Critical Casimir Effect in superfluid wetting films

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

CriticalCasimirE ectinsuper uidwetting lms

A.Macio lek,1,2,3A.Gambassi,1,2andS.Dietrich1,2

1Max-Planck-Institutf¨urMetallforschung,

arXiv:0705.1064v1 [cond-mat.stat-mech] 8 May 2007Heisenbergstr.3,D-70569Stuttgart,Germany2Institutf¨urTheoretischeundAngewandtePhysik,Universit¨atStuttgart,Pfa enwaldring57,D-70569Stuttgart,Germany3InstituteofPhysicalChemistry,PolishAcademyofSciences,Kasprzaka44/52,PL-01-224Warsaw,Poland(Dated:February1,2008)AbstractRecentexperimentaldataforthecompletewettingbehaviorofpure4Heandof3He-4Hemixturesexposedtosolidsubstratesshowthatthereisachangeofthecorresponding lmthicknessesLuponapproachingthermodynamicallytheλ-transitionandthetricriticalendpoint,respectively,whichcanbeattributedtocriticalCasimirforcesfC.Wecalculatethescalingfunctions offCwithinmodelsrepresentingthecorrespondinguniversalityclasses.Forthemixturesouranalysisprovidesanunderstandingoftherichbehaviorof deducedfromtheexperimentaldataandpredictsthecrossoverbehaviorbetweenthetricriticalpointandtheλ-transitionofpure4Hewhichareconnectedbyalineofcriticalpoints.Theformationofa’soft-mode’phasewithinthewetting lmsgivesrisetoapronouncedmaximumoffCbelowthetricriticalpointasobservedexperimentally.Nearthetricriticalpointwe ndlogarithmiccorrections～L 3(lnL)1/2fortheleadingbehaviorof dominatingthecontributionsfromthebackgrounddispersionforces.PACSnumbers:05.50.+q,,64.60.Kw,67.40.Kh

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

I.INTRODUCTIONThereisgrowingexperimentalevidencefortheanalogueoftheelectromagneticCasimire ect[1]invariouscriticalcondensedmattersystems[2,3,4,5,6,7].Inwettingexperimentsthecon nementofcritical uctuationswithinanadsorbedliquid lmgivesrisetoane ectiveCasimirforcefCbetweenthesubstrate-liquidandtheliquid-vaporinterfacesoftheliquid lm[8,9,10].NearthecriticalendpointoftheliquidtheemergingCasimirforceaddstotheomnipresentdispersionforcesandthusleadstoachangeofthethicknessofthecompletewetting lm.FromthisresponseonecaninfertheCasimirforcebysubtractingthee ectofthebackgroundforceswhichvariessmoothlynearthecriticalendpointwithtemperatureTc.Inaccordancewith nite-sizescalingtheory[11]thisforcefCperunitareaandinunitsofkBTccanbeexpressedintermsofauniversalscalingfunction ;itsshapedependssensitivelyonthetypeofboundaryconditions(BC)[9]andthusonthesurfaceuniversalityclassesthecon ningsurfacesbelongto[12].

Capacitancemeasurementsoftheequilibriumthicknessof4Hewetting lmsnearthesuper uidtemperatureTλofthecriticalendpointoftheλ-line[2,7]quantitativelysupportthetheoreticalpredictionsoffCforthebulkuniversalityclassoftheXYmodelwithsym-metricDirichlet-DirichletBC(O,O)formingtheso-calledordinary(O)surfaceuniversalityclass[12].SuchBCcorrespondtothecasethatthequantum-mechanicalwavefunctionofthesuper uidstatevanishesatbothinterfaces,givingrisetoanattractiveCasimirforce(fC<0)[9,10].However,theavailabletheoreticalresultshavealimitedrangeofapplicabil-ity,i.e.,T≥TλandT Tλ.AboveandatTλexplicit eld-theoreticalcalculationswithinthe -expansionschemeareavailable[13,14].FortemperatureswellbelowTλtherearecal-culationswhichtakeintoaccountcapillary-wavelikesurface uctuationsintheasymptoticlimitofthick lms,predictingalevellingo ofthescalingfunctionforlargenegativescalingvariables[15],i.e.,T Tλ,inqualitativeagreementwiththeexperimentalobservations.SofartherearenotheoreticalresultsavailableforthecriticalregionbelowTλwhichprovideanunderstandingofthedeepminimumoftheexperimentalscalingfunction(ca.20timesdeeperthanitsvalueatTλ).

3He-4Hemixturesneartheirtricriticalendpoint(seeFig.12inRef.[14])areanothercriticalsystemforwhichwettingexperimentshavebeenperformedrecently[4,5].Thetri-criticalendpointwithtemperatureTtisthepointinthe3He-4Hephasediagramwherethe

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

linesignallingtheonsetofsuper uidityjoinsthetopofthetwo-phasecoexistenceregionforphaseseparationintoa4He-richsuper uidphaseanda3He-richnormalphase.Themixturebelongstoabulkuniversalityclassdi erentfromthatoneofpure4Heand,becauseitsuppercriticalspatialdimensiond equals3,theactualphysicalsystemischaracter-izedbyrationalmean- eldcriticalexponents(uptologarithmiccorrections)[16,17].Thecapacitancemeasurementsofthewetting lmthicknessofthemixturerevealarepulsiveCasimirforcefCaroundthetricriticalendpointwhichsuggestsnon-symmetricBCforthesuper uidorderparameter(OP).TheprobablephysicalmechanismbehindsuchaBCisthatwithin3He-4Hewetting lmsa4He-richlayerformsnearthesubstrate-liquidinterface,whichmaybecomesuper uidalreadyabovethelineofonsetofsuper uidityinthebulk

[18]whereasthelighter3Hehasapreferencefortheliquid-vaporinterface.Thusthetwointerfacesimposeanontrivialconcentrationpro lewhichinturncouplestothesuper uidOP.

