CondensedMatterPhysics,2014,Vol.17,No4,43601:1–14 DOI:10.5488/CMP.17.43601 http://www.icmp.lviv.ua/journal The mechanism of domain-wall structure formation in Ar-Kr submonolayer films on graphite 5 A.Patrykiejew,S.Sokołowski 1 0 DepartmentfortheModellingofPhysico-ChemicalProcesses,FacultyofChemistry,MCSUniversity,20031 2 Lublin,Poland n a ReceivedJuly18,2014,infinalformOctober23,2014 J 2 Using Monte Carlo simulation method in the canonical ensemble, we have studied the commensurate- 1 incommensuratetransitionintwo-dimensionalfinitemixedclustersofArandKradsorbedongraphitebasal planeatlowtemperatures.Ithasbeendemonstratedthatthetransitionoccurswhentheargonconcentration ] l exceedsthevalueneededtocovertheperipheriesofthecluster.Theincommensuratephaseexhibitsasimi- l a lardomain-wallstructureasobservedinpurekryptonfilmsatthedensitiesexceedingthedensityofaperfect h (p3 p3)R30◦ commensuratephase,butthesizeofcommensuratedomainsdoesnotchangemuchwith × - theclustersize.Whentheargonconcentrationincreases,thecompositionofdomainwallschangeswhilethe s e commensuratedomainsaremadeofpurekrypton.Wehaveconstructedasimpleone-dimensionalFrenkel- m Kontorova-likemodelthatyieldstheresultsbeinginagoodqualitativeagreementwiththeMonteCarloresults obtainedfortwo-dimensionalsystems. . t a Keywords:adsorptionofmixtures,commensurate-incommensuratetransitions,MonteCarlosimulation, m finitesystems,Frenkel-Kontorovamodel - d PACS:68.35.Rh,64.70.Rh,64.60.an n o c 1. Introduction [ 1 The commensurate-incommensurate (C-IC) transitionsinadsorbed films have been experimentally v observedinavarietyofsystems[1–15]andstudiedbytheoreticalmethods[16–27]andcomputersimu- 6 6 lations[28–37]. 5 Inalargenumberofsystems,theincommensuratephaseformsacollectionoffinitecommensurate 2 regions(domains)separatedbydomainwalls[21].ThedomainwallnetworksappearwhentheCphase 0 isdegenerated,i.e.,whentheatoms(ormolecules)intheCphaseoccupyoneofseveralequivalentlat- . 1 tices.ThestabilityofICphaseswithdomainwallsnetworksisconsiderablyenhancedbylargeentropic 0 effects[16,21].Themostprominentexampleofsuchasystemisthekryptonmonolayeradsorbedonthe 5 graphitebasalplane[38–41].Atlowtemperatures,theadsorptionofkryptonleadstotheformationofthe 1 : three-folddegeneratecommensurate(p3 p3)R30◦ (p3 p3)R30◦ structure,inwhicheverykrypton v atomoccupiesoneofeverythreehexagons×ofthegraphite×surface,hasthelatticeconstant p3timesthe i X graphitelatticeconstant(a 2.46Å)andisrotatedwithrespecttothegraphitelatticeby30degrees. gr = r Upontheincreaseofpressure,thecommensuratephaseundergoesatransitionintotheincommensurate a phaseofhigherdensity,inwhichthecommensuratedomainsareseparatedbywalls,andtheatomsin neighboring domains are shifted to a different sublattice [40, 42–44]. In this case, the walls are called heavyandcarryalltheexcessofdensity.AtthedensitiesjustabovetheC-ICtransitionthedomainsare verylargeandtheirsizegraduallydecreaseswhenthedensityincreasestowardsthemonolayercomple- tion.Thishasbeenclearlydemonstratedbyalargescalecomputersimulationstudy[30]. The C-IC transition has also been experimentally observed in the mixed Kr-Xe and Ar-Xe films on graphite[45–48].Ontheotherhand,thestructureofmixedAr-Krfilmsongraphitehasnotbeenexper- imentallystudiedyet.TheonlyavailableexperimentalresultswerepublishedbySingletonandHalsey ©A.Patrykiejew,S.Sokołowski,2014 43601-1 A.Patrykiejew,S.Sokołowski nearly60yearsago[49]anddemonstratedthatmixedliquid-likefilmsexhibitcompletemixing,indepen- dentlyofthecomposition.Theonlyinformationaboutthestructureandpropertiesofsolid-likemixed Ar-Krfilmsongraphitehavebeenobtainedusingcomputersimulationmethods[50,51].Inparticular,it hasbeendemonstratedthatalreadyatsubmonolayercoveragesandatsufficientlylowtemperatures,the additionofargonintokryptonfilmtriggerstheC-ICtransition,andleadstotheformationoffiniteclus- tersinwhichsmallcommensuratedomainsconsistingofpurekryptonareseparatedbyheavywallsbuilt ofargonandkrypton[51].TheC-ICtransitioninone-componentfiniteclustersofLennard-Jonesatoms adsorbedongraphitehasalsobeenstudiedbyHoulriketal.