Ju n 20 06 Phase transition in a static granular system

We have measured the force needed by a rod to penetrate a static column of grains for different well-defined particle volume fractions φ. Measurements on our system reveal a phase transition at φ = 0.590 ± 0.003. It was first noted long ago, at least as far back as the investigations of Reynolds (1885, 1886), that there are distinct fluid and solid states of static granules such as sand. The distinction between fluid and solid for static states is based on the degree of resistance to shear, e.g. to motion of an inserted rod: if a hard rod is easily inserted and/or withdrawn, the material is acting as a fluid, while if it difficult to insert and/or withdraw, the material is acting as a solid. There was even a well-known magic trick a century ago based on this change of state (Branson, 1922; Aste & Weaire, 2000). A pot with a narrow neck is filled with rice and a knife is repeatedly shown to be easily inserted and removed. Then the knife is inserted and the pot is shaken (compactifying the rice). The whole pot can then be lifted by the knife handle and swung around the performer’s head. While the existence of fluid and solid phases has been widely discussed, a sharp distinction between the two phases has remained elusive (Jaeger et al., 1996; Kadanoff, 1999; de Gennes, 1999). The distinction between different states of a granular system is hampered by the difficulty in preparing a well-defined initial state (de Gennes, 1999). The effort to overcome this was advanced significantly by Nowak et al. (1998), who used a mechanical tapping protocol to obtain well-defined densities in the range 0.628 to 0.658. Recently Schröter et al. (2005) showed that a different protocol, based on expanding the granular medium by pulses of fluid from below, could be used to prepare a column of grains with a well-defined particle volume fraction. The volume fraction was obtained to 0.1% in a broad range, 0.566 to 0.609. Using this technique, we have found a phase transition at density 0.590. Experiment. Measurements are performed with a home built granular penetrometer: A translation stage driven by a stepper motor with a step size of 2.5 μm moves a stainless steel rod with a diameter of 6.3 mm downwards into a granular sample. The force needed for penetration is measured with a load cell with a full range of 10 N (Honeywell, Model 31). The sample consists of soda lime glass beads from Cataphote with a diameter of 265 ± 15 μm and a density of 2.484 ± 0.002 g/cm (measured with a Micromeritics gas pycnometer AccuPhys 1330). The beads are contained in a water-fluidized bed where flow pulses of different flow rates allow us to control their volume fraction (Schröter et al., 2005). The beads are fluidized inside a square bore glass tube (39.9 × 39.9 mm). The ratio of inner tube diameter to rod diameter is 6.3, larger than the value of 5 that Stone et al. (2004) found to be the ratio where the influence of the vessel diameter vanishes. Flow pulses are generated using a digital gear pump (Barnant Co., model no. 75211). The volume fraction is determined from bed height measurements with a CCD camera at a resolution of 39.6 μm/pixel. Results. Figure 1 shows that the force on the penetrating rod increases monotonically with its depth; however, the slope of the curve depends strongly on the volume fraction of the sample. The dependence of the ∗ schroeter@chaos.utexas. edu radin@math.utexas. edu