ahardspherefluid
1
arXiv:cond-mat/0301337v1 [cond-mat.stat-mech] 19 Jan 2003DepartmentofPhysics,2DepartmentofChemistryandBiochemistry,
and3InstituteforPhysicalScienceandTechnology,UniversityofMaryland,CollegePark,Maryland20742
(Dated:February2,2008)
Yng-gweiChen1,3andJohnD.Weeks2,3
Thispaperdeterminestheexcessfreeenergyassociatedwiththeformationofasphericalcavityinahardspherefluid.Thesolvationfreeenergycanbecalculatedbyintegrationofthestructuralchangesinducedbyinsertingthecavityusinganumberofdifferentexactthermodynamicpathways.Weconsiderthreesuchpathways,includinganewdensityroutederivedhere.Structuralinformationaboutthenonuniformhardspherefluidinthepresenceofageneralexternalfieldisgivenbytherecentlydevelopedhydrostaticlinearresponse(HLR)integralequation.UseoftheHLRresultsinthedifferentpathwaysgivesagenerallyaccuratedeterminationofthesolvationfreeenergyforcavitiesoverawiderangeofsizes,fromzerotoinfinity.Resultsforarelatedmethod,theGaussianFieldModel,arealsodiscussed.
I.
INTRODUCTION
Thesolvationfreeenergydetermineshowreadilyaso-lutecanbedissolvedinagivensolventfluid.Thisplaysanimportantroleinmanychemicallyandbiologicallyim-portantprocesses,perhapsmostnotablyinhydrophobicinteractionsinwater.Asignificantpartofthesolvationfreeenergyarisesfromtherequiredexpulsionofsolventmoleculesfromtheregionoccupiedbytheharshlyre-pulsivemolecularcoreofthesolute.Theseverystrong“excludedvolume”interactionscansignificantlyperturbthelocaldensityaroundthesoluteandcausesimpleap-proachesbasedongradientexpansionstofail.
Theseeffectscanbeseenmostclearlyinthesimplemodelsystemtreatedinthispaper.WewillcalculatetheexcessorsolvationfreeenergyassociatedwiththeinsertionofasphericalcavitywithradiusRvintoahardspherefluid,whosemoleculeshavediameterσ.Bydefini-tion,thecentersofthesolventmoleculesarecompletelyexcludedfromtheregionofthecavity,whichthusactslikeahardcoreexternalfield.
Thissystemhasmanyinterestinglimits.Whentheex-clusionfieldorcavityradiusRvequalsσ,thenthecavityactslikeanothersolventparticleandthesolvationfreeenergyisdirectlyrelatedtothechemicalpotentialofthesolvent.Asthecavityradiustendstoinfinityiteffec-tivelyturnsintoahardwallandtherelevantthermody-namicquantityisthesurfacefreeenergyorsurfaceten-sionassociatedwithahardwallinahardspherefluid.AcavitywithradiusRv=σ/2actslikeahardcore“pointsolute”ofzerodiameter.Evenshorter-rangedhardcorefieldsor“tinycavities”withRv≤σ/2arealsoofinter-est,sincetheinducedstructureandsolvationfreeenergyofatinycavitycanbecalculatedexactly.Thislimitcanthusserveasanontrivialcheckonapproximatemethods.Themostcommonlyusedmethodtodayforsuchprob-lemsisweighteddensityfunctionaltheory(DFT)[1].Hereoneattemptstodescribethefreeenergydirectlyasafunctionalofsomekindofsmoothedorweightedaverageofthenonuniformandoftenrapidlyvaryingsin-
gletdensity.Thishastheadvantagethatthefreeen-ergyisobtaineddirectlyandbyconstructiontheassoci-atedfluidstructure(obtainedbyfunctionallydifferenti-atingthefreeenergy)isconsistentwiththeapproximatefreeenergy.Howeverthechoiceofappropriateweightingfunctionsisbynomeansobviousandanumberofdiffer-entandoftenhighlyformalschemeshavebeenproposed.Wefocusinsteadinthispaperonmakingdirectuseofstructuralinformationaboutthenonuniformsolventfluidtoobtainthesolvationfreeenergy.Webelievethisallowsphysicalintuitiontoplayamorecentralroleandwecantakeadvantageoftherecentdevelopmentofagenerallyveryaccuratetheoryrelatingthestructureofanonuniformhardspherefluidtotheassociatedexternalfield[2].
Aswewillseebelow,thefreeenergycanthenbecal-culatedbyintegration,startingfromaninitiallyknownstate(e.g.,theuniformfluid)anddeterminingthefreeen-ergychangesasthesolute-solventinteraction(thehardcoreexternalfield)is“turnedon”,oralternatively,asthedensityischangedfromtheinitialtothefinalstate.Thereexistmanypossibleroutesfromtheinitialtothefinalstate,andwewillgenerallyrefertothemasther-modynamicpathways.Ifexactresultsareusedfortheintermediatevaluesofthestructureandassociatedfields,thenallthesedifferentpathwayswillgivethesame(ex-act)resultforthefreeenergy.
