Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glass Transition

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

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aNonlinearhydrodynamicsofahardsphere uidneartheglasstransitionLisaM.LustandOriolT.VallsSchoolofPhysicsandAstronomyandMinnesotaSupercomputerInstitute,UniversityofMinnesota,Minneapolis,Minnesota55455-0149andChandanDasguptaDepartmentofPhysics,IndianInstituteofScience,Bangalore560012,India

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

Abstract

Weconductanumericalstudyofthedynamicbehaviorofadensehardsphere uidbyderivingandintegratingasetofLangevinequations.ThestaticsofthesystemisdescribedbyafreeenergyfunctionaloftheRamakrishnan-Yussou form.We ndthatthesystemexhibitsglassybehaviorasevidencedthroughstretchedexponentialdecayandtwo-stagerelaxationofthedensitycorrelationfunction.ThecharacteristictimesgrowwithincreasingdensityaccordingtotheVogel-Fulcherlaw.Thewavenumberdependenceofthekineticsisextensivelyexplored.Theconnectionofourresultswithexperiment,modecouplingtheory,andmoleculardynamicsresultsisdiscussed.

1992PACSnumbers:64.70Pf,61.20Ja,61.20Lc

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

I.INTRODUCTION

Despiteextensiveexperimental,numericalandtheoreticalinvestigations1,2overseveraldecades,thepresentunderstandingoftheslownon-exponentialdynamicsofdenseliquidsneartheglasstransitionremainsincomplete.Whenaliquidiscooledrapidlyenoughtotemperaturesbelowtheequilibriumfreezingtemperature,crystallizationisbypassedandthesystemundergoesatransitionintoanamorphoussolidstatecalledaglass.Thecharacteristicrelaxationtimet ,asre ectedinalargenumberofexperimentallymeasuredquantitiessuchasviscosityanddielectricrelaxation,growsrapidlyinthesupercooledstateasthetemperatureisdecreasedorthedensityincreased.Theglasstransitiontemperature(oralternatively,density)Tg(ρg)isconventionallyde nedasthetemperature(density)wheretheviscosityreachesavalueof1013poise.

Inrecentyears,considerableprogressinthedevelopmentofatheoreticalandex-perimentalunderstandingofthephenomenaassociatedwithglassformationandtheglasstransitionhasbeenachieved.Inparticular,theso-calledmodecoupling(MC)theoriesoftheglasstransition3 10haveledtoaframeworkforunderstandingandinterpretingmanyexperimentalresults.InMCtheories,theslowingdownofthedynamicsneartheglasstransitionisattributedtoanonlinearfeedbackmechanismarisingfromcorrelationsofdensity uctuationsintheliquid.TheMCequationsforthedynamicsneartheglasstransitionwereoriginallyderived4,5fromthekinetictheoryofdense uidsandwerelater

6,7generalizedandextendedbyG¨otzeandco-workers

theories4,5.IntheoriginalversionofMC,thecharacteristictimescalesoftheliquidarepredictedtoexhibitapowerlawdivergenceatan‘idealglasstransition’orcrossovertemperatureTc,higherthanTg.However,thisdivergenceisnotfoundexperimentally:thepowerlawformbreaksdown,andrelaxationtimesatTcaretypicallyoforder10 8s.Morerecentcalculations

9,10have

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

uncoveredacuto mechanismwhichissupposedtoroundo thepredicteddivergenceandtorestoreergodicityoveramuchlongertimescale.MostoftheexistingMCstudiesoftheglasstransitiondonottakeintoaccounttheequilibriumshort-rangestructureofthedenseliquid.Asaresult,thesetheoriesdonotleadtoanypredictionforthewavenumberdependenceoftherelaxation.SomeattemptstowardstheincorporationofinformationaboutliquidstructureintheMCformalismhavebeenmaderecently8,10.MCtheories3,7andexperimentalinformationcanbecombinedinaconsistentdescriptionofthedecayofS(q,t),thespatialFouriertransformofthedynamicdensitycorrelationfunction,atsuf- cientlylowtemperaturesorhighdensities,whenthedecaytimeshavegrownverylarge.Thedecayaccordingtothisschemetakesplaceinasuccessionofseveralregimes:afterafastdecayintimesoforderoftheinversephononfrequency,a rstslowdecayoccurswhichMCtheoriespredicttobeaninversepowerlawintime.Thisiscalledtheβ-relaxationregime.Evidenceforpower-lawdecayofcorrelationsintheβregimeisprovidedbylight11andneutronscatteringexperiments12.Thisdecayistoanonzerovalue(anapparentnonergodicphase)fromwhichthesystemeventuallymovesawayleadingtotheprimaryorα-relaxationregime.Therelaxationintheαregimeisfoundtofollowtheso-calledKohlrausch-Williams-Watts‘stretchedexponential’form13.Boththedurationoftheβrelaxationandthetimescaleofthestretchedexponentialdecayintheαregimearefoundtoincreasesharplyasthe‘glasstransition’isapproached.Insomecases,theβandtheαregimesareseparatedbyaregionofso-calledvonSchweidlerrelaxationwhichhasapowerlawform.ThestretchedexponentialbehaviorintheαregimeandthevonSchweidlerrelaxationareseenindielectricmeasurements