Forthissystem,recently[19]webrie yreportedexplicitmodelcalculationswhichdemon-stratethattheconcentrationpro leindeedinducesindirectlynon-symmetricBCforthesu-per uidOP.Forsymmetry-breaking(+)BCatthesubstrate-liquidinterfaceandDirichlet(O)BCattheliquid-vaporinterfacewecalculatedtheCasimirforceandfoundasemiquan-titativeagreementwiththeexperimentaldatagiveninRef.[4].Moreover,weformulatedtheoreticalpredictionsforthebehavioroffCinthecrossoverregimebetweenthetricriticalpointandtheλ-transitionofpure4HewhichareconnectedbyalineofcriticalpointsandprovidedtheuniversalleadingbehavioroftheCasimirforceatthetricriticalpoint.

ThepurposeofthepresentstudyistoelucidatethedetailsofthetwocomplementaryapproachesusedinRef.[19]andtoextendtheminordertoobtainnewresultsbothforthetricritical3He-4Hemixtureandthecriticalpure4He.Thepresentationisorganizedasfollows:InSec.IIwediscusstheuniversalpropertiesoftheCasimirforce.Asalreadymentionedabove,forthepresenttricriticalbehaviortheuppercriticaldimensiond equals3andthereforethethermodynamicfunctionsofthree-dimensionalsystemsexhibitpower-lawbehaviorswithcriticalexponentstakingtheirclassicalvalues.However,logarithmiccorrectionstothemean- eld(MF)behaviorareexpectedunderexperimentalconditions

[17].Using eld-theoreticalmethodsandrenormalization-group(RG)analysesweobtaintheleadingasymptoticbehavioroftheCasimirforceatthetricriticalpoint.Asafunctionofthe lmthicknessLithastheformofapowerlawmultipliedbyafractionalpower

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

ofalogarithmandbytheuniversalCasimiramplitude.Inaddition,wealsoderivetheformofthe nite-sizescalingfortheCasimirforceinthevicinityofthetricriticalpoint.Asexpected[17],alsotheargumentsoftheassociatescalingfunctionacquirelogarithmiccorrections.Thesescalingfunctionsarecomparedwiththeonesdeducedfromtheexperi-mentaldatainRef.[4].InSec.IIIwestudywithinmean- eldtheory(MFT) lmsofthelatticevectoralizedBlume-Emery-Gri ths(VBEG)model[20]whichbelongstothesameuniversalityclassasthe3He-4HemixturebutissimpleenoughtoallowforsystematicstudiesoffCalongallthermodynamicpathsfollowedinthewettingexperimentsofRef.[4].ThisfacilitatestheexplorationofthecrossoverbetweenthetricriticalpointTtandthelineofcriticalpointsandthecoexistenceregionbelowTt.ThisenablesustofollowtheCasimirforceuponcontinuouslyswitchingthebulkuniversalityclass(fromtricriticaltocritical)bychangingtheconcentrationofthe3He-4Hemixture.Thescalingfunctionscorrespondingtothermodynamicpathsofconstantconcentrationofthetwocomponentsofthe3He-4HemixturesarecalculatedandcomparedwiththecorrespondingexperimentaldatainRef.[4].AsalimitingcasetheVBEGmodelcandescribealsoa lmofpure4HewhichisstudiedinSec.IVwithinMFT.ThescalingfunctionofthecorrespondingCasimirforceisobtainedinthecriticalregionbelowTλandcomparedwiththatoneextractedfromtheexperimentaldatainRef.[2].Wealsocomparetheseresultswiththemean- eldpredictionswhichfol-lowfromtheLandau-Ginzburgtheoryinthe lmgeometrywithsuitableBC.InSec.VwediscussthetheoreticalresultsobtainedwithintheVBEGmodelandassesstheirrelevanceforinterpretingtheexperimentaldata.WeconcludewithasummaryandanoutlookinSec.VI.

II.UNIVERSALPROPERTIES

For lmgeometries,inthissectionweinvestigatetheuniversalpropertiesoftheCasimirforceneartricriticality.Ingeneraltwo-componentsystemsarecharacterizedbytheorderingdensityΦanditsconjugate eldh,andbyanon-orderingdensityxanditsconjugate eld .Forliquid3He-4Hemixtures,Φ,x,and correspondtothesuper uidOP,tothe3Heconcentrationandtothedi erencebetweenthechemicalpotentialsofthe3Heand4Hecomponents,respectively,whereasthe eldhconjugatetothesuper uidOPisexperimentallynotaccessible.

