Measurement of Growing Dynamical Lengthscales and Prediction of the Jamming Transition in a Granular Material
Supercooled liquids and dense colloids exhibit anomalous behaviour known as “spatially heterogeneous dynamics” (SHD), which becomes increasingly pronounced with approach to the glass transition Ediger (2000); Kegel and van Blaaderen (2000); Weeks et al. (2000). Recently, SHD has been observed in confined granular packings under slow shear near the onset of jamming, bolstering speculation that the two transitions are related Pouliquen et al. (2003); Marty and Dauchot (2005); Dauchot et al. (2005). Here, we report measurements of SHD in a system of air-driven granular beads, as a function of both density and effective temperature. On approach to jamming, the dynamics become progressively slower and more spatially heterogeneous. The rapid growth of dynamical time and length scales characterizing the heterogeneities can be described both by mode-coupling theory Gotze and Sjogren (1992) and the Vogel-Tammann-Fulcher (VTF) equation Vogel (1921), in analogy with glass-forming liquids. The value of the control variable at the VTF transition coincides with point-J O’Hern et al. (2002, 2003) the random close-packed jamming density at which all motion ceases, indicating analogy with a zero temperature ideal glass transition. Our findings demonstrate further universality of the jamming concept and provide a significant step forward in the quest for a unified theory of “jamming” in disparate systems.
At low temperature, high density, and low driving, the constituent particles in supercooled liquids Ediger (2000), dense colloids Kegel and van Blaaderen (2000); Weeks et al. (2000), and granular packings Pouliquen et al. (2003); Marty and Dauchot (2005); Dauchot et al. (2005), respectively, are nearly locked into a single disordered configuration in which motion is heterogeneous in space and time. Dynamics in these systems may be governed by proximity to a generic “jamming transition Liu and Nagel (1998)”, beyond which rearrangements cease and the viscosity diverges. Key features of SHD on approach to the transition include unusual correlations Bennemann et al. (1999) in which particles move in one-dimensional paths (“strings Donati et al. (1998)”) that aggregate into clusters Donati et al. (1999), and dynamical correlations as measured by a dynamic four-point susceptibility Donati et al. (2002); Glotzer et al. (2000); Chandler et al. (2006). Clusters of strings arise naturally in dynamic facilitation Garrahan and Chandler (2002, 2003) theory and the random first-order theory of glasses Stevenson and Wolynes (2005); their shape reflects the fractal nature of dynamical motion in these systems Stevenson et al. (2006). Strings are also a crucial ingredient in a recent theory of liquid dynamics near the glass transition Langer (2006).
Recent studies demonstrate that close-packed granular systems under slow shear exhibit SHD as well Pouliquen et al. (2003); Marty and Dauchot (2005); Dauchot et al. (2005), bolstering speculation that liquids and granular matter share dynamical similarities on approach to the jammed state. However, the universality of the jamming hypothesis has not yet been tested in terms of variation in the hallmark dynamical heterogeneities as a function of the control parameter. Here, we present the first simultaneous measurements in any experimental system of the growth of the cluster correlation length, string length, four-point correlation length, and their characteristic timescales by varying the control parameter. We show that the SHD observed in a far-from-equilibrium, athermal system of air-fluidized granular beads is essentially indistinguishable from that observed in thermal systems like supercooled liquids and dense colloidal suspensions. Moreover, we show that theoretical models developed for the glass transition can be used to describe our granular system, and predict a mode-coupling like transition and, more importantly, the jamming transition packing fraction, known as point-J O’Hern et al. (2002, 2003) from quantities characterizing SHD.
