Cond-Mat.Soft

Home , Sharon Glotzer

arXiv:cond-mat/9810060v1 [cond-mat.soft] 6 Oct 1998 pta orltoso oiiyadimblt naglassf a in immobility and mobility of correlations Spatial ain fprilsasge acrigt hi displace corre- their spatial to for (according present assigned directly the particles test of To of goals lations (1) principle twofold: The are data paper system. other same to [11] the (MCT) on theory coupling mode ideal the clciia temperature critical ical eae yaiswr bevdi eieo tempera- of regime a in ture observed cor- were spatially dynamics These super- related mixture. a (LJ) of Lennard-Jones simulations [8] cooled dynamics motion molecular molecular extensive cooperative in also and [7] erogeneity liquid glass-forming [2–6]. and to in dynamics relaxation heterogeneity” slow exponential of “dynamical stretched have origin of microscopic years the study recent understand the in simulations on experimental particular, focused and In theoretical complement efforts. increas- to are micro- used available, this ingly immediately which is in information computer of liquids, consequence, scopic motions a supercooled As dynamical of liquid. simulations microscopic the the of molecules phenom- of bulk the terms these for in account theories to ena of transition goal central glass the a the to been of long approach has the It of “signature” transition. glass a as discussed often is ue n w-tp tece xoeta ea fthe of decay function exponential scattering stretched tempera- intermediate two-step, decreasing with and time ture, relaxation melt- of and their increases viscosity non-Arrhenius below rapid cooled exhibit behavior liquids temperature temperature many ing example, high For their expected [1]. of be might extrapolation what from from dramatically differ liquids 1 eetyw eotdteosraino yaia het- dynamical of observation the reported we Recently h ukdnmclpoete fmn od dense cold, many of properties dynamical bulk The oyesDvso n etrfrTertcladComputati and Theoretical for Center and Division Polymers T density , 2 4 eateto ple ahmtc,Uiest fWsenO Western of University Mathematics, Applied of Department aallCmuainlSine eatet adaNatio Sandia Department, Sciences Computational Parallel lui Donati Claudio 3 ASnmes 27.s 12.c 61.43.Fs 61.20.Lc, po 02.70.Ns, local numbers: the PACS in fluctuations equilibrium the that to show related We the are and temperature. ments c temperature, critical the mode-coupling decreasing of the with at scale strongly length grow par cooperat and to immobile move strength found The particles The mobile clusters. interval. the string-like time while chosen clusters, e suitably compact or a (“mobile”) over large motions ments extremely atomic undergoing local particles the detail among in characterize we mixture, sn xesv oeua yaissmltoso nequil an of simulations dynamics molecular extensive Using ntttfu hsk oansGtnegUniversität, St Gutenberg Johannes für Physik, Institut ρ .INTRODUCTION I. n pressure and T 1 c hrnC Glotzer C. Sharon , band[,0 rmfisby fits from [9,10] obtained P for F ( q T t , bv h dynam- the above .Sc behavior Such ). 1 ee .Poole H. Peter , Fbur ,2008) 1, (February s - 20899 1 nlMtrasSine IT atesug ayad USA Maryland, Gaithersburg, NIST, Science, Materials onal oadsrbto frlxto ie 1] hsinterpre- This [12]. times relaxation i.e., of distribution constants, a time ex- to different local independent with many relaxations of sum ponential a to attributed be can rgniy n nlyi e.I ecnld ihadis- a with conclude we IX het- cussion. Sec. dynamical in dependent observed finally time and the certain erogeneity, the examine with of we associated composition VIII quantities Sec. local alter- In the or in energy, liquid. cor- fluctuations potential are the to mobility of natively local fluctuations the to of related fluctuations that show we eairehbtdb h ogtm eaainof relaxation time long the by exhibited behavior ilycreae.Fo hsw r bet dniya identify to able decreasing with are grows we that this scale spa- From are length displacement low correlated. that or show tially high the and extremely section, on of previous achieve the particles they in defined displacement accord- scale maximum subsets time into the particles to group we ing VI study Sec. to In use the paper. we of remainder which the scale throughout heterogeneity time dis- dynamical a we particle define of V to distribution the Sec. placements time-dependent analyze In the and of displacement shape dynamics liquid. square bulk mean the the the of examine examine structure we simulation equilibrium IV, computer and Sec. and Sec- In model in the techniques. describe and we III information, tion background relevant present structure. local or and this mobility between heterogeneity” extreme connections “dynamical establish of to subsets (2) to and immobility, time) some over ment thsbe rpsdta h tece exponential stretched the that proposed been has It hsppri raie sflos nScinI we II Section In follows. as organized is paper This 2 uigrWg7 -59 an,Germany Mainz, D-55099 7, Weg audinger atrKob Walter , a aoaoy luuru,N 87185-1111 NM Albuquerque, Laboratory, nal eso htsailcreain exist correlations spatial that show We . reain ewe oieprilsare particles mobile between orrelations tro odn nai 6 B,Canada 5B7, N6A Ontario London, ntario, eta nryadlclcomposition. local and energy tential teeysal(imbl” displace- (“immobile”) small xtremely vl n omquasi-one-dimensional, form and ively ecreain ntepril displace- particle the in correlations se bim ls-omn Lennard-Jones glass-forming ibrium, encutrsz per odiverge to appears size cluster mean ilsfr h oe frelatively of cores the form ticles I BACKGROUND II. rigLnadJnsliquid Lennard-Jones orming 3 n tvnJ Plimpton J. Steven and , T nSc VII Sec. In . 4 F ( q t , ) tation is one form of the so-called “heterogenous” sce- whether the molecules in a subset are randomly scattered nario for relaxation [6,12–17]. A number of recent experi- through the sample or tend to cluster in a characteristic ments [13–15] have shown that in liquids such as orthoter- way. phenyl and polystyrene within 10 K of their glass tran- The explicit connection between dynamical hetero- sition temperature Tg, subsets of molecules rotate slowly geneity and cooperative motion is only recently being in- relative to the rest of the molecules on time scales long vestigated experimentally in detail [20]. However, there compared with collision times, but shorter than the re- have been a number of recent computational investiga- laxation time of density fluctuations. These liquids were tions addressing these issues. For example, Muranaka thus termed “dynamically heterogenous.” None of these and Hiwatari [2] showed that displacements of particles experiments were able to explicitly demonstrate whether measured over a timescale of the order of 5 collision slow molecules are spatially correlated, but typical dis- times are correlated within a range of about two inter- tances over which slow molecules may be correlated were particle distances in a two-dimensional binary mixture inferred [13]. of soft disks below the freezing point. Wahnström [29] There have been numerous attempts to indirectly mea- showed that hopping processes in a strongly supercooled sure a characteristic length scale over which molecular binary mixture are cooperative in nature. Hurley and motions are correlated at the glass transition both in ex- Harrowell [4] identified fluctuating local mobilities in a periments [18–21] and in simulations [3,22]. Donth [18] supercooled two-dimensional (2-d) soft-disk system, and relates the distribution of relaxation times in systems ap- showed an example of correlated particle motion on a proaching their glass transition to equilibrium thermo- timescale of the order of 20 collision times. Mountain dynamic fluctuations having a characteristic size of ∼ 3 [3] demonstrated similar correlated particle motion in a nm at Tg. Thermodynamic measurements on orthoter- 2-d supercooled Lennard-Jones mixture. By examining phenyl [19], and dielectric measurements on salol [20], N- the time at which two neighboring particles move apart methyl-ǫ-caprolactan and propylene glycol [21], showed in 2-d and 3-d simulations of a supercooled soft-sphere a shift in Tg due to confinement in pores of the order mixture, Yamamoto and Onuki demonstrated the growth of a few nanometers. Mountain [3] showed that the size of correlated regions of activity [5]. They further stud- of regions that support shear stress in a simulation of a ied the effect of shear on these regions [5], and showed glass-forming mixture of soft spheres grows with decreas- that the size of the regions diminished in high shear. The ing temperatures. Monte Carlo simulations of polymer clusters of “broken bonds” (denoting pairs of neighboring chains in two dimensions demonstrated strong finite size particles that separate beyond the nearest neighbor dis- effects on diffusion [22]. A number of experiments and tance) identified in that work are similar in some respects simulations on polymers confined to thin films all found to the clusters of highly mobile particles in a 3-d binary a shift of Tg due to confinement [23–27]. These effects Lennard-Jones liquid reported previously by us [7], and have all been attributed to the presence of cooperatively described in detail in the present paper. The connec- rearranging regions that grow with decreasing T . How- tion between the clusters of Ref. [7], which demonstrate ever, the origin of this characteristic length has never a form of dynamical heterogeneity, and cooperative par- been shown explicitly. In particular, the connection of ticle motion, was shown in Ref. [8]. the characteristic length to a cooperative mechanism of molecular motion has not been experimentally demonstrated. III. SIMULATION DETAILS The intuitively-appealing picture of cooperative molecular motion was proposed in 1965 by Adam and We performed equilibrium molecular dynamics (MD) Gibbs [28]. In their classic paper, they proposed that simulations of a binary mixture (80:20) of N = 8000 significant molecular motion in a cold, dense fluid can particles in three dimensions. The simulations were per- only occur if the molecules rearrange their positions in a formed using the LAMMPS molecular dynamics code [30] concerted, cooperative manner. They postulated that a which was designed for use on distributed memory par- glass-forming liquid can be viewed as a collection of in- allel machines. LAMMPS partitions particles (atoms or dependently relaxing subvolumes within which the mo- molecules) across processors via a spatial decomposition tion of the particles is cooperative. As the temperature [31] whereby each processor temporarily “owns” parti- of the liquid is lowered, the number of particles involved cles in a small fixed region of the simulation box. Each in cooperative rearrangements increases. If structural re- processor computes the motion of its particles and ex- laxation occurs through the cooperative rearrangement of changes information with neighboring processors to com- groups of molecules, the liquid observed over a time-scale pute forces and allow particles to migrate to new proces- shorter than the structural relaxation time will appear as sors as needed. a collection of regions of varying mobility. These predic- The 6400 particles of type A and 1600 particles of type tions can be tested by selecting subsets of molecules that B interact via a 6-12 Lennard-Jones potential, relax slower (or faster) than the average, and determining

