the ligand. There is an alternative way of thinking in terms of the physical forces which are a consequence of binding and dissociation processes. These forces cause changes in curvature that result in the transformation of induced cell membrane pits to protein-coated vesicles, the new cubosomal structures described in Chapters 5 and 6 and the fusion of those vesicles. Further changes in curvature are evident in the formation and separation of the curl, and in the subsequent fusion of the tubular assembly into the cell membrane. Chemical and physical processes are coupled. The bewildering diversity of assemblies that biology confronts us with is so vast that it seems impossible to make sense of the matter. Yet the patterns are so repeatable and precise that there must be an underlying unity that derives from the forces that determine curvature and guide the necessary chemical reactions. It is only recently that some of that unity has been revealed. This Chapter outlines some of the nature, delicacy and specificity of molecular forces, and how it is that these forces conspire with the geometry of molecules to organise self-assembled molecular aggregates. The shapes and topologies that set the physico-chemical environments for biochemistry are the subject of following chapters. The references provide a sufficient guide to the literature for the reader interested in exploring further complex technical issues.

88

Chapter3

Figure 3.1: (Top:) Schematicview of processes controlled by curvature in endocytosis. Adapted from [2]. (Bottom:) Representative of the simplest surfactant aggregates: spherical miceUes (v/al between 1/3 (spheres) and 1/2 (cylinders) ; planar bilayers (v/al = 1); inverted micelles (v/al > 1).

3.2.1

The nature of force, shape and size

Geometry deals with the disposition of objects, originally idealised as points, and time labels the evolution in position of those objects. The two idealisations, distance and time were linked by Newton through the separate concept of mass to give us the concepts of momentum and force. Matter, and later charge, both the measure of inertia and the source and sink of force, were once seen as immutable properties of matter. They became inextricably part of the idealised, matterless space-time continuum after Gauss developed the concept of potential energy, after Maxwell's theory of the electromagnetic field, Lebedev's proof of radiation pressure exerted by the field, and finally Einstein's synthesis in the theory of relativity. Boltzmann's statistical mechanics gave us entropy, temperature and chemical potential in a large collection of interacting molecules. Planck imposed the condition of discreteness on the electromagnetic field that led to the wave-particle duality and the muddled grandeur that is modern quantum mechanics. The rest is history, and the soaring flights of imagination that have taken us to m o d e m theories of the universe and the search for a unified field theory embracing

Force, size and shape

89

gravitational, weak and strong interactions appear to far transcend the more m u n d a n e concerns of chemists. But the point particles of physics ignore shape and size that are the axiomatic attributes of the subject of chemistry, be they atoms, molecules, proteins joined in a supposedly particular configuration by "chemical bonds", or transient lipid vesicles or micelles. And where one object ends and another begins is not so self-evident. The notion of a bond that emerges from a quantum mechanical theory of two interacting atoms is not so obvious if those objects are immersed in a sea of their neighbours, forming a solid or liquid. To make sense of a very complex matter which can easily bog down in either recondite mathematical theories, or be relegated and s u b s u m e d u n d e r qualitative, non-predictive characterisations embraced by terms like "ionbinding" to surfaces, we dissect the problem in stages. The ensuing discussion attempts first to come to grips with what we mean by the size and shape of a molecule. Thereafter we explore in general, historical, terms the conceptual background to what is known about forces between surfaces in solution to set the b a c k g r o u n d for the specific technical discussion that follows. Subsequently we show how a knowledge of these forces enables us to change the curvature of aggregates of self assembled molecules. The different sections are self contained and can be read independently.

3.2.2 Self-energy, molecular size and shape In most discussions of molecular forces [3-10], atoms have been treated as if they are point particles, and size or shape has been invoked as a separate concept. For example, the van der Waals energy of interaction V(r) between two atoms a distance r apart is V(r) or - 1 / r 6. It becomes infinite on contact, unless some preassigned hard core contact size is invoked, and so too for physisorption of an atom to a surface. This convention is necessary, but its origins lie deep. In fact no distinction can be made between energy, size and shape, which are inextricably linked. The linking concept of self-energy takes on significance whenever an object is considered to have a finite extent or is delocalised: for then the abstraction that the object can be considered separately from its surroundings becomes philosophically tenuous, as one part of the object can consider its other parts to belong to the rest of the world; hence, perhaps, the uncertainty principle. No difficulty occurs if the environment or object are immutable. If the opposite obtains, as indeed it always does at some level, the reaction of the (changed) environment to the object will be different, and the self-energy due to this reaction field will be different. The shift in self-energy due to radiative corrections to energy levels is a central problem of quantum electrodynamics. The Born electrostatic selfenergy of an ion is important in electrolyte theory, physical adsorption and in the migration of ions through membranes. The Debye-Htickel theory of electrolytes results from the change in Born self-energy of each ion due to all

90

Chapter 3

other ions in the solution. The self-energy of a dipole e m b e d d e d in a dielectric sphere is the key to Onsager's theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechanics from which the adsorption isotherm is derived. The idea is clear, but the details are complicated [3-12]. We omit details and focus on the results. For a molecule of finite extent its dispersion self-energy can be defined as the change in its energy due to its coupling with the electromagnetic field, or equivalently, as the change in q u a n t u m mechanical zero point energy of the field due to its coupling with the oscillating dipole moment it induces on the molecule. If we consider a single atom centred at position R interacting with the radiation field, we can write d o w n Maxwell's equations to describe the process. Their solution requires a closure relation, between the induced dipole moment density p(r, ca) and the electric field E(r,r at all points in space r, and at all frequencies of oscillation co of the field. For point molecules within the framework of linear response theory, this relation is

p(r, co) = a(co) E(R,o0) 8(r- R)

where ~ is the polarisability tensor of the atom and 5(r) the Dirac delta function. This relation is strictly incorrect, because the dipole is spread out over a region of space (of the order of the "volume" of the molecule), and in a semi-classical formulation of q u a n t u m mechanics it can be shown that the actual relation is p(r, co) = cx(r- R, co) E(R, ca) where the components of the tensor a ( r - R, co) are peaked around R with a range which defines the size of the molecule. It can be given an explicit expression in terms of a sum over matrix elements of atomic wavefunctions [11, 12]. That is the end of the matter, and the solution can then be effected in principle. We can assume for computational convenience that the atom is isotropic and that ~(r, ca)=a (co)f( r ) / w h e r e / is the unit tensor, and the form factor f(r) is a distribution function for the electrons peaked at the centre of the atom. A gaussian is a good approximation, and we take

/ r2) , ff(r)

f(r)= n3/21 ag.exp - ~ -

d3r = l

Self-energy

91

where a can be taken as the size (radius) of the atom. In the n o n - r e t a r d e d limit i.e. i g n o r i n g the finite velocity of light, the dispersion self-energy computed with this form factor can be shown to be [11, 12]*

Es _ ~ 3 /2h2 a 3 Jo d~a(i~) _=~ Rydbergs (Hydrogen atom)

(3.~)

The dispersion self-energy is of the same order of m a g n i t u d e as the binding energy for a hydrogen atom, but of opposite sign. The same kind of semi-classical formalism permits extension to two or more atoms. The interaction energy, the difference between the complete energy of the coupled system and the s u m of the dispersion self-energies of two isolated atoms, reduces to the van der Waals interaction energy at large distances, but n o w remains finite at zero separation. For like atoms this energy is of the order of the binding energy of the molecule that w o u l d be formed by them. (For point atoms, the van der Waals interaction energy is infinite.) The same concepts can be used to develop a simple semi-classical estimate [3-10, 11, 12] of the Lamb shift in hydrogen, and to explain the differences in binding energies [3-10, 13] (e.g. face-centred cubic versus hexagonal close packed) of rare gas crystals. Extension to include quadruple and octopole effects explains the shape of the interaction potential necessary to fit the thermodynamic properties of simple liquids like argon, calculated by Monte Carlo methods. Further extensions to include surrounding media [310], polymers or the presence of surfaces [3-10], or to physisorption [3-10], reconcile a number of difficulties.

3.2.3

Self-energy and adsorption

As an illustration consider the a d s o r p t i o n of molecules into zeolites, discussed in Chapter 2. If an adsorbate molecule or atom is immersed in a m e d i u m of dielectric constant ez(rO) instead of a vacuum, the dispersion selfenergy is modified, and now takes a form corresponding to eq. (3.1)

* Here the polarisability a(co) as a function of the complex frequency variable co=TI+i~is evaluated on the imaginary frequency axis ~. The reasons are technical [3-10] and need not concern us. The simple classical form of the polarisability for a one-electron atom is Or(CO)= e2 where e is the electronic charge, in the mass of the atom, and -h-co0 is the m~0~- ~ ~) ground state energy, coo= e2, a0 the Bohr radius. 2a0

Chapter3

92

Es(medium)

_ 2h = lt3/2 a3

d~ a(i~) r

(3.2)

The expression immediately gives an estimate of the enthalpy of adsorption in taking an atom from the gaseous (vacuum) state to a liquid, or to a composite medium like a zeolite, characterised by its measured dielectric frequency dependent response &(c0). It is, exactly as for the electrostatic Born self-energy in taking an ion from vacuum to water:

AH(adsorption) = [Es(vac) - Es(medium)!

= 2/~ -11:3/2 a3

l" [

d{ a(i{) 1 - - 1 ] ez(i~ ]

The measured enthalpies of adsorption of alkanes into zeolites are very large, and can be calculated from this formula [13]. The dielectric constants for the open networks formed by the aluminium silicate frameworks of zeolites can be related to the known and measured dielectric and optical properties of bulk quartz. The calculation assumes that the zeolite framework tessellates a hyperbolic surface as described in Chapter 2. The polarisabilities of different adsorbate alkanes are also known. When the calculations are carried out for a whole range of alkanes as adsorbate molecules and for different zeolites (with differing pore structure and size) the agreement between measured and predicted heats of adsorption is excellent (cf. Fig. 3.2). The results depend explicitly on the Gaussian curvature (AH=~I~/2) , and involve only a single parameter, a, the effective radius of a single CH2/CH3 group, which turns out to be 2.9~. The approach using the self-energy concept is very different to the simpleminded two-body potential summation method of Section 2.5. It connects the Gaussian curvature of the hyperbolic surface underlying the zeolite framework quantitatively and directly to the many body interactions and excitations of all the molecules involved. These are included through the measured dielectric properties in a way not accessible to two body potentials. The approach provides a key to the mechanism of catalysis in zeolite frameworks. Thus, from a consideration of the self energies of say a single dodecane, and two hexane molecules in vacuo, we can conclude that the spontaneous decomposition of dodecane into two hexane molecules is most improbable. (The general formula for self-energy includes temperature effects also.) But inside the zeolite a different story obtains. The vibration and other excited modes of the dodecane molecule built into the self-energy expression are very different, and self-energy calculations show that it becomes

Self-energy

93

favourable for such a molecule to split into smaller parts - just as for example a salt molecule held together by an ionic bond, will dissociate into its component ions when dissolved (adsorbed) in water, thermodynamically favourable if one considers entropy and electrostatic self energies. (The water, from our viewpoint can be considered to be a dynamic self intersecting hyperbolic surface tessellated by the hydrogen-bonded network.) The selfenergy concept has been shown to be quantitatively correct in predicting the 0-temperatures of polymers in simple liquids [3,61].

aHax p (Jg")

4O

.i

o

30"

20

ill

~

U

31"

0

"

II

2OO

~

"

d

~~

~

r

II

4O0

"

o ZeoliteY

l

6OO

"

l

800

"

l

lO00

, ~ X ~ ' (J~,'s")

Figure 3.2: Theoretical (x-axis) vs. measured (y-axis) heats of adsorption in zeolite Y and silicalite for a range of alkanes. The theoretical heats are multiplies by the "volume" of a methyl group (a3).

In organic chemistry, phenomenological rules that predict reactions and reactivity that begin to break down, e.g., with increasing hydrocarbon chain length, also makes sense once the connection between dispersion self-energy, the environment, size and shape are recognised. While the notion of size and shape, the geometry of molecules or objects, emerges naturally in semiclassical theories, and is intimately related to energy, these concepts appear to disappear in full quantum mechanical descriptions of molecules. Thus the full quantum mechanical description of molecules in the gas phase gives excellent agreement with observed spectra, but fails to explain optical isomerism. Yet the question of chirality is vitally important to the understanding of biological activity of organic molecules. The much cruder Born-Oppenheimer approximation- which assumes that nuclei are fixed in space - does build in shape, the concept of a potential energy surface, and the idea of a chemical bond. Quantum mechanics in its present form misses the notion of shape [14].