We characterize the spatiotemporally heterogeneous nature of dynamics in an athermal, far-from-equilibrium system of air-driven steel spheres on approach to jamming. Compared to sheared or shaken granular systems, in which energy is injected at the boundaries, in air-driven systems the energy input is uniform in space and time. Our granular system consists of a 1:1 bidisperse mixture of steel beads of diameters cm and cm, with respective masses of 0.130 gm and 0.266 gm, confined to a circular region of diameter 17.7 cm. The packing density is varied from an area fraction of to by changing the total number of beads from 1470 to 1904. Bead motion is restricted to rolling within a horizontal plane, and is excited by an upward flow of air at a fixed superficial flow speed of 545 cm/s. Bead positions are identified by reflecting light from their chrome surface to a camera three feet above. The duration of experimental runs is 20 minutes. By contrast with the molecules in a supercooled liquid, here the particles are macroscopic objects driven at random by a continuous input of energy. Consequently the speed distributions are non-Maxwellian, and the average kinetic energies of the two bead species are unequal. Nevertheless, as reported previously Abate and Durian (2006), the system mimics a simple liquid for low and exhibits tell-tale changes in the average structure and dynamics at increasing packing densities.
Dynamical characteristics for an example case, , are displayed in Fig. 1a-d. The mean-squared displacement versus time interval (delay time), Fig. 1a, averaged over all start times and all beads, is perhaps the most familiar quantity. It shows ballistic motion at short times, diffusive motion at late times, and a plateau of sub-diffusive motion at intermediate times that is indicative of caging. While informative, the mean-squared displacement cannot distinguish uniform from heterogeneous dynamics. For this, we perform three measurements developed for supercooled liquids. The first involves clusters of beads that are “mobile,” i.e. which have displacements ranking among the top 10% of all bead displacements in a given delay time Kob et al. (1997) (in this system, 10% gives the largest distinction between mobile and immobile beads at all packing densities). The configuration displayed in Fig. 1e, where beads are colour-coded according to mobility, demonstrates that the mobile beads are not distributed at random; rather, mobile and immobile beads are clustered and spatially separated, indicating spatially heterogeneous dynamics. One measure of SHD is thus the average size of mobile clusters, , defined as the average number of neighbouring mobile beads for a given time interval . The motion within a mobile cluster, shown by the displacement vectors in Fig. 1e, tends to be correlated into quasi-1d paths called “strings Donati et al. (1998).” Thus a second measure of SHD is the average string length, , defined in terms of the number of beads that, within a time interval t, replace the initial positions of neighbouring beads to within a tolerance of Donati et al. (1998). Yet a third measure of SHD may be constructed from a four-point susceptibility , which measures the extent to which the dynamics at any two points in space are correlated within a time interval Donati et al. (2002). The self-contribution dominates the general result Glotzer et al. (2000) and is computed from the variance of the self overlap order parameter , which decays from one to zero:
Here, is the total number of beads, and is defined as:
The overlap parameter is defined by:
Here, is the position of bead at time ; averages are taken over all beads and over all start times.
The example results in Fig. 1b-d for the cluster size , the string length , and the four-point susceptibility , all exhibit well-defined peaks as a function of time interval, as found in glass-forming liquids. The locations of the peaks indicate the time interval over which the dynamics are most heterogeneous, and the heights of the peaks indicate the spatial extent or “strength” of the heterogeneities. As with glass-forming liquids Donati et al. (1999); Gebremichael et al. (2001); Gebremichael (2004); Vogel and Glotzer (2004) and colloids Weeks et al. (2000), the cluster size and string length are largest at the crossover between caged and diffusive motion, while (and ) peaks later, in the so-called alpha or structural relaxation regime Glotzer et al. (2000); Lačević et al. (2003). The athermal air-fluidized beads therefore exhibit spatially heterogeneous dynamics that is identical to thermal glass-forming systems with respect to these three measures.
