Geminate Delayed Fluorescence by Anisotropic Di ff usion-Mediated Reversible Singlet Fission and Triplet Fusion - Seki et al. - 2021 - U

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pubs.acs.org/JPCCArticleGeminateDelayedFluorescencebyAnisotropicDiffusion-MediatedReversibleSingletFissionandTripletFusionKazuhikoSeki,*TomomiYoshida,TomoakiYago,MasanobuWakasa,andRyuziKatohCiteThis:J.Phys.Chem.C2021,125,3295−3304ReadOnlineACCESSMetrics&MoreArticleRecommendations*sıSupportingInformationABSTRACT:Singletfission(SF)occursinmolecularsolidsandisexpectedtoincreasetheconversionefficiencyofphotonsintoelectronsbytransferringexcitonsorchargesintolow-band-gapsemiconductors.ThefundamentalprocessesofSFanditsreverseprocessoftripletfusion(TF)wouldhavetobeelucidatedtocharacterizetheeffectofSFontheconversionefficiency.InsituationsinwhichTFoccurssubsequenttoSFwithinthesamepair,acertainfractionoftheexcitedsingletscouldberegeneratedbyTFandthismayevenhappenrepeatedly.Consequently,theexcitedsingletsexhibitdelayedfluorescencebygeminatefusionofthetriplets.Inthisstudy,weshowthattheexponentintheasymptoticpowerlawdecayofthegeminatedelayedfluorescencereflectsthediffusionalanisotropy.Inmolecularcrystals,thedifferenceintheratiobetweenthelargestandsmallestexcitondiffusionconstantscanbeordersofmagnitude.Bycomparingthetheoreticalresultsandthefluorescencedecayintetraceneandrubrene,weestimatedtherangeofanisotropyinthetripletdiffusionconstantsinthesematerials.Wealsoshowthattheexponentofthepowerlawdecaychangesasafunctionoftimeforcertainvaluesoftheanisotropyratiointhetripletdiffusionconstants.1.INTRODUCTIONthreedimensionsandβis1withaweaklogarithmiccorrectionSingletfission(SF)convertsasingletexcitonintotwotripletintwodimensions;theexponentreflectsthedimensionalityin9−12excitons,andtripletfusion(TF)convertstwotripletexcitonsthediffusionalmigrationoftriplets.Althoughthepreciseintoasingletexciton.Thisoccurswhentheenergyreactionschemesaredifferent,thesameexponentisobtainedconservationissatisfiedbetweentheenergyofthesingletforthekineticsofreversiblegeminaterecombinationby13−17excitonandthesumoftheenergiesofthetwotripletexcitonsdiffusion.Inbothprocesses,theexponentβreflectstheDownloadedviaBUTLERUNIVonMay16,2021at06:34:24(UTC).includingtheenergyshiftsresultingfromtheinteractionofthedifferenceinthetimedistributiontoregeneratethepairbyexcitonswiththeirsurroundings;thetransitionbetweenadimensionality-dependentdiffusionalmigrationafterdissocia-singletexcitonandanentangledstateofapairoftripletsistion.Inadditiontotheasymptoticpowerlawkinetics,1,2spin-allowed.SFoccursinmolecularsolidsandisexpectedunexpectedmagneticfieldeffectsareobtained;thetwotoincreasetheefficiencyofconvertingphotonsintoelectronsdecaycurvesatdifferentmagneticfieldstrengthscrosseachSeehttps://pubs.acs.org/sharingguidelinesforoptionsonhowtolegitimatelysharepublishedarticles.bytransferringexcitonsorchargesintolow-band-gapsemi-otherdependingonwhethereitherTForSFoccursconductors.1−6CharacterizationoftheeffectofSFonthe9,11effectively.Althoughdifferentinterpretationswerepro-conversionefficiencywouldrequirethefundamentalprocessesposed,themagnetic-field-inducedcrossingofthedecaycurvesofSFanditsreverseprocessofTFtobeelucidatedandthewasobservedexperimentally.181,2,7,8kineticparameterstobeobtained.Inthisstudy,weexaminedtheeffectsofdiffusionalIngeneral,thelifetimeoftripletstatesislongerthanthatofanisotropyongeminatedelayedfluorescence.Inliquidsandsingletstatesowingtoaspin-forbiddentransitiontothesingletamorphoussolids,themotionofreactantsisusuallyisotropic,groundstates.TheoccurrenceofTFsubsequenttoSFwithininwhichcasetheexponentβoftheasymptoticdecaycanbethesamepaircouldresultintheregenerationofacertaineasilyinterpretedusingthespatialdimension.However,thefractionoftheexcitedsingletsbyTF,andthiscouldevenoccurrepeatedly.Asaresult,theexcitedsingletsexhibitdelayedfluorescencebythegeminatefusionoftriplets.InthisReceived:November25,2020manuscript,werefertothisprocessasgeminatedelayedRevised:January25,2021fluorescence.TheSFandTFrateconstantscanbeestimatedPublished:February5,2021fromkineticsstudiesonthegeminatedelayedfluorescence.Recently,thegeminatedelayedfluorescencehasbeenshowntoundergopowerlawdecayC/tβ,whereβis3/2inoneand©2021AmericanChemicalSocietyhttps://dx.doi.org/10.1021/acs.jpcc.0c105823295J.Phys.Chem.C2021,125,3295−3304

1TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleexistenceofdiffusionalanisotropyinmolecularcrystalsresultsprofilecouldbeobtained.Thetimeresolutionoftheintheinterpretationoftheexponentβbecomingnontrivial.Inspectrometerwasapproximately0.5ns.molecularcrystals,thedifferenceintheratiobetweentheThefluorescencedecayprofilesoftetracenecrystalswerelargestandsmallestexcitondiffusionconstantscanbeordersofrecordedatSaitamaUniversitywithacustom-builttime-magnitude.Forexample,theratioofthetripletdiffusionresolvedfluorescencespectrometer,whichisbasedonthe19photon-countingtechnique.Asamplespecimenwasexcitedconstantsinanthraceneis10−20.AsspecificexamplesthatexhibitSF,weselectedtostudyrubreneandtetracenewithalaserdiodeat470nm(Hamamatsu,PLP-10,M10306).20−22Therepetitionrateandpulsedurationwere20MHzand80crystals.Intetracene,theelectronicπ-orbitaloverlapisestimatedtobesmallinonedirectioncomparedwiththatinps,respectively.Thefluorescencesignalfromthesampletheotherdirections.Theoppositesituationprevailsinspecimenwasdetectedwithafastphotosensormoduleorthorhombicrubrene,wheretheπ-orbitaloverlapisestimated(Hamamatsu,H7422-20)afterbeingdispersedwithatobedominantinonedirectioncomparedwiththeothermonochromator.Thearrivaltimeofafluorescencephotondirections.Inbothmaterials,theratioofthetripletdiffusionwasmeasuredandanalyzedwithatime-correlatedsingle-constantscouldbecomparabletothatofanthraceneorevenphotoncountingmodule(Becker&HickelGmbH,SPC-larger.19−22Thewayinwhichthediffusionalanisotropy130EM).influencesthekineticsofgeminatedelayedfluorescence,inparticular,theβvalue,hasnotyetbeenclarified.3.THEORETICALFORMULATIONTheoretically,theratioofthetripletdiffusionconstantscanWedenotethedensityofexcitedsinglets(S1)attimetafterbetakenintoaccountusingthelatticeGreenfunction.InotherphotoexcitationbyS(t).Forsimplicity,weassumethattheSFwords,thetimedistributiontoregeneratethecorrelatedtripletandthereverseprocess[tripletfusion(TF)]occurbetweenpairbydiffusionalmigrationafterthedissociationcanbemoleculeslocatedattheoriginofthelattice,asshowninexpressedusingthelatticeGreenfunction.Thus,itisnecessaryFigure1a−c.ThepairoftripletsoccupyingtheorigintoaccuratelyevaluatethelatticeGreenfunctiontostudythelargeratiooftripletdiffusionconstants.AlthoughtheanalyticalformofthelatticeGreenfunctionoftheanisotropiccubiclatticeisknownandcanbeaccuratelyevaluated,thisformis23−25tedious;theanalyticalequationwasthereforeconfirmedbynumericallyevaluatingtheotherformsofthelatticeGreenfunction.WeconsideredthelatticeGreenfunctionoftheanisotropiccubiclatticetostudythecaseinwhichthediffusionconstantinonedirectionissmallerthanthoseintheotherdirections.Thissituationcorrespondstothetetracenecrystal.ThelatticeGreenfunctionoftheanisotropicsquarelatticeisalsoconsideredinthestudyofplaneanisotropy,asituationthatcorrespondstotherubrenecrystal.AcomparisonFigure1.(a)Kineticmodeldescribingthedissociationofanassociatedtripletpairtoaseparatetripletpairandregenerationofanofthetheoreticalresultsandthefluorescencedecayinassociatedtripletpairbydiffusion.S1andS0indicatetheexcitedtetraceneandrubreneenabledustoestimatetherangeofsingletstateandthesingletgroundstate,respectively.TPindicatesanisotropyintripletdiffusionconstantsinthesematerials.Wetheassociatedtripletpair.(b)Kineticmodeldescribinganassociatedalsoshowthattheβvaluesundergocrossoverforcertainvaluestripletpairandtheregenerationofanassociatedtripletpair.(c)oftheanisotropyratiointhetripletdiffusionconstantsandSchematicoftheanisotropictransitionrateconstants.Thehoppingexaminethecrossovertimethoroughly.transitionrateconstanttotheoneofthesitesonthex-ory-axisisgivenbyγ⊥/2,andthattotheoneofthesitesonthez-axisisgivenbyγ∥/2.Thedissociationrateconstantoftheassociatedtripletpairto2.EXPERIMENTALMETHOD(ζ)theoneofthenearestneighborsitesisdenotedbykdis.TheCommerciallyavailablepurifiedrubrene(Sigma-Aldrich,dissociationrateconstanttotheoneofthesitesonthex-ory-axisissublimedgrade99.99%)wasusedasreceived.Purifiedgivenbykdis(γ⊥/2)/(2γ⊥+γ∥),andthattotheoneofthesitesonthetetracenesinglecrystalswereobtainedfromcommerciallyz-axisisgivenbykdis(γ∥/2)/(2γ⊥+γ∥).availabletetracene(Sigma-Aldrich,98%)byaphysicalvaportransportmethodatatmosphericpressure.26representsanassociatedtripletpair.ThispairisdenotedasaThefluorescencedecayprofilesofrubrenecrystalswereseparatetripletpairwhenthetwotripletsoccupydifferentrecordedwithacustom-builttime-resolvedfluorescencelatticepoints.ThedensityoftheassociatedtripletpairsatthespectrometerbasedonthephotoncountingtechniqueatoriginisdenotedbyC(t).Thesimplestmodelthatincludes27−29NihonUniversity.AsamplespecimenplacedbetweentwospinselectivityistheJohnson−Merrifieldmodel,whichisquartzplateswasexcitedwithapulsedlaserat355nm(thirdextendedtotakeintoaccountthediffusioneffectanalytically.harmonicoftheoutputfromaNd3+:YAGlaser,TeemCorrelatedtripletpairstatescanberepresentedby9spinPhotonics,Powerchip).Therepetitionrateandpulsedurationstates.WemaywritetheHamiltonianofthetripletpairaswere1kHzand400ps,respectively.ThefluorescencesignalL=++HHH(1)(2)(1)+H(2),whereH(i)andZFSZFSZeemanZeemanZFSfromthesamplespecimenwasdetectedwithafastH(i)describethezerofieldsplittingandtheZeemanZeemanphotomultiplier(Hamamatsu,H7422P-50)afterbeingdis-splittingofi=1,2,respectively,andeachtripletinthepersedwithamonochromator.Thearrivaltimeofaassociatedtripletpairisdenotedby1and2.The9spinstatesfluorescencephotonwasmeasuredwithafastoscilloscopecanbeexpressedonthebasisofthezerofieldsplitting(LeCroy,HDO4022)andanalyzedwithacomputer.AfterHamiltonianoronthebasisoftheZeemanHamiltonian.Theaccumulationofthearrivaltimesignals,afluorescencedecayquantumspindynamicscanbestudiedusingtheLiouville3296https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

2TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleequationforthedensitymatrix∂∂=ρρ/1ti/(ℏ[),L],wherepairwhenbothtripletsoccupytheoriginagain,andwecan[,]denotesacommutator.Inthepresenceoftriplet−tripletexpresstheregenerationrateasfusiontothesingletexciton,theLiouvilleequationcanbet(,)jζ()()ζjexpressedas∂∂=ρρ/1ti/(ℏ[]−{}),LO,ρ,where{,}Vr()tt≡−∫d(,11Rrζtt)kdisC()t10(4)denotesananticommutator,OistheHabercornreactionoperatorgivenby(kTF/2)Pŝ,andP̂s=|S⟩⟨S|istheprojectionwhererζindicatesoneofthenearestneighborsoftheorigin.tothesingletspinstate.30GenerationoftheassociatedtripletTheregenerationkernelR(rζ,t)reflectsthemigrationofseparatetripletsbysuccessivehoppingtransitions.Weassumepairbysingletfissioncanbetakenintoaccountbythatanassociatedtripletpairisimmediatelyformedwhen∂∂=ρρ/1ti/(ℏ[]−{}+),LO,ρkSFS(t)Pŝ,wherekSFin-tripletsoccupytheorigin.Underthiscondition,thedicatesthesingletfissionrateconstantoftheexcitedsingletregenerationkernelR(rζ,t)isthefirstpassagetimedistributionexcitonintotheassociatedtripletpair.IntheJohnson−requiredtoregenerateanassociatedtripletpairattheoriginMerrifieldmodel,off-diagonalelementsofthedensitymatrixfromtheseparatetripletpairinitiallylocatedatrζ;rζistheareignored.Wedenotethediagonalelementofthedensity(j)initiallocationofaseparatetripletpairafterthedissociationofmatrixoftheassociatedtripletpairsbyC(t)withtheindexoftheassociatedtripletpair.Theright-handsideofeq4canbethespinstate(j)onthebasisofthezerofieldsplittingunderstoodbynotingthatthedissociationoccursatanytimeHamiltonianortheZeemanHamiltonian.C(t)isgivenbyC(t)9(j)afterthephotoexcitationuntiltimetwiththerategivenbythe=∑j=1C(t).Therateconstantoffissiontothejthspinstatedissociationrateconstantkdis(ζ)timesthedensityoftheoftheassociatedtripletpairisgivenbyk|C(j)|2,andtherateSFsassociatedtripletpairs.Theoveralldissociationrateconstantconstantoffusionfromthejthspinstateoftheassociated(ζ)(j)2(j)2isgivenbykdis=∑ζkdis.tripletpairisgivenbykTF|Cs|,where|Cs|indicatestheEquation4canbeevaluatedbynoticingthatR(rζ,t)canbeprojectionofthespinstatesoftheassociatedtripletpairtothe9(j)2expressedusingtheGreenfunctiong(r0,t),whichisthesingletstateandisnormalizedby∑j=1|Cs|=1.Thedensityprobabilityoffindingtheparticleoriginallylocatedatr0attheofseparatetripletpairswithspinstate(j)isdenotedbyf(j)(t),originduringtandt+dt.Thefirstpassagetimedistributionatandthetotaldensityofseparatetripletpairsisgivenbyf(t)=9(j)theoriginwhentheparticlestartsfromr0satisfiestherecurrent∑j=1f(t).relationgivenbyThekineticequationforthemodelshowninFigure1acanbeexpressedastg(,)r00tt=−∫d(,11gtt)(,)Rr0t10(5)9∂()2()jjSt()=−(kSF+kStrad)()+∑kCCtTF|s|()whereg(r0,t)representstheGreenfunction,whichisthe∂tj=1(1)probabilityoffindingtheparticleoriginallylocatedatr0atthe34originduringtandt+dt.TheLaplacetransformofg(r0,t)is34∂()jj()2()2jj()givenbyCtkCSt()=||−||+SFs()(kCTFskCtdis)()∂t1(,)jζg(,)̂r0s=3+∑Vr()t(2)πζ(2)π3cos(kr·0/)b∫∫···dk∂ftkCt()jjj()=−()()V(,)ζ()ts++−2(γγγ⊥⊥λ⊥kk)−γλ()dis∑r−π∂tζ(3)(6)wherebindicatesthelatticespacing,andwedefineλ⊥(k)wherekrad,kSF,kTF,andkdisindicatetheradiativerateconstant,∞singletfissionrateconstant,tripletfusionrateconstant,andthe=cos(kx)+cos(ky)andλ∥(k)=cos(kz),andĥ(s)=∫0dth(t)exp(−st)indicatestheLaplacetransformofh(t).Theoveralldissociationrateconstantofassociatedtripletpairs,hoppingtransitionrateconstantalongthex-ory-axisisgivenrespectively,andζistheindexforcoordinationaroundthebyγ⊥,andthatalongthez-axisisgivenbyγ∥.Equation5origin.Thesumextendsupto2dforthed-dimensionalrepresentsthatthecarrierexecutinghoppingtransitionsandhypercubiclattice,thatis,ζextendsto6forthecubiclatticearrivingattheoriginshouldhavebeenthereforthefirsttime(,)jζand4forthesquarelattice.Vr()tindicatestheregenerationattimet1equaltoorshorterthant.Usingtheintegralrateoftheassociatedtripletpairwithspinstatejfromtherepresentationgivenbyeq6and∫πdk∫πdk∫πdk/(2π)3=−πx−πy−πzseparatetripletpairwhoseinitialpositionisoneofthenearest1,wecanderivetheidentityneighborsoftheoriginindicatedbyrζ.Theassociatedtripletpairdissociatesintoaseparatetriplet1pair.Theeffectofdissociationoftheassociatedtripletpairinto(2sg++γγ⊥)(0,s)−3∫···(2)πtheseparatetripletpairhasalreadybeentakenintoaccountbyextendingtheoriginalJohnson−Merrifieldmodel.18,31−33πγλ()kk+γλ()Here,weconsidertheanisotropicdiffusionofseparatetriplets∫d3k⊥⊥byasequenceofanisotropichoppingtransitions.Fors++−2(γγγλkk)−γλ()⊥⊥⊥−πsimplicity,oneofthetripletexcitonsofthepairisassumedtobealwayslocatedattheorigin.Thekeyquantityisthe=1(7)regenerationrateoftheassociatedtripletpairwithspinstatejWeconsiderthecaseinwhichboththedissociation(,)jζfromtheseparatetripletpair,denotedbyVr()t.Theanisotropyandhoppinganisotropyarecharacterizedbyγ⊥andassociatedtripletpairisregeneratedfromtheseparatetripletγ∥;thedissociationrateconstantalongthex-ory-axisisgiven3297https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

3TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticlebykdisγ⊥/(2γ⊥+γ∥),andthatalongthez-axisisgivenbyGw(,)α=1kdisγ∥/(2γ⊥+γ∥),wherekdisindicatestheoveralldissociation(2)π3rateconstantsatisfyingtherelationgivenby∑k(ζ)=k[seeζdisdisπFigure1b,c].ByapplyingtheLaplacetransformation,we31obtainR̂(r,s)=ĝ(r,s)/ĝ(0,s),and∫∫···dk00wkk−−−cos()xyzcos()αcos()k−πk/(2γγ+)(16)()ζdis⊥∑Rs(,)rζkdis=[((,)gsgŝrrxy+(,))̂γ⊥TheanalyticalexpressionforG(α,w)isgivenby23−25gs(,)̂0ζ2+gs(,)̂rzγ](8)wG(,)w=2ijj2yzzKkKk()()αjz+−11−+−ββ22kπ{−+kdis/(2γγ⊥+)(17)=[−+1(2sg+γγ+)(,)0s]gs(,)̂0⊥whereβ±=(2±α)/w,(9)222−3wheretheidentitygivenbyeq7isusedforthelastequality.211k±≡−[−+−][+βββ−+111−+−βWealsointroducedrx=(b,0,0),ry=(0,b,0),andrz=(0,0,b):+−+]11ββ−+weusedĝ(r0,s)=ĝ(−r0,s)forthehyper-cubiclattice,andthetransitionrateconstantineachdirectionisgivenbyγ⊥/2or22{±(4/)ww+1(/)−αγ∥/2.Thelasttermineq3canthereforebeexpressedas2[+11ββ++−11ββ−]}−+−+(,)jζijjsgs−1/(,)0yzz()j(18)∑V̂r()sk=disjjjj+1(zzzzCŝ)2γγ+andK(k)denotesthecompleteellipticintegralofthefirstkindζk⊥{(10)12−1/2withmoduluskdefinedbyK(k)=∫0dt(1−t)(1−∞kt2)−1/2.35IntheSupportingInformation,wepresentthewhereĥ(s)=∫0dth(t)exp(−st)denotestheLaplacetransformofh(t).TheresultcanbecomparedtoalternativeformofG(α,w)thatisusedtoconfirmeq17.TheLaplacetransformofS(t)/S(0)canbeobtainedusinĝ(,)jζ=̂()ζ̂()jtheLaplacetransformofeq2expressedas∑∑Vr()sR(,)rζskdisC()sζζ(11)()2jj()2()jkCSsSF||s()̂=[+skCTF|s|−kRsCsdiŝ()]̂()(19)obtainedfromtheLaplacetransformofeq4.Equation10whereR̂(s)isgivenbyeq13.Bysubstitutingeq19intotheindicatesthatwemayintroducetheLaplacetransformoftheLaplacetransformofeq1,weobtaintheLaplacetransformofscalarregenerationkerneldenotedbyR̂(s)usingS(t)/S(0)expressedaŝ(,)jζ=̂̂()ĵ∑Vr()sR()skdisCs()Ss()=ζ(12)S(0)(ζ)ÄÅÅÉÑÑ−1wherekdis=∑ζkdisissubstitutedandR̂(s)isdefinedbyÅÅ9kkC||()4jÑÑÅÅSFTFsÑÑÅÅskk++−∑ÑÑÅÅradSFskC+||()2j+−k(1Rŝ())ÑÑ̂=sgs−1/(,)0ÅÅÇj=1TFsdisÑÑÖR()s+12γγ+(20)⊥(13)wheretheLaplacetransformofthescalarregenerationkernelTheeffectofanisotropichoppingtransitionsofseparategivenbyeq13canbeexpressedastripletpairscanbetakenintoaccountbycalculatingg(0,s),ÄÅÅÉÑÑsubstitutingtheresultintoeq12witheq13,andfinallyusinĝ()=1ÅÅÅÅ1ÑÑÑÑ1(,)jζRss2+ÅÅ̅−+Gw(,)ÑÑeq3.Here,theregenerationrate,∑Vr()t,isgivenintermsααÅÇÑÖ(21)ζofthethefirstpassagetimedistributionattheorigin,andthewheres̅=s/γ⊥andw=s̅+2+αinthreedimensions.TheinfluenceofexcitonmotionisgivenbyR̂(s).InR̂(s),thefirstpassagetimedistributionisobtainedusingtherecurrentLaplacetransformoftheasymptotictimedependencecanberelationgivenbyeq5.Analternativederivationisgiveninthestudiedfrom1/G(α,w)−s̅inR̂(s).SupportingInformation.Whenwestudythein-planeanisotropybyignoringtheTheanisotropyintransitionrateconstantscanbetransitionsinthez-direction,weintroduceaparametertocharacterizedbycharacterizethetwo-dimensionalanisotropydenotedbyα2d=α=γγ/⊥(14)γx/γy,whereγx=γ∥andγy=γ⊥indicatethehoppingrateconstantsalongthetwoaxesofthesquarelattice,andweThecaseinwhichα=0inthecubiclatticereducestotheconsiderthecaseα2d<1.Intwodimensions,eq21shouldbetwo-dimensionalsquarelattice.Fortheanisotropiccubicmodifiedtolattice,g(0,s)isknownandcanbeexpressedasÄÅÅÉÑÑ1ÅÅ1ÑÑR̂()ss=ÅÅ̃−+ÑÑ1g(,)0sGw=(,)/αγ⊥(15)1+αα22ddÅÅÅÅGw(,)2d2dÑÑÑÑ(22)ÇÖwherew=(s+2γ⊥+γ∥)/γ⊥andG(α,w)isgivenbywheres̃=s/γyandeq16isalsomodifiedto3298https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

4TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleπtimeobtainedforthecubiclattice,andα=γ/γgivenbyeq1∥⊥Gw222ddd(,)α=2∫dkx14isvaried.Here,α=1correspondstoanisotropiccubic(2)πlattice,wheretheasymptotictimedependenceisgivenby−π∼t−3/2.Further,α=0correspondstoanisotropicsquareπ1lattice,wheretheasymptotictimedependenceisapprox-∫dkyimatelygivenby∼1/t.Whenα=0.1,wefindthatthewk22ddx−−αcos()cos()ky−3/2−πasymptotictimedependencefollows∼t,whichischaracter-(23)isticofthethree-dimensionalrandomwalk.Whenα=10−5,wewithw2d=s̃+1+α2dintwodimensions.TheGreenfunctionfindthattheasymptotictimedependencefollows∼1/t,whichgivenbyeq23canbeexpressedas36ischaracteristicofthetwo-dimensionalrandomwalk.Later,inFigure4a,b,weshowthatthecharacteristicasymptoticdecay2owingtothetwo-dimensionalrandomwalkcanbefoundwhenGw222ddd(,)α=−3παw2−−(1)20<α<10.2dd2InFigure3,weshowthenormalizedexcitedsingletdensityijj4α2dyzzasafunctionoftimeobtainedforthecubiclatticewhenα=Kjjzzjw2−−(1α)2zk2dd2{(24)whereK(k)denotesthecompleteellipticintegralofthefirstkindwithmoduluskgivenbyK(k)=∫1dt(1−t2)−1/2(1−0kt2)−1/2.35Wesubstituteeq21witheq17intoeq20tostudytheanisotropyinthecubiclattice,whichischaracterizedbytheratiooftheout-of-planehoppingtransitionrateconstantagainstthein-planehoppingtransitionrateconstant[eq14].Wesubstituteeq22witheq24intoeq20tostudytheanisotropyinthesquarelattice,whichischaracterizedbyα2d=γx/γy.WeusetheStehfestalgorithmfortheinverseLaplace37,38transformation.4.THEORETICALRESULTS4.1.AnisotropyintheCubicLattice.InFigure2,weshowthenormalizedexcitedsingletdensityasafunctionoftheFigure3.Normalizedexcitedsingletdensityshownasafunctionofdimensionlesstime[kSFt].Theredcirclesindicateα=γ∥/γ⊥=0.005.Theparametervaluesarekrad/kSF=1,kTF/kdis=1,kdis/kSF=1,andk/γ=2+α.Thereddashedlinerepresents∼t−3/2[0.078/dis⊥(tk)3/2]timedependence,andtheblackdashed-dottedlineSFrepresents∼t−1[0.0073336/(tk)]timedependence.Thelong-SFtimeasymptoticdecayinisotropicsystemsisexpressedas∼t−β,whereβis3/2inthreedimensionsand1apartfromalogarithmicweaktime-dependenceintwodimensions.0.005,whichisbetweentheαvaluesindicatingthecharacteristicdecayinthreedimensionsandthatintwodimensions.Thedecayoftheexcitedsingletdensityinitiallyfollowsthe1/tlaw,characteristicofatwo-dimensionalrandomwalk,andfinallyfollowsthe∼t−3/2law,whichischaracteristicofathree-dimensionalrandomwalk.Thecrossoverbetweenthedifferentpowerlawdecayscouldbeobservedforlayeredmolecularcrystalswithweakinteractionsbetweenlayers.TheaboveresultsobtainedforanisotropiccubiclatticesFigure2.NormalizedexcitedsingletdensityshownasafunctionofcouldbestudiedindetailusingthelatticeGreenfunction.Indimensionlesstime[kSFt].Theredcurve,bluecrosses,andblackthreedimensions,theasymptotictimedependencecanbesquares,indicateα=γ/γ=1,α=0.1,α=10−5.Theparameter∥⊥studiedbyexpandingvaluesarekrad/kSF=1,kTF/kdis=1,kdis/kSF=1,andkdis/γ⊥=2+α.Thereddashedlinerepresents∼t−3/2[0.0383/(tk)3/2]timeFs3dim()1/(,)̅=−GwsGαα̅−+1/(,2α)(25)SFdependence,andtheblackdashed-dottedlinerepresents∼t−1inthelimitofs̅→0obtainedfromeq21.Whentheinverse[0.0142/(tkSF)]timedependence.Thelong-timeasymptoticdecay−3/2inisotropicsystemsisexpressedas∼t−β,whereβis3/2inthreeLaplacetransformofs̅−1/G(α,w)+1/G(α,2+α)hastdimensionsand1apartfromalogarithmicweaktime-dependenceinasymptotictimedependence,s̅−1/G(α,w)+1/G(α,2+α)twodimensions.shouldbeapproximatedbythes̅-dependenceaccordingto3299https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

5TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticle(,)jζFigure4.Themaintermof∑Vr()sshowingsdependenceisplottedasafunctionofs̅=s/γ⊥.Thethicksolidline,thinsolidline,dottedline,ζdashedline,circles,andcrosses[inpanel(b)]representα=γ/γ=1,10−1,10−2,10−3,10−4,and10−5,respectively.(a)F−1(s)=[1/̅G(α,w)−s]̅−1∥⊥2dimisplottedasafunctionofs.Theredsolidlineindicatestheresultof̅α=0.Thereddashedlineindicatesthesignatureoftwo-dimensionaldiffusiongivenbyΔF−1(s)̅≈[1/(2π)]ln(16/s)-dependence[̅eq28].(b)F(s)=1/̅G(α,s̅+2+α)−s̅−1/G(α,2+α)isplottedasafunctionofs.The̅2dim3dimreddotsindicatethesignatureofthree-dimensionaldiffusiongivenbys̅-dependence.Moreprecisely,wefindthes̅-dependenceusingΔFsAs()̅*=̅*tofindAbychoosingasmallvaluefors*̅thatcorrespondstotheasymptotictimedecay.Wechooses*̅=10−4forα>10−53dimands*̅=10−5forα=10−5.theTauberiantheorem.39Therefore,thet−3/2asymptotictimesquarelattice,whereasα=1correspondstothethree-dependenceinthegeminatedelayedfluorescencecorrespondsdimensionalisotropiccubiclattice.tos̅-dependenceintheexpansionofF3dim(s)when̅s̅→0.AsshowninFigure4a,F−1(s)exhibits1/(2̅π)ln(16/s)-̅2dimThelimitingcaseofα=0correspondstotheisotropicdependencewhen0<α<10−3intheintermediateregionofs.̅squarelattice;G(0,w=s̅+2)=G2d(α2d=1,w2d=s̅+2),Theresultssuggest1/ttimedependenceintheintermediatewhereG2d(α2d,w2d)isgivenbyeq24.Thetwo-dimensionaltimeregimewhen0<α<10−3.Interestingly,the1/(2π)ln(16/characteristictimedependenceinthelatticeGreenfunctions)-dependenceisabsentif10̅−3≤α.canbestudiedbythefactorinthesquarelatticegivenbyeq22expressedasInFigure4b,weshowF3dim(s)forvariousvaluesof̅α.F3dim(s)exhibits̅s̅-dependenceass̅approacheszeroforΔFs2dim()1/(,)̅=−Gwsα̅(26)three-dimensionalcubiclatticesystems.AsshowninFigure4b,Thecaseinwhichα=0intheanisotropiccubiclatticeisthetheasymptotics̅-dependenceofF3dim(s)existsaslongas̅isotropicsquarelattice.Byintroducingtheapproximationα>0,althoughtheasymptoticcomponentappearsatsmaller34givenbyvaluesofs̅asαdecreases.Theasymptotics̅-dependencecanÄÉÄÉbefoundevenwhenαisassmallas10−5.ÅÅ2ÑÑÅÅ−1ÑÑÅÅÅÅijj21yzzÑÑÑÑÅÅÅÅijj2yzzÑÑÑÑBasedontheresultsinFigure4a,b,wecanconcludethatKÅÅjjzzÑÑ≈−ln81ÅÅjjzzÑÑÅÅjww2ddzÑÑ2ÅÅj2zÑÑintermediate1/tdependence,whichischaracteristicoftheÅÅÇk{ÑÑÖÅÅÇk{ÑÑÖ(27)two-dimensionalrandomwalk,canbeobtainedwhen0<α<10−3,whereasthelongtime1/t3/2trueasymptoticineq24withα2d=1,ΔF2dim(s)givenby̅eq26inthelimitofs̅→0canbeexpressedasdecay,whichischaracteristicofthethree-dimensionalrandomwalk,can,inprinciple,beobtainedaslongas0<α.Onthe−11ΔFs2dim()̅≈ln(16/)s̅otherhand,1/ttimedependenceispossibleonlyforstrongly2π(28)anisotropictransitionrateconstantsexpressedbyα=γ∥/fortheisotropicsquarelattice.Therefore,theasymptoticγ<10−3.Theseconclusionsareobtainedsolelyfromthetime⊥componentofF−1(s)intwo-dimensionalisotropicsystems̅2dimdistributiontoregeneratethecorrelatedtripletpairaftercanbeexpressedbythelogarithmicfunctiongivenby1/dissociationandareindependentoftherateconstant,suchas(2π)ln(16/s).The1/̅tasymptotictimedependenceinthegeminatedelayedfluorescencefortheisotropicsquarelatticekrad,kSF,andkTF.4.2.AnisotropyintheSquareLattice.Now,wecorrespondstothe−ln(s)-dependenceintheexpansionof̅ΔF2dim(s)when̅s̅→0.considertheanisotropyinthesquarelattice.IndimensionsInFigure4,weshowF−1(s)and̅F(s)forvariousvalues̅smallerthanorequaltotwo,1−R̂(s)→0ass→0,andthe2dim3dimofα.Here,α=0correspondstothetwo-dimensionalisotropicasymptoticexpansionofeq20canbeobtainedas3300https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

6TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleÄÅÅÉÑÑ−1wherez()tkkt=γ/(92kk),andthelimitof̂ÅÅÅÅ9()2jÑÑÑÑradTFdisSFSs()≈+−ÅÅkkkCSF||sÑÑγtk>(92)22k2/(k2k2)isintroducedtoobtainthelastÅÅradSF∑kÑÑdisSFradTFS(0)ÅÅ1(+−dis1Rŝ())ÑÑapproximateexpressiongivenbythepowerlawdecaywiththeÅÅj=1kC||()2jÑÑÅÇTFsÑÖexponent−3/2.Thepowerlawasymptoticdecaywiththe(29)exponent−3/2iscommonbetweensystemswithone-ÄÅÅÉÑÑ−1dimensionalhoppingtransitionsandthosewiththree-dimen-ÅÅ9ijjkyzzÑÑ≈+−ÅÅÅÅkk∑k||−C()2jjj1(dis1−Rŝ())zzÑÑÑÑsionalisotropichoppingtransitions.However,thethresholdÅÅradSFSFsjj()2jzzÑÑtimetoreachtheasymptotictimeregioninvolvesalargefactorÅÅÇj=1kkCTF||s{ÑÑÖ2(30)givenby(92)=162,whichisabsentfromsystemswith9three-dimensionalisotropichoppingtransitions.Therefore,ÄÅÅÉÑÑ−1ÅÅkdisÑÑcontrarytoisotropicthree-dimensionalhoppingmedia,=ÅÅÅÅkkrad+−9(SF1(Rŝ))ÑÑÑÑasymptoticpowerlawdecaywiththeexponent−3/2couldÅÇkTFÑÖ(31)hardlybeobservedinone-dimensionalhoppingmedia.AsEquation31indicatesthatthefielddependenceowingtoshowninFigure5,theinitialexponentialdecayisfollowedby|C(j)|2droppedoutforasymptotictimedependence,whereassthefielddependenceoriginatingfrom|C(j)|2remainsinthree-sdimensionalexcitondiffusionaslongasthediagonalcomponentsinthedensitymatrixaretakenintoaccountasintheMerrifieldmodel.Inthetwo-dimensionalsquarelattice,bysubstitutingeq24witheq27intoeq21,weobtainR̂()ss=−1πγ/ln(16/)(32)⊥Usingeq31,theasymptoticdecayofthenormalizedexcitedsingletpopulationisobtainedasSs()̂9kkSFdis1≈S(0)ktTF∞exp(−w)dw∫[+]ktln(16γπ/)w9k(k/k)22+πk2rad⊥SFdisTFradFigure5.Normalizedexcitedsingletdensityasafunctionof0(33)dimensionlesstime[kSFt].Thereddotsandblacksquaresareobtainedbyassumingone-dimensionalhoppingtransitionsand9kkSFdis11isotropictwodimensionalhoppingtransitions,respectively.The≈222thinredandthinblacklinesindicatetheasymptoticdecaygivenbyktTF[+ktradln(16γπ⊥)9kSF(kdis/kTF)]+πkradC/t3/2[1dim.]andC/t[2dim.],respectively,whereCandCcan1212(34)beobtainedfromeqs39and34,respectively.Theblackcircles,wherewetakethelimitofttobelargeasshowninthetriangles,andbluediamondsrepresentα2d=0.1,α2d=0.01,andα2d=SupportingInformation.0.001,respectively.Theparametervaluesarekrad/kSF=1,kTF/kdis=1,Inonedimension,wehavekdis/kSF=1,andkdis/γ⊥=1+α2d.Thereddashedcurveindicatestheintermediatetoasymptoticdecaygivenbyeq38obtainedbyπassumingone-dimensionalhoppingtransitions.Theasymptoticdecay112222Gw21dd(0,)∫dkyisobtainedwhenγt>162kdiskSF/(kradkTF)issatisfied.=(2)πwk−cos()1dy(35)−πthecharacteristictransientdecayofaone-dimensionalrandom1=walkforα2d≤0.01.Carefulanalysisisrequiredtorevealthew2−1smallcomponentofthetrueasymptoticpowerlawdecaywith1d(36)exponent−1,whichshouldbepresentiftheexcitonmotionwherew1d=s/γ+1andγindicatesthehoppingtransitioncanbeapproximatedbytwo-dimensionalhoppingtransitions34rate.Inonedimension,theLaplacetransformofS(t)/S(0)inwithintheplane.Forα2d>0.01,thedecayoftheexcitedsingletthelimitofsmallscanbeobtainedbysubstitutingeq36intodensityfollowsthe1/tlaw,whichischaracteristicofatwo-eq31asdimensionalrandomwalk.Clearly,theone-dimensional−1asymptoticpowerlawwithexponent−3/2isobtainedforaSs()̂ijjkdisyzzlongertimeregimecomparedwiththethresholdtimeof1/t≈+jjjkkrad92SFs/γzzz−3/2S(0)kkTF{(37)decay.Therefore,crossoverfromthepowerlawdecay,∼t,whichischaracteristicofaone-dimensionalrandomwalk,toTheinverseLaplacetransformofeq37isobtainedas∼1/tdecay,whichisthetrueasymptoticdecayforatwo-ÄÅÅÉÑÑdimensionalrandomwalk,couldhardlybeobserved.ThisSt()zt()ÅÅ12ÑÑ≈−ÅÅzt()exp(())erfc(())ztztÑÑsituationisinsharpcontrasttothecaseofasymmetryintheS(0)ktradÅÅÅÇπÑÑÑÖ(38)cubiclattice,wherecrossoverfromthepowerlawdecay,which9kkischaracteristicofatwo-dimensionalrandomwalk,tothetruedisSF≈asymptoticdecayobtainedforthethree-dimensionalrandom23/2kkradTF2πγt(39)walkisshowninFigure3.3301https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

7TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticle5.EXPERIMENTALRESULTSnotinfluencedbypumpexcitationintensityandchoosetheThetripletexcitondiffusionconstantsinrubrene,providedinlowestpossibleexcitationintensityasshowninFigure6.Table1,arestronglyanisotropiccomparedwiththoseinBesidestheintensitydependence,thefluorescencedecaykineticsisalterediftriplet−tripletfusionoccursbybimolecular40−42Table1.AnisotropyinDiffusionCoefficientsofTripletreactions.SuchaninfluenceisnotvisibleinourExcitonsinRubreneandTetracene,theCrystalsofwhichexperimentaldata;fortypicalcases,thefluorescencedecaysAreOrthorhombicandTriclinic(NearlyMonoclinic),exponentiallywhendelayedfluorescenceoriginatesfrom40−42Respectivelytriplet−tripletfusionbybimolecularreactions.Figure6showsthefluorescencedecaycurvesofrubrene(a)rubreneD(cm2s−1)tetraceneD(cm2s−1)andtetracene(b).Thelongdashedlineinpanel(a)representsexp.20cal.22exp.21cal.22theasymptoticpowerlawC/t3/2.Byconsideringthediffusionalongthe1.22×10−51.35×10−33.51×10−4constantsoftheexcitedtripletsinrubreneinTable1,thea-axisresultsinpanel(a)indicatethatthediffusionofexcitedtripletsalongthe1.6×10−32.51×10−32.28×10−35.45×10−4inrubreneisessentiallyone-dimensionalratherthantwo-b-axisdimensional.AsindicatedinTable1,thediffusionconstantofalongthe03.1×10−41.47×10−7c-axistheexcitedtripletexcitonsalongthea-axisisapproximately10−3timessmallerthanthatalongtheb-axis.InFigure5,theasymptoticdecaycloselyapproximatesthatofapurelyone-tetracene.Thediffusionconstantofthetripletexcitonsindimensionallatticewhentheasymmetryinthein-planerubrenewouldbeexpectedtobepracticallyzeroalongthedirectioninwhichthemolecularwavefunctionshardlyhoppingrateconstantsisgivenbyα2d=0.001.TheasymptoticdecaygivenbyC/t3/2inFigure6isconsistentwiththelarge22overlap.Therefore,weinvestigatedwhetherthelong-time22asymmetryinthecalculateddiffusionconstantsinTable1.asymptoticdecayinrubreneisgivenby∼1/twithaThisdecayisalsoconsistentwiththedecaylineofthesingletlogarithmiccorrectionresultingfromin-planeanisotropyorby∼1/t3/2correspondingtoone-dimensionaldiffusion.excitondensityinFigure5forα2d=0.001,whichapproximatesThediffusionconstantsofthetripletexcitonsintetracenethedecayunderone-dimensionaldiffusion.TheseresultslistedinTable1areofthesameorderalongthea-andb-axes,reflectthattheelectronicoverlapsalongtheb-axisinthewhereasthatalongthec-axisislowerthanthoseintheotherrubrenecrystalsarestrongcomparedwiththoseinother22twodirections.Therefore,weaimedtodeterminewhetherthedirections.InFigure5,weshowtheasymptoticpowerlaw3/2C/t3/2associatedwiththeone-dimensionalexcitondiffusionlong-timeasymptoticdecayintetraceneisgivenby∼1/t,whichcorrespondstothree-dimensionaldiffusionorby∼1/t,thatoccursatthelonger-timeregimescomparedwiththetimewithalogarithmiccorrectionduetoin-planediffusion.regimesinwhichthepowerlawdecayisassociatedwithSofar,wehaveassumedgeminatetripletfusionandshowndiffusioninhigherdimensions.ThemeasurementsofthedatathatdelayedfluorescenceafterpulsedexcitationisdescribedbyshowninFigure6awerecarriedoutforaslongas1μs,whichpowerlawdecaycausedbypairwiseregeneratedsingletsunderismorethan30timeslongerthanthetimerequiredtoshowdiffusion.ThoughgeminatetripletfusionmainlyoccursshortlythepowerlawdecayinFigure6b.Measurementsrecordedforafterphotoexcitation,tripletfusionmayprogressivelyproceedalongtimeperiodallowustoconfirmtheasymptoticpowerlawC/t3/2associatedwithone-dimensionalexcitondiffusion.bynongeminatebimolecularreactionsatlongtimeregimes.Toruleoutthepossibleinfluenceofbimoleculartriplet−tripletInFigure6b,thelongdashedlinerepresentstheasymptoticfusion,weconfirmthatthenormalizedfluorescencedecayispowerlawC/t3/2.AccordingtoTable1,thediffusionconstantsFigure6.(a)Fluorescencedecaycurvesofrubrenemeasuredat650nmfollowingexcitationwitha355nmflashlight.Theexcitationlightintensitiesare0.07and0.25μJcm−2.Theinitialdecaywasfittedbyanexponentialdecayfunction,andtheasymptoticdecaywasfittedbythepowerlawdecayfunction,withtheexponent−3/2givenbyC/t3/2.(b)Fluorescencedecaycurvesoftetracenemeasuredat580nmfollowingexcitationwitha470nmflashlight.Theintensityoftheexcitationlightislessthan2nJcm−2.Theasymptoticdecaywasfittedbyapowerlawdecayfunctionwiththeexponent−3/2givenbyC/t3/2.3302https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

8TheJournalofPhysicalChemistryCpubs.acs.org/JPCCArticleofthetripletexcitonsintetracenealongthea-axisaresimilarcomparedwiththetransitionfromthree-totwo-dimensionaltothosealongtheb-axis,whereasthatofthetripletexcitonsdecay.alongthec-axisissmallerincomparison.Specifically,theWeexperimentallyexaminedthedelayedfluorescenceofexperimentalvalueisapproximately10timessmaller,andtherubreneandtetracene.Inrubrene,theelectronicwavecalculatedvalueisapproximately3ordersofmagnitudefunctionshardlyoverlapalongonedirectionandthetripletsmaller.BasedontheresultsinFigure2,ifthediffusionexcitonsmigratewithinaplane.TheexperimentaldatarelatingtothedelayedfluorescenceinrubreneexhibitclearC/t3/2-constantalongthec-axisis10timessmallerthanthein-planediffusionconstants,theasymptoticdecayofthesingletexcitondependenceinonedimensionratherthanC/t-dependenceindensityexhibitsC/t3/2-dependenceratherthanC/t-depend-twodimensions.Theseresultssuggestvirtualone-dimensionalence.Ontheotherhand,ifthediffusionconstantalongthec-migrationofthetripletexcitonsinrubrene.Intetracene,theaxisis5×10−3timessmallerthanthein-planediffusiontransitionrateconstantinonedirectionislowerthanthoseinconstant,thedecayofthesingletexcitondensityexhibitsaC/theplane,andthein-planetransitionrateconstantsareofthet-dependencebeforeshowingthetrueasymptoticdecaygivensameorder.Theexperimentaldataofthedelayedfluorescence3/2intetraceneexhibitC/t3/2decayofthethree-dimensionalbyC/t-dependenceinFigure3.TheresultsshowninFigure6bareconsistentwiththeexperimentalvaluesofthediffusionmigrationoftripletexcitons.ThisresultisconsistentwiththeconstantsofthetripletexcitonsintetracenesummarizedinexperimentallydeterminedanisotropyinthediffusionTable1.Theresultsarealsoconsistentwithrecentconstantsofthetripletexcitons,wheretheout-of-planeexperimentalobservationsofC/t3/2asymptoticdecayfounddiffusionconstantisapproximately1orderofmagnitudeintetracenedepositedonSi/SiO_x.43smaller.Tosummarize,wethoroughlyinvestigatedtheinfluenceofanisotropyinthediffusivemigrationoftripletexcitonsonthe6.DISCUSSIONANDCONCLUSIONSgeminatedelayedfluorescencewithpowerlawdecay.TheTheasymptotictimedependenceofgeminatedelayedtheoreticalresultswereverifiedbyanalyzingthefluorescencefluorescencedecaycanbeexpressedbytheC/tβlawunderfromrubreneandtetracene,withthereportedvaluesofthethediffusion-mediatedreversiblegeminatefusionoftriplets.diffusionconstantsofthetripletexcitonsdependingontheTheβvalueis3/2inoneandthreedimensionsand1withamethodsofmeasurementsandcalculations.Weshowedthat10logarithmicweaktimedependenceintwodimensions.thegeminatedelayedfluorescencecanbeusedtocharacterizeHowever,thediffusionconstantsoftripletsinmostorganictheanisotropyinthediffusionconstantsoftripletexcitons.Wecrystalsarenotperfectone-,two-,andthree-dimensionalalsopredictthatthegeminatedelayedfluorescencemayexhibitsystems.WeinvestigatedtheA/tβlawofdelayedfluorescenceacrossoveroftheexponentsinthepowerlawdecay:theusingalatticemodel,inwhichtheanisotropyinthediffusioncrossoverreflectsadimensionalreductioninthemigrationofoftripletexcitonsistakenintoaccountbyusingthelatticetripletexcitonsbeforereachingtrueasymptoticdecay.Greenfunction.Thisfunctionissuitabletostudythelargeanisotropyassociatedwithtransitionrateconstants,butthe■ASSOCIATEDCONTENTexpressionistedious.Thevalidityoftheexpressionwas*sıSupportingInformationthereforeexaminedbynumericallyevaluatingalternativeformsTheSupportingInformationisavailablefreeofchargeatofthelatticeGreenfunctionbyalteringtheanisotropyinthehttps://pubs.acs.org/doi/10.1021/acs.jpcc.0c10582.transitionrateconstantswithin1orderofmagnitude.Then,theeffectoftheanisotropyonthetransitionrateconstantswasAlternativederivationofeq12witheq5,alternativeexaminedbychangingtheratiobetweenthemaximumandexpressionofG(α,w),andinverseLaplacetransformofminimumtransitionrateconstantsbyasmanyas5ordersofeq33(PDF)magnitude.Weshowedthatthethree-dimensionaldecaygivenbyC/t3/2■AUTHORINFORMATIONholdswhenthetransitionrateconstantintheout-of-planeCorrespondingAuthordirectionis10timeslowerthanthatinthein-planedirection.KazuhikoSeki−NanomaterialsResearchInstitute,NationalThedelayedfluorescencecanbeapproximatedbythetwo-InstituteofAdvancedIndustrialScienceandTechnologydimensionaldecaygivenbyC/twhenthetransitionrate(AIST),,Tsukuba,Ibaraki305-8565,Japan;orcid.org/constantintheout-of-planedirectionis5ordersofmagnitude0000-0001-9858-2552;Email:k-seki@aist.go.jplowerthanthatinthein-planedirection.Interestingly,thedelayedfluorescencefollowstheC/tlawandthereaftertheC/Authorst3/2lawwhenthetransitionrateconstantintheout-of-planeTomomiYoshida−DepartmentofChemistry,Graduatedirectionis3−4ordersofmagnitudelowerthanthatinthein-SchoolofScienceandEngineering,SaitamaUniversity,planedirection.Saitama338-8570,JapanWealsoshowedthatthetwo-dimensionaldecaygivenbytheTomoakiYago−DepartmentofChemistry,GraduateSchoolC/tlawwithalogarithmicweaktimedependenceholdswhenofScienceandEngineering,SaitamaUniversity,Saitamathetransitionrateconstantinonedirectionis10timeslower338-8570,Japan;orcid.org/0000-0002-4507-245Xthanthatintwodimensions;thedelayedfluorescencecanbeMasanobuWakasa−DepartmentofChemistry,Graduateapproximatedbytheone-dimensionaldecaygivenbyC/t3/2SchoolofScienceandEngineering,SaitamaUniversity,whenthetransitionrateconstantinonedirectionis3ordersofSaitama338-8570,Japan;orcid.org/0000-0001-9084-magnitudelowerthanthatintwo-dimensionaldiffusive8459migration.WefoundthatthetransitionofthepowerlawRyuziKatoh−DepartmentofChemicalBiologyandApplieddecayfromtheone-dimensionaldecaygivenbyC/t3/2totheChemistry,CollegeofEngineering,NihonUniversity,two-dimensionaldecaygivenbyC/twashighlyunlikelyKoriyama,Fukushima963-8642,Japan3303https://dx.doi.org/10.1021/acs.jpcc.0c10582J.Phys.Chem.C2021,125,3295−3304

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