Competing Hexagonal and Square Lattices on a Spherical Surface.

The structural properties of packed soft-core particles provide a platform to understand the cross-pollinated physical concepts in solid-state and soft-matter physics. Confined on a spherical surface, the traditional differential geometry also dictates the overall defect properties in otherwise regular crystal lattices. Using molecular dynamics simulation of the Hertzian model as a tool, we report here the emergence of new types of disclination patterns: domain and counter-domain defects, when hexagonal and square patterns coexist.

Neural-network-based solver for vesicle shapes predicted by the Helfrich model.

That a three-dimensional vesicle morphology can be modeled by an artificial neural network is proposed and demonstrated. In the phase-field representation, the Helfrich bending energy of a membrane is equivalently cast into field-based energy, which enables a more direct representation of a deformable, three-dimensional membrane surface. The core of our method is incorporating recent machine-learning techniques to perform the required energy minimization.

Forced extension of a wormlike chain in the Gibbs and Helmholtz ensembles.

A semiflexible polymer can be stretched by either applying a force to it or by fixing the positions of its endpoints. The two approaches generally yield different results and correspond to experiments performed in either the Gibbs or Helmholtz statistical ensembles. Here, we derive the Helmholtz force-extension relationship for the commonly used wormlike-chain model in the strongly stretched regime.

Defect patterns of two-dimensional nematic liquid crystals in confinement.

A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.

Determining the nonequilibrium criticality of a Gardner transition via a hybrid study of molecular simulations and machine learning.

Apparent critical phenomena, typically indicated by growing correlation lengths and dynamical slowing down, are ubiquitous in nonequilibrium systems such as supercooled liquids, amorphous solids, active matter, and spin glasses. It is often challenging to determine if such observations are related to a true second-order phase transition as in the equilibrium case or simply a crossover and even more so to measure the associated critical exponents. Here we show that the simulation results of a hard-sphere glass in three dimensions are consistent with the recent theoretical prediction of a Gardner transition, a continuous nonequilibrium phase transition.

General liquid-crystal theory for anisotropically shaped molecules: Symmetry, orientational order parameters, and system free energy.

A general theory of liquid crystals is presented, starting from the group-theory symmetry analysis of the constituting molecules. A particular attention is paid to the type of elastic free-energies and their relationships with the molecular symmetries. The orientational order-parameter tensors are identified for each molecular symmetry, in a consideration of consistently keeping the leading, characteristic elastic free energies in a model.

Rodlike molecules in extreme confinement.

A unique feature of colloid particles and biopolymers is the molecule's intrinsic rigidity characterized by a molecular-level length scale. Under extreme confinement conditions at cellular scales or in nanodevices, these molecules can display orientational ordering accompanied by severe density depletion. Conventional liquid-crystal theories, such as the Oseen-Frank or Landau-de Gennes theories, cannot capture the essential molecular-level properties: the boundary effects, which extend to a distance of the rigidity length scale, and the drastic variations of the inhomogeneous molecular density.

Construction of a Pathway Map on a Complicated Energy Landscape.

How do we search for the entire family tree of possible intermediate states, without unwanted random guesses, starting from a stationary state on the energy landscape all the way down to energy minima? Here we introduce a general numerical method that constructs the pathway map, which guides our understanding of how a physical system moves on the energy landscape. The method identifies the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau-de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in a square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

Orientationally ordered states of a wormlike chain in spherical confinement.

One of the basic characteristics of a linear dsDNA molecule is its persistence length, typically of order 50 nm. The DNA chain inflicts a large energy penalty if it is bent sharply at that length scale. Viruses of bacteria, known as bacteriophages, typically have a dimension of a few tens of nanometers.

Structural Prediction and Inverse Design by a Strongly Correlated Neural Network.

Macromolecules contain molecular units as the coding information for their correlated structures in physical dimensions. The relationship between these two features is governed by the interaction energies of the involved molecular units and their encoded sequences. We present a neural network algorithm that treats molecular units themselves as neural networks, which has the flexibility to allow each unit to respond to its own environment and to influence others in the system.

Machine learning topological defects of confined liquid crystals in two dimensions.

Supervised machine learning can be used to classify images with spatially correlated physical features. We demonstrate the concept by using the coordinate files generated from an off-lattice computer simulation of rodlike molecules confined in a square box as an example. Because of the geometric frustrations at high number density, the nematic director field develops an inhomogeneous pattern containing various topological defects as the main physical feature.

Generating the conformational properties of a polymer by the restricted Boltzmann machine.

In polymer theory, computer-generated polymer configurations, by either Monte Carlo simulations or molecular dynamics simulations, help us to establish the fundamental understanding of the conformational properties of polymers. Here, we introduce a different method, exploiting the properties of a machine-learning algorithm, the restricted Boltzmann machine network, to generate independent polymer configurations for self-avoiding walks (SAWs), for studying the conformational properties of polymers. We show that with adequate training data and network size, this method can capture the underlying polymer physics simply from learning the statistics in the training data without explicit information on the physical model itself.

Self-Avoiding Wormlike Chain Confined in a Cylindrical Tube: Scaling Behavior.

Within a confining tube section, the multithreads of a strongly confined, backfolding polymer exert the excluded-volume repulsions on each other and produce physical properties that are very different from those of a confined ideal chain. The conformational properties of a such confined wormlike chain are of fundamental interest and are also practically useful in understanding the DNA confinement problems. Here, the excluded-volume effects are added to the standard wormlike-chain model by a self-consistent field theory.

Formation of three-dimensional colloidal crystals in a nematic liquid crystal.

We investigate the possible structures of three-dimensional colloidal crystals formed when these spherical particles are dispersed in a liquid crystal. The case of a strong homeotropic boundary condition is considered here. Their corresponding defect structures in the space-filler nematic liquid crystal are induced by the presence of the spherical surface of the colloids and produce an attraction between colloidal particles.

