We present a calculation of a fourth-order, time-dependent density
correlation function that measures higher-order spatiotemporall correlations of
the density of a liquid. From molecular dynamics simulations of a glass-forming
Lennard-Jones liquid, we find that the characteristic length scale of this
function has a maximum as a function of time which increases steadily beyond
the characteristic length of the static pair correlation function $g(r)$ in the
temperature range approaching the mode coupling temperature from above

A. Arbe

A. van Blaaderen

A.I. Mel’cuk

B. Doliwa

C. Bennemann

C. Dasgupta

C. Donati

E.R. Weeks

F. W. Starr

G. Johnson

G. Parisi

H. Sillescu

H.W. Spiess

J. Colmenero

J. Qian

J.P. Garrahan

K. Schmidt-Rohr

M. Hurley

M.D. Ediger

M.R. Ernst

N. Lačević

N.E. Moe

R. Ahluwalia

R. Yamamoto

R.D. Mountain

R.L. Leheny

S. C. Glotzer

S. Franz

S.C. Glotzer

S.F. Edwards

T. B. Schrøder

T. Muranaka

T.B. Schrøder

U. Tracht

V. N. Novikov

W. Götze

W. Kob

W.K. Kegel

Y. Gebremichael

English

arXiv

We present a calculation of a fourth-order, time-dependent density correlation function that measures higher-order spatiotemporal correlations of the density of a liquid. From molecular dynamics simulations of a glass-forming Lennard-Jones liquid, we find that the characteristic length scale of this function has a maximum as a function of time which increases steadily beyond the characteristic length of the static pair correlation function g(r) in the temperature range approaching the mode coupling temperature from above. This length scale provides a measure of the spatially heterogeneous nature of the dynamics of the liquid in the alpha-relaxation regime

Lačević, N.

Starr, F. W.

Schrøder, Thomas

Novikov, V. N.

Glotzer, S. C.

