Helical Ribbons: Simple Chiral Sedimentation

TL;DR

This paper addresses how chiral particle shapes couple translation and rotation in Stokes flow, identifying co-centered helical ribbons as a simple, highly coupled class with a reduced 9-parameter mobility description. The authors fabricate four ribbons, measure full 3D translation-rotation dynamics, and extract the mobility tensors $\mathbb{a}$ and $\mathbb{b}$, finding good agreement with bead-model simulations. They observe quasi-periodic tilt-spin dynamics with closed orbits in tilt-spin space and report a length-dependent bifurcation near $2\pi L/s \approx 4\pi/3$ where fixed points swap stability, with axisymmetric coupling ($b_{11}=b_{22}$) at special lengths. These results offer a benchmark for designing geometries that optimize translation-rotation coupling in viscous flows and lay groundwork for extending the roadmap to more complex, non-co-centered shapes.

Abstract

We investigate the sedimentation of chiral particles in viscous fluid flow. We identify helical ribbons as simple particles with strong translation-rotation coupling whose symmetry ensures that the centers of mass, buoyancy, resistance, and mobility coincide. Experimental measurements of both relevant mobility tensors show excellent agreement with simulations of ribbons made of interacting spheres. We observe quasi-periodic angular dynamics causing complex spatial trajectories. In tilt-spin phase space, orbits are closed due to time-reversal and reflection symmetry. Changing the helical ribbon length reveals a bifurcation at which the stable sedimentation orientations switch.

Figures (7)

• Figure 1: (a) Body coordinate system and dimensions for a helical ribbon with Length $2\pi L/s=5\pi/4$ (b) A measured trajectory in space (Helix is not to scale, length $3\pi/4$) (c) top view. Red dots are measurements. Yellow line is a numerical integration using measured parameters for this particle. Blue line continues the numerical trajectory. (d) Same measured trajectory in tilt-spin space. Supplemental video is available sm.
• Figure 2: Angular dynamics for different particle lengths, $2\pi L/s$: (a) $3\pi/4$, (b) $5\pi/4$, (c) $4\pi/3$, and (d) $3\pi/2$. Particle images are above each plot. Solid black lines show numerically integrated trajectories. Color symbols indicate distinct experimental trajectories.
• Figure 3: Diagonal elements of the mobility tensors for $\mathbb{a}$ (top) and $\mathbb{b}$ (bottom) as functions of helical ribbon length. Diamonds indicate experimentally measured values. Solid lines are numerical simulations using the bead-model described in the text.
• Figure 4: Normalization factors for mobility tensors as a function of helical ribbon length. (Blue) Average of the eigenvalues of $\mathbb{a}$ (Red) Largest eigenvalue of $\mathbb{b}$.
• Figure 5: Spatial trajectories from numerical simulations for selected initial conditions. Insets show the same trajectories (matched by color) in tilt-spin phase space. Helical ribbons lengths are (a) $3\pi/4$, (b) $5\pi/4$, (c) $4\pi/3$, and (d) $3\pi/2$. Videos showing the full range of trajectories are available sm.