Main

In a seminal paper dated nearly fifty years ago, Adam and Gibbs8 associated the rapid growth of a supercooled liquid’s relaxation time with a decrease in its configurational entropy Sc. Within RFOT, Sc is related to the number of metastable minima in the free energy landscape of the liquid that can be explored by the system at a given temperature. This theory further claims that the supercooled liquid freezes into a mosaic whose domains correspond to configurations in these metastable minima1. The typical domain size is expected to diverge at the ‘ideal’ glass transition temperature, where Sc vanishes. The existence of a growing static ‘mosaic’ length scale that serves as a clear indicator of the glass transition is therefore intrinsic to RFOT (ref. 2), although a systematic procedure for measuring it from point-to-set correlations was established much later9. Since the findings of ref. 9, a variety of growing static length scales have been identified and computed in numerical simulations10,11,12,13. Of these, the point-to-set correlation length ξPTS (ref. 10) is of central importance, as it follows directly from the mosaic picture. ξPTS is measured by freezing a subset of particles in the liquid’s equilibrium configuration, and examining their influence on the configuration of the remaining free particles. As such, when evaluated for appropriate pinning geometries, ξPTS can provide an estimate of the typical domain size of the mosaic10. In addition, it has been shown analytically that a divergence in the relaxation time is indeed associated with a diverging ξPTS (ref. 14). ξPTS was first extracted in simulations by pinning all particles outside a spherical cavity and examining the configurations of free particles inside the cavity10. Subsequently, the growth of ξPTS has been studied for various pinning geometries15 as well as in various simulated glass-formers16. Of particular importance is the case in which the pinned particles form a single amorphous wall. Using this geometry, in addition to ξPTS, recent simulations17 have computed a dynamic correlation length ξdyn that evolves non-monotonically with temperature across the mode coupling crossover. It was surmised that this non-monotonicity reflects a change in the morphology of cooperatively rearranging regions (CRRs), which are string-like at high temperatures and compact close to the glass transition6. However, the crucial and long-standing microscopic predictions of RFOT pertaining to growing point-to-set correlations and the morphology of CRRs remain untested in experiments. Point-to-set correlations cannot be investigated in atomic and molecular glass-formers, because the dynamics of their constituent particles cannot be traced, and it is not possible to freeze a subset of particles in an equilibrium configuration. These problems can be alleviated in colloidal glass-formers; in fact, the random pinning geometry was realized in a very recent experiment18. Given this advance in colloid experiments, testing the key predictions of RFOT directly in colloidal glass-formers would constitute a major step in unravelling the nature of the glass transition.

We performed optical video microscopy experiments on a binary mixture of small and large polystyrene colloids of diameters σS and σL, respectively (see Methods for experimental details). As mentioned before, measuring point-to-set correlations requires pinning particles in an equilibrium configuration of the liquid, which is experimentally challenging. In colloidal systems, this can be realized by manipulating light fields using holographic optical tweezers18,19. Here, we demonstrate the power of this technique by pinning an amorphous wall of particles in a two-dimensional colloidal glass-forming liquid. We first captured a bright-field image of the sample and extracted particle coordinates within a strip of width ∼2σL along the longer dimension of the field of view. We then calculated the hologram and fed it to a spatial light modulator (SLM), which in turn created traps at the desired positions. Further, the use of a SLM ensured that all the particles constituting the wall were frozen simultaneously (see Supplementary Movie 1). To ensure that the particles thus pinned are indeed a part of the liquid’s equilibrium configuration, we superimposed the coordinates of these particles on time-averaged images of the sample in the presence of the amorphous wall, for two different area fractions ϕ (Fig. 1). Particles forming the wall appear bright in the time-averaged images owing to their negligible mobility, and can be easily identified. We observe that, by and large, the initial set of particle coordinates and the centres of pinned particles acquired from the time-averaged images are separated by a distance smaller than the cage size, which shows that particles forming the amorphous wall are pinned in an equilibrium configuration of the colloidal liquid. Interestingly, from the time-averaged images in Fig. 1, we see that clusters of immobile particles extend further away from the wall for ϕ = 0.76 than for ϕ = 0.68, suggesting that the influence of the wall is felt over longer distances with increasing area fraction.

Figure 1: Visualization of the amorphous wall.
figure 1

The underlying grey scale images have been generated by time-averaging snapshots over 30τα for ϕ = 0.68 (a) and ϕ = 0.76 (b) for a pinned wall. τα = 12 s and 274 s for a and b, respectively. The red circles correspond to the coordinates of the trapped particles that form the amorphous wall. The spheres at the top of the images in a and b constitute the pattern whose fast Fourier transform was fed into the spatial light modulator. Spheres are colour coded according to the distance between the input coordinates for creating traps and time-averaged particle positions in units of σL.

