Defects in chiral columnar phases: Tilt-grain boundaries and iterated moiré maps

Geodesic fibrations for packing diabolic domains

We describe a theory of packing hyperboloid "diabolic" domains in bend-free textures of liquid crystals. The domains sew together continuously, providing a menagerie of bend-free textures akin to the packing of focal conic domains in smectic liquid crystals. We show how distinct domains may be related to each other by Lorentz transformations and that this process may lower the elastic energy of the system. We discuss a number of phases that may be formed as a result, including splay-twist analogues of blue phases. We also discuss how these diabolic domains may be subject to "superluminal boosts," yielding defects analogous to shock waves. We explore the geometry of these textures, demonstrating their relation to Milnor fibrations of the Hopf link. Finally, we show how the theory of these domains is unified in four-dimensional space.

How geometric frustration shapes twisted fibres, inside and out: competing morphologies of chiral filament assembly

Chirality frustrates and shapes the assembly of flexible filaments in rope-like, twisted bundles and fibres by introducing gradients of both filament shape (i.e. curvature) and packing throughout the structure. Previous models of chiral filament bundle formation have shown that this frustration gives rise to several distinct morphological responses, including self-limiting bundle widths, anisotropic domain (tape-like) formation and topological defects in the lateral inter-filament order. In this paper, we employ a combination of continuum elasticity theory and discrete filament bundle simulations to explore how these distinct morphological responses compete in the broader phase diagram of chiral filament assembly. We show that the most generic model of bundle formation exhibits at least four classes of equilibrium structure-finite-width, twisted bundles with isotropic and anisotropic shapes, with and without topological defects, as well as bulk phases of untwisted, columnar assembly (i.e. 'frustration escape'). These competing equilibrium morphologies are selected by only a relatively small number of parameters describing filament assembly: bundle surface energy, preferred chiral twist and stiffness of chiral filament interactions, and mechanical stiffness of filaments and their lateral interactions. Discrete filament bundle simulations test and verify continuum theory predictions for dependence of bundle structure (shape, size and packing defects of two-dimensional cross section) on these key parameters.

Observation of the Chiral and Achiral Hexatic Phases of Self-assembled Micellar polymers

We report the discovery of a thermodynamically stable line hexatic (N + 6) phase in a three-dimensional (3D) system made up of self-assembled polymer-like micelles of amphiphilic molecules. The experimentally observed phase transition sequence nematic (N)  N + 6  two-dimensional hexagonal (2D-H) is in good agreement with the theoretical predictions. Further, the present study also brings to light the effect of chirality on the N + 6 phase. In the chiral N + 6 phase the bond-orientational order within each "polymer" bundle is found to be twisted about an axis parallel to the average polymer direction. This structure is consistent with the theoretically envisaged Moiré state, thereby providing the first experimental demonstration of the Moiré structure. In addition to confirming the predictions of fundamental theories of two-dimensional melting, these results are relevant in a variety of situations in chemistry, physics and biology, where parallel packing of polymer-like objects are encountered.

Helical buckling in columnar assemblies of soft discotic mesogens

We investigate the emergence of chiral meso-structures in one-dimensional fluids consisting of stacked discotic particles and demonstrate that helical undulations are generated spontaneously from internal elastic stresses. The stability of these helical conformations arises from an interplay between long-ranged soft repulsions and nanopore confinement which is naturally present in columnar liquid crystals. Using a simple mean-field theory based on microscopic considerations we identify generic scaling expressions for the typical buckling radius and helical pitch as a function of the density and interaction potential of the constituent particles.

Elasticity-dependent self-assembly of micro-templated chromonic liquid crystal films

We explore micropatterned director structures of aqueous lyotropic chromonic liquid crystal (LCLC) films created on square-lattice cylindrical-micropost substrates. The structures are manipulated by modulating the LCLC mesophases and their elastic properties via concentration through drying. Nematic LCLC films exhibit preferred bistable alignment along the diagonals of the micropost lattice. Columnar LCLC films, dried from nematics, form two distinct director and defect configurations: a diagonally aligned director pattern with local squares of defects, and an off-diagonal configuration with zig-zag defects. The formation of these states appears to be tied to the relative splay and bend free energy costs of the initial nematic films. The observed nematic and columnar configurations are understood numerically using a Landau-de Gennes free energy model. Among other attributes, the work provide first examples of quasi-2D micropatterning of LC films in the columnar phase and lyotropic LC films in general, and it demonstrates alignment and configuration switching of typically difficult-to-align LCLC films via bulk elastic properties.

