View Online

HIGHLIGHT

www.rsc.org/softmatter | Soft Matter

Recent advances in DEM simulations of grains in a rotating drum Patrick Richarda and Nicolas Taberletb

Downloaded by Bibliotheque de L’Universite de Rennes I on 05 July 2011 Published on 14 May 2008 on http://pubs.rsc.org | doi:10.1039/B717129C

DOI: 10.1039/b717129c

The discrete elements method (DEM) has been widely used in the past decade to study a wide variety of granular systems. The use of numerical simulations constitutes an interesting alternative to the experiment as they can shed new light on a phenomenon as they can overcome experimental obstacles. A lot of granular phenomena can be studied in 2D or with a limited number of grains but the peculiar phenomenon of axial segregation (or banding) is 3-dimensional by nature and requires a large number of grains. Only very recently has it been made possible to simulate 3D systems on a large scale. This highlight reviews recent work on this topic and attempts to show what knowledge is gained from DEM numerical simulations. The perspectives on the future benefit of this method as well as the challenges it faces are discussed.

1 Introduction Despite their seeming simplicity, granular media exhibit many striking properties. One property of forced granular matter is the tendency of mixtures of different types of material to unmix (e.g. during avalanching on a static pile1,2 or in a container under vertical vibrations).3 Among these phenomena, axial segregation4 or banding is one of the least understood. Due to the fundamental interest as well as the numerous industrial applications,5 a great deal of both experimental6–10 and theoretical6,11–13 work has a Institut de Physique de Rennes, UMR CNRS 6251, Universite´ de Rennes I, Campus de Beaulieu, F-35042 Rennes, France b Universite´ de Lyon, E´cole Normale Supe´rieure, Laboratoire de Physique, 46 alle´e d’Italie, F-69007 Lyon, France

been devoted to this type of segregation. It has been suggested that banding occurs when the frictional properties of the grains are different. In particular there can exist a noticeable difference in the slopes of the bands of small or large grains as the drum is rotating. Yet recent work has invalidated this explanation10 and as of today the understanding of this phenomenon still remains matter of debate. Experimentally, due to the opacity of the grains, it is difficult to study individual trajectories or to measure local properties. A few techniques have been developed but they require long exposure times and heavy equipment (MRI,8 X-ray tomography)14 or fail to track individual grains (light-projection).15 In the past two decades discrete element methods (DEM) simulations have been used to numerically study granular media.16 DEM

Dr Patrick Richard is a lecturer (Maıˆtre de Confe´rences) at the Universite´ Rennes I. He is a member of the Institut de Physique de Rennes (IPR). His current research focuses on gravity flows of granular material, on avalanche precursors and on the dynamics of compaction of granular packing under vibration.

Patrick Richard

This journal is ª The Royal Society of Chemistry 2008

Nicolas Taberlet

simulations consider individual grains (often spherical) which interact through frictional collision and/or enduring contacts. The grains are in general free to rotate and the collisions inelastic. The position and displacement of each grain is obtained by integrating the equations of motion.17 There are three major advantages of the simulations over the experiment. First, the simulation allows one to measure any physical quantity (velocity field, rotation of individual grains, granular temperature, forces acting between grains, stress tensor, local packing fraction.) without perturbing the system. In particular the stresses within a static of flowing granular medium can be probed. Second, all parameters can be varied independently: size and mass distribution, friction properties, coefficient of elasticity, rigidity (i.e. Young modulus), etc.

Dr Nicolas Taberlet is a lecturer (Maıˆtre de Confe´rences) at the Ecole Normale Supe´rieure (ENS) of Lyon and is a member of the Laboratoire de Physique, and a visiting scientist at the DAMTP, University of Cambridge. His current research topics include the formation of washboard road on granular surfaces and the rheology of complex fluids, studied experimentally through ultrasound velocimetry. Soft Matter, 2008, 4, 1345–1348 | 1345

Downloaded by Bibliotheque de L’Universite de Rennes I on 05 July 2011 Published on 14 May 2008 on http://pubs.rsc.org | doi:10.1039/B717129C

View Online

One can for instance block the rotation of the grains or use perfectly elastic grains, which is impossible experimentally. Last but not least, DEM simulations are obviously free of undesirable effects such as humidity, static electricity or misalignment. The very fact that a phenomenon can be reproduced in DEM simulations constitutes an interesting result. The appeal of numerical simulations is obvious but until recently very few numerical studies dealing with a large number of grains over a long period of time have been reported. This is mainly due to the large amount of CPU time needed to carry out such calculations. Nowadays, recent computers allow such simulations in a reasonable time (typically a few weeks on one processor). A powerful tool is now available to carry out extensive large scale studies of axial segregation of a binary mixture of grains in a rotating drum. This highlight reviews a group of published studies and presents some perspectives on the future benefit of this method.

