Nearly-logarithmic decay of correlations in glass-forming liquids

0.26 MB·English

by  W. Gotze

Nearly-logarithmicdecay ofcorrelations inglass-formingliquids
W. Go¨tze and M. Sperl
Physik-Department, Technische Universita¨t Mu¨nchen, 85747 Garching, Germany
(Dated:January23,2004)
Nearly-logarithmicdecayofcorrelations,whichwasobservedforseveralsupercooledliquidsinoptical-Kerr-
4 effectexperiments[G.Hinzeetal.Phys.Rev.Lett.84,2437(2000),H.Cangetal.Phys.Rev.Lett.90,197401
0 (2003)],isexplainedwithinthemode-couplingtheoryforidealglasstransitionsasmanifestationoftheb -peak
0 phenomenon. A schematic model, which describes the dynamics by only two correlators, one referring to
2 densityfluctuationsandtheothertothereorientationalfluctuationsofthemolecules,yieldsforstrongrotation-
translationcouplingresponsefunctionsinagreementwiththosemeasuredforbenzophenoneandSalolforthe
n
timeintervalextendingfrom2picosecondstoabout20and200nanoseconds,respectively.
a
J
PACSnumbers:64.70.Pf,61.20.Lc,33.55.Fi,61.25.Em
7
2

Optical-Kerr-effect(OKE)spectroscopyisapowerfultech- critical value T. Using e =(T −T)/T as a small parame-
] c c c
n nique for the study of the dynamics of supercooled liquids ter,thelong-timebehaviorofthecorrelatorscanbeevaluated
n [1]. The experiment provides a response function c (t) for byasymptoticexpansions[7]. Theleading-orderformulases-
A
-
s times t exceeding a fraction of a picosecond. The function tablish the universalfeatures of the dynamics. Comparisons
i c (t)isproportionaltothenegativetimederivativeofacor- of experimental results with these formulas [8] and tests by
d A
. relator f A(t)=hA∗(t)Ai/h|A|2i. Here, hi denotes canonical molecular-dynamicssimulation [8, 9] have shown thatMCT
t
a averaging, and the probingvariable A is the anisotropic part isaseriouscandidateforanexplanationofglassydynamics.
m ofthedielectricfunction. Theinstrumentationwasimproved Nearly-logarithmicdecayofcorrelationswaspredictedwithin
- recently by applicationof heterodynedetection. As a result, MCT for states close to bifurcationsof the cusp type as can
d itwaspossibletomeasurec (t)fortimesupto500nanosec- be inferredfromRef. [10] and the paperscited there. Itwas
n A
onds,i.e.,glassydynamicswasdocumentedfortheenormous pointed out [2, 5] that the OKE data might be fitted by the
o
c timeintervalstartingattheendofthetransientandextending universalresultsfor relaxationnear cusp bifurcations. How-
[ overmorethanfiveordersofmagnitude[2].Itwasshownthat ever,onehastovaryatleasttwo controlparametersinorder

theevolutionoftheglassydynamicsofSalolupondecreasing to approach such singularity. A fit of the OKE data, which
1
v thetemperatureT canbeinterpretedbytheuniversalformulas onlydependonthesinglecontrolparameterT,wouldrequire
6 derived within the mode-couplingtheory of ideal glass tran- implausible fine tuning of the coupling coefficients entering
3 sitions (MCT) [1, 2]. The fit values for the variousparame- theMCTequations.Theexplanationproposedbelowisbased
5
tershavebeenfoundtobe consistentwith thoseobtainedby ontheexistenceofthesimpleliquid-glass-transitionsingular-
1
otherlight-scatteringtechniques[3,4]. However,theuniver- ity. OurresultsarevalidforresponsefunctionsofvariablesA
0
4 salformulasdonotdescribethedataforalltimesoutsidethe thatcouplestronglytodensityfluctuationssothataso-called
0 transient regime. Rather, if T decreases, there opens a time b -peakcanoccurinthesusceptibilityspectra[11,12].