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.

Formation of impeller-like helical DNA-silica complexes by polyamines induced chiral packing

The helicity of DNA and its long-range chiral packing are widespread phenomena; however, the packing mechanism remains poorly understood both in vivo and in vitro. Here, we report the extraordinary DNA chiral self-assembly by silica mineralization, together with circular dichroism measurements and electron microscopy studies on the structure and morphology of the products. Mg(2+) ion and diethylenetriamine were found to induce right- and left-handed chiral DNA packing with two-dimensional-square p4mm mesostructures, respectively, to give corresponding enantiomeric impeller-like helical DNA-silica complexes. Moreover, formation of macroscopic impeller-like helical architectures depends on the types of polyamines and co-structure-directing agents and pH values of reaction solution. It has been suggested that interaction strength between negatively charged DNA phosphate strands and positively charged counterions may be the key factor for the induction of DNA packing handedness.

Extrinsic curvature, geometric optics, and lamellar order on curved substrates

When thermal energies are weak, two-dimensional lamellar structures confined on a curved substrate display complex patterns arising from the competition between layer bending and compression in the presence of geometric constraints. We present broad design principles to engineer the geometry of the underlying substrate so that a desired lamellar pattern can be obtained by self-assembly. Two distinct physical effects are identified as key factors that contribute to the interaction between the shape of the underlying surface and the resulting lamellar morphology. The first is a local ordering field for the direction of each individual layer, which tends to minimize its curvature with respect to the three-dimensional embedding. The second is a nonlocal effect controlled by the intrinsic geometry of the surface that forces the normals to the (nearly incompressible) layers to lie on geodesics, leading to caustic formation as in optics. As a result, different surface morphologies with predominantly positive or negative Gaussian curvature can act as converging or diverging lenses, respectively. By combining these ingredients, as one would with different optical elements, complex lamellar morphologies can be obtained. This smectic optometry enables the manipulation of lamellar configurations for the design of materials.

Structure of toroidal DNA collapsed inside the phage capsid

The structure of DNA toroids made of individual DNA molecules of various lengths (3,000 to 55,000 bp) was studied, by using partially filled bacteriophage capsids in conjunction with cryoelectron microscopy. The tetravalent cation spermine was diffused through the capsid to condense the DNA under conditions that were chosen to produce a hexagonal packing. Our results demonstrate that the frustration arising between chirality and hexagonal packing leads to the formation of twist walls; the correlation between helices combined with their strong curvature impose variations of the DNA helical pitch.

Braided bundles and compact coils: the structure and thermodynamics of hexagonally packed chiral filament assemblies

Molecular chirality frustrates the two-dimensional assembly of filamentous molecules, a fact that reflects the generic impossibility of imposing a global twisting of layered materials. We explore the consequences of this frustration for hexagonally ordered assemblies of chiral filaments that are finite in lateral dimension. Specifically, we employ a continuum-elastic description of cylindrical bundles of filaments, allowing us to consider the most general resistance to and preference for chiral ordering of the assembly. We explore two distinct mechanisms by which chirality at the molecular scale of the filament frustrates the assembly into aggregates. In the first, chiral interactions between filaments impart an overall twisting of filaments around the central axis of the bundle. In the second, we consider filaments that are inherently helical in structure, imparting a writhing geometry to the central axis. For both mechanisms, we find that a thermodynamically stable state of dispersed bundles of finite width appears close to but below the point of bulk filament condensation. The range of thermodynamic stability of dispersed bundles is sensitive only to the elastic cost and preference for chiral filament packing. The self-limited assembly of chiral filaments has particular implications for a large class of biological molecules--DNA, filamentous proteins, viruses, and bacterial flagella--which are universally chiral and are observed to form compact bundles under a broad range of conditions.

Chirality and equilibrium biopolymer bundles

We use continuum theory to show that chirality is a key thermodynamic control parameter for the aggregation of biopolymers: chirality produces a stable disperse phase of hexagonal bundles under moderately poor solvent conditions, as has been observed in in vitro studies of F actin [O. Pelletier et al., Phys. Rev. Lett. 91, 148102 (2003)]. The large characteristic radius of these chiral bundles is not determined by a mysterious long-range molecular interaction but by in-plane shear elastic stresses generated by the interplay between a chiral torque and an unusual, but universal, nonlinear gauge term in the strain tensor of ordered chains that is imposed by rotational invariance.

