Phase transition dynamics

725 Pages·2007·6.51 MB·English

by  Akira Onuki

PhaseTransitionDynamics
Phase transition dynamics is of central importance in current condensed matter physics.
Akira Onuki provides a systematic treatment of a wide variety of topics including
critical dynamics, phase ordering, defect dynamics, nucleation, and pattern formation by
constructing time-dependent Ginzburg–Landau models for various systems in physics,
metallurgy,andpolymerscience.
Thebookbeginswithasummaryofadvancedstatistical–mechanicaltheoriesincluding
therenormalizationgrouptheoryappliedtospinandfluidsystems.Fundamentaldynami-
caltheoriesarethenreviewedbeforethekineticsofphaseordering,spinodaldecomposi-
tion,andnucleationarecoveredindepthinthemainpartofthebook.Thephasetransition
dynamics of real systems are discussed, treating interdisciplinary problems in a unified
manner. New topics include supercritical fluid dynamics, boiling near the critical point,
stress–diffusioncouplinginpolymers,patternsandheterogeneitiesingels,andmesoscopic
dynamics at structural phase transitions in solids. In the final chapter, theoretical and ex-
perimentalapproachestoshearflowproblemsinfluidsarereviewed.
Phase Transition Dynamics provides a comprehensive treatment of the study of phase
transitions. Building on the statics of phase transitions, covered in many introductory
textbooks, it will be essential reading for researchers and advanced graduate students in
physics,chemistry,metallurgyandpolymerscience.
AKIRA ONUKIobtainedhisPhDfromtheUniversityofTokyo.Since1983hehasheld
apositionatKyotoUniversity,takinguphiscurrentprofessorshipin1991.Hehasmade
important contributions to the study of phase transition dynamics in both fluid and solid
systems.