94

Chapter 3

3.2.4 The shape of bonds This paradox, of physics, is more apparent than real, and the chemists have persisted with the fiction that objects exist. The concept of a chemical bond, ionic, van der Waals, covalent, is taken for granted and is essential to chemistry. The first two make no sense except in the context of an infinite crystal. (An "ion pair" in solution, or a "hydrophobic bond" in water between two methane molecules is due to complex statistical mechanical solvent mediated association behaviour, to be discussed below.) If the positions of the atoms are assigned in a regular lattice by X-ray crystallography one can calculate the electrostatic energy of formation of an ionic crystal like CsC1. This energy, per CsC1 pair in the lattice, is the bonding energy of the pair. If the Cs +, C1- ions that form the lattice structure are modelled as hard, non-interpenetrating systems (of different radii), there is no real bond; just an attractive Coulomb potential, cut off, and infinite at a "hard core" contact. In practice, the ionic electron clouds are not sharp, and the hard core can be replaced by a softer short-range repulsive potential that gives a minimum in the combined potential at the "equilibrium" radius of contact. The pairs oscillate with zero point energy, more with increasing temperature, about this minimum, or bond length. At best such a "bond" is an effective one, since the Coulomb potential is of infinite range, and the energy per bond depends on a sum of interactions of all the ions of the crystal. A similar situation holds for the van der Waals interaction or bond, in a non-ionic crystal. Here the attractive (V(r)o~-1/r6) potential is again opposed by the short-range repulsive potential due to electron cloud overlap to give a potential minimum, which if we like, we can call a bond. As already remarked, even for this case (short-range attractive forces) discrimination between the theoretically calculated bonding energies of rare-gas crystals requires that the dispersion energy be calculated to all orders of perturbation theory [3]. It is, like the ionic crystal case, a many body problem. The entire electromagnetic radiation field due to inter-oscillations of all the atoms of the crystals must be taken into account to explain observed energies of formation. So, even for the simplest conceivable cases of ionic and van der Waals crystals, global properties, many-body forces that depend on the arrangement of atoms, and local force properties are linked. Minimal surfaces appear here in a very natural way. To see this, consider an array of electrostatic point charges arranged in the different crystallographic symmetries of say NaC1, CsC1 crystals of Chapter 2. Suppose that the particular space-group symmetry is given, and for the moment admit the approximation of a hard-sphere model for the ions to set the lattice parameters. The Ewald sum of electrostatic energies provides the major contribution to the binding energy. Now consider surfaces of zero electrostatic potential traced out inside the lattice [15-18]. These can be

The shape of bonds

95

calculated, and look very much like minimal surfaces. If the crystal zero potentials are truly minimal surfaces, then phase changes can occur easily via the Bonnet transformation. Excited states of the crystal, including anharmonic states, are then included and counted as the (infinity) of different allowed minimal surfaces that we might suppose forms a complete set. But these surfaces are not exactly minimal, although appearing close to such. The reasons seem clear enough: it can be proved that no electrostatic set of charges can be in an equilibrium state. Presumably if one built in quantummechanical zero-point energies and then calculated the new equipotential surfaces, the new zero potential surfaces would be minimal surfaces of the field. With that idea, we can turn the argument around and say that since the crystal exists as an equilibrium system, it must be permeated by surfaces of zero stress of the entire electromagnetic field. Imposition of this (mechanical) requirement, together with the condition that the Poynting vector (E x H) ( m o m e n t u m transport) is zero, p r e s u m a b l y will give back q u a n t u m mechanics. The argument implies the emergence of quantum mechanics as a consequence of minimal surfaces, a necessary Pythagorean imperative that effects the bridge between geometry and arithmetic, discrete and continuum, the particle and field points of view [18]. For covalent bonds we are likewise on tenuous ground and the lone pair concept so useful to chemists has been hotly disputed. Consider only the ground state of a covalently bonded few-atomic system, and presume that an ab initio quantum mechanical calculation gives us a lowest energy state. Let us ignore the complication that different isomers of the covalently bonded system could be degenerate. There will be a shape associated with this lowest energy configuration, set by the electron cloud distributions about the fixed positions of the nuclei. The squared amplitude of the calculated Schr6dinger wave function gives this electronic distribution. A frustration for chemists has been that this probability distribution, while appearing to mimic vaguely what one thinks a bond should look like, is too smeared out to give any real confidence beyond its mathematical expression. Some remarkable recent work [19, 20] has taken up ideas that were already a part of quantum mechanics, accessible at least thereto since the discovery of the Pauli principle, itself a fundamental building block in chemistry through Hundt's rule. By calculating an electron localisation function from the wave function, the probability that if an electron is in a given spin state another will be a prescribed distance away can be calculated, and the positions of the electron in a bonded system can indeed be pinned down. Various calculations confirm that the intuitive covalent bond picture of chemistry, and the lone pair concept, are very precise and sharp. What is more the 'shape' of the electron localisation distribution functions turns out invariably to give precisely the shape traced out by sections of periodic minimal surfaces [21].

96 3.3

Chapter3

The background to surface forces

We turn now to the nature of molecular forces between surfaces separated by a liquid. The main question is the nature of forces at distances greater than molecular dimensions, and less than those at which matter can be treated as a continuum. Very clearly, its answer must be intimately connected with the meaning assigned to the word surface. The (molecularly smooth) mica, or mercury - water interfaces are one thing, the liquid vapour or immiscible liquid liquid interface is another. And it might reasonably be imagined that the thesis that the world is fiat would be considered extravagant madness by an anthropomorphic huddle of water molecules near a biological membrane or surfactant interface. However, we persist with the idealisation of a surface. Consider first planar solid surfaces. -

Two themes emerge in dealing with the force between two such surfaces:(see Appendix 3): (i) an intervening liquid can be thought of as a structureless continuum with bulk liquid properties, up to a molecular distance from the surface; (ii) any object (surface) must perturb proximal liquid structure (density, dipolar orientation, hydrogen bonding) so that the transmission of force is propagated via a stress field passing from molecule to molecule in much the same way that the electromagnetic field is carried through the vacuum or a dielectric in MaxweU's theory of the electromagnetic field. Once the forces between rigid surfaces are understood, we can proceed to the more complex question of how an object reorganises its shape in response to the change in chemical potential, of force induced by a neighbouring object. The classical intuition on molecular forces is embodied in the famous Derjaguin-Landau-Verwey-Overbeek theory of colloid stability. It blends themes (i) and (ii) above in a contradictory way that has taken until the last few years to sort out. The best elementary account of the theory is contained in the book by Israelachvili [4]. But there have been very many developments since, as outlined below. These developments are very complicated, but cannot be ignored, and the subtleties have to be recognised to make sense of many phenomena in colloid science and biology. Imagine a suspension of colloidal particles in water. What causes stability, and what, under changing solution conditions like addition of salt causes flocculation (precipitation of the suspension)? Two opposing forces were considered to operate between two such particles. The one, attractive, is the quantum mechanical van der Waals force and treats an intervening liquid as if it has bulk liquid properties up to the interfaces of the particles (theme (i)). The other, repulsive, due to charges formed by dissociation of ionisable surface groups, is electrostatic in origin, and depends on salt concentration.

Surfaceforces: background

97

Here by contrast, it is the overlap of the inhomogeneous profiles of electrolyte concentrations induced by the charged surface of the particle that gives rise to the osmotic (double-layer) force (theme (ii)). When the repulsive double-layer forces win out, the suspension is stable. On addition of sufficient salt, the range of these forces decreases, the attractive forces take over, and the system of particles flocculates. While the picture is (usually) qualitatively correct, it is deficient in very many respects, and an understanding of how these deficiencies show up is essential to our thesis. Before launching into detail we first outline how this theory [3-10] has changed. We remark in passing that the more sophisticated m o d e m theories have been confirmed by direct measurement, or so the story goes. Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these: van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of socalled retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. For the double-layer (electrostatic) forces, extensions beyond the point ion approximation [22] to allow a hard core size to the "hydrated" cations and anions, inclusion of image interactions and higher order correlations in the inhomogeneous electrolyte profile show that these forces cannot be treated independently of the van der Waals forces. For multivalent counter-ions, like Ca 2+, the "new double-layer" forces can actually be strongly attractive at small distances ~ >c/OOo, where c is the velocity of light, this energy goes over to the weaker (retarded) form V(r)~B/rT). These results are obtained by treating the atoms as if they have zero size. The form of the interaction due to induced dipoledipole correlations is the leading term in a power series in 1/r 6 and higher order terms (induced d i p o l e - q u a d r u p o l e interactions, q u a d r u p o l e quadrupole ....) contribute at smaller distances. At distances of the order of an atomic size, where the electron clouds of the atoms begin to overlap, the expansion breaks down, and a different form obtains. Solution of that problem is the province of quantum chemistry. For species that do not form a 'chemical bond', the problem is avoided by invoking the approximation of a

van der Waals forces

99

phenomenological, sharply repulsive, soft core or hard core, with the potential V(r) arbitrarily set to infinity for distances less than the atomic size. If both atoms have a permanent dipole m o m e n t ~t, the thermally averaged rotating

dipole-dipole

e n e r g y is V(r)Keeson = -

correlation

2/~4 9 In 3kT r 6 condensed media, the form for the interaction energy is quite wrong, as the interactions are strictly non-additive and must be dealt with differently as below. If one atom has a p e r m a n e n t dipole moment and the other is polarisable, the dipole-induced dipole (Debye) correlation energy has a similar form, again proportional to 1 / r 6. Three- and higher-body interaction energies [3-10] are much more complicated. Then if only two-body forces act between the atoms of an assembly, the energy of formation at zero temperature is simply

V(rij)--

1 Jri- r]6

where i,j run over all lattice sites of, say, a crystal. If such an idealised material is split, and the two halves separated by an infinite distance, thus creating surface area (cf. Fig. 3.3) then the difference in energy between the two states is twice the surface energy.

000000 000000 000000 000000 000000 000000 -I~I~I~

0000 0000 0000! 0000 0000

0000 0000 0000 0000 0000

--~

Figure 3.3: Surface energy related to interaction energy

If only two-body dispersion forces operate, a rough estimate of the surface energy can be found by adding up the energies of interaction on contact. The energy of interaction per unit area at a distance l has the form E=

A 12tc [2

(3.3)

where A is called the Hamaker constant, and the factor 12~ survives as a convention for historical reasons of no importance. At "contact" d, a separation of the order of a molecular diameter,

lOO

Chapter 3

2Es =

A

12v~d2

(3.4)

where E s is the surface energy per unit area. If we consider an atom interacting with a half space, the energy of interaction is proportional t o A / l 4.

The Hamaker constant is obtained by elementary integration and is related to the molecular polarisability through A - 372 ~'r p20~2(0) , p~_ number density of atoms 4 For hydrocarbons, a typical ultraviolet ionisation potential is h~a= l016 radians/sec and with a choice of 2~ for the size of the CH 2 group one obtains E s ~ 20 d y n e s / c m which is about what is obtained. The corresponding absorption energy depends on the inverse fourth power of the cut-off distance, d. If we deal with the absorption of a molecule to the surface of a crystalline inorganic material, these phenomenological estimates vary enormously depending on precisely where the adsorbed molecule sits; If the adsorbed molecule can nestle into and between the surface atoms, the physisorption energy is as large as a chemical bond. As we have already seen for the example of adsorption in zeolites, the distinction between a bond and physical interactions becomes tenuous.

