Recently, there has been renewed interest in the coupling between geometry and topological defects in crystalline and striped systems. Standard lore dictates that positive disclinations are associated with positive Gaussian curvature, whereas negative disclinations give rise to negative curvature. Here, we present a diblock copolymer system exhibiting a striped columnar phase that preferentially forms wrinkles perpendicular to the underlying stripes. In free-standing films this wrinkling behavior induces negative Gaussian curvature to form in the vicinity of positive disclinations.
Aspects of the present invention describe soft imprint lithography methods capable of preparing structural features on surfaces. Disclosed methods include surmounting a deformable substrate, having an original form, with a composition, wherein the deformable substrate is capable of achieving at least one predetermined deformed state; predictably deforming said deformable substrate from its original form to the at least one predetermined deformed state; and transferring at least a portion of the composition surmounting the deformed substrate to a receiving substrate.
The homotopy theory of topological defects is a powerful tool for organizing and unifying many ideas across a broad range of physical systems. Recently, experimental progress was made in controlling and measuring colloidal inclusions in liquid crystalline phases. The topological structure of these systems is quite rich but, at the same time, subtle. Motivated by experiment and the power of topological reasoning, the classification of defects in uniaxial nematic liquid crystals was reviewed and expounded upon. Particular attention was paid to the ambiguities that arise in these systems, which have no counterpart in the much-storied XY model or the Heisenberg ferromagnet.
Matsumoto EA. Patterns on a Roll. In: Experience-centered Approach and Visuality in the Education of Mathematics and Physics. Kaposvár, Hungary: Kaposvár University ; 2012. pp. 175-176.
Exploiting elastic instability in thin films has proven to be a robust method for creating complex patterns and structures across a wide range of lengthscales. Even the simplest of systems, an elastic membrane with a lattice of pores, under mechanical strain, generates complex patterns featuring long-range orientational order. When we promote this system to a curved surface, in particular, a cylindrical membrane, a novel set of features, patterns and broken symmetries appears. The newfound periodicity of the cylinder allows for a novel continuous method for nanoprinting.
Riemann’s minimal surfaces, a one-parameter family of minimal surfaces, describe a bicon- tinuous lamellar system with pores connecting alternating layers. We demonstrate explicitly that Riemann’s minimal surfaces are composed of a nonlinear sum of two oppositely handed helicoids.
Nonlinear elastic phenomena appear time and again in the world around us. This work considers two separate soft matter systems, instabilities in an elastic membrane perforated by a lattice of circular holes and defect textures in smectic liquid crystals. By studying the set of singularities characterizing each system, not only do the analytics become tractable, we gain intuition and insight into complex structures.
Under hydrostatic compression, the holes decorating an elastic sheet undergo a buckling instability and collapse. By modeling each of the buckled holes as a pair of dislocation singularities, linear elasticity theory accurately captures the interactions between holes and predicts the pattern transformation they undergo. The diamond plate pattern generated by a square lattice of holes achieves long ranged order due to the broken symmetry of the underlying lattice. The limited number of two dimensional lattices restricts the classes of patterns that can be produced by a flat sheet. By changing the topology of the membrane to a cylinder the types of accessible patterns vastly increases, from a chiral wrapped cylinder to pairs of holes alternating orientations to even more complex structures.
Equally spaced layered smectics introduce a plethora of geometric constraints yielding novel textures based upon topological defects. The frustration due to the incompatibility of molecular chirality and layers drives the formation of both the venerable twist-grain-boundary phase and the newly discovered helical nanofilament (HN) phase. The HN phase is a newly found solution of the chiral Landau-de Gennes free energy. Finally, we consider two limiting cases of the achiral Landau-de Gennes free energy, bending energy dominated allows defects in the layers and compression energy dominated enforces equally spaced layers. In order to minimize bending energy, smectic layers assume the morphology of minimal surfaces. Riemann's minimal surface is composed of a nonlinear sum of two oppositely handed screw dislocations and has the morphology of a pore. Likewise, focal conic domains result from enforcing the equal spacing condition. We develop an approach to the study of focal sets in smectics which exploits a hidden Poincaré symmetry revealed only by viewing the smectic layers as projections from one-higher dimension.
Liquid crystalline systems exhibiting both macroscopic chirality and smectic order experience frustration resulting in mesophases possessing complex three-dimensional order. In the twist-grain-boundary phase, defect lattices mediate the propagation of twist throughout the system. We propose a new chiral smectic structure composed of a lattice of chiral bundles as a model of the helical nanofilament (B4) phase of bent-core smectics.
Focal conic domains are typically the “smoking gun” by which smectic liquid crystalline phases are identified. The geometry of the equally spaced smectic layers is highly generic but, at the same time, difficult to work with. In this Letter we develop an approach to the study of focal sets in smectics which exploits a hidden Poincaré symmetry revealed only by viewing the smectic layers as projections from one-higher dimension. We use this perspective to shed light upon several classic focal conic textures, including the concentric cyclides of Dupin, polygonal textures, and tilt-grain boundaries.
Recent experiments have exploited elastic instabilities in membranes to create complex patterns. However, the rational design of such structures poses many challenges, as they are products of nonlinear elastic behavior. We pose a simple model for determining the orientational order of such patterns using only linear elasticity theory which correctly predicts the outcomes of several experiments. Each element of the pattern is modeled by a “dislocation dipole” located at a point on a lattice, which then interacts elastically with all other dipoles in the system. We explicitly consider a membrane with a square lattice of circular holes under uniform compression and examine the changes in morphology as it is allowed to relax in a specified direction.
We report on a simple yet robust method to produce orientationally modulated two-dimensional patterns with sub-100 nm features over cm2 regions via a solvent-induced swelling instability of an elastomeric film with micrometer-scale perforations. The dramatic reduction of feature size (∼10 times) is achieved in a single step, and the process is reversible and repeatable without the requirement of delicate surface preparation or chemistry. By suspending ferrous and other functional nanoparticles in the solvent, we have faithfully printed the emergent patterns onto flat and curved substrates. We model this elastic instability in terms of elastically interacting “dislocation dipoles” and find complete agreement between the theoretical ground-state and the observed pattern. Our understanding allows us to manipulate the structural details of the membrane to tailor the elastic distortions and generate a variety of nanostructures.