Phase Transition Dynamics
AKIRA ONUKI
KyotoUniversity
         
The Pitt Building, Trumpington Street, Cambridge, United Kingdom
  
The Edinburgh Building, Cambridge CB2 2RU, UK
40 West 20th Street, New York, NY 10011-4211, USA
477 Williamstown Road, Port Melbourne, VIC 3207, Australia
Ruiz de Alarcón 13, 28014 Madrid, Spain
Dock House, The Waterfront, Cape Town 8001, South Africa
http://www.cambridge.org
©A. Onuki 2004
First published in printed format 2002
ISBN 0-511-03974-3 eBook (netLibrary)
ISBN 0-521-57293-2 hardback
Contents
Preface page ix
Part one: Statics 1
1 Spin systems and fluids 3
1.1 Spinmodels 3
1.2 One-componentfluids 10
1.3 Binaryfluidmixtures 23
Appendix1A Correlationswiththestresstensor 30
References 32
2 Critical phenomena and scaling 34
2.1 Generalaspects 34
2.2 Criticalphenomenainone-componentfluids 45
2.3 Criticalphenomenainbinaryfluidmixtures 53
2.4 4Henearthesuperfluidtransition 66
Appendix2A Calculationinnon-azeotropiccases 74
References 75
3 Mean field theories 78
3.1 Landautheory 78
3.2 Tricriticalbehavior 84
3.3 Bragg–Williamsapproximation 90
3.4 vanderWaalstheory 99
3.5 Meanfieldtheoriesforpolymersandgels 104
Appendix3A Finite-straintheory 119
References 122
4 Advanced theories in statics 124
4.1 Ginzburg–Landau–Wilsonfreeenergy 124
4.2 Mappingontofluids 133
4.3 Staticrenormalizationgrouptheory 144
4.4 Two-phasecoexistenceandsurfacetension 162
4.5 Vorticesinsystemswithacomplexorderparameter 173
Appendix4A Calculationofthecriticalexponentη 178
Appendix4B Randomphaseapproximationforpolymers 179
v
vi Contents
Appendix4C Renormalizationgroupequationsfor n-componentsystems 180
Appendix4D Calculationofafree-energycorrection 181
Appendix4E Calculationofthestructurefactors 182
Appendix4F Specificheatintwo-phasecoexistence 183
References 184
Part two: Dynamic models and dynamics in fluids and polymers 189
5 Dynamic models 191
5.1 Langevinequationforasingleparticle 191
5.2 NonlinearLangevinequationswithmanyvariables 198
5.3 Simpletime-dependentGinzburg–Landaumodels 203
5.4 Linearresponse 211
Appendix5A DerivationoftheFokker–Planckequation 217
Appendix5B Projectionoperatormethod 217
Appendix5C Timereversalsymmetryinequilibriumtime-correlation
functions 222
Appendix5D Renormalizationgroupcalculationinpurelydissipative
dynamics 222
Appendix5E Microscopicexpressionsforthestresstensorandenergycurrent 223
References 224
6 Dynamics in fluids 227
6.1 Hydrodynamicinteractioninnear-criticalfluids 227
6.2 Criticaldynamicsinone-componentfluids 237
6.3 Pistoneffect 252
6.4 Supercriticalfluidhydrodynamics 265
6.5 Criticaldynamicsinbinaryfluidmixtures 271
6.6 Criticaldynamicsnearthesuperfluidtransition 281
6.7 4Henearthesuperfluidtransitioninheatflow 298
Appendix6A Derivationofthereversiblestresstensor 307
Appendix6B Calculationinthemodecouplingtheory 308
Appendix6C Steady-statedistributioninheatflow 309
Appendix6D Calculationofthepistoneffect 310
References 311
7 Dynamics in polymers and gels 317
7.1 Viscoelasticbinarymixtures 317
7.2 Dynamicsingels 335
7.3 Heterogeneitiesinthenetworkstructure 351
Appendix7A Single-chaindynamicsinapolymermelt 359
Appendix7B Two-fluiddynamicsofpolymerblends 360
Appendix7C Calculationofthetime-correlationfunction 362
Appendix7D Stresstensorinpolymersolutions 362
Contents vii
Appendix7E Eliminationofthetransversedegreesoffreedom 363
Appendix7F Calculationforweaklychargedpolymers 365
Appendix7G Surfacemodesofauniaxialgel 366
References 366
Part three: Dynamics of phase changes 371
8 Phase ordering and defect dynamics 373
8.1 Phaseorderinginnonconservedsystems 373
8.2 Interfacedynamicsinnonconservedsystems 389
8.3 Spinodaldecompositioninconservedsystems 400
8.4 Interfacedynamicsinconservedsystems 407
8.5 Hydrodynamicinteractioninfluids 421
8.6 Spinodaldecompositionandboilinginone-componentfluids 432
8.7 Adiabaticspinodaldecomposition 437
8.8 Periodicspinodaldecomposition 440
8.9 Viscoelasticspinodaldecompositioninpolymersandgels 444
8.10 Vortexmotionandmutualfriction 453
Appendix8A GeneralizationsandvariationsofthePorodlaw 469
Appendix8B Thepaircorrelationfunctioninthenonconservedcase 473
Appendix8C TheKawasaki–Yalabik–Guntontheoryappliedtoperiodic
quench 474
Appendix8D Thestructurefactortailfor n =2 475
Appendix8E Differentialgeometry 476
Appendix8F CalculationintheLanger–Bar-on–Millertheory 477
Appendix8G TheStefanproblemforasphereandacircle 478
Appendix8H Thevelocityandpressureclosetotheinterface 479
