On the reversibility of granular rotations and translations

Transcription

1 On the reversibility of granular rotations and translations Anton Peshkov, Michelle Girvan, Derek C. Richardson, 3 and Wolfgang Losert IREAP, University of Maryland, College Park, Maryland, USA Departments of Physics, IPST and IREAP, University of Maryland, College Park, Maryland, USA 3 Department of Astronomy, University of Maryland, College Park, Maryland, USA The experimental setup, sketched in figure, is composed of a container with transparent walls filled with transparent acrylic beads. The container is. cm wide (y direction) and approximately cm long (x direction). The side walls and top wall are of transparent polycarbonate. The back aluminum wall is displaced by a motor to compress the whole system. The amplitude of the compression was varied between % and. % of the container length []. At this compression amplitude, our experimental system is still below the conarxiv:9v [cond-mat.soft] 30 Oct 0 We analyze reversibility of both displacements and rotations of spherical grains in threedimensional compression experiments. Using transparent acrylic beads with cylindrical holes and index matching techniques, we are not only capable of tracking displacements but also, for the first time, analyze reversibility of rotations. We observe that for moderate compression amplitudes, up to the bead diameter, the translational displacements of the beads after each cycle become mostly reversible after an initial transient. By contrast, granular rotations are largely irreversible. We find a weak correlation between translational and rotational displacements, indicating that rotational reversibility depends on more subtle changes in contact distributions and contact forces between grains compared with displacement reversibility. I. INTRODUCTION Flows of granular matter are an important subject of study in many fields, including geology (where they are relevant for avalanches and earthquakes), engineering (where they play a role in industrial processes and construction), and astronomy (where they affects the formation of asteroids and planets). In these contexts, it is particularly important to understand how the granular material transitions from a jammed state to a state of flow with irreversible rearrangements. A common driver of granular flow are cyclic perturbations, which can caused by a wide range of driving forces, such as vibrating apparatus (construction), periodic loads (roads and rails), earthquakes, cyclic gravitational fields (tidal forces), or thermal expansion and contraction due to day/night temperature variations (on planets and asteroids). Beside the question of flows, the cycling compression or shearing of the granular system can change the properties of the packing, leading to compaction[] and ultimately crystallization in some cases []. Most past numerical and experimental work in this area has concentrated on the translational motion of particles and the associated force fields. However, granular systems are frictional and often aspherical, which implies the presence of rotations of the grains. Rotations and torques can play an important role in mesoscopic granular motions, as has been shown in two-dimensional investigations[3]. Thus bulk granular mechanics is affected by both force networks and torques, though rotations and torques are often ignored, with the notable exception of the Cosserat continuum elastic models []. Furthermore, rotations are likely more important in real three-dimensional systems than in two-dimensional model systems. In a three-dimensional frictional system (i.e., a system where rotations matter), the number of constraints increases by 3 to a total of, as compared to a non frictional system, while it only increases from to 3 in two dimensions. Thus frictional, rotating systems in three dimensions could become isostatic with only four contacts, long before the jamming point at contacts per particle[]. A typical frictional packing is thus overconstrained due to the frictional forces. The distribution of forces, between normal and frictional, is then dependent on the history of contacts[], i.e., the method of preparation of the system. Periodic compression of a frictional system will alter the contact distribution and could lead to a different state of the system, called a random loose packing, in contrast to the random close packing with six contacts per particle that we would find for a frictionless system. Thus understanding rotations is important to elucidate the nature of different jammed states of frictional spherical particles as well as of aspherical particles. Despite the importance of rotations for granular systems, only a handful of experimental investigations have studied them, especially in three dimensions[ ]. In this study of reversibility we build on our first experimental measurements of particle-scale rotations in a three-dimensional granular flow [9], and systematically measure both rotations and translations together under cyclic forcing. We focus on small enough driving amplitudes that the translational motion of the system is reversible. We had previously shown that in this small forcing regime, convective flows and segregation two hallmarks of bulk granular flow are also suppressed[0]. II. EXPERIMENTAL SETUP

