DTIC ADA550062: Correlation of Nonexponentiality with Dynamic Heterogeneity from Four-Point Dynamic Susceptibility X4(t) and Its Approximation XT(t)

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THEJOURNALOFCHEMICALPHYSICS133,124507(cid:2)2010(cid:3)
Correlation of nonexponentiality with dynamic heterogeneity
from four-point dynamic susceptibility (cid:2) t and its approximation (cid:2) t
4„ … T„ …
C. M. Roland,1,a(cid:2) D. Fragiadakis,1 D. Coslovich,2 S. Capaccioli,3 and K. L. Ngai1
1NavalResearchLaboratory,Washington,DC20375-5342,USA
2InstitutfurTheoretischePhysik,TechnischeUniversitatWien,WiednerHauptstraße8-10,A-1040Wien,
Austria
3DipartimentodiFisica,UniversitàdiPisa,LargoPontecorvo3,I-56127Pisa,Italy
(cid:2)Received 9 June 2010; accepted 1August 2010; published online 24 September 2010(cid:3)
Variouspropertiesofvitrifyingliquidsarecorrelatedwiththedispersityofthedynamics,thelatter
reflected in the magnitude of the nonexponentiality parameter, (cid:2), describing the distribution of
K
relaxation times. These properties include the mean relaxation time, (cid:3)(cid:4), the fragility, and the
dynamiccrossover.Thecorrelationswith(cid:2) areobservedinbothexperimentaldataandtheresults
K
frommoleculardynamicssimulationsonLennard-Jones(cid:2)LJ(cid:3)typesystems.Another,ratherobvious
propertytocorrelatewith(cid:2) isthedynamicheterogeneity,whichcanbequantifiedfromthenumber
K
of molecules, Nc, dynamically correlated over a time span (cid:3)(cid:4). For a given LJ system, Nc can be
rigorously calculated and we find that it does indeed correlate with (cid:2) over a range of
K
thermodynamic conditions. However, the analysis of experimental data for a broad range of real
materials,whereinanapproximationisrequiredtoobtainN ,revealstheabsenceofanyrelationship
c
between N and (cid:2) among different materials. © 2010 American Institute of Physics.
c K
(cid:4)doi:10.1063/1.3481355(cid:5)
I. INTRODUCTION Since the conventional description of the dynamics in-
volves the correlation of a variable at two times (cid:2)for ex-
Viscous liquids are sufficiently dense for molecules to ample, (cid:6)(cid:5)(cid:2)˜k,0(cid:3)(cid:5)(cid:2)˜k,t(cid:3)(cid:7), where (cid:5)is the density and ˜k is the
exertareciprocalinfluence,andthesecorrelated,many-body
wave-vector(cid:3), it follows that a four-point, time-dependent
interactions distinguish the supercooled regime from the
correlation function is required to describe dynamic
much simpler dynamics at high temperature. The degree of
heterogeneities.21,22 Thus, the susceptibility defined in terms
intermolecular cooperativity varies spatially, with the dy-
of spatial and temporal correlations,
namiccorrelationspersistingovertimescalesontheorderof (cid:8)
the relaxation time. Since dynamic heterogeneity is intrinsic
(cid:6)(cid:2)t(cid:3)= (cid:6)(cid:5)(cid:2)r ,0(cid:3)(cid:5)(cid:2)r +r ,0(cid:3)(cid:5)(cid:2)r ,t(cid:3)(cid:5)(cid:2)r +r ,t(cid:3)(cid:7) dr ,
to the supercooled liquid state, it is an obvious quantity to 4 1 1 2 1 1 2 r1 2
characterize dynamic properties.1–4 One consequence of dy-
(cid:2)1(cid:3)
namic heterogeneity is deviation of the linear susceptibility
from the Debye or exponential behavior, with the relaxation has a maximum at t(cid:9)(cid:3)(cid:4)that is proportional to Nc, the num-
functionbecomingmoredispersivetoreflecttheheterogene- ber of correlating molecules22–24
ity of the local dynamic environments. A variety of experi-
mental methods have been employed to demonstrate that N =max(cid:10)(cid:6)(cid:2)t(cid:3)(cid:11)(cid:12)(cid:6)max. (cid:2)2(cid:3)
c 4 4
