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Comparison of Dynamical Heterogeneity in Hard-Sphere and Attractive Glass Formers David R. Reichman Department of Chemistry, Columbia University, 3000 Broadway, New York, NY 10027 Eran Rabani School of Chemistry, The Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel 5 0 Phillip L. Geissler 0 2 Department of Chemistry, University of California, and Physical Biosciences and Materials Sciences Divisions, n Lawrence Berkeley National Laboratory, Berkeley, CA 94720 a (Dated: February 2, 2008) J 5 Usingmolecular dynamicssimulations, wehavedeterminedthatthenatureof dynamicalhetero- geneity in jammed liquids is very sensitive to short-ranged attractions. Weakly attractive systems ] differ little from dense hard-sphere and Lennard-Jones fluids: Particle motion is punctuated and t f tendsto proceed in steps of roughly asingle particle diameter. Both of thesebasic features change o in the presence of appreciable attractions. Transient periods of particle mobility and immobility s cannot be discerned at intermediate attraction strength, for which structural relaxation is greatly . t enhanced. Strong attractions, known to dramatically inhibit relaxation, restore bimodality of par- a ticle motion. But in this regime, transiently mobile particles move in steps that are significantly m morebiasedtoward largedisplacementsthaninthecaseofweakattractions. Thismodifiedfeature - ofdynamicheterogeneity,which cannotbecapturedbyconventionalmodecouplingtheory,verifies d recent predictions from a model of spatially correlated facilitating defects. n o c [ Dynamical heterogeneity is perhaps the most reveal- Eventually,arare,collectiverearrangementfreesaparti- ing feature of relaxation in deeply supercooled liquids. cle from its cage, transiently allowing it to move rapidly 2 Ahigh-temperaturefluidisdynamicallyhomogeneousin over distances comparable to a particle diameter. Since v 6 the sense that the local environment restricting fluctua- suchadisplacementitselffacilitateslocalrearrangement, 3 tions of any given particle is, for all important purposes, transientmobilityappearstopropagatecontinuouslyand 1 identical to that surrounding any other particle, even with some degree of directionality. Schematic models of 6 on the short time scales of basic microscopic motions. glassinesshavebeenconstructedwithonlythesefeatures 0 ThedistributionP(δr,t)ofparticledisplacementsδrasa in mind[11, 12, 13]. They account for a surprising va- 4 0 functionoftimetprovidesaquantitativemeasureofsuch riety of anomalous behaviors and yield unique scaling / uniformity. Results of molecular dynamics simulations predictions for the length and time scales characterizing at indicatethatP(δr,t)isGaussianoverawiderangeofdis- relaxation[13, 14, 15]. m placements,aswouldbe expectedfroma mean-fieldper- These basic features of dynamical heterogeneity have - spective, for typical dense fluids[1]. The microscopic en- d little to do with the identity of particles comprising a vironments constraining particle motion in a glassy ma- n glassy material or with the interactions between them. terialarebycontrastprofoundlynonuniform,evenonthe o They have been reported for many model atomic liquids c longtimescalesoflarge-wavelengthrelaxation[2,3,4,5]. as well as for viscous silica, a network-forming liquid : This fact has been clearly demonstrated by experiments v which vitrifies with qualitatively different temperature i that focus on dynamics of single probe molecules[6] or X dependence[16]. Itisthereforetempting topresumethat subsets of molecules in a pure liquid that relax more the scenario sketched above is universal among super- r slowly than the average[7, 8]. As a result, P(δr,t) de- a cooled molecular liquids[12, 17]. velops substantial weight in the wings, reflected in ap- preciably nonzero values of the non-Gaussian parameter When interactionsbetweenparticlesin asimple liquid α (t) = 3hδr4(t)i/hδr2(t)i2 −1. Microscopy studies of are augmented by strong, short-ranged attractions, spa- 2 5 colloidal suspensions have confirmed this expectation[9]. tiallyaverageddynamicalquantitieschangeinnontrivial ways[18, 19, 20, 21]. This situation has been realized Through extensive computer simulations, a detailed experimentally by adding linear chain