Defect formation and coarsening in hexagonal 2D curved crystals

Stability and Elasticity of Ultrathin Sphere-Patterned Block Copolymer Films

Sphere-patterned ultrathin block copolymer films are potentially interesting for a variety of applications in nanotechnology. We use self-consistent field theory to investigate the elastic response of sphere monolayer films with respect to in-plane shear, in-plane extension, compression deformations, and bending. The relations between the in-plane elastic moduli are roughly compatible with the expectations for two-dimensional elastic systems with hexagonal symmetry, with one notable exception: The pure shear and the simple shear moduli differ from each other by roughly 20%. Even more importantly, the bending constants are found to be negative, indicating that free-standing block copolymer membranes made of only a sphere monolayer are inherently unstable above the glass transition. Our results are discussed in view of the experimental findings.

Droplet-like Defect Annihilation Mechanisms in Hexagonal Cylinder-Forming Block Copolymers

The annihilation of typical individual defects in hexagonal cylinder-forming block copolymers is investigated using the self-consistent field theory (SCFT) in conjunction with the string method. Usually, defect removal in two-dimensional hexagonal patterns involves reorganizing the cylindrical domains. Unlike atoms in solid crystals, the self-assembled cylindrical domains of block copolymers are "soft". Thus, the kinetic motions of the cylindrical domains resemble liquid droplets. Dislocations in hexagonal patterns are eliminated via creating and removing cylindrical domains. Our results show that new cylindrical domains are created via either a nucleation-like process or a fission-like process, whereas excessive domains are eliminated via a fusion-like or evaporation-like process. For weakly segregated block copolymers, the nucleation-like and evaporation-like processes are preferred.

Two-dimensional crystalization on spheres: Crystals grow cracked

Here we study how curvature affects the structure of two-dimensional crystals growing on spheres. The mechanism of crystal growth is described by means of a Landau model in curved space that accounts for the excess of strain on crystal bonds caused by the substrate's curvature (packing frustration). In curved space elastic energy penalization strongly dictates the geometry of growing crystals. While compact faceted crystals are observed when elastic energy contribution can be neglected, cracked crystals with ribbonlike forms appear as the main mechanisms to reduce elastic frustration for highly curved systems.

The energy landscapes of bidisperse particle assemblies on a sphere

The interplay between crystalline ordering, curvature, and size dispersity make the packing of bidisperse mixtures of particles on a sphere a varied and complex phenomenon. These structures have functional significance in a broad range of systems, such as cellular organisation in spherical epithelia, catalytic activity in binary colloidosomes, and chemical activity in heterofullerenes. In this contribution, we elucidate the potential energy landscapes for systems of repulsive, bidisperse particles confined to the surface of a sphere. It is commonly asserted that particle size dispersity destroys ordered arrangements, leading to glassy landscapes. Surprisingly, across a range of compositions, we find highly ordered global minima. Moreover, a minority of small particles is able to passivate defects, stabilising bidisperse global minima relative to monodisperse systems. However, our landscape analysis also reveals that bidispersity introduces numerous defective, low-lying states that are expected to cause broken ergodicity in corresponding experimental and computational systems. Probing the global minimum structures further, particle segregation is energetically preferred at intermediate compositions, contrasting with the approximate icosahedral global packing at either end of the composition range. Finally, changing the composition has a dramatic effect on the heat capacity: systems with low-symmetry global minima have melting temperatures an order of magnitude lower than monodisperse or high-symmetry systems. This observation may provide a further example of the principle of maximum symmetry: higher symmetry global minima exhibit a larger energy separation from the minima that define the high-entropy phase-like region of configuration space, raising the transition temperature.