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We first extracted ξPTS by calculating the total overlap function, qc(t, z), at various distances z from the pinned wall17. Although this analysis was initially developed for three-dimensional systems, we have used its two-dimensional analogue for our colloidal glass-former. We divided the field of view into boxes of size 0.25σS, and computed qc(t, z) for all boxes that lie at a fixed distance away from the wall, using the equation

where 〈〉 correspond to time averaging, i is the box index, ni(t) = 1 if the box contains a particle at time t and ni(t) = 0 otherwise. Figure 2a shows qc(t, z) at various z for ϕ = 0.74. The box size 0.25σS was chosen to be larger than the cage size, ∼0.14σS, to avoid spurious overlap fluctuations due to cage rattling. Moreover, the chosen size is small enough to provide sufficient spatial resolution for the computation of ξPTS and ξdyn. By definition, qc(t, z) (equation (1)) measures the overlap between configurations at two different times at a given distance from the wall. As qc(t, z) is insensitive to particle exchanges, in the limit of long times and large distances from the wall, it attains a finite asymptotic bulk value qrand = qc(t → ∞, z → ∞) corresponding to the probability of occupation of a box. Consistent with simulations17, we observe that the presence of the wall influences the asymptotic value of qc(t → ∞, z) = q∞(z), such that q∞(z) > qrand (Fig. 2a). In our experiments, q∞(z) is estimated by averaging the saturation value of the overlap function over a 5–10 min time window20. We note that qc(t, z) does not saturate for all z within the experimental duration and hence, to extract ξPTS, we consider only those values of z for which qc(t, z) attains saturation. As expected, we observe that q∞(z) decreases with z in the vicinity of the wall. This is also evident from Fig. 2a, where the qc(t, z) profiles for large z overlap almost completely. We observe that q∞(z) − qrand decays exponentially with z (Fig. 2b and Supplementary Fig. 1), which allowed us to extract ξPTS from the relation

Having computed ξPTS, we computed ξdyn from the self part of the overlap function, qs(t, z):

where, once again, 〈〉 correspond to time averaging, i is the box index, and nis(t) = 1 if the box is occupied by the same particle at time t and nis = 0 otherwise17. qs(t, z) is similar to the self-intermediate scattering function calculated for the wavevector corresponding to the box size. Unlike qc(t, z), qs(t, z) (equation (3)) is sensitive to particle exchanges and reaches zero at long times, when all the particles undergo a displacement larger than the box size. Owing to its similarity with the self-intermediate scattering function, qs(t, z) yields relaxation times τs(z) at different distances z from the wall17. Owing to the limited temporal resolution in our experiments, we defined τs(z) as the time taken for qs(t, z) to decay to 0.2 (ref. 21). Figure 2c shows qs(t, z) at various z for ϕ = 0.74. As expected, τs(z) approaches its bulk value τsbulk for large z. In accordance with simulations17,22, we find that the dynamic length scale ξdyn (Fig. 2d) can be extracted from the equation

Figure 2: Overlap functions and relaxation times.
figure 2

a, The total overlap qc(t, z) for ϕ = 0.74. Different colours represent different z. The horizontal grey solid lines indicate q∞(z), the value at which qc(t, z) saturates. b, q∞(z) − qrand versus z for ϕ = 0.68 (black open circles), ϕ = 0.71 (blue left triangles), ϕ = 0.74 (brown down triangles), ϕ = 0.75 (olive green diamonds), ϕ = 0.76 (red squares) and ϕ = 0.79 (green triangles). c, The self overlap qs(t, z) for ϕ = 0.74. Different colours correspond to different z. The horizontal black dashed line corresponds to qs(t, z) = 0.2. d, In (τs(z)/τsbulk) as a function of z. The colours and symbols in d are identical to those in b. In b and d, the solid lines are exponential fits of the forms given in equations (2) and (4), respectively. In d, for ϕ = 0.76 (red squares), ξdyn was extracted from the asymptotic slope17. The red dashed line is a guide to the eye.