Triply periodic smectic liquid crystals

Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a pi2 twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk's first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and that the pi2 and pi3 grain boundary structures are particularly simple.

Liquid crystal helical ribbons as isometric textures

Deformations that conserve the parallelism and the distances--between layers, in smectic phases; between columns, in columnar phases--are commonplace in liquid crystals. The resulting isometric deformed textures display specific geometric features. The corresponding order parameter singularities extend over rather large, macroscopic, distances, e.g., cofocal conics in smectics. This well-known picture is modified when, superimposed to the 1D or 2D periodicities, the structure is helical. However isometry can be preserved. This paper discusses the case of a medium whose structure is made of 1D modulated layers (a lamello-columnar phase), assuming that the modulations rotate helically from one layer to the next. The price to pay is that any isometric texture is necessarily frustrated; it consists of layers folded into a set of parallel helicoids, in the manner of a screw dislocation (of macroscopic Burgers vector), the modulations being along the helices, i.e. double-twisted. The singularity set is made of two helical disclination lines. We complete this geometric analysis by a crude calculation of the energy of a helical ribbon. It is suggested that the helical ribbons observed in the B7 phase of banana-like molecules are such isometric textures. As a side result, let us mention that the description of double-twist, traditionally made in terms of a partition of the director field into nested cylinders, could more than often be profitably tested against a partition into nested helicoids.

Textural analysis of a mesophase with banana shaped molecules

Observed under the polarizing microscope, the B7bis phase in the banana compound D14F3 [J.P. Bedel et al., Liq. Cryst. 27, 1411 (2000)] displays two types of textures of defects, namely (a): helical ribbons, that nucleate in large quantities when the samples are quenched from a sufficiently high temperature in the isotropic phase (b)- shapes with no helicity having the structure of developable domains much akin to those observed in columnar phases, either resulting from the annealing of the helical ribbons or nucleating under slow cooling processes. The existence of these two kinds of defects points toward the complex nature of the structure of the B7 phase, which is at the same time a columnar and a smectic phase. Our observations fit the model [M. Kleman, J. Phys. France 46, 1193 (1985)] according to which the geometry of a helical ribbon is that one of the central region of a screw dislocation with a giant Burgers vector, split into two helical disclination lines of strength k = 1/2 which bound the ribbon. Textures and defects, already partly documented, and growth features and annealing processes, not yet reported in the literature, are analyzed. We conclude that the helical ribbons and the developable domains with no helicity are textures of two different B7 states, namely a metastable state and the ground state respectively. Comparative textural analysis is performed for two other banana compounds exhibiting B2 phases.

Chiral discotic columnar germs of nucleosome core particles

In concentrated solution and in the presence of high concentrations of monovalent cations, nucleosome core particles order into a discotic columnar mesophase. This phase is limited to finite-sized hexagonal germs that further divide into six coiled branches, following an iterative process. We show how the structure of the germs comes from the competition between hexagonal packing and chirality with a combination of dendritic facetting and double-twist configurations. Geometrical considerations lead us to suspect that the chirality of the eukaryotic chromosomes may originate from the same competition.

Structure of DNA-cationic liposome complexes: DNA intercalation in multilamellar membranes in distinct interhelical packing regimes

Cationic liposomes complexed with DNA (CL-DNA) are promising synthetically based nonviral carriers of DNA vectors for gene therapy. The solution structure of CL-DNA complexes was probed on length scales from subnanometer to micrometer by synchrotron x-ray diffraction and optical microscopy. The addition of either linear lambda-phage or plasmid DNA to CLs resulted in an unexpected topological transition from liposomes to optically birefringent liquid-crystalline condensed globules. X-ray diffraction of the globules revealed a novel multilamellar structure with alternating lipid bilayer and DNA monolayers. The lambda-DNA chains form a one-dimensional lattice with distinct interhelical packing regimes. Remarkably, in the isoelectric point regime, the lambda-DNA interaxial spacing expands between 24.5 and 57.1 angstroms upon lipid dilution and is indicative of a long-range electrostatic-induced repulsion that is possibly enhanced by chain undulations.