Defects and defect engineering in Soft Matter

Soft matter covers a wide range of materials based on linear or branched polymers, gels and rubbers, amphiphilic (macro)molecules, colloids, and self-assembled structures. These materials have applications in various industries, all highly important for our daily life, and they control all biological functions; therefore, controlling and tailoring their properties is crucial. One way to approach this target is defect engineering, which aims to control defects in the material's structure, and/or to purposely add defects into it to trigger specific functions. While this approach has been a striking success story in crystalline inorganic hard matter, both for mechanical and electronic properties, and has also been applied to organic hard materials, defect engineering is rarely used in soft matter design. In this review, we present a survey on investigations on defects and/or defect engineering in nine classes of soft matter composed of liquid crystals, colloids, linear polymers with moderate degree of branching, hyperbranched polymers and dendrimers, conjugated polymers, polymeric networks, self-assembled amphiphiles and proteins, block copolymers and supramolecular polymers. This overview proposes a promising role of this approach for tuning the properties of soft matter.

Defect patterns on the curved surface of fish retinae suggest a mechanism of cone mosaic formation

The outer epithelial layer of zebrafish retinae contains a crystalline array of cone photoreceptors, called the cone mosaic. As this mosaic grows by mitotic addition of new photoreceptors at the rim of the hemispheric retina, topological defects, called "Y-Junctions", form to maintain approximately constant cell spacing. The generation of topological defects due to growth on a curved surface is a distinct feature of the cone mosaic not seen in other well-studied biological patterns like the R8 photoreceptor array in the Drosophila compound eye. Since defects can provide insight into cell-cell interactions responsible for pattern formation, here we characterize the arrangement of cones in individual Y-Junction cores as well as the spatial distribution of Y-junctions across entire retinae. We find that for individual Y-junctions, the distribution of cones near the core corresponds closely to structures observed in physical crystals. In addition, Y-Junctions are organized into lines, called grain boundaries, from the retinal center to the periphery. In physical crystals, regardless of the initial distribution of defects, defects can coalesce into grain boundaries via the mobility of individual particles. By imaging in live fish, we demonstrate that grain boundaries in the cone mosaic instead appear during initial mosaic formation, without requiring defect motion. Motivated by this observation, we show that a computational model of repulsive cell-cell interactions generates a mosaic with grain boundaries. In contrast to paradigmatic models of fate specification in mostly motionless cell packings, this finding emphasizes the role of cell motion, guided by cell-cell interactions during differentiation, in forming biological crystals. Such a route to the formation of regular patterns may be especially valuable in situations, like growth on a curved surface, where the resulting long-ranged, elastic, effective interactions between defects can help to group them into grain boundaries.

Thermodynamics and ordering kinetics in asymmetric PS-b-PMMA block copolymer thin films

The ordering kinetics of standing cylinder-forming polystyrene-block-poly(methyl methacrylate) block copolymers (molecular weight: 39 kg mol^-1) close to the order-disorder transition is experimentally investigated following the temporal evolution of the correlation length at different annealing temperatures. The growth exponent of the grain-coarsening process is determined to be 1/2, signature of a curvature-driven ordering mechanism. The measured activation enthalpy and the resulting Meyer-Neldel temperature for this specific copolymer along with the data already known for PS-b-PMMA block copolymers in strong segregation limit allow investigation of the interplay between the ordering kinetics and the thermodynamic driving force during the grain coarsening. These findings unveil various phenomena concomitantly occurring during the thermally activated ordering kinetics at segmental, single chain, and collective levels.

Substrate curvature governs texture orientation in thin films of smectic block copolymers

Self-assembly of ordered nanometer-scale patterns is interesting in itself, but its practical value depends on the ability to predict and control pattern formation. In this paper we demonstrate theoretically and numerically that engineering of extrinsic as well as intrinsic substrate geometry may provide such a controllable ordering mechanism for block copolymers films. We develop an effective two-dimensional model of thin films of striped-phase diblock copolymers on general curved substrates. The model is obtained as an expansion in the film thickness and thus takes the third dimension into account, which crucially allows us to predict the preferred orientations even in the absence of intrinsic curvature. We determine the minimum-energy textures on several curved surfaces and arrive at a general principle for using substrate curvature as an ordering field, namely that the stripes will tend to align along directions of maximal curvature.

