Defects in crystalline packings of twisted filament bundles. II. Dislocations and grain boundaries

Stiffness heterogeneity of small viral capsids

Nanoindentation of viral capsids provides an efficient tool in order to probe their elastic properties. We investigate in the present work the various sources of stiffness heterogeneity as observed in atomic force microscopy experiments. By combining experimental results with both numerical and analytical modeling, we first show that for small viruses, a position-dependent stiffness is observed. This effect is strong and has not been properly taken into account previously. Moreover, we show that a geometrical model is able to reproduce this effect quantitatively. Our work suggests alternative ways of measuring stiffness heterogeneities on small viral capsids. This is illustrated on two different viral capsids: Adeno associated virus serotype 8 (AAV8) and hepatitis B virus (HBV with T=4). We discuss our results in light of continuous elasticity modeling.

Mechanical stress relaxation in molecular self-assembly

Molecular self-assembly on a curved substrate leads to the spontaneous inclusion of topological defects in the growing bidimensional crystal, unlike assembly on a flat substrate. We propose in this work a quantitative mechanism for this phenomenon by using standard thin shell elasticity. The Gaussian curvature of the substrate induces large in-plane compressive stress as the surface grows, in particular at the rim of the assembly, and the addition of a single defect relaxes this mechanical stress. We found out that the value of azimuthal stress at the rim of the assembly determines the preferred directions for defect nucleation. These results are also discussed as a function of different defect combinations, like dislocations and grain boundaries or scars. In particular, the elastic model permits us to compare quantitatively the ability of various defects to relax mechanical stress. Moreover, these findings allow us to understand the progressive building-up of the typical disclination and grain boundary pattern observed for ground states of large 2D spherical crystals.

Growth of curved crystals: competition between topological defect nucleation and boundary branching

Topological defect nucleation and boundary branching in crystal growth on a curved surface are two typical elastic instabilities driven by curvature induced stress, and have usually been discussed separately in the past. In this work they are simultaneously considered during crystal growth on a sphere. Phase diagrams with respect to sphere radius, size, edge energy and stiffness of the crystal for the equilibrium crystal morphologies are achieved by theoretical analysis and validated by Brownian dynamics simulations. The simulation results further demonstrate the detail of morphological evolution governed by these two different stress relaxation modes. Topological defect nucleation and boundary branching not only compete with each other but also coexist in a range of combinations of factors. Clarification of the interaction mechanism provides a better understanding of various curved crystal morphologies for their potential applications.

Topological vacancies in spherical crystals

Understanding geometric frustration of ordered phases in two-dimensional condensed matter on curved surfaces is closely related to a host of scientific problems in condensed matter physics and materials science. Here, we show how two-dimensional Lennard-Jones crystal clusters confined on a sphere resolve geometric frustration and form pentagonal vacancy structures. These vacancies, originating from the combination of curvature and physical interaction, are found to be topological defects and they can be further classified into dislocational and disclinational types. We analyze the dual role of these crystallographic defects as both vacancies and topological defects, illustrate their formation mechanism, and present the phase diagram. The revealed dual role of the topological vacancies may find applications in the fabrication of robust nanopores. This work also shows the promising potential of exploiting richness in both physical interactions and substrate geometries to create new types of crystallographic defects, which have strong connections with the design of crystalline materials.

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 Ω 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_{eff}=3Ω^{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.

Neutral versus charged defect patterns in curved crystals

Characterizing the complex spectrum of topological defects in ground states of curved crystals is a long-standing problem with wide implications, from the mathematical Thomson problem to diverse physical realizations, including fullerenes and particle-coated droplets. While the excess number of "topologically charged" fivefold disclinations in a closed, spherical crystal is fixed, here we study the elementary transition from defect-free, flat crystals to curved crystals possessing an excess of "charged" disclinations in their bulk. Specifically, we consider the impact of topologically neutral patterns of defects-in the form of multidislocation chains or "scars" stable for small lattice spacing-on the transition from neutral to charged ground-state patterns of a crystalline cap bound to a spherical surface. Based on the asymptotic theory of caps in continuum limit of vanishing lattice spacing, we derive the morphological phase diagram of ground-state defect patterns, spanned by surface coverage of the sphere and forces at the cap edge. For the singular limit of zero edge forces, we find that scars reduce (by half) the threshold surface coverage for excess disclinations. Even more significant, scars flatten the geometric dependence of excess disinclination number on Gaussian curvature, leading to a transition between stable "charged" and "neutral" patterns that is, instead, critically sensitive to the compressive vs tensile nature of boundary forces on the cap.

