Colloquium: Geometry and optimal packing of twisted columns and filaments

This Colloquium presents recent progress in understanding constraints and consequences of close-packing geometry of filamentous or columnar materials possessing nontrivial textures, focusing, in particular, on the common motifs of twisted and toroidal structures. The mathematical framework is presented that relates spacing between linelike, filamentous elements to their backbone orientations, highlighting the explicit connection between the interfilament metric properties and the geometry of non-Euclidean surfaces. The consequences of the hidden connection between packing in twisted filament bundles and packing on positively curved surfaces, like the Thomson problem, are demonstrated for the defect-riddled ground states of physical models of twisted filament bundles. The connection between the “ideal” geometry of fibrations of curved three-dimensional space, including the Hopf fibration, and the non-Euclidean constraints of filament packing in twisted and toroidal bundles is presented, with a focus on the broader dependence of metric geometry on the simultaneous twisting and folding of multifilament bundles.

Figure 1

(a)–(e) Twisted bundles (a)of filaments or columns in biological and synthetic materials [electron microscope (EM) images in (b)–(e))]: (b) fibrin bundles (diameter $\sim 100\mathrm{nm}$) ([87]); (c) twisted collagen fibrils derived from tendon (diameter $\sim 100\mathrm{nm}$) ([70]); (d) twisted fibers of chiral organogel assemblies (diameter $\sim 100\mathrm{nm}$) ([26]); and (e) mesoporous silica templated by twisted columnar assemblies of wormlike surfactant micelles, with schematic in the inset (diameter $\sim 100\mathrm{nm}$) ([91]). (f)–(i) Toroidal bundles [schematic of twisted toroidal bundle in (f)] of filaments or columns from biological and synthetic materials: (g) EM images of twisted toroidal fibers of collagen ([20]); (h) EM images of toroidal condensates of dsDNA ([40]); and (i) schematic and optical microscopy of faceted columnar droplets of chromonic liquid crystals ([44]).

Figure 2

Local distance of closest approach ${\mathbf{\Delta }}_{*}$ between filament $\alpha$ at $s_{\alpha }^0$ to filament $\beta$, where $s_{\beta }^{*}$ is the closest point to ${\mathbf{R}}_{\alpha }^0$.

Figure 3

Examples of filament textures with positive $K_{\mathrm{eff}}>0$ and negative $K_{\mathrm{eff}}<0$ effective curvatures, whose equivalent surface geometry is shown schematically with spherical and saddlelike surface patches.

Figure 4

(a) A triplet of three twisted filaments, with lines indicating distances of closest approach between them. (b) The mapping of interfilament spacing onto the geodesic separation between three points that form vertices of a geodesic triangle on a positively curved (spherical) surface patch. The Gauss-Bonnet relates the sum of interior angles (labeled as ${\theta }_v$) to the integrated Gaussian curvature within the triangular patch; see Eq. (6).

Figure 5

A (double-)twisted bundle, where the color gradient highlights the radial distance of filaments from the central filament. A single, helical filament is shown in the upper portion to highlight the local tilt angle $\theta (\rho )$ between the filament at radius $\rho$ and the pitch axis.

Figure 6

(a) The packing of finite-diameter filaments at a radial distance $\rho$ from the bundle center. The amount of space available for packing filaments at $\rho$ is determined by the length $\ell (\rho )$ of a curve between to two points of “self-contact” along the same filament. (b) The 2D axisymmetric surface that carries the interfilament metric properties of a twisted bundle. The lines of latitude of length $\ell (\rho )$ as defined by the geometry in (a).

Figure 7

Equivalence of finite-diameter filament packing in twisted bundles, and finite-diameter disk packing on a “domelike” surface carrying the metric of a twisted bundle. (a) The side view and (b) the top view of a twisted bundle, highlighting noncircular shapes of the filament intersections with the plane perpendicular to the pitch axis. (c) A side view of the equivalent disk packing on the “dome” shown in Fig. 6, and (d), (e) two “polar” projections of the disk packing. The orthographic projection in (d), a view from the top down, preserves distances along the azimuthal direction while compressing distances along the radial direction. (e) The azimuthal equidistant projection preserves radial distances while stretching azithumal distances, producing the identical image of the filament intersections (azimuthally stretched disks) shown in (b).

Figure 8

A simulated ground state of an $N=70$ twisted bundle from [12] is shown in side view in (a), along with the corresponding disk packing on the bundle-equivalent surface in (b). Triangulation of the packing on the curved surface yields the nearest-neighbor “bond network,” identifying defects in the packing as deviations from sixfold coordination of the bond network (i.e., disclinations). Filaments with fivefold, sixfold, and sevenfold neighbor coordination are highlighted in different shades. (c) The total topological charge of the ground-state packing $Q$, defined in Eq. (20), plotted as a function of the twist angle of the outermost filament in the bundle $\theta =\arctan (\mathrm{\Omega }R)$ with the colored data points showing results from simulated ground states and the solid line showing the geometric prediction for the ideal topological charge given by Eq. (24).

