We report a comprehensive analysis of the ground state properties of axisymmetric toroidal crystals based on the elastic theory of defects on curved substrates. The ground state is analyzed as a function of the aspect ratio of the torus, which provides a non-local measure of the underlying Gaussian curvature, and the ratio of the defect core-energy to the Young modulus. Several structural features are discussed,including a spectacular example of curvature-driven amorphization in the limit of the aspect ratio approaching one. The outcome of the elastic theory is then compared with the results of a numerical study of a system of point-like particles constrained on the surface of a torus and interacting via a short range potential.
We investigated the nematic ordering on a torus by means of analytic method and the method of simulated annealing, the Frank free energy, both in the standard form and covariant form, were used in the study. The defect free state was found to be the ground state in both cases. However, in the case of the standard model, there are two kinds of defective free ordering and a transition between the two occurs at a critical value of radius ratio $k=\frac{r}{R}$. The first one is $θ=0$ in the small $k$ regime and the second one is a variable $θ$ with position of the torus. In the case of the covaria…
We analyse here the problem of large deformation of dielectric elastomeric membranes under coupled electromechanical loading. Extremely large deformations (enclosed volume changes of 100 times and greater) of a toroidal membrane are studied by the use of a variational formulation that accounts for the total energy due to mechanical and electrical fields. A modified shooting method is adopted to solve the resulting system of coupled and highly nonlinear ordinary differential equations. We demonstrate the occurrence of limit point, wrinkling, and symmetry-breaking buckling instabilities in the s…
Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planar degenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic environment or cavities carved out of a solid substrate. A universal solution of Frank's elastic free energy is an equilibrium configuration for the nematic director field, irrespective of the values of the elastic constants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of this configuration and identify a range of parameters where the main drive towards …
Programming the non-uniform growth of a responsive polymer gel has emerged as a powerful tool to shape sheets into prescribed three dimensional shapes. We demonstrate that shapes with zero Gaussian curvature, except at singularities, produced by the growth-induced buckling of a thin elastic sheet are the same as those produced by the Volterra construction of topological defects in which edges of an intrinsically flat surface are identified. With this connection, we study the problem of choosing an optimal pattern of growth for a prescribed developable surface, finding a fundamental trade-off b…
Toroidal nematics are droplets of nematic liquid crystals in the form of a circular torus. When the nematic director is subject to planar degenerate boundary conditions, the bend-only director field with vector lines along the parallels of all internal torodial shells is an equilibrium solution for all values of the elastic constants. Local stability analyses have shown that an instability is expected to occur for sufficiently small values of the twist elastic constant. It is natural to conjecture that in this regime the global equilibrium would be characterized by a maximum twist deflection o…
On microscopic scales, the crystallinity of flexible tethered or cross linked membranes determines their mechanical response. We show that by controlling the type, number and distribution of defects on a spherical elastic shell, it is possible to direct the morphology of these structures. Our numerical simulations show that by deflating a crystalline shell with defects, we can create elastic shell analogs of the classical Platonic solids. These morphologies arise via a sharp buckling transition from the sphere which is strongly hysteretic in loading-unloading. We construct a minimal Landau the…
Conducting polymers have wide technological applications in sensors, actuators, electric and optical devices, solar cells etc. To improve their operational performance, mechanical, thermal, electrical and optical properties, such polymers are doped with carbon allotrope nanofillers. Functionality of the novel nanocomposite polymers may be stipulated by size characteristics of nanoparticles and the polymer, different physical effects like charge transfer in such objects etc. We characterize and analyze structure, elastic, electric properties and of novel polymer nanocomposites, isotactic polypr…
We previously observed that an intrinsic staking fault shrunk through a glide of a Shockley partial dislocation terminating its lower end in a hard-sphere crystal under gravity coherently grown in <001> by Monte Carlo simulations [Mori et al., Molec. Phys. 105, 1377 (2007)]; it was an answer to a one-decade long standing question why the stacking disorder in colloidal crystals reduced under gravity [Zhu et al., Nature 387, 883 (1997)]. Here, we present an elastic energy calculation; in addition to the self-energy of the partial dislocation [Mori et al., Prog. Theor. Phys. Suppl. 178, 33 (2009)…
In complex crystals close to melting or at finite temperatures, different types of defects are ubiquitous and their role becomes relevant in the mechanical response of these solids. Conventional elasticity theory fails to provide a microscopic basis to include and account for the motion of point-defects in an otherwise ordered crystalline structure. We study the elastic properties of a point-defect rich crystal within a first-principles theoretical framework derived from microscopic equations of motion. This framework allows us to make specific predictions pertaining to the mechanical properti…
Elucidating the interplay of defect and stress at the microscopic level is a fundamental physical problem that has strong connection with materials science. Here, based on the two-dimensional crystal model, we show that the instability mode of vacancies with varying size and morphology conforms to a common scenario. A vacancy under compression is fissioned into a pair of dislocations that glide and vanish at the boundary. This neat process is triggered by the local shear stress around the vacancy. The remarkable fractionalization of vacancies creates rich modes of interaction between vacancies…
In this paper, we introduce a methodology applicable to a wide range of localized two-dimensional sources of stress. This methodology is based on a geometric formulation of elasticity. Localized sources of stress are viewed as singular defects---point charges of the curvature associated with a reference metric. The stress field in the presence of defects can be solved using a scalar stress function that generalizes the classical Airy stress function to the case of materials with nontrivial geometry. This approach allows the calculation of interaction energies between various types of defects. …
We present a novel, self-consistent and robust theoretical model to investigate elastic interactions between topological defects in liquid crystal shells. Accounting for the non-concentric nature of the shell in a simple manner, we are able to successfully and accurately explain and predict the positions of the defects, most relevant in the context of colloidal self-assembly. We calibrate and test our model on existing experimental data, and extend it to all newly observed defects configurations in chiral nematic shells. We perform new experiments to check further and confirm the validity of t…
An experimental and theoretical study of lyotropic chromonic liquid crystals (LCLCs) confined in cylinders with degenerate planar boundary conditions elucidates LCLC director configurations. When the Frank saddle-splay modulus is more than twice the twist modulus, the ground state adopts an inhomogeneous escaped-twisted configuration. Analysis of the configuration yields a large saddle-splay modulus, which violates Ericksen inequalities but not thermodynamic stability. Lastly, we observe point defects between opposite-handed domains and explain a preference for point defects over domain walls.
