Friction-Controlled Entropy-Stability Competition in Granular Systems

Using cyclic shear to drive a two-dimensional granular system, we determine the structural characteristics for different interparticle friction coefficients. These characteristics are the result of a competition between mechanical stability and entropy, with the latter’s effect increasing with friction. We show that a parameter-free maximum-entropy argument alone predicts an exponential cell order distribution, with excellent agreement with the experimental observation. We show that friction only tunes the mean cell order and, consequently, the exponential decay rate and the packing fraction. We further show that cells, which can be very large in such systems, are short-lived, implying that our systems are liquidlike rather than glassy.

Figure 1

(a) The stadium shear device is mounted on a horizontal glass plate. Particles occupy the region within the stadium, including areas under the sprockets. A constant boundary confining pressure is maintained by two aluminium bars, mounted on four air bearings with constant forces pushing against the lateral walls of the belt. The imaging system includes a CCD camera and a LED panel above and below the device, respectively. (b) A cell is defined as the smallest loop of contacts. Two cells are drawn in the diagram. (c) A part of the gear system, containing cells, shaded blue.

Figure 2

Cell order distributions of six different systems: Gears (triangles), gears-stainless steel cylindrical rings mixture (circles), photoelastic disks (squares), plastic cylindrical rings (inverted triangles), and Teflon-coated photoelastic disks (diamonds). The solid lines are the exponential functions predicted by Eq. (2). Inset: The variance of the COD for the different systems, with the same symbols as in the main panel and stars for photoelastic disks under pure shear. All measurements fall exactly on the analytical function predicted from the PDF in Eq. (2) The small spread of symbols within a given cluster of same-type particle represents variations of strain amplitudes from 3% to 10% and boundary pressure of $\pm 30\%$.

Figure 3

The conditional CODs $p(n|m)$ for different systems. For clarity, data of stainless-steel particles, plastic particles, photoelastic particles and gears have been shifted vertically by factors of 5, 25, 125, and 625, respectively.

Figure 4

The conditional PDFs of the internal cell angles, defined in the inset, for (a) gears, (b) photoelastic particles, and (c) Teflon-coated photoelastic particles.

Figure 5

The ratio $\gamma =V_c(n)/V_{\mathrm{rp}}(n)$ increases to 1 as cells are more round and vice versa. Increasing entropy reduces $\gamma$, opposing the effect of mechanical stability. The monotonic decrease of $\gamma (n)$ at high friction indicates that entropy dominates over stability, while the up-turn of $\gamma (n)$ at low friction is evidence to the increasing restraining effect of stability. The latter behavior was also observed in Ref. [20] (included with permission), albeit under different dynamics.

Figure 6

(a)–(c) The survival probability of cells of different orders $n$, and rattlers, in the systems of (a) gears, (b) photoelastic particles, and (c) coated photoelastic particles. (d) The corresponding characteristic relaxation times $\tau (n)$, estimated from an exponential fit of the data in the main panel.