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LETTERdoi:10.1038/nature24062
Granular materials flow like complex fluidsbinquan Kou1, Yixin Cao1, Jindong Li1, Chengjie Xia1, Zhifeng Li1, Haipeng Dong2, Ang Zhang2, Jie Zhang1,3, Walter Kob4 & Yujie Wang1,5,6
Granular materials such as sand, powders and foams are ubiquitous in daily life and in industrial and geotechnical applications1–4. These disordered systems form stable structures when unperturbed, but in the presence of external influences such as tapping or shear they ‘relax’, becoming fluid in nature. It is often assumed that the relaxation dynamics of granular systems is similar to that of thermal glass-forming systems3,5. However, so far it has not been possible to determine experimentally the dynamic properties of three-dimensional granular systems at the particle level. This lack of experimental data, combined with the fact that the motion of granular particles involves friction (whereas the motion of particles in thermal glass-forming systems does not), means that an accurate description of the relaxation dynamics of granular materials is lacking. Here we use X-ray tomography to determine the microscale relaxation dynamics of hard granular ellipsoids subject to an oscillatory shear. We find that the distribution of the displacements of the ellipsoids is well described by a Gumbel law6 (which is similar to a Gaussian distribution for small displacements but has a heavier tail for larger displacements), with a shape parameter that is independent of the amplitude of the shear strain and of the time. Despite this universality, the mean squared displacement of an individual ellipsoid follows a power law as a function of time, with an exponent that does depend on the strain amplitude and time. We argue that these results are related to microscale relaxation mechanisms that involve friction and memory effects (whereby the motion of an ellipsoid at a given point in time depends on its previous motion). Our observations demonstrate that, at the particle level, the dynamic behaviour of granular systems is qualitatively different from that of thermal glass-forming systems, and is instead more similar to that of complex fluids. We conclude that granular materials can relax even when the driving strain is weak.
Static and driven granular systems behave in many ways like thermal glassy systems3. However, they also exhibit phenomena such as arching and force chains2, and shear thickening and ava-lanches, which demonstrates that their properties are also affected by many-particle effects, friction and dissipation. Although there are many studies on the macroscale properties of granular systems, only a few investigations have probed the structure and dynam-ics of granular systems at the particle level and in three dimen-sions, owing to the experimental difficulty of determining the position and orientation of individual particles7–9. However, this information is crucial for obtaining a fundamental understanding of the properties of these systems at the macroscale2. Dynamic infor-mation about the system can be obtained through cyclic shear exper-iments and simulations that enable direct insight into compaction and ageing dynamics10,11, memory effects12, dynamic heterogeneity13, plastic deformation14, avalanches15 and the role of friction11,16,17. Our X-ray tomography study shows that this relaxation dynamics gives rise to a distribution of the particle displacements that is well described
by a Gumbel law6, and to marked memory effects. Surprisingly, we find that the cage effect—the mechanism that slows the dynamics in glass-forming systems, whereby particles are temporarily trapped by their nearest neighbours5—is absent in the system that we study, thus indicating a marked difference in the microscale relaxation dynamics of thermal glass-formers and granular systems.
The particles that we study are hard plastic prolate ellipsoids (minor axis 2b = 12.7 mm) with an aspect ratio of 1.5. The particles are poured into a rectangular box with side walls that can be tilted to impose a cyclic shear on the system and a heavy plate on the top of the particles (see Methods). We used four strain amplitudes, γ = 0.07, 0.10, 0.19 and 0.26, and a strain rate of around 1.7 × 10−2 s−1. This strain rate implies that the inertial number is about 1.4 × 10−4, meaning that the experiment can be considered as quasi-static18. Before taking any meas-urements, we performed hundreds of oscillatory shear cycles to bring the system to a stationary state (see Methods). After each cycle we did a tomography scan and measured the positions and orientations of all particles19 (Fig. 1a; Supplementary Videos 1 and 2 show the dynamics of the particles and Extended Data Fig. 1 shows some trajectories). In the following we measure time t in units of complete cycles and length in units of b.
