The IMA Volumes in Mathematics and its Applications Volume 103 Series Editors A vner Friedman Robert Gulliver Springer New York Berlin Heidelberg Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo Institute for Mathematics and its Applications IMA The Institute for Mathematics and its Applications was estab lished by a grant from the National Science Foundation to the University of Minnesota in 1982. The IMA seeks to encourage the development and study offresh mathematical concepts and questions of conc~rn to the other sciences by bringing together mathematicians and scientists from diverse fields in an atmosphere that will stimulate discussion and collaboration. The IMA Volumes are intended to involve the broader scientific com munity in this process. A vner Friedman, Director Robert Gulliver, Associate Director ********** IMA ANNUAL PROGRAMS 1982-1983 Statistical and Continuum Approaches to Phase Transition 1983-1984 Mathematical Models for the Economics of Decentralized Resource Allocation 1984-1985 Continuum Physics and Partial Differential Equations 1985-1986 Stochastic Differential Equations and Their Applications 1986-1987 Scientific Computation 1987-1988 Applied Combinatorics 1988-1989 Nonlinear Waves 1989-1990 Dynamical Systems and Their Applications 1990-1991 Phase Transitions and Free Boundaries 1991-1992 Applied Linear Algebra 1992-1993 Control Theory and its Applications 1993-1994 Emerging Applications of Probability 1994-1995 Waves and Scattering 1995-1996 Mathematical Methods in Material Science 1996-1997 Mathematics of High Performance Computing 1997-1998 Emerging Applications of Dynamical Systems 1998-1999 Mathematics in Biology 1999-2000 Reactive Flows and Transport Phenomena Continued at the back Stuart G. Whittington De Witt Sumners Timothy Lodge Editors Topology and Geometry in Polymer Science With 67 Illustrations Springer Stuart G. Whittington De Witt Sumners Deparlment of ChemistJy Department of Mathematics University of Toronto Florida State University Toronto, ON M5S IAI, CANADA Tallahassee, FL 32306, USA Timothy Lodge Department of Chemistry University of Minnesota Minneapolis, MN 55455, USA Series Editors: A vner Friedman Robert Gulliver Institute for Mathematics and its Applications University of Minnesota Minneapolis, MN 55455, USA Mathematics Subject Classifications (1991): U5C 10, 05C80, 52A22, 53A04, 53A05, 57M25, 60D05, 60115, 82B27, 82B41, 82B80, 82D60, 92EIO, 92-08 Library of Congress Cataloging-ill-Publication Data Topology and geometry volume I [edited byl Stuart G. Whittington, De Witt Sumners, Timothy Lodge. p. cm. - (The IMA volumes in mathematics and its applications; 103) Papers from a workshop on the topology and geometry of polymers held at the IMA in June 1996. Includes hibliographical references and index. ISBN 0-387-98580-8 (alk. paper) I. Polymers-Mathematical models-Congresses. 2. Topology Congresses. 3. Geometry-Congresses. I. Whittington. Stuart G. II. Sumners, De Witt L. III. Lodge, Timothy. IV. Series: IMA volumes in mathematics and its applications; v. 103. QD381.9.M3T66 1998 547'.7'015 14-dc2 I 98-24449 Printed on acid-free paper. eel 1998 Springer-Verlag New York, Inc. All rights reserved. This work may not he translated or copied in whole or in part without the written permission of the puhlisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. 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The copyright owner's consent does not include copying for general distribu tion, promotion, new works, or resale. In these cases. specific written permission must first be obtained from the publisher. Production managed by Karina Mikhli; manufacturing supervised by Jacqui Ashri. Camera-ready copy prepared by the IMA. ~ 8 7 6 5 4 3 2 I ISBN 0-387-985S0-R Springer-Verlag New York Berlin Heidelherg SPIN 10661183 FOREWORD This IMA Volume in Mathematics and its Applications TOPOLOGY AND GEOMETRY IN POLYMER SCIENCE is based on the proceedings of a very successful one-week workshop with the same title. This workshop was an integral part of the 1995-1996 IMA program on "Mathematical Methods in Materials Science." We would like to thank Stuart G. Whittington, De Witt Sumners, and Timothy Lodge for their excellent work as organizers of the meeting and for editing the proceedings. We also take this opportunity to thank the National Science Foun dation (NSF), the Army Research Office (ARO) and the Office of Naval Research (ONR), whose financial support made the workshop possible. A vner Friedman Robert Gulliver v PREFACE This book is the product of a workshop on Topology and Geometry of Polymers, held at the IMA in June 1996. The workshop brought together topologists, combinatorialists, theoretical physicists and polymer scientists, who share an interest in characterizing