Stationary state volume fluctuations in a granular medium

A statistical description of static granular material requires ergodic sampling of the phase space spanned by the different configurations of the particles. We periodically fluidize a column of glass beads and find that the sequence of volume fractions $\varphi$ of postfluidized states is history independent and Gaussian distributed about a stationary state. The standard deviation of $\varphi$ exhibits, as a function of $\varphi$, a minimum corresponding to a maximum in the number of statistically independent regions. Measurements of the fluctuations enable us to determine the compactivity $X$, a temperaturelike state variable introduced in the statistical theory of Edwards and Oakeshott [Physica A 157, 1080 (1989)].

Figure 1

(a) Experimental setup. The inset shows the correlation between the height changes $\Delta h$ during a flow pulse to the fluidized bed, as seen by the two cameras (800 flow pulses, $Q=60\mathrm{ml}∕\min$). (b) Development of the bed height during flow pulses of $Q=40\mathrm{ml}∕\min$. The arrows $(\uparrow )$ indicate measurements of $h$.

Figure 2

History-independent steady state: (a) The volume fraction $\varphi$ of the sedimented bed reaches a steady state that depends only on the flow rate $Q$. (b) The inset shows the increase in volume fraction as $Q$ was decreased in 14 steps from $60\mathrm{ml}∕\min$ to $23\mathrm{ml}∕\min$, followed by a decrease in volume fraction as $Q$ was increased again; each step consisted of 50 flow pulses. The main graph (b) shows that the steady-state volume fraction is the same for increasing (엯) and decreasing (●) $Q$; the dashed line is a fit to (1).

Figure 3

Volume fraction fluctuations around the steady states shown in Fig. 2a are Gaussian distributed. The upper row shows examples of the volume fraction measurements as a function of the number of flow pulses; ${\varphi }_{\mathrm{avg}}$ is indicated above each column. The lower row shows the corresponding probability distributions obtained from 800 flow pulses; the curves are Gaussian fits.

Figure 4

(a) Volume fraction fluctuation measurements fit a parabolic curve, (b) number of particles in a statistically independent region, and (c) compactivity as a function of the average volume fraction. At ${\varphi }_m$ the system has a maximum in the number of statistically independent regions [cf. (2)].

Figure 5

Influence of particle surface roughness. On the left are scanning electron microscopy images of the surface of new and aged glass beads. The plot on the right shows the volume fraction fluctuations of new beads (∎) together with a parabolic fit (solid line); to minimize aging, measurements were made with only 200 flow pulses. The dashed line is the parabolic fit for aged beads from Fig. 4.