3.4.2

Lifshitz forces

The idealised calculations described above presume that interacting surfaces remain unperturbed as they come into contact, i.e. that surface energies are infinite (or for finite sized atoms very large) as compared with interaction energies. If a surface is in contact with a liquid, or two surfaces are separated by a liquid, we immediately run into a conceptual difficulty. Thus if the liquid is, say, water, at a single solid surface we might expect the surface to order proximal molecules, which evidently will have a different arrangement of dipole moment, hydrogen bonding, density, etc. from that of bulk water. The range and nature of this ordering depends on both the solid (which induces changes in liquid structure) and the liquid. We return to this central question later. For the moment, suppose that the intervening liquid has bulk properties up to the surface. The solution to this interaction problem, valid at "large" distances, was obtained by Lifshitz [3-10]. Instead of adding two-body atomic forces as above, the method used is as follows. Consider the totality of electromagnetic fluctuations due to all the atoms and molecules of the interacting materials. These fluctuations satisfy Maxwell's equations with boundary conditions set by the dielectric properties (as a function of frequency) of the materials. Fourier analysis of these equations and their

Lifshitz forces

101

solution leads to a set of allowed normal modes of frequencies co and wave vector k. Then assignment of a harmonic oscillator free energy

Fto= l kT ln{2sinh~k T} 2 to each mode, and addition of these free energies gives an expression that includes the retardation of fluctuation correlation contributions due to the finite velocity of light. The expression for the resulting free energy of interaction per unit area is extremely complicated and analysed completely in

[3]. It is, for media 1 and 3 interacting across a medium, m oo

F([,T)= k T

~t~n,

8if, ]2 n=O where

R

=

l" R

R

with

A mR=Slem - pc1 Ama 1= sl - p sl C.m + pel " sl + p s1=~/p2-1+el/Sm

,

e=e(i~n)

and the prime on the summation indicates that the term in n=O to be taken with a factor 1/2, and the sum is over imaginary frequencies ton = i~n,

~,~ = 2~rn kT/h, and c is the velocity of light. This formidable formula is not so forbidding as it seems, and has been analysed in great detail, reduced to tractable forms that make sense in [3]. Pairwise summation emerges as a very special case, valid for gases only, and even then is a bad approximation to the full m a n y - b o d y interaction. Calculation of the interaction free energy for particular cases is not difficult [310] and requires a k n o w l e d g e of m e a s u r e d dielectric properties and adsorption frequencies in the infrared, visible and ultraviolet, all known, in principle.

lO2

Chapter3

If the dielectric properties are known, the expression represents a complete solution to the interaction problem, provided that a liquid separating the two interacting solid surfaces is itself not perturbed in structure by the surfaces. These forces can be calculated and measured. We make the following remarks. The assumption of two-body forces is completely misleading and qualitatively erroneous for condensed media interactions. The sum in the full expression above includes contributions from all frequencies. A contribution to the sum from a frequency ran is retarded, in effect behaves as the non-retarded component of the total interaction-frequency multiplied by an exponential factor ex~-~n/~s), ~s = c/2! C~- which has the nature of a cut-off factor. Different frequencies lock in at different distances. The zero frequency, temperature dependent contributions are extremely important in oil-water or biological systems, about half or more of the total interacting free energy. The behaviour of the interaction free energy depends strongly on geometry. Sometimes, the geometric factor can be evaluated by taking a "Hamaker function" [3-10] calculated by Lifshitz theory, multiplied by an appropriate geometric distance factor evaluated from pairwise summation. (This assumption fails for the temperature dependent contribution, i.e. in water, the most interesting liquid!) Loosely speaking, the theory shows that two bodies (or macromolecules) sense and feel their different frequencies of inter-oscillations at different distances. The zero frequency (microwave, non-retarded) classical or non q u a n t u m mechanical coupled p e r m a n e n t dipole fluctuations ~ u i d e orientation and interactions at large distances. At distances -- 200-500A, the infra-red locks in and at distances below about 50A, the strong ultraviolet correlations take over. At smaller distances still chemistry takes over, as the far ultraviolet correlations lock in. At one level the very existence of such an aesthetically unpleasing formula for the attractive interaction energy between surfaces is discouraging. It describes the interaction problem in the simplest possible idealised case, and the reader is advised that matters are going to get worse as we deal with more realistic situations. There is not much one can do about it. At another level it is encouraging, because it indicates that nature has available to it some very subtle forces to steer self-assembly. The simpler intuition concerning molecular forces in inorganic materials that are the preoccupation of colloid scientists does not map over to biological systems. (The strength, range, and subtlety of these forces in biological materials are more delicate, because of their different dielectric properties.) For unlike materials interacting across a liquid, the forces can be repulsive as well as attractive, depending on material dielectric properties and distance, a situation which also obtains for like anisotropic media, depending on orientation of dielectric axes [3-10].

Lifshitz forces

103

For inorganic materials in vacuum or in a liquid such as water, the van der Waals interactions are dominated by ultraviolet contributions. Typically, below the retardation cut-off we have A--10-20 Joules. For organic materials in water, the interaction is qualitatively different. All frequencies contribute to the strength of the interaction, about a factor of 10 or more lower in magnitude than for inorganic materials. The temperature dependent (n = 0 term of the sum) dominates in biological or oil-water systems, being at least half the total interaction. This particular contribution has the form F,,=o (/,T) = kT I.el(0)- e.m(0) 'e2(0) - em(0) 16~ ~el(0) + e.m(0) ~2(0) - era(0) where the dielectric constants are evaluated at zero frequency (static values). It is a cooperative many-body dipole-dipole correlation force that reduces to a sum of individual dipole-dipole forces only for dilute media (gases).

Salt effects

A further complication appears when the intervening liquid contains salt. The temperature dependent contribution above is then modified [3-10] and at "large" distances is damped by a factor exp(-2~/) where, for a 1:1 electrolyte ~= ~ ek T

e= era(O)

and w is the inverse of the Debye length A D of the solution. We shall return to this fundamental observation later. Salt solutions can also change the form of the long range interaction dramatically [3-10] within coiled polymers or macromolecules.

3.4.3

Double-layerforces

We turn now to the other side of the colloidal particle interaction problem idealised to the case of two half spaces separated by salt water [3-10, 22-24]. Typically such particles will contain ionisable groups at their surfaces, so that the surfaces are charged. Imagine that the water, as before, retains its bulk properties up to the surface of the half spaces. The charged surfaces create an inhomogeneous profile of cationic and anionic density. For an isolated surface at the simplest level of approximation and schematically only, this distribution follows from the equation: V. E = V 2 0(x) =

4~p(x)

104

Chapter 3

where E is the electric field, ~ the potential, p the charge density and the boundary conditions are elE 1 -e2E 2 = 4~cr, where cr is the surface charge density, and el and e2 are the static dielectric constants of adjoining media. The charge density profile p(x) is described by the Boltzmann distribution p(x)+ = p o e

-e~

-e~p

where e denotes unit ionic charge and Po is the salt concentration in the bathing medium far from the surface. Subject to the surface charge (or potential) boundary condition assumed, the potential follows from the non-linear Poisson Boltzmann equation:

V2tp= (87rP0)sinn[k-~-]

9

(The ions of the (symmetric) electrolyte are assumed to be point ions.) It is the overlap of the profiles that gives rise to a repulsive "osmotic pressure" which gives a form: Force ~ f(o)e -~, where f(6) is a function of surface charge (or potential) and ~ is the inverse Debye length, defined above. Combination of the attractive van der Waals potential --- l//z and the repulsive double-layer potential (~ e'r~, gives rise to the famous DLVO theory of colloid stability. Depending on salt, the net potential can be such as to induce flocculation directly, pose a barrier to flocculation, or lead to a stable suspension. The theory is deficient in many respects, e.g. in any real case the degree of surface charge dissociation will be affected by the presence of a neighbouring surface. This can be taken into account in a self-consistent manner [22, 27]. A further inconsistency is that the treatment of temperature-dependent

attractive fluctuation forces given by the extension of Lifshitz theory outlined above ignores the inhomogeneous profile of ionic charge distribution at the (charged) surface, whereas the double-layer force depends on that profile. In a more complete theory these must be treated at the same level, and the inhomogeneous profile is essential in treating fluctuation forces correctly. A full theoretical description must allow ionic density fluctuations about the inhomogeneous profile, a difficult problem of statistical mechanics only recently solved [3-10, 23-29]. When this is done, it turns out that no distinction can be made between the van der Waals and double-layer forces. Indeed, in solutions of di- or trivalent salts at sufficiently high concentration the (formerly repulsive) forces have a deep attractive well due to ionic correlations [23-25].

Double-layer forces

105

These forces have been measured. A further consequence of the new theories is that inferences of surface charge, "ion-binding" to surfaces as adduced from force measurements a n d / o r electrophoretic mobility studies or NMR which involve the crude Poisson-Boltzmann theory are wrong. If the simple analytic Poisson-Boltzmann theory is used to interpret direct force measurements, experimental results can be forced into that analytic form, but the real binding constants are quite different.

3.5

A gallimaufry offorces

Our brief outline so far calls in the full a p p a r a t u s of q u a n t u m electrodynamics, q u a n t u m and classical statistical mechanics, applied to inhomogeneous liquids at interfaces. Generally "agreement" between theory and experiment has been confirmed at "large" distances [3-10], and, even at small distances [23-29] - provided a hydrated size for the ions of electrolyte solutions is allowed. But the "agreement" between theory and experiment claimed in even the most recent literature is often more apparent than real. An open question is: at what distances does the molecular granularity of matter, like a liquid, show up? The answer depends on the surfaces, the materials, their interaction strength, and the liquid. It is not too surprising then that its resolution presents difficulties. Nonetheless some real progress has been made.

3.5.1 Forces due to liquid structure Such a force has been called the "structural component of the disjoining pressure" by Derjaguin [36]. These forces have usually been considered as additional to those deduced from continuum approximations. (In reality the continuum theories should be considered as asymptotic approximations to the forces, which hold only at "large" distances.) Two kinds of structural forces can be considered. If the surface does not perturb bulk liquid structure, we should still expect continuum theories to break down at some point. For example, between molecularly smooth, rigid mica surfaces separated by simple liquids, hard sphere van der Waals liquids like octamethyltetrasiloxane, (a r o u g h l y spherical molecule of d i a m e t e r ~8A),forces with large oscillations, of period equal to the molecular size, have been measured up to surprisingly large distances. Beyond about 8 molecular diameters, the measured force merges into the van der Waals-Lifshitz forces. The same occurs for hard-sphere-dipole molecular liquids, which are not hydrogen-bonded like propylene carbonate [33-35]. For soft surfaces, or surfaces rough on a molecular scale (like lipid bilayers), the oscillations tend to be smoothed and can be ignored.

lO6

Chapter3

3.5.2 Surface-induced liquid structure The second type, of more importance, is a force due to surface-induced liquid structure. At a conceptual level we have already encountered such a force in the double-layer. There the electrolyte can be regarded as the "liquid", with the suspending water a continuum background that affects the problem only through its dielectric constant. The bulk "liquid" electrolyte has a uniform distribution of cations and anions. In the presence of the charged surface, that uniform distribution changes. The overlap in profile of the surface induced liquid structure causes the force. In general, a surface will always induce re-ordering (e.g. of hydrogen bonds) resulting in dipole orientations which extend over a number of layers, as for the double-layer interactions. These are called "hydration", or "solvation forces" [37-41] and can dominate interactions in precisely the distance regimes of most interest, below about 50tL Between lipid bilayers with large hydrated zwitterionic head-groups, the forces become extraordinarily large and repulsive. These forces ought properly to be called dehydration forces since work must be done to remove proximal bound water from the head-groups. They dominate electrostatic forces below 30,/L

3.5.3 Hydration forces in

phospholipids

A vast amount of pioneering work has been devoted to measurements of forces between lipid bilayers [37-39]. Quantification of these forces represented the first real advance in understanding the subtleties of biological systems, which do not fit really into the scheme of things provided by classical colloid and surface chemistry. The repulsive forces are very large; they dominate double-layer and van der Waals interactions below distances of about 30./~. The forces prevent fusion of bilayers, an eminently satisfactory result, as otherwise cells would not exist. There is no theory of such forces beyond the simple statement which goes back to Poisson 150 years ago, and was quantified by J. Clerk Maxwell in 1876 using mean field theory [41] - that the overlap of profiles of surface-induced liquid structure gives rise to the force. The theory and exponential decay length of approximately 3/~, (the size of a water molecule), were predicted by Maxwell, rediscovered by Marcelja 100 years later [40], following earlier measurements by Parsegian and co-workers [37-39] in a set of pioneering investigations. Oscillations with a periodicity of 3/~, are smoothed because the surfaces provided by phospholipid head-groups are bumpy, at least on a molecular scale. -

For multilamellar liposomes, the hydration forces should balance the predicted longer range attractive van der Waals forces, to give an equilibrium lamellar phase spacing of about 30,/~ in water. They do not, but once surface dipole-dipole correlations are taken into account [31, 32], theory and experiment do agree.

