Statistical mechanics of membranes and surfaces

443 Pages·2004·5.339 MB·English

by  David Nelson, Steven Weinberg, T Piran

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Published by
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A catalogue record for this book is available from the British Library.
STATISTICAL MECHANICS OF MEMBRANES AND SURFACES
2nd Edition
Copyright © 2004 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,
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PREFACE TO THE FIRST EDITION
The Fifth Jerusalem Winter School on the Statistical Mechanics of Membranes
and Surfaces was held from December 28, 1987 to January 6, 1988. The School
focused on the theory of the highly convoluted surface fluctuations which appear
in such diverse condensed matter systems as microemulsions, wetting and growth
interfaces, bulk lyotropic liquid crystals, chalcogenide glasses and sheet polymers.
Thedelicateinterplaybetweengeometryandstatisticalmechanicsinthesesystems
canbedescribedusingtoolsfromthefieldsofpolymerphysics,differentialgeometry,
and critical phenomena. Our theoretical understanding of these problems can be
testedbyawidevarietyoflaboratoryexperiments,whichprobefluctuationsranging
fromrelativelybenigncapillarywavesatinterfaces,towildundulationsinbiological
membranes.TheSchoolwasfortunatetohavemanylecturerswhowereoutstanding
teachers as well as distinguished scientists: Ju¨rg Fro¨hlich spoke on the roughening
transition, as well as on his extensive and pioneering work on random surfaces.
Michael Fisher lectured on the wetting transition and on interfacial wandering.
Stanislas Leibler discussed fluctuations in liquid membranes, lyotropic smectics,
and other lipid systems. David Andelman spoke about Langmuir–Blodgett films
and the physics of microemulsions. Yacov Kantor gave a very thorough review of
the theory of polymerized surfaces; I followed with a discussion of the crumpling
transition. Francois David gave a beautiful series of talks on differential geometry,
anditsapplicationtoliquidandhexaticmembranes.BertrandDuplantierdescribed
importantrecentworkonepsilonexpansionsforpolymerizedmembranes.Virtually
allthelecturerscontributedmanuscriptstothisvolume,whichcanserveasauseful
introduction for theorists and experimentalists who wish to learn more about this
rapidly developing field. I would in conclusion like to thank Steven Weinberg, who
made this School possible, and Tsvi Piran, who helped make the School a reality.
David Nelson
Cambridge, Massachusetts
December, 1988
v
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PREFACE TO THE SECOND EDITION
IwasverypleasedwhenK.K.PhuaatWorldScientificPublishingsuggestedissuing
a second edition of “Statistical Mechanics of Membranes and Surfaces”, a book I
edited (with Tvsi Piran and Steve Weinberg) and contributed to in 1988. Over the
intervening 15 years, I received many compliments on the excellent work described
by Michael E. Fisher, Stanislas Leibler, David Andelman, Yacov Kantor, Franc¸ois
DavidandBertrandDuplantierinthisaccountofaJerusalemWinterSchoolwhich
took place in 1988. My students still consult this book frequently, and the sales
during the past decade and a half suggest that there is still considerable interest in
the relevant theory and experimental systems.
To capture important additional developments in the statistical mechanics of
membranes and surfaces, I was fortunate to persuade four excellent researchers
to write three new chapters for the second edition. Leo Radzihovsky contributed a
chapteronthefascinatingeffectswhicharisewhenanisotropyandheterogeneityare
incorporatedinpolymerizedmembranes.MarkBowicksurveyedthephysicsoffixed
connectivity membranes in general, including very recent theory and experiments
probing crystalline ground states on curved surfaces. This second edition concludes
with an authoritative survey of triangulated surface models of fluctuating mem-
branes (including studies of liquid and hexatic phases) by Gerhard Gompper and
Dan Kroll.
IamparticularlygratefultoBetrandDuplantier,whoprovidedaveryextensive
update for his chapter on self-avoiding crumpling manifolds. Thanks are also due
toMichaelE.FisherwhokindlyprovidedsomeadditionalreferencesforChapter3.
Althoughseveralpreviousauthorstooktheopportunityofasecondeditiontorevise
or correct their contributions, most of the older chapters should not be viewed as
comprehensive updates. Rather, they are “snapshots” of progress in a field which
was just beginning to emerge and confront real experiments. Nevertheless, I feel
thereisatimelessqualityaboutalltheearlychapterswhichmakesthemasrelevant
today as when the first edition was published.
David R. Nelson
Cambridge, Massachusetts
November, 2003
vii
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CONTENTS
Preface to the First Edition v
Preface to the Second Edition vii
Chapter 1 The Statistical Mechanics of Membranes and Interfaces 1
David R. Nelson
1 Flat Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Roughening Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Wetting Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Crumpled Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Experimental Realizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Plaquette Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Perturbation Theory for Tethered Surfaces . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Chapter 2 Interfaces: Fluctuations, Interactions and Related Transitions 19
Michael E. Fisher
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1 Interface Models, Mean Field Theory, and Wetting . . . . . . . . . . . . . . . . . 20
1.1 Levels of Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 Mean Field Theory for Order Parameters . . . . . . . . . . . . . . . . . . . 20
1.3 Derivation of Interface Models . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 External Forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 The Complete Wetting Transition . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Wall Effects and the Interface Hamiltonian . . . . . . . . . . . . . . . . . . 23
1.7 Wetting Transitions with Short-range Forces:
Mean Field Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Fluctuations and Steric Repulsions . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1 The Wandering Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 Interfaces in Two-dimensions: Random Walks . . . . . . . . . . . . . . . . . 27
2.3 Correlations and Correlation Lengths . . . . . . . . . . . . . . . . . . . . . 28
2.4 The Stiffening or Roughening Transition. . . . . . . . . . . . . . . . . . . . 28
2.5 Random Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6 Fluctuations at Complete Wetting under Long-range Forces . . . . . . . . . 31
2.7 Constrained Interfaces and the Wall-interface Potential . . . . . . . . . . . 31
2.8 Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.9 Checks of the Wall-interface Potential . . . . . . . . . . . . . . . . . . . . . 33
ix