Granular coarsening: Phase space and evolution analogies

Various aspects of axial banding of size-varying bidisperse granular mixtures in cylindrical tumblers have been documented repeatedly over a decade or so, but the dependence of surface band formation on the relative concentration of particles and rotation rate has not been thoroughly examined. Coarsening patterns analogous to nucleation and spinodal decomposition occur as the relative concentration of small and large particles and the rotation rate of the tumbler are varied. A phase diagram with a portion analogous to a miscibility gap can be constructed from the space-time plots. A dynamic scaling approach similar to that for reacting lamellae can be applied to the coarsening patterns as a result of large bands growing at the expense of neighboring smaller bands.

Figure 1

The matrix of space-time portraits for the liquid granular system shows band formation and evolution as a function of small-particle percentage and angular velocity for 600 tumbler revolutions. In each cell, time progresses from top to bottom. White regions correspond to small particles; dark regions correspond to large particles. The granular nucleation region occurs for small-particle percentages of 20%–40%, 80%, and 90%. The granular spinodal region, bounded approximately by the dotted line, occurs for lower angular velocities in small-particle percentages of 40%–80%.

Figure 2

(a) Space-time plot for a granular mixture with moderate small-particle fraction (2500 rotations, $c_s=60\mathrm{\%}$ at 10 rpm) typical of those bounded by the dotted line for low angular velocities with small-particle percentages of 40%–80%. (b) The percent surface area of small particles rises sharply and remains constant. (c) The number of bands reaches a maximum and then decreases logarithmically.

Figure 3

Typical space-time plot (600 rotations) for a granular mixture in the granular nucleation region. (a) For low small-particle fractions ($c_s=20\mathrm{\%}$ at 70 rpm), there is nucleation of small-particle bands. (b) For high small-particle fractions ($c_s=90\mathrm{\%}$ at 10 rpm), there is nucleation of large-particle bands. (c) The percent surface area of small or large particle bands changes slowly in time. (d) The total number of bands increases slowly in time and no merging occurs.

Figure 4

Example of granular dynamic scaling that applies to any space-time plot bounded by the dotted line in Fig. 1. (a) Histograms of band thickness $s$ at every 400 revolutions. (b) The six curves for the scaling anzatz, $s^{2}f\left(s\right)=s/S\left(t\right)$, corresponding to 400 (○), 800 (◻), 1200 (△), 1600 (▽), 2000 $(◁)$, and 2400 $(▷)$ revolutions collapse for ten different experiments under the same conditions of $c_s=60\mathrm{\%}$ at 10 rpm over a duration of 2500 revolutions.