Ordering in granular-rod monolayers driven far from thermodynamic equilibrium

The orientational order in vertically agitated granular-rod monolayers is investigated experimentally and compared quantitatively with equilibrium Monte Carlo simulations and density functional theory. At sufficiently high number density, short rods form a tetratic state and long rods form a uniaxial nematic state. The length-to-width ratio at which the order changes from tetratic to uniaxial is around $7.3$ in both experiments and simulations. This agreement illustrates the universal aspects of the ordering of rod-shaped particles across equilibrium and nonequilibrium systems. Moreover, the assembly of granular rods into ordered states is found to be independent of the agitation frequency and strength, suggesting that the detailed nature of energy injection into such a nonequilibrium system does not play a crucial role.

Figure 1

Sketch of the experimental setup. The closed cylindrical container of height $H$ and radius $R$ is driven sinusoidally against gravity with an electromagnetic shaker. The rods have a length $L$ and a diameter $D$. The embedded image shows a close view of the detected particles.

Figure 2

Raw experimental images (topview) showing typical configurations of rods with aspect ratio $L/D=3.3$ (a) and $L/D=10.0$ (b) at high global area fractions. The yellow (light gray) dashed circle indicates the region of interest.

Figure 3

Typical snapshots of tetratic (left column, $L/D=3.3$) and uniaxial nematic (right column, $L/D=10.0$) states. Panels (a) and (d) are reconstructed from the positions and orientations of the particles detected in the center region of the container. Panels (b) and (e) are from MC simulations with periodic boundary conditions. The particles are color coded according to their orientations. Panels (c) and (f) show the orientational distribution functions $h\left(\gamma \right)$ of the particles in experiments (gray bars) and simulations (solid line). $\gamma$ is the angle with respect to the director.

Figure 4

Wall-rod angular correlation function $g_4$ as a function of the rescaled distance $s/R$ to the wall for various $L/D$. A sketch with various definitions is shown in the inset. The data are obtained through an average over all $\mathrm{\Gamma }$, global area fractions ${\mathrm{\Phi }}_{\mathrm{g}}$, and experimental runs. The typical error for $g_4$ ($\sim 5\times {10}^{-3}$) is comparable to the size of the symbols for $s/R<0.5$. Note the logarithmic scale on the $y$ axis.

Figure 5

Tetratic $q_4$ and uniaxial $q_2$ order parameters in the ROI as a function of the area fraction $\mathrm{\Phi }$ in a system of rods with aspect ratio $L/D=3.3$: (a) For three ranges of $\mathrm{\Gamma }$ at $f=50$ Hz and (b) for four $f$ with $q_k$ accumulated over all $\mathrm{\Gamma }$. The threshold ${\mathrm{\Phi }}_{\mathrm{c}}$ is obtained through fits to the data (straight lines) accumulated over all $\mathrm{\Gamma }$ for $f=50$ Hz (see text for details). The order parameters are not exactly zero in the isotropic state due to the finite size of the system [40].

Figure 6

Tetratic and uniaxial order parameters $q_{\mathrm{k}}$ as a function of the area fraction $\mathrm{\Phi }$ for (a) $L/D=3.3$ and (b) $L/D=10.0$. The insets show the corresponding results from MC simulations. The experimental data is an accumulation over all $\mathrm{\Gamma }$ at $f=50$ Hz. Straight lines are linear fits to determine the threshold ${\mathrm{\Phi }}_{\mathrm{c}}$.

Figure 7

State diagram in the plane of aspect ratio $L/D$, and area fraction $\mathrm{\Phi }$ obtained via experiments of agitated granular rods (circles) and MC simulations of thermally driven hard rectangles (triangles). The labels denote the states: isotropic (I), uniaxial nematic (U), and tetratic (T). They are colored according to the experimental data. The diameter of the rods used in the experiments is $D=3.0$ mm for $L/D\le 5$ (gray symbols) and $D=1.5$ mm for $L/D\ge 5$ (black symbols). Closed and open symbols indicate the I-T and I-U transitions, respectively. The inset shows the state diagram of equilibrium hard rods according to DFT in comparison to MC simulations in an extended region of $L/D$. Dashed lines are continuous transitions and solid lines denote first-order transitions.