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

A.ScalingfunctionfromLandau-Ginzburgtheory

InordertocaptureuniversalpropertiesweconsiderthestandarddimensionlessO(n)-symmetricLandau-Ginzburg(LG)Hamiltonianforatricriticalsysteminthe lmgeometry: L 1u0H[Φ]=dd 1xdz(1)Φ2+(Φ2)3,26!0

whereListhe lmthickness,Φisthen-componentorderparameterOP(n=2correspondstotheXYuniversalityclass),andzisthecoordinatenormaltothecon ningsurfaces;r0,u0,andv0arebarecouplingconstantsdepending,interalia,onthetemperatureTandthenon-ordering eld .r0(u0)=0andu0>0de nethecriticalline,whereasatthetricriticalpointonehasr0=u0=0,v0>0.Thesemi-in niteversionofEq.(1)hasbeenstudiedinthecontextofsurfacecriticalbehavior[21].Inthe lmgeometrytheCasimirforceperareaAofthecrosssectionofthe lmandinunitsofkBTt,

fC≡ ( fex/ L)= Tzz ,

isgivenbythethermalaverageofthestresstensorcomponentTzz[9]:

fex(L)≡(f fb)L/(kBTt)(3)(2)

wherefisthetotalfreeenergyofthe lmpervolumeV=LAandfbisthebulkfreeenergydensity.ForlargeLtheexcessfreeenergycanbedecomposedintosurfaceand nite-sizecontributions:fex(L)=fs,1+fs,2+δf(L).Thestresstensorisgivenby[9]

Tij= iΦ· jΦ δijL (d 2)/(4(d 1))( i j δij 2)Φ2,(4)

whereListheintegrandinEq.(1).InwhatfollowsweassumeΦ=(m(z),0,...,0),i.e.,weneglecthelicity.Fornon-symmetricBCitsrelevanceforthebehavioroftheCasimirforceisnotclearbecausetheOPhastheadditionalfreedomtorotateacrossthe lmbyapositiondependentangleφ(z);theanalysesoftheroleofhelicityisleftforfutureresearch.WithinMFTfortheLGHamiltonian,thedeterminationofthetricriticalCasimirforceinthe lmgeometrystartsfromtheEuler-Lagrangeequation

m′′(z)=r0m(z)+u0

120m5(z).(5)

AsdiscussedinSec.I,(+,O)boundaryconditions,withthesubstrateatz<0andvaporatz>L,

m(0)=+∞andm(L)=0

(6)

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

aresupposedtomimictheexperimentalsystemof3He-4Hewetting lmsasstudiedinRef.[4].AccordingtoEq.(4)thestresstensorcomponentTzzevaluatedwithinMFTandwithΦ=(m(z),0,...,0)fortheOP(inthepresentMFapproachweomitthebrackets · indicatingthethermalaverage)yields

Tzz=1

12m4+,O(z)+v0

360

wherev0isdimensionless.Similarly,withinthisapproachthenormalizedCasimirforcecanbeexpressedintermsofadimensionlessscalingfunction +,O:

v0Tzz=fC(L,r0,u0,v0)=

1/2

6u0L 4+,O(x)+ +,O(x), 1/4L 1/2 +,O(z/L,r0L2,u0L;v0),(9)2v0(11)

wherex=z/L.Inturn,Eq.(11)canbeintegrateddirectlyyieldingtheimplicitequation ∞d 1= 0MF +,O+r0L2 2+5

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

fortheintegrationvariableonecanrewriteEq.(12)inthemoreconvenientbutstillimplicitform MF1/3 +,O= ∞

0dp1+ap2+bp4+p6,(14)

wherethedimensionlessparametersaandbaregivenby

2/3a=r0L2 MF+,Oandb= 5

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

wherea′istheslopeofthelinetangentialtothephaseboundaryatthetricriticalpoint.Thusfort→0withg=0thetricriticalpointisapproachedtangentiallytothephaseboundary.Insteadoftonecouldalsouseascalingvariablewhichisorthogonaltothelocig=0;thiswouldnota ecttheleadingsingularbehaviorfort,g→0[17].NearthetricriticalpointB( ),A( ),andTλ( )ingEq.(17)onehasT Tλ( )=T Tt+(a′kB) 1( t)+O(( t)2)=(a′) 1Ttg+O(( t)2).Expressing andTasafunctionoftandgone nds:

r0=A1g+A2t2+O(g2,gt)andu0=B1t+B2g+O(gt,g2,t2)(18)

whereA1>0,B1>0,A2,andB2areconstants.DuetotheanalyticstructureofEq.(16)andbecause( t)=kBTt(g a′t)thecoe cientr0doesnotcontainatermlinearintsothatu0～t+O(t2)ifr0=0.Ontheotherhandr0～g+O(g2)ifu0=0.

B.LogarithmiccorrectionsatT=Tt

Atthetricriticalpointa=b=0Eq.(14)reducesto

( MF)1/3= ∞

dp/

0

90

wherebotht 1/2tzz+ Tzz 1+O(v1/20)(21)zz

singularitiesvia≡renormalization2.75684L 3and Tzz(R)the 1donotdependonv0.AfterremovingultravioletasymptoticscalingbehavioroffCfollowsfromsubstitutingtherenormalizedvbytheappropriate xed-pointvaluev

underspatialrescalingbyadimensionlessfactor ,v owstoitsRG xed∝point .Atvalued=d ,andv =0accordingto[21]

v¯( )=240π2ln |

|ln |+cln|

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

wherev¯( )istherunningcouplingconstantwiththeinitialconditionv¯R( =1)=vR.Withtherescalingfactor =l0/L,wherel0isamicroscopiclengthscaleoftheorderofafew A,thisyieldsalogarithmiccorrectiontothepower-lawdependenceonLofthetricriticalCasimirforce:

fCt 0.54(3n+22)1/2(ln(L/l0))1/2L 3 1 c

|ln(L/l(23)

0)|+... .