Geometry and kinetics determine the microstructure in arrested coalescence of Pickering emulsion droplets

Arrested coalescence occurs in Pickering emulsions where colloidal particles adsorbed on the surface of the droplets become crowded and inhibit both relaxation of the droplet shape and further coalescence. The resulting droplets have a nonuniform distribution of curvature and, depending on the initial coverage, may incorporate a region with negative Gaussian curvature around the neck that bridges the two droplets. Here, we resolve the relative influence of the curvature and the kinetic process of arrest on the microstructure of the final state. In the quasistatic case, defects are induced and distributed to screen the Gaussian curvature. Conversely, if the rate of area change per particle exceeds the diffusion constant of the particles, the evolving surface induces local solidification reminiscent of jamming fronts observed in other colloidal systems. In this regime, the final structure is shown to be strongly affected by the compressive history just prior to arrest, which can be predicted from the extrinsic geometry of the sequence of surfaces in contrast to the intrinsic geometry that governs the static regime.

Topologically-guided continuous protein crystallization controls bacterial surface layer self-assembly

Many bacteria and most archaea possess a crystalline protein surface layer (S-layer), which surrounds their growing and topologically complicated outer surface. Constructing a macromolecular structure of this scale generally requires localized enzymatic machinery, but a regulatory framework for S-layer assembly has not been identified. By labeling, superresolution imaging, and tracking the S-layer protein (SLP) from C. crescentus, we show that 2D protein self-assembly is sufficient to build and maintain the S-layer in living cells by efficient protein crystal nucleation and growth. We propose a model supported by single-molecule tracking whereby randomly secreted SLP monomers diffuse on the lipopolysaccharide (LPS) outer membrane until incorporated at the edges of growing 2D S-layer crystals. Surface topology creates crystal defects and boundaries, thereby guiding S-layer assembly. Unsupervised assembly poses challenges for therapeutics targeting S-layers. However, protein crystallization as an evolutionary driver rationalizes S-layer diversity and raises the potential for biologically inspired self-assembling macromolecular nanomaterials.

Bacteria assemble the surface layer (S-layer), a crystalline protein coat surrounding the curved surface, using protein self-assembly. Here authors image native and purified RsaA, the S-layer protein from C. crescentus, and show that protein crystallization alone is sufficient to assemble and maintain the S-layer in vivo.

Coarsening dynamics in the Swift-Hohenberg equation with an external field

We study the Swift-Hohenberg equation (SHE) in the presence of an external field. The application of the field leads to a phase diagram with three phases, i.e., stripe, hexagon, and uniform. We focus on coarsening after a quench from the uniform to stripe or hexagon regions. For stripe patterns, we find that the length scale associated with the order-parameter structure factor has the same growth exponent (≃1/4) as for the SHE with zero field. The growth process is slower in the case of hexagonal patterns, with the effective growth exponent varying between 1/6 and 1/9, depending on the quench parameters. For deep quenches in the hexagonal phase, the growth process stops at late stages when defect boundaries become pinned.

Curvature as a Guiding Field for Patterns in Thin Block Copolymer Films

Experimental data on thin films of cylinder-forming block copolymers (BC)-free-standing BC membranes as well as supported BC films-strongly suggest that the local orientation of the BC patterns is coupled to the geometry in which the patterns are embedded. We analyze this phenomenon using general symmetry considerations and numerical self-consistent field studies of curved BC films in cylindrical geometry. The stability of the films against curvature-induced dewetting is also analyzed. In good agreement with experiments, we find that the BC cylinders tend to align along the direction of curvature at high curvatures. At low curvatures, we identify a transition from perpendicular to parallel alignment in supported films, which is absent in free-standing membranes. Hence both experiments and theory show that curvature can be used to manipulate and align BC patterns.

Block copolymer crystalsomes with an ultrathin shell to extend blood circulation time

In water, amphiphilic block copolymers (BCPs) can self-assemble into various micelle structures depicting curved liquid/liquid interface. Crystallization, which is incommensurate with this curved space, often leads to defect accumulation and renders the structures leaky, undermining their potential biomedical applications. Herein we report using an emulsion-solution crystallization method to control the crystallization of an amphiphilic BCP, poly (l-lactide acid)-b-poly (ethylene glycol) (PLLA-b-PEG), at curved liquid/liquid interface. The resultant BCP crystalsomes (BCCs) structurally mimic the classical polymersomes and liposomes yet mechanically are more robust thanks to the single crystal-like crystalline PLLA shell. In blood circulation and biodistribution experiments, fluorophore-loaded BCCs show a 24 h circulation half-life and a 8% particle retention in the blood even at 96 h post injection. We further demonstrate that this good performance can be attributed to controlled polymer crystallization and the unique BCC nanostructure.