[37].TheyhaveusedMonteCarlomethodto studytheeffectsofthecorrugationpotentialonthestructureoffinitesystemsandhaveshownthatfora givenamplitudeofthecorrugationpotentialthepresenceoffreesurfacesenhancedthestabilityoftheC structure. Inthispaperwestudythemechanism oftheC-ICtransitionofsubmonolayer mixed filmsmadeof argonandkryptonusingMonteCarlosimulationmethodsandappropriatelymodifiedone-dimensional (1D) Frenkel-Kontorova (FK) model [52, 53]. The original FK model handles an infinite chain of atoms interactingviaharmonicpotentialandsubjectedtoaperiodic(sinusoidal)externalfieldatzerotemper- ature.TheFKmodelcanbeusedtodescribethebasisfeaturesofC-ICtransitions[17].Atthispoint,we should mention thatthe FKmodel hasbeenextended totwo-dimensional systems [54, 55] tomixtures [56],systemswithdisorder[57,58]andhasalsobeenusedtostudyfinitechains[59–63]. Inoneofourrecentworks[52],westudiedtheimpuritydrivencommensurate-incommensuratetran- sitionsinone-dimensionalfinitesystemsusingtheFKmodel.Weassumedthatthepurechainiscommen- surateanddiscussedtheeffectsofimpuritieslocatedeitheronlyatoneendorattwoendsofthechain. It is shown that in both situations the C-IC transition occurs when the amplitude of the external field experiencedbytheimpurityatomsfallsintotheregionbetweenthelowerandupperthresholdvalues. Theselimiting values of theamplitude depend uponthe parameterscharacterizingthe interactionbe- tweentheatomsinthemainchain,theamplitudeofexternalfieldactingonthemainchainatomsand theinteractionbetweenthemainchainandtheimpurities.Thenumberofsolitons(domainwalls)inthe ICstructurewerefoundtodependuponthechainlength,andtheparametersdescribingtheinteractions inthesystem.OurfindingshavebeenconfirmedbyMonteCarlosimulationintwo-dimensionalsystems. Here, we also considered finite chains of Kr atoms with impurities (Ar atoms) located atthe chain endsaswellasinthechaininterior. Thepaperisorganizedasfollows.Inthenextsectionwepresentedthetwo-dimensionalmodeland describedtheMonteCarlomethodused.TheresultsofMonteCarlosimulationarepresentedinsection3. Then, insection 4 we presented the modified 1D FK model and demonstrated that ityields the results qualitativelyverysimilartothoseemergingfromthetwo-dimensionalMonteCarlosimulation. 2. The two-dimensional model and simulation method We have considered strictly two-dimensional mixed submonolayer films consisting of Ar and Kr atoms.Theinteractionbetweentheatomsismodelledviathetruncated(12,6)Lennard-Jonespotential 12 6 4ε σ /r σ /r , r r , uij(r) ij ij − ij É max (2.1) =( 0, h¡ ¢ ¡ ¢ i r rmax, > wherer isthedistancebetweenapairofatoms,thesubscriptsi and j markthespeciesArandKr,and thepotentialiscutatthedistancermax 3.0σij.Thepotentialparametersforapairofunlikeatomsare = givenbytheLorentz-Berthelotrelations: 1 σArKr = 2(σAr+σKr) and εArKr= pεArεKr. (2.2) Theexternalfieldduetothegraphitesubstrateisassumedtobegivenby[33,51] v(x,y) Vb,i cos(q1r) cos(q2r) cos[(q1 q2)r] , (2.3) =− + + − © ª 43601-2 Domain-wallformationinAr-Krsubmonolayerfilmsongraphite Table1.Lennard-JonesparametersforArandKrusedinthiswork i,j σi,j,Å εi,j,K Ar,Ar 3.4 120.0 Kr,Kr 3.6 171.0 Ar,Kr 3.5 142.83 whereq1andq2arethereciprocallatticevectorsofthegraphitebasalplaneandtheamplitudesVb,i (i = Ar,Kr)determinethepotentialbarriersbetweenadjacentminima.WeassumedherethatVb∗,Ar=8.4K andVb∗,Kr=14.4K[51]. Thevaluesofparametersenteringthepotential(2.1)aregivenintable1.Thegraphitelatticeconstant a 2.46Åisassumedtobeaunitoflengthandε istakenasaunitofenergy. gr Ar = The Monte Carlo simulation is carried out in the canonical ensemble [64] for systems of different (submonolayer)density,ofdifferenttotalnumberofatoms, N,andofdifferentmolefractionofargon, x ,between0and0.55.WeusedastandardMetropolissamplingandtwotypesofmovesaretakeninto Ar account,i.e.,thetranslationofarandomlychosenatombyarandomlychosenvectorwithinthecircleof radiusd andtheidentityexchange.Themagnitudeofd isupdatedevery1000MonteCarlosteps max max (each Monte Carlo step consists of N attempts to move a single atom as well as N attempts to change the identity of randomly chosen atoms) in order to keep the acceptance rate equal to about 0.3. The equilibrationandproductionrunsconsistof5 106 and5 107 MonteCarlosteps,respectively.Standard · · periodicboundaryconditionswereusedtherein. 