DeterminingtheconstantcrequirestoextendtheanalysisinRef.[21]whichisleftforfutureresearch.Gaussian uctuationsgivecontributionsofatleastO(v0)whicharethereforeoforderL 3andthussubdominant(seeEq.(23)).WecompareEq.(23)forn=2withthedataobtainedbyGarciaandChan[4]fortheirexperimentalvalueofL≈520 Aandforl0ζ(t)/|t| νtwithν≈1.3 A,theexperimentalvalueofthecorrelationlengthamplitudeζ0=t=1forconcentration uctuationsbelowTtinthesuper uidphase[23].ForthesevaluesEq.(23)predicts

t≡fCtL3≈6.96(24)

whereas expt=8.4±1.7.Thevalueofthetheoreticalfunction tatTt,withl0between1and2 A,isinreasonableagreementwiththemeasured expt.InordertoextractthevalueoftheuniversalCasimiramplitude(i.e.,thenumericalprefactor0.54√actual

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

systemisnotrenormalized[11].Second,intherenormalized(R) nite-sizecontributiontothefreeenergyδf(L)(seethetextbeforeEq.(4))thecontributionsfromtheadditivecountertermscancelandonehas[12,25]:

δfR(r,u,v;µ,L)=δf(r0,u0,v0;L)(25)

wherethebarequantitiesu0,r0,andv0areexpressedintermsofrenormalizedonesr,u,andv;µisanarbitrarymomentumscale.Sincewearenotconsideringcorrelationfunctionsatthesurface,allrenormalizationfactorsZarethesameasthoseinthebulk[12,21]:

r0=Zrr+u2µ 2 P,u0=Zuu,v0=2π2Zvv,(26)

wherethedimensionsofthecouplingconstantare[r0]=µ2,[u0]=µ1+ and[v0]=µ2 .ExplicitperturbativeresultsforthetricriticalbulkrenormalizationfunctionsZr,P,Zu,andZvareknown(see,e.g.,Refs.[17,21]).FromEq.(25)theRGequationcanbederivedinastandardfashionbyexploitingthefactthatδf(r0,u0,v0;L)isindependentofµ.BecauseinEq.(25)therearenoadditiverenormalizationtermsitfollowsthatδfR(L)satis esthefollowinghomogeneousRGequation[12]:

µ µ+ βκ κ δfR(L)=0(27)

κ=r,u,v

whereβκ(r,u,v; )≡µ µ

interactionconstantsforκ|0κand µ=r,u,v.|0denotesderivativeswithrespecttoµat xedbareTheRGequationissolvedbyusingthemethodofcharacteristics(see,e.g.,Ref.[26]):

δfR(r′,u,v,µ;L)=δfR(¯r′( ),u¯( ),v¯( );µ ;L)(28)

where isagainadimensionlessspatialrescalingfactor,κ¯( )aretherunningcouplingconstantswiththeinitialconditionκ¯(1)=κ,andduetotheformoftherenormalizationofr0(seeEq.(26))thenewvariabler′isgivenby[17,21]

r′=r+w(v,µ)u2.(29)

Foranexplicitexpressionofw(v,µ)seeRefs.[17,21].Equation(28)ingdimensionalanalysisoneobtains

δfR(r′,u,v,µ;L)=(µ )(d 1)δfR r¯′( )v¯( )

(µ )4 d,

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

ThedesiredasymptoticscalingbehaviorofδfRfollowsbysubstitutingontherhsofEq.(30)theappropriate xed-pointvaluesfortherunningcouplingconstantsr¯′,u¯,andv¯.Theinfraredstable xedpointliesatv =(240/(3n+22)) +O( 2)[21].Uponapproachingtheuppercriticaldimensionv

theclassicalexponentsaregenerated→0andbythefor ow→0ofthetherelevantcouplinglogarithmicconstantsundercorrectionstheRGtotransformation →0.Inthelimit →0,v¯( )isgivenbyEq.(22).Therunningvariablesr¯′(l)andu¯(l)canbewrittenasr¯′( )=Er( ;v)r′andu¯( )=Eu( ;v)u.Astraightforwardanalysis[17,21]showsthatEr( ;v)→constandEu( ;v)～|ln | 2(n+4)/(3n+22)for →0.Choosingµ=1/l0,µ L= (L/l0)=1,andomittingtheconstantfactorErweobtainthefollowingscalingformforδf:

δfR(r′,u,v,µ;L)=L 2δfR(r′L2,uL|ln(L/l0)| 2(n+4)/(3n+22),|ln(L/l0)| 1;1,1).(31)DuetoEq.(2)thescalingformfortheCasimirforcefollowsfromEq.(31)as:

fC(r′,u,v;L) L 3θ(r′L2,uL|ln(L/l0)| 2(n+4)/(3n+22),|ln(L/l0)| 1).(32)

ThescalingfunctionθisgivenintermsofδfR(z1,z2,z3,1)asθ=2δfR+2z1( δfR/ z1) z2( δfR/ z2).Thehigher-ordertermsneglectedinEq.(32)areoftheformL 3(ln(L/l0)) 1(2(n+4)/(3n+22))z2( δfR/ z2)+L 3(ln(L/l0)) 1z3( δfR/ z3)+L 3( 1+2c(ln|ln(L/l0)|)/(ln3|(L/l0)|)( δfR/ z3).Thethirdterminthelatterexpressionstemsfromthecorrectiontoz3(seeEqs.(30)and(22)).