nonexponentialrelaxationresultsfromasuperpositionofdy-
namicallydistinguishablecontributions.5Thisnonexponenti- Efforts to study dynamic heterogeneities have been stymied
by the difficulty of determining (cid:6)(cid:2)t(cid:3) since conventional re-
ality, commonly assessed from the magnitude of the Kohl- 4
rausch stretch exponent, (cid:2), is known to correlate with laxation spectroscopies yield the linear susceptibility. How-
relaxation properties such Kas fragility6 and the dynamic ever, molecular dynamics simulations20,25–28 and multidi-
crossover (cid:2)change in dynamics above T (cid:3).7Additionally, (cid:2) mensional NMR29–32 have been employed to obtain (cid:6)4(cid:2)t(cid:3)
g K
respectively for model particle systems and a few real mate-
is related to phenomena such as rotational-translational
decoupling8–11 and the confinement-induced enhancement of rials (cid:2)data for the latter limited to low temperatures(cid:3).
molecularmobilities.12–14Thus,quantifyingthedynamichet- A development in quantifying dynamic heterogeneities
in real materials was the derivation by Berthier et al.21 of
erogeneity and nonexponentiality of liquids can provide in-
(cid:6)(cid:2)t(cid:3) in terms of the temperature derivative of two-point lin-
sight into the fundamental variables governing the dynamics 4
ear susceptibilities, (cid:7)(cid:2)t(cid:3),
nearT .Indeed,modelsoftheglasstransitiongenerallyposit
g (cid:13) (cid:14)
a growing dynamic correlation length as causing the marked k k (cid:3)(cid:7)(cid:2)t(cid:3) 2
liantciroenasreesouflt(cid:3)s(cid:4)siungvgietrsitfyainngalltieqruniadtsiv15e–1v9ie(cid:2)awltphooiungt2h0(cid:3)r.ecentsimu- (cid:6)4(cid:2)t(cid:3)(cid:8) (cid:9)cBPT2(cid:6)T2(cid:2)t(cid:3)= (cid:9)cBPT2 (cid:3)T , (cid:2)3(cid:3)
where (cid:9)c is the isobaric heat capacity change at T and k
P g B
a(cid:3)Electronicmail:[email protected]. is the Boltzmann constant. This offers the possibility of ob-
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4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Correlation Of Nonexponentiality With Dynamic Heterogeneity From
5b. GRANT NUMBER
Four-Point Dynamic Susceptibility X4(t) And Its Approximation XT(t)
5c. PROGRAM ELEMENT NUMBER
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14. ABSTRACT
Various properties of vitrifying liquids are correlated with the dispersity of the dynamics, the latter
reflected in the magnitude of the nonexponentiality parameter, K, describing the distribution of relaxation
times. These properties include the mean relaxation time, , the fragility, and the dynamic crossover. The
correlations with K are observed in both experimental data and the results from molecular dynamics
simulations on Lennard-Jones LJ type systems. Another, rather obvious property to correlate with K is
the dynamic heterogeneity, which can be quantified from the number of molecules, Nc, dynamically
correlated over a time span. For a given LJ system, Nc can be rigorously calculated and we find that it does
indeed correlate with K over a range of thermodynamic conditions. However, the analysis of experimental
data for a broad range of real materials, wherein an approximation is required to obtain Nc, reveals the
absence of any relationship between Nc and K among different materials.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
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124507-2 Rolandetal. J.Chem.Phys.133,124507(cid:4)2010(cid:2)
tainingthenumberofmoleculeswithcorrelateddynamicsin
terms of readily accessible experimental quantities,
k