The translational mean squared displacement (TMSD; ⟨ d2(t)⟩ = ⟨ | rj(t) − rj(0)| 2⟩ , where rj(t) is the position of particle j at time t) of the particles is shown in Fig. 1b. We see that the TMSD has no plateau at intermediate times even if the driving is weak, which demonstrates that the cage effect observed in thermal glassy systems5 is absent. This is also confirmed by the intermediate scattering function, which does not show any sign of a two-step relaxation (see Methods). For γ = 0.07, ⟨ d2(t)⟩ is proportional to t for all times accessed, indicating normal diffusive behaviour. If γ is increased to 0.26, then the t dependence of ⟨ d2(t)⟩ can be well described by a power law with an exponent of around 0.75; that is, the diffusion is anomalous. This result is surprising because it could be expected that a stronger driving would induce more noise to the system and hence lead to a diffusive motion. However, this is obviously not the case, and we discuss the origin of the anomalous diffusion below. For γ = 0.10 and γ = 0.19, we find a power law with an exponent close to 1.0 (normal diffusion) for early times, but at t ≈ 100 and t ≈ 10, respectively, the t dependence changes to one that can again be well described by a power law with exponent close to 0.75 (the same exponent we found for γ = 0.26). Some possible ways of explaining the presence of anomalous diffusion are to assume that the underlying configuration space has a fractal nature20, that the dynamics exhibits aspects of a Levy flight21, or that the motion of the particles is collective22–25. However, we show that here the anomalous diffusion can be rationalized by memory effects in the dynamics.
The rotational dynamics at early times also shows anomalous diffusion for γ = 0.26 (Fig. 1c). However, at t ≈ 40 the dynamics changes to a linear t dependence, indicating that for this strain amplitude the particles start the rotational diffusion well before the translational diffusion sets in. For γ = 0.19 the anomalous diffusion in the rotational
1School of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China. 2Department of Radiology, Ruijin Hospital, Shanghai Jiao Tong University School of Medicine, Shanghai 200025, China. 3Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China. 4Laboratoire Charles Coulomb, University of Montpellier and CNRS, UMR 5221, 34095 Montpellier, France. 5Materials Genome Initiative Center, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai 200240, China. 6Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China.
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dynamics is less pronounced (the exponent is larger), and for γ ≤ 0.1 we find normal diffusion for all times. From these observations we conclude that the manner in which the system explores its phase space depends on γ and the time lag considered.
More detailed information on the relaxation dynamics can be obtained from the distribution function for the displacements, known as the self part of the van Hove function5 (Fig. 2a):
∑ δ= ⟨ − − ⟩=
r rG d tN
d t( , ) 1 ( ( ) (0) ) (1)j
N
j js1
where N is the number of particles and δ is the Dirac delta function. Although the curves in Fig. 2a suggest that Gs(d, t) has a similar shape to that found for thermal glass-forming systems5,26, a plot of the dis-tribution of the displacements along a given axis demonstrates that this distribution is not Gaussian, but can instead be well fitted by a Gumbel law6 (Fig. 2b; see Methods section ‘Comparison of Gumbel and q-Gaussian fitting’):
λλ λ
=
−| |−
−| |
f d Ad d
( ) ( )exp exp (2)yy y
Here λ is a length scale, the nature of which we discuss in Methods, dy is the displacement of the particle in the y direction and A(λ) is a normali-zation constant. We also show in Methods that this Gumbel law can be interpreted as the consequence of the interplay between two relaxation mechanisms: diffusion on small length scales induced by the roughness of the particles and by friction, and irreversible relaxation events on larger scales. Because this functional form seems to be compatible with observations from other granular systems (see, for example, ref. 15), we conclude that our findings apply not only to the system that we study, but also more generally. Most remarkable is the fact that the shape of Gs(d, t) is independent of t and γ (see Fig. 2b), and of the direction of the displacement, which suggests that the mechanism that leads to this distribution is very general; we present arguments for why this is the case in Methods.
That the shape of Gs(d, t) is independent of γ and t indicates that the anomalous diffusion is not related to the details of this distribution and, more generally, that an observable that depends on only a single time difference is not able to explain the anomalous TMSD. We there-fore look at a generalization of this dynamic observable and define a correlation function that depends on two time correlation functions:
δ δ=⟨ − − − − ⟩G d d t d r t r d r t r t( , , ) ( [ ( ) (0)]) ( [ (2 ) ( )])y y y y4 1 2 w 1 w 2 w w
where ry is the y component of the position of a particle. This function defines the probability of a particle that moves in the y direction by d1 in the time window [0, tw] moving by d2 in the window [tw, 2tw]. For a Markovian process the joint probability G4(d1, d2, tw) will factorize; this property can therefore be used to test for the presence of non-Markovian behaviour such as memory effects27.
Scatter plots for this joint distribution for γ = 0.10 (Fig. 3a) reveal a reflection symmetry around the horizontal axis, indicating that for any given value of d1 the probability of finding a positive d2 is the same as that of finding a negative d2. (The tw dependence of these plots is shown in Fig. 4.) By dividing the d1 axis into three ranges (see Fig. 3a) we can compute for these values of d1 the conditional probability that a particle moves a distance d2. The resulting three distributions are shown in Fig. 3b, and we see that they are independent of the range considered, which demonstrates that the joint distribution can indeed be factorized. In Fig. 3c, d we show the same quantities as in Fig. 3a, b, but for γ = 0.26. In this case, the joint distribution is aligned with the negative diagonal, thus indicating the presence of a memory effect: a particle that has moved in the time interval [0, tw] by an amount d1 is likely to move in the time interval [tw, 2tw] by an amount − d1 (refs 13, 27). The memory effect can be seen clearly in the coarse-grained distributions (Fig. 3d), which are no longer symmetric but become increasingly skewed with increasing d1. Because the motion during the interval [tw, 2tw] is back-wards, the particles advance a bit slower than expected from Gs(d1, t), thus giving rise to anomalous diffusion.
