Epitaxial Growth of Granular Single Crystals Y. Nahmad-Molinari and J. C. Ruiz-Sua ´ rez * Departamento de Fı ´sica Aplicada, CINVESTAV-IPN, Unidad Me ´rida, A.P. 73 ‘‘Cordemex,’’ Me ´rida, Yucata ´n 97310, Me ´xico (Received 30 August 2002; published 11 December 2002) Compaction from a random-loose-packed to a random-close-packed phase is observed when mono- disperse granular beds are shaken, but beyond this packing, the system freezes up in a jammed structure. Here we report a technique to grow large hard-sphere granular crystals, with perfect stacking and no defects by means of a ‘‘gas phase’’ epitaxial procedure. We study the growth mechanism and provide evidence that the observed granular crystallization is driven by gravity and energy dissipation. DOI: 10.1103/PhysRevLett.89.264302 PACS numbers: 45.50.–j, 45.70.Cc Hard spheres, both in computer simulations and in the laboratory, have been for many years useful model entities to advance the understanding of fundamental problems in statistical mechanics, such as nucleation, crystallization, and the so-called glass transition. In the last decade the study of the dynamics of hard-sphere colloidal assemblies has produced a significant number of reports in the scientific literature (see [1] for a review). One of the main conclusions of this intensive research is that the phase diagram of colloidal hard spheres is ather- mal and governed completely by entropy. In addition, techniques to grow large colloidal crystals have emerged: among them an epitaxial technique based on the slow sedimentation of hard spheres onto a patterned substrate [2] and a temperature gradient driven method [3]. These two techniques will, very likely, contribute soon to tech- nological applications in the incipient industry of colloi- dal crystals [4]. Going from hard-sphere colloidal suspensions to granular beds, a very different scenario appears: while in hard-sphere colloidal systems temperature does not play a role, and gravity precludes any crystallization beyond a volume fraction of 0.59 [5]; in granular packings both gravity and ‘‘granular temperature’’ are essential variables to seek for structural order. Granular structures have important applications as photonic and phononic crystals [6–9], in model systems to study stress propa- gation [10,11], in the study of lattice dynamics [12], among others. Nevertheless, attempts to avoid the frus- trated relaxation of granular packings toward defect- free crystal structures have failed [13,14]. Until now, the closest packing achievable by vibration or shaking is around 0.68 of volume fraction, still far from the ideal 0.74 corresponding to a perfect single hexagonal- close-packed (hcp) structure [15]. A purely mechani- cal method has been reported to circumvent this problem and make large hcp granular crystals [10]. Such a method consists in vacuum holding a complete hexagonal layer of beads on a perforated plate and the subsequent release of one layer after the other until a perfect structure is formed. Upon construction, a small number of defects are removed by hand before depositing the next layer. The aim of this Letter is to report a technique to grow large hard-sphere granular crystals with perfect stacking and no defects. The technique is based on a gas phase epitaxial procedure. We provide evidence that granular crystallization is driven by an interplay of gravity and energy dissipation. Our experimental setup consists of a Plexiglas cell of an equilateral triangular base in which an exact number of particles fit in the first layer (the dimensions of the cells we used range from 5 to 15 cm per side and 20 cm of height). The cell is mounted on top of a vertical shaking system with an accelerometer attached to it. A particle feeder drops particles, one at a time, into the cell, at a given and controlled frequency. We used steel ball bear- ings of 3.175 mm in diameter. Dropping the balls at slow rates while vertically shaking the cell at a starting accel- eration a just above g (a=g  1:09) leads to the formation of a seed or an inelastic collapselike state (a cluster of motionless particles in constant contact [16]). The term ‘‘inelastic collapse’’ refers to the state in which a group of particles undergoes an infinite number of collisions in a finite period of time. As a result, this group of particles comes into contact without any attrac- tive forces between them. The main condition to have an inelastic collapse is that the energy dissipation rate should be large enough to promote self-stabilizing clusters [17]. Normally, a granular system of hard spheres will show inelastic collapse dynamics if the restitution coefficient between particles is less than unity. The restitution coeffi- cient of the ball bearings has been reported to be about 0.90 [18]. Convergent velocity fluctuations, in the granular gas- like phase [19] within the cell, increase the density of a region, which then acts as a condensation nucleus. Clearly, the walls favor the appearance of these nuclei due to the lower restitution coefficients. The particles fed into the cell join one of the two coexisting phases, the collapse or the gas, until the density is high enough that the entire first layer becomes hexagonally close packed VOLUME 89, NUMBER 26 PHYSICAL REVIEW LETTERS 23 DECEMBER 2002 264302-1 0031-9007= 02=89(26)=264302(4)$20.00  2002 The American Physical Society 264302-1