t/ intervallargerthantwoordersofmagnitude,whichprecedes Let us consider an MCT model describing schematically
a the interval of validity of the universal formulas. Here, the the density fluctuations by a single correlator f (t). It obeys
m
response for Salol follows closely a 1/t-law, i.e., the corre- theZwanzig-Moriequationofmotion
- lator exhibits nearly a logarithmic decay: f (t)(cid:181) −ln(t/t )
d A
t
on [v2a]n.-dTehri-sWinatarlisguliiqnugidfes,atluikreewbeanszoobpsheernvoendea(lsBoZfPo)r.sTohmisehoothldesr ¶ t2f (t)+¶n tf (t)+W 2f (t)+W 2Z0 m(t−t′)¶ t′f (t′)dt′=0,
c with the reservation that the heretofore unknown relaxation (1)
: whereW andn arefrequenciesparameterizingtheshort-time
v process can be described more adequately by a power law,
i f (t)− f (cid:181) −tb′, albeit with a rather small exponent b′ [5]. asymptote, f (t)=1−1/2(W t)2+1/6Wn 2t3+O(t4). The
X A
kernel m(t), which represents the interactions of the density
Gaininganunderstandingof theindicatedfindingsisa chal-
r fluctuations,ismodeledas
a lenge to all theories aiming to unlock a comprehensive de-
scription of liquids. In this Letter, it will be shown that the
m(t)=v f (t)+v f 2(t). (2)
measured complex relaxation scenario [2, 5] is a generic, 1 2
thoughnotuniversal,implicationofthestandardMCT.
Here, v1 >0 and v2 >0 are the coupling coefficients. The
TheMCTwasproposedasamathematicalmodelforglassy state ofthe systemis specifiedbya pointV=(v ,v )in the
1 2
dynamics[6], whosefascinatingfeaturesareobtainedasim- v -v plane. For small V, the model describes liquid states
1 2
plication of bifurcationpoints of nonlinearequationsof mo- wheref (t →¥ )=0. ForlargeV,onegetsanontriviallong-
tionderivedforthedensityfluctuations.Thebasicbifurcation time limit f =f (t →¥ ), 0< f <1. Parameter f quantifies
is a fold singularity describing a transition from ergodic to thearrestofdensityfluctuationsinthenon-ergodicglassstate.
non-ergodicbehavior if the temperature decreases througha Therearelinesofliquid-glasstransitionpointsVc=(vc,vc),
1 2
2
each characterized by some number l ,1/26 l < 1. This The VH-light-scattering spectra of Salol [3] or orthoter-
numberdeterminesthecriticalexponenta,0<a<1/2,and phenyl [17] show that fc is about 0.9. This means there is
A
the von Schweidler exponent b, 0 <b 61, by the relation strong arrest of the reorientationalmotion at the ideal glass-
G (1−a)2/G (1−2a)=l =G (1+b)2/G (1+2b). We con- transition points in these van-der-Waals systems. Integrat-
sider the line, which can be parameterized by vc = (2l − ing c (t) over time, one can determine f (t) and read off
1 A A
1)/l 2, vc =1/l 2, 0.56l <1.Crossingthe line, thelong- the plateau. We find fc =0.90±0.05 from the T =260K-
2 A
time limit of f (t) jumpsfromzero to the criticalvalue fc = dataforBZP[5]and fc=0.93±0.03fromthoseforSalolat
A
1−l . ThedistanceofthestateVfromVc isspecifiedbythe T=257K[2],corroboratingtheprecedingconclusion.There-
separationparameters =[(v −vc)+(v −vc)fc]fc(1−fc), fore,wefocusonc (t)forlargetranslation-rotationcoupling
1 1 2 2 A
whichisnegativeforliquidstatesandpositiveforglassstates. v .
