P.g. De Gennes' Impact On Science - Volume Ii: Soft Matter And Biophysics: Soft Matter and Biophysics 9789814280648, 9789814280631

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SERIES ON DIRECTIONS IN CONDENSED MATTER PHYSICS Published Vol. 1:

Directions in Condensed Matter Physics — Memorial Volume in Honor of Shang-Keng Ma eds. G. Grinstein and G. Mazenko

Vol. 2:

Ionic Solids at High Temperatures ed. A. Stoneham

Vol. 3:

Directions in Chaos (Vol. 1) ed. B.-L. Hao

Vol. 4:

Directions in Chaos (Vol. 2) ed. B.-L. Hao

Vol. 5:

Defect Processes Induced by Electronic Excitation in Insulators ed. N. Itoh

Vol. 6:

Spin Glasses and Biology ed. D. Stein

Vol. 7:

Interaction of Electromagnetic Field with Condensed Matter eds. N. N. Bogolubov, Jr., A. S. Shumovsky and V. I. Yukalov

Vol. 8:

Scattering and Localization of Classical Waves in Random Media ed. P. Sheng

Vol. 9:

Geometry in Condensed Matter Physics ed. J. F. Sadoc

Vol. 10: Fractional Statistics & Anyon Superconductivity (also published as a special issue of IJMPB) ed. F. Wilczek Vol. 11: Quasicrystals — The State of the Art eds. D. DiVincenzo and P. Steinhardt Vol. 12: Spin Glasses and Random Fields ed. A. P. Young Vol. 13: The Superconducting State in Magnetic Fields ed. C. Sa de Melo Vol. 14: Morphological Organization in Epitaxial Growth & Removal eds. Z.-Y. Zhang and Max G. Lagally Vol. 15: Thin Films: Heteroepitaxial Systems eds. Amy W. K. Liu and Michael B. Santos Vol. 16: Quasicrystals — The State of the Art (2nd Edition) eds. D. DiVencenzo and P. Steinhardt Vol. 17: Insulating and Semiconducting Glasses ed. P. Boolchand Vol. 18: P.G. de Gennes’ Impact in Science – Vol. I Solid State and Liquid Crystals eds. J. Bok, J. Prost and F. Brochard-Wyart

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World Scientific

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

The editors and publisher would like to thank the following publishers of the various journals and books for their permission to include the selected reprints found in this volume: Academic des Sciences, Institut de France (C. R. Acad. Sci.); American Chemical Society (J. Phys. Chem.); American Institute of Physics (J. Chem. Phys.); EDP Sciences (Le Journal de Physique); Elsevier Science Publishers (Phys. Lett. A); National Academy of Sciences (Proc. Natl. Acad. Sci. USA).

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

Series on Directions in Condensed Matter Physics — Vol. 19 P.G. DE GENNES’ IMPACT ON SCIENCE — Volume II Soft Matter and Biophysics Copyright © 2009 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4280-65-5 (Set) ISBN-10 981-4280-65-8 (Set) ISBN-13 978-981-4280-63-1 ISBN-10 981-4280-63-1 ISBN-13 978-981-4291-05-7 (pbk) (Set) ISBN-10 981-4291-05-6 (pbk) (Set) ISBN-13 978-981-4291-04-0 (pbk) ISBN-10 981-4291-04-8 (pbk)

Printed in Singapore.

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Pierre-Gilles de Gennes

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PREFACE

In the Pantheon of Science, a few giants emerge: they opened avenues in our quest for more knowledge of our surrounding world. Among them, a tall, elegant man is still very present in our minds and hearts — Pierre-Gilles de Gennes. He has influenced many fields of science and many of us. The ideas Pierre-Gilles introduced were almost always very novel, creating many possibilities. Hundreds of people would start working right away on a new suggestion of PierreGilles’ ! His ideas bear important consequences in the present as well. Thus, it became natural to pick out a few of these contributions and ask some of the leaders in their fields first to explain the ideas, and then to show how it led to the current state of the art. This is what World Scientific asked Francoise Brochard-Wyart, Julien Bok and myself to organise in two volumes. The first one is dedicated to solid state and liquid crystal physics. The second deals with soft condensed matter and biological physics. The general title “P.G. de Gennes’ Impact on Science” reflects the goal that we would like to attain: to give the reader an idea of Pierre-Gilles’ contribution to science. Rather than just “an idea”, perhaps “a measure” would be better but how is it possible to gauge more than 500 original papers in more than 15 different areas from solid state physics to biology? Three aspects of his contributions will be missing though. The first one concerns PierreGilles’ impact in chemistry: he had a great admiration for the inventiveness of chemists and was able to make many relevant suggestions during his conversations with synthetic chemists. His papers on asymmetric synthesis, written to commemorate Pierre and Jacques Curie’s discovery of piezoelectricity, are in Pierre Curie’s vein: they are important for theorists but do not reflect the real influence he had on liquid crystal, polymer and colloidal chemistry. The second one concerns the impact he had on the industry. He has been the scientific advisor of companies such as General Electric, Exxon, Rhˆ one Poulenc and Rhodia. He was also on the board of directors of several other companies. I know he had a significant influence in many important decisions, but by their nature they cannot be public knowledge. The third important aspect is Pierre-Gilles’ dedication and gift for communicating his passion for science. To give an example, after being awarded his Nobel Prize, Pierre-Gilles went throughout France to high schools for several years and gave lectures on soft matter which triggered many vocations in this area of science. In the following paragraphs, I try to explain how Pierre-Gilles’ scientific life helped us narrow the scope of the present books.

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When he starts his career in 1955, Jacrot and Cribier at Saclay have just observed antiferromagnetic order with neutron scattering. The young Pierre-Gilles works out the theory, which is still a landmark nowadays. Philippe Monod tells me that Pierre-Gilles is also the first to suggest the use of neutron scattering to show the existence of vortex lattices in the superconducting Schubnikov phase. During his postdoc in Charles Kittel’s group at Berkeley in 1957, he keeps working on magnetism, in particular, on double exchange and ferromagnetic resonance in rare-earth garnets. Jacques Friedel has fully understood the scientific interest motivating all this work and attracts him to Orsay. In 1958, they write a paper together discussing resistive anomalies in magnetic metals: early “spintronics”! We are very grateful to Jacques Friedel for accepting to describe this work along with its consequences. All throughout his life, Pierre-Gilles will be eager to learn new fields. He quickly gets interested in superconductivity. He sets up a very active group where theories and experiments are simultaneously investigated. Superconductivity is a very competitive field, which blooms after the BCS theory provides the long expected microscopic understanding of the phenomenon. Here, Pierre-Gilles provides us with an original perspective. He understands that the gap between the phenomenological Landau–Ginzburg theory and the microscopic BCS theory must be bridged, which he does simultaneously with Gork’ov in Moscow. This allows him and his collaborators to discuss surface effects and vortices. With Saint James he predicts the existence of a surface critical field Hc3 , which is subsequently observed by Alexis Martinet, one of his young collaborators. He also predicts with Saint James, one year before Andreev what is now known as the Andreev–Saint James reflection: electrons impinging on a normal superconductor interface from the normal side can be reflected as holes when a Cooper pair is transmitted on the superconducting side. The experimental activity is also brilliant. Pierre-Gilles has built a very active young group world-renowned as the “Orsay superconductor group”. In 1967, he summarises his thoughts on superconductivity in the book Superconductivity in Metals and Alloys which is still a reference in the domain. It is time for him to sail towards new continents. Yet, he will always keep an eye on superconductors and superfluids, and once in a while publish a paper. In 1988, when high temperature superconductors are discovered, he proposes a mechanism based on magnetic interactions. At the same time, he attracts his friend Julien Bok to ESPCI (Ecole Sup´erieure de Physique et Chimie Industrielles), asking him to develop the activity on “High Tc s”. Julien is a brilliant proponent of the Van Hove singularity with conventional electron-phonon interactions, Pierre-Gilles is not a man of cliquishness; he is only concerned with the quest for understanding and he knows Julien will do well. It was thus natural to ask Julien Bok to write a contribution on high Tc s in this volume. He discusses the high Tc problem in the light of Pierre-Gilles’ very last paper, which he published in 2007 with Guy Deutscher who was one of his early collaborators. In turn, Guy Deutscher has played an important role in convincing the international community that the Andreev effect should be named the Saint James–Andreev effect. He was indeed a witness at Orsay when this important work was made. Guy tells us that he asked Pierre-Gilles if he wanted the effect to be called “de Gennes–Saint James–Andreev” but Pierre-Gilles declined

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the offer although he was a co-author in the first paper. Guy Deutscher’s contribution deals naturally with elementary excitations in the vicinity of a normal metal-superconducting metal contact. One cannot switch to another field of interest of Pierre-Gilles without mentioning uncharged superfluids. He contributes to the description of the dynamics of triple lines involving superfluid helium, to that of the symmetries of the helium 3 superfluid order parameter, and in these last years becomes very interested in supersolids. This leads him to describe what is, to my knowledge, the first theoretical description of the motion of a dislocation in the quantum regime. He has long discussions on supersolids with S´ebastien Balibar. In the book, we asked S´ebastien to discuss Pierre-Gilles’ suggestions together with the stateof-the-art results. In 1967, the new continent is soft matter. Pierre-Gilles starts by describing the dynamics of dilute polymer suspensions. Neutron scattering provides a check of his theory on conventional polymers. Nowadays, light scattering in dilute DNA solutions allows us to observe all predicted regimes with excellent accuracy and to confront theory with essentially no adjustable parameter. He also discusses the “coil-stretch” transition of polypeptides. In the following few years, Pierre-Gilles focuses his attention on liquid crystals. The community wavers between a completely obsolete “short range” view and a mathematically formal continuum description. In 1968, Pierre-Gilles simplifies the continuum description, showing where the relevant physics is. He explains the strong light scattering by nematics, and much of their dynamical behaviour. He draws around him again a large number of bright physicists. The “Orsay liquid crystal group” soon becomes as famous as the “Orsay superconductor group” was. One of the young experimentalists working on nematodynamics at that time is Pawel Pieranski in Etienne Guyon’s lab. We asked him to give his view on this aspect. Pierre-Gilles also used the Landau approach to describe phase transitions in liquid crystals. He is first very successful with the isotropic-nematic transition which he argues should be mean field-like. All his predictions are born out by experiment. He then shows a beautiful analogy between the nematic-smectic A transition and the normal conductor-superconductor transition. The smectic density modulation is analogous to the superconductor order parameter, the nematic director analogous to the vector potential, the dislocations analogous to vortices; gauge coupling is an expression of rotational invariance. The only difference is that there is no gauge invariance in general. Among the key predictions is Helium-like critical behaviour and the existence of a phase equivalent to the Schubnikov vortex phase. The success of the isotropic-nematic transition theory has been so impressive that laboratories all around the world jump to work on the problem. The critical behaviour turns out to be only qualitatively in agreement with the predictions. In particular, critical exponents seem to be anisotropic. It takes about fifteen years to observe the equivalent of the vortex phase and to work out a complete theory but the initial prediction is beautifully confirmed! Tom Lubensky, who coined the name Twist Grain Boundary to that phase is no doubt the physicist who contributed most to the fine understanding of the nematic-smectic A transition: his contribution is the penultimate of the first volume. Pierre-Gilles’ strong ties with liquid crystals ends soon after the publication of the book The Physics of Liquid Crystals in

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1972. As for superconductors, he keeps on following new developments in the field and from time to time introduces one more original idea like that of “artificial muscle” or fracture and spreading in smectics. Somewhat arbitrarily, we have included in the first volume a discussion on Pierre-Gilles’s impact in the physics of macroscopic random media, by Jean-Pierre Hulin, Etienne Guyon and Stephane Roux. This could have appeared in the second volume as well. Etienne, who was a member of the early superconductor and liquid crystal groups, was asked by Pierre-Gilles to reorganise the ESPCI hydrodynamics laboratory, opening its research areas to a wide variety of problems including dynamic instabilities, turbulence, magnetic fluids, wetting, and the physics of macroscopic random media. Indeed Pierre-Gilles had taken responsibility of ESPCI a few years after being appointed at the College de France. The style introduced by Etienne is in perfect harmony with Pierre-Gilles’ conception of research. Coming back to the early seventies, the years of 1971–1972 are Anno Mirabilis for PierreGilles. While making major contributions to liquid crystals, he provides fundamental input in polymer science. Improving on the tube image introduced earlier by Sam Edwards, he introduces the reptation concept which provides an elegant and powerful tool for discussing polymer dynamics. This opens the way to a large number of theoretical and experimental works which Michael Rubinstein discusses in the second volume dedicated to soft matter and biological physics. A long-standing problem is that of a self-avoiding random walk. It describes, among other things, the statistics of polymer chains in good solvents. K. Wilson has just invented a renormalization group expansion, which explains the origin of the non-trivial critical exponents characterising continuous phase transitions. He sends his preprint to friends. This is not an easy paper to read and understand. Pierre-Gilles sees right away that when the number of components of the vector is set to zero, the described situation is that of a self-avoiding random walk, a conceptual tour de force. He is so fast that his paper on the self-avoiding walk comes out before K. Wilson’s paper on the renormalisation group approach to phase transitions! Tom Witten, who has worked on the renormalisation group description of polymers during his postdoc with Pierre-Gilles, provides an analysis of this exceptional contribution to science. The renormalisation group techniques are not easy to handle: Pierre-Gilles invents the image of “blobs” which allows anyone to use the corresponding concepts without knowing it! The polymer community can now discuss very complex situations in simple terms and understand the emerging scaling laws! The price to pay is the absence of prefactors in the formulae, but the physics comes out nicely. In the second volume, Fran¸coise BrochardWyart and Karine Guevorkian illustrate the use of this beautiful tool in the context of polymers in confined geometries. Some aspects of polyelectrolytes can be discussed with scaling laws, some others like the semi-dilute polyelectrolyte solutions require keeping track of prefactors. Pierre-Gilles’ long time friend Philip Pincus discusses these cases together with Omar Saleh. In 1979, Pierre-Gilles publishes the book Scaling Concepts in Polymer Physics. Contrarily to what happened with superconductors and liquid crystals, the publication of this book does not really slow down his activity on polymers. Among some of the important areas he

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tackles a few years later are those of adhesion and friction in which polymers are essential. All the knowledge acquired on polymer dynamics proves to be important in understanding these phenomena — the use of which is ubiquitous in every day’s life. We asked Hugh Brown to expose Pierre-Gilles’ contributions. Although Pierre-Gilles keeps in touch with the development on polymers, he sails towards new continents. After the first oil crisis, oil recovery motivates strong research activity on micro-emulsions. Pierre-Gilles again contributes with a few seminal papers, introducing an important length characterising interfaces, and describing the main features of complex phase diagrams. He also becomes interested in wetting and dewetting phenomena, which play an important role in areas as diverse as textile and printing industries on the one hand, and automobile and aeronautical industries on the other. Spreading of a fluid on a wettable surface is surprisingly universal, independent of the substrate wettability. Macroscopic theories fail to explain experimental observations. Pierre-Gilles shows how the idea of a precursor film reconciles observations and theory. First, a film of microscopic thickness spreads on the substrate and then the bulk fluid spreads on the film. As always, he investigates a large number of situations: he shows the importance of van der Waals forces, discusses the spreading of polymers, smectics, magnetic fluids as well as superfluid Helium IV. He also discusses the influence of volatile impurities on the spreading velocity, the importance of the substrate topographic and chemical heterogeneities, describing in detail the case of a pinning point. Dewetting in situations mimicking either aquaplaning or printing is also investigated. A book comes naturally to summarise this activity. The French version Gouttes, bulles, perles et ondes comes out in 2002, the English version Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves in 2004. For a change, there are three authors: Pierre-Gilles de Gennes, Fran¸coise Brochard-Wyart, and David Qu´er´e. We asked Lyderic Bocquet to write the article on wetting/dewetting phenomena. Looking just at Pierre-Gilles’ scientific production, one easily overlooks the fact that he was awarded the 1991 Nobel Prize! Answering an enormous amount of mail, giving television and radios interviews, interacting with high-level politicians and visiting high schools does not really slow him down. After retiring from Coll`ege de France and ESPCI, Pierre-Gilles joins the Curie Institute and learns a great deal of biology. In his amazingly elegant style, he shows an analogy between bacterial chemotaxis and gravitational interactions! He predicts the existence of a number of original phenomena, which are currently being tested experimentally. Naturally, Pierre-Gilles is interested in cell adhesion and cell spreading. He shows that the collective dynamics is important and predicts scaling regimes. Pierre Nassoy, who collaborated on the experimental side, describes the current situation. In 2006, Pierre-Gilles gives a “biologist” series of lectures on the brain function: almost no equations, but a fascinating description of the state-of-the-art knowledge. He works on olfactive memory storage and neuronal growth. In May 2007, he is working simultaneously on superconductivity with Guy Deutscher and on the creeping behaviour of a cell subjected to a uniaxial tensile stress. He has just published a few papers on solid friction with Jacques Friedel, and the analysis of the

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quantum motion of a dislocation! On the 18th of the same month, we lose one of the brightest scientists of the second half of the twentieth century and a dear friend. Fran¸coise Brochard-Wyart, Julien Bok and myself are extremely grateful to the authors, who did not hesitate one second to accept the difficult task that we were asking for. I am personally very grateful to Fran¸coise and Julien who have been doing all of the editorial work and to Jacqueline Bouvier whose help has been essential in all aspects of the preparation of the two volumes. Jacques Prost

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ACKNOWLEDGMENTS

The editors thank the Foundation Pierre-Gilles de Gennes and ESPCI ParisTech for their financial support.

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CONTENTS

Preface

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Acknowledgments

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The n = 0 Discovery Thomas A. Witten

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P. G. de Gennes, Phys. Lett. A 38, 339–340 (1972) Exponents for the excluded volume problem as derived by the Wilson method

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Dynamics of Entangled Polymers: The Three Key Ideas Michael Rubinstein

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P. G. de Gennes, J. Chem. Phys. 55, 572–579 (1971) Reptation of a Polymer Chain in a Presence of Fixed Obstacles

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P. G. de Gennes, J. Phys. France 36, 1199–1203 (1975) Reptation of Stars

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Polyelectrolytes: The de Gennes Legacy Philip Pincus and Omar A. Saleh

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P. G. de Gennes, P. Pincus, R. M. Velasco and F. Brochard, J. de Phys. 37, 1461–1473 (1976) Remarks on polyelectrolyte conformation

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Polymers in Confined Geometries Karine Guevorkian and Francoise Brochard-Wyart

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F. Brochard and P. G. de Gennes, J. Chem. Phys. 67, 52–56 (1977) Dynamics of confined polymers chains

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Pierre-Gilles de Gennes, PNAS 96, 7262–7264 (1999) Passive entry of a DNA molecule into a small pore

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Adhesion and Friction Hugh Brown

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E. Rapha¨el and P. G. de Gennes, J. Phys. Chem. 96, 4002–4007 (1992) Rubber-rubber adhesion with connector molecules

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F. Brochard-Wyart, P. G. de Gennes, L. L´eger, Y. Marciano and E. Rapha¨el, J. Phys. Chem. 98, 9405–9410 (1994) Adhesion promoters

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An Approach to Cell Adhesion Inspired from Polymer Physics Pierre Nassoy

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Fran¸coise Brochard-Wyart and Pierre-Gilles de Gennes, C. R. Physique 4, 281–287 (2003) Unbinding of adhesive vesicles

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Spreading Made a Splash Lyd´eric Bocquet

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Pierre-Gilles de Gennes, C. R. Acad. Sci. 298, 111–115 (1984) Dynamique d’´etalement d’une goutte

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The n = 0 Discovery

Thomas A. Witten James Franck Institute, University of Chicago Chicago, Illinois 60637, USA [email protected] We describe Pierre-Gilles de Gennes’ 1972 letter explaining polymer swelling as a form of critical phenomenon. We trace the impact of this “n = 0” discovery on polymer theory and experiment. We discuss later developments in mainstream statistical physics that reflect the n = 0 insight of this paper. We collect the views of several leading statistical physicists on the significance of the discovery.

1.

Introduction

During the midwinter holidays of 1971–1972, the statistical physics community was abuzz with rumors of a breakthrough. A professor named Kenneth Wilson had announced1 a new way to understand the famous problem of critical fluctuations of certain dense gases. Under just the right conditions these critical gases have regions of high and low density. The regions range from molecular size to thousands of times larger, giving them a distinctive opalescent appearance. Wilson called his method the renormalization group. It embodied a symmetry relating the patterns of density seen at a large spatial scale to the sub-patterns within those patterns on smaller scales. This symmetry under spatial dilation had been qualitatively appreciated by others.2 However, Wilson’s method of expressing it gave unprecedented power. It allowed one to account for the peculiar measured power laws describing the growth of the fluctuations as the critical conditions of temperature and pressure were approached. Explaining these “critical exponents” was the nub of the problem of critical fluctuations. The new theory got its power by connecting the intractable critical fluctuations to a simpler regime of weak fluctuations. The new approach had a generality that went beyond critically opalescent gases. Since the anomalous power-law behavior arose from a symmetry, any system with the same form of symmetry should show the same power laws. Moreover, fluctuating quantities more general than a scalar density field could be explained. For

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example, the theory could also explain the vector fluctuations of the local magnetization in a metal on the verge of becoming a ferromagnet. Indeed, it could handle analogous fluctuations of a general vector field with an arbitrary number of components n. For the gas, n = 1; for the magnet, n = 3. The vector dimension n is often called a spin index, adopting the language of magnetic phase transitions. As these rumors about the Wilson theory were circulating, Pierre-Gilles de Gennes was on a skiing vacation. But other things were on his mind. As reported by Fran¸coise Brochard, 3 he had been in the United States the previous summer and had discussed Wilson’s ideas with physicists there. On his return to France in Autumn 1971, de Gennes had lectured on these ideas in his course at the College de France (lecture 15). He had seen a connection between Wilson’s theory and a very different puzzle — the famous “excluded volume problem” of polymer physics (lecture 16). Polymers are long chain molecules whose links twist randomly under thermal fluctuations when they are dissolved in a suitable liquid. Such a chain is a type of random walk, whose spatial extent should vary as the square root of its length. However, the measured properties of real polymers were different from this prediction. The spatial extent grew too quickly with chain length, and there was no systematic explanation for the anomalous power law describing its growth. There were only ad hoc arguments that attempted to account for the excluded volume or self avoidance of the chain. These attempts tried to extrapolate from the simple cases of one contact between parts of the chain, then two contacts, and so forth. De Gennes and others had noticed a resemblance between this counting of contacts and an analogous form of counting in a dense gas. The gas molecules — like the polymer subunits — cannot intersect one another. This collective mutual repulsion must be treated by a painstaking calculational scheme in order to avoid double counting. It was this scheme which resembled the perturbations of a polymer owing to self avoidance. Both schemes kept track of the accounting by means of diagrams representing classes of self-intersecting configurations to be corrected for. As de Gennes puzzled over the two forms of diagrams, he had noticed closer and closer correspondences, especially when he used Wilson’s style of accounting. In Wilson’s method, the same set of diagrams described fluctuating fields with arbitrary numbers of vector components n. In this scheme there was a perfect correspondence between each polymer diagram and a corresponding vector field diagram. Moreover, the mathematical expression associated with a given polymer diagram was identical to that for the vector field, provided one chose the proper value of n. What was this magical value of n? Zero!4 Now, these diagrams were the only input needed by the Wilson theory in order to determine its renormalization-group symmetry, and thence its critical exponents. The approximate formulas for these exponents were expressed as explicit functions of n. By setting n = 0 in these formulas, one was discovering information about self-avoiding polymers. Interpreting these power laws required another insight of interpretation. How could the critical exponents describing thermodynamic effects involving density and temperature apply to the purely geometric polymer properties of chain length and spatial extent? The field theory language of the Wilson theory allowed one to make the proper correspondence, by recalling the physical situations that each diagram represented. Now de Gennes could explain the

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excluded volume exponent of polymers merely by transcribing from the Wilson theory of critical phenomena. Armed with this insight, making this transcription was a simple matter. It could be explained in a 1 1/2-page paper. It was this paper that the journal Physics Letters received on January 10, 1972.5 It appeared, peppered with misprints, in the issue of February 28. With this paper two of the most important puzzles of statistical physics were recognized as aspects of a single phenomenon. Often when two very different phenomena are unified in this way, the implications are profound. De Gennes n = 0 discovery carried such implications. First, it cast important light on Wilson’s renormalization symmetry. Up to that point, renormalization symmetry had been viewed as an exclusive feature of phase transitions. As such, the notion of renormalization was entangled with the separate notion of spontaneous symmetry breaking, a necessary feature of phase transitions. The new polymer realization of renormalization symmetry disentangled these two features. Polymers were not a phase transition and had no spontaneous symmetry breaking. Thus the polymer case provided insight into the essence of renormalization symmetry. As important as this insight was for renormalization symmetry, the implications for polymer physics were no less profound. Transcribing well-developed arguments from phase transitions to their polymeric counterparts led to clear cut predictions for a range of polymer properties where previous understanding had been murky and ambiguous. In this chapter I explore the implications of the n = 0 paper. I first describe the paper itself, providing a gloss on each paragraph in turn. Then I review the impact on polymer physics. Next I review the impact on mainstream renormalization theory and statistical physics. Finally I report the reflections of several prominent physicists whose work was influenced. Since our task is to judge the importance of an idea, the outcome is necessarily subjective. The account below comes from an avowed devotee of the n = 0 paper. Other observers might relegate the n = 0 paper to a much smaller historical importance. A thoughtful weighing of the evidence might prove them right. With these caveats, I give my own subjective and tentative impressions below.