To quantify the way in which the memory effect changes with γ and tw, we divide the d1 axis into intervals of width . ⟨ ⟩ /d0 5 y
2 1 2 and compute
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SD
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2 )
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2 )
cb
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D/t
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SD
/t
X-ray tube
Camera
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y
z
x
Pressure
Shear direction
Time, t
Figure 1 | Experimental set-up and time dependence of the translational (TMSD) and rotational (RMSD) mean squared displacement for different strain amplitudes (γ), showing the presence of anomalous diffusion. a, Schematic of the experimental set-up, showing the computational tomography (CT) scanner (which includes an X-ray tube and a camera) and the shear cell that contains the granular media (red ellipsoids) to be probed. The plate used to generate the pressure on the sample is also shown. The directions of the coordinate axis (x, y, z) used in our study are also indicated; the shear is applied in the y direction and the pressure is applied in the z direction. b, The TMSD for γ = 0.07, 0.10, 0.19 and 0.26 exhibits power-law behaviour with an exponent that depends on the time interval considered. For γ = 0.10 we have two set of curves from two independent measurements: the first one (green) probes the early-time dynamics; the second one (orange; labelled ‘L’) uses a larger time grid and thus shows the TMSD at late times. From the fact that the two curves agree very well at intermediate times, we infer
that the precision of the data is high. Inset, TMSD/t as a function of t, demonstrating that for small γ and early times the TMSD is indeed linear in time, whereas for late times and large γ the dynamics is subdiffusive. The kink in the curves for γ = 0.19 (γ = 0.10) at t ≈ 10 (t ≈ 100) also indicates that the relaxation dynamics of the system changes from normal diffusion to anomalous diffusion. The anomalous diffusion sets in once the particles start to move distances that are of the order of their size. This behaviour can therefore be caused by the shear during one cycle with large strain (such as for γ = 0.26) or by the diffusive motion after many cycles (such as for γ ≤ 0.19). For times later than those accessed in these experiments, the TMSD will become linear in time for all values of γ; that is, the particle will ultimately become diffusive. c, RMSD for γ = 0.07, 0.10, 0.19 and 0.26, showing that at early times the exponent of the power law depends on γ. Inset, RMSD/t as a function of t, showing that at late times the RMSD is linear in time for all shear amplitudes.
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for each interval the average value of d2. If this average is zero, then the joint probability factorizes; a non-zero value signals non-Markovian behaviour. In Fig. 4a we show this mean value for different times and see that for γ = 0.10 and early times the memory effect is very small (the process is Markovian). In contrast, we find that for γ = 0.10 and late times, and for γ = 0.26 for all times, the memory effect is quite pronounced. This finding confirms the onset of non-Markovian dynamics as the particles start to explore a region that is comparable in size to that of the individual particles.
To study the duration of the memory effect we calculate the mean value of d2 (considering only d1 > 0 because there is a point symmetry with respect to the origin), and in Fig. 4b we show its dependence on tw. The curve for γ = 0.10 demonstrates that the memory effect starts to become noticeable at tw ≈ 100, which is also the time at which the TMSD changes from simple diffusion to anomalous diffusion (see Fig. 1b), indicating that these two phenomena are related. The curve for γ = 0.26 shows that the memory effect lasts much longer than 200 cycles—a timescale on which the particles have undergone a transla-tional motion that is a large fraction of their size (Fig. 1a).