2 vection regime, which will invariably appear at higher amplitudes[]. The compression speed was mm s for all amplitudes, i.e. a constant shear rate. We verified that this speed is slow enough to expect granular rearrangements that resemble dry systems[3]. All the experiments contained between 00 and 000 cycles of compression, with between and three-dimensional images captured during each cycle. All figures in the manuscript are presented for the case of. % compression, unless noted otherwise. The container is filled with 0,000 transparent acrylic beads of radius R =. mm, forming a height (z direction) of approximately twenty layers. Each bead has a mm diameter cylindrical cavity running through it, allowing us to detect the orientation of the bead. A. kg transparent acrylic weight is placed on the beads to maintain a constant pressure. The beads and the weight are submerged in an index-matched solution of Triton X-00, which additionally includes a fluorescent Nile blue dye as well as % of hydrochloric acid (3 % concentration) for effective dilution of the dye[]. The whole system is illuminated with two 0 nm[] scanning sheet lasers from both sides of the container. We should note that a single hole in the grains does not allow us to detect rotations when the orientation vector is along the hole of the bead. Only rotation components perpendicular to the hole can be resolved. Thus, our current experimental setup captures only a part of the rotation amplitude. However, in this work we focus on the statistical analysis of amplitudes of rotations, not taking into account such information as the axis of rotation or its correlation with other quantities. Typically four full scans of the system are performed per compression cycle, as shown in figure.b), i.e., a scan before compression (time A), a scan at half-compression (B), a scan at full compression (C), a scan at halfdecompression (D) and a scan at full decompression (E). The first and last 0 cycles are recorded with scans per cycle. Positions and rotations are then extracted and tracked using in-house developed algorithms [9]. To directly compare the rotations and translations in the same units, we multiply all rotation angles by the radius of the bead. In the case of non-slip rolling of a bead on a flat surface, this will make the values of rotational and translation displacements identical. We have estimated the standard error on particle position detection to be 0 µm and the standard error on angle detection to be 3 rad ( ) or µm. For the statistical analysis, we omit all particles located within one particle diameter of the border, as our algorithm fails to correctly detect particle orientation in this region. We also omit particles in the bottom four layers, where optical aberrations lead to larger detection errors. Finally, for the statistical analysis, we systematically omit particles whose motion during the compression cycle is less than the standard error mentioned previously. III. TRANSLATIONAL AND ROTATIONAL DISPLACEMENTS We study both the mean displacements and rotations of beads as compared with their initial position in the experiment as well as compared with the beginning of the cycle. We denote by x T i the position of particle i at the temporal position T (one of A E) in the cycle and by ˆq qt i its orientation expressed as a unit vector. We introduce the displacements of particles at time T : the translational µ T i = x T i x A i and the rotational θ T i = sin ( T ˆq i ˆq i A ), as well as their mean values: µ T = N N i= µt i and θ T = N N i= θt i. Figure d) shows a snapshot of typical translational displacements of the particles at time C. The gradient angle matches the symmetry of the system, with a compressing right wall and moveable top wall kept at constant pressure. As compared with translations, the rotations do not present any apparent shear zones, with a larger concentration of rotations near the bottom of the cell (see supplemental figures.c-d). This suggests that there is no direct correlation between translations and rotations, the latter being a more complicated phenomenon including effects from both translations and contacts between particles. This is corroborated by statistical analysis as presented below and in supplemental part IV. We now turn to the study of the reversibility of granular displacements. Note that due to the nature of our experiments, a small but measurable compaction (following a power law) is found as expected in our system. While the particle positions are not perfectly reversible, we focus on small amplitudes where particles retain their neighbors, and where (as shown in [0]) convection and segregation are suppressed. In this regime, where the position of most particles with respect to their neihbors is reversible, we investigate rotations, as well as their evolution with cycle number. To examine the irreversibility of the granular motion we can study the translational and rotational displacements of particles at the end of the cycle: µ E i and θi E. Looking at the snapshots of these at the end of the cycle (supplemental figures.b and.d), translations are reversible as expected, while rotations are mostly irreversible. This can best be seen in scatter plots of translations and rotations (figure ) comparing the displacements of each particle at time C to the displacements at time E. For the translations, almost all of the displacement is reverted at the end of the cycle. However this is not the case for rotations, in which a lot of particles do not return to their initial orientation.