N (cid:2)T,P(cid:3)(cid:15) B T2(cid:10)max(cid:6)(cid:2)t,T,P(cid:3)(cid:11)2. (cid:2)4(cid:3)
c (cid:9)cP(cid:2)P(cid:3) t T
A number of works have appeared in which the (cid:6)(cid:2)t(cid:3) ap-
T
proximation was used to evaluate N for different
c
materials33,34 and different thermodynamic conditions.35 A
comparisonofsimulationresultsfor(cid:6) and(cid:6) indicatedrea-
4 T
sonable correspondence between the two methods for longer
(cid:3)(cid:4).27
Conceptually the spatial extent and amplitude of the dy-
namic correlations are expected to have a connection to the
dispersion in the relaxation times since dynamic heterogene-
ity is defined by molecular mobilities. By the same token
properties that correlate with (cid:2) should also correlate with FIG.1. Kohlrauschexponentasafunctionoftheproductvariable(cid:5)5.07/T
K forsimulatedLJparticles.Eachsymbolrepresentsadistinctstatepoint.Fits
N andviceversa.Upuntilnow,anyrelationbetween(cid:2) and
c K totheself-intermediatescatteringfunctionatlongertimeareshownfortwo
N hasnotbeenevaluated.Inthispaper,weprovidethetest, temperaturesintheinset.SimulationdataarefromRef.28.
c
using simulations of the Kob–Andersen Lennard-Jones (cid:2)LJ(cid:3)
mixture36 as a function of temperature T and density (cid:5). The
nonexponentiality parameter (cid:2) is obtained from the self- (cid:5)(cid:10)/T in Ref. 28.Within the experimental error, (cid:2)K is invari-
K ant for state points for which the scaling variable (cid:5)(cid:10)/T is
intermediate scattering function and N from the four-point
dynamicsusceptibility,(cid:6)(cid:2)t(cid:3).Correlatiocnbetween(cid:2) andN constant.InFig.2thesesame(cid:2)Kareplottedasafunctionof
4 K c (cid:3)(cid:2), and the results indicate that (cid:2) is a unique function of (cid:3)(cid:2)
isassessedbycomparingthechangesofthesetwoquantities K
on varying the thermodynamic conditions, i.e., at different for any thermodynamic condition in the regime where den-
combinations of T and (cid:5).To test the correlation between (cid:2)K sityscalingholds.Thisconstancyof(cid:2)Kforallstatepointsat
and N obtained from (cid:6) via Eq. (cid:2)4(cid:3), we use experimental any fixed (cid:3)(cid:4) has been shown from experimental data on
c T
data of 45 glass-formers including polymers, oxide glass- many glass-formers.39,40
formers, selenium, hydrogen bonded materials, and van der Tocomparedynamiccorrelationtononexponentiality,in
Waals glass-formers. Fig.3(cid:6)maxisplottedversus(cid:2).Thecorrespondenceisgood
4 K
and in accord with the usual interpretation of dynamic
II. RESULTS heterogeneity—an increasing correlation (cid:2)larger Nc(cid:3) associ-
ated with an increasing breadth of the relaxation function
A. Correlation between stretching parameter (cid:3) and
K (cid:2)smaller (cid:2)(cid:3). Also shown in Fig. 3 are the values of N
N from (cid:2)max K c
c 4 calculatedfrom(cid:6)(cid:2)t(cid:3)(cid:4)Eq.(cid:2)4(cid:3)(cid:5).Theagreementisgoodatlow
T
Previously it was shown28 from molecular dynamics temperaturesandhigherdensities,butotherwisetheapproxi-
simulation of the Kob–Andersen binary mixture with par- mationunderestimatesthenumberofdynamicallycorrelated
ticlesinteractingaccordingtoa12-6LJpotentialthatthefull particles. This agrees with previous simulation results com-
t-dependence, and thus the maximum in (cid:6)4(cid:2)t(cid:3)(cid:2)=Nc(cid:3), were paring (cid:6)4(cid:2)t(cid:3) and (cid:6)T(cid:2)t(cid:3) as a function of temperature at zero
invariantoverarelevantrangeofT and(cid:5)forstatepointsfor pressure.27
which the scaling variable (cid:5)(cid:10)/T is constant with (cid:10)=5.07.
