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Mesoscopic model of temporal and spatial heterogeneity in aging colloids

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 505102 (http://iopscience.iop.org/0953-8984/26/50/505102) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 505102 (7pp)

doi:10.1088/0953-8984/26/50/505102

Mesoscopic model of temporal and spatial heterogeneity in aging colloids Nikolaj Becker1 , Paolo Sibani1 , Stefan Boettcher2 and Skanda Vivek2 1 2

FKF, University of Southern Denmark, Campusvej 55, DK5230 Odense M, Denmark Department of Physics, Emory University, 201 Dowman Dr, Atlanta, GA 30322, USA

E-mail: [email protected] Received 28 July 2014, revised 16 October 2014 Accepted for publication 31 October 2014 Published 24 November 2014 Abstract

We develop a simple and effective description of the dynamics of dense hard sphere colloids in the aging regime deep in the glassy phase. Our description complements the many efforts to understand the onset of jamming in low density colloids, whose dynamics is still time-homogeneous. Based on a small set of principles, our model provides emergent dynamic heterogeneity, reproduces the known results for dense hard sphere colloids and makes detailed, experimentally-testable predictions for canonical observables in glassy dynamics. In particular, we reproduce the shape of the intermediate scattering function and particle mean-square displacements for jammed colloidal systems, and we predict a growth for the peak of the χ4 mobility correlation function that is logarithmic in waiting-time. At the same time, our model suggests a novel unified description for the irreversible aging dynamics of structural and quenched glasses based on the dynamical properties of growing clusters of highly correlated degrees of freedom. Keywords: dense hard-sphere colloids, intermittency and heterogeneity, glassy dynamics, mesoscopic clusters (Some figures may appear in colour only in the online journal)

quantities, and precisely what changes in aging colloids at the macroscopic level [22, 23] remains an open question. Irreversible pace-setting events associated with crossings of free-energy barriers [24] are often called ‘cage breakings’ in colloidal suspensions [14, 18]. Similar intermittent events are numerically detected via non-Gaussian fluctuations of the energy in spin glasses [6] and as large density fluctuations in vibrated compactifying granular piles [25]. In spinglasses, increasingly rare energy barrier crossings require everlonger chains of spin-flip events [26] and tapped granular piles slowly increase their density by expelling free volume over growing domains. In colloidal suspensions, growing domains of particles need to find a collective arrangement allowing for increasingly rare irreversible cage-breakings [18]. The ‘parking lot model’ [25, 27, 28] and the ‘East model’ [29–31] provide simple conceptualizations of the situations just described. There, each irreversible event decreases free volume and is preceded by a certain sequence of rearrangements within a domain that becomes exponentially unlikely with the size of the latter, followed by an even longer

1. Introduction

Aging entails a slow but persistent change of thermodynamic averages. In quenched disordered systems (‘glasses’) [1–6], measurable quantities, e.g. the thermo-remanent magnetization [7] and the thermal energy [5, 8–10] change, on average, at a rate which decreases as the system ages. Similarly, in dense colloidal suspensions, the free energy decreases at a decelerating rate [11] and, concomitantly, a gradual slowing down of the rate at which particles move is observed by light scattering [12, 13] and by particle tracking techniques [14–20]. Experiments also find that the intermittent motion is associated with clustered activity of particles [18] and with abnormal fluctuations of the free surface of oil in water emulsions [20]. Intermittency suggests a hierarchical dynamics (in contrast to the gradual coarsening of, say, a quenched Ising ferromagnet [21]) and heterogeneity implies strong spatial variations, a combination one might refer to as spatio-temporal heterogeneity. The latter property is not easily seen in averaged 0953-8984/14/505102+07$33.00

