Equilibrating Temperaturelike Variables in Jammed Granular Subsystems

Although jammed granular systems are athermal, several thermodynamiclike descriptions have been proposed which make quantitative predictions about the distribution of volume and stress within a system and provide a corresponding temperaturelike variable. We perform experiments with an apparatus designed to generate a large number of independent, jammed, two-dimensional configurations. Each configuration consists of a single layer of photoelastic disks supported by a gentle layer of air. New configurations are generated by cyclically dilating, mixing, and then recompacting the system through a series of boundary displacements. Within each configuration, a bath of particles surrounds a smaller subsystem of particles with a different interparticle friction coefficient than the bath. The use of photoelastic particles permits us to find all particle positions as well as the vector forces at each interparticle contact. By comparing the temperaturelike quantities in both systems, we find compactivity (conjugate to the volume) does not equilibrate between the systems, while the angoricity (conjugate to the stress) does. Both independent components of the angoricity are linearly dependent on the hydrostatic pressure, in agreement with predictions of the stress ensemble.

Figure 1

Schematic of apparatus showing (a) two walls biaxially compressing an array of disk-shaped particles composed of an outer subsystem (black, high $\mu$) and an inner subsystem (red, low $\mu$) and (b) reflective photoelasticity on air-floated particles. Light shines from green LEDs through a linear polarizer ($P$) a wavelength-matched quarter wave plate ($Q$) before entering the photoelastic material. A mirrored surface on the bottom of each particle reflects light back through the particle. A second quarter-wave plate and linear polarizer are mounted on the camera to resolve the photoelasticity. Three images of each configuration are recorded: (c) unpolarized white light for locating particle positions, (d) polarized green light showing isochromatic fringes for calculating contact forces, and (e) an ultraviolet light for identifying the low-$\mu$ particles.

Figure 2

(a) Volume histograms $\mathcal{P}(V)$ for $\Phi =0.776$ (filled upward triangle), 0.784 (filled square), and 0.802 (filled circle) with $m=48$. (b) A semilogarithmic plot of the ratio each histogram with respect to the $\Phi =0.784$ distribution, i.e., ${\mathcal{P}}_i(V)/{\mathcal{P}}_{i=2}(V)$. (c) The inverse compactivity given by Eq. (3) plotted as a function of the inverse volume fraction where ${\mu }_B$ are shown as black filled circle and ${\mu }_S$ are red filled diamond. Large (small) symbols denote jammed (unjammed) configurations, respectively. Error bars shown are uncertainties in $\mathcal{P}(V)$ and propagated through the calculation. The inverse compactivity given by the FDT method [Eq. (4)], is shown with the solid line for comparison. (d) The ratio of the number of jammed (unjammed) configurations recorded at each $\Phi$.

Figure 3

(a) Distribution of ${\sigma }_p$ where $m=8$ and $\Gamma =0.0007$ (filled downward triangle), 0.0010 (filled square), 0.0015 (filled circle), 0.0020 (filled diamond), and $0.0024\mathrm{N}\mathrm{m}$ (filled upward triangle). A semilogarithmic plot of the (b) ratio ${\mathcal{P}}_i({\sigma }_p)/{\mathcal{P}}_j({\sigma }_p)$ where the reference system $j$ is $\Gamma =0.0015\mathrm{N}\mathrm{m}$. The pressure angoricity $A_P$ and shear angoricity $A_{\tau }$ are shown as a function of $\Gamma$ where the results using overlapping histograms for ${\mu }_B$ and ${\mu }_S$ are shown as black open circle and are red open triangle, respectively. The solid line is the angoricity calculated using FDT. The gray dashed lines provide a visual reference of the slopes 0.15 and 0.45, respectively. Inset: The scaled variance $⟨\delta {\sigma }_p^2⟩$ of the ${\mathcal{P}}_j({\sigma }_p)$ distribution, as a function of the cluster size $m$.