Our results demonstrate that the relaxation dynamics of cycled granular systems is qualitatively different from that of thermal glass-formers in that it does not exhibit the cage effect. Instead our obser-vations indicate that the presence of small-scale relaxation events and friction during the cycle give rise to a distribution of the displacements that is well described by a Gumbel law. These results, combined with the fact that for intermediate-sized displacements of the particles we see marked memory effects, suggest that granular materials are similar to complex fluids—liquids that have non-Debye relaxation dynamics (non-exponential in time) and often involve memory effects28. Because the mechanism that leads to the observed relaxation dynamics is very general, we conclude that in general granular systems will relax even if they are perturbed only slightly, despite the fact that if not driven they undergo a jamming transition and become solids. A perturbation thus makes a granular system behave like a continuously evolving solid that, as it relaxes with time, steadily explores the part of phase space that corresponds to mechanically stable configurations. This insight should enable the relaxation dynamics at the particle level to be con-nected to mesoscale phenomena such as secondary creep and non-local
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a b Figure 2 | Time evolution of the self part of the van Hove function Gs(d, t) and probability density function of the translational displacement along the y direction. a, Gs(d, t) (equation(1)) as a function of displacement d at different times t (see legend) for γ = 0.26. b, Probability density functions of particle displacements in the shear (y) direction (dy) (see Fig. 1a) as a function of /⟨ ⟩ /d dy y
2 1 2 for different values of γ ; coloured symbols correspond to the same times as in a. For the sake of clarity, the data for γ = 0.19, 0.10 and 0.07 have been shifted horizontally by 2.0, 4.0 and 6.0, respectively. A fit to the data with a Gumbel law (equation (2), with λ = 0.605 and A(λ) = 1.313) is also shown for each γ (solid black lines). The same quantitative behaviour as shown here for the y direction is found for the x and z directions.
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c = 0.26, tw = 5
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(I) (II) (III)
(I) (II) (III)
Figure 3 | Scatter plots of the displacements for two consecutive time intervals and the conditional probability density function of d2, demonstrating the presence of memory effects for large γ. a, Scatter plot of d2, the displacement in the time interval [30, 60], as function of d1, the displacement in the time interval [0, 30], for γ = 0.10. The vertical dashed lines at d1 = 0, ⟨ ⟩ /d1
2 1 2, ⟨ ⟩ /d2 12 1 2 and ⟨ ⟩ /d3 1
2 1 2 are used to define the regions (I), (II) and (III). The colour scale (shown as contours) corresponds to the density of points. b, The conditional probability density function of d2 for d1 in regions (I), (II) or (III) in a. The solid line is the full probability density function of d2, that is, the self part of the van Hove function in y direction. c, Scatter plot of d2, the displacement in the time interval [5, 10], as function of d1, the displacement in the time interval [0, 5], for γ = 0.26 (tw = 5). The time intervals in a and c are chosen so that the corresponding TMSDs are comparable (see Fig. 1b). As in a, the vertical dashed lines correspond to d1 = 0, ⟨ ⟩ /d1
2 1 2, ⟨ ⟩ /d2 12 1 2
and ⟨ ⟩ /d3 12 1 2. d, Same as b, but for d1 in the regions
1 6 N O V E m b E R 2 0 1 7 | V O L 5 5 1 | N A T U R E | 3 6 3
rheology29, thus laying the groundwork for rationalizing the macroscale behaviour of granular systems in terms of their microscale properties.
Online Content Methods, along with any additional Extended Data display items and Source Data, are available in the online version of the paper; references unique to these sections appear only in the online paper.
received 27 September 2016; accepted 18 August 2017.
Published online 1 November 2017.
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Supplementary Information is available in the online version of the paper.
Acknowledgements Some of the preliminary experiments were carried out at BL13W1 beamline of Shanghai Synchrotron Radiation Facility. The work is supported by the National Natural Science Foundation of China (numbers 11175121, 11675110 and U1432111), Specialized Research Fund for the Doctoral Program of Higher Education of China (grant number 20110073120073) and ANR-15-CE30-0003-02. W.K. is member of the Institut Universitaire de France.
Author Contributions Y.W. and W.K. designed the research. B.K., Y.C., J.L., C.X., Z.L., H.D., A.Z., J.Z. and Y.W. performed the experiment. B.K., W.K. and Y.W. analysed the data and wrote the paper.
Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Readers are welcome to comment on the online version of the paper. Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Correspondence and requests for materials should be addressed to Y.W. ([email protected]) or W.K. ([email protected]).
reviewer Information Nature thanks R. Behringer and the other anonymous reviewer(s) for their contribution to the peer review of this work.
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= 0.19 = 0.26tw = 1
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tw = 100
Figure 4 | Memory effect as a function of particle displacement and time. a, Average of d2 (⟨ d2⟩ ) as a function of d1 for different tw and γ. For fixed time tw, the averages increase (in absolute value) with γ, showing that large γ gives rise to stronger memory effects. The data for γ = 0.10 show that even for small γ the memory effect is seen, although at later times than those for larger γ. For displacements larger than about 0.6b, the curves start to flatten (such as for γ = 0.26 and tw = 100), indicating
that once a particle has been displaced to about this distance it starts to lose its memory. Qualitatively the same behaviour is found in the x and z directions. b, The tw-dependent average of d2 for d1 > 0 (⟨ d2(d1 > 0)⟩ ) showing that the memory effect increases with time, before it starts to decrease again at late times. Error bars are calculated as the standard deviation of d2 in the three directions.