Now that spatially heterogeneous dynamics are established for gas-fluidized beads, we turn to their variation as a function of control parameter. The distribution of cluster sizes at the peak time interval, shown in Fig. 2a for three different packing densities , approaches a power-law as is increased. This is consistent with the percolation of mobile bead clusters; similar power-laws have been observed in colloids Weeks et al. (2000) and in simulations of supercooled liquids Donati et al. (1999); Gebremichael et al. (2001); Gebremichael (2004) near the mode-coupling temperature. Furthermore, the distribution of string lengths at the peak time interval, shown in Fig. 2b, is exponential, , at all values of , where is set by the average string length. This is consistent with behaviour reported in simulations of several supercooled liquids Donati et al. (1998); Gebremichael (2004); Aichele et al. (2003). We note the average cluster size is not much larger than the average string length, although the largest clusters observed ( 100 particles) are substantially larger than the largest string observed ( 30 particles) (not shown).
Results for , , and vs are displayed in Fig.3a-c for a sequence of different packing densities . When beads are added to the system, the average effective temperature also decreases, resulting in a trajectory in the ( ,) phase diagram that heads towards point-J, the zero-temperature jamming transition previously found for this system at , which is coincident with the packing density at which the system is random close-packed. As the motion becomes more restricted, the peaks in all three measures of SHD grow and move to later times. Therefore, the dynamics not only slow down but also become more heterogeneous on approach to point-J. Since the SHD functions have approximately the same shape when viewed on a log-log plot (see data collapse in insets of Fig. 3a,b), this behaviour is fully characterized by the -dependence of the characteristic or peak time scales and and length scales and . The length is a correlation length for stringlike motion, is a correlation length of mobile particle clusters, and is a correlation length Dauchot et al. (2005) of clusters of caged particles. On approach to point-J, both the characteristic times and the correlation lengths appear from Fig. 3 to grow without bound. This is reminiscent of behavior for supercooled liquids as temperature is lowered. Though very different, both types of systems appear to approach an unusual critical point where the growing length scale is purely dynamical, such that there is no macroscopic change in instantaneous structure Ediger (2000); Stone et al. (2004); Corwin et al. (2005); Silbert et al. (2005).
To further quantify this analogy, the growth of the characteristic timescales and dynamical length scales is shown in Fig. 4a-d as a function of packing density. Motivated by recent studies Biroli and Bouchaud (2004); Biroli et al. (2006); Biroli and Bouchaud (2007) predicting a power law divergence of dynamical lengthscales from mode coupling theory Gotze and Sjogren (1992) (MCT), as well as earlier applications of MCT to liquids and colloids, we fit all data to a power-law of the form , where both and are adjustable parameters. As seen in Fig. 4a,b, excellent fits are obtained to all data for a single value . This value of lies well above the onset of caging and is less than the jamming packing fraction, in analogy with well-established findings that the mode-coupling temperature is below the caging transition but above the glass transition temperature Gotze and Sjogren (1992); Reichman and Charbonneau (2005), and demonstrates for the first time a mode-coupling-like transition in a granular system. In addition to MCT, the glass transition can also be well-described by the Vogel-Tammann-Fulcher (VTF) equation Vogel (1921); therefore, we also fit the characteristic time and length scales to the form , where and are adjustable parameters. As seen in Fig. 4c,d, excellent fits are obtained to all data for a single value , consistent with the value of random-close packing for the bead system and the value of point-J. Since random close packing is the densest random packing possible and the point at which all motion ceases, a VTF packing fraction of is analogous to an effectively zero-temperature ideal glass transition, consistent with the definition of point-J. This is the first prediction of point-J in a granular system from analysis of spatially heterogeneous dynamics.
Our study implies that the behaviour of jammed systems, both thermal and athermal alike, may be understood using the theoretical tools developed for liquids. This, in turn, highlights the importance of packing in the underlying physics of the glass transition and jamming. Our results open the door to future theoretical insight into the relationship between granular materials and supercooled liquids, which might be united by a unified theory of jamming.
Acknowledgements: This work was supported by the National Science Foundation under grant no. NSF-DMR0514705 (A.R.A. and D.J.D.), NASA under grant no. NNC04GA43G (A.S.K. and S.C.G.) and the Department of Education GAANN fellowship program (A.S.K.).
- M. D. Ediger, Annual Review of Physical Chemistry 51, 99 (2000).