3 3 (c) Moving compression wall z x (d) y Z Scanning laser sheet Wall displacement C B D A E X Time Figure. Schematics of the setup. Schematics of the positions of the wall at which 3D scans are performed. Note that position E of the current cycle correspond to position A of the next cycle. (c) One example image of one hole beads - typically a sequence of 00 images is captured in each 3D scan. A whole scanning sequence can be seen in supplemental movies. (d) Displacement of the beads at full compression of the cell. Projection into the x-z plane. The color indicates the amplitude of displacement µe i as compared with the beginning of the cycle..0 E x -x A C A x -x E A R(q -q ) C A R(q -q ) Figure. Scatter plot of translations between the middle of the cycle and the end of cycle. Same for rotations. The color indicates the beads density at the particular location. All units in mm. To obtain a better statistical understanding of the phenomena, we compare the mean value of end-of-cycle displacement of the particles to that of their mid-cycle displacement. Figure 3 shows a comparison between the expected end-of-cycle displacements µe and θe for given values of mid-cycle displacements µc and θc. These are well characterized by power-law behaviors with approximate exponents of 0.3 for translations and 0.3 for rotations. As seen in the figure, these exponents do not depend on the amplitude of compression of our system, indicating a universal behavior for the amplitudes we investigated. Both of these exponents are smaller than, which implies that the relative reversibility improves with an increase of particle translation or rotation. We introe duce relative irreversibility parameters Iµ,i = µi /µci for E translations and Iθ,i = θi /θic for rations which plotted against the mid-cycle displacements µc and θc give the expected mid-cycle exponents of 0. and 0. (see supplemental figure ). Returning to the question of correlations between the rotational and translational displacements, we plot the expected middle or end-of-cycle rotational displacements θc,e as a function of the translational displacements µc,e or vice-versa (see supplemental part IV). These plots show very weak correlations as compared with the correlations between mid-cycle and end-of-cycle displacements. Note that this result is in contrast to the case of ellipsoidal particles, where a strong correlation between rotations and translations was experimentally found[]. The difference can be explained by the fact that for ellipsoidal particles rotations imply a change in the volume occupied by the particles and thus can simply couple to translations of surrounding particles.

4 <µ E > 9 7 %.0%.%.0%.% R<θ E > 9 %.0%.%.0%.% <µ C > R<θ C > Figure 3. The expected end of cycle translational and rotational displacements as compared with the mid-cycle displacements of these grains. The solid lines illustrate power laws with exponents of 0.3 and 0.3. All units in mm. Beyond the reversibility to its initial position and orientation, we investigate the reversibility of the whole particle trajectory, both in terms of translation and rotation. We plot the displacements of particles during a cycle of compression averaged over similar values of midcycle displacements (Fig. ). These results are based on the actual motion of the moving wall, corrected for slight backlash present in our experimental system. Figure highlights the difference between the behavior of granular translations and rotations. Translations are very reversible, with beads following much of the same path during compression and dilation, especially at high values of the wall displacement. On the other hand, the rotation trajectories are much less reversible. In this case, the compression path, which begins at maximum wall displacement and goes back to zero wall displacement, diverges almost immediately from the dilation path. In other words, in contrast to the translations, the forward and backward paths for rotations differ significantly even close to point of wall reversal (maximum wall displacement). The reversibility of both translations and rotations improves after many cycles as compared with the beginning of the experiment (see supplemental part III), suggesting that, after an initial transient period, the system self-organizes into a more reversible configuration. IV. CONTACT DYNAMICS For particles to revert back to their original arrangement relative to their neighbors without reversal of rotations requires some change in frictional contacts. To this end, we define particles that are within a cutoff distance r c of each other as being likely in contact. To capture all likely contacts we chose r c as the distance within which particles have on average contacts per particle, significantly above the isostatic limit at the jamming point for frictionless particles[]. We have verified that varying r c such that the mean number of contacts per particles is between, the isostatic limit for frictional particles, and, the jamming point, does not change the quantitative results presented here. Figure shows the mean orientation of created and broken links. Created (broken) links are ones that exist at step t(t ) but not at t (t). Mean orientations are plotted in the x-y plane, as compared with the direction of compression, for the compression and decompression parts of the cycle. We can see a clear asymmetry between the phase of compression, where created links are mostly oriented along the axis of compression and broken links perpendicular to it and the phase of decompression, where this tendency is inverted. Thus the contact networks during compression and dilation are distinct. This is a logical result. When we compress the system, links will mostly be created along the axis of compression, and beads can only escape to the sides. During the dilation, the creation of links along the dilation axis will be less likely. V. SUMMARY AND CONCLUSIONS We conducted a novel experiment allowing us to track translations and rotations of particles in three dimensions during cycles of uniaxial compression. We found that the overall reversibility of translations is much higher than that of rotations. This is explained by the absence of direct correlations between the rotations and translations: the reversible motion of particles trapped in cages[] does not imply the reversibility of rotations. However, given the available results of a presence of correlations between translations and rotations for ellipsoid particles[], a larger correlation between translations and rotations

