Main

Dynamical heterogeneity is a key ingredient in many of the most advanced attempts to understand and rationalize the glass and the jamming transitions. The recent observation of a critical-like growth of temporal and spatial dynamical fluctuations in a two-dimensional athermal system approaching jamming10, similar to that hypothesized for glass-formers12, has raised hope that the glass and the jamming transitions may be unified, calling at the same time for further, tighter experimental verifications. Here, we investigate temporal dynamical heterogeneity in a three-dimensional thermal system, concentrated colloidal suspensions close to the maximum packing fraction. Temporal and spatial dynamical heterogeneity are usually closely related: the former can be quantified by a ‘four-point dynamical susceptibility’ χ4 (the variance of a time-resolved correlation function describing the system relaxation), the amplitude of which is proportional to ξ3 (refs 11, 13, 14). Surprisingly, we find that very close to jamming, temporal and spatial dynamical heterogeneity decouple: whereas ξ continuously grows with volume fraction, the amplitude of temporal fluctuations drops sharply close to the maximum packing fraction. These findings challenge current scenarios where the slowing down of the dynamics on approaching jamming is accompanied by enhanced temporal fluctuations of the dynamics.

The dynamics of colloidal hard spheres slows down markedly close to ϕ=ϕg≈0.58; beyond ϕg, ultraslow relaxations on short length scales are still observed3, until dynamics freeze at the maximum (random) packing fraction, ϕmax. We study concentrated suspensions of polyvinyl chloride xenospheres15 suspended in dioctyl phthalate (DOP) at volume fractions larger than ϕg and close to ϕmax. The particles have a typical radius of ≈5 μm and 0.33 size polydispersity, to avoid crystallization; they behave as slightly deformable hard spheres. The dynamics are measured by dynamic light scattering in the highly multiple scattering limit (diffusing wave spectroscopy16, DWS). This technique allows particle motion to be probed on length scales of the order of a few tens of nanometres (see the Methods section), which match well the restrained motion of our tightly packed suspensions. A CCD (charge-coupled device) detector is used to record the speckle pattern scattered by the sample. The evolution of the speckle images is quantified by the two-time degree of correlation17 cI(t,τ)=〈Ip(t)Ip(t+τ)〉p/(〈Ip(t)〉p〈Ip(t+τ)〉p)−1, where Ip(t) is the scattered intensity at pixel p and time t and 〈⋯〉p is an average over the CCD pixels.

To follow the evolution of the dynamics, we calculate g2(t,τ)−1, the two-time intensity correlation function obtained by averaging cI(t,τ) over 10–20 CCD frames. Figure 1a shows representative g2(t,τ)−1 functions for all ϕ. Owing to the limited acquisition rate of the CCD, the initial decay of g2(t,τ)−1, corresponding to the rattling of particles in the cages formed by their neighbours, is not captured; in the accessible time window, a plateau followed by a final relaxation is observed, indicative of very slow rearrangements. Figure 1b shows a typical example of the time dependence of τ0, the characteristic time of the final relaxation obtained by fitting g2−1 to a stretched exponential a(t)exp{−[τ/τ0(t)]β(t)}. Initially, τ0 grows nearly linearly with t, as observed in many glassy systems. However, a stationary regime is eventually reached (the dynamics on length scales larger than those probed here may still slow down, because the sample is unlikely to be fully equilibrated). In this regime, τ0 exhibits surprisingly large fluctuations but no overall increasing trend. A similar behaviour is observed for all volume fractions; all data presented in the following refer to the stationary regime, the duration of which is denoted by texp. We first investigate the ϕ-dependence of the average dynamics, as quantified by and , where denotes a time average over the duration of the stationary regime. As shown in Fig. 1c, continuously increases with ϕ. Data taken for freshly prepared samples (filled circles) can be fitted by a critical law, , with ϕmax=0.752, consistent with expectations for highly polydisperse samples18, and x=1.01±0.04, similarly to ref. 10. We also study samples that have been aged for several days and the dynamics of which is re-initialized by vigorously stirring and outgassing them (half-filled circles). Their effective volume fraction is higher than the nominal one, owing to the slight swelling of polyvinyl chloride particles suspended in DOP for very long times15, allowing them to achieve an even tighter packing. To compare the dynamics of both fresh and aged samples, we assign an effective volume fraction, ϕeff, to the latter so that their average relaxation time falls on the critical-like curve determined for the fresh samples (Table 1). Figure 1d shows : at the lowest volume fraction, the shape of g2 is slightly stretched (), similarly to what is observed for correlation functions in many glassy systems1. Surprisingly, as ϕ increases grows above one, finally saturating around 1.3. A similar ‘compressed’ exponential relaxation has been observed in single19,20 and multiple21 scattering experiments on systems close to jamming, usually associated with ultraslow ballistic motion.