3. The results of Monte Carlo study of Ar-Kr submonolayer films Sinceourstudywasperformedforfiniteclustersatlowtemperaturesitseemsreasonabletobegin withtheestimationoftheeffectsoffreeboundariesonthegroundstatepropertiesoffinitecommensu- ratepatchesmadeofpurekrypton. WeperformedMonteCarlocalculationsatlowtemperaturesusingdifferentstartingconfigurations. Atfirst,therectangular simulationcellsofthesizeLx Ly L Lp3/2withL 30, 60and120,were × = × = used.ThenumberofKratoms inaperfectandfully covered (p3 p3)R30◦ commensuratestructure wasequaltoNKr L2/3.Then,wemodifiedthesimulationcellby×makingLx larger(Lx L ∆),while keepingthenumb=erofKratomsunchanged,i.e.,equaltoN L2/3.Inthisway,twofr=eeb+oundaries Kr = runningalongthey-axiswerecreated.Ofcourse,thedistancebetweenthesetwoboundaries,givenby∆, shouldbelargerthanthecutoffdistanceoftheLennard-Jonespotential[equation(2.1)].Thesecondseries of calculations were donefor finite hexagonal clusters of the commensurate phase builtof a different number of kryptonatoms. Thesimulation was carried out atreduced temperatures between 0.01 and 0.1 and the energies obtained were extrapolated to T 0. In this way, we estimated the ground state = energies. In the case of a fully occupied rectangular box, we did not observe any finite size effects, and the groundstateenergy(perparticle)ofthecommensuratephasewasestimatedtobeequaltoe0,c 4.617 =− (seefigure1).Similarcalculationscarriedoutforthesystemswithfreeboundariesalongthe y-axisled to the ground state energy equal to e0,f 4.562 (see figure 1) for three different systems of the size =− (60 15p3)(N 600),(60 45p3)(N 1800)and(90 30p3)(N 1200). Kr Kr Kr × = × = × = Inthegroundstate,theexcessinterfacialenergyperunitlengthduetothepresenceoffreebound- ariesofthetotallengthL canbecalculatedasfollows: int Eint (E0,f E0,c)/Lint, (3.1) = − whereE0,f NKre0,fandE0,c NKre0,carethetotalenergiesofthesystemswithandwithoutfreebound- = = aries.Forthethreesystemsmentionedabove,weobtainedthesamevalueoftheexcessinterfacialenergy atT 0equaltoabout0.635.Thisindicatesthatfinitesizeeffectsarenegligiblysmallatverylowtem- = peratures. 43601-3 A.Patrykiejew,S.Sokołowski Τ 0.00 0.02 0.04 0.06 0.08 0.10 -4.4 e = -4.562 T)] o,f e( N), [Kr-4.5 e(hex,o -4.6 e = -4.618±0.001 o,c 0.00 0.02 0.04 0.06 0.08 0.10 1/N1/2 Figure1.Thetemperaturechangesoftheenergy(perparticle)offullycovered(opencircles)andwith freeboundaries (opendiamonds) commensuratephaseobtained using arectangularslabscontaining 600,1200and1800atomsandofdifferenttotallengthofthefreeinterface,Lint=Lyp3,withLy=30,60 and90.ThefilledcirclesarethegroundstateenergiesoffinitehexagonalpatchesplottedagainstN−1/2. On the other hand, the ground state energies of finite hexagonal clusters are bound to show large finitesizeeffects.Insuchclusters,thefractionofatomsatthepatchboundariesisproportionalto1/pN. InordertoextrapolatetheresultstothelimitN ,wemadetheplot(giveninfigure1)oftheground Kr →∞ stateenergies(peratom)fortheclustersofdifferentsizee (N )against1/pN andextrapolatedthe hex,0 Kr resultsto1/pN 0.Inthislimit,theboundaryeffectsshouldvanishandtheenergyforN should Kr = →∞ beequaltoe0,c.Theobtainedvalueisequalto 4.169andagreesverywellwiththevalueofe0,cobtained − forthefullycoveredsurface.Theabovepresentedresultsdemonstratethatoursimulationmethodgives reliableresults. Itwasdemonstratedinourearlierwork[51],thatfinitesubmonolayerclustersmadeofthemixtureof