2.

Gloss

The note is titled “Exponents for the excluded volume problem as derived by the Wilson method.” Clearly this terse account is addressed to experts. The excluded volume problem is the polymer physicist’s name for explaining how the size R of a self-repelling polymer grows with its length N . This amounts to finding the exponent ν in the formula R ∼ N ν . The Wilson method refers specifically to a technique of calculating the diagrams mentioned above. Since the numerical value corresponding to a given diagram depends only on distances between selected interaction points, each diagram has a simple, explicit dependence on the dimension d of space. The critical point where the interesting behavior appears is just the point where the numerical values diverge. The technical achievement of the renormalization group is that it provides a way to make sense of the diagrams despite this divergence. Still, concrete predictions using the diagrams require definite numbers, not divergent integrals. Here a final observation by Wilson was crucial. The divergences

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went away above a special dimension: d = 4. If one pretended that the dimension of space were very close to four, one could obtain finite quantities by the technique of performing a Taylor expansion in the small parameter  ≡ 4 − d. (Thus Wilson’s famous renormalization group paper — still unpublished at the time of de Gennes’ writing — was titled “Critical exponents in 3.99 dimensions.”)1 The Wilson paper had calculated critical exponents by expanding to second order in this , and then obtaining the prediction for three-dimensional space by evaluating the expansion at  = 1. De Gennes’ two-sentence abstract reports the result of this procedure using his n = 0 realization to calculate ν. The body of the paper begins by defining the partition function needed to determine ¯ of a self-avoiding polymer. It is the number of distinct the average end-to-end separation R ¯ ∼ N ν , this self-avoiding walks of length N with ends separated by R, denoteda ΓN (R) If R Γ quantity must show it. In order to make contact with Wilson’s quantities, de Gennes immediately introduces the quantity G, a transform of Γ. The spatial variable is replaced by its Fourier transform variable k. The chain length variable is likewise replaced by its Laplace transform P , nowadays known as the monomer chemical potential. At this point, G is claimed to have a divergence as this potential approaches a certain value P C : G(0, P ) ∼ P (P − Pc )−γ . This amounts to saying ZN ≡ R ΓN (R) ∼ N γ−1 eN Pc . This kind of scaling behavior had been suspected on numerical grounds12, 13 but not established. Thus this claim is justified by the arguments to follow. A second claim follows, regarding the dependence of G(k, P ) on k when P takes its critical value Pc . G is said to vary as a power of k involving a second exponent denoted η. The justification for this claim is also found in the arguments to follow. Now the desired ¯ of the chain is claimed to be related to the γ and η just exponent ν governing the size R defined. The “usual scaling arguments” are invoked to justify it. These arguments amount to saying that G(k, P ) has a fixed functional form as ∆P ≡ P − Pc → 0. The fixed functional form means that the only difference between G(k, ∆P ) and the same quantity evaluated λ times farther from the critical point is a difference of scale factors in the k and G dependence: G(k, ∆P ) = µG G(µk k, λ(∆P )), where µG (λ) and µk (λ) are the scale factors needed to reflect the expansion of ∆P by a factor λ. This homogeneity or scaling law connects the claimed behavior of G for k = 0 to that for ∆P = 0. From the k = 0 behavior we infer. µG ∝ λγ , (by taking λ = 1/∆P ). From the ∆P = 0 behavior, we infer µG ∝ µk 2−η . Thus µk ∝ λγ/(2−η) . That is, a change in ∆p (or inverse chain length) by a factor λ is equivalent to a change in k (or inverse spatial distance) by a factor λγ/(2−η) . This is the content of the statementb ν = γ/(2 − η) in de Gennes’ note. Such “usual scaling arguments” were common and accepted in the phase transition literature, 11 but not among polymer physicists.12 The reason for this difference may be experimental. For phase transitions, experimental support for these scaling arguments in the vicinity of the critical a Here b Here

an apparent misprint was corrected. an apparent misprint was corrected.

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temperature Tc spanned 4 to 5 decades in T − Tc in 1971.11 The corresponding range of N in polymer systems was scarcely two decades, and was subject to technical uncertainties in determining the chain length N . Thus, de Gennes’ invocation of the “usual scaling arguments” rested chiefly on the correspondence to phase transitions that was to be argued in the sequel. Shortly de Gennes will describe the diagrammatic correspondence between the polymer problem and the phase transition problem. The structure of the polymer diagrams, which accounted for a self-repulsion between monomers with a strength v0 had been articulated e.g. by Fixman.14 The n = 0 correspondence with phase-transition diagrams is to be asserted below. But first de Gennes makes a digression to comment on the “renormalized repulsion” denoted vR . The digression is aimed at explaining a widely believed property of polymers in Wilson’s renormalized language. The strength of the repulsion v0 was believed not to affect the value of the exponent ν. This fact is naturally explained if the diagrams can be cast into a form in which v0 is replaced by a fixed quantity. The renormalization procedure aims to perform this recasting. Renormalization aims to describe the calculation of a quantity such as G(k, P ) by replacing the original lattice degrees of freedom such as monomer density by local averages. One may account for the overall effects of self-avoidance using only these local averages. Moreover, the locally averaged calculation has the same form as the original one, except that the input parameters such as v0 have different values. The quantity playing the role of v0 in the locally averaged calculation is denoted vR . One may determine vR from v0 for a given type of averaging. On the other hand, one may use the anticipated scaling dependence of the calculated quantity e.g. G ∼ k −2+η to determine how vR must vary with the averaging length scale. Equation (3) is an expression for vR of this form. It does not refer to an averaging length scale directly; rather, it uses P − Pc the monomer chemical potential that would generate polymers of the desired scale. The aim of this equation is to show that vR reaches a fixed value. In the following paragraph de Gennes sketches a way to reach the same conclusion using a variant of the polymer problem in which the polymer is confined to a finite volume with a fixed, small, average monomer density. Both of these arguments about vR are sketchy and obscure. In any case they address a side issue. They aim to justify a point that will be better justified by the simple observation in the following paragraph. The next paragraph gives the central observation of the paper: the G function for polymers is a special case of a corresponding G function for phase transitions. The argument is based on the perturbation diagrams described above. Figure 1 shows a typical diagram for the counterpart of G for an n-vector field. Each line in the diagram accounts for the correlations of the field at the two end points of the line. The lines terminate at interaction points shown as heavy rectangles. The field interacts with itself at every point; the statistical weight of the field at any point differs from what it would be without the interaction. This altered statistical weight influences the field elsewhere because of its correlations, in particular, the field at other interaction points is altered. By allowing the interaction points to be at arbitrary positions one accounts for part of the effects of the interaction on G. This diagram represent a mathematical expression that contributes to the Taylor expansion for

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Fig. 1. Top: Perturbation diagram for the n-vector field shown in two equivalent drawings. Meaning of the lines and rectangles is explained in the text. Light lines denote loops. Bottom: Perturbation diagram for a self-repelling polymer, also shown by two equivalent drawings. The lower pair is obtained from the upper pair by removing the loops.

G in powers of v0 . Since there are five interaction points, this diagram is part of the v0 5 term in this expansion. By connecting the interaction points in all possible ways and adding the corresponding mathematical expressions together, one obtains the full v0 5 contribution. One feature of the illustrated diagram is the appearance of the two loop lines shown as narrow lines. These lines indicate field correlations that pass from one interaction point to another and eventually return to the original point. The light and dark lines have the same mathematical meaning. Such a loop disturbance can occur for any of the n degrees of freedom of the vector field. Thus the overall disturbance carried by such a loop is proportional to n. A factor n for this loop must appear in the mathematical expression for the diagram. The same is true for both loops in this diagram and for all other loops in other diagrams. Now we consider a polymer diagram for the G of Eq. (1). Its interpretation is simpler. The thick line represents the polymer, following some arbitrary path through space. As pictured in the diagram this polymer intersects itself at two points indicated by black rectangles. The statistical weight of such paths is altered by the self repulsion of strength v 0 . This diagram, with its two interaction points, thus contributes to the power series expansion in v0 in order v0 2 . We now observe that the polymer diagram may be obtained from the n-vector field diagram by simply removing the two loops. This fact, illustrated with this diagram, is true in general. If one considers an arbitrary n-vector diagram and removes the loops, one obtains a polymer diagram. Conversely, every polymer diagram is contained (exactly once) within the set of loop-removed n-vector diagrams. As noted above, the mathematical expression corresponding to the polymer diagram is identical to that for the n-vector diagram. This equivalence arises partly from a fundamental similarity between the two systems. In the polymer diagrams, the lines between interaction points represent random walks (segments of noninteracting polymer) between the points. These random walks are equivalent to a diffusive motion, as Einstein noted in his famous paper relating Brownian motion and diffusion.6 In the n-vector diagrams the corresponding line represents spatial correlations between the fluctuating vector field at two different points. In effect, these correlations

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also arise by diffusion: a statistical weight tending to make the field at a point equal to the average of the field at neighboring points. The equivalence between the polymer and the n-vector diagrammatic expressions also required some mathematical jiggering. The quantity G constructed in de Gennes’ Eq. (1) has been devised to make the mathematical correspondence exact. Once one knows the mathematical expression for the n-vector diagrams, it is absurdly simple to extract the expression for the polymer diagrams. This simplicity arises from our observation above that each loop in an n-vector diagram contributes a factor n to the mathematical expression for that diagram. We have noted that the sum of contributions from all the (infinite number of) diagrams completely accounts for the effects of the interaction v 0 on G. Once one has the expression encompassing all n-vector diagrams, the corresponding expression encompassing all loopless diagrams is thus obtained by simply setting n = 0 in the n-vector expression. Thus the G function for polymers is a mere special case of the G for the n-vector field. Naturally this equivalence extends to the critical exponents of the n-vector model. De Gennes’ paragraph points out this diagrammatic correspondence, without pictures, in a way that an expert in these diagrams can follow. The next step is to make use of the correspondence. He quotes Wilson’s calculation for the exponents γ and η expanded to next leading order in the the difference of spatial dimensionality d from 4. He evaluates these expressions for n = 0 and d = 3, then uses the exponent relations cited above to get the desired exponent ν. This ν is then compared with the best available value and observed to be in good agreement. He also compares with two theoretical hypotheses which, like the Wilson values, can be evaluated for general dimension d. He notes that their expansions in d − 4 do not agree with the Wilson counterparts. Though de Gennes cites a precise value of ν and compares it closely with other values, one should note that the d − 4 expansion provides only ambiguous information about the three-dimensional world. For example, if one expands 2ν = γ/(2 − η) to order (4 − d) 2 , and evaluates it for d = 3, one obtains 2ν = 1.1835 . . ., rather than de Gennes’ value of 1.195. By using such schemes, one could obtain a wide range of estimates for ν. The Wilson prescription does not provide a way to choose among such alternative estimates. The claims of this brief paper seem to have been immediately accepted by the statistical physics community. The n = 0 finding is reported as established in the Cargese lectures on renormalization group7 in 1973 and in the review article by Wilson and Kogut8 in 1974.

3. 3.1.

Polymer Impact Antecedents

The excluded volume problem addressed in Ref. 5 is at the heart of much of polymer physics. Any property related to a concentration or molecular weight dependence in a good solvent necessarily involves the exponent ν cited above. In order to deal with these properties decisively, one must thus have a framework for understanding this ν and hence the excluded volume problem.

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The importance of the excluded volume problem had been recognized since early 20th century, when it was first established that polymers were flexible chains9 of many independent statistical segments. By mid-century this common feature of polymers had given rise to a powerful body of theory led by Paul Flory.10 Flory devised a simple and appealing explanation of why self-repulsion or excluded volume causes a polymer’s size R to grow as its molecular weight M to a power ν greater than 1/2. The Flory theory predicted ν = 3/(d+2) in d dimensions, and this value was completely consistent with experiments. The explanation was deemed adequate by those who worked with polymers daily. Yet efforts 12, 14 to justify its assumptions and develop the idea into a systematic theory remained at the level of ad hoc attempts. Meanwhile, the analogous statistical problem of the self-avoiding walk on a lattice was recognized as fundamental. Statistical theorists like Fisher15 and Domb16 were devising systematic ways to infer the asymptotic functional form of R(N ) and the number of walks Z N of length N . One approach was to enumerate all self-avoiding walks up to a given size, and then examine the consistency of these exact results with a given asymptotic form. A second approach was to view the self-avoidance constraint as a perturbation on the ensemble of unrestricted random walks. Since the large-M behavior of unrestricted walks was completely understood, the limitations of the small-M enumerations were removed. One could then calculate the effects of this perturbation systematically by assigning a statistical penalty v 0 for each intersection and then performing an expansion in powers of v0 .14 This approach leads to the diagrams discussed in the last section. However, the limit of interest is one where the unrestricted walks have many self-intersections and hence many complicated diagrams. Thus the potential of this method seemed very limited. Only for dense and strongly interpenetrating polymer solutions could these diagrams be reduced to a tractable form,17 in this concentrated regime all signs of excluded volume swelling had vanished. Yamakawa’s18 widely-read 1971 monograph reviewed these different approaches. It was clear that understanding of the excluded-volume effect was fragmentary and unsatisfactory. 3.2.

Early impact

Almost immediately after the n = 0 discovery, de Gennes’ colleague Des Cloizeaux proved an important generalization.19 The n = 0 paper concerns the critical state of a gas or magnet at a temperature Tc at which a symmetry begins to be spontaneously broken. Here the spatial correlations grow to infinity in their range. For systems with n > 1, a form of long-range correlation persists in the symmetry-broken, magnetized state below Tc . Spontaneously broken symmetry means that all directions of magnetization are equal in energy. Thus the magnetization is free to rotate throughout the sample with no energy cost. This freedom of rotation is known as a Goldstone mode. It amounts to an infinite range correlation. This range is only made finite by applying a symmetry-breaking magnetic field. Remarkably, an analogous phenomenon occurs in the polymer domain. To find the analogy, one has only to ask what the perturbation diagrams for T < Tc look like when n = 0. The diagrams consist of many polymer lines interacting with each other. The density of ends is proportional to the applied magnetic field. Evidently this is equivalent to the concentration of polymer chains. When this end concentration is low for a given

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monomer concentration, the polymers must be long. As the magnetic field goes to zero, the length and size of these polymers evidently diverge. This is the polymer analog of the divergent correlation length below Tc for a magnet. Again, a complicated system of fluctuating spins has been replaced by a prosaic liquid containing flexible chain molecules. A concrete interpretation of these so-called semi-dilute solutions soon emerged in the neutron scattering group connected with de Gennes. The solution could be viewed as a dense packing of self-avoiding “blobs” linked end-to-end to form chains. These chains should show no self-avoidance beyond the scale of a blob. This picture led to several direct predictions for how the scattered neutron intensity should depend on wave-vector, chain length and concentration. Soon after the n = 0 paper, these predictions were satisfyingly confirmed. 20 A year before the n = 0 paper de Gennes had conceived the notion of reptation to describe the kinetics of relaxation of an entangled polymer solution.21 The notion of blobs complemented the reptation idea and extended its predictive power. 3.3.

Further implications

Theoretical reflection turned up further striking analogies. One was an extension of the critical amplitude ratios seen in phase transitions. For example one may observe the divergence of the correlation length ξ by approaching the critical temperature Tc from either above or below. The rate of divergence is governed by the same critical exponent ν in both cases. Thus the ratio of the two lengths ξ(Tc + ∆T )/ξ(Tc − ∆T ) must remain finite as ∆T → 0. It was observed that this ratio took the same value for all critical points of the same type. The Wilson theory justified this similarity. Like the critical exponents, these amplitude ratios were calculable by the renormalized theory, which was independent of the specifics of the critical system being studied. Similar universal ratios should exist in polymer solutions. The most accessible ones were not comparisons between the quantities above and below T c . But they did preserve the general feature of being ratios of diverging quantities known to diverge with the same power.22 A variant of critical phase transitions proved to have a striking polymer analog. It can happen that magnetism can set in at the surface of a metal slightly before magnetism appears in the bulk. The correlations (the lines in the perturbation diagrams) have altered mathematical expressions in the vicinity of the surface, owing to the boundary conditions it imposes on the field. The result is a critical point with a new renormalization symmetry, known as the “special critical point.” By examining the corresponding diagrams for n = 0 one can divine the polymer analog. Polymers are weakly attracted to the surface of the solution — so weakly that the adsorbed polymers extend arbitrarily far into the solution. The new exponents appear in the form of the concentration profile.23 Again an esoteric phenomenon in the domain of phase transitions proves to have a prosaic and accessible polymer analog. Soon polymer theorists found further variants. Further new exponents arose when the polymers were adsorbed in a wedge or a corner.24 Without motivation from phase transitions, polymer theorists sought to find how exponents such as ν would be influenced by changing the architecture of the polymers. The chemists could make star-shaped polymers with a desired number f of arms. They could also make loops. These each showed instructive behavior when one examined the quantity

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ZN giving the number of configurations. As noted above, this ZN for an ordinary polymer is governed by the exponent γ. However, for loops, one need only know the exponent ν governing the size. For the stars, the exponent γf depended on the number of arms f in a nontrivial way. These exponents γf are experimentally relevant. They dictate the thermodynamic work required to assemble the star from separated arms. Analogous quantities in phase transitions were soon found. By comparing the diagrams, one could identify the γ f exponents as “anomalous dimensions of composite operators.”7 visible only by evaluating multi-point correlation functions in special limits. In the polymer case, all this subtlety was sidestepped simply by synthesizing a polymer with the proper architecture. Like polymers, diffusing particles are random walks. Thus it is easy to represent the interaction between diffusing particles and polymers by simple modifications of the n = 0 diagrams. One had only to remove the self-interaction lines from the diagram lines representing the diffusing particles.25 If f of these particles meet the polymer at a point, one may again ask how the number of configurations varies with f . The corresponding exponents γ˜f describe the interaction of a self-avoiding polymer that absorbs a diffusing substance. These γ˜f exponents again prove to have a nontrivial dependence on f . This fact had conceptual importance. It showed that the probability distribution of absorption on a dilation-symmetric structure like a polymer is a multifractal or fractal measure.26 These fractal measures had been postulated in other phenomena, such as the distribution of vorticity in a highly turbulent fluid. A fractal measure has a complicated form of dilation symmetry that requires a one-index set of critical exponents to characterize it. The polymer absorber example showed in a straightforward way how fractal measures were a natural consequence of diffusion onto a dilation-symmetric object. By the early nineties, the methodology of renormalization was fully accepted by polymer theorists as the way to treat excluded volume effects. It had grown beyond its origins in phase transition theory as an independent realization of the basic renormalization ideas. 27 The methodology was explained in textbooks and monographs by polymer theorists. 28–30 Polymer theorists could readily extract the polymer implications from discoveries in the phase-transition domain. They could also generalize the formalism to discover new forms of dilation symmetry in polymer phenomena with no obvious counterpart in phase transitions. An example is the new exponent governing the demixing of immiscible polymers in a common solvent.31 The n = 0 paper opened the door to this culmination.

4.

Field Theory Implications: A Paradoxical Limit

The aim of the n = 0 paper was to address polymer phenomena. Still, the n = 0 discovery inevitably had an influence on the statistical field theory that made the discovery possible. Some of this influence has been indicated above. The polymeric realization of renormalization symmetry sharpened understanding of this symmetry by decoupling it from the domain of phase transitions and spontaneously broken symmetry. It added a new level of concreteness to our idea of what it means for a system to embody renormalization symmetry. The polymer aspect of phenomena like composite operators enabled insight into field theory that would otherwise have been more difficult.32 The polymer example was the first instance of

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self-organized criticality33 where dilation symmetry is attained without the need to tune a parameter like temperature to its critical value. In this section we deal with a deeper implication of n = 0. The idea of n = 0 forces one to embrace a paradox. How could a vector field with zero components represent anything but empty space? How could one imagine a broken-symmetry field with minus one Goldstone modes? The perturbation diagrams give a perfectly unambiguous answer to these questions. But since the diagrammatic correspondence is complete, n = 0 represents a polymer beyond the confines of any calculational technique. The passage to the paradoxical state reveals a completely new kind of object. The n = 0 paper does not dwell on this paradox. Still, it establishes a clear meaning for a state where no meaning would have been thought possible beforehand. Since the n = 0 paper, several other n = 0 states have been discovered. Like the polymer case, these states, when viewed naively, represent nothing but empty space. However, when they are treated as de Gennes treated the polymer case, they too reveal surprising states. The simplest case was the generalization from de Gennes’ self-avoiding curves to arbitrary self-avoiding lattice clusters, the so-called lattice animals. In 1975 Lubensky 34 found that these animals emerge from another field theory in just the way that polymers emerge from the conventional critical field theory. His discovery established the equivalence between lattice animals and the randomly-branched polymers long considered by the polymer community.35 The broken symmetry phase of this theory represented another object important in stochastic geometrical objects: the spanning clusters of a randomly-filled lattice at its percolation threshold. Following the polymeric counterpart of this regime led to striking experimental discoveries about the how branched polymers interpenetrate and compress one another.36 Striking differences from linear polymers were thus revealed. A second and more pervasive instance of n = 0 is Edwards’ replica trick.37 It was invented to represent fields such as the vector fields above in the presence of a frozen, disordered environment. One must account for the frozen randomness of the environment differently from the fluctuating randomness of the fields. The replica trick is a way of accounting for the frozen randomness. One imagines n copies or replicas of the vector field, all having the same state of the random environment. Next one allows the random variables representing the environment to fluctuate as the field variables do. With the frozen variables removed, one may calculate the properties of the n-replica state using ordinary methods of statistical physics. Finally, to represent the frozen randomness, one takes the limit in which the number of replicas goes to zero. Edwards had certainly thought about de Gennes’ n = 0 paper before he formulated the replica trick.4 However, there is no evidence that the two discoveries were causally related. The replica trick reveals an n = 0 state, like the cases above, with no apparent relation to the n 6= 0 states that gave rise to it. Here the limit reveals not a new geometry but a new state of randomness. As with the cases above, the ordered phase yields additional insight. By construction all the replicas are equivalent, yet at low temperature the replicas may become spontaneously different. This spontaneously broken replica symmetry 38 persists to n = 0 and represents an important transition in the state of frozen disorder.

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5.