Rotating crystals of magnetic Janus colloids

Monodisperse magnetic colloids are found to self-assemble into unusual crystals in the presence of rotating magnetic fields. First, we confirm a predicted phase transition (S. Jäger and S. H. L. Klapp, Soft Matter, 2011, 7, 6606-6616), directly coupled to the dynamic transition of single particle motion, from a disordered state to a hexagonal crystal. Next, going beyond what had been predicted, we report how hydrodynamic coupling produces shear melting, dislocations, and periodically mobile domain boundaries. These uniform magnetic colloids, whose structures are modulated in situ using the protocols described here, demonstrate a strategy of stimulus-response in the colloid domain with potential applications.

Emergent structure of multidislocation ground States in curved crystals

We study the structural features and underlying principles of multidislocation ground states of a crystalline spherical cap. In the continuum limit where the ratio of crystal size to lattice spacing W/a diverges, dislocations proliferate and ground states approach a characteristic sequence of structures composed of radial grain boundaries ("neutral scars"), extending radially from the boundary and terminating in the bulk. Employing a combination of numerical simulations and asymptotic analysis of continuum elasticity theory, we prove that an energetic hierarchy gives rise to a structural hierarchy, whereby dislocation number and scar number diverge as a/W→0 while scar length and dislocation number per scar become independent of lattice spacing. We characterize a secondary transition occurring as scar length grows, where the n-fold scar symmetry is broken and ground states are characterized by polydisperse, forked-scar morphologies.

Universal collapse of stress and wrinkle-to-scar transition in spherically confined crystalline sheets

Imposing curvature on crystalline sheets, such as 2D packings of colloids or proteins, or covalently bonded graphene leads to distinct types of structural instabilities. The first type involves the proliferation of localized defects that disrupt the crystalline order without affecting the imposed shape, whereas the second type consists of elastic modes, such as wrinkles and crumples, which deform the shape and also are common in amorphous polymer sheets. Here, we propose a profound link between these types of patterns, encapsulated in a universal, compression-free stress field, which is determined solely by the macroscale confining conditions. This "stress universality" principle and a few of its immediate consequences are borne out by studying a circular crystalline patch bound to a deformable spherical substrate, in which the two distinct patterns become, respectively, radial chains of dislocations (called "scars") and radial wrinkles. The simplicity of this set-up allows us to characterize the morphologies and evaluate the energies of both patterns, from which we construct a phase diagram that predicts a wrinkle-scar transition in confined crystalline sheets at a critical value of the substrate stiffness. The construction of a unified theoretical framework that bridges inelastic crystalline defects and elastic deformations opens unique research directions. Beyond the potential use of this concept for finding energy-optimizing packings in curved topographies, the possibility of transforming defects into shape deformations that retain the crystalline structure may be valuable for a broad range of material applications, such as manipulations of graphene's electronic structure.

Non-euclidean geometry of twisted filament bundle packing

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.

Defects in crystalline packings of twisted filament bundles. I. Continuum theory of disclinations

We develop the theory of the coupling between in-plane order and out-of-plane geometry in twisted, two-dimensionally ordered filament bundles based on the nonlinear continuum elasticity theory of columnar materials. We show that twisted textures of filament backbones necessarily introduce stresses into the cross-sectional packing of bundles and that these stresses are formally equivalent to the geometrically induced stresses generated in thin elastic sheets that are forced to adopt spherical curvature. As in the case of crystalline order on curved membranes, geometrically induced stresses couple elastically to the presence of topological defects in the in-plane order. We derive the effective theory of multiple disclination defects in the cross section of bundle with a fixed twist and show that above a critical degree of twist, one or more fivefold disclinations is favored in the elastic energy ground state. We study the structure and energetics of multidisclination packings based on models of equilibrium and nonequilibrium cross-sectional order.