Other forces

107

3.5.4 Surface dipole correlations Recognition that a further correlation contribution exists, due to twodimensional p e r m a n e n t dipole (zwitterionic) head-group fluctuations confined to a surface accounts for the discrepancy in equilibrium lamellar spacings [31, 32]. Elegant force measurements have been made between adsorbed monolayers of the protein cytochrome-c, and insulin on mica, immersed in water [26]. Hydration forces here play no role. If the full armoury of theoretical predictions is invoked, the complicated force curves measured all seem to fall into place. It is possible, indeed probable that with real biological membranes that contain up to 50% proteins, the hydration forces that prevent fusion of pure phospholipid membranes do not always operate. Agreement between theory and experiment is often illusory. Consider the double-layer forces measured between mica or silica in an electrolyte, under conditions where hydration forces are absent. The curves certainly decay with the predicted exponential forces (e-Kl) of the double-layer theory, at least for 1:1 or 2:2 electrolytes. But agreement with the basic Poisson-Boltzmann theory has been achieved by fitting theory with an assumed surface charge density, from which an "ion-binding" constant [46] can be inferred. The full theory [22-30] which includes ionic correlations has the form of the PoissonBoltzmann theory, it will give a different degree of binding. It is impossible to infer ion-binding constants which have any meaning, important to considerations of specific ion effects in biology, unless the full theory is used. Usually this is argued away on the grounds that if the simpler (wrong) theory gives a consistent set of parameters for a series of ions, e.g. Li +, Na +, K*, that should be sufficient. This is not so, and even for those situations where classical theory appears to work well there are problems. A further example can be seen in force measurements between ionic bilayers. The double-chained cationic surfactants didodecyl, dehexadecyl, dioctadecyl dimethyl ammonium, bromides or acetate, can be adsorbed, or deposited by Langmuir trough techniques onto molecularly smooth mica. For bromide as counter-ion, with or without added NaBr, the force curves fit nicely to Poisson-Boltzmann theory provided one postulates that 80% of the headgroups are neutralised by bound Br- [46]. This agrees with the binding deduced from NMR and other studies on miceUes with the same head-group. There is a fitted phenomenological parameter, the assumed surface charge necessary to secure agreement between theory and experiment. On the other hand, with acetate as counter-ion, the forces are an order of magnitude larger, the fit to theory is perfect, with no free parameters [46]. There are no bound counter-ions. The Poisson-Boltzmann theory here can be shown to provide an upper bound to the magnitude of the double-layer interaction. If the more refined theory [23-25] is used, the predicted result is somewhat less than the measured curve, less still if there exists any real ionbinding. (How much of a difference exists depends on the presumed hydrated

lO8

Chapter3

ion size invoked, i.e. again involving water structure, which is not taken account of in the theory. With large counter-ions like acetate, the deviations from Poisson-Boltzmann theory are less than those with smaller counterions like bromide.) The differences must be attributed to additional hydration forces, hidden if one insists on a simple Poisson-Boltzmann description [47]. Specific counter-ion effects are critical to biological function, in determining forces between individual sub-units of macromolecules and in the consequent shapes they take up. How much these effects can be attributed to physics and how much to specific chemistry can only be revealed by a reanalysis of all data in the light of the new theories of molecular forces. Until that reanalysis is done, present experimental inferences on binding surface potential and charge remain phenomenological curve fitting. A very large amount of sophisticated NMR studies and other work on specific ion-binding to proteins and membranes has been interpreted in terms of a phenomenological ion-binding model equivalent to the ionbinding model for micellisation discussed below. The model "works" and is equivalent to the Poisson-Boltzmann theory of the double-layer as we have already mentioned, but only in the special limit of strong binding [48]! Why then give up a good simple theory for a more sophisticated one? The simple theory based on the Poisson-Boltzmann description does provide some useful qualitative predictability as we shall see. The ever-present proviso is that it must not be taken too far. It is an old story, and ultimately comes back to the fact that we have no real molecular theory of water. Langmuir's view, that water is itself a giant connected molecule is probably closer to reality than a molecular picture.

3.5.5 Secondary hydration forces and

ion-binding

A different kind of hydration force occurs between hard surfaces bearing dissociable groups, like mica in water. Competition between two competing ionic species of different hydration - (e.g. H +, Na +) for the surface (and for the hydronium ion sub-surface sites) can lead to equally large, indeed ubiquitous "secondary" hydration forces. At low salt concentrations, the classical DLVO theory appears to work well. But at higher salt content, the law of mass action demands that the less easily bound Na § ions must win out over H § in competition for the surface. Work must be done to dehydrate these ions as two surfaces approach. The onset of secondary hydration forces occurs for a given pH at a critical salt concentration, and these forces appeared to dominate double-layer forces, again below about 30-40/~, separation [27, 28]. The overall effective decay length of secondary hydration forces is 10]~ (univalent), 20]~ (di-valent), 30A (tri-valent), as opposed to lipids, where the decay length is 3~. Later work has shown that while these secondary hydration forces exist, their range is probably the same as that for lipid interactions, i.e. characteristic of the size of a water molecule. The apparent long-range of secondary ionic hydration forces is an artefact of earlier

Other forces

109

theoretical analysis [29, 30]. (This analysis assigns an area per ionic adsorption site in the lateral direction, and, inconsistently, treats the completing ionic species as of zero extent in the perpendicular direction. But the "apparent" phenomenological force remains real.)

3.5.6 Range of the double-layer force and implications Just to complicate matters further, we remark on the range of double-layer forces. All theories take it as axiomatic that the force between surfaces or between charged molecules should decay as exp(-Ic/) where K"1 is the Debye length defined by

~o = (4~.(vlz12+ v2z22)e2c) kT where c is the concentration of a salt, CZ] A v2z2(e.g. Ca 2+ C~). In fact, it can be shown, and has been confirmed experimentally, that the actual decay length is given by

Ir

-l

A D = K0

( 1 + 71n (3) 4c (vl z~ + v2 z ~ + O(c2/31n (c)) + ... ) (24)~r2

(vl zl + v2 z 2)

where c is electrolyte concentration in moles per litre [26]. This result affects profoundly our intuition on double-layer forces, especially in mixed electrolytes [26]. (It has been confirmed by direct m e a s u r e m e n t of the forces acting between molecularly smooth mica surfaces in solutions of the p r o t e i n c y t o c h r o m e - c [48]. This protein has a charge of 12 + or 8 + d e p e n d i n g on pH, so the effects show up very strongly. The experiments have an interesting consequence. In many applications a trace a m o u n t of a highly charged ion like V s+, Cr 3~ added to a 1:1 salt solution like NaC1 can have a quite dramatic effect on the stability of a colloidal dispersion. The reason is due to the surface concentration, n s, of a species of charge z near a charged surface, which is determined by the formula ns = ns exp(+ze~0/kT) where n B is the bulk solution concentration far from the surface, ~ is the surface potential and z the charge of the counter-ion species. The exponential factor means that near the surface such a system becomes more like a z:l electrolyte. Hence the double-layer force b e t w e e n two such surfaces, determined by the overlap of the surface-induced ionic profiles, resembles a z:l electrolyte more than a 1:1, and can be of much shorter range than expected. The same effect occurs in many biological situations, where a m e d i u m between two interacting membranes often contains a small amount

110

Chapter3

of highly charged species of proteins or RNA that affect interactions strongly by this mechanism. The Boltzmann factor, depending on charge, also explains why it is that trace amounts of charges Se4+,V 5+, etc., are of such importance in agriculture and to animal metabolism. At the surface of a (charged) membrane, the concentration of such ions - which determine the local ionic concentrations, and hence the conformation of a membrane bound protein involved in recognition is orders of magnitude higher, than in the bulk solution far from the membrane. Biological activity will be critically dependent on trace element concentration. If the solution between two surfaces contains surfactants that form highly charged micelles, a different effect occurs. Theory, confirmed by measurement [49-51], shows that the Debye length is then to be calculated as if the micelles and their "bound" counter-ions are simply ignored. The doublelayer force is here much longer ranged than it would be on the basis of standard theory. These forces are sometimes called "depletion" forces. The mechanism is the same as that for the oscillatory forces discussed in section 3.5.1.

3.5.7 Hydrophobic interactions The hydrophobic interaction between oil-like surfaces or between oil-like molecules in water is the driving force for self-assembly. In biological systems it is the most important force of all. While the word "hydrophobic" is ubiquitous in the chemical and biological literature, its use disguises ignorance. That ignorance occurs because we have no real quantitative knowledge of water structure and the influence of solutes, be they hydrocarbons, gases or ions, on water structure, let alone at surfaces. In the brief account below we summarise what we knew of hydrophobic interactions [42-45] in the context of this overview of complex molecular forces until 1993, so as not to deviate too far from our main theme. Recent work to be discussed in Appendix 3C, has opened up a whole new perspective. The consequences of that work affect our understanding in profound ways not yet fully comprehended. The hydrophobic interaction between two small molecules (e.g. methane in water), is generally considered to be short ranged, virtually a contact force. With systems of large surface area, as opposed to small molecules, a completely different situation occurs. Measurements of forces between modified hydrophobic mica, silica, or between surfaces on which monolayers of surfactants have been deposited show that the hydrophobic interaction is extremely long ranged [4245]. These forces are 10 to 100 times larger than any conceivable van der Waals force. The contact free energy (adhesive energy) is what one expects from hydrocarbon-water interfacial energies. If fully hydrophobic surfaces (contact angle > 90") brought to contact are pulled apart, spontaneous cavitation occurs. A hydrophobic surface prefers contact with a

Other forces

111

vacuum or dissolved gas and water vapour over contact with water, and a thin film of water between a hydrophobic surface is metastable. With fluorocarbon surfaces, cavitation occurs before contact. Although well documented, and repeatable, the mechanism of this hydrophobic interaction has not been fully explained at this time. Addition of salt changes the range of the forces, and this provides a clue to the mechanism. Electrolyte correlation effects can be ruled out, as any force would depend exponentially on the Debye decay length, and this is not observed. The most recent experiments suggest that the phenomenon, along with observed cavitation effects, is related to specific ion adsorption and nucleation of gas at the hydrophobic surfaces, which induce fluctuation states in the metastable water film. The fluctuations, and hence the consequent force, are dependent on specific ions and the dampening of the forces with salt has the same kind of critical switching property present in gas bubble-bubble interactions discussed in Appendix 3C. This force, acting between biological macromolecules (with varying degrees of hydrophobicity over their surfaces, such as proteins, and hence varying gas and ion adsorption) is of extreme importance in biology. Changes in the amounts of dissolved gases like carbon dioxide and oxygen materially affect these processes and the interactions described in detail in the Appendix.

3.5.8 Non-ionic surfactant forces The forces [52] between two monolayers of non-ionic polyoxyethylene (PEO) head-groups deposited on hydrophobed mica exhibit peculiar properties. At temperatures below the so-called cloud point, the forces are repulsive. Above the cloud point they become attractive. Simultaneously with the onset of attractive forces, the measured head-group area decreases, in agreement with NMR measurements on micelles formed by these surfactants above and below the cloud point. (The head-groups give up two molecules of hydration per PEO group.) Such changes at the molecular level between head-group interactions produce large changes in the self-organisational properties of solutions of these surfactants. Therefore it ought not to be too surprising to see, as we do, dramatically sensitive changes in biological activity of enzymes and other macromolecules over small temperature ranges. The fact that such effects have been made explicit and quantified underlines the real dangers of modelling structure of macromolecules with universal potential parameters currently in vogue [53].

3.5.9 Forces of thermodynamic origin Two further forces operate in any colloidal suspension or self-organised system. The first is due to Onsager and to Langmuir [54-56] who explained colloidal stability of clays and cylindrical particles in terms of purely repulsive

112

Chapter 3

forces, due to either double-layer effects, or hard rod or hard plate repulsion. In a many-particle system, an ordered phase will always be in equilibrium with the disordered phase of plates or cylinders. The equilibrium between the two coexisting phases is set by the volume fractions of each. The "force" that causes ordering is entropic. Recent work of Lekkerkerker and colleagues has explored such effects extensively.

3.5.10 The Helfrich force Another force [57, 58] occurs in a multilayered system, like a swollen lamellar phase of surfactant bilayers or phospholipid vesicles. Shape fluctuations in the bilayers can give rise to steric effects that are supposed to stabilise such systems where the van der Waals and double-layer forces are very weak, as they often are. The magnitude of such fluctuations depends on the "stiffness" of the bilayer. The status of these forces is the subject of an active debate and unclear. This potential force occurs in microstructured fluids like microemulsions, in cubic phases, in vesicle suspensions and in lamellar phases, anywhere where an elastic or fluid boundary exists. Real spontaneous fluctuations in curvature exist, and in liposomes they can be visualised in video-enhanced microscopy [59]. Such membrane fluctuations have been invoked as a mechanism to account for the existence of oil" or water-swollen lamellar phases. Depending on the natural mean curvature of the monolayers bounding an oil region - set by a mixture of surfactant and alcohol at zero these swollen periodic phases can have oil regions up to 5000A thick! With large fluctuations the monolayers are supposed to be stabilised by steric hindrance. Such fluctuations and consequent steric hindrance play some role in these systems and in a complete theory of microemulsion formation. The status of such forces is an open and unclear situation. They exist, but cannot be too specific in nature.