Assessment

The ultimate measure of the importance of an idea is its impact on people’s thinking. How would physicists have thought differently about phase transitions and polymers if the n = 0 paper had not been written? Potential answers to this question range widely. At one extreme, one might view the discovery as a simple bookkeeping trick, like the Dyson formula for summing repeated insertions into a diagram.39 Such a trick would likely have been quickly discovered elsewhere if the original discovery had not been reported. At the other extreme are profound notions such as the duality linking high temperature and low temperature behavior of a phase transition system, discovered by Kramers and Wannier in 1941.40 No student first learning of duality can avoid a sense of wonder that interacting fields can embody such a unifying symmetry. In order to gauge the effect of the n = 0 paper on people’s thinking, the author sought the opinion of several experts within the polymer field and outside it. These scientists gave their permission for their remarks to be used in this chapter. Still, they should not be held responsible for the author’s misinterpretations of their statements. Sir Sam Edwards’ impact spans both polymer physics and statistical field theory. His development of the diagrammatic description set the stage for de Gennes’ insight. He is one of the fathers of the paradigmatic state of frozen disorder: the spin glass.37 In his reply to the author’s query, he focussed on the central problem of computing the exponent ν. He summarized the various ad hoc theories of ν such as Flory’s and noted that he had found a more systematic argument to justify the Flory result. He recalls the series expansion methods noted above. Then he comments on the n = 0 paper: “The paper of dG left me puzzled. Clearly the mean field approach did not give correct indices even though it was pretty accurate so for complex problems one had to work with methods like mean field but there seemed no way of getting that last bit of accuracy. RG [the Renormalization Group] is hopelessly too complicated for real life problems even though, as in Karl [Freed]’s book,28 it offers a rigorous basis for polymer theory. My own view is [that] this whole area is overtaken by the work of Baumgartner and Muthukumar,41 who showed that one can write a program that allows the series expansion to be computerized and the (divergent) series can be summed by Pade summation easily and quickly. It is enormously easier than RG.”

Edwards addresses the n = 0 paper on its own terms, as a means to determine the numerical value of ν. He notes that if this is the goal, the most efficient means need not be based on the renormalization symmetry that ν represents. Edwards views the paper as a clever idea to address an academic issue about the means to achieve arbitrary accuracy for a fundamental number that was well-enough known for practical purposes. The implications of deeper understanding argued above were not the point of the n = 0 paper. Giorgio Parisi is another giant of statistical physics. He was an early participant in developing the renormalization group for phase transitions.7 He co-discovered the notion of replica symmetry breaking in the spin glass.38 Parisi writes: “I do not remember exactly when I learned the de Gennes paper. Quite soon, I guess, because it was used in the framework of the epsilon [4 − d] expansion

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to confirm the prediction for the critical exponents. I think that this paper has played a very important role and opened the way to others people to make analytic continuation in n. I think that also the formalism for localization originates for that paper. I do not know how much Edwards and Anderson37 have been influenced by that paper. There was also a paper by Brout in the 50’s42 [titled] “Statistical Mechanical Theory of a Random Ferromagnetic System,” . . . but I do not know if it has influenced the developing of the replica method. Certainly I arrived to spin glasses because I wanted to understand better the n = 0 limit for branched polymers that was developed by Lubensky (I guess).”

Parisi goes farther than Edwards in acknowledging a conceptual advance in the n = 0 paper. Yet it does not seem to have exerted a pivotal influence on his own thinking. Instead, he cites the more direct influence of Lubensky’s use of n = 0 in his treatment of branched polymers, summarized above. This impact was evidently strong. It led from a polymeric form of n = 0 to its replica realization to implement frozen randomness. Parisi also raises the possibility of previous awareness of the replica trick in Brout’s early paper. (The author has not investigated this possibility.) Tom Lubensky, the inspiration for Parisi’s work, offered his own views about the importance of the n = 0 paper. As noted above, Lubensky showed that branched polymers, random lattice animals and percolation clusters are realizations of renormalization symmetry, using an n = 0 representation. He is the coauthor of the leading textbook about soft matter statistical physics.43 Lubensky writes: “I don’t remember the exact moment I heard about the n = 0 paper. I believe I saw it in preprint form before it was published. I found it to be a major discovery — so simple yet so powerful. It opened up the statistics of polymers to the entire arsenal of renormalization group techniques. In the end, other RG techniques not relying on the n = 0 trick produced equivalent results in a more physical context, but the n = 0 idea showed that all of this was possible. This paper and the subsequent ones by Des Cloizeaux and by de Gennes on adding an external field to treat semi-dilute systems did have a great impact on my thinking though not immediately. In the late 70’s I extended the n = 0 ideas to branched polymers, where the number of components of the field was merely the fugacity for polymer number, so n → 0 allowed me to develop field theory techniques to study the statistics of individual branched polymers and lattice animals. The de Gennes work also had an impact on my thinking about percolation and the statistics of clusters near the percolation threshold. I think that the n = 0 approach has fallen from favor. The modern polymer focus more on scaling and the Direct RG approach (developed by Des Cloizeaux if memory serves me) and discussed in de Gennes’ book.44 The original concept, however, stands as a very important idea that brought modern RG ideas to polymers.”

Lubensky accords the n = 0 paper a fundamental role. He views it as leading the way to understand renormalization symmetry on a deeper and more general basis than was previously seen via phase transitions. He calls this “direct renormalization,” and notes that it is no longer dependent on the phase-transition context that gave birth to it. Thus the n = 0 connection is no longer of great relevance for calculations of polymers. In this sense it has “fallen out of favor.” The cross-fertilization between polymer and phase transition

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phenomena is no longer primary in his view. Still its fundamental significance of unifying two dissimilar forms of dilation symmetry is acknowledged. The author queried Elihu Abrahams, a father of the theory of waves in disordered media.45 The purpose was to judge the impact of the n = 0 paper on this domain. Abrahams writes: . . . “I would not say that the paper (or rather, note) affected my own work, since I was interested in an entirely different sector of critical phenomena. On the other hand, I think it does represent a substantial contribution, when one recognizes how influential it was on the work of lots of other people.”

Bernard Nienhuis is another father of modern statistical physics. The power of dilation symmetry has proven much greater in two dimensions than in three, and much of this power was brought to light by Nienhuis.46 Nienhuis writes: “I remember precisely how I learned of this result. It was quoted and used by H. Hilhorst in a lecture, in 1976. I was very pleased to learn it. Not many years later I heard de Gennes speak about the subject in Vlieland, a small island of the Netherlands, at national theoretical physics meeting. Indeed it influenced my thinking. Especially the usefulness of generalizing models and parameters beyond their original definition. Later I worked with the O(n) [n-vector] model including the limit n → 0, in a paper with Domany Mukamel and Schwimmer (Nucl Phys B190). Soon it enabled me to calculate critical exponents of the O(n) model in two dimensions (PRL 49, 1982). I think de Gennes indeed had an essential influence on my thinking, and his work attracted me to the subject of polymers. Concerning the question if this was a unexpected breakthrough or a follow up on earlier work, I tend to lean to the latter opinion. I think de Gennes’ major contribution was that he understood and used the full consequences of the connection, and developed it into a complete scaling theory.”

Nienhuis acknowledges de Gennes’ influence and the power of the n = 0 thinking to draw new implications from field theory. He views the n = 0 finding as one element of an extended process rather than a pivotal discovery. At the time of the n = 0 paper Russia was an active center for polymer field theory, as it is today. The author consulted Alexander Grosberg to learn of its impact there. Grosberg went on to co-author a highly respected monograph on polymer statistical physics. 29 Grosberg writes: “I was a student in 1972, and I could understand nothing in this n = 0 paper until after I learned a little bit of Wilson concurrently with de Gennes. Ilya M. Lifshitz, my teacher, was busy at the time working on the collapsed state of chains, on globules. When the strength of de Gennes approach became evident, there was a huge excitement about the fact that de Gennes approach to what happens above theta-point and Lifshitz approach to what happens below are complementary and together cover the whole range in a way that was seen as a uniformly solid physics (not solid state physics, but solidly built physics). In the subsequent years, there was a huge effort in Moscow to learn the polymer papers by de Gennes and his group. Special seminar was organized, chaired by Lifshitz and working every Tuesday for several years. A significant fraction of the seminar was about reviewing and discussing the French group’s papers. De Gennes’ book44

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was translated and published in Russian very soon after the original. In this sense, n = 0 and subsequent work very much shaped our thinking exactly along the lines of complementarity between this approach to good solvent and Lifshitz approach to the bad solvent. This is how our book with Khokhlov 29 is written.”

Grosberg and his colleagues clearly found the n = 0 notion and its implications exciting. For them it was an integral part of the “scaling concepts”44 that gave de Gennes work on polymers their major impact. The author himself was greatly influenced by the paper: “Like several of the others quoted above, I learned of the n = 0 discovery secondhand. In this period I was learning about the renormalization group as a postdoc. I viewed the n = 0 discovery as a profound realization. On the one hand, it meant that all the powerful results from phase transition renormalization were immediately applicable to the polymer domain, where understanding was much more primitive. On the other hand, the polymer interpretation of the diagrams revealed the renormalization symmetry as a phenomenon of striking and concrete simplicity. It was much more accessible than its phase transition counterpart. These implications were clear from the statement of the n = 0 correspondence, even though none of these implications were stated in the paper. The implications that this paper brought to light were the seed leading to the advances much celebrated above.”

6.

Conclusion

Almost forty years have passed since the n = 0 paper of Ref. 5 was published. The strong activity of exploring the implications of anomalous scaling properties like the exponent ν have largely subsided. The main activity in polymers concerns dense fluids and solids where these good-solvent scaling properties are of little importance. Likewise the search for analogous “anomalous scaling” phenomena in the domain of phase transitions has subsided. How should we interpret this decline in activity? Were the n = 0 paper and those that followed it a distraction from the problems of ongoing importance? Another interpretation seems more likely. These papers about excluded volume effects, both theoretical and experimental, transformed our understanding of this issue. What had been a central puzzle of polymers is now a settled problem. We now have the means to understand why exponents such as ν appear and to calculate these exponents as accurately as needed. We know how to devise well-founded scaling laws that show the consequences of these exponents for measured properties. Finally, we know how to recognize combinations of measured quantities that must approach universal, system-independent values for sufficiently long chains. For polymer physics the n = 0 paper revealed the key idea that enabled this framework of understanding to be built for polymers. But the benefits of the paper do not stop there. The recognition that polymers are a realization of the renormalization group gave profound insight into the nature of renormalization symmetry. Using the polymer example, the process of establishing the dilation symmetry of a system via renormalization was shown to be simpler and more direct than in the phase transition phenomena where renormalization was invented. One could infer the symmetry merely by examining a single long polymer

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configuration. As noted above, the scope of dilation symmetric phenomena has expanded greatly since that time. Stochastic geometric objects like random animals, disordered quantum states, stochastic growth processes and chaotic dynamics have all shown dilation symmetry. For all such phenomena, the polymer example served as an early inspiration and continues to serve as one of the simplest paradigms of nontrivial dilation symmetry.

References 1. K. G. Wilson and M. E. Fisher, Critical exponents in 3.99 dimensions, Phys. Rev. Lett. 28, 240 (1972). 2. L. Kadanoff, Scaling laws for Ising models near Tc , Phys. 2, 263 (1966). 3. Fran¸coise Brochard, private communication. 4. S. F. Edwards (private communication) recalls that de Gennes had recognized the n = 0 connection years before submitting the n = 0 paper. He had seen that the n-vector diagrams for n = 0 were those needed to account for polymer excluded volume. Such diagrams for general n had been known before Wilson used them to calculate exponents. However de Gennes had not published this observation since at that time he saw no concrete use to be made of it. 5. P. G. de Gennes, Exponents for excluded volume problem as derived by Wilson method, Phys. Lett. A 38, 339 (1972). 6. A. Einstein, Ann. Physik 17, 549 (1905); 19, 371 (1906). 7. E. Br´ezin and J. M. Charap (eds.), Carg`ese Lectures in Field Theory and Critical Phenomena (Gordon and Breach, New York, 1975). 8. K. G. Wilson and J. Kogut, The renormalization group and the epsilon expansion, Phys. Repts. 12, 75–200 (1974). ¨ 9. H. Staudinger, Uber Polymerisation, Berich. Deut. Chem. Ges. 53, 1073 (1920). 10. Paul J. Flory, Statistical Mechanics of Chain Molecules (Interscience, New York, 1969). 11. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomona (Clarendon Press, Oxford, 1971). 12. H. Yamakawa et al., J. Chem. Phys. 45, 1938 (1966). 13. F. T. Wall and J. J. Erpenbeck, J. Chem. Phys. 30, 634 (1959). 14. M. Fixman, J. Chem. Phys. 23, 1657 (1955). 15. M. E. Fisher and M. F. Sykes, Excluded-Volume Problem and the Ising Model of Ferromagnetism, Phys. Rev. 114, 45 (1959). 16. C. Domb and M. F. Sykes, Use of series expansions for Ising model susceptibility and excluded volume problem, J. Math. Phys. 2, 63 (1961). 17. S. F. Edwards, Statistical mechanics of polymers with excluded volume, Proc. Phys. Soc. London 85, 613 (1965). 18. H. Yamakawa, Modern Theory of Polymer Solutions (Harper and Row, New York, 1971). 19. J. des Cloizeaux, Lagrangian theory of polymer-solutions at intermediate concentrations, J. de Phys. 36, 281 (1975). 20. M. Daoud, J. P. Cotton, B. Farnoux, G. Jannink, G. Sarma, H. Benoit, R. Duplessix, C. Picot and P. G. de Gennes, Solutions of flexible polymers — neutron experiments and interpretation, Macromolecules 8, 804 (1975). 21. P. G. de Gennes, Reptation of a polymer chain in presence of fixed obstacles, J. Chem. Phys. 55, 572 (1971). 22. T. A. Witten and L. Sch¨ afer, Two critical ratios in polymer solutions, J. Phys. A 11, 1843 (1978). 23. E. Eisenriegler, K. Kremer and K. Binder, Adsorption of polymer-chains at surfaces: scaling and Monte-Carlo analyses, J. Chem. Phys 77, 6296 (1982).

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24. Z. G. Wang, A. M. Nemirovsky and K. F. Freed, Polymers with excluded volume in various geometries — renormalization-group methods, J. Chem. Phys. 86, 4266 (1987). 25. M. E. Cates and T. A. Witten, A family of exponents for Laplace’s equation near a Polymer, Phys. Rev. Lett. 56, 2497 (1986). 26. T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia and B. I. Shraiman, Fractal measures and their singularities — the characterization of strange sets, Phys. Rev. A 33, 1141 (1986). 27. J. des Cloizeaux, Polymers in solutions — principles and applications of a direct renormalization method, J. de Phys. 42, 635 (1981). 28. K. F. Freed, Renormalization Group Theory of Macromolecules (Wiley, New York, 1987). 29. A. Yu. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules (AIP Press, New York, 1994). 30. L. Sch¨ afer, Excluded Volume Effects in Polymer Solutions as Explained by the Renormalization Group (Springer, New York, 1999). 31. J. F. Joanny, L. Leibler and R. Ball, Is chemical mismatch important in polymer-solutions? J. Chem. Phys. 81, 4640 (1984). 32. B. Duplantier, Polymer network of fixed topology — renormalization, exact critical exponent gamma in 2 dimensions, and d = 4-epsilon, Phys. Rev. Lett. 57, 941 (1986). 33. P. Bak, C. Tang and K. Wiesenfeld, Self-organized criticality, Phys. Rev. A 38, 364 (1988). 34. T. C. Lubensky, Critical properties of random-spin models from epsilon expansion, Phys. Rev. B 11, 3573 (1975); T. C. Lubensky and J. Isaacson, Statistics of lattice animals and dilute branched polymers, Phys. Rev. A 20, 2130 (1979). 35. W. H. Stockmayer, Theory of molecular size distribution and gel formation in branched-chain polymers, J. Chem. Phys. 11, 45 (1943). 36. M. Daoud, F. Family and G. Jannink, Dilution and polydispersity in branched polymers, J. de Phys. Lett. 45, L199 (1984). 37. S. F. Edwards and P. W. Anderson, Theory of spin glasses, J. Phys. F 5, 965 (1975). 38. M. Mezard, G. Parisi, N. Sourlas, G. Toulouse and M. Virasoro, Replica symmetry-breaking and the nature of the spin-glass phase, J. de Phys. 45, 843 (1984). 39. L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962). 40. R. Savit, Duality in field-theory and statistical systems, Rev. Mod. Phys. 52, 453 (1980). 41. The author could not find a reference to this work. 42. R. Brout, Statistical mechanical theory of a random ferromagnetic system, Phys. Rev. 115, 824 (1959). 43. P. Chaikin and T. Lubensky, Principles of Condensed Matter Physics (Cambridge Press, Cambridge, 1995). 44. P.-G. de Gennes, Scaling Concepts in Polymer Physics (Cornell Univ. Press, Ithaca NY, 1979). 45. E. Abrahams, P. W. Anderson, D. C. Licciardello and T. V. Ramakrishnan, Scaling theory of localization — absence of quantum diffusion in 2 dimensions, Phys. Rev. Lett. 42, 673 (1979). 46. B. Nienhuis, Exact critical-point and critical exponents of O(n) models in 2 dimensions, Phys. Rev. Lett. 49, 1062 (1982).

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Volume 38A, number 5

28 February 1972

PHYSICS L E T T E R S

E X P O N E N T S FOR THE E X C L U D E D VOLUME P R O B L E M AS D E R I V E D BY T H E W I L S O N M E T H O D P. G. DE GENNES College de France, pi. M. Berthelot, 75 Paris 5e, France Received 10 January 1972 By an expansion to second order in e = 4-d, we derive the mean square extension fl2 for a random, self excluding walk of N jumps on a d-dimensional lattice. The result is: R% = const. JV^-^^(for d = 3).

Let Tn(R) be the number of non intersecting walks of N steps connecting the sites 0 and R on the lattice, and: 00

G(P,k)=

£ 2 TAR) N^OR "

exp(ift-/?) exp(--tfP) . (1)

When P decreases, on the real axis, down to a certain value P c , we reach a singularity of G: limp _

p

G(P, k =0) = const. {P-Pc)'y

.

N -» °°, N - °°, N/N = p o o

(2)

The total number of non-intersecting walks of N steps starting from the origin is: 71

Zn = const. N '

Define the thermodynamic functions S(P), S(P, p), b y :

exp (S(P,p)) = Hw exp (-NP)\N=NQP

Just at P = Pc, we expect that G(Pc,k) = const, k -2+f?. From the usual scaling arguments, the mean square distance traveled during a self-avoiding walk of N steps is i?2 = const. N%vwithy =7/2-77. In the present note we compute y and 77 using the method of Wilson [1]. We expand G in powers of the excluded volume parameter v0. The corresponding diagrams are described in the literature [2]. The scaling results are known to be independent of the magnitude of v0, provided that v0 > 0. We then choose v0 so that the renormalised coupling constant r satisfies the scaling requirement for the 4-point vertex*: r=

finite.

exp(S(P)) = EH M exp(-JV-p) N

exp(NPc) .

const. r£ " 2?j/2 -r\,

Eq. (3) may also be obtained as follows: for real values of P below P c , it is possible to define a continuation to the problem of eq. (1). Consider a lattice of N0 sites, and call H the number of ways of drawing on this lattice one self excluding chain of lenght N. The limit of interest is:

(P-

Pp)r

* This derivation of eq. (3) was suggested by P. Martin.

(3)

.

These functions are also singular at P = P c . The funtionS (P, p) may be expanded in powers of p. With the usual scaling assumption and notation, the expansion is: S(P, p) = a ( P - P ) 2 - « ' 0 c +

+

a 1 (P c -P)^'p + ia2(Pc-P) y '" 2 V

(4)

The coefficient of £p2 is the renormalised coupling constant. Again through scaling relations, it coincides with eq. (3). The diagrams for the spin problem of ref. [1] and the diagrams for the chain problem have the same topology, if the interactions are drawn as point-like. But, if the interactions are r e presented by dotted lines, a distinction appears. For instance, in the first order corrections to G, the "direct" diagram with one closed particle loop, which contributes a term propor339

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Volume 38A, number 5

PHYSICS L E T T E R S

28 February 1972

this would give y = 1 + e / 6 , in disagreement with eq. (5). A similar criticism applies to the Gaussian variational method [5] where 2vG = 4/d -> i + e/4 .

tional to the spin index n in ref. [ 1], has no counterpart for the chain problem: only the "exchange" diagram remains. Finally the Wilson formulae may be used provided that (a) the index n is set equal to zero; (b) the Wilson interaction constant uQ is replaced by jv0- The results a r e :

It is a pleasure to thank P. Martin, J. des Cloiseaux and P. Hohenberg for various discussions on related subjects.

y = l + e / 8 + 1 3 e 2 / 2 8 + 0(e 3 ) (5) T/ = (e 2 /64)[l + 17e/l6] + 0(e 4 ) .

References

For e = 1 (d= 3), y = 1.176, 77 = 0.032 and 2v = 1.195, in very good agreement with the series results of Fisher and Hiley [3]. The result for 2v is close the Flory value (2VY = I), but this is somewhat fortuitous: for an arbitrary d ( 100) have an overlap parameter P > 10. These melts are liquids, but have elastic properties at short time scales similar to polymer networks. For example, silly putty is a polymer liquid (melt), but it bounces off the floor like a rubber ball. This “bouncing” capability is reflected by the characteristic rubbery plateau region in the stress relaxation function of a melt.1, 2 Elastic properties of polymer melts on short time scales were attributed to topological interactions (called entanglements) between chains as early as in 1930’s and 1940’s. 3, 4 These entanglements arise because polymers cannot cross each other and impose topological constraints on polymer motion [see Fig. 1(a)]. Topological constraints in polymer melts were P ∼

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Fig. 1. (a) An entanglement is often visualized as a pairwise topological constraint between two chains. (b) Pairwise entanglements due to non-crossability of chains were represented by a temporary cross-link in early models (shown in the figure by associations between green reversible “stickers”).

called temporary entanglements because they put temporary restrictions on the motion of polymers (as opposed to permanent entanglements as in an Olympic gel).5 Entanglements between polymer chains in a melt are arguably the most important and the most characteristic property of the dynamics of polymer melts. The precise definition of a temporary entanglement between chains in a melt was not given in 1930’s and unfortunately still does not exist now. Early models of entanglements treated them as temporary cross-links [see Fig. 1(b)].6–8 On time scales shorter than the lifetime τ of such cross-links, properties of polymer melts were similar to those of permanent networks. Cross-links were assumed to break and re-connect on time scales on the order of τ , allowing a polymer melt to flow. The problem with this class of models is that in order for them to agree with experiments, one would need to assume that the life-time of crosslinks depends on the degree of polymerization of chains in a melt. This assumption was hard to justify as entanglements were visualized as local constraints. This class of models is currently extensively used to describe unentangled reversible networks with temporary cross-links representing reversible associations between the groups along the chains. 9–12 An alternative model of the dynamics of entangled polymer melts13 was proposed by Bueche in 1952. Bueche introduced the idea of entanglement friction and analyzed it by studying snaking circular motion of chains around entanglements. This idea is closest in spirit to the de Gennes’ reptation model that was proposed two decades later and revolutionized the field of polymer dynamics.

2.

Tube Model

The first major breakthrough in modeling polymer entanglements was made by Edwards 14 for polymer networks in which such entanglements are permanent.15 Topological restrictions of surrounding network strands imposed on a given chain were represented by the topological potential applied to every monomer of the chain. This potential effectively restricts fluctuations of the chain to a confining tube (Fig. 2). Edwards’ tube model reduces a many-chain problem with complicated topological interactions to a much simpler problem of a single chain in a potential. Each entangled chain in a network fluctuates around the axis of its confining tube, called the primitive path.1 The diameter a of the confining tube determines the length scale at which the confining potential restricts free fluctuations of the chain. On smaller length scales the chain “does not know” that it is entangled, as the

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Fig. 2. Topological constraints imposed by neighboring chains (left sketch) are replaced

Fig. 2. Topological constraints imposed by neighboring chains (left sketch) are replaced by a potential that restricts chain fluctuations to a confining tube (right sketch) in the Edwards tube model. The center line of the tube is called primitive path (dashed line).

effect of a confining potential on chain fluctuations is less than thermal energy kB T . On length scales larger than tube diameter a, chain fluctuations are strongly suppressed by the confining potential and the chain is forced to follow the trajectory of the primitive path. The primitive path is a random walk with the same end-to-end vector as the chain, but with step size on the order of the tube diameter a and with an average contour length of the tube hLi ≈ b2 N/a ,

(2)

hR2 i ≈ b2 N ≈ hLia ,

(3)

where

is the mean square end-to-end distance of the chain. The confining tube model is consistent with the experimental observations16 of the nonzero limiting value of the modulus of polymer networks with increasing molecular weight between cross-links. This limiting value of the elastic modulus, Ge , is related to the strength of the confining potential as expressed by the entanglement strand with degree of polymerization, Ne , and size defined by the tube diameter a ≈ bNe1/2 .

