《Accelerated Pseudo-Spectral Method of Self-Consistent Field Theory via Crystallographic Fast Fourier Transform - Qiang, Li - 2020 - Unkn》由會(huì)員上傳分享，免費(fèi)在線閱讀，更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫。

pubs.acs.org/MacromoleculesArticleAcceleratedPseudo-SpectralMethodofSelf-ConsistentFieldTheoryviaCrystallographicFastFourierTransformYichengQiangandWeihuaLi*CiteThis:Macromolecules2020,53,9943?9952ReadOnlineACCESSMetrics&MoreArticleRecommendations*s?SupportingInformationABSTRACT:Self-consistent?eldtheory(SCFT)hasbeenprovenasoneofthemostsuccessfulmethodsforstudyingthephasebehaviorofblockcopolymers.Inthepastdecades,anumberofnumericalmethodshavebeendevelopedforsolvingSCFTequations.Recently,thepseudo-spectralmethodbasedonfastFouriertransform(FFT)hasbecomeoneofthemostfrequentlyusedmethodsduetoitsversatilityandhighe?ciency.However,thecomputationalcostisstillratherhigh,especiallyforsomecomplexstructuresorinthestrong-segregationcase.Toacceleratethecalculation,weintroducecrystallographicFFTintothepseudo-spectralmethod,whichutilizesthesymmetryoforderedphases.Thus,ageneralalgorithmisdevelopedbymakingpartialuseofsymmetryoperationscommonlycontainedbymanydi?erentspacegroups,leadingtoaspeed-upofaboutsixtimesformostofthethree-dimensionalorderedmorphologiesobservedinAB-typeblockcopolymers,includingBCC,FCC,HCP,G,D,O70,andPLphasesaswellascomplexFrank?Kasperphases(σ,A15,C14,C15,andZ).Inaddition,wedemonstratethatmoree?cientalgorithmscanbespeci?callydesignedbyfullyconsideringsymmetryoperationsforsomecomplexstructures.Forinstance,averylargespeed-upofabout30timesisachievedwithaspeci?calgorithmforthecomplexFrank?KasperσphasewiththeP42/mnmspacegroup.Besidesacceleration,thememoryusedbythepseudo-spectralmethodwithcrystallographicFFTisconcomitantlysavedbymanytimes.■INTRODUCTIONbyexpandingthefreeenergyfunctionalaroundthe25Sinceself-consistent?eldtheory(SCFT)basedonthehomogeneousstate.TheexpansionwithinlimitedtermsGaussian-chainmodelwasexplicitlyadaptedtotreatmakesthismethodvalidonlyintheweak-segregationregime.polymer?polymerinterfacesandmicrodomainstructureofAfewyearslater,anotherusefulanalyticalmethodwas1introducedbySemenovforthestrong-segregationcase.26blockcopolymersbyHelfandin1975,ithasbeenwidelyusedforthestudyofmanyphysicalproblemsofinhomogeneousTherewasnoanalyticalmethodsuitablefortheintermediate-polymericsystems.2,3Inparticular,SCFThasbecomeasegregationregime.Accordingly,manye?ortshavebeenDownloadedviaUNIVOFCAPETOWNonMay14,2021at15:13:26(UTC).standardtoolforstudyingthephasebehaviorofblockdevotedtodevelopingnumericalmethodsforsolvingSCFT23,24,27?36copolymersduetoafewadvantages.Firstofall,itcanequations.Ascomputersadvancerapidly,thefast-calculatethefreeenergyofeachorderedstructureandthusgrowinge?ciencyofnumericalmethodsleadstothe1identifytheequilibriumstructureforagivenblockcopolymer.broadeningoftheirapplicationsaswellastheapplicationsofSeehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.Second,SCFTcanreadilydealwithallkindsofarchitecturesSCFTsigni?cantly.Withhighlye?cientnumericalmethods,4?7ofblockcopolymers.Third,su?cientinformationcanbeSCFTcannotonlyrationalizetheexperimentalresultsbutalsoachievedfromSCFTforanalyzingtheself-assemblymecha-promoteexperimentalresearchbypredictingnewre-nismofblockcopolymers,suchasdi?erentcontributionstosults.5,14,18,37?39thefreeenergyandthespatialdistributionofeachTheearliestattemptstoobtainnumericalsolutionsofSCFT8?11segment.Ofcourse,themostimportantadvantageisthefordiblockcopolymersweremadebyHelfandandWasser-reliabilityofSCFTcon?rmedbythegoodagreementbetweenman.40In1992,Shulldevelopedanapproximatenumericalitsresultsandexperimentalresults,especially,excludingthetechniqueforone-dimensionalbulkandthin-?lmsystemsofinaccuracyoftheinputparametersmeasuredinexperimentsdiblockcopolymermelts,41whileVavasourandWhitmore(e.g.,theFlory?Hugginsinteractionparameter,thesegment9,12?22length,andthedistributionofmolecularweight).However,SCFTequationsofblockcopolymers,eventheReceived:August26,2020simplestABdiblockcopolymer,aretoocomplextobesolvedRevised:October18,2020exactly.23,24Intheearlyyears,someanalyticalorsemi-Published:November13,2020analyticalmethodsweredevelopedtoobtainapproximatesolutionstoSCFTequationsundersomeextremeconditions.In1980,Leiblerproposedanapproximateanalyticalmethod?2020AmericanChemicalSocietyhttps://dx.doi.org/10.1021/acs.macromol.0c019749943Macromolecules2020,53,9943?9952

