The mechanism of long-term coarsening of granular mixtures in rotating drums

Three fundamental segregation and pattern formation processes are known in granular mixtures in a rotating cylindrical drum: radial segregation, axial banding, and coarsening of the band pattern. While the mechanism for the first effect is well understood and for the second effect, several models have been proposed, the long-term coarsening mechanism remained unexplained so far. We demonstrate that the unidirectional flow between the bands in an axially segregated pattern is driven by small differences in size of the small beads at the band edges. Due to a process of microsegregation inside each band of small particles, which was so far unrecognized, this difference in diameter will be effective in all experiments with polydisperse beads. In consequence the stability of individual bands can be easily controlled by minor alterations of their composition. Our results make evident that a new mechanism as the driving force behind the axial particle flow has to be sought. We suggest possible hypotheses for such a mechanism.

Apart from the fundamental scientific interest in the dynamical features of this simple multiparticle system, impact on technological processes cannot be underestimated. In most applications, segregation of mixtures of grains is unwanted, in some others it may be highly desired. General understanding of the underlying physical mechanisms is therefore of enormous practical interest.
The segregation of a binary granular mixture typically happens in three stages. First, radial segregation leads to the formation of a core region consisting of the smaller particles [9,16,18,37]. This phase is well understood as the consequence of a sieving effect. In many systems it is followed by axial segregation, the formation of a banded pattern [8, 11-14, 19-21, 36, 39]. Several models have been proposed for this effect, and solid experimental evidence has been collected for the evaluation of the proposed mechanisms (see, e.g. discussion in Ref. [4] and references. therein).
Finally, coarsening of this stripe patterns [5,6,[30][31][32][33][34] leads to a logarithmic decay of the number of axial bands with increasing number of rotations of the drum. While the reason for coarsening has been clearly iden-tified as the transport of small-sized beads through the core channel of small beads from one band of small grains to a neighboring band of small grains [32,40], the mechanism driving this flux has not been explained up to now. It cannot be explained intuitively, nor does it seem to have relations to any other known dynamic segregation phenomena in granular matter.
Finger et al. [33] studied the elementary process of this exchange optically and with NMR imaging. The mixer was initially prepared with two separate bands of smallsized beads, enclosed by bands of the large-sized grain component (cf. Fig. 1a). It was established that the transport between two adjacent bands is unidirectional. The narrower band loses material to the larger one at a constant rate, until it is extinguished [33,34]. Once this transport sets in, its direction is irreversible. The large beads do not contribute to the pattern dynamics. When a band of small beads separating two large-grained bands is extinguished, the latter combine to a broader region with the sum width of the original bands. If the experiment is continued for sufficiently long time (up to weeks), the result is the complete segregation into one or two bands of large grains and a band containing the smaller grains, plus the core channel. This feature, albeit described in numerous papers, is unexplained so far. Particularly, the following aspect is remarkable: The agitation of the grains is a revolution about the cylinder axis, while the directional transport is axial. It breaks the axial symmetry of the horizontal tube. Consequently one has to search for a mechanism that drives a unidirectional flow of the small grains, preferentially from smaller to broader segregation bands. The process reminds of a hydrodynamic situation where one container empties into a communicating vessel by pressure differences. Thus, one potential hypotheses for the driving mechanism of the flux could be an effective The flux of small particles between two stripes is controlled by both the stripe width ratio and the diameter of the particles at the opposing stripe edges. a) image of a mixer with two prepared stripes of small particles embedded in large-size beads. Panels b-e show space-time plots of four coarsening scenarios. The central stripe of large beads (1.5 mm grain diameter, appearing white) is always 8 cm broad and separates two (darker) stripes of either 355 to 500 µm (S1) or 500 to 630 µm (S2) particle diameter. Top graphs sketch the preparation conditions, qualitatively. b) For same grain types, here S2, the narrow stripe (10.5 cm versus 14 cm) dissolves. c) If the narrower stripe is prepared from the larger grains S2 (same widths as in b), the broader one with S1 grains dissolves. Panel d) demonstrates the dominance of grain diameter over stripe width (4 cm S2 versus 30 cm S1). e) Even a small amount (0.5 cm) of S2 beads on the inner edge of the left stripe with 7.5 cm S1 beads, triggers dissolution of the broad stripe (20 cm) with purely S1 beads.
surface tension between the regions composed of large and small grains. Such a free surface energy could originate from an decrease in a configurational entropy [41,42] at the interface, as discussed below.
