你觉得老罗的情怀最终会值多少钱 A.3000以上 B. C.400-80 D.一_百度知道
你觉得老罗的情怀最终会值多少钱 A.3000以上 B. C.400-80 D.一
C你觉得老罗的情怀最终会值多少钱A.400-80D.3000以上B
当然我觉得不是你们这种人我还不知道老罗卖手机，我也不知道情怀卖过多少钱，未来会卖多少，我只知道老罗是个人名，连老罗哪根葱都不知道
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出门在外也不愁91Directed Copolymer Assembly on Chemical Substrate Patterns
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91Directed Copolymer Assembly on Chemical Substrate Patterns
1284Langmuir4；DirectedCopolymerAssembl；oftheInfluenceofRoughnes；KostasCh.Daoulas*andMarc；Institutfu¨rTheoretische；MarkP.Stoykovich,?Huiman；DepartmentofChemicalandB；Madis
1284Langmuir4-1295DirectedCopolymerAssemblyonChemicalSubstratePatterns:APhenomenologicalandSingle-Chain-in-Mean-FieldSimulationsStudyoftheInfluenceofRoughnessintheSubstratePattern?KostasCh.Daoulas*andMarcusMu¨llerInstitutfu¨rTheoretischePhysik,Georg-AugustUniVersita¨t,37077Go¨ttingen,GermanyMarkP.Stoykovich,?HuimanKang,JuanJ.dePablo,andPaulF.NealeyDepartmentofChemicalandBiologicalEngineering,UniVersityofWisconsin-Madison,Madison,WisconsinReceiVedAugust10,2007.InFinalForm:October13,2007Thedirectedassemblyoflamella-formingcopolymersystemsonsubstrateschemicallypatternedwithroughstripeshasbeenstudiedusingaHelfrich-type,phenomenologicaltheoryandSingle-Chain-in-Mean-Field(SCMF)simulations.Thestripeperiodmatchesthatofthelamellarspacinginthebulk.Theeffectofthelineedgeroughness(LER)ofthesubstratepatternonthemicrophase-separatedmorphologywasinvestigatedconsideringtwogenerictypesofsubstrateLERwithasinglecharacteristicwavelengthimposedontheedgesofthestripes:undulationandperistalticLER.Inbothcases,thedomaininterfacesarepinnedtotheroughstripeboundaryatthesubstrateand,thus,aredeformed.Westudyhowthisdeformationdecaysasafunctionofthedistancefromthesubstrate.ThesimpletheoryandtheSCMFsimulationsdemonstratethatoneofthebasicfactorsdeterminingthedecayoftheroughnesstransferredintotheself-assembledmorphologyisthecharacteristicLERwavelengthoi.e.,thedistanceoverwhichtheroughnesspropagatesawayfromthesubstrateincreaseswithwavelength.However,bothapproachesrevealthat,foraquantitativeunderstandingoftheconsequencesofsubstrateLER,itisimportanttoconsidertheinterplayofthepatternwavelengthwiththeothercharacteristiclengthscalesofthesystem,suchasthefilmthicknessandthebulklamellarspacing.Forinstance,inthinfilms,theinduceddeformationofthelamellarinterfacedecaysslowerwithdistancefromthepatternedsurfacethaninthickerfilms.Itisshownthatthephenomeno-logicaltheorycancapturemanyofthesamequalitativeresultsastheSCMFsimulationsforcopolymerassemblyonsubstratepatternswithLER,but,atthesametime,islimitedbyanincompletedescriptionoftheconstraintsonthepolymerchainconformationsimposedbythesubstrate.I.IntroductionThenaturaltendencyofblockcopolymerstoself-organizeonthemolecularscaleintodense,regulararraysofstructuresmakesthemattractiveforvariousnanotechnology-relatedapplications.1However,thestructuresformedduringspontaneousself-assemblycommonlycontainmanydefectsthatlimittheirtechnologicalapplication.Thus,variousstrategiessuchaschemicallypatternedsurfaces,2-4a,4bgraphoepitaxy,5,6solventevaporation,7,8shear,9,10externalelectricalfields,11-13anddirectionalsolidification14,15ahavebeenproposedtocontrolorientationandlong-rangeorderingofthedomains.DirectedassemblyofsymmetriccopolymersinPartoftheMolecularandSurfaceForcesspecialissue.*Correspondingauthor.Electronicaddress:daoulas@theorie.physik.uni-goettingen.de.?Presentaddress:DepartmentofMaterialsScienceandEngineering,UniversityofIllinoisatUrbana-Champaign,Urbana,IL61801.(1)Xia,Y.;Rogers,J.A.;Paul,K.E.;Whitesides,G.M.Chem.ReV.3.(2)Kim,S.O.;Solak,H.H.;Stoykovich,M.P.;Ferrier,N.J.;dePablo,J.J.;Nealey,P.F.Nature.(3)Edwards,E.W.;Montague,M.F.;Solak,H.H.;Hawker,C.J.;Nealey,P.F.AdV.Mater.5.(4)(a)Stoykovich,M.P.;Mu¨ller,M.;Kim,S.O.;Solak,H.H.;Edwards,E.W.;dePablo,J.J.;Nealey,P.F.Science2.