2 σαβ 12 σαβ 6 0.03% of the desired P and T at low temperatures. Vαβ(r)=4ǫαβ − , (1) r r At the lowest T studied (T = 0.4510), the total run time following equilibration is 1.2×104 time units. Thus, where αβ ∈{A, B}. The interaction forces between par- assuming Argon values for the parameters in Eq. 3.1, the ticles are zero for all r > rc = 2.5σAA. Both types data presented here extend up to 25.8 ns. of particles are taken to have the same mass m. The Lennard-Jones interaction parameters ǫα,β and σα,β for this mixture are: ǫAA = 1.0, ǫAB = 1.5, ǫBB = 0.5, IV. STRUCTURE AND BULK RELAXATION σAA = 1.0, σAB = 0.8, σBB = 0.88. Both the relative concentration of particle types and the interaction pa- In this section, we show that the simulated liquid rameters were chosen to prevent demixing and crystal- exhibits the characteristic features of an atomic glass- lization [9]. Throughout this paper, lengths are defined forming liquid. in units of σAA, temperature T in units of ǫAA/kB, and Structural relaxation may be probed experimentally 2 time t in units of σAAm/ǫAA. by the intermediate scattering function F (q,t), which is The simulationsp for each state point (P,T,ρ) are per- both the spatial Fourier transform of the van Hove corre- formed in three stages. First, a constant NPT adjust- lation function G(r,t) and the inverse time transform of ment run is performed by coupling the system to stochas- the dynamic structure factor S(q,ω) [34]. In a computer tic heat and pressure baths to bring the system from a simulation, the self (incoherent) part of the intermediate nearby state point (usually the previously simulated state scattering function Fs(q,t) may be calculated directly point) to the desired state point [32]. Second, a constant from NVT equilibration run is performed to test for unwanted 1 iq·(rj (t)−rj (0)) drifts in pressure P or potential energy U [33]. If no drift Fs(q,t)= e , (2) N is observed, the final state of the system is considered to α D Xj∈α E represent an equilibrium state of the system. Third, a where rj (t) is the position of particle j at time t, and constant NVE data-gathering run is performed using the h· · ·i indicates an average over independent configura- final equilibrated state obtained from the second stage, tions. This quantity describes the relaxation of density and snapshots containing the particle coordinates and fluctuations due to single particle displacements on an velocities are taken at logarithmic time intervals during inverse length scale 2π/q, where q ≡ |q|. If we assume the run. In this stage, the equations of motion are in- rotational invariance of the system, Fs(q,t) depends only tegrated using the velocity Verlet algorithm with a step on q. The time dependence of Fs(q,t) for the A particles size of 0.0015 at the highest temperature, and 0.003 at for q = qmax is shown in Fig. 1. (Throughout this paper, all other temperatures. All quantities presented here are q is chosen as qmax, the position of the first maximum of calculated in this third stage. The analysis is performed the static structure factor S(q, 0)). At high T , Fs(q,t) by post-processing the snapshot files, which number as decays to zero exponentially. As the system is cooled, many as several thousand for the lower temperatures. Fs(q,t) develops a plateau that separates a short time We simulated nine state points along a path in P,T,ρ relaxation process from a long time relaxation process. that is linear when projected in the (P,T ) plane. This This plateau indicates a transient “localization” of par- path was chosen so that we would approach, from high ticles in the “cages” formed by their neighbors, and is a temperature, the mode-coupling critical point Tc = characteristic feature of all glassforming liquids. 0.435, Pc =3.03,ρc =1.2 [9] along a path different from The mode-coupling theory developed for supercooled that used in Ref. [9]. Table 1 shows the values of P,T, liquids by Götze and Sjögren makes a number of predic- and ρ for each state point. tions concerning the decay of the intermediate scattering For state points far above Tc (e.g. runs 1–5), the data- function [11]. These predictions have been tested and gathering runs required more cpu time than the equilibra- verified for the LJ potential used here in a regime of P , tion. For state points nearer Tc, the equilibration stage T , and ρ similar but not identical to that simulated here was the most time consuming. In these cases the NVT [9]. There it was shown, e.g., that the early and late stage of the simulations showed a slight drift of the pres- β-relaxation regimes are well described by power laws, sure over time. To shorten the time required for complete and that the late time behavior of Fs(q,t) exhibits time- equilibration, we estimated a new volume or temperature temperature superposition with a time constant τα that to create a nearby state point that we expected would be diverges as a power law as T approaches Tc ≃ 0.432, with very nearly equilibrated with the current particle config- exponent γ ≃ 2.7. The diffusion constant was found to uration. Then we instantaneously scaled the positions or −γ scale as D ∼ (T −Tc) , with γ =2.0 for the A particles, velocities of the system (ie. adjusted the volume or tem- γ =1.7 for the B particles, and Tc =0.435. perature) and began another constant NVT run to test The simulations performed in the present work extend for equilibration. By iterating this procedure a few times from a point in the phase diagram where two-step re- we were able to find an equilibrated state point within laxation begins to emerge, down to a state point that