ArandKratomsexhibitdomain-wallstructures,inwhichthecommensuratedomainsmadeofkrypton areseparatedbythedomainwallsofvaryingcomposition.Here,weconsideredsuchsystemsinamore systematic way, aiming at the determination of conditions necessary for the domain-wall structure to appear. TheMonteCarlosimulationswerecarriedoutforaseriesofhexagonal clustersofadifferenttotal numberofatomsequalto343,601,931and1333andofalsodifferentargonmolefractionbetween0.08 and0.55.Thecalculationswereperformedoveraratherwiderangeofreducedtemperaturesbetween 0.01and0.45.Figure2showsthetemperaturechangesofthepotentialenergy(perparticle)inthesystems ofdifferentclustersizeandofdifferentcomposition.Onlyinthecaseofx 0.1andtheclusterswith Ar = N up to 931, the energy smoothly changes with temperature, and the inspection of snapshots showed that all kryptonatoms form a commensurate patch while all argon atoms are located along the patch boundary[seefigure3(a)]. Inordertodistinguishthecommensurateandincommensurateatoms,weusedthefollowingorder parameter[31,37,51] φ(r) cos(q1r) cos(q2r) cos[(q1 q2)r], (3.2) = + + − andassumedthattheatomiscommensurate(incommensurate)whenφ 0(φ 0). > É In all other cases, the domain-wall structures were observed at sufficiently low temperatures. Fig- ure3(b)showstheexampleofasnapshotrecordedatT 0.02andx 0.1forthelargerpatchmade Ar = = of1333atoms.ItcanbeclearlyseenthatunlikeinthesmallerclusterofN 931atoms,theclusterwith = N 1333exhibitsthedomain-wallstructure,andthewallsaremadenearlyentirelyofkryptonatoms. = Theargoncontributestothewallsonlyintheareasclosetothepatchboundaries.Thepatchboundaries 43601-4 Domain-wallformationinAr-Krsubmonolayerfilmsongraphite T T 0,0 0,1 0,2 0,3 0,4 0,0 0,1 0,2 0,3 0,4 -3,6 -3,4 N=343 -3,8 N=601 -3,6 N=931 N=1333 -3,8 -4,0 > > e e < -4,0< -4,2 x = 0.2 xAr = 0.1 Ar -4,2 -4,4 -4,4 -3,2 -3,0 -3,4 -3,6 e> -3,5e> < < -3,8 -4,0 x = 0.3 x = 0.4 Ar Ar -4,0 -4,2 0,0 0,1 0,2 0,3 0,4 0,0 0,1 0,2 0,3 0,4 T T Figure2.(Coloronline)Thetemperaturechangesofthepotentialenergy(perparticle)obtainedforfinite hexagonalpatchesofdifferentsizeandofdifferentargonmolefraction.Theverticalbluelinesmarkthe locationsoftransitionbetweenthemixedcommensuratestructure(stableathightemperatures)andthe domain-wallstructure(stableatlowtemperatures)inthelargestclustersofN 1333atoms,obtained = fromannealingandfreezingruns. arecoveredbyasinglelayerofargonatoms.ThestructureoftheICphaseisqualitativelysimilartothe ICphaseinpurekryptonmonolayersofthedensityexceedingthatofafullyfilledCstructure.However, themechanismleadingtothecommensurate-incommensuratetransitionisquitedifferentinbothsitua- tions.Inthecaseofafullyfilledmonolayer,thedensityexcessoverthevaluecorrespondingtoaperfect Cphasetriggerstheformationofdomainwalls.Upontheincreaseofdensity,thesizeofcommensurate domains gradually decreases [30, 65]. Ultimately, a dense IC phase without commensurate domains is formedclosetothemonolayercompletion. Infinitemixedclusterstheatomsclosetotheboundariesarestrainedandthemagnitudeofthisstrain dependsonthenumber ofargonatomsinthesystem. Itisshown thattheexcessinterfacialenergyin purekryptonclustersispositiveanditinducesaforcenormaltotheinterfaceandpointingtowardsthe patchinterior.Evidently,thisforceistooweaktoleadtotheC-ICtransitioninthepurekryptonclusters. Thepresenceofalayerofargonatomsattheboundariescausestheinterfacialenergytobeconsiderably higherandhenceinducesaconsiderablylargerforce,whichissufficienttocompressthepatchandto triggertheC-ICtransition.Thenumberofargonatomsnecessarytocovertheentirepatchboundaryis proportionaltothetotallengthoftheinterface,whichinturnisproportionalto N1/2.Forasmalland Kr fixedargonmolefractioninthesystem,theconcentrationofargonalongtheinterfaceincreaseswiththe clustersize.Figure3(a)showsthatinthecaseofN 931and x 0.1,argonatomshardlycoverthe Ar = = entireinterface.Ontheotherhand,whenN 1333[figure3(b)],theentireinterfaceisdenselycovered = byargonandthereissomeexcessofargonatomswhichtrytopenetratethepatchandtrigger thede- velopmentofdomainwalls.Whentheargonconcentrationincreases, thewallsbecomemixedandfor alargeargonmolefractionarepredominantlybuiltofargon (seefigure4).Westudiedthechanges of thewallcompositionresultingfromthechangesofargonconcentrationinthesystemfortheclustersof differentsizewithN 601,931and1333.Wedidnottakeintoaccounttheincommensurateatomswith = lessthan5nearestneighbors.Suchatomsarelocatedalongthepatchboundaries,andtheinspectionof snapshots(seefigure3and4)allowedustoassumethatthereisasingleatomiclayerofincommensurate atomsatthepatchboundaries.Usingsuchaprocedurewewereabletomonitorthechangesofthenum- berofargonandkryptonatomsincorporatedwithinthewallsandtoestimatetheargonconcentration withinthewallandalongthepatchboundaries.WeperformedaseriesofMonteCarlosimulationsovera 43601-5 A.Patrykiejew,S.Sokołowski Figure3.(Coloronline)TheexamplesofconfigurationsforthefinitepatcheswithxAr 0.1atT 0.02 = = consistingof931(parta)and1333(partb)atoms.Blueandredcirclesmarkkryptonandargon,respec- tively.Theopencirclescorrespondtothecommensurateatomswiththeorderparameterφ(ri) 0,while Ê thefilledcirclesmarktheincommensurateatomswithφ(ri) 0. < certaintemperaturerangebetween0.01and0.3fordifferentclustersizesanddifferentargonmolefrac- tioninthefilm.Theexamplesoftemperaturechangesofthewallcompositionaregiveninfigure5(a), whichshowsthenumbersofargonandkryptonatomsformingthewallsintheclustersof1333atoms and different argon mole fraction.The results demonstrate that for x 0.1, the patch is commensu- Ar = rate at sufficiently high temperatures, and undergoes a transition to the incommensurate structure at T 0.14 0.02.Thetransitionseemstobeofthefirstorderassuggestedbyratherlargehysteresisalong = ± the freezing and annealing runs. In the case of x 0.2, the transition occurs at higher temperature Ar = ofaboutT 0.20 0.02 andisrounded.Forthepatcheswithahigherargonconcentration,theincom- = ± mensuratestructure occursover the entirerangeof temperaturesstudied.Itisalso evidentthat upon theincreaseofargonconcentrationinthefilm,thecompositionofwallschanges fromkryptonrichto argonrich.Havingthetemperaturechangesofthewallcompositionwecouldestimatethegroundstate behaviorofthesystemsstudied. The results are given in figure 5 (b), which shows the changes of the argon mole fraction within thewallsandatthepatchboundariesversusthetotalargonmolefractioninthesystem.Theseresults demonstratethatwhentheclustersizeincreases,theonsetoftheC-ICtransitionoccursatalowerargon concentration in the film. In the cluster with N 601, it occurs when x 0.125, in a larger cluster Ar = ≈ consistingof931atomsitstartswhen x isslightlyhigherthan0.1,whileinthelargestclusterofN Ar = Figure4.(Coloronline)TheexampleofconfigurationfortheclusterwithN 1333atomsandxAr 0.3 = = atT 0.06.Thelabelingofatomsisthesameasinfigure3. = 43601-6 Domain-wallformationinAr-Krsubmonolayerfilmsongraphite 500 1,0 a ArKr - x =0.1 - xAr =0.2 Ar 400 -- xxAr ==00..34 0,8 N=601 Ar N=931 N=1333 N(Ar), N(Kr)ICIC230000 x [x]Ar,wAr.p00,,46 100 0,2 0 0,0 0 0.05 0.1 0.15 0.2 0.25 0 0,1 0,2 0,3 0,4 0,5 T xAr Figure5.(Coloronline)(parta)Thetemperaturechangesofthenumbersofargonandkryptonatoms withinthewallsinfiniteclustersofN 1333forthesystemswithdifferentargonmolefractionequal = 0.1(circles),0.2(squares)0.3(diamonds)and0.4(triangles).Blackandredsymbolsmarkthenumbers of argon and krypton atoms,respectively. (Part b). Themole fractionof argon within thewall (black symbols)andthemolefractionofargonatomsatthepatchboundaries(redsymbols)againstthetotal molefractionofargonatomsobtainedforclustersofdifferenttotalnumberofatoms(giveninthefigure). Thelinesserveonlyasaguidance. 