Roskilde Universitet

RoskildeUniversityGrowing correlation length on cooling below the onset of caging in a simulated glass-forming liquidLaevi, N.; Starr, F. W.; Schrøder, Thomas; Novikov, V. N.; Glotzer, S. C.Published in:Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)DOI:10.1103/PhysRevE.66.030101Publication date:2002Document VersionPublisher's PDF, also known as Version of recordCitation for published version (APA):Laevi, N., Starr, F. W., Schrøder, T., Novikov, V. N., & Glotzer, S. C. (2002). Growing correlation length oncooling below the onset of caging in a simulated glass-forming liquid. Physical Review E (Statistical, Nonlinear,and Soft Matter Physics), 66(3). https://doi.org/10.1103/PhysRevE.66.030101General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain.            • You may freely distribute the URL identifying the publication in the public portal.Take down policyIf you believe that this document breaches copyright please contact rucforsk@ruc.dk providing details, and we will remove access to thework immediately and investigate your claim.Download date: 27. Mar. 20218109nology,RAPID COMMUNICATIONSPHYSICAL REVIEW E 66, 030101~R! ~2002!Growing correlation length on cooling below the onset of caging in a simulated glass-forming liquidN. Lačević, 1,2,3 F. W. Starr,2 T. B. Schro”der,2,4 V. N. Novikov,2,* and S. C. Glotzer1,21Departments of Chemical Engineering and Materials Science and Engineering, University of Michigan, Ann Arbor, Michigan 42Center for Theoretical and Computational Materials Science and Polymers Division, National Institute of Standards and TechGaithersburg, Maryland 208993Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 212104IMFUFA, Roskilde University, DK-4000 Roskilde, Denmark~Received 30 March 2001; revised manuscript received 19 June 2002; published 13 September 2002!We present a calculation of a fourth-order, time-dependent density correlation function that measures higher-order spatiotemporal correlations of the density of a liquid. From molecular dynamics simulations of a glass-forming Lennard-Jones liquid, we find that the characteristic length scale of this function has a maximum as afunction of time which increases steadily beyond the characteristic length of the static pair correlation functiong(r ) in the temperature range approaching the mode coupling temperature from above. This length scaleprovides a measure of the spatially heterogeneous nature of the dynamics of the liquid in the alpha-relaxationregime.DOI: 10.1103/PhysRevE.66.030101 PACS number~s!: 64.70.Pf, 61.20.Lcesleeeoreee-r--.oon-aionrolinnteteidsgasdraor-tnthek,ina-if-r--off,indtrRelaxation in liquids near their glass transition involvthe correlated motion of groups of neighboring partic@1–3#. This correlated motion results in spatially heterogneous dynamics, which becomes increasingly heterogenas the liquid is cooled. Much remains to be understoodgarding the nature of this heterogeneity, and how to bmeasure and quantify it. The traditional two-point, timdependent, van Hove density correlation functionG(r ,t),provides information about the transient ‘‘caging’’ of paticles upon cooling@4#, but does not provide local information about correlated motion and dynamical heterogeneityparticular, the static correlation length associated with twpoint density fluctuations remains relatively constant upcooling @5,6#. Instead, other correlation functions, which ivolve, e.g., spatial correlations of the displacements of pticles in the liquid, and other measures of correlated mothave been used to demonstrate the heterogeneous natuthe liquid dynamics in molecular dynamics~MD! simula-tions @7,8#. These ‘‘measures’’ are readily accessible in cloidal suspensions, where microscopy provides detailedformation on particle trajectories similar to the informatioobtained from MD simulations. Indeed, such experimehave confirmed simulation predictions of increasingly herogeneous dynamics near the glass transition@9#.In this paper, we evaluate a fourth-order, time-dependdensity correlation functiong4(r ,t) that is more sensitive tospatially heterogeneous dynamics thanG(r ,t). This four-point function was first investigated in supercooled liquby Dasguptaet al. @10#, but they did not detect a growincorrelation length in their simulations. Recently, it wshown @11,12# that it is possible to define a generalizetime-dependent susceptibilityx4(t) which ~i! is proportionalto the volume integral ofg4(r ,t) in the same way that theisothermal compressibility is related to the volume integof the static pair correlation function@13#, ~ii ! is nonzero inthe caging regime, and~iii ! increases with decreasingT.*Permanent address: Institute of Automation and ElectromeRussian Academy of Science, Novosibirsk 630090, Russia.1063-651X/2002/66~3!/030101~4!/$20.00 66 0301s-us-stIn-nr-,e of-n-s-nt,lWhile this indirectly suggests the presence of a growing crelation length, neitherg4(r ,t) nor its rangej4(t) was cal-culated in those works. Here we calculateg4(r ,t) in a simu-lation of 8000 Lennard-Jones~LJ! particles, and show thaj4(t) grows slowly but steadily beyond the correlatiolength j of the static pair correlation functiong(r ) as tem-peratureT is decreased from the onset of caging towardsmode coupling temperatureTMCT @4#.We first briefly review the general theoretical frameworsome of which was previously discussed in Refs.