A
Thismodelisthesimplestonereproducingallvaluesforthe
anomalousexponents,whichcanoccurinthe generaltheory 0
[13]. The parameter fc appearsas plateauvalue ofthe f (t)- -1
versus-logt curvesforVnearVcorasrelativestrengthofthe -0.05
a -peakofthelossspectrum[7]. s
-0.10
Theprobing-variablecorrelatorf (t)obeysthesamegen- -2
A
emr(atl)ehqauvaetiotonboefrmepoltaiocnedanbdyitnhietiaclodrreecsapyonasdifn(gt)q,ubaunttWiti,ensWanAd, (t)A 240260T28[K0]300320
n A andmA(t). Themodelforthekernelis c10-3
g
o
l 30
mA(t)=vAf (t)f A(t), (3) 20 v
A
10
-4 sc
wherev >0 quantifiesthe couplingofthe probingvariable v
A 0.8 1
tothedensityfluctuations. IfvAfc>1,thelong-timelimitof 0.6
f A(t)jumpsfromzerotothecriticalvalue fAc=1−1/(vAfc) 1 1.5 2 v2
as V crosses the transition line. Originally, this model was -5
0 1 2 3 4
motivatedforthedensityfluctuationsofataggedparticle[14]. log t[ps]
10
Also,theMCTequationsforreorientationaldynamicsofalin-
earmoleculesuggestanexpressionlikeEq.(3)[15]. Within FIG. 1: OKE response c A(t) measured for BZP for temperatures
the microscopic theory, the coupling coefficients depend on T/K=251,260,290,320 (fulllinesfrombottom totop) [5]. The
the equilibrium structure functions, which depend smoothly dashedstraightlinehasslopex=0.80. Thedottedlinesexhibitre-
on T. Therefore, v1, v2, and vA are smooth functions of sultsN·c A(t),withc A(t)calculatedfortheMCTmodeldefinedby
thetemperature. Thespecifiedmodelwasappliedrepeatedly Eqs.(1–3)withW =W A=0.2n =0.2n A. ThefactorsN andW are
chosen(frombottomtotop)as3.5,6,9.5,8,and1.67,1.43,2.0,2.0
for the description of experiments, as can be inferred from
THz,respectively.Theleftinsetexhibitsthevariationofthecoupling
Ref.[16]andthepaperscitedthere.
constantswithchangesofthetemperature,andtherightinsettheone
The long-time decay of the correlators at the transition oftheseparation parameters . Thedashedlinemarked scshowsa
pointisgivenby[f (t)− fc]/h=[f A(t)− fAc]/hA=(t0/t)a+ scaling-lawresult, Eq.(4),calculatedforl =0.70. Thelineinthe
O(t−2a). The time t depends on the transient dynamics. It lowerleftinsetexhibitstransitionpointsVc; thepoints =−0.003
0
isdeterminedbymatchingthenumericalsolutionoff (t)for forl =0.70,ismarkedbyacross.
large times to the asymptotic formula. The amplitudes are
h=l ,hA=l /(vAfc2).Inthelimitofsmall|s |,thereappears Figure 1 reproducesthe OKE response functionsfor BZP
alargetimeintervalwhere|f (t)−fc|issmall. Here,onegets [5]. A dashed straight line of slope x = 0.80 shows that
inleading-order[f (t)−fc]/h=[f A(t)−fAc]/hA=cs g(t/ts ). the data for T = 251 K exhibit power-law decay c (t) (cid:181)
This is a scaling law with a correlation scale cs = |s | 1/tx for 2 ps6t 620 ps. There is a von Schweidler-law-
and a time scale ts =t0/|s |1/2a. The function g(tˆ) is dpeter- like variation: f (t)−const. (cid:181) −tb′,b′ = 1−x = 0.20. In
minedsolelybyl . Itexhibitsthecriticalpowerlawforsmall thedouble-logarithmicrepresentation,thescaling-lawresult,
rescaledtimestˆ,g(tˆ≪1)=1/tˆa,andvonSchweidler’spower Eq. (4), appears as interpolation between a straight line of
lawforlargerescaledtimesintheliquid,g(tˆ≫1)=−Btˆb[7]. slope(1+a)>1,andoneofslope(1−b)<1. Changingthe
Thescalinglawimpliesonefortheresponsefunctions scales ss hA andts is equivalentto a translationof the curve.
