Model for Glass Transition in a Binary fluid from a Mode Cou

We consider the Mode Coupling Theory (MCT) of Glass transition for a Binary fluid. The Equations of Nonlinear Fluctuating Hydrodynamics are obtained with a proper choice of the slow variables corresponding to the conservation laws. The resulting model equa

arXiv:cond-mat/0103420v1 [cond-mat.stat-mech] 20 Mar 2001ModelforGlassTransitioninaBinary uidfromaModeCouplingapproachUpendraHarbolaandShankarP.DasSchoolofPhysicalSciences,JawaharlalNehruUniversity,NewDelhi110067,IndiaABSTRACTWeconsidertheModeCouplingTheory(MCT)ofGlasstransitionforaBinary uid.TheEquationsofNonlinearFluctuatingHydrodynamicsareobtainedwithaproperchoiceoftheslowvariablescorrespondingtotheconservationlaws.Theresultingmodelequationsaresolvedinthelongtimelimittolocatethedynamictransition.ThetransitionpointfromourmodelisconsiderablyhigherthanpredictedinexistingMCTmodelsforbinarysystems.ThisisinagreementwithwhatisseeninComputerSimulationofbinary uids.

PACSnumber(s):64.70P,05.60.+w,64.60C,47.35.+i,05.20.-y

We consider the Mode Coupling Theory (MCT) of Glass transition for a Binary fluid. The Equations of Nonlinear Fluctuating Hydrodynamics are obtained with a proper choice of the slow variables corresponding to the conservation laws. The resulting model equa

Thephenomenonofglasstransitionhasbeenstudiedwidelyinrecentyearsusingthe

modecouplingmodelsfordenseliquids.Thisinvolvesafeedbacktothetransportpropertiesoftheliquidfromthecouplingofhydrodynamicmodesintheliquid.ThekeyaspectoftheModeCouplingTheory[1](MCT)ofGlasstransitiontakesintoaccounttheenhancedfeedbacke ectstotheviscosityduetothecouplingoftheslowlydecayingdensity uctuations.Asthedensityincreasesbeyondacriticalvaluethesystemundergoesadynamictransitiontoanon-ergodicstate.Thelongtimelimitofthetimecorrelationofdensity uctuationsistreatedasanorderparameterforstructuralrelaxation.Thetransitiontothenonergodicstateischaracterizedbyanonzerovalueoftheorderparameterortheso-calledNonErgod-icityParameter(NEP).ForabinarymixturetheselfconsistentMCTforGlassTransitionandthedynamicinstabilityresultingfromthefeedbackmechanismhasbeenstudiedbyseveralauthors[2]-[4]inthepast.Themodelequationsintheseworkspredictthedynamictransitionatamuchlowerdensitycomparedtotheonewhereastructuralarrestisseenincomputersimulationofsimilarsystems.ThisaspectoftheModecouplingmodelforthebinary uidwasindicatedinthecomputersimulationsresultsreportedinRef.[5].InthepresentworkwehaveconstructedtheMCTforthebinarysystemfromtheequationsoftheNonlinearFluctuatingHydrodynamics(NFH)[6,7]withproperconservationlaws.Weconsiderthecorrectiontothetransportcoe cientusingthemodecouplingapproximationofdominantdensity uctuations.Thisgivesrisetothefeedbackmechanismonthetransportproperties.Thepossibledynamictransitionallowedbythemodelequationsisanalyzedintermsofthesolutionoftheresultingintegralequations.Ourmain ndingsareverydi erentfromtheexistingresults[3].Thetheoreticalpredictionforthecriticaldensityofthedynamictransitionnowshiftstomuchhigherdensitybringingitinbetterquantitativeagreementwiththecomputersimulationresults.

Theslowvariablesforabinary uidarepartialdensitiesρs( x),s=1,2andtotalmo-

We consider the Mode Coupling Theory (MCT) of Glass transition for a Binary fluid. The Equations of Nonlinear Fluctuating Hydrodynamics are obtained with a proper choice of the slow variables corresponding to the conservation laws. The resulting model equa

mentumdensity g( x).Thesecanbede nedmicroscopicallyas,

ρs( x)=m Ns

sδ( x R sα(t))

α=1

gi( x)= 2 Ns

P iαs(t)δ( x R sα(t))(1)

s=1α=1

whereR sα(t)andP iαs(t)arerespectivelythepositionandi-thcomponentofthemomentumoftheα-thparticleinthes-thspecies.Nsisthenumberofparticlesinthes-thspeciesinthemixtureandthetotalnumberofparticlesisgivenby,N=

equationfortheslowvariablesare sNs.GeneralizedLangevin

ρs

ρ g]+γδFuss′

gigj

t+ jδρ+LδF

ijs

We consider the Mode Coupling Theory (MCT) of Glass transition for a Binary fluid. The Equations of Nonlinear Fluctuating Hydrodynamics are obtained with a proper choice of the slow variables corresponding to the conservation laws. The resulting model equa

contribution,wefollowthestandardprocedureduetoLangerandTurski[9].

F

k

=

1ms

d

xρs( x)[lnρs( x)

2msms′ d xd x′css′( x x′)δρs( x)δρs′( x′)(5)

Fromthedynamicequations(2)and(3),wecomputethetimecorrelationfunction[11]Css′( x x′,t t′)=1

χssχs′s′whereχss′istheequaltimedensitycorrelationfunction

betweeni-thandj-thspecies.ThepartialstructurefactorSss′[12]isde nedfromthe

√χss′throughtherelationχss′=asas′Sss′whereas=ms

2ρ0q2 d k

2css′(q)isrelatedtothestructurefactorSss′(q)throughtheOrnstein

Zernikerelation[δls c ls(q)]Sls′(q)=δss′.Inobtainingtheaboveexpression(6)wehaveconsideredonlythecouplingofthedensity uctuationscomingfromthenonlinearityin

We consider the Mode Coupling Theory (MCT) of Glass transition for a Binary fluid. The Equations of Nonlinear Fluctuating Hydrodynamics are obtained with a proper choice of the slow variables corresponding to the conservation laws. The resulting model equa

thePressuretermthatappearsinthemomentumconservationequation(3).Thisinvolvetakingthedensity uctuationsasdominantandusingtheoneloopcorrectionortheso-calledKawasakiapproximationtothefourpointfunctionstothetransportcoe cients.Thisisdoneinthesamespiritoftheself-consistentmodecouplingapproximationoftakingdensity uctuationstobedominantasinthecaseofonecomponent uid.Weliketopointoutherethatintheappropriatelimittheseresultsreducetotheonecomponent uidresultthatisincompleteagreementwithallotherwavevectordependentmodels[14]foronecomponentsystems.

Nextweconsidertheimplicationsonthedynamictransitionasaresultofthenewsetof

integralequationsfortheNEP.Theidealglassphaseischaracterizedbythenoner …… 此处隐藏：11629字，全部文档内容请下载后查看。喜欢就下载吧 ……