@11,12,14#,and extend it to obtain a form forg4(r ,t) suitable for calcu-lation in our simulations. Consider a liquid ofN particlesoccupying a volumeV, with density r(r ,t)5( i 51N d„r2r i(t)…. Extending an idea originally proposed for spglasses@15#, one may construct a time-dependent ‘‘order prameter’’ that compares the liquid configuration at two dferent times@11,12,14#,Q~ t !5E dr1dr2r~r1,0!r~r2 ,t !w~ ur12r2u!5(i 51N(j 51Nw~ ur i~0!2r j~ t !u!. ~1!Here,r i in the second equality refers to the position of paticle i, and w(ur i2r j u) is an ‘‘overlap’’ function which isunity for ur i2r j u<a and zero otherwise, where the parametera is associated with the typical vibrational amplitudethe particles@11,12#. For the present work, we takea50.3particle diameters as in Refs.@11,12#, since this value maxi-mizes the effect studied here@16#. As defined,Q(t) is thenumber of ‘‘overlapping’’ particles when configurations othe system att50 and at a later timet are compared; that isQ(t) counts the number of particles that either remain witha distancea of their original position, or are replaced~withina distancea) by another particle in an intervalt.The fluctuations inQ(t) are described by a generalizesusceptibilityx4~ t !5bVN2@^Q2~ t !&2^Q~ t !&2#, ~2!y,©2002 The American Physical Society01-1et-th--otliqemomnc-e--xedreb--n-ionatofula-ichusti-lib--eef-tRAPID COMMUNICATIONSLAČEVIĆ, STARR, SCHRO”DER, NOVIKOV, AND GLOTZER PHYSICAL REVIEW E66, 030101~R! ~2002!whereb51/kBT, kB is Boltzmann’s constant, and̂•••& in-dicates an ensemble average as in Ref.@12#. Note that at veryearly times, whenQ(t)5N because no particle has ymoved beyond a distancea, x4(t) is identical to the isother-mal compressibilitykT @17#. Substituting Eq.~1! into Eq.~2!, we obtainx4~ t !5bVN2E dr1dr2dr3dr4G4~r1 ,r2 ,r3 ,r4 ,t !, ~3!whereG4~r1 ,r2 ,r3 ,r4 ,t !5^r~r1,0!r~r2 ,t !w~ ur12r2u!3r~r3,0!r~r4 ,t !w~ ur32r4u!&2^r~r1,0!r~r2 ,t !w~ ur12r2u!&3^r~r3,0!r~r4 ,t !w~ ur32r4u!&. ~4!As written,G4 is a function of four spatial and two temporal coordinates, constrained in a particular way byoverlap functions. To investigate the behavior ofG4, weconsider a functiong4(r ,t) such thatx4(t)5b*drg4(r ,t).We integrate first over 2 and r4 in Eq. ~3!, define r[r32r1, and then integrate overr3 to obtaing4~r ,t !5K 1Nr (i jkl d„r2r k~0!1r i~0!…w„ur i~0!2r j~ t !u…3w„ur k~0!2r l~ t !u…L 2 K Q~ t !N L 2. ~5!We investigate the behavior ofg4(r ,t), which is the angularaveraged function of a single variabler . With the abovechoice of integration variables,g4(r ,t) describes spatial correlations between overlapping particles at the initial time~us-ing information at timet to label the overlapping particles!.The first term ing4(r ,t) is a pair correlation function restricted to the subset of overlapping particles,g4ol(r ,t). Thesecond term represents the ‘‘bulk’’ probability of any twparticles overlapping. We can rewriteg4(r ,t) asg4~r ,t !5g4ol~r ,t !2 K Q~ t !N L 25 K Q~ t !N L 2F g4ol~r ,t !K Q~ t !N L 2 21G[ K Q~ t !N L 2g4* ~r ,t !. ~6!Since^Q(t)/N& is a function of time only, information abouspatial correlations is contained ing4* (r ,t), which we inves-tigate in the rest of the paper. In comparingg4(r ,t) withother correlation functions used to study glass-forminguids, we note that neutron scattering studies, such as thosColmenero and Richter and co-workers on polymer syste@18#, measure at most two-point spatiotemporal density crelation functions. 4-D NMR methods used to probe dyna03010e-bysr-i-cal heterogeneity measure multiple time correlation futions @19#. In contrast to those,g4(r ,t) contains additional,higher order spatiotemporal information.To evaluateg4* (r ,t), we perform MD simulations of amodel LJ glass-forming liquid. The system is a thredimensional~50:50! binary mixture of 8000 particles interacting via LJ interaction parameters@12,20#. We simulateeight state points in the microcanonical ensemble at fidensityr51.29, in a temperature range 0.5922.0; we esti-mateTMCT50.5760.02 and the Vogel-Fulcher temperatuT050.4860.02. The error bars are confidence intervals otained as a result of fittingta(T) to a power law and exponential form, respectively. The onset of caging and noexponential relaxation of the intermediate scattering functoccur atTcage'1.0. Our simulations thus span a range thincludes the onset of caging and the initial slowing downthe liquid approachingTMCT . The isochore along which thesimulations are performed was chosen to reproduce simtions of a smaller system size performed earlier, whshowed~i! that this model exhibits spatially heterogeneodynamics @12#, and ~ii ! that transitions between inherenstructures close toTMCT occur through the collective, quasone-dimensional motion of strings of particles@21#. All dataevaluated in the present work are in thermodynamic equirium above the glass transition temperature.We plot g4* (r ,t) for severalt at our second coldest temperatureT50.60 in Figs. 1~a! and 1~b!. The positions ofthe peaks ing4* (r ,t) are identical to the positions of thpeaks ing(r ) ~not shown!. We confirm thatg4* (r ,t)5g(r )21 in the ballistic and diffusive regime. Note that in thdiffusive regimeg4ol(r ,t) is the pair correlation function othe random overlaps normalized by^Q(t)/N&2 to yield g(r ).We plot x4(t) and ^Q(t)/N& at T50.60 in Fig. 1~c!; asfound in Refs.@11,12#, x4(t) has a maximum at an intermediate timet4* . References@11,12# showed that this maximumincreases and shifts to longer time asT decreases towardTMCT , and we further find that4* is in the a-relaxationregime at eachT. g4* (r ,t) deviates fromg(r )21 when^Q(t)/N& deviates from unity andx4(t) becomes nonzero@22#. The amplitude and range ofg4* (r ,t) increase with in-creasingt until a timet4* . At t4* , g4* (r ,t4* ) @indicated by thesolid curve in Figs. 1~a!,1~b!