Defect scars on flexible surfaces with crystalline order

Dynamics of fluid bilayer vesicles: Soft meshes and robust curvature energy discretization

Continuum models like the Helfrich Hamiltonian are widely used to describe fluid bilayer vesicles. Here we study the molecular dynamics compatible dynamics of the vertices of two-dimensional meshes representing the bilayer, whose in-plane motion is only weakly constrained. We show (i) that Jülicher's discretization of the curvature energy offers vastly superior robustness for soft meshes compared to the commonly employed expression by Gommper and Kroll and (ii) that for sufficiently soft meshes, the typical behavior of fluid bilayer vesicles can emerge even if the mesh connectivity remains fixed throughout the simulations. In particular, soft meshes can accommodate large shape transformations, and the model can generate the typical ℓ^{-4} signal for the amplitude of surface undulation modes of nearly spherical vesicles all the way up to the longest wavelength modes. Furthermore, we compare results for Newtonian, Langevin, and Brownian dynamics simulations of the mesh vertices to demonstrate that the internal friction of the membrane model is negligible, making it suitable for studying the internal dynamics of vesicles via coupling to hydrodynamic solvers or particle-based solvent models.

Buckling transitions and soft-phase invasion of two-component icosahedral shells

What is the optimal distribution of two types of crystalline phases on the surface of icosahedral shells, such as of many viral capsids? We here investigate the distribution of a thin layer of soft material on a crystalline convex icosahedral shell. We demonstrate how the shapes of spherical viruses can be understood from the perspective of elasticity theory of thin two-component shells. We develop a theory of shape transformations of an icosahedral shell upon addition of a softer, but still crystalline, material onto its surface. We show how the soft component "invades" the regions with the highest elastic energy and stress imposed by the 12 topological defects on the surface. We explore the phase diagram as a function of the surface fraction of the soft material, the shell size, and the incommensurability of the elastic moduli of the rigid and soft phases. We find that, as expected, progressive filling of the rigid shell by the soft phase starts from the most deformed regions of the icosahedron. With a progressively increasing soft-phase coverage, the spherical segments of domes are filled first (12 vertices of the shell), then the cylindrical segments connecting the domes (30 edges) are invaded, and, ultimately, the 20 flat faces of the icosahedral shell tend to be occupied by the soft material. We present a detailed theoretical investigation of the first two stages of this invasion process and develop a model of morphological changes of the cone structure that permits noncircular cross sections. In conclusion, we discuss the biological relevance of some structures predicted from our calculations, in particular for the shape of viral capsids.

Dislocation screening in crystals with spherical topology

Whereas disclination defects are energetically prohibitive in two-dimensional flat crystals, their existence is necessary in crystals with spherical topology, such as viral capsids, colloidosomes, or fullerenes. Such a geometrical frustration gives rise to large elastic stresses, which render the crystal unstable when its size is significantly larger than the typical lattice spacing. Depending on the compliance of the crystal with respect to stretching and bending deformations, these stresses are alleviated either by a local increase of the intrinsic curvature in proximity of the disclinations or by the proliferation of excess dislocations, often organized in the form of one-dimensional chains known as "scars." The associated strain field of the scars is such as to counterbalance the one resulting from the isolated disclinations. Here we develop a continuum theory of dislocation screening in two-dimensional closed crystals with genus one. Upon modeling the flux of scars emanating from a given disclination as an independent scalar field, we demonstrate that the elastic energy of closed two-dimensional crystals with various degrees of asphericity can be expressed as a simple quadratic function of the screened topological charge of the disclinations, at both zero and finite temperature. This allows us to predict the optimal density of the excess dislocations as well as the minimal stretching energy attained by the crystal.

Complex Self-Assembly from Simple Interaction Rules in Model Colloidal Mixtures

Building structures with hierarchical order through the self-assembly of smaller blocks is not only a prerogative of nature, but also a strategy to design artificial materials with tailored functions. We explore in simulation the spontaneous assembly of colloidal particles into extended structures, using spheres and size-asymmetric dimers as solute particles, while treating the solvent implicitly. Besides rigid cores for all particles, we assume an effective short-range attraction between spheres and small monomers to promote, through elementary rules, dimer-mediated aggregation of spheres. Starting from a completely disordered configuration, we follow the evolution of the system at low temperature and density, as a function of the relative concentration of the two species. When spheres and large monomers are of same size, we observe the onset of elongated aggregates of spheres, either disconnected or cross-linked, and a crystalline bilayer. As spheres grow bigger, the self-assembling scenario changes, getting richer overall, with the addition of flexible membrane sheets with crystalline order and monolayer vesicles. With this wide assortment of structures, our model can serve as a viable template to achieve a better control of self-assembly in dilute suspensions of microsized particles.