and predicting the microscopic en tanglement properties of polymers, and their effect on macroscopic physical properties. Linear polymer molecules in dilute solution are highly flexible and self entangled. In more concentrated solutions, or in the melt, there can be important entanglement effects both within and between polymers, and t.hese entanglements can influence the rheological properties of the sys tem as weB as the crystallization properties, and hence the properties of the polymeric system in more ordered states. Although polymer scientists have been aware of these problems for more than forty years, it is only recently that the powerful methods of algebraic topology have been used systematically to characterize and describe these entanglements. Starting from the simplest possible system (a ring polymer in dilute solution) one can ask how badly knott.ed the polymer will be, as a function of the degree of polymerization, the stiffness, the solvent quality, etc. To some extent these questions have been answered by a combination of rigorous mathe matical arguments (combining ideas from combinatorics and from algebraic topology) and numerical methods such as Monte Carlo techniques. For in stance, the paper in this book by OrIandini et al addresses the question of the relative probability of different knot types in a ring polymer. One can also ask about the differential geometry of the polymer and the pa per by Bickis investigates this question for a random walk model. As the concentration increases, linking between rings becomes possible and these links will influence the static and dynamic properties of the solution. Sev eral problems associated with linking are treated in this book by Diao and J anse van Rensburg and by Uberti et al. As we pass from dilute solutions to melts the characterization of the entanglements becomes more difficult. The paper by Edwards addresses these problems. Having characterized the entanglement complexity one then needs to know how it will affect rheological properties. For instance, what is the contribution of entanglements to the elastic properties of a rubbery polymer? How do the dynamics of polymers, in solution or in the melt, depend on entanglement? The conformations of polymers are strongly influenced by any applied geometrical constraints. Polymers behave quite differently in pores or when confined in a slab geometry and their properties in these environments in fluence their behaviour for instance, stabilizers of colloidal dispersions. 3.<;, Approximate theories of colloidal stability have been available for many years, but it is only recently that simple models of polymers in confined geometries have been analysed rigorously. There can be interesting inter- vii Vlll PREFACE actions between topological properties and these geometrical constraints. E.g. how does t.he knot probability in a ring polymer change when the polymer is confined to a pore or slab'? These questions are treated in the papers by Soteros and by Tesi et al. Topological problems also occur in the modelling of polymeric mem branes. These are closely related to self-avoiding random surfaces, an area in which rapid progress has recently been made, although many important questions still remain open. Closely related are the properties of vesicles where the topology of the surface can have an important influence on the behaviour of the vesicle. The papers by Einstein and Stella, and by Beichl and Sullivan, discuss numerical approaches to these problems, while J anse van Rensburg presents some corresponding rigorous results. An area where ideas from physics have had a direct influence on knot theory is the topic of knot energies. A knot is a simple closed curve in R3 (which could be a smooth curve or a finite polygon). One associates an energy functional with the curve so that each embedding has an associated energy. The minimum energy over all embed dings with a fixed knot type is a knot invariant. The book contains four articles on this subject. The paper by Kauffman et al describes one such functional and the search for its energy minima. Of course, there is tremendous flexibilit.y about the choice of the energy functional and the paper by Ernst et at considers desirable properties of the knot energy and compares different choices for it. The papers by Simon, and by Kusner and Sullivan discuss aspects of a closely relat.ed issue, the thickness of knots. This is an area which is attracting considerable interest at the present time, with recently observed correspon dences between the electrophoretic mobility of knotted DN A circles and the energy of "ideal" knot representatives. The book gives an account of recent progress in our understanding of polymer geometry and topology, and the physical ramifications of polymer entanglement. We trust that it will be of int.erest to specialists in this area. 