3 is within approximately 4% of Tc. Over this range, we (see Fig. 5). In the short time limit (regime I) the MSD 2 2 find that τα increases by 2.4 orders of magnitude, and is ballistic, i.e. hr (t)i ∝ t . For longer times (regime fits well to the power law form found in Ref. [9], with III), the MSD is diffusive, i.e. hr2(t)i∝ t. As the system approximately the same critical temperature and critical is cooled, an intermediate regime (II) between these two exponent. limiting behaviors develops. Before entering the diffu- It is well known that although relaxation becomes sive regime, hr2(t)i exhibits a plateau, analogous to the strongly nonexponential and relaxation times increase by plateau in the intermediate scattering function, that like- many orders of magnitude as a supercooled liquid ap- wise arises from a transient “caging” of each particle by proaches a glass transition, changes in the static structure its neighbors. As seen in the figure, the time the sys- of most liquids are far less remarkable. To demonstrate tem spends in the plateau depends strongly on T , and this for our system, we examine the pair correlation func- increases with decreasing T . The MSD for the B par- tions gαβ(r) given by ticles (not shown) exhibits qualitatively the same time dependence as shown in Fig. 5 , but the diffusive regime V gαβ(r)= δ(r + rj − ri) , (3) is reached at shorter times, and the diffusion constant is NαNβ D Xi∈α E larger, than for the A particles [9]. This difference can j∈β be explained by the different sizes of the A and B parti- for α 6= β and cles and by the fact that the interaction constant ǫBB is smaller than ǫAA. V g (r)= δ(r + r − r ) , (4) In this paper, we are interested in whether spatial cor- αα N (N − 1) j i α α D i,jX∈α E relations exist between particles that exhibit either extremely large or extremely small displacements over some where Nα (Nβ) is the total number of particles of species time interval. To determine this, we must first define the α (β). With this normalization, gαβ(r) converges to unity time interval over which the particle displacements will for r → ∞ in the absence of long range correlations. be monitored. Obviously, displacements may be mon- Assuming rotational invariance, the correlation functions itored over any time interval, from the ballistic regime do not depend on the direction of the vector r, but only to the diffusive regime. To see whether there is a natu- on the distance r = |r|. ral time scale on which the particle displacements might In Figs. 2,3 and 4 we show the pair correlation func- exhibit a particularly strong correlation, we turn to the tions gAA(r), gAB(r), and gBB(r) for three temperatures. self part of the van Hove correlation function, Gs(r, t), The figures show that these functions do not change dra- which gives the probability to find a particle at time t at matically as a function of the state point. As the tem- a distance r from its position at t = 0 [34]: perature is lowered, the main effect on all three func- 1 tions is that the maxima and the minima become slightly G (r,t)= δ(r + r (0) − r (t)) . (5) s N i i more pronounced. Additionally, the second maximum of α D Xi E gAA(r) and gAB(r) at low T shows a splitting that has Due to the rotational symmetry of the system, Gs(r,t) commonly been interpreted as a signature of an amor- is a function of the modulus r of the vector displacement phous solid, although at these state points our system 2 r. The quantity 4πr Gs(r, t), which gives the number of is an equilibrium liquid. Recently, evidence has been re- particles located a distance r from their original position ported [35] that in a 2-d system of hard-disks the splitting at time t, is shown in Fig. 6 for the A particles for three of the second peak in the pair correlation function is due different times at the lowest T . Also shown in Fig. 6 is to the formation of regions with hexagonal close-packed the Gaussian approximation 4πr2G0(r, t) , where [34] order. 3 2 2 0 3 3r G (r, t)= 2 exp 2 (6) V. SINGLE PARTICLE DYNAMICS 2πhr (t)i 2hr (t)i and where hr2(t)i is equal to the measured one. The Having established that the model liquid studied here Gaussian form appears to be a good approximation to exhibits the characteristic bulk phenomena of a glass- Gs(r, t) at both short and long times. However, it is forming liquid, we examine in this section the distribu- apparent from the figure that Gs(r, t) is significantly dif- tion of individual particle motions. ferent from G0(r, t) at intermediate times. In particu- The most basic dynamical bulk quantity that is easily lar, while many of the particles have traveled less than accessible to simulation is the particle mean square dis- would be expected from the knowledge of hr2(t)i alone, a placement (MSD), hr2(t)i. Because we are investigating small number of particles have traveled significantly far- a binary mixture, we refer in the following to a MSD for ther. As a result, at intermediate times Gs(r, t) displays the A particles and a MSD for the B particles. At high T , a long tail that extends beyond one interparticle distance the MSD for both species exhibits two distinct regimes at T =0.4510 (cf. Fig. 7).