The dashed line marked sc shows an example calculated for
c (t)/h=c A(t)/hA=ss k(t/ts ), (4) l =0.70 (a=0.33,b=0.64) shifted to match the data for
T =251 K. It providesa proper description of the observa-
wheress =cs /ts . Themasterfunctionk(tˆ)=−¶ gˆ/¶ tˆinter- tionsfor60ps<t<6ns. However,since1+a>1>x,the
polatesbetweenthecriticalpowerlaw,k(tˆ≪1)=−a/tˆ(1+a), scaling law cannot account for the data for t <60 ps. This
and,intheliquidstate,thevonSchweidlerpowerlaw,k(tˆ≫ conclusion [5] is not altered by choosing l or ts , ss differ-
1)=−bB/tˆ(1−b). ently. Thedotsarethetheoreticalresultsforc (t)scaledby
A
3
amplitudefactorsN andscalesW ,whichvarysomewhatwith wherethefitvalueT =1000Kisnotunreasonableforavan-
0
T. Notice that the calculated functions exhibit transient os- der-Waalsliquid.
cillationsfortimesbelowandupto2ps,asdothemeasured
curves. ThisshowsthattheparametersW ,n ,W ,n arecho- 3 2
A A +
senreasonably.Thestrongvariationofc (t)withchangesof 6
T isdueto the changesof thecouplingcAoefficientsasdocu- W(t)/ 4 log10c (t)ts’
c0 0
mentedintheleftinset. ThestateV=(v1,v2)shiftstowards og1 2
thetransitionlineifT decreases. Anextrapolationtoa tran- l
sition point for l =0.70, which is near the cross, is consis- -2 cc 0
tent with the fit values for v1 and v2. Parameter s exhibits W(t)/A n=2 -2
an almost linear temperature dependence, and extrapolation c0 -8 -6 -4 -2 0
tos =0suggestsTc≈235K. Uncertaintyestimatesforl and og1 -4 log10t/ts’
l
Tc cannot be given yet, since we did not study in detail the 3
possibilityfordatafitsbyotherchoicesforv ,v ,andv . 1/w
1 2 A -6 b 2
0
0 4
-8
-1 -0.01 3
6
s log c (t)t’
-2 -0.02 4 10 A s
-10 cc
240260280300320340 2
) T[K] 4
(tA-3 0
60
cg10-4 vA -12 -2 log10t/ts’
o 30
l
-8 -6 -4 -2 0
-5 v
0.9 1 0 2 4 6 8 10
log tW
-6 0.8 10
1.6 1.8 2 v
2 FIG.3:ResponsefunctionsforthemodeldefinedbyEqs.(1–3)cal-
-7
-2 -1 0 1 2 3 4 5 culatedwithW =W A=n /5=n A/5andvA=20forthreestatesspec-
log10t[ps] ifiedbyl =0.70,(v1,v2)=(vc1,vc2)(1+e ),e =−10−n,n=2,3,4,
yielding s =3e /10. The results for c (t) are shifted upwards by
FIG.2:AnalogresultsastheonesinFig.1fordatameasuredforSa- 3 decades. The dashed lines are the scaling-law results, Eq. (4),
lolforT/K=247,257,270,300,340[2]. Thedashedstraightline wheret0=0.755. The dotted line shows the leading correction to
hasslopex=1.15.TheMCTresultsarecalculatedfor7.9THz=W = thescaling-lawforshorttimes. Thedottedlinemarkedccexhibits
2W A=10n A,n =0,andfactorsN=25,37.3,27.8,17.9,19,respec- theCole-Coleresponsefunctionc cAc(t). Theinsetsdemonstratethe
tively. Theopencirclesintheinsetsarethefitvaluesfortheother superpositionprincipleforthelong-timedynamics.