Ordered Surface Nanostructures Self-Assembled from Rod-Coil Block Copolymers on Microspheres

An ordered surface nanostructure endows materials advanced functions. However, fabricating ordered surface-patterned particles via the polymer self-assembly approach is a challenge. Here we report that poly(γ-benzyl-l-glutamate)-block-poly(ethylene glycol) rod-coil block copolymers are able to form uniform-surface micelles on polystyrene microspheres through a solution self-assembly approach. The size of the surface micelles can be varied by the molecular weight of the block copolymers. These surface micelles are arranged in a manner consistent with the Euler theorem. Most of the micelles are six-fold coordinated, and the number difference between the five-fold and the seven-fold coordination is 12. Simulations on model systems qualitatively reproduced the experimental findings and provided direct observations for the surface-patterned particles, including the polymer chain packing manner in surface micelles at the molecular level and the array feature of the surface micelles through 2D projections of the surface patterns.

Shape Morphologies of Icosahedral Two-Component Vesicles

What are the features of partitioning of crystalline materials on the surface of a two-component icosahedral vesicle? We model the response of the rigid hardly stretchable crystalline icosahedra upon addition of a softer component on its surface. We demonstrate how the soft phase "invades" the shell regions with the highest elastic energy density around 12 5-fold topological defects. We explore the phase diagram of these inhomogeneous shells as a function of the soft material fraction, shell radius, and elastic moduli of the two phases. The findings are compared with the recent computer simulation findings, and their biological relevance, for example, for the structure of icosahedral viruses, is also discussed.

How faceted liquid droplets grow tails

Liquid droplets, widely encountered in everyday life, have no flat facets. Here we show that water-dispersed oil droplets can be reversibly temperature-tuned to icosahedral and other faceted shapes, hitherto unreported for liquid droplets. These shape changes are shown to originate in the interplay between interfacial tension and the elasticity of the droplet's 2-nm-thick interfacial monolayer, which crystallizes at some T = Ts above the oil's melting point, with the droplet's bulk remaining liquid. Strikingly, at still-lower temperatures, this interfacial freezing (IF) effect also causes droplets to deform, split, and grow tails. Our findings provide deep insights into molecular-scale elasticity and allow formation of emulsions of tunable stability for directed self-assembly of complex-shaped particles and other future technologies.

Monte Carlo simulations of fluid vesicles

Lipid vesicles are closed two dimensional fluid surfaces that are studied extensively as model systems for understanding the physical properties of biological membranes. Here we review the recent developments in the Monte Carlo techniques for simulating fluid vesicles and discuss some of their applications. The technique, which treats the membrane as an elastic sheet, is most suitable for the study of large scale conformations of membranes. The model can be used to study vesicles with fixed and varying topologies. Here we focus on the case of multi-component membranes with the local lipid and protein composition coupled to the membrane curvature leading to a variety of shapes. The phase diagram is more intriguing in the case of fluid vesicles having an in-plane orientational order that induce anisotropic directional curvatures. Methods to explore the steady state morphological structures due to active flux of materials have also been described in the context of Monte Carlo simulations.

Fluctuating shells under pressure

Thermal fluctuations strongly modify the large length-scale elastic behavior of cross-linked membranes, giving rise to scale-dependent elastic moduli. Whereas thermal effects in flat membranes are well understood, many natural and artificial microstructures are modeled as thin elastic shells. Shells are distinguished from flat membranes by their nonzero curvature, which provides a size-dependent coupling between the in-plane stretching modes and the out-of-plane undulations. In addition, a shell can support a pressure difference between its interior and its exterior. Little is known about the effect of thermal fluctuations on the elastic properties of shells. Here, we study the statistical mechanics of shape fluctuations in a pressurized spherical shell, using perturbation theory and Monte Carlo computer simulations, explicitly including the effects of curvature and an inward pressure. We predict novel properties of fluctuating thin shells under point indentations and pressure-induced deformations. The contribution due to thermal fluctuations increases with increasing ratio of shell radius to thickness and dominates the response when the product of this ratio and the thermal energy becomes large compared with the bending rigidity of the shell. Thermal effects are enhanced when a large uniform inward pressure acts on the shell and diverge as this pressure approaches the classical buckling transition of the shell. Our results are relevant for the elasticity and osmotic collapse of microcapsules.

Dynamics of vesicle unbinding under axisymmetric flow

The competition between adhesion and external flow to unbind settled vesicles from substrates is investigated. An experimental setup is developed to apply a hydrodynamic pulling force in the range of a few piconewtons to a vesicle with retained axisymmetry. In the limit of a small excess of membrane area, vesicles are found to transit during unbinding from a process of fluid film thickening at constant contact area to a finite-time process of contact radius drop to zero with an exponent 1/2. Both characteristic times vary linearly with the inverse flow rate. On the contrary, deflated vesicles under a moderate pulling force exhibit a decrease of contact area at a constant film thickness before a film thickening.

Curvature-induced frustration in the XY model on hyperbolic surfaces

We study low-temperature properties of the XY spin model on a negatively curved surface. Geometric curvature of the surface gives rise to frustration in local spin configuration, which results in the formation of high-energy spin clusters scattered over the system. Asymptotic behavior of the spin-glass susceptibility suggests a zero-temperature glass transition, which is attributed to multiple optimal configurations of spin clusters due to nonzero surface curvature of the system. It implies that a constant ferromagnetic spin interaction on a regular lattice can exhibit glasslike behavior without possessing any disorder if the lattice is put on top of a negatively curved space such as a hyperbolic surface.