Non-Euclidean geometry of twisted filament bundle packing

Abstract

Densely packed and twisted assemblies of filaments are crucial structural motifs in macroscopic materials (cables, ropes, and textiles) as well as synthetic and biological nanomaterials (fibrous proteins). We study the unique and nontrivial packing geometry of this universal material design from two perspectives. First, we show that the problem of twisted bundle packing can be mapped exactly onto the problem of disc packing on a curved surface, the geometry of which has a positive, spherical curvature close to the center of rotation and approaches the intrinsically flat geometry of a cylinder far from the bundle center. From this mapping, we find the packing of any twisted bundle is geometrically frustrated, as it makes the sixfold geometry of filament close packing impossible at the core of the fiber. This geometrical equivalence leads to a spectrum of close-packed fiber geometries, whose low symmetry (five-, four-, three-, and twofold) reflect non-Euclidean packing constraints at the bundle core. Second, we explore the ground-state structure of twisted filament assemblies formed under the influence of adhesive interactions by a computational model. Here, we find that the underlying non-Euclidean geometry of twisted fiber packing disrupts the regular lattice packing of filaments above a critical radius, proportional to the helical pitch. Above this critical radius, the ground-state packing includes the presence of between one and six excess fivefold disclinations in the cross-sectional order.self-assembly | topological defects | geometric frustration P acking problems arise naturally in a multitude of contexts, from models of crystalline and amorphous solids to structure formation in tissues of living organisms. Though often easy to state, packing problems are notoriously difficult to solve. The cannonball stacking of spheres, for example, conjectured by Kepler to be the densest packing in three-dimensional space (1), was only proven so in 1999, and even then with the aid of a computer algorithm (2). In some cases, like the densest packing of spheres, the face-centered cubic lattice-or its two-dimensional analogue, the densest packing of discs, the hexagonal lattice-the optimal structure is a regular, periodic partition of space. In many problems, however, perfect lattice packings are prohibited, and optimal structures necessarily lack the translational and rotational symmetry of periodic order. A well-known example is the problem of finding the densest packing of N discs on a sphere of a given radius, known alternately as the Tammes or the generalized-Thomson problem (3). In this problem, spherical topology requires a variation in the local packing symmetry: All states possess at minimum 12 discs whose nearest-neighbor geometry is fivefold coordinated (4). Beyond its relevance to structural studies of such diverse materials as spherical viral capsids (5) and colloid-stabilized emulsions (6, 7), the problems involving point or disc packing on spheres serve as an important example of systems where topological de...

Geometric frustration and compatibility conditions for two-dimensional director fields

“…In this work we provide an explicit formula for the Gaussian curvature associated with given splay and bend fields, thus mapping the geometrically frustrated problem of bent-core liquid crystals (or any other local splay/bend tendency) in two dimensions to the much studied realm of optimal embedding of manifolds of mismatched Gaussian curvatures. A problem encountered in elasticity [13][14][15], crystal growth of curved surfaces [11,12] and the bundling of twisted filaments [16,17].…”

Perspective: Geometrically frustrated assemblies

“…31,88,104,108 Excess 5-fold defects, characteristic of positive Gaussian curvature surfaces, have been observed in multiple numerical simulation models of twisted bundles (see Fig. 7(c)), both when twist emerges from chiral interfilament forces 88,98 and when twist is imposed mechanically on otherwise achiral bundles.…”

“…102,104 Specifically, any 2D cross section of a filament assembly maps directly on a 2D curved surface, whose geodesic distances encode the distance of closest approach (or metric) between filament backbones. The Gaussian curvature of this surface K eff derives directly from the tilt gradients in the cross section,…”

Ideal geometry of periodic entanglements

“…The extension is similar to that developed for tight open knots [67][68][69], however, it goes further to support three-periodicity. As an initial test of the algorithm, it replicates tight knot embeddings, comparable to the SONO algorithm, and we present tight embeddings of entangled θ-, tetrahedron-and cube-graphs to demonstrate its efficacy at dealing with vertices in the structure.…”