measuredtemperatures[2];thecorrespondingresponsefunctionsare
notshowntoavoidoverloading. Figure 3 shows the two response functions, c (t) =
−f¶ (t)/¶ t and c (t) = −f¶ (t)/¶ t, for three liquid states
A A
Figure2showstheOKEresponseforSalolforthehighest near the transition point with l =0.70 and the correspond-
and lowest temperature measured and three temperatures in ing scaling-law results. There is a second scaling law, the
between[2]. Thedataarenormalizedtounityfort=0.01ps. superposition principle for the a process, dealing with the
A dashed straight line of slope x=1.15 shows that the data dynamics for times within the von Schweidler-law regime
forT =247Kcanbedescribedbyc (t)(cid:181) 1/txfortheinterval and longer: c (t) = c˜(t/ts′)/ts′,c A(t) = c˜A(t/ts′)/ts′, ts′ =
2ps<t<100ps. Equivalently,thecorrelatorshowsadecay t /|s |g ,g =1/(2a)+1/(2b). Theinsetsdemonstratethisre-
0
similartothecriticalone,f (t)(cid:181) 1/ta′ witha′=x−1=0.15. sult. ThesolutionatVcincludingtheleadingcorrectionreads
For the T = 257 K data, a similar behavior is found with [f (t)− fc]=h (t /t)a[1+K (t /t)a] and a corresponding
A A A 0 A 0
x≈0,i.e.,f (t)exhibitslogarithmicdecay[2]. TheMCTfits formula with index A dropped. The number K = −1.5 is
A
havebeenevaluatedfortemperature-independentparameters evaluatedfromthecouplingconstantsv ,v ,v ;itisnegative
1 2 A
W ,W ,n ,n . ThenormalizationconstantsN exhibitsomeT and largesince fc is high [7]. The resultis shownas dotted
A A A
dependence. The path V=(v ,v ) extrapolates to a critical lineforc .Thecorrespondingcorrectionforc (t),K=0.020,
1 2 A
pointforl =0.73(a=0.31,b=0.59). The s -versus-T re- is so smallthat it cannotbe made visible in Fig. 3. Thisex-
sultssuggestT ≈245K.Thenumbersforl andT areconsis- emplifies that the range of validity of the universalformulas
c c
tentwiththoseobtainedpreviously[1,2,3,4].Fortheidenti- canbequitedifferentfordifferentfunctions. Thescaling-law
ficationoftheidealtransitionpointitwouldbehelpfultohave formulasexplainquantitativelytheevolutionoftheglassydy-
data available also for T below the estimated T. The varia- namicsasexhibitedbyc (t)andthisforthecompleteregime
c
tionofv closelyfollowsanArrheniuslaw: v (cid:181) exp(T /T), outside the transient, W t >2. The state n=2 is marked by
A A 0
4
a cross in the left inset of Fig. 1. The states of interest for equations.Theleading-ordercorrectionformulasexplainthat
thedatainterpretationhaveamuchlargerseparationfromthe for strong probing-variable-environment coupling there ap-
critical point. One finds here, as in previous work [7, 16], pearsalargetimeintervaladjacenttotheoneforthetransient
thatthescaling-lawdescriptioncanexplainallthequalitative motion where the scaling laws cannot explain the data. The
features of c (t), but that a quantitative description does not nearly-logarithmicdecay within this interval is explained as
workfor|s |>0.05.Thesameconclusionsarevalidforc (t), b -peakdynamicsandisfittedperfectlybythesolutionsofthe
A
albeit only for times with tW '3000. The size of the cor- schematicmodel.Ourtheoryimpliesthepredictionofnearly-
rectiontermK issolarge,thatthereappearsathree-decade logarithmicdecayofreorientationalcorrelationsforliquidsof
A
timeintervaloutsidethetransientwheretheuniversalformu- molecules with large elongation. Molecular-dynamics sim-
lascannotaccountfortheMCTsolution. Withintheinterval ulations can test this and provide the structure factors for a
26log (W t)63.5,theleading-order-correctionformulade- microscopic calculation within MCT. Such work could sub-
10
scribestheresults. Butfor0.56log (W t)62,eventhisfor- stantiateorfalsifytheprecedingexplanation.Itremainstobe
10
mulaisinsufficient.Withintheinterval0.56log (W t)63.5, explainedwhytheb -peakphenomenonhasnotbeendetected
10
nearly-logarithmicdecayisexhibitedbyf (t). inearlierstudies.