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Having extracted ξPTS and ξdyn from overlap functions, we studied the variation of these length scales with the area fraction ϕ on approaching the glass transition (Fig. 2b, d). We find that, in concord with simulations17, ξPTS grows monotonically with ϕ (Fig. 3a and Supplementary Fig. 1). This finding constitutes the first experimental evidence of growing point-to-set correlations in glass-forming liquids. We note that, as in simulations, the prefactor B changes with ϕ (Fig. 2b). To ensure that the trend of growing amorphous order is not influenced by the variation in B, we defined a second static length scale ξPTS−Int = BξPTS, which is also observed to increase with ϕ (ref. 17; Supplementary Fig. 1). Turning towards the dynamic length scale, ξdyn, we observe that it grows faster than ξPTS, as expected17. Most strikingly, however, ξdyn exhibits a non-monotonic dependence on ϕ (Fig. 3a). This result is remarkable, as it is the first experimental observation of non-monotonicity in dynamic correlations. With the exception of the numerical results of Kob et al.17, all dynamic length scales reported in the literature were seen to grow monotonically on approaching the glass transition23,24,25,26. The paucity in observations of the aforementioned non-monotonicity is due to the fact that not all dynamic length scales are expected to show non-monotonicity and the presence of a pinned wall seems to be crucial to this observation. Even in the presence of a pinned wall, it has been shown that the existence of non-monotonicity is dependent on the interaction potential27. An important point to note is that in the simulations of Kob and coworkers17, as well as our experiments, the maximum in ξdyn occurs in the vicinity of the mode coupling crossover (Fig. 3a and Supplementary Fig. 2), strongly suggesting that our observations correspond to the same dynamic crossover seen in refs 27, 28. In ref. 17, it has been speculated that the observed non-monotonicity is a consequence of a change in the morphology of CRRs across the mode coupling crossover, and is therefore consistent with RFOT. In particular, the authors of ref. 17 claim that the spatial inhomogeneity introduced by the wall makes ξdyn sensitive not only to the number of particles in a CRR, but also to their arrangement into string-like or compact structures. A closer look at the relaxation profiles τs(z) gives a preliminary indication in support of this claim. Whereas the profiles for ϕ < 0.76 as well as for ϕ = 0.79 are described well by a single exponential decay, τs(z) exhibits two slopes for ϕ = 0.76 (Fig. 2d), which indicates the presence of multiple relaxation mechanisms associated with the morphology and internal structure of CRRs (ref. 17).

Figure 3: Static and dynamic length scales and morphology of CRRs.
figure 3

a, Point-to-set length scale, ξPTS, (red circles) and dynamic length scale, ξdyn, (blue triangles). The error bars have been obtained from the exponential fits. The dotted black line indicates the mode coupling crossover ϕMCT. b–d, Representative cluster morphologies for 25-particle clusters for ϕ = 0.74, ϕ = 0.76 and ϕ = 0.79 respectively. Core-like particles are shown in red and string-like particles are shown in light blue. e, Distribution of the number of mobile nearest neighbours, P(NN), for ϕ = 0.68 (black open circles), ϕ = 0.71 (blue left triangles), ϕ = 0.74 (brown down triangles), ϕ = 0.75 (olive green diamonds), ϕ = 0.76 (red squares) and ϕ = 0.79 (green triangles). f, Fraction of string-like, ns, (open green circles) and core-like, nc, (red circles) particles as a function of ϕ for 25-particle clusters. The grey shaded areas correspond to 50% confidence bands. g, Maximum of the mobility transfer function M(Δt), Mmax, for small (blue circles) and large (red triangles) particles computed by considering the top 10% most mobile particles.

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To test whether the non-monotonicity indeed stems from a change in the shapes of CRRs, we examined the nature of dynamical heterogeneities in our system. To ensure that the analysis is not influenced by the presence of a wall, we examined the shapes of CRRs in the corresponding unpinned system (see Methods for details). To define CRRs, we first identified the top 10% most mobile particles over various time intervals Δt and clustered them based on nearest neighbour distances. As expected, the mean cluster size is maximal at a characteristic time t∗. On observing the shapes of these clusters defined over Δt = t∗, we find that, remarkably, the clusters are predominantly string-like at low ϕ and compact at high ϕ (Fig. 3b, d). To quantify this change in morphology, we computed the distribution P(NN) of the number of mobile nearest neighbours of a mobile particle, following the protocol used in refs 23, 29 (Fig. 3e). Interestingly, we observe that for ϕ < 0.76, P(NN) exhibits a peak at NN = 2, indicating string-like morphology30. For ϕ ≥ 0.76, the distribution becomes increasingly broader and the maximum shifts to NN = 3, consistent with Fig. 3b–d. To illustrate this point better, we have plotted the percentage of mobile particles that have greater than two mobile neighbours, given by (inset to Fig. 3e). We observe that the trend in this quantity indeed reflects the change in morphology of CRRs near ϕMCT anticipated in the Fuzzy Sphere Model. We repeated the above analysis in the presence of a pinned wall, for regions located sufficiently far away from the wall, and observed a similar behaviour in P(NN) (see Supplementary Fig. 3). These findings are consistent with RFOT, which predicts that string-like CRRs occur with greater frequency on decreasing ϕ and eventually become dominant below ϕMCT (ref. 7).