(4)

The elastic modulus Ge , called the plateau modulus, is on the order of thermal energy kB T per entanglement strand Ge ≈

kB T . b 3 Ne

(5)

Typical values of tube diameter are a ∼ 5 nm and of plateau modulus are Ge ∼ 1 MPa. 3.

Reptation Model

The giant step of extending the confining tube idea to uncross-linked chains for which topological interactions are not permanent was made by P. G. de Gennes17 (see the first reprint paper at the end of the present chapter). In his classic paper that started a new era of polymer dynamics, de Gennes considered a single chain in an array of fixed obstacles1 (e.g. free chain diffusing through cross-linked network). He represented topological constraints of these obstacles by a confining tube. The confining tube that was permanent in Edwards

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Fig. 3. An elementary step of de Gennes’ reptation model. (a) A small unentangled loop (defect) is formed at one end of the chain, “erasing” the section of the tube there. (b) The unentangled loop diffuses along the tube. (c) The loop (defect) is released at the other end of the chain, creating a new section of the tube at that end.

model of entangled polymer networks14 became temporary in the de Gennes model of a free chain diffusing through an array of fixed obstacles17 as it was moving with the chain. The process by which the chain moves along its confining tube was called reptation by analogy with the snaking movement of reptiles. The main idea of the reptation model is that although chain motion perpendicular to the axis of the confining tube is topologically restricted by the confining potential, its motion along the contour of the tube remains unconstrained.5 De Gennes analyzed the main mode of this motion — chain diffusion along the primitive path of the confining tube. He attributed this motion to the diffusion of small unentangled loops, that he called “defects” (see first reprint paper for details). These small loops of stored length are created by fluctuations at one end of the tube [Fig. 3(a)] and diffuse along the contour of the tube [Fig. 3(b)] to the other end, where they are released and form a new section of the tube [Fig. 3(c)]. Since the motion of the chain along the contour of the tube is unconstrained, the corresponding curvilinear friction coefficient of the chain is the same as the friction coefficient of an unentangled chain. In a melt with screened hydrodynamic interactions, this friction coefficient is linearly proportional to the number of monomers per chain ζ = ζ0 N ,

(6)

where ζ0 is the monomeric friction coefficient. The related curvilinear diffusion coefficient is obtained from the fluctuation-dissipation theorem (Einstein relation) Dc =

kB T D0 kB T = = , ζ ζ0 N N

(7)

where D0 is the monomeric diffusion coefficient. The time it takes for the chain to reptate out of its original tube is called the reptation time τrep ≈

hLi2 N3 ≈ τ0 , Dc Ne

(8)

where the monomeric relaxation time is τ0 ≈

ζ0 b2 . kB T

(9)

During reptation time τrep the chain diffuses an average distance hLi along the contour of the tube and a distance R ≈ bN 1/2 in three-dimensional space with the corresponding 3-d diffusion coefficient D3d ≈

Ne R2 ≈ D0 2 . τrep N

(10)

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The success of the reptation model is that its two main predictions, reptation time and diffusion coefficient, obtained in an elegant and relatively simple way were found to be in reasonable agreement with experiments. The experimentally measured dependence of the diffusion coefficient on the degree of polymerization18 D ∼ N −2 was found to be in agreement with the prediction of the reptation model [Eq. (10)], while the measured N-dependence of the relaxation time of a polymer melt is slightly stronger19 τrelax ∼ N 3.4

(11)

than that predicted by reptation. We address the possible reason for this disagreement in Sec. 6.2 below. Despite the crude approximations made by the reptation model, which describes the motion of a single chain in an array of fixed obstacles and ignores the motion of surrounding chains, the agreement between its predictions and results of experiments and computer simulations20 is astonishing.

4.

Constraint Release

It was not at all obvious why and how the reptation model could be extended from describing the motion of a chain in an array of fixed obstacles to a polymer melt in which all chains are free to diffuse and therefore constraints they impose on a given chain change with time.21 The confining tube of a chain in polymer melts is renewed not only at chain ends by reptation, but also all along its contour.1 The first model describing the effect of the motion of surrounding chains on the dynamics of a given entangled polymer was proposed by de Gennes22 in 1975 (see the second reprint paper at the end of the present chapter). When a neighboring chain reptates away, the constraint it used to impose on a given polymer is released and the confining tube of this polymer can locally rearrange (see Fig. 4). This rearrangement corresponds to the local change of the primitive path of this tube — its local “hop” by a distance on the order of tube diameter a. De Gennes proposed to use the Rouse model to describe this rearrangement process of a confining tube of a chain due to reptation of surrounding polymers. This approach remains the state of the art even today. In his 1975 paper (see second reprint paper) de Gennes assumed that the rate of the constraint release process (jump rate of primitive path segments) is proportional to the fraction of chain ends.22 This assumption was corrected and replaced a couple of years later23, 24 by a more physical assumption that the lifetime of

Fig. 4. Constraint release process. As chain B reptates away, it releases a constraint it used to impose on chain A and allows its tube to locally rearrange. The lower set of drawings represents local Rouse-like hop of the primitive path of chain A due to this event.

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a constraint is proportional to the reptation time of the corresponding neighboring chains P3 τP ≈ τ 0 , (12) Ne where P is the degree of polymerization of surrounding polymers. The resulting diffusion time of the primitive path of a test N -mer due to the constraint release process, caused by the reptation of surrounding P -mers, is calculated as the Rouse time of a chain consisting of N/Ne primitive path segments  2 N P 3N 2 τcr ≈ τP (13) ≈ τ0 Ne Ne3 where τP is the hop time of a primitive path segment [Eq. (12)]. The constraint release time is much longer than reptation time of an N -mer for sufficiently well-entangled surrounding P -mers which is certainly the case in a monodisperse melt (P = N > Ne ). The above estimate suggests that contribution of constraint release to the diffusion of a test N -mer is negligible in comparison to the contribution of reptation [Eq. (10)] for (P/Ne )3  N/Ne . This justifies ignoring the motion of surrounding chains in the calculation of polymer diffusion in monodisperse melts (P = N ) and validates the assumption of fixed obstacles made in the original reptation model. See Sec. 6.3 for a more recent modification of the constraint release model.

5.

Arm Retraction

Chain ends play an essential role in the de Gennes reptation model.17 Small loops of stored length (defects) are formed at one end of the chain and released at the other end (Fig. 3). Branched polymers, such as stars, present the first conceptual challenge for the reptation model, as branch points cannot slide along the confining tube. If they did, several strands of the molecule would be pulled into the confining tube of one of the arms. This process is entropically unfavorable and is therefore exponentially suppressed (for stars with more than three arms). Entangled stars in an array of fixed obstacles diffuse by the arm retraction process that was proposed by de Gennes in the same 1975 paper (see second reprint paper).22 The arm retraction process requires for an arm of a branched polymer to double-fold, forming a huge loop. This double-folding process can be visualized by the end of an arm moving up along its confining tube towards the branch point (Fig. 5). Any “mistake” in this retraction process would lead to an unacceptable conformation and the chain end would have to go back and start the retraction process again. The probability of a successful arm retraction is therefore exponentially low in the number of primitive path steps N/Ne which the end has to make in order to reach the branch point p ∼ exp(−νN/Ne ), where ν is a constant on the order of unity. The resulting arm retraction time is exponentially large in the number of primitive path steps τarm ∼ exp(υN/Ne ) This prediction of the de Gennes’ arm retraction model ments.25

(14) 22

has been confirmed by experi-

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Fig. 5.

Arm retraction process of a branched polymer in an array of fixed obstacles (represented by circles).

De Gennes did not consider rearrangements of the rest of the chain while the end is retracting along the confining tube towards the branch point and therefore he assumed a linear retraction potential.22 Rearrangements of the arm in the form of large unentangled loops lead to a roughly quadratic potential for arm retractions, as was shown by Doi and Kuzzu.26 The resulting arm retraction time obtained for the quadratic retraction potential still has the same form as predicted by de Gennes [Eq. (14)]. The constraint release process in the melt of entangled stars seems to be more complex due to an exponentially broad distribution of constraint release rates.27 6. 6.1.

Main Developments in the Wake of de Gennes Ideas Doi–Edwards stress relaxation model

In order to calculate the stress relaxation function G(t) for entangled polymer melts, Doi and Edwards assumed28 that all chains in the melt undergo de Gennes reptation17 and ignored constraint release effects due to reptation of surrounding polymers on the stress relaxation of a given chain. The stress due to a step-strain imposed on a melt at time t = 0 is stored in the segments of the primitive path at times longer than the relaxation time of an entanglement strand τe ≈ τ0 Ne2 .

(15)

The main idea of the Doi–Edwards model is that at longer times t > τe the stress is supported only by the sections of the confining tube that have not been vacated by the reptating chain between times 0 and t. As the chain reptates and vacates sections of the originally occupied tube, stress in the melt imposed by a step-strain relaxes. The Doi–Edwards stress relaxation function21 calculated for the de Gennes reptation model is   X 1 p2 exp − t (16) G(t) = Ge p2 τrep p odd

The Doi–Edwards model qualitatively explains the origin of the rubbery plateau in entangled polymer melts, as it predicts the delay in the stress relaxation between relaxation time of

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an entanglement strand τe and the reptation time of the chain τrep . The resulting melt viscosity is proportional to the reptation time5, 21 Z ∞ ηrep = G(t)dt ≈ Ge τrep ∼ N 3 (17) 0

and slightly disagrees with the experimentally observed scaling 1, 19 ηexp ∼ N 3.4 .

(18)

The Doi–Edwards stress relaxation model28 is the first and the simplest in the class of tube models proposed over the last 30 years. 6.2.

Doi tube length fluctuation model

The reptation model17 proposed by de Gennes in 1971 as well as the Doi–Edwards stress relaxation model28 consider only the main diffusion mode of the chain along the contour of its confining tube and completely ignore all other modes of the chain motion along the tube. The reptation process was described by de Gennes as the creation of “defects” of stored length at one end of the chain (shortening the tube) and their destruction at the other end (lengthening the tube). Many of such “defect” creation and destruction events occurring simultaneously inevitably lead to fluctuations of tube length (Fig. 6). The length of the confining tube should undergo fluctuations around its average value hLi [Eq. (2)] with a root-mean-square amplitude ∆L equal to the chain size R ≈ bN 1/2 [Eq. (3)] L = hLi ± ∆L = hLi ± bN 1/2 .

(19)

These tube length fluctuations were first described by Doi29 who demonstrated that the stress relaxation function is dominated by the tube length fluctuation modes on time scales up to the Rouse relaxation time of the chain τRouse ≈ τ0 N 2 with a fraction  1/2 Ne ∆L (20) ≈ L N of the stress relaxing by these modes. The stress relaxation function at the Rouse time is thus lower than the plateau value r ! Ne G(τR ) ≈ Ge 1 − γ , (21) N where γ is a numerical coefficient on the order of unity.

Fig. 6.

Tube length fluctuations lead to partial relaxation of stress.

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In order to relax the remaining stress, the chain has to reptate along the tube not its full length hLi, but a smaller distance hLi − ∆L. Therefore the chain is able to relax this stress in a shorter time r !2 Ne . (22) τDoi ≈ τrep 1 − γ N leading to lower viscosity with stronger N -dependence1, 29 r !3 r !3 N3 Ne Ne ∼ 2 1−γ . ηDoi ≈ ηrep 1 − γ N Ne N

(23)

This prediction is in reasonable agreement with experimentally observed19 scaling of viscosity with the degree of polymerization [Eq. (18)]. It turns out that tube length fluctuation modes introduced by Doi make a similar correction to the diffusion coefficient predicting a stronger dependence than the reptation model [Eq. (10)]. This stronger molecular weight dependence was observed in experiments 30 and computer simulations.31 6.3.

Self-consistent constraint release

Although constraint release does not make a significant contribution to polymer diffusion in monodisperse melts, it is very important for stress relaxation. A significant fraction of stress is relaxed on the reptation time scales τrep by the highest Rouse mode of the constraint release motion of the primitive path. The constraint release process in polydisperse (or bidisperse) melts is essential for obtaining a qualitatively correct stress relaxation function. The idea of the self-consistent constraint release model32 is to use the spectrum P (ε) of the disentanglement rates ε of the motion of a single chain along its tube (rates at which the chain abandons its entanglements with neighbors) for the calculation of the constraint release process. The first step is to find the single-chain-in-a-tube stress relaxation function µ(t) that includes contributions from reptation and tube length fluctuations. This stress relaxation function is the Laplace transform of the spectrum of disentanglement rates Z ∞ µ(t) = P (ε)e−tε dε . (24) 0

The spectrum of disentanglement rates P (ε) obtained from the single-chain-in-a-tube stress relaxation function µ(t) is used as the distribution of rates of elementary steps (mobility of Rouse monomers) in the constraint release motion of the tube itself. The resulting constraint release component R(t) of the stress relaxation function is calculated. This procedure assumes the duality of the single-chain-in-a-tube and the constraint release motion (e.g. loss of an entanglement due to reptation of chain B corresponds to the constraint release for its entanglement partner — chain A in Fig. 4). Therefore the rates of these complimentary processes are assumed to be identical. Although it makes a number of uncontrolled approximations (e.g. the combined stress relaxation function G(t) is the product of the single-chain-in-a-tube µ(t) and the constraint release R(t) components G(t) = Ge µ(t)R(t)), the self-consistent constraint release model,32 based on the de Gennes idea of constraint release,22 leads to predictions in very good agreement with experiments.33

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6.4.

Polymers with no ends or with too many ends

The ideas of reptation and arm retraction introduced by de Gennes in 1970’s were extended to entangled branched polymers by McLeish and collaborators.34 For example, reptation of a comb or a pom-pom polymer (Fig. 7) along the tube confining its backbone is slowed down by the necessity to retract side chains (arms). These arm retraction events (similar to arm retraction of a star — Fig. 5) dominate the curvilinear friction coefficient of the comb or pom-pom along the confining tube of its backbone. Thus, both reptation17 and arm retraction22 processes are essential for understanding the dynamics of entangled branched polymers in an array of fixed obstacles. In melts of such polymers one needs to also take into account constrain release processes.22, 34 The only polymer topology that does not seem to follow the mechanisms of reptation and arm retraction is the cyclic (ring) polymer. Non-concatenated ring polymers maximize their entropy in an array of fixed obstacles in double-folded randomly branched conformations (Fig. 8).35 However, the dynamics of these ring polymers is qualitatively different from the arm retraction of branched polymers because double-folded loops freely pass through “branch points” resulting in a self-similar dynamics with relaxation time36  5/2 N , (25) τring ≈ τ0 Ne2 Ne shorter than that of entangled linear chains with the same degree of polymerization N . Extension of this model to melts of non-concatenated entangled cyclic polymers predicts a

Fig. 7.

Reptation of the backbone of a pom-pom is slowed down by arm retraction.

Fig. 8. Double-folded randomly branched conformation of a non-concatenated entangled cyclic polymer in an array of fixed obstacles (represented by circles).

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power law stress relaxation function1 G(t) ≈ Ge



t τe

−2/5

(26)

without a rubbery plateau, which is in good agreement with recent experiments.37 7.

Open Questions and Future Directions

In spite of the major progress in our understanding of the dynamics of entangled polymers based on the three ingenious ideas of de Gennes described above (reptation, constraint release, and arm retraction) — there are still a number of remaining open questions that are left as challenges to present and future researchers. 7.1.

The nature of entanglement and confining tube

The first and foremost of the remaining challenges is the quantification of the concept of entanglements and of the confining tube. In spite of numerous efforts, our present understanding of this important concept remains rather sketchy. The most successful principle of determining the tube diameter in polymer melts is the Kavassalis–Noolandi conjecture 38 of a constant overlap parameter P [see Equation (1)] for an entanglement strand P ≈ 20. 1 Attempts to derive this very useful result from the first principles based on the ideas of topological constraints have not been successful so far. Similar difficulties are encountered in the attempts to derive concentration dependence of the tube diameter in theta solutions of entangled polymers where chains are also almost ideal, as in melts. A simple scaling argument39 based on the assumption that the tube diameter a is proportional to the distance between pairwise contacts between monomers at concentration c a ∼ c−2/3

(27)

40

is in very good agreement with experiments. The problem with this scaling argument is that it ignores both chain connectivity and correlations between chains due to three body repulsion. Computer simulations suggest that these correlations are important to obtain the scaling exponent −2/3 [Eq. (27)] and different scaling exponent results from simulations of ideal chains without correlations,41 a ∼ c−1 .

(28)

Quantitative calculations of the tube diameter and the primitive path length from computer simulations also turned out to be a very difficult task. Most currently employed methods simultaneously reduce polymer chains to their primitive paths following various protocols.42, 43 The problem with all these methods is that in the process of reducing chains to their primitive paths, the fluctuations of all chains are modified. Therefore the confining potential that defines both tube length and tube width is changed, leading to modified values of these lengths. A successful method has to determine the confining tube of a polymer while leaving the fluctuations of all surrounding chains unperturbed.44

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How do the confining potential and related parameters, such as tube diameter, change upon bulk deformation? Some recent theories of entangled polymer networks45, 46 predict that the confining potential softens upon extension and stiffens upon compression, resulting in a non-affine deformation of tube diameter. Extension of these ideas to the non-linear deformation of polymer melts under strong extensional or shear flows using the convective constraint release model47 still remains mostly unexplored. 7.2.

Constraint release revisited

Are entanglements in polymer melts due to predominantly pairwise or collective topological constraints? The most likely answer to this question is that they are due to collective constraints imposed simultaneously on many chains, but a number of currently employed models, such as the slip-link model,48, 49 rely on entanglements being pairwise. If entanglements are collective rather than pairwise, the duality of the self-consistent constraint release model32 needs to be revised to estimate the actual effect of a single chain on the constraint it imposes on a neighboring polymer. On one hand, the effective displacement of a primitive path due to a single neighbor reptating away could be much smaller than the tube diameter a (in Fig. 4 this displacement is assumed to be a). On the other hand, chains A and B are typically entangled in multiple places (there are (N/Ne )1/2 such common entanglements). Therefore there would be an amplification of the effect of a single chain reptating away on the displacement of the primitive path of its neighbor.50 These two effects — collective nature of entanglements (reducing the influence of a single chain on the constraint release of its neighbor) and coherence of constraint release (increasing this influence) — could partially compensate each other. Another concern is that most constraint release models, such as self-consistent constraint release, assign mobility to a particular section of the chain that does not change as the tube is modified by constraint release jumps. The problem with this approach is that it ignores the fact that if a fast constraint is released and the section of the tube jumps, the next constraint it encounters could be a longer-lived constraint (or vice versa). Some attention has been devoted in the literature to the relationship between constraint release and another process, called tube dilation.51, 52 Tube dilation was introduced53 as an alternative way of taking into account many chain effects in the dynamics of entangled polymers. If the surrounding constraints are relatively short-lived, a chain is not confined to the original narrow tube, but instead effectively samples a wider tube at longer times scales (as if short-lived constraints were not topologically effective). The tube dilation model was extensively used in theories of dynamics of entangled branched polymers.34 Nevertheless, the applicability of tube dilation as a special case of constraint release is still not completely clear.52 Another uncertainty is related to the actual value of the tube dilution exponent. As sections of polymers that do not provide effective topological constraints are treated as solvent, one needs to determine how the diameter a of the effective diluted tube depends on the concentration c of the remaining topologically effective parts of polymers. Is the dependence similar to that for polymers in theta solvents [Equation (27)] because in both cases the statistics of chains is ideal-like, or different [Equation (28)] because three-bodyinduced correlations between polymer sections are different in melts and theta solvents?

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Thus a qualitatively correct way of treating many-chain effects — the next generation of the constraint release models — remains a major challenge in the dynamics of entangled polymers. 7.3.

Asymptopia

Most ideas of polymer physics are based on scaling arguments and become more exact in the asymptotic limit of very long chains. For example, tube length fluctuations (Sec. 6.2) become less important in comparison to reptation for longer chains [their role is proportional to (Ne /N )1/2 — see Equation (20)]. Does this imply that reptation becomes exact in the limit of very long polymer chains N → ∞ and viscosity of melts consisting of very long linear chains follows a simple reptation law [Equation (17)]? An alternative view of dynamics of melts of very long linear polymers was proposed by Deutsch,54, 55 who suggested that as chain ends become sparse (the distance between them increases as bN 1/3 ) the reptation process becomes more difficult, as it effectively moves material from one end of the tube to another end, thereby swelling an entanglement network at the receiving end of the tube. The extra stress due to this swelling can be released only by pumping this extra mass through the neighboring end of one of the surrounding tubes, which is typically located at distance ∼ bN 1/3 . Thus in order to avoid creating excessive stresses of an entanglement net, very long chains in a melt have to follow each other in a process, called collective or activated reptation.56, 57 Both the original Deutsch’s estimate54, 55 and the more recent collective reptation models56, 57 predict exponential growth of viscosity of ultra high linear melts ηcol rep ∼ exp(βN 2/3 ) ,

(29)

where coefficient β depends on the details of the model (in reference57 β ∼ Ne−2 ). This exponential growth of melt viscosity [Equation (29)] has not yet been observed experimentally and, therefore, it is still an open question whether ultra-high molecular weight melts follow the collective activated reptation [Eq. (29)] or the simple reptation [Eq. (17)] model. Another unsolved problem is melts dynamics of high molecular weight non-concatenated cyclic polymers. The theory based on self-similar dynamics in a melt of rings36, 38 is valid only for intermediate molecular weights. The ultra-high molecular weight non-concatenated rings do not heavily overlap with each other, while following double-folded randomly branched statistics as assumed in Ref. 38 because the fractal dimensionality 4 of their structures is not compatible with 3-dimensional space. A change in the conformations of high molecular weight rings would lead to a change in their dynamics. There is also a possibility of loops penetrating and trapping other loops significantly slowing down dynamics of the melt of rings.58 This glass-like or gel-like slow-down has not yet been observed in experiments, but melts of rings studied so far have not had a very large number of entanglements per polymer (N/Ne ∼ 20).38 Dynamics of melts of rings with much larger number of entanglements is still an open question and a challenge both theoretically and experimentally.

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Acknowledgments I am grateful to S. Panyukov for insightful discussions and to L. Cai and J. Brock for their help with the manuscript. I would like to acknowledge financial support of National Science Foundation under grants CHE-0616925 and CBET-0609087, National Institutes of Health under grant 5-R01-HL077546-01-03.