- W. K. Kegel and A. van Blaaderen, Science 287, 290 (2000).
- E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schofield, and D. A. Weitz, Science 287, 627 (2000).
- O. Pouliquen, M. Belzons, and M. Nicolas, Physical Review Letters 91, 14301 (2003).
- G. Marty and O. Dauchot, Physical Review Letters 94, 15701 (2005).
- O. Dauchot, G. Marty, and G. Biroli, Physical Review Letters 95, 265701 (2005).
- W. Gotze and L. Sjogren, Reports on Progress in Physics 55, 241 (1992).
- H. Vogel, Physikalische Zeitschrift 22, 645 (1921).
- C. S. O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel, Physical Review Letters 88, 75507 (2002).
- C. S. O’Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, Physical Review E 68, 11306 (2003).
- A. J. Liu and S. R. Nagel, Nature 396, 21 (1998).
- C. Bennemann, C. Donati, J. Baschnagel, and S. C. Glotzer, Nature 399, 246 (1999).
- C. Donati, J. F. Douglas, W. Kob, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Physical Review Letters 80, 2338 (1998).
- C. Donati, S. C. Glotzer, P. H. Poole, W. Kob, and S. J. Plimpton, Physical Review E 60, 3107 (1999).
- C. Donati, S. Franz, S. C. Glotzer, and G. Parisi, Journal of Non-Crystalline Solids 307, 215 (2002).
- S. C. Glotzer, V. N. Novikov, and T. Schrøder, The Journal of Chemical Physics 112, 509 (2000).
- D. Chandler, J. P. Garrahan, R. L. Jack, L. Maibaum, and A. C. Pan, Physical Review E 74, 51501 (2006).
- J. P. Garrahan and D. Chandler, Physical Review Letters 89, 35704 (2002).
- J. P. Garrahan and D. Chandler, Proceedings of the National Academy of Sciences 100, 9710 (2003).
- J. D. Stevenson and P. G. Wolynes, Journal of Physical Chemistry B 109, 15093 (2005).
- J. D. Stevenson, J. Schmalian, and P. G. Wolynes, Nature Physics 2, 268 (2006).
- J. S. Langer, Physical Review E 73, 41504 (2006).
- A. R. Abate and D. J. Durian, Physical Review E 74, 31308 (2006).
- W. Kob, C. Donati, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Physical Review Letters 79, 2827 (1997).
- Y. Gebremichael, T. B. Schrøder, F. W. Starr, and S. C. Glotzer, Physical Review E 64, 51503 (2001).
- Y. Gebremichael, The Journal of Chemical Physics 120, 4415 (2004).
- M. Vogel and S. C. Glotzer, Physical Review Letters 92, 255901 (2004).
- N. Lačević, F. W. Starr, T. B. Schrøder, and S. C. Glotzer, The Journal of Chemical Physics 119, 7372 (2003).
- M. Aichele, Y. Gebremichael, F. W. Starr, J. Baschnagel, and S. C. Glotzer, The Journal of Chemical Physics 119, 5290 (2003).
- M. B. Stone, D. P. Bernstein, R. Barry, M. D. Pelc, Y. K. Tsui, and P. Schiffer, Nature 427, 503 (2004).
- E. I. Corwin, H. M. Jaeger, and S. R. Nagel, Nature 435, 1075 (2005).
- L. E. Silbert, A. J. Liu, and S. R. Nagel, Physical Review Letters 95, 98301 (2005).
- G. Biroli and J. P. Bouchaud, Europhysics Letters 67, 21 (2004).
- G. Biroli, J. P. Bouchaud, K. Miyazaki, and D. R. Reichman, Physical Review Letters 97, 195701 (2006).
- G. Biroli and J. P. Bouchaud, Journal of Physics: Condensed Matter 19, 205101 (2007).
- D. R. Reichman and P. Charbonneau, Journal of Statistical Mechanics: Theory and Experiment 2005, P05013 (2005).