5 Mean translational displacement Probability 0 Created A C C E Broken Mean rotational displacement Wall displacement Links orientation (radians) Figure. The mean orientation angle (relative to the axis of compression) in the x-y plane of created and broken links during the compression and decompression parts of the cycle. work. We are grateful to Zackery Benson, Jacob Prinz, Charlotte Slaughter and Dara Storer for their help with the experiments. This work was supported by National Science Foundation grant DMR Wall displacement 3. Figure. Mean translational displacement µ T trajectories and rotational displacement R θ T trajectories, averaged over particles with comparable mid-cycle displacements. All units in mm. The mid-cycle displacements go from µ C = 0 mm to 3 mm for translations and from Rθ C = 0 mm to 3 mm for rotations. may also be observed for larger friction and different frictional properties of the particles. Extensive numerical investigations would be needed to probe the vast array of possible friction effects. The contact network analysis highlights a likely cause of the rotational irreversibility. Even though particles retain their neighbors, their contact-point dynamics follows a more hysteretic path and rotational motion is not reversible. It is highly probable that the irreversibility of rotations is also due to a parameter which we cannot track in our experiments: the force distributions. To elucidate this hypothesis, further work is needed to track the rotations and the forces on the particles at the same time, either through simulations or different experimental methods[7]. An important aspect of granular flow is the vector of rotation compared to the direction of shear/compression. It has been predicted[] that particles will rotate perpendicular to the shearing motion. However, to verify this hypothesis experimentally, we need to access all three degrees of rotations, which will be the basis of our future [] M. Hecke, Journal of Physics: Condensed Matter, 0330 (00). [] A. Panaitescu, K. Reddy, and A. Kudrolli, Physical Review Letters 0, 000 (0). [3] A. Tordesillas, D. Walker, and Q. Lin, Physical Review E, 030 (00). [] L. Srinivasa Mohan, K. Kesava Rao, and R. Nott Prabhu, Journal of Fluid Mechanics 7, 377 (00). [] A. Kasahara and H. Nakanishi, Physical Review E 70, 0309 (00). [] F. Guillard, B. Marks, and I. Einav, Scientific Reports 7, (07). [7] B. Kou, Y. Cao, J. Li, C. Xia, Z. Li, H. Dong, A. Zhang, J. Zhang, W. Kob, and Y. Wang, Nature, 30 (07). [] B. Kou, Y. Cao, J. Li, C. Xia, Z. Li, H. Dong, A. Zhang, J. Zhang, W. Kob, and Y. Wang, PRL, 000 (0). [9] M. Harrington, M. Lin, K. Nordstrom, and W. Losert, Granular Matter, (0). [0] M. Harrington, J. Weijs, and W. Losert, Physical Review Letters, 0700 (03). [] Note that we can not go beyond 3% with our setup, as it corresponds to the diameter of a single bead. Beyond this limit, the beads will jam between the weight placed on the beads and the moving wall. [] J. Royer and P. Chaikin, Proc Natl Acad Sci USA, 9 LP (0). [3] J. Dijksman, F. Rietz, K. Lőrincz, M. van Hecke, and W. Losert, Review of Scientific Instruments Review of Scientific Instruments 3, 030 (0), doi: 03/.3773 DO - 03/.3773.

