Avalanches mediate crystallization in a hard-sphere glass

Significance

Glasses are formed from the supercooled liquid state when motion is arrested on the scale of the particle diameter. Such states are thermodynamically unstable: An apparently deeply arrested amorphous material can transform into a crystal (devitrification) without large-scale particle motion. The prediction and avoidance of devitrification represent major formulation issues in materials science, hence the importance of understanding its mechanism. Using molecular-dynamics simulations, we study the crystallization of the simplest glass-forming system, a hard-sphere glass. We find that crystallization is caused by a subset of particles cooperatively undergoing large rearrangements in an intermittent stochastic fashion (avalanches). Particles involved in an avalanche are not themselves crystallizing, but they induce crystallization in nearby regions that already show incipient local order.

Abstract

By molecular-dynamics simulations, we have studied the devitrification (or crystallization) of aged hard-sphere glasses. First, we find that the dynamics of the particles are intermittent: Quiescent periods, when the particles simply “rattle” in their nearest-neighbor cages, are interrupted by abrupt “avalanches,” where a subset of particles undergo large rearrangements. Second, we find that crystallization is associated with these avalanches but that the connection is not straightforward. The amount of crystal in the system increases during an avalanche, but most of the particles that become crystalline are different from those involved in the avalanche. Third, the occurrence of the avalanches is a largely stochastic process. Randomizing the velocities of the particles at any time during the simulation leads to a different subsequent series of avalanches. The spatial distribution of avalanching particles appears random, although correlations are found among avalanche initiation events. By contrast, we find that crystallization tends to take place in regions that already show incipient local order.

Glasses are formed from the supercooled liquid state when motion is arrested on the scale of the particle diameter. Such states are thermodynamically unstable and may crystallize during, or shortly after, the initial quench. (This is the usual fate of so-called “poor” glass formers.)

Computer simulations have shown that, in such cases, crystallization readily proceeds by a sequence of stochastic micronucleation events that enhance the mobility in neighboring areas, leading to a positive feedback for further crystallization (1). Importantly, however, crystallization can also arise in mature, well-formed glasses after a long period of apparent stability. The microscopic mechanism of this process, known as “devitrification,” remains elusive. Here, we simulate the dynamics of a mature hard-sphere glass and find that crystallization is associated with a series of discrete avalanche-like events characterized by a spatiotemporal burst of particle displacements on a subdiameter scale. The locations of these avalanches cannot be predicted from the prior structure of the glass, and they vary among replicate runs that differ only in initial particle velocities. Each avalanche leads to a sharp increase in crystallinity, but remarkably the crystallizing particles are primarily not those that participated in the avalanche itself. Instead, they tend to lie in nearby regions that are already partially ordered. We argue that a structural propensity to crystallize in these regions is converted into actual crystallinity by small random disturbances provided by the displacement avalanche. Although spontaneous rather than externally imposed, this pathway may relate to designed crystallization protocols such as oscillatory shear.

Devitrification is a phenomenon of both fundamental interest (2, 3) and practical importance (4–10). Indeed, the prediction and avoidance or control of devitrification represent major formulation issues in materials science, arising for both metallic (4–6) and network glasses (7, 8) as well as glass ceramics (9, 10). So far, however, there is limited understanding of the mechanisms whereby an apparently deeply arrested amorphous material can transform itself into a crystalline packing without the large-scale, diffusive particle motions whose absence [stemming from the formation of cages (11)] is a defining property of glasses.

To gain such a mechanistic understanding, we study here by molecular-dynamics (MD) simulation what is probably the simplest model of a glass: a metastable, amorphous assembly of equal-sized hard spheres in thermal motion. These systems undergo a glass transition at a volume fraction of (12). However, when the glass is prepared by rapid compression to a density just above , crystallites develop and grow almost immediately (1, 12). Put differently, monodisperse glasses normally crystallize before reaching maturity, where we define “maturity” by persistence of the glass for decades beyond the molecular time. This has so far precluded using hard spheres as a model system for studying the devitrification of a mature glass.

Recently, however, we have shown that mature monodisperse glasses can be created by a numerical protocol called “constrained aging” (13), in which motions that increase the global crystallinity are actively suppressed. This protocol can be viewed as selecting only the minority of dynamic trajectories in which the fresh (newly quenched) glass accidentally outlives the quench.