scattering1614andinlight15andneutronexperiments.TheparameterswhichdescribethedecayofS(q,t)areknowntobeq-dependent,butthenatureofthisdependencehasnotbeenstudiedindetail.Thus,

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

theMCtheoriesprovideaqualitativeunderstandingofanumberofexperimentallyob-servedfeaturesofglassyrelaxation.However,someofthedetailedMCpredictionsarenotinagreementwithexperiments17andtheMCdescriptionclearlyfailstoaccountforthebehaviorobservedattemperaturesclosetoandlowerthanTc.Itisgenerallybe-lieved3thatthisfailurearisesfromthefactthatMCtheoriesdonottakeintoaccountactivatedprocessesinvolvingtransitionsbetweendi erentlocalminimaofthefreeenergywhicharesupposedtodevelopasthetemperatureisloweredbelowtheequilibriumfreezingtemperature.

IncontrasttoMCtheorieswhichportraytheglasstransitionasbeingpurelydy-namicinnature,therehavebeenanumberofattempts18 21todevelopa‘thermody-namic’theoryinwhichsomeoftheinterestingbehaviorobservedneartheglasstransitionisattributedtoanunderlyingcontinuousphasetransition.Theseattemptshavebeenmotivatedbythefactthattheobservedgrowthoftherelaxationtimeinso-called‘fragile’liquids2iswell-describedbytheVogel-Fulcherlaw

t=τse aT022:

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

explicitlydemonstratestheexistenceofsuchatransitioninaphysicallyrealisticsystemisnotyetavailable.Thus,the‘thermodynamicglasstransition’scenarioremainsessentiallyspeculative.Inthisapproach,thegrowthoftherelaxationtimeisattributedtoagrowingcorrelationlengthwhichwoulddivergeattheidealglasstransition.Tworecentnumericalstudies25,26whichlookedforsuchagrowingcorrelationlengthdidnot ndanyevidence

27foritsexistence.Arecentexperiment,ontheotherhand,haspresentedevidencefor

theexistenceofalengthscalethatgrowsasthetemperatureisloweredbelowthecrossovertemperatureTc.

Thestaticanddynamicpropertiesofdenseliquidshavealsobeenstudiedexten-sivelybymoleculardynamics(MD)simulations28 32.Theverynatureofthesimulationmethodrestrictssuchstudiestosimplemodelsystemsandtotimescaleswhicharerathershort(oftheorderof10 9sec).Inspiteoftheselimitations,MDsimulationshavepro-ducedanumberofinterestingresultswhichareinqualitativeagreementwithpredictionsofMCtheoriesandresultsofexperiments(suchasthoseusingtheneutronspin-echotech-nique33)whichhavetimescalescomparabletothoseofthesimulations.Thisobservationleadstotheinterestingandusefulconclusionthatastudyofsimplemodelsystemsmaybesu cientforunderstandingthebasicphysicsoftheglasstransition.

Itisevidentfromthisbriefsurveyofthecurrentstatusoftheglasstransitionproblemthatanobviousneedexistsforthedevelopmentofnewanalyticandnumericalmethodswhichmayaddresssomeoftheoutstandingissuesrelatedtothisproblem.Inthispaper,wedescribetheresultsobtainedfromtheapplicationofanewnumericalmethodtoastudyofthedynamicbehaviorofadensehard-sphereliquidneartheglasstransition.ThemethodweuseconsistsofdirectnumericalintegrationofasetofLangevinequationswhichdescribethenonlinear uctuatinghydrodynamics(NFH)

rmation

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

aboutthestaticstructureoftheliquidisincorporatedintheLangevinequationsthroughafree-energyfunctionalwhichhasaformsuggestedbyRamakrishnanandYussou (RY).35Recently,ithasbeenshown36thattheRYfreeenergyfunctionalprovidesacorrectmean- elddescriptionofthestaticsoftheglasstransitioninthissystem.Inthatwork,anumericalprocedurewasusedtolocatelocalminimaofadiscretizedversionoftheRYfreeenergyappropriateforthehardspheresystem.Alargenumberofglassylocalminimawithinhomogeneousbutaperiodicdensitydistributionwerefoundtoappearastheaveragedensitywasincreasedabovethevalueatwhichequilibriumcrystallizationtakesplace.Athigherdensities,thefreeenergiesoftheseminimawerefoundtodropbelowthatoftheminimumrepresentingtheuniformliquidsignalingamean- eldglasstransition.ThesuccessoftheRYfreeenergyfunctionalinprovidingacorrectdescriptionofthestaticsoftheglasstransitionofthehardspheresystemsuggeststhatagoodstartingpointforastudyofthedynamicsofthissystemwouldbeobtainedbyincorporatingthisfreeenergyintheappropriateNFHequations.

Anumberofimportantissuesareaddressedinourstudyofthedynamics.BycomparingtheresultsofourcalculationswithexistingMDresults28onthesamesystem,weareabletotestthevalidityoftheNFHdescriptionwhichiscastintermsofcoarse-grainednumberandcurrentdensityvariablesinsteadofthecoordinatesandmomentaofindividualparticles.ThecorrectnessoftheNFHequationsweuse,althoughusuallytakenforgranted,isnotobviousinviewofthefactthatthehydrodynamictermsintheseequationsdescribethephysicsatrelativelylonglengthscales,whereasthetermsarisingfromthefreeenergyfunctionalinvolvelengthscalesoftheorderof(orsmallerthan)theinterparticlespacing.IntheRYfreeenergyfunctional,informationaboutthemicroscopicinteractionsisincorporatedintheformoftheOrnstein-Zernikedirectpaircorrelation

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

function37oftheliquid.Thisappearstobeadequateforacorrectdescriptionofthestaticsofthefreezingoftheliquidintobothcrystalline35andglassy36states.Oneofthequestionsweaddressinthepresentstudyiswhetherthisisalsosu cientforacorrectdescriptionofthedynamicbehavior.TwonumericalstudiesofNFHequationsdescribingthedynamicsofdenseliquidshavebeenreportedrecently38,39.Themaindi erencebetweenthesestudiesandthepresentoneisthattheequilibriumstructureoftheliquidwastreatedonlyinanapproximatewayintheearliercalculations.ConsequentlytheresultsofthesecalculationscouldnotbecompareddirectlywithMDdata.Theresultsofthesecalculationsdidnotexhibitsomeofthefeatures(suchastwo-regimedecayofcorrelations)generallyassociatedwithglassydynamics,althoughnon-exponentialdecayofcorrelationsoftenassociatedwithitsonsetwasfound.InRef.(39),itwassuggestedthatthisfailurearisesfromtheapproximationsmadeinthetreatmentofliquidstructure.Thepresentstudyprovidestheopportunitytocheckwhetherthisexplanationiscorrectandtoinvestigateingeneraltheroleofstaticstructureinthedynamicsofadenseliquid.Further,perturbativetreatmentsoftheNFHequationssimilartotheonesweuseareknown9,10toleadtoresultswhichareveryclosetothoseobtainedfromtheMCapproach.Apartfromnumericalerrorsarisingfromspatialdiscretizationandtheintegrationprocedure,ourtreatmentoftheseequationsisexact.Inparticular,ournumericalsolutionoftheseequationsisobviouslynonperturbative,therefore,acomparisonoftheresultsofourcalculationwithMCpredictionsprovidesawaytotestthevalidityofsomeoftheapproximationsmadeintheanalyticstudies.Sinceourcalculationtakesfullaccountoftheshortrangestructureoftheliquid,ithasthemostdirectconnectionwiththeMCstudiesofKirkpatrick8andDas10.Finally,bymonitoringwhichminimaofthefreeenergyarevisitedduringthetimeevolutionofthesystemweareabletodeterminewhethertheobserveddynamicbehavior

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

arisesfromnonlinearitiesofdensity uctuationsintheliquidorfromtransitionsamongdi erentglassyminimaofthefreeenergy.Itisnotpossibletodistinguishbetweenthee ectsofthesetwokindsofprocessesinconventionalMDsimulations.