References 1. M. Rubinstein and R. H. Colby, Polymer Physics (Oxford University Press, Oxford, New York 2003). 2. W. W. Graessley, Polymeric Liquids and Networks: Dynamics and Rheology (Taylor & Francis Group, New York, 2008). 3. W. F. Busse, J. Phys. Chem. 36, 2862 (1932). 4. L. R. G. Treloar, Trans. Faraday Soc. 35, 538 (1940). 5. P. G. de Gennes, Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca NY, 1979). 6. M. S. Green and A. V. Tobolsky, J. Chem. Phys. 14, 80 (1946). 7. A. S. Lodge, Trans. Faraday Soc. 52, 120 (1956). 8. M. Yamamoto, J. Phys. Soc. Jpn. 11, 413 (1956). 9. M. Rubinstein and A. V. Dobrynin, Current Opinion in Colloid & Interface Sci. 4, 83 (1999). 10. M. Rubinstein and A. N. Semenov, Macromolecules 31, 1386 (1998). 11. M. Rubinstein and A. V. Dobrynin, Trends in Polymer Sci. 5, 181 (1997). 12. L. Leibler, M. Rubinstein and R. H. Colby, Macromolecules 24, 4701 (1991). 13. F. Bueche, J. Chem. Phys. 25, 599 (1956). 14. S. F. Edwards, Proc. Physical Society of London 92, 9 (1967). 15. W. W. Graessley, Polymeric Liquids and Networks: Structure and Properties (Garland Science, New York, 2004). 16. S. K. Patel, S. Malone, C. Cohen, J. R. Gillmor and R. H. Colby, Macromolecules 25, 5241 (1992). 17. P. G. de Gennes, J. Chem. Phys. 55, 572 (1971). 18. M. Tirrell, Rubber Chemistry and Technology 57, 523 (1984). 19. J. D. Ferry, Viscoelastic Properties of Polymers, 3rd edn. (Wiley, New York, 1980). 20. K. Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990). 21. M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Clarendon Press, New York, 1986). 22. P. G. de Gennes, J. de Phys. 36, 1199 (1975). 23. J. Klein, Macromolecules 11, 852 (1978). 24. M. Daoud and P. G. de Gennes, J. Polymer Sci. Part B: Polymer Phys. 17, 1971 (1979). 25. L. J. Fetters, A. D. Kiss, D. S. Pearson, G. F. Quack and F. J. Vitus, Macromolecules 26, 647 (1993). 26. M. Doi and N. Y. Kuzuu, J. Polymer Sci. Part C: Polymer Lett. 18, 775 (1980). 27. R. C. Ball and T. C. B. Mcleish, Macromolecules 22, 1911 (1989). 28. M. Doi and S. F. Edwards, J. Chem. Soc.: Faraday Transactions II 74, 1789 (1978). 29. M. Doi, J. Polymer Sci. Part C: Polymer Lett. 19, 265 (1981). 30. T. P. Lodge, Phys. Rev. Lett. 83, 3218 (1999). 31. K. Hagita and H. Takano, J. Physical Soc. of Japan 72, 1824 (2003). 32. M. Rubinstein and R. H. Colby, J. Chem. Phys. 89, 5291 (1988). 33. A. E. Likhtman and T. C. B. McLeish, Macromolecules 35, 6332 (2002).

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34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58.

T. C. B. McLeish, Adv. Phys. 51, 1379 (2002). M. Rubinstein, Phys. Rev. Lett. 57, 3023 (1986). S. P. Obukhov, M. Rubinstein and T. Duke, Phys. Rev. Lett. 73, 1263 (1994). M. Kapnitos, M. Lang, D. Vlassopoulos, W. Pyckhout-Hintzen, D. Richter, D. Cho, T. Chang and M. Rubinstein, Nature Materials 7, 997 (2008). T. A. Kavassalis and J. Noolandi, Macromolecules 22, 2709 (1989). R. H. Colby and M. Rubinstein, Macromolecules 23, 2753 (1990). M. Adam and M. Delsanti, J. de Phys. 45, 1513 (1984). M. Chhajer and M. Rubinstein, to be published. R. Everaers, S. K. Sukumaran, G. S. Grest, C. Svaneborg, A. Sivasubramanian and K. Kremer, Science 303, 823 (2004). Q. Zhou and R. G. Larson, Macromolecules 38, 5761 (2005). M. Lang, S. Panyukov and M. Rubinstein, to be published. M. Rubinstein and S. Panyukov, Macromolecules 30, 8036 (1997). M. Rubinstein and S. Panyukov, Macromolecules 35, 6670 (2002). G. Marrucci, J. Non-Newtonian Fluid Mechanics 62, 279 (1996). S. F. Edwards and T. Vilgis, Polymer 27, 483 (1986). S. F. Edwards and T. A. Vilgis, Rep. Prog. Phys. 51, 243 (1988). F. Brochard-Wyart, A. Ajdari, L. Leibler, M. Rubinstein and J. L. Viovy, Macromolecules 27, 803 (1994). J. L. Viovy, M. Rubinstein and R. H. Colby, Macromolecules 24, 3587 (1991). T. C. B. McLeish, J. Rheology 47, 177 (2003). G. Marrucci, J. Polymer Sci. Part B: Polymer Phys. 23, 159 (1985). J. M. Deutsch, Phys. Rev. Lett. 54, 56 (1985). J. M. Deutsch, J. de Phys. 48, 141 (1987). M. Rubinstein and S. P. Obukhov, Phys. Rev. Lett. 71, 1856 (1993). A. N. Semenov and M. Rubinstein, European Phys. J. B 1, 87 (1998). S. P. Obukhov, M. Rubinstein and R. H. Colby, Macromolecules 27, 3191 (1994).

rubinstein

35

THE

JOURNAL

OF

CHEMICAL

PHYSICS

VOLUME

55,

NUMBER

2

15

JUL

Reptation of a Polymer Chain in the Presence of Fixed Obstacles P. G. DE GENNES

Laboratoire de Physique des Solides, Faculti des Sciences, Ql—Orsay, France (Received 18 January 1971) We discuss possible motions for one polymer molecule P (of mass M) performing wormlike displacements inside a strongly cross-linked polymeric gel G. The topological requirement that P cannot intersect any of the chains of G is taken into account by a rigorous procedure: The only motions allowed for the chain are associated with the displacement of certain "defects" along the chain. The main conclusions derived from this model are the following: (a) There are two characteristic times for the chain motion: One of them (Td) is the equilibration time for the defect concentration, and is proportional to M2. The other time (T,) is the time required for complete renewal of the chain conformation, and is proportional to M3. (b) The over-all mobility and diffusion coefficients of the chain P are proportional to M^. (c) At times t M JF/(.Va) 2 ]Pz.

(III.4)

(HI.5)

The center of gravity g of the chain moves with a velocity fN g = -Y-1 l dnin. •'o We can transform r„ by Eq. (11.10) and insert (III.5) as the current, obtaining J

%=

~fbdn=~V.

(HI.6)

Averaging over the values of P we are left with one nonvanishing velocity component:

where the over-all mobility p.tot is explicitly given by Mtot

= M[p&2aV>/(A7a)3] = wb2/N2a.

(III.7)

Equation (III.7) may also be written in terms of the self-diffusion coefficient Dtot=kBTfxtot. Both coefficients are seen to decrease like N~2 (or M~2). This is to be compared with the case of a free Rouse chain, where

38

REPTATION

OF

A

IV. DISPLACEMENTS OF ONE MONOMER The quantity

POLYMERCHAIN coil. Thus we may write ([_in(t)-rn(0)J)

o*, and (c) the terminal opening hf. 11.1. A Single Promoter in Air. The situation is described on Figure 3. The two blocks A and B have been separated by an air gap of thickness h. A connector bridges this gap: the bridging portion contains a certain number of monomers (n). We do not, for the moment, assume that the bridge is fully stretched. We describe the bridge as a "pillarn of diameter d and height h. The volume fraction in the pillar is Our model4describes the polymer inside the pillar as a semidilute solution in a poor solvent which is just in equilibrium with the s ~ l v e n t . ~For such a solution, the correlation length is [ = a / $ and the interfacial energy with the solvent is We shall often use, instead of y,, the dimensionless parameter:

-

Small values of KA mean fl solvents. The realistic situation with air (a very bad solvent !) is KA 1. But we focus our attention on small values of KA for three reasons: (a) they may be of interest when we fracture in a liquid and not in air; (b) the discussion gives more insight; (c) in any case, the limit KA 1 for air may still be taken on our scaling formulas without harm. Let us now discuss the pulling force f necessary to maintain the pillar in Figure 3. This is a combination of two terms: a capillary force: an elastic force which in our case has the ideal chain form:

(4) A somewhat related analysis concerning the deformation behavior of

a collapsed coil in a poor solvent may be found in: Halperin, A.; Zhulina, E. B. Europhys. Lett. 1991, IS, 417. (5) de Gennes, P. G.J . Phys. Lrtt. (Poris) 1978, 39, L 299.

0022-3654/92/2096-4002$03.00/0O 1992 American Chemical Society

The Journal of Physical Chemistry, Vol. 96, No. 10, 1992 4003

Rubber-Rubber Adhesion with Connector Molecules The minimum off = f,

+ f,]is reached when RUBBER A

or equivalentely (from eq 2.3)

d r ad-' r E

(2.7)

The corresponding value of the force is

This is the threshold force required to pull out one connector. 11.2. The Threshold Stress. Suppose that a uniform stensile stress a is applied to the rubber blocks (Figure 2). The energy g per unit area (as a function of the distance h between the two blocks) has the aspect shown on Figure 4, where Wdenotes the DuprE work of separation of the two rubbers in the absence of promoter:

W = YA + YB - YAB For h

> a, the energy g(h)

Figure 1. A model for a weak rubber (A)/rubber (B) junction. The two parts are connected by long A polymer chains (adhesion promoters) grafted to the network B and entering freely in the A block.

(2.9)

is linear (see eq 2.8):

As long as u is smaller than the critical value

the energy g(h) is minimal for h = 0 and the system remains closed. But as soon as u becomes greater than 6,the energy minimum is not at h = 0. It is true that there remains an energy barrier: but, in the fracture processes to be discussed below, the fracture tip acts as a nucleation center and removes this barrier. Thus a* appears as a threshold stress for opening. When a > a*, the promoters are progressively sucked out of the rubber A. This suction process ends only when all the promoter is extracted, i.e., for h = hf with

RUBBER B

m

1

1

Figure 2. The two rubber blocks submitted to a uniform tensile stress u.

Here N is the polymerization index of the promoter. We now analyze the dissipation involved in this suction process. 11.3. The Suction Process. Assume that a > a*. When the distance between the two rubber blocks increases by dh, the chain is sucked by a length The work performed by the stress a is the sum of two terms:

The first term is the work of extraction of the connectors. The second term represents the viscous losses inside the rubber, due to slippage of the connector chains. In the simplest picture

where [, r t l N is a tube friction coefficient (we ignore for the moment the complications related to the fact that when the connector is half pulled out, the friction is also reduced by a factor 1/21. Equations 2.14 and 2.15 give us a suction law of the form

where Q = p{,KA-' is the friction coefficient of the junction and is, for the moment, taken to be independent of h. 111. Fracture

III.1. General Features. We now consider the propagation of a fracture along the rubber (A)/rubber (B) interface. The general

Figure 3. A connector bridging the air gap (of thickness h) between the two blocks A and B. We describe the bridge as a "pillar" of diameter d and height h. The pulling force necessary to maintain the pillar is a combination of a capillary force f, yAird and an elastic force f,,.

-

aspect of the junction is shown on Figure 5. Following ref 6, we describe the elastic field associated with the junction by a superposition d(x) of elementary sources extending over all the active part of the junction (i.e., from Z(x=O) to J(x=L)). The normal stress distribution a(x) and the crack opening h(x) are then given by ( 6 ) de Gennes, P.G. Can. J . Phys. 1990, 68, 1049.

4004 The Journal of Physical Chemistry, Vol. 96, No. 10, 1992

RaphaBl and de Gennes

(3.1)

a(x) =

i

W-ua

2 ( 1 - v ) l L dy $(y)Ol - x)ln (x < L )

h(x) =

0

-

(3.2)

(x> L)

(We assume a mode I plane strain loading7) At large distances (1x1 >> L), the eqs 3.1 and 3.2 reduce to the standard scaling laws for a crack in a purely elastic medium:' a(x) = K ( 2 ~ x ) - l / ~ (x

>> L)

(3.3)

+

where p = E/2(1 v) is the shear modulus and K is called the stress intensity factor. We now impose the suction law (2.16) in all the active region IJ (apart from a narrow region of size -a near the tip (J) of the junction). Assuming a steady-state propagation of the fracture (velocity V), we obtain

I

a

wh

Figure 4. Energy (per unit area) versus gap for the system depicted in Figure 2.

Figure 5. A global view of the advancing fracture. The junction corresponds the IJ.

d x )

I

where X is defined by

Equation 3.6 must be supplemented by the boundary condition: h(x=O) = hf (3.8) where hf is the terminal value of the junction opening (eq 2.12). ITI.2. Distribution of Sources. The distribution of sources d(x) can be written as the sum of two contributions: (a) the "initiation distribution" dini(x)determined in the Appendix 1 and corresponding to the elastic field in the absence of connectors; (b) a (yet unknown) 'viscous distributionn dVi,(x)associated with the suction process:

and 9 (x) is a perturbation. The system (3.14, 3.1 5) has been solv$"r'ecently by Hui et al. in their thorough investigation of cohesive zone model^.^ The result is ~H(x= ) r - ~ a *2223 x-(1/2+t)[ ( L - a) - x]' X

Accordingly, the stress a(x) and the "density of dislocations" -dh/dx can be written as .(x) = ( ~ i n i (+ ~ )~ v i s ( ~ )

(3.10)

dh = --dhinl - dhvis -(3.1 I ) dx dx dx Since .ini(x) = 0 for x < L - a (see Appendix I), eq 3.6 reduces to dhvis dhini r-'aVi,(x) - p-la* = -A- A(O No should not enter at all. The only pieces which enter freely are the two ends of the chain. From

At the crude level of scaling laws, we begin to have a certain perception of the main controlling factors for rubber adhesion in the presence of adhesion promoters, or "connector molecules". (1) For rubbedrubber contacts, strengthened by mobile chains, the history of the rubber is crucial: if it has been cross-linked prior to the insertion of mobile chains, the maximum concentration of connectors should be severely limited, and short connectors are preferable, as seen from eqs 3 and 18. (2) For simple grafted layers, we predict a regime of partial interdigitation, where the adhesion energy increases very slowly with the grafting density. At higher surface concentrations ( a > No-Il2), the brush should not interdigitate any more. All this should be amenable to experiments, possibly by using block copolymers rather than grafted chains. (3) With a typical model system such as PDMS against silica, the most readily available situation corresponds to the Guiselin brush. Our peeling data on these brushes show an interesting maximum in G(o). However, no complete theory is available yet for this case. (4) There are other limitations to our discussion, such as the following: (a) Rubberlrubber contacts (or latexllatex contacts) will be, in practice, extremely sensitive to the polydispersity of the mobile connectors. This can, in fact, be taken into account rather easily in our picture. (b) We have not incorporated the possibility of chemical scission for the connectors. In fact, our discussion of section I shows that, for most loose systems, the force on the connectors (at low velocity) i s p = U,,la, where U,, is a van der Waals energy. This is far below the chemical rupture forces f, U,/ a. Thus, in the low-velocity regime, scission is indeed negligible for loose systems of unbranched connectors.

-

Acknowledgment. We have benefited from very stimulating exchanges with M. Aubouy, J. C. Daniel, M. Deruelle, H. Hervet, C. Ligoure, H. Brown, G. Schorsch, and P. Silberzan. The experimental part of this work has benefited from support by the RhBne-Poulenc Co. References and Notes (1) Raphael, E.; de Gennes, P. G. J . Phys. Chem. 1992, 96, 4002. (2) Brown, H. R.; Hui,C . Y.; Raphael, E. Macromolecules 1994, 27, 608. (3) de Gennes, P. G. C. R. Acad. Sci. Paris 1994, 318 (II), 165. (4) Lake, G. J.; Thomas, A. G. Proc. R. Soc. London, A 1967, 300, 108. (5) Ahagon, A.; Gent, A. J. Polym. Sci., Polym. Phys. Ed. 1975, 13, 1285.

9410 J. Phys. Chem., Vol. 98, No. 38, 1994 ( 6 ) Ji, H.; de Gennes, P. G. Macromolecules 1993, 26, 520. ( 7 ) Ellul, M. D.; Gent, A. J. Polym. Sci., Polyrn. Phys, Ed. 1984, 22, 1953.

( 8 ) de Gennes, P. G. C. R . Acad. Sci. Porir 1988, 307, 11 (9) Cohen Addad. J. P.; Viallat, A. M.; Pouchelon, A. Polymer 1986, 27, 843. (10) Guiselin, 0. Europhys. Lett. 1992, 17, 225. ( I I ) Auvray, L., Auroy, P.; Cmz. M, 3. Phys. (Paris) 1992.2 (6). 943. (12) Demelle, M.; Ldger, L. To be published.

Brochard-Wyart et al. 1131 f h i s law was fust described bv M. Daoud. See for instance: P. G. he kennes, Scaling laws in p01yme;~hysics;Cornell Univ. Press, 2nd printing, 1985; Chapter m. (14) Bastide, 1.; Candau, S.; Leibler, L. Macromolecules 1980, 14, 719. (15) Bmchard, F. J . Phyr. (Paris) 1981, 42, 505. (16) de Gennes. P. G Macromolecules 1980, 13. 1069. 117) Lieoure. C. These Univ. Paris 6 . 1991. i18) ~ i b o u y ;M.; Raphael. E. Macrornolecule~,robe published. (19) Marciano, Y.: Ldger, L To be published.

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An Approach to Cell Adhesion Inspired from Polymer Physics

Pierre Nassoy Institut Curie, Centre de Recherche, CNRS, UMR 168 Universit´e Pierre et Marie Curie, F-75248 Paris, France [email protected]

1.

Introduction

When P.-G. de Gennes was asked about his reasons for joining the Curie Institute in 2002 and turning to biology, he answered that he “could envision some natural bridges between (his) fields of research and cell adhesion” (L’Express, 12th November 2002). As an immediate consequence, he was approached by local cell biologists who wished to have him explain the adhesive properties of their favorite proteins. Despite the wealth of information collected over the last 20 years about the structural and biochemical properties of cell adhesion molecules,1 an integrated view of cell adhesion was still poorly understood. While most biologists put forward the peculiarity of each single adhesive protein to rationalize the lack of any general mechanism, de Gennes, like other biophysicists, aimed to define some primary parameters that could provide a quantitative description of some universal features of cell adhesion. For instance, in the paper related to this chapter, he proposed that cell-tosubstrate or cell-to-cell adhesivity should be fundamentally described by an adhesion energy instead of a separation force, as it was usually done in the vast majority of classical adhesion assays. Quite remarkably, de Gennes had already attempted to clarify this methodological approach in 1994, in the preface of a book dedicated to cell adhesion,2 while he was still working on polymer adhesion. Back then, his maybe too formal message had zero impact in the cell biology community. The main merit of the selected publication lay in the fact that de Gennes concretely proposed an assay, which was directly inspired from the Johnson–Kendall–Roberts test in polymer adhesion, to measure cell-cell of cell-substrate adhesion energies. Not only can this cellular JKR test be of practical importance for designing biomaterials, implant surfaces, and tissue-engineering scaffolds, but it also enables to correlate the molecular parameters

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derived from force measurements at the single bond level with the cellular response upon enforced unbinding. To better realize the relevance of this paper, we will start by describing the most prevalent cellular adhesion assays used in the last 20 years and by highlighting some related pending issues. Then, we will detail and discuss the soft matter approach to cell adhesion developed by de Gennes. Finally, despite the lack of sufficient hindsight, we will tentatively wonder whether de Gennes’ contribution to cell adhesion can be taken up by the cell biologists’ community. 2. 2.1.

On the Specificity of Cell Adhesion Nature of cell-cell and cell-substrate interactions

The peculiarity and complexity of cell adhesion processes arise when it is compared to adhesion between colloids for instance. Colloidal interactions are basically non-specific (or generic), in a sense that they are mostly governed by an interplay between Van der Waals attraction and electrostatic double-layer and steric repulsion, which leads either to colloidal aggregation or to colloidal stabilization.3 In contrast, interactions between cells are intrinsically specific, meaning that they are controlled by expression of receptors at the cell surface that bind specifically to the corresponding ligands of target cells by the lock-and-key principle. If the appropriate receptors and ligands are borne by opposing cells, their exquisite recognition properties allow them to initiate adhesive contacts. These specific interactions however add to all generic colloidal forces. In consequence, if receptors and ligands do not match, anarchic intercellular adhesion is inhibited due to the steric repulsion induced by the brushy glycocalix, which overcomes Van der Waals attraction between naked membranes. This repulsive barrier between cells is further enhanced by the electrostatic repulsion mediated by cell surface receptors carrying a large number of negatively charged residues.4 Even though the ionic strength of cell culture media is often relatively high (of order of 0.1 M), electrostatic interactions can indeed not be neglected, especially for cell-substrate adhesion. For instance, for efficient cell plating, polylysinecoated petri dishes are commonly used by cell biologists. In this case, the polymer-induced steric barrier is clearly overcome by non-specific electrostatic attraction between negatively charged cells and the positively charged surface. Figure 1 is a cartoon that summarizes the main possible interactions between colloids and between cells. 2.2.

A variety of cell adhesion molecules for specialized biological functions

Cell adhesion is an ubiquitous phenomenon. It is involved in most cellular physiological responses such as survival, growth, differentiation, proliferation, migration, as well as in many pathological situations such as metastasis formation, tissue inflammation and in hostbiomaterial interactions. All these biological functions are fulfilled and finely regulated by a plethora of cell adhesion molecules. To-date hundreds of proteins have been identified as adhesive molecules, and classified into four groups (integrins, selectins, cadherins and Ig superfamily1 ) even though they generally exhibit distinct structural and/or functional

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Fig. 1.

Schematic view of colloidal and intercellular interactions.

features. At most, one may itemize three robust common features. (i) Adhesion molecules are relatively long (of order of 20 nm or more) and contain several repeats. They can thus extent into the extracellular space, beyond the repulsive barrier due to the glycocalix; (ii) their density at the cell surface is quite low (typically 1000 molecules per cell), which indicates that adhesive contacts may be reinforced by diffusion of the adhesion molecules along the cell membrane; (iii) the majority of adhesion molecules contains an intracellular moiety that interacts with the cytoskeleton via one or several protein complexes, which suggests that the strength of cell-cell adhesion is not only due to the rupture of bonds between receptors and ligands, but involves the overall compliance of the cell. 3.

Attempts to Measure the Mechanical Strength of Cell Adhesion: Advances, Discrepancies and Missing Scales

One of the most fascinating features in intercellular adhesion relates to the dual property of cell adhesion molecules that participate to the cohesion and stability of tissues by establishing robust cell bridges, but authorize active morphogenetic rearrangements during embryo development through facile bond dissociation. How does this Janus-like property of noncovalent bonds between surface receptors and ligands impact on the mechanical strength of cell-cell or cell-substrate adhesion? How is cell adhesivity to a substrate or to another cell affected by the nature of the adhesion molecules that are involved? Two seemingly complementary approaches have been pursued over the last decades to address these issues. The first one is based on the development of assays to quantify the strength of adhesion at a cellular scale, mostly for practical biomedical purposes such as implant design. The second one is a fundamental approach, which aims at determining the relationship between the structures and the rupture forces of receptor-ligand bonds at a single molecule scale. However, since both the scales of investigation and the ultimate goals are significantly different, a consistent and integrated multi-scale understanding of the mechanics of cell adhesion is difficult to reach.

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3.1.

Classical assays to quantify cell adhesivity from separation force measurements

The ability of soluble adhesion proteins, such as vitronectin and fibronectin, which are present in blood plasma and tissue culture solutions, to adsorb on biomaterial surfaces may ultimately direct some adhesion-dependent cellular physiological responses. Consequently, there has been a great deal of research over the last three decades to quantify cell adhesivity to surfaces, and further optimize the design of artificial biocompatible materials used as implants, artificial organs or vascular grafts. Numerous attempts have been made towards this goal. The oldest and simplest attachment assay is based on rinsing the surface to remove weakly attached cells from the substrate and counting the remaining cells.5 Cells adhering with a force less than the rinsing force are removed, but this force is difficult to control and therefore ill-defined. More quantitative methods have been developed. While their working principle is similar and consists of measuring the ability of adherent cells to remain attached when exposed to a detachment force, they differ by the geometry of the experimental arrangement, and thus by the nature of the distractive force. As depicted in Fig. 2, the most prevalent techniques may be classified into three main categories: centrifugation, hydrodynamic shear, and micromanipulation. Here, we shall detail the basics of each experimental procedure in order to highlight their advantages and drawbacks. (i) Taking advantage of the slight difference in density between a cell and the surrounding medium, centrifugation was utilized to apply a normal force and separate the cell from the substrate [Fig. 2(a)].6–8 By progressively increasing the centrifugal acceleration and counting the number of remaining cells, an average separation force can be derived from the median of the obtained probability distribution. (ii) Another method to dislodge cells attached to a substrate consists of applying an hydrodynamic shear [Fig. 2(b)].9–11 In the radial flow detachment assay for instance, cells are sheared, and thus submitted to an hydrodynamic force, which is inversely proportional to the radial distance from the central stagnation point of the laminar flow. Further variation of the volumetric flow rate enables to cover a large range of hydrodynamic forces. Cell-tosubstrate adhesivity is then characterized by the critical shear stress that is sufficient to peel off half of the cell population. (iii) In contrast with the previous two methods that assess some average adhesive properties of an ensemble of adhering cells, an alternative approach is to test single-cell adhesion

Fig. 2. tion.