In what follows, we present MD results for crystallization in these mature glasses at . This enables us to give a detailed mechanistic analysis of the devitrification process, in what is arguably the simplest model system available. We work at fixed volume (1, 12, 13) to match the conditions in colloidal glasses, which are the nearest experimental realization of the hard-sphere model system and have long formed a key testing ground for glass physics concepts (11, 14).

Our first finding is that particle dynamics in a mature glass are intermittent: quiescent periods of intracage motion are punctuated by “avalanches” in which a correlated subset of particles undergo cage-breaking displacements. Dynamic heterogeneities in glasses (15–21) [as opposed to supercooled liquids (22–27)] have been reported previously, but avalanches have not been investigated in detail and no link has yet been made with crystallization dynamics. Importantly, therefore, our second finding is that crystallization is intimately associated with these avalanches. This connection is, however, subtle: crystallinity increases during the avalanche, but most of the crystallizing particles are not among those taking part in the avalanche itself. Third, both the avalanche sequence and final crystallization pattern are stochastically determined: they depend not only on the initial particle coordinates but on their velocities, and change if these are reassigned (following ref. 28) in midsimulation. Finally, we nevertheless find that crystallization preferentially occurs in regions already showing semicrystalline correlations or “medium-range crystalline order” (MRCO) (29–31).

Although certain of the above features can be individually discerned in our previous study of crystallization in fresh glasses (1), only for mature glasses, which evolve more slowly, is the chain of causality between these events resolvable.

Results

Avalanches.

Using the constrained aging method (13), we generated a mature monodisperse hard-sphere glass of . This had an initially low crystallinity, , where crystallinity is defined as the fraction of solid-like particles (the latter identified as described in Materials and Methods). Starting from the same initial particle coordinates, we launched 15 MD runs, each having a different random (Maxwellian) set of particle momenta. We have repeated the procedure for different starting configurations, all producing similar results.

In Fig. 1A, we show the growth of crystallinity for these 15 trajectories. One might expect that, because crystallization in a glass takes place with only small (subdiameter) particle motions (1), its course should depend only on the starting configuration of the particles and not on their velocities. However, Fig. 1A shows that the 15 replicas have strongly dissimilar profiles. This establishes a key role for stochasticity in the devitrification of mature glasses, like that reported previously for the crystallization of freshly formed ones (1). However, the curves seen here for devitrification differ qualitatively from those of fresh glasses (figure 1A of ref. 1), which show slow monotonic growth from the beginning of the run. By contrast, in the mature samples, stays constant for between two and five decades of time (measured in microscopic units; Materials and Methods) before steep upward jumps in are seen. (These features depend on system size, as we discuss later.) Because the crystal is locally denser than the glass, each such upward step in increases the free volume and speeds the approach of the next step. Under this feedback, the system finally crystallizes catastrophically and goes rapidly to 1.

Fig. 1.

(A) Fraction of solid-like particles versus time for a system of equal-sized hard spheres at volume fraction . Fifteen trajectories are started from the same spatial configuration of particles but with different randomized momenta. (B) Crystallinity X (in black) and MSD (in red) versus time around the step-like crystallization event shown in the black curve of Fig. 1 at . The green curve, X^avl, is the fraction of avalanche particles defined in time interval that are solid-like.

Key mechanistic insights are gained when we analyze one of these step-like crystallization events in more detail. The black curve in Fig. 1B is a close-up of the crystallinity jump shown in the black curve of Fig. 1A at . The mean-square displacement (MSD) (Materials and Methods) is also plotted (red curve). First, we notice that and the MSD are strongly correlated: both quantities jump simultaneously. To understand the MSD jump, we compute displacement vectors u of individual particles over chosen time intervals and select those with , with σ the particle diameter σ. (This threshold is justified in SI Appendix.)

Fig. 2 shows these vectors as red arrows for the time windows indicated in Fig. 1B. In window , the system is largely immobile; most particles rattle locally in their cages and less than 1% undergo significant displacements. During window , which spans the jump, a burst of displacements is recorded, with around 25% of all particles moving more than . After the jump (window ), the system returns to quiescence, with again less than 1% of all particles moving significantly. We call such a sequence an “avalanche” and denote those particles that move by more than during the jump “avalanche particles” (see SI Appendix for a justification of this cutoff alongside a more quantitative statistical analysis of the avalanches). It is clear from the red arrows in the second frame of Fig. 2 that these particles are not homogeneously distributed, but cluster into “avalanche regions,” resembling in exaggerated form the milder dynamic heterogeneities often reported on the fluid side of the glass transition (32–34).