According to refs 6, 7, the morphological changes in CRRs can be described by the ‘Fuzzy Sphere Model’. This model assumes CRRs to be composite objects that contain a compact core surrounded by a ramified string-like shell. Further, the string-like shell dominates at low ϕ and the compact core dominates at large ϕ, with a smooth crossover between the activation barrier distributions for the two morphological types near ϕMCT. Interestingly, we observe that CRRs of a fixed size indeed contain a large fraction of string-like particles at low ϕ and a large fraction of core-like particles at high ϕ (Fig. 3b–d, also see Supplementary Movies 2–4). For further analysis, we quantitatively extracted the fractions ns and nc of string-like and core-like particles respectively, within CRRs of various sizes (see Methods for a definition of string-like and core-like particles). Figure 3f shows the dependence of ns and nc on ϕ for CRRs containing N = 25 particles (also see Supplementary Movies 2–4). We see that ns and nc indeed cross over near ϕ = 0.76, even for the bulk unpinned system (see Supplementary Fig. 4 for similar analysis for different values of N). Although we could sample only a limited number of ϕ values owing to experimental difficulties, it is evident from the data that the maximum in ξdyn (Fig. 3a) coincides with the crossover in the morphology of CRRs (Fig. 3e, f). These findings therefore provide direct confirmation that the non-monotonicity in ξdyn results from a change in the shape as well as internal structure of CRRs. Further, we observe that the average number of particles per cluster increases monotonically (Supplementary Fig. 5). This is consistent with previous studies on colloidal glass-formers23,24 and strongly suggests that length scales that grow monotonically on approaching the glass transition are sensitive only to the number of particles in a CRR and not to their arrangement within it. In a broader context, our results provide the first direct verification for the change in morphology and internal structure of CRRs across the mode coupling crossover6,7.

As a final point, we discuss our findings in the light of the dynamical facilitation (DF) approach3, a purely kinetic theory of glass transition that has recently garnered experimental support18,31. The DF theory states that structural relaxation takes place via the coordinated motion of localized mobile defects, whose concentration decreases on approaching the glass transition. Within the DF approach, string-like cooperative motion over t∗ arises hierarchically from the dynamics of these defects32. However, in its current form, the DF theory for atomistic glass-formers does not anticipate a crossover in the morphology of CRRs. A major difference between the RFOT and DF approaches is that the former emphasizes the importance of activated events that grow in size, whereas in the latter, relaxation is dominated by the facilitated dynamics of localized defects. The importance of facilitated dynamics can be quantified by evaluating the mobility transfer function M(Δt) (ref. 33; see Supplementary Information for a definition of M(Δt) and Supplementary Fig. 6 for the evolution of M(Δt) with ϕ). The maximum Mmax of M(Δt) is expected to increase monotonically with ϕ if facilitation dominates structural relaxation. In the presence of non-facilitated activated processes that become dominant close to the glass transition, Mmax is expected to exhibit a maximum at the ϕ corresponding to the crossover from the facilitation-dominated regime to the activated-hopping regime34. Remarkably, we observe that, in our system, Mmax shows a maximum at ϕ = 0.76 (Fig. 3g and Supplementary Fig. 6), the very same ϕ at which ξdyn shows a maximum (Fig. 3a) and the morphology of CRRs changes from string-like to compact (Fig. 3e, f). We note that a similar dependence of Mmax on ϕ has been observed in a completely different colloidal system as well18. The observed ϕ dependence of Mmax is qualitatively consistent with RFOT, where facilitation is a secondary relaxation process that diminishes in importance on approaching the glass transition35, but is in stark contrast with predictions of the DF theory.

