Symmetry breaking in smectics and surface models of their singularities

The homotopy theory of topological defects in ordered media fails to completely characterize systems with broken translational symmetry. We argue that the problem can be understood in terms of the lack of rotational Goldstone modes in such systems and provide an alternate approach that correctly accounts for the interaction between translations and rotations. Dislocations are associated, as usual, with branch points in a phase field, whereas disclinations arise as critical points and singularities in the phase field. We introduce a three-dimensional model for two-dimensional smectics that clarifies the topology of disclinations and geometrically captures known results without the need to add compatibility conditions. Our work suggests natural generalizations of the two-dimensional smectic theory to higher dimensions and to crystals.     

Deformation and failure of curved colloidal crystal shells

Designing and controlling particle self-assembly into robust and reliable high-performance smart materials often involves crystalline ordering in curved spaces. Examples include carbon allotropes like graphene, synthetic materials such as colloidosomes, or biological systems like lipid membranes, solid domains on vesicles, or viral capsids. Despite the relevance of these structures, the irreversible deformation and failure of curved crystals is still mostly unexplