TheScalingofRelaxation Processes
123
Editors
FriedrichKremer
Molekülphysik,Peter-Debye-Institutfür
PhysikderweichenMaterie
UniversitätLeipzig Leipzig Germany
AloisLoidl
InstitutfürPhysik
UniversitätAugsburg Augsburg Germany
ISSN2190-930XISSN2190-9318(electronic)
AdvancesinDielectrics
ISBN978-3-319-72705-9ISBN978-3-319-72706-6(eBook) https://doi.org/10.1007/978-3-319-72706-6
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Preface
Duringthepasttwodecades,theexperimentalcapabilitiestostudymaterialsinan extraordinarywidespectralrangeatlargelyvaryingtemperaturesaswellaspressureshaveenormouslydeveloped.ThisisespeciallytrueforBroadbandDielectric Spectroscopy(BDS)whichnowadayscoversthewholefrequencyrangefrom 10 6 Hzuptothefarinfrared(FIR) withoutanygap.Inaddition,othermethods likephotoncorrelationspectroscopy(PCS),nuclearmagneticresonance(NMR), viscosimetryandmechanicalspectroscopyandevencalorimetryhavealsobecome broadband.Consequently,knowledgeconcerningthescalingofrelaxationprocesseshastremendouslygrown,anditisnowcommontocombinedifferent techniquesinordertodeterminethedistinctcorrelationsandtheirmutualinteractionsinamaterialunderstudy.Thisdevelopmenthasbeennicelyexemplifiedfor amorphoussystemsbeingcharacterizedbyglassydynamicswhichisextendedfrom verylong(106 s)toshort(picoseconds)timescales,includingthestructural a-relaxation(dynamicglasstransition),secondary(slowandfast) b-relaxations, andtheBosonpeakinthefarinfrared.
ItiswellestablishedthatstructuralrelaxationsroughlyfollowaVogel–Fulcher–Tammanntypeofthermalactivation,whilesecondaryrelaxations,atleastbelowthe glasstransitiontemperature,canbedescribedbyanArrhenius-liketemperature dependence.FastprocessesintheGHzregimeseemtofollowscalingpredictions, whilethebosonpeakandintramolecularvibrationsareonlyweaklytemperature dependent.Thestrikingsimilaritiesoffrequencyandtemperaturedependenciesfor thelargeclassofsupercooledliquidsleadtothedevelopmentofscalingapproaches andtheoriesthatcanmodeltheseuniversalities.However,oftenacloserinspection revealsseverediscrepanciesanddeviations.
v
Itistheobjectiveofthisissueof “AdvancesinDielectrics ” onthe “Scalingof RelaxationProcesses” tosummarizethecurrentknowledgeandtheenormous amountofhigh-qualitydataonglassydynamicsofsupercooledliquidsandto discussitwithrespecttotheoftencompetingtheoreticalconcepts.
Leipzig,GermanyFriedrichKremer Augsburg,GermanyAloisLoidl March2018
vi
Preface
TheScalingofRelaxationProcesses Revisited 1 FriedrichKremerandAloisLoidl
GlassyDynamics:FromMillihertztoTerahertz 23
P.LunkenheimerandAloisLoidl
GlassyDynamicsasRe flectedintheInter-andIntra-molecular Interactions .............................................. 61
FriedrichKremer,WilhelmKossackandArthurMarkusAnton
UniversalityofDensityScaling ................................ 77 AndrzejGrzybowskiandMarianPaluch
ScalingofSuprastructureandDynamicsinPureandMixed DebyeLiquids 121
SebastianPeterBierwirth,JenniferBolle,StefanBauer, ChristianSternemann,CatalinGainaru,MetinTolanandRolandBöhmer
DynamicHeterogeneitiesinBinaryGlass-FormingSystems
173 D.Bock,Th.Körber,F.Mohamed,B.PötzschnerandE.A.Rössler
DepolarizedDynamicLightScatteringandDielectricSpectroscopy: TwoPerspectivesonMolecularReorientation inSupercooledLiquids ...................................... 203
J.Gabriel,F.Pabst,A.Helbling,T.BöhmerandT.Blochowicz
RelaxationProcessesinLiquidsandGlass-FormingSystems:What CanWeLearnbyComparingNeutronScatteringandDielectric SpectroscopyResults? ...................................... 247
ArantxaArbeandJuanColmenero
TheScalingoftheMolecularDynamicsofLiquidCrystalsas RevealedbyBroadbandDielectric,SpecificHeat,andNeutron Spectroscopy
AndreasSchönhals,BernhardFrickandReinerZorn
279
Contents
vii
TheCalorimetricGlassTransitioninaWideRangeofCoolingRates andFrequencies 307
T.V.Tropin,J.W.P.Schmelzer,G.SchulzandC.Schick
DipolarCorrelationsin1,4-PolybutadieneAcrosstheTimescales: ANumericalMolecularDynamicsSimulationInvestigation .......... 353 MathieuSolarandWolfgangPaul
LinearViscoelasticityofPolymersandPolymerNanocomposites: Molecular-DynamicsLargeAmplitudeOscillatoryShearandProbe RheologySimulations ....................................... 375
TheodorosDavris,AlexeyV.Lyulin,ArletteR.C.Baljon, VictorM.Nazarychev,IgorV.Volgin,SergeyV.Larin andSergeyV.Lyulin Index
405 viii Contents
TheScalingofRelaxation Processes—Revisited
FriedrichKremerandAloisLoidl
Abstract Glassydynamicscoverstheextraordinaryspectralrangefrom10+13 to 10 3 Hzandbelow.Inthisbroadfrequencywindow,fourdifferentdynamicprocessestakeplace:(i)theprimaryor α-relaxation,(ii)(slow)secondaryrelaxations (β-relaxations),(iii)fastabsorptionprocessesintheGHzand(iv)theboson-peak intheTHzrange.Thedynamicglasstransitionisassignedtofluctuationsbetween structuralsubstatesandscaleswellwiththecalorimetricglasstransitiontemperature. Itshowsasimilartemperaturedependenceastheviscosityandfluctuationsofthe densityorheatcapacity.Thetemperaturedependenceofthemeanrelaxationrateof thedynamicglasstransitionfollowsatfirstglancetheempiricalVogel–Fulcher— Tammannlaw,albeitafurtheranalysisunravelsclear-cutdeviations.The(slow) secondaryrelaxationsareassignedtolibrationalrelaxationsofmolecularsubgroups hencehavingastraightforwardmolecularassignment.Theymayalsoshowupasa wingonthehigh-frequencysideofthedynamicglasstransition.ThefastabsorptionprocessesatGHzfrequenciescanformallybedescribedwithintheframework ofthemode-couplingtheory(MCT).Theboson-peakresemblesthePoleyabsorptionandoriginatesfromoverdampedoscillations.Inthischapter,especiallythefirst threecontributionswillbediscussedindetailandcomparedwithexistingtheoretical models.
F.Kremer(B)
Molekülphysik,Peter-Debye-InstitutfürPhysikderweichenMaterie,UniversitätLeipzig, Linnéstr.5,04103Leipzig,Germany
e-mail:kremer@physik.uni-leipzig.de
A.Loidl
UniversityofAugsburg,ExperimentalPhysicsV,Universitätsstrasse2,86135Augsburg, Germany
e-mail:alois.loidl@physik.uni-augsburg.de
©SpringerInternationalPublishingAG,partofSpringerNature2018 F.KremerandA.Loidl(eds.), TheScalingofRelaxationProcesses, AdvancesinDielectrics,https://doi.org/10.1007/978-3-319-72706-6_1
1
1Introduction
Theglassystateisubiquitousininorganicandorganicmatter.Itischaracterized bythelackoflongrangeorderandshowsarefineddynamicsincludingprocesses spanningaspectralrangefrom10+13 to10 3 Hzandbelow.Despiteconcentrated efforts[1–9],acommontheoreticalunderstandingoftheglassystatedoesnotexist andavarietyofdifferentandoftencontroversialviewscompete.Theglassystate isfurthermorereflectedinmanydifferentphysicalquantities,e.g.theheatcapacity, theviscosity,themechanicalmoduli,thedensity,ultrasonicabsorption,magnetization,thecomplexindexofrefractionandthecomplexdielectricfunction.Hence,a multitudeofexperimentaltechniqueshavebeenemployedtostudyglassymaterials, suchasfrequency-dependentanddifferentialscanningcalorimetry[10],dynamic mechanicalspectroscopy[11],ultrasoundattenuation[12],light[13]andneutron scattering[14],NMRspectroscopy[15]andespeciallybroadbanddielectricspectroscopy[16–38].
Themeanrelaxationrate ν(T )ofthe α-relaxationischaracterizedbytheempirical Vogel–Fulcher–Tammann(VFT)-equation[39–41]:
where v∞ (2πτ∞ ) 1 isthehightemperaturelimitoftherelaxationrate, D isa constant,and T 0 denotestheVogel–Fulchertemperature.The“fragility”parameter D [42]describesherebythedeviationfromanArrhenius-typetemperaturedependence
where E A istheactivationenergyandktheBoltzmannconstant.Atthecalorimetric glasstransition T g ,themeanrelaxationrate ν (T g )andtheviscosity η (T g )have reachedtypicalvaluesof~10 3 Hzand~1013 Poise,respectively.Ingeneral, T 0 is foundtobeapproximately40Kbelow T g .Thus,thechangeinthedynamicsofthe glass-formingprocessesspansmorethan15decades.
ThedivergenceofEq.(1)at T=T 0 isalsosupportedbytheso-calledKauzmann paradoxoccurringintheentropydeterminedbymeasurementsofthespecificheat [43, 44]:iftheentropyofthesupercooledliquidisextrapolatedtolowtemperatures, itseemstobecomeidenticaltothatofacrystalorevensmaller.Insometheories (liketheGibbs–DiMarziomodel[45]forpolymers),theKauzmannparadoxis resolvedbyaphasetransition.Butthephysicalmeaningofthedivergenceof ν (T ) at T=T 0 remainsunclear.BecauseoftheuniversalityofEq.(1), T 0 isconsidered asacharacteristictemperature,wherethemeanrelaxationrateextrapolatestozero, albeitlittleevidencecouldbefoundforadynamicdivergence[46].
Qualitatively,glassydynamicsisoftendiscussedasfluctuationofamoleculein thecageofitsneighbours.Thelibrationalmotionsofthelattergiverisetofastsecondary β -relaxationswhichtakeplaceonatimescaleof10 10 –10 12 s,whilethe
2F.KremerandA.Loidl
ν ( T ) 1 2πτ ( T ) v∞ exp DT0 T T0 (1)
ν ( T ) ν∞ exp E A kT (2)
reorientationsofthemoleculesformingthecageareassignedtothedynamicglass transitionor α -relaxationobeyingaVFT-temperaturedependence.Thisrelaxation processmusthavecooperativecharacter;i.e.thefluctuationsofthemoleculesformingthe“cage”cannotbeindependentfromeachother.Theextensionofthesize ofsuch“cooperativelyrearrangingdomains”[3, 6, 7, 45]isoneofthecentral(and controversial)topicsofglassresearch.
Therelaxationfunctionofthe α-relaxationisusuallybroadened.Itshighfrequencysideexhibitsoftentwopowerlaws.Inthecaseofglycerol,thiswas observedalreadybyDavidsonandCole[47]andinterpretedascausedbyhighfrequencyvibrations.Itisnowadaysestablishedforavarietyofglass-forming(low molecularweightandpolymeric)materials[s.alsothechapterofP.Lunkenheimer andA.LoidlandF.Kremeretal.inthisbook]andconsideredtobethehigh-frequency contributionofasecondaryrelaxation.
Manysystemsshowadditionallyaslowsecondary β -relaxation(withan Arrhenius-typetemperaturedependence).Thisprocessbeingobservedforrelaxationrates~<108 Hzcanoftenbeassignedtointramolecularfluctuations.Butthere areseveralexampleslikethelowmolecularweightliquidortho-terphenyl(OTP) [19, 20]orthemainchainpolymerpoly(ethyleneterephthalate)(PET)[38]where suchaninterpretationisnotimmediatelyobvious.Therefore,itwassuggestedby GoldsteinandJohari[11, 12]thattheslow β-relaxation“isintrinsictothenatureof theglassystate”[12].IntheTHzregimeafurthermolecularprocessisobserved, the“bosonpeak”[4]whichhassimilaritieswiththePoleyabsorption[48],which wasinterpretedasbeingcausedbystronglocalfieldsexertedonamoleculebyits immediateneighboursintheglassystate(Fig. 1).
Indetailinthischapter,thefollowingquestionswillbeaddressed:(i)istherea scalingfunctionwhichdescribesthetemperaturedependenceofthemeanrelaxation rateintheentirespectralrangefrom10+11 to10 3 Hzandbelow?(ii)Howdoesthe relaxationtimedistributionfunctionchangewithtemperature,orinotherwords,is time–temperaturesuperpositioningeneralvalidfor(dielectric)relaxationprocesses? (iii)Howdoesthestrength ε ofarelaxationprocesschangewithtemperatureinthe courseofthedynamicglasstransition?(iv)Whatisthemolecularoriginofthe“highfrequency”wing,sometimestermedexcesswing,whichisobservedinthedynamic glasstransitionofmany(lowmolecularweightandpolymeric)systems?(v)Isthere amodel-freecharacteristictemperature,whereglassydynamicsundergoesachange?
2TheoriesDescribingtheScalingofRelaxationProcesses inGlassySystems
Numerousapproaches[49–69]havebeendevelopedtodescribethedynamics ofglassysystems.Inthefollowing,twomostimportantapproachesarebriefly described.
TheScalingofRelaxationProcesses—Revisited3
Fig.1 Schemeofthedynamicalprocessestakingplaceinthespectralrangebetween10 6 and 1014 Hz,(i)the α-relaxation,(ii)(slow)secondaryrelaxations,(iii)fastabsorptionprocessesand (iv)thebosonpeak.ThetemperatureshiftisdepictedfortwodifferenttemperaturesT1 <T2
TheexperimentallyobservedVFT-dependence(Eq.(1))canbefoundedbytwo approaches:theAdam–Gibbsmodel[52]andthefreevolumetheoryasdevelopedby Doolittle[53]andCohenandTurnbull[54, 55].Thelatterisbasedonfourassumptions:
(i)Alocalvolume V isattributedtoamoleculeorpolymersegment.
(ii)Thedifference V f V V c canbeconsideredas“free”,if V islargerthana criticalvalue V c .
(iii)Ifforthefreevolume V f ≥ V * ≈ V M holds,moleculartransporttakesplacein jumpsoveradistancecorrespondingtothesizeofthemolecule V M . V * isthe minimalfreevolumerequiredforajumpofamolecule.
(iv)Themolecularrearrangementoffreevolumedoesnotrequirefreeenergy.
FollowingBoltzmannstatistics,amoleculeorapolymersegmentcarriesout positionaljumpsonlyifthenecessaryfreevolumeisprovided.Henceforthejump rate1/τ
4F.KremerandA.Loidl
1 τ ∼ ∞ V ∗ exp Vf Vf d V f ∼ exp V ∗ Vf (3)
isobtainedwhere V f istheaveragedfreevolume.Assumingthattherelativeaveraged freevolume f V f / V (V :totalvolume)dependslinearlyontemperature
while f ∗ V ∗ V istemperature-independentresultsinaVFT-equation. α f isthe thermalexpansioncoefficientofthefreevolumeand f g therelativefreevolumeat T g .ComparisonwithEq.(1)delivers
Atthetemperature T 0 ,thevolume V f vanishes.Withinthisapproach,noinherent lengthscaleisinvolvedandalltransportpropertiesshouldhavethesametemperature dependencebecausethejumpbetweenholesistheonlytransportmechanism.Cohen andGrest[55]extendedthisapproachbyconsideringsolid-andliquid-likeclusters inapercolationapproach.
ThemodelofAdamandGibbs[52]suggeststheexistenceof“Cooperatively RearrangingRegions(CRR)”beingdefinedasthesmallestvolumewhichcanchange itsconfigurationindependentfromneighbouringregions.Itrelatestherelaxationtime tothenumbersofparticles(moleculesforalowmolecularliquid,segmentsfora polymer) z (T )perCRRby
where E isafreeenergybarrierforonemolecule. z (T )canbeexpressedbythe totalconfigurationalentropy S c (T )as z (T ) S c (T )/(N kB ln2)whereNisthetotal numberofparticles,kB theBoltzmannconstantandln2theminimumentropyofa CRRassumingatwo-statemodel.Usingthermodynamicconsiderations, S c (T )can belinkedtothechangeoftheheatcapacitance cp attheglasstransitionby
With T 2 =T 0 and cp ≈ C/T fromEqs.(6)and(7),theVFT-dependencefollows. At T 0 ,theconfigurationalentropyvanishesandthesizeofaCRRdivergesas z ( T ) ∼ 1 C ( T T0 ) .TheAdam–Gibbsmodeldoesnotprovideinformationabouttheabsolute sizeoftheCRRatTg
Donth[3, 6, 7]suggestedathermodynamicfluctuationmodelleadingtoaexpressionwhichconnectstheheightofthestepin cp andthetemperaturefluctuation δT ofaCRRat T g withthecorrelationlength ξ as
TheScalingofRelaxationProcesses—Revisited5
f f g + αf T Tg (4)
DT0 f ∗ αf , T0 Tg f g αf (5)
1 τ ∼ exp z ( T ) E kT (6)
Sc ( T ) T T2 cp T d T (7)
where ρ isthedensityand (1/cp )thestepofthereciprocalspecificheat(if cV ≈ cp is assumed).Experimentally, δTcanbeextractedfromthewidthoftheglasstransition [6, 7]orfromthermalheatspectroscopymeasurements[56, 57].
Withinthefluctuationapproachforthetemperaturedependenceof ξ
isobtained.AsimilarequationwasderivedbyKirkpatrickandTirumalai[58]using scalingarguments.
BasedontheAdam–Gibbsequation(6)andanexpressionproposedbyWaterton [59]asearlyas1932,Mauroetal.[60]suggestedanapproach,whichavoidsthe divergenceoftheVFT-formula(1)at T T 0
K and C arerelatedtoactivationenergiesdeducedthrougha“physicalrealistic modelforconfigurationalentropybasedonaconstraintapproach”.
Incomparingviscousliquidswithspinglasses,SouletieandBertrand[61]suggestedforthemeanrelaxationrate
where γ >0and T c areconstants.
TheshovingmodeldevelopedbyDyreetal.[62]isbasedessentiallyonthree assumptions.
1.Theactivationenergyismainlyelasticenergy.
2.Thiselasticenergyismainlylocatedinthesurroundingsoftheflowevent.
3.Theelasticenergyismainlyshearelasticenergy.
Itrelatesthemeanrelaxationratetothemeansquarevibrationaldisplacement u 2 ( T )andacharacteristicmolecularlengtha,whichisassumedtobeconstant.
In[63, 64],itisshownthatthetemperaturedependenceoftheshearmodulus dominatesthetemperaturedependence,leadingto
6F.KremerandA.Loidl ξ3 ∼ VCRR kT 2 g (1/cp ) ρ(δT)2 . (8)
ξ ( T ) ∼ 1 ( T T0 )2/3 (9)
ν ( T ) v∞ exp K T exp C T (10)
τ 1 ∼ ( T Tc ) T γ (11)
ν ( T ) v∞ exp a 2 u 2 ( T ) (12)
u 2 ( T ) ∝ T G ∞ ( T ) (13)
where G ∞ ( T )istheelasticshearmodulus.Theshovingmodeldoesnotmakeaspecificpredictionofthetemperaturedependenceofthemeanrelaxationrate,except thatitcannotdivergeatanyfinitetemperature.Themodel,however,relatestwo independentlymeasurablequantitiesinapredictionthathasbeenconfirmedforseveralglass-formingliquids;seeforexample,thereviewoftheexperimentalsituation givenin[65].
Themode-couplingtheory(MCT)[9, 66–69]isahardspheremodelbasedona generalizednonlinearoscillatorequation
where (t )q isthenormalizeddensitycorrelationfunctiondefinedas
ρ q (t )aredensityfluctuationsatawavevectorq, isamicroscopicoscillator frequency,and ς describesafrictionalcontribution.ThefirstthreetermsofEq.(14) describeadampedharmonicoscillator;thefourthtermcontainsamemoryfunction mq (t τ ).Asaconsequence,thetotalfrictionallossesinthesystembecometimedependent.
InordertosolveEq.(14),anansatzfor mq (t )isrequired.AlreadyasimpleTaylor expansionof m leadstoarelaxationalresponseof q havingsomesimilaritywith thedynamicglasstransition[66, 67].Assuming m q (t ) v1 Φq (t )+ v2 Φ 2 q (t )(F12model,[67])deliversatwo-stepdecreaseofthecorrelationfunction q (t ).The fastercontributionisinterpretedintermsofa(fast) β -relaxationwhiletheslower componenttothedynamicglasstransition(α -relaxation).Atacriticaltemperature T c ,therelaxationtimediverges;thisisinterpretedasaphasetransitionfroman ergodic(T>T c )toanon-ergodic(T<T c ).Furthermore,MCT(intheidealized version)makesthefollowingpredictions:
(i)for T>T c therelaxationtime τ α ofthe α -relaxationscalesaccordingto
where γ isaconstant.
(ii)therelaxationfunctionofthe α -relaxationcanbedescribedby
TheScalingofRelaxationProcesses—Revisited7
d2 Φq (t ) dt 2 + Ω 2 Φq (t )+ ζ d Φq (t ) dt + Ω 2 t 0 m q (t τ ) d Φq (τ ) dτ dτ 0(14)
Φq (t ) ρq (t ) ρq (0) ρ 2 q (15)
τα ∼ η ∼ Tc T Tc γ (16)
Φq (t ) ∼ exp t τα βKWW (17)
with(0< β KWW <1),where 0 istheamplitudeofthe α -relaxation.For T > T c ,therelaxationtimedistributionshouldbetemperature-independent;i.e. time–temperaturesuperpositionshouldhold.
(iii)aboveandclosetothecriticaltemperature T c ,theminimumofthesusceptibility
εmin ,ωmin betweenthe α -relaxationandthe β -relaxationshouldfollowa powerlaw
Glassydynamicsspansatimescaleofmorethan15decades.Inordertounravel theevolutionofthetemperaturedependenceindetail,itismostadvantageousto calculatethederivativesofthemeanrelaxationratewithrespectto1/Tofthedifferenttheoreticalapproaches.Bythat,oneobtainsfortheVFT-equation(Eq.(1)) theArrheniusdependence(Eq.(2)),theMauroequation(Eq.(10)),theapproachby SouletieandtheMCT(Eq.(11))for T > T c thefollowingexpressions: VFT:
Arrhenius:
Mauro:
Souletie:
Henceinaplotofthe differential quotientd( logv/d(1/T )) 1/2 versus1/T, theVFTdependenceshowsupasastraightline.Thederivativeplotsenabletoanalyseindetail thescalingwithtemperature(Fig. 2).Thisisespeciallytrueforthehightemperature regime.Bythatthe difference quotient, ( logv)/ (/1/T )oftheexperimentaldata canbedeterminedandcomparedwiththeanalyticalderivatives.
8F.KremerandA.Loidl
εmin ∼ T Tc Tc 1/2 (18)
dlog ν d(1/ T ) −( DT0 ) log e 1 T0 T 2 (19)
dlog ν d(1/ T ) E A k · log e (20)
dlog ν d(1/ T ) K · log e · exp C T · C T +1 (21)
dlog v d(1/ T ) γ log e TC · T Tc T (22) MCT: dlog v d (1/ T ) γ · log e · T 2 TC T (23)
(a) (b)
Fig.2a ThescalingbehaviouraspredictedbytheArrheniusequation(Eq. 2),theVogel-FulcherTammannequation(VFT)(Eq. 1),theMauroapproach(Eq. 10),thatofSouletie(Eq. 11)andof themode-couplingtheory(MCT)(Eq. 16).Theglasstransitiontemperature T gasthetemperature, wherethemeanrelaxationrateaccordingtotheVFT-functionhasreachedavalueof10 2 Hzis indicated. b Differentialquotient( d(log(ν)/(d(1/T)) 1/2 × 100forthefunctionalitiesshownin a
TheScalingofRelaxationProcesses—Revisited9
3TheScalingoftheDynamicGlassTransitioninLow MolecularWeightandPolymericOrganicGlasses
Salolisoneofthemostexploredorganicglass-formingliquids.Itisconsideredas avanderWaalsglass,despitethefactthatitcanformH-bonds,presumablymainly withinthesamemolecule.InFig. 3,dielectricmeasurements[70]extendedovera broadspectralrangefromabout10 2 Hzupto1011 Hzaredisplayedfortemperatures 211and361K.Thechartsarecharacterizedbyapronounceddynamicglasstransition (α-relaxation)havinganexcesswing,whichappearsasasecondpowerlawonthe high-frequencyflankofthe α-relaxation[71].Thelatterisinterpretedasasubmerged slowsecondaryrelaxationshowingupasashoulderwithasignificantcurvaturefora sampleagedat211Kfor6.5daysasdiscussedindetailinref.[72].Forfrequencies ν >1010 Hz,ashallowlossminimumisfound;itcanbeinterpretedintermsofthe fast β-relaxationofthemode-couplingtheory(s.below)butalsootherexplanations havebeenproposed[73].
ThespectracanbedescribedbyasuperpositionofaHavriliak–Negami(HN) andCole–Cole(CC)[16]functionfortheprimary α-processorforthesecondary β-process,respectively:
Fig.3 Dielectriclossasafunctionoffrequencyforaseriesoftemperaturesfrom211Kupto 361Kforsalol.ThesolidlinesarefitswithaHavriliak–Negami(HN)functionfor T ≥ 243Kand withthesumofaHNandCole–Cole(CC)functionfor T ≤ 238K.ThedashedlinesshowtheCC components.Thedash-dottedlinethroughthe211Kdataisaguidetotheeyes.Takenandmodified from[70]withkindpermissionofTheEuropeanPhysicalJournal(EPJ)
10F.KremerandA.Loidl
ε ∗ total (ω ) ε∞ + εHN (1+(i ωτHN )βHN )γHN + εCC (1+(i ωτCC )βCC ) (24)
where εHN and εCC aretherelaxationstrengths, τHN and τCC therelaxationtimes, and βHN , γHN and βCC thespectralwidthparametersoftheHNandCCfunction, respectively,and ω isthecircularfrequency.FortemperaturesT ≥ 243K,thesecondary β-peakhascompletelymergedwiththe α-peak.
ByfittingthedielectricspectrawiththeempiricalrelaxationfunctionofEq.(24), anactivationplotisobtained,wherethemeanrelaxationrateversustheinverse temperatureisdisplayed(Fig. 4a).Thechartsattemperatures>300Kcanbeequally welldescribedbytheArrheniusequation,theformulasuggestedbyMauro(Eq.(10)) andSouletie(Eq.(11))andtheMCT-ansatz(Eq.(16)).Comparingtheexperimentally determineddifferencequotientswiththederivativesofthedifferentscalingfunctions withrespectto1/Thoweverproofsthat none ofthesuggestedformulaedescribesthe datawithinthelimitsofexperimentalaccuracyintheentiretemperaturerangeand thatitisfurthermore not possibletodescribetheexperimentaldataadequatelyby useof one VFT-functionortoreplacetheVFT-dependencebyanArrheniusfunction asonemightexpectfromtherawdatainFig. 4a.Thisissupportedaswellbyan analysis[74]basedonthesecondderivativeofthetemperaturedependenceofthe structuralrelaxationtime τ α (T )withrespectto T g /T .
Glycerol(Fig. 5a/b)isanH-bondformingliquid.Itsmeanrelaxationrateshows apronounceVFT-dependence;thedatafortemperatures ≥270Kseemtofollow equallywellaVFT-functionordependenciesassuggestedbyMauro,Souletieorthe MCT.Butfromthederivativeplot(Fig. 5b)againonemustconcludethatnoneof thesuggestedformulaefitsthetemperaturedependencecorrectlywithinthelimits ofexperimentalaccuracy.Similarlyasforsalol,twoVFT-equations(VFT1and VFT2)arerequiredtodescribethedatawithinexperimentalaccuracyintheentire temperaturerange.Fromthederivativeplots,itcanbededucedthatattemperatures above270KneithertheArrheniusequationnortheMCT-ansatzisadequate.
Thedynamicglasstransitionforpropyleneglycol,tripropyleneglycolandits polymericcounterpartpoly(propyleneglycol)havingameanmolecularweightof M w 4000g/molarecomparedinFig. 6a.BothchartsdisplayaVFT-dependence, butduetotheconnectivityofthechainforthelattertherelaxationisslower,especially atlowerrelaxationrates.Inthederivativeplots(Fig. 6b),itisshownagainthata singleVFT-dependenceis not sufficienttodescribethedataadequatelyintheentire temperaturerange.
Adielectricrelaxationprocessisnotonlycharacterizedbytherelaxationratebut alsobyitsdielectricstrengthandbytheshapeoftherelaxationtimedistribution function.AccordingtotheDebyeformula,theproduct T ε shouldbeindependent ontemperaturebesidestheweaktemperatureeffectonnumberdensityofdipoles. Insteadoneobserves(Fig. 7a)forallmaterialsthat T ε increaseswithdecreasing temperature;thismightbeinterpretedascausedbyagrowinglengthscale,where polarfluctuationsbecomemorecooperativeandhenceits effective dipolemoment increases.ThetemperaturedependenciessuggestedbytheMCT T ε ~(T c T )1/2 for T < T c and T × ε ≈ const.for T > T c arenotfulfilled.However,onehastobe awarethatthereportedvaluesofdeltaepsiloninmostcasesexhibitlargeexperimentaluncertaintiesandsometimesdifferconsiderablywhenreportedbydifferent
TheScalingofRelaxationProcesses—Revisited11
Fig.4a Activationplotforsalol. Solidlines:VFT-fits(VFT1):logν∞ 23.5,DT0 4618K,T0 141.6K;(VFT2)logν∞ 10.4,DT0 333K,T0 224.7K. Dashdoubledottedline:Arrhenius-fit logν∞ 12.1,EA /kB 2283K. Dashedline:MCTfitlogν∞ 10.4, γ 2.6,Tc 254K; dotted line:Souletiefitlogν∞ 12.1, γ 5.25, T c 239K; dash-dottedline:Maurofitlogν∞ 10.5, K 17.1K, C 1301K.Thedataaretakenfrom[37b, 75];theerrorbarsaresmallerthanthesizeofthe symbolsifnotindicatedotherwise. b Differencequotient(– (log(νmax ))/ (1000/T)) 1/2 )versus 1000/Tforthedatashownin a .Forcomparison,thedifferentialquotientsfortheVFT-equationand thetemperaturedependenciesassuggestedbythemode-couplingtheory(MCT),Souletie(SOU), andMaurotheory(MAU)usingthefitparametersshownin a
12F.KremerandA.Loidl
(a) (b)
Fig.5a Activationplotforglycerol. Solidline:VFT-fit(VFT1):logν∞ 14.3, DT 0 2448K, T 0 126.0K. Dashdoubledotted :VFT-fit(VFT2):logν∞ 12.0, DT 0 1331K, T 0 183.1K. Dashedline:MCTfitwithlogν∞ 10.4, γ 3.65, T c 248.8K. Dottedline:Souletiefitwithlogν∞ 12.8, γ 3.69, T c 215.1. Dash-dottedline:Maurofitwithlogν∞ 12.8, K 517K, C 471K. Datatakenfrom[37b, 75].The errorbars aresmallerthanthesizeofthesymbolsifnotindicated otherwise. b Experimentallydetermineddifferencequotient(– (log(νmax ))/ (1000/T )) 1/2 versus 1000/T .Thelinesdescribethefitsshownin a .Forcomparison,thedifferentialquotientsforthe VFT-fitsandthetemperaturedependenciesassuggestedbythemode-couplingtheory(MCT), Souletie(SOU),andMauro(MAU)theoryusingthefitparametersshownin a
TheScalingofRelaxationProcesses—Revisited13
(a) (b)
Fig.6a Activationplotforpropyleneglycol(opencircles),tripropyleneglycol(opentriangles) andthepolymericpendant(M w 2000g/mol)poly(propyleneglycol)(opendiamonds).Theerror barsaresmallerthanthesizeofthesymbolsifnotindicatedotherwise.Solidlines(VFT1 ):VFTfitswithlogν∞ 12.1, DT 0 793K, T 0 166Kforpropyleneglycolandlogν∞ 12.1, DT 0 833K, T 0 179Kforpoly(propyleneglycol)).Dashedlines(VFT2 ):VFT-fitsforthelower temperaturerangewithlogν∞ 14.1, DT 0 1956K; T 0 115Kforpropyleneglycol,logν∞ 13.1, DT 0 1343K; T 0 151Kfortripropyleneglycol,andlogν∞ 12.8, DT 0 1041K, T 0 169Kforpolypropyleneglycol. b Differencequotient(– (log(νmax ))/ (1000/T )) 1/2 versus 1000/T .ForcomparisonthedifferentialquotientsfortheVFT-fitsusingthefitparametersfrom a aredepicted.Thedataforpropyleneglycolandpoly(propyleneglycol)aretakenfrom[16]andfor tripropyleneglycolfrom[75]
14F.KremerandA.Loidl
(a) (b)
Fig.7a Theproductofrelaxationalstrength ε andtemperature T ,(T ε)versus T ;forsalol, propyleneglycol(PG),poly(propyleneglycol)(PPG)andglycerolasindicated.ForsalolandPPG, T ε isnormalizedby100andforPGandglycerolby1000.Theerrorbarsaresmallerthanthesize ofthesymbolsifnotindicatedotherwise.ThecriticaltemperaturesTcoftheMCTareindicatedfor thedifferentmaterials.Thedataforsalolaretakenfrom[76],forPGfrom[16],forPPGfrom[77] andforglycerolfrom[78]. b Shapeparameter β fromtheCole–Davidsonfunctionforthematerials shownin a
TheScalingofRelaxationProcesses—Revisited15 (a
(b)
)
16F.KremerandA.Loidl groups.Therearealsosomereportswhichshowatleastaroughagreementwiththe predictionsofMCT[76].
ForallexaminedmaterialsshowninFig. 7b,theshapeparameter β oftheCole— Davidsonfunctionshowsastrongtemperaturedependence.Thisholdsingeneral forthevastmajorityofglass-forming(lowmolecularweightandpolymeric)materialsandprovesthatrelaxationprocessesdo not obeytheruleoftime–temperature superpositionwhichisoftenemployedinmechanicalspectroscopy.
Schönhals[79]analysedthescalingofthedynamicglasstransitionforavarietyof glassymaterialsandsuggestedtodisplaythetwo measured quantities,therelaxation strengthversusthemeanrelaxationrate.Bythat,hefoundunambiguouslythatas differentmaterialsassalol,glycerol,propyleneglycol,dipropylenglycol,tripropylenglycolandpoly(propyleneglycole),apronouncedchangeintheslopeofthe correlationbetweenthetwodependentquantitiesexists.Thiscrossovertakesplace atameanrelaxationrateofabout108 Hzandmarksperhapsthebeginningofcooperativedynamics.Forallmaterials,therelaxationstrengthincreasesstronglywith decreasingtemperature.Extrapolatedtohightemperatures,themeanrelaxationrate isintherangebetween1011 and1013 Hzwhichistypicalforhighlyactivatedlibrationalfluctuations.Thefactthatacrossovertemperature T B existscanbeinterpreted inseveralways;(i) T B andthecriticaltemperature T c oftheMCThavesomeresemblance,hencethecrossovermightreflectatransitionfromanergodictoanon-ergodic state.(ii) T B canbealsocomprehendedastheonsetofacooperativedynamicsas suggestedbyDonth[6, 7].Itischaracterizedbycooperativelyrearrangingdomains havingasize ξ (T )whichincreaseswithdecreasingdiameter.Atthecalorimetric glasstransitiontemperatureTg ,avaluebetween2and3nmcanbeestimatedbased onmultiplestudies[80]ofglassydynamicsinnanometricconfinement(Fig. 8).
Themode-couplingtheorymakesdetailedpredictionsfortheminimumregion betweenthe“microscopicpeak”andthedynamicglasstransitionfollowingamaster function:
withtemperature-independentexponentsaandbbeinginterrelatedas
2 (1+ b )
(1+2b )
(1 2a ) (26)
where isthe -function.Theexponentscanbeaswelldeterminedfromthetemperaturedependenceofthefrequencyoftheminimumofthesusceptibilityandof thefrequencyofthemaximum ω max ofthedynamicglasstransition
ε (ω ) εmin a + b b ω ωmin a + a ωmin ω b (25)
Γ
Γ
λ Γ
Γ
2 (1 a )
Fig.8 Relaxationalstrength ε,normalizedwithitsmaximumvalueversusthemeanrelaxation ratelog νmax forsalol,propyleneglycol(PG),poly(propyleneglycol)(PPG)andglycerolasindicated.AtthetemperatureTB ,theslopeofthecorrelationbetween ε and νmax changes.Thedata forsalolaretakenfrom[22],forPGandPPGfrom[16]andforglycerolfrom[81]
Carryingoutsuchananalysisdeliversforglycerolavalueof a 0.325and b 0.63.Forthelowesttemperatures,theincreasetowardsthebosonpeakapproachesa powerlaw ε ~ ν3 asindicatedbythedashedlineinFig. 9.Theinsetdemonstratesfor twotemperaturesthatthesimplesuperpositionansatzof,Eq.(25),isnotsufficient todescribetheshallowminimum.
TheScalingofRelaxationProcesses—Revisited17
ωmin ∼ T Tc Tc 1 2a
ωmax ∼ T Tc Tc ( 1 2a + 1 2b ) (28)
(27)
Fig.9 Dielectriclossofglycerolintheminimumandbosonpeakregion.Thesolidlinesarefitswith theMCTprediction,Eq.(10),with a 0.325, b 0.63forglycerol.Forthelowesttemperatures, theincreasetowardsthebosonpeakapproachespowerlaws ε ~ ν3 forglycerolasindicatedbythe dashedline.Notethat,incontrasttoPC,thebosonpeakseemstobesuperimposedtotheshallow minimuminglycerol.Theinsetdemonstratesfortwotemperaturesthatthesimplesuperposition ansatz,Eq.(9),isnotsufficienttoexplaintheshallowminimum.Takenfrom[82]withpermission
4Conclusions
Inthespectralrangebetween10 3 and1013 Hz,fourdynamicprocessestakeplacein thedynamicglasstransition,slowandfastsecondaryrelaxationsandtheboson-peak. Thequestionsformulatedintheintroductioncanbeansweredindetail:
(i)Isthereascalingfunctionwhichdescribesthetemperaturedependenceofthe meanrelaxationrateintheentirespectralrangefrom10+13 to10 3 Hzand below?
Forallmaterialsunderstudy, none ofthesuggestedscalingfunctionsisableto describetheobservedtemperaturedependenceofthemeanrelaxationratein theentirespectralrange.Theanalysisofthedatausingderivativeplotsreveals furthermore,thateveninthehigh-frequencylimitanArrheniusdependence doesnotdescribethemeasurementswithinthelimitsofexperimentalaccuracy.TheempiricalVogel–Fulcher–Tammanndependenceturnsouttobea coarse-graineddescriptiononlywithinalimitedtemperaturerange.Thereis noindicationpointingtowardsadivergenceattheVogeltemperature T 0 .
(ii)Howdoestherelaxationtimedistributionfunctionchangewithtemperatureor inotherwords,istime–temperaturesuperpositioningeneralvalidfor(dielectric)relaxationprocesses?
Therelaxationtimedistributionfunctionwithitsshapeparameters β and γ showsapronouncedtemperaturedependence.Hence,time–temperaturesuperpositionis not validingeneral.
18F.KremerandA.Loidl 910111213 -1 0 1 log10 [ν (Hz)] log 10 ε " ~ν 3 323K 295K 273K 363413K K 184K 234K glycerol 253K 101112 -1 0 253K 273K
(iii)Howdoesthestrength ε ofarelaxationprocesschangewithtemperaturein thecourseofthedynamicglasstransition?
Therelaxationstrength ε ofthedynamicglasstransitiondecreaseswith increasingtemperature,aneffectwhichcanbenotexplainedbythetemperature dependenceofthedensity.Insteadanincreasinglengthscaleofthedynamic glasstransitionseemstobelikelyresultinginanincreasedeffectivedipole moment.
(iv)Whatisthemolecularoriginofthe“high-frequency”wingwhichisobserved inthedynamicglasstransitionofmany(lowmolecularweightandpolymeric) systems?
Severalglassformersshowahigh-frequencywing;itisconsideredasaslow secondaryrelaxationwhichmightbecoupledtothedynamicglasstransition.
(v)Whatistheassignmentofthe“fastsecondaryrelaxation”?
Inthespectralrangebetween109 and1012 Hz,afastsecondaryrelaxationis observed.ItcanbequantitativelydescribedbytheMCT.
(vi)Isthereacharacteristictemperature,whereglassydynamicsundergoesa change?
AssuggestedbyA.Schönhals,oneobservesbydisplayingthecorrelation betweenthetwodependentvariables,relaxationstrengthandmeanrelaxation rate—withoutanyassumptions—atransitionatabout108 Hz.Thismightbe interpretedastheonsetofcooperativedynamicswithdecreasingtemperature.
Acknowledgements SupportbyM.Antoninpreparingsomeofthefiguresishighlyacknowledged.
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