molecules to sus- pictureofdynamicalheterogeneityinsimplejammedliq- pensions of colloidal particles, whose direct interactions uids (e.g., a binary mixture of Lennard-Jones spheres) are almost purely repulsive[22]. By varying φ one can p has developed[10]. At low temperatures, the majority of tune the fluid from purely repulsive to strongly attrac- particles are confined within cages, composed of neigh- tive. Forlargecolloidvolume fraction,φ , andvanishing c boring particles, that may persist for very long times. attractionstrength(φ =0),thesuspensionisinessence p 2 a dense hard-sphere fluid with the basic phenomenology 10-2 (a) of simple supercooled liquids. As φ increases, however, p relaxationacceleratessignificantly,sothatahard-sphere glass can be “melted” by adding attractions[18]. Be- 10-3 yond a certain value of φp, relaxation becomes instead )p φ 5 more sluggish as more polymer is added, leading to re- ( D vitrification at large φp. 10-4 V(r)0 In this paper we examine whether the peculiar behav- -5 ior of attractive colloids reflects fundamental changes in 10-5 1.9 2 r / a2.1 2.2 the nature of dynamical heterogeneity. Although exten- sive computational work has been done to characterize 0 0.1 0.2 0.3 0.4 φ suchheterogeneityinsimpleliquids,previoussimulations p of materials with short-ranged attractions have not fo- 12.5 cused on correlated microscopic motions. For this pur- (b) φp=0.05 φ=0.15 p pose we have adopted the model of Puertas et al. for 10 φ=0.05 p polymer-mediated interactions between colloids[19, 20]. φ=0.05 p The effective interaction potential between a pair of col- φ=0.25 7.5 p l2o1i-d2s2s,eipsapralottetdedbyindiFstiagn.c1efro,rdtehscerivbaeludeisnodfetφapilwineRheafvse. α(t )2 φφpp==00..33575 5 studied. Interactions consist of a short-ranged repulsion parameterizedbythesumofparticleradii,ashort-ranged attractionthatmimicsthepolymer-induceddepletionin- 2.5 teraction, and a slowly varying, long-ranged repulsion designed to prohibit phase separation [23]. The effective 0 rangeandformofeachtermarepreciselyasgiveninRefs. 100 102 104 t 21-22. We have focused on weakly polydisperse systems, in which these radii are drawn from a uniform distribu- tion, p(R)=(2δa)−1θ[R−(a−δa)]θ[(a+δa)−R], with FIG. 1: (a) Diffusion constant D as a function of polymer meanaandhalf-widthδa=a/10. TheHeavisidefuncion volume fraction φ . In each case, D was computed from the p θ(x)isdefinedasusualtobe1forx≥0,and0forx<0. mean squared displacement averaged over particle size. In- Inthispaperallquantitieswithunitsoflengthhavebeen set: Potential energy of interaction V(r) as a function of the scaled by the mean radius a, and quantities with units distance r between two particles with radius a for the values of time have been scaled by t = 8a2. We have used of φp we have studied numerically. From the least negative 0 q3T minimum of V(r) to the most negative minimum, these val- standard methods of molecular dynamics to propagate ues are φ = 0.05, 0.15, 0.2, 0.25, 0.3, 0.35, and 0.375. (b) p equilibrated systems of 1000 periodically replicated col- Non-Gaussian parameter α (t) as a function of time t. 2 loidal particles, stochastically rescaling colloid momenta every 101 time steps to maintain a Boltzmann distribu- tionattemperatureT = 4. Wehaveinvestigatedasingle extentofrelaxationiscomparableforeachstate. Weuse 3 colloid volume fraction φc = 43πVNa3(1+(δa/a)2) = 0.55 severaldifferent measures of the extent of relaxation,in- and several polymer volume fractions (i.e., attraction cluding α (t), mean squared particle displacement, and 2 strengths as plotted in Fig. 1) ranging from φp = 0.05 the dynamic structure factor. Although these choices to φp =0.375. are somewhat arbitrary, they paint a consistent picture In order to contrastfeatures of dynamic heterogeneity of changes in dynamic heterogeneity induced by short- unique to attractive systems with those generic to sim- ranged attractions. ple supercooled liquids, we have selected a colloid den- The non-Gaussian parameter, plotted in Fig. 1(b) as sity for which relaxationis already very slow for φ =0. a function of time for several values of φ , indicates p p The self-diffusion constant D(φ ), plotted in Fig. 1 as a that short-ranged attractions effect changes more pro- p function of φ , thus evinces