6 [] Up to % of water in volume can be added to the final solution to tune the index matching between the solution and the beads. [] 0 nm approximately corresponds to the maximum excitation fluorescence frequency of nyle blue. [] M. Hanifpour, N. Francois, V. Robins, A. Kingston, S. Vaez Allaei, and M. Saadatfar, Physical Review E 9, 00 (0). [7] R. Hurley, S. Hall, J. Andrade, and J. Wright, PRL 7, 0900 (0). [] T. Halsey, Physical Review E 0, 0303 (009).

7 Supplemental material for On the reversibility of granular rotations and translations Anton Peshkov, Michelle Girvan, Derek C. Richardson, 3 and Wolfgang Losert IREAP, University of Maryland, College Park, Maryland, USA Departments of Physics, IPST and IREAP, arxiv:9v [cond-mat.soft] 30 Oct 0 University of Maryland, College Park, Maryland, USA 3 Department of Astronomy, University of Maryland, College Park, Maryland, USA

8 I. SNAPSHOTS OF DISPLACEMENTS We can compare the snapshots of the amplitude of displacement between the fully compressed state C and the fully decompressed state E. We can see in figure that while most particles go back to their original position under translation, a lot do not under rotation. We also note the absence of any apparent shear bands for the rotations. (c) (d) z x Figure. Snapshots of translational (a and b) and rotational displacements (c and d) at times C (a and c) and E (b and d). Videos of full compression cycles are available in the supplemental material. II. IRREVERSIBILITY OF DISPLACEMENTS In addition to studying the translational and rotational displacements, we can introduce the relative irreversibility parameters I µ,i = µe i /µ C i for translations and I θ,i = θe i /θi C for rotations. As seen in figure, both of these show decaying power laws as a function of the mid-cycle mobility, with exponents of 0. and 0. for translations and rotations respectively.

9 <I µ > %.0%.%.0%.% 3 R<I θ > %.0%.%.0%.% <µ C > R<θ C > Figure. Expected relative irreversibility parameters I µ and I θ as a function of the mid-cycle mobility µ C and θ C. III. CYCLES OF DISPLACEMENT In addition to plotting the cycles of translational and rotational displacements at the end of the experiments we can plot the same quantities at the beginning of the experiment to compare the change in the irreversibility. Mean translational displacement Mean rotational displacement Wall displacement Wall displacement 3. Figure 3. Bead translational and rotational displacement cycles averaged over identical mean mobilities for the first five cycles. Compare with figure in the article where the same quantities are plotted for the last ten cycles of the experiment. The mid-cycle mobility goes from 0 mm to 3 mm for translations and from 0 mm to 3 mm for rotations. All units in mm. 3

10 IV. CORRELATIONS BETWEEN TRANSLATIONS AND ROTATIONS To investigate the presence of correlations between rotations and translations, we plot averaged values of one against the other in figure. While small variations are present they are negligible compared to the decades of variation of mobilities seen in figure 3 of the main article R<θ C > <µ C > (c) <µ C > (d) R<θ C > 0. R<θ E > 0.3 <µ E > 0 0. <µ E > R<θ E > Figure. Averaged value of rotational displacement binned over values of translational displacement at state C. Averaged value of translational displacement binned over values of rotational displacement at state C. (c) and (d) same as and at state E. All units in mm.