Attheuppercriticaldimensiontheasymptoticcriticalbehaviorobtainedfromthepertur-bativeRGcalculationswithintheGaussianapproximationisexpectedtobeexact.However,atthelowestorder,oftenreferredtoasrenormalizedmean- eldtheory(RMF)–whichyieldsthefreeenergycorrectlywiththeleadinglogarithms–oneneglectsthecontributionsstem-mingfromtheGaussian uctuationsandreplacesthescalingfunctionbyitsmean- eld-likeformbutwiththerescaledarguments.

Applyingthisreasoningtothefreeenergyweusethemean- eldresultgivenbyEqs.(10)and(12)withr0replacedinfavorofr′accordingtoEq.(29)withw(¯v( ),µ( ))→constas →0,u0replacedbyu|ln(L/l0)| 2(n+4)/(3n+22),andv0replacedby((240π2)/(3n+

22))|ln(L/l0)| 1toobtainatlowestorder:

fCRMF 3n+22

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

InthefollowingwewanttocomparethebehavioroftheMFandRMFexpressionfortheCasimirforce.Aswehavealreadystressedbefore,fCcalculatedwithintheMFapproachdependsonthenon-universalanddimensionlessparameterv0(seeEq.(10)).Uponcompar-ingwithexperimentaldatathisparametercanbeusedto ttheamplitudeoftheCasimirforce,becausev 1/2

0appears(albeitnotexclusively)asaprefactorofthescalingfunction.

Thefactorv 1/2

0,whichmultipliesthecouplingconstantu0(seethetextafterEq.(12)),is

absorbedinthede nitionofthescalingvariable.

InFig.1wehaveplottedtwocurves:(1) ¯MF(r0L2=0,yMF)=fCL3(v0/90)1/2asafunctionofyMF=(5/(2v0))1/2u0L(seeEq.(12)).Here,thenon-universalfactorv 1/2

0is

absorbedinthede nitionsofthescalingfunctionandofthescalingvariable.Asalreadymentionedbeforeu0

(28/(8π2/3))1/2(ln(L/l～tifr0=0,sothatu0L～tL.(2)fCL3≡ ¯RMF(0,yRMF)=0))1/2 MF(0,yRMF)(forn=2),whereyRMF=uL(ln(L/l0))1/14.Here,renormalization xestheamplitudeoftheCasimirforcereplacingthe

prefactor non-universal

v0

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

III.VECTORALIZEDBLUME-EMERY-GRIFFITHSMODEL.

BasedonthemotivationprovidedintheIntroduction,inthissectionweextendtheVBEGmodeltothe lmgeometryandstudy3He-4Hemixtures.

A.Themodel

¯parallelWeconsiderathree-dimensionalslabofasimplecubiclatticeconsistingofL

¯.EachlayerhasA¯=A/a2sites,(100)latticelayerswithlatticespacingasothatL=La

labeledi,j,...,whichareassociatedwithanoccupationvariableti=0,1andaphaseθi(0≤θi<2π)whichmimicsthephaseofthe4HewavefunctionandthusrenderstheXYbulkuniversalityclass(n=2).A3He(4He)atomatsiteicorrespondstoti=0(1)sothatinthebulkX=1 ti isthe3Heconcentration.Unoccupiedsitesarenotallowedsothatthemodeldoesnotexhibitavaporphase.Accordinglythismodeldoesnotallowfortheoccurrenceofatricriticalendpoint.However,weexpectthattheuniversalpropertiesweareinterestedinarethesamefortricriticalpointsandtricriticalendpoints.TheHamiltonianconsistsofbulkandsurfacecontributionsH=Hb+Hswith

Hb= J ij titjcos(θi θj) K ij titj+ iti,(34)

wherethe rsttwosumsrunovernearest-neighborpairsandthelastoneisoveralllatticesites,exceptthoseatthesurface.Inthislatticegasmodelof3He-4HebinarymixturesthecouplingconstantKandthe eld arerelatedtothee ectiveαHe-βHeinteractionsKαβ(see,e.g.,Ref.[27]),

K=K33+K44 2K34,

andtothechemicalpotentialsµ3andµ4of3Heand4He,respectively,

=µ3 µ4+2q(K33 K34),(36)(35)

whereqisthecoordinationnumberofthelattice(q=2d,wheredisthespatialdimensionofthesystem;q=6inthepresentcase).Intheliquidthee ectiveinteractionsKαβaredi erentfordi erentαandβduetothedi erencesinmassandofstatisticsbetween3Heand4Heatoms.