1Macromoleculespubs.acs.org/MacromoleculesArticleconstructedthephasediagramofdiblockcopolymermelts,DespitethegreatsuccessinsolvingSCFTequations,theconsistingoflamellar,cylinderandsphere,usinganothere?ciencyofthepseudo-spectralmethodstillneedstobeapproximatenumericaltechniquethatignoresthedetailedenhanced,especiallywhenthecalculationneedstobe42structurale?ects.Lateron,anaccuratenumericaltechniqueperformedwithlargeMandNsforhighaccuracy.Usually,wasdevelopedbyMatsenandSchick,whichwasreferredtoaslargerMandNsareneededforthepseudo-spectralmethodtothespectralmethodorreciprocalspacemethodasitexpandsmaintaintheaccuracyofthefreeenergyofagivenstructureat23allofthespatialfunctionsintermsofasetofbasisfunctions.strongersegregationduetosharperinterfacesandlargerThemostingeniousideaofthespectralmethodconsistsindomainperiods.Iftheperiodsofthecalculatedstructuresarereconstructingthebasisfunctionsbygroupingallplanewavesverylarge,thecomputationalcostcanstillbehigh.Forofequalmagnitudestogetheraccordingtothegroupsymmetryexample,Cochranetal.stillusedlargeMandNstodetermineofagivenorderedphase,whichreducesthenumberofbasisthestableregionofdoublegyroidsofadiblockcopolymerfunctionsdramaticallyandthusenhancesthecomputationalaccuratelythoughtheydevelopedthefourth-orderpseudo-e?ciency.Thishighlye?cientmethodcancalculatethefreespectralmethod.6Recently,aclassofcomplexsphericalphases,energyofeventhree-dimensional(3D)orderedstructuresi.e.,Frank?Kasperphasesincludingσ,A15,C14,C15,andZ,accurately.Accordingly,ithasbeenwidelyusedtoidentifythehavebecomeparticularlyappealinginthecommunityofblockequilibriumorderedstructuresofvariousblockcopolymersbycopolymers.6,13,14,16,17,19?21,46?50Ontheonehand,SCFT5,23constructingphasediagrams.Forexample,thespectralstudiesdeepentheunderstandingoftheformationmechanismmethoddistinguishedtheequilibriumdouble-gyroidphaseofFrank?Kasperphases.14,16,20Ontheotherhand,SCFTfromthedouble-diamond,andperforatedlamellarphasesincalculationsbroadlypredictstableFrank?Kasperphasesindiblockcopolymermeltsforthe?rsttimeandquantitativelyvariousblockcopolymersystemsthatarepurposelydesigned23analyzedtheirrelativestability.Althoughpriorknowledgeofaccordingtotheirformationmechanism.6,19,50,51Evenso,thesymmetryisrequiredandthusrestrictsthespectralmethodtocomputationalcostofthecalculationsoftheseFrank?Kasperthecalculationofknownstructures,thespectralmethodcanbephasesisstillhighbecausemostoftheFrank?Kasperphasesgeneralizedforexploringunknownstructuresatthecostof27havecomplexstructuralunitsandveryclosefreeenergy.Ine?ciency.Itisnecessarytostressthatthecomputationalcost3otherwords,largeMandNsmustbeutilizedtocalculatetheofthespectralmethodisproportionaltoM,withMbeingthe43freeenergyofeachFrank?Kasperphaseatsuchahighnumberofbasisfunctions.Asaresult,thee?ciencyoftheaccuracythattherelativestabilitybetweendi?erentFrank?spectralmethodwoulddecreaserapidlyasMincreases,e.g.,forKasperphaseswithsmallfreeenergydi?erencescanbethestrong-segregationcaseorlow-symmetrystructures.distinguished.Forinstance,M=128×128×64andNs=100Complementarytothereciprocalspacemethod,anwereusedfortheFrank?KasperσphaseformedinABdiblockalternativewayistosolveSCFTequationsusingthe?nite14orABnmiktoarmstarcopolymermelts,whileM=256×256di?erencemethodinrealspace.Inparticular,Droletand×128andNs=450wereusedinourveryrecentworktodealFredricksondevelopedaniterativealgorithmforsolvingSCFTwiththemultiblockarchitectureandconcaveinterfacesoftheequationsinrealspace,inwhichonecriticalstepistosolvethe6extremelyenlargedsphericaldomains.modi?eddi?usionequationsforthepropagatorfunctionsusing10,24Itiswell-knownthatthemosttime-consumingpartoftheanalternatingdirectionimplicitscheme.Theyproposed45pseudo-spectralmethodisFFT.Generally,FFTcalculatesthatthisrealspacemethodcouldbeusedtodiscovernewdiscreteFouriertransform(DFT)e?cientlybydecomposingastructuresstartingfromrandominitialconditions.ThelargeDFTintoaseriesofsmallpiecesofDFTsviaCooley?e?ciencyoftherealspacemethodislimitedforcomplexTurkeyfactorizationbutdoesnotmakeuseofthespatialblockcopolymersystemsduetotheroughfreeenergy10symmetryofthedatapointsinrealspace.Infact,somedatalandscapeswithmanycompetingmetastablelocalminima.pointsintheunitcelloforderedstructuresareequivalentdueIn2002,anewnumericalmethod,referredtoasthepseudo-tospacesymmetry,andthusthecalculationsofsomespectralmethod,wasdevelopedbyTzeremesandco-workers.44Thepseudo-spectralmethodtakesadvantageofdecomposedpiecesofDFTscanbeskipped,leadingtoaboththereciprocalandrealspacemethods.28Speci?cally,itmoree?cientalgorithmthatisreferredtoascrystallographicsolvesthemodi?eddi?usionequationsbyperformingforwardFFT.CrystallographicFFTwasproposedbyTenEyckfora52andbackwardfastFouriertransforms(FFTs)foreachstepofsubsetofcrystallographicsymmetries,andwasfurtherintegrationwhilecomputingtheotherspatialfunctionsaswelldevelopedforall230spacegroupsinaseriesofsubsequent53?58asperformingtheiterationinrealspace.Accordingly,theworks.Inthispaper,wewillincludeacrystallographiccomputationalcostofthismethodscalesasNsMlogM,whereFFTintothepseudo-spectralmethodtoenhanceitse?ciency,NsandMrepresentthenumberofpointsdividingthechainfocusingonitsapplicationstothecommon3DorderedcontourandthenumberofgridpointsdiscretizingthestructuresaswellascomplexFrank?Kaspersphericalphasescomputationalbox.45DuetotheuseofFFT,thepseudo-formedbyAB-typeblockcopolymers.Inprinciple,sincespectralmethodisrathere?cient.However,thepseudo-di?erentorderedstructureshavedi?erentspacegroups,eachspectralmethod,especiallythesecond-orderalgorithm,haslessofthemneedsaspeci?callydesignedcrystallographicFFTaccuracythanthespectralmethodbecauseofthediscretealgorithmtoutilizeallofitssymmetryoperations.Fortunately,integrationalongthechaincontour;varioushigher-ordermostofthese3Dstructures,alongwithmanyFrank?Kasperalgorithmshavebeendevelopedtoimprovetheaccuracyofthephases,sharesomesimilarsymmetryoperations.Bymaking31,32,35useofthesecommonsymmetries,ageneralalgorithmsuitablepseudo-spectralmethod.Moreover,someusefulalgo-rithmshavealsobeendevelopedtoacceleratetheconvergenceforallofthese3Dstructurescanbedeveloped.Althoughthe29,30,34,36oftheiterationprocessforsolvingSCFTequations.pseudo-spectralmethodisnotacceleratedtothemaximuminAsaresult,thepseudo-spectralmethodhasbecomeoneofthethisgeneralalgorithm,itstillachievesconsiderablespeed-up.mostfrequentlyusedmethods.Inaddition,wewilldemonstratethatalargerspeed-upcould9944https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

2Macromoleculespubs.acs.org/MacromoleculesArticleFigure1.Freeenergyerror(a)andinterfacialenergyerror(b)oftheA15phaseasafunctionof1/Δswithdi?erentspatialgridsizesMfortheAB3copolymeratf=0.31andχN=40.beobtainedbydevelopingmorespeci?calgorithmsforsomeq(,0)1r=Acomplexorderedstructures,e.g.,theσphase.??nqfq(,)rr=[(,0)]■ABSCFTANDPSEUDOSPECTRALMETHOD?AsastandardtooltostudytheequilibriumphasebehaviorofqfB(,(1r?=)/)1nblockcopolymers,SCFTcanbeeasilyappliedtoblock??n1qq(,0)rr=[](,)(,0)fqrcopolymerswithdi?erentarchitectures.Here,the?exibleABnBAB(5)miktoarmstarcopolymeristakenasanexampletoformulateIntheaboveequations,thechaincontourisrescaledbyN,andtheSCFTequationsinacanonicalensemble.Forsimplicity,AandBsegmentsareassumedtohavethesamelengthbandthespatiallengthisrescaledbyNb/6.Moreover,s=0andsvolumeρ?1.EachcopolymerconsistsofNsegmentsintotal,=faresetatthefreeendoftheA-blockandatthejunctionpoint,respectively,forq(r,s)andq?(r,s),whiles=0ands=whichiscomposedoffNA-segmentsand(1?f)NB-AAsegments.TheimmiscibleinteractionbetweenAandB(1?f)/naresetatthejunctionpointandthefreeendofoneB-block,respectively,forq(r,s)andq?(r,s).segmentsischaracterizedbytheFlory?Hugginsparameterχ.BBConsideringanincompressiblemeltconsistingofnpidenticalMinimizationofthefreeenergyfunctionalleadstotheABncopolymerchainswithinavolumeofV,thefreeenergystandardSCFTequationsfunctionalperchaincanbeexpressedaswNA()rr=+χ?B()ξ()rF1=?lnQ+∫d(rr{χ??NwAB)(rr)(?A)?A(r)wNB()rr=+χ?A()ξ()rnkTpBV1f???wB()()rrr?ξ??BA()1[?()r?B()r]}?A()rr=Q∫qsAA(,)(,)dqssr0(1)n1/?fnwhere?A(r)and?B(r)arethespatialdistributionsofvolume?()rr=∫qs(,)(,)dqss?rBQBBfractionsofAandBsegments,andwA(r)andwB(r)aretheir0(6)conjugatemean?elds.ξ(r)istheLagrangemultipliertoForotherblockcopolymermeltingsystems,theSCFTenforcetheincompressibilityconditionsequationsareslightlydi?erent,whilethemodi?eddi?usion??()rr+=()1(2)equationsaresimilarforeachblock.Inthesecond-orderABalgorithmofthepseudo-spectralmethod,themodi?edIneq1,scalarQisthepartitionfunctionofasinglecopolymerdi?usionequationsareintegratedalongthechaincontour28chaininteractingwiththemean?eldswA(r)andwB(r),anditviaatwo-stepFFTascanbecalculatedby2?Δsw/2()rr?Δ??Δssw/2()qss(,rr+Δ≈)eeeqs(,)1?Qqs(,)(,)drrqsr2=∫?Δsw/2()rhr?1Δsk()?Δsw/2()VKK(3)=[eFFTeFFTe[]qs(,)r](7)whereq(r,s)andq?(r,s)(K=A,B)arethepropagatorInotherhigher-orderalgorithms,FFTalsoplaysakeyroleinaKK31,32,35functions,whichcanbeobtainedbysolvingthefollowingsimilarway.Thus,thee?ciencyofthepseudo-spectralmodi?eddi?usionequationsmethodmainlydependsonthee?ciencyofFFT.AsthetimecostbyonenormalFFTisproportionaltoMlogM,thetotal?qsH(,)rr=??qs(,)timecostbythewholeintegrationscalesasNsMlogM,where?sKKKNindicatesthenumberofpointsusedtodiscretizethechainscontour.Inthepseudo-spectralmethod,theaccuracyoffree????qsH(,)rr=??qs(,)energyissensitivetoMandΔs～1/Ns.Usually,largeMand?sKKKNsareneededtoobtainhighaccuracyoffreeenergy.InFigureHw?=??+2()r1,weshowthedependenceoffreeenergyerrorsonMandΔsKK(4)withtheFrank?KasperA15phaseformedbyAB3miktoarmwiththeinitialconditionsasstarcopolymer,wherethefreeenergyiscalculatedusingthe9945https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

3Macromoleculespubs.acs.org/MacromoleculesArticleFigure2.Schematicillustrationofone-stepsymmetryreductionof2Ddatawithp2mmsymmetry(a)andrecursivesymmetryreductionofdatawiththediagonalre?ectionplane(b).Solidblacklinesrepresentre?ectionplanes,anddoublearrowsindicatethattwosubmatricescanbeconvertedtoeachotherbycertainsymmetryoperation.fourth-orderalgorithmbyRanjan,Qin,andMorse(RQM4)de?nesanactionS#whichappliestotheperiodicfunctionf(x)whilethereference-freeenergyofhighaccuracyiscalculatedtoobtainusingM=1283and1/Δs=2000.32Itisnecessarytonotethat#?1thespectralmethodshouldbeanidealchoicetocancelthe(Sxff)()′=(Rxt(′?))(9)numericalerrorinducedbythediscreteintegrationalongtheThenDFTof(S#f)(x′)canbedenotedby(S*F)(h′),wherechaincontour.However,forcomplexFrank?KasperphasesTwithrelativelylowsymmetries,thenumberofbasisfunctions()S*′Ff()h=∑(xRhx)(,)(,)eA′eRAR?1ht′ofthespectralmethodbecomeslargeandthusmakesitsx∈Γcomputationexpensive.T=′′eFRAR?1(,)()htRh(10)Thefreeenergyerrordecreasesas1/Δsincreasesandapproachesaplateau.Inparticular,theplateauappearsatHereAisanintegralmatrixdescribingtheperiodicity,whosesmaller1/ΔsforsmallerM.Ourresultsareconsistentwithcolumnsareprimitivetranslationvectors45thosebyStasiakandMatsen.ThisobservationindicatesthatthenumericalerrormainlyresultedfromthespatialAa=[123aa](11)discretizationwhen1/Δsislargeenough,con?rmingthat33Γ=Z/ZAistheunitcell,andF(h)istheDFTofthebothlargeMand1/Δs(orNs)areneededforthecalculationoriginalinput.ThetwiddlefactoreA(h,x)isde?nedasoffreeenergywithhighaccuracy.Forexample,todistinguishtherelativestabilitybetweentheA15andσphaseswithafree?1eA(,)exp(2hx=?