Wehavestudiedthetime-decayofS(q,t),thesphericallyaveragedspatialFouriertransformofthetime-dependentdensitycorrelationfunctionofthehard-sphereliquid,fordi erentvaluesofwavevectorqandforanumberofvaluesofthereduceddensityn (n ≡ρ0σ3,whereρ0istheaveragenumberdensityandσisthehardspherediameter)intherange0.75 0.93.ItwasfoundinRef.(36)thatahardspheresystemdescribedbyadiscretizedversionoftheRYfreeenergyexhibitsacrystallizationtransitionnearn =0.83.SincethepresentcalculationusesthesamediscretizedfreeenergyasthatofRef.(36),thesystemmaybeconsideredtobeinthe‘supercooled’regimeforalargepartofthedensityrangeconsideredbyus.Thenumericale ciencyoftheLangevindynamicsarisingfromtheuseofcoarsegrainedvariablesenablesustoverifythatthestaticsofthesystemremainstationarythroughoutthelongtimeintervalsthatweconsider.Theob-serveddynamicbehaviordescribedlaterindetailexhibitsanumberofcharacteristicglassyfeatures.Theseincludestretchedexponentialdecayofcorrelations,two-stagerelaxation,andVogel-Fulchergrowthofrelaxationtimes.OurresultsareinagreementwithexistingMDdata28onthedynamicsofthehard-sphereliquidandinqualitativeagreementwithotherMDresultsobtainedforsimilarbutdi erentsystems.ThisobservationestablishesthecorrectnessoftheNFHdescriptionusedinthisworkandalsodemonstratesthattheRYfreeenergycontainstheessentialphysicsofthedynamicsofthissystem.Ourcalcu-lationsalsoreproducequalitativelyanumberofpredictionsofMCtheories,however,we ndsigni cantdeviationsfromsomeofthequantitativeMCpredictions40fortheglassykineticsofthehard-spheresystem.Ourstudyindicatesthattheonsetofglassyfeaturesin

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

thedecayofS(q,t)occursatrelativelylowerdensitiesforwavevectorsclosetothe rstandsecondpeaksinthestaticstructurefactor.Thisresultclearlyillustratestheimportantroleplayedbytheequilibriumstructureinthelong-timedynamicsoftheliquid.Theobservedq-dependenceofthedecayofS(q,t)alsosuggeststhattheglassybehaviorsetsinearlier(atlowerdensities)atshorterlengthscales.Finally,thesystemisfoundto uctuateabouttheliquid-stateminimumofthemean eldfreeenergyinalloursimulations.

Therestofthepaperisorganizedasfollows.SectionIIcontainsadescriptionofthemodelconsideredbyus.We rstde nethemodel,discussitsstaticsandderivetheappropriateNFHequations.Byde ningappropriateunitsoflength,timeandmass,theseequationsarethenwrittenindimensionlessform.ThemethodusedbyustointegratetheseequationsforwardintimeisdescribedinSectionIII.WealsodiscussinthisSectionthephysicalquantitiesmeasured,thedatacollectionprocedure,testsofequilibrationandotherrelatedmatters.TheresultsobtainedfromthenumericalworkaredescribedindetailinSectionIV.WealsocompareandcontrastourresultswiththoseobtainedfromMDsimulationsofthehard-sphereandsimilarsystemsandthepredictionsofMCtheories.

II.THEMODEL

WehaveexplainedintheIntroductionthatwewillanalyzethenumericalsolutionofasetofLangevinequationsappropriatetoadensehard-sphere uid.Webeginherebydiscussingthestaticsofthemodel.Thesearegivenintermsofafreeenergywhichisafunctionalofthetwo eldsintheproblem:thedensity eldρ(r,t)andthecurrent eldg(r,t).Thisfreeenergyhastwoterms:

g2

FTot[ρ,g]=(m0/2)dr

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

whereF[ρ]isoftheRYform

F[ρ]=Fl[ρ0]+kBT 35: drdr′C(r r′)δρδρ′ (2.2)dr(ρlog(ρ/ρ0) δρ) (1/2)

