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. The versatile ability of the method, to compute axisymmetric and nonsymmetric shapes, is discussed.

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. By analyzing it in comparison with the Gibbs ensemble result, we show that equivalence between the two relationships is achieved only in the long-chain thermodynamic limit.

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. Using a hybrid molecular simulation-machine learning approach, we obtain scaling laws for both finite-size and aging effects and determine the critical exponents that traditional methods fail to estimate. Our study provides an approach that is useful to understand the nature of glass transitions and can be generalized to analyze other nonequilibrium phase transitions.

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.

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. Through a deep neural network and a self-consistent procedure, molecular units in the network establish a strong correlation to produce the desirable features in the physical world. The proposed framework is applied to the HP model. Both the forward problem of predicting folded structures from given sequences and the inverse problem of predicting required sequences for a given structure are examined.

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. We critically examine how the trained Boltzmann machine can generate independent configurations that are not in the original training data set of SAWs.

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. The numerical solutions are examined in light of their scaling properties.

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. Here, a standard Landau-de Gennes free energy model for a spatially inhomogeneous liquid crystal is numerically minimized to yield an optimal configuration of both spherical colloids and the orientational field. The stable and metastable structures obtained in this work are described and analyzed according to the type of periodic liquid-crystal defect lines that couple the colloidal spheres together. A large range of the spherical size is covered in this study, corresponding to a 5CB-liquid-crystal comparison for assembling micron- to nano-sized colloidal spheres. Multiple configurations are found for each given particle size and the most stable state is determined by a comparison of the free energies. From large to small colloidal particles, a sequence of structures, which range from quasi-one-dimensional (columnar), to quasi-two-dimensional (planar), and to truly three-dimensional, are found to exist.

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. Accurate representations of the solutions by simple interpolation formulae are suggested.

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. As the overall density increases in such a lyotropic system, first order phase transitions between uniaxial and biaxial states, biaxial and condensed states can occur.

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. We identify important parameters in the system which are used as axes of the four-dimensional phase diagram. The numerical study demonstrates a strong correlation between the liquid-crystal defect-free and defect structures and the geometry of the liquid-crystal embedded surface.

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. The latter yields profound effects on systems consisting of short wormlike block copolymers. The noticeable shift of the order-disorder transition point is discussed. Attention is also paid to the strong-segregation regime, where low molecular weight polymers are subject to finite stretchability.

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. Examples of recent applications are also briefly demonstrated: AB homopolymer interface, AB diblock copolymers, and rod-coil copolymers.

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. We suggest that tuning the flexibility parameter, which reduces to the concept of degree of polymerization in the coil-like limit and characterizes the chain-persistency in the rod-like limit, provides a promising approach that can be used to design the resulting microphase-separated structures in semiflexible copolymer melts.

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. The range of the potentials, σ, controls the strength of correlations due to non-bonded interactions. Increasing σ (which can be seen as an increase of coarse-graining) while preserving the integrated strength of the potentials reduces correlations. The model is studied with particle-based Monte Carlo (MC) simulations and SCF theory which uses partial enumeration to describe discrete WLC. In MC simulations the Helmholtz free energy is calculated as a function of strength of MS interactions to obtain reference thermodynamic data. To calculate the free energy of the nematic branch with respect to the disordered melt, we employ a special thermodynamic integration (TI) scheme invoking an external field to bypass the first-order isotropic-nematic transition. Methodological aspects which have not been discussed in earlier implementations of the TI to LC are considered. Special attention is given to the rotational Goldstone mode. The free-energy landscape in MC and SCF is directly compared. For moderate σ the differences highlight the importance of local non-bonded orientation correlations between segments, which SCF neglects. Simple renormalization of parameters in SCF cannot compensate the missing correlations. Increasing σ reduces correlations and SCF reproduces well the free energy in MC simulations.

The phase diagram and radial collapse of an inflated soft tube under twist.

When an inflated soft tube such as a cylindrical balloon is twisted, mechanical instability can arise and produces a kink-like radius collapsing in the middle of the tube. Here this phenomenon inspires us to theoretically analyze a standard non-linear model of rubber elasticity for soft tubes. We show that there exists a critical pressure beyond which such instability arises. The critical pressure depends on the elastic properties of the tube material and the geometric dimensions of the thin-walled tube. This general theory covers a large class of soft materials and explains why twist-induced collapsing is observable in soft and thin elastic tubes such as balloons, but not in hard and thick tubes such as water hoses.

A model of vesicle tubulation and pearling induced by adsorbing particles.

We study the basic theoretical model of a deformable vesicle immersed in a solution of particles that can adsorb onto one of the two surfaces of a membrane. The model consists of an adsorption energy gain for the adsorbing particles and the Canham-Helfrich membrane bending energy, in which the spontaneous curvature is coupled with the adsorption area. We demonstrate that bud, pearling, and tube conformations can be stabilized after minimizing the free energy and that the pearling-tubulation transition has the characteristics of an abrupt structural transition.

Conformation of a charged vesicle.

We report on the first systematic study of vesicle conformational change caused by Coulomb interaction between surface charges on a lipid vesicle. The equilibrium configuration of a charged vesicle is found, as the result of the competition between the local bending elastic energy and the long-range electrostatic interaction within the membrane where the counter-ion effects are neglected. Because of the Rayleigh instability, a charged vesicle undergoes conformational transitions as a function of the surface charge density.

Phase transitions in semiflexible-rod diblock copolymers: a self-consistent field theory.

We investigate the phase behavior of semiflexible-rod diblock copolymers in a parameter range where the system displays columnar and lamella structures, using a self-consistent field theory based on the wormlike-chain model. Both Flory-Huggins and Maier-Saupe orientational interactions are incorporated in the formalism, which allows us to explore microphase separation and liquid-crystal ordering simultaneously. Order-to-order phase transitions induced by chain rigidity and orientational interaction are both reported and analyzed. Coupled orientational ordering and spatial inhomogeneity of the four microphase-separated states are discussed in this work: hexagonal column, ellipse column, smectic-A, and smectic-C.

Free energy of a long semiflexible polymer confined in a spherical cavity.

The free energy and conformational properties of a wormlike chain confined inside a spherical surface are investigated. We show that in the weak-confinement limit, the wormlike chain model exactly reproduces the confinement properties of a Gaussian chain; in such a case the confinement entropy dominates the free energy; in the strong-confinement limit, the free energy is dominated by the bending energy of the chain, which is forced to wrap around the confining surface. We also present a numerical solution within the crossover region between the two limits, solving the differential equation that the probability distribution function satisfies.