(b)Stoykovich,M.P.;Kang,H.;Daoulas,K.Ch.;Liu,G.;Liu,C.;dePablo,J.J.;Mu¨ller,M.;Nealey,P.F.ACSNano.(5)Segalman,R.A.;Yokoyama,H.;Kramer,E.J.AdV.Mater.2.(6)Sundrani,D.;Darling,S.B.;Sibener,S.J.NanoLett..(7)Kim,S.H.;Misner,M.J.;Russell,T.P.AdV.Mater.9.(8)Kim,S.H.;Misner,M.J.;Xu,T.;Kimura,M.;Russell,T.AdV.Mater..(9)Angelescu,D.E.;Waller,J.H.;Adamson,D.H.;Deshpande,P.;Chou,S.Y.;Register,R.A.;Chaikin,P.M.AdV.Mater.6.(10)Angelescu,D.E.;Waller,J.H.;Register,R.A.;Chaikin,P.M.AdV.Mater.8.?thinfilmsonchemicallystripedsubstrateswasrecentlydem-onstratedtoformdefect-freearraysofalignedlamellaeoverarbitrarilylargeareasandtopreciselypositionorregisterthecopolymerdomainswithrespecttothesubstratepattern.2-4aThisapproachalsocanfabricatedevice-orientedstructuresatthenanoscale,includingthoserequiredforapplicationssuchasmagneticstoragemedia,flashmemorydevices,capacitors,andcompleteintegratedcircuitlayouts.4bLocalpropertiesoftheself-assembledmorphology,however,canbeequallyimportantasthelong-rangeorderforthesuitabilityofcopolymernanostructuresinmanyapplications.Forinstance,thelineedgeroughness(LER)offeaturesproducedbyconventionallithographicprocessesutilizingchemicallyamplifiedphotoresistsisanimportantissue.Whenpatterningastripeservingasatransistorgate,16,17theLERontheleftandrightedgesleadstovariationsinthegatelength.Thevariationsonsmalllengthscalesgeneratedbyshort-wavelengthLERcanseriouslyimpairthefunctionalityofindividualtransistors,whilelargerlength-scalefluctuations,duetolongwavelengthLER,causevariationsintheperformanceofentiretransistorgroups(see,e.g.,Figure1ofref17).(11)Morkved,T.L.;Lu,M.;Urbas,A.M.;Ehrichs,E.E.;Jaeger,H.M.;Mansky,P.;Russell,T.P.Science.(12)Li,H.W.;Huck,W.T.S.NanoLett.3.(13)Thurn-Albrecht,T.;Schotter,J.;Ka¨stle,G.A.;Emley,N.;Shibauchi,T.;Krusin-Elbaum,L.;Guarini,K.;Black,C.T.;Tuominen,M.T.;Russell,T.P.Science6.(14)Bodycomb,J.;Funaki,Y.;Kimishima,K.;Hashimoto,T.Macromolecules5.(15)Rosa,C.D.;Park,C.;Thomas,E.L.;Lotz,B.Nature.(16)Yamaguchi,A.;Komuro,O.Jpn.J.Appl.Phys.3.(17)Yamaguchi,A.;Fukuda,H.;Arai,T.;Yamamoto,J.;Hirayama,T.;Shiono,D.;Hada,H.;Onodera,J.J.Vac.Sci.Technol.B1.10.1021/la702482zCCC:$40.75?2008AmericanChemicalSocietyPublishedonWeb12/08/2007DirectedCopolymerAssemblyonPatternedSubstratesInthecaseofdirectedcopolymerassemblyonchemicallypatternedsubstrates,theboundariesbetweenthechemicallydistinctregionsofthesubstratepatternalsohavesomeLER(denotedas“substrateLER”todistinguishitfromthedistortionoftheinternalinterfacesofthesoftcopolymermaterial).Thus,itisofinteresttounderstandtowhatextenttheroughnessofthesubstratepatternpropagatesintothecopolymermorphol-ogy.Sincetheequilibriumpropertiesoftheinterfacesinacopolymermorphologyaredictatedbythermodynamics,theinterfacialtensionandrigidityfavorssmoothinterfaces,andtheeffectsofsmallpatternimperfei.e.,themorphologyisexpectedtoexhibitadegreeofself-healing.Copolymersshouldsmooth-outthelocalLERofthesubstratepattern,andthesubstrateLERisexpectednottoaffectthecopolymermorphologyfartherawayfromthesubstrate.Thus,blockcopolymerlithography,inprinciple,couldofferbettermeansforcontrollingthedimensionsofthepatterneddevicefeaturesthanconventionalphotoresist-basedmethods.Ontheotherhand,thefree-energycostsofdeformingthemorphologyareonlyontheorderofkBTpermoleculeinthesesoftmattersystems.Therefore,thestronginteractionswiththesubstratepatternhaveapronouncedinfluenceonthereconstruc-tionofthemorphology.Infact,forthefilmthicknessesconsideredhereandpreviously,2-4a,18thesubstrateinteractionscaninfluencethemorphologyovertheentirefilmthickness.Thissoftnessofthemorphologiesenablesustofabricateirregular(e.g.,non-periodic)patternsinfilmsthathavenoanaloginthebulk.Inviewofthesubtlebalancebetweenthestrongforcesatthechemicallypatternedsubstrate,thesoftnessofthecopolymermorphology,andthespatialconfinement,aquantitativeapproachiswarranted.Thesmallfree-energycostsassociatedwiththedeformationofthesoft,self-assembledmorphologyinevitablybringsforththeroleofthermalfluctuations.EvenifassembledonsubstrateswithnosubstrateLER,theinstantaneousshapeoftheinterfacesexhibitsacertaindegreeofroughnessinducedbythermalfluctuations.Recentdevelopmentsinin-situscanningforcemicroscopy19-22allowformonitoringoftheshapechangesofthesoftdomainsintimeandexploringtowhatextenttheinterfacialroughnessgeneratedbythethermalfluctuationscaninfluencethemorphology.Inthepast,theinterfacebetweentwosimpleliquidspreferentiallywettingthetwodifferentregionsofapatternedsubstrateseparatedbyawavyboundaryhasbeenconsideredbytheory.23Atthesubstrate,theinterfaceispinnedtotheroughboundaryanddeforms.Thisdeformationincreasestheareaoftheinterfaceand,likecapillarywaves,itraisesthefreeenergybyanamountthatisproportionaltotheinterfacetensionandtheexcessarea.Thedeformationexponentiallydecaysfartherawayfromthesubstrate,andthecharacteristicdecaylengthperpen-diculartothesubstrateisproportionaltothecharacteristicwavelengthofthewavyboundaryalongthesubstrate.Thus,weexpectthataroughnesswithasmallwavelengthpropagateslessintotheliquidthanonewithalargewavelength.Although,thislimitingcaseofasimpleliquidoffersanimportantinsightintotheself-healingpropertiesofliquidinterfaces,inthecaseofstantopoulos,(18)Daoulas,K.Ch.;Mu¨ller,M.;Stoykovich,2006Y.J.;dePablo,J.J.;Nealey,P.F.;M.Solak,P.;Park,H.H.S.Phys.M.;RePapakon-(19),96V.Lett.(20)Knoll,,36104.A.;Magerle,R.;Krausch,G.Macromolecules9.Zvelindovsky,(21)Tsarkova,Knoll,A.;L.;Lyakhova,Knoll,A.;K.Magerle,S.;R.NanoLett.4.Zvelindovsky,(22)Horvat,A.V.;Magerle,R.Nat.Horvat,Mater.A.;2004Krausch,,3,886.G.;Sevink,G.J.A.;A.A.;V.;Knoll,Magerle,A.;Krausch,R.MacromoleculesG.;Lyakhova,2007K.,S.;Sevink,G.J.A.;Phenomena(23)deGennes,;Springer:P.G.;NewBrochard-Wyart,York,2004.F.;Quere,D.Capillarity40,6930.andWettingLangmuir,Vol.24,No.4,copolymermorphologies,additionalfactorssuchasthecouplingofneighboringinterfaces,interfacialbendingrigidity,andthegeometricalconstraintsimposedontheextendedmacromoleculesmustbeaccountedfortoobtainaquantitativedescription.VariousaspectsoftheinfluenceofsubstrateLERinthecopolymersystemscouldbeaddressedthroughawealthoftheoreticalmethods,includingLandau-Ginzburgtheory,24-26self-consistentfield(SCF)theory,27-31dynamicself-consistentfieldtheory,22,32-35andanalyticalapproaches.36-40ArathersimpleapproachconsistsinconsideringtheelasticpropertiesofthecopolymerinterfaceswithinHelfrich-typeHamiltonians.41Inthisframework,onedescribestheinterfaceasaninfinitelythinsheetthatischaracterizedbyafewmaterialparameters:tension,spontaneouscurvature,andbendingrigidities.Thesecoarse-grainedpara-meterscanbeestimatedfromstrongsegregationtheory(SST)42-44orSCFtheory45,46forelasticpropertiesofadi-blockcopolymermonolayer.Approachesofthiskindhavebeenusedtodescribeinstabilities(undulations)instrainedlamellarphases,wherethestrainwasgeneratedeitherbyuniaxialstress44orbytheassemblyoflamellaeonastripedsurfacewithamismatchbetweenstripeperiodicityandbulklamellarspacing.47Withinasimilarphenomenologicaldescription48andexperi-ments/Single-Chain-in-Mean-Field(SCMF)simulations,49theeffectofsubstratepatternscomprisingasymmetricstripewidthsonthecopolymermorphologyhasbeenconsidered.Morere-cently,aphenomenologicalapproachwasusedincombinationwithMCsimulationstostudythemorphologyofcopolymermeltsconfinedinnanopores.50Inalltheabovecases,theedgesofthesubstratepatternwereperfectlystraight,whichcor-respondstothelimitwherethewavelengthofthesubstrateLERdiverges.InthisworkwepresentacombinedphenomenologicalandsimulationstudyoftheeffectsofsubstrateLERontheaverageshapeofcopolymerinterfacesforthecaseoflamella-formingcopolymermeltsassembledonsubstrateschemicallypatternedwithstripes.ThesubstratepatternwithnoLERisasequenceofalternatingA-andB-attractingstripeswithperiodL.IfLmatcheswithinapproximately10%thelamellarspacingLointhebulk,thepolymercanregisterperfectlywiththepattern.2SubstrateLERconsistsofsinusoidalfluctuationsofthestripeLett(24)(25).2001Tsory,,53,Y.;Andelman,D.J.Chem.Phys.0;Europhys.