A
Theequationofmotionforf A(t)isequivalenttoc A(w )= WethankM.D.FayerandG.Hinzeforprovidingtheirdata
−W 2A/[w (w +in A)−W 2A+W Aw mA(w )], where c A(w ) and sets. WethankthemaswellasH.Z.Cummins,V.N.Novikov,
mA(w ) are the Fourier transforms of c A(t) and mA(t), re- R. Schilling and Th. Voigtmann for stimulating discussions.
spectively, for frequency w . For low-frequency phenom- OurworkwassupportedbytheDFGGrantNo. Go154/13-2.
ena, |w (w +in )|≪W 2, this simplifies to c (w )=1/[1−
A A A
w mA(w )]. Fort/t0≫1andt/ts ≪1,onegetsuptonext-to-
leadingorder: m (t)= f +h (t /t)a[1+K (t /t)a], where
A m m 0 m 0
f = v fcfc, h = v (fch + fch), K h = v [fch K +
m A A m A A A m m A A A
fchK+hh ]. There is a trend that the negative contribution [1] R. Torre, P.Bartolini, and R.M. Pick, Phys.Rev. E57, 1912
A A (1998).
ofthefirstterminthebracketiscanceledbythepositiveone
[2] G. Hinze, D. D. Brace, S. D. Gottke, and M. D. Fayer,
due to the last term. Hence, because the correctionterm K
A Phys.Rev.Lett.84,2437(2000),4783(E);J.Chem.Phys.113,
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is small. Thus, the critical law is a good approximation for [3] G.Li,W.M.Du,A.Sakai,andH.Z.Cummins,Phys.Rev.A
m (t). ThisleadstotheCole-Coleformulafortheb -process: 46,3343(1992).
A
c cAc(w )=c c0c/[1+(−iw /w b )a], c c0c =1/[1+ fcfAcvA], w b = [4] Y.YangandK.A.Nelson,Phys.Rev.Lett.74,4883(1995).
[(1+fm)/hmG (1−a)]1/a/t0. Thelimitw /w b ≪1reproduces [5] H19.7C4a0n1g(,2V0.0N3).;NJ.oCvihkeomv,.aPnhdysM.1.1D8.,F2a8y0e0r,(2P0h0y3s)..Rev.Lett.90,
theuniversalcriticaldecay:c (t)(cid:181) 1/t1+a. Thelimitw /w b ≫ [6] U.Bengtzelius,W.Go¨tze,andA.Sjo¨lander,J.Phys.C17,5915
1leadstoavonSchweidler-law-likedecay: c (t)(cid:181) 1/t1−a. It (1984).
dependsontherelativesizeoft0−1,w b andts−1,whichpartof [7] T.Franosch,M.Fuchs,W.Go¨tze,M.R.Mayr,andA.P.Singh,
theb -peakspectrumdominatesthenearly-logarithmicdecay, Phys.Rev.E55,7153(1997).
c (t)(cid:181) 1/tx,1−a6x61+a.ThedottedlineinFig.3marked [8] W.Go¨tze,J.Phys.:Condens.Matter11,A1(1999).
[9] W. Kob, in Slow Relaxations and Nonequilibrium Dynamics
ccexhibitsc cc(t)forourmodel. Theprecedingreasoningfor
A in Condensed Matter, edited by J.-L. Barrat, M. Feigelman,
usingtheleadingasymptoticexpansionform (t)ratherthan
A J.Kurchan,andJ.Dalibard(Springer,Berlin,2003),p.199.
for f A(t) is valid also for the microscopic version of MCT. [10] W.Go¨tzeandM.Sperl,Phys.Rev.E66,011405(2002).
Theschematicmodelusedhereismerelythesimplestexam- [11] G.Buchalla, U.Dersch, W.Go¨tze, andL.Sjo¨gren,J.Phys.C
pleillustratingourderivationoftheb -peakphenomenon. 21,4239(1988).
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(1989).
measured by OKE spectroscopy for two liquids can be de-
[13] W.Go¨tze,Z.Phys.B56,139(1984).
scribed reasonably by the solutions of a standard schematic
[14] L.Sjo¨gren,Phys.Rev.A33,1254(1986).
MCT model, and this for time intervalsextending up to five
[15] S.-H.ChongandW.Go¨tze,Phys.Rev.E65,051201(2002).
ordersofmagnitude.Thelong-timepartoftheresponsefunc- [16] W.Go¨tzeandT.Voigtmann,Phys.Rev.E61,4133(2000).
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lawsreflectingleading-orderasymptoticsolutionsoftheMCT Prog.Theor.Phys.Suppl.126,21(1997).