Our observation of a growing point-to-set correlation length, ξPTS (Fig. 3a), is consistent with the prediction of RFOT, although it does not rule out competing theoretical scenarios36,37. However, at present, the non-monotonic density dependence of ξdyn (Fig. 3a) and the concomitant change in the morphology and internal structure of CRRs (Fig. 3e, f) can be rationalized only within the framework of RFOT. Crucially, the morphological crossover in CRRs as well as the maximum in the ϕ dependence of Mmax find no natural explanation within the prominent competing framework of the DF theory. The non-monotonicity in Mmax (Fig. 3g) observed here as well as in ref. 18 may be associated with the diminishing role of facilitated dynamics in governing structural relaxation, rather than finite size effects34. Indeed, reconciling these findings within the facilitation paradigm poses an exciting challenge for future theory, experiments and simulations. A promising course in this direction would be to examine the influence of a pinned wall on facilitated dynamics of localized defects. Given that we see signatures of increasing cluster size of immobile particles with ϕ in the time-averaged images shown in Fig. 1a, it would be fascinating to explore connections between the regions of slow dynamics and the static and dynamic length scales extracted here. It would also be instructive to investigate whether our results are consistent with other thermodynamic frameworks, such as geometric frustration-based models4. We expect our findings to engender future research aimed at addressing these unresolved issues on the theoretical, numerical as well as experimental fronts.

Methods

Experimental details.

Our system consisted of a binary mixture of NS small and NL large polystyrene particles of diameters σS = 1.05 μm and σL = 1.4 μm, respectively. The particle size ratio σL/σS = 1.3 and number ratio NS/NL = 1.23 provided sufficient geometric frustration to prevent crystallization. The samples were loaded into a wedge-shaped cell18 and the desired area fractions ϕ were attained by sedimentation of the sample to the monolayer-thick region of the wedge. The typical waiting time before data collection was ∼8–10 h, which is several times larger than τα for all ϕ < 0.79. Samples were imaged using a Leica DMI 6000B optical microscope with a × 100 objective (Plan apochromat, NA 1.4, oil immersion) and images were captured at frame rates ranging from 3.3 fps to 5 fps for 1–1.5 h, depending on the value of ϕ. The size of the viewing region is 44σ × 33σ, with σ = (σS + σL)/2. The holographic optical tweezers set-up consisted of a linearly polarized constant power (800 mW) CW laser (Spectra-Physics, λ = 1,064 nm) and the optical traps were created using a SLM (512 × 512, 100 fps refresh rate, Boulder Nonlinear Systems). Standard Matlab algorithms38 were used to generate particle trajectories and subsequent analysis was performed using codes developed in-house. The particle tracking resolution is 0.08 μm and the mean squared displacement curves for all ϕ lie above the tracking resolution (Supplementary Fig. 7). For all ϕ ≤ 0.76, the duration of the experiment corresponds to several times τα, where τα is the time at which the self-intermediate scattering function Fs(q, t) drops to 1/e for q = 2π/σ. For ϕ = 0.79, the available experimental data are not sufficient to extract τα, although the waiting time is six times longer than the duration of the experiment. The presence of a pinned wall does not allow for sample drift corrections. To overcome this issue, we have ensured that the cell was left undisturbed for 8–10 h before recording each data set. The typical drift in our data over the entire duration of the experiment (1–1.5 h) is small—∼0.5σL in the x-direction and ∼0.1σL in the z-direction. Further, for every ϕ, we have also recorded data in the absence of pinning, immediately after capturing the data in the presence of a pinned wall. We find that the estimate for ϕMCT for the data in the absence of pinning, after appropriately subtracting the drift, is almost identical to that for data captured far away from the wall in the presence of pinning (see Supplementary Fig. 2).

Reference for measuring z.

To set the reference for measuring z, we first calculated overlap functions for all the boxes lying within strips of width 0.5σS, parallel to the wall—that is, along the x-axis—for the entire image. From these overlap functions, we estimated the z-coordinate for which the overlaps exhibit no decay and labelled that as the centre of the wall. Given that the wall is approximately two particle diameters wide, the overlap does not decay with time only for a few strips of 0.5σS away from the centre of the wall. We have chosen the centre of the last strip for which the overlap function does not exhibit a decaying profile with time as the reference from which to measure z. The error in estimating the z = 0 line is thus 0.5σS.

Procedure for identifying string-like and core-like particles in a CRR.

We labelled particles in a given cluster as string-like or core-like based on the number of nearest and next-nearest neighbours using the procedure described below. First, we identified all the particles that have more than two nearest neighbours, iNN>2, where i is the particle index. This set contains particles that form the core of the CRR as well as the particles that connect the compact core to the strings emerging from it. A particle ‘j’ is labelled core-like, only if it has at least two iNN>2 neighbours. The remaining particles are considered to be string-like. Supplementary Fig. 4 shows the fraction of core-like particles, nc, and string-like particles, ns, for different cluster sizes. Remarkably, we find that for all cluster sizes nc increases, whereas ns decreases with increasing ϕ. For the range of cluster sizes studied here, we see that the crossover in nc and ns systematically shifts to lower ϕ with increasing N, although it continues to remain in the vicinity of ϕ = 0.76.