Inpractice,ofcourse,approximationswillhavetobemadeandthedifferentpathwayswillgenerallyyielddif-ferentresults.Thisissometimesreferredtoasthe“ther-modynamicinconsistency”ofstructurallybasedmethods[1].Butthiscanbeviewedmorepositivelyasgivingonethefreedomtochooseparticularpathwaysthatcouldberelativelyinsensitivetotheerrorsthatexistinthestruc-turaltheory,andwewilltrytousethisflexibilitytoobtainthemostaccurateresults.Moreover,thereisaninherentsmoothingofthestructuralinformationintheintegrationusedtoobtainthefreeenergy.Thediffer-encesinfreeenergypredictedbydifferentpathwayswillalsogiveussomeindicationabouttheoverallqualityof
thetheory.
Thisapproachgenerallyrequiresthedensityprofilesandassociatedfieldsofalltheintermediatestatesalongthevariouspathways,andthusafastandaccuratemethodfordeterminingthesequantitiesiscrucialforcomputationefficiency.Wewilluseherethegenerallyaccuratehydrostaticlinearresponse(HLR)equation[2]proposedbyKatsovandWeeks.AdifferentphysicallymotivatedderivationoftheHLRequationisgivenbe-low.
Wewillalsoexaminethealternativefreeenergypre-dictionsthatarisefromatheorycloselyrelatedtotheHLRequation,theGaussianfieldmodel(GFM)devel-opedbyChandler[3].ForasolutewithahardcoretheGFMproposesanapproximatepartitionfunctionfromwhichtheassociateddensityresponsecanbederived.Intheparticularcasewherearigidcavityisinsertedintoahardspherefluid,theHLRandtheGFMapproachesturnouttomakeidenticalpredictionsfortheinducedstruc-ture.Thusstructurallybasedroutestothefreeenergyinvolvingonlyhardcorefieldswillgivethesameresults.Inaddition,onecanusetheapproximateGFMpartitionfunctiontoevaluatethesolvationfreeenergydirectly.However,aswewillshowlater,thelatterapproachtendstoproducelessaccurateresults.ThisdeficiencyshowsupevenmorestronglyinthetinycavitylimitwherethestructuralpredictionsoftheHLRandtheGFMareex-act,andseveralpathwaysgivingtheexactfreeenergycanbefound.Thisillustratestheadvantageofconsider-ingavarietyofthermodynamicpathwaysthatcanmakebestuseoftheavailablestructuralinformation.
II.
DENSITYRESPONSETOANEXTERNAL
FIELD
A.
TheHLRequation
Wedescribethesystemusingagrandcanonicalen-semble,andthuswanttodeterminetheexcessgrandfreeenergyarisingfrominsertionofasphericalcavityorhardcoreexternalfieldofradiusRv.ToderivetheHLRequation[2]westartwiththebasiclinearresponseequation[4]foranonuniformhardspheresysteminageneralexternalfieldφ(r),withchemicalpotentialµB,inverseBtemperatureβ=(kBT)−1andassociateddensityρ(r;µ,[φ])≡ρ(r):
−βδφ(r1)=
dr2χ−1(r1,r2;[ρ])δρ(r2).(1)Thisrelatessmallperturbationsinthedensityandfield
throughthe(inverse)linearresponsefunctionχ−1(r1,r2;[ρ])≡δ(r1−r2)/ρ(r1)−c(r1,r2;[ρ]).
(2)
Herec(r1,r2;[ρ])isthedirectcorrelationfunctionofthenonuniformhardspheresystem.Thenotation[ρ]indi-catesthatthesecorrelationfunctionsarenonlocalfunc-tionalsofthedensityρ(r).
2
Sincewewanttofocusontheeffectsoftheperturbingfield,wehaveusedtheinverseformoflinearresponsetheory[5]inEq.(1),wherethefieldappearsexplicitlyonlyonthelefthandside,evaluatedatr1.Thispro-videsmanyadvantagesindealingwithlargefieldper-turbations,aswillsoonbecomeapparent.Inmostcaseswewillconsiderperturbationsaboutauniformsystemwithchemicalpotentialµanddensityρ(µ)≡ρ(r;µ,[0]).Whenusingthissimplifiednotationρ(µ)shouldnotbeconfusedwithρ(r)≡ρ(r;µB,[φ]).Similarly,wewillletµ(ρ)denotethechemicalpotentialoftheuniformfluidasafunctionofdensityρ.Inauniformsystemthedirectcorrelationfunctioncwilltakethesimpleformc(r12;ρ),wherer12≡|r1−r2|.
ButhowcanweuseEq.(1)todescribethedensityresponsetoalargefieldperturbationsuchasthehardcorefieldofinteresthere?Thislinearrelationbetweena(possiblyinfinite)externalfieldperturbationonthelefthandsideandthefiniteinduceddensitychangeontherightmustcertainlyfailforvaluesofr1wherethefieldisverylarge.Conversely,Eq.(1)shouldbemostaccurateforthosevaluesofr1wherethefieldissmall—inparticularwherethefieldvanishes—andthenthroughtheintegrationoverallr2itrelatesdensitychangesinregionswherethefieldvanishestodensitychangesintheregionswherethefieldisnonzero.