1333thetransitionisfoundalreadywhenx 0.09.Ofcourse,whentheargonmolefractionislower Ar ≈ thanthelimitingvaluenecessarytotriggertheC-ICtransition,allargonatomsarelocatedalongthepatch boundaries.Theargonmolefractionwithinthewallsincreaseslinearlywiththeargonmolefractionin thefilmandforsufficientlylargeargonconcentrationthewallsareentirelyformedbyargon.Afurther increase of the argon mole fraction leads to a gradual decrease of the size of krypton commensurate domains[51]. The inspection of configurations showed that when the walls are predominantly made of krypton atoms(atlowargonconcentrations)theyarewiderthanthewallsformedinthefilmswithhighargon concentration.Thedomainwallsmadeofargonatomsonlywereobservedtousuallyconsistofjusttwo rowsofargonatomsandcanbeclassified asheavywalls.Theneighboringdomainsareshiftedbythe displacementvectorofthelengthequaltoa [21].Theresultsobtainedfordifferentclustersizesshowed gr thatthedomainsarerathersmall,andusuallycontainupto10commensuratekryptonatomsalongthe linejoiningoppositewalls.ThisisquitedifferentfromthealreadymentionedpurekryptonICphase,in whichthedomainsareverylargeclosetotheC-ICtransitionandtheirsizegraduallydecreaseswhenthe densitybecomeshigher. Wealsostudiedsemi-infinitesystemsconsistingoftherectangularcommensuratedomainofthesize Lx Ly withperiodicboundaryconditionsappliedalongthex and y-axes,butwithfreeinterfacesrun- × ningalongthey-axis.ThesimulationcellwasofthesizeLx 60 ∆,with∆ 30,andLy 30p3/2. = + = = StartingfromaperfectCstructure,weestimatedthegroundstateenergiesforthreedifferentconfig- urationswiththefixednumbersofArandKratomsequaltoN 60and N 540.Thesimulations Ar Kr = = were carried out at low temperatures, between 0.01 and 0.1, using two types of algorithms, with and withouttheidentityexchangeattempts. Threedifferentsystemswereconsidered.Inthefirst(i),tworowsofkryptonatomsadjacenttoeach boundaryrunningalongthey-axiswerereplacedbyargonatoms.Thesecondsystem(ii)containedonly singlerowsofargonatomsatbothfreeboundariesandtherestofAratomsformedasinglewallconsist- ingoftworowsofatomsinsidetheCpatchatacertaindistancerw fromoneoftheinterfaces.Thethird system(iii)alsoconsistedofsinglerowsofargonatomsalongthetwofreeboundaries,buttherestofAr atomsformedtwowalls,eachmadeofonerowofatomslocatedatthedistancerw fromthecloserfree interface.Thecalculationsshowedthatinthesystems(ii)and(iii)energybecomesindependentofrw as soonasrw islargerthantheassumed cutoff distancermax.Theresultsgivenbelow wereobtained for 43601-7 A.Patrykiejew,S.Sokołowski rw 10. = The simulation without identity exchange attempts demonstrated that the system (ii) has a lower groundstateenergy(perparticle)(e 4.450)thanthesystems(i)(e 4.425)and(iii)(e 4.398). i ii iii ≈− ≈− ≈− Moreover,theentirekryptonpatchinthesystem (i)wasobserved toformtheC structureatlowtem- peratures,whilethevastmajorityofargonatomsweredisplacedtoincommensuratepositionsduetoa certaincontractionresultingfromthepresenceoffreeinterfaces.Inthecaseofsystem(ii),argonatoms wereobservedtoformanincommensuratewallwiththedisplacementvectorbetweenthecommensu- ratedomainsatbothsidesofthewallcorrespondingtoaheavywall.Thestructureofsystem(iii)was foundtobecommensurate,justthesameasinthecaseofsystem(i)AnotherMonteCarlorun,inwhich theidentityexchangeattemptedwastakenintoaccount,performedforthesystem(i)demonstratedthat alreadyatverylowtemperaturesthesystemstructurespontaneouslychangestothatcorrespondingto thesystem (ii).Moreover, the twosingle rows ofargonatoms insidethe patchof thesystem (iii)were observedtomergeintoasingleheavywall consistingoftworowsofargonatoms. Wealsoperformed arunusing thealgorithminvolving theidentityexchange attempts andthestaringconfigurationwith threerowsofargonatomsateachfreeboundary.Inthiscase,weobservedtheformationoftwoheavy wallsrunningalongthey-axisandmadeoftworowsofargonatomsagain. Wecalculatedthegroundstateenergiesofperfectlycommensurate(Ec,k)andrelaxed(Erel,k)struc- turesforthesystemsk (i),(ii)and(iii)aswellastheenergygain(pertheunitlengthoffreeinterfaces) = duetotherelaxation eint,k=(Erel,k−EC,k)/Lint. (3.3) Theenergies(peratom)ofperfectlyorderedcommensuratestructuresareverysimilarandequalto eC,1 4.3748,eC,2 4.3699andeC,3 4.36707.Ontheotherhand,theenergygainsduetorelaxation =− =− =− areconsiderablydifferentandequaltoeint,1 1.16,eint,2 1.85andeint,3 0.76. ≈− ≈− ≈− Figure6.