Classical cell detachment assays: (a) centrifugation; (b) hydrodynamic shear; (c) micromanipula-

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with various micromanipulation techniques [Fig. 2(c)], including AFM cantilever,12–14 magnetic tweezers,15, 16 or micropipettes.17, 19 In this kind of assay, which was also applied to intercellular adhesion, the maximal cantilever deflection during cell sweeping, the intensity of the magnetic field at cell-substrate unbinding or the aspiration pressure applied in the pipette prior to separation yield a measure of the separation force. These three classes of widely used assays advantageously provide a quantification of cell adhesion that integrates both the contribution of the specific bonds between adhesion molecules and the elasticity (or viscoelasticity) of cells. Although separation forces or stresses are measured precisely, closer inspection of the results obtained with the different techniques in similar conditions (same cell line, same surface) surprisingly reveals strong discrepancies. Whereas centrifugal forces required to displace cells attached to surfaces usually fall in the 1–10 nN range, separation forces measured by single-cell micromanipulation are typically 1 or 2 orders of magnitude higher, and hydrodynamic peeling forces are at least 10-fold lower. Consequently, even though each of these adhesion assays was proved to be highly reliable for comparative studies and practical purposes, the fact that the measured separation forces are assay-dependent suggests that additional care must be taken to reach a rigorous and intrinsic quantification of cell adhesion.

3.2.

Rupture forces of individual specific bonds

More recently, with the advent of ultra-sensitive force probes (AFM, optical tweezers, biomembrane force probe) and their use in biology,20 a molecular approach of the mechanics of cell adhesion has been undertaken. As seen in paragraph 2.2, the regulation of adhesiondependent physiological processes mostly originates from the expression of specialized adhesive molecules. One may thus anticipate that cell adhesivity is also mostly determined by the sole mechanical strength of these specific bonds. In 1999, E. Evans pioneered a new force technique, named Dynamic Force Spectroscopy, which was originally dedicated to the study of individual bonds between adhesion molecules immobilized on solid surfaces (Fig. 3).21 He thus showed that (i) the rupture force of individual non-covalent bonds depends on the loading rate, or equivalently on the velocity at which the bond is pulled apart, and (ii) the force spectrum provides a direct map of the energy landscape of the bond. The main merit of this approach is to gain insight with unprecedented precision in the relation between strength, lifetime and structure of specific molecular bridges. However, if this molecular approach is aimed to be considered as a minimalist tool for exploring the mechanics of cell adhesion, one has to recognize several difficulties. First, by grafting isolated specific molecules on solid surfaces, all cell signaling processes tightly coupled to bond formation are automatically suppressed, even though they might be of primary importance in setting the strength of adhesion.22 Second, even in the smallest adhesives structures, such as focal contacts, which are observed in vivo, hundreds of bonds are commonly involved. The impact of collective effects on the rupture force of clusters of bonds is not trivial to tackle experimentally.23

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Fig. 3. Dynamic Force Spectroscopy of specific molecular bonds: (a) the Biomembrane Force Probe used to measure rupture forces at a single bon level; (b) Energy landscape derived from force measurements.

3.3.

About the unbridged gap between molecular and cellular scales

Fundamentally, the mechanics of cell adhesion is a multi-scale problem, which needs to integrate the molecular details of specific bonds to the overall cellular response upon detachment. So far, the two routes that have been investigated did not permit to bridge the gap between molecular and cellular scales. The abovementioned detachment assays could be envisioned as a first step of a top-down approach. However, the inconsistency in the obtained results between different methods for the separation force makes them only reliable for comparative and semi-quantitative studies. Conversely, single bond force spectroscopy could be taken as the initial stage of a bottom-up approach. But, adding further complexity to design some realistic cell mimics seems to be out of reach for now.

4.

Refreshing Old Soft Matter Concepts in Cell Biology

Throughout his whole career, Pierre-Gilles de Gennes successfully demonstrated that general concepts can be applied to analyze very different problems in physics. When he became interested in biology, he also proposed surprising analogies between some complex biological processes and some well-understood physics issues (we think for instance of the one between self-trapping of bacteria in their cloud of chemo-attractants and the polaron problem 24 ). Actually, almost 10 years before actively contributing to the field of cell adhesion, de Gennes proposed that the methodology to address rigorously and quantitatively cell adhesion should be identical to the one that has been pursued for studying bonding between soft objects in polymer physics. Probably because cells are living, and thus complex objects, and because cell adhesion is mediated by specific molecules, which do not exist anywhere else in dead inert matter, the two caveats made by de Gennes in the preface of a book entitled “Studying Cell Adhesion” were partly ignored by the biology/biophysics community. It is however worth recalling these two enlightening messages taken from polymer physics that he has made more concrete later.

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4.1.

Adhesion energy versus separation force

Instead of attempting to rephrase what de Gennes stated clearly, let us reproduce the first sentences of the book preface. “We know that cellular adhesion plays an essential role in tissue build up and tissue function. But it is hard to measure. Indeed, it is even hard to define. When we join two solid blocks via a glue, our first (na¨ıve) mode of thinking focuses on the critical stress σc required to separate them. However, this is not the material parameter defining the quality of the glue. The true material parameter is the adhesion energy G, i.e. the energy per unit area required to separate the two partners. Of course, the critical stress σc is related to G, but it also depends on the thickness W of the glue layer: σc ∼ (µG/W )1/2 , where µ is the elastic modulus of the glue. Thus σc is not a material parameter. Cell/cell adhesion is a case of bonding between soft objects. But, here also, it is not well expressed in terms of separation force; it is defined more fundamentally by an adhesion energy. (. . .) For future quantitative studies, the definition of adhesion in terms of the energy G will have to be kept in mind.” Figure 4(a) shows a cartoon of the industrial test mentioned by de Gennes to assess the performances of a glue. By making this remark, de Gennes settled the basis for a proper analysis of cell adhesion. The choice of the separation force as a parameter to characterize cell adhesivity in most detachment assays was unfortunate and may explain the observed discrepancies depending on the experimental geometry. Nonetheless, the ideal cell adhesion assay still remained to be designed, in such a way that the adhesion energy, which is the intrinsic parameter of the problem, may be easily derived from the detachment force, which is the measurable parameter. Before discussing de Gennes’ concrete proposal for such measurements, we shall mention the second caveat that he made about cell adhesion in the book preface.

Fig. 4.

Classical polymer adhesion assays: (a) industrial test; (b) Johnson-Kendall-Roberts (JKR) test.

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4.2.

The role of connectors

In paragraph 2.2, we mentioned that adhesion receptors are often quite long. This extended length allows them to protrude out of the glycocalix and perform an efficient tip-to-tip recognition with the partner ligand on the opposing cell or substrate. However, de Gennes pointed out that the presence of a spacer, which contributes to the overall length of the adhesion molecule, may also strengthen cell adhesion. He considered “a connector firmly bound at both ends to the cytoskeleton of the two partner cells. The connector is made up of one recognition site C plus two spacers. The spacers are assumed to be more or less linear (not globular): they are formed of N repetitive units (each of them can be either flexible or rigid: this is not the problem). When the two complementary pieces, emerging from both cells, establish contact at point C, this implies a certain binding energy Ub . The equilibrium interfacial energy is γ = νUb , where ν is the number of connectors per unit area. But the adhesion energy G is much larger: G ∼ νN U b . This was first understood, in a different context, by Lake and Thomas. The point which they made may be stated as follows: when I hold an elastic rubber band under tension, if somebody comes with a pair of scissors and cuts it into two halves, the two halves snap back on my hands, and the energy which I (painfully) experience on my hands is much larger than the energy required for the scissors to clip the band.” (Fig. 5). Here, the message is that understanding how the kinetics of bond formation and dissociation impacts on the rupture force of individual specific bonds (as it is usually done in Dynamic Force Spectroscopy) is insufficient to account for the true mechanics of cell adhesion: spacers, such as multiple immunoglobulin domains in series, do not only permit facilitated recognition, but may serve as powerful amplifiers of the adhesion energy. This is a direct lesson from previous works in polymer adhesion by de Gennes and others. 25, 26

Fig. 5. De Gennes’ lecture note presenting the Lake and Thomas argument (private archives — PierreGilles de Gennes).

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5.

The PGG Cellular Version of the JKR Test to Probe Cell Adhesion

In their paper “Unbinding of vesicles” published in the Comptes Rendus de Physique in 2003, F. Brochard–Wyart and P.-G. de Gennes have proposed a complete mechanical analysis for a well-defined adhesion assay that provides for the first time (i) a facile determination of the adhesion energy from the measurement of the unbinding force, and (ii) a way to link the molecular parameters of receptor-ligand interactions (as derived from Dynamic Force Spectroscopy) with the dynamics of cell detachment. At first sight, the proposed experimental arrangement is nothing else but a classical micromanipulation adhesion test. However, Brochard–Wyart and de Gennes suggested subtle, yet decisive, changes in the experimental procedure. Their point was to reproduce the Johnson–Kendall–Roberts adhesion test, which is widely employed in the field of polymer adhesion, with the only difference that the solid elastic body used as a test surface in the JKR test is replaced here by a cell. Later, F. Brochard–Wyart dubbed this assay the “Cellular JKR test”. The beauty of the JKR geometry [Fig. 4(b)] is that the adhesion energy is simply given by the product of the pull off force and the size of the elastic sphere (if one omits some numerical prefactors). Similarly, by analyzing the deformation of the adhering cell, modeled

Fig. 6.

De Gennes’ calculation of the “lemon problem” (private archives — Pierre-Gilles de Gennes).

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as a simple liquid shell, and thus solving the “lemon problem” (Fig. 6), Brochard–Wyart and de Gennes found a JKR-like relation (with different numerical prefactors). As for the JKR assay, the characteristic length of the problem is the radius of the cell, or more precisely, the equatorial radius of the stretched cell prior to detachment. Although this finding may seem obvious for experts in polymer adhesion, it may look quite counter-intuitive for biologists or biophysicists, which would anticipate the size of the adhesive contact to be the relevant length scale. Beyond the rigorous determination of the adhesion energy between a cell and a solid surface, this short selected paper contains a second key idea related to the dynamics of detachment. By extending the Evans theory,21 they found that the time required for a cell to unbind from a substrate at a given force (close to the threshold separation force) is directly related to the molecular parameters of the specific bond (in particular, its activation energy). In brief, the proposed Cellular JKR test yields a direct measure of the separation force and time, which are unambiguously related to the macroscopic adhesion energy, G, and the molecular activation energy. In this perspective, to the best of our knowledge, this approach is quite unique, since previous adhesion assays failed both in determining G, which is yet the intrinsic parameter of the problem, and in accounting for the thermodynamic properties of specific molecular bonds.

6.

Perspectives

Both aspects of the methodology and analysis proposed by Brochard–Wyart and de Gennes have been experimentally checked on a well-characterized system, namely a red blood cell decorated with a prototypical ligand (biotin) adhering to a solid surface coated with the corresponding receptor (streptavidin).27 Additionally, in this paper, Brochard–Wyart and coworkers have shown that spacers between connectors greatly influence the dynamics of unbinding, as expected from the Lake and Thomas argument. The JKR approach was also successfully tested with eukaryotic cells adhering to each other via soluble polymer depletion.28 It might even seem a posteriori surprising that the simplest form of the JKR theory works so well for cell adhesion, since cells are intrinsically viscoelastic bodies. Yet, one has to keep in mind that the procedure recommends to apply step tensile forces, which thus correspond to high frequency deformations. In this regime, the apparent oversimplification, which consists of neglecting any viscous dissipation, is therefore fully valid. From a theoretical point of view, other groups did follow up with the same approach and developed interesting variants.29 Even though the time since the original publication is too short to faithfully evaluate the impact of the discussed paper, we may attempt to extract some trends. The fact that the cellular JKR configuration is a single-cell method, which is time consuming if statistical results are desired, is a severe limitation for bioengineers interested in implant biocompatibility or tissue engineering. The development of qualitative high-throughput screening assays exploiting the emergence of sophisticated microfluidic chips should remain a priority.30, 31 On the contrary, the cellular JKR assay proposed by de Gennes and Brochard–Wyart may become

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a standard methodology to address fundamental issues in cell adhesion. Our belief is that it might even reconcile two opposing viewpoints: the one of biologists, who wish to highlight the molecular peculiarities of their adhesion proteins, and the one of (bio)physicists, whose current effort is often to provide general material descriptions of cells. To this respect, we think that the present publication is a good illustration of two main features in de Gennes’ way of working: his wish to abolish the borders between disciplines and his way to capture apparently complex phenomena with simple unifying ideas or pictures. “A map where one mile equals one mile never ever helped anybody” (quote by L. Mahadevan). The maps drawn by de Gennes during his career did indeed help us stroll safely in the vast landscape of physics, and in some engaging areas at the border with biology.

References 1. J.-P. Thiery, Cell adhesion in cancer, C. R. Physique 4, 289–303 (2003). 2. P.-G. de Gennes, in Studying Cell Adhesion, eds. P. Bongrand, P. M. Claesson and A. Curtis (Springer, Heidelberg, 1994). 3. J. N. Israelachvili, Intermolecular and Surface Forces (Academic Press, London, 1992). 4. E. Sackmann and R. Bruinsma, Cell adhesion as wetting transition, Chem. Phys. Chem. 3, 263–269 (2002). ¨ 5. B. T. Walther, R. Ohman and S. Roseman, A quantitative assay for intercellular adhesion, Proc. Nat. Acad. Sci. USA 70, 1569–1573 (1973). 6. D. R. McClay, G. M. Wessel and R. B. Marchase, Intercellular recognition: quantitation of initial binding events, Proc. Nat. Acad. Sci. USA 78, 4975–4979 (1981). 7. L. Chu, L. A. Tempelman, C. Miller and D. Hammer, Centrifugation assay of IgR-mediated cell adhesion to antigen-coated gels, AIChE J. 40, 692–703 (1994). 8. O. Thoumine, A. Ott and D. Louvard, Critical centrifugal forces induce adhesion rupture or structural reorganization in cultured cell, Cell Motil. Cytosk. 33, 276–287 (1996). 9. H. M. Kowalczynska and M. Nowak, Effect of cell-substrate interaction time and shearing force on adhesion of L1210 cells to collagen and glass, J. Cell Sci. 66, 321–333 (1984). 10. C. Cozens-Roberts, J. A. Quinn and D. A. Lauffenburger, Receptor-mediated cell attachment and detachment kinetics: II. Experimental model studies with the radial-flow detachment assay, Biophys. J. 58, 857–872 (1990). 11. E. Decav´e, D. Garrivier, Y. Br´echet, B. Fourcade and F. Br¨ uckert, Shear flow-induced detachment kinetics of Dictyostelium discoideum cells from solid substrate, Biophys. J. 82, 2383–2395 (2002). 12. G. Sagvolden, I. Giaever, E. O. Pettersen and J. Feder, Cell adhesion force microscopy, Proc. Nat. Acad. Sci. USA 96, 471–476 (1999). 13. C.-C. Lee, C.-C. Wu and F.-C. Su, The technique for measurement of cell adhesion force, J. Med. Biol. Eng. 24, 51–56 (2004). 14. M. Benoˆıt and H. E. Gaub, Measuring cell adhesion forces with the atomic force microscope at the molecular level, Cells Tissues Organs 172, 174–189 (2002). 15. C.-M. Lo, M. Glogauer, M. Rossi and J. Ferrier, Cell-substrate separation: effect of applied force and temperature, Eur. Biophys. J. 27, 9–17 (1998). 16. N. Walter, C. Selhuber, H. Kessler and J. P. Spatz, Cellular unbinding forces of initial adhesion processes on nanopatterned surfaces probed with magnetic tweezers, Nanolett. 6, 398–402 (2006). 17. A. T¨ ozeren, K.-L. P. Sung, L. A. Sung, M. L. Dustin, P.-Y. Chan, T. A. Springer and S. Chien, Micromanipulation of adhesion of a Jurkat cell to a planar bilayer membrane containing

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Lymphocyte function-associated antigen 3 molecules, J. Cell Biol. 116, 997–1006 (1992). 18. E. Evans, D. Berk, A. Leung and N. Mohandas, Detachment of agglutinin-bonded red blood cells: II. Mechanical energies to separate large contact areas, Biophys. J. 59, 849–860 (1991). 19. Y.-S. Chu, W. A. Thomas, O. Eder, F. Pincet, E. Perez, J.-P. Thiery and S. Dufour, Force measurements in E-cadherin-mediated cell doublets reveal rapid adhesion strengthened by actin cytoskeleton remodeling through Rac and Cdc42, J. Cell. Biol. 167, 1183–1194 (2004). 20. P. Nassoy, How accurate are ultra-sensitive biophysical force probes?, Biophys. J. 93, 361–362 (2007). 21. R. Merkel, P. Nassoy, A. Leung, K. Ritchie and E. Evans, Energy landscapes of receptor-ligand bonds explored with dynamic force spectroscopy, Nature 397, 50–53 (1999). 22. E. A. Evans and D. A. Calderwood, Forces and bond dynamics in cell adhesion, Science 316, 1148–1153 (2007). 23. T. Erdmann, S. Pierrat, P. Nassoy and U. Schwarz, Dynamic force spectroscopy on multiple bonds: experiments and model, Europhys. Lett. 81, 48001 (2008). 24. Y. Tsori and P.-G. de Gennes, Self trapping of a single bacterium in its own chemoattractant, Europhys. Lett. 66, 599–602 (2004). 25. G. J. Lake and A. G. Thomas, The Strength of Highly Elastic Materials, Proc. Royal Soc. London A 300, 108–119 (1967). 26. F. Brochard-Wyart and P.-G. de Gennes, Adhesion between rubbers and grafted solids, J. Adhesion Sci. 57, 21–30 (1996). 27. S. Pierrat, F. Brochard-Wyart and P. Nassoy, Enforced detachment of red blood cells adhering to surfaces: statics and dynamics, Biophys. J. 87, 2855–2869 (2004). 28. Y.-S. Chu, S. Dufour, J.-P. Thiery, E. Perez and F. Pincet, Johnson-Kendall-Roberts theory applied to living cells, Phys. Rev. Lett. 94, 028102 (2005). 29. Y. Lin and L. B. Freund, Forced detachment of a vesicle in adhesive contact with a substrate, Int. J. Solids Structures 44, 1927–1938 (2007). 30. C. D. Reyes and A. J. Garcia, A centrifugation cell adhesion assay for high-throughput screening of biomaterial surfaces, J. Biomed. Mater. Res. 67A, 328–333 (2003). 31. H. Lu, L. Y. Koo, W. M. Wang, D. A. Lauffenburger, L. G. Griffith and K. F. Jensen, Microfluidic shear devices for quantitative analysis of cell adhesion, Anal. Chem. 76, 5257–5264 (2004).

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C. R. Physique 4 (2003) 281–287

Hydrodynamics and physics of soft objects/Hydrodynamique et physique des objets mous

Unbinding of adhesive vesicles Détachement des vésicules adhésives Françoise Brochard-Wyart ∗ , Pierre-Gilles de Gennes Institut Curie, section de physique et chimie, laboratoire de physico-chimie des surfaces et interface, 11, rue Pierre et Marie Curie, 75231 Paris cedex 05, France Presented by Guy Laval

Abstract We consider a vesicle, bound on one side to a pipette and sticking on the other side to a flat plate. When a pulling force f is applied to the pipette, the radius R c of the contact patch decreases, and jumps to zero at a critical value of the force. We present here an extension of the Evans theory for these processes. Then we discuss the dynamics of separation for two distinct cases: (a) nonspecific adhesion; and (b) specific adhesion induced by mobile proteins. To cite this article: F. Brochard-Wyart, P.-G. de Gennes, C. R. Physique 4 (2003).  2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. All rights reserved. Résumé On considère une vésicule qui, aspirée par une pipette d’un côté, adhère de l’autre sur une surface plane. Lorsqu’on tire sur la pipette avec une force f , le rayon du contact adhésif décroît, et s’annule brusquement à une valeur critique de la force. On présente ici une extension de la théorie d’Evans pour interpréter ces processus de détachement. Puis l’on discute la dynamique de la séparation pour deux cas distincts : (a) adhésion non spécifique ; et (b) adhésion spécifique par des protéines mobiles. Pour citer cet article : F. Brochard-Wyart, P.-G. de Gennes, C. R. Physique 4 (2003).  2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés.

1. Introduction A common way to test the strength of an adhesive contact of a cell on a substrate is to measure the force of detachment. The experiment is shown in Fig. 1 (here, the cell adheres to a plate, but it can also adhere on another cell). The experiment is performed as follows: the suction pressure P is increased step by step. At each step, the pipette is pulled. If P < Pc , the contact is maintained, and the vesicle is extracted from the pipette: adhesion wins. At P = Pc , the cell separates from the substrate and remains attached to the pipette. The force fc = Pc πRp2 , where R p is the pipette radius, is called the ‘breaking force’. We discuss here in Section 2 how fc is related to the separation energy W . To model the cell, we consider a vesicle, which sticks on a substrate. The adhesion may be nonspecific (i.e., depletion forces [1,2], electrostatic [3], or van der Waals attraction), or specific [4–7]: a vesicle, decorated with mobile cellular adhesion proteins (‘stickers’), is facing a wall grafted with the corresponding receptors.

* Corresponding author.

E-mail address: [email protected] (F. Brochard-Wyart). 1631-0705/03/$ – see front matter  2003 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés. doi:10.1016/S1631-0705(03)00048-3

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(c) Fig. 1. (a) A vesicle, held to the pipette under an increasing aspiration pressure P , sticks to a wall. As the pipette is moved outwards: (b) if P < Pc , the vesicle pipette junction becomes too weak and the finger is pulled out; or (c) if P > Pc , the junction is strong and the vesicle is ultimately detached from the wall. We show how this method is applied to break nonspecific bonds between cells (courtesy of Yeh-Shiu Chu and S. Dufour).

The paper is organised as follows: we first reconstruct the equilibrium state with no applied external force. Then, always assuming equilibrium, we see how the contact resists to an applied force, and suddenly breaks. This analysis is classical: the principles can be found in papers by E. Evans and coworkers [8,9]. However, we have attempted to present it in very simple terms. In the last sections, we describe the dynamics of detachment above fc .

2. Unstressed contact (f = 0) The free vesicle (Fig. 2(a)) maintained with the pipette before attachment is composed of a ‘finger’ of length h0 and a spherical part (radius Rv0 ). The tension γ is imposed by the suction pressure P = P0 − Pp [10]: 
1 1 γ ∼ P = 2γ − (1) =2 . Rp Rv0 Rp

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Fig. 2. (a) Free vesicle under aspiration by a micropipette maintained at a prescribed membrane tension γ ; (b) adhesion of the vesicle to a wall in the absence of external forces: the vesicle is spherical; (c) vesicle stretched under a force f : the shape (reminiscent of droplets hung on a fiber), is onduloidal.

The ‘bound’ vesicle (Fig. 2(b)) includes again a finger of length h plus a truncated sphere. The contact angle θE can be deduced from a capillary force balance γSV − γSL = γ cos θE ,

(2)

where γij are the substrate/vesicle, the substrate/liquid and the vesicle surface tension. The separation energy W = γ + γSL − γSV is directly related to θE from Eq. (2) W = γ (1 − cos θE ).