(26)Kielhorn,722;L.;MacromoleculesMuthukumar,2001M.J.,34Chem.,2719;Phys.Interface1999,Sci..(27)Wu,(28)Petera,X.D.;F.;Dzenis,Muthukumar,Y.A.M.J.Chem.111,2259.J.Chem.Phys.707.(29)Matsen,(30)Chen,(31)Miao,H.M.B.;Y.;W.Yan,Fredrickson,J.Chem.Phys.Phys.1.D.;Wickham,G.R.H.8.A.;J.Chem.Shi,A.Phys.C.Polymer,,8.(32)Huang,Katz,A.K.;A.;Balazs,Fredrickson,A.Phys.G.ReH.VMacromolecules.Lett..2007Langmuir(33)Lyakhova,K.S.;Horvat,A.;Zvelindovsky,A.V.;Sevink,,40,4075.G.J.A.(34)(35)Morita,2006,H.;22,Kawakatsu,5848.T.;Doi,M.Macromolecules7.(36)Hasegawa,(37)Stepanow,S.;R.;Doi,M.Macromolecules6.(38)Fredrickson,Fedorenko,A.A.Europhys.Lett.Halperin,A.;G.Sommer,H.MacromoleculesJ.5.6210.(39)Angerman,H.J.;Johner,A.;U.;Semenov,Daoud,M.A.Europhys.N.MacromoleculesLett.,297.,39,(40)(41)Turner,(42)Helfrich,M.W.S.;Z.Joanny,Naturforsch.,J.F.MacromoleculesC2,25,6681.(43)Wang,(44)Wang,Z.Z.G.;G.;Safran,Safran,S.S.A.A.J.J.Phys.(Paris693.).(45)Wang,(46)Matsen,Z.M.G.W.J.Chem.Chem.Phys..J.Chem.Phys.8.(47)MuChem.(48)Perreira,¨ller,M.;Gompper,G.Phys.Phys.1999ReV.,E8.,66,041805.Wang,Q.;G.Nath,G.;Williams,S.K.;Graham,D.R.M.M.D.;Europhys.Nealey,Lett.P.F.;.J.J.;(49)Phys.Edwards,2000E.,112W.;,9996.dePablo,J.J.J.Mu¨ller,M.;Stoykovich,M.P.;Solak,H.(50)Nealey,Wang,P.Q.F.J.MacromoleculesChem.Phys.,40H.;dePablo,,,2007.90.1286Langmuir,Vol.24,No.4,2008Daoulasetal.Figure1.SketchofanABcopolymermorphologyassembledonpatternswithundulation(left)andperistalticroughness(right).Theleftandtherightinsetsshowthesubstratepatternforthecasesofundulationandperistalticroughness,respectively.TheAandtheBmonomer-attractingregionsareshowninyellowandblue,respectively.Thetwomainpanelspresentthedetailedmorphologiesassembledonpatternswithundulation(left)andperistalticroughness(right).Forclarity,onlytheA-richphaseisshowninyellow.Thewavylinesareexamplesofthepolymerchaintrajectories,whichareassumedtobeparalleltothelocalnormaloftheABinterface,whilethestraightlinesclarifythegeometricconstructionusedforthephenomenologicaldescriptionofchainstretching(seemaintext).Fortheperistalticroughness,thetwoshadedsurfacesmarktheverticalmidplanesofanA-richandaB-richslab.edgeswithaspecificwavelength,LLER.TwomodelcasesofsubstrateLERsperistalticandundulational(seeinsetofFigure1)sareconsidered,and,fortheseexamples,aphenomenologicaltheoryofthedeformationofthecopolymerinterfacesgeneratedbytheroughstripeboundaryisformulated.Phenomenologicalapproachesareattractivebecauseoftheirsimplicityandabilitytohighlightthebasic,qualitativebehaviorofcomplexinterfaceassembliesinpolymericsystems.Ontheotherhand,theyinvokesignificantsimplificationsandimportantfactorsdeterminingthebehaviorofthestudiedsystemsmightbeignored.Additionally,acoarse-grained,particle-basedmodelforstudyingtheeffectsofsubstrateLERisinvestigated,andahighlyaccurate,yetapproximatesolutionofthismodelisobtainedviaSCMFsimulations.51,52BasicaspectsoftheinfluenceofthesubstrateLERontheassembledpolymermorphologiesderivedbybothapproachesarepresented.ParticularemphasisisplacedupontheroleofsubstrateLERwavelengthanditsinterplaywithothercharacteristiclengthscalesofthesystem,includingthefilmthicknessandthedimensionsofthenaturalperiodicityofthebulkmorphology.Ourmanuscriptisorganizedasfollows:sectionIIpre-sentsabriefdescriptionofthecoarse-grainedpolymermodelandtheSCMFsimulationtechnique,andsectionIIIpresentsaphenomenologicalapproachtodescribetheeffectsoftheLERofthesubstratepattern.SectionIVdiscussestheresultsofthephenomenologicalapproachandSCMFsimu-lations