Measuring geometric frustration in twisted inextensible filament bundles

We investigate with experiments and mapping the structure of a hexagonally ordered filament bundle that is held near its ends and progressively twisted around its central axis. The filaments are free to slide relative to each other and are further held under tension-free boundary conditions. Measuring the bundle packing with micro x-ray imaging, we find that the filaments develop the helical rotation $\mathrm{\Omega }$ imposed at the boundaries. We then show that the observed structure is consistent with a mapping of the filament positions to disks packed on a dual non-Euclidean surface with a Gaussian curvature which increases with twist. We further demonstrate that the mean interfilament distance is minimal on the surface, which can be approximated by a hemisphere with an effective curvature $K_{\mathrm{eff}}=3{\mathrm{\Omega }}^2$. Examining the packing on the dual surface, we analyze the geometric frustration of packing in twisted bundles and find the core to remain relatively hexagonally ordered with interfilament strains growing from the bundle center, driving the formation of defects at the exterior of highly twisted bundles.

Figure 1

(a) A schematic of a fiber bundle held within two hexagonal clamps that are twisted through a prescribed angle $\alpha$. Bundle transects obtained with x-ray scanning corresponding to (b) $\alpha =0^{\circ }$, (c) $\alpha ={129}^{\circ }$, and (d) $\alpha ={176}^{\circ }$.

Figure 2

[(a)–(c)] Reconstructed filament bundles color coded according to the tilt angle $\theta$ of the filament shown in the color bar. The maximum tilt angle ${\theta }_{\max }=0^{\circ }$ (a), ${\theta }_{\max }={20}^{\circ }$ (b), and ${\theta }_{\max }={30}^{\circ }$ (c). (d) $\theta$ as a function of the helical radius $\rho$ binned in $d/2$ intervals. The lines are linear fits. (e) The average helical rotation $\mathrm{\Omega }$ as a function of the helical radius $\rho$ binned in $d/2$ intervals. The bars correspond to the root-mean-square of $\mathrm{\Omega }$ measured for filaments in that bin, and the horizontal dashed lines to the mean values ${\mathrm{\Omega }}_m$.

Figure 3

Disks plotted on a surface given by Eq. (1) for (a) ${\theta }_{\max }={20}^{\circ }$ and (b) ${\theta }_{\max }={30}^{\circ }$. [(c) and (d)] The probability distribution function (PDF) of distances shown in the legend. ${\mathrm{\Delta }}_{ff}$ and ${\mathrm{\Delta }}_{dd}$ are observed to be similar and distinct from the PDF of ${\mathrm{\Delta }}_{pp}$ for both (c) ${\theta }_{\max }={20}^{\circ }$ and (d) ${\theta }_{\max }={30}^{\circ }$.

Figure 4

(a) Comparison of Eq. (1) (solid surface) with spherical approximation for ${\theta }_{\max }={30}^{\circ }$ (mesh) for ${\theta }_{\max }={30}^{\circ }$. (b) ${\delta }^2$ as a function of the Gaussian curvature $K_{\mathrm{eff}}$ is observed to reach a minimum when $K_{\mathrm{eff}}/{\mathrm{\Omega }}^2=3$ for both twist angles consistent with the quasihemispherical mapping.

Figure 5

Planar transect of the bundle corresponding to (a) ${\theta }_{\max }={20}^{\circ }$ and (b) ${\theta }_{\max }={30}^{\circ }$ in a plane perpendicular to the bundle axis labeled according to the strain ${\epsilon }_n$. [(c) and (d)] The corresponding disk packing on the dual surface given by Eq. (1) viewed from above. ${\epsilon }_n$ is observed to be significantly lower by comparison.

Figure 6

(a) A schematic of the 3-point flexural test system. (b) The measured load versus deflection graph can be described by a linear fit.

Figure 7

Helical fit to experimental data corresponding to ${\theta }_{\max }={30}^{\circ }$. The color of the helix which is extended beyond the data for clarity corresponds to the angle $\theta$ that the filament subtends with the vertical axis of the filament bundle.