Nematic Order in Small Systems: Measuring the Elastic and Wall-Anchoring Constants in Vibrofluidized Granular Rods

We investigate nematic order in vibrated granular rods confined to a small quasi-2D container less than 10 rod lengths in diameter. As rod density $\rho$ increases, patterning shifts from bipolar to uniform alignment. We find that a continuum liquid crystal free energy functional captures key patterning features down to almost the particle size. By fitting theory to experiments, we estimate the relative values of bend and splay elastic constants and wall anchoring. We find that splay is softer than bend for all $\rho$ and rod lengths tested, while the ratio of the average elastic constant to wall anchoring increases with $\rho$.

Figure 1

Rod patterning in a circular container. For all images, $L/D=40$. (a),(b) Snapshots of granular experiments for $\rho =15$ and 30, respectively. Shading indicates local orientational order, $S(x,y,p=L)$. (c),(d) Local $\mathbf{n}$ around continuum defects for experiments (black lines) and minimized free energy $\mathcal{F}$ (red or gray lines) for $\rho =15$ and 30, respectively (see text). Dashed arcs are at $1.5L$ and $2.0L$ from defect cores. Bars show rod length. (e) Snapshot depicts defects dissimilar to the continuum free energy minimum: line defect (arrow 1), a close pairing of two point defects (arrows 2,3), shown for $\rho =15$. (f) Probability density of finding $S(x,y,p=L)<0.2$ at a given location in the container for $\rho =15$ (average of $>500$ images).

Figure 2

Changes in nematic director orientation around point defects. For all plots, $L/D=40$. (a) Probability distribution of finding equilibriumlike point defects in granular experiments as a function of the normalized $d$ distance from the wall, $d/L$. (b) Schematic of a point defect at a boundary. We measure the orientation of the local nematic director, $\phi$, on arcs located $r$ distance from the center of the defect, where $\theta$ is with respect to the wall normal. (c) Variation of $\phi$ with $\theta$ at $r=1.75L$ of all point defects at $\rho =15$. (d) Variation of $\phi$ with $\theta$ at $r=1.75L$ averaged over all point defects, for $\rho =12.5$ (○), 15 (×), 25 (□), and 30 ($+$), with their corresponding fits to $\mathcal{F}$ (lines). For each $L/D$ and $\rho$, we collected $>40$ defects, with an average of $\approx 150$ defects. Error bars, estimated using a bootstrapping method, fall within the symbol size.

Figure 3

Relative strength of elastic constants and wall anchoring in granular experiments. In all figures, open symbols [$L/D=40$ (○), $60$ (△)] indicate a two parameter fit, $ϵ$ and $\gamma$, of $\mathcal{F}$, while filled symbols show a single parameter fit, $\gamma$, for $\rho =30$ where $ϵ=-0.66$ (dotted line), $-0.76$ (dashed line) for $L/D=40$ (●), 60 (▴), respectively (see text). (a) Elastic constant anisotropy $ϵ$ as a function of $\rho$. (b) Dimensionless ratio of the average elastic constant to wall anchoring, $\gamma$, versus $\rho$. The line is a linear least squares fit of the data (all open symbols). (c) ${\mathcal{F}}_{\mathrm{bulk}}/{\mathcal{F}}_{\mathrm{wall}}$ versus $\rho$. Lines indicate the iso-$ϵ$ curve for $ϵ=-0.66$ (dotted), $-0.76$ (dashed), and the linear relation of $\gamma$ versus $\rho$ found in (b). Error bars in all graphs are determined by bootstrapping.