MethOdSExperimental sample and set-up. The granular particles are prolate ellipsoids with an aspect ratio of 1.5 and size polydispersity of around 0.9%, made from polyvinyl chloride (PVC) by injection moulding. The major and minor axes are 2a = 19 mm and 2b = 12.67 mm, respectively. During the course of the experiment we found no evidence of crystallization (see the static structure factor S(q) in Extended Data Fig. 2). This good glass-forming ability is probably related to the fact that ellipsoids with this aspect ratio can be packed into an amorphous phase that is relatively dense30.
Using a materials testing machine (Zwick/Roell Z100) we determined the Young’s modulus E of the particles to be 4.13 ± 0.18 GPa, close to the value found in the literature for PVC31. From this value and the applied pressure, we estimate the deformation of the particles32:
=
/
= . µ = . ×
/−d F
b Em b9
16( 2)1 778 2 81 10
2
2
1 34
Here F = 0.735 N is the force on a particle, which we estimate as the total weight (24.3 kg × 9.8 m s−2 = 238.14 N) divided by the total number of particles in the bottom layer (324). We therefore conclude that under our experimental conditions the particles keep their shape and so can be considered as hard ellipsoids22.
The rectangular shear cell (see Fig. 1a) is made of acrylic plates and has a size of 40.2b × 43b × 22.6b (25.5 cm × 27.25 cm × 14.3 cm). The front and back plates are permanently fixed on the baseplate. A plate is laid on top of the particle packing and is constrained by linear guides on the front and back plates, with the result that it can undergo motion only in the vertical direction. The total weight of the particles is 8.3 kg and the weight of the top plate is 16 kg, thus providing a constant normal pressure of 2.3 kPa. This extra weight from the top plate helps to minimize the relative pressure gradient in the packing along the direction of gravity.
The side plates are linked by hinges to the bottom plate, which in turn is attached to a linear motor stage, the horizontal displacement of which generates the shear on the shear cell. The shear direction is defined as the y direction and the vertical direction as the z direction. During a shear cycle, the cell is first sheared in one direction to the designated shear strain; then, the shear is reversed and the cell is sheared in the other direction to reach the same negative strain; finally, the whole cell is returned to its original geometry to complete the cycle.Experimental details. There are about 4,100 ellipsoids in the shear cell. Before we started the measurements with the CT scanner to obtain the positions and orientations of the particles we cycled the system for a long time to allow it to reach a steady state. To estimate the time it takes the system to reach a steady state we monitored the height of the movable top plate as a function of time; this type of measurement enables a reliable estimate of the total volume occupied by the particles to be obtained. We found that the average height decreases quickly during the first hundred cycles and then shows a very slow and weak decrease. Because the region that we used for analysing our CT measurements is no longer changing in time after these first hundred cycles (see Extended Data Fig. 3), we conclude that the central part of the system has indeed reached the steady state, meaning that the changes in its structure are sufficiently small that they can be neglected. The number of cycles used for the preparation and for the measurements are given in Extended Data Table 1.
The three-dimensional structural information (position and orientation of all particles) was acquired by a medical CT scanner (SOMATOM Perspective, Siemens) with a spatial resolution of 0.6 mm. One entire scan took about 10 s. We took a tomography scan after each cycle for the first 10 cycles and then a scan every 5 cycles. For γ = 0.10 we conducted a second experiment in which we scanned only every 50 cycles, thus enables us to reach later times. In the figures, the data from this run are labelled ‘L’.
We followed similar imaging processing steps as in previous studies19. Through a marker-based watershed image segmentation technique, we extracted the centre position r(t) and orientation e(t) of each ellipsoid. By making two consecutive tomography scans of the same static packing and analysing them independently, the precision of the extracted centre position r(t) and orientation e(t) were estimated to be 5.3 × 10−3b and 8.4 × 10−3 rad, respectively. To minimize the influence of boundary effects on our results we considered only those particles that have a distance of at least 5b from the cell boundaries. This condition means that in practice we used about 1,300 particles for the analysis.Structural properties of the sample. Effect of the walls on the structure. In Extended Data Fig. 4 we plot the normalized local particle number density ρi/⟨ ρi⟩ (i ∈ {x, y, z}) of the system. Because clear layering effects are found near the boundary region along all three axes, we did not include the particles close to the wall in the analysis. However, as shown in Extended Data Fig. 4c, a weak layering effect still exists along the z direction in the centre region. To estimate the influence of this small layering effect, we divided the central part of the box along the z direction into three parts of equal height. We carried out the same analyses