4 This “long tail” behavior is most pronounced at a VI. ANALYSIS OF SPATIAL CORRELATIONS ∗ OF PARTICLE DISPLACEMENTS time t when Gs(r, t) deviates most from a Gaussian (cf. Fig. 7) as characterized by the “non-Gaussian” parameter [36], In this paper, we are interested in studying the extreme behavior of the individual particle motion, from the ex- 4 3hr (t)i tremely mobile to the extremely immobile. In Refs. [7] α2(t)= − 1, d =3. (7) 5hr2(t)i2 and [8], we defined the magnitude of the displacement ∗ ∗ ui(t,t ) ≡ |ri(t + t) − ri(t)| of particle i in a time inter- While one can define many different non-Gaussian pa- ∗ val t , starting from its position at an arbitrarily chosen rameters, this particular one involves the lowest possible time origin t, as a measure of the mobility of the i-th par- moments. With this definition, α2(t) is zero if Gs(r, t) is ∗ ticle. At t , the distribution of ui values is given by the Gaussian. If a distribution has a tail that extends to dis- ∗ self part of the van Hove correlation function, G (r, t ), tances exceeding those for a Gaussian distribution with s where r ≡ u (cf. Fig. 7). In Refs. [7,8] a subset of the same second moment, all higher order moments of “mobile” particles was defined by selecting all the par- this distribution will exceed those of the corresponding ∗ ticles that in the interval [0,t ] had traveled beyond the Gaussian, and consequently α (t) will assume positive ∗ ∗ ∗ 2 distance r where G (r, t ) exceeds G0(r, t ). With this values. In Fig. 8 we show α (t) for various T for the A s 2 definition, “mobile” particles are those that contribute particles. As expected, α (t) is zero at short times, then 2 to the long tail of the van Hove distribution function at becomes positive, exhibits a maximum, and finally goes ∗ the time t (cf. Fig. 7). In Refs. [7,8], it was shown that to zero at long times. As T decreases, the position of the mobile particles selected according to this rule tend to maximum t∗ shifts towards longer times, and the height cluster [7], and move cooperatively [8]. This definition of the maximum α∗ increases. For all T , we find that 2 of mobility given by the magnitude of particle displace- t∗ corresponds to times in the late-β/early-α relaxation ment is thus sufficient to establish the phenomena of both regime. Furthermore, by dividing α (t) by α∗, and divid- 2 2 dynamical heterogeneity and cooperative motion. ing t by t∗, one can show [37] that all curves collapse onto Intuitively, we think of immobile particles as those par- a single master curve for all times larger than the micro- ticles which are trapped in cages formed by their neigh- scopic time (where they already collapse before scaling) bors. Nevertheless, particles do not sit at one position; [4,10,38]. This data collapse, which is likely related to they essentially “oscillate” back and forth within the cage the time-temperature superposition exhibited by the in- formed by their neighbors. To study correlations between termediate scattering function (but not trivially related, the most immobile particles, we need a definition of mo- since t∗ does not scale linearly with τ ), suggests that t∗ α bility which allows us to select the particles for which the is in some sense a characteristic time for this system [37]. amplitude of this oscillation (ie. the maximum displace- Note that t∗ is orders of magnitude larger than the micro- ment of the particle) is the smallest. In this paper, we scopic “collision time” [39] τ; for example, at T =0.4510 therefore define the mobility µ (t) of the i-th A particle t∗ = 155.5 and τ =0.09. i as the maximum distance reached by that particle in the We see from an analysis of the temperature-dependent ∗ time interval [t,t + t ]: distribution of particle displacements at various times, and the calculation of the time-dependence of the non- ′ µi(t) = maxt′∈[0,t∗]{|ri(t + t) − ri(t)|} (8) Gaussian parameter, that the single particle dynamics is most non-Gaussian — and displays the widest range of This new definition of mobility, which we use throughout ∗ possible behaviors — on the timescale t . The interval this paper, allows us to examine different subsets of par- ∗ from zero to t thus provides a convenient choice over ticles, from the most to the least mobile, in the same way. which to monitor the particle displacements and study As a compromise between examining the most extreme ∗ their correlations because (i) since t is the time at which behavior and including enough particles to obtain good the distribution of particle displacements is broadest, it statistics when examining their spatial correlation (i.e. may also be when the liquid is likely to be most “dy- maximizing the signal-to-noise ratio), we will examine ∗ namically heterogeneous”; and (ii) t is well-defined and the 5% most mobile and 5% least mobile particles. Thus easily calculated. Thus, throughout this paper we will we define as “mobile” the 5% of all particles having the ∗ use the time window from zero to t as the time interval highest values of µ(t), and “immobile” the 5% having the over which the particle displacements are calculated, and lowest value. Note that this new definition of mobility over which we investigate dynamical heterogeneity. does not qualitatively change the results obtained previously in [7,8], provided that the new definition selects approximately the same fraction of the sample as the definition previously used (approximately 5.5% in Ref. [7]) Compare, for instance, Figs. 11 and 12 with Fig. 3 of Ref. [7].

5 The subsets of mobile particles selected using the defi- than a decade on either axis, the figure shows that the nition of Ref. [7] and that used here have a large overlap, temperature dependence of S is consistent with a diver- −γ since particles that have moved relatively far at some gence at Tc of the form S ∼ (T − Tc) , with γ ≈ 0.618. time in the interval [0,t∗] are likely to remain relatively Note that MCT makes no predictions about clustering or far at the end of the interval. However, subsets of im- the divergence of any length scales as the critical point mobile particles selected with the two different rules do is approached [42]. not have as large an overlap, since a particle with a small To test the sensitivity of the apparent percolation tran- displacement at some time may have previously traveled sition at the mode-coupling temperature, we repeat the far, and then returned to its original position. The dis- cluster size distribution analysis for the 3% and 7% most tribution 4πµ2P (µ, t∗) at t∗ is shown in Fig. 9. For com- mobile particles. For each subset, the mean cluster size 2 ∗ parison, the probability distribution 4πr Gs(r, t ) is also S is shown vs. T − Tc in Fig. 14. The best fit of ∗ −γ shown. Note that, although at t particles can be found S ∼ (T − Tp) to each set of data gives Tp = 0.440 arbitrarily close to their position at t = 0, P (µ, t∗) is zero for the set containing the 3% most mobile particles, for µ< 0.17. Tp = 0.431 for the set containing the 5% most mobile In Fig. 10, we show the 320 mobile particles (light particles, and Tp = 0.428 for the set containing the 7% spheres) and the 320 immobile particles (dark spheres) most mobile particles. However, within the accuracy of at the beginning of an arbitrary time interval [t,t + t∗] the data the three sets are also consistent with a diver- for one configuration at T = 0.4510. The other 7360 gence at Tc. If we further increase the fraction of mobile particles are not shown. The figure shows that particles particles beyond the fraction corresponding to a random of similar mobility are spatially correlated and that par- close-packed percolation transition [43], the mobile parti- ticles with different mobility tend to be anticorrelated. cles percolate and most of the mobile particles are found These correlations can be quantitatively studied by cal- in a single cluster that spans the whole simulation box. culating static pair correlation functions between parti- In Fig. 15 we show one of the largest clusters of mobile cles belonging to the different subsets. particles found in our coldest simulation. It is evident In Fig. 11 we show the pair correlation function from the figure that these clusters cannot be described gMM (r) between mobile particles for four different tem- as compact, as often supposed either implicitly or explic- peratures. gMM (r) is defined by Eq. 4 with the sum itly in phenomenological models of dynamically hetero- restricted to the mobile particles.. For all T , gMM (r) is geneous liquids [13,44]. Instead, the clusters formed by appreciably higher that the average gAA(r) (cf. Fig. 2) the mobile particles appear to have a disperse, string-like for all r. The “excess” correlation given by the ratio nature. As discussed in [8], a preliminary calculation of Γ(r) = [gMM (r)/gAA(r)] − 1 is plotted as a function of the fractal dimension of the clusters, although hampered r in Fig. 12. With the exception of the excluded volume by a lack of statistics, indicates that the clusters have a sphere of the LJ potential, Γ(r) > 0 at intermediate dis- fractal dimension close to 1.75, similar to that for both tances and converges to zero for large r. It is clear from self-avoiding random walks and the backbone of a ran- the figure that the total excess correlation, given by the dom percolation cluster in three dimensions [45]. area under the curve, increases with decreasing T . In Ref. [8], it was shown that this quasi-one- We can obtain an estimate of the typical distance over dimensionality appears to arise from the tendency for which mobile particles are correlated by identifying clus- mobile particles to follow one another. This is demon- ters of nearest-neighbor mobile particles [40]. To do this, strated in Fig. 16, where we plot the time-dependent pair ∗ we use the following rule: two particles belong to the correlation function for the mobile particles, gMM (r, t ) same cluster if their distance at t = 0 is less than rnn, for different temperatures. At t = 0, this function co- the radius of the nearest neighbor shell, which is defined incides with gMM (r) in Fig. 11. For t > 0, the nearest by the first minimum in gAA(r) and has a weak temper- neighbor peak moves toward r = 0, demonstrating that a ature dependence. In our hottest run rnn = 1.45, while mobile particle that at t = 0 is a nearest neighbor of an- in the coldest run rnn = 1.40. The distribution P (n) of other mobile particle tends to move toward that particle clusters of size n is shown in Fig. 13. Although most of at later times. We find that the peak at r = 0 is highest the clusters have only a modest size, the data show that a near t = t∗, and decreases for later times. A small but significant fraction of the mobile particles, which them- discernable peak at r = 0 is also present in g(r, t∗) [46]. selves make up only 5% of the sample (320 particles), Fig. 17 shows a cluster of mobile particles at two dif- are part of big clusters. For instance, at T = 0.4510, ferent times, t = 0 and t = t∗, to demonstrate the coop- there is typically at least one cluster in each configuration erative, string-like nature of the particle motion. that contains ≈ 100 particles. For that T , P (n) ∼ n−τ In a manner identical to our analysis of the mobile par- with τ = 1.86. In the inset we show the mean cluster ticles, we define as immobile the 5% of the A particles size S = n2P (n)/ nP (n) [41], plotted log-log versus that have the lowest value of µ. The pair correlation func- T −Tc, whereP Tc =0.P435 is the fitted critical temperature tion gII (r) between immobile particles shown in Fig. 18 of the mode coupling theory [9,10]. Although there is less shows that these particles also tend to be spatially corre-