'We hope that it will also serve to introduce polymer scientists to topological approaches, and topologists to interesting applications of their subject in polymer science. Stuart G. Whittington De Witt Sumners Timothy Lodge CONTENTS Foreword ............................................................. v Preface ............................................................. Vll 1. Entanglement Complexity of Polymers Entanglements of polymers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 S.F. Edwards Entropic exponents of knotted lattice polygons. . . . . . . . . . . . . . . . . . . . . . .. 9 Enzo Or/andini, E.J. Janse van Rensburg, Maria Carla Tesi, and S. G. Whittington The torsion of three-dimensional random walk. . . . . . . . . . . . . . . . . . . . . . .. 23 Mike/is G. Bickis 2. Knot Energies Self-repelling knots and local energy minima .......................... 29 Louis H. Kauffman, Mi/ana Huang, and Robert P. Greszczuk Properties of knot energies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37 Y. Diao, C. Ernst, and E.J. Janse van Rensburg Energy and thickness of knots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 49 Jonathan Simon On distortion and thickness of knots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 67 Robert B. Kusner and John M. Sullivan 3. Random Linking Percolation of linked circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 79 Y. Diao and E.J. Janse van Rensburg Minimal links in the cubic lattice ..................................... 89 R. Uberti, E.J. Janse van Rensburg, E. Orlandini, M. C. Tesi, and S. G. Whittington ix x CONTENTS 4. Effect of Geometrical Constraints Knots in graphs in subsets of Z3 .................................... 101 C.E. Soteros Topological entanglement complexity of polymer chains in confined geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135 Maria Carla Tesi, E.l. lanse van Rensburg, Enzo Orlandini and Stuart G. Whittington 5. Surfaces and Vesicles Survey of self-avoiding random surfaces on cubic lattices: Issues, controversies, and results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 159 T.L. Einstein and A.L. Stella Computational methods in random surface simulation.. . . . . . . . . . . . .. 175 Isabel Beichl and Francis Sullivan A model of lattice vesicles ........................................... 193 E. l. lanse van Rensburg ENTANGLEMENTS OF POLYMERS S.F. EDWARDS' Abstract. The interaction of polymers must not allow them to pass through each other. This poses a topological problem which strictly speaking is not solved. Never theless much progress can be made using the Gaussian invariant, and the way to do this is reviewed below. It turns out to give a renormalisable quantum field theory, and is effective in understanding experimental results. 1. The mathematics of polymers. Polymer molecules are totally understood in their specification, but their long chain nature introduces highly complex dynamics, for the chains not only have the dynamics of connection but also cannot pass through one another. Much progress has been made on dense systems, but the topological integrity of the molecules is ignored in dilute solutions, both in self or mutual interactions for no better reason than it is difficult to handle. Suppose we study the configuration of a chain of N segments of length I, freely hinged, and of no thickness with no interactions. This problem is well known and of course completely soluble. For N large i.e. N2 » N the distribution of one end with the other end at the origin tends to 3 3R2 (1.1) (27r N f2 )23 exp( - 2N [2 ) which is the solution of (1.2) (;s - ~V'2)P(R,s) = b(R)b(s) = were s Ni. To get P, this Fick equation is fine, but it cannot solve the problem of a polymer threading its way through a hole in a plane barrier, for the condition that at some s the polymer is at the hole, and at all other points of the plane the probability of tending the polymer is zero (for it cannot cross the plane except at the hole), and these two conditions are incompatible when using (1.2). A real polymer of freely hinged segments is difficult to handle mathe matically for its mathematical structure is of a continuous function of its = arc length B.(s), which is not differentiable at s mi, m integral, where the tangent R' changes direction abruptly. A model much studied is to allow the locus R( s) to be differentiable, but have its second derivative R" to change abruptly. In terms of Wiener integrals, the Fick equation model is to use a prob- • Polymers and Colloids. Cavendish Laboratory, Cambridge CB3 OHE, UK. 1 S. G. Whittington et al. (eds.), Topology and Geometry in Polymer Science © Springer-Verlag New York, Inc. 1998