Fig. 2.

Displacement vectors with modulus larger than (red arrows with yellow heads) and solid-like particles (turquoise spheres) for time intervals , , and shown in Fig. 1B. The lengths of the arrows correspond to the modulus of the displacements. Solid-like particles are defined at the beginning of each time interval.

By interrogating the dynamics across narrower time intervals, we have observed that avalanches start to build in localized regions, then grow to peak activity, and finally die out (Movie S1). From start to finish, an avalanche typically takes about time units. Highly cooperative movements can be seen during the main avalanche phase, including particles moving in rows or circles (Fig. 3). Turquoise spheres in Fig. 2 correspond to solid-like particles. As expected from Fig. 1B, the avalanche leaves behind an increased population of solid-like particles.

Fig. 3.

Displacement field for a typical avalanche in which cooperative motion where particles follow each other are highlighted.

Avalanches Mediate Crystallization.

Figs. 1B and 2 show one representative example of a jump in crystallinity partnered with a displacement avalanche. This is a general phenomenon: in none of the runs do we see crystallinity jumps that are not associated with avalanches. The question thus arises: do avalanches cause crystallization, or vice versa? If avalanches cause crystallization, one obvious hypothesis is that the particles that move to become crystalline are the ones that form the avalanche. However, this hypothesis can be ruled out by visually inspecting Fig. 2 and realizing that there is no clear overlap between avalanche regions and regions where new crystalline particles appear. The fraction of crystalline particles is ≃4% before the avalanche and ≃9% afterward. Of the new crystalline particles, only 25% were directly involved in the avalanche, as one can infer from the green curve in Fig. 1B. (The proportion depends somewhat on the exact threshold of displacement used to define avalanche particles.) We conclude that the particles that crystallize are mainly not the ones that participated in the avalanche.

An alternative hypothesis is that avalanches are caused by crystallization in the sense of being triggered by the small rearrangements (12, 13) needed to achieve local crystallinity. If so, avalanches would be absent whenever crystallization is suppressed by size polydispersity. Fig. 4A shows the MSD and of a glass with 6% polydispersity at volume fraction . As expected from our earlier work (12, 35), the crystallinity stays flat throughout the run; yet we see that the MSD jumps in a way that, by the methods already described, can be identified as avalanches. Moreover, avalanche-like dynamic heterogeneity (in less extreme form) was previously seen for other noncrystallizing glassy systems in 2D and 3D simulations (16, 18, 19, 22, 24, 25) and in colloid experiments (36, 37). Therefore, we can discard the hypothesis that crystallization causes avalanches, rather than vice versa.

Fig. 4.

(A) Crystallinity (black) and MSD (red) versus time for a 6% polydisperse system at . (B) Red curve: MSD versus time for a trajectory of the monodisperse system showing an avalanche. Blue, green, and black curves: MSDs for the same system when the particle velocities are randomized immediately before the avalanche and in the middle of the avalanche.

The stochastic nature of avalanches was already shown in Fig. 1, where the trajectory of each replica has a different crystallinity evolution . A further illustration is given in Fig. 4B, where we compare a trajectory undergoing an avalanche with three systems started from a common configuration just before the avalanche. Each replica is launched with a different set of particle velocities, and in all three cases the avalanche is averted. This finding shows that the triggering of an avalanche from the quiescent state does not depend on particle coordinates alone, but rather on the appearance of a successful combination of positions and momenta. We speculate that these rare events involve emergence of cooperative motions such as those illustrated in Fig. 3. In contrast, if velocities are reassigned midway through an avalanche (Fig. 4B), the avalanche does not stop, but continues along an altered path. This implies that the “activated” state is structurally distinguishable from the quiescent one, although we have not yet found a clear static signature for it.

The requirement of an unlikely combination of positions and velocities to trigger an avalanche, combined with the fact that avalanches cause crystallinity to grow (explored further below), explains the stochasticity of devitrification in our mature samples and is likely also implicated in the stochastic crystallization in fresh glasses (1). That displacement avalanches mediate crystallization in hard-sphere glasses is the central finding of this paper.

Heterogeneities.