6 lated. It is interesting to note that while the maxima in In Fig. 24 we show the distributions of the potential gII (r) are higher at all T than the corresponding max- energies of the 5% most mobile, 5% least mobile, and all ima in gAA(r), the depth of the minima does not change particles at T =0.4510, calculated at the beginning of an appreciably for the lowest temperatures. Fig. 19 shows arbitrary time interval [t,t + t∗]. The distributions have the ratio Γ(r) = [gII (r)/gAA(r)] − 1 as a function of r. been normalized such that the area under each curve is In contrast to what we find for the most mobile parti- one. The distributions differ by a small relative shift of cles, the correlation between immobile particles does not the mean value, approximately 3% for the high mobil- show any evidence of singular behavior as T decreases. ity distributions and somewhat less for the low mobility Instead, the correlation appears to grow and then “satu- distribution. We find that the magnitude of the shift in- rate” to some limiting behavior for all T < 0.468. More- creases with decreasing T , but the relative shift appears over, Fig. 19 shows that the local structure of the liquid to be independent of T . Since the liquid is in equilibrium, appears to be more ordered in the vicinity of an immobile this shift will vanish for t → ∞. Thus, not suprisingly, A particle than in the vicinity of a mobile A particle. mobile particles are those that in a time t∗ are able to In Fig. 20 we show the size distribution of the clusters rearrange their position so as to lower their potential en- of immobile particles, formed with the same rule used ergy. It is worth noting that the mobility does not show for the mobile ones. One of the largest clusters found at any correlation with the kinetic energy of the particles T =0.4510 is shown in Fig. 21. In the inset of Fig. 20 we measured at t = 0. The kinetic energy distributions of show the mean cluster size S versus T − Tc. We find that the subsets with different mobility coincide exactly with the mean cluster size of immobile particles is relatively the average distribution, showing that the mobility can- constant with T . This may be because immobile parti- not be related to the presence of “hot spots” in the liquid. cles are relatively well-packed, and cannot grow beyond We next divide the entire population of A particles into some limiting size [47]. Or, these clusters may be the 20 subsets, each composed of 5% of the particles. In the “cores” of larger clusters of particles with small displace- first subset we put the 5% of the particles with the highest ments, that may grow with decreasing T . To elucidate values of µ (the mobile particles defined above), in the this, more particles (e.g. the next 5% higher mobility) second subset the next 5%, and so on. The last subset should be included in the analysis. We will return to this thus contains the 5% most immobile particles. In Fig. 25 important point and provide further relevant data in the we plot (on the x-axis) the average mobility of each subset next section. versus (on the y-axis) the average potential energy of that The correlation between mobile and immobile parti- subset at t = 0. We find that the subset with the lowest cles, measured by the pair correlation function gMI (r) mobility is also the one with the lowest potential energy. (Fig. 22), shows that mobile and immobile particles We also find that as the potential energy increases, the are anti-correlated. A comparison between gMI (r) and mobility increases. We see from the figure that the mobile gAA(r), shown in Fig. 23, demonstrates that, over several particles are the subset with the highest average potential interparticle distances, the probability to find an immo- energy at t = 0. bile A particle in the vicinity of a mobile one is lower Two more points are worth noting in Fig. 25. First, than the probability to find a generic A particle. The at all T the mobile particles move, on average, approx- figure also shows that the characteristic length scale of imately one interparticle distance in the time interval the anticorrelation grows with decreasing T . This length [0,t∗]. Second, for all T the difference in both mobil- scale does not show a tendency to diverge as Tc is ap- ity and potential energy between the 5% most mobile proached. In particular, the curves for the two coldest particles and the next subset is significantly larger than runs (and closest to Tc) are almost coincident. between any other two consecutive subsets. This observation suggests that the choice of 5%, while arbitrary, is a reasonable one. As shown in the figure, the separation VII. LOCAL ENERGY AND LOCAL between the 5% most mobile particles and the next sub- COMPOSITION VS. MOBILITY set shows a tendency to grow with decreasing T . Note however, that the distance between the lowest mobility We have seen in the previous section that despite the subset and the next subset decreases with decreasing T , lack of a growing static correlation, a growing dynamical making it very difficult in the current approach to define correlation — characterizing spatial correlations between an appropriate subset containing particles whose mobil- particles of similar mobility — does exist. These corre- ity is distinctly lower than the rest. This, together with lations must therefore arise from subtle changes in the the result that the mean cluster size of immobile parti- local environment that are not completely captured by cles is relatively constant over the range of temperatures the usual static pair correlation function. In this Sec- studied, suggests that our analysis of the lowest subset tion, we calculate several quantities to elucidate whether is inadequate to fully characterize clusters of particles the mobility of a particle is related to its potential energy, which do not move a substantial distance [48]. and to the composition of its local neighborhood. Thus we see that the gross structural information con-

