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Granular materials flow like complex fluids

Abstract

Granular materials such as sand, powders and foams are ubiquitous in daily life and in industrial and geotechnical applications1,2,3,4. 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.

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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.
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.
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.
Figure 4: Memory effect as a function of particle displacement and time.

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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.

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Authors

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.

Corresponding authors

Correspondence to Walter Kob or Yujie Wang.

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The authors declare no competing financial interests.

Additional information

Reviewer Information Nature thanks R. Behringer and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

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 2 The static structure factor S(q) (equation (3)) of the system in steady state, for different γ.

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).

Extended Data Table 1 Experimental protocol used to prepare the system and to measure its properties

Supplementary information

The dynamics of the particles during the cycling experiment (γ=0.26): Top view.

The video shows a view of the system from the top. It shows a horizontal cut through the middle of the sample and only the particles in the lower half of the central region of the box are displayed. The value of γ is 0.26 and the video covers the 615 cycles of the measurement with the CT scanner. Note that on the time scale considered, most of the particles move less than their long axis (see Fig. 1b of the main text) (MOV 6585 kb)

The dynamics of the particles during the cycling experiment (γ=0.26): Side view.

The video shows a side view of the same system of Supplementary Video 1. We made a vertical cut through the middle of the sample and show only the particles in the sector with large x. Note that the system shows no evident signature for convective motion. (MOV 3672 kb)

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Kou, B., Cao, Y., Li, J. et al. Granular materials flow like complex fluids. Nature 551, 360–363 (2017). https://doi.org/10.1038/nature24062

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