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Three-Dimensional Direct Imaging of Structural Relaxation Near the Colloidal Glass Transition

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Science  28 Jan 2000:
Vol. 287, Issue 5453, pp. 627-631
DOI: 10.1126/science.287.5453.627

Abstract

Confocal microscopy was used to directly observe three-dimensional dynamics of particles in colloidal supercooled fluids and colloidal glasses. The fastest particles moved cooperatively; connected clusters of these mobile particles could be identified; and the cluster size distribution, structure, and dynamics were investigated. The characteristic cluster size grew markedly in the supercooled fluid as the glass transition was approached, in agreement with computer simulations; at the glass transition, however, there was a sudden drop in their size. The clusters of fast-moving particles were largest near the α-relaxation time scale for supercooled colloidal fluids, but were also present, albeit with a markedly different nature, at shorter β-relaxation time scales, in both supercooled fluid and glass colloidal phases.

As a glass-forming liquid is cooled, its viscosity smoothly but rapidly increases by many orders of magnitude (1–4). This macroscopic viscosity divergence is related to the divergence of the microscopic structural relaxation time (α-relaxation time). Microscopically, a glass still has liquid-like structure; no structural change has been found which would explain the glass transition (3–5). Instead, theories for the glass transition focus on microscopic dynamical mechanisms (14, 68). The underlying concept of many of these theories is the Adam and Gibbs hypothesis (6), which states that flow in a supercooled fluid involves cooperative motion of molecules and that the structural arrest at the glass transition is due to a divergence of the size of cooperating regions. Some support for this hypothesis comes from experiments that found dynamical heterogeneity in the relaxations of supercooled fluids (9): at a given time, different regions relax with different rates. The size of these regions has been inferred from indirect evidence (10), but their spatial structure remains unknown. Thus, definitive corroboration of these concepts, and indeed a detailed theoretical understanding of the glass transition, has remained elusive.

Some evidence for cooperative motion in structural relaxation was found very recently in computer simulations of supercooled liquids, which showed that structural relaxation occurred through the motion of relatively few, fast-moving particles (11–17). Surprisingly, the positions of the particles were highly correlated and string-like clusters formed, whose size increased as the glass transition was approached (12). Unfortunately, however, there has been no direct experimental observation of these three-dimensional (3D) clusters; moreover, it is unlikely that experiments with molecular glasses will directly observe these structures.

Cooperative motion in structural relaxation can, however, be observed in colloidal suspensions, using a microscope to directly image particles (5, 1820). Sterically stabilized colloids are an excellent model of hard spheres (5, 2123), perhaps the simplest system with a glass transition. Although the microscopic, short-time motion of colloidal particles differs from that of model hard spheres because of the suspending fluid, the phase behavior is nevertheless in excellent agreement with predictions for hard spheres (21, 23). The thermodynamic variable for hard spheres is the volume fraction φ, rather than the temperature. Monodisperse hard spheres form crystals for φ ≥ 0.494, with coexistence between crystal and liquid domains for 0.494 ≤ φ ≤ 0.545, and form glasses for φ > φG ≈ 0.58 (21). The liquid disorder of low volume fraction colloidal suspensions can be quenched into a glass by centrifugation. In a concentrated hard-sphere system, individual particles are trapped in transient cages formed by their neighbors. Structural relaxation is due to the rearrangement of these cages, and the time scale for cage rearrangement, the α-relaxation time, diverges at φG (18, 23). Although this relaxation has been extensively studied with light scattering, direct 3D visualization of these dynamics has not been reported, precluding any detailed study of the structure and dynamics of the relaxing clusters.

We have now used confocal microscopy to follow the motion of several thousand colloidal particles in order to determine directly how motion occurs before and at the α-relaxation time scale. The faster-moving particles move cooperatively in supercooled fluids and form large extended clusters whose size increases dramatically as the glass transition is approached. We have characterized the sizes and structures of these clusters. In addition, at shorter time scales (β-relaxation) the clusters are much smaller, and similar clusters persist even for glassy samples. The use of 3D, time-resolved confocal microscopy is essential for these studies; 2D time-resolved experiments observed some cooperative motion but could not provide any insight into the structure and distribution of clusters (19, 20), whereas 3D static images only determined average structure of the glass (5).