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

ThepropertiesofthemodeldescribedbythebulkHamiltonianHbhavebeenstudiedwithinMFTandbyMonteCarlosimulationsind=3[20].Incontrasttoitstwo-dimensionalversion,forwhichthereisnotruetricriticalpointforanyvalueofthemodelparameters,ind=3forreasonablevaluesoftheinteractionparameterstheresultingphasediagramresemblesthatobservedexperimentallyfor3He-4Hemixtures,forwhichphaseseparationoccursasaconsequenceofthesuper uidtransition(seeFig.4).TheformofthesurfaceHamiltonianHsshouldcapturethephenomenonofsuper uid lmformationnearawallin3He-4Hemixtures[18]whichgeneratesane ectiverepulsionof3Heatomsbythewall.ThevanderWaalsinteractionsbetweenthewalland3Heor4Heatomsareequal.However,3Heatomsoccupyalargervolumebecauseoftheirlargerzero-pointmotions.Thisgivesrisetothepreferentialadsorptionof4Heatomsatthesubstrate- uidinterface,whichmayinducealocalsuper uidorderingandanenrichmentof3Heneartheopposing uid-vaporinterface.HerewechoosethefollowingformforHs:

Hs=δ (l)(l)

iti+δ (r)(r) iti,(37)

wherethe rstsumrunsoverthesitesofthe rstlayerandthesecondoverthoseintheL-ththerelativepreferencesof4Heatomsforthetwosurfacessuchthatδ (l)<yerofthelattice.Thedi erencesδ (l)≡ (l) andδ (r)≡ (r) aremeasuresof

B.Mean-FieldTheory

Wehavestudiedtheabovemodelforthe lmgeometrywithinmean- eldtheory.Wehaveemployedthevariationalmethodbasedonapproximatingthetotalequilibriumdensitydistributionbyaproductoflocalsitedensitiesρi(see,e.g.,Ref.[28]).Thecorrespondingvariationtheoremforthefreeenergyreads

F≤Fρ=Tr(ρH)+(1/β)Tr(ρlnρ),(38)

whereFistheexactfreeenergyandFρisanapproximatefreeenergyassociatedwiththedensitydistributionρ;β=1/(kBT).TheminimumofFρwithrespecttoρsubjecttotheconstraintTrρ=1isattainedfortheequilibriumdensitydistributionρ=e βH/Tr(e βH).

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

Withinmean- eldtheorythedensitydistributioninthe lmgeometryisapproximatedby

¯Lρ=ρ0=AL i=1ρi,(39)

i.e.,thedensitydistributionisconstantwithineachlayerparalleltothesurfacesbutvariesfromlayertolayer.Wetreatthelocallayerdensityρiasavariationalansatz,andthebest

¯+ηTr(ρi)withrespectfunctionalformintermsoftiandθiisobtainedbyminimizingFρ0/A

toρiandwithηasaLagrangemultiplierinordertoimplementTrρ=1.Thisleadsto

ρi=e βhi/Tr(e βhi)

wherehiisthesingle-layermean eldgivenby

hi= J(Mi 1+q||Mi(1)(1)(40)+Mi+1)ticosθi J(Mi 1+q||Mi(1)(2)(2)+Mi+1)tisinθi

(41)(2) K(Qi 1+q||Qi+Qi+1)ti+ (i)ti,

¯,and (i)= (l)( (r))fori=1(L¯).Wehaveintroducedthewhere (i)= fori=1,L

followingorderparameters:

Qi≡1 X(i)=Tr(tiρi)

and

Mi

(1)(1)(42)=Tr(ρiticosθi),Mi(2)=Tr(ρitisinθi).(43)Qi= ti correspondstotheconcentrationpro leof4He,X(i)=1 ti totheconcentrationpro leof3He,andMi,Mi(2)arethecomponentsofthetwo-componentsuper uidOPpro le

Mi.q||isthein-layercoordinationnumberwhileeachsite(butnotinthe rstandlastlayer)isconnectedtoq′atomsineachadjacentlayerandq=q||+2q′isthecoordinationnumberinthebulkofthelattice.Withinourmodelq′=1andq||=2(d 1).Thisyieldsthefollowingsetofself-consistentequationsfortheOPMi=(Mi,0)≡(mi,0)intheithlayer:

β(Kai (i))Qi=I0(βJbi)/e+I0(βJbi),

(i)mi=I1(βJbi)/e β(Kai )+I0(βJbi).(1)(44)and(45)

I0(z)andI1(z)arethemodi edBesselfunctionsofthe rstkind,Tisthetemperature.Wehaveintroduced

bi≡mi 1+q||mi+mi+1

for¯i=1,L,(46)

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

b1≡q||m1+m2,andbL¯,andanalogously¯ 1+q||mL¯≡mL

ai≡Qi 1+q||Qi+Qi+1for¯i=1,L,(47)

a1=q||Q1+Q2,andaL¯.ThecoupledsetsofequationsforQiandmiare¯ 1+q||QL¯=QL

solvednumericallybystandardmethodsofmultidimensionalroot nding.Theequilibrium

¯:solutionminimizesthefreeenergypernumberoflaterallatticesitesF≡Fρ/A0

f=¯ 1 L J

i=22

ln(1 Qi)+f1+f2,(Qi 1Qi+q||Q2i+Qi+1Qi)+kBT

where¯L i=1 (48)

f1=

and

f2=JJ2(q||Q21+Q2Q1)(49)

22(QL¯+q||QL¯ 1QL¯).