×πihAx)(12)energydi?erenceofaround10?4kTperchain,thegridsizesBfortheA15andσphasesaretypicallychosenasM=643andEquation10indicatesthatafterapplyingasymmetryoperationM=128×128×64,respectively,and1/Δs≥100ischosen.14tothedatapointsinrealspace,therotationalpartoftheFurthermore,largerMandNsshouldbeconsideredfortheseoperationresultsinacorrespondingrotationinthereciprocalcomplexphasesatstrongersegregation.6Lastbutnotleast,space,whilethetranslationalpartbringsinanadditionaltwiddlefactor.largerMandNsarerequiredforthecalculationofinterfacialTheotheressentialpropertiesusedtoconstructcrystallo-energyandentropiccontributionsincetheirerrorsaregraphicFFTistheCooley?TukeyFactorization.Supposethatconsiderablyhigherthanthatofthefreeenergyitself(Figure1b).AcanbedecomposedintointegralmatricesA0andA1■AAA=01(13)CRYSTALLOGRAPHICFFTthecoordinatesinrealspacecanbedecomposedasCrystallographicFFTtakesadvantagesofsymmetryofdatax=+Ax01x0(14)pointsduringDFT,yieldingmoree?cientandmemory-savingFFTroutines.ComparedtothenormalFFTthatusesdataThenthereciprocalcoordinatescanbedecomposedaswellpointsfromthefullunitcellasinput,thecrystallographicFFTTrequiresonlythenonequivalentpoints(e.g.,pointsfromanhAhh=+101(15)asymmetricunit),suchthatthepseudo-spectralmethodcouldTheDFToftheoriginalinputcanbeexpressedasbesigni?cantlyacceleratedbyreplacingnormalFFTwiththecrystallographicone.Followingthenotationsinaseriesofloo|ooooworkbyKudlickietal.,53?58thecrystallographicFFTisFe()hh=+∑∑mooAA01(,)10xfe(x0A01x)(,)AA10h11x}ooe(,)h00xooanchoredtotwousefulpropertiesofDFT.First,supposethatx00∈Γnx11∈Γ~(16)anactionS,de?nedbyindicatingthattheDFTofalargematrixcanbedecomposedS()xR=+xt(8)intoseveralDFTsofitssubmatrices,followedbyalinearcombination.Whenonesubmatrixcanbeconvertedfromactsonthecoordinatesofsamplepoints,whilematrixRistheanotherbyacertainsymmetryoperation,eq10canbeusedtorotationalpart,andvectortisthetranslationalpart.ItalsoskipitsDFT,thusreducingthecomputationalcost.Thesetwo9946https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

4Macromoleculespubs.acs.org/MacromoleculesArticleTable1.SymmetryPlanesfor2x2y2zAlgorithminDi?erentSpaceGroupsaspacegroupphasessymmetryplanesFdddO70111111dx(,,0),,0ydx(,0,),0,zdy(0,,)0,,z444444P42/mnmσmx,y,0mx,x?,zmx,x,zP6/mmmZmx,y,0m2x,x,zm0,y,zP63/mmcHCP,PL,C14mx,y,1m2x,x,zc0,y,z4Pm3?nA15mx,y,0mx,0,zm0,y,zPn3?mDnx(,,0),,11y1nx(,0,),,111zny(0,,),,111z224224224Fm3?mFCCmx,y,0mx,0,zm0,y,zFd3?mC15dx(,,0),,13y1dx(,0,),,311zdy(0,,),,131z444444444Im3?mBCC,Pmx,y,0mx,0,zm0,y,zIa3?dGax,y,1cx,1,zb1,y,z444a59ThesymbolsofsymmetryplanescorrespondtothoseintheInternationalTablesforCrystallography.Figure3.ComputationalboxforP63/mmc(a)andP42/mnm(b).Theblackframerepresentsthenormalunitcellwhiletheblueframerepresentsthecomputationalboxwiththeprimitivetranslationvectorsnormaltosymmetryplanes.TheredframeindicatestheasymmetricunitofgroupP42/mnmin(b)anditsenlargedview(c).propertiesleadtoone-stepsymmetryreduction(Figure2a)ortheoriginalinputf(x)canbesplitintoeightsubmatricesfj(x1),recursivesymmetryreduction(Figure2b),dependingonthej=0,1,...,7,byextractingevenoroddelementsineachsymmetryofdata.Ingeneral,one-stepsymmetryreductioncandimensionbeviewedasthesimplestcaseoftherecursivesymmetry57f()xA=+fn(xeee++ml)reduction,whichasksfortheleaste?orttoimplement.j101123(18)Althoughthespacegroupsofdi?erentperiodicmorphologiesinblockcopolymersystemsusuallydi?er,eachofthesewherenmlisthebinaryexpressionoftheintegerjandeiistheunitvectorsalongai(i=1,2,and3).Onlyoneoffj(x1)isrequiresaspeci?callydesignedalgorithmtoachieveafullindependentsincealloftheothersevensubmatricesarerelatedsymmetryreduction;mostofthe3Dmorphologiesobservedintoitbythesymmetryplanes.Forexample,are?ectionplaneblockcopolymersystemssharesomesimilarsymmetryperpendiculartoe1indicatesthatsubmatrixf4(x1)canbeoperations.Thereforeitispossibletogeneratearelativelyobtainedbysimplyreversingthepointsinf0(x1)inthee1generalalgorithmthatmakesuseofonlythesesimilardirection.Accordingly,F(h)canbesplitintoeightsubmatricessymmetryoperationsbutappliestomostoftheusualaswell,bysplittingeachdimensioninthemiddle,formingvectorF={F(h)}7withF(h)=F(h+AT(ne*+me*+morphologies.j1j=0j11112First,weimplementthe2x2y2zalgorithmforspacegroupsle*3)).Heree*iistheunitvectordualtoeiinthereciprocallistedinTable1,54,57allofwhichhavethreesymmetryplanesspace.TheDFToffj(x1)isdenotedbyGj(h1)andde?neG={G(h)}7.Fromeq16wehaveperpendiculartoeachother,includingre?ectionplanesandj1j=0glideplanes.Forsimplicity,wechosethecomputationalboxF=tcG(19)withthreeprimitivetranslationvectorsai(i=1,2,and3)perpendiculartothesymmetryplanes,respectively,andtheHereTcisa8×8matrixrepresentingthetwiddlefactorsfromoriginattheinversioncenter.NotethatthecomputationalCooley?Turkeyfactorizationboxeschosenherearedi?erentfromthoseusuallyusedfortheT7spacegroupsP4/mnm,P6/mmm,orP6/mmc(Figure3).Tc1={enA((hAeeee+1123123*+m*+l*),n′+′+′}meel)jk,=023Suppose(20)???é?????é??wheren′′ml′isthebinaryexpressionoftheintegerk.??200N????200??Moreover,fromeq10????????AA==????020M????and0????020????????????GeG()hh=(,)(tRTh)(21)???002P?????002??(17)jj11A0j19947https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