3.5.11 Forces of very long range A further curious force operates when charged polymeric molecules are in a stretched out state, or indeed with any thin cylindrical structure. Fluctuations of the ions in the associated double-layer can lead to forces of extremely long range. Their existence is not in dispute, and they probably play a large role in the organisation of DNA in cell biology [3-10]. A schematic representation of the evolution of concepts of molecular forces is given in Appendices 3A,3B.

Other forces

113

3.5.12 Summary From a historical perspective, our understanding of molecular forces derives from colloid science, concerned with interactions between solid particles. The opposing forces between particles, one attractive, the other repulsive, which form the core of the DLVO theory which has been the mainstay of that subject, sit in uneasy juxtaposition. The attractive van der Waals forces were derived assuming the liquid between two interacting surfaces is a bulk continuum medium. On the other hand, if we regard an electrolyte as the operative liquid, the double-layer force is the first example of a structural force and surface-mediated interaction. We have seen h o w m o d e r n refinements have changed that simple view, especially in oil-water and biological systems of low Hamaker constant; and how in the new theories the conceptual distinction made between the opposing forces disappears. At one level, the litany above presents us with an appalling situation. The reader will observe that except for hydration forces, for which there is no real theory, we have treated water as if it is a continuum, and the ions of electrolytes as hydrated hard spheres. That approximation is not always valid [60], and one simply has to be aware of the limitations of present physical theories. Again the hydrophobic "surfaces", as discussed in section 3.5.7, the forces, and their salt dependence, depend critically on the amount of dissolved gas, and nature of the charged, or polar species. Behind that litany, at even the most elementary model approximation sits the full apparatus and complexity of quantum electrodynamics, statistical mechanics of inhomogeneous fluids which lie at the foundations of modern physics. That is comforting in a way at least in that the direct traces of a unifying thread between physics through chemistry to biochemistry can be discerned. In another way it is disconcerting. The biologist and biochemist have enough concerns of their own not to be bothered with the subtleties of physical chemistry. It is evident that one can go a long way towards understanding phenomena at the exquisitely refined level of molecular biology knowing nothing of molecular forces. Indeed too facile application of the older simpler theories of colloid and surface science fails in complex situations. Nonetheless, awareness of the new forces does enable us to make sense of a variety of issues not comprehensible without that awareness, and provides mechanisms for the extensive geometric rearrangements needed in biology.

3.6

Self-organisation in surfactant solutions: the Euclidean desert

We now move away from rigid surfaces and their interactions, and allow the interactions to dictate the formation of bodies and their surface shapes. In general the role of molecular forces in self-assembly is extremely subtle. This can be seen from the following simple examples. The simplest example

ll4

Chapter3

is a polyelectrolyte. At very low ionic strength the Debye length is large and the screened electrostatic repulsion between individual charged units along the backbone is very large. The polyelectrolyte takes up an extended linear conformation, resulting in a stiff viscous solution. With addition of salt the segment-segment interaction is reduced and the polyelectrolyte collapses to a random coiled state, with the solution becoming fully flowing. Another example is the 0-point of polymers in solvents, characterised by an abrupt change to an effectively non-interacting polymer at a particular temperature, well explained by calculations involving the dispersion self-energy and for polyelectrolytes, using the Poisson-Boltzmann equation [61]. More subtle are specific ion effects on conformation. The repulsive forces between bilayers of quaternary ammonium surfactants adsorbed onto mica differ markedly depending on counter-ion, as we have seen. With Br-as counter-ion the forces are weak, with acetate and a range of carboxylates, Fand OH- the forces are strong. In the language of the Poisson-Boltzmann theory about 80% Br- is "bound", no acetate is "bound". In a totally insoluble (hydrophobic) material like glycosamine (e.g. chitosan, the essential ingredient of crustacean shells and the most common polymer on earth), the chloride counter-ion is strongly bound. After immersion in weak acetic acid, the acetate ions exchange with chloride ions; the electrostatic interactions between polymer units increase, the polyelectrolyte dissolves easily to form a clear solution. Knowledge of the counter-ion effect is sufficient to understand this phenomenon, but further effects remain unclear: addition of further acetic acid ought to screen interactions and collapse the polymer again. It does not, so that extra interactions such as hydration effects and dissolved gas are involved (cf Appendix 3C). (Addition of NaC1 will collapse the polymer.) The acute sensitivity of many biological phenomena to temperature can in some cases be correlated with forces. As we have already discussed, the cloud point behaviour of polyoxyethylene surfactants can be understood in terms of forces. If we take a cellulose acetate polymer, ethoxylate some of the groups, then dissolve the polymer in water and add an ionic single-chained surfactant above the critical micelle concentration an extraordinary phenomenon is observed. Hydrophobic portions of the polymer act as "nucleation sites" for micelles, leading to a complex assembly, resulting in a string of charged beads. Below the cloud point both the charged micelles along the string and repulsive interactions between ethoxylated groups conspire to give a clear freely flowing solution. Above the cloud point, the solution becomes immediately extremely rigid. (The effect is reversible.) Evidently, the repulsive and (now) attractive forces are in tension, with the ethoxylated groups coiling up. This kind of behaviour is a simple system which can be understood in terms of the molecular forces operating. It allows us to understand why and how it is that, e.g. enzymes can be so sensitive to small variations in temperature. The present section gives a brief overview of ideas and the physical notions behind self-assembly of surfactant-water systems.

Self-assembly

115

A schematic listing of ideas in self-assembly is also indicated in Appendix 3B. These theoretical developments were until recently confined to spheres, cylinders and planes, a Euclidean desert. New developments that allow access to the full variety of shapes used by nature are the subject of other chapters. The word "surfactant" (sometimes "tenside") is a mnemonic for surfaceactive-agent. In a less limited sense we should take it to mean any molecule, one part of which is hydrophobic (i.e. water-hating) with one or more flexible hydrocarbon (or other oil soluble) moiety. This part of the molecule is chemically bound to a hydrophilic (water-loving) head-group, e.g. a salt, or other predominantly water-soluble group, like a polyoxyethylene. This dual or schizophrenic character is responsible for the self organisational capacity. Depending on a loosely defined hydrophilic-lipophilic balance ("HLB", lipophilic - f a t - or oil-loving) the solubility in water of m o n o m e r i c surfactants can range from a maximum of typically 10-2M for ionic singlechained chemical soaps, to the extremely low value of 10-16 M for the doublechained phospholipids that form biological membranes. Beyond the solubility limit, the surfactant molecules self-assemble into aggregates as follows. At first, as surfactant is added to water, the free energy per molecule is dictated by entropy, F = E - T S , - T S = -kTln X1 --~ + ~, X1 = mole fraction. With increasing concentration, the entropy decreases, and the solution can minimise its free energy if some of the molecules migrate to the air (or oil)water interface to form a monolayer. (The hydrophobic tails prefer association with each other or with any hydrophobic interface due to the "hydrophobic interaction".) The surface tension drops continuously until a complete monolayer is formed. Thereafter, at the so-called critical micelle concentration (cmc) the surface tension curve (and the electric conductivity for ionic surfactants) exhibits a sharp break, indicating the onset of selfassembly. The interface no longer being available, the molecules satisfy their dual requirements, and that of entropy, protected as far as possible from water by the hydrophilic head-groups by arranging their tails together into monodisperse micelles, i.e. spherical or globular aggregates. Double-chained surfactants cannot usually pack into spheres, and are forced immediately into closed bilayers which then associate under the influence of attractive van der Waals forces to form e.g. multi-walled vesicles (liposomes), or more complicated phases. (For single-chained surfactants typical micellar aggregation numbers are of the order of 25-50, set by the minimum geometric size that packs the molecules favourably.) If the chains are sufficiently flexible, and repulsive (hydration or other) forces between aggregates sufficiently large and the chains sufficiently hydrophobic, as for double-chained surfactants like phospholipids, single-walled vesicles can occur. The process of self-assembly is a dynamic one. With some single-chained ionic surfactants, (e.g. sodium dodecyl sulphate), the residence time of a monomer in a micelle is of the order of 10-6 sec, and the lifetime of the micelle as a whole is of the order of a millisecond. At the other extreme, for highly insoluble phospholipids that form the bulk of bilayer membranes, the

116

Chapter 3

time taken even for flip-flop of a molecule from one side of a bilayer to the other can vary from seconds to months, depending on chain length. That stability is necessary to preserve the integrity of biological membranes. Alcohols, at least those of short chain length, have moieties that are neither extremely hydrophilic, nor extremely hydrophobic, and partition in a complex way between oil and water, or between the oil-like interior of a micelle or membrane, and water. This property, together with their small head-group area, enables them to be used with surfactant-water, or surfactantoil-water systems to produce a rich diversity of microstructured solutions through changing curvature of the interface, as does cholesterol, for the same reasons. The self-assembly of biological aggregates is further complicated by the presence of amphiphilic proteins.

Figure 3.4: Schematic representation of surfactant-water phase diagrams, for non-ionic surfactants (left) and ionic surfactants (right).

Typical surfactant-water-phase diagrams are shown in Fig. 3.4 for singlechained ionic, and non-ionic surfactants respectively. Below a "Krafft" temperature characteristic of each surfactant, the chains are crystalline and the surfactant precipitates as a solid. Increased surfactant concentration (Fig. 3.4) results in sharp phase boundaries between micellar rod-shaped (hexagonal), bilayer (lamellar) and reversed hexagonal and reversed micellar phases. (The "cubic" phases, bicontinuous, will be ignored in this section and dealt with in Chapters 4, 5 and 7.) 3.6.1 Aggregate structure in the Euclidean desert

If we confine ourselves to simple geometries, the "explanation" of the gross phase behaviour indicated is immediate: given the maximisafion of entropy

Self-assembly

117

that accompanies the formation of the smallest possible aggregates and given that this free energy minimisation accrues to minimal exposure of the hydrocarbon chains, then the implication is that for a given volume and head-group area, micelles of smallest size satisfying these requirements will form first. When these spherical globules can no longer pack (maximum volume fraction 0.74 for a face-centred-cubic array) they have no alternative except to form a hexagonal array of cylinders (maximum volume fraction 0.91), thereafter a lamellar phase, and so on. That is, global packing, volume fractions of the several components, are and must be one requirement that accounts for the features of phase changes that occur with increased concentration. And if inter-aggregate forces, e.g. between ionic micelles, are operating, they might reasonably be accommodated by invoking an "effective" micellar radius that includes the Debye length, the range of the double-layer repulsion between aggregates, as an adjustable parameter. At first sight the multi-molecular, multi-component assemblies of biology occur in dilute solutions where global packing constraints are not a problem. In fact that is not so. Compartmentalisation of components occurs naturally, and within those compartments global packing is a main issue. Even so their transformations in response to biochemical processes require more detailed accounting of the interplay between intra- and inter-molecular forces. A clear and delicate example can be seen for the case of non-ionic surfactants. On increasing temperature the dilute micellar phase shows a sharp cloud point (section 3.5.8, cf. Fig. 3.4). Two factors are involved. At and above this temperature, the surfactant head-group gives up water of hydration. The head-group area is reduced and the surfactants can no longer pack geometrically into spheres. They increase in size to become rod shaped objects. Simultaneously the forces between aggregates change from repulsive to attractive. However weak these long range forces, they will induce a phase separation as long as the micelles are sufficiently long (rod-like). (No phase separation is possible if the attractive forces acted between small spherical micelles- the solution is too dilute for the forces to have such an influence.)