Moreover, the reduced relaxation time (cid:3)(cid:2), defined as the
Kohlrausch decay time for the self-intermediate scattering
function F (cid:2)k,t(cid:3) multiplied by(cid:5)1/3T1/2,37 is also invariant for
s
statepointsforwhich(cid:5)(cid:10)/T isconstant.Thus,thesamevalue
ofthematerialconstant(cid:10)=5.07superposesboth(cid:6)maxand(cid:3)(cid:2)
4
of the system versus (cid:5)(cid:10)/T, which means that the number of
dynamicallycorrelatedparticlesisauniquefunctionof(cid:3)(cid:2),at
least in the regime where density scaling holds.
Values of (cid:2) were obtained by nonlinear fitting of the
K
stretchedexponentialfunction(cid:7)(cid:2)t(cid:3)=(cid:7) exp(cid:2)−(cid:2)t/(cid:3)(cid:3)(cid:2)K(cid:3) tothe
0
long-time portion of F (cid:2)k,t(cid:3) at a reduced wave-vector k(cid:5)1/3
s
=7.44 over the same range of T and (cid:5)as in Ref. 28. The fit
was cut off at short times where the contribution of the fast
initialdecayofF (cid:2)k,t(cid:3) becamesignificant;thatis,wherethe
s
shape of the curve begins to deviate from a stretched
exponential.38 Representative fits are shown in the inset of
Fig. 1. This figure shows (cid:2)K plotted as a function of (cid:5)(cid:10)/T FfrIoGm.2fi.tsKoofhlFra(cid:2)uksc,th(cid:3)eaxtploonnegnetratsimaef.unEcatciohnsoyfmtbhoelrerelapxraetsieonntstimaedidsetitnecrmtsitnaetde
using (cid:10)=5.07, the same exponent superposing (cid:6)max versus point.SimulatsiondataarefromRef.28.
4
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124507-3 Nonexponentialityanddynamicheterogeneity J.Chem.Phys.133,124507(cid:4)2010(cid:2)
FIG.5. FragilitycorrespondingtodatainFig.4.Correlationcoefficientis
FIG.3. N from(cid:6) (cid:4)Eq.(cid:2)2(cid:3)—opensymbols(cid:5)andfrom(cid:6) (cid:4)Eq.(cid:2)4(cid:3)—filled
c 4 T indicated.