regarding the relaxation dynamics described in the main text for each of these three regimes and found no quantitative differences among them. We therefore conclude that our results are not affected by wall effects.Evolution of the volume fraction Φ under shear. The volume fraction Φ is deter-mined by Φ=∑ /∑V Vj j j j,cell, where Vj and Vj,cell are the volume of particle j and its Voronoi volume, respectively, and the sums are taken over all particles in the central volume. The time dependence of Φ is shown in Extended Data Fig. 3. The relatively constant values of Φ during measurements indicate that the system has indeed reached the steady state for all γ investigated. Also, for the second run for γ = 0.10 we find a volume fraction that is compatible with that found for the first run with this γ. This indicates that our results are reproducible and do not show large sample-to-sample fluctuations.The static structure factor S(q). We calculated the static structure factor S(q) from the particle position via
∑∑= − ⋅ −= ≠
q r rS qN
i( ) 1 exp[ ( )] (3)k
N
k j
N
k j1
As shown in Extended Data Fig. 2, we find that for all values of γ, S(q) shows the typical form found in disordered systems5, thus indicating that there is no sign of crystallization. (The peak at very small q is related to the boundary effect of the window function used for the Fourier transform.)Determination of the rotational dynamics. To quantify the rotational motion of the particles, we calculated the time integral of the angular increment of each cycle,
∫θ ω= ′ ′t t t( ) ( )dj
t
j0
where the modulus and direction of ωj(t) are cos−1[ej(t) · ej(t + 1)] and ej(t) × ej(t + 1), respectively. From this we define an unbounded RMSD ⟨ | θ(t)| 2⟩ (Fig. 1c).Intermediate scattering function. From Fig. 1b we conclude that our system does not exhibit a cage effect for any γ. To test whether this conclusion is correct we calculated from the particle positions rj the self intermediate scattering function Fs(q, t) and the collective intermediate scattering function F(q, t) (the density correlation function for wave vector q; refs 5, 33):
∑
∑ ∑
= − ⋅ −
= − ⋅ −
=
= =
q r r
q r r
F q tN
i t
F q tN
i t
( , ) 1 exp[ ( ( ) (0))]
( , ) 1 exp[ ( ( ) (0))]
(4)j
N
j j
j
N
k
N
k j
s1
1 1
The time dependence of these two correlation functions is shown in Extended Data Fig. 5 for two wave vectors: q = 3.49b−1, the position of the main peak in the static structure factor S(q) (Extended Data Fig. 5a); and q = 4.46b−1, the location of the first minimum in S(q) (Extended Data Fig. 5b; Extended Data Fig. 2). We recall that for thermal glass-forming systems the cage effect means that the self and collective intermediate scattering functions show a two-step relaxation process, whereby at intermediate times they show a plateau before decaying at late times to zero5. The curves in Extended Data Fig. 5 show no indication of the presence of such a plateau for any γ, thus demonstrating that for this system the cage effect is absent. A closer inspection of the data shows that for q = 3.49b−1 the collective function decays slower than the self one, whereas for q = 4.46b−1 it is the other way around. This behaviour is also found in the relaxation dynamics of liquids and can be directly related to the so-called de Gennes narrowing33.Particle trajectories. To obtain a qualitative idea of the nature of the motion of the particles, we show in Extended Data Fig. 1 the trajectories of 20 particles. These trajectories show that most particles explore a region of space that grows continu-ously with increasing time. These particles are probably the ones that give rise to the Gaussian core in the self part of the van Hove function (see Fig. 2). There are also particles that move substantially farther; it is probably these particles that give rise to the exponential tail found in the self part of the van Hove function and hence the Gumbel distribution.Origin of the Gumbel law. Here we outline the arguments for why the probability density function (PDF) for the displacements of the particles is well described by a Gumbel law, provided that the displacements of the particles are not so large that the particles have not yet changed a substantial fraction of their nearest neighbours (see Fig. 2b). For a granular system with no friction, every mechanically stable configura-tion corresponds to an arrangement of the particles in which the total gravitational energy of the system is in a local minimum. Although this is similar to the situation for a glass at finite temperature (that is, for a thermal system), there is one impor-tant difference: in thermal systems every particle vibrates around its equili brium
position with an amplitude that is finite and proportional to the square-root of the temperature. An arrangement of particles in which such a vibrational motion would allow any one of the particles to overcome the local barrier (saddle point) in the potential energy is therefore unstable. In other words, every particle in the thermal system must have a typical distance from its closest local saddle point that is finite and not too small. In contrast, a granular system at rest is at zero temperature and so there is no need for a particle to have a minimum distance from its nearest saddle point. As a consequence, even the slightest perturbation of a granular system will destabilize some of the particles, in contrast to thermal systems. The presence of friction does not substantially alter this conclusion.