In this letter, we demonstrate that two cooperative processes are responsible for the coarsening. First, we show that if two adjacent stripes contain grains of slightly different sizes near their edges, the transport of grains through the core channel is always directed from the edge containing smaller grains to the edge containing larger grains. Secondly, we report a new microsegregation mechanism inside the band of small beads: When the size distribution of the small beads is not exactly monodisperse, the larger fraction accumulates at the borders of the segregation stripes. The center of the stripes contains smaller grains.
These results have three important consequences: (1) they provide an explanation why directed transport proceeds from narrower to broader neighboring stripes containing initially the same grain compositions, (2) they allow to control the stability of individual stripes by adding a small amount of slightly larger grains, and (3) they lead to the prediction that bands of monodisperse small grains do not undergo merging or coarsening. We verify this prediction experimentally.
The experiments are performed in a 66 cm long tube of 37 mm diameter rotated at 20 revolutions per minute (not critical). The tubes are half filled with bands composed of large (1.62 ± 0.062 mm diameter) and smaller glass spheres. The latter are obtained by sieving mixtures, we use two fractions with size distributions between 0.355 mm and 0.500 mm ('S1') and between 0.500 mm and 0.630 mm ('S2'), resp. The tube is filled up with water after preparation of the stripes (Fig. 1a). It has been shown that the interstitial fluid has only quantitative but no qualitative consequences for the coarsening process [34]. We use water primarily to avoid static friction effects and to improve sample transparency. The tube is illuminated from the top and images are recorded automatically in intervals of 1 min. From long-term observations (between 15,000 and 200,000 rotations), we construct space-time plots of the tube axis profiles, in which regions with small beads S1, S2 appear darker than those of large beads. Figure 1 shows four typical scenarios with differently prepared initial configurations. The plots show that (1) if two bands consist of the same particle types, the narrower one vanishes, see Fig. 1b, (2) if one band contains slightly smaller grains than the other, it vanishes even if it is broader, see Fig. 1c,d, and (3) it is sufficient that the band edge contains larger particles to stabilize it, see Fig. 1e. This observation allows us to stabilize a dissolving band by adding a few S2 grains (Fig. 2, left): After a narrow band of S1 grains had already lost half of its content to the neighboring S1 band, we added roughly 15 % of S2 material to it. This led to an immediate reversal of the transport through the core channel, even though the competing band contained almost 3 times as many S1 material.
The second process active in the coarsening is microsegregation inside each individual band of small particles. It moves the S2 beads in a mixed S1/S2 region to the band edges where they effectively control the material flow. Even when the band grows by incoming S1 grains, these are transported into the center of the axial coordinate [cm] S1 S1 S2 S1 FIG. 2: Coarsening control and microsegregation. Left: After the narrower of two prepared pure S1 stripes starts to dissolve, we add an small amount S2 material (equivalent to 1 cm band) to the dissolving 6.5 cm broad S1 band (circled region), which reverses the transport and the 17.5 cm broad S1 stripe starts to decay immediately. Right: Stripes of white S1 and red S2 were prepared. A core quickly forms by radial segregation, from material of both stripes (thus, the regions of large grains also adopt a pink appearance). The band of smaller beads S1 transfers its material to the S2 band, where it is not simply deposited at the edge but forms a clearly segregated central band, surrounded by S2 beads. stripe and the edge region remains occupied by S2 grains. This is demonstrated in Fig. 2, right, where we used colored (0.63-0.71 mm) and transparent (0.5-0.63 mm) small spheres. Initially, two bands were prepared where one contains only S1, the other one contains only S2 grains. The S1 material is transported into the growing stripe and in there, a microsegregation places the S1 fraction in the stripe center. The edges between S2 and S1 regions are astonishingly sharp, even though both species are neighboring sieving fractions. This result agrees qualitatively with the segregation experiments by Newey et al. [43] where they used three bead sizes of 0.6 mm, 1 mm, and 2 mm and found that the bands of large and small grains were separated by small bands of the medium sized grains.