In(2.1)and(2.2)ρisthenumberdensity eldandδρ≡ρ ρ0thedeviationofthat eldfromitsaveragevalueρ0.Flisthefreeenergyoftheuniformliquid,TisthetemperatureandkBtheBoltzmannconstant.Inthe rstequationm0denotesthemassofahardsphere.Finally,C(r r′)isthedirectcorrelationfunction.Theinclusionofthisfunctioninthefreeenergyensuresthatuponlinearizationofthelogarithminthe rsttermontherighthandsideof(2.2)oneobtainstheusualexpressionforthestaticstructurefactorofasimple uidintermsofC.ForhardspheresasimpleexpressionforC(r r′)canbeobtainedinthePercus-Yevickapproximation37:

ξ<1(2.3a)

(2.3b)C(ξ)= λ1 6ηfλ2ξ (1/2)ηfλ1ξ2;C(ξ)=0;ξ>1

whereξ≡|r r′|/σ,ηfisthepackingfraction:

ηf=(π/6)ρ0σ3≡(π/6)n

and:

λ1=(1+2ηf)2/(1 ηf)4

λ2= (1+ηf/2)2/(1 ηf)4(2.4)(2.5)(2.6)

Wehavewrittenρ0inthedenominatoroftheg2terminEq.(2.1),ratherthanρasinRef.(39),sothatthefulldensitydependenceofthefreeenergyisgivenbytheRYform.AspointedoutinRef.(41),ifweusedρwewouldobtainuponfunctionalintegrationwithrespecttothegaussianvariableg2alog(ρ)contributioninvolvingdensity uctuationsassociatedwiththekineticenergy.ThesearealreadyincludedinEq.(2.2).

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

Itisconvenienttochooseunitsatthispoint,whichwillsimplifysubsequentexpres-sions.Wetakem0astheunitofmass.Wechoosealsoaunitoflength,h,whichweshalllateridentifywiththelatticeconstantofthecomputationallattice.Fortheunitoftimewechooset0suchthat:

t0=h/c(2.7)

wherecisthespeedofsound.Thischoiceoftheunitoftimeismotivatedasfollows:thecharacteristicphonontimeforthesystemistp=1/(cq)whereqisawavevector.Hencetp/t0=1/(qh).Weshallchoosebelowtheratioσ/hsothatqhisoforderunityinthewavevectorrangeofinterest.Hencethechoice(2.7)correspondstoacharacteristictimeofordertp.ThesamechoicewastakeninRef.(39).

Wethende nethedimensionlessquantities:

x=r/h(2.8)

n=ρh3

j=gh3/c(2.9)(2.10)

andintroducealsothedimensionlessfreeenergyF[n,j]intermsoftheabovequantities:

F[n,j]=(1/2) dxj2

m0c2

(2.13)

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

ThequantityKisforhardspheresafunctionofonlythedensity.Thisfollowsfrom:

c2=1

πn g(σ)√

t=Vα[ψ] ΓδFα,β

β

δjj(x)]

(2.18)

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

Vji= n(x) iδFn

n0 jjj(x) ijj(x)

t

and:

ji

δn (1/n0) j+(1/n0) (nj)=0(2.20) j(jijj) (1/n0) jjj ijj+(1/n0)η 2ji+Θi(2.21)

whereηisthebareshearviscosityinourunits45.Thenoise eldsΘi(x,t)satisfythesecond uctuation-dissipationtheoremintheform:

<Θi(x,t)Θj(x′,t′)>= 2Kληn0δi,j 2δ(r r′)δ(t t′).(2.22)

Werecallthatintheseequations,andintheremainderofthepaper,thetimeismeasuredinunitsasgivenin(2.16).In(2.22)theangularbracketsdenotethethermody-namicaverage.Thequantityλisadimensionlessmeasureoftheequilibrium uctuations.Theaveragevaluen0isrelatedton =nσ3throughn0=n (h/σ)3.Forhardspheres,onecanwriteηintermsofKandthedensityintheform:

η=(h/σ)246√π)][g(σ) 1+0.8(2πn /3)+0.761(2πn /3)2g(σ)](2.23)Theequationsofmotion(2.20)and(2.21)areslightlymorecomplicatedthanthecor-respondingequationsforthemodelinRef.(39).Thereisanadditionalconvectionterminthesecondequationandsomeadditionalfactorsofn/n0.Thesecomplicationscanbedirectlytraceddowntothedi erentdensitydependenceofthe rsttermin(2.11)asdis-cussedabove,thatis,tothefactthatakineticenergycontributionisnowincludedinthe

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

rsttermofFn.Thesedi erencesthenarisebecauseofthedi erentformofδF/δjwhichresultsinmorecomplicatedexpressionsforthestreamingvelocities.Ourexpressionscor-rectlyreducetothecontinuityandNavier-Stokesequationsinthelinearlimit.Ingeneral,onecaninterpret(2.20)asthecontinuityequationifthe eldgisthoughtofasρ0timesthevelocity.Thecurrentdensityisthennotgivensimplybyg,however.Thesequestionsdonota ect,obviously,thevalidityoftheconclusionsobtainedfromthismodel.