2 DEM numerical simulations of axial segregation An initial exploratory study was performed by Shoichi18 with a very limited number of grains (1000) and only premises of axial segregation were observed. Simulating axial segregation has only been made possible recently since such simulations require heavy computation time both because a large number of grains is needed (typically 105 grains) and long times are needed for the pattern to develop (typically 100 rotations, typically corresponding to 108 time steps). A large scale numerical study of axial segregation was performed by Rapaport19 who reports remarkable results, but the interactions between grains were derived from the Lennard-Jones potential, which allows to speed up the program but is not well-suited for granular material. The first study of axial segregation using an adequate soft-sphere DEM was performed by Taberlet et al.20 and followed by Rapaport.21,22 The DEM simulations employed in ref. 20 deal with spherical soft spheres colliding with one another. The grains are frictional and free to rotate. The collisions are inelastic and a solid-friction model 1346 | Soft Matter, 2008, 4, 1345–1348

was implemented following the techniques described in ref. 16,17. The drum is filled to approximately 30% with an initially well-mixed binary mixture of small and large grains (the large ones being twice as big in diameter), and the volumes of the two species are equal. The end-walls of the drum are capped. Within the first five rotations, the small beads disappear from the free surface, forming a radially segregated centre core of semi-cylindrical shape. After typically one hundred rotations, a striped pattern appears, forming the well-known bands of axial segregation.

Fig. 1 shows a snapshot of the final steady state for a typical simulation. It contains roughly 50 000 grains and was run for over 200 full rotations of the drum (corresponding to 108 time steps) and took 6 weeks of CPU time. The drum length is 200d and its radius 25d, where d is the diameter of the large grains. All the friction coefficients used for this run are equal (mab ¼ 0.5). This means that the two species of grain made of identical material differ in size but not in frictional properties or restitution coefficient. Fig. 2a is a space–time diagram showing the evolution of this system. The

Fig. 1 Snapshot of the axial segregation pattern obtained with DEM simulations using 50 000 grains. The small grains are colored red, the large ones (twice the diameter) yellow. The volume concentrations of the two species are equal. Initially the material is well mixed but after typically 100 rotations of the drum, alternating bands of small and large grains spontaneously appear.

Fig. 2 a) Typical space–time plot showing regions of high concentration of small grains. b) Corresponding snapshots of the drum during the coarsening. Red ¼ small grains, yellow ¼ large grains.

This journal is ª The Royal Society of Chemistry 2008

Downloaded by Bibliotheque de L’Universite de Rennes I on 05 July 2011 Published on 14 May 2008 on http://pubs.rsc.org | doi:10.1039/B717129C

View Online

pixel is black if the concentration of small beads is higher than its average value. The space–time plot shows rich dynamics with bands appearing shortly after the rotation starts, with bands disappearing and merging with one another. The system seemingly reaches a steady state consisting of bands somewhat regularly spaced. Fig. 2b shows snapshots of the drum at different times during the coarsening. The birth and merging of bands can be clearly seen. The last snapshot corresponds to the final steady state and shows 5 bands of large grains (L-bands), and 4 of small grains (S-bands). A plot of the concentration of small beads shows that in this final state, the bands are pure, i.e. the radial core has disappeared. This timeevolution is very similar to that observed experimentally (see ref. 23 for a comparison). This clearly demonstrates that axial segregation can be obtained with two species of ideally spherical grains, rotated at a constant speed in an ideally horizontal cylinder, differing only in their radius (the densities, friction coefficients and inelasticity being exactly equal). Since the micromechanical properties of grains may depend on their size, it is very difficult to experimentally study segregation by size only. A difference in frictional (or collisional) properties can certainly affect the dynamics of segregation. Yet, our

simulations show that such a difference is not necessary for axial segregation. The DEM simulations gave other interesting results on axial segregation.20 Although a difference in frictional properties between the two species is not necessary to observe axial segregation, it leads to the onset of oscillations in the band position or width. This suggests that the oscillations observed in experiments may originate from a difference in the frictional properties between species of grains. The mechanisms of band merging were also elucidated by tracking the positions of individual grains during a merging event. They consist of two complementary events: on the one hand, the disappearance of a band of small grains (S-band): its grains being shared between the two neighboring S-bands through the radial core, on the other hand, the collapse of two L-bands: the two L-bands surrounding the decaying S-band mix together and are oblivious to the other L-bands. Finally, a segregation function that measures the degree of axial segregation in the medium is introduced. This function is a powerful tool to study the dynamics of banding. It allows precise measurement of the period of band oscillations and shows that the coarsening process can stop or slow dramatically when the radial core breaks (see ref. 20 for details).