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in models of fragmentation [33] and coalescence [34]. In all cases, the coarse-graining of local microscopic interactions results in an effective description framed in terms of emerging mesoscopic objects, e.g. our clusters, whose existence is due to the global spatial constraints the system is subjected to. Our mesoscopic model of hard colloids enables us to calculate key quantities as the internal energy, the intermediate scattering function and the mean-square displacement, and to compare with previous experiments [15] and simulations [35]. We take special care to empirically determine and theoretically justify what we propose is the correct scaling form of two-time variables and we suggest that the mobility correlation function χ4 commonly used for supercooled liquids [36, 37], when measured for times t > tw as a function of the waiting time tw in the aging regime, can be used to experimentally verify our predictions for the logarithmic growth of the effective cluster size in aging systems. The rest of the paper is organized as follows: In the next section, we fix our notation and discuss the rejection free method used to simulate the dynamics. We then describe our results, and we finally conclude with a discussion, also sketching how our description might be generally useful for other glassy systems.

sequence of re-arrangements for the next event to happen. In [18] the contiguous degrees of freedom that need to move in order for a cage breaking to occur have been termed ‘clusters’. The similarity of aging phenomenology in glassy systems with vastly different microscopic interactions calls for a description that relies on a minimal set of principles independent of microscopic details. In this work we argue that such description should be probabilistic and framed in terms of the growth and collapse of mesoscopic objects we here call ‘clusters’. These consist of a number of microscopic degrees of freedom, whose motion is highly correlated within each cluster, but uncorrelated across clusters. Real-space objects of this kind have been identified, with different labels, in many glassy systems [3–5, 18], but are not treated as the protagonists of glassy dynamics, which is what we are about to do. Our analysis is tailored to hard sphere colloids, but the results are, we believe, of more general relevance. We build our treatment on a recent theoretical approach by two of the authors [32], where colloidal dynamics is controlled by irreversible changes consisting of the sudden collapse of clusters of particles bound together in spatially localized domains. Specifically, the dynamics is there described heuristically by the collapse-probability per unit of time, P (h), of a cluster of size h. The choice P (h) ∼ 1/(1 + h) leads to time-homogeneous dynamics where colloidal particles diffuse through the system, which fits the behaviour of fluid colloids well above the jamming. The exponential cluster-collapse probability, P (h) ∼ e−h , (1)

2. Simulation method

In our Monte Carlo (MC) simulations, particles reside on a lattice with periodic boundary conditions, with each site being occupied by one particle. Particles are either mobile singletons (cluster-size h = 1), indicating that they possess a disproportionate amount of local free volume, or they form immobile, contiguous clusters of size h > 1, representing the domain which would have to be re-arranged to facilitate the next irreversible event. At each MC update, a particle, say ‘a’, is picked at random. If ‘a’ is a singleton, it exchanges position with a randomly selected neighbor, say ‘b’, and joins its cluster, whose size thereby increases by one unit. If ‘b’ is a singleton as well, then a two-particle cluster is formed. It is important that ‘a’ and ‘b’ exchange their positions for e.g. mobility measures and for calculating the mean square displacement, but the exchange has no effect on cluster sizes. If ‘a’ is not a singleton, with probability P (h), given in equation (1), its entire cluster ‘shatters’ into h newly mobile particles, while with probability 1 − P (h) no action is taken. Following initial investigations which showed similar results in two and three dimensions, the bulk of our simulations was carried out in 2D for simplicity. Although the initial state consists of singletons only (i.e. it lacks any structure), the system develops spatially heterogeneous clusters with a length scale growing logarithmically in time. The state of the system on a square lattice with L = 256 after 1015 sweeps is depicted in figure 1(a) while figure 1(b) shows the logarithmic growth of the average cluster size, reminiscent of the domain growth in the East model at T → 0 [29]. With P (h) 1/(1 + h), a stationary dynamics with simple diffusive behaviour is obtained [32]. As previously shown [32], with P (h) given in equation (1), the rate of events decelerates as 1/t, see the inset of figure 1(b), which makes random-sequential updates

