Onset of mechanical stability in random packings of frictional spheres
Using sedimentation to obtain precisely controlled packings of noncohesive spheres, we find that the volume fraction of the loosest mechanically stable packing is in an operational sense well defined by a limit process. This random loose packing volume fraction decreases with decreasing pressure and increasing interparticle friction coefficient . Using X-ray tomography to correct for a container boundary effect that depends on particle size, we find for rough particles in the limit a new lower bound, .
pacs:83.80.Fg, 46.65.+g, 45.70.Cc, 47.57.ef
Introduction. – If granular materials such as sand, sugar, or snow are excited strongly (e.g. by shaking or shearing), they exhibit fluid-like behavior. However, after the excitation stops, dissipation quickly produces a static packing that is mechanically stable under its own weight. Experiments Scott (1960); Rutgers (1962); Onoda and Liniger (1990); Ojha et al. (2000); Valverde et al. (2004); Valverde and Castellanos (2006); Dong et al. (2006); Umbanhowar and Goldman (2006) and simulations Makse et al. (2000); O’Hern et al. (2002); Zhang and Makse (2005); Shundyak et al. (2007); silbert:08 () have shown that the volume fraction has a well defined lower limit, , called Random Loose Packing (RLP).
The value of depends on the particle-particle interactions. Packings of cohesive particles like fine powders are stable under their own weight for values of as low as 0.15 Valverde et al. (2004); Valverde and Castellanos (2006); Dong et al. (2006); Umbanhowar and Goldman (2006). However, many granular materials do not exhibit cohesive forces. Simulations of frictionless elastic noncohesive spheres have found the onset of a finite bulk modulus at the jamming point, Makse et al. (2000); O’Hern et al. (2002); Zhang and Makse (2005). Real spheres have friction and then it has been suggested that depends on the density difference between the particles and the surrounding fluid Scott (1960); Rutgers (1962); Onoda and Liniger (1990); Ojha et al. (2000). The lowest volume fraction reported thus far, , was observed for slowly sedimenting spheres in a liquid of nearly the same density Onoda and Liniger (1990).
Here we demonstrate a limit process that yields well defined values of that depend on pressure and coefficient of friction. The results are discussed in the context of a statistical mechanics approach based on the ensemble of all mechanically stable configurations Edwards and Blumenfeld (2007).
Experiment. – Mechanically stable packings of glass spheres were prepared by allowing the particles to sediment following flow pulses in a water fluidized bed. The fluidized bed was contained in a vertical polycarbonate tube with an inner diameter of 12.8 mm and a length of 230 mm. The tube’s bottom end was closed by a distributor consisting of a porous bronze disc (height, 8 mm; nominal pore size, 25 m). A programmable syringe pump (Harvard Apparatus) created pulses of constant flow rate . During a flow pulse of 2 minutes length the granular medium fluidized and expanded until it reached a steady state height. After each flow pulse, the particles sedimented onto the distributor and formed a mechanically stable packing whose volume fraction depended on , as shown in Fig. 1. A higher value of resulted in a more expanded fluidized bed, longer sedimentation time, and lower of the sediment. Packings created in this way are independent of the state of the sample prior to the last flow pulse Schröter et al. (2005). This property is important for any statistical mechanics approach aste:07b (). The volume fractions in Fig. 1 are averaged over the whole sample: , where is the particle density, is the total mass of all the spheres, is the cross sectional area of the tube, and is the height of the sedimented sample determined from images.
The properties of the different samples of particles are given in Table 1. The density of the particles in each sample was measured with an accuracy of 0.06 % using a Gay-Lussac specific gravity bottle and a Micromeritics AccuPyc 1330 gas pycnometer; the average was 2.48 g/cm. To characterize the frictional properties of the samples we measured the angle of repose under water: a beaker containing a layer of particles about 5 mm high was tilted until the particles started to move. To obtain an especially rough sample (F) we soaked part of sample C for 3 hours in hydrofluoric acid. Sample E consists of spheres that were smoothed by exposure to more than 45,000 flow pulses in a fluidized bed Schröter et al. (2005).
|sample||(m)||(%)||supplier||angle of repose|
RLP is defined by a limit process. – The main improvement over earlier studies using sedimenting particles Onoda and Liniger (1990); Dong et al. (2006) is that our control of allows us to change the sedimentation time independent of the liquid density. This procedure reveals the convergence of to . The observation that the slowest relaxing preparation yields the loosest packings agrees with simulations of frictional discs and spheres Zhang and Makse (2005); Shundyak et al. (2007). Figure 1 shows that is well described by the fit function used in Schröter et al. (2005),
which we use to determine .
Dependence on particle diameter. – The inset of Fig. 1 indicates that decreases with particle diameter; however, this decrease is due to lower volume fraction near a container wall, an effect known since the earliest studies Scott (1960). This effect is explained in Fig. 2(a): since particles cannot penetrate the container wall, voids are larger there and the volume fraction of the layer adjacent to the boundary is lower than measured in the core of the sample. The difference between averaged over the whole container and increases with the ratio / and produces the trend displayed in the inset of Fig. 1.