# exhibits a long tail which de-creases slowly to zero with increasing distance. Fort greaterthan t4* , the amplitude and range ofg4* (r ,t) decrease, andg4* (r ,t)5g(r )21 whenx4(t) decays to zero~not shown!.The positions of the peaks ing4* (r ,t) do not appear tochange with decreasingT.Figure 2 shows theT dependence ofg4* (r ,t) at the peakcharacteristic timet4* when the correlations at eachT aremost pronounced, as measured byx4(t). The inset of Fig. 2shows the ‘‘four-point’’ structure factor S4* (q,t4* )5r*drg4* (r ,t4* )sin(qr)/qr, and static structure factorS(q)215r*dr @g(r )21#sin(qr)/qr. We find that while S(q)hows no change at smallq, S4(q,t4* ) develops a peak asmall q that grows with decreasingT, indicating a growingrange of correlations.1-2ig.se.ed,itying,fc-arn-csingRAPID COMMUNICATIONSGROWING CORRELATION LENGTH ON COOLING BELOW . . . PHYSICAL REVIEW E66, 030101~R! ~2002!FIG. 1. Time dependence ofg4* (r ,t) at T50.60, showing~a!the amplitude and range ofg4* (r ,t) growing in time and~b! theamplitude and range ofg4* (r ,t) decaying in time. We multiplyg4* (r ,t) with 4pr2 to better reveal its long decaying tail. The fration ^Q(t)/N& indicates the average fraction of overlapping pticles present at timet. ~c! Time dependence ofx4(t) and^Q(t)/N&at T50.60. The long time value of̂Q(t)/N&5rVa ~solid line!,whereVa54pa3/3 corresponds to the probability of finding a radom overlapping particle. The error bars inQ(t) and x4(t) arecalculated by averaging over 100 successive independent bloand are representative of the error in all points.FIG. 2. Temperature dependence ofg4* (r ,t) at six values ofT,indicated in the legend. Insets show the structure factorS4* (q,t4* )and static structure factorS(q) for the same values ofT. We notethat g4* (r ,t) is analogous tog(r )21, andS4* (q,t) is analogous toS(q)21.03010Figure 3 provides a closeup of the behavior ofg4* (r ,t4* )for 1.7,r ,7, for several values ofT. To extract a value forj4(t), we fit peaks ofg4* (r ,t) in the range shown to theexponential functiony(r )5a*exp@2r/j4(t)#. We refer to thismethod as an ‘‘envelope fit.’’ The time dependence ofj4(t)obtained from this fit is plotted for several state points in F4. We see that the qualitative behavior ofj4(t) is similar tothat of x4(t): j4(t) has a maximum in time, and asT de-creases, the amplitude and time of this maximum increaThe length scalej4(t) characterizes the typical distancover which ‘‘overlapping’’ particles are spatially correlateand includes contributions from the static two-point denscorrelations. At temperatures above the onset of cagwhere the dynamics is everywhere homogeneous,j4(t4* ) andj coincide. Below the onset of caging,j4(t4* ) begins to growlarger thanj; over the limitedT range of our simulations, wefind that j4(t) increases from 0.860.1 particle diametersabove Tcage to 1.560.1 particle diameters within 5% o-ks,FIG. 3. Semilogarithmic plot ofg4* (r ,t4* ) in the range 1.7,r,7 at four values ofT. The solid lines are the exponential curveobtained from an ‘‘envelope fit.’’FIG. 4. Time andT dependence ofj4(t). The inset showsj4(t4* ) vs T. The data shown are smoothed by performing a runnaverage over successive groups of five points.1-3.cesu-re-ingg,ilethepa-ne-RAPID COMMUNICATIONSLAČEVIĆ, STARR, SCHRO”DER, NOVIKOV, AND GLOTZER PHYSICAL REVIEW E66, 030101~R! ~2002!TMCT ~Fig. 4 inset!. In contrast, we find thatj, which iscoincident with j4(t) at short times when̂ Q(t)/N&51,changes less, from 0.860.1 to 1.160.1 particle diametersThus, over theT range studied, the characteristic distanover which particle motion is most correlatedgrows toexceed the static correlation length. While we cannotreliably predict the behavior ofj4(t4* ) at lower T, we findno tendency for slowing down of its growth, unlike thlow-T behavior ofj which is known to remain finite andsmall. The behavior ofj4(t4* ) belowTMCT, if extrapolated tolower T, appears consistent with measured length scaledynamical heterogeneity of several nanometers close toTg@1,2#..C..J.n03010eofThe relatively small but growing correlation length calclated here from the four-point spatiotemporal density corlation function should be contrasted with that characterizthe size of highly mobile regions within the fluid@7#. Thatlength was shown to grow much more rapidly on coolinapproaching the size of the simulation box close toTMCT .Interestingly, whereas the correlation length of highly mobregions was found to be largest on a time scale inb-relaxation regime, the length calculated in the presentper is largest in thea regime. We note thatg4(r ,t) is domi-nated by ‘‘caged’’ particles, and thusj4(t) may be related tolength scales calculated in Ref.@23#. The relationship of thedifferent length scales characterizing dynamical heterogeity will be explored in a subsequent publication., J..,lble@1# H. 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Growing correlation length on cooling below the onset of caging in a simulated glass-forming liquid

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Growing Correlation Length on Cooling Below the Onset of Caging in a
  Simulated Glass-Forming Liquid

Growing Correlation Length on Cooling Below the Onset of Caging in a Simulated Glass-Forming Liquid

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