leads instead to non-homogeneous decelerating dynamics, where particle motion is (nearly) diffusive on a logarithmic time scale, as found in our analysis [32] of previous tracking experiments [15]. In this case, irreversible cluster-collapses or quakes follow a log-Poisson process. This is a Poisson process with average proportional to the logarithm of time, and a process which describes the aging phenomenology of a wide class of glassy systems [5, 7–10]. In the model introduced in [32], clusters are placed on a one-dimensional periodic array, and the two fragments resulting from a quake can only join the neighboring clusters. To simulate actual particle motion, ‘test particles’ were added to the lattice which would take a random step of unit size if and only if a cluster happened to collapse at the site where they resided. In the present model, clusters in any dimension are space filling objects endowed with a shape which changes dynamically through the motion of the contiguous particles each cluster contains. Once clusters break down in a quake, their particles move randomly and independently in space, and can either join neighboring clusters or join together to create new clusters, which can then repeatedly break up and re-form.This aspect was absent from the previous model and is closely related to spatial heterogeneity, a key property of glassy dynamics. Microscopic re-arrangements preceding a quake are all coarse-grained away and the collective effect of in-cluster particle motion is still described through the cluster collapse probability P (h). Incidentally, this approach is commonplace 2

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Figure 1. The left panel is a snapshot of a 256 × 256 system after t = 1015 sweeps. Random colours are assigned to different clusters for visibility. Cluster sizes are peaked around an average value h ∼ log t, as shown in the right panel. The decelerating rate λ(t) ∼ 1/t of cluster break-up events that emerges from the probability in equation (1) is shown in the inset.

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Figure 2. Average interface energy per particle, eInt , for L  = 64. For large t, the interface energy follows the form eInt  ∼ 1/ log(t) derived in equation (2), as confirmed by the inset.

anneals away excess free volume. In the East model, energy e derives from the number of frustrated spins which scales with ¯ since d¯ ∼ ln t [29], the inverse of the average domain size d; the energy is this time e ∼ 1/ ln t. Available through light scattering experiments, the selfintermediate scattering function (SISF) fs assesses two-time correlations used to resolve dynamical characteristics of nonequilibrium systems. Formally, it is defined as the spatial Fourier transform,        fs q , tw , t = dr Gs r, tw , t exp −i q · r , (3) of the self-part of the distribution function,    van Hove   Gs r, tw , t = N1 , with rj (tw , t) = rj (t)− δ r −  r j j rj (tw ) as the displacement of particles j in the time interval between tw and t. In general, SISF is a measure of the average tendency of particles to stay confined in cages whose size scales with the inverse magnitude of the wave vector q . Using symmetry and the integer values of the positions, the discrete version of SISF reduces to

 N   1 fs (q, tw , t) = (4) cos q · rj . N

3. Results

Using a Lennard–Jones potential in their molecular dynamics simulation, El-Masri et al [35] were able to determine the evolution of the internal energy of a colloidal system in terms of its pressure. The interface between our clusters decreases as they grow, and its shrinking can be interpreted as a decline in free volume concomitant with the particles within clusters optimizing their mutual interactions. We therefore treat the interface between clusters as a proxy of the internal energy. The average number of clusters n can be written in terms of average cluster size h as n = L2 /h. Since in two dimensions the interface-length of a compact cluster of size h √ scales as S(h) ∝ h, and since h ∼ log(t), see figure 1(b), the average energy per particle eInt  is estimated as n 1 1 ∼√ ∼ . 2 L h log t

0.5

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inefficient. In our simulations, we use the Waiting Time Method [38, 39], where a random ‘lifetime’ is assigned to each cluster based on the geometric distribution associated with P (h); the cluster with the shortest remaining lifetime is shattered and lifetimes for other pre-existing or newly formed clusters are adjusted or newly assigned, following the Poisson statistics. With this event-driven algorithm, we have followed the evolution of our simulations over 15 decades in time, far exceeding current experimental time windows. Important aspects of aging are described by observable quantities with two time arguments. Here, we denote by t the current time and by tw the waiting time before measurements are taken for a system initialized at t = 0. To conform to common usage, the lag time τ ≡ t − tw is used as abscissa in the main plot of relevant figures. However, we also provide a collapse of the data, which is best accomplished when the global time t is scaled by tw as an independent time variable.