In block copolymer vesicles, crystallization often leads to defects and renders the structures leaky that undermines their potential biomedical application. Here the authors use an emulsion solution method to control the crystallization of an amphiphilic block copolymer at the curved liquid/liquid interface to improve the blood circulation time.

Defect formation dynamics in curved elastic surface crystals

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 validity for non-thermal transitions in curved and topologically nontrivial systems still poses open questions. Here, we use recent experimentally confirmed theory to investigate topological defect formation in curved elastic surface crystals formed by stress-quenching a bilayer material. For both spherical and toroidal crystals, we find that the defect densities follow KZ-type power laws. Moreover, the nucleation sequences agree with recent experimental observations for spherical colloidal crystals. Our results suggest that curved elastic bilayers provide an experimentally accessible macroscopic system to study universal properties of non-thermal phase transitions in non-planar geometries.

Polyethylene nano crystalsomes formed at a curved liquid/liquid interface

Crystallization is incommensurate with nanoscale curved space due to the lack of three dimensional translational symmetry of the latter. Herein, we report the formation of single-crystal-like, nanosized polyethylene (PE) capsules using a miniemulsion solution crystallization method. The miniemulsion was formed at elevated temperatures using PE organic solution as the oil phase and sodium dodecyl sulfate as the surfactant. Subsequently, cooling the system stepwisely for controlled crystallization led to the formation of hollow, nanosized PE crystalline capsules, which are named as crystalsomes since they mimic the classical self-assembled structures such as liposome, polymersome and colloidosome. We show that the formation of the nanosized PE crystalsomes is driven by controlled crystallization at the curved liquid/liquid interface of the miniemulson droplet. The morphology, structure and mechanical properties of the PE crystalsomes were characterized using scanning electron microscopy, transmission electron microscopy, X-ray diffraction, and atomic force spectroscopy. Electron diffraction showed the single-crystal-like nature of the crystalsomes. The incommensurateness between the nanocurved interface and the crystalline packing led to reduced crystallinity and crystallite size of the PE crystalsome, as observed from the X-ray diffraction measurements. Moreover, directly quenching the emulsion below the spinodal line led to the formation of hierarchical porous PE crystalsomes due to the coupling of the PE crystallization and liquid/liquid phase separation. We anticipate that this unique crystalsome represents a new type of nanostructure that might be used as nanodrug carriers and ultrasound contrast agents.

Frustrated crystallization of a monolayer of magnetized beads under geometrical confinement

We present a systematic experimental study of the confinement effect on the crystallization of a monolayer of magnetized beads. The particles are millimeter-scale grains interacting through the short range magnetic dipole-dipole potential induced by an external magnetic field. The grains are confined by repulsing walls and are homogeneously distributed inside the cell. A two-dimensional (2d) Brownian motion is induced by horizontal mechanical vibrations. Therefore, the balance between magnetic interaction and agitation allows investigating 2d phases through direct visualization. The effect of both confinement size and shape on the grains' organization in the low-energy state has been investigated. Concerning the confinement shape, triangular, square, pentagonal, hexagonal, heptagonal, and circular geometries have been considered. The grain organization was analyzed after a slow cooling process. Through the measurement of the averaged bond order parameter for the different confinement geometries, it has been shown that cell geometry strongly affects the ordering of the system. Moreover, many kinds of defects, whose observation rate is linked to the geometry, have been observed: disclinations, dislocations, defects chain, and also more exotic defects such as a rosette. Finally, the influence of confinement size has been investigated and we point out that no finite-size effect occurs for a hexagonal cell, but the finite-size effect changes from one geometry to another.