Figure 9

Simulated ground states of twisted filament bundles. (a) Optimal packings of a 34-filament bundle, with increasing twist angle showing an increase in the number of fivefold (disclination) defects. (b) The total topological charge of simulated ground states for bundles of variable twist angle $\theta =\arctan (\mathrm{\Omega }R)$ and filament number $N$. (c) The number of disclinations per charge $N_{\mathrm{d}\text{isclination}}/Q$ is shown for simulated ground states, with dark lines drawn to guide the eye to regions of roughly constant value. (d) A series of simulated ground states at fixed $\theta \approx 30°$ (corresponding to $Q=1$) with increasing $N$, showing the transition from compact disclinations to extended “charged scars” of alternating 5–7 defect pairs. Adapted from [12], [13].

Figure 10

(a) The bulk energy density (total energy minus excess energy of surface filaments) of simulated ground states of twisted bundles vs twist angle, for large bundle sizes $N=166-193$ as computed by [13]. Energy curves are overlaying the results for $Q$, highlighting the coincidence of multiple minima in the energy density with stepwise transitions in an optimal value of $Q$. (b) The shape of the simulated bulk energy density is compared to continuum elasticity theory calculations for twisted bundles possessing only fivefold disclinations, calculated by [31], [32], with optimal arrangement of defects shown for each distinct branch (corresponding to distinct $Q$ values) of energy minimal. The dashed lines show the metastable branches of defect-free and $Q=1$ elastic energy density, which meet at the transition point $(\mathrm{\Omega }R{)}_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}=\sqrt{2/9}$ (corresponding to ${\theta }_{\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{c}}\simeq 25°$).

Figure 11

(a) Stability phase diagram for defects in twisted bundles, calculated from continuum elasticity theory ([3]), showing regions where dislocations (neutral 5–7 disclination pairs) and charged defect configurations possessing at least one excess fivefold disclination are stable relative to the defect-free bundle. Here $a$ is the interfilament lattice spacing. (b) The “scarred,” multidislocation ground state at intermediate twist for sufficiently large bundles (i.e., $R/a\gg 1$), with fivefold and sevenfold coordinated filaments.

Figure 12

(a) Optimal number of dislocations vs bundle twist in a $R=100a$ bundle, from continuum theory of twisted bundles ([3]). The integer pair $(m,n)$ refers to structures with $m$ “scars” each possessing $n$ dislocations. (b) The collapse of the total number of defects (from bundles $R/a=20-700$) with parameter $R/a[(\mathrm{\Omega }R{)}^2-(\mathrm{\Omega }R{)}_{*}^2]$ where $(\mathrm{\Omega }R{)}_{*}$ is the critical twist for stable dislocations. The inset of (b) shows the proportionality between the total dislocation number and optimal scar number, roughly predicting six dislocations per scar independent of $R/a$. Here the color scale indicates the gradient in bundle sizes corresponding to large and smaller $R/a$, respectively.

Figure 13

(a) The toroidal coordinate system of fibrations in $S^3$, showing the dimensions of the $T^2$ unit cell at fixed $\mathrm{\Theta }$. Fibers (filament backbones) wind along the dark solid lines at an angle $\theta$ with respect to the $\widehat{\psi }$ axis. The perimeter is defined as the distance of closest contact between the fiber at $\varphi$ and its periodic image at $\varphi +2\pi$. (b) Surfaces (bases) carrying the interfilament metric of fibrations in $S^3$.

Figure 14

The ideal packings of twisted and equally spaced filaments (diameter $d$) in $S^3$ whose positions correspond to vertices of Platonic solids projected on $S^2$, where $\varphi$, $\mathrm{\Omega }d$, and $z$ denote packing fraction, reduced twist, and coordination number of the packing. From [49].

Figure 15

Toroidal coordinates under stereographic projection to $R^3$. Surfaces of constant $\mathrm{\Phi }$ are concentric tori, and fibers and filaments wind along these surfaces around both the minor and major axes of the tori, where $\theta$ is the (constant) angle between tangents and the circular axis of the torus. The circular fiber shown here corresponds to $\alpha =1$, a projection of the Hopf fibration.

Figure 16

(a) Stereographic projections of twisted filament packings in $S^3$ to $R^3$, based on the Hopf vibration ($\alpha =1$). Filament positions on the $S^2$ base derive from the icosodeltahedral tesselations, where the value of $\mathrm{\Omega }d$ is chosen based on the distance between the central filament (at the pole in $S^2$) and its first shell of neighbors. (a) $\{2,2\}$ packings are shown, both on $S^2$ (top) and projections to $R^3$ (bottom). Left to right shows examples with increasingly larger maximum $\mathrm{\Phi }$, corresponding to larger polar distance on $S^2$, larger toroidal thickness, and larger twist angles of the outermost filaments. (b) The strain energy density defined by Eq. (39) and calculated numerically and plotted vs the twist angle of the outer filaments for icosodeltahedral tesselations, for a range of tesselations from small twist ${\mathrm{\Omega }}_{\{4,2\}}=0.0895d^{-1}$ to large twist ${\mathrm{\Omega }}_{\{1,1\}}=0.325d^{-1}$.