We used poly-(methylmethacrylate) particles, sterically stabilized by a thin layer of poly-12-hydroxystearic acid (24). The particles have a radius a= 1.18 μm, a polydispersity of ∼5%, and were dyed with rhodamine and suspended in a cycloheptylbromide/decalin mixture which nearly matches both the density and the index of refraction of the particles. We used a confocal microscope to rapidly acquire images (15 images per second) in a viewing volume of 69 μm × 65 μm × 14 μm; we focused at least 25 μm away from the cover slip to avoid wall effects. We identified particle positions with a horizontal accuracy of 0.03 μm and a vertical accuracy of 0.05 μm, and tracked every particle for the entire duration of the experiment (25). We determined φ for each sample by measuring the volume per particle directly with the microscope; the φ found by this method agrees with the known phase behavior of hard spheres at coexistence. Samples were stirred several hours before observation.

We determined the characteristic relaxation times by calculating the ensemble-averaged mean square displacement (MSD) for different volume fractions, plotted in Fig. 1A. The MSD decreased as the volume fraction increased. The initial plateau in the MSD in Fig. 1A reflects the cage-trapping, and the slow rise is due to the β-relaxation (8). The end of this plateau where the MSD rises corresponds to cage rearrangement (18) and occurs at larger lag times Δt as the glass transition is approached. For fluid samples (open symbols), the longer time rise in the MSD is due to the α-relaxation. The nature of this motion is illustrated by the particle track shown in Fig. 2; the cage-breaking rearrangement corresponds to rarely occurring large steps in the particle displacement. The long-time diffusion coefficient decreases with increasing volume fraction φ and signals the approaching glass transition (8, 26). To better characterize the α-relaxation, we determined the distribution of particle displacements Pxt)] (Fig. 3). Although this distribution is gaussian for purely diffusive particles, it is expected to be considerably broader near the α-relaxation (11, 18, 19). Deviations from a gaussian are quantified by a nongaussian parameterEmbedded Image(1)the simplest combination of the second and fourth moments of a 1DPxt)], which is zero for a gaussian distribution (27). Broader distributions result in large values of α2. As in Fig. 1B, for supercooled fluids (open circles), α2 is largest for lag times corresponding to the end of the cage-trapping plateau in the MSD. We see a clear indication of the approach to the glass transition in the rise of the peak value of α2 as φ increases toward φG. Note that the magnitude of α2 may be increased due to the slight polydispersity (5% by radius) of the particles (17).

Figure 1

Relaxation behavior. (A) Mean square displacement 〈Δx 2t)〉 for several volume fractions φ. Open symbols are “supercooled fluids,” which form crystals after a few hours (except for φ = 0.46, which remains a fluid). Closed symbols are “glasses,” which do not form crystals even after several weeks. Although the particles are tracked in three dimensions, only the one-dimensional 〈Δx 2〉 is shown because thez resolution is poorer. The straight line shows a slope of 1. There is inherent uncertainty in these data due to the difficulty in averaging over the temporally and spatially inhomogeneous relaxation processes; see, for example, the data for φ = 0.53. (B) Nongaussian parameter α2 calculated from displacements Δx. (C) Average cluster size (number of particles) N c. The dashed line shows the expected result for a random distribution of fast particles (31). The symbols for (B) and (C) correspond to the data shown in (A).

Figure 2

A typical trajectory for 100 min for φ = 0.56. Particles spent most of their time confined in cages formed by their neighbors and moved significant distances only during quick, rare cage rearrangements. The particle shown took ∼500 s to shift position. The particle was tracked in 3D; the 2D projection is shown.

Figure 3

Distribution functionPx) for φ = 0.56, at Δt* = 1000 s, corresponding to the peak in α2t) (Fig. 1B). The dashed line is the best fit gaussian, and the solid line is a fit of a stretched exponential to the tails of the distribution [P ∼ exp(−|x/x 0|β) with β = 0.8; we found that 0.8 < β < 1.5 for different choices of φ and Δt. Smaller values of β coincide with larger values of α2]. The data within the dotted lines are the slowest 95%; particles in the fastest 5% have |Δx| > 0.2 μm.

A dramatic change in the behavior of α2 occurred at φ ≈ 0.58; at lower φ, α2 exhibited a distinct peak near the α-relaxation, whereas at higher φ, the peak in α2 was much broader but not as high. We identify this sharp change as the glass transition and determine φG = 0.58 ± 0.01, in agreement with previous work (18, 21). For glasses (closed symbols), α2 drops at longer lag times, even at lag times when the MSD begins to rise (19). The upturn in the MSD at longer lag times for the glasses has been seen in other experiments (18, 19,28) and may be due to activated processes (1).