3 Diffusion of a pulse of small grains Several models have assumed a diffusion law for grains within the drum. A first step in trying to develop further theories is to study the simple case of one unique band of small grains in a drum otherwise filled with large grains. This system was studied experimentally by Khan and Morris.15 An initial pulse entirely made of small grains was placed in the middle of a long drum (whose length is 400 times the width of the pulse), the remaining space was filled with large grains. The overall filling fraction was about 28%. After a few rotations, the initial pulse formed a radially segregated core and disappeared beneath the surface. The bulk visualization technique described in ref. 10 was used. From the projected shadow of the radial core, the authors could compute the concentration profiles along the axis of the drum, c(x,t). Note that their technique assumes that the core has a cylindrical symmetry but the authors ensured its validity by checking that the total volume of the radial core (defined as the integral of c(x)) remained constant in time throughout a single run.15 Using this technique, they found that the process is sub-diffusive, with a diffusion exponent, a, always near 1/3 (Fig. 3a). Indeed, Fig. 3(ii) shows a clear

Fig. 3 Diffusion of a pulse of small grains in a drum otherwise filled with large grains. a) From experiments15 and b) from DEM simulations.24 (i) Concentration profiles at various times. (ii) Log–log plot of the width of the shadow (i.e. the extension of the radial core along the axis) vs. time. (iii) Rescaled concentration profiles using an exponent a ¼ 0.37 showing an excellent collapse. (iv) and (v): Linear and semi-log plots of the rescaled concentration profiles using an exponent a ¼ 1/2 showing an excellent collapse. (vi) and (vii) Failure to obtain a decent collapse with a ¼ 1/3.

This journal is ª The Royal Society of Chemistry 2008

Soft Matter, 2008, 4, 1345–1348 | 1347

Downloaded by Bibliotheque de L’Universite de Rennes I on 05 July 2011 Published on 14 May 2008 on http://pubs.rsc.org | doi:10.1039/B717129C

View Online

power law between the width of the concentration distribution and time with an exponent a ¼ 0.38, and all profiles are nicely collapsed when rescaled using a ¼ 0.37. This result remains true for every pair of grains (glass, bronze, sand, salt) with only small variations of the exponent a. Later, Taberlet and Richard24 developed a DEM simulation of the same system. An initial pulse of small grains is placed in a drum of average filling fraction 37%. The concentration profiles of small grains are ‘‘recorded’’ as the drum is rotated at a constant speed. Very surprisingly, all data clearly show that the diffusion is not sub-diffusive but normal instead. Fig. 3(iv) and (v) display a very nice collapse using an exponent a ¼ 1/2 whereas (vi) and (vii) show that the same data cannot be collapsed with a ¼ 1/3. The mean square displacement increases linearly in time, confirming the normal diffusion process.24 The results from the experiment of ref. 15 and from the DEM simulations of ref. 24 are in strong contradiction and the origin of the discrepancy remains unknown. It could originate from the force models used in the simulations. In particular, it would be very interesting to test the consistency of the results using the Hertz model or a tangential spring friction model.16 The discrepancy might also originate in the projection method used by Khan and Morris which is an indirect measurement of the concentration. However, it seems unlikely that the details of the projection technique would change the diffusion power laws. There are a number of physical differences between the two systems that might lead to different results. One is the ratio of the drum diameter to the average particle diameter: d ¼ D/d. In the simulations d ¼ 26 and in the experiments d ¼ 100. Another possibility (although not very likely) is the filling fraction (28% for ref. 10 and 37% for ref. 24). The discrepancy might also be caused by the difference in the shape of the grains although a sub-difussive process was observed experimentally using bronze beads. Finally, the experiments were conducted in a humidity-controlled room and the capillary bridges between the grains might modify the diffusion process. 1348 | Soft Matter, 2008, 4, 1345–1348