Figure 1: Time and volume fraction dependence of the relaxation of concentrated colloidal suspensions, as probed by time-resolved DWS.
figure 1

a, Symbols: representative two-time intensity correlation functions for all volume fractions investigated. Data are taken in the stationary regime described below and in the text. From bottom to top and left to right, curves refer to samples A–I in Table 1. The lines are stretched exponential fits to the data, g2(t,τ)=a(t)exp[−(τ/τ0(t))β(t)], with t being the time elapsed since loading the cell. For each curve, the fitting parameters τ0 and β are close to the average values shown in c and d. b, Time dependence of τ0, for sample B (ϕ=0.664). After an initial ageing regime where the dynamics slows down, the system reaches a dynamically heterogeneous stationary state, where τ0 fluctuates significantly without any overall growing trend. c,d, Volume fraction dependence of the relaxation time and the stretching exponent, respectively, averaged over time in the stationary regime. The filled symbols refer to freshly prepared samples; the half-filled circles refer to old samples, the dynamics of which has been re-initialized. The bars are the standard deviations of the distributions over time of τ0 and β in the stationary regime. The solid line in c is a critical-law fit to the growth of τ0 for the fresh samples, yielding a critical exponent x=1.01±0.04 and a critical packing fraction ϕmax=0.752, indicated by the dashed line in c and d. The solid line in d is a guide to the eyes.

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Table 1 Sample volume fractions and experiment duration. Samples A, B, C, E and G are fresh samples prepared at a volume fraction ϕ, for which measurements start right after preparation. Samples D, F, H and I are aged samples initially prepared at a volume fraction ϕ, the effective volume fraction of which is slightly larger owing to particle swelling. An effective volume fraction ϕeff is assigned to these samples as described in the text. is the average relaxation time of the two-time intensity correlation function g2(t,τ)−1 measured in the regime where the dynamics is stationary. texp is the duration of the stationary regime.
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We quantify the temporal fluctuations of the dynamics by calculating χ(τ,ϕ), the relative variance of cI, defined by

where the factor accounts for the ϕ-dependence of the amplitude of the final relaxation of g2−1; data are furthermore corrected for experimental noise arising from the finite number of probed speckles17. The variance introduced above corresponds to the dynamical susceptibility χ4 much studied in simulations of glass-formers11,13. Figure 2a shows both the average correlation function (open circles) and χ (filled circles) for ϕ=0.738. The dynamical susceptibility exhibits a marked peak around τ0, a direct manifestation of dynamical heterogeneity also found in many other glassy systems5,10,11,13,14,22.

Figure 2: Non-monotonic volume fraction dependence of the dynamical susceptibility close to jamming.
figure 2

a, Average correlation function (open symbols and left axis) and dynamical susceptibility (filled symbols and right axis), for sample G (ϕ=0.738). A peak of height χ* is observed for τ≈τ0 (see the Methods section for the normalization of χ). b, Volume fraction dependence of χ* (same symbols as in Fig. 1c,d). The solid line is a critical-law fit to the initial growth of χ*, yielding an exponent y=1.5±0.2. Note the unexpected drop of χ* near the maximum packing fraction, shown by the dashed line.

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Figure 2b shows the height of the peak of the dynamical susceptibility, χ*, as a function of ϕ. At the lowest volume fractions, χ* increases with ϕ; using ϕmax=0.752 as obtained from the fit of , the data can be fitted by a critical law χ*∼1/|ϕ/ϕmax−1|y with y=1.5±0.2 (line in Fig. 2b), close to y=1.70 recently reported for driven grains10. This growing trend is also analogous to that observed in simulations of glass-formers6 and colloids3,4,9 (albeit at lower ϕ) and has been interpreted as due to a growing dynamical length scale on approaching dynamical arrest. At higher volume fractions, however, an opposite trend is observed: the amplitude of dynamical fluctuations markedly decreases close to ϕmax. This striking behaviour represents our central result, which challenges current views of dynamical heterogeneity close to dynamical arrest. The unexpected drop of dynamical fluctuations very close to jamming is confirmed by the non-monotonic behaviour of the width of the temporal distributions of τ0 and β, shown by the vertical bars in Fig. 1c,d. The dispersion of both parameters initially increases with ϕ, but eventually is reduced close to ϕmax, further demonstrating reduced dynamical heterogeneity.