the re-entrant behavior we foundthansimplyarenormalizedaveragetime orlength p have described. Specifically, D increases by two orders scale[19, 20]. The peak of α (t) roughly locates the time 2 of magnitude as φ approaches 0.25 and then decreases ofmaximumdynamicheterogeneity. Wedenotethetime p ∗ ∗ sharply for higher polymer concentrations. Any mean- corresponding to this peak as t . The dependence of t ingful comparison of relaxation mechanisms at different on φ closely mirrors that of the diffusion constant, re- p φ must take into account the disparate time scales cor- flectingaglobalchangeinrelaxationtime. Buttheshape p responding to this range of diffusivity. Here we examine and scale of α (t) also change significantly with attrac- 2 ∗ time evolutionover intervals scaled such that the overall tion strength. Most notably, the peak height α (t ) de- 2 3 clines by nearly an order of magnitude as φ approaches p 0.25, then grows rapidly for larger values of φp. This 0.3 φp=0.375 φ=0.25 evolution strongly suggests a change in the character of p microscopic dynamics[19]. τ)0 φp=0.05 , Brownian ) theTirhelogfualrlitdhimstr(iPb˜u[ltoigo1n0(o|δfrp|)a,rtt]i)c,leovdeirspalaspceemcifiencttsi,moerino-f δ(|r| 0.2 0 terval (of duration t) provides a more detailed picture 1 g of microscopic rearrangements[24]. Fig. 2 shows a plot o of P˜[log (|δr|),τ ] for states representative of weak at- ~P(l 0.1 10 0 tractions (φ = 0.05), strong attractions (φ = 0.375), p p and intermediate attraction strength (φ = 0.25). Since p relaxation rates vary greatly among these three states, 0 0.1 1 10 we have followed Cates et al. [24] in comparing mo- |δr| tion over time intervals τ (φ ) yielding the same mean 0 p squared displacement, 10a2. As reflected by α (t), the 2 distributions at φp = 0.05 and φp = 0.375 are highly FIG.2: DistributionsP˜[log10|δr|,τ0]ofthelogarithmofpar- non-Gaussian, exhibiting distinct populations of espe- ticle displacements |δr| at time τ . Results are shown for 0 cially mobile and especially immobile particles. Cates et polymer volumefractions φp =0.05, 0.375, and 0.25, and for al. [24] have reported a similarly bimodal distribution of a Gaussian distribution of δr. Times τ0(φ) and the width of log(|δr|2) for strongly attractive colloids at much lower the reference distribution were chosen to obtain a consistent mean squared displacement, h|δr|2i=10a2. densities (φ = 0.4) and have suggested that such bi- c modality is a unique feature of attractive glasses. Multi- peaked van Hove distribution functions, however, have been reported several times for systems lacking short- attractions[28]. Heterogeneity associatedwith formation ranged attractions[25, 26, 27]. Indeed, Fig. 2 demon- anddecay of a cluster wouldnot have a strongsignature stratesthatthedistinctionbetweenmobileandimmobile in P˜[log |δr|,τ ] due to translation of the cluster as a 10 0 particlesisinfactmorepronouncedwhenattractionsare whole. almost negligibly weak. Bimodally distributed particle Theessenceofadynamicheterogeneityperspectiveon displacements thus appear to be a feature common to glassyliquids is that relaxationoverany shortintervalis many sluggish systems. drivenbyasmallsubsetofparticlesthataretemporarily Theglassystateswithweakandstrongattractionsalso much more mobile than the average. The statistics of shareadegreeofstructureinP˜[log (|δr|),τ ]withinthe 10 0 extreme displacements should therefore be revealing of subpopulation of mobile particles. For φ = 0.05 parti- p basicrelaxationmechanisms[10]. Herewefocusonparti- cles clearly tend to move in discrete steps of approxi- clesamongthe5%mostmobileoveranintervaloflength mately integer multiples of a typical particle diameter, ∗ t ≪ τ . We judge mobility in this case by monitoring 0 2a. This feature highlights the decoupling of diffusion thelargestdisplacement|δr| ofaparticlefromitspo- max and structural relaxation in jammed liquids. Although sition at the beginning of each interval. Distributions of fluctuations in local environment may permit a particle maximum displacement magnitudes for these especially to move out of its cage, density correlations persist such mobile particles, P(>)(|δr| ,t∗), are plotted in Fig. 3 max that the newly formedcage has a well-definedspatialre- for each of the polymer volume fractions