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Due to spatial isotropy, the SISF only depends on the √ magnitude q, with qmin = 2π/L  q  π 2 = qmax . Evaluating it, as we do, figure 3 shows results for fs (q, tw , t) averaged over 2000 instances for waiting times varying from 210 to 218 in powers of two. We used q = qmax to facilitate comparison with persistency data which describe the degree to which particles stay put. Panel (a) depicts the SISF as a function of lag time. For large tw , to a good approximation the data can be represented by a power law of the scaling variable tˆ = log (t/tw ), fs ∼ C tˆ−A , where C is a constant and A a positive, non-universal exponent, see panel (b) of figure 3. Some curvature remains, nevertheless, and data do not completely collapse. In contrast, panel (c) achieves excellent scaling and collapse using the form (discussed later)   fs ∝ exp − Atˆ 1 −  tˆ , (5)

(2)

Figure 2 shows that the approximation holds after a more rapid initial decay. The slow decay matches that of the Lennard– Jones simulations in figure 1 of [35], and it is reminiscent of granular compactification [25], where noisy tapping slowly 3

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√ 2π and system size L = 64 for tw = 2 , k = 10, 11, . . . , 18, using three different forms of independent variable: (a) the lag time τ , (b) the total run time t scaled by waiting time tw , and (c) the correction to tˆ = log(t/tw ) in equation (5) (see text for details). The values A and  remain approximately constant for different tw , i.e. A ≈ 0.1 and  ≈ 1%. Figure 3. Decay of SISF at qmax = k

Figure 4. Persistence on a 64 × 64 lattice. For times t from tw = 212

to 4tw activations are recorded. Within a dominant inert background (white), domains are encircled with darker to lighter contours to mark one to ten activations. This mobility pattern demonstrates dynamic heterogeneity and the associated preferential return of activity to the same sites.

with  ≈ 0.012. The power-law exponent A ≈ 0.1 weakly depends on tw and changes systematically by about 40% over two decades of tw , likely reflecting the effect of higher order corrections in equation (5). Note that A is comparable to the same exponent found in expensive Lennard–Jones simulations, see figure 2 of [35]. We recall that in our model reversible particle fluctuations have been removed from the outset, and particles can only move through quakes, or, equivalently, cage breaking events. If we neglect the small  correction, the persistency data collapse is entirely consistent with our quakes being a log-Poisson process, and with the particle mean square displacement (MSD) growing with log(t/tw ), see figure 5. Persistence alternatively characterizes immobility by the average fraction of particles that never move [32, 40]. Conceptually simple and easily accessible in simulations, persistence must be deduced in experiments from the SISF at its peak wave-vector. In terms of the Van Hove function, persistence is defined as   P (tw , t) = Gs r  d, tw , t . (6)

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This is the fraction of particles whose coordinates at times tw and t satisfy |r(t) − r(tw )|  d, where d is a threshold representing the largest undetectable movement. For small d, the SISF in equation (4) reduces to persistence for | q| → ∞. To avoid over-counting particles that return to their original position, in simulations we only count particles that have been activated since tw . Our results for persistence with d = 1 are virtually indistinguishable from those for SISF at qmax in figure 3. Figure 4 illustrates the recurrence of quake activity to sites on a L = 64 lattice for a time interval [tw , 4tw ] with tw = 212 . While most particles persist in their position, mobility concentrates in areas scattered about the system (‘dynamic heterogeneities’ [14]) with future activity favoring previously mobile sites. The MSD between times tw and t is computed by averaging the square displacement, first over of all particles and then over

the ensemble. Using · for the ensemble average and | · | for the Euclidean norm, the MSD is written as

 N 1 r 2 (tw , t) = (7) |rj (t) − rj (tw )|2 . N j =1 Figure 5 shows the MSD for a system of size L = 64 with waiting times tw = 2k for k = 10, 11 . . . 18. In analogy with figure 3, the MSD is plotted versus three different variables: panel (a) uses the lag time, panel (b) the scaling variable t/tw and panel (c) uses the  same type of correction as in equation (5), that is, r 2 ∝ A tˆ 1 −  tˆ with the same  = 0.012 and the ‘log-diffusion’ constant A ≈ 0.2. Note that a system aged up to time tw has a ‘plateau’ of inactivity for lag times up to τ ∼ tw . These plateaus, often associated with the ‘caging’ of particles [14], are easily removed with t/tw as an independent 4