5Macromoleculespubs.acs.org/MacromoleculesArticlewhereRjandtjaretherotationalandtranslationalpartsofTable2.Speed-UpofCrystallographicFFTComparedwitheightsymmetryoperationsderivedfromthreesymmetryNormalFFTplanes.DenoteG′={G(RTh)}7andT=diag({e(h,0j1j=0sA17usedsymmetryspeed-tj)}j=0),wherediag(v)representsthediagonalmatrixwithvasspacegroupoperationsasizet/msbupitsdiagonalelements.Combinedwitheq19P1none6433.51.0F=TTGcs′(22)Pm3?n,Pn3?m(26),(27),(28)6430.595.9P6/mmm(16),(20),(24)6430.605.8whichrecoverstheDFToftheoriginaldata(i.e.,F(h))from3P63/mmc(16),(20),(24)640.595.9G′,whichreliesonlyontheDFTofsubmatrixf0(x1),i.e.,I4132t(1,1,1),(2),(3),(21)6430.3510G0(h1).Accordingly,theFFTofsizeAisreplacedbyFFTsof222Im3?mt(1,1,1),(26),(27),(28)6430.2813sizeA1,whichisone-eighthoftheoriginalsize,leadingtoa222muchfasteralgorithm.InmanySCFTinstances,afullrecoveryIa3?dt(1,1,1),(26),6430.2315ofF(h)isnotnecessary.Asanexample,wetakethesecond-222(27),(28),(39)orderalgorithmofthepseudo-spectralmethod(eq7).TheFdddt(0,1,1),t(1,0,1),(6),6430.1523modi?eddi?usionequationsaretypicallysolvedbyrepeatinga2222(7),(8)forwardFFT→elementwisemultiplication→backwardFFTFm3?m,Fd3?mt(0,11116430.1523triplet.Supposethattheresultofthetripletisfnew(x),and,),t(,0,),(26),2222Fnew(h)istheDFToffnew(x),sincefnew(x)hasthesame(27),(28)c3symmetryasf(x),fromeq22,wehavePm3?n(26),(27),(28)640.854.1d3Pm3?n(26),(27),(28)640.2912newnewF=′TTGcs(23)P1none2562×1282111.0P4/mnmd(10),(11),(15),(16)2562×1282.972Fromeq7,Fnew(h)=exp(?Δsk2(h))F(h),then2aTheindicesandsymbolscorrespondtothoseintheInternationalFnew=KF(24)59bTablesforCrystallography.Onlygeneratorsarelisted.TimewhereK=diag({exp(?Δsk2(h+AT(ne*+me*+le*)))}7)consumptionforaforwardtransform-elementwisemultiplication-11123j=02backwardtransformtriplet.Alltestsareperformedwithasingleisobtainedbysplittingexp(?Δsk(h))intoeightsubmatrices.threadonanIntel(R)Xeon(R)CPUE5-2690v4@2.60GHzNotethatbothTcandTsarereversible,thusprocessorsinarealSCFTinstance.cImplementationbasedonDCTnew??11dG′=′=QGQTTKTT,sccs(25)routinesfromFFTW3.ImplementationbasedonourownDCTroutines.ForaconstantΔs,Kisconstant;thereforematrixQbecomesconstantforacertainspacegroupandcanbecalculatedinadvance.The?rstelementofG′isobtaineddirectlyfromaoperationnumber(2),(3),and(21)intheInternational59normalFFT,whilealloftheotherscanbeobtainedbyTablesforCrystallography.SincetherotationalpartRiisrearrangingthematrixelementsaccordingtotheconsidereddi?erentfromthoseofthesymmetryplanes,thememorysymmetryoperations(Figure2).Furthermore,onlythe?rstaccesspatternisdi?erentfromthoseofthespacegroupslistedrowofQisneededtocalculatethe?rstelementofG′new,i.e.,inTable1,probablyin?uencingtheperformanceoftheGnew(h).Thenfnew(x)canbeobtainedbyconductingaalgorithm.0101normalbackwardFFTofGnew(h),andfnew(x)canbeTomakethealgorithmsuitableforasmanyordered01recoveredbymakinguseofthesymmetryoperationsinrealmorphologiescommonlyformedinblockcopolymersasspace.Byimplementingthealgorithmdescribedbyeq25,apossible,theimplementationdescribedabovebecomesalittlesigni?cantspeed-upofaboutsixtimesisobtainedforgroupcomplicated.Inotherwords,thealgorithmcanbesimpli?ed,Pm3?nona643grid(Table2).Besidestheacceleration,theespeciallywithrespecttocoding,ifitisappliedtothecrystallographicFFTalsoallowsustoreplacethe?eldsandsupergroupsofPmmm,suchasP42/mnm,P6/mmm,Pm3?n,propagatorsinthepseudo-spectralmethodwiththeirFm3?m,andIm3?m.Comparedwiththeglideplanes,thesubmatrices.Asthesubmatrixfnew(x)alreadycontainsallofoperationofeachre?ectionplaneinthesespacegroupsalters01thenonequivalentelementsoffnew(x),itisnotnecessarytothecoordinatesofthesamplepointsinonlyonedirection.recoverfnew(x)duringthewholeSCFTcalculations,suchthatConsideraone-dimensionalarrayf(x),x=0,1,...,2N?1thememorycostcanbecutdowntoonlyaboutone-eighthofwiththere?ectionplaneinthemiddle,i.e.,satisfyingthenormalone.Itisnecessarytoemphasizethatthef()xfN=?(21?x)(26)operationsofsymmetryplanescontainedindi?erentspacegroupslistedinTable1havedi?erenttranslationalparts,butApartfromCooley?Turkeyfactorization,itsDFTcanalsobethesetranslationalpartsa?ectonlythevalueofpre-calculatedrepresentedasmatrixQbutnotthememoryaccesspattern.AsaN?1iconsequence,asimilarspeed-upcanbeachievedforallspaceFhf()=?∑()exp2xjjijjπihxyzz2,DFTNjjjzgroupsinTable1onthesamegrid.x=0kk2N{Asthedeductionofeqs22or25doesnotrelyontheexactihN(2??1x)yyformofRiandti,theabovealgorithmcanbeappliedtoother+?expjjjj2πizzzzzzzzspacegroupswithoutsymmetryplanesaswell,suchasthek2N{{spacegroupI4132.NotethatI4132possessesthealternateijj1yzzdouble-gyroidmorphologycommonlyformedinABC-type=<2exp2jπizFhhN,DCTII()ifNk4N{(27)blockcopolymersaswellasthesingleGyroidmorphologypossiblyformedinAB-typeblockcopolymers.ForspacegroupHeresymbolFP,X(h)indicatestheXtransformoff(x),x∈[0,I4132,aproperchoiceofsymmetryoperationscouldbeP).ConsideringthatF2N,DFT(N)=0,theHermitiansymmetry9948https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

6Macromoleculespubs.acs.org/MacromoleculesArticleoftherealDFTF2N,DFT(h)canberecoveredfromFN,DCTII(h),f(,,)xyz=?fN(21?yxM,,2?1?z)whichisthetype-IIdiscretecosinetransform(DCTII)ofthe?rsthalfoff(x).Accordingly,a2N-pointDFTcanbereplaced=?fyN(,21??xM,21?z)byanN-pointDCT,thusreducingthecomputationalcost.=?fN(21??xN,21?yz,)(29)Similartoeq25,thefullrecoveryoftheDFTresultisunnecessarywhensolvingdi?usionequations.Inaddition,thewhichmeans[0,2N)×[0,N)×[0,M)isaproperchoiceoftwiddlefactorexp(2πi/4N)canbesimplyomittedsinceitjusttheasymmetricunitinrealspaceforgroupP42/mnm(Figurecancelsitselfwhenconductingthebackwardtransform,further3c).UsingthefollowingpropertiesofDCTssimplifyingthealgorithm.Analogously,theDFTof3DdatahwithPmmmsymmetrycanbeobtainedfromthe3DDCTIIofFhFh2,DCTIIN()(1)=?′2,DCTIIN()one-eighthoftheoriginaldata.Forthebackwardtransform,if()fx=′fN(2??1x)type-IIIdiscretecosinetransform(DCTIII)canbeusedtoreplacetheoriginalbackwardFFT,whichisthereversetransformofDCTII.Fromthecodingaspect,onecansimplyFh2,DCTIIN()switchtotheDCTroutinestoobtainthespeed-upeasilyforloo2(FhN,DCTII/2)evenh=msupergroupsofPmmm.Inourtest,weusetheDCTroutinesoon0oddhfromtheFFTW3libraryandobtainaspeed-upofaboutfour3if()fx=?fN(21?x)timesfora64grid.Notethatinourtest,theperformanceisslightlyinferiortothealgorithmdescribedbyeq25,probablybecauseFFTW3actuallyindirectlyimplementstheDCTfromFh()2,DCTIIN60FFT.Toobtainbetterperformance,wecouldeitherloo0evenhcustomizethelibraryorimplementDCTbyourselves.Here=mweimplementourownversionofDCTs,61,62whichfullyoo2(FhN,DCTIV(?1)/2)oddhnmakesuseoftheAVX2SIMDinstructionslikeotherlibrariesif()fx=?fN(2??1x)(30)andobtainaspeed-upofabout12times(Table2).Obviously,moree?cientalgorithmscanbedesignedbytheDCTIIofsizeofA1canbereplacedbyAlgorithm1,whereconsideringmoresymmetryoperationsforaspeci?cspaceonlysamplepointswithintheasymmetricunit[0,2N)×[0,group.IthasbeenshownthatafullsymmetryreductionisN)×[0,M)willbeaccessed.TheoutputofAlgorithm1ispossibleforall230spacegroups,meaningthatallsymmetrysplitintoseveralmatricesforthereasonthattheasymmetricoperationscouldbeusedtoaccelerateFFT.57However,itisaunitinthereciprocalspaceisnotacuboid.However,thetotalsizeofallofthesematricesisexactly2MN2,whichisthesamenontrivialworktorealizethefullyacceleratedalgorithmforthesespacegroupsownedbyorderedmorphologiesinblockasthesizeoftheasymmetricunitinrealspace.Itisworthcopolymers.Oneofthemostimportantreasonsisthatusually,noticingthatallofthesubroutinesandproceduresinrecursivesymmetryreductionisrequiredtotacklesomeAlgorithm1arereversible,indicatingthattheinversesymmetryoperations,increasingthecomplexityofthetransformcanbeimplementedaccordinglyaswell.Di?erent57algorithm,especiallywhenmanycodingdetailslikeparalleliza-fromthealgorithmbyKudlickietal.,norecursivesymmetrytionandvectorizationmustbeconsidered.Anotherreasonisreductionisusedhere,andthustheimplementationisexpectedtobemucheasier.Sinceallofthesymmetrythatthesesymmetryreductionmethodsareoriginallydesignedoperationsareused,theperformanceofouralgorithmisforcomplexdata,andthusanadditionalHermitiansymmetry57expectedtobecompetitivewiththatintheliterature.Weusemustbetakenintoaccountwhenmigratedtorealdata.ToourownversionofDCTstoimplementthisalgorithm,leadingfullyshowthepotentialofcrystallographicFFT,wemanagetotoanastonishingspeed-upofabout72timesfora256×256×designanothere?cientcrystallographicFFTalgorithmforthe128grid,comparedtothenormalFFTinFFTW3.spacegroupP42/mnmoftheσphasethatisoneofthemostConsideringthatthevolumeofourcomputationalboxiscomplicatedperiodicmorphologiesinblockcopolymertwicethatofthenormalone,thee?ectivespeed-upbecomessystems.about30times.Itisnecessarytostressthatthespeci?cAsmentionedbefore,thecomputationalboxischosen,suchalgorithmnotonlyleadstoamuchhigherspeed-upbutalsothateachofitsprimitivetranslationvectorsisnormaltothesavesmuchmorememorythanthegeneralalgorithm.Asthere?ectionplanes,respectively,whosevolumebecomestwicedataofallspatialfunctionsarereducedtothoseinthethatofthenormalunitcellofgroupP42/mnm(Figure3b).asymmetricunit,thememoryusageisreducedbyafactorofSupposethat32,ore?ectivelyafactorof16afterexcludingthee?ectofthe???400Né?????é??doubledvolumeofthecomputationalbox.??????200N??????????Todemonstratethee?ciencyofAlgorithm1,weapplyittoAA==????040N????and1????020N????aspeci?cexample,i.e.,theσphaseformedintheAB3???????????004M??????002M???miktoarmstarcopolymerwithf=0.31andχN=40.We(28)choosearatherlargegridof256×256×128andalargenumberofcontourstepswithastepsizeofΔs=0.002.HereOnthebasisofourpreviousdiscussion,3DDFToftheweimplementthecrystallographicFFTspeci?callyforP42/originaldatawiththesizeofAcanbereplacedbyaDCTIImnmintotheRQM4pseudo-spectralmethod,andweusethewiththesizeofA1bymakinguseofthesymmetryoperationsAndersonmixingschemetoacceleratetheiterationprocessofofthere?ectionplaneswhileleavingthe“42”screwaxiswithSCFTsolutiontowardconvergence.After856iterations,boththecenterofsymmetryunexploited.Representingf(x)byf(x,theincompressibilityconditionsandtherelative?elderrory,z)andthescrewaxisindicatesbetweensuccessiveiterationstepsareconvergedtolessthan9949https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