Appendix8I Calculationofvortexmotion 480
References 482
9 Nucleation 488
9.1 Dropletevolutionequation 488
9.2 Birthofdroplets 499
9.3 Growthofdroplets 506
9.4 Nucleationinone-componentfluids 518
9.5 Nucleationatverylowtemperatures 530
9.6 Viscoelasticnucleationinpolymers 533
9.7 Intrinsiccriticalvelocityinsuperfluidhelium 538
Appendix9A Relaxationtothesteadydropletdistribution 543
Appendix9B Thenucleationratenearthecriticalpoint 544
Appendix9C Theasymptoticscalingfunctionsindropletgrowth 545
Appendix9D Movingdomainsinthedissipativeregime 546
Appendix9E Pistoneffectinthepresenceofgrowingdroplets 547
viii Contents
Appendix9F Calculationofthequantumdecayrate 547
References 548
10 Phase transition dynamics in solids 552
10.1 Phaseseparationinisotropicelastictheory 556
10.2 Phaseseparationincubicsolids 577
10.3 Order–disorderandimpropermartensiticphasetransitions 584
10.4 Propermartensitictransitions 593
10.5 Macroscopicinstability 615
10.6 Surfaceinstability 622
Appendix10A Eliminationoftheelasticfield 625
Appendix10B Elasticdeformationaroundanellipsoidaldomain 629
Appendix10C AnalysisoftheJahn–Tellercoupling 630
Appendix10D Nonlocalinteractionin2Delastictheory 631
Appendix10E Macroscopicmodesofasphere 632
Appendix10F Surfacemodesonaplanarsurface 635
References 635
11 Phase transitions of fluids in shear flow 641
11.1 Near-criticalfluidsinshear 642
11.2 Shear-inducedphaseseparation 668
11.3 Complexfluidsatphasetransitionsinshearflow 684
11.4 Supercooledliquidsinshearflow 686
Appendix11.A Correlationfunctionsinvelocitygradient 700
References 701
Index 710
Preface
This book aims to elucidate the current status of research in phase transition dynamics.
Because the topics treated are very wide, a unified phenomenological time-dependent
Ginzburg–Landau approach is used, and applied to dynamics near the critical point. Into
the simple Ginzburg–Landau theory for a certain order parameter, we introduce a new
propertyorsituationsuchaselasticityinsolids,viscoelasticityinpolymers,shearflowin
fluids,orheatflowin4Henearthesuperfluidtransition.Bydoingso,weencounterarich
class of problems on mesoscopic spatial scales. A merit of this approach is that we can
understandsuchdiverseproblemsindepthusinguniversalconcepts.
The first four chapters (Part one) deal with static situations, mainly of critical phen-
omena,andintroducesomenewresultsthatwouldstandbythemselves.However,themain
purposeofPartoneistopresentthedefinitionsofmanyfundamentalquantitiesandintro-
ducevariousphasetransitions.SoitshouldbereadbeforePartstwoandthreewhichdeal
with dynamic situations. Chapter 5 is also introductory, reviewing fundamental dynamic
theories,theschemeofLangevinequationsandthelinearresponsetheory.Chapter6treats
critical dynamics in (i) classical fluids near the gas–liquid and consolute critical points
and (ii) 4He near the superfluid transition. Chapter 7 focuses on rather special problems
in complex fluids: (i) effects of viscoelasticity on composition fluctuations in polymer
systems; and (ii) volume phase transitions and heterogeneity effects in gels. Chapters 8
and 9 (in Part three) constitute the main part of this book, and consider the kinetics of
phaseordering,spinodaldecomposition,andnucleation.Motionsofinterfacesandvortices
areexaminedintheGinzburg–Landaumodels.Chapter10focusesondynamicsinsolids,
includingphaseseparation,order–disorderandmartensitictransitions,shapeinstabilityin
hydrogen–metalsystems,andsurfaceinstabilityinmetalfilms.Theseproblemshavehith-
erto been very inadequately studied and most papers are difficult to understand for those
outsidethefield,soitwasimportanttowritethischapterinacoherentfashion,thoughithas
turnedouttobeamostdifficulttask.Ibelievethatmanyinterestingdynamicalproblems
remainvirtuallyunexploredinsolids,becausesuchphenomenahavebeenexaminedeither
toomicroscopicallyinsolid-statephysicswithoutgivingduerespecttolong-rangeelastic
effectsorwithtechnologically-orientedobjectivesinengineering.Chapter11isonshear
flowproblemsinfluids,atopiconwhichagreatnumberoftheoreticalandexperimental
papers appeared in the 1980s and 1990s. This book thus covers a wide range of phase
transitiondynamics.Ofcourse,manyimportantproblemshadtobeomitted.
I have benefited from discussions with many people working in the fields of low-
temperature physics, statistical physics, polymer science, and metallurgy. Particularly
ix