.Finally,sectionVsummarizesthebasicconclusionsofourstudy.lamella-formingsystemsaredescribedthroughaminimalcoarse-grainedmodelaimedatreproducinguniversalfeaturesofpolymericbehavioronamesoscopicscale.ThechainconnectivityismodeledthroughadiscretizedEdwardsHamiltonian,Ho:Ho(m)[r(s)]kBTN(m)-1)∑s)13(N(m)-1)ri(s+1)-ri(s)]2(1)2Re(m)2II.Coarse-GrainedModelingofSubstrateLEREffectsonCopolymerAssemblyA.MolecularModelandLERRepresentation.TheeffectofsubstrateLERontheassembledpolymericstructuresisstudiedforthecaseoftwocopolymersystems.First,symmetricdiblockABcopolymermeltswillbeconsideredallowingadirectcom-parisonwithaphenomenologicaltheorypresentedinthefollow-ingsection.Second,inthespiritofrecentexperiments,2-4a,4b,18symmetricternaryABcopolymerblendswiththeirrespectiveAandBhomopolymerswillbebrieflyinvestigated.Theabove(51)Mu¨ller,M.;Smith,G.D.J.Polym.Sci.,PartB:Polym.Phys..(52)Daoulas,K.Ch.;Mu¨ller,M.J.Chem.Phys.904.whereri(s)isthecoordinateofsthbeadofpolymeri,andmstandsforthechemicalspeciesofthebeadandthetypeofthemolecule,respectively.Re(m)2denotesthemeansquaredend-to-enddistanceoftheunperturbedpolymercoil,whilekBandTaretheBoltzmannconstantandtemperature,respectively.Theend-to-enddistance,Re(m)istheonlyparameterdefiningthechainarchitecture,throughwhichthelengthscaleofthecoarse-grainedrepresentationandanexperimentalsystemcanberelated.Inthefollowing,alllengthsarereferredtoinunitsoftheend-to-enddistanceofthediblockcopolymer,Re.Althoughthenumbersofbeads,N(m),permoleculeexplicitlyappearineq1,theydonothaveaspei.e.,differentcontourdis-cretizationsdescribethesamephysicalrealization.Here,inbothpurediblockmeltsandternaryblends,thediblockchainsarediscretizedwithN)32beads.Inpurediblockmelts,themoleculesareperfectlysymmetric,andtheasymmetryparameterofthediblock,f,issettof)0.5.Inthecaseofternaryblends,weselectasystemthathaspreviouslybeenstudiedbyexperimentsandsimulation.4a,18Avalueoff)0.468fortheasymmetryparameterischosen,whilethenumberofbeadsintheAandBblocksissetto15and17,respectively.Theternaryblendscontain60%ABdiblockcopolymers,20%Aand20%Bhomopolymers,composedofNA)12andNB)13beads,respectively.ExcludedvolumeeffectsaredescribedviaHelfand’spoten-tial,53andtheincompatibilitybetweentheAandBspeciesofthediblockischaracterizedbyaFlory-Hugginsapproach.Thepolymer-substrateinteractionsaremodeledasexternalfields.TheHamiltonianofthenonbondedpolymer-polymerandpolymer-substrateinteractionsisgivenby(53)Helfand,E.;Tagami,Y.J.Chem.Phys.1971,56,3592.DirectedCopolymerAssemblyonPatternedSubstratesHnbκNk)Fod3r2φ?BTN∫[??NA+φ?B-1)2-4(φ?A-φ?B)2]+FoN∫d3rNUW(r)(φ?A-φ?B)(2)Intheaboveequation,Fodenotestheaveragemonomernumberdensity,while??istheFlory-Hugginsparameter.ThelocalAandBmonomervolumefractionsaredenotedasφ?A(r)andφ?B(r).LiketheReforthechainconnectivity,onlycoarse-grainedparametercombinationswillbeimportantfordefiningthenonbondedinteractions:the“invariants”oftherepresentation.Sincetheyhavebeendiscussedelsewhere,53-56itisonlymentionedthat??NsetstheincompatibilitybetweentheAandtheBblocks,whileκNcontrolstheamplitudeofthelocaldensityfluctuations,beingproportionaltotheinversecompressibility.Here,weset??N)37.6,whichistypicalforpolystyrene-block-poly(methylmethacrylate)(PS-b-PMMA)meltsthathavebeenusedintemplatedblockcopolymerassemblyexperiments.2-4aInadensepolymermelt,localdensityflucttherefore,κNshouldbechosenhighenoughtomimictheincompressiblebehavioronthemesoscopiclengthscale.52HerewechooseκN)50.Thisvalueissufficienttosuppresstotaldensityfluctuationsonthescaleofasmallfractionofthemolecularsizeandsimultaneouslyallowsforcomputationalefficiency.Theroleoffluctuationsindeterminingthephysicalpropertiesofapolymersystemiscontrolledbytheinvariantdegreeofpolymerization,Nh≡(F0Re3/N)2,characterizingthedegreeofinterdigitationofpolymerchains.Itconstitutesanadditionalinvariantofthecoarse-grainedmodeland,inthecurrentwork,issettoNh)14884,whichcorrespondstothetypicalmolecularweightinexperiments.2-4aForthesesystems,theequilibriumbulklamellarspacingisLo)1.72ReforthepurecopolymermeltandLo)2.27Refortheternaryblend.Thelasttermineq2accountsforthepolymer-substrateinteractions.Sincethecoarse-grainedmodeldoesnotincorporatelocalfluidstructureordetailsofchainarchitecture,itcannotaddresspropertiesonlengthscalessmallerthanasmallfractionofthemolecularsize.Thus,itisconsistenttoadoptthesamelevelofdescriptionwithrespecttothepatternedsubstrate,ignoringthefinedetailsoftheinteractionswiththepolymeronthelengthscaleofangstroms.WerathertrytoelucidatetheuniversaleffectsofLERofthesubstratepattern,consideringwavelengthsontheorderoforabovetheRescale.Specifically,thesubstrateisplacedatx)0,anditsinteractionswiththeAandBmonomersaredescribedthroughthepotentialUW(r):Ur))Λf(y,z)x2W(??/Rexp-e[2??2](3)Itcanbeseenthat,inthexdirectionperpendiculartothesubstrate,thedetailsofmonomer-substrateinteractionsareneglected,andtheyarerepresented,inacoarse-grainedway,throughagenericfunction.Thisfunctionisshort-ranged,anditsdecayonthesub-Rescaleisdefinedbyasinglecharacteristiclength,??.Here,thevalue??)0.15Reisused.TheparameterΛdescribesthestrengthofthepolymer-substrateinteractions,andthechemicalpatternisencodedthroughthefunctionf(y,z),takingvaluesofRor-??dependingonwhetherthelateralyandz(54)Weinheim,(55)Helfand,Mu¨ller,M.E.;InSapse,SoftMatterA.M.;Gompper,J.Chem.Phys.G.,Schick,1975,M.,62,Eds.;1327.Y.(56)B:J.;Polym.deDaoulas,Germany,Pablo,K.Ch.;2005;Mu¨Vol.ller,M.;1.WileyVCH:Stoykovich,M.P.;Papakonstantopoulos,Phys.J.2006J;Nealey,,43,3444.P.F.;Park,S.M.;Solak,H.H.JPolym.Sci.,PartLangmuir,Vol.24,No.4,coordinatesofthebeadcorrespondtoanA-orB-preferentialregionofthepattern.ThesubstrateLERisintroducedasaperturbationtotheedgesofthestripes.ThesimplestcaseofLERwithasinglecharacteristicwavelengthwillbeconsidered,andtheperturbationimposedonastripeedgetakestheformofasimpleperiodicfunction:Aocos(ωoy+φ),withωπo)2L(4)LERwhereyisthecoordinatealongtheedge,AoistheLERamplitude,ωoistheinversewavelength,andLLERisthecorrespondingwavelength.ThecasesofperistalticandundulationalLERarerealizedthroughthephaseshiftfactorφ(seeinsetofFigure1).Inparticular,forthecaseofundulationalLER,foralledges,φ)0.FormodelingperistalticLER,theedgesofthestripesarenumberedsuchthatφ)0forevenedges,andφ)πforoddedges.Theamplitude,Ao,ofthepatternLERisAo)0.2Re.Generally,theregistrationofthepolymerwithapatternthatdiffersfromthenaturalsymmetryandgeometryofthebulkmorphologyiscontrolledbythestrengthoftheinteractions.Ifthepolymer-substrateinteractionsarestrongenough,asurfacereconstructionofthesoftmorphologywilltakeplace.18Theassemblyofalamella-formingcopolymermeltonstripeswithLERcanbealsoconceivedasa“local”surfacereconstructionbecausesubstrateLERdeformstheABinterfacefromitsplanargeometry,whichisthermodynamicallyfavoredinthebulk.Inthecaseofweakpolymer-substrateinteractionsand/orpro-nouncedsubstrateLER,oneexpectsaninteri.e.,thepolymermorphologydoesnotcloselyfollowtheroughpattern.Substitutionofeq3intoeq2showsthatΛNistheparametersettingtheenergyscaleofthepolymer-substrateinteractions.Earlierstudies4a,18,56haveshownthatanorderofmagnitudeΛN～1isarealisticchoiceforachievinggoodagreementwiththeexperimentallyobservedmorphologiesinPS-b-PMMAmeltsassembledonchemicallypatternedsubstrates.HerewechooseΛN)3,inaccordwitharecentstudy18ofsurface-inducedreconstruction,yieldingexcellentagreementwithexperiments.Inpurediblockmelts,theinteractionsoftheAbeadswiththeA-preferentialregionsofthesubstrateandtheinteractionsoftheBbeadswiththeB-preferenti.e.,R)-1and-??)1.Thissymmetricchoiceallowsforadirectcomparisonwiththephenomenologicaltheory.Inthecaseofternaryblends,theinteractionsoftheAbeads4a,18arechosentobestronger,andthevaluesR)-1and-??)0.6areused.Finally,theupperlayerofthefilmisconsideredtobeexposedtoair.AssumingthatthereisnopreferentialAorBsegregation,itismodeledasahardwall,positionedatadistanceHabovethesubstrate.Weconsidertwofilmthicknesses,H)1.43ReandH)2.86Re,theformerbeingatypicalvalueforthethicknessofspin-coatedfilmsusedinpreviousstudies.56Withtheseingredients,thecoarse-grainedmodelincorporates(a)thecharacteristicfreeenergyscalesthatinfluencethethermodynamicsofthedirectedassemblyofthesoftmatteronachemicallypatternedsubstratesetby??NandΛN,(b)thestrengthoffluctuations,characterizedbyN,and(c)therelevantlengthscales,LLER,L,Lo,andH.Thefirst,LLER,characterizesthelengthoftheroughnessofthesubstratepattern.Inprinciple,patternscombiningmanycharacteristicwavelengthscouldbeconsidered,inthesamespirit.Second,themodelaccountsfortheequilibriumperiodicity,Lo,ofthebulklamellarstructureand,third,theperiodicityofthepatternL.Aswasdiscussedabove,theirmismatchmustnotbeexcessiveforachievingperfectlong-rangeorder.However,evenasmallmismatchcreatesastress1288Langmuir,Vol.24,No.4,2008inthestructure,whichcouldaffectthepropagationofsubstrateLER.Finally,thefinitefilmthickness,H,whichwillturnouttobeanimportantlengthscalewhenconsideringsubstrateLEReffectsisalsoaccountedfor.B.SCMFSimulations.Withintheabovecoarse-grainedmodel,theequilibriumpropertiesofthesystemcanbeobtainedviaSCMFsimulations.Thisisanapproximate,particle-basedsimulationtechniqueinspiredbythestandardSCFtheory.ItretainsthecomputationalefficiencyofSCFtheory,including,atthesametime,fluctuationeffects.Amoredetaileddescriptionofthistechniquecanbefoundinref52.Wementionthat,fortheimplementationofSCMFsimulations,thesimulationboxisdiscretizedwithagridofmeshsize?L,suchthattheintegralsineq2areapproximatedbyasummationovergridpoints.Toeachgridpointm,volumefractionsφ?R,mareassignedbycountingthebeadsofRspecieswithinthecorrespondinggridcell.Thespatialdiscretizationintroducesanadditionalparameter,thegridmeshsize?L,intothecoarse-grainedmodel.Themeshsizeplaystheroleofaninteractionrange.Here,weuseagridsizeof?L～Re/6,whichiscomparabletotheamplitudeofthesubstrateLERandmuchsmallerthanitswavelength.Periodically,duringtheSCMFsimulations,thepolymerchainsareconsideredindependent,beingplacedinfluctuatingexternalfields,wR,m,definedbywR,mN?HnbNk)F?L3?φR,m≡δHnbBTkTkBTF(5)BooδφRTheseexternalfieldsapproximatetheinstantaneousinteractionsofanR-typesegmentwithitssurroundings,andtheyarefrequentlyrecalculatedfromtheinstantaneousspatialdistributionofthecomponents.Betweenfieldupdates,thechainsevolveinthesebackgroundfieldsviaashortMonteCarlo(MC)simulation.TokeeptheSCMFsimulationsaccurate,52onetypicallyuseslocalMCmovessuchastherandomsegmentaldisplacementorslithering-snakemoves.ThemovesareacceptedaccordingtoaMetropoliscriterion,pacc)min[1,e-?Hb-?W],where?Hbisthedifferenceinbondedenergybetweentheproposedandtheoldconfiguration,while?Wisthechangeintheinteractionswiththeexternalfields(includingthepolymer-substrateinteractions).AfterasmallnumberofMCsteps,thedensitiesarerecalculatedfromthemonomerpositions,andthefieldsareupdatedaccordingtoeq5.ThisshortMCsimulationandthedensity/fieldupdateconstitutesoneSCMFsimulationcycle.Itispointedoutthattherecalculationofthefieldsfromthedensitydistributionoftheensemblerecoversthecorrelationsbetweenthepolymerchains,and,inthelimitthattheexternalfieldsfollowthepolymerspatialdistributioninstantaneously,theSCMFsimulationsbecomeexact.InthiscasetheyareequivalenttoastandardMCsimulation57,58ofHamiltonians1and2.However,ashortdecouplingofchainsisadvantageousbecauseitallowsforanefficientimplementationonparallelcomputers.Thisshortdecouplingofinteractionsconstitutesthequasi-instantaneousfieldapproximation52whichisthebasisoftheSCMFsimulations.Thisapproximationiscontrolledbyasmallparameter??)(V/nN2?L3))(1/N2??