(1)(50)Theaboveequationsneglectthehelicity,i.e.,Mi=(Mi,0)≡(mi,0).Ingeneralthehelic-

itymightbenon-zerobecausetheBCforthesuper uidOParee ectivelynon-symmetric, lm.TherelevanceofthehelicityontheCasimirforcewillbeanalyzedelsewhere.i.e.,M1=0whereasML=0sothatthesuper uidOPcaninprinciplerotateacrossthe

C.Resultsfor3He-4Hemixtures

First,wehaveanalyzedthesemi-in nitesystem.Closetothelineofbulkcriticalpointswehavefoundahigher4Heconcentrationnearthesurface(chosentobetheleftsideofthesystem),whichinducesalocalsuper uidordering.ByvaryingTand oneobtainsalineofcontinuoussurfacetransitionscorrespondingtotheonsetoftheformationofthissuper uid lmnearthewall;itmeetsthelineofbulkcriticalpointsataso-calledspecialtransitionpoint,thepositionofwhichdependsonthevalueof (l)(seeFig.4).These ndingsareinagreementwiththeresultsofaMigdal-Kadano analysis[29].

¯)Inthe lmgeometrytheCasimirforcefC(Eq.(2))isobtainedbycalculatingfex(L

¯andL¯+1andtakingthedi erence.(Notethatinthelatticemodelf(seeEq.(3))forL

¯A¯oflatticesitesandfbisthebulkfreeisthetotalfreeenergyofthe lmpernumberL

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

¯A¯.Accordinglyfex(L¯)=(f fb)L/¯(kBTt),fC= fex/ L¯,aswellasenergydensityperL

¯dwithd=3neartricriticalityandd=4neartheλ-transitionaredimensionless. =fCL

Inordertoavoidaclumsynotationwedonotintroducedi erentsymbolsforthelatticeandthecontinuumversionsofthefreeenergies.)Figure5summarizesourresultfora lmof¯=20,K/J=0.5, (l)/J= 3,and (r)= t/J 0.61,whichisthetricriticalthicknessLbulkvalue.Suchachoiceofthesurfacecouplingconstantscorrespondstonon-symmetricBCandisconsistentwiththeassumptionmadeinRef.[4]fortheconcentrationpro leacrossthewetting lm,whereuponattheinterfacewiththevaporthe3Heconcentrationtakesthebulkvalue.Fortemperaturesabovethebulkcoexistencelineat rst-orderdemixingtransitionsfCiscalculatedalongthethermodynamicpathsindicatedinFig.4whichcorrespondto xed3HeconcentrationsX.OurselectionofXcoversthetricriticalregionaswellasthecrossovertothecriticalsuper uidbehaviorofpure4He,i.e.,X=0.Inordertocalculatetheforceata xedvalueX0we rstdetermine (X=X0,T)bysolvingthetwocoupledself-consistentequationsforQ( ,T)=1 XandM( ,T)inthebulk(Eqs.(12)and(13)inRef.[20]).ForeachtemperaturealongthethermodynamicpathsindicatedinFig.4wesolveEqs.(44)and(45)withthisvalue (X=X0,T).Thisrendersthepro lesQ(l)andm(l)andallowsustocalculatethefreeenergyfromEq.(48).WhenuponloweringthetemperaturethepathsofconstantXreachthecoexistencelineoftwo-phasecoexistence(seeFig.4)wecontinueourcalculationsalongthecoexistenceline,in nitesimallyonthesuper uidbranchofbulkcoexistence.InFig.5thisleadstothefulllineforT<Tt,i.e.,y<0.

ContrarytotheLGmodel,forthepresentmicroscopicmodelitisnaturaltoexpressthepropertiesofthesystemasfunctionsoftheexperimentalthermodynamic eldstand( t)/(kBTt)orthescaling eldstandg(seeEq.(17)).Accordingly,wepresentourresultsfortheCasimirforceintermsofthescalingfunctionde nedthroughtherelation

¯3fCasafunctionofonlyasinglescalingvariabley¯1/ν=((L/a)/(ξ/ξ+))1/ν. ≡L¯≡tL0

¯above¯+aistheamplitudeoftheorderparametercorrelationlengthξ=ξ+t ν=ξaξ+=ξ000

¯2alsovariesalongaTtandν(d=3)=1.Thesecondrelevantscalingvariablex≡gL

pathof xed3Heconcentration(seeFig.6)andaproperscalingdescriptionhastoaccountforit.However,inordertobeabletocompareourresultswiththepresentationofthecorrespondingexperimentalones[4],wefollowRef.[4]wherethevariationofxhasbeenneglected.AscanbeinferredfromthephasediagraminFig.6,theg-componentsofthe

