ARTICLES Coarsening of granular segregation patterns in quasi-two-dimensional tumblers STEVEN W. MEIER 1 , DIEGO A. MELANI BARREIRO 2 , JULIO M. OTTINO 1,3,4 AND RICHARD M. LUEPTOW 3 * 1 Department of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, Illinois 60208, USA 2 Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Ave, Cambridge, Massachusetts 02139, USA 3 Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Rd, Evanston, Illinois 60208, USA 4 Northwestern Institute on Complex Systems (NICO), Northwestern University, Chambers Hall, 600 Foster St, Evanston, Illinois 60208, USA *e-mail: r-lueptow@northwestern.edu Published online: 10 February 2008; doi:10.1038/nphys881 A fundamental characteristic of granular flows is segregation on the basis of particle size or density. For bidisperse mixtures of particles, revolutions of the order of 10 produce a segregation pattern of several radial streaks in quasi-two-dimensional rotating tumblers with fill fractions between 50% and 70%. By extending the duration of the experiments to the order of 10 2 –10 3 tumbler revolutions, we have found the first evidence of coarsening of the radial streak pattern to as few as one streak, resulting in an unexpected wedge-shaped segregation pattern. This phenomenon occurs for a wide range of conditions including several fill fractions, particle sizes and mixtures of particles varying in both size and density in circular tumblers as well as for particles varying in size in square tumblers. Coarsening seems to be driven by transport of small (or dense) particles from streak to streak through the semicircular radial core, leading to new questions about the physics of coarsening of granular segregation patterns. Coarsening is a much studied subject in physics, but the coarsening of granular matter when subjected to flow is poorly understood. Theoretical understanding is lacking but, as we will show, even the experimental boundaries of what is possible—under what conditions does granular matter coarsen—are unclear as well. Granular matter segregates during flow 1–7 on the basis of particle properties such as size and density. Examples are axial segregation in long tumblers 5,8–21 and the various types of segregation in quasi-two-dimensional (quasi-2D) tumblers, the simplest of these being ‘classical’ radial segregation 2–7 , but also pattern formation in time-periodically rotating circular tumblers 22 and steadily rotating non-circular tumblers 23,24 . Of these, axial segregation stands out as one of the most studied and least understood granular phenomena. When a long cylinder with a circular or square cross-section is rotated about its axis, a mixture of large and small particles will separate into bands of mostly large particles and mostly small particles in O(10 2 ) revolutions of the tumbler 5,8,9,13–17,21 . Over O(10 3 ) tumbler revolutions, these bands merge or coarsen 5,10–13,16,18–20 . The details of how bands form and why coarsening occurs are not fully understood, but it is generally agreed that the process begins with radial segregation. Classical radial segregation (a semicircular core of small particles surrounded by large particles) in steadily rotated quasi- 2D circular tumblers is well understood. In a rotating tumbler with a surface flow in the continuous-flow or rolling regime 25 , particles originally in the bed of particles in solid-body rotation with the tumbler enter the upstream portion of the flowing layer and fall out of the flowing layer further downstream. In the slightly dilated flowing layer, small particles percolate through the interstitial spaces of large particles. Size segregation results in small particles drifting towards the lower portion of the flowing layer and large particles drifting towards the upper portion of the flowing layer. As a result, the small particles quickly fall out of the flowing layer to occupy the inner radial core region in the bed of solid-body rotation, whereas the large particles fall out later to occupy the outer regions near the tumbler wall. Under reasonably general conditions, there is very good agreement between Poincar´ e sections derived from continuum models and experimental results for the cases of time-periodically rotating circular tumblers and steadily rotating non-circular tumblers 22–24 . In fact, a two-species model based on an inter-penetrating continuum—an unchanging underlying flow with segregation of the two species riding on top of this continuum—seems to capture the essential physics from a modelling viewpoint 3,23 . Roughly, the flow affects segregation, but segregation does not affect the flow. Initial condition 100 revolutions 10 revolutions 200 revolutions 50 revolutions 400 revolutions Figure 1 Radial streak coarsening in a bidisperse size-varying mixture. Images of a 55%-full quasi-2D circular tumbler rotated at 2 revolutions per minute (r.p.m.). The mixture is 50/50 by volume 1 mm painted black glass particles and 3 mm clear glass particles. The initially homogeneous mixture segregates into radial streaks in 10 revolutions. These streaks coarsen into one over several hundred revolutions. 244 nature physics VOL 4 MARCH 2008 www.nature.com/naturephysics © 2008 Nature Publishing Group