we have stud- lationship with the original. Stepwise motion is much ied. For purposes of comparison, we have scaled |δr| max less pronounced at high φ . p by its most likely value r (φ ) for each φ . By contrast, the shape of P˜[log |δr|,τ ] at φ =0.25 0 p p 10 0 p For reference we have included in Fig. 3 an extreme is nearly that of a Gaussian, plotted for reference in Fig. 2. There is evidence neither of a distinct popula- value distribution PB(>)(|δr|) that would be obtained for tion of immobile particles nor of stepwise motion. The a Brownian analogue of our system. Because maximum statisticsofsingle-particledisplacementsatintermediate displacements are not easily defined for fractal trajecto- attractionstrength thus more strongly resemble those of ries, we consider in this case the displacement of a par- a dynamically homogeneous fluid at lower density than ticle |δr| from its initial position only at the end of an those of a non-attractive fluid at the same density. The interval. The interval length is in fact arbitrary, deter- simplest conclusion is that the φ = 0.25 system is mining only the overall scale of displacements, which is p in essence dynamically uniform. Evidence exists, how- irrelevantforthecomparisoninFig.3. Indetail,wecom- ever, that correlatedmotions of neighboring particles do puted P(>)(|δr|) by repeatedly drawing N = 1000 dis- B show signs of dynamic heterogeneity[28]. This situation placements from a Gaussian distribution with unit vari- could be expected if particle displacements were domi- anceinallthreedimensions,eachtimeaddingthe0.05N nated by movement of clusters transiently stabilized by largest values to a histogram. P(>)(|δr|) is therefore a B 4 superposition of extreme value distributions: 1 φ=0.05 0.05N )p φp=0.15 PB(>)(|δr|)= Xj=1 gj(|δr|). φ(max φφpp==00..235 P p Here,gj(|δr|)istheprobabilitydensityforobserving|δr| ) / φφp==00..33575 asthejth largestvalueinasampleofsizeN. Inthelimit *,tx0.5 Bprownian N →∞[29], a m r| gj(x)= (j−jj1)!exp[jxj(x−x)−jex(x−xj)], (>)δP(| where x = 2ln(j/N). Because convergence to this j p 0 asymptoticlimitisslow,however,wehavechosentocon- 0 1 2 3 |δr| / r (φ ) struct P(>)(|δr|) from direct sampling. max 0 p B The scaled distributions in Fig. 3 reveal a monotonic trend toward broadly distributed mobile particle dis- FIG.3: DistributionsP(>)(|δr| ,t∗)ofmaximumdisplace- placementsasattractionstrengthincreases. Forweakat- max tractions, the large-displacementtail of P(>)(|δr|max,t∗) mofemntosbfiolirtyth|δer5|%m,otshtemlaorbgeilsetpdaisrptilcalceesm. Wenetuaspeaarstiaclemueansduerre- max is attenuated relative to a simple Brownian fluid. For goes(fromitspositionattimet)duringthetimeintervaltto strong attractions, this tail is greatly enhanced. Al- t+t∗. Distributions and displacements have been scaled to though the overall shape of the full displacement dis- coincide at thepeak of P(>). A scaled extreme displacement tribution for intermediate attraction strength is roughly distributionisalsoplottedforaperfectlyBrowniansystemof Gaussian, statistics of the extreme subensemble demon- 1000 particles. strate that mobile particles nonetheless execute larger jumps(relativetotheaverage)thaninaBrownianrefer- ence system. This fact is consistent with the attraction- andforφp =0.05(c),evenoverthislongtimescale. The enhanced large-displacement tail of P˜[log |δr|,τ ] plot- cages that constrain these particles’ motion are clearly 10 0 ted in Fig. 2. smaller in the case of strong attractions. The 5% most The comparison in Fig. 3 emphasizes changes in the mobile particles, on the other hand, move several parti- shape of displacement distributions. Because we have cle diameters during the same interval. For the case of scaled distance by different lengths for different values weakattractionsin panel(d), the discrete nature ofpar- ofφ ,theseresultsdonotdirectlyimplythattransiently ticle motion and correlations between subsequent cages p mobileparticlesexecutelargerjumpsinspaceinthecase are immediately evident. Trajectories are qualitatively of strong attractions. They instead reflect the breadth different for the case of strong attractions in panel (b). of particle motion relative to the mean, a feature that Here,mobileparticlesexploremuchmorediffuseregions. couldreflecttighteningofparticlecagesaswellasgrowth