3.6.2

Curvature as the determinant of microstructure

These ideas can be formalised in terms of statistical mechanics to some extent, and an outline of the main ideas is given in the following section. We remark parenthetically that there are profound difficulties confronting the definition of an aggregate. The nature of the hydrophobic free energy of transfer of a hydrocarbon from water to the hydrophobic core of a micelle can be measured, but its temperature dependence is not understood because it depends on water, an u n k n o w n quantity. For the same reasons, solution theory, does not even tell us whether mole fractions or mole volumes are the correct ratios to use to determine entropy. However, provided certain assumptions are allowed [62-65], then simple rules emerge. The rules are: if v is the hydrocarbon chain volume, a the head-group area, and I of an optimal

Chapter 3

118

chain length, a "surfactant parameter" v/al characterises and determines the particular aggregation states which form. This molecular parameter is a function of the curvature at the surfactant-water interface. For a single component system, the volume v is fixed, and the area per head-group can change as solution conditions change intermolecular forces. The rules, in the limited class of Euclidean shapes are: v = 1_ micelles; 1 < v < 1_ rod shaped al 3 3 al 2 micelles: 12__< val < 1 vesicles, (closed single walled bilayers) or lamellar phases, v > 1 reversed phases. Indeed that progression is readily observed by adding al salt to a solution of ionic micelles, the head-groups of which interact predominantly by electrostatic rather than steric forces. Addition of salt screens the electrostatic repulsion, reducing head-group area appropriately to produce the corresponding phase. Again, simultaneously, inter-aggregate forces are reduced, and assist in causing the phase changes to be abrupt. The same progression can be observed if one adds progressively a singlechained surfactant (S) (v/al = 1/3) to a lamellar phase of a double-chained surfactant D (v/al=l) of the same head-group area and chain length. Here it is the effective volume which changes according to,

{•-}

Xs + I(XD)

ff

Xs +

9(Xs, XD mole fractions)

If curvature so prescribed is the major determinant of self-assembly, then phase diagrams ought to exhibit universality, yet ionic and non-ionic phase diagrams are different (Fig. 3.4). The latter exhibit an upper consolute point (phase separation beyond the cloud point). That issue can be resolved if we recognise that for the non-ionic surfactants we might equally replace the temperature axis by the curvature v/al. It can be shown that for ionic surfactants v/al decreases with increasing temperature. To achieve universality one need then only to reverse this direction. This can be done by taking a combined surfactant system (as above), admixing double-chained v/al = 1 to a single-chained v/al = 1/3 surfactant, both with the same area and chain length, and replacing the temperature axis by (v/al)eff. The resulting phase diagram, with the x-axis now the volume fraction of the combined surfactant in water exhibits the cloud point phenomenon. (Some unusual single-chained ionic surfactants do in fact show cloud point phenomena in any event and this can be understood only in terms of as yet unknown surfactant-induced water structure.) These arguments suggest that since the x-axis is some measure of interactions, and the y-axis in the phase diagram a measure of curvature, that it is curvature vs. interactions that provide a better characterisation of phase behaviour for such systems than the usual Gibbsian parameters (T,V,N). These ideas and observations have been put on a sound basis linking molecular forces quantitatively to the prediction of cmc's as a function of salt

Self-assembly

119

and temperature ion-binding parameters, and chain length [47], and the universality so found gives confidence that curvature, as set by molecular forces can be used to elucidate structure.

3.6.3

Genesis of the surfactant parameter

An outline of the ideas is as follows [62-65]. It suffices to assert that a dilute solution of surfactant molecules can be considered to consist of water plus monomers, dimers, trimers, and larger allowed aggregates (micelles, vesicles, liposomes .... ). The concentration is assumed to be so low that aggregates can be considered to be non-interacting. The probability distribution of aggregates, is then determined from the law of mass action

p~ + ~-Z In (-~)=/zl~ + k T l n X 1

(3.5)

where the chemical potential of an aggregate of size N has been written as NIZN 0 + kTIn(XN/N) and X N is the concentration (mole fraction, volume fraction...) of surfactant molecules in the N-aggregate. (The theory also allows for the possible formation of infinite aggregates, i.e. separate phases.) The glib assertion eq. (3.5) represents a beginning to a chemist and an essential stumbling block to a physicist who can go no further without questioning foundations. If aggregates of given size N were distinct, identical well-defined chemical species there would be no problem, apart from the vexed question of concentration units. They are not: even within a given N-aggregate, if such can be defined, there exist an infinite diversity of shapes or configurations which the association of surfactant molecules could take up. Implicit in eq. (3.5) is the understanding that for any N, there is a shape of optimal energy that exists, e.g. a sphere, cylinder, bilayer, and is overwhelmingly more probable than its fellows. These problems are profound, and unresolved. But under certain conditions and assumptions that can be spelt out [62-64] an aggregate can be defined, and an equilibrium partition function of statistical mechanics written down, from which it is possible to extract the thermodynamics eq. (3.5). The difficult question of how even to write down a partition function that averages over size and shape of aggregates, rather than a restricted set like spheres, cylinders.., is identical to that of the general theory of phase transitions. The same problem occurs in the problem of nucleation theory, e.g. gas-liquid condensation. The usual models are physically comprehensible, but give erroneous critical behaviour. The more sophisticated renormalisation group methods of condensed matter physics gives correct critical behaviour, but completely avoid shape, which remains a hidden variable. In any event, extant theories fail to provide a quantitative description of first-order gas-liquid transitions, or of melting of a solid.

120

Chapter3

If we accept these limitations, eq. (3.5) can be rewritten as

=

~ _.o )/kTl

aggregates are allowed to occur at any concentration, albeit with infinitesimal probability, even below the cmc. Above the cmc, defined by

All

oo

xl=

xN, N>I

X1 increases slowly with concentration. It can be shown that if ]~N0, the chemical potential of a monomer in an aggregate is sharply distributed about some N, then the distribution of aggregates peaks at a value of N just less than the N with minimum #N 0 and is also sharply distributed. Otherwise, pronounced polydispersity may occur (e.g. for long cylindrical micelles). Thus reduced to bare bones it can be seen that the use of the word "theory" is dubious. We have simply characterised the observation of miceUes, and claim that it can be shown that the law of mass action is an appropriate vehicle for this characterisation. The entropic term ( k T / N ) In ( X N / N ) has considerable nuisance value. In the pseudophase approximation- valid for finite disconnected aggregates- it can be dropped. Then in this approximation no micelles occur below the cmc. This is now the value X 1 of monomer concentration for which ]~N0 = ]al 0 + k T lnX 1. Above this cmc, all additional surfactant molecules form micelles or whatever aggregate has the minimum lUN0. No other aggregates form until activity coefficients, i.e. interactions between aggregates, become significant. Keeping in mind possible complications due to phase transitions and interactions, the strategy is then to compare the chemical potential of different aggregates to see which has the minimum free energy ]~N0. We can write for the free energy difference f between a monomer in a miceUe and in solution #N - #1 - f = fB + fs + "packing term"

Here fB is the bulk (hydrophobic) free energy of transfer of monomer hydrocarbon tails from water to the oil-like interior of a micelle. It is the same for all aggregates, f s is the surface free energy per monomer in the aggregate. It depends on the size of the aggregate and the curvature of the interface. Then there is a packing term: the assumption that the interior of our micelle is fluid-like, and is to first approximation incompressible, has an immediate consequence, provided we require that aggregates can obtain no holes. (The occurrence of an interior vacuum or a water-filled region inside the (oil-like) interior of an aggregate would result in a large unfavourable increase in free energy, which possibility must be excluded from

Self-assembly

121

consideration.) This can be taken into account if we assume ~N 0 = oo when the packing criterion is violated. For spherical and cylindrical micelles this criterion is usually R < Ic where R is the radius of the micelle and l c is a critical chain length which is usually about 80% of the fully extended chain length for bilayers. The packing criteria is an extreme simplification. Sometimes, e.g. for vesicles, it needs to be relaxed or extended. The welding of two notions, of a fluid-like interior for the micelle, and of "packing" is at first sight contradictory. However the two notions can be shown to be compatible in a first-order theory [48, 62]. Possible candidates for aggregates can now be examined. For surfactant-water systems these have been restricted in the past to spherical micelles, nonspherical micelles (globular, cylindrical), vesicles, liposomes, bilayers, and for oil-water-surfactant systems spherical drops, normal or inverted (water in oil) or (oil in water). The analysis requires some assumptions concerning the surface free energy of the aggregate. Under quite general assumptions, the story as outlined above does emerge. (Vesicles, single-walled bilayers represent more complicated structures [63-65], because the chains on the inner surface are subject to compression, rather than extension, as for the outer layer. Further, all closed aggregates including liposomes, have different interior solution conditions to those outside. This difference is especially marked for ionic surfactants. This effect leads to supra-self-assembly, a higher class of structures to be discussed in (Chapters 5 and 6). In the case of insoluble double-chained surfactants, the interior of a liposome can collapse to a cubic phase or micelles because the interior solution conditions that set c u r v a t u r e t h r o u g h molecular interactions are different inside [63]. Nonetheless vesicles can be shown to be thermodynamically stable structures sometimes, with aggregation numbers, size, depending on chain compressibility) [63-65]. A crucial parameter-free test of the theory is provided by its application to

micelle formation from ionic surfactants in dilute solution [47]. There, if we accept that the Poisson-Boltzmann equation provides a sufficiently reasonable description of electrostatic interactions, the surface free energy of an aggregate of radius R and aggregation number N can be calculated from the electrostatic free energy analytically. The whole surface free energy can be decomposed into two terms, one electrostatic, and another due to short-range molecular interactions that, from dimensional considerations, must be proportional to area per surfactant molecule, i.e. fs = f e s ( N , R , a ) + y a

where ~, is a constant that subsumes all other intramolecular interactions like head-group steric repulsion, hydration, and chain interactions. The optimal micelle can be shown to be given by

122

Chapter 3

Ofs = o = Ores +7' , N v = 41r Oc ~c 3

,Na=4~R 2

Hence ~ris determined - electrostatic forces must balance all others. The older pseudophase, or ion-binding model had characterised ionic micellisation through a "chemical" equilibrium reaction XM ~ X~ X~ where X1 represents the monomer surfactant concentration, X2 the counterion concentration s u p p o s e d to be in equilibrium with micelles at concentration XM. N is the aggregation number of micelle, and Q the number of "bound" counter-ions. As such it is a phenomenological description with no predictability. This model emerges as a special case of the more general theory, and the number of "bound" counter-ions emerges as the calculated physisorption excess of counter-ions at the micellar surface. In this theory, cmc and aggregation numbers all are given correctly as a function of chain length, added salt, and temperature-up to 160~ [65]. The term ~, is constant for a given chain length with salt variation, and the predicted value agrees with independent statistical mechanical calculations of chain interactions in a micellar configuration. More refined calculations that evaluate the statistical mechanics of chain packing give essentially the same result. Of more interest perhaps is the fact that from the analysis emerges a reconciliation of the older pseudophase chemical equilibrium model and the more general statistical mechanical model.

3.6.4

The tyranny of theory

The importance of advances in basic science is not so much in the advances themselves. It lies rather in the breakdown of dogma, the removal of the tyranny of theory which tends to become set in stone, and the tyranny of disciplinary boundaries so inhibiting to the opening of new vistas. That is so equally for the Ptolemaic planetary system, for Darwinism, for q u a n t u m mechanics, or even the new molecular biology. Darwinism in retrospect is in part tautological, and q u a n t u m mechanics shot through with logical inconsistencies despite its astonishing successes. Even the new molecular biology, focused on linear sequences of DNA, and beset by X-ray structure determination of the individual atoms of proteins to the neglect of the shapes of the hydrophobic regions (of hyperbolic geometry) that guide and drive RNA replication, is so beset. The neglect of the environment, and its interaction with shape, i.e. forces, has inhibited the joining to advantage of what mathematics and physics and chemistry have to offer biochemistry and life processes. D'Arcy Thompson had argued ever so gently in his book "On Growth and Form" [66] that classification and evolutionary lineage through

Self-assembly

123

morphology alone might be fraught with danger. The equiangular spiral that underlies similarity in morphology of shells and the shape of plants and other things derives from processes driven by constant rates of growth. They are not necessarily an indication of relatedness. And the present vogue of assigning evolutionary connectedness depending on statistically correlated amounts of conserved partial DNA sequences is as dangerous as phrenology. On the other side, in colloid and surface chemistry, which borders most closely from the physical sciences on biology, the DLVO theory of stability that held sway so successfully for 50 years and more, has been equally or more inhibiting to the joining of structure and function. The gallimaufry of forces outlined above removes that disjunction, somewhat, because it is now clear that nature has at her disposal a richer, more specific and stronger diversity of forces than we had imagined. And we now understand better how they can be turned on or off to advantage through shape and topology triggered by biochemistry in a changing environment engineered itself by those shapes. Awareness of the new forces, and how they can be called in to play via hydration, specific ion effects, the Debye length in mixed electrolytes or micellar systems, the new hydrophobic forces, dipole- dipole and ionic fluctuation effects (and others) to affect the state of phospholipid curvature is necessary. The older inhibitions have been removed. Experiment and theory now begin to be consistent, at least for model systems, and the rich diversity of shapes and structures available, and controllable t h r o u g h c u r v a t u r e set by molecular forces becomes comprehensible, perhaps even accessible. We move on now out of the Euclidean desert to explore these matters further. In Chapter 4 we shall see how curvature together with global packing constraints conspires to p r o d u c e and predict the rich diversity of bicontinuous cubic phases and others of constant mean curvature which can be prescribed by variations of solution conditions or temperatures.