7 tained in the potential energy is sufficient to establish a one-dimensional regions of high mobility, and relatively general correlation between energy and mobility. How- compact regions of low mobility. ever, as seen in Fig. 24, the distribution of potential en- To measure how long a mobile particle will continue M ergies of mobile particles overlaps for most of the range to be mobile, we define a variable νi (t) as 1 if the i- of the abscissa with the distribution for the generic A th particle belongs to the subset of the 5% most mobile particles. Thus, it is not possible to decide if a certain particles in the interval [t,t + t∗], and 0 otherwise. The particle is mobile on the basis of that particle’s instan- function σM (t) taneous potential energy alone. Other factors, such as 2 defects in the local packing, and the relative potential 1 M M nM σM (t)= n2 hνi (t)νi (0)i− , (9) energy of neighboring particles, must contribute as well. M NA nM − N Xi The relation between mobility and potential energy A suggests a relation between mobility and local compo- measures the fraction of particles that are mobile in the sition. Indeed, a calculation of the pair correlation func- interval [0,t∗] and still mobile in the interval [t,t + t∗], tions gMA(r) and gMB (r) (see Figs. 26 - 29) between a when the time origin is shifted by t. Here nM is the mobile particle and a generic A or B particle, respec- number of mobile particles (320 in the present case), and tively, shows that, on average, a mobile A particle tends NA is the total number of A particles (6400). The second to have less B particles, and more A particles, in its near- term on the right-hand side of Eq. 9 is the number of est neighbor shell than a generic A particle. particles that by random statistics would be classified A correlation can also be found between immobility as mobile in both time intervals. The normalization of and small composition fluctuations of the mixture. A σM (t) is chosen so that σM (0) = 1. The results for σM (t) comparison of gAB(r) to the pair correlation function for the coldest T are shown in Fig. 31. gIB(r), which measures the number of B particles a dis- We have also measured the fraction of particles that are tance r from a test immobile A particle, shows that an immobile in the interval [0,t∗] and still immobile in the immobile A particle has, on average, more B particles in interval [t,t + t∗]. Analogous to the case for the mobile its nearest neighbor shell than does a generic A particle. particles, we define σI (t) as As shown in Fig. 30, where the ratio [gIB/gAB(r)] − 1 2 is plotted as a function of r, this enhanced correlation is 1 I I nI σ (t)= 2 hν (t)ν (0)i− , (10) I n i i independent of T , and therefore does not suggest any ev- I N nI − Xi A idence of A − B phase separation (recall that the chosen NA energy parameters preclude A − B phase separation). where nI is the number of immobile particles (320) and From these results it is clear how a correlation between I νi (t) is a function that is 1 if the i-th particle is an immobility and local composition causes a correlation be- mobile one in the interval [t,t+t∗], and 0 otherwise. The tween mobility and potential energy. Since the attractive function σI (t) is also shown in Fig. 31. interaction between A and B particles is stronger than The functions σM (t) and σI (t) are memory functions of either the attractive AA or BB interaction, the presence mobility. When they have decayed to zero, there are no of a B between two A’s reduces their potential energy. particles that have retained memory of their mobility in A particles in a B-rich region can thus be expected to the initial time interval. Because a particle’s mobility is have a reduced mobility. A particles in a B-poor region, based upon a criterion that depends on t∗, certain time- however, will have a higher potential energy, resulting in dependent functions measured for these subsets will have a higher mobility. some “kink” at t∗. If a different t∗ is chosen, the kink will move to the new t∗. In this respect, there is no “natural” lifetime for these clusters — by definition, they survive VIII. STRUCTURAL RELAXATION OF ∗ PARTICLE SUBSETS AND DYNAMICAL for a time t [50]. HETEROGENEITY Nevertheless, we can obtain information from the form of the decay both before and after t∗. Because the data was stored not less than every 3 time units, we are un- We have shown that it is possible to select subsets of able to calculate σM (t) and σI (t) for t< 3. However, we particles according to their maximum displacement over see that these functions decay substantially before this a timescale in the region of the late β-early α relaxation. time, since already at t = 3 both functions are signif- We have also shown that the particles belonging to sub- icantly smaller than one. After this initial short-time sets selected at the extrema of the mobility spectrum relaxation, a second decay of both functions is observed are spatially correlated, and are related to small fluctu- up to t = t∗. At this time, a third decay process appears ations in the local potential energy, and, consequently, for the mobile particles, and possibly also for the immo- in the local composition of the mixture. All of the data bile particles. The main point of Fig. 31 is that beyond presented here suggest that this supercooled liquid con- t∗, both functions are less than 0.1. Thus there is only tains fluctuations in local mobility, with diffuse, quasi-

8 a small tendency for particles to retain memory of their Finally, we show in Fig. 33 the fraction φ of particles mobility beyond the initial time interval [51]. that at time t have not yet been labeled mobile. This Thus, mobile and immobile regions do not persist be- function is calculated by labeling the mobile particles in yond the time t∗ over which the particle mobility is mon- the first interval [0,t∗], and then shifting the interval by itored. After the observation time, mobile and immobile t and reassigning the particle mobilities. Thus at t = 0, particles maintain little memory of their previous state. 95% of the particles have not been labeled mobile. In the Therefore the strong correlations found between particles interval [t,t + t∗], more particles will have been labeled must arise from the motion itself. If, for instance, the mo- mobile, so φ will decrease. We have normalized φ(t) such bility of a particle and its spatial correlation with other that φ(0) = 1. Also shown in Fig. 33 is the long-time particles of similar mobility could be explained solely by α-relaxation part of the bulk Fs(q,t) for q = qmax. Fits local fluctuations in quantities like density or composi- to both functions are also shown. Both functions fit well tion, the mobility should persist until these fluctuations to a stretched exponential y(t) = Aexp[(−t/τ)β] with decay to zero. Instead, the dependence of the lifetime on β = 0.75 and τα = 655 for the intermediate scattering the observation time can be explained if one assumes that function, and with β = 0.78 and τα = 475 for φ. That particles can move only in a cooperative manner. Indeed, both functions have a similar form (similar β), and sim- as was shown in Ref. [8], clusters of mobile particles like ilar time constants, suggests that the process by which that shown in Fig. 15 can be decomposed into numerous, immobile particles become mobile governs the long time smaller string-like clusters (“strings”) of particles which structural relaxation of density fluctuations at wavevec- follow one another in a cooperative fashion. tors corresponding to the peak of the static structure Fig. 32 shows the intermediate scattering functions factor. Moreover, it demonstrates that the “arbitrary” M I Fs (q,t) and Fs (q,t), defined as the spatial Fourier choice of 5% represents a physically meaningful fraction transform of the self part of the van Hove correlation of the system . M I function Gs (r, t) or Gs(r, t) of the mobile and immobile particles, respectively. Both functions are identical to IX. DISCUSSION the bulk Fs(q,t) for times less than the “collision time” τ = 0.09 [39]. The figure shows that a two-step relaxation process occurs for the mobile particles, although In this paper, we have described an investigation of the the height of the plateau is smaller than for the bulk. individual particle dynamics of a cold, dense Lennard- M The presence of the plateau in Fs (q,t) indicates that Jones mixture well above the glass transition in an effort the mobile particles are subject to the same “cage effect” to discover if the liquid is dynamically heterogenous, and experienced by the other particles, although the effective if so to determine the extent and nature of the dynami- cage “size” and “lifetime” are different. Thus clusters of cal heterogeneity. Since there were no quantitative the- mobile particles should not be thought of as “fluidized” oretical predictions regarding this matter, the approach regions of the liquid in the simple sense that those re- we have taken is exploratory; particle trajectories were gions might behave like high temperature or low density saved during the course of the simulation and then an- liquids. Instead, the difference between the mobile par- alyzed and visualized in numerous ways. We find that ticles and the rest of the sample appears to be, from the this supercooled liquid is “dynamically heterogeneous” point of view of the single particle dynamics, that they because particles with similar mobility are spatially cor- “escape” the cage earlier than the other particles. related. Note that our definition of heterogeneity is dif- We also see that the three curves in Fig. 32 cannot be ferent from the one used, for instance, in 4-D NMR ex- superimposed by scaling the time axis in the same way as periments, where the system is defined as heterogeneous one can superimpose F (q,t) curves for different temper- if a slow subset remains slow for times longer than the atures. Again, this indicates that the mobile and immo- average relaxation time [14]. We further find that highly bile subsets are not simply “hotter” or “colder” subsets of ramified clusters of mobile particles grow with decreasing the sample, in agreement with the perfect superposition T and appear to percolate at the mode-coupling temper- of the kinetic energy distributions. ature. This is the first evidence for a percolation transi- M In contrast to the bulk average Fs(q,t), Fs (q,t) is not tion coincident with Tc, and it is very different from the a monotonically decreasing function of time. For times type of percolation transition proposed in free volume longer than t∗, a small but clearly detectable increase of theory [52]. It is especially interesting since MCT does the function can be noticed in Fig. 32. This behavior can not make any predictions regarding clusters or diverging be interpreted as a tendency of a small fraction of the length scales. We also find that particles of low mobil- particles that we have selected to return towards their ity form relatively well-ordered, compact clusters which position at the beginning of the selection interval. These do not appear to grow with decreasing T if the num- particles may also be those that contribute to the small ber of immobile particles included in the subset is kept memory effect observed in σM (t) in Fig. 31, but further constant. Although mobile and immobile clusters are analysis is required to establish this connection. anti-correlated, there is no tendency towards bulk phase