We examined the finite size effect using X-ray tomography Sakellariou et al. (2004), Fig. 2. In each run positions of spheres were measured with a resolution of better than 0.1% of a sphere diameter Aste et al. (2007). Figure 2(c) shows the difference between the apparent volume fraction using all particles and the bulk volume fraction (measured using the Voronoi volumes Aste et al. (2007) of all particles that are at least away from the container walls). For ml/min, values from the tomographic measurements agree with the results (dotted curve) for the 277 m particles in Fig. 1, but for ml/min the tomographic values are larger due to unavoidable vibrations during the recording of the tomogram.
The inset of Fig. 2(c) shows the difference between the bulk and apparent as a function of . A linear fit yields
For all further experiments we used only spheres with diameter 261 or 257 m (sample E or F), and we corrected for the effect of finite container size using (2).
Influence of pressure. – The stress inside a column of grains differs from the hydrostatic case in two ways: (i) anisotropy – the horizontal stress in the column differs from the vertical stress . (ii) wall friction – the part of the load carried by the frictional sidewalls increases with depth below the surface. Consequently, and saturate with . In our analysis we use a pressure dependence on height given by the Janssen model Sperl (2006), which assumes a constant stress ratio everywhere in a sample. Experiments show that this model is a fair approximation in the absence of external loads Vanel and Clément (1999); Vanel et al. (2000). The model gives a saturation of pressure () with depth,
to a constant value , where is the coefficient of particle-wall friction, is the gravitational acceleration, is the density difference between spheres and surrounding liquid, and . Equation (3) indicates two ways of controlling the pressure distribution inside the column:
I) Increasing the sample height, which increases the fraction of the sample at . If increases with , then the average measured by our method should increase with sample height. This behavior is confirmed in Fig. 3.
II) Decreasing the density difference , which decreases but keeps the pressure profile unchanged. We increased the fluid density to as high as 2.39 g/cm, close to the 2.48 g/cm particle density, by adding sodium polytungstate to the water. Results for different (Fig. 4) again confirm that decreases with decreasing .
A limit would be given by matching the fluid and particle densities, but in this limit there would be no sedimentation and no connected granular packings would form. Therefore, we extrapolate to determine : In the absence of theory we follow Valverde et al. (2004) and use the pressure dependence close to the jamming point known for frictionless static soft spheres Makse et al. (2000); O’Hern et al. (2002); Zhang and Makse (2005); Majmudar et al. (2007) and frictionless thermal hard spheres Brito and Wyart (2006):
where we identify . A fit of for smooth particles (sample E) in Fig. 4 yields . The value of is approximate because our derivation of (4) did not take into account the dependence of Vanel and Clément (1999).
Influence of frictional properties. – Figure 4 shows that for the rough spheres was lower than for the smooth spheres. For the rough spheres a fit to (4) yields and . The decrease of and with increasing friction agrees with another experiment men (), model Srebro and Levine (2003), and simulations Zhang and Makse (2005); Shundyak et al. (2007); silbert:08 (); pica_ciamarra:08 ().
Discussion. – Our experimental results and numerical simulations Zhang and Makse (2005); Shundyak et al. (2007) both show that RLP is well defined in an operational sense: in the limit of infinitesimally slow preparation, the volume fraction of a sample converges to , independent of the details of preparation. Care should be taken in comparing theory for frictionless hard spheres with the experimental results, in part because of the different possible ways of defining mechanical stability Torquato et al. (2001); Torquato and Stillinger (2007).
The observation of a well-defined can be considered within the framework of a statistical mechanics of static granular material Edwards and Blumenfeld (2007), where a configurational entropy is defined as the logarithm of the number of mechanically stable configurations for a given , , and friction coefficient. Two different approaches can explain RLP using two different assumptions of how depends on . The first approach assumes that RLP is the smallest where becomes larger than zero. This is compatible with the existence of looser, highly ordered configurations Torquato and Stillinger (2007), as their number seems not to grow exponentially with system size, so .
The second approach is supported by numerical results on the number of stable configurations of frictional discs pica_ciamarra:08 (), where has a maximum at RLP. This idea agrees with slow sedimentation leading to RLP: it is simply the most probable configuration. If the sedimentation speed is increased, the additional kinetic energy allows the system to explore the local energy landscape and find rarer but lower potential energy (denser) configurations. Further, the maximum of and therefore RLP moves to higher values of for decreasing friction pica_ciamarra:08 (). This agrees with our results and with simulations of frictionless disks that have a maximum of at Random Close Packing Gao et al. (2006). Our results indicate also that increasing shifts the maximum of in a similar way.
Conclusions. – Mechanically stable packings of spheres prepared with increasing sedimentation time display a lower bound of their volume fraction, , which depends on the pressure and the coefficient of friction but not on the diameter of the spheres. In the limit of zero pressure we have found a new lowest value of , .
Acknowledgments. – We thank Massimo Pica Ciamarra and Antonio Coniglio for sharing their unpublished results, and Brandon McElroy for his assistance with the Camsizer. Further we thank W.D. McCormick, Narayanan Menon, Charles Radin, Leo Silbert, Jack Swift, and participants of the 2007 Aspen Jamming workshop for helpful discussions. This work was supported by Robert A. Welch Foundation Grant F-0805.
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