We propose that the non-monotonic behaviour of χ* results from a competition between the growth of ξ on approaching ϕmax and the reduced particle displacement associated with rearrangement events close to jamming, due to tighter packing23,24,25,26. Indeed, as ξ increases, fewer statistically independent dynamical regions are contained in the sample, leading to enhanced fluctuations22. Conversely, as the particle displacement per event decreases, χ* is reduced, because more events are required to significantly decorrelate the scattered light and fluctuations on a timescale tend to be averaged out. These competing mechanisms should be quite general and should be observable in a variety of systems, provided that dynamical heterogeneities are probed close enough to dynamical arrest.

We have incorporated these ideas in a simple model for DWS for a dynamically heterogeneous process, significantly extending previous work on the intermittent dynamics of foams27 and gels28. The dynamics is assumed to be due to discrete, random rearrangement events each affecting a volume ξ3; however, in contrast to ref. 27, we assume that several events will be in general necessary to fully decorrelate the phase of scattered photons, because in concentrated suspensions the particle displacement associated with one single event may be much smaller than the wavelength of the light. The two-time field correlation function for a photon crossing the cell along a path of length s may then be written as

(see the Methods section). Here, ns(t,τ) is the number of events along the path between time t and t+τ and σφ2 is the variance of the change of phase of a photon due to one single event, proportional to the particle mean squared displacement (MSD) associated with such an event. For a totally uncorrelated change of photon phase due to distinct events, we have p=1, whereas in the opposite limit of a perfectly correlated change of phase p=2.

We implement our model in Monte Carlo simulations where photon paths are random walks on a square lattice. The lattice sites are affected by random rearrangement events of volume ξ3, occurring at a constant rate per unit volume. The simulated degree of correlation is calculated from , where the sum is over Ns=200,000 photon paths and g1(s) is calculated according to equation (1). The two-time intensity correlation function, g2,sim(t,τ), and its fluctuations, χsim, are then calculated from cI,sim as for the experiments. We vary the control parameters in the simulation, ξ3, p and σφ2, to reproduce the experimental ϕ-dependence of and χ*. As the particle displacement resulting from one rearrangement—and thus σφ2—is expected to decrease as ϕ grows, owing to tighter particle packing23,24,25,26, we choose 1/σφ2 as the control parameter against which simulation results are presented, corresponding to increasing volume fractions in Fig. 1b,c. Figure 3a shows versus 1/σφ2. The data are obtained using the values of ξ3 shown in Fig. 3c; however, we find that depends only very weakly on ξ3. For large particle displacements (small 1/σφ2), in fair agreement with the experimental value at the lowest ϕ. As particle displacements become increasingly restrained, grows and saturates at , close to the experimental values at the highest ϕ. The saturation value depends on the choice of p: here, p=1.65, showing that the change of phase of a photon due to distinct rearrangements is partially correlated. It is unlikely that such a correlation exists for events occurring in non-overlapping regions; in contrast, successive events in the same region will lead to partially correlated changes of phase when the direction of displacement persists during several events. Thus, p=1.65 indicates intermittent supradiffusive motion, a behaviour close to the ballistic motion reported for many jammed systems19,20.

Figure 3: Results of simulations reproducing the non-monotonic behaviour of the dynamical susceptibility observed experimentally.
figure 3

a,b, Time-averaged stretching exponent (a) and peak of the dynamical susceptibility χ* (b) as a function of 1/σφ2, as obtained from the simulations described in the text. The parameter 1/σφ2 on the x axis increases with ϕ. In a, the bars indicate the standard deviation of the distribution of stretching exponents. c, The volume of the rearranged regions used as an input in the simulations to reproduce the ϕ-dependence of and χ* observed in the experiments. ξ saturates at the value of the smallest dimension of the scattering cell.

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Figure 3b shows the 1/σφ2-dependence of χ*. The simulations reproduce well both the non-monotonic trend and the absolute values of the experimental χ*. They reproduce also the non-monotonic ϕ-dependence of the dispersion of β (bars in Fig. 3a), once again matching closely the experimental data. In spite of the drop of dynamical fluctuations close to ϕmax, ξ3 grows steadily with ϕ (Fig. 3c), until rearrangement events extend over the thickness of the cell, corresponding to ξ≈2,000 particle diameters. System-spanning ‘earthquakes’ have been reported in simulations of both thermal29 and athermal26 systems close to dynamical arrest, but have never been observed experimentally.

Our results show that the behaviour of dynamical heterogeneity very close to the jamming transition is much richer and complex than expected. On the one hand, dynamically correlated regions can extend over distances much larger than those previously reported, suggesting that confinement effects, usually observed on the scale of tens of particles at most30, may eventually become relevant macroscopically. On the other hand, temporal fluctuations of the dynamics are suppressed, challenging current views of dynamical heterogeneity close to jamming and calling for new theories.