124 Appendix 3A: Evolution of concepts on long range molecular forces responsible for organisation and interactions in colloidal systems

(A) Original theories (1930's) involved (1) steric repulsion, "hydration shells" of water to stabilise (i.e. prevent coagulation) biological aggregates and emulsions, and later invoked; (2) entropy (two-phase equilibrium) opposed to repulsive electrostatic (double-layer) and hard-core interaction between anisotropic particles, clay plates, tobacco mosaic virus ordering (Langmuir, Onsager). These ideas are coming back into vogue and rediscovered in the steric fluctuation forces of Helfrich between biological membranes and in surfactant-water surfactant-water-oil systems. The current status of these ideas, presently under intense investigation by new low angle x-ray scattering and neutron scattering techniques, is unclear. Hydrodynamic forces also play a role, but depend on short range forces at an interface (boundary layers). (B)

Forces between solid particles.

Two Themes: (1) YOUNG-LAPLACE

(2) POISSON

4--1~ water

action at a distance Intervening liquid has bulk liquid properties to atomic dimensions from an interface

surface induced liquid structure Forces due to overlap of profiles of surface induced liquid structure

odispute reviewed by Challis in report to British Association (1833, 1834) "On the Present State of Theories of Capillary Action". The dispute between Laplace and Poisson there reviewed taken up by J. Clerk Maxwell (1876) in "Theory of Capillary Action" [41].

Appendix 3A

125

1930"s

9double-layer theory and theory of van der Waals forces developed. 9attractive Van der Waals Forces by pairwise summation plus double-layer theory gives potential of interaction.

vl(I) =

r6

l2

+

v2(I)

kl ;

Ir = Debye length of salt solution yields

1940"s

9Derjaguin Landau Vervey Overbeek (DLVO Theory): explains salt dependence of lyophilic colloid stability, fails to account for biological interactions oil-water systems. 1960's

9Lifshitz theory of attractive forces across vacuum, measurements of long range forces.

9measurement of repulsive forces via soap film thickness.

1970"s

9van der Waals forces, effects of interacting media, non-additive, infrared, ultravolet, microwave, temperature dependent, salt d e p e n d e n t , r e t a r d a t i o n show extreme subtlety of forces even in continuum approximation.

9surface charge regulation, surface charge responds to presence of neighbouring surfaces to regulate interactions and re-organise structures.

9very long range forces discovered. 9awareness of recognition. 9forces even at primitive level guide recognition processes. 9van der Waals self-energy concept parallels self-energy for charged ions. 9interactions and surface energies intimately coupled.

126

Chapter 3

9measurements of long-range forces (Israelachvili et al.).

9measurements and "theories" of hydration forces between phospholipids (Parsegian and

Rand). 9measurements of short-range forces (Maxwell rediscovered).

9charge regulation, ionic hydation-~ 9"secondary hydration" forces. 9oscillatory forces of surprisingly long range in simple liquids and ionic solutions.

1980"s

9beyond Poisson-Boltzmann theories. 9ion-fluctuation correlations explain failure of DLVO theory for divalent salts. 9ion-binding inferences need reinterpretation. 9van der Waals and double-layer forces now inextricably entangled. Distinction once made disappears, especially for oil-water and biological systems. 9measurements of specific ion effects show extreme subtlety of biological interactions e.g. acetate vs, Br-. 9temperature dependence of polyoxyethylene surface interactions from repulsive to attractive in narrow temperature range show further subtlety, in biological situations. 9very long range hydrophobic interactions discovered.

1990

9surface dipole-dipole forces explain discrepancies between long-range forces predicted and measured for biomembranes, phospholipids and proteins. 9oscillatory forces in micellar and lamellar phase systems provide further mechanisms for organisation and recognition.

9forces in asymmetric electrolytes explain failure of theories in many situations.

Appendix 3A

127

1991

9awareness that measurements of short-range (< 20.~) forces may be erroneous due to experimental limitations, as for lubrication, viscosity, adhesion, friction. Similar conclusions for scanning tunnelling and atomic force microscopy "imaging" of macromolecules. New techniques available. .awareness of how to control forces and their subtlety with temperature, salt, hydration removes disjunction between colloid science and membrane science.

1993

*hydrophobic interactions shown to be critically dependent on amount of dissolved gas, salt and salt types. This completely changes our understanding of the role of "hydrophobicity" in self-assembly.

1993

9It now turns out that there is not one, but many hydrophobic interactions. Developing theories recognise that earlier ideas have ignored the role of dispersion forces acting on ions, that drive ionic adsorption. Evidence is mounting for ultrastructure in water, perhaps due to dissolved gas; recognition of the interplay between dissolved gas organisation and electrolyte concentration and type. Free radical production due to formation of nanometric sized cavities connects hydrophobicity, reactivity and recognition in fundamentally new ways. The shape of the bridge linking the molecular forces of surface and colloid science and biochemistry and selforganisation begins to emerge.

128

Appendix 3B: Modern concepts of self-assembly 1936

oconcept of micellar and lameUar phases oself organisation of surfactants (Hartley). 1945 omicroemulsions (oil-water-surfactant-cosurfactant) oequilibrium phases. 1960"s

obeginnings of microscopic theories (Tanford). oprinciple of opposing forces at surfactant-water interface + hydrophobic free energy of transfer). ocubic phases discovered (Luzzati, Fontell). 1970's osimple unified theories give good description of phase diagrams of surfactant-water systems. Restricted to simple geometries. Spontaneous vesicles discovered. 1980"s

obicontinuous cubic phases shown to be ubiquitious. ,structure of microemulsions explored and put on firm basis theoretically using concepts of curvature set by molecular forces (inter- and intra-aggregate interactions) and global packing constraints. *discovery of rich diversity of cubic and intermediate phases in phospholipids and other systems, easily transformed from one state to another. orecognition that lipids and membranes and the topology and shapes taken up play a crucial role in guiding biochemical processes. 1990"s ocubosomes, supra self-assembly and quasi-crystalline structures.

129

Appendix 3C

Remarks on the nature o f the hydrophobic interaction and water structure

For the all-important hydrophobic interaction, the problem of water structure cannot be easily swept away. Recent work raises the spectre that most of our progress has been illusory. This can be seen from studies of the effect of salts and other solutes in reducing bubble coalescence in aqueous solutions. We refer to the original papers for more detail [67, 68]. The reasons for the effects of salt (or other solutes) are not understood. Salts generally increase the surface tension of water and are desorbed from the airwater interface. These factors might be expected to destabilise bubbles. Salts would also be expected to reduce any electrostatic repulsion produced by charge build up on bubble surfaces. The mechanisms involved in the enhanced bubble stability produced by salts have yet to be elucidated. The phenomenon of interest is about as simple to demonstrate as it is dramatic. If we imagine bubbles formed by passing a gas through a glass frit at the base of a column of water, the bubbles fuse on collision and grow in size as they ascend the column. However, on addition of an alkali halide, beyond a certain critical salt concentration the bubbles will not fuse and remain the same size. That phenomena has been explored by several authors. On the other hand, increasing concentrations of HCI do not have any effect on bubble coalescence (cf. Fig. 3.5) A whole range of cations and anions in different combinations have been explored. The results are surprising. Measurements of coalescence rates for a range of typical electrolytes as a function of electrolyte concentration are shown in Fig. (3.5). There is a correlation between valency of the salt and transition concentration, defined as 50% bubble coalescence, with more highly charged salt effective at lower concentration. The effect is independent of gas flow rate. All the results scale with Debye length (ionic strength). Some salts and acids have no effect at all on bubble coalescence, a situation summarised in Table 3.1. There is no known mechanism that can account for these effects. Water structure has to be implicated. But there is clearly a remarkable correlation between the ions present in a salt and their effect on the coalescence phenomenon. A property a or I~ can be assigned to each anion or cation. The combination a a or [~1~results in inhibition of bubble coalescence at a critical salt concentration, whereas the combinations a13 or [3a produces no effect at all. Different gases of widely different molecular size, from helium to sulphur hexafluoride, affect the transition concentration a little, but do not change the phenomenon.

130

Chapter 3

Figure 3.5: Plot of bubble coalescence as a function of electrolyte concentration for a range of common salts. Beyond a fairly sharply defined concentration range, it is impossible for bubbles to fuse. The sugars sucrose, fructose and glucose have also been found to affect bubble coalescence. On addition to water these sugars raise the surface tension and are d e s o r b e d from the air-water interface. Thus their effect on bubble coalescence equally cannot be described in terms of surfactant-like behaviour and certainly no charge effects are involved. Hence, even if an "explanation" could be found within the confines of the primitive model of electrolytes, that explanation could not accommodate this observation. The reduction in bubble coalescence achieved with increasing concentration is s h o w n in Fig. 3.7. Although earlier work had focussed on those electrolytes which exhibited coalescence inhibition, it has now been s h o w n that some other salts and mineral acids have no effect whatsoever. For those electrolytes inhibiting coalescence there does appear to be a correlation with the ionic strength, which brings the results into a relatively narrow band. However, and to repeat, as yet there is no obvious explanation why some electrolytes produce no effect on coalescence.

Appendix 3C

131

Table 3.1: Effect of added ions on bubble coalescence in water.

CATION

H+ ~Mg2+ Na+

ANION

OH- , X

0~ 9

I. Sol

9

9

c,-

X

J

J

9

Br-

0~

X

NO39

9

) CIO3-

J

X

9

m

Ca2+

K+

I. Sol

J

J m

C104- j C !-I.~00-

of.

J

oxanate2- X

X

X

X

I. Sol I. Sol

Combining Rules: ~

9

9

9

9

9

9

9

ji

Li+

X

J

Unavail 9

9

I. Sol

J m

Unavail

9

Unavail

9

J

l

=

X

Unavail I. Sol

I. Sol

9

X I. Sol

J

or 1313gives J

!. Sol=Insufficiently soluble

9

1

9

X

9

UnavailI I. Sol Unavail I. Sol

!J 9

9

J,J

9

X

9

J

Unavail UnavaiI X

SO42-

Me4N+

Cs+

j=jij j

9

J

NH4+

~.

X

Unavail =Unavail ,,

X

9

J

I. Sol Unavail Unavail 0t13or 13orgives X Unavail=Salt unavailable

Addition of salt: Prevents coalescence JI-las no effect on coalescence

X

In seeking an explanation we can definitely rule out any correlation with changes in surface tension at the air-water interface that are responsible for the foaming of surfactant solutions. The electrolytes studied increase the surface tension of water. Changes in the h y d r o d y n a m i c force caused by viscosity can also be excluded because some electrolytes decrease the viscosity, yet inhibit coalescence. The t e m p e r a t u r e d e p e n d e n c e if any is weak. Predictions of conventional DLVO theory go in the wrong direction - added salt reduces repulsive double-layer forces, to suggest enhanced coalescence. There is no evidence of significant bubble charging in salt solutions. A

132

Chapter 3

different kind of double-layer force that increases with salt concentration can be considered. This can come about through the differing cation and anion hydrated sizes that affect the electrostatic potential at the water surface. Detailed statistical mechanical calculations of this repulsion force have been carried out assuming a range of ion sizes and surface charge. However, the predicted short-range (10nm) interaction involves some type of micro-bubble cavitation between the approaching hydrophobic surfaces; the formation of such cavities between hydrophobic surfaces is energetically favourable. Further, spontaneous cavitation has been observed between hydrophobic surfaces at close separations. However, the formation of a small vapour cavity is a high energy process, a circumstance that poses a problem for the cavitation mechanism. A modification of this approach might eliminate this difficulty. In all cases of long-range (>10nm) hydrophobic force measurements, reported so far, the aqueous solutions were apparently in equilibrium with the atmosphere. Hence, they contain about 25ml per litre of dissolved nitrogen and oxygen gases, close to saturation. These dissolved gases might accumulate in the vicinity of hydrophobic surfaces, so releasing high energy water surrounding the gas molecules back into the bulk state. Fluctuations in