Most importantly however, microsegregation does also occur in stripes made of particles of a much narrower size distribution. This can be demonstrated by Xray tomography in a down-sized system (tube diameter 24 mm, length 60 mm) filled with beads of diameters 1.01 ± 0.01 mm and 423 ± 23 µm (roughly Gaussian diameter distribution, measured with a Retsch Camsizer). Measurements were performed in a Nanotom (GE Sensing and Inspection) with 40 µm voxel size. After detection of the individual small and large particles we measure their average diameter as a function of the axial coordinate. A detailed description of this measurement can be found in the supplemental material. The analysis of the small particle diameter in homogeneously prepared (d-d mean )/d mean · 100% 1,000 rot. single stripes of small grains shows that beads are redistributed and microsegregated by the rotation of the mixer. Figure 3 shows the axial profile of the average diameter of the small grains in such a stripe. Initially, the mixture is uniform (black, horizontal curve). After 1,000 rotations of the mixer, a stable segregation is established and beads near the edges of the stripe are on average 2 % larger than those in the stripe center. It is this microsegregation which explains also the extinction of the narrower one of two neighboring bands of beads with nominally the same diameters (c.f. figure 1 b). This is demonstrated with another experiment shown in figure 4. Using again X-ray tomography, we show that in a rotating drum prepared with two stripes of the same small particles two combined features are observed: the narrower stripe dissolves and the dissolving stripe also displays a smaller mean bead diameter at its boundaries. This confirms that the direction of the flux between two stripes is pointing again from the relatively seen smaller particles to the larger ones.
Two considerations complete this argument. First the smaller particle diameter at the boundary of the narrower stripe can be understood as a consequence of the smaller reservoir inside its bulk which can not provide the same number of largest particles as they are present in the broader stripe. Second, this mechanism is self-sustaining. Once the flow from the narrower stripe to the broader sets in, the narrower stripe preferentially looses its largest particles which had accumulated at the boundary. If the above interpretation is correct, then the coarsening should be absent in systems where the small particles are monodisperse. Thus, we prepared drums with mixtures of large-size beads of 2.62 ± 0.065 mm and monodisperse small spheres of diameter 1.01 ±0.01 mm. As shown in Fig. 5, the initially prepared bands remain either stable or start to drift or jitter. However, there is indeed no coarsening on the timescales of our experiments, which is an order of magnitude longer than e.g. the experiment displayed in Fig. 1 b). A sign that our experimental timescales are sufficient is the positional jitter of the thin stripe in Fig. 5b, which evidences that the dynamics of large beads tunneling through the thin stripe becomes comparable to the small particle dynamics.
At this point the crucial question remains, why the transport in the core channel between two stripes is related to the size difference of the small particle at the interfaces. We propose three different hypotheses, that will need experimental verification.
One possible explanation is the existence of an effective free surface energy σ at the interface of each stripe where σ increases with the difference between small and large particle diameters d l − d s [44]. It is always the stripe with the larger (d l − d s ) and therefore the larger σ which dissolves. In order to explain the sequence of stripes evolving in Fig. 2 b), we need to assume that σ depends on (d l − d s ) α with an exponent α larger one. An argument for the origin of σ can be derived from the assumption of an Edwards ensemble [41,42,[45][46][47] where a configurational entropy S conf is defined as the logarithm of the number of possible mechanically stable configurations for the given volume fraction and boundary stresses. The important point is that the number of possible packings of monodisperse spheres does not depend on their diameter. Therefore in the bulk of both large and small stripes, the entropy per particle S conf /N is approximately identical. However, at an interface between two sizes of particles, the number of mechanically stable configurations will depend on the diameter ratio. This could provide an effective surface tension σ(d l − d s ). However, our experiment cannot provide direct evidence for this.
The second hypothesis presupposes that the linking channel contains a concentration gradient dc s /dx of small beads that drives their directed motion. We have established experimentally by varying the size ratios of large and small beads that the equilibrium concentration of the smaller species in the core channel, c s (number of particles in the cross section) grows with the size ratio r = d l /d s . Different r at the interfaces of two neighboring bands of small beads (caused by different compositions of the stripe edges) will therefore lead to a concentration gradient, c s becomes larger at the end where r is larger (narrower segregation band or S1 beads). An effective particle flow towards the opposite channel end (large segregation band or S2 beads) is the consequence, until the band which provides the particles is completely extinguished.
None of this two hypotheses can be directly proven with the experiments presented here, and detailed measurements of individual grain dynamics will be necessary for a validation.
Summarizing, it has been shown that two mechanisms govern the coarsening of axially banded pattern of bidisperse granular mixtures in a horizontal rotating drum: a microsegregation process accumulates larger spheres of the small component of the mixture at the edges of the bands, and material transport generally sets in from bands that contain smaller particles at their edges to neighboring bands containing larger particles at the edge. When the small-sized beads are monodisperse, no coarsening of the pattern is observed, even over long experimental timescales. Exploiting this finding, we demonstrated that the stripe stability can be controlled by addition of a few percent of different sized grains. Furthermore, the observed microsegregation might be exploitable in technological processes where particles are to be sorted efficiently by size.