Methods

Experiments

DWS16 experiments are carried out in the transmission geometry, in a cell of thickness L=2 mm. The photon transport mean path is ℓ*≈L/10. The intensity correlation function g2−1 decays to zero on the timescale it takes particles to move on a length scale16 Λ≈λℓ*/L, with λ being the wavelength of the laser in the solvent. Here, Λ≈35 nm for all samples. Note that, as shown, for example, in refs 5, 14, 28, the dynamical susceptibility usually depends on the length scale at which the dynamics are probed. It is therefore important to keep this length scale fixed when comparing different samples, as in our experiments. After loading the cell, the samples are vigorously stirred and outgassed to remove any air bubbles. We set t=0 at the end of the loading procedure. The particles have a typical radius of ≈5 μm and a polydispersity of 0.33 (standard deviation of the number distribution of radii normalized by the mean value). The size distribution was obtained in single scattering measurements on very diluted samples, applying the constrained regularization method (CONTIN) to intensity correlation functions measured at scattering angles θ=45∘,60∘ and 90∘. We check that in moderately concentrated suspensions (ϕ=0.28 and 0.46) the particle MSD measured by DWS increases linearly with time, indicating that on the length scales probed here the particles are brownian. The diffusion coefficient D=9.02×10−16m2s−1 extracted from the MSD using the viscosity of DOP (η=0.08 Pa s) is consistent with what is expected from DWS measurements on polydisperse particles of radius ≈5 μm (ref. 16).

As pointed out in refs 8, 17, the experiment duration texp should be long enough to fully sample dynamical fluctuations, because χ would be underestimated if texp was shorter than the characteristic time of the fluctuations of the dynamics. By dividing our data into segments of duration tseg<texp and by calculating χ as a function of tseg, we have checked that for all samples measurements last long enough for χ to be essentially independent of texp. In particular, we find that the relative deviation of the maximum of χ when processing only half of the available data, |χ*(texp)−χ*(texp/2)|/χ*(texp), is on average ∼8% and does not exceed 19%.

In simulations on glass-forming systems, the dynamical susceptibility χ4 is usually normalized by multiplying the variance of the intermediate scattering function—or a similarly defined correlation function— by the number of particles, Np. In our DWS experiments, the number of probed particles is not known precisely, because the scattering volume does not have sharp boundaries (owing to the diffusive nature of the photon paths) and because the particles are polydisperse. In addition, the Np normalization factor introduced in simulations originates from the fact that in that case the intermediate scattering function is a sum over the contributions of Np particles, whereas in DWS the field correlation function—and thus g2−1—is expressed as a sum over the contributions of photon paths, rather than particles16. We have thus omitted the Np normalization factor in our definition of the dynamical susceptibility χ, in agreement with previous work17. The values of χ reported here are therefore much smaller than those typically obtained in simulations of dynamically heterogeneous glassy systems.

Monte Carlo simulations

Photon paths are simulated as random walks on a square lattice with lattice parameter ℓ* and dimensions that match those of the experimental cell. In our model of DWS from a temporally intermittent process, the relevant parameter that controls the dynamics is ns(t,τ), the number of rearrangement events along a path of length s that occurred between times t and t+τ. In analogy with ref. 16, we thus write g1(s)=exp[−(1/2)〈Δφ2(ns)〉], where Δφ2(ns) is the change of phase due to ns events along a path of length s. By adapting the formalism of ref. 16 to our case, we find Δφ2(ns)=(2/3)N k02(ℓ/ℓ*)〈Δr2(ns)〉, where N is the number of scatterers along the path that have been displaced owing to a rearrangement event and 〈Δφ2(ns)〉 is the MSD of a particle after ns events. k0 and ℓ are the wave vector of the incident light and the photon scattering mean path, respectively. If the particle displacements due to successive events are totally uncorrelated, 〈Δφ2(ns)〉∝ns, whereas for perfectly correlated displacements (for example, if the direction of motion persists over successive events) 〈Δφ2(ns)〉∝ns2. For the sake of generality, we write 〈Δφ2(ns)〉=σ2nsp, with 1≤p≤2 and where σ2 is the MSD for one single rearrangement event (except for a numerical factor of order one). The field correlation function then reduces to equation (1). The parameter p governs the shape of the correlation function for : in this limit, has a compressed exponential shape with a compressing exponent equal to p. Summing over all paths with different length s—and thus different decay rates—results in an effective compressing exponent .