Totreatlargefields,wenotethatforanygivenr1wecanlocallyimposetheoptimalconditionthatthefieldperturbationvanishesbyintroducingashiftedchemicalpotential
µr1≡µB−φ(r1),
(3)
andashiftedexternalfield
φr1(r)≡φ(r)−φ(r1).
(4)Sincethereisanarbitraryzeroofenergyandaconstantexternalfieldactslikeashiftofthechemicalpotentialinthegrandensemble,wemakenophysicalchangesifweshiftbothfunctionsbythesameamount.Inparticularρ(r;µB,[φ])=ρ(r;µr1,[φr1]).
Thesuperscriptr1inµr1indicatesaparticularvalueofthechemicalpotential,whichfromEq.(3)dependsparametricallyonr1throughthelocalvalueofthefield.Whenφ(r1)vanishes,thenµr1reducestoµB.Wedefineρr1,thehydrostaticdensity,by
ρr1≡ρ(r;µr1,[0])=ρ(µr1).
(5)
Thusρr1isthedensityoftheuniformfluidinzerofieldattheshiftedchemicalpotentialµr1;equivalentlyρr1satisfies
µ(ρr1)=µr1=µB−φ(r1).
(6)
Thusfar,wehavehavemerelyintroducedanequiva-lent(andapparentlymorecomplicated!)wayofdescrib-ingthesystemintermsofashiftedfieldandashifted
chemicalpotential.Howeverthisperspectiveimmedi-atelysuggestsaverysimplefirstapproximationtothedensityresponsetoaslowlyvaryingexternalfield.Sinceφr1(r)byconstructionvanishesforr=r1,ifφr1(r)issuf-ficientlyslowlyvarying,thentheregionaroundr1withinacorrelationlengthisessentiallyinzerofield.Inthatcasetheuniformhydrostaticdensityρr1isclearlyagoodapproximationtoρ(r1),theexactinduceddensityatr1.Moreover,whenthefieldismorerapidlyvarying,itisnaturaltointroduceasecondandevenmoreaccurateapproximationtothedensityresponse.
Thehydrostaticdensityρr1takesaccountonlyofthelocalvalueofthefieldatr1byashiftofthechemicalpo-tential.TheHLRequationimprovesonthis“localfield”approximationbyusinglinearresponsetheorytodeter-minethedensitychangefromthehydrostaticdensityin-ducedbynonlocalvaluesoftheshiftedfieldφr1(r).Thusstartingfromtheuniformdensityρr1,weassumealinearresponse1totheshiftedfield,replacingχ−1(r1,r2;[ρ])byχ−(r12;ρr1)inEq.(1)andsettingδφ(r)=φr1(r)andδρ(r2)=ρ(r2)−ρr1.ThentheleftsideofEq.(1)van-ishes(givingtheoptimallinearresponsecondition),andwehave
0=
dr2χ−1(r12;ρr1)[ρ(r2)−ρr1],(7)whichcanberewrittenexactlyusingEq.(2)as
ρ(r1)=ρr1
+ρr1dr2c(r12;ρr1)[ρ(r2)−ρr1].
(8)
Thisisourfinalresult,whichwerefertoastheHLRequation.Weviewthisasanintegralequationrelatingthehydrostaticdensityρr1tothefulldensityρ(r)andsolveitself-consistentlyforallr1.Whenφ(r1)isknown,wecanimmediatelydetermineρr1ateachr1fromthelocalrelationinEq.(6),andthensolveEq.(8)byit-erationforallr1todeterminethefulldensityresponseρ(r).Conversely,foragivenequilibriumdensitydistri-butionρ(r)wecanuseHLRequationtodeterminetheassociatedfieldφ(r).ThisinversesolutionofEq.(8)isparticularlyeasytocarryout,sincewecandeterminethelocalfieldateachr1separately,withoutiteration.Accurateresultshavebeenobtainedformanytestcaseswithstrongrepulsiveorattractivefields[2,6].
Thisrequiresinparticularexpressionsforµ(ρ)andforthedirectcorrelationfunctionc(r12;ρ)oftheuniformhardspherefluid.InthispaperwewillusethePercus-Yevick(PY)[7]approximationforc(r12;ρ).Thissamefunctionalsoarisesfromaself-consistentsolutionoftheHLRequation,wherethedensityresponsetoahardcorefieldwithRv=σ(equivalenttofixingasolventparticleattheorigin)isrelatedtotheuniformfluidpaircorre-lationfunction.Thusthisself-consistentuseoftheHLRequationprovidesaphysicallysuggestivewayofderivingthePYresultforc(r12;ρ)[2].ThePYc(r12;ρ)hasaverysimpleanalyticalformandprovessufficientlyaccu-rateforourpurposeshere.Evenbetterresultscanbefoundifoneusestheveryaccurateexpressionsforthe
3
bulkc(r12;ρ)andµ(ρ)asgivenbytheGMSAtheory[8]
asinputstotheHLRequation.