4 Conclusion In this highlight we have reviewed recent studies that use DEM numerical simulations to investigate granular flows in a rotating drum. Different perspectives can be drawn. Taberlet et al.20 show that axial segregation can occur in a mixture of grains differing by size only and that a difference in frictional properties leads to the onset of oscillations in band position or width. Since the micromechanical coefficients (coefficient of friction, coefficient of restitution) may strongly influence the behaviour of granular media, an extensive study of their effect on the banding dynamics, on the oscillations of the band position and on diffusion properties should be carried out. Such kind of study is of course impossible experimentally. Moreover, numerical simulations allow one to measure forces between grains. The evolution of these forces during segregation should be complex. Numerical simulations can also be used to precisely tune external forces such as cohesion forces and to study the influence of these forces on the behaviour of granular media. Finally, numerical simulations can probably shed some light on the formation of complex patterns in rotating drums.25 Let us point out that although numerical simulations constitute a powerful alternative to experiments, they carry inherent flaws. Among them, they are time consuming methods. This is why the number of particles used in numerical simulations often remains small compared to real systems. Typically, with a standard computer it seems difficult to simulate more than 106 over a long period of time whereas a litre of fine sand typically contains 107 or 108 grains. Moreover in order to use a large time step (and to speed up the program) it is common to use a Young modulus smaller than the actual values (by several orders of magnitude). Finally the forces used in simulations remain model-forces. The fact that DEM simulations mimic very accurately the experiments seems to indicate that these flaws are unimportant but one has to be aware of their existence. Another future potential advancement of the use of numerical simulations is the inclusion of fluid phases within the media. Recent experimental work26 reports a large range of complex behaviour for

such systems. The DEM simulations reported here would probably facilitate the understanding of these phenomena but modelling the fluid–grain interaction remains an obstacle yet to be overcome.

References 1 H. A. Makse, R. C. Ball, H. E. Stanley and S. Warr, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1998, 58(3), 3357–3367. 2 N. Thomas, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 2000, 62(1), 961–974. 3 J. B. Knight, H. M. Jaeger and S. R. Nagel, Phys. Rev. Lett., 1993, 70(24), 3728–3731. 4 Y. Oyama, Bull. Inst. Phys. Chem. Res. Rep., 1939, 5, 600. 5 H. Jaeger, S. Nagel and R. Behringer, Rev. Mod. Phys., 1996, 68, 1259. 6 O. Zik, D. Levine, S. Lipson, S. Shtrikman and J. Stavans, Phys. Rev. Lett., 1994, 73(5), 644–647. 7 K. Choo, T. C. A. Molteno and S. W. Morris, Phys. Rev. Lett., 1997, 79(16), 2975–2978. 8 K. Hill, A. Caprihan and J. Kakalios, Phys. Rev. E, 1997, 56, 4386. 9 M. Newey, J. Ozik, S. van der Meer, E. Ott and W. Losert, Europhys. Lett., 2004, 66, 205–211. 10 Z. Khan, W. Tokaruk and S. Morris, Europhys. Lett., 2004, 66, 212–218. 11 S. Savage, in Disorder and Granular Media, ed. D. Bideau and A. Hansen, NorthHolland, Amsterdam, 1993, p. 255. 12 I. Aranson and L. Tsimring, Phys. Rev. Lett., 1999, 82, 4646. 13 S. Puri and H. Hakawa, Phys. A, 2001, 290, 218. 14 P. Richard, P. Philippe, F. Barbe, S. Bourle`s, X. Thibault and D. Bideau, Phys. Rev. E, 2003, 68, 020301. 15 Z. S. Khan and S. W. Morris, Phys. Rev. Lett., 2005, 94(4), 048002. 16 J. Scha¨fer, S. Dippel and D. E. Wolf, J. Phys. I, 1996, 6. 17 D. Frenkel and B. Smit, Understanding Molecular Simulation, Academic Press, San Diego, 1996. 18 S. Shoichi, Mod. Phys. Lett. B, 1998, 12, 133. 19 D. Rapaport, Phys. Rev. E, 2002, 65, 061306. 20 N. Taberlet, W. Losert and P. Richard, Europhys. Lett., 2004, 68, 522–528. 21 D. C. Rapaport, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 75(3), 031301. 22 D. C. Rapaport, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2007, 76(4), 041302. 23 N. Taberlet, M. Newey, P. Richard and W. Losert, J. Stat. Mech.: Theory Exp., 2006, 2006(07), P07013. 24 N. Taberlet and P. Richard, Phys. Rev. E, 2006, 73(4), 041301. 25 J. Gray and K. Hutter, Continuum Mech. Thermodyn., 1997, 9, 341–345. 26 S. J. Fiedor, P. Umbanhowar and J. M. Ottino, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2006, 73(4), 041303.

This journal is ª The Royal Society of Chemistry 2008