Domains of facile particle motion are clustered in space of facilitated regions. More direct evidence for both of and markedly elongated, with asymmetries apparently these trends is provided by scrutinizing the trajectories correlated over several particle radii. of individual particles. Most of the qualitative changes in dynamical hetero- The routes traced by several particles are plotted in geneity we have reported can be understood as con- Fig. 4 for the smallest and largestvalues of φ . We have sequences of changing spatial patterns of structural p chosen the duration of displayed trajectories to be τ , defects[31]. Models based on dynamical heterogeneity α the time required for correlation of density fluctuations typically assume that the subtle defects whichenable lo- over length scales comparable to a particle diameter to calrelaxationare sufficiently sparseas to be statistically decay by a factor of e. Dynamics over this time scale, independent(despitesignificantcorrelationsindefectdy- ∗ which is much larger than t in both cases, exhibit the namics). We have proposed that an important effect of full character of particle motion. This choice also al- short-ranged attractions is to introduce non-negligible lows a straightforwardcomparisonof dynamics for weak spatial correlations among such facilitating entities[30]. and strong attractions,since the correspondingvalues of In particular, defects should aggregate with increasing τ are very similar (τ ≈ 8 × 103 for φ = 0.05 and attraction strength as an indirect result of particle clus- α α p τ ≈ 104 for φ = 0.375). The depicted excursions of tering. Microscopicregionsofmobilitythus growinsize, α p extremely immobile (toppanels of Fig.4) andextremely but become still more sparse if their overall concentra- mobile(bottompanels)particlesreinforcethedynamical tion(looselyanalogoustofreevolume)isheldfixed. This features we have gleaned from probability distributions. picture accounts for the broadening distributions of mo- The 5%leastmobile particlesexhibit only small flucuta- bile particle displacements we have computed. Particle tionsabouttheirinitialpositions,bothforφ =0.375(a) trajectories depicted in Fig. 4 make the agreement es- p 5 10 suggest instead very fluid and uniform motion. The role (a) (c) ofattractionsin this regime,we suggest,is to bind small 5 transientclusterswhichmoveonatimescalecomparable to their lifetimes. Strong attractions restore some dis- y/a 0 creteness of particle displacements, presumably because transient clusters are too large to move appreciably on −5 pertinenttimescales. Wethusconcludethat“attractive” glassinessisdrivenbyspatialredistributionoffacilitating defects. Although extended loose domains permit large 10 (b) (d) particle displacements, their growth depletes mobility in surrounding areas, which in turn inhibits relaxation of 5 domain interfaces. We are pursuing further calculations y/a 0 tmoencotsnfiarnmdttohcishacrluasctteerriinzegtohfemcoorbrielliatytedinflduecntuseateionnvsiruonn-- derlying fluidity at intermediate attraction strength. −5 We would like to thank Laura Kaufman for useful −10 discussions. We acknowledge the NSF (D.R.R.), DOE −10 −5 0 5 10 −5 0 5 10 (P.L.G.), and the United States-Israel Binational Sci- x/a x/a ence Foundation (grant number 2002100 to D.R.R and E.R.) for financial support. D.R.R. is a Camille Drey- FIG. 4: Representative trajectories of particles whose dis- fus Teacher-Scholar and an Alfred P. Sloan Foundation placements in the xy-plane over a time interval τα are much Fellow. smaller or much larger in magnitude than average. Particle traces havebeen projected onto this plane for graphical sim- plicity. Left panels depict dynamics of a strongly attractive system, φ =0.375. Particles among the5% least mobile are p shown in (a), while particles among the 5% most mobile are [1] U.BalucaniandM.Zoppi,Dynamics of the LiquidState shownin(b). Rightpanelscorrespond toaweaklyattractive (Oxford, New York,1994). system, φ = 0.05. Again, trajectories exhibiting extremely p [2] W. Kob et al.,Phys. Rev.Lett. 79, 2827 (1997). smalldisplacements(c)areplottedinthetoppanel. Trajecto- [3] C. Donati et al.,Phys. Rev.Lett. 80, 2338 (1998). ries exhibiting extremely large displacements (d) are plotted [4] D. N. Perera