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applicable above the critical density, where the system does not equilibrate within experimentally accessible time scales. The present work addresses hard sphere colloidal relaxation in the dense aging regime and considerably extends an earlier model [32] where colloidal dynamics is described at a coarse grained level in terms of objects called ‘clusters’. These clusters grow in a continuous fashion but break up in sudden events called quakes akin to cluster breaking events. The fast in-cluster rattling of particles is eventually responsible for the quakes but is not formally part of the model. Its effect is described through the cluster break-up probability per unit time, P (h). An exponentially decaying P (h) leads to temporal intermittency and to a decelerating quake rate. These features remain in the current version, but clusters are now geometrical objects whose particles can move in space after each quake. They can be absorbed by nearby clusters or re-form (smaller) clusters, leading to a cascade of spatially confined dynamical events, which produces a dynamical heterogeneity similar to what is observed in glassy system. Our simulation results correctly reproduce available key data and throw some light on their scaling form. Two-point averages have been plotted versus the lag time, τ = t − tw , to adhere to tradition and, in insets, versus the scaled variable t/tw . The first choice lacks a theoretical basis in the absence of time translational invariance. The second indicates that the distinction between an early dynamical regime, τ < tw , and an asymptotic aging regime, τ > tw , is moot. Deviations from log(t/tw )-scaling are visible at long times in both the MSD and the SISF. Interestingly, experimental data show a similar behaviour, see figure 1 in [32] and figure 4 in [35]. As shown in the insets, these deviations can be eliminated by a new scaling variable with a small ( ≈ 1%) correction in log(t/tw ), originating as follows: consider first the fraction of persistent particles pn after n quakes (marked white in figure 4), and neglect that the size of a quake slowly increases with cluster size and that particle hits are not uniformly spread throughout the system. That persistence curve pn then decays exponentially with n. Averaging pn over the Poisson distribution of n gets

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variable, see figure 5(b), leading to the approximate scaling behaviour r 2 ∼ log(t/tw ) and to a reasonable data collapse. The residual curvature is removed in panel (c) using the same correction as in figure 3,  = 0.012, but a constant A ≈ 0.2 that is unrelated to A. The deviation from t/tw scaling in both MSD and SISF data are connected to the spatial heterogeneity of the dynamics, as discussed further below. To explore correlated spatial fluctuations and dynamical heterogeneity, we follow [37] and gauge the displacement of the j ’th particle between times tw and t using the function cj (tw , t) = exp −||rj (t) − rj (tw )||1 where || · ||1 denotes the Manhattannorm. We then define the mobility measure M(tw , t) = j cj (tw , t) and construct the 4-point susceptibility χ4 (tw , t) = M(tw , t)2  − M(tw , t)2 .

(8)

Figure 6 shows χ4 versus t for a range of waiting times tw . We note that using the lag time τ as abscissa produces qualitatively similar curves. While the data does not allow a global collapse, the logarithmic increase in peak-height versus tw is clearly visible. This is expected since, by construction, heterogeneous dynamics in the model is mainly due to the collapse of clusters, locked in at a typical size h ∼ log tw , that re-mobilizes a corresponding number of particles at a time tpeak > tw , which is reflected in the height of the peak. Figure 6 is reminiscent of results in [36, 41] for experiments and simulations, where χ4 is measured for a sequence of supercooled equilibrium states prepared ever closer to jamming, either by increasing density or decreasing temperature.

P (tw , t) = exp(−Aµq (tw , t)) = exp(−Atˆ ),

(9)

see equation (5), where A is a small constant and µq (tw , t) ∝ log(t/tw ) as the average number of quakes occurring between tw and t. Due to heterogeneity, quakes tend to repeatedly hit the same areas, see figure 4, and their effect on persistence hence gradually decreases. Heuristically, this effect is accommodated by correcting the exponent with the O()term in µq , as in equation (5). Technically, all moments of the quaking process can be expressed in terms of µq , and the correction is the first term of a Taylor expansion of the actual exponent. Furthermore, the dependence of cluster size distribution on tw leads to a similar dependence of A. The downward curvature of the MSD plotted versus tˆ is analogously explained, and the weak curvature seen in our data hence appears to be directly related to heterogeneity. The mobility correlation function defined in equation (8) reveals a logarithmically growing length scale in colloidal systems, here, the average linear cluster size.