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

pathsX=constinthephasediagramaresmallerthanthet-components,sothattheformofthescalingfunctionforthesepathsareexpectedbecloseto (x=0,y).Alsoexperimentallythevariationofthescalingvariablegalongthepathof xedXcannotbedeterminedeasily.NearthetricriticalpointpathsofconstantXcrossthreedi erentphasetransitionlines:thesurfacetransitionline,thelineofbulkcriticalpoints,andthelineof rst-orderphasecoexistence.AsshowninFig.5,closetothesurfacetransitionfCissmallandthistransitiondoesnotleaveavisibletraceinitsbehavior.fCremainssmalluptothecoexistencelineortothelineofbulkcriticalpointsforX>XtorX<Xt,respectively.Thereitincreasesverysteeplyandfor3HeconcentrationsX<Xtuponcrossingthelineofbulkcriticalpointsthereisabreakinslope(seethedotsinFig.5)givingrisetotheformationofshoulderswhicharesimilartothoseobservedexperimentally[4].WhenTreachesthetemperatureof rst-orderphaseseparation,fCisgivenbythecurve(fulllinefory<0inFig.5)commontoallvaluesofX.ThesecurvesofconstantXmeetthefulllinewithdi erentslopes.TheaforementionedcommoncurveexhibitsapronouncedmaximumbelowTtaty thistemperatureregioncanbeattributedtopurelyinterfaciale ects.Indeed,weobservecommoncurvedisplayaninterface-likestructureseparatingtwodomainsofthecoexistingbulkphases(seethecaset= 0.0633inFig.7).This lmphaseissoftwithrespecttoshiftsoftheinterfacepositionandissimilartotheoneoccurringinIsing-like lmswithoppositeBC[30]fortemperaturesbelowthebulkcriticaltemperaturebutabovethewettingtemperatureofthecon ningwalls,inwhichcasetheCasimirforceisrepulsivewithapronouncedmaximumoccurringbelowthebulkcriticaltemperature[31].Ingeneralapositivesignoftheforcecanberegardedasaconsequenceofentropicrepulsion[32].TypicallythemaximumoftheforceoccursatthattemperatureTatwhichtheinterfacialwidth,whichisproportionaltothebulkcorrelationlengthξoftheorderparameter,becomescomparablewiththewidthLofthe lm.Inthepresentcaseboththeconcentrationandthesuper uidOPpro lecontributetothefreeenergyandhencetotheCasimirforce.Theirinterfacialwidthsareproportionaltocorrelationlengthζassociatedwithconcentration uctuationsandtotheOPcorrelationlengthξ,respectively.AscanbeseenfromFigs.7and8,withinMFTtheseinterfacialwidthsandthereforeζandξarecomparable.Accordingly,byanalogywithIsing-likesystems[30]weexpectthatwithinMFTthemaximumofthe

0.74andgraduallydecreasestozerofory→ ∞.ThepropertiesoftheCasimirforceinthatbelowTtboththeconcentrationandthesuper uidOPpro lecorrespondingtothis

Recent experimental data for the complete wetting behavior of pure 4He and of 3He-4He mixtures exposed to solid substrates show that there is a change of the corresponding film thicknesses L upon approaching thermodynamically the lambda-transition and the

forceoccurswhenξ(or,equivalently,ζ ξ)isoftheorderofL,whichisactuallyconsistentwithwhatisobservedinFig.5,wherethemaximumofthescalingfunctionislocatedaty 1.WemayexpectthatalsointheactualsystemtheoccurrenceofthemaximumoftheCasimirforcebelowthetricriticalpointcanbeattributedtosuchinterfaciale ects.However,sincethecorrelationlengthofthesuper uidOPξ=∞inthesuper uidphaseitisnotyetclearwhichlengthscalegovernstheinterfacialwidthofthesuper uidOPpro leinthe’softmode’phasebelowTtandhencewhatlengthscaledeterminesthepositionoftheforcemaximum.

4Heandagradualformationofasecond,lesspronouncedlocalmaximumlocatedslightlyForX Xt 0.05weobserveacrossovertothecriticalsuper uidbehaviorofpurebelowthelineofbulkcriticalpoints(y>0inFig.5).ThislocalmaximumdecreasesupondepartingfromXtand nallyfCbecomesvanishinglysmallalongpathswhichcrossthelineofbulkcriticalpointsabovethespecialtransitionS(seeFig.4).Thisisexpected,becauseaboveSthereisnolongerasuper uid lmformationnearthesolidsubstrateforthermodynamicstatescorrespondingtothebulk“normal”phaseofa uidclosetothelineofbulkcriticalpoints.Thismeansthatthesuper uidOPinthe lmisidenticallyzerouptothelineofbulkcriticalpointsandtheBCe ectivelyturnintothetype(O,O)forwhichfCvanisheswithinMFT.(For(O,O)BC uctuationsbeyondMFTgenerateanattractiveCasimirforcefC<0[10].)ForlowerT,fCincreasessteeplyuponapproachingbulkcoexistencerevealingthatinterfaciale ectsassociatedwiththe’softmode’leadtoamuchstrongerCasimire ectthanthecritical uctuationsnearthelineofbulkcriticalpoints.

IV.

A.RESULTSFORPURE4HEThelimitingcaseoftheVBEGmodel

Inthissectionweconsiderthelimitingcase → ∞inwhichalllatticesitesareoccupied,i.e.,ti→1.Inthiscasethe rsttermofthebulkHamiltonianHbinEq.(34)correspondstotheclassicalXYmodel(theplanarrotatormodel)forpure4Heandtherefore,asfarasthebulkcontributionisconcerned,thepartitionfunctionoftheVBEGmodelreducestothatoftheXYmodeluptoafactoreKzNwhereNisthenumberoflatticesites.

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