7Macromoleculespubs.acs.org/MacromoleculesArticle10?10.Thefreeenergyperchainconvergesto6.38015006kT,Bandthechangesofthefreeenergyandcorrespondingentropicorenthalpiccontributionarelessthan10?9kTinthelast100Bsteps.UnderthesameconditionsasthoseinTable2,thetotalmemoryconsumptionislessthan2GB,andthetotalelapsedtimeisabout85min.Mostofthetime,67minisusedbythecrystallographicFFTs,whiletheremainingismainlyconsumedbytheintegrationofpropagatorsandAndersonmixing.ItisworthmentioningthatAndersonmixingcanbefurther45acceleratedbyreusinghistoricalinformation.Forsuchalargesystem,thecomputationalcostincreasesbyabout30timesintime(i.e.,2500min)and16timesinmemory(i.e.,32GB)andbecomesveryhighwithoutaccelerationfromtheFigure4.Evolutionofthe?elderrorduringiterationofSCFTcrystallographicFFT.Forothermorphologies,similarsolutiontotheA15phaseformedbytheAB3copolymerwithf=0.31techniquescanbeappliedtoutilizemoresymmetryoperationsandχN=40,wherethemodi?eddi?usionequationsaresolvedbythethanourgeneralalgorithm(Table2)withoutrecursiveRQM4pseudo-spectralmethodcoupledwiththeAndersonmixingsymmetryreduction.Forexample,aspeed-upofabout23schemeandvariablecellmethod.RedandgreensymbolsindicatethetimesisobtainedforgroupsFddd,Fm3?m,andFd3?m,byresultsobtainedbythecrystallographicFFTandthenormalFFT,makinguseofthebody-centeredsymmetryalongwiththerespectively.symmetryplanes.moreconvenientforonetoselectthenonequivalentdatapointswithevenindicesinthegeneralalgorithm(eq18)suchthatthenumberofselecteddatapointsisone-eighthofthetotalnumberofdatapointsinthefullunitcell.■CONCLUSIONSInsummary,wehavedemonstratedthatthee?ciencyofthepseudo-spectralmethodcanbesigni?cantlyenhancedbyreplacingthenormalFFTwithacrystallographicFFTthatmakesuseofthespatialsymmetryoftheconsideredperiodicstructures.Speci?cally,wehaveincludedageneralalgorithmofcrystallographicFFT,whichmakesuseoftheplanesymmetriescommonlycontainedbymanyspacegroups,suchasFm3?m,Im3?m,Ia3?d,Pn3?m,P63/mmc,Pm3?n,Fd3?m,Fddd,P6/mmm,andP42/mnm,intothepseudo-spectralmethod.Asaresult,thepseudo-spectralmethodwiththiscrystallographicFFTcanItisessentialtomentionthatcrystallographicFFTcanbebeappliedtosolvetheSCFTequationsformostof3DcombinedwithmostoftheothernumericalalgorithmsthatstructuresobservedinAB-typeblockcopolymers,includingtheclassicalBCC,FCC,HCP,G,D,O70,andPLphasesasweredevelopedtoimprovetheaccuracyortoacceleratetheconvergenceoftheiterationprocess,suchasthehigher-orderwellasthecomplexFrank?Kasperphases(σ,A15,C14,C15,pseudo-spectralalgorithm,31,32,35Andersonmixing,29andandZ).Ourresultsindicatethataspeed-upofaboutsixtimesvariablecellmethod.34InFigure4,wecomparetheiterationisachievedwiththisgeneralalgorithm.InadditiontoprocessesofsolvingSCFTequationswithRQM4,32whichacceleration,thepseudo-spectralmethodwiththisgeneraladoptsthenormalFFTandcrystallographicFFT,respectively,crystallographicFFTsavesthememoryusedbyeighttimes.ItfortheA15phaseformedbyAB3miktoarmstarcopolymeratfisworthmentioningthattheadditionale?ortforcoding=0.31andχN=40.Withinbothiterationprocesses,therequiredbythealgorithmislittle,butthespeed-upisAndersonmixingschemeisappliedforacceleration,andtheconsiderable.variablecellmethodisusedtooptimizethesizesoftheInaddition,wehavealsodevelopedaspeci?calgorithmofcomputationalbox.ThetwoiterationprocessesareexactlythecrystallographicFFTbyfullyconsideringthesymmetriesofthesameuntiltheAndersonmixingschemeisenabledatthe100thP42/mnmspacegroupofthecomplexFrank?Kasperσphasestep,whiletheybecomeslightlydi?erentsincethespatialtoacceleratethepseudo-spectralmethod.TomakefulluseofpointsofthecrystallographicFFTadoptedbytheAndersonthespacesymmetry,wereconstructthecomputationalboxbymixingalgorithmarelessthanthoseofthenormalFFT.rotatingthetetragonalsectionoftheunitcellby90°andSinceallnonequivalentpointsarepurposelychosentobelengtheningthetetragonalsidebyafactorof√2.Byincludingcontinuouslylocatedwithinanasymmetricunitinthespeci?cthisspeci?calgorithmofFFT,about30×speed-upisachievedalgorithm,theideaismorestraightforwardthanthatintheafterexcludingtheincreasedcomputationbytheexpandedgeneralalgorithm.However,formanyorderedstructures,itiscomputationalbox.Thisexampleshowsthatalargespeed-upnotpracticallyconvenienttouseanasymmetricunitdirectly.canbeachievedwiththepseudo-spectralmethodforagivenInstead,wejustneedtoallowallofthenonequivalentdatacomplexorderedstructurebyspeci?callydevelopinganpointstobedispersedintheentireunitcell.Forexample,itisalgorithmofcrystallographicFFT.9950https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943?9952