As previously stated, the different trajectories in Fig. 1 lead to different final crystallization patterns from the same initial configuration. Visual inspection of these patterns shows only limited similarity between them. Nonetheless, one might expect some regions to be more likely to crystallize than others. The crystallization propensity is assessed by superimposing the crystalline particles (XP) of all trajectories as these first cross a fixed crystallinity threshold (we choose ). To quantify any heterogeneity in the resulting superimposition, we divide the simulation box in 3 × 3 × 3 equal subvolumes and evaluate the density in each, normalizing by the overall density. The resulting normalized densities, , are plotted as a function of subvolume index in Fig. 5A. By computing the fluctuations of around the average value, 1, we get a quantitative measure of the degree of heterogeneity, . For crystalline particles in our replicated runs, we find , more than four times above the background level, , computed by superposing crystalline particles for 15 runs starting from independent initial configurations rather than from the same one. We can conclude that there are some regions in the initial configuration that are more prone to crystallize than others. It has been found in supercooled liquids that these regions correlate with a partial ordering known as MRCO, which is quantified by an averaged local bond order parameter (30, 31, 38). Fig. 5A compares the density of XP particles in our simulations with the density of MRCO, identified as those particles with in the top 10%. As with the earlier work on supercooled liquids (30, 31), there is a clear, although not complete, correlation between MRCO in the initial configuration and subsequent crystallization.

Fig. 5.

Normalized density, , as a function of the index i identifying each subvolume of the simulation box for various particle types (see text). (A) Red: crystalline particles (XP) are those of all of the trajectories in Fig. 1A as they first cross the crystallinity threshold X = 0.1. Black: medium-range crystalline order (MRCO) particles are those in the initial configuration with bond order parameter in the top 10%. (B) Dark green (solid line): avalanche particles (AP) are those participating in the first avalanche of all trajectories. Light green (dashed line): avalanche initiator particles (AIP) are those involved in initiating the first avalanche of all trajectories as defined in SI Appendix.

We also investigate whether there are regions where avalanches have a higher propensity to take place by doing a similar analysis as that described above but for particles involved in the first avalanche (AP) instead. As seen in Fig. 5B, the density of these particles shows only small variations between subvolumes, suggesting that avalanches occur almost at random throughout the system in mature glasses (whereas the crystallinity induced by these avalanches has a significantly higher propensity to appear in some regions than in others).

It has been found that dynamic heterogeneities in supercooled fluids, involving large-scale rearrangement of the particle positions (the α process), tend to grow from regions of high displacement in low-frequency quasilocalized phonon modes (so-called soft spots) (18, 23, 39–41). In view of our result that avalanches occur almost at random throughout the system, one would be tempted to conclude that avalanches and dynamic heterogeneities are fundamentally different dynamic events. However, a closer study does reveal a clear correlation across trajectories among avalanche initiator particles (AIP) (those involved in the first steps of avalanche formation). In fact, the density heterogeneities plot of AIP shown in Fig. 5B shows large density variations between subboxes (in SI Appendix, we show that this is a statistically significant result). Therefore, AIP and dynamic heterogeneities share the tendency to develop in certain regions of the system. Whether or not these regions also correspond to soft spots for the case of mature glasses requires further investigation beyond the scope of this paper. Nevertheless, we show some preliminary analyses in SI Appendix, alongside a more detailed account of heterogeneities, including pictorial representations.

Discussion and Conclusions

We have investigated the mechanism by which crystals develop in amorphous glasses composed of equal-sized hard spheres. In contrast with our previous work on freshly prepared samples, we addressed here mature glasses, whose arrest is characterized by a MSD that stays flat for several decades in time before the onset of crystallization. We have shown that crystallization is intimately associated with particle displacement avalanches (Figs. 1B and 2) and that crystallization is caused by these avalanches and not vice versa. However, the majority of avalanche participants do not become crystalline (green curve in Fig. 1B), and most crystallizing particles move little during the avalanche. Thus, the displacement avalanche is not, of itself, the sequence of motions needed to transform an amorphous region into crystal.