To study structural relaxations in supercooled fluid samples, we examined the fastest moving particles: For Δt* when α2 is a maximum, the fastest particles are precisely the particles contributing to the tails ofPxt*)], thus making α2 large (Fig. 3). We chose a cutoff Δr* for a given sample such that over time, 5% of the particles had displacements |Δr⃗| ≥ Δr* (11, 14, 29), although at any given time, the exact fraction may not be 5%. On average, these particles had moved five times farther than the ensemble of particles (Δr*/x2 ≈ 5). The 5% most mobile particles were also examined in simulations (14), but the cutoffs Δr* were typically larger than the particle radius a, whereas for our data, Δr* is typically 0.4a – 0.8a. The difference may be due to the binary size distribution or to the different particle interaction potential used in the simulations. To look for spatial correlations of these fast particles, we constructed the 3D Delaunay triangulation of the particle positions (30), which provides the nearest neighbor connectivity, and we identified the clusters of connected fast particles.

For the supercooled fluid, the fast particles were strongly spatially correlated and exhibited large extended clusters (Fig. 4A). This result is a dramatic demonstration that the α-relaxation in colloidal fluids occurs by means of cooperative particle motion: when one particle moves, another particle moves by closely following the first (12, 13, 19,20). We calculated the angles between displacement vectors of neighboring fast particles; the distribution of these angles is strongly peaked at 0° and shows that neighboring particles move in parallel directions. Moreover, we found that the displacement vectors are more likely to point toward other fast particles than elsewhere, confirming that the motion is cooperative.

Figure 4

The locations of the fastest particles (large spheres) and the other particles (smaller spheres). The spheres are drawn smaller for clarity; the particles all have the same physical size, which is the size of the large spheres shown in this figure. (A) “Supercooled” sample with φ = 0.56, Δt* = 1000 s; the fastest particles had a displacement >0.67 μm. The red cluster contained 69 particles; the light blue cluster contained 50 particles. (B) “Glassy” sample with φ = 0.61, Δt* = 720 s; the fastest particles had a displacement >0.33 μm. The largest cluster (red) contained 21 particles. The “speed” of a particle was determined over a time Δt* corresponding to the α-relaxation for (A) and the β-relaxation for (B); see text for details.

We characterized the nature of these clusters by observing the samples in 3D for several hours. The distribution of cluster sizes for a given volume fraction is broad (Fig. 5A);P(n c) ∼n c –μ with μ = 2.2 ± 0.2 for the supercooled fluids, similar to the value μ = 1.9 ± 0.1 seen in simulations (14). An exponent μ < 3 implies that quantities which depend on 〈n c 2〉, such as average cluster size, will be dominated by the largest clusters; thus, structural relaxation occurs because of a small number of large clusters of cooperative fast particles, rather than many individual fast particles moving independently. It is likely that the distributions of cluster sizes are even broader than indicated because the largest clusters extend out of the viewing volume.

Figure 5

(A) Typical probability distribution functions for cluster sizes (number of particlesN c) for a supercooled fluid (φ = 0.56, open circles) and a glass (φ = 0.60, filled triangles). The lines are least-squares fits to the data, with slopes −2.0 and −3.1, respectively. (B) Average size of clusters 〈N c〉 at different volume fractions φ. The horizontal dashed line indicates the average cluster size that would be expected for a random distribution of fast particles (31). For this graph, the clusters for the glasses are defined by choosing the time scale Δt*, which maximizes the average cluster size. (C) Cluster size plotted against the radius of gyration R g. The power-law behavior indicates fractal scaling, NR g dF, with a sloped F = 2.0 for the data shown (φ = 0.56, a supercooled fluid). (D) Typical distribution functions for N f, the number of nearest neighbors of a fast particle that are also fast. These functions were computed only for particles within clusters containing at least 15 particles, with the definition of “fast” as the fastest 5%. For randomly distributed fast particles,N f ≥ 5 for less than 1% of the fast particles (31).