The cyclic shear makes the particles explore their neighbourhood, and if they encounter a mechanical instability then they relax to a new local minimum. The details of how this exploration occurs depends on the properties of the particles in that, for example, for perfectly smooth particles (with or without dynamic friction) the exploration is completely deterministic, thus allowing for the existence of limit cycles17. In contrast, we expect that particles with some roughness will explore their neighbourhood in a way that is similar to a random walk, because the roughness induces a weak stochastic component to their motion. Static friction also introduces random noise into the trajectories of the particles: every time the system arrives at the maximal strain of the oscillation, the motion stops and the presence of static friction means that the return path of the particles will not necessarily coincide with the trajectories that lead to the turning point. Because our particles do have some roughness and static friction, we expect that for weak γ their motion is diffusive, with diffusion constant D, and hence the TMSD depends linearly on time. This expectation is in agreement with what we find (Fig. 1b) and with the Gaussian shape of the PDF for small distances (Fig. 2b). This Gaussian distribution can be written as
∆ ∆=
π
−
P d tt
dt
( , ) 12 ( )
exp2 ( )g 2
2
2
where Δ(t) = ⟨ d2⟩ 1/2 = (2Dt)1/2 is the square-root of the TMSD at time t. (For the sake of simplifying the notation we consider here only one-dimensional motion characterized by a displacement d.) This random walk of the particles means that after some time certain particles will reach a saddle point in the energy land-scape and so will relax, flipping over or falling down into a ‘hole’, thus representing an irreversible relaxation event. We define ε as the distance that a given particle has to move from its starting position to reach this saddle point. We argued above that for a granular system a particle in a stable packing can be arbitrarily close to a saddle point, which implies that the probability distribution of this distance, W(ε), is not small even for small ε. As a first approximation we assume that W(ε) is essentially constant for ε ≤ σ, where σ is the size of the particles, and becomes zero for ε = O(σ) (that is, there are no cavities larger than the size of the particles). Because of the normalization of W(ε) we can approximate it by 1/σ.
Owing to the diffusive motion of the particles, ε(t) is a random variable that represents the same type of diffusive dynamics as the position of the particles: [ε(t) − ε(0)]2 = 2Dt. Destabilization of the particle means that ε(t) has become zero; that is, after a time t a particle for which ε(0) ≤ (2Dt)1/2 is likely to have undergone an irreversible relaxation event. Let h0 be the typical distance that a particle moves because of an irreversible relaxation event (h0 < σ because we assume that there are no cavities and typically h0 ≈ σ/2). Let us assume that the probability that a particle has undergone an irreversible relaxation event is small,
σ/ =/ �Dt p(2 ) 11 20 . The probability that a particle has undergone k such events
(k = 1, 2, 3, …) in time t is therefore ∝ − /p k pexp( )k0 0 , and in doing so it has moved
d = O(h0k). We therefore have
σ σ∆
≈ − /
≈
−
=
−
/
P d t k p t
dh Dt
dh t
( , ) exp[ ( )]
exp(2 )
exp( )
(5)IRE 0
01 2 0
The probability that a particle has moved by a distance d in time t is given by the sum of these two processes:
∆σ∆
= +
≈
−
+
−
P d t P d t P d t
dt
dh t
( , ) ( , ) ( , )
exp2 ( )
exp( )
T g IRE
2
20
(For the sake of simplicity, we assume here that the two processes have the same weight.) Because σ/h0 ≥ 1, the first term on the right hand side will dominate the second for small d (with respect to Δ(t)) and the PDF will essentially be Gaussian. For d ≈ Δ(t) the two terms on the right hand side become comparable. Consequently, a plot of PT as a function of d/Δ(t) will show a crossover from Gaussian behaviour to an exponential dependence at d/Δ ≈ h0/σ = O(1), in agree-ment with our finding (Fig. 2b).
We now compare the slopes of the two distributions at d = Δ (or rather of the logarithm of the distributions). We find
∆=−P d
dddlog[ ( )]
dg
2
and
σ∆
=−P d
d hdlog[ ( )]
dIRE
0
At the crossing point d = Δ, the two distributions have the same slope − 1/Δ because σ/h0 ≈ 1, which means that there is no kink in the total distribution. As a consequence the total PDF is well approximated by a Gumbel law because this functional form has also a Gaussian dependence at short distances and an exponential tail at large distances. Equation (5) shows that the parameter λ of the Gumbel distribution (see Fig. 2 and equation (2)) is given by h0Δ(t)/σ and so has a simple geometrical interpretation. Other functional forms can be used to describe these two limiting behaviours, such as the q-Gaussian discussed below. However, because of its simplicity and the fact that it has only one free parameter, here we give preference to the Gumbel law.
The result regarding the presence of two relaxation mechanisms (diffusion and jump) holds for all values of time: once the PDF has been normalized by Δ(t) its shape is independent of time, in agreement with our results shown in Fig. 2b. Furthermore, we recall that the shape of the PDF is independent of γ (Fig. 2b). This result can be rationalized by noticing that our argument for the observation of the Gumbel law depends on the presence of diffusive dynamics on short length scales that is independent of γ, provided that the distribution is normalized by Δ(t). Thus, even if the dynamics is not perfectly diffusive, owing to, for example, the presence of memory effects, the final result will not change substantially because of symme-try arguments, and the central part of the PDF can always be well approximated by a parabola, which for small arguments is similar to a Gaussian.