Instead, avalanches within the mature glass appear to be autonomous structural rearrangements, involving cooperative particle motion. These mesoscopic avalanches have a strongly stochastic character, and are triggered by unlikely local combinations of particle positions and momenta. An individual avalanche can be averted entirely by reassigning momenta just before its inception; once underway, however, such reassignment only diverts it along a different path (Fig. 4). Although no obvious propensity to occur in particular positions can be seen in the statistics of avalanche participants, this can be detected among AIP. This finding implies a correlation with static structure (explored further in SI Appendix), possibly including “soft spots” of the type known to be linked to dynamic heterogeneity in supercooled liquids (18, 39, 40) and some glasses (18, 23, 41). If so, our avalanches might be viewed as a limiting type of dynamic heterogeneity, arising as the system’s density or age increases so that activity becomes rare. However, the stochastic character of the avalanches might also be taken as support for suggestions (18, 24) that a qualitatively different type of dynamics takes over in systems, such as ours, that are deep into the glassy state. In addition, and in common with supercooled liquids, we find that the crystals tend to grow in regions of MRCO [which seem to be themselves anticorrelated with the soft spots (29–31); SI Appendix, Fig. S8].

The likely role of avalanches in crystallization is to create the small disturbances required to accomplish ordering in regions that, as noted above, already have a propensity to crystallize. Avalanche-induced disturbances might shake a nearly ordered region into order, but could also facilitate growth of an established crystallite at its perimeter. This avalanche-mediated mechanism for devitrification somewhat resembles the breakdown dynamics of an attractive colloidal gel (42). The process could also be closely related to protocols such as shearing in which mature glasses are induced to crystallize by gentle agitation (43–45). In contrast to those protocols, here the required agitation is spontaneously generated. Indeed, the intrinsic avalanche dynamics remain present even when crystallization itself is prevented by polydispersity.

In keeping with previous findings for fresh glasses (1, 12), the ordering induced by an avalanche reduces the pressure in the system and creates positive feedback for further avalanches. This process gives rise to a nontrivial system size dependence for the time evolution of global properties such as the mean crystallinity, as explained in SI Appendix. However, it does not qualitatively change the mesoscopic mechanism of avalanche-mediated devitrification that we have described.

To confirm that our findings are not some special feature of systems prepared by constrained aging, we have additionally performed simulations on fresh glasses prepared by rapid compression to a higher concentration, , where there is no need to resort to constrained aging to obtain a mature glass. We found that these glasses show similar behavior to that reported above for the constrained-aged systems at : long quiescent periods and sudden coincident jumps in the crystallinity and MSD (SI Appendix). Therefore, this devitrification mechanism is evident for mature glasses, either prepared by constrained aging or by quick compression . By contrast, a glass prepared by quick compression at (1) crystallizes while still fresh and does not clearly show the avalanche mechanism.

Our work suggests several avenues for future research. One is to study hard-sphere devitrification at constant pressure. A second is to address by our methods mixtures of different-sized hard spheres. This would represent a first step toward modeling bulk metallic glasses, which are generally multicomponent alloys (46, 47). Mechanistic insights along the lines pursued in this paper might then shed light on the devitrification of such glasses during processing, which is a major issue in technology (5, 6).

Materials and Methods

Simulation Details.

We perform event-driven MD simulations in the NVT ensemble with cubic periodic boundary conditions for a system of n = 3,200 monodisperse hard spheres (48, 49). We also simulate a polydisperse system of n = 2,000 particles where the particle diameters are chosen according to a discrete Gaussian distribution with relative standard deviation . Mass, length, and time are measured in units of particle mass m, particle diameter σ (or for the polydisperse case), and , where is the Boltzmann constant and T is the temperature, and we set . The packing fraction is defined as (with V the system’s volume).

To generate the initial configuration, we follow the “constrained aging” procedure described previously (13). We use a configuration resulting from constrained aging as a starting point for unconstrained MD runs.

Analysis Details.

The MSD is calculated as , where r_i is the position of particle i.

The crystallinity, X, is defined as the number of solid-like particles divided by the total number of particles. As in previous work (35), we identify solid-like particles according to a rotationally invariant local bond order parameter d₆ (50, 51). To compute it, we first identify the number of neighbors of each particle i using the parameter-free SANN algorithm (52). Next, for every particle i, we compute the complex vector q₆ whose components are given by (with ), where Y_6m are sixth-order spherical harmonics. Then we compute the rotationally invariant bond order parameter d₆ by calculating the scalar product between each particle’s q₆ and its neighbors, , and consider particles i and j as having a “solid connection” if their exceeds the value of 0.7. A particle is labeled as solid-like if it has at least six solid connections.