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The Chaotic Dynamics of Jamming LETTERS PUBLISHED ONLINE: 24 MARCH 2013 | DOI: 10.1038/NPHYS2593 The chaotic dynamics of jamming Edward J. Banigan1,2, Matthew K. Illich1, Derick J. Stace-Naughton1 and David A. Egolf1,3* Granular materials are collections of discrete, macroscopic Beginning about 15 years ago, a series of seminal experiments particles characterized by relatively straightforward interac- on jamming were performed using an annular two-dimensional tions. Despite their apparent simplicity, these systems exhibit (2D) Couette shear cell packed with small, bidisperse, birefringent a number of intriguing phenomena, including the jamming elastic discs at a packing density φ (refs6 –8). The inner and transition, in which a disordered collection of grains becomes outer walls of the cell were covered in treads and the inner rigid when its density exceeds a critical value1,2. Many aspects wall was rotated to shear the material at a constant rate. As of this transition have been explored, but an explanation of the φ was increased over a narrow range, the total pressure in underlying dynamical mechanisms for the transition remains the system increased by many orders of magnitude—the system elusive. Here, applying nonlinear dynamical techniques3–5 to jammed. The grains formed a network of stress chains6–8, and simulated two-dimensional Couette shear cells6–8, we reveal the system exhibited behaviour that was sometimes flowing and the mechanisms of jamming and find that they conflict with the sometimes almost static. Measurements of a variety of quantities prevailing picture of growing cooperative regions. In addition, in experiments6–8 and simulations22,23 have provided insight into at the density corresponding to random close packing9,10, we this transition, but the dynamical mechanisms for the observed find a dynamical transition from chaotic to non-chaotic states behaviours remain unclear. accompanied by diverging dynamical length- and timescales. To elucidate these mechanisms, we perform molecular dy- Furthermore, we find that the dominant cooperative dynamical namics simulations of a 2D Couette shear cell and quantify the modes are strongly correlated with particle rearrangements behaviour using nonlinear dynamical techniques3–5. As shown and become increasingly unstable before stress jumps, provid- in Fig.1 a, we simulate N soft discs interacting through fric- 22,23 ing a way to predict the times and locations of these striking tional and dissipative elastic forces in an Lx × Ly rectangular stress-release events in our simulations. cell with periodic boundaries in the x-direction, parallel to the Even after over a century of study, granular materials are counter-moving, treaded, shearing walls. Our system exhibits be- not fully understood, probably owing to the unusual importance haviours qualitatively similar to the corresponding experiments as of dynamical aspects of their behaviour. Interestingly, granular φ is varied. systems at zero temperature undergo a transition from a fluid- The field of nonlinear dynamics provides tools for untangling like state to a rigid, disordered state known as the jamming the complicated behaviour of a nonlinear system into a hierarchy of transition1,2,11 when the granular density is increased. These spatiotemporal modes, organized by how much each mode drives disordered solids are intriguing (and difficult to understand) owing the dynamics. This hierarchy consists of Lyapunov vectors δu(i)(t) to the presence of structural heterogeneities, with the internal and associated Lyapunov exponents λi, indexed by i D 1;2;:::; and 2 (i) forces varying strongly with the material . Under weak stress such ordered by decreasing λi. Each Lyapunov vector, δu (t), represents as slow shear or compression, the behaviour of the disordered, a particular set of time-dependent, infinitesimal, translational δ (i) δ (i) heterogeneous networks of internal forces is fairly static over perturbations, f rj (t)g, and velocity perturbations, f vj (t)g, for extended periods of time but is occasionally punctuated by striking, all particles. The first Lyapunov vector, δu(1)(t), is the set of localized rearrangements of the particles. perturbations that grows the fastest (or shrinks the slowest) on In recent years, researchers have made progress in understanding average over the evolution of the system—the dominant collective the structure of granular systems near the jamming transition dynamical mode. The first Lyapunov exponent, λ1, is the average by, for example, quantifying force12 and contact13 networks, exponential growth rate of δu(1)(t) and can be computed as the 13–15 λ1t δ (1) measuring quantities such as the dynamic susceptibility and average of