The Language of Shape: The Role of Curvature in Condensed Matter: Physics, Chemistry and Biology PDF
Preview 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 RESEARCH INSTITUTE S. LJ~NGGATAN 27 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 (cid:14)91 997 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 VUo, 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 self- assembly 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 modem 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 1 The Mathematics of Curvature 1.1 ................. Introductory remarks ............................................................................................... 1 1.2 ................. Curvature ................................................................................................................... 2 1.3 ................ Differential geometry of surfaces ........................................................................... 4 1.4 ................ The Gauss map .......................................................................................................... 6 1.5 ................. Geodesic curvature and geodesics ......................................................... . ............... 7 1.6 ................ Torsion ........................................................................................................................ 8 1.7 ................. The Gauss-Bonnet theorem ..................................................................................... 10 1.8 ................. Topology .................................................................................................................... 11 1.9 ................. A provisional catalogue of surface forms .............................................................. 14 1.10 ............... A historical perspective ............................................................................................ 18 1.11 ............... Periodic minimal surfaces ....................................................................................... 21 1.12 ............... The Bonnet transformation: the P-surface, the D-surface and the gyroid ........ 27 1.13 .............. Parallel surfaces ......................................................................................................... 32 1.14 ............... Future directions ....................................................................................................... 32 Appendix: ..... A catalogue of some minimal surfaces .................................................................. 33 Mathematical Bibliography ............................................................................................................. 40 References ........................................................................................................................................ 41 Chapter 2 The Lessons of Chemistry 43 Inorganic Chemistry-. From the discrete lattice of crystal symmetry to the continuous ....... ...................... manifolds of differential geometry ......................................................................... 43 2.1 ................. The background ........................................................................................................ 43 2.2 ................. The unravelling of complex structures .................................................................. 44 2.3 ................. Defects ........................................................................................................................ 46 2.4 ................. The intrinsic curvature of solids ............................................................................. 49 2.5 ................. Hydrophobic zeolites and adsorption ................................................................... 52 2.6 ................. Phase transitions, order and disorder .................................................................... 55 2.7 ................. Quantitative analysis of hyperbolic frameworks: silicate densities .................. 58 2.8 ................. Tetrahedral frameworks: Three- or two-dimensional structures? ..................... 63 2.9 ................. Quasicrystals ............................................................................................................. 66 Organic Chemistry : The Shape of Molecules ............................................................................. 73 2.10 ............... The hyperbolic nature of sp 3 orbitals ..................................................................... 3'3 2.11 ............... Organic sculptures: carcerands, crowns, etc ......................................................... 75 2.12 ............... Beyond graphite: fuUerenes and schwarzites ....................................................... 78 Appendix: ..... The problem of quasicrystals .................................................................................. 80 References ........................................................................................................................................ 84 Chapter 3 Molecular Forces and Self-Assembly 87 3.1 ................. The background ........................................................................................................ 87 3.2.1 .............. The nature of force, shape and size ........................................................................ 88 3.2.2 .............. Self-energy, molecular size and shape ................................................................... 89 3.2.3 .............. Self-energy and adsorption ..................................................................................... 91 3.2.4 .............. The shape of bonds ................................................................................................... 94 3.3 ................. The background to surface forces ........................................................................... 96 3.4 ................. Molecular forces in detail ......................................................................................... 98 3.4.1 .............. van der Waals forces ................................................................................................. 98 3.4.2 .............. Lifshitz forces ............................................................................................................. 100
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