Accelerated Pseudo-Spectral Method of Self-Consistent Field Theory via Crystallographic Fast Fourier Transform - Qiang, Li - 2020 - Unkn

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pubs.acs.org/MacromoleculesArticleAcceleratedPseudo-SpectralMethodofSelf-ConsistentFieldTheoryviaCrystallographicFastFourierTransformYichengQiangandWeihuaLi*CiteThis:Macromolecules2020,53,9943−9952ReadOnlineACCESSMetrics&MoreArticleRecommendations*sıSupportingInformationABSTRACT:Self-consistentfieldtheory(SCFT)hasbeenprovenasoneofthemostsuccessfulmethodsforstudyingthephasebehaviorofblockcopolymers.Inthepastdecades,anumberofnumericalmethodshavebeendevelopedforsolvingSCFTequations.Recently,thepseudo-spectralmethodbasedonfastFouriertransform(FFT)hasbecomeoneofthemostfrequentlyusedmethodsduetoitsversatilityandhighefficiency.However,thecomputationalcostisstillratherhigh,especiallyforsomecomplexstructuresorinthestrong-segregationcase.Toacceleratethecalculation,weintroducecrystallographicFFTintothepseudo-spectralmethod,whichutilizesthesymmetryoforderedphases.Thus,ageneralalgorithmisdevelopedbymakingpartialuseofsymmetryoperationscommonlycontainedbymanydifferentspacegroups,leadingtoaspeed-upofaboutsixtimesformostofthethree-dimensionalorderedmorphologiesobservedinAB-typeblockcopolymers,includingBCC,FCC,HCP,G,D,O70,andPLphasesaswellascomplexFrank−Kasperphases(σ,A15,C14,C15,andZ).Inaddition,wedemonstratethatmoreefficientalgorithmscanbespecificallydesignedbyfullyconsideringsymmetryoperationsforsomecomplexstructures.Forinstance,averylargespeed-upofabout30timesisachievedwithaspecificalgorithmforthecomplexFrank−KasperσphasewiththeP42/mnmspacegroup.Besidesacceleration,thememoryusedbythepseudo-spectralmethodwithcrystallographicFFTisconcomitantlysavedbymanytimes.■INTRODUCTIONbyexpandingthefreeenergyfunctionalaroundthe25Sinceself-consistentfieldtheory(SCFT)basedonthehomogeneousstate.TheexpansionwithinlimitedtermsGaussian-chainmodelwasexplicitlyadaptedtotreatmakesthismethodvalidonlyintheweak-segregationregime.polymer−polymerinterfacesandmicrodomainstructureofAfewyearslater,anotherusefulanalyticalmethodwas1introducedbySemenovforthestrong-segregationcase.26blockcopolymersbyHelfandin1975,ithasbeenwidelyusedforthestudyofmanyphysicalproblemsofinhomogeneousTherewasnoanalyticalmethodsuitablefortheintermediate-polymericsystems.2,3Inparticular,SCFThasbecomeasegregationregime.Accordingly,manyeffortshavebeenDownloadedviaUNIVOFCAPETOWNonMay14,2021at15:13:26(UTC).standardtoolforstudyingthephasebehaviorofblockdevotedtodevelopingnumericalmethodsforsolvingSCFT23,24,27−36copolymersduetoafewadvantages.Firstofall,itcanequations.Ascomputersadvancerapidly,thefast-calculatethefreeenergyofeachorderedstructureandthusgrowingefficiencyofnumericalmethodsleadstothe1identifytheequilibriumstructureforagivenblockcopolymer.broadeningoftheirapplicationsaswellastheapplicationsofSeehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.Second,SCFTcanreadilydealwithallkindsofarchitecturesSCFTsignificantly.Withhighlyefficientnumericalmethods,4−7ofblockcopolymers.Third,sufficientinformationcanbeSCFTcannotonlyrationalizetheexperimentalresultsbutalsoachievedfromSCFTforanalyzingtheself-assemblymecha-promoteexperimentalresearchbypredictingnewre-nismofblockcopolymers,suchasdifferentcontributionstosults.5,14,18,37−39thefreeenergyandthespatialdistributionofeachTheearliestattemptstoobtainnumericalsolutionsofSCFT8−11segment.Ofcourse,themostimportantadvantageisthefordiblockcopolymersweremadebyHelfandandWasser-reliabilityofSCFTconfirmedbythegoodagreementbetweenman.40In1992,Shulldevelopedanapproximatenumericalitsresultsandexperimentalresults,especially,excludingthetechniqueforone-dimensionalbulkandthin-filmsystemsofinaccuracyoftheinputparametersmeasuredinexperimentsdiblockcopolymermelts,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,usinganotherefficiencyofthepseudo-spectralmethodstillneedstobeapproximatenumericaltechniquethatignoresthedetailedenhanced,especiallywhenthecalculationneedstobe42structuraleffects.Lateron,anaccuratenumericaltechniqueperformedwithlargeMandNsforhighaccuracy.Usually,wasdevelopedbyMatsenandSchick,whichwasreferredtoaslargerMandNsareneededforthepseudo-spectralmethodtothespectralmethodorreciprocalspacemethodasitexpandsmaintaintheaccuracyofthefreeenergyofagivenstructureat23allofthespatialfunctionsintermsofasetofbasisfunctions.strongersegregationduetosharperinterfacesandlargerThemostingeniousideaofthespectralmethodconsistsindomainperiods.Iftheperiodsofthecalculatedstructuresarereconstructingthebasisfunctionsbygroupingallplanewavesverylarge,thecomputationalcostcanstillbehigh.Forofequalmagnitudestogetheraccordingtothegroupsymmetryexample,Cochranetal.stillusedlargeMandNstodetermineofagivenorderedphase,whichreducesthenumberofbasisthestableregionofdoublegyroidsofadiblockcopolymerfunctionsdramaticallyandthusenhancesthecomputationalaccuratelythoughtheydevelopedthefourth-orderpseudo-efficiency.Thishighlyefficientmethodcancalculatethefreespectralmethod.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−Kasperphasesindiblockcopolymermeltsforthefirsttimeandquantitativelyvariousblockcopolymersystemsthatarepurposelydesigned23analyzedtheirrelativestability.Althoughpriorknowledgeofaccordingtotheirformationmechanism.6,19,50,51Evenso,thesymmetryisrequiredandthusrestrictsthespectralmethodtocomputationalcostofthecalculationsoftheseFrank−Kasperthecalculationofknownstructures,thespectralmethodcanbephasesisstillhighbecausemostoftheFrank−Kasperphasesgeneralizedforexploringunknownstructuresatthecostof27havecomplexstructuralunitsandveryclosefreeenergy.Inefficiency.Itisnecessarytostressthatthecomputationalcost3otherwords,largeMandNsmustbeutilizedtocalculatetheofthespectralmethodisproportionaltoM,withMbeingthe43freeenergyofeachFrank−Kasperphaseatsuchahighnumberofbasisfunctions.Asaresult,theefficiencyoftheaccuracythattherelativestabilitybetweendifferentFrank−spectralmethodwoulddecreaserapidlyasMincreases,e.g.,forKasperphaseswithsmallfreeenergydifferencescanbethestrong-segregationcaseorlow-symmetrystructures.distinguished.Forinstance,M=128×128×64andNs=100Complementarytothereciprocalspacemethod,anwereusedfortheFrank−KasperσphaseformedinABdiblockalternativewayistosolveSCFTequationsusingthefinite14orABnmiktoarmstarcopolymermelts,whileM=256×256differencemethodinrealspace.Inparticular,Droletand×128andNs=450wereusedinourveryrecentworktodealFredricksondevelopedaniterativealgorithmforsolvingSCFTwiththemultiblockarchitectureandconcaveinterfacesoftheequationsinrealspace,inwhichonecriticalstepistosolvethe6extremelyenlargedsphericaldomains.modifieddiffusionequationsforthepropagatorfunctionsusing10,24Itiswell-knownthatthemosttime-consumingpartoftheanalternatingdirectionimplicitscheme.Theyproposed45pseudo-spectralmethodisFFT.Generally,FFTcalculatesthatthisrealspacemethodcouldbeusedtodiscovernewdiscreteFouriertransform(DFT)efficientlybydecomposingastructuresstartingfromrandominitialconditions.ThelargeDFTintoaseriesofsmallpiecesofDFTsviaCooley−efficiencyoftherealspacemethodislimitedforcomplexTurkeyfactorizationbutdoesnotmakeuseofthespatialblockcopolymersystemsduetotheroughfreeenergy10symmetryofthedatapointsinrealspace.Infact,somedatalandscapeswithmanycompetingmetastablelocalminima.pointsintheunitcelloforderedstructuresareequivalentdueIn2002,anewnumericalmethod,referredtoasthepseudo-tospacesymmetry,andthusthecalculationsofsomespectralmethod,wasdevelopedbyTzeremesandco-workers.44Thepseudo-spectralmethodtakesadvantageofdecomposedpiecesofDFTscanbeskipped,leadingtoaboththereciprocalandrealspacemethods.28Specifically,itmoreefficientalgorithmthatisreferredtoascrystallographicsolvesthemodifieddiffusionequationsbyperformingforwardFFT.CrystallographicFFTwasproposedbyTenEyckfora52andbackwardfastFouriertransforms(FFTs)foreachstepofsubsetofcrystallographicsymmetries,andwasfurtherintegrationwhilecomputingtheotherspatialfunctionsaswelldevelopedforall230spacegroupsinaseriesofsubsequent53−58asperformingtheiterationinrealspace.Accordingly,theworks.Inthispaper,wewillincludeacrystallographiccomputationalcostofthismethodscalesasNsMlogM,whereFFTintothepseudo-spectralmethodtoenhanceitsefficiency,NsandMrepresentthenumberofpointsdividingthechainfocusingonitsapplicationstothecommon3DorderedcontourandthenumberofgridpointsdiscretizingthestructuresaswellascomplexFrank−Kaspersphericalphasescomputationalbox.45DuetotheuseofFFT,thepseudo-formedbyAB-typeblockcopolymers.Inprinciple,sincespectralmethodisratherefficient.However,thepseudo-differentorderedstructureshavedifferentspacegroups,eachspectralmethod,especiallythesecond-orderalgorithm,haslessofthemneedsaspecificallydesignedcrystallographicFFTaccuracythanthespectralmethodbecauseofthediscretealgorithmtoutilizeallofitssymmetryoperations.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/ΔswithdifferentspatialgridsizesMfortheAB3copolymeratf=0.31andχN=40.beobtainedbydevelopingmorespecificalgorithmsforsomeq(,0)1r=Acomplexorderedstructures,e.g.,theσphase.††nqfq(,)rr=[(,0)]■ABSCFTANDPSEUDOSPECTRALMETHOD†AsastandardtooltostudytheequilibriumphasebehaviorofqfB(,(1r−=)/)1nblockcopolymers,SCFTcanbeeasilyappliedtoblock†−n1qq(,0)rr=[](,)(,0)fqrcopolymerswithdifferentarchitectures.Here,theflexibleABnBAB(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=+χϕB()ξ()rF1=−lnQ+∫d(rr{χϕϕNwAB)(rr)(−A)ϕA(r)wNB()rr=+χϕA()ξ()rnkTpBV1f†−−wB()()rrrϕξϕϕBA()1[−()r−B()r]}ϕA()rr=Q∫qsAA(,)(,)dqssr0(1)n1/−fnwhereϕA(r)andϕB(r)arethespatialdistributionsofvolumeϕ()rr=∫qs(,)(,)dqss†rBQBBfractionsofAandBsegments,andwA(r)andwB(r)aretheir0(6)conjugatemeanfields.ξ(r)istheLagrangemultipliertoForotherblockcopolymermeltingsystems,theSCFTenforcetheincompressibilityconditionsequationsareslightlydifferent,whilethemodifieddiffusionϕϕ()rr+=()1(2)equationsaresimilarforeachblock.Inthesecond-orderABalgorithmofthepseudo-spectralmethod,themodifiedIneq1,scalarQisthepartitionfunctionofasinglecopolymerdiffusionequationsareintegratedalongthechaincontour28chaininteractingwiththemeanfieldswA(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,theefficiencyofthepseudo-spectralmodifieddiffusionequationsmethodmainlydependsontheefficiencyofFFT.AsthetimecostbyonenormalFFTisproportionaltoMlogM,thetotal∂qsH(,)rr=−̃qs(,)timecostbythewholeintegrationscalesasNsMlogM,where∂sKKKNindicatesthenumberofpointsusedtodiscretizethechainscontour.Inthepseudo-spectralmethod,theaccuracyoffree∂††−qsH(,)rr=−̃qs(,)energyissensitivetoMandΔs∼1/Ns.Usually,largeMand∂sKKKNsareneededtoobtainhighaccuracyoffreeenergy.InFigureHw̃=−∇+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)andrecursivesymmetryreductionofdatawiththediagonalreflectionplane(b).Solidblacklinesrepresentreflectionplanes,anddoublearrowsindicatethattwosubmatricescanbeconvertedtoeachotherbycertainsymmetryoperation.fourth-orderalgorithmbyRanjan,Qin,andMorse(RQM4)definesanactionS#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,confirmingthat33Γ=Z/ZAistheunitcell,andF(h)istheDFTofthebothlargeMand1/Δs(orNs)areneededforthecalculationoriginalinput.ThetwiddlefactoreA(h,x)isdefinedasoffreeenergywithhighaccuracy.Forexample,todistinguishtherelativestabilitybetweentheA15andσphaseswithafree−1eA(,)exp(2hx=−×πihAx)(12)energydifferenceofaround10−4kTperchain,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,yieldingmoreefficientandmemory-savingFFTroutines.ComparedtothenormalFFTthatusesdataThenthereciprocalcoordinatescanbedecomposedaswellpointsfromthefullunitcellasinput,thecrystallographicFFTTrequiresonlythenonequivalentpoints(e.g.,pointsfromanhAhh=+101(15)asymmetricunit),suchthatthepseudo-spectralmethodcouldTheDFToftheoriginalinputcanbeexpressedasbesignificantlyacceleratedbyreplacingnormalFFTwiththecrystallographicone.Followingthenotationsinaseriesofloo|ooooworkbyKudlickietal.,53−58thecrystallographicFFTisFe()hh=+∑∑mooAA01(,)10xfe(x0A01x)(,)AA10h11x}ooe(,)h00xooanchoredtotwousefulpropertiesofDFT.First,supposethatx00∈Γnx11∈Γ~(16)anactionS,definedbyindicatingthattheDFTofalargematrixcanbedecomposedS()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.SymmetryPlanesfor2x2y2zAlgorithminDifferentSpaceGroupsaspacegroupphasessymmetryplanesFdddO70111111dx(,,0),,0ydx(,0,),0,zdy(0,,)0,,z444444P42/mnmσmx,y,0mx,x̅,zmx,x,zP6/mmmZmx,y,0m2x,x,zm0,y,zP63/mmcHCP,PL,C14mx,y,1m2x,x,zc0,y,z4Pm3̅nA15mx,y,0mx,0,zm0,y,zPn3̅mDnx(,,0),,11y1nx(,0,),,111zny(0,,),,111z224224224Fm3̅mFCCmx,y,0mx,0,zm0,y,zFd3̅mC15dx(,,0),,13y1dx(,0,),,311zdy(0,,),,131z444444444Im3̅mBCC,Pmx,y,0mx,0,zm0,y,zIa3̅dGax,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,whichasksfortheleastefforttoimplement.j101123(18)Althoughthespacegroupsofdifferentperiodicmorphologiesinblockcopolymersystemsusuallydiffer,eachofthesewherenmlisthebinaryexpressionoftheintegerjandeiistheunitvectorsalongai(i=1,2,and3).Onlyoneoffj(x1)isrequiresaspecificallydesignedalgorithmtoachieveafullindependentsincealloftheothersevensubmatricesarerelatedsymmetryreduction;mostofthe3Dmorphologiesobservedintoitbythesymmetryplanes.Forexample,areflectionplaneblockcopolymersystemssharesomesimilarsymmetryperpendiculartoe1indicatesthatsubmatrixf4(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)anddefineG={G(h)}7.Fromeq16wehaveperpendiculartoeachother,includingreflectionplanesandj1j=0glideplanes.Forsimplicity,wechosethecomputationalboxF=tcG(19)withthreeprimitivetranslationvectorsai(i=1,2,and3)perpendiculartothesymmetryplanes,respectively,andtheHereTcisa8×8matrixrepresentingthetwiddlefactorsfromoriginattheinversioncenter.NotethatthecomputationalCooley−TurkeyfactorizationboxeschosenherearedifferentfromthoseusuallyusedfortheT7spacegroupsP4/mnm,P6/mmm,orP6/mmc(Figure3).Tc1={enA((hAeeee+1123123*+m*+l*),n′+′+′}meel)jk,=023Suppose(20)ÄÅÅÉÑÑÄÅÅÉÑÑwheren′′ml′isthebinaryexpressionoftheintegerk.ÅÅ200NÑÑÅÅ200ÑÑMoreover,fromeq10ÅÅÑÑÅÅÑÑAA==ÅÅÅÅ020MÑÑÑÑand0ÅÅÅÅ020ÑÑÑÑÅÅÑÑÅÅÑÑGeG()hh=(,)(tRTh)(21)ÅÅÇ002PÑÑÖÅÇ002ÑÖ(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)Pm3̅n,Pn3̅m(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,theFFTofsizeAisreplacedbyFFTsof222Im3̅mt(1,1,1),(26),(27),(28)6430.2813sizeA1,whichisone-eighthoftheoriginalsize,leadingtoa222muchfasteralgorithm.InmanySCFTinstances,afullrecoveryIa3̅dt(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),6430.1523modifieddiffusionequationsaretypicallysolvedbyrepeatinga2222(7),(8)forwardFFT→elementwisemultiplication→backwardFFTFm3̅m,Fd3̅mt(0,11116430.1523triplet.Supposethattheresultofthetripletisfnew(x),and,),t(,0,),(26),2222Fnew(h)istheDFToffnew(x),sincefnew(x)hasthesame(27),(28)c3symmetryasf(x),fromeq22,wehavePm3̅n(26),(27),(28)640.854.1d3Pm3̅n(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.ThefirstelementofG′isobtaineddirectlyfromaoperationnumber(2),(3),and(21)intheInternational59normalFFT,whilealloftheotherscanbeobtainedbyTablesforCrystallography.SincetherotationalpartRiisrearrangingthematrixelementsaccordingtotheconsidereddifferentfromthoseofthesymmetryplanes,thememorysymmetryoperations(Figure2).Furthermore,onlythefirstaccesspatternisdifferentfromthoseofthespacegroupslistedrowofQisneededtocalculatethefirstelementofG′new,i.e.,inTable1,probablyinfluencingtheperformanceoftheGnew(h).Thenfnew(x)canbeobtainedbyconductingaalgorithm.0101normalbackwardFFTofGnew(h),andfnew(x)canbeTomakethealgorithmsuitableforasmanyordered01recoveredbymakinguseofthesymmetryoperationsinrealmorphologiescommonlyformedinblockcopolymersasspace.Byimplementingthealgorithmdescribedbyeq25,apossible,theimplementationdescribedabovebecomesalittlesignificantspeed-upofaboutsixtimesisobtainedforgroupcomplicated.Inotherwords,thealgorithmcanbesimplified,Pm3̅nona643grid(Table2).Besidestheacceleration,theespeciallywithrespecttocoding,ifitisappliedtothecrystallographicFFTalsoallowsustoreplacethefieldsandsupergroupsofPmmm,suchasP42/mnm,P6/mmm,Pm3̅n,propagatorsinthepseudo-spectralmethodwiththeirFm3̅m,andIm3̅m.Comparedwiththeglideplanes,thesubmatrices.Asthesubmatrixfnew(x)alreadycontainsallofoperationofeachreflectionplaneinthesespacegroupsalters01thenonequivalentelementsoffnew(x),itisnotnecessarytothecoordinatesofthesamplepointsinonlyonedirection.recoverfnew(x)duringthewholeSCFTcalculations,suchthatConsideraone-dimensionalarrayf(x),x=0,1,...,2N−1thememorycostcanbecutdowntoonlyaboutone-eighthofwiththereflectionplaneinthemiddle,i.e.,satisfyingthenormalone.Itisnecessarytoemphasizethatthef()xfN=−(21−x)(26)operationsofsymmetryplanescontainedindifferentspacegroupslistedinTable1havedifferenttranslationalparts,butApartfromCooley−Turkeyfactorization,itsDFTcanalsobethesetranslationalpartsaffectonlythevalueofpre-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)ofthefirsthalfoff(x).Accordingly,a2N-pointDFTcanbereplaced=−fyN(,21−−xM,21−z)byanN-pointDCT,thusreducingthecomputationalcost.