Demonstrating Science with Soap Films - Frontispiece F2. Photographs sh o wing the jump between the two soap-film configurations within a framework s haped as a pe ntagonal prism. 1_- Demonstrating Science with Soap Films David Lovett Department of Physics University of Essex Computer Programs John Tilley and David Lovett Institute of Physics Publishing Bristol and Philadelphia © lOP Publishing Ltd 1994 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 permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Lovett, D. R. Demonstrating science with soap films / David R. Lovett; computer programmes, John Tilley and David Lovett. p. cm. Includes bibliographical references and index. ISBN 0-7503-0270-4. -- ISBN 0-7503-0269-0 (pbk.) 1. Soap-bubbles. 2. Soap-bubbles--Study and teaching. I. Tilley, John, Ph. D. II. Title. QC I83.L85 1994 530.4'275--dc20 94-30453 CIP ISBN 0 7503 0269 0 (pbk) o7503 0270 4 (hbk) The computer programs given in Appendix II were written using Mathematica. The programs were prepared for versions up to 2.2. Mathematica is a registered trademark of Wolfram Research, Inc., 100 Trade Center Drive, Champaign, IL 61820- 7237, USA (phone: 1-217-398-0700; fax: 1-217-398-0747). Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Techno House, Redcliffe Way, Bristol BS I 6NX, UK US Editorial Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, Independence Square, Philadelphia, PA 19106, USA Typeset in TJY( by Institute of Physics Publishing Printed in the UK by J W Arrowsmith Ltd, Bristol Contents Frontispiece iii Preface Xl 1 Introduction 1 1.1 What are soap films? 1 1.2 Surface tension and surface energy 4 1.3 A brief history of soap-film studies 5 1.4 Plateau's problem 7 1.5 The shape of minimal surfaces 9 1.6 The saddle-point bifurcation 11 2 Two-dimensional soap-film patterns 12 2.1 Introduction 12 2.2 Simple two-dimensional patterns 12 2.3 Bubbles within the two-dimensional patterns 21 2.4 Symmetry of patterns, objects and soap films 23 2.5 Further discussions on geometry 25 2.6 The general Steiner problem 27 3 Soap films and first- and second-order phase transitions 30 3.1 A 20 model for a first-order phase transition 30 3.2 Thermodynamics of the phase transition 38 3.3 The water-vapour phase transition 41 3.4 Physics of the four-pin soap-film model 42 3.5 A 20 model for a second-order phase transition 44 3.6 The Ehrenfest classification 46 3.7 The order parameter and symmetry 48 4 Soap-film models and catastrophe models 50 4.1 Introduction to catastrophe theory 50 4.2 The fold catastrophe 51 Vll VIII CONTENTS 4.3 The cusp catastrophe 52 4.4 The Zeeman catastrophe machine 54 4.5 Illustrating double-cusp catastrophes with soap films 57 4.6 Chapter appendix; phase transitions and Landau theory 67 5 Film within a wedge-the catenoidal surface 72 5.1 Film between three pins joining non-parallel plates 72 5.2 Film within a wedge 72 5.3 Setting the wedge apex upwards in a gravitational field 75 5.4 Interference colours 76 5.5 Theory of thin-film interference 78 5.6 Films between two cones 81 5.7 The H -surfaces of revolution 81 5.8 Altering pressure on one side of a film 83 5.9 Films within a helix-the helicoid 84 5.10 Chapter appendix; mathematics of the catenoidal surface 85 6 Soap films within three-dimensional frameworks and minimal surfaces 91 6.1 Films within simple prism-shaped frameworks 91 6.2 Films within simple variable frameworks 93 6.3 Film patterns within the regular Platonic polyhedra 98 6.4 Great arc configurations 100 6.5 Minimal surfaces and computers 104 6.6 Examples of minimal surfaces suitable for representation by computer graphics 105 7 Fermat's principle and refraction 112 7.1 Fermat's principle 112 7.2 Modelling ~imple refraction 114 7.3 Prisms and lenses 115 7.4 Mirages 117 7.5 P-waves through the earth 121 7.6 Curvature of space-the Schwarzschild geometry 127 7.7 Bending of light near a black hole 129 8 Bubbles 131 8.1 Simple bubbles 131 8.2 Bubbles within fixed frameworks 132 8.3 Bubbles within variable frameworks 135 8.4 Coalescing bubbles 137 CONTENTS IX 8.5 More about bubble patterns 141 8.6 The bubble raft and crystal structure 144 9 Analogies within the scientific world 149 9.1 Introduction 149 9.2 Biological unicellular forms and radiolarians 149 9.3 Forms of carbon and the fullerenes 152 9.4 The honeycomb 154 9.5 Cracks and fissures 157 