(R/?L)3)eo,whichplaysarolesimilartothatoftheGinzburgparameterinmeanfieldtheory.Thesmallvalueof??)0.002guaranteesthatthesimulationsareaccurate.Inthiswork,defect-freelamellarmorphologiesassembledonapatternwithnoroughnesswereutilizedasaninitialconfigurationSoga,(57)Laradji,M.;Guo,H.;Zuckermann,M.J.J.Phys.ReV.E9.L.;(58)Guo,K.G.;Detcheverry,H.;Zuckermann,Zuckermann,M.F.A.;M.J.;Daoulas,J.Guo,MacromoleculesH.Macromolecules8.Miao,K.Ch.;Mu¨ller,1996M.;,de29,Pablo,2289.J.J.,inpress.Daoulasetal.forSCMFsimulations.Theassemblywasfacilitatedbyanexternalfield,actinginthebulkofthefilm,incombinationwithnonlocalMCmovessuchaschaintranslation.Thepurposewastoachievetheideallimitofnearlyequaldistributionofmoleculesamongthestripes.Ifatthisstage,oneusedSCMFsimulations(whichonlyutilizelocalmoves),onewouldmimicthedynamicsofthespatialorganizationoftherealpolymersystem.Duringtheprocessofphaseseparation,alocalsurplusofpolymerchainsinastripeareacanbecreated.Then,oncethestructureisformed,itisdifficulttoequalizethenumberofchainsamongdifferentlamellaethroughdiffusionbecauseofthecheme.g.,theAmonomersareslowto“tunnel”throughtheB-richdomainsandviceversa.Inarathersmallsystem,thiscouldaffecttheshapeofthelamellaeandwouldcreatedifficultiesinelucidatingtheeffectofsubstrateLER.TheassembledlamellaeweresubjectedtoSCMFsimulationsinvokinglocaldisplacementandslitheringsnakemoves.Forlocalmonomerdisplacements,theexternalfieldsareupdatedafter100%ofthemonomerswereattemptedonaveragetobemoved.Forslithering-snakemoves,weupdatedafterattemptingtoreptate10%ofallchains.Tocalculatetheaverageshapeoftheassembledpolymermorphol-ogiesontheroughpattern,wetypicallyemployed500configurations,savedevery5000SCMFsimulationcyclesaftertheinitialmorphologyisequilibrated.Theequilibrationstagelaststypically5×105SCMFcycles.Becauseofperiodicboundaryconditionsinlateraldirections,thedimensionsofthesimulationcellareamultipleofthestripespacingofthepatternandthecharacteristicwavelengthoftheLER.Thesystemgeometryis1.43(2.86)Repurediblockmeltsand1.43(2.86)R×5.16Re×10.32Refore×6.81Re×13.62Reforternaryblends.Thecorrespondingnumberofgridnodeswas9(18)×36×64and9(18)×48×96,respectively.Thenumberinparenthesesdenotesthevalueemployedforthickerfilms.Toobtainthedesiredvalueoftheinvariantpolymerizationdegree,N,thesimulationcellcontains)ABcopolymerchainsinthecaseofpurediblockmelts,while)copolymers,)Ahomopolymers,and)Bhomopoly-mersareusedinternaryblends.ThesenumbersillustratethecomputationalefficiencyoftheSCMFsimulationstechnique.Forexample,thegenerationof2.5×106MCstepsforthelargeternaryblendsystem,typicallytakes6dayson10IBMp690,Power4+,1.7GHzprocessors.Additionally,wealsousedlargersystemsizestoexplorefinitesizeeffects(cf.Figure6).AlthoughthepresentworkfocusesontheeffectofLERonthesubstratepatternontheaVerageshapeofthecopolymerinterfaces,weemphasizethatSCMFsimulationsdoincorporateinterfacefluctuations(seediscussionofFigure3).TheinterplaybetweensubstrateLER,themismatchbetweenthesubstratepatternandbulklamellarspacing,andthermalinterfacialfluctuationswillbeaddressedinafuturestudy.III.PhenomenologicalTheoryofLERPropagationA.FreeEnergyofModulatedInterfaces.ThestartingpointofthephenomenologicaltheoryonhowtheLERonthesubstratepatternpropagatesintothepolymerfilmistheassumptionthat,atthesubstrate,theABcopolymerinterfaceispinnedtotheroughboundaryofthetwochemicallydifferentregions.Thus,theroughpatternedgeactsasaboundaryconditionfortheABinterfaceatthesubstrate,andtheproblemisreducedtodescribinghowthisboundaryconditionaffectstheshapeoftheinternalinterfacefartherfromthesubstrate.Thefreeenergy,F,perchaininamonolayerofamodulatedlamellacanbeexpressedasasum50ofchainstretching(Fel),interfacial(FAB),bending(Fbend),andGaussiancurvature(FGauss)terms:包含各类专业文献、生活休闲娱乐、中学教育、外语学习资料、文学作品欣赏、91Directed 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