Many systems in nature and the synthetic world involve ordered arrangements of units on two-dimensional surfaces. We review here the fundamental role payed by both the topology of the underlying surface and its detailed curvature. Topology dictates certain broad features of the defect structure of the ground state but curvature-driven energetics controls the detailed structured of ordered phases. Among the surprises are the appearance in the ground state of structures that would normally be thermal excitations and thus prohibited at zero temperature. Examples include excess dislocations in the…
Twisted assemblies of filaments in ropes, cables and bundles are essential structural elements in wide use in macroscopic materials as well as within the cells and tissues of living organisms. We develop the unique, non-linear elastic properties of twisted filament bundles that derive from generic properties of two-dimensional line-ordered materials. Continuum elasticity reveals a formal equivalence between the elastic stresses induced by bundle twist and those induced by the positive curvature in thin, elastic sheets. These geometrically-induced stresses can be screened by 5-fold disclination…
A systematic computational and scaling analysis of defect textures in polygonal arrangement of cylindrical particles embedded in a cholesteric (Ch) liquid crystal matrix is performed using the Landau de Gennes model for chiral self assembly, with strong anchoring at the particles' surface. The defect textures and LC phases observed are investigated as a function of chirality, elastic anisotropy, particle polygonal arrangement and particle size. The presence of a polygonal network made of N circular inclusions results in defect textures of a net charge of -(N-2)/2 per unit polygonal cell. As th…
We find that crystalline states of repulsive active Brownian particles at high activity melt into a hexatic state but this transition is not driven by an unbinding of bound dislocation pairs as suggested by the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory. Upon reducing the density, the crystalline state melts into a high-density hexatic state devoid of any defects. Decreasing the density further, the dislocations proliferate and introduce plasticity in the system, nevertheless maintaining the hexatic state, but eventually melting into a fluid state. Remarkably, the elastic constan…
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 z…
Defect Kinematics in 2D Nematics: Contributions from Surface Topology, Intrinsic and Extrinsic Geometry, Solitons, Defect Orientations, and Elastic Anisotropy
School of Physics, UNSW（新南威尔士大学物理学院）；EMBL Australia Node in Single Molecule Science, School of Medical Sciences, UNSW（UNSW医学院单分子科学澳大利亚EMBL节点）；ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems, UNSW Node（UNSW节点，ARC细胞系统数学分析卓越中心）
We consider three-dimensional reshaping of thin nemato-elastic sheets containing half-charged defects upon nematic-isotropic transition. Gaussian curvature, that can be evaluated analytically when the nematic texture is known, differs from zero in the entire domain and has a dipole or hexapole singularity, respectively, at defects of positive or negative sign. The latter kind of defects appears in not simply connected domains. Three-dimensional shapes dependent on boundary anchoring are obtained with the help of finite element computations.