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The contact radius is Rc = Rv sin θE and the equilibrium distance between the plate and the pipette is de = Rv (1 + cos θE ). 2.1. Remarks on the free energy The contribution of the finger to the volume balance is negligible. The finger provides a reservoir of area: in the presence of a finger, we may change the area A of the truncated sphere without any work performed in the finger (because of the pressure balance equation (1)). We may thus write the free energy (at constant volume) in the form: F = γ A − πRc2 W + constant.

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For small θ the area A of the truncated sphere differs from the area Ai without contact (and with the same volume) by

 4 θ4 2 1+ θ A = Ai 1 + = 4πRv0 . (5) 16 16 We also have Rc = Rv θ . Inserting this into Eq. (4) we arrive at: θ4 − πW Rv2 θ 2 + const. 16 Optimizing this with respect to θ gives 2 f = 4πγ Rv0

1 2 γθ =W 2 and this is the Young equation (3) (for small θ).

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3. Unbinding = statics The pipette is now used not only to create a finger, but also to stretch the vesicle and to unbind it. The pipette is pulled from the plate with a force f (Fig. 2(c)). The free part becomes elongated, with a length d = de + δ. The extension δ(f ) has

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been calculated in detail in [8,9]. Using this calculation, red blood cells have been operated as soft spring to pull on one single molecular bond [8]. Very roughly δ ≈ f/γ (omitting logarithmic factors). Our aim here is to study how Rc decreases with the external force f . In [9], both the pipette radius Rp and the contact radius Rc were kept constant. For the sticking vesicle, what is maintained constant is the pipette radius Rp and the Young angle θE (Eq. (2)), imposed by a balance of forces. The contour of the vesicle is a surface of constant curvature (because the pressure inside is uniform) and looks like the profile of a droplet deposited on a fiber [10,11]. The profile can be derived from a balance of forces. On any section of the vesicle (radius r, angle θ shown in Fig. 2(c)), the projection of the force along the symmetry axis is constant and equal to f . It contains a surface term and a bulk pressure term: 2πrγ sin θ − πr 2 γ C = f.

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For f = 0, the solution is a sphere (C = C0 = 2/Rv ). For f = 0, we write Eq. (8) at both ends, and at the apex where the cross section radius is maximal (r = R, θ = π/2). f = 2πRp γ sin θp − πRp2 Cγ ,

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f = 2πRc γ sin θE − πRc2 Cγ , f = 2πRγ − πR 2 γ C.

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We set f¯ = f/(2πRγ ). Eq. (11) gives C = R2 (1 − f¯) and Eq. (10) becomes: R2 Rc sin θE − c2 (1 − f¯) ∼ f¯ = = Ψ θE − Ψ 2 (1 − f¯), R R where we put Ψ = Rc /R. The relation f¯(Ψ ) is Ψ θE − Ψ 2 f¯ = . (12) 1−Ψ2 The result is plotted in Fig. 3(a). For small θE , Ψ is small and we may replace the denominator by unity f (Ψ ) as a maximum for: Ψ ∼ = θE /2 1 (13) fmax = πR γ θE2 = πRW. 2 The maximal force is related to the maximal radius R (not Rc !) and to the adhesion energy W . Eq. (15) defines the rupture force. It is the intersection of f (R) by the line of slope 2πW (Fig. 3(b)). (i) If W γ , f¯ = W/γ is very small and the profile is almost spherical. From [9], R ∼ = Rv − fmax /(4πγ ), i.e., R/Rv ≈ 1 − W/(2γ ). This leads to: 
W ∼ frupt ∼ = πRv W. = πRv W 1 − 2γ (ii) If W  γ , the relation R(f ) must be calculated numerically.

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Fig. 3. (a) Plot of the force versus contact radius (in reduced units (Eq. (12))). Above f¯max = 41 θE2 , the contact is destroyed abruptly; (b) the intersection of the curve relating the force to the apex radius R with the line of slope π W gives the rupture force fR .

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Detachment of contact/versus extraction from the pipette As shown in Fig. 1, what is measured is not the force f acting on the pipette, but the critical pressure Pc . If P < Pc , the vesicle is extruded because the aspiration force fa fa = πRp2 P = 2πRp γ is smaller than the rupture force. On the other hand, as soon as P
Pc , the break arises at the wall. Thus fa (Pc ) = πW R allows us to measure W .

4. Dynamics of unbinding: the case of nonspecific adhesion 4.1. The rate equation If we pull on the pipette with a certain force f¯ (in reduced units), we induce a contact angle θ(t) different from the equilibrium value θe . The radius of the contact patch Rc (t) = Rv Ψ (t) decreases. The dissipation by viscous flow near the contact line creates a force opposing the noncompensated Young force F  1  F = γ (cos θe − cos θ) ∼ (14) = γ θ 2 − θE2 . 2 The viscous force is of the form [11] 1 dΨ η dRc = −kηRv , (15) −k θ dt θ dt where the factor θ −1 describes the strong dissipation present for narrow wedges. η is the solution viscosity and k is a numerical constant (ignoring logs)   Rv dΨ = −θ θ 2 − θE2 , (16) V ∗ dt where V ∗ = γ /kη2. We express θ in terms of ψ by the force equation (12), which can be expressed θ = Ψ + f¯/Ψ. (17) 4.2. The case of weak forces If f¯ < θe2 /4 = f¯c , there is an equilibrium patch. We can linearize Eq. (16) in the vicinity of the equilibrium conditions. We then find an exponential relaxation, with a relaxation time −1/2   Rv  ¯ τ= fc − f¯ f¯ → f¯c (18) 8V ∗ θE when f¯ < ˜ 0.9f¯c , this time is short (seconds). But when f¯ becomes very close to f¯c , there is a critical slowing down. 4.3. The case of strong forces We now assume that the pulling force is far beyond threshold f¯ > ˜ 2f¯c . Then the patch shrinks to zero: we focus our attention on the late stages, where ψ is small, and Eq. (16) reduces to f¯3 Rv dΨ =− 3. − ∗ V dt Ψ The law of decay in this regime is: 
t 1/4 , Ψ = Ψ0 1 − τl where
3 1 Rv Ψ04 1 Rc fc ≈ τl = 4 V ∗ f¯3 4 V ∗θ 3 f E since Ψ0 = θE . Thus the whole process is completed in a finite time. For usual conditions, τl is equal to a few seconds.

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5. Dynamics of unbinding with specific stickers We now discuss the case of specific adhesion, where a population of ‘stickers’ have built a dense adhesive patch, with an internal concentration (number of stickers/unit area) Γi which is high and fixed. The adhesion energy is then large: to observe an unbinding, we must choose a high surface tension γ (through the aspiration pressure (Eq. (1))). The stickers are torn out at the periphery of the adhesive contact. The gain of mechanical energy f dδ/dt is transferred into heat, when sites near the contact line are detached. On the other hand, the viscous dissipation due to the flow of surrounding water is now negligible. For each binder/receptor beginning to be separated by a vertical distance z, we expect a rate equation of the form [12] dz = V0 e−(B−ϕa)κT = V1 eϕa/κT dt

(22)

where V0 is a typical thermal velocity (of order 10 m·s−1 ), B the barrier energy (B ≈ 10 kT), ϕ is the pull out force on one binder and ‘a’ molecular length. Eq. (22) can be rewritten in the form aϕ = kT ln

Vz , V1

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dz = − dR dz . Following [12], we can construct the entropy loss due to the retraction of the where the vertical velocity Vz = V dx dt dx patch as an integral over all sites near the line that are partially detached. Per unit length,  (24) TS˙ = γ (1 − cos θ)V = Γi dx ϕVz .

Omitting coefficients of order unity, and again assuming (for simplicity) that θ is small, the balance of force can be written as: zm V 1 = γ θ 2, kT Γi *n a V1 2 where zm is the maximum length of a bonded pair, and a is a molecular diameter. We extract θ from Eq. (12). For small ψ , this reduces to θ = f¯/ψ . The rate equation is then
¯2  f Rv dΨ − = exp , V1 dt εΨ 2 where ε = 2kT Γi zm /γ a is a parameter of order unity (since γ is large) We set u = f¯2 /2εψ 2 and t˜ = t/τ , with τ = (Rv /V1 )(f¯/(ε1/2 ). The solution of Eq. (31) (neglecting logarithmic corrections) is:

 1 f¯2 1 t − = ln 1 − , ε ψ2 ψ2 τs i

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(26)

(27)

where τs =

2 2 2 2 f e−f /4π γ Rci ε . 2πγ V1 ε1/2

Here also, the time for detachment is finite. τs is maximal for f ≈ fc ε/θE . Remark 1. Eq. (26) can be understood if one assumes that the force f is distributed on the stickers at the periphery of the contact, on a band of width *s ≈ a/θ , proportional to f −1 . Remark 2. The tear out process controls the dynamics of unbinding if the specific time τs is larger than the hydrodynamic time τ* . One must compare the ‘wetting’ velocity W/η to V1 : viscous dissipation is dominant if W/η < V1 , i.e., for weak adhesion or small energy barriers.

6. Conclusions (i) It would be most useful to monitor not only the aspiration pressure but also the force f on the pipette. The threshold force fc for separation is: fc = πRv W

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and allows for a direct measurement of W . (ii) For nonspecific adhesion, or weak links (small activation energy), if we impose a force f significantly larger than fc , we predict a separation time τ* controlled by hydrodynamic friction (Eq. (21)). (iii) For strong specific adhesion, the separation time τs should be extremely sensitive to the pulling force (Eq. (27)). This regime is expected if τs > τ* , i.e., for a ‘wetting’ velocity W/η > V1 = V0 e−B/kT , where B is an activation energy. Tear out processes limit the rupture if the activation energy is large, while hydrodynamic losses are dominant for small adhesion or viscous solutions.

Acknowledgements We thank J.P Thiery, S. Dufour, S. Chu, P. Nassoy, P.H. Puech and A. Buguin for stimulating questions and discussions on the detachments of cells.

References [1] T.L. Kuhl, A.D. Berman, S.W. Hui, J.N. Israelachvili, Macromolecules 31 (1998) 8250–8257. [2] E. Evans, D.J. Klinggenberg, W. Rawicz, F. Szoka, Langmuir 12 (1996) 3031–3037; E. Evans, D. Needham, Macromolecules 21 (1988) 1822–1831; E. Evans, B. Kukan, Biophys. J. 44 (1983) 255–260. [3] A.L. Bernard, M.A. Guedeau, L. Julien, J.M. di Meglio, Langmuir 16 (2000) 6809. [4] A.L. Bernard, M.A. Guedeau, L. Julien, J.M. di Meglio, Europhys. Lett. 46 (1999) 101. [5] A. Boulbich, Z. Gutenberg, E. Sackmann, Biophys. J. 81 (2001) 2743. [6] J. Nardi, R. Bruinsma, E. Sackmann, Phys. Rev. E 58 (1998) 6340. [7] B.J. Carroll, J. Colloid Interface Sci. 57 (1976) 488; B.J. Carroll, Langmuir 2 (1986) 248. [8] E. Evans, D. Berk, A. Leung, Biophys. J. 59 (1991) 838. [9] P.A. Simson, F. Ziemann, M. Strigl, R. Merkel, Biophys. J. 74 (1998) 2080. [10] F. Brochard, J. Chem. Phys. 84 (1986) 4664. [11] P.G. de Gennes, Rev. Modern Phys. 57 (1985) 827. [12] F. Brochard-Wyart, P.G. de Gennes, PNAS 99 (2002) 7854. [13] T.L. Kuhl, Y. Guo, J.L. Alderfer, A.D. Berman, D. Leckband, J.N. Israelachvili, S.W. Hui, Langmuir 12 (1996) 3003–3014.

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Spreading Made a Splash

Lyd´eric Bocquet Laboratoire PMCN Universit´e Lyon 1 and UMR CNRS 5586 69622 Villeurbanne, France [email protected] . . .I would say to myself : “I will understand this, too, I will understand everything, but not the way they want me. I will find a shortcut, I will make a lock-pick, I will push open the doors.” It was enervating, nauseating, to listen to lectures on the problem of being and knowing, when everything around us was a mystery pressing to be revealed: the old wood of the benches, the sun’s sphere beyond the windowpanes and the roofs, the vain flight of the pappus down in June air. Would all the philosophers and all the armies of the world be able to construct this little fly? No, nor even understand it: this was a shame and an abomination, another road must be found. Primo Levi, The Periodic Table (1975)

As one reads these beautiful lines of Primo Levi, the personality of Pierre-Gilles de Gennes comes naturally to the mind. Undeniably, these two imposing figures shared a fascinating acuity of observation, a prolific creativity and a sobriety in expressing complex and esoteric ideas. In many domains, de Gennes built the new roads of understanding that Primo Levi was dreaming of. This “de Gennes’ touch” is particularly evident in his analysis of wetting and capillarity phenomena, that we will shortly discuss in this chapter. Capillarity is indeed a fascinating playground, where the obvious simplicity (and beauty) of the wetting phenomena — rain drops rolling down a windowpane, droplets on a spider’s web, soap bubbles, . . . — sharply contrast with the complexity and richness of the mechanisms involved. The understanding of apparently simple phenomena often raises considerable difficulties, both experimentally and theoretically, but also in the industrial domain. As de Gennes started to work on wetting in the 80’s, “intrigued by the process of assisted oil recovery”,1 many aspects of wetting were still a mystery pressing to be revealed (paraphrasing the words of Primo Levi). Of course a lot of work had already been performed,

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in particular by the Russian school, but “for the spreading of liquids on solids, progress has been slow” (citing the own words of de Gennes2 ). During the year 1984, de Gennes published a serie of short but illuminating papers on the statics and dynamics of wetting,4–14 where he was able to disentangle the various physical regimes at play and thereby draw the main lines of understanding of complex wetting phenomena. He provided straight and insightful arguments and suggested many “easy experiments” to explore their pertinence.1 He first summarized his views in his 1985’s Review of Modern Physics paper,2 and more recently in his textbook with Fran¸cois Brochard-Wyart and David Qu´er´e.3 It is interesting to note that although considerable progress has been made over the past 20 years, the field of wetting and capillarity remains very lively, with new questions and novel phenomena being constantly discovered (see for example the Lotus effect and its derivatives,3 or other phenomena that we will discuss in this chapter). In this chapter we shall focus on the contributions of de Gennes on the dynamics of wetting, starting from his prolific 1984’s publications on the subject.4–10, 12–14 We will start this exploration from one of his communications in the Comptes Rendus de l’Acad´emie des Sciences, Ref. 4 (published in french), in which he analyses the spreading dynamics of a droplet on a surface. Of course, we do not aim here at developing a thorough review of wetting and spreading questions, and we refer the reader to the various published reviews on the subject for more details: see e.g. the classical review by de Gennes himself,2 the one by L´eger and Joanny15 and more recently by Bonn et al.;16 see also the pedagogical book by de Gennes et al.3 Rather we shall merely recall the key ideas put forward by de Gennes in this 1984 paper and its companions, where he discusses the subtle balances between capillary driving, longrange forces and viscous dissipation. In particular, we shall more specifically emphasize on the intrinsically multiscale nature of the problem, from the microscopics to the macroscopics. Such scale bridges occur quite naturally in wetting problems and we will discuss some recent progress and perspectives along these lines.

1. 1.1.

Spreading Dynamics and Moving Contact Lines A few words on the statics

Before turning to the dynamics, some basic considerations on equilibrium are in order. Discussions on wetting usually starts with a drawing of a droplet deposited on a solid surface, such as in Fig. 1. Various cases occur, depending on the affinity of the liquid with the surface: cases (a) and (b) correspond to partial wetting, while case (c) correspond to perfect wetting (for water as a liquid, we would say case (a) corresponds to a ‘hydrophobic’ surface, while (b) to a rather ‘hydrophilic’ surface). A key quantity when drawing these pictures is the contact angle θe , between the liquid-vapor interface and the solid. As recognized a long time ago by Thomas Young, this angle is fixed by the free energy balance between the various interfaces, liquid-solid (SL), solid-vapor (SV ) and liquid-vapor (LV ): γLV cos θe = γSV − γSL

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Fig. 1.

Geometries of droplets deposited on a solid surface. Figure from Ref. 2.

This leads to the introduction of another key parameter: the spreading parameter, S, defined as S = γSV − γSL − γLV .

(2)

The solid is completely wetted by the liquid when S ≥ 0 and a droplet of liquid deposited on the surface will spread to develop a wetting liquid film on the solid surface.2 For negative spreading coefficients, the droplet does not spread completely and takes an equilibrium shape with a contact angle θe at the contact line between the liquid, vapor and solid. In practice however, as one tries to measure a contact angle, it happens that the contact line remains pinned and immobile for a finite range of angles around the equilibrium value. The bounds of this interval define the so-called ‘advancing’ and ‘receding’ contact angles.2 The origin of this hysteresis is attributed to chemical or physical heterogeneities on the solid surface, as discussed in the seminal work by de Gennes and Joanny.2, 14 While the above free energy balance does characterize the macroscopic behavior, the microscopic details in the region very close to the contact line actually play an important role in the wetting process. The structure of the meniscus in a “core” region close to the contact line is quite complex in general and depends strongly on the nature of interactions between the liquid and the solid (electrostatic or van der Waals forces, etc.).2, 9, 11–14 In the following, we will leave these questions aside (see Refs. 2 and 16 for detailed discussions on the subtleties of the statics of wetting), and focus on the dynamics of spreading.

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1.2. 1.2.1.

Starting to get moved Spreading dynamics

Equilibrium is not the end of the story and many practical situations involve displacement of the contact line with respect to the solid. The example of a droplet spreading down a car windowshield comes immediately to our mind; but actually many industrial processes involve coatings of surfaces with various (e.g. polymeric) liquids, requiring the hopefully homogeneous spreading of the liquid over the solid. As the contact line starts to move, (liquid) life becomes quite complex. The displacement of the contact line involves the motion of a liquid wedge moving with respect to the solid. Since according to the hydrodynamic dogma the flow velocity should obey the noslip boundary condition at the solid surface, the liquid will roll in the form of a caterpillar vehicle, as sketched in Fig. 2. This rolling motion was indeed nicely demonstrated in an experiment by Dussan and Davis,20 by vizualizing the motion of fluorescent spots deposited on the liquid surface. However, a difficulty raised by such a motion, as was first pointed out by Huh and Scriven,17 is that it leads to a divergence of the viscous dissipation in the corner. This follows from a simple estimate:2 the energy dissipated per unit volume by viscous channels is the viscosity (η) times the square of the velocity gradients, typically (U/h)2 , with U the wedge velocity and h the local fluid height. Altogether,  2
U (3) P = h(x) · dx × η h(x)

Fig. 2. The contact line motion (a) bares some similarities with that of a caterpillar vehicle (b). Figure from Ref. 2.

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The integral runs over the lateral size of the liquid wedge, say L. For a liquid wedge with slope θa , the meniscus height vanishes as h(x) = θa x, and the dissipation obviously diverges at its lower bound (in the corner). Imposing a lower cut-off at a distance xmin to get an idea of the behavior, one gets a logarithmic divergence of the dissipation as the distance xmin goes to zero:   L U2 P = 3η × log (4) θa xmin The divergence is weak, but still, considering literaly this behavior, moving the contact line costs an infinite energy and it then shouldn’t be able to move! Huh and Scriven concluded — with a somewhat strange word — that “the total force exerted on the solid surface is logarithmically infinite: not even Herakles could sink a solid” (to which they add . . . “if the physical model were entirely valid, which it is not”). Some mechanism has to be invoked to avoid the divergence of the dissipation in the corner. In other words, what would justify the existence of a minimum cutoff xmin and what is its physical signification? A bit later in the 70’s, a number of systematic experimental studies of contact line dynamics appeared in the litterature. In particular two studies, by Hoffman18 and Tanner,19 had an important impact. These authors considered two different spreading situations: while Hoffman studied the forced flow of various liquids in thin capillaries, Tanner considered the spreading dynamics of a droplet on a completely wetting surface (with a positive spreading parameter S ≥ 0), see Fig. 3.

Fig. 3. Two experimental geometries probing the spreading dynamics: in the Hoffman experiments, a fluid meniscus is forced inside a thin capillary (left); in the Tanner experiments, a droplet spreads over a completely wetting surface (right). Figure from Ref. 2.

Some key results emerged from these studies. In particular Hoffman showed that: (i) as the contact line moves, an apparent contact angle θd , larger than the static one, is measured;18 (ii) for perfectly wetting liquids, the apparent contact angle θd is related to the contact line velocity U , via the capillary number Ca = ηU/γLV , according to a rather universal relation Ca = F (θd ), while for partial wetting, experimental data could be described as Ca = F (θd ) − F (θe ). On his side, Tanner obtained that (i) the spreading dynamics of a (perfectly wetting) droplet is characterized by a slow expansion of the droplet radius scaling like R(t) ∝ tα with α  0.1, a relationship known as “Tanner law”; (ii) by an analysis of the equations for the shape of the contact line under viscous stress, he showed that the apparent contact angle is expected to scale like Ca1/3 .a a An

‘anecdotic’ but conceptual difficulty from the calculations by Tanner is that the viscous stress singularity in the liquid corner, along the point made by Huh and Scriven, forbids the contact line to truely reach the solid surface: as a consequence its profile cannot decrease below a minimum height. We shall come back on this point.

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This was basically the situation in the litterature as de Gennes entered the subject:b most actors were on stage but there was a bit of confusion in the scenario and especially a lack of distance and global view. In his first 1984 paper,4 de Gennes was able to reconcile the various points of view with an amazing economy of means. First he could recover the Tanner law for the spreading dynamics of a wetting droplet (and the associated relationship between the apparent contact angle and capillary number). But most importantly, he also made the key point that a precursor film develops in front of the contact line, thereby avoiding the viscous loss singularity. Let us discuss in more details the contents of the paper. De Gennes starts by reconsidering the energy balance at the contact line from a “macroscopic view”, meaning he first forgets about the fine structure of the meniscus near the contact line. Accordingly, he takes the above logarithmic term as a constant  = log(L/xmin ) when considering the dissipation balance (in a subsequent publication he makes the rather provocative statement that the “logarithmic singularity has caused a lot of agitation among hydrodynamicists — but is of minor scientific interest”22 ). At a “macroscopic” scale, the viscous loss should be compensated by the work supplied by the (uncompensated) capillary force F = γLV (cos θe − cos θa ) 

1 γLV (θa2 − θe2 ) . 2

(5)

De Gennes thus writes the energy balance as 3η

U2 1 = γLV (θa2 − θe2 ) × U . θa 2

(6)

This ‘simple’ balance gives immediately the relationship between the apparent contact 1 angle and the capillary number in the form Ca = 6/ θa (θa2 − θe2 ). For wetting surfaces (θe = 0), one recovers the Tanner result Ca ∝ θa3 , which is indeed measured experimentally.c Furthermore when applied to the spreading of a droplet, he recovers Tanner’s law for the spreading dynamics of a droplet 1  γLV 3 10 tΩ0 (7) R(t) ∼ η (R(t) the droplet radius and Ω0 its volume). Then, going deeper in the fine structure of the meniscus close to the triple line, de Gennes analyses the effect of long range forces on the profile. A remarkable prediction which comes out of this analysis is the existence of a thin precursor film, moving in front of the contact line. The origin of this film is quite clear: it takes its origin in the large viscous stress close to the contact line, which is balanced by the long-range interactions between the liquid and the solid surface. This film is expected to be very thin (but spreads over long b It

appears that de Gennes missed the work by Voinov published in 197621 in the Russian literature, in which Voinov obtains the meniscus shape from the detailed analysis of the hydro-capillary couplings in the liquid corner. c For partially wetting surfaces however, Eq. (6) slightly differs from predictions obtained from the detailed calculation of the meniscus profile, which merely takes the form θa3
θe3 + 9
Ca, a relationship known as the Cox-Voinov law.16, 21, 23

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Fig. 4. As a droplet spreads on a prefectly wetting surface, a precursor film of nanometric thickness develops in front of the droplet. Figure from Ref. 2.

distances). Its thickess is typically fixed by a microscopic scale a constructed on the basis of the surface tension γLV and  a Hamaker constant A associated with the long-range van om range, the thickness der Waals interactions: a = A/γLV . Although a is in the Angstr¨ h ≈ a/θa is much larger for wetting surfaces and typically in the tens of nanometers range. It is thus not visible by eye, but the reality of these nanometric precursor films has indeed been demonstrated in various clever experiments,2 in agreement with the predictions of de Gennes. The concept of a precursor film thus successfully remove the singularity problem at the contact line. Its existence furthermore provides the solution to another mystery: in the Tanner law, the spreading parameter S is absent, although this should be the driving force for spreading. The reason is actually that the free energy excess S is fully dissipated in the precursor film, as he demonstrated in a follow-up paper with H. Hervet.8 In the end, the spreading parameter S disappears from the apparent contact angle θa measured at the macroscopic scale. De Gennes applied similar ideas to describe the spreading of polymer melts in a subsequent publication with F. Brochard-Wyart.10 An important difference with ‘simple’ liquids is that polymers may exhibit very large slippage at the solid surface. As he predicted in a 1979 paper,31 the contrast between the entangled dynamics of the polymer in the bulk (highly viscous) and the friction at the monomer scale on the surface leads to a large slip effect at the surface. These effects were indeed confirmed experimentally by L. L´eger and coworkers.32 1.2.2.

