Dipolar coupling molecular level mobility - PDF document

(tistheinternucleardistanceand)=(3cos1)/2isthesecond-orderLegendrepolynomial(forsimplicity,wehereneglectthecontributionsofhigherorderLegendrepolynomials/Wignerfunctions).Forisotropicsamplesandxedintermolecularvectors,equation(1)simpliestoC(
) isthecorrelationtime.If,however,themotionisfastbutanisotropic,theaveragingisincomplete,leavinganorderparameterplateau,,relatedtoaresidualcouplingfromwhichthegeometryofthemotionalprocesscanbe DIPOLARCOUPLING:MOLECULAR-LEVELMOBILITY beendeveloped.Fordipolarcoupling,onebasicallyneedstodistinguishdifferenttimeregimesthataredenedbytheratiooftherateofreorientation(thatisinverselyproportionaltothecorrelationtimeofmotion,)versusthedipolarcouplingA.slowmotion:(B.intermediatemotion:(C.fastmotion:(Inturn,theseconditionsroughlytranslateintotherelationbetweenthetransverserelaxationtimeinthestaticcaseandtheinverseofthedynamicrate,i.e.,A.slowmotion:B.intermediatemotion:C.fastmotion:approximatelyequalsthelifetimeoftheNMRsignal(lengthofthefreeinductiondecay,(FID)time),thedifferentregimescorrespondtothefollowing:(A)TheFIDdecaysfasterascomparedtotheaveragelifetimeofacertainmolecularorientation,(C)therateofreorientationisfasterascomparedtoanyfrequencycomponentoftheFID,or(B)anappreciablenumberofreorientationshappensduringthelifetimeoftheFID.FastprocessesintherangeoftheinverseLarmorfrequencyefcientlystimulatetransitionsofspinsbetweentheirZeemanlevelsanddriveNMRrelaxation.Thus,theseexperimentsprovideaccurateinformationaboutfastmotionswithcorrelationtimesontheorderoftheinverseLarmorfrequency,whileintheabsenceofmoleculardynamicprocesses,onlyspindynamicslikespindiffusionthatdonotcontaininformationaboutthemoleculardynamicscontributesubstantiallytotherelaxationdataandmakeafaithfuldeterminationofmoleculardynamicparametersdifcult.SincethereareexcellenttextsonNMRrelaxation,wewillnotdealwithsuchexperimentshere(seeRelaxation:AnIntroduc-).Also,thesefastprocessesleadtoanaveragingoftheexperimentallydetermineddipolarcouplingthatprovidesdynamicinformation,seebelow.Slowermotion,however,canbeeitherinvestigatedbyrelaxationexperimentsinlowereffectiveeldsoronecanmakeuseofthefactthatprocesseswithcorrelationtimesontheorderofthelifetimeofthesolid-stateNMRsignal,i.e.,microseconds,haveaneffectontheappearanceoftheNMRspectrumduetoamodulationoftheresonancefrequenciesorinterferencewithexternalperturbations.EvenslowerprocessesarethedomainofexchangeexperimentsthatbasicallyprovideaccesstoprocesseswithcorrelationtimesaslongastherelaxationHistorically,therst‘dipolar’experimentswerecarriedoutonabundantspinswithlargegyromagneticratio,i.e.,HandF.InparticularforH,thedominantinteractionisthehomonuclearHinteractionthatleadstoanoverallspectralwidthofabout40kHzinatypicalorganicsolid,incontrasttothechemicalshifteffectsthatareassmallasafewtensofhertzattheexternaleldstrengthsusedinthosedays.Basically,itispossibletocalculatetheeffectsofamolecularreorientationontheexperimentaldipolarspectrumusingstandardprocedureslikeexchangetheory;however,forthecaseofdipolarinteractioninsolids,thisisstrictlytrueonlyforisolatedspinsformingaspinpair.Theproblemin 2wD3 Stationary pairRotating pairFigure2Theoreticallineshapesforadipolar-couplednuclearspinpairwithlikespins1/2whenstationaryandwheninmotionaboutanaxisperpendiculartotheinternuclearaxis.(ReproducedfromRef.AmericanInstituteofPhysics,1950)realsystemsisthatmanycloselyspacedprotonsinteractwitheachother;therearehardlyanyisolatedHpairsthatwouldyieldspectrathatmatchtheoryandenableaquantitativecalculationofthelineshape.Thoughtheoreticalattemptsweremadetoincreasethenumberofspinsinthecalculation,theusualapproachwastoconvolutethetheoreticaltwo-spinspectrumwithabroadGaussianorLorentzianfunctionthatresultsinaratherfeaturelessbroadline(seeFigure2)fromwhichinmostcasesthesecondmomentofthelineshapewasextracted.InFigure3,temperature-dependentdipolarspectraandtheextractedlinewidthareshownforThe‘onset’oftheringrotation,i.e.,thetemperatureatwhichtheinversecorrelationtimeoftherotationbecomessimilartothedipolarcouplingconstantandasubstantiallinenarrowinghappens,isalsodetectedintheDSCtrace.Theapplicationoftemperature-dependentlineshapesandthemeasurementsofsecondmomentinmorecomplexorganicsolidslikepolymersfollowedsoonafterPolymerDynamicsandOrderfromMultidimensionalSolidStateNMR).Inthesetemperature-dependentexperiments,thislineisnarroweddownclosetoatemperaturewhenthemotionalrateofthethermallyactivatedprocessapproachesandeventuallyexceedsthedipolarcouplingconstant;however,thisbasicallyprovidesasingledatapointintheactivationdiagram.Inaddition,thetransitionfromthestaticlineshapetothenarrowlinehappensoveracertaintemperaturerange,sotheaccuracyinbothtemperatureandrateisfairlylimited.Nodetailedinformationaboutthegeometryoftheprocessisavailableeither,exceptaroughestimateabouttheanisotropyofthereorientation:ifthedynamicprocesshappenstobeisotropic,thelimitisthenaturallinewidthrelatedtotheappliedexternaleld,whileforanisotropicmotions,alargerlinewidthisobtained.Qualitatively,thesmallertheamplitudeofthereorientation,thecloserthefast-limitlinewidthtothatoftherigidlimit.Itshouldbenotedthatthereisexperimentalevidencethatdynamicparameterscanbeobtainednotonlybyrecordingspectraatdifferenttemperatures(andthusmotionalratesofthermallyactivatedprocesses)butalsobychangingparametersoftheNMRexperimentlikethedecouplingefciency,inducingan‘exchangenarrowing’analogoustotheclassical DIPOLARCOUPLING:MOLECULAR-LEVELMOBILITY i.e.,themotionoccursinaplaneperpendiculartothereferenceorientation.Forspecialtopologiesofmotion,aquantitativerelationbetweenthereorientationangleandcanbedrawn,asshowninFigure4wherethisrelationisdisplayedfortwo-andthree-sitejumpmotionsaswellasplanardiffusionanddiffusionwithinaconeandonacone,respectively.Ifsuchmodelsarenotapplicable,atleastanisotropicjumpmotionscanbedistinguishedfromrotationaldiffusionsorrelativeinformation,comparingdifferentsamples,etc.,canbeobtained.However,itshouldbenotedthatfromaverageddipolarcouplingsmeasuredinthefastlimit,noinformationaboutthetimescaleofthemotioncanbeobtained.4DYNAMICLINESHAPE–INTERMEDIATEIftherateoftheprocessdoesnotsatisfytheconditionforafastprocess,i.e.,iftherearemotionallyinducedfrequencyalterationsduringthelifetimeoftheFID,equation5cannotbeappliedanymoreand,therefore,easydeterminationofdynamicparametersisnotpossible.However,theexperimentalparameterslikethedipolarspectrumnowcontaininformationaboutthetopologyoftheprocessandadditionallyaboutthetimescaleoftheprocess.Inprinciple,theNMRtime-domainsignal()cannowbewrittenas(t)(twhereinadynamicsystem,theresonancefrequencythatisdeterminedbyanisotropicinteractionsnowisarandomfunctionoftimeasaresultofthemotion.Forsinglespinpairs,andinparticulartherelatedcaseHNMRexperimentswithmodelsrelyingonjumpsbetweenalimitednumberofsites,theoreticalsolutionsbasedonexchangematricesarestraightforwardandwellestablishedintheliterature.Moregenerally,followingthewell-knownAndersonandWeiss(A–W)approachof‘randomfrequencymodulation’,approximateandrathersimpleanalyticalsolutionsarepossible.ThetheoryisbasedonaGaussianresonancefrequencydistribution,whichisduetothedistributionofthedipolarlocalelds,andrequiresthattheresonancefrequencyvariesintimeduetoarandomthermalprocess.Thesimplestmodelusuallyadoptedisisotropicrotationaldiffusion,whichhasaparticularlysimple,monoexponentialcorrelationfunction.WhiletheGaussianfrequencydistributionnaturallyarisesforthemultispinscenarioofaprotonsystem,itturnsoutthattheshort-timebehaviorofjustaspinpair(Pake-likefrequencydistribution)performingN-sitejumpscanstillreasonablywellbedescribedbysuchanDetailsareaddressedbelow.TheA–Wapproachisanunderestimatedandpowerfultool,asitprovidessimpleanalyticalequationstodescribefrequencyortime-domaindata,andeventhetimedependenceinMASexperimentscanbeaccommodated.Thereareanumberofextensionstothemodelandapplicationstoextractdynamicparametersfromorganicsolids,inparticularrubberymaterialsandofadsorbedmolecules.5SITE-RESOLVEDHETERONUCLEARANDHOMONUCLEARDIPOLARCOUPLINGSInadditiontothedisturbingmultispineffectsdescribedabove,experimentsonabundantnuclei(likeH)intypicalorganicsolidsareoftentoounselective:oneofthepowersofNMR:itschemicalselectivitythusmolecularresolution,islostduetotheunfavorableratioofthestrongdipolarinteractionbetweenthespinsthatbroadensthespectraandthesmallchemicalshiftthatprovidesthespectralresolution.Itisthereforegenerallydesirabletoseparatetheinformationfromstructure(chemicalshift)anddynamics(dipolarcoupling)inatwo-dimensionalmanner.Fororganicsolidsatnaturalabundance,dipolarexperimentsshouldbecombinedwithhigh-resolutionconditionsprovidedbyMAS,andawealthofhomonuclearandheteronucleardipolarrecouplingexperiments,providingprecisevaluesforthecouplingconstants,havebeendesignedinthepastHowever,mostoftheseexperimentsareusedtoobtainstructuralinformation(distances),andonlyafew(infactlesssophisticated)approacheshavegainedimportanceindynamicstudiesanddeservespecialattention:cross-polarization(CP)dynamics,separated-localeld(SLF)andrelatedrecouplingtechniques,andhomonucleardouble-quantum(DQ)experiments.Theformertwoapproachesrelyontheheteronucleardipolarcouplingbetweenabundant(likeH)andrarespins(likeC).InaCPexperiment,theincreaseofrare-spinmagnetizationoccurswiththebuilduptimethatis—similartotherelaxationtime—sensitivetodynamicprocessesinthekilohertzrangeandcorrelatestoimportantmechanicalparameterslikedynamicmoduli.Themeasurementofbecameratherpopularindynamicinvestigationsoforganicsolids;however,caremustbetakenintheinterpretationofthedata.SLFexperimentsmeasurechemicalshiftsanddipolarcouplingseparatelybutinawaythattheycanbecorrelated.versionsofSLFexperimentsaredipolar-rotationalspinechoesandwide-lineseparation(WISE)experiments.AnearlyexampleofDRSEisshowninFigure5,wherethedipolarspectraofabackboneandaringcarboninpolystyrene(PS)arecompared:thelatterexhibitsasmallerdipolarcoupling,duetotheaveragingbytheipmotion.TheWISEapproachwasrstsuggestedasadynamicltertoseparatemobileandrigidcomponentsCspectraandhaslaterfoundbroadapplicationinpolymerscience.WISEdatafromablock-copolymerofPSandpoyldimethylsiloxane(PDMS)areshowninFigure6:thedifferentwidthsinthedipolardimensionrevealdifferentmobilitiesoftheglassyPSandtherubberyPDMS.FurtherdevelopmentsofWISEincludethesuppressionoftheH-spindiffusionduringtheCPstepthatwouldtransfertheHmagnetizationtodifferentcarbonsandthusspoilsthemolecularresolutionofthemethod.TheworkofFrydmandeservesparticularattention;ithasshownthatSLFexperimentsarepossibleevenwithouthomo-decouplingbuildingblocksandformultispinsystems:whensufcientlyfastMASratesareemployed,quantitativedipolarsidebandpatternsfromdirectlybondedspinpairscanbeacquiredintheabsenceofHmultiple-pulsehomonucleardecouplingevenfor‘real’organicsolids.AspecialSLFversionistheso-calledPISEMAtechnique:itworksforstaticsamplesandisusuallyappliedtoorientedbiologicalmaterialsinwhichtheX DIPOLARCOUPLING:MOLECULAR-LEVELMOBILITY 1000100kHz 1000100kHz RRRRRR13Core-CHCore-CH n-C12H25-n-C12HSolid phasetrcpl = 4 TRtrcpl = 2 TR- (a)(b)160ppm20160ppmFigure10CREPT-HMQCexperimentsonhexabenzocoronenes(HBC).ThetechniqueisanearliervariantofREPT-HDOR(Figure9f),withinterchanged90pulsesankingandanadditional180pulseonCinthemiddleof—ithasadditional(hereunnecessary)Hchemicalshiftinformationin,Forthegivenpurpose,REREDOR(Figure9b)isnowadaysmorerecommendable.(a)MolecularstructureoftheHBC.(b)SchematicdisplayofmolecularstackingofHBCs.Eachgraydiscisamoleculeasshownin(a).(c)ThepatternsarethesumprojectionsoverthecoreCHsignalsinthe2-DspectraofHBC-andHBCPh-,respectively.Experimentaltemperatureswere35Cforroomtemperatureand120Cforthehigh-temperatureLCphases.TheMASspectraontopwererecordedataMASrateof12kHzwhilefortheREPT-HMQCexperiments,25kHzwasused.Theinsertsaresimulatedspectra.(d)Visualizationofmotionalmodesofasinglediscinthestacks.WhiletheHBCwiththePh-sidechainsperformsarotationaroundanaxisperpendiculartotheplaneofthemoleculeonly,theonewiththe-sidechainsappearstodoanadditionalwiggleintheplaneofthemolecule.ThiswasconcludedfromthereducedCcoupling,whichcanbequalitativelyinferredfromthereducedintensityoftheouterdipolarsidebands DIPOLARCOUPLING:MOLECULAR-LEVELMOBILITY morereliabletreatmentandextendthefrequencyrangeofthemolecularmotionsaccessiblebySLFexperiments.Also,theeffectofmolecularmotiononLG-CP(asshownbynumericalsimulationsinFigure12a)aswellasontheimportantissueofCPefciencieswasinvestigatedinthisarticle.Inparticular,ifcross-polarizationisusedastheexcitationmethod,thepresenceofadistributionofmotionalcorrelationtimesleadstoabiasinthetteddistributionbecauseofthesignaldecayduringthecontacttime(CT).Thisholdsinparticularforsegmentsdeepintheintermediate-motionalregime(withratesofto10Hz),whicharesuppressedintheresultingcarbonspectrum,whilesegmentswithlittle(orfastbutanisotropic)mobilityareoverrepresented(seeFigure12b).Weliketoplaceashortcommenthere:althoughtheDIP-SHIFTexperimentismoreefcientintermsofexperimentaltime,itseemstohavedifcultieswhenthereisoverlapofsignalsresultingfromdifferentmoleculardynamicsleadingtodifferentdipolardephasings.Inthesecases,itappearsdif-culttoseparatethedifferentcontributionsintheDIPSHIFTdephasingcurves,since—atleastinthefastlimitwhenthesignalisperiodicwith—twocomponentswithdifferentlyaveragedcouplingcanhardlybedistinguishedfromasinglecomponentwithanin-betweencoupling.Incontrast,inLG-CPexperiments,thesecomponentswouldappearinthedipolardimension(intimedomain)withdifferentinitialslopesanddipolaroscillations(oralternativelywithdifferentwidthsofthewell-structuredlineshapeswhenaFouriertransforminisperformed)andcaneasilybetoldapart.6DOUBLE-QUANTUMEXPERIMENTSOnealternativewayofdealingwiththecomplexcou-plingtopologyinabundantdipolar-coupledspinsystemsistheexcitationofhigher-quantumcoherences(seeMultipleQuan-tumCoherenceinSpin-1/2DipolarCoupledSolidsQuantumCoherencesinExtendedDipolarCoupledSpinMultipleQuantumNMRinSolids).Manyappli-cationsinsolid-stateNMRwerepioneeredbyPinesintheThebuildupofcoherencesinparticularservestoextractinformationonthedipolarcouplingbetweenthespinsinquestion.Effectsduetothetime-dependentmodula-tionsofpaircouplingswererstreportedforDQexperimentsAdvancesinNMRhardwarerecentlypavedthewayforultrafastMASexperimentswithcommerciallyavailableMASratesof35kHz,andrecentlyevenupto70kHz.Suchspin-ningrates,incombinationwithhighelds,enablesolid-stateHexperimentswithconsiderablyimproved(ascomparedtowide-lineexperiments)butstilllimitedspectralresolu-tion.TheseMASratesefcientlysuppressthehomonucleardipolarinteractionbetweenmoreremotenucleibutaffectthosebetweennext-neighbournucleimuchless.experimentsselectonlythosenucleiwithnearbylikespinsandrevealspatialcorrelationsbetweensuchspins.Thepre-cisevalueofthedipolarcouplingthatcarriesinformationonstructureand,importantly,thegeometryoffastmotions,canreliablybeextractedfromDQsidebandintensities.MAStechniquesprovedparticularlyusefulincharacterizingtheanisotropyofmesogenicmoleculesinliquidcrystals,fast-movingpolymerchainsinelastomers,polymermelts,andblockcopolymers.Anumberofapplicationsofsuchexperimentstodifferentcomplexmacro-andsupramolecularsystemshasrecentlybeenreviewed.However,thelimitedresolutionstillleavesthemfarfrombeinggenerallyappli-cablemethods—theyarerestrictedtocasesinwhichthemolecularmoietyinquestioniswellresolvedintheHMASspectrum.Moreover,apartfromapioneeringscalinganaly-sisoftherelationshipbetweenpolymerchaindynamicsandDQintensities,DQ-MAStechniqueshavetodatenotbeenusedtoelucidatetheactualoffast-to-intermediateInrelativelysimplesystemssuchassingle-componentpolymermeltsorelastomers,chemicalselectivityisoftennotanimportantissue.Then,staticDQmethodshaveinfactproventobemorequantitativewithrespecttothepreciseabsolutedeterminationofweakmotionallyreduced(residual)Hdipolarcouplings.TheabsenceofcomplicationsduetotheadditionalMAStimescalethenevenallowsforamuchsimplerquantitativetreatmentofintermediate-motionaleffectsinDQexperiments(intermsofAnderson–Weisstheory),andanapplicationtochaindynamicsinelastomersisdescribedinRef.34.7DIPOLAREXCHANGENMR—SLOWMOTIONSTheslowmotionregime(A)iscommonlycalledthestaticcase,i.e.,althoughtheremightbeanappreciableamountofmolecularmobility,itistooslowtobeobservedinNMRspectra(‘itisslowontheNMRtimescale’).Suchdynamicprocessescan,however,beobservedinso-calledexchangeexperimentsorspecicNMRrelax-ationexperiments.Theapplicationofthedipolarinteractionintheframeworkofexchangeexperiments,i.e.,theobserva-tionofthereorientationofadipolarvectorconnectingtwonucleiduringtimeperiodsmuchlongerthanthelifetimeoftheFIDandquantitativeinterpretationisinprinciplepossible;however,itrequirestheexistenceofspinpairs,asother-wise,magnetizationexchangeprocesses(spindiffusion,etc.)dominateandseverelyrestrictthedynamicrange.Thiscanbefacilitatedbyisotopiclabeling,insertingthedesiredspinsintothemolecules,orbyexperimentalconditionsthatsup-pressmultispininteractions.Suchsituationscanbefound—forexample—inH-depletedsubstanceswhereonlyfewHspinsarearound,andtheinteractionbetweenremotenucleiisfur-thersuppressedbyveryfastMAS.Experimentsofthelatterkindareinprogressandwillcertainlybereportedsoon.ThemethodologyofDQexperimentshasbeenextendedtoDQ–DQexchangeexperimentspermittingtheinvestigationofslowmotion.Generally,applicationsofexchange-typeexperimentsutilizingthedipolarcouplingarerareandarethusnotdiscussedhereatlength.8SUMMARYHomo-andheteronucleardipolarcouplingsbelongedtothetoolboxofdynamicsolid-stateNMRfromtheearlydaysofNMR.OwingtotheprogressinNMRhard-ware,thedevelopmentprogressedfromlow-resolutionwide-linespectraviaseparated-localeldexperiments,employing