Role of Fluctuation-Induced Interactions in the Axial Segregation of Granular Materials I. Zuriguel, J.F. Boudet, Y. Amarouchene, and H. Kellay Centre de Physique Moleculaire Optique et Hertzienne, UMR 5978, U. Bordeaux1 351 cours de la Liberation, 33405 Talence cedex France (Received 11 May 2005; published 16 December 2005) The movement of a few large diameter spheres immersed in a granular medium composed of smaller beads in a rotating cylinder is studied. We evidence attractions and repulsions between the large spheres depending on the rotation frequency. The large spheres also show relative position ﬂuctuations which are Gaussian. A complete study of this problem sheds new light on the problem of size segregation in granular materials and points to the importance of ﬂuctuation-induced interactions. DOI: 10.1103/PhysRevLett.95.258002 PACS numbers: 45.70.Mg, 64.75.+g, 83.80.Fg Size segregation in granular materials is an intriguing phenomenon that poses many fundamental physics ques- tions [1,2]. The issues raised cross the frontiers between statistical mechanics, nonlinear physics, and out of equi- librium physics [3,4]. Much industrial processes also suffer from or use such segregation. Segregation occurs in natural granular ﬂows such as avalanches and land slides giving rise to a change of landscape. There are different situations where segregation by size occurs. One of these situations, studied for over 70 years, is known to occur in cylindrical tumblers [5]. Here, a long cylinder is ﬁlled halfway with grains of two different sizes and is rotated slowly around the cylinder long axis. After a few rotations, axial segre- gation starts and the two species separate in bands that are equally spaced: regions consisting mostly of the smaller grains are intercalated by regions containing the larger di- ameter grains. Such an instability has been widely studied both experimentally and theoretically [6 –9]. A difference in the repose angles of the two species against the surface of the cylinder has been invoked as a possible mechanism for such axial segregation [8]. Other mechanisms such as depletion [10,11] have been invoked recently for horizon- tally and vertically vibrated two-dimensional layers. For three-dimensional granular layers subjected to vertical vi- bration three mechanisms have been identiﬁed in a recent experiment: convection, inertia, and buoyancy [12]. Here we present results that shed new light on this phenomenon and point to the relevance of induced particle interactions for axial segregation. We use a long cylinder that is half ﬁlled with small glass beads to which we have added a few steel spheres of much larger diameter. The steel spheres are sufﬁciently heavy to be almost completely covered by the small grains and remain at a ﬁxed distance from the surface during the course of the experiment. Our main observation is that at high rotation speeds the larger spheres are equally spaced from each other as seen in Fig. 1: the assembly of large spheres shows long range order along the cylinder long axis. This situation is stable for very long times. Upon reducing the rotation speed below a threshold frequency, these spheres start to form aggregates: couples, triplets, and higher order aggregates can be obtained as seen in Fig. 1. By studying what hap- pens for just two spheres we show that they experience either a mutual attraction below the threshold frequency or a mutual repulsion above this threshold. An essential fea- ture of the induced interactions is the ﬂuctuation of the small beads. The experimental apparatus consists of a long cylinder of 19 cm in length with a diameter of 1.8 cm. This cylinder was attached with a gasket to a continuous motor deliver- ing stable rotation frequencies from 0.1 to 10 Hz. The cylinder was half ﬁlled with small diameter glass beads (0.05 cm) and the larger diameter (0.6 cm) steel spheres. Video imaging, both at a standard rate (25 images per second) and at faster rates (typically 250 images per sec- ond), was used to track the movement of the steel spheres. The cylinder was usually illuminated with a broad white light and images were taken in transmission so the steel spheres appeared black in a white background. A home- made particle tracking program was used to extract the coordinates of the steel spheres versus time. What are the mechanisms leading to the observed ag- gregation and what is its kinetics? To answer these ques- tions we studied what happens for just two spheres. In a typical experiment, we ﬁll the cylinder to a controlled a b c FIG. 1. Photographs of the particle conﬁgurations in a cylin- drical cell half ﬁlled with small glass beads of 0.05 cm in diameter including 4 steel spheres of 0.6 cm in diameter for two different frequencies. (a) Above the threshold frequency, the large spheres (the dark spots) are equidistantly separated from each other. (b) and (c) Below threshold, aggregation occurs showing a triplet and the formation of a quadruplet. PRL 95, 258002 (2005) PHYSICAL REVIEW LETTERS week ending 16 DECEMBER 2005 0031-9007= 05=95(25)=258002(4)$23.00 258002-1  2005 The American Physical Society