Recent experiments on vesicles formed from block copolymers with liquid-crystalline side chains reveal a rich variety of vesicle morphologies. The additional internal order ("structure") developed by these self-assembled block copolymer vesicles can lead to significantly deformed vesicles as a result of the delicate interplay between two-dimensional ordering and vesicle shape. The inevitable topological defects in structured vesicles of spherical topology also play an essential role in controlling the final vesicle morphology. Here we develop a minimal theoretical model for the morphology of the membrane structure with internal nematic/ smectic order. Using both analytic and numerical approaches, we show that the possible low free energy morphologies include nano-size cylindrical micelles (nano-fibers), faceted tetrahedral vesicles, and ellipsoidal vesicles, as well as cylindrical vesicles. The tetrahedral vesicle is a particularly fascinating example of a faceted liquid-crystalline membrane. Faceted liquid vesicles may lead to the design of supramolecular structures with tetrahedral symmetry and new classes of nano-carriers.amphiphilic block copolymers | bending energy | Frank free energy | liquid crystalline polymers | self-assembled bilayer A mphiphilic block copolymers in water, like natural phospholipids, can self-assemble into various monolayer or bilayer structures, such as micelles and vesicles (1, 2). In particular, rodcoil block copolymers, with a flexible hydrophilic chain and one or more rod-like hydrophobic blocks, exhibit a rich morphology of structures, and therefore have significant potential to advance fundamental science and drive technological innovations (3-12). Among these rod-coil block copolymers, we are especially interested in liquid crystalline (LC) block copolymers in which the hydrophobic block is a nematic or smectic liquid crystal polymer (13)(14)(15)(16)(17)(18)(19)(20). The in-plane LC order that results from molecular pair interactions in these systems, and the associated defect structure, play very important roles in determining the preferred intermediate and final shapes of vesicles. The tailor-design of both material properties and vesicle morphology by controlling the molecular structures of the block polymers is state-of-the-art research in the fields of polymer science, materials science, and chemical engineering.Some of the structures formed by these LC side-chain block copolymers in aqueous solution are rather counterintuitive, such as faceted vesicles, nanotubes and compact vesicles with tiny inner space (15,20). In all these structures, the in-plane smectic order is clearly visible under Cryo-TEM. In this article we develop a theoretical explanation of the geometric structures of vesicles with in-plane nematic or smectic order. We present a simple model free energy as a functional of both the membrane geometry and the in-plane nematic order. Using both analytic and numerical methods, we then analyze the low free energy morphologies in various parameter regimes. Results ...
Topological defects are found in particles confined to planar disks interacting via the 1/r Coulomb potential. The total interior topological charge is found to monotonically converge to a negative value as the energy decreases during the relaxation process regardless of initial configurations; it is more negative in a larger cluster. The comparison with a uniform hyperbolic tessellation reveals an underlying hyperbolic structure in a low-energy configuration where the particle density increases from the center of the disk to its boundary. An elliptic structure is identified in an opposite particle distribution where the particle density decreases from the center to the edge of the disk. The novel mechanism of density inhomogeneity driven topological defects as well as the underlying geometric structure may shed light in understanding a wide variety of relevant systems.
We analyze the stability and dynamics of toroidal liquid droplets. In addition to the Rayleigh instabilities akin to those of a cylindrical droplet there is a shrinking instability that is unique to the topology of the torus and dominates in the limit that the aspect ratio is near one (fat tori). We first find an analytic expression for the pressure distribution inside the droplet. We then determine the velocity field in the bulk fluid, in the Stokes flow regime, by solving the biharmonic equation for the stream function. The flow pattern in the external fluid is analyzed qualitatively by exploiting symmetries. This elucidates the detailed nature of the shrinking mode and the swelling of the cross-section following from incompressibility. Finally the shrinking rate of fat toroidal droplets is derived by energy conservation.
Understanding and controlling vesicle shapes is a fundamental challenge in biophysics and materials design. In this paper, we design dynamic protocols for enlarging the shape space of both fluid and crystalline vesicles beyond the equilibrium zone. By removing water from within the vesicle at different rates, we numerically produced a series of dynamically trapped stable vesicle shapes for both fluid and crystalline vesicles in a highly controllable fashion. In crystalline vesicles that are continuously dehydrated, simulations show the initial appearance of small flat areas over the surface of the vesicles that ultimately merge to form fewer flat faces. In this way, the vesicles transform from a fullerene-like shape into various faceted polyhedrons. We perform analytical elasticity analysis to show that these salient features are attributable to the crystalline nature of the vesicle. The potential to use dynamic protocols, such as those used in this study, to engineer vesicle shape transformations is helpful for exploiting the richness of vesicle geometries for desired applications.
We study the defect structure of crystalline particle arrays on negative Gaussian curvature capillary bridges with vanishing mean curvature (catenoids). The threshold aspect ratio for the appearance of isolated disclinations is found and the optimal positions for dislocations determined. We also discuss the transition from isolated disclinations to scars as particle number and aspect ratio are varied.