and P. Harrowell, Phys. Rev. E 54, 1652 in thebottom panel. (1996). [5] R. Yamamoto and A. Onuki, Phys. Rev. E 58, 3515 (1998). pecially vivid. The limited spatial extent of facilitating [6] L. A. Deschenes and D. A. Vanden Bout, Science 292, defects in a hard sphere glass cuts off the range of avail- 255 (2001). able displacements. Clustering of these defects as at- [7] E.V.RussellandN.E.Israeloff,Nature408,695(2000). tractions are introduced provides increasingly extended [8] S.A.Reinsberget al.,J.Chem.Phys.114, 7299(2001). loose regions for mobile particles to explore. Our nu- [9] E. R.Weeks et al.,Science 287, 627 (2000). [10] C. Donati et al.,Phys. Rev.E 60, 3107 (1999). mericalresultsprovidestrongevidenceforsegregationof [11] S. Butler and P. Harrowell, J. Chem. Phys. 95, 4454 jammed and unjammed regions of attractive liquids at (1991). high density. Such segregation is an obvious feature of [12] J. P. Garrahan and D. Chandler, Proc. Nat. Acad. Sci. attractive colloids at low φc, which form stable (though 100, 9710 (2003). likelynonequilibrium)gel-likenetworks. Inthatcase,the [13] J. P. Garrahan and D. Chandler, Phys. Rev. Lett. 89, spacingbetweendense regionsofthe networkestablishes 035704 (2002). a minimum length scale for dynamic heterogeneity. It [14] S.Whitelam,L.Berthier,andJ.P.Garrahan,Phys.Rev. Lett. 92, 185705 (2002). is remarkable that remnants of this behavior persist at [15] S. Whitelam and J. P. Garrahan, Facilitated spin mod- high packing fraction, where spatial heterogeneityof liq- els inonedimension: areal-spacerenormalization group uid structure is subtle. study,cond-mat/0405647. Insummary,thechangeindynamicsinducedbyshort- [16] M. Vogel and S.C. Glotzer, Temperature dependenceof ranged attractions in a dense model liquid is dramatic spatially heterogeneous dynamics in a model of viscous even on the microscopic scale. Our results reveal three silica, cond-mat/0404733. [17] L. Berthier and J. P. Garrahan (unpublished). distinct regimes of dynamical heterogeneity. For weak [18] K. Dawson et al.,Phys.Rev. E 63, 011401 (2001). attractions, mobilized particles make discrete jumps be- [19] A. M. Puertas, M. Fuchs, and M. E. Cates, Phys. Rev. tweencagestructures,whichmayremaincorrelatedover Lett. 88, 098301 (2002). many jump times. For a range of intermediate attrac- [20] A. M. Puertas, M. Fuchs, and M. E. Cates, Phys. Rev. tion strengths, Gaussian particle displacement statistics E 67, 031406 (2003). 6 [21] L. Fabbian et al., Phys.Rev.E 59, R1347 (1999). tions in jammed liquids: How cooling can melt a glass, [22] K.N. Pham et al.,Science 296, 104 (2002). cond-mat/0402673. [23] The effect of such a term on the type of heteroge- [31] AfterthisworkwassubmittedtotheLosAlamosarchive neousmotiondiscussedherehasnotbeenunambiguously server, we became aware of related work in press by demonstrated.Norisitclearthatsucharepulsionexists Puertas, Fuchs and Cates (which appeared as J. Chem. between colloidal particles studied in experiments. Phys.121,2813(2004)).Thatworkalsoconfirmstheear- [24] M. E. Cates et al., Theory and simulation of lier central ideas contained in the theory of Geissler and gelation, arrest and yielding in attracting colloids, Reichman (cond-mat/0402673). Their conclusions differ cond-mat/0403684. from those presented in thispaperin that they viewthe [25] T.B. Schroder,S.Sastry,J.C. Dyre,andS.C. Glotzer, bimodal distribution of particle displacements as indica- J. Chem. Phys. 112, 9384 (2000). tive of a form of dynamical heterogeneity unique to sys- [26] R. Yamamoto and A. Onuki, Phys. Rev. Lett. 81, 4915 tems with attractive interactions. We contend that this (1998). bimodality is generic to glassy systems. The study of [27] R. K. Murarka and B. Bagchi, Phys. Rev. E 67, 051504 Puertas et al. further differs from our own in focusing (2003). on a much lower volume fraction, for which there is no [28] D. R. Reichman, E. Rabani, and P. L. Geissler (unpub- corresponding repulsiveglassy statetocompare. Itisin- lished). teresting that much of thebehavior they reported exists [29] E. J. Gumbel, Statistics of Extremes (Columbia Univer- even at the veryhigh volumefractions studied here. sity Press, New York,1958). [30] P.L.Geissler andD.R.Reichman,Short-rangedattrac-

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