=−fN(21−−xN,21−yz,)(29)Similartoeq25,thefullrecoveryoftheDFTresultisunnecessarywhensolvingdiffusionequations.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,moreefficientalgorithmscanbedesignedbytheDCTIIofsizeofA1canbereplacedbyAlgorithm1,whereconsideringmoresymmetryoperationsforaspecificspaceonlysamplepointswithintheasymmetricunit[0,2N)×[0,group.IthasbeenshownthatafullsymmetryreductionisN)×[0,M)willbeaccessed.TheoutputofAlgorithm1ispossibleforall230spacegroups,meaningthatallsymmetrysplitintoseveralmatricesforthereasonthattheasymmetricoperationscouldbeusedtoaccelerateFFT.57However,itisaunitinthereciprocalspaceisnotacuboid.However,thetotalsizeofallofthesematricesisexactly2MN2,whichisthesamenontrivialworktorealizethefullyacceleratedalgorithmforthesespacegroupsownedbyorderedmorphologiesinblockasthesizeoftheasymmetricunitinrealspace.Itisworthcopolymers.Oneofthemostimportantreasonsisthatusually,noticingthatallofthesubroutinesandproceduresinrecursivesymmetryreductionisrequiredtotacklesomeAlgorithm1arereversible,indicatingthattheinversesymmetryoperations,increasingthecomplexityofthetransformcanbeimplementedaccordinglyaswell.Different57algorithm,especiallywhenmanycodingdetailslikeparalleliza-fromthealgorithmbyKudlickietal.,norecursivesymmetrytionandvectorizationmustbeconsidered.Anotherreasonisreductionisusedhere,andthustheimplementationisexpectedtobemucheasier.Sinceallofthesymmetrythatthesesymmetryreductionmethodsareoriginallydesignedoperationsareused,theperformanceofouralgorithmisforcomplexdata,andthusanadditionalHermitiansymmetry57expectedtobecompetitivewiththatintheliterature.Weusemustbetakenintoaccountwhenmigratedtorealdata.ToourownversionofDCTstoimplementthisalgorithm,leadingfullyshowthepotentialofcrystallographicFFT,wemanagetotoanastonishingspeed-upofabout72timesfora256×256×designanotherefficientcrystallographicFFTalgorithmforthe128grid,comparedtothenormalFFTinFFTW3.spacegroupP42/mnmoftheσphasethatisoneofthemostConsideringthatthevolumeofourcomputationalboxiscomplicatedperiodicmorphologiesinblockcopolymertwicethatofthenormalone,theeffectivespeed-upbecomessystems.about30times.ItisnecessarytostressthatthespecificAsmentionedbefore,thecomputationalboxischosen,suchalgorithmnotonlyleadstoamuchhigherspeed-upbutalsothateachofitsprimitivetranslationvectorsisnormaltothesavesmuchmorememorythanthegeneralalgorithm.Asthereflectionplanes,respectively,whosevolumebecomestwicedataofallspatialfunctionsarereducedtothoseinthethatofthenormalunitcellofgroupP42/mnm(Figure3b).asymmetricunit,thememoryusageisreducedbyafactorofSupposethat32,oreffectivelyafactorof16afterexcludingtheeffectoftheÄÅÅ400NÉÑÑÄÅÅÉÑÑdoubledvolumeofthecomputationalbox.ÅÅÑÑÅÅ200NÑÑÅÅÑÑÅÅÑÑTodemonstratetheefficiencyofAlgorithm1,weapplyittoAA==ÅÅÅÅ040NÑÑÑÑand1ÅÅÅÅ020NÑÑÑÑaspecificexample,i.e.,theσphaseformedintheAB3ÅÅÑÑÅÅÑÑÅÅÇ004MÑÑÖÅÅÇ002MÑÑÖmiktoarmstarcopolymerwithf=0.31andχN=40.We(28)choosearatherlargegridof256×256×128andalargenumberofcontourstepswithastepsizeofΔs=0.002.HereOnthebasisofourpreviousdiscussion,3DDFToftheweimplementthecrystallographicFFTspecificallyforP42/originaldatawiththesizeofAcanbereplacedbyaDCTIImnmintotheRQM4pseudo-spectralmethod,andweusethewiththesizeofA1bymakinguseofthesymmetryoperationsAndersonmixingschemetoacceleratetheiterationprocessofofthereflectionplaneswhileleavingthe“42”screwaxiswithSCFTsolutiontowardconvergence.After856iterations,boththecenterofsymmetryunexploited.Representingf(x)byf(x,theincompressibilityconditionsandtherelativefielderrory,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.EvolutionofthefielderrorduringiterationofSCFTcrystallographicFFT.Forothermorphologies,similarsolutiontotheA15phaseformedbytheAB3copolymerwithf=0.31techniquescanbeappliedtoutilizemoresymmetryoperationsandχN=40,wherethemodifieddiffusionequationsaresolvedbythethanourgeneralalgorithm(Table2)withoutrecursiveRQM4pseudo-spectralmethodcoupledwiththeAndersonmixingsymmetryreduction.Forexample,aspeed-upofabout23schemeandvariablecellmethod.RedandgreensymbolsindicatethetimesisobtainedforgroupsFddd,Fm3̅m,andFd3̅m,byresultsobtainedbythecrystallographicFFTandthenormalFFT,makinguseofthebody-centeredsymmetryalongwiththerespectively.symmetryplanes.moreconvenientforonetoselectthenonequivalentdatapointswithevenindicesinthegeneralalgorithm(eq18)suchthatthenumberofselecteddatapointsisone-eighthofthetotalnumberofdatapointsinthefullunitcell.■CONCLUSIONSInsummary,wehavedemonstratedthattheefficiencyofthepseudo-spectralmethodcanbesignificantlyenhancedbyreplacingthenormalFFTwithacrystallographicFFTthatmakesuseofthespatialsymmetryoftheconsideredperiodicstructures.Specifically,wehaveincludedageneralalgorithmofcrystallographicFFT,whichmakesuseoftheplanesymmetriescommonlycontainedbymanyspacegroups,suchasFm3̅m,Im3̅m,Ia3̅d,Pn3̅m,P63/mmc,Pm3̅n,Fd3̅m,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.ItfortheA15phaseformedbyAB3miktoarmstarcopolymeratfisworthmentioningthattheadditionaleffortforcoding=0.31andχN=40.Withinbothiterationprocesses,therequiredbythealgorithmislittle,butthespeed-upisAndersonmixingschemeisappliedforacceleration,andtheconsiderable.variablecellmethodisusedtooptimizethesizesoftheInaddition,wehavealsodevelopedaspecificalgorithmofcomputationalbox.ThetwoiterationprocessesareexactlythecrystallographicFFTbyfullyconsideringthesymmetriesofthesameuntiltheAndersonmixingschemeisenabledatthe100thP42/mnmspacegroupofthecomplexFrank−Kasperσphasestep,whiletheybecomeslightlydifferentsincethespatialtoacceleratethepseudo-spectralmethod.TomakefulluseofpointsofthecrystallographicFFTadoptedbytheAndersonthespacesymmetry,wereconstructthecomputationalboxbymixingalgorithmarelessthanthoseofthenormalFFT.rotatingthetetragonalsectionoftheunitcellby90°andSinceallnonequivalentpointsarepurposelychosentobelengtheningthetetragonalsidebyafactorof√2.ByincludingcontinuouslylocatedwithinanasymmetricunitinthespecificthisspecificalgorithmofFFT,about30×speed-upisachievedalgorithm,theideaismorestraightforwardthanthatintheafterexcludingtheincreasedcomputationbytheexpandedgeneralalgorithm.However,formanyorderedstructures,itiscomputationalbox.Thisexampleshowsthatalargespeed-upnotpracticallyconvenienttouseanasymmetricunitdirectly.canbeachievedwiththepseudo-spectralmethodforagivenInstead,wejustneedtoallowallofthenonequivalentdatacomplexorderedstructurebyspecificallydevelopinganpointstobedispersedintheentireunitcell.Forexample,itisalgorithmofcrystallographicFFT.9950https://dx.doi.org/10.1021/acs.macromol.0c01974Macromolecules2020,53,9943−9952

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