AsalientfeatureofEq.(2.21)ofconsiderableimportanceisthattheterminδFn/δnwhichinvolvesthedirectcorrelationfunctionCisanintegraloverspacewithrangeσ.Thus,wehavetosolveinthiscasenotmerelyasetofpartialdi erentialequationswithstochastictermsbutasfarasthespatialdependenceisconcernedanintegrodi erentialequation.Thiscomplicationmakesthisadi cultproblemtosolvenumerically.Themethodsusedwillbeexplainedbelow.

III.METHODS

InthisSectionwediscussthemethodsweusetosolveourequationsofmotion,thephysicalquantitieswefocuson,thedatacollectionmethodsandstatistics,therangeofparametervalueswehavestudied,andrelatedmatters.

Ourobjectiveistostudythedynamiccorrelationsofoursystemasde nedbelow.InordertodosowesolvenumericallyEqns.(2.20)and(2.21)onathreedimensionalcubiclatticeofsizeN3.ThemaincomplicationwemustconsideristhespatialintegralinvolvingthedirectcorrelationfunctionC(r).ForeachintegrationstepthisintegralmustbedoneN3timessinceitiscomputedoveraspherewithitsoriginateachoneofthelatticesites.Toavoidrepeatedcalculationswecreate,intheinitializationofourprogram,atablelistingforeachlatticesitethelocationofallneighboringsitestobeintegratedoverandtheircorrespondingvaluesofC(r).Additionally,sincethesphereisimbeddedona

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

coarsediscretelatticeweusea nermeshthande nedontheoriginallatticetoimprovetheaccuracyoftheintegration.Theproceduresemployedtointegratetheremainingsetofdi erentialequationsovertimeandgeneratethegaussiannoiseareidenticaltothoseusedinRef.(39)andreferencescitedthere.

WewillfocusouranalysisonthetimedependenceofthedynamicstructurefactorS(q,t):

S(q,t)=dxe3iq·(x x′)<δn(x,0)δn(x′,t)>(3.1)

Speci cally,wewillconsidertheangularaverageofS(q,t).Onacubiclattice,itisappropriatetode nethee ectivelengthofq,q,as:

q2=2(3 cosqx cosqy cosqz),(3.2)

andweperformangularaveragesofq-dependentquantitiesbyaveragingovervaluesofqinthe rstBrillouinzone,inasphericalshellofmeanradius(asgivenbyq)correspondingtothatofthevector(πQ/N,0,0)andthicknessπ/N.ThevalueofQrangesfrom1toapproximately31/2N,althoughonlyasmallerrangeisfreeof nitesizee ects.Wewill,forsimplicityofnotation,denotequantitiesaveragedinthiswaybysimplydroppingthevectorsymbolfromthewavevectorargument:S(q,t),andoftenwewillindicatethevaluesofqbythe‘shellnumber’Q.

Nextweturntoourchoicesforparametervalues.Thecorrelationfunctionsthatweareinterestedinarespatiallyshortranged.Itisthereforenotnecessarytouseextremelylargelatticesizes.TheresultspresentedwereobtainedusingN=15.AsinRef.(39),thisprovedtobeadequate.Wecheckedthat nitesizee ectsdonota ectthedynamicsinthewavevectorregionforwhichresultsarepresentedhere,(5<Q<15,seeSection