Finally, we note that the existence of the Gumbel law is directly connected to the fact that the probability of a given particle undergoing an irreversible relaxation event is (statistically) related to its displacement since the last time it made such a jump, which in turn is proportional to the square-root of the TMSD. This is in contrast to the case of thermal systems, for which the timescale on which hopping occurs is independent of time and hence no Gumbel law is observed34.Comparison of Gumbel and q-Gaussian fitting. In previous studies it has been found that Gs(d, t) can be well fitted by a ‘q-Gaussian’ with a parameter q that depends on γ. This q-Gaussian distribution is given by35,36
Γ
Γ=
− −
π − −
− −
−
−
/ −
G dq q
A q qq d
A( )
( 1) (( 1) )((3 )[2( 1)] )
1 (1 )q
q1
1
2 1 (1 )
where A and q are fit parameters and Γ(x) is the Gamma function. We fitted our data with this functional form and found that the resulting fit does not give a very satisfactory fit of the data in the tails (see Extended Data Fig. 6a). However, close to the peak the q-Gaussian does give a better description of the data (see Extended Data Fig. 6b). In Extended Data Fig. 6c, d we show the difference between the data fitted with the two functional forms. Although the q-Gaussian provides a better description of the data overall, it has two fit parameters whereas the Gumbel distribution has only one. In addition, the choice of the Gumbel distribution can be rationalized by the arguments in the preceding section, whereas for granular materials the q-Gaussian law does not have a solid foundation. In view of these problems it is therefore difficult to tell which functional form is more appropriate. Although the difference between our results and previous results11,15 might be related to the fact that we have a three-dimensional system whereas previous investigations considered two-dimensional ones, the mechanism leading to the Gumbel distribution is general and so might also be present in two-dimensional systems.Data availability. The data that support the findings of this study are available from the corresponding authors on reasonable request.
30. Donev, A. et al. Improving the density of jammed disordered packings using ellipsoids. Science 303, 990–993 (2004).
31. Titow, M. PVC Technology 1189–1191 (Springer, 1984).32. Johnson, K. L. & Johnson, K. L. Contact Mechanics 92–95 (Cambridge Univ.
Press, 1987).33. Hansen, J. P. & McDonald, I. R. Theory of Simple Liquids (Elsevier, 1990).34. Chaudhuri, P., Berthier, L. & Kob, W. Universal nature of particle displacements
close to glass and jamming transitions. Phys. Rev. Lett. 99, 060604 (2007).35. Umarov, S., Tsallis, C. & Steinberg, S. On a q-central limit theorem consistent
with nonextensive statistical mechanics. Milan J. Math. 76, 307–328 (2008).36. Picoli, S., Mendes, R. S., Malacarne, L. C. & Santos, R. P. B. q-Distributions in
complex systems: a brief review. Braz. J. Phys. 39, 468–474 (2009).
Extended Data Figure 1 | Trajectories of 20 particles in the central region of the sample. The strain amplitude is γ = 0.26 and the length of the trajectory is 615 cycles, the total length of the experiment for this γ. Time is represented by the colour scale. The short axis of the particles is 2b.
Extended Data Figure 3 | Evolution of the volume fraction Φ of the system during the cyclic shear measurements at different γ. For γ = 0.10 we have two curves that stem from two completely independent measurements. The fact that the corresponding packing fractions are compatible shows that sample-to-sample fluctuations are small.
Extended Data Figure 4 | The normalized local number density ρi/⟨ρi⟩ in the system as a function of i, for i ∈ {x, y, z}, for different γ. The vertical dashed lines denote the boundaries of the central region that we chose for the subsequent analysis.
Extended Data Figure 5 | Time dependence of the self (Fs(q, t); large open symbols) and collective (F(q, t); small closed symbols) intermediate scattering functions (equation (4)), for different γ.
a, q = 3.49b−1. b, q = 4.46b−1. The correlators decay without any sign of a two-step relaxation, thus showing that in this system the particles are not caged.
Extended Data Figure 6 | PDF in the y direction for γ = 0.26. a, Comparison of the fits with the Gumbel distribution (black line) and the q-Gaussian distribution (pink line) for the whole accessible range in the displacement. b, An enlarged view of the area indicated by the blue
rectangle in a. c, d, The ratio between the logarithm of the PDF for the data (ln(P(dy))) and for the Gumbel distribution (ln(Pg); c), and for the data and the q-Gaussian distribution (ln(Pq); d).