We use density--functional theory to study the structure of two-dimensional defects inside a circular nematic nanocavity. The density, nematic order parameter, and director fields, as well as the defect core energy and core radius, are obtained in a thermodynamically consistent way for defects with topological charge $k=+1$ (with radial and tangential symmetries) and $k=+1/2$. An independent calculation of the fluid elastic constants, within the same theory, allows us to connect with the local free--energy density predicted by elastic theory, which in turn provides a criterion to define a defe…
Nuclei of ordered materials emerging from the isotropic state usually show a shape topologically equivalent to a sphere; the well-known examples are crystals and nematic liquid crystal droplets. In this work, we explore experimentally and theoretically the nuclei of columnar lyotropic chromonic liquid crystal coexisting with the isotropic phase that are toroidal in shape. The geometry of toroids depends strongly on the molecular concentrations and presence of a crowding agent, polyethylene glycol. High concentrations result in thick toroids with small central holes, while low concentrations yi…
Employing the fracton-elastic duality, we develop a low-energy effective theory of a zero-temperature vortex crystal in a two-dimensional bosonic superfluid which naturally incorporates crystalline topological defects. We extract static interactions between these defects and investigate several continuous quantum transitions triggered by the Higgs condensation of vortex vacancies/interstitials and dislocations. We propose that the quantum melting of the vortex crystal towards the hexatic or smectic phase may occur via a pair of continuous transitions separated by an intermediate vortex superso…
Active matter is characterized by its ability to induce motion by self-generated stress. In the case of a solid, such motion can lead to shape transformations. The stress-generating components can be anisotropic endowing the material with mesoscopic orientational order. It is currently unknown how the specific postions and orientations of these active constituents influence morphological changes. We study theoretically the effects of imposing topological point defects in the arrangements of the stress-generating components on the morphology of elastic active nematic shells. We show that topolo…
In this work a periodic crystal with point defects is described in the framework of linear response theory for broken symmetry states using correlation functions and Zwanzig-Mori equations. The main results are microscopic expressions for the elastic constants and for the coarse-grained density, point-defect density, and displacement field, which are valid in real crystals, where vacancies and interstitials are present. The coarse-grained density field differs from the small wave vector limit of the microscopic density. In the long wavelength limit, we recover the phenomenological description …
Topological defects shape the material and transport properties of physical systems. Examples range from vortex lines in quantum superfluids, defect-mediated buckling of graphene, and grain boundaries in ferromagnets and colloidal crystals, to domain structures formed in the early universe. The Kibble-Zurek (KZ) mechanism describes the topological defect formation in continuous non-equilibrium phase transitions with a constant finite quench rate. Universal KZ scaling laws have been verified experimentally and numerically for second-order transitions in planar Euclidean geometries, but their va…
Self-assembly of colloidal particles is poised to become a powerful composite material fabrication technique, but remains challenged by a limited control over the ensuing structures. We develop a new breed of nematic colloids that are physical analogs of a mathematical surface with boundary, interacting with the molecular alignment field without inducing defects when flat. However, made from a thin nanofoil, they can be shaped to prompt formation of self-compensating defects that drive pre-programmed elastic interactions mediated by the nematic host. To show this, we wrap the nanofoil on all t…
We use an elastic model to explore faceting of solid-wall vesicles with elastic heterogeneities. We show that faceting occurs in regions where the vesicle wall is softer, such as areas of reduced wall thicknesses or concentrated in crystalline defects. The elastic heterogeneities are modeled as a second component with reduced elastic parameters. Using simulated annealing Monte Carlo simulations we obtain the vesicle shape by optimizing the distributions of facets and boundaries. Our model allows us to reduce the effects of the residual stress generated by crystalline defects, and reveals a rob…
We investigate the influence of curvature and topology on crystalline wrinkling patterns in generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elast…
An isolated positive wedge disclination deforms an initially flat elastic sheet into a perfect cone when the sheet is of infinite extent and is elastically inextensible. The latter requires the elastic stretching strains to be vanishingly small. In this paper, rigorous analytical and numerical results are obtained for the disclination induced deformed shape and stress field of a bounded F{ö}ppl-von K{á}rm{á}n elastic sheet with finite extensibility, while emphasising the deviations from the perfect cone solution. In particular, the Gaussian curvature field is no longer localised as a Dirac sin…
Colloidal systems find important applications ranging from fabrication of photonic crystals to direct probing of phenomena typically encountered in atomic crystals and glasses. New applications - such as nanoantennas, plasmonic sensors, and nanocircuits - pose a challenge of achieving sparse colloidal assemblies with tunable interparticle separations that can be controlled at will. We demonstrate reconfigurable multiscale interactions and assembly of colloids mediated by defects in cholesteric liquid crystals that are probed by means of laser manipulation and three-dimensional imaging. We find…
We investigate the behavior of non-Euclidean plates with constant negative Gaussian curvature using the Föppl-von Kármán reduced theory of elasticity. Motivated by recent experimental results, we focus on annuli with a periodic profile. We prove rigorous upper and lower bounds for the elastic energy that scales like the thickness squared. In particular we show that are only two types of global minimizers -- deformations that remain flat and saddle shaped deformations with isolated regions of stretching near the edge of the annulus. We also show that there exist local minimizers with a periodic…
Achieving control and tunability of lyotropic materials has been a long-standing goal of liquid crystal research. Here we show that the elasticity of a liquid crystal system consisting of a dense suspension of semiflexible biopolymers can be manipulated over a relatively wide range of elastic moduli. Specifically, thin films of actin filaments are assembled at an oil-water interface. At sufficiently high concentrations, one observes the formation of a nematic phase riddled with $\pm1/2$ topological defects, characteristic of a two-dimensional nematic system. As the average filament length incr…
Plasticity in amorphous solids is mediated by localized quadrupolar instabilities, but the mechanism by which an amorphous solid eventually fails or melts is debated. In this work we argue that these phenomena can be investigated in the model problem of an elastic continuum with quadrupolar defects, at finite temperature. This problem is posed and the collective behavior of the defects is analytically investigated. Using both renormalization group and field-theoretic techniques, it is found that the model has a yielding/melting transition of spinodal type.