8Macromoleculespubs.acs.org/MacromoleculesArticle■(13)Lee,S.;Bluemle,M.J.;Bates,F.S.DiscoveryofaFrank-KasperASSOCIATEDCONTENTσPhaseinSphere-FormingBlockCopolymerMelts.Science2010,*s?SupportingInformation330,349?353.TheSupportingInformationisavailablefreeofchargeat(14)Xie,N.;Li,W.;Qiu,F.;Shi,A.-C.σPhaseFormedinhttps://pubs.acs.org/doi/10.1021/acs.macromol.0c01974.ConformationallyAsymmetricAB-TypeBlockCopolymers.ACSSCFTPScrys?tSamplecode(ZIP)MacroLett.2014,3,906?910.(15)Shi,W.;Lynd,N.A.;Montarnal,D.;Luo,Y.;Fredrickson,G.H.;Kramer,E.J.;Ntaras,C.;Avgeropoulos,A.;Hexemer,A.Toward■AUTHORINFORMATIONStrongThermoplasticElastomerswithAsymmetricMiktoarmBlockCorrespondingAuthorCopolymerArchitectures.Macromolecules2014,47,2037?2043.WeihuaLi?StateKeyLaboratoryofMolecularEngineeringof(16)Liu,M.;Qiang,Y.;Li,W.;Qiu,F.;Shi,A.-C.StabilizingthePolymers,KeyLaboratoryofComputationalPhysicalFrank-KasperPhasesviaBinaryBlendsofABDiblockCopolymers.Sciences,DepartmentofMacromolecularScience,FudanACSMacroLett.2016,5,1167?1171.University,Shanghai200433,China;orcid.org/0000-(17)Kim,K.;Schulze,M.W.;Arora,A.;Lewis,R.M.,III;Hillmyer,M.A.;Dorfman,K.D.;Bates,F.S.ThermalProcessingofDiblock0002-5133-0267;Email:weihuali@fudan.edu.cnCopolymerMeltsMimicsMetallurgy.Science2017,356,520?523.Author(18)Duan,C.;Zhao,M.;Qiang,Y.;Chen,L.;Li,W.;Qiu,F.;Shi,YichengQiang?StateKeyLaboratoryofMolecularA.-C.StabilityofTwo-DimensionalDodecagonalQuasicrystallineEngineeringofPolymers,KeyLaboratoryofComputationalPhaseofBlockCopolymers.Macromolecules2018,51,7713?7721.(19)Zhao,M.;Li,W.LavesPhasesFormedintheBinaryBlendofPhysicalSciences,DepartmentofMacromolecularScience,AB4MiktoarmStarCopolymerandA-Homopolymer.MacromoleculesFudanUniversity,Shanghai200433,China2019,52,1832?1842.Completecontactinformationisavailableat:(20)Cheong,G.K.;Bates,F.S.;Dorfman,K.D.SymmetryBreakinghttps://pubs.acs.org/10.1021/acs.macromol.0c01974inParticle-FormingDiblockPolymer/HomopolymerBlends.Proc.Natl.Acad.Sci.U.S.A.2020,117,16764?16769.Notes(21)Lindsay,A.P.;Lewis,R.M.;Lee,B.;Peterson,A.J.;Lodge,T.Theauthorsdeclarenocompeting?nancialinterest.P.;Bates,F.S.A15,σ,andaQuasicrystal:AccesstoComplexParticlePackingsviaBidisperseDiblockCopolymerBlends.ACSMacroLett.■2020,9,197?203.ACKNOWLEDGMENTS(22)Miyamori,Y.;Suzuki,J.;Takano,A.;Matsushita,Y.PeriodicThisworkwassupportedbytheNationalNaturalScienceandAperiodicTilingPatternsfromaTetrablockTerpolymerSystemFoundationofChina(NSFC)(GrantNo.21925301).oftheA1BA2CType.ACSMacroLett.2020,9,32?37.(23)Matsen,M.W.;Schick,M.StableandUnstablePhasesofa■REFERENCESDiblockCopolymerMelt.Phys.Rev.Lett.1994,72,2660?2663.(1)Helfand,E.TheoryofInhomogeneousPolymers:Fundamentals(24)Drolet,F.;Fredrickson,G.H.CombinatorialScreeningofoftheGaussianRandom-WalkModel.J.Chem.Phys.1975,62,999?ComplexBlockCopolymerAssemblywithSelf-ConsistentField1005.Theory.Phys.Rev.Lett.1999,83,4317?4320.(2)Shi,A.-C.DevelopmentinBlockCopolymerScienceand(25)Leibler,L.TheoryofMicrophaseSeparationinBlockTechnology;Wiley:NewYork,2004.Copolymers.Macromolecules1980,13,1602?1617.(3)Fredrickson,G.H.TheEquilibriumTheoryofInhomogeneous(26)Semenov,A.ContributiontotheTheoryofMicrophasePolymers;OxfordUniversityPress:Oxford,2006.LayeringinBlock-CopolymerMelts.Sov.Phys.JETP1985,88,1242?(4)Tyler,C.A.;Qin,J.;Bates,F.S.;Morse,D.C.SCFTStudyof1256.NonfrustratedABCTriblockCopolymerMelts.Macromolecules2007,(27)Guo,Z.;Zhang,G.;Qiu,F.;Zhang,H.;Yang,Y.;Shi,A.-C.40,4654?4668.DiscoveringOrderedPhasesofBlockCopolymers:NewResultsfrom(5)Matsen,M.W.EffectofArchitectureonthePhaseBehaviorofaGenericFourier-SpaceApproach.Phys.Rev.Lett.2008,101,AB-TypeBlockCopolymerMelts.Macromolecules2012,45,2161?No.028301.2165.(28)Rasmussen,K.?.;Kalosakas,G.ImprovedNumerical(6)Qiang,Y.;Li,W.;Shi,A.-C.StabilizingPhasesofBlockAlgorithmforExploringBlockCopolymerMesophases.J.Polym.CopolymerswithGiganticSpheresviaDesignedChainArchitectures.Sci.,PartB:Polym.Phys.2002,40,1777?1783.ACSMacroLett.2020,9,668?673.(29)Thompson,R.B.;Rasmussen,K?.;Lookman,T.Improved(7)Xie,Q.;Qiang,Y.;Chen,L.;Xia,Y.;Li,W.SynergisticEffectofConvergenceinBlockCopolymerSelf-ConsistentFieldTheorybyStretchedBridgingBlockandReleasedPackingFrustrationLeadstoAndersonMixing.J.Chem.Phys.2004,120,31?34.ExoticNanostructures.ACSMacroLett.2020,9,980?984.(30)Ceniceros,H.D.;Fredrickson,G.H.NumericalSolutionof(8)Matsen,M.W.;Bates,F.S.BlockCopolymerMicrostructuresinPolymerSelf-ConsistentFieldTheory.MultiscaleModel.Simul.2004,theIntermediate-SegregationRegime.J.Chem.Phys.1997,106,2,452?474.2436?2448.(31)Cochran,E.W.;Garcia-Cervera,C.J.;Fredrickson,G.H.(9)Matsen,M.W.TheStandardGaussianModelforBlockStabilityoftheGyroidPhaseinDiblockCopolymersatStrongCopolymerMelts.J.Phys.:Condens.Matter2002,14,R21?R47.Segregation.Macromolecules2006,39,2449?2451.(10)Drolet,F.;Fredrickson,G.H.OptimizingChainBridgingin(32)Ranjan,A.;Qin,J.;Morse,D.C.LinearResponseandStabilityComplexBlockCopolymers.Macromolecules2001,34,5317?5324.(11)Chen,L.;Qiang,Y.;Li,W.TuningArmArchitectureLeadstoofOrderedPhasesofBlockCopolymerMelts.Macromo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