symbols(cid:5)vstheKohlrauschexponentforLJparticles.Eachsymbolrepre-
sents a distinct state point. Increasing dynamic heterogeneity is associated (cid:13) (cid:14) (cid:13) (cid:14)
fwroitmhathberoraidgeorroduisspecraslicounlaotifornelaatxahtiigohnetrimteems.pTerhaetu(cid:6)rTesapapnrdoxliomwaetriodnednespitaiertss. N = kB (cid:2)K 2 dln(cid:3)(cid:4) 2, (cid:2)5(cid:3)
SimulationdataarefromRef.28. c (cid:9)c e dlnT
P
where e is Euler’s number. One of the most common dy-
B. Lack of correlation between (cid:3)K and Nc from (cid:2)T„t… namic properties of supercooled liquids is their fragility, m,
approximation proportionaltotheapparentactivationenthalpyatT normal-
g
In Fig. 4 plotted versus (cid:2) is N calculated at T from ized by kBT; that is, m is proportional to the last factor in
Eq. (cid:2)4(cid:3) using experimental meKasuremcents for 45 matgerials. brackets in Eq.(cid:2)5(cid:3). It is well accepted thatm correlates with
The N data were taken from Ref. 34 and (cid:2) are as reported the breadth of the relaxation time distribution; this has
in thecoriginal papers (cid:2)references in Ref. 34K(cid:3). Since a larger even been quantified as m=250(cid:2)(cid:11)30(cid:3)−320(cid:2)K (cid:2)Ref. 6(cid:3) or
m(cid:12)(cid:2)−2(cid:2)Ref.34(cid:3).Thisimpliesthatasmaller(cid:2) inEq.(cid:2)5(cid:3)is
numberofdynamicallycorrelatedmoleculesshouldcoincide K K
compensated by a larger fragility. The inference is that the
withagreaterdispersityofrelaxationtimes,weexpectthese
variationofN amongdifferentmaterialsisalmostentirelya
quantities to be inversely related. However, a linear correla- c
consequence of differences in (cid:9)c . Thus, both the form of
tion of the data in Fig. 4 yields a Pearson correlation coeffi- P
Eq.(cid:2)5(cid:3)andthedatainFig.4indicatethatthereisessentially
cient R=−0.22, indicating an absence of dependence. If we
no relationship between the dispersion of relaxation times
replaceN bythecorrespondingmolarvolumeorthenumber
c
of chemical groups (cid:2)e.g., “beads” as defined by Stevenson and the degree of correlation of the dynamics as quantified
and Wolynes41(cid:3), the correlation deteriorates further, with the using (cid:6)T(cid:2)t(cid:3).
absolute value of R becoming smaller. Since (cid:2)K and m are correlated (cid:2)inversely(cid:3), it follows
If(cid:7)(cid:2)t(cid:3)hastheKohlrauschform,toaverygoodapproxi- from the results in Fig. 4 that Nc and m would exhibit no
mation Eq. (cid:2)4(cid:3) can be rewritten so that (cid:2) appears mutual dependence. Previously Berthier et al.21 examined
K
explicitly,34 data for 15 materials and concluded: “Dynamic correlations
revealed by (cid:6)(cid:2)t(cid:3) increase weakly with fragility.” Extending
T
the data set threefold (cid:2)Fig. 5(cid:3) emphasizes the weakness of
this putative correlation. Hong et al.42 recently reported that
the dynamic heterogeneity length scale deduced from the
frequency of the Boson peak for various polymers and mo-
lecular liquids is not correlated with fragility.This is consis-
tent with our results herein. Moreover, at T N from Eq. (cid:2)5(cid:3)
g c
shows an inverse proportionality to the configurational en-
tropy (cid:2)assuming the latter is related to the heat capacity
change at T (cid:3),34 which implies no correlation with (cid:2).
g K
III. CONCLUSIONS
Wefindthatmax(cid:10)(cid:6)(cid:2)t(cid:3)(cid:11)correlateswith(cid:2) asTand(cid:5)are
4 K
variedforagivensystem,andalthoughthe(cid:6) approximation
T
FIG.4. PlotofKohlrauschexponentvsNcobtainedfromthe(cid:6)T(cid:2)t(cid:3)approxi- is inaccurate away from Tg, using the latter to obtain Nc this
mationfor45glass-formersatT andatmosphericpressure(cid:2)dataarefrom correlation with (cid:2) is maintained for a given material. The
Ref.34(cid:3):polymers(cid:2)diamonds(cid:3),ogxideglass-formersandselenium(cid:2)circles(cid:3), problem arises whKen comparing different systems at their
hydrogen bonded materials (cid:2)triangles(cid:3), and van der Waals glass-formers
(cid:2)squares(cid:3).Theabsenceofcorrelationisindicatedbythesmallvalueofthe respective Tg. Different materials exhibit different Nc (cid:2)from
Pearsonlinearcorrelationcoefficient. (cid:6)T(cid:3) for a given (cid:2)K because of differences in (cid:9)cP. The form
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