For details, see Nature 397 (1999) pp. 675-8 (available online to subscribers).

Figure 1 - Comparison between regular (left) and chaotic (right) flow of identical
grains in continuum model. (click "reload" in your browser to restart animations).
Applet simulation available here.

In the stick-slip model, the free surface is deformed into a skewed parabola, as shown in the figure below. The material in the red layer flows downhill, to the right, and as it does so the layer itself slowly creeps downstream. This continues until a slip event occurs, at which point the layer rapidly slides upstream.

Figure 2- Illustration of stick-slip model. Particles in red region flow downstream, to the right,
whereas particles in blue region rotate clockwise with the drum.

In the figure below, we compare the results of this model with experiments. In both cases, snapshots of the mixing of the same segregated initial states of identical fine grains are taken at 1/4 revolution increments, starting at the initial state: T = 0, and ending at 1 revolution: T = 1. The experimental snapshots are obtained by rotating a partially filled drum of grains at 8 rpm for the prescribed duration, after which the grains are infiltrated with glue. Once the glue has set, the contents are sliced open near the axial center of the drum, revealing the structures shown below.

Figure 3- Comparison between experimental and model snapshots of mixing of identical
beads after 0, 1/4, 1/2, 3/4 and 1 revolution of the drum.

The intricate mixing patterns shown are seen during the blending of fine grains, which are prone to stick-slip behavior (in our experiments, ~100 µm glass spheres were used). It can be shown that this flow is properly characterized by an exponential, chaotic, mixing rate. Blending of coarse, freely flowing grains produces the regular mixing behavior shown to the left of Fig. 1. The transition from regular to chaotic mixing is revealed below in identical experiments in which identical glass particles ranging in mean diameter from 700 µm down to 120 µm are tumbled through 1 revolution at 8 rpm. Significantly, blending of coarse grains can be shown to occur linearly with time, while blending of fine grains can be exponential in time, leading to dramatic improvements in blending rates.

Figure 4 - Effect of particle size on mixing patterns.