Mapping the limits of equilibrium in sheared granular liquid crystals

Abstract

Athermal elongated particles are well-known to follow Jeffery orbits when sheared in viscous fluids. It is less clear if similar orbits appear in dense granular flows. We show that when sheared for long enough, sufficiently elongated frictionless granular rods, rather than following noisy Jeffery-like orbits, exist in a quasi-equilibrium state, whose orientational statistics are quantitatively described by classical liquid crystal theory, where the noise is provided by collisions due to shear. At the same time, we demonstrate a systematic breakdown of this equilibrium analogy at two distinct limits: at low aspect ratios, where the equilibrium theory incorrectly predicts an isotropic state, and as inter-particle friction is introduced, where the system moves from steric screening to frictional gearing. Even within this frictionally geared state, the rotational dynamics remain distinct from classical Jeffery orbits. We link this frictional breakdown directly to the system being driven far from equilibrium, as quantified by an effective Ericksen number that compares non-equilibrium rotational driving to steric ordering. Our results provide a quantitative map of the transition from a quasi-equilibrium to a far-from-equilibrium steady state in a dense, driven system, defining the limits of applicability for thermal liquid crystal theory in athermal matter.

Figures (12)

• Figure 1: (a) Shear cell for $\alpha=5.0$, $\mu_p=0.01$, with particles colored by their normalized flow velocity, $v_f$, with respect to the uniform gradient $\dot \gamma L_g /2$, $L_g$ being the average box length in the gradient direction. (b) Nematic order parameter $S_2$ and (c) packing fraction $\phi$ as a function of aspect ratio $\alpha$. The plots show that increasing inter-particle friction $\mu_p$ (from dark blue to yellow) reduces nematic ordering (lowering $S_2$) and induces dilatancy (lowering $\phi$). The gray line marks the theoretical prediction for Jeffery orbits, without noise or steric interaction, starting from an isotropic distribution, as done in talbot_exploring_2024. All the data lie above this prediction, confirming that Jeffery orbits are not sufficient to describe the dense granular flows. The color scheme for $\mu_p$ is used in subsequent figures.
• Figure 2: Quantifying the "screening-to-gearing" transition. (a) Normalized average angular velocity $\tilde{\omega}$ extracted from simulations across varying friction coefficients (markers). Colors denote the particle aspect ratio $\alpha$. Solid lines represent the theoretical Jeffery prediction for an isolated particle in a viscous fluid, which fails to capture the dense granular dynamics. Instead, the data are well-described by a two-state phenomenological model (dashed lines) capturing the transition between a low-friction "screening" state and a high-friction "gearing" state. In the gearing state, grains rotate faster than the isolated particle prediction, whereas in the screening state, their rotation is severely hindered. The inset schematic illustrates the shear setup, defining the in-plane angle for a single particle $\vartheta$. (b) Normalized angular velocity $\langle \omega (\vartheta)\rangle / \dot{\gamma}$ vs. the angle with respect to the flow direction in the flow-gradient plane, $\vartheta$, for $\alpha=5.0$. The dashed black line represents the theoretical prediction for Jeffery orbits, while the vertical dotted lines indicate the measured director angle ($\eta$) in the flow-gradient plane. In the low-friction "screening" regime (dark curves, $\mu_p \le 0.01$), $\omega \approx 0$ as steric forces (director at $\eta$, dotted lines) cancel the shear torque (Jeffery orbit, black dashed line), meaning the aligned grains effectively lock into place and slide past one another without spinning. In the high-friction "gearing" regime (yellow curves, $\mu_p \ge 1.0$), $\omega$ is large, showing strong rotational driving. (c) The effective Ericksen number $\Pi$ shows the consequence. The system is in the quasi-equilibrium limit ($\Pi \ll 1$) for low friction. As $\mu_p$ increases, $\Pi$ grows by orders of magnitude, crossing the $\Pi=1$ threshold (dashed black line) and signaling a transition to a far-from-equilibrium, drive-dominated state.
• Figure 3: (a1-a3) Log-scale orientational distributions $f(\theta)$ (markers) compared to the equilibrium Maier-Saupe theory (Eq. \ref{['eq:eq_3D']}, solid lines). The data reveals the breakdown of the equilibrium analogy as friction increases. (Left) In the low friction, screening regime ($\mu_p=0.001$), the theory accurately predicts the simulation data at high elongation, but fails at small $\alpha$. (Middle) In the intermediate friction case ($\mu_p=0.1$), the simulated distributions become noticeably broader than predicted, though the theory still qualitatively captures the overall trends. (Right) In the high-friction, gearing regime ($\mu_p=10.0$), the theory fails, underestimating the broadening of the distribution even at high $\alpha$. Within each panel, colors represent aspect ratio $\alpha$, with higher $\alpha$ (yellow) leading to stronger ordering (sharper peaks). (b) Nematic order parameter $S_2$ as a function of aspect ratio $\alpha$, comparing experimental measurements wegner_alignment_2012 (black circles) with equilibrium theory predictions (red squares) and average for simple shear rod simulations with $\mu_p \in (0.1, 0.7)$, (blue hollow diamonds). Theoretical points are shifted slightly along the horizontal axis for visual clarity. At small elongations, the theory predicts an isotropic state ($S_2 \approx 0$) rather than the observed nematic distribution. Conversely, at high friction the order parameter is remarkably close to the one from equilibrium theory. At moderate and high elongation there is a great correspondence between split-bottom shear cell experiments and simple shear simulations. Red error bars represent the upper theoretical bounds assuming a $+10\%$ uncertainty (underestimation) in the experimentally measured volume fraction. (c) Comparison of the experimental in-plane orientation distribution $p(\theta_{2d})$ for $\alpha=5$ (black markers) wegner_alignment_2012 against a theoretical reconstruction (red solid line). The reconstruction represents a 2D projection of the uniaxial 3D distribution estimated from the measured nematic order parameter, shifted to align with the experimentally measured average in-plane angle. The red shaded region reflects a plausible $+10\%$ uncertainty in the experimentally measured volume fraction due to binarization.
• Figure 4: The relative deviation between the measured nematic order parameter and the equilibrium prediction, $|S_2 - S_2^{\text{eq}}|/S_2$, plotted against the effective Ericksen number $\Pi$. Data points include both spheroids (squares) and rods (hollow triangles) simulations for different aspect ratios. The collapse confirms that $\Pi$ acts as a control parameter: for $\Pi \ll 1$ (steric screening), the system remains in a quasi-equilibrium state with minimal deviation, while for $\Pi \gtrsim 1$ (frictional gearing), the system is driven far from equilibrium, leading to a systematic power-law divergence from the theoretical prediction.
• Figure 5: Transient evolution of (a) the nematic order parameter $S_2$, (b) the director angle $\eta$, and (c) the biaxiality $r$ as a function of accumulated strain $\gamma$in simulations. Data is shown for aspect ratios $\alpha=3.0$ (left column) and $\alpha=7.0$ (right column). All systems converge to stable plateaus, confirming the absence of director tumbling. While increasing elongation slows down the dynamics, increasing friction (lighter colors) accelerates the convergence to the steady state.