134

Chapter 3

density near hydrophobic surfaces due to the "adsorption" of gas could give rise to the long-range attraction. For hydrophobic surfaces in close proximity, this accumulation appears to develop into bubble nucleation sub-critical fluctuations in density of the liquid film. Support for this notion comes from the simple observation of the effect of dissolved gas on emulsion stability. It can be shown that de-gassing a mixture of dodecane and water increases the stability of an oil-in-water emulsion produced by shaking. This simple observation suggests that the hydrophobic interaction responsible for oil droplet fusion is reduced by the removal of dissolved gases. Measurements of the hydrophobic attraction between solid polypropylene surfaces show that on degassing the water the force drops back to van der Waals forces predicted by Lifshitz theory - by one or two order of magnitude. Measurements of the hydrophobic attraction between solid surfaces using an atomic force microscope have also been performed in solutions of NaCl and NaC103. The attraction was found to be equivalent to that in water in 0.2M NaCIO3, whilst in 0.2M NaCl the attractive force was much reduced, supporting expectations. (NaCIO3, has no effect on bubble coalescence, NaC! does, cf. Table 3.1.) It has also been observed previously that the range of the attractive force acting between bubbles is substantially reduced from about 100nm to 40nm on addition of KCI above the transition concentration. The effect of NaCl on bubble nucleation in the presence of hydrophobic surfaces has also been examined. Excess nitrogen gas was dissolved in solution by equilibration under 25 atmospheres of pressure. Immediately following decompression the solution was supersaturated with nitrogen gas. In water and 0.02M NaCl, it was found that bubbles nucleated quickly ( 1, cross-section through a pore of a saddle-shaped bilayer, whose mid-surface is a minimal surface; (centre:) v/al = 1, a planar bilayer; (right:) v/al < 1, a "blistered" bilayer, containing a vacuous region. In the last case, a reversed bilayer (Fig. 4.7) is favoured over the bilayer configuration illustrated. We have seen that a symmetric bilayer composed of chemically identical monolayers will spontaneously buckle so that the mid-surface of the bilayer wraps onto a minimal surface. Since the average Gaussian curvature, , of a minimal surface must be non-positive, eq. 4.9 implies the constraint on the surfactant parameter of the molecules which make up the bilayer: ~ > I. This is a severe constraint on the molecular shape of bilayer-forming surfactants: their chains must be bulky compared with their head-groups (e.g. doublechain surfactants or lipids). If the surfactant parameter is exactly unity, a planar bilayer results. However, once it exceeds unity, the bilayer must buckle, resulting in some negative Gaussian curvature. The data for threeperiodic minimal surfaces suggest that this buckling results in a bilayer whose global structure is similar to those of IPMS, with an array of catenoidal pores. The density of these pores depends intimately on the molecular shape, as the molecule becomes less cylindrical (Y-- = l) and increasingly wedged

5 urfactan t monolayers

153

shaped (~ > 1), this density increases. The possibility of tuning the pore density and consequent transport properties through adjustment of the molecular architecture (e.g. varying the head-group area, a, by varying the electrolyte concentration at the head-groups) is a real one, with obvious ramifications for the functioning of biological membranes. If the surfactant parameter is less than unity, a reversed bilayer can form, where the constituent surfactant molecules are placed head-to-head, rather than the chain-to-chain configuration characteristic of normal bilayers. If a chain-to-chain configuration occurs, the bilayer must be "blistered", to accommodate the bulkier head-groups (Fig. 4.6).

Figure 4.7: Images of (left) a portion of a surfactant bilayer wrapped onto the P-surface, a triply-periodic minimal surface, with two interwoven polar labyrinths and (right) a reversed bilayer on the P-surface, with interwoven lipophilic labyrinths.

The molecular shape is not the sole determinant of the structure of the aggregate. If the surfactant-water mixture is to form a single phase, the surface and volume requirements set by the composition of the mixture must be satisfied. Introducing the global constraint set by the composition leads to an estimate of the relation between the local geometry (expressed by the surfactant parameter) and the composition at which the surfactant mixture is expected to form a bilayer - or reversed bilayer - wrapped onto an IPMS (illustrated in Fig. 4.7). The details of the calculations for both reversed and normal bilayers (for which the tunnels are filled with water and surfactant respectively) are given elsewhere [7-9]. We characterise the concentration of the surfactant by the The relation between volume fraction of the hydrophobic region, r composition and molecular shape for hyperbolic bilayers is:

154

Chapter 4

v {3(~/"1))1'2 Ochains = 4~[~H-~ (3~-1)3/2

(4.10)

A similar relation can be derived between the local and global geometry of hyperbolic reversed bilayers, for which v/al varies between I and 2. x

3

The regions within the local/global "phase diagram" for which these hyperbolic bilayer structures can be realised within a surfactant-water mixture are plotted in Fig. 4.8.

Figure 4.8: Plot of the relations between the surfactant-water composition (characterised by (hint) and the surfactant parameter for normal (v/aiM) and reversed (v/alc2D), is the diversity of lipid composition and the effect of environmental factors, like temperature, pressure, and lipophilic agents on function. Related proposals [32] for "non-bilayer" lipid structures in membranes invoke the occurrence of the transition Lot---> H I I . The data from which such a proposal is deduced are also consistent with an Lr 2D ---> C 2D transition. As discussed in Chapter 4, cubic structures are intermediate between Lot and HII. The difference is that for the La 2D ---> C 2D phase transition the bilayer conformation alone is involved, whereas any membrane conformation related to the HII structure seems improbable in the extreme because of the resulting exposure of hydrophobic tails to the aqueous medium. Pressure increases the trans/gauche ratio of the chains, which is reflected in increased bilayer thickness and decreased molecular cross-section. A pressure increase of 100 atmospheres is considered to be equivalent to the effect of a temperature decrease in the range 2-8~ Reported effects of pressure as well

Lipid composition control

217

as of temperature on lipid composition of membranes are consistent with this phase transition model [33]. Any mechanism proposed to control lipid composition must take account of enzyme system activity monitoring by such diverse factors as temperature, pressure, pH, ionic strength, variations in content of cis-double bonds and trans/gauche ratio, and the presence of sterol skeletons. A phase transition can provide a direct conformation related switch, with the ability to control all these factors [34]. Asymmetry in the lipid distribution over the bilayer could also be controlled in a similar way by the lateral packing pressure, which is likely to differ between constituent monolayers, due to the distinct chemical environments inside and outside the membrane. The enzymes involved m a y also be distributed asymmetrically. A configuration with constant, but nonzero, mean curvature, shown in Fig. 5.7, reflects such a situation. A membranespanning protein can then be viewed as a sensor of the lateral packing pressure in both monolayers. This speculation has some experimental justification. In a recent study of chromaffin granules, trans-membrane lipid asymmetry was shown to be induced by an ATP-dependent "flippase" [35]. The "fixing" of the proteins that fill holes in the C 2D structure can be considered to take place in the following way. In an L0~ phase all molecules are subject to the same packing conditions. In the cubic structures, however, there is a continuous variation in packing conditions, from fiat points to the points with highest Gaussian curvature due to the necessary inhomogeneity of hyperbolic surfaces. The disorder must increase with increasing Gaussian curvature. In a membrane bilayer the lipid molecules in a L(x2D conformation are statistically distributed about a mean shape, whereas a C 2D conformation must exhibit continuous variations. A higher Gaussian curvature might correspond to a higher relative proportion of lipids with cis-double bonds, for example. It should be pointed out also that a C 2D bilayer consisting of only one lipid type (like the cubic phases discussed in Chapter 4) must have v a r y i n g molecular disorder c o r r e s p o n d i n g to the varying Gaussian curvature. The lateral lipid diffusion is almost identical to that of the La phase [3]. Studies of lipid mobility have shown that the molecules close to membrane proteins are more disordered than those lipids further removed. In a lipid bilayer with variation in lateral disorder, the proteins should therefore be expected to go into positions of highest disorder. Thus, there is a driving force for proteins to locate in the "holes" in the C 2D bilayer structure, which are surrounded by maximally curved membrane. There is also a lateral packing pressure to adopt the protein according to the average hydrocarbon chain direction. Ideally, the proteins filling the "hole" of the C 2D lipid bilayer should be wedge-shaped (see Fig. 5.7). Thus the C 2D conformation (contrary to the Lr 2D one) provides a periodic variation in the degree of lipid molecular disorder, or "fluidity", and this

218

Chapter5

might have functional significance. Different studies show that the activity of membrane-bound enzymes can be directly related to the bilayer fluidity, e.g. p h o s p h o l i p a s e s , N a + / K + A T P a s e or Ca2+-ATPase. It is obvious that a transition Lot2D ---) C 2D, with the enzyme molecules in the "fixed" locations, will result in an enzyme environment of lipid molecules of higher fluidity than that of the L~ 2D conformation.

Fig. 5.7: Periodically curved membrane conformations. Proteins filling the "holes" towards the corresponding three-dimensional phase are indicated by filled units, whereas open units are free to diffuse along the bilayers. Successive changes towards asymmetric and constant (nonzero) mean curvature are shown from top to bottom. All the other protein molecules of the membrane in the C 2D conformation, which are not in the "fixed" or "hole" positions discussed above, will possess lateral diffusional freedom just as in the Lr 2D conformation.

5.2.4 The nerve membrane, signal transmission and anaesthesia The ion fluxes and the functions of ion channels and p u m p s in nerve signal transmission have been studied and are k n o w n in considerable detail. The lipid bilayer of the nerve membrane is highly specific in composition with well-controlled asymmetry. In spite of this, no other role has been assigned to the lipid bilayer except to provide a barrier and a passive carrier function for the membrane proteins involved in the p u m p s and channels. Arguments will be given below that during the propagation of the action potential there occurs a bilayer phase transition involving a periodically curved bilayer.

Bilayer conformation during the action potential Several independent studies indicate that a lipid bilayer phase transition takes place in conjunction with transmission of the action potential. Forty

219

Nerve membranes

years ago it was reported that heat was associated with the passage of the action potential, and detailed thermal data were later reported in studies of impulses through the pike olfactory nerve [36]. Two regimes were observed first a positive value (at 0~ 44.2 ~tcal/g/ impulse), followed by a negative value (48.9 ~cal/g/impulse). Calculation shows that these values are much too high to correspond to free energy changes associated with the membrane capacitance. The natural interpretation is therefore that the thermal transition is related to changes in the lipid bilayer. Studies of optical properties of the squid giant axon have shown that the pulse gives a change in the birefringence [37]. It was proposed that these changes reflect changes in membrane thickness. Also, in a fluorescence study of an olfactory nerve fibre, a reduced fluidity of the hydrocarbon chains of the membrane on the passage of the action potential was found [38], which is consistent with reduction in entropy of the lipid. Taken together the observations imply that transmission of the action potential may involve the propagation of a phase transition "wave". In that case the spike

can be d u e

to the transitions

L~D'-~C2D--~I-~D

or

C 2D .._)I~D --~ C 2D .

As periodic structures have not been reported from electron microscopy studies of freeze-fractured axon membranes, it seems probable, if this phase transition is involved, that the region of spike exhibits the C 2D conformation. The inner lipid monolayer of the axon membrane (with PE and C22:6 acyl chains dominating) is expected to favour a periodically curved structure, due to the high average wedge-shape associated with these lipids, whereas the outer monolayer (sphingomyelin and PC dominating) should favour a planar conformation. From this perspective it seems reasonable that the membrane bilayer can exhibit both an L~2D as well as a C 2D conformation. At the neural membrane region there is a requirement for regularly packed protein molecules, p r e s u m a b l y the ion channels, located at the hole positions. The regularity in packing might be related to the required narrow spread in the speed of signal transmission. The m i n i m u m time between action potential spikes is so short that the protein order is not lost due to diffusion during the period of existence of the L~ 2D conformation of the bilayer. The existence of a curved C 2D conformation associated with the action potential is supported by the fact that the ion influx at the spike will induce an increased average wedge-shape of the molecules, due to electrostatic screening of the lateral repulsion of p h o s p h a t i d y l s e r i n e molecules. Furthermore a C 2D conformation associated with the spike would directly relate action potential propagation to the mass-cooperative vesicular fusion, involved in the chemical signal transfer by transmitter molecules at the presynaptic membrane. Experimental support for this concept has been recently reported [39]. This well-controlled fusion process of numerous "vesicles" with the presynaptic membrane must take place as a phase transition. The

220

~ter 5

process is a beautiful example of cubic phase function, as will be described below. This membrane fusion process (outside the brain) is known to involve thousands of single membrane units, previously thought of as vesicles, assembled into units that have been termed "boutons". We have examined the EM texture of the boutons and found that they are in fact a cubic phase. The synaptic signal transmission can take place as frequently as hundreds of times per second. A fusion process involving a hyperbolic membrane can be well controlled, and the calcium ion i n f l u x - which induces fusion - is expected to change the conformation of the cubosome surface membrane from its planar bilayer conformation to the fusogenic saddle-saddle conformation. (It is known phase transitions of membrane lipids can occur when exposed to calcium, e.g. [40]). In this connection it can also be mentioned that the lipid composition of synaptic membranes responds to temperature changes (fish acclimatised in the range 2-37~ as well as pressure changes, expected if the membrane bilayer must balance on the edge of an La2D