(Coloronline)Theexampleofconfigurationfortherectangularslabofthesize60 30p3with × xAr 0.2atT 0.03.Blueandredcirclesmarkkryptonandargonrespectively.Circleswiththinlines = = correspondtocommensurateatomswiththeorderparameterφ(ri) 0,whilecircleswiththicklinesare Ê forincommensurateatomswithφ(ri) 0. < Theabovepresentedresultsdemonstratethattheformationofheavywallsconsistingoftworowsof argonatomsinsidethekryptonpatchleadstothemoststablestructure.Itshouldbealsonotedthatsim- ulationforthesystemswithlargerargonconcentrationsleadtothedevelopmentofnetworksofheavy wallslikethatgiveninfigure 6,ratherthantoaseriesofparallelwalls runningalong the y-axis.This demonstrates that wall crossings contribute to the system stability and suggests that the wall crossing energyisnegative.Ofcourse,atfinitetemperaturesthenetworkofhexagonalwallsisalsostabilizedby entropiceffects,inparticular,bytheso-calledbreathingentropy[22,23,66]. Concludingthissection,weshouldmentionthatfromourearlierstudy[51]itfollowsthatuponthe increaseoftemperaturethedomainwallsstructuredisappearsandthemixedcommensurate phaseis formedatthetemperaturesbelowthemeltingpoint. 43601-8 Domain-wallformationinAr-Krsubmonolayerfilmsongraphite 4. A simple one-dimensional Frenkel-Kontorovamodel Inthissection,weproposeasimplemodifiedone-dimensionalFrenkel-Kontorovamodel[52,53]that leadstoqualitativelysimilarresultsasthosediscussedintheprevioussection. Taking into account that the domain-wall structures that appear for submonolayer coverages are stableonlyatverylowtemperatures,weconsiderfinitechainsatzerotemperature. In general, the energy of the chain consisting of N atoms of two different species A and B can be writtenas 1 N 1 N E=2(i−1Ki,i+1 xi+1−xi−bi,i+1 2+i 1vi[1−cos(2πxi/a)]). (4.1) X= £ ¤ X= In the above, we assumed that the interaction between a pair of nearest neighbors is harmonic and characterized by the force constant Ki,i 1 K(A,A), K(A,B) or K(B,B) and the equilibrium distance bi,i 1 b(A,A), b(A,B) or b(B,B), depend+ing=on the composition of the pair. The second sum in equa- tion+(4=.1)representsthecontributiontothepotentialenergyduetotheperiodicsubstratefield,withthe distancebetweenadjacentminimaequaltoa,andtheamplitudeoftheexternalfieldvi isequaltovAor v . B Inordertofind theequilibriumconfigurationofsuch amixed chainofthelength N,oneneedsto specify the numbers of atoms A and B and their positions along the chain, and then to minimize the energywithrespecttothesetof{xi}. Here,weconsideredafewsituationsthataresupposedtomimicthebehaviorofthepreviouslydis- cussedtwo-dimensionalsystems.WeassumedthatthecomponentAisKr-like,whilethecomponentBis Ar-like.TheequilibriumconfigurationofthechainmadeofatomsAonlyshouldcorrespondtothecom- mensuratestructure.Ontheotherhand,thechainmadeofonlyBatomsshouldleadtoanincommensu- ratefloatingstructure.AssumingthattheelasticconstantK(A,A)istheunitofenergyandthesubstrate latticeconstantaistheunitoflength,wefoundoutthatthemodelwiththeparametersv 0.0055and A = b(A,A) 1.9describesreasonablywellthecommensuratestructure.Notethatbytakingb(A,A) 1.9we = = assumedthatinthecommensuratestructureeverysecondpotentialwellisoccupiedbyanatomandthat themisfitm b(A,A) 2 0.1isnegative.ThisvalueofmisfitislargerthaninthecaseofKr/Graphite AA = − =− system, for which it is equal to about 0.06 [51], but assuming the value of the misfit equal to 0.06, − − theFKmodelleadstothestablecommensuratephaseevenforunreasonablylowamplitudesofthesur- facefield.TomodelAratomsweassumedthatv 0.003,b(B,B) 1.8andK(B,B) 0.9Theparameters B = = = K(A,B)andb(A,B)areassumedtobegivenbytheLorentz-Berthelotmixingrules,sothat K(A,B) K(A,A)K(B,B) and b(A,B) 0.5[b(A,A) b(B,B)]. (4.2) = = + p We considered the chains that begin and terminate with atoms