Mode-Coupling Theory (MCT) Lecture Notes

arXiv:cond-mat/0511407v1[cond-mat.soft]16Nov2005Mode-CouplingTheoryDavidR.Reichman∗PatrickCharbonneau†DepartmentofChemistryandChemicalBiologyHarvardUniversity12OxfordStreet,Cambridge,Massachusetts,02138AbstractInthissetoflecturenoteswereviewthemode-couplingtheoryoftheglasstransitionfromseveralperspectives.First,wederivemode-couplingequationsforthedescriptionofdensityfluctuationsfrommicroscopiccon-siderationswiththeusetheMori-Zwanzigprojectionoperatortechnique.Wealsoderiveschematicmode-couplingequationsofasimilarformfromafield-theoreticperspective.Wereviewthesuccessesandfailuresofmode-couplingtheory,anddiscussrecentadvancesintheapplicationsofthetheory.1ImportantPhenomenologyforMCTSinceourobjectivewillbetosketchaderivationofwhatwewillcallMode-CouplingTheory(MCT),wewillfocusourattentionononeobservableinparticular,namelydensityfluctuations.Forthiswewillfirstdefinesomeoftheconceptsneededtodoso.Wewanttocalculateaspecifictimecorrelationfunction.Ingeneral,suchafunctionisexpressedasfollows,C(t)=hA(t)A(0)i.(1)Itisanensembleaverageoftheevolutionofthefluctuationsofavariableintime,atequilibrium.AsseeninFig.1,A(t)fluctuatesarounditsaveragevalueinequilibrium,whileC(t)measuresthecorrelationofAatonetimewiththevalueofAatanothertime.ThedensityorparticlesinaliquidcanbeoneexampleofA(t),ρ(r,t)=Xiδ(r−ri(t)),(2)∗Currentaddress:DepartmentofChemistry,ColumbiaUniversity3000Broadway,NewYork,[e-mail]†[e-mail]1LFigure1:Left:timeevolutionoftheinstantaneousfluctuationsofthequantityA.Theproductoffluctuationsseparatedbytimet′,averagedoverallt0’sgivesthecorrelationfunctionC(t′),atequilibrium.Right:fluctuationsofthedensityonlengthscaleL∼2π/k;ifkissmall,theareaofdensityfluctuationsislarge.whichwecanFouriertransform,ρk(t)=XiZdreik·rδ(r−ri(t)),=Xieik·ri(t).(3)InthiscasethecorrelationfunctionwillbelabelledF(k,t),whichiscanbeexpressedasfollows.F(k,t)=1Nhρ−k(0)ρk(t)i=1NXijDe−ik·ri(0)eik·rj(t)E.(4)NotethatweneedtohavePiki=0(i.e.−k+k=0!)toconservemomentum,otherwisethecorrelationfunctionisequaltozero.Thevariableslabeledbykmeasuredensityfluctuationsinreciprocal(“k=|k|”)space,whichcanbethoughtastheinverselength.Whenkissmallwearelookingatlonglengthscales,aswecanseeinFig.1.Whenitislarge,weareprobingveryshortscales.ThefunctionF(k,t)isessentiallywhatscatteringexperimentsmea-sure.Att=0,F(k,t=0)=1Nhρ−k(0)ρk(0)i≡S(k),(5)whereS(k)iscalledthestaticstructurefactoroftheliquid.Whythatname?Considertheradialdistributionfunctionofaliquidg(r).Thefunctiong(r)isproportionaltotheprobabilitythataparticleisadistancerawayfromaparticleattheorigin.Inadenseliquidg(r)showsthestructureofthesolvationshellsasdepictedinFig.2.Also,itcanbeshownthat[1,2,3]S(k)=1+ρZdre−ik·rg(r),(6)2s2s3srg(r)S(k)k»p/s2Figure2:Left:radialdistributionfunctiong(r)forasimpleliquidofsizeσ.Right:thecorrespondingstructurefactorS(k).Asamplestructureisalsode-pictedwherethesolvationshellsareindicatedbythedottedlines.Theexclusionradiuscanbeseenintheabsenceofamplitudeofg(r)forr≪σ.whereρ=N/Visthedensityofthesystem,andthusS(k)isalsoindi-catingsomethingabouttheliquidstructure[1,2,3].AnexampleforasimpleliquidisdepictedinFig.2.ButhowdoweexpectF(k,t)tobehave?Forhightemperatures–abovethemeltingpoint–F(k,t)willdecaylikeasingleexponentialfunc-tionintimeforfork≥2π/sigmaasplottedinFig.3.Forsupercooledliquids,thesituationisdifferentandacharacteristicdecaypatterncanalsobeseeninFig.3.Weobserveamulti-steprelaxation.1.Atshorttimesdecayiscomingfromfreeandcollisionaleventsthatinvolvelocalparticlemotion.Consistentwithashort-timeexpan-sion,F(k,t)∼S(k)−A(k)t2+...inthisregime[1,2,3].Thiswillbetrueatanytemperature.Wewillnotbeconcernedmuchwiththispartofthedecay.2.Intermediatetimesencompassaperiodduringwhichparticlesap-peartrappedincagesformedbyotherparticles.Thisregimeistheβ-relaxationregime.Thedecaytotheplateau(IIa)maybefittedasf+At−aandthedecayfromtheplateau(IIb)asf−Btb.Also,theexponentshaveascalingconsistentwiththerelationshipΓ(1−a)2Γ(1−2a)=Γ(1+b)2Γ(1+2b).(7)3.Atlongtimes,intheα-relaxationregime,thedecaymaybefittedtoastretchedexponentiallaw[4]F(k,t)∼e−(tτ)β.(8)with0β1.Donotbeconfusedwiththenotation.Itistheβpowerthatappearsintheα-relaxationregime!Ingeneralβandτwillbekandtemperaturedependent.3IIIIIIIIbIIafln(t)F(k,t)ln(t)Figure3:Left:F(k,t)exhibitingexponentiale−t/τdecayforanormalliq-uid.Right:supercooledliquidsdonothavesuchasimpledecay.Thevarioustemporalregimesaredescribedinthetext.Noticethelogarithmicscale.Inalatersection,wewillreturntothiskindofphenomenology.Fornow,wejustmakeafewsuperficialremarksaboutthingsthatwillbecoveredinmoredepthbyothersintheselectures.Theconstantτthatappearsinthestretched-exponentialdecaylawisstronglytemperaturedependent.Alltransportscoefficients–D(diffu-sion),η(viscosity),etc.–arestronglytemperaturedependentaswell.Overaratherwiderangeoftemperatures,afittothistemperaturede-pendencemaybe[4]η∼eET−T0.(9)Clearly,asT0isapproached,relaxationtimesbecomesolargethatthesystemcannotstayinequilibrium.Otherfittingforms,somethatdonotimplyadivergenceatfinitetemperatures,maybeusedtofitthedataaswell.Somesyste