B.
RelationtothePYapproximationforahard
coresolute
Asphericalcavityactslikeahardcoreexternalfieldφthatexcludesthecentersofallsolventmoleculesfromthecavityregion.Wetakethecenterofthecavityastheoriginofourcoordinatesystem,sothatalldistancesaremeasuredrelativetothecavitycenter.Notethatboththehydrostaticdensityρr1fromEq.(5)andthefulldensityresponseρ(r1)fromEq.(8)vanishwheneverr1islocatedinthecavity.Thisexact“hardcorecondition”comesoutnaturallyfromthetheory,anddoesnothavetobeimposedbyhandasintheGFMortheGMSAapproaches.
TomakecontactwiththePYapproximation,re-callthatthecavity-solventdirectcorrelationfunctionC(r1;ρB,Rv)forthissystemexactlysatisfies
C(r1;ρB
,Rv)=ThusC(r
dr2χ−1(r12;ρB)[ρ(r2)−ρB].(9)
1)isthefunctionthatreplaces−βδφ(r1)sothatthelinearresponseequation(1)givesexactresultswhenthefulldensitychangerelativetothebulkisusedontherighthandside.WhenthisiscomparedtotheHLRequation(7)forr1outsidethecavityregion(whereρr1=ρBandφ=0)weseethattheHLRequationpredictsthatC(r1)vanishes.ThusfortheHLRequationρ(r1)vanishesinsidethecavityregionandC(r1)vanishesoutside.ThisisthesameasthePYapproximationforthehardcorecavity-solventsystem[2,9].
IfRvisgreaterthanσ/2,withσthesolventhardcorediameter,thenanequivalentexclusionisachievedbyre-placingthehardcoreexternalfieldbyahardcoresoluteparticlewith(additive)diameter
σv≡2Rv−σ.
(10)
FromthisitfollowsthatifthePYapproximationforthebulkcorχ−1isused,thedensityρ(r)predictedbytheHLRequationisidenticaltothatgivenbythePYequa-tionforthesolute-solventpaircorrelationfunctionforabinaryhardspheremixtureinthelimitthattheconcen-trationofthesolutespeciesgoestozero[10].SinceanexactanalyticalsolutionofthePYequationforabinaryHSmixtureatarbitraryconcentrationsisknown[11],wecantakeadvantageoftheseresultswhencomputingtheexcessgrandfreeenergy.
ThisequalityofsolutionsoftheHLRequationandthePYmixtureequationholdsonlyforhardcorecavityfieldswithradiusRv≥σ/2orσv≥0.AsdiscussedbelowinSec.V,fortinycavitieswithRv≤σ/2theHLRequa-tioncanbesolveddirectlyandgivesexactresultsforthedensityresponseifexactbulkcorrelationfunctionsareused,andveryaccurateresultswhenthePYapproxima-tionforthebulkcisused.However,thecorresponding
PYmixturesolutionsinthisrangeofRv(arrivedatfor-mallybytakingσvinEq.(10)tobenegative)aremuchlessaccurate.ThisinaccuracyarisesfromusingthePYmixturesolutionsfornegativeσv.ThedirectsolutionofthePYcavity-soluteequationforatinycavity,whereagivenapproximationforthebulkcisusedalongwiththePYapproximationthatC(r1)vanishesoutsidethecavityandρ(r1)vanishesinside,givesthesameaccurateresultsastheHLRequation.However,formoregeneralexternalfields,theHLRequationisquitedistinctfromthePYapproximation,andisgenerallymoreaccurate;ithasgivengoodresultsforawiderangeoffields[2,6,9].ThisadditionalflexibilityoftheHLRequationwillberequiredlaterinthispaperwhenwediscussalternatedensityroutestothefreeenergy.
III.
THERMODYNAMICPATHWAYSTOTHE
FREEENERGY
Inthissectionwediscussthreedifferentexactthermo-dynamicpathwaysforobtainingtheexcessfreeenergyofinsertingacavityintoahardspherefluid.Thefirsttwoarewellknown,andthethirddescribesanewdensityroutethatmayhavesomecomputationaladvantagesinotherapplications.WeusetheHLRequationtoprovidetheneededstructuralinformationinallcases.Webelieveourcalculationhererepresentsthefirstuseofadensityroutetoobtaintheexcessfreeenergyforthissystem.WethendescribeasimplebutlessaccurateroutetothefreeenergybasedonuseofthepartitionfunctionfortheGFM.
A.
Compressibilityroute
Inthisroutetheexcessfreeenergyisdeterminedby
varyingthechemicalpotentialofthesystemwhiletheexternalfieldφ(r)producingthecavitywithradiusRvremainsconstant.InthegrandcanonicalensembletheaveragenumberofparticlesNisgivenby
∂Ω
4
dρ
ρ=vpB,
(14)
onusingthethermodynamicrelationρ(∂µ/∂ρ)T=(∂p/∂ρ)T.ThisexactleadingordertermforlargevisdeterminedBwhenusingthecompressibilityroutesothatpistheuniformfluidpressurecalculatedbythecom-pressibilityroute[12].