The cluster size increased dramatically as φ increased (Fig. 5B) (31), consistent with the increased size of the cooperatively rearranging regions of the Adam and Gibbs hypothesis (6). Moreover, there is a pronounced drop in the average cluster size at φG ≈ 0.58 (vertical dashed line inFig. 5B), which signifies the onset of the colloidal glass transition.

The larger clusters are generally extended structures (Fig. 4); thus, we plot the number of particles in a cluster against the cluster's radius of gyration R g (Fig. 5C). The power law scaling observed is indicative of fractal structure with a fractal dimension d f = 1.9 ± 0.4 for all volume fractions, comparable to the preliminary valued f ≈ 1.75 seen in the simulations (14). We further characterized the structure by measuring the number of neighbors, N f, of each fast particle. For supercooled fluids, the distributionP(N f) exhibited a broad peak, with ∼10% of the particles having N f ≥ 7, indicating dense regions of cooperating particles (32). A typical P(N f) for fluids is shown by the open symbols in Fig. 5D.

What are the dynamical properties of these cooperative clusters? The clusters of fast particles persisted for time scales comparable to Δt*. Clusters of fast particles appeared in different parts of the sample at different times, so that after many Δt*, most particles have been “fast” at some time. At all times, clusters of fast particles were present, although at any particular time, the fraction of fast particles in our viewing volume ranged from 2 to 8%. These 3% fluctuations are significantly greater than random fluctuations N≈ 0.3%. Presumably with a larger viewing volume, the fraction of fast particles at any given time would approach the average value of 5%; the large temporal fluctuations we see are evidence of the large-scale inhomogeneity of fast particles. Further, the presence of these clusters shows that ensemble averaged quantities, such as those shown in Fig. 1 and those obtained in scattering experiments, provide an incomplete picture of the dynamics. For example, 〈Δx 2〉 increased smoothly (Fig. 1A), whereas the particle motion was in fact temporally and spatially localized (Fig. 4).

The behavior of the clusters of fast particles was markedly different as φG was crossed. In the supercooled fluid (Fig. 4A), almost all of the fast particles formed a few, very large clusters. In sharp contrast, in the glass (Fig. 4B) there were no large clusters at all, but instead, a large number of smaller clusters. We emphasize however, that the clusters shown for the supercooled fluid correspond to structural (α-) relaxations. In contrast, there was no discernible α-relaxation in our glass data, and instead, the clusters corresponded to the β-relaxation. Indeed, it is not obvious which time scale Δt* should be used for the glasses, because the average cluster size exhibited only a weak dependence on Δt (Fig. 1C, closed symbols). Moreover, the average cluster size was not correlated with α2. Thus, for Fig. 4B, we chose Δt* to correspond to near the middle of the plateau in the MSD (Δt* ≈ 700 s), clearly reflecting the β-relaxation; however, there was virtually no change in the behavior of the clusters with Δt*. On all time scales accessible in these experiments, particles remained confined to their cages (Δr* < 0.4 μm = a/3). We found that neighboring particles still moved in similar directions, confirming that the motion was cooperative. In addition, we found no φ dependence for the cluster sizes for φ > φG (Fig. 5B). The distribution of clusters was more narrow, but nevertheless power-law in shape, with the exponent μ > 3 for all glasses (Fig. 5A, closed symbols). The clusters had far fewer compact regions, as indicated by the more narrowly peakedP(N f) (Fig. 5D, solid symbols).

To properly compare supercooled fluids and glasses, we measured cluster properties for the supercooled fluids at much shorter time scales that correspond to the β-relaxation (Δt = 30 to 100 s). Their behavior was nearly identical to that of the glass samples, and thus markedly different than the behavior observed at much longer times. The cluster behavior evolved smoothly from β- to α-relaxation over several decades of time for the supercooled fluids. Moreover, the cluster size had nearly the same Δtdependence as the nongaussian parameter α2 (Fig. 1C, open symbols); this result suggests that at any time scale, the motion of the anomalously fast particles is cooperative. However, only a fraction of the smaller clusters at short time scales ultimately became part of the larger clusters; this evolution was not observed for glasses.

We emphasize that the existence and behavior of the clusters indicates that the relaxations are very inhomogeneous, both temporally and spatially (9–16). This correlated motion can play a critical role in the dynamics of the sample near the glass transition, and its consequences must be incorporated in any theoretical treatment.

  • * To whom correspondence should be addressed. E-mail: weeks{at}deas.harvard.edu

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