A perversion in an otherwise uniform helical structure, such as a climbing plant tendril, refers to a kink that connects two helices with opposite chiralities. Such singularity structures are widely seen in natural and artificial mechanical systems, and they provide the fundamental mechanism of helical symmetry breaking. However, it is still not clear how perversions arise in various helical structures and which universal principles govern them. As such, a heterogeneous elastic bistrip system provides an excellent model to address these questions. Here, we investigate intrinsic perversion properties which are independent of strip shapes. This study reveals the rich physics of perversions in the 3D elastic system, including the condensation of strain energy over perversions during their formation, the repulsive nature of the perversion-perversion interaction, and the coalescence of perversions that finally leads to a linear defect structure. This study may have implications for understanding relevant biological motifs and for use of perversions as energy storers in the design of micromuscles and soft robotics.S pontaneous symmetry breaking provides a unifying conceptual understanding of emergent ordered structures arising in various condensed matters (1). In an elastic medium, which is one of the simplest organizations of matter, symmetry-breaking instabilities via buckling can lead to extraordinarily rich patterns and generate a wealth of shapes at multiple length scales that can be exploited in many scientific disciplines (2). A prototype of elastic buckling is the Euler instability of a homogeneous elastic rod under uniaxial compression at the ends that finally breaks the rotational symmetry (3). Introduction of extra structures in an elastic medium like mechanical heterogeneities (4), nonlinearity of materials (2), geometric asymmetry (5), or intrinsic curvature (6) provides new dimensions that can produce even richer buckling modes, including helices and perversions (6, 7), wavy structures (8), regular networks of ridges (9), and even selfsimilar fractal patterns (2, 10). Of these emergent symmetry broken structures, the helical shapes are of particular interest due to their ubiquitousness in nature and the strong connection with biological motifs, as noticed by Darwin in his 1875 book describing the curl of plant tendrils (11). Remarkably, biological helical structures permeate over several length scales from the developed helical valve on opening seed pods (12), to the regular chiral structures in the flagella of bacteria (13), the spiral ramps of rough endoplasmic reticulum (14), and the chromosome of Escherichia coli (15, 16).The proliferation of perversions in an otherwise uniform helical structure can further break the helical symmetry (Fig. 1A shows a typical perversion in the helix) (4, 6, 17). Here, a perversion refers to a kink that connects two helices with opposite chiralities. Therefore, perversions belong to a large class of fundamental defects in systems with discrete symmetry which have the n...
In this paper we derive the general equilibrium equations of a polymer chain with a noncircular cross section by the variation of the free energy functional. From the equilibrium equation of the elastic ribbon we derive analytically the equilibrium conformations both of the helical ribbons and the twisted ribbons. We find that the pitch angle of the helical ribbons depends on the ratio of the torsional rigidity to the bending one. For the twisted ribbons, the rotation rate depends on the spontaneous torsion, which is determined by the elastic properties of the polymers. Our results for helical and twisted ribbons strongly indicate that the formation of these structures is determined by their elastic properties.PACS number͑s͒: 36.20.Fz, 46.25.Cc, 36.20.Ey The equilibrium conformation of biopolymers and filaments is an important issue in molecular and cellular biology, and polymer physics ͓1͔. For example, earlier theoretical models used elastic theory to describe the conformation of vital biomolecules such as proteins and DNA ͓2-6͔. In general, such models of polymer chains involve the use of elastic coefficients, which describe the elastic properties of materials. However, up to now most models of polymer chains have only considered circular cross sections. The need to improve on these models by introducing the significant effects of geometry is obvious. For example, recently there have been numerous experimental observations of polymers with noncircular cross sections such as chemically defined lipid concentrate ͑CDLC͒ ͓7-10͔, gemini surfactants ͓11͔, synthetic nonracemic chiral polymers ͓12,13͔, and carbon nanotubes ͓14͔. During the past decade there have been some theoretical attempts made to account for the effects of geometry in order to improve models. The helical structures of tilted chiral lipid bilayers were investigated by using a free energy model without considering the cross section ͓15͔. The stretching instability of helical ribbons was discussed by using continuum phenomenological elastic models ͓16,17͔. The transition between the helical and twisted ribbon structures of chiral materials has been studied both using continuum elastic theory and lattice Monte Carlo simulations ͓18,19͔. For polymer chains with noncircular cross sections, the thermal fluctuations on the statistical properties of thin elastic filaments have also been investigated ͓20,21͔. However, their equilibrium conformation equations have yet to be presented in the literature.In this paper, we focus on polymers characterized by a nonzero thickness and a noncircular cross section. We present a general elastic model of polymer chains with noncircular cross sections and use it to discuss elastic ribbons. First we consider the free energy density as a general functional of the curvature, the torsion, and the twist angle of a polymer chain. Then by calculating the variation of the free energy functional, we obtain the equilibrium conformation equations of a polymer chain with a noncircular cross section. Finally we consider the case ...
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