ThetermincurlybracketsinEq.(13)canberewritteninamoreconvenientformforcalculationsbyusingtheinverserelationtoEq.(9)forageneralchemicalpotentialµ:
ρ(r1;µ,[φ])−ρ(µ)=
dr2χ(r12;µ)C(r2;ρ(µ),Rv),
2
(15)
whereχ(r12;µ)≡ρδ(r1−r2)+ρ[g(r12)−1]istheusuallinearresponsefunctionoftheuniformsolventfluidandg(r)istheradialdistributionfunction.SubstitutingintoEq.(13)andcarryingouttheintegrationoverr1,wehavetheformallyexactresult[13]
µB
β∆Ωv=−β
dµχˆ(0,µ)
dr2C(r2;ρ(µ),Rv)
−∞
ρB
=−
dρC
ˆ(0;ρ,Rv).(16)
0
Hereχˆ(0,µ)isthek=0valueoftheofχ,withasimilardefinitionforC
ˆFouriertransform
(0;ρ,Rv).Inthelastequalityweusedtheuniformfluidcompressibilityrelationβχˆ(0,µ)=dρ(µ)/dµtochangevariablestoanintegrationoverdensity.WewillexplicitlysolvetheHLRequationforRv≤σ/2inSec.Vbelow,andfromtheequivalencebetweentheHLRequationandPYmixtureequationforRv≥σ/2,wecanusethethePYmixtureequationtoobtainC
ˆexactsolutionof
atlargerRv.ThuswecananalyticallycarryouttheintegrationinEq.(16)forallRv.
B.
Virialroute
Wenowconsideradifferentthermodynamicpathway,whichwasfirstusedinscaledparticletheory[14,15].HerewekeepthechemicalpotentialfixedatµBandvarytherangeoftheexternalhardcorefieldbyascalingparameterλ,definingφλ(r)≡φ(r/λ).Forthehardcorecavityfieldofinteresthere,asλisvariedfrom0to1theradiusoftheexclusionzonethenvariesfrom0toRv.Sincethedensityisgenerallyrelatedtotheexternalfield
inthegrandensembleby
δΩ
∂λ
.(18)
Hereρλ(r)≡ρ(r;µB,[φλ]).Thisformulaisquitegeneralandholdsforanyλ-dependentpotentialthatvanishesforλ=0.Byexploitingspecialpropertiesofthescaledhardcorepotential(thederivativeoftheBoltzmannfactorofahardcorepotentialisadeltafunction)itiseasytoshowthatEq.(18)canbeexactlyrewrittenas
β∆Ωv=4πR3
v1dλλ2ρλ(λRv).(19)
0
Hereρλ(λRv)isthecontactdensityatthesurfaceofthe
scaledexclusionzonewithradiusλRv.Asinthecom-pressibilityroute,wecananalyticallycarryouttheinte-grationinthevirialroutetoobtainsolvationfreeener-giesforcavitiesforallRv.TheequivalentPYsolutionforbinaryhardspheremixturesisusedforthecontactdensitiesforallλRv’slargerthanσ/2,whiletheexplicitsolutionoftheHLRequationisusedfortheλRv’ssmallerthanσ/2.
C.
Densityroutes
Inadditiontotheseparticularpathways,wecanalsoimaginedirectlychangingtheequilibriumdensityfromρBtoρ(r)oversomeconvenientpathwayspecifiedbyacouplingparameterλ,whiletakingaccountoftheasso-ciatedchangesinΩandφ(r).IntegratingEq.(18)bypartstomakeρλ(r)explicitlythecontrollingvariable,wehaveexactly
β∆Ωv=drρ(r)φ(r)−
1dλ
drφ∂ρλ(r)λ(r)
0
5
ρλ(r),etc.Forthispathwaywehave∂ρλ(r)
∂λ
.
(23)
BothfactorsontherightsideofEq.(23)areeasytodeterminefromEq.(22).ThenumericalintegrationinEq.(20)cannowbecarriedoutstraightforwardlysince
theρ1/2
λ(r)factorinEq.(23)willcause∂ρλ(r)/∂λtotendtozeroexponentiallyfastwhereverφλ(r)becomeslarge.Resultsusingthispatharereportedbelow.Otherpathsimplementingthisideaexistandwehavenottriedtomakeanoptimalchoice.
D.
GaussianFieldModel
Finallyweconsideranalternativeapproach,theGaus-sianfieldmodel(GFM)[3],thatforhardcorefieldshas
manycommonelementswiththeHLRmethod.TheGFMdescribesdensityfluctuationsinauniformfluidwithaveragedensityρbyaneffectivequadraticHamil-tonianHB=
kBT
Fourierrepresentationfortheδ-functionsandformallyintegratingρˇ(r)from−∞to∞yieldsaGaussianap-proximationforthepartitionfunction,asdiscussedbe-low.