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

IV)byperformingaportionofthecalculations(withreducedstatistics)atN=25.Ourchoiceoftheratioσ/hwhich xesthelengthscalefortheproblem,isdictatedbytwoconcerns.The rstisthatwewishtobeabletostudythedependenceofthedynamicsonwavevectorinthemainregionofinterestfromthepointofviewofthestaticstructurefactorS(q)≡S(q,t=0).Thus,wewishtochooseourunitoflengthsothatthemainpeakinS(q)fallsinthemiddlepartoftherangeofwavevectorswithinthe rstBrillouinzoneofthecomputationallattice.Secondly,toavoidcrystallizationatthehigherdensitiesstudied,wehavefoundthatweneedNandσtobeincommensurate.Selectingσ/h=4.6leavesqmaxwellawayfromthezoneedgeforalldensities,neartheQ=8shell,andisclearlyincommensuratewithN.Thedensityrangewehaveinvestigatedincludesn =0.5,0.75≤n ≤0.90at0.05intervals,andn =0.93.The rstoftheseiswellwithinthediluteliquidregion,andwasusedonlytocheckthatlimit.AsexplainedinRef.(39),itisnecessarytoincludetheparameterλinordertorepresenttheactual uctuationsthroughgaussiannoise.Itsprecisevalueisnotcrucial,sinceitessentiallyamountstoachoiceofthenormalizationofthestatic uctuationsS(q),butitclearlymustbesmallsincethedensitymustalwaysbepositive.Wehavetakenhereλ=0.001.TheremainingparametersKandηarefunctionsofn andmaybecalculatedasdiscussedinthepreviousSection.

Weturnnowtotheveryimportantquestionofdatacollection.Weareparticularlyconcernedwithensuringthatwithinstatisticalerrortheaveragescollectedbeequilibrated,thatis,stationaryinthetimescalesstudied.We ndthat,withthedatacollectionprocedureasoutlinedbelow,theinitialconditionsthatweusetobegintheintegrationoftheequationsofthemotionareunimportant(exceptinthattheydeterminetosomeextentthedurationoftransientbehavior)andweusuallytakethemtobea atdistributionofnequaltoitsaveragevalue,n0,andvanishingcurrents.Wethenmonitorthecurrent-current

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

correlationsasafunctionofrunningtimet0.AfterarelativelyshorttimetKoforder10thecurrentcorrelationsreachtheirequilibriumvalueasgivenbytheequipartitiontheorem.ItmightbetemptingtoassumethatthedensitycorrelationshavealsoequilibratedbythattimeandsuchanassumptionissometimesmadeinMDwork,butwe ndthatforoursystematleastthisassumptiondoesnothold.

Tostudythedensitycorrelations,westore,forrunningtimest0≥tK,wheret0isthetimemeasuredfromtheinitiationofthecomputation,atalargenumberofperiodictimebins,theproductsoftheformδn(x,t0)δn(x′,t0+t)forallxx′.WethenmonitorthesphericallyaveragedspatialFouriertransform,S(q,t,t0)ofthequantity:

S(x,x′,t,t0)=<δn(x,t0)δn(x′,t0+t)>(3.3)

wheretheaverageisunderstoodtobeoveranumbernboftimebinsseparatedbyaninterval t.ThetimerangecoveredbytheaveragingprocessistR=nb t.InorderforS(q,t,t0)tobeanadequateapproximationtothethermodynamicaverageS(q,t)itisrequiredthatitbenotonlyindependentoft0,butalsoindependentoftRwithinstatisticalerror.Dependenceonthet0indicatesthepresenceofatransient.DependenceontRindicatesthattheaveragingtimeistooshortforergodicitytohold.Averyimportantpointisthatwe ndthattheminimumvalueofthetransienttimefordensity uctuationsisnottK,butitisoftheorderoftheslowestcharacteristicdecaytimet inthesystem.AsdiscussedinthenextSection,t isastronglyincreasingfunctionofdensity,andismuchlongerthantheequilibrationtimeforthekineticenergy.Similarly,itisnecessaryfortheaveragetoincludearangetRoforderofseveraltimest .We ndthatS(q,t,t0)athigherdensitieshasconsiderableoscillationsoverttimerangessmallerthant .Asoneincreases

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

thedensityanestimatefort canbefoundbyextrapolationfromthelowerdensities,sincethebehavioroft withdensityturnsouttoobeytheVogel-Fulcherlaw22.

Workingwithintheseconstraints,obtainingstatisticallyreliableresultsstillrequiresaveragingoveralargenumberoftimebinswhichmeansverylargetotalrunningtimes.Inaddition,inordertoeliminateanypossibilityofspuriouscorrelationsduetoapeculiartransient,wehaverepeatedthewholeprocedurethreeto vetimesateachdensity.Theresultspresentedherecorrespondtoacombinedtotalofbetween1000and3900bins,dependingonthedensity,atalldensitieswepresentresultsforexceptn =0.75whereatotalof600wastaken.Theseverylargenumbers,muchlargerthanthecorrespondingnumbersinRef.(39),shouldbeconsideredascomparabletothe‘numberofruns’inastandardsimulationforanonequilibriumproblemsuchasspinodaldecompositionandleadtoverygoodqualitydata.Thecostofobtainingthedatarisesaccordingly,ofcourse:atotalof200hoursofCray2andCrayX/MPtimewererequired.

Wehavealsoveri edthatthequantityS(q,t=0)calculatedfollowingtheaboveprocedureandoverthetimerangesjustdescribedisconsistentwiththepurelystaticresult.Todothis, rstwecalculatethestaticresult:weevaluatenumerically,usingfastFouriertransforms,thediscreteFouriertransformC(q)of(2.3)forasystemofthesizeconsidered,andweobtainfromthatresultthestaticresultSs(q)∝1/(1 n C(q)).Toobtainthisequationonemustexpandthe rsttermin(2.12)tosecondorderinδn.ThecomparisonbetweenS(q)andSs(q)isshowninFig.1.Wecanseethatthetworesultsareinverygoodagreementatthedensitiesplotted.ThePYstaticvaluesareknowntobe(seee.g.Fig2inRef.(42))ingoodagreementwithMDresultsinthisdensityrange.OurresultsarethenalsoinagreementwithMDinthislimit.Athigherdensities,thecomputedresultrepresentsahigherdegreeoforderthanthatcalculatedfromthestatics.

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

Thisisduetothefactthattheexpansionofthefreeenergyjustalludedtoisnotaswelljusti edatthosedensities.Whenaveragedovertimescalesoflessthanseveraltimest ,S(q,0)oscillatesbroadlyaboutitsstationaryvalue.OurresultsforS(q,0)atalldensitiesstudiedshowthatwearedealingwithaliquid-likestatehere.Ifweuseacommensuratevalueofσ,orifweincreasen to0.95,we ndindicationsthatcrystallizationbegins.Weplantostudythiscrystallizationquestioninfuturework.

Thestabilityofourdynamicallyobtainedresultsoverlongcomputationalruns,andtheiragreementwithstaticresultsconstituteaverystringentcheckofthestabilityofournumericalalgorithms.

Toanalyzeourresults,itisconvenienttointroducethenormalizedquantityC(q,t)de nedas:

C(q,t)=S(q,t)

We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

alldatauptoeitherthemaximumtimeforwhichwehavedataattheparticularvaluesofn andqunderconsiderationoruptothetimewhereC(q,t)issosmallthatitfadesintothenoise.ThisoccurswhenC(q,t)<0.025exceptinsomeveryfewcaseswherethedatabecomesnoisyatthe0.05level.AsinRef.(39)thestatisticalnoiseseemstohaveanadditivecomponentwhichcausestherelativeerrorstoincreasewhenC(q,t)becomessmaller.WerecallthattherelationbetweenourtimeunitsandEnskogcollisiontimeisgivenby(2.16).Thefactorrelatingthetwoinversetimesvariesfrom1.52atn =0.75to

1.90atn =0.93.Therelationbetweenourunitoflengthandσisthetrivialfactorofσ/h=4.6.

Webeginbyattemptinga ttoastretchedexponentialform:

C(q,t)=e (t/τ0)β(4.1)

wheretheparametersτ0andβarefunctionsofn andq.Thisstepismotivatedinpartbytheexpectationfrompreliminaryinspectionofthedatathatthereisawideregionofqandn valuesforwhichthisformisadequate.Indeedwe ndthatforaconsiderablepartofthedataparticularlyinthelowerdensityregionthisturnsouttobeasatisfactory t.Eveninthecaseswherethedataisnotwell ttedbytheformofEq.(4.1),thenumberτ0(q,n )stillgivesauseful gureofmeritoroverallestimateofthedecaytime.Theresultsofthis tareinTablesIandII.WehaveindicatedintheseTablesthecaseswherethe ttotheaboveformisagood ttothedataandthoseinwhichitisactuallynotthebest tbyenclosingthelattercasesinparentheses.Thequalityofthe tscanbejudgedvisuallyandbytheχ2valuesthatweobtain.

ThesalientpointsoftheresultsinTableIareapparent:thecharacteristictimeisatconstantdensityaverystrongfunctionofq,andithasamaximumatthewavevectorshellsclosetowherethestaticstructurefactorS(q)hasitsmaximum.Alesswellde ned

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