9.6 Equilibrium shapes of crystals 157 9.7 Langmuir-Blodgett films 160 9.8 Application in technology; communications networks 163 9.9 Architectural applications 163 9.10 Concluding comments 164 Appendices 166 A.I Construction of the models 166 A.L1 The two-dimensional models and the wedge 166 A.I.2 The frameworks 169 A.II Soap-film patterns by computer using Mathematica, by John Tilley and David Lovett 171 A.II.I-A.II.16 Computer programs 172 Bibliography 192 Index 197 Preface For many, the blowing of soap bubbles as children formed a significant part of their introduction to science. By generating these bubbles, children meet ideas of shapes and forms which can be of considerable surprise even within very simple situations. They encounter the idea of stability, or perhaps more importantly the idea of instability as the bubbles burst. And they see something of the beauty which science can produce as they watch the bubbles catch the light to produce startling colours as a result of the interference of light within the surfaces of the bubbles. Throughout scientific history, soap films have held the interest of scientists and mathematicians with an equal intensity to that of children. A topic which, on the one hand, is very simple with few variables and constraints proves, on the other hand, to be difficult to solve in its entirety by mathematics and to embrace fundamental principles which underlie all of physical SClence. This book attempts to pass on some of the fun which can be obtained from studying soap films and soap bubbles. At the same time, it has a much more serious objective, that of using films to help one understand principles of physics in particular and other aspects of science in general. This is possible because soap films are forever attempting to minimize their energy. In the same way, minimization of energy is fundamental to the setting up of equilibrium physical situations. So this book is about soap The author demonstrating the films; but it is also about the importance of energy minimization wonder of soap films to a captivated to the establishment of the world around us. And it is about the audience. Picture by courtesy of patterns which are set up as a result of this minimization. Essex County Newspapers Ltd. In most topics in physical science it is difficult to avoid mathematics. The study of soap films is no exception and simple calculations and algebraic equations are dispersed throughout the book. Where the amount of mathematics is more significant, the relevant section has been boxed and can be left out if desired. Although the balance of content reflects very much my own emphases and interests, the development of the material has arisen from considerable interaction with colleagues and friends. I first became interested in the topic after attending, in close Xl XlI PREFACE succession, public lectures by Dr Cyril Isenberg on soap films and soap bubbles and by Professor Christopher Zeeman on catastrophe theory. Following this, the interest of colleagues in the Depa.rtment of Physics here at Essex University has been a great encouragement. In particular, I must acknowledge the considerable input from Dr John Tilley. Not only has he collaborated with me over many of the ideas, but he is making a major contribution to one of the appendices, and has provided numerous helpful suggestions concerning the overall text. Over the years, Dr Stephen Smith, Professor Rodney Loudon and Professor David Tilley have made many helpful suggestions. I should thank Brian Diamond and John Bartington who made the models, modified them with considerable originality, and sometimes repaired them. The staff at Institute of Physics Publishing have been most helpful in the preparation of the text. I am indebted to Jim Revill for his enthusiastic encouragement of the project and to AI Troyano for his contribution to the layout and his efficient transformation of the manuscript into a most attractive book. Above all, this book is a result of the fun and excitement I have had with hundreds of enthusiasts who have marvelled at the amazing patterns set up by soap films. David Lovett University of Essex December 1993 1 Introduction The nature of soap films and films produced by other surfactants is described. The importance of surface energy is explained, and minimal-area surfaces are introduced.