9 separation of mobile and immobile regions because of the [13] F. R. Blackburn, M. T. Cicerone, G. Hietpas, P. A. Wag- highly ramified, extended nature of the mobile regions. ner, M. D. Ediger, J. Non-Cryst. Solids 172-174, 256 In our analysis, we find no evidence to support a pic- (1994). ture in which the system can be thought of as a collection [14] K. Schmidt-Rohr and H.W. Spiess, Phys. Rev. Lett. 66, of subvolumes that each relax independently and simul- 3020 (1991); J. Leisen, K. Schmidt-Rohr and H. W. 172-174 taneously with their own time constant. Instead, it ap- Spiess, J. Non-Cryst. Solids 737 (1994); A. Heuer, M. Wilhelm, H. Zimmermann and H.W. Spiess, pears that at any given time, most particles are localized Phys. Rev. Lett. 95, 2851 (1995) in cages and a small percentage of particles form large [15] M.T. Cicerone, F.R. Blackburn and M.D. Ediger, J. clusters of smaller, cooperatively rearranging “strings.” Chem. Phys. 102, 471 (1995); M.T. 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10 sign. Run T P ρ [40] Statically defined clusters in liquids have been reported 1 0.5495 0.4888 1.0859 in a number of computational studies. See for example: 2 0.5254 1.0334 1.1177 Y. Hiwatari, J. Chem. Phys. 76, 5502 (1982); R. M. J. 3 0.5052 1.4767 1.1397 42 Cotterill, Phys. Rev. Lett. , 1541 (1979); S.-P. Chen, 4 0.4899 1.8148 1.1553 37 T. Egami and V. Vitek, Phys. Rev. B , 2440 (1988); 5 0.4795 2.0488 1.1651 A. I. Mel’cuk, R. A. Ramos, H. Gould, W. Klein and 6 0.4737 2.1746 1.1705 Raymond D. Mountain, Phys. Rev. Lett. 75, 1995); T. 7 0.4685 2.2959 1.1757 Tomida and T. Egami, Phys. Rev. B 52, 3290 (1995); V. P. Voloshin, Y. I. Naberhukin, N. N. Medvedev and 8 0.4572 2.5490 1.1856 M. S. Jhon, J. Chem. Phys. 102 (1995); G. Johnson, A. 9 0.4510 2.6800 1.1910 I. Mel’cuk, H. Gould, W. Klein,R. D. Mountain, Phys. Table 1:Temperature T, pressure P and den- 57 Rev. E 5707 (1998); M. Li, preprint. sity ρ of the nine state points simulated. [41] To take into account the possibility that the mobile particles percolate in our simulation box at T >Tp, we discard the largest cluster in each configuration in the calculation of S; D. Stauffer and N. Jan, private communication. [42] However, there have been several computational studies which suggest some type of growing dynamical length 1 scale as the glass transition is approached. See, e.g. Ref. [3,22]. [43] Barry D. Hughes, Random Walks and Random Environ- ments (Clarendon Press, Oxford, 1996). [44] F. H. Stillinger and J. A. Hodgdon, Phys. Rev. E 50, 2064 (1994). (q,t)

[45] D. Stauﬀer, Introduction to Percolation Theory (Taylor s 0.5 and Francis. 1985). F T = 0.5495 [46] J.L. Barrat, J.N. Roux, J.P. Hansen, Chem. Phys. T = 0.5254 149 T = 0.5052 , 197 (1990); H. Miyagawa, Y. Hiwatari, B. Bernu, T = 0.4795 J.P.Hansen, J. Chem. Phys. 86, 3879 (1988). T = 0.4685 [47] D. Kivelson and G. Tarjus, Phil. Mag. B 77, 245 (1998), T = 0.4572 and references therein. T = 0.4510 [48] A more promising approach in this direction is to 0 −2 −1 0 1 2 3 4 group particles according to their ﬁrst passage time; see 10 10 10 10 10 10 10 Ref. [49]. t [49] P. Allegrini, C. Donati, J.F. Douglas and S.C. Glotzer, FIG. 1. Incoherent (self) intermediate scattering function in preparation. Fs(q,t) for qmax = 7.12. [50] However, there is a natural time at which the spatial correlation of particle mobilities is the most pronounced. See S.C. Glotzer, C. Donati and P.H. Poole, “Spatially- 4 Correlated Dynamics in Glass-forming Systems: Corre- lation Functions and Simulations”, in Computer Simula- tions in Condensed Matter Physics XI, ed. D.P. Landau, T = 0.5495 et al. Springer-Verlag, in press. 3 T = 0.4795 [51] We have observed that a small fraction of mobile particles T = 0.4510 have a tendency to return to their initial positions at ∗ times longer than t . These particles may be responsible

for this small memory eﬀect, but further analysis needs (r) 2

to be done. AA [52] G.S. Grest and M.H. Cohen, Adv. Chem. Phys. 48, 454 g (1981). 1

0 0 1 2 3 4 5 6 r

FIG. 2. Pair correlation function gAA(r) of the A particles for three diﬀerent temperatures.

11 101 5

T = 0.5495 0 10 T = 0.4795 4 T = 0.4510 III 10−1 3 (r) (t)> 2 AB

g −2 II T = 0.5495

0 0 2 4 6 10−2 10−1 100 101 102 103 104 r t 2 FIG. 3. Pair correlation function gAB (r) between A and B FIG. 5. Mean square displacement hr (t)i of the A particles particles for three diﬀerent temperatures. vs. time.

1.5 2 6 π 4 r Gs(r,t) 2 0 t = 0.363 4πr G (r,t)

1 4 (r) BB g

T = 0.5495 t = 155.5 0.5 T = 0.4795 T = 0.4510 2 t = 5965

0 0 2 4 6 r 0 0 0.5 1 1.5 2 FIG. 4. Pair correlation function gBB (r) of the B particles r for three diﬀerent temperatures. 2 FIG. 6. Solid line: 4πr Gs(r, t) of the A particles for three times at T = 0.4510. Dashed line: Gaussian approximation calculated using the measured hr2(t)i for the same three times.

12 4 4 πµ2 µ 2 * 4 P( ) π 2 4 r Gs(r,t ) 4πr G (r,t*) 3 π 2 0 * s 4 r G (r,t ) 3

2 2

r* 1 1

0 0 0 0.5 1 1.5 2 0 0.5 1 µ , r r ∗ FIG. 9. Probability distribution 4πµ2P (µ, t ) (dashed line) FIG. 7. Same intermediate time data as in previous ﬁgure, 2 of a particle having a maximum displacement of magnitude µ but enlarged. Solid line: 4πr Gs(r, t) of the A particles at ∗ 2 ∗ at t . For comparison, the distribution 4πr Gs(r, t ) is also t = 155.5 at T = 0.4510. Dashed line: Gaussian approx- 2 shown. imation calculated using the measured hr (t)i for the same time.

2.0

T = 0.5495 T = 0.5254 1.5 T = 0.5052 T = 0.4795 T = 0.4685 T = 0.4572 T = 0.4510 2 1.0 α

0.5

0.0 10−1 100 101 102 103 104 t

FIG. 8. Non-Gaussian parameter α2(t) vs. time for diﬀer- ent temperatures. FIG. 10. The 320 mobile particles (light spheres) and the 320 immobile particles (dark spheres) in a conﬁguration at an arbitrarily chosen time.

13 20 101

100 T = 0.5495 T = 0.5052 10 15 −1 S T = 0.4795 10 T = 0.4510 10−2

(r) 10 −1 10 P(n) −3

MM 10 T − Tc

g T = 0.5495 T = 0.5254 10−4 T = 0.5052 5 T = 0.4795

−5 T = 0.4685 10 T = 0.4572 T = 0.4510 10−6 0 1 10 100 1000 0 1 2 3 4 r n FIG. 13. Distribution of the size n of clusters of mobile FIG. 11. Pair correlation function gMM (r) between mobile particles. Inset: Mean cluster size S plotted versus T − Tc, A particles at four different temperatures. where is the fitted MCT critical temperature Tc = 0.435. The −γ straight line is a power law fit S ∼ (T −Tc) , with γ = 0.618.

6.0

3% T = 0.5495 5% T = 0.5052 7% 4.0 T = 0.4795 0.397 30 S=1.17/(T−0.440) T = 0.4572 0.687 S=0.975/(T−0.431) T = 0.4510 0.741 (r)]−1 S=1.56/(T−0.428) AA

S 20 (r)/g 2.0 MM [g

0.0 0 2 4 6 8 10 r 0 0.44 0.46 0.48 0.50 0.52 0.54 0.56 FIG. 12. Γ(r) = [gMM (r)/gAA(r)] − 1 vs. r for different temperatures. T FIG. 14. Mean cluster size S plotted versus T , for subsets containing 3%, 5% and 7% of the most mobile particles. The data for the 5% are the same as those shown in the inset of the −γ previous figure. The lines are power law fits S ∼ (T − Tp) . Best fit parameters are Tp = 0.440, 0.431 and 0.428, respectively, and γ = 0.397, 0.687, and 0.741, respectively.

14 T = 0.5495 T = 0.5052 10 T = 0.4795 T = 0.4572 T = 0.4510 (r) II g

0 0 2 4 6 r

FIG. 18. Pair correlation function gII (r) between immobile particles.

T = 0.5495 FIG. 15. One of the largest clusters of mobile A particles 1 T = 0.5052 T = 0.4795 found at T=0.4510. The cluster is composed of 125 particles, T = 0.4572 which are represented here as spheres of radius r = 0.5σaa. T = 0.4510

(r)]−1 0 AA 40 (r)/g II [g T = 0.5495 −1 30 T = 0.5052 T = 0.4795 T = 0.4572 T = 0.4510

) −2 * 0 2 4 6 20 (r,t r MM

g FIG. 19. Γ(r) = [gII (r)/gAA(r)] − 1 vs. r for diﬀerent temperatures. 10

0 0 1 2 r FIG. 16. Time-dependent pair correlation function ∗ gMM (r, t ) vs. r, for diﬀerent temperatures.

FIG. 17. A cluster of mobile particles at t = 0 (light ∗ spheres) and t = t (dark spheres), for T = 0.4510.

15 1 10 2 3

0 T = 0.5495 10 T = 0.5052 S 1.5 T = 0.4795 −1 T = 0.4572 10 T = 0.4510

−2

10 (r) 1 MI P(n) 2 −1

10 g T − Tc 10−3 T = 0.5495 T = 0.5052 T = 0.4795 0.5 10−4 T = 0.4685 T = 0.4572 T = 0.4510 −5 10 0 1 10 100 0 2 4 6 8 10 n r

FIG. 20. Distribution of the size n of clusters of immobile FIG. 22. Pair correlation function gMI (r) between mobile particles. Inset: mean cluster size S plotted vs. T − Tc. and immobile A particles.

0.0 (r)]−1 −0.5

AA T = 0.5495 T = 0.5052 T = 0.4795 (r)/g T = 0.4572 MI T = 0.4510 [g

−1.0

0 2 4 6 8 10 r

FIG. 23. Γ(r) = [gMI (r)/gAA(r)] − 1 vs. r for diﬀerent temperatures.

FIG. 21. One of the largest clusters of immobile A particles found at T=0.4510. The cluster is composed of 70 particles, which are represented here as spheres of radius r = 0.5σaa

16 0.04 4 A particles

immobile A T = 0.5495 T = 0.5052 mobile A 3 0.03 T = 0.4795 T = 0.4572 T = 0.4510

(r) 2 0.02 MA g

1 0.01

0 0 2 4 6 0 −10 −8 −6 −4 r U FIG. 26. Pair correlation function gMA(r) between mobile FIG. 24. Distribution of the potential energy of all the A A and generic A particles. particles, of the mobile A particles and of the immobile A particles for T = 0.4510. 5

−6.95 −6.40 4 T = 0.550 T = 0.505 T = 0.480 T = 0.457 −7.15 −6.60 3 T = 0.451 > > i i (r)

−7.35 T=0.4510 −6.80 g 2 T=0.5495

1 −7.55 −7.00 0.0 0.2 0.4µ 0.6 0.8 1.0 < i >

FIG. 25. Potential energy hUii as a function of the mobility 0 hµii for the A particles. The A particles have been divided 0 2 4 6 ∗ into 20 subsets according to their mobility at t . Each subset r is represented by a point in the graph. The energy scale for FIG. 27. Pair correlation function gMB (r) between mobile T = 0.4510 is on the left hand side y axis, while the energy A and generic B particles. scale for T = 0.550 is on the right hand side y axis.

17 0.2

0.2

T = 0.5495 T = 0.5052 0.1 T = 0.4795 T = 0.4572 0

T = 0.4510 (r)]−1 (r)]−1 AB AA T = 0.5495 (r)/g (r)/g IB T = 0.5052 0 [g MA −0.2 T = 0.4795

[g T = 0.4572 T = 0.4510

−0.4 −0.1 0 2 4 6 0 2 4 6 r r FIG. 30. Γ(r) = [gIB /gAB(r)] − 1 vs. r for diﬀerent tem- FIG. 28. Γ(r) = [gMA/gAA(r)] − 1 vs. r for diﬀerent temperatures. peratures.

0.8 0.1

σ 0.6 i(t) 0.0 σ m(t) (t) m

σ 0.4 t* −0.1 (t), i σ (r)/g(r)]−1

MB T = 0.5495

[g T = 0.5052 0.2 −0.2 T = 0.4795 T = 0.4572 T = 0.4510

0 10 100 1000 10000 −0.3 0 2 4 6 t r FIG. 31. σM (t) (dot-dashed line) and σI (t) (solid line) for FIG. 29. Γ(r) = [gMB /gAB(r)] − 1 vs. r for diﬀerent tem- T=0.4510. peratures.

18 1

0.5 F(q,t) A particles mob A particles imm A particles

0 10−2 100 102 104 t FIG. 32. Self intermediate scattering function for T=0.4510 for all the A particles (solid line), for the mobile particles (dashed line) and for the immobile particles (dot-dashed line).

1.0

0.8

0.6

0.4 φ(t) F (q,t) 0.2 s

0.0 101 102 103 104 t

FIG. 33. φ(t) (dashed line) and Fs(q,t) (dotted line) for T=0.4510. The solid lines are ﬁts to a stretched exponential.