Topological defects in two-dimensional liquid crystals confined by a box.

When a spatially uniform system that displays a liquid-crystal ordering on a two-dimensional surface is confined inside a rectangular box, the liquid crystal direction field develops inhomogeneous textures accompanied by topological defects because of the geometric frustrations. We show that the rich variety of nematic textures and defect patterns found in recent experimental and theoretical studies can be classified by the solutions of the rather fundamental, extended Onsager model. This is critically examined based on the determined free energies of different defect states, as functions of a few relevant, dimensionless geometric parameters.

Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles.

We present numerical solutions to the Landau-de Gennes free-energy model under the one-constant approximation for systems of single and double spherical colloidal particles immersed in an otherwise uniformly aligned nematic liquid crystal. A perfect homeotropic surface anchoring of liquid-crystal molecules on the spherical surface is considered. A large parameter space is carefully examined, including those in the free-energy model and those describing the dimer configurations and the background liquid-crystal orientation.

Conformational Properties of a Back-Folding Wormlike Chain Confined in a Cylindrical Tube.

When a semiflexible chain is confined in a narrow cylindrical tube, the formation of a polymer hairpin is a geometrical conformation that accompanies an exponentially large local free energy and, hence, is a relatively rare event. Numerical solutions of the hairpin distribution functions for persistence-length-to-tube-radius ratios over a wide range are obtained in high precision, by using the Green's function approach for the wormlike-chain model. The crossover region between the narrow and moderately narrow tubes is critically investigated in terms of the hairpin free energy, global persistence length, mean hairpin-tip distance from the tube axis, and hairpin-plane orientational properties.

Identifying polymer states by machine learning.

The ability of a feed-forward neural network to learn and classify different states of polymer configurations is systematically explored. Performing numerical experiments, we find that a simple network model can, after adequate training, recognize multiple structures, including gaslike coil, liquidlike globular, and crystalline anti-Mackay and Mackay structures. The network can be trained to identify the transition points between various states, which compare well with those identified by independent specific-heat calculations.

Thermodynamics of a Compressible Maier-Saupe Model Based on the Self-Consistent Field Theory of Wormlike Polymer.

This paper presents a theoretical formalism for describing systems of semiflexible polymers, which can have density variations due to finite compressibility and exhibit an isotropic-nematic transition. The molecular architecture of the semiflexible polymers is described by a continuum wormlike-chain model. The non-bonded interactions are described through a functional of two collective variables, the local density and local segmental orientation tensor.

Maier-Saupe model of polymer nematics: Comparing free energies calculated with Self Consistent Field theory and Monte Carlo simulations.

Self Consistent Field (SCF) theory serves as an efficient tool for studying mesoscale structure and thermodynamics of polymeric liquid crystals (LC). We investigate how some of the intrinsic approximations of SCF affect the description of the thermodynamics of polymeric LC, using a coarse-grained model. Polymer nematics are represented as discrete worm-like chains (WLC) where non-bonded interactions are defined combining an isotropic repulsive and an anisotropic attractive Maier-Saupe (MS) potential.

Complex liquid-crystal nanostructures in semiflexible ABC linear triblock copolymers: A self-consistent field theory.

We show that two series of ABC linear triblock copolymers possess sequences of order-to-order phase transitions between microphase-separated states, as the degree of flexibility of the semiflexible middle B-blocks varies. The spatial and orientational symmetries of these phases, some of them containing liquid-crystal ordering, are analysed in comparison with related structures previously determined experimentally and theoretically. A theoretical framework based on the self-consistent field treatment of the wormlike-chain model, which incorporates the Flory-Huggins and Maier-Saupe interactions in the free energy, is used here as a basic foundation for numerical calculations.

Perspective: parameters in a self-consistent field theory of multicomponent wormlike-copolymer melts.

We review a formalism that can be used to calculate the microphase-separated crystallographic structures of multi-component wormlike polymer melts. The approach is based on a self-consistent field theory of wormlike polymers where the persistence length of each component is an important parameter. We emphasize on an analysis of the number of independent parameters required to specify a problem in general, for a system that includes Flory-Huggins and Maier-Saupe energies.

Nematic ordering of semiflexible polymers confined on a toroidal surface.

We study the isotropic-like and nematic states of wormlike liquid-crystal polymers embedded on the surface of a torus. The role played by surface curvature, which couples to the molecular rigidity, is reported as the main reason that causes the weak nematic ordering in an otherwise ordinary isotropic phase. The same coupling has a profound effect on the nematic states as well, which are stabilized by the Onsager excluded-volume interaction; the latter has been frequently used to study lyotropic liquid crystal polymers and is used here as an example of the physical mechanisms that drive the system to make orientational ordering.

Surface-induced phase transitions of wormlike chains in slit confinement.

On the basis of a self-consistent field theory treatment of semi-flexible polymer chains, we analyze the effects of the flexibility on the structure of polymers sterically confined between two parallel, structureless walls separated by a distance. The model is built from a wormlike chain formalism which crosses over from the rod limit to the flexible limit, and the Onsager-type interaction which describes the orientation-dependent excluded-volume interaction. Three surface states were obtained from the numerical solution to the theory: uniaxial, biaxial, and condensed.

Microphase separation of short wormlike diblock copolymers with a finite interaction range.

We investigate several structural properties of low-molecular weight AB diblock copolymer melts, focusing on a number of features that substantially deviate from those of high-molecular weight copolymer melts. The study is based on the wormlike chain formalism aided by random phase approximation and self-consistent field theory. We examine the effects that stemmed from both the finite molecular weight and the finite interaction range between unlike AB monomers.