4. Discussion

Slightly below their critical density, hard sphere colloids can equilibrate through a very slow α-relaxation process. El Masri et al [13] experimentally studied this process using time resolved correlation spectroscopy. The above authors could fit the correlation between two images taken near tw and a time τ apart by a stretched exponential with an age dependent relaxation time τα (tw ) ∝ tw . The approach demonstrates that the equilibration dynamics is age dependent, but is not 5

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We suggest three measurements to further probe colloidal dynamics. The first uses the χ4 susceptibility function, as we presently do, to investigate a growing correlation length as a function of the age. The second collects the PDF of fluctuations in particle positions over short time intervals of length t = atw , where a is a small constant, uniformly covering the interval (tw , 2tw ). If the rate of intermittent quakes decreases as 1/tw , the log-Poisson statistics in equation (9) predicts PDFs which are independent of tw , as shown in [6] for spin-glass simulations. Finally, we suggest that the slight curvature seen in experiments for MSD versus log(t/tw ) [32] and in the tail of the SISF [35] reflects spatial heterogeneity, as discussed above. The correction producing our data collapse might have a similar effect on experimental data. Our approach coarse-grains away the ‘in-cage rattling’ of particles and incorporates intermittency and heterogeneity into a probabilistic description of the growth and breakup of mesoscopic objects, our clusters. Qualitatively similar considerations can be applied to relaxation in other disordered systems. For instance, our logarithmic cluster growth compares well with scaling theories of droplets in spin glasses [42, 43] and with experimental results on the compactification of granular materials [25]. Intermittency is well-documented both in colloids [14, 18, 35] and in systems with quenched disorder [26, 43–46] and can be described analytically using the log-Poisson statistics of cluster collapses [32]. The important question remaining is the origin of clusters [18] in real colloids, where particles do perform in-cage rattling. As long as a cluster persists, its particles are forced to move back and forth in a more or less coordinated fashion. We should not think of these correlations as due to (non-existing) long-ranged interactions between the participating particles. Instead, the considerable rearrangement of neighborhood relations which is implied by a cluster collapse is likely to require one specific sequence of small re-arrangements out of many, an event which becomes less and less probable as the cluster size grows. The closest theoretical model with these properties is the already mentioned parking lot model [25, 27, 28]. In the model, an additional car can be squeezed in and free volume can be taken out once the random fluctuations in the number of cars allow this to happen. The denser the parking lot is packed, the more cars must be rearranged in order for to make room for a new car. Interestingly, preliminary investigations indicate the number of car insertions in a denely packed parking lot is a log-Poisson process. In summary, we argue that the temporal intermittency and spatial heterogeneity of aging dynamics deep in the glassy phase should be described in a probabilistic fashion based on the properties of mesoscopic real-space objects which contain a growing number of strongly correlated microscopic degrees of freedom. Our analysis is tailored to hard sphere colloids, where it is strongly supported by experimental evidence, but has, we believe, a larger domain of applicability.