B. This assumption arises directly fromtheobservationthatintwo-dimensionalsystems,thedomain-wallstructuresappearingonlyafter theperipheriesofthecluster madeofKratomsarecovered withafilled singlelayerofAratoms. The situationinwhichargonatomsarelocatedonlyattheendsofthechaincorrespondstothemodellabeled asModelI.Weconsideredtwoversionsofthismodelassumingthatthereisonlyoneargonatomateach chainend(ModelI1)andthateachendisdecoratedbytwoargonatoms(ModelI2).Then,intheModel II,weassumed thatatacertainpositionafter theatom N1,i.e., attheposition N1 1, amixedwall of + thethicknessequaltokatomsappears.TheversionofthemodelwithagivenvalueofkisnamedModel II(k,N1). Inordertoconsidermixedwalls,weassumedthatwithinthewall,theparametersrepresentingthe elasticconstant,Kw,theequilibriumdistance,bw,andtheamplitudeofthesubstratepotential,vw,are alldependentonthewallcompositionmeasuredbythemolefractionofcomponentB,x ,andaregiven B by: Kw xBK(B,B) (1 xB)K(A,A), (4.3) = + − bw xBb(B,B) (1 xB)b(A,A) (4.4) = = − and vw xBvB (1 xB)vA. (4.5) = + − 43601-9 A.Patrykiejew,S.Sokołowski Thepairs(N1,N1 1)and(N1 k,N1 k 1),i.e.,atbothsidesofthewall,consistofoneAatomand + + + + onewallatom,sothattherespectiveelasticconstantsandequilibriumdistancesaregivenby K(A,w) x K(A,B) (1 x )K(A,A) and b(A,w) x b(A,B) (1 x )b(A,A). (4.6) B B B B = + − = = − TheaboveassumptiontakesintoaccountthefactthatwhenthewallismadeonlyofatomsB,theinter- actionbetweenthepairs(N1,N1 1)and(N1 k,N1 k 1)isthesameasbetweenthepairs(A,B)and + + + + (N 1,N),whichareequaltoK(A,B)andb(A,B),respectively. − Ofcourse, theaboveassumed additivityrelationsaretoosimpletoproperlydescribetheeffectsof wall composition on the behavior of adsorbed films, but we will demonstrate that even such asimple modelleadstotheresultsqualitativelyverysimilartothoseobtainedfortwo-dimensionalfilms. Then,weintroducedthedisplacementsui xi/b 2i,withbbeingtheequilibriumdistanceforthe = − pairi andi 1,andrewroteequation(4.1)intheform + 1 N 1 N E=2(i−1Kˆi,i+1 ui+1−ui−mi,i+1 2+i 1vˆi[1−cos(2πui)]), (4.7) X= £ ¤ X= wheretheenergyisexpressedinunitsofK(A,A)a2,andKˆi,i 1 Ki,i 1/[K(A,A)a2]andvˆi vi/[K(A,A)a2] andthemisfitsmi,i 1 aredefinedasmi,i 1 b(α,β)/a 2+wi=th(α,+β) (A,A),(A,B),(A,w=)or(w,w). + + = − = InthecaseofModelI,theenergygivenbyequation(4.7)dependsonthenumberofargonatomsat theendsofthechain,whileintheModelsII(k,N1)itdependsonboththethicknessofthewall(k)andits position(N1). OneshouldnotethatthesystemsconsideredintheprevioussectionwerestudiedusingMonteCarlo methodallowingfortheexchangeofatomsidentity,sothatthedistancesbetweenthepatchboundary andthewallsandthewall-wall distancesinsidethepatchcorrespondtotheequilibriumstates. Inthe presentone-dimensionalmodel,thelocationofthewallispredeterminedbyN1,andoneneedstomin- imizetheenergywithrespecttoalldisplacements{ui}aswellaswithrespecttoN1 inordertofindthe equilibriumstateofthechain.Ontheotherhand,thewallthicknessandcompositionarecontrolledby theargonmolefractioninthechain.Thefirstseriesofcalculationsisaimedatdeterminingwhetherthe assumed model predictsthe formation of solitons when B atoms are located only at the chain bound- aries,i.e., when x 0.0withinthewall. Thecalculations carriedoutfor differentchainlengths, with B = N rangingfrom11to101,andusingModelI1alwaysledtoasinglecommensuratedomain,i.e.,tothe same qualitative results as for pure A chain. The only effect of the B atoms located at the chain ends wasanincreaseofatomicdisplacementsnearthechainends(seefigure7).Thisresultagreesverywell with two-dimensional Monte Carlo simulations carried out for low x , up to the value for which the Ar 2 Model I1 Model I2 Pure Kr 1 ui 0 -1 -2 50 100 x i Figure7.(Coloronline)Atomicdisplacementsversusatomicpositionsforfinitechainsconsistingof41 atoms.BlackcirclesandlinecorrespondtopurechainofBatoms,whileredandbluecirclesandlines correspondtothechainsdecoratedbyoneandtwoatomsAateachendofthechain,respectively. 43601-10