short-time growth rates 1 (t) of u (t) measured over 16 performing critical point scaling analyses . Other work has focused short intervals 1t (ref.5 ). If λ1 > 0, the system is chaotic and thus 17–19 on characterizing the vibrational modes , although there is sensitive to small perturbations. If λ1 < 0, any perturbations decay considerable controversy over the relevance of normal mode exponentially quickly. analysis in the vicinity of the jamming transition19,20. This type of The time-dependent Lyapunov vectors and their short-time analysis has been used well above the jamming density to predict growth rates provide detailed information about the heteroge- the locations of soft spots where rearrangement events occur18. neous dynamics of our system. Figure1 b shows the spatial dis- Yet another type of structure-based analysis, applied to a related tribution of the magnitude-squared of the first Lyapunov vector, δ (1) 2 δ (1) 2 δ (1) 2 granular system—a free-standing pile of beads—also shows a small, j uj (t0)j D j rj (t0)j Cj vj (t0)j , for each particle j at a time t0. but measurable, degree of predictive power21. We take a different The largest components of the Lyapunov vector (red) are localized δ (1) 2 approach and use the mathematics of nonlinear dynamics to reveal to a small region of the cell, and j uj (t0)j correlates strongly with 2 and analyse the behaviours of the dominant dynamical modes of a the square of the velocity of the jth particle, jvj (t0)j , as can be seen sheared granular system. by comparing Fig.1 b to Fig.1 c. 1Department of Physics, Georgetown University, Washington DC 20057, USA, 2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA, 3Institute for Soft Matter Synthesis and Metrology, Georgetown University, Washington DC 20057, USA. *e-mail: [email protected]. 288 NATURE PHYSICS j VOL 9 j MAY 2013 j www.nature.com/naturephysics © 2013 Macmillan Publishers Limited. All rights reserved NATURE PHYSICS DOI: 10.1038/NPHYS2593 LETTERS abc Lx v Ly v Figure 1 j Visualizations of the stress, Lyapunov vector and velocity fields. Colour-coding denotes the total stress on each disc, from low (blue) to high (red). a, Schematic of an Lx ×Ly simulated Couette cell with counter-moving treads (white discs) at time t0 during a stress-release event. The cell is D D D φ D : jδ (1) j2 periodic in the x-direction and Lx Ly 25ds. N 440 discs are packed with density 0 850. b, Example of the Lyapunov vector contribution uj (t0) 2 for each disc j at a time t0. c, The velocity-squared, jvj(t0)j , for each disc at the same time t0. a 1.0 To quantify the correlations between the Lyapunov vectors and the disc velocity fields, we compute the particle-by-particle 0.9 cross-correlation: 0.8 (1) Cδu;v (t) t 0.7 〉 ) t ( N (1) (1) P 2 2 2 2 v jδ j −hjδ j i j j −hj j i , 0.6 jD1 uj (t) uk (t) k vj (t) vk (t) k u δ D C 〈 q 2q 0.5 PN jδ (1) j2 −hjδ (1) j2i PN j j2 −hj j2i 2 jD1 uj (t) uk (t) k jD1 vj (t) vk (t) k 0.4 dP/dt < 0 Overall As can be seen in Fig.2 a (black symbols), the two fields are 0.3 dP/dt > 0 well correlated, on average. During periods of stress release 0.2 (blue symbols), the dynamics is dominated by a sequence of 0.83 0.84 0.85 0.86 0.87 0.88 rearrangement events and the correlation is remarkably strong; φ a single dynamical mode typically dominates, at least for dense b (1) 4 systems, and Cδu;v (t) is almost perfect at many times, differing from 1 by less than 0:0001%. (For an example, see Supplementary 1 Fig. S1.) The correlations are higher at larger values of φ because 2 fewer modes contribute to the dynamics (as quantified by the set of fλ g ) Lyapunov exponents i ). Altogether, the connection between the Δ ¬1 P ( fastest moving particles and the most important dynamical modes (s t 1 )/ λ provides direct evidence that these localized clusters are responsible 0 Δ P ) ¬ (0) t for cooperative rearrangements. Interestingly, a similar connection ( t Δ 24 1 was found in supercooled colloidal fluids near the glass transition . λ ¬2 A striking feature of sheared granular systems is the way stress builds in the whole system and is then released through particle 0 rearrangement events in these dynamically unstable regions. These ¬4 events appear in the time-series of the total pressure P(t) as small ¬4 ¬3 ¬2 ¬1 0 jumps in P(t) when the rearranging particles re-jam. The blue line 1 t (s) in Fig.2 b shows the average behaviour of P(t) about a jump at t D 0. We find that the particles involved in a rearrangement event Figure 2 j Correlations between Lyapunov vectors and exponents and are in a dynamically unstable configuration for a period of time system behaviour. a, Time-averaged particle-by-particle cross-correlations before the instability is triggered.
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