the V. Kann Rasmussen Foundation for support. SB is further supported by the NSF through grant DMR-1207431. References [1] Struik L 1978 Physical Aging in Amorphous Polymers and Other Materials (New York: Elsevier) [2] Nordblad P, Svedlindh P, Lundgren L and Sandlund L 1986 Phys. Rev. B 33 645 [3] Rieger H 1993 Phys. Rev. A 26 L615 [4] Kob W, Sciortino F and Tartaglia P 2000 Europhys. Lett. 49 590 [5] Crisanti A and Ritort F 2004 Europhys. Lett. 66 253 [6] Sibani P and Jensen H J 2005 Europhys. Lett. 69 563 [7] Kenning G G, Rodriguez G F and Orbach R 2006 Phys. Rev. Lett. 97 057201 [8] Sibani P 2007 Eur. Phys. J. B 58 483 [9] Sibaniand P 2008 Phys. Rev. E 77 041106 [10] Christiansen S and Sibani P 2008 New J. Phys. 10 033013 [11] Zargar R, Nienhuis B, Schall P and Bonn D 2013 Phys. Rev. Lett. 110 258301 [12] Cipelletti L, Manley S, Ball R C and Weitz D A 2000 Phys. Rev. Lett. 84 2275–8 [13] El Masri D, Pierno M, Berthier L and Cipelletti L 2005 J. Phys. A: Condens. Matter 17 S3543 [14] Weeks E R, Crocker J, Levitt A C, Schofield A and Weitz D 2000 Science 287 627 [15] Courtland R E and Weeks E R 2003 J. Phys.: Condens. Matter 15 S359 [16] Lynch J M, Cianci G C and Weeks E R 2008 Phys. Rev. E 78 031410 [17] Candelier R, Dauchot O and Biroli G 2009 Phys. Rev. Lett. 102 088001 [18] Yunker P, Zhang Z, Aptowicz K B and Yodh A G 2009 Phys. Rev. Lett. 103 115701 [19] Yunker P, Zhang Z and Yodh A G 2010 Phys. Rev. Lett. 104 015701 [20] Kajiya T, Narita T, Schmitt V, Lequeux F and Talini L 2013 Soft Matter 9 11129 [21] Biroli G 2005 J. Stat. Mech. P05014 [22] Hentschel H G E, Ilyin V, Makedonska N, Procaccia I and Schupper N 2007 Phys. Rev. E 75 050404 [23] Cianci G C, Courtland R E and Weeks E R 2006 Solid State Commun. 139 599 [24] Hunter G L and Weeks E R 2012 Rep. Prog. Phys. 75 066501 [25] Nowak E R, Knight J B, Ben-Naim E, Jaeger H M and Nagel S R 1998 Phys. Rev. E 57 1971 [26] Dall J and Sibani P 2003 Eur. Phys. J. B 36 233 [27] Krapivsky P L and Ben-Naim E 1994 J. Chem. Phys. 100 6778 [28] Ben-Naim E, Knight J, Nowak E, Jaeger H and Nagel S 1998 Phys. D: Nonlinear Phenom. 123 380 (Annual Int. Conf. of the Center for Nonlinear Studies (Los Alamos, NM, USA, 12–16 May 1997)) [29] Sollich P and Evans M 1999 Phys. Rev. Lett. 83 3238 [30] Sollich P and Evans M R 2003 Phys. Rev. E 68 031504 [31] Faggionato A, Martinelli F, Roberto C and Toninelli C 2012 Commun. Math. Phys. 309 459 [32] Boettcher S and Sibani P 2011 J. Phys.: Condens. Matter 23 065103 [33] Cheng Z and Redner S 1988 Phys. Rev. Lett. 60 2450 [34] Krapivsky P 1991 J. Phys. A: Math. Gen. 24 4697 [35] El Masri D, Berthier L and Cipelletti L 2010 Phys. Rev. E 82 031503 [36] Berthier L, Biroli G, Bouchaud J-P, Cipelletti L, El Masri D, L’Hote D, Ladieu F and Pierno M 2005 Science 310 1797 [37] Berthier L 2011 Physics 4 42 [38] Bortz A B, Kalos M H and Lebowitz J L 1997 J. Comput. Phys. 17 10

Acknowledgments

NB thanks the Physics Department at Emory University and SB thanks SDU for their hospitality. The authors are indebted to 6

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[39] Dall J and Sibani P 2001 Comput. Phys. Commun. 141 260 [40] Pastore R, Ciamarra M P, de Candia A and Coniglio A 2011 Phys. Rev. Lett. 107 065703 [41] Glotzer S C 2000 J. Non-Cryst. Solids 274 342 [42] Fisher D S and Huse D A 1988 Phys. Rev. B 38 373

[43] Fischer K H and Hertz J A 1991 Spin Glasses (Cambridge: Cambridge University Press) [44] Palmer R G, Stein D L, Abraham E and Anderson P W 1984 Phys. Rev. Lett. 53 958 [45] Ogielski A T and Stein D L 1985 Phys. Rev. Lett. 55 1634 [46] Sibani P and Hoffmann K H 1989 Phys. Rev. Lett. 63 2853

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Mesoscopic model of temporal and spatial heterogeneity in aging colloids.

We develop a simple and effective description of the dynamics of dense hard sphere colloids in the aging regime deep in the glassy phase. Our descript...
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