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Cond-Mat.Soft Spatial correlations of mobility and immobility in a glassforming Lennard-Jones liquid Claudio Donati1, Sharon C. Glotzer1, Peter H. Poole2, Walter Kob3, and Steven J. Plimpton4 1 Polymers Division and Center for Theoretical and Computational Materials Science, NIST, Gaithersburg, Maryland, USA 20899 2Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7, Canada 3 Institut f¨ur Physik, Johannes Gutenberg Universit¨at, Staudinger Weg 7, D-55099 Mainz, Germany 4 Parallel Computational Sciences Department, Sandia National Laboratory, Albuquerque, NM 87185-1111 (February 1, 2008) Using extensive molecular dynamics simulations of an equilibrium, glass-forming Lennard-Jones mixture, we characterize in detail the local atomic motions. We show that spatial correlations exist among particles undergoing extremely large (“mobile”) or extremely small (“immobile”) displace- ments over a suitably chosen time interval. The immobile particles form the cores of relatively compact clusters, while the mobile particles move cooperatively and form quasi-one-dimensional, string-like clusters. The strength and length scale of the correlations between mobile particles are found to grow strongly with decreasing temperature, and the mean cluster size appears to diverge at the mode-coupling critical temperature. We show that these correlations in the particle displace- ments are related to equilibrium fluctuations in the local potential energy and local composition. PACS numbers: 02.70.Ns, 61.20.Lc, 61.43.Fs I. INTRODUCTION ment over some time) to subsets of extreme mobility or immobility, and (2) to establish connections between this The bulk dynamical properties of many cold, dense “dynamical heterogeneity” and local structure. liquids differ dramatically from what might be expected This paper is organized as follows. In Section II we from extrapolation of their high temperature behavior present relevant background information, and in Sec- [1]. For example, many liquids cooled below their melt- tion III we describe the model and computer simulation ing temperature exhibit rapid non-Arrhenius increases of techniques. In Sec. IV, we examine the bulk dynamics viscosity and relaxation time with decreasing tempera- and equilibrium structure of the liquid. In Sec. V we ture, and two-step, stretched exponential decay of the examine the mean square displacement and analyze the intermediate scattering function F (q,t). Such behavior shape of the time-dependent distribution of particle dis- is often discussed as a “signature” of the approach to the placements to define a time scale which we use to study glass transition. It has long been a central goal of theories dynamical heterogeneity throughout the remainder of the of the glass transition to account for these bulk phenom- paper. In Sec. VI we group particles into subsets accord- ena in terms of the microscopic dynamical motions of ing to the maximum displacement they achieve on the the molecules of the liquid. As a consequence, computer time scale defined in the previous section, and show that simulations of supercooled liquids, in which this micro- particles of extremely high or low displacement are spa- scopic information is immediately available, are increas- tially correlated. From this we are able to identify a ingly used to complement theoretical and experimental length scale that grows with decreasing T . In Sec. VII efforts. In particular, simulations in recent years have we show that fluctuations of the local mobility are cor- focused on the study of “dynamical heterogeneity” to related to fluctuations of the potential energy, or alter- understand the microscopic origin of slow dynamics and natively to fluctuations in the local composition of the stretched exponential relaxation in glass-forming liquids liquid. In Sec. VIII we examine certain time dependent [2–6]. quantities associated with the observed dynamical het- erogeneity, and finally in Sec. IX we conclude with a dis- arXiv:cond-mat/9810060v1 [cond-mat.soft] 6 Oct 1998 Recently we reported the observation of dynamical het- erogeneity [7] and also cooperative molecular motion [8] cussion. in extensive molecular dynamics simulations of a super- cooled Lennard-Jones (LJ) mixture. These spatially cor- II. BACKGROUND related dynamics were observed in a regime of tempera- ture T , density ρ and pressure P for T above the dynam- ical critical temperature Tc obtained [9,10] from fits by It has been proposed that the stretched exponential the ideal mode coupling theory (MCT) [11] to other data behavior exhibited by the long time relaxation of F (q,t) on the same system. The principle goals of the present can be attributed to a sum of many independent local ex- paper are twofold: (1) To test directly for spatial corre- ponential relaxations with different time constants, i.e., lations of particles assigned (according to their displace- to a distribution of relaxation times [12]. This interpre- 1 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. Wahnstr¨om [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 demon- strated. III. SIMULATION DETAILS The intuitively-appealing picture of cooperative molec- ular 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
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