Mesostructure of Ordered Corneal Nano-nipple Arrays: The Role of 5-7 Coordination Defects

Corneal nano-nipple structures consisting of hexagonally arranged protrusions with diameters around 200 nm have long been known for their antireflection capability and have served as biological blueprint for solar cell, optical lens and other surface designs. However, little is known about the global arrangement of these nipples on the ommatidial surface and their growth during the eye development. This study provides new insights based on the analysis of nano-nipple arrangements on the mesoscale across entire ommatidia, which has never been done before. The most important feature in the nipple structures are topological 5- and 7-fold coordination defects, which align to form dislocations and interconnected networks of grain boundaries that divide the ommatidia into crystalline domains in different orientations. Furthermore, the domain size distribution might be log-normal, and the domains demonstrate no preference in crystal orientation. Both observations suggest that the nipple growth process may be similar to the nucleation and growth mechanisms during the formation of other crystal structures. Our results are also consistent with the most recently proposed Turing-type reaction-diffusion process. In fact, we were able to produce the key structural characteristics of the nipple arrangements using Turing analysis from the nucleation to the final structure development.

Responsive Shape Change of Sub-5 nm Thin, Janus Polymer Nanoplates

Responsive shape changes in soft materials have attracted significant attention in recent years. Despite extensive studies, it is still challenging to prepare nanoscale assemblies with responsive behaviors. Herein we report on the fabrication and pH-responsive properties of sub-5 nm thin, Janus polymer nanoplates prepared via crystallization-driven self-assembly of poly(ε-caprolactone)-b-poly(acrylic acid) (PCL-b-PAA) followed by cross-linking and disassembly. The resultant Janus nanoplate is comprised of partially cross-linked PAA and tethered PCL brush layers with an overall thickness of ∼4 nm. We show that pronounced and reversible shape changes from nanoplates to nanobowls can be realized in such a thin free-standing film. This shape change is achieved by exceptionally small stress-a few orders of magnitude smaller than conventional hydrogel bilayers. These three-dimensional ultrathin nanobowls are also mechanically stable, which is attributed to the tortoise-shell-like crystalline domains formed in the nanoconfined curved space. Our results pave a way to a new class of free-standing, ultrathin polymer Janus nanoplates that may find applications in nanomotors and nanoactuators.

Highly robust crystalsome via directed polymer crystallization at curved liquid/liquid interface

Lipids and amphiphilic block copolymers spontaneously self-assemble in water to form a plethora of micelles and vesicles. They are typically fluidic in nature and often mechanically weak for applications such as drug delivery and gene therapeutics. Mechanical properties of polymeric materials could be improved by forming crystalline structures. However, most of the self-assembled micelles and vesicles have curved surfaces and precisely tuning crystallization within a nanoscale curved space is challenging, as the curved geometry is incommensurate with crystals having three-dimensional translational symmetry. Herein, we report using a miniemulsion crystallization method to grow nanosized, polymer single-crystal-like capsules. We coin the name crystalsome to describe this unique structure, because they are formed by polymer lamellar crystals and their structure mimics liposomes and polymersomes. Using poly(L-lactic acid) (PLLA) as the model polymer, we show that curved water/p-xylene interface formed by the miniemulsion process can guide the growth of PLLA single crystals. Crystalsomes with the size ranging from ∼148 nm to over 1 μm have been formed. Atomic force microscopy measurement demonstrate a two to three orders of magnitude increase in bending modulus compared with conventional polymersomes. We envisage that this novel structure could shed light on investigating spherical crystallography and drug delivery.

Smectic block copolymer thin films on corrugated substrates

In this work we study equilibrium and non-equilibrium structures of smectic block copolymer thin films deposited on a topographically patterned substrate. A Brazovskii free energy model is employed to analyze the coupling between the smectic texture and the local mean curvature of the substrate. The substrate's curvature produces out-of-plane deformations of the block copolymer such that equilibrium textures are modified and dictated by the underlying geometry. For weak curvatures it is shown that the free energy of the block copolymer film follows a Helfrich form, scaling with the square of the mean curvature, with a bending constant dependent on the local pattern orientation. On substrates of varying mean curvature simulations show that topological defects are rapidly expelled from regions with large curvature. These results compare well with available experimental data of poly(styrene)-co-poly(ethylene-alt-propylene) smectic thin films.