Forced wetting

In all the above discussions, the contact line is implicitly assumed to advance on the solid surface. What does occur when the contact line recedes: is this situation symetric? Actually not. An interesting “forced wetting” transition occurs when the velocity is reversed. A typical configuration is a vertical plate pulled out of a bath. This situation occurs in many coating operations, or in the process of depositing Langmuir–Blodgett monolayers on a surface. Experimentally the contact line is found to become unstable above a threshold pulling velocity, and a film with finite thickness is entrained on the solid surface. De Gennes discussed this question in a short communication in 1986.24 His analysis follows the lines

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of the 1984 paper discussed above. The relationship between velocity and apparent contact angle is obtained by simply reversing the sign of the velocity in the balance equation, Eq. (6), giving: U=

γLV θa (θe2 − θa2 ) . 6η

(8)

An interesing feature of this relationship is that no solution for the apparent contact angle θa exists when the velocity is above a critical velocity, obtained as: 1 γLV 3 U = √ θe . 9 3 η

(9)

In other words the contact line disappears and a wetting film coats the solid surface. The properties of this entrained film have been described in the famous work by Landau and Levich.3 This transition has been critically revisited recently by a number of authors, see e.g. Refs. 26 and 27. While there is no doubt that a dynamical transition indeed exists, alternative transition scenario have been put forward. In particular, the detailed calculation of the meniscus shape rather leads to a continuous transition, in the sense that the apparent contact angle is vanishing at the transition threshold.25, 26 This alternative picture has been confirmed in detailed experiments, showing that the meniscus profile at the transition matches the corresponding equilibrium profile (for a perfectly wetting surface).27 Another interesting refinement of the transition scenario has also been put forward. While up to now we considered a straight contact line, some wiggling of the contact line has been observed close to the transition,2, 16 which for a droplet running down a surface may even lead to the formation of a corner at the backside and the ejection of small droplets.28 These behaviors are still the object of intense research.16

2.

Echoes from the Contact Line

The views of de Gennes on wetting have generated a tremendous amount of work, as stated by the huge number of quotations of his 1985 review paper: more than 20 years later, his review, and the ideas it contains, remains a reference and a guide for the understanding of wetting dynamics. We will definitely not discuss all the work done hereafter, but rather choose a few points in connection to the problems discussed above. 2.1.

Slippage effects

As explained above, de Gennes successfully showed that the singularity problem at the contact line raised by Huh and Scriven is removed by the existence of the precursor film. But actually the existence of such films becomes doubtful as the contact angle (apparent or equilibrium) strongly departs from zero, since their thickness becomes molecular and the validity of the above approach breaks down. Therefore the singularity problem remains in the absence of long range interactions between the liquid and the solid.

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A recurrent idea is the possibility of fluid slippage at the contact line. Fluid slippage is generally described by a slip length, say b, entering the Navier boundary condition: b∂n v|s = vs .

(10)

This condition thus replaces the no-slip boundary condition at the wall surface. It is quite obvious that this new “microscopic” scale will remove the singularity by providing a lower cutoff in the viscous dissipation in the corner. In practice the cutoff length in the logarithmic term, Eq. (4), is xmin = b/θa .10 This point was actually discussed by Huh and Mason and Hocking already in the late 70’s.29, 30 de Gennes was apparently reticent to this idea for ‘simple’ (not polymeric) liquids.2 Indeed while large slippage effects are indeed expected for polymer melts as he predicted,31, 32 slip lengths for water or other ‘simple’ liquids were a priori expected to be in the molecular range, so that the hypothesis of slippage appeared at that time merely as a concept, rather than a true physical reality. However over the years, a number of results have given substance to the slip phenomena and the question of hydrodynamic boundary conditions at solid surfaces has been extensively explored over the past 20 years. Powerful experimental tools have been developped which are now able to reach the appropriate resolution to measure nanometric slip lengths. The combination of such increasingly accurate measurement methods and of insights from simulation and theory has now produced a rather complete understanding.36 One important result emerging from these works is that hydrodynamic slippage is indeed a reality. The slip length is found to depend strongly on the strength of the liquid-solid interaction, e.g. measured by wettability. Slip lengths in the range of a few tens of nanometers are typically measured for water on hydrophobic surfaces (silanized surfaces). But on hydrophilic surfaces, a no slip boundary condition (i.e. a slip length below the nanometer) rather applies at the solid surface. Some aspects of slippage are still strongly debated and in some specific systems, like hexadecane on saphire,32 much larger slip length are measured. However the contact line is quite specific in this context. Using molecular dynamics simulations — where the equations of motion of a whole assembly of atoms are solved exhaustively — Thompson and Robbins have shown that a localized slippage effect indeed occurs at the contact line. This fact gives strong support to the slip idea put forward by Huh and Mason and Hocking. The question of the specific form for the boundary condition to be applied at the solid surface close to the contact line however remains open. Recent simulation results suggest that the Navier boundary condition is not appropriate to describe the slip velocity in the vicinity of the contact line. Alternative boundary conditions for the fluid velocity at the surface have been proposed in the form of a “generalized Navier boundary condition”, balancing the liquid friction at the surface — proportional to a velocity slip — with the viscous stress plus an unbalanced Young stress integrated across the contact line.34 Interestingly, the latter contribution is reminiscent of interfacial transport mechanisms occuring at diffuse interfaces.35 As shown by Qian et al., a “long-ranged” slip velocity decaying like one over the distance to the contact line is accordingly demonstrated in the simulations. The consequences of such effects on the spreading dynamics have not been explored.

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Fig. 5. Detailed spreading dynamics of alcohol droplets impacting a surface. The ‘splash’ (associated here with the corona in front of the spreading film) can be suppressed for sufficiently low ambient air pressure. Figure from Ref. 37.

2.2.

When spreading makes a splash

Up to now we were considering situations in which the fluid is gently forced to move on the surface. What if we violently force the liquid to spread? This is what occurs during the impact of droplets on solid surfaces, like rain drops falling on a windowpane. This situation has been studied in the group of Nagel.37 A remarkable feature emerging from these experiments is the existence of a threshold for splashing, which crucially depends on the pressure of the surrounding gas. This observation is summarized in Fig. 6. The role of the surrounding gas is quite surprising. Nagel and co-workers proposed to interprete this splashing transition in terms of a balance between a compression-induced stress, associated with the compressibility of the gas in front of the spreading liquid, and a capillary stress, which they estimate as the surface tension divided by the layer thickness. This criterion appeared to reproduce correctly the various trends measured in the experiment, but remains puzzling. A detailed theory explaining the origin of this splashing phenomenon is however still missing. As an alternative to the splash of a drop, one may consider the impact of a solid body in the liquid, i.e. the reverse situation . . . Dropping stones in the water is a game and a pastime that we all played once: an entertaining feature is the ‘splash’, that one may hear after the stone has impacted water. This sound is caused by the rapid closure of the air cavity created by the impact (if any). Although quite obvious, this problem brings a few surprises and the forced wetting transition we have discussed above comes unexpectedly into play in this problem. A first suprise which is illustrated on Fig. 6 is that one may fully avoid the splash by proper coatings of the impacting body: under the same throw conditions, a hydrophobic stone splashes while a hydrophilic one gently enters water without creating an air cavity, thereby producing a tiny ‘plop’. This influence of coating is not expected at all: in such

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Fig. 6. A solid body impacting a liquid surface can either produce a tiny ‘plop’ (left) or a big ‘splash’ (right). This is tuned by modifying the wettability of the impacting object (left: hydrophilic, θe = 10◦ ; right: hydrophobic, θe = 115◦ ).

throws indeed, the impact velocity is so large that both capillarity and viscosity effects should a priori play no role. But here it does: experiments show that there is a threshold velocity V  from the ‘plop’ to the ‘splash’ configurations (i.e. the threshold for air entrainement), and that this threshold is proportional to the capillary velocity. It is furthermore strongly dependent on the static contact angle θe on the surface body, as illustrated by Fig. 6. Altogether, the experimental results are well described by a law of the type: V =

γLV × f (θe ) η

(11)

with f a function of the contact angle, which, in the limit of ‘superhydrophobic’ impacting bodies with θe ∼ π, is well described by f (θe ) ∼ (π − θe )3 .38 The dependence on viscosity is really puzzling in this splashing situation where the inertia largely dominates all other forces (high Reynolds and Weber number regimes): the liquid should only enter via its mass density. All these observations are strikingly reminiscent of the discussion of the forced wetting transition by de Gennes, see above: compare the observed experimental threshold for the splash, described by Eq. (9) with the threshold velocity for the contact line destabilization, as predicted in Eq. (11). The main difference is that it involves the contact angle with respect to the vapor phase π − θe . The reason why the forced wetting transition plays a role here is hidden in the detailed dynamics of the splash. As the body enters the water, a thin film develops and spreads over its surface, as shown on Fig. 7. This film takes its origin in inertia (a large pressure excess developping at the interface between the sphere and the water surface). But its stability is a matter of forced wetting. The only difference to the situation described in the previous paragraph is that it is the air phase which is forced between the liquid film and the sphere surface, and we expect the complementary angle π − θe to enter the threshold criterion, as it does indeed. It is amusing that the contact line instability, as discussed by de Gennes, plays a role in such an extreme and violent spreading situation. It is only via a unforeseen bridge over scales that wetting and viscous effects do matter. De Gennes made some interesting

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Fig. 7. Zooming in on the impact dynamics. A liquid film develops and spreads over the impacting object surface. Below the velocity threshold (top), the film gathers at the top, but is ejected from the surface above the threshold (bottom).

Fig. 8.

The eleven bulls by Picasso.

comments on this splash phenomenon, pointing to questions concerning viscoelastic fluids, or the dynamics of surfactants if a Langmuir Blodgett film is deposited on the surface.d

3.

Some Final Words

A leitmotiv in the dynamics of spreading and wetting is the broad range of spatial scales which it does involve: from the macroscopic wedge, described by an apparent contact angle, down to the microscopic scale of the precursor film, 5 to 6 orders of magnitude separate these two extreme scales . . . It is a tour de force of de Gennes to have disentangled these intimately connected effects, extracting the key driving mechanisms (and thereby minimizing the use of d As

a sad story, I received the message from P.-G. de Gennes a few days after he passed away.

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the technical artillery). As he quotes in his 1985 review: “It is only by a patient separation of different physical regimes that one may hope to reach general laws. . .” This quest towards sobriety is a constant in the work by de Gennes: by drawing a few main lines, he catched the key point, leaving aside unnecessary details. Such an elegance calls another big hero to our mind. In his celebrated serigraphy “The Bull”, Picasso started with an animal described in his full complexity and realism . . . but, state after state, stripped the drawing of superfluous ornaments to reach its essence. Pierre-Gilles de Gennes had this astounding talent to draw our complex world in a few strokes. References 1. F. Brochard-Wyart, Nature 448, 149 (2008). 2. P.-G. de Gennes, Rev. Mod. Phys. 57, 827 (1985). 3. P.-G. de Gennes, F. Brochard-Wyart and D. Qu´er´e, Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves (Springer, 2003). 4. P.-G. de Gennes, C. R. Acad. Sci. 298II, 111 (1984). 5. P.-G. de Gennes, C. R. Acad. Sci. 298II, 439 (1984). 6. P.-G. de Gennes, C. R. Acad. Sci. 298II, 475 (1984). 7. P.-G. de Gennes, C. R. Acad. Sci. 300II, 129 (1984). 8. H. Hervet and P.-G. de Gennes, C. R. Acad. Sci. 299II, 499 (1984). 9. B. Legait and P.-G. de Gennes, J. Phys. Lett. 45, 499 (1984). 10. F. Brochard-Wyart and P.-G. de Gennes, J. Phys. Lett. 45, L597 (1984). 11. P.-G. de Gennes, C. R. Acad. Sci. 297II, 9 (1983). 12. J.-F. Joanny and P.-G. de Gennes, C. R. Acad. Sci. 299II, 605 (1984). 13. J.-F. Joanny and P.-G. de Gennes, C. R. Acad. Sci. 299II, 279 (1984). 14. J.-F. Joanny and P.-G. de Gennes, J. Chem. Phys. 81, 552 (1984). 15. L. L´eger and J.-F. Joanny, Rep. Prog. Phys. 55, 431 (1992). 16. D. Bonn et al., Rev. Mod. Phys. (in press, 2009). 17. C. Huh and L. E. Scriven, J. Colloid Interface Sci. 35, 85 (1971). 18. R. Hoffman, J. Colloid Interface Sci. 50, 228 (1975). 19. L. Tanner, J. Phys. D 12, 1473 (1979). 20. V. E. Dussan and S. Davis, J. Fluid Mech. 65, 71 (1974). 21. O. V. Voinov, Fluid Dynamics 11, 714 (1976), translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, 5, 76 (1976). 22. F. Brochard-Wyart and P. G. de Gennes, Adv. in Colloids and Interface Sci. 39, 1 (1992). 23. R. G. Cox, J. Fluid Mech. 168, 169 (1986). 24. P. G. de Gennes, Colloids Polymer Sci. 264, 463 (1986). 25. O. V. Voinov, J. Colloid Interface Sci. 226, 5 (2000). 26. J. Eggers, Phys. Rev. Lett. 93, 094502 (2004). 27. J. H. Snoeijer, G. Delon, M. Fermigier and B. Andreotti, Phys. Rev. Lett. 96, 174504 (2006); G. Delon, M. Fermigier, J. H. Snoeijer and B. Andreotti, J. Fluid Mech. 604 (2008). 28. T. Podgordski, J. M. Flesselles and L. Limat, Phys. Rev. Lett. 87, 036102 (2001). 29. C. Huh and S. G. Mason, J. Colloid Interface Sci. 60, 11 (1977). 30. L. M. Hocking, J. Fluid Mech. 79, 209 (1977); J. Mech. App. Math. 34, 37 (1981). 31. P.-G. de Gennes, C. R. Acad. Sci. 288, 219 (1979). 32. H. Hervet and L. L´eger, C. R. Physique 4, 241 (2003). 33. P. A. Thompson and M. O. Robbins, Phys. Rev. Lett. 63, 766 (1989). 34. T. Qian, X.-P. Wang and P. Sheng, Phys. Rev. E 68, 016306 (2003); Phys. Rev. Lett. 93,

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35. 36. 37. 38.

094501 (2004). A. Ajdari and L. Bocquet, Phys. Rev. Lett. 96, 186102 (2006). L. Bocquet and J.-L. Barrat, Soft Matter 3, 685 (2007). L. Xu, W. W. Zhang and S. R. Nagel, Phys. Rev. Lett. 94, 184505 (2005). C. Duez, C. Ybert, C. Clanet and L. Bocquet, Nature Phys. 3, 180 (2007)

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P H Y S I Q U E D E S S U R F A C E S E T D E S I N T E R F A C E S . — Dynamique d'etalement d'une goutte. N o t e de Pierre-Gilles de Gennes, Membre de 1'Academic Remise le 12 decembre 1983, acceptee le 9 Janvier 1984. O n analyse la progression d ' u n coin fiuide sur u o solide ( d a n s le cas o u Tangle d e contact t h e r m o d y n a m i q u e Qe est mil : regie d ' A n t o n o v satisfaite) en t e n a n t c o m p t e des interactions V a n der W a a l s a longue portee. O n t r o u v e : (a) u o angle de contact apparent60 relie a la vitesse d ' a v a n c e e U p a r 0 * — U q / y ( q = viscosite, y tension superficielle d u liquide) d ' o u u n e loi r a y o n / t e m p s d'etalement p o u r u n e g o u t t e r(t}~tlno; (b) u n film precurseur d'epaisseur £ ( x , f) decroissant asymptotiquement c o m m e x _ 1 oil x est l a distance a la ligne triple. L'epaisseur h* d u film a u voisinage d e l a ligne triple est / i * ~ a / 6 a ( o u a est u n e distance atomique). Ceci permet d e c o m p r e n d r e le fait (reconnu) q u e le film precurseur est bien visible seulement si Tangle d e contact t h e r m o d y n a m i que est nul. SURFACE

AND INTERPHASE

PHYSICS.

— T h e Dynamics of a Spreading Droplet.

We analyse the shape of the liquid-air interface for a droplet spreading on a solid, in a regime where the Antonov rule is satisfied, taking into account the long range Van der Waals interactions between liquid and solid. We find: (a) an a p p a r e n t contact angle 9„ related to the velocity U of the triple line by 9 f ~ U q / y ( q = viscosity, y surface tension of the liquid). This leads to a law of spreading (radius rj'time t) for a droplet r~t110; (b) a precursor film of thickness E, decreasing asymptotically likex"1, where x is the distance from the triple line. The thickness h* of the film at this line is h*~ajQa where a is an atomic length: this explains why the precursor films are observed only when the thermodynamic contact angle vanishes.

I. DESCRIPTION ELEMENTAIRE DE L'ETALEMENT.

— L a dynamique des phenomenes

de

mouillage est complexe, et controverseefl]. N o u s considerons ici une petite goutte (rayon r, epaisseur maximum e) s'etalant sur un solide plan sous l'effet des seules forces capillaires. Contrairement a Y. Pomeau[2] mais en accord avec d'autres auteurs[l a] nous pensons que la forme macroscopique de la goutte est une calotte spherique dans le regime de temps interessant : dans la region epaisse, les pressions s'equilibrent rapidement, la difference de pression entre fiuide et air est constante, d ' o u une courbure constante definie p a r la condition de Laplace. N o u s allons discuter ici la dynamique en supposant q u e les seuls effets importants sont la capillarite et la viscosite du fiuide : (a) pas de rugosite ou d'inhomogeneite de la surface; (b) viscosite de la phase gaz negligeable p ] . Commencons p a r une discussion elementaire, qui permet de cerner les phenomenes priscipaisx. Postulons pres de la ligne triple u n coin liquide d'angle constant ^ = 8 a x (fig. 1) et s u e vitesse de ligne — U . (Pour simplifier, nous supposerons tous les angles petits 0CT YSL + YLG e t aboutit a des cinetiques beaucoup plus rapides. Avec un support solide, l'analyse de Cahn[7] suggere que Fegalite d'Antonov est en general satisfaite. D'apres l'equation (6), l'aire A = 7 t r 2 croit comme t0' 2 , ce qui est conforme a u n certain nombre d'observations, mais pas a toutes [1 b\ La loi d'echelle deduite de (6) a la forme : (7)

A~Q3/5(V*r)

1,'S

ou Q ( ~ e 0 r o ) est le volume de la goutte. Cette dependance en Q. est peu differente de celle proposee p a r M a r m u r [1 b] (Q 2 / 3 ). L a dependance en t a ete, semble-t-il, comprise en premier par Tanner [11]. II.

STRUCTURE FINE AU VOISINAGE DE LA LIGNE TRIPLE. — N o u s allons m a i n t e n a n t decrire

le profil liquide ^(x, t) pres de la ligne triple, dans l'approximation dite de lubrication (qui est acceptable si l'angle de contact apparent 8 n est petit). N o u s incluons maintenant l'interaction a longue portee fluide/mur : pour un element de fluide de volume dv a une

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113

distance z du mur, cette interaction vaut dv W (2). Avec des forces de va: - der Waals non retard&, la forme explicite est : (8)

W(z)= - A (A =Cte de Hamaker). 6n:z3

Le champ de pression dans l'kpaisseur du liquide ( 0 t z t > S la pente est A peu p r h constante, et l'angle de contact apparent est :

avec I = In r / 4 b + Cte. Nous avons pris arbitrairement x = r/4 comme point de raccord (Ce raccord est ktudik avec beaucoup plus de soin dans les rkfkrences ( [ 5 ] , [Ill), mais l'effet sur 1 est assez faible). Noter l'analogie entre (3') et (3) pris pour 8,=0 : les

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coefficients diff6rent seulement de 15 %. Donc les conclusions macroscopiques (tquation 6) sont pratiquement iuchangks. 2. Rkgime du $Em (x t 0). - Lorsque W' est dominant dans (11) on a une solution simple :

(c)

qui dkcrit un film d'kpaisseur lentement dkcroissante pour x t o . Le paramGtrex, est (pour le moment) indktermink. 3. Raccord. - Celui-ci pourra etre effectuk par une mkthode systtmatique de perturbations singulieres. Nous nous limiterons ici a une discussion nayve, obtenue en raccordant un coin simple ,C = 0, x a la solution (12') en un point x = x * a dkterminer. L'accord des valeurs et des pentes impose :

et l'epaisseur du film au point de raccord vaut : (15)

h * r c ( x * ) z a 0,''

06 a =(Aly) ' f 2 a l'ordre de grandeur d'une distance interatomique. 111. DISCUSSION- 1. Le rksultat(l5) montre que le film est significatif seulement pour 8, faible. Par exemple 8,= conduit a h* = 100a, ce qui est une valeur raisonnable pour l'utilisation de la formule Van der Waals non retard& dans I'kquation (8). Par contre, pour les angles de contact finis A 1'6quilibre (8, f 0) on attent h* --a,donc le film devient une monocouche. (Dans ce dernier cas, la discussion de Pomeau [2] sur la diffusion en monocouches d k r i t probablement le prkurseur). ExpCrimentalernent il est reconnu qu'un film relativement ipais existe en avant de la goutte seulement dans le cas ce mouillage parfait [ l b]. Les mesures klectriques suggerent effectivement un film d'kpaisseur lentement dkcroissante 1121. 2. Au total, il semble qu7une description hydrodynamique simple, mais incorporant les forces i longue portke, permet de comprendre la dynamique de la ligne triple. Certaines limitations sont kvidentes toutefois; (a)passage a un regime de monocouche pour 6 -a; (b) modifications possibles dues aux interactions de van der Waals retardkes pour 6 > 300 A. A cause de cette deuxi&mecomplication, I'ktude complete du raccord entre le rkgime > et le rkgime > sera lourde, mais toujours justiciable de l'kquation gknkrale ( 11). 3. I1 semble que l'incorporation du film prkcurseur permette d'kviter les divergences logarithmiques qui apparaissent dans le traitement macroscopique usuel et qui sont levkes dans la littkrature par adjonction d'un ltger glissement hydrodynamique ii la surface solide ([I], [5]). Par exemple, on peut montrer que la longueur xmi, (introduite dans l'kquation 1) ou la longueur b de l'kquation (l2), sont comparables a x*. Physiquement, la dissipation dans le film n7est pas singulikre : elle kquilibre juste le travail des forces i longue portee. Le pr6sent travail a bknkficik #&changes tres utiles avec B. Legait, Y. Pomeau et J. F. Joanny.

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R E F E R E N C E S BIBLIOGRAPHIQUES

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College de France, Laboratoire 11, place Marcellin-Berthelot,

de Physique theorique, 75231, Paris Cedex 05.