Moreover,usingthesameapproximations,byfunction-allydifferentiatingΞvwithrespecttothefield,oneob-tainsthenonuniformsingletdensityintheGFM.Inthecaseofapurehardcorefieldwithφ1=0,thedensityresponsetoacavitywithradiusRvisgivenρ(r1)=ρB−ρB
dr2
by
dr3χ(r12;µB)χ−1in(r2,r3).
v
v
(26)
Theintegrationsarerestrictedtothecavityregion,asin-dicatedbythesubscriptvontheintegralsymbols.Hereχ−1inistheinverseBoftherestrictedlinearresponsefunc-tionχin(r12;µ),whichequalsχ(r12;µB)ifbothr1andr2areinthecavityregionandequalszerootherwise.
Thusχ−1
when
inisnonzeroonlyinsidethecavityandsatisfies
dr2χ(r12;µB)χ−1
(27)v
in(r2,r3)=δ(r1−r3),
bothr1andr3areinthecavityregion.ComparingEq.(26)toEq.(15),onecanidentifythecavity-solventdirectcorrelationfunctionintheGFMas
C(r2)=−ρB
dr3χ−1in(r2,r3).
(28)v
Bypropertiesofχ−1
regionandin,theGFMC(r)vanishesoutsidethecavityρ(r)inEq.(26)vanishesinside.ThustheGFMgivesexactlythesamesolutionforthedensityresponsetoahardcoreexternalfieldasthePYortheHLRequations.(Inthemoregeneralcasewherethereisanadditionalperturbationpotentialφ1,thevar-iousapproachesdiffer.TheGFMcanbeshowntotreatthesoftertailusingthemeansphericalapproximation,whichisdifferentfromandgenerallylessaccuratethanthehydrostaticshiftusedintheHLRequation.)
ThusforcavitiesorhardcoresolutesallstructurallybasedroutestotheexcessfreeenergywillgivethesameresultswhenusingtheGFMortheHLRequation.Inad-dition,theGFMpartitionfunctionalsoprovidesadirectandverysimpleroutetothefreeenergy[3].HoweverthisrouteisinherentlyapproximatebecauseEq.(25)isnotreallyafreeenergyfunctionalforthewholeconfigura-tionspace,butratherarestrictedonedescribingonlythespaceoutsideofthespecifiedcavityregion.Thisfunc-tionalmaylegitimatelydescribesubsequentsmallpertur-bationsofφ1outsideofthecavity,butitdoesnotcontainenoughinformationaboutthefunctionaldependenceonthecavityvolumeinthefirstplace.Moreovertheapprox-imationsmadeinevaluatingtheGFMpartitionfunctiondonotbuildinthefactthatingrandcanonicalensemble,thethermodynamicpropertiesshoulddependonµ−φratherthanonµandφindividually.Thusitisalsonotconsistentwiththefreeenergypredictionfromthecom-pressibilityroute,whichintegratesoverstatesatdiffer-entchemicalpotentialsbutwithafixedhardcorealwayspresent.
6
EvaluatingtheGaussianintegralsinEq.(25),theex-cessgrandfreeenergyarisingfromacavitywithradius
Rvisgivenby
β∆Ωv=−logΞv/ΞB
=−
1HereΞv[N]istheconstrainedpartitionNmax
N=0
Ξv[N]
.(30)
functionwhen
Nparticlesareinthespecifiedvolume.Thisformulahasbeensuccessfullyusedintheinformationtheoryap-proachdevelopedbyHummer,Prattandcoworkers[17].IftheGFMisusedtoapproximatethepartitionfunctionsinEq.(30)byreplacingtheproductofδ-functionsδ[
inEq.(25)bythesingleaverageconstraintvdrρ
ˆ(r)−N],onearrivesataGaussianapproxima-tion[17,18]forΞv[N].This“discrete”approximationforβ∆ΩvbasedonthisuseoftheGFMis
¯2/2χv
β∆Ωv=−log
e−N
nodefixedatr=Rv,butmadeorthogonaltothefirst(constant)basisfunction.Thesecondbasisfunctionisthusalinearcombinationofj0(Rvr/π)andaconstant.Thiswasintroducedtotesttheaccuracyoftheoneba-sisfunctionapproximationpreviouslyusedandhopefullywillgiveimprovedresultsforlargerRv.
IV.
RESULTSFORLARGERCAVITIESWITH
Rv>σ/2
WenowdiscussthesolvationfreeenergiesgivenbythevariouspathwaysforacavitywithRv>σ/2,equiv-alenttoaphysicallyrealizablehardcoresoluteparticlewithdiameterσv>0.(ResultsfortinycavitieswithRv≤σ/2arediscussedinSec.Vbelow.)Weusethesimplestversionofthetheory,wherethePYapproxima-tionisusedfortheuniformfluidcorrelationfunctions.Fig.(1)givethesolvationfreeenergyβ∆Ωvfromthedifferentpathwaysasafunctionofthepackingfractionη=πρBσ3/6forRv/σ=1,1.5,1.75,wheretheresultscanbecomparedtocomputersimulations[19]ofCrooksandChandler.Notethatthevolumeofasphericalcav-itywithRv=1.75σisover42timesgreaterthanthatofasolventparticle.ForRv=σtheresultsalsogivetheexcesschemicalpotentialasafunctionofdensityfortheuniformhardspherefluid.
Asdiscussedabove,wecanobtainanalyticalexpres-sionsfor∆Ωvforboththecompressibilityandvirialroutes.Thecompressibilityroutegivesβ∆Ω(−2+7η−11η2)
Rv=
ηv
(1−η)3
Rv2
Rv3
(1−
η)3
(1−
η)3
2(1−η)3−log(1−η)−
18η2(1−η)
8η(1+η−2η2)
σ2
+
σ3
.(34)
ResultsforthedensityrouteandfortheGFMarecom-putednumerically.
InthisrangeofRvthereisgoodagreementexceptatthehighestdensitiesbetweenthecompressibility,virialanddensityroutes,withbestresultsoverallarisingfromthecompressibilityroute.ThedirectGFMpredictionsinFig.(2)fromthepartitionfunctionarelesssatisfactory.BoththediscreteandthecontinuumversionsoftheGFMgiveresultsthatapproachzeroincorrectlyasρB→0,andthecontinuumvaluesareconsistentlytoolargeathighdensitywhilethediscretevaluesaretoosmall.ThediscreteversionoftheGFMusesaGaussianapproxi-mationfortheconstrainedpartitionfunctionsandgiveslessaccurateresultsthancouldbeobtainedfromafittoaccuratevaluesofNandN2asintheinformationtheoryapproach[17].
7
20compressibility routevirial routeRv=1.75σ15density routesimulationΩ∆Rv=1.5σβ105Rv=σ000.10.2η0.30.4FIG.1:Theexcessfreeenergypredictedbythethreether-modynamicroutesforcavityradiiRv=σ,1.5σand1.75σ,comparedwithsimulationdata.ηisthepackingfractionandisequaltoπρBσ3/6.
20GFM: discreteRv=1.75σGFM: continuum15simulationΩ∆βRv=1.5σ105Rv=σ000.10.2η0.30.4FIG.2:Theexcessfreeenergypredictionsbyboththedis-creteandthecontinuumversionsoftheGFM,plottedforthesameRvvaluesandonthesamescaleasinFig.1.
AsRv→∞,thesurfaceofthecavityapproachesthatofaplanarwall.AsshowninEq.(14)thereisadivergingtermintheexcessfreeenergygivenbythecavityvolume
v=4πR3
v/3timesthebulkpressurepB,andthemoreinterestingquantitytocalculateisthesurfacetermγv,givenby
βγβ∆Ωv=
v−βpBv
15density routecompressibility∞virial routeγπ10simulation fitting formulaβ4-5000.10.2η0.30.4FIG.3:Surfacetensionpredictedbythethermodynamicroutescomparedwiththesimulationfittingformula.
yieldsthesamebulkpressureasgivenbytheaccurateuniformfluidPYcompressibilityequationofstate.Thevirialroutedoesnotautomaticallybuildinthisconsis-tency,andthepressurepredictedfromthecoefficienta3islessaccuratethantheuniformfluidPYvirialequationofstate.
Thecoefficientofthequadratictermthengivesthesurfacetensionβγ∞=a2/4πσ2.Usingthecompressibil-ityroutewefind
−4πβγ∞σ2
=
18η2(1+η)
(1−η)3
.(37)
Theγ∞obtainedbythecompressibilityroutecoincideswiththatgivenbyscaledparticletheory[14,15,20].Weobtainedγ∞numericallyforthedensityroute.
Theseresultscanbecomparedtothequasi-exactfor-mula[21]
−4πβγ∞σ2
=
18η2(1+
44
5η
2
)
8
(1−ρBv)
(43)
forroutsidev,withρ(r)=0forrinside.Hereg(r)istheexactradialdistributionfunctionfortheuniformsolventfluid.BNotethatthecontactdensityρ(Rv)=ρB/(1−ρv)isexactlydeterminedindependentofthedetailsofg(r),sincethecorrespondingg(|r′−r|)inEq.(43)vanishesforallr′insidev.Thisresultisvalidaslongastheinsertedregionvcanholdnomorethanonesolventparticle,soEq.(43)alsoholdsforsolventswithahardcorepairpotentialplusasoftertail.
B.
StructuralpredictionsoftheGFMandHLR
methods
NowletusexaminetheGFMresultinEq.(26)forthecaseofadensityresponsetoatinycavity.Sinceg(|r1−r2|)=0whenbothr1andr2areinv,χin(r1,r2)=ρBδ(r1−r2)−(ρB)2
then
insidev.ItiseasyseefromEq.(27)thattheinversefunctionχ−1
to
inthenhasthesimpleform[24]
χ−1in(r1,r2)=
δ(r1−r2)
1−ρBv
.(44)
exact0.50.05density route0.04GFM: discrete0.40.03GFM: continuum0.02Rv=0.3σΩ0.01∆0.3β000.10.20.30.40.2Rv=0.5σ0.1000.10.2η0.30.4FIG.4:Thecompressibilityandvirialroutesareexactforthetinycavityregime.ThedensityrouteisplottedalongwiththeGFMresultstocomparewiththeexactfreeenergypredictionsforRv=0.3σand0.5σ.
WhenEq.(44)isinsertedintoEq.(26)toobtaintheGFMdensityresponsetoatinycavity,werecovertheexactexpressionforρ(r)giveninEq.(43),providedthattheexactuniformfluidg(r)orc(r)isused.Ifapproxi-mate(sayPY)resultsareusedtodescribetheuniformfluidresponsefunctionsthenstrictlyspeakingtheGFMandHLRpredictionsforρ(r)willnotbeexactforallr.However,thecontactdensityρ(Rv)isexactinanycase,since,asnotedearlier,thisrequiresonlythattheap-proximateg(r)vanishinsidethecavityregion.BecauseoftheequivalencebetweenthestructuralpredictionsoftheGFMandtheHLRequation,thesesameconclusionsholdfortheHLRequation.Inparticular,thedensityre-sponseoutsideatinycavityisexactlydescribedbylinearresponsetheoryabouttheuniformbulksystem.
C.
Solvationfreeenergyfromthermodynamic
pathways
Moreover,thesetheoriesgivetheexactresultofEq.(42)forthesolvationfreeenergyβ∆Ωv,independentofpossibleerrorsing(r),forallstructurallybasedthermo-dynamicpathwaysthatuseonlytinyhardcorefields.Inparticular,thevirialrouteinEq.(19)givesexactresultsforβ∆Ωvsincethisrequiresonlythe(exact)contactval-uesρλ(λRv).ThecompressibilityrouteaswritteninEq.(16)requiresC(r1),whichtheexactχ−1
fromEq.(28)dependsonlyon
ininEq.(44).Bothresultsrequireonlythatthebulkg(r)vanishforr<σ,andareunaffectedbyanyerrorsatlargerr.Thisisinaccordwithourgeneralsup-positionthatparticularthermodynamicpathwayscanberelativelyinsensitivetoerrorsinthestructuraltheory.Howeverthechoiceofpathwayisimportant.Thusthedensityroutesdonotgiveβ∆Ωvexactlyeveninthetinycavityregime.Thisisbecauseasρλ(r)isvaried,thecorrespondingφλ(r)ingeneralisnotapurehardcore
9
fieldandspreadsoutsidethecavityregion.NeithertheGFMnortheHLRtheoriescantreatthesesofterandlonger-rangedfieldsexactlyevenifexactuniformfluidcorrelationfunctionsareused.
D.
SolvationfreeenergyfromtheGFMpartition
function
Theβ∆ΩvobtaineddirectlybytakingthelogarithmofthepartitionfunctioninEq.(29)canbeexpressedanalyticallyfortinycavitiesas
β∆Ω1
ρBv
v=
2
e−N¯/[2(1−N¯)]+e−(1−N
¯)/[2N¯],(46)
whereN
¯≡ρBv.Thishasthepeculiarbehaviorasv→0thatallderivativesvanish,andsoissignificantlyinerrorinthisregime.SeeFig.4forcomparisonwiththeexactanswer.
VI.
CONCLUSION
Wehavediscussedseveraldifferentthermodynamicroutesthatcanbeusedtodeterminethesolvationfreeenergyforinsertingbothsmallandlargecavitiesintoahardspherefluid.GenerallyaccurateresultsarefoundbyusingtheHLRequationtorelatethedensitiesandas-sociatedfieldsovertheintermediatestatesofthediffer-entpathways.WealsoconsideredtheGFMandshowedthatitgivesresultsequivalenttotheHLRequationforthedensityresponseinducedbyarigidcavity.HowevertheGFMcannotdescribethesofterexternalpotentialsandintermediatedensitiesneededforthedensityroutesandformoregeneralthermodynamicpathways.DirectuseoftheapproximatepartitionfunctionoftheGFMtodeterminethesolvationthefreeenergyofacavityalsogiveslessaccurateresults.
BestresultsusingtheHLRequationforthesolvationfreeenergyofacavityarefoundfromthecompressibilityanddensityroutes.Thiscanbeunderstoodsincemoststatesalongtheseroutesrequirethedensityresponseatintermediatedensitiesanddistancesawayfromthecav-itywheretheHLRequationismostaccurate.TheHLRequationcanalsobeusedformoregeneralsoluteswithdifferentshapesorlongerrangedattractiveinteractionsandinapplicationswhereotherpathwaysmaybemoreuseful.Combinedwithanappropriatepathwayitrepre-sentsaversatileandcomputationallyefficientmethodfor
10
determiningboththestructureandthethermodynamicsofnonuniformfluids.
WethankKirillKatsov,MichaelFisher,JimHender-son,andLawrencePrattforhelpfulcommentsandGavinCrooksforsendingusthesimulationdatathatisusedintheplotsoffinitesizecavities.ThisworkwassupportedbytheNationalScienceFoundationthroughGrantCHE-0111104.
APPENDIX
Inthe“wall”limitwhereRv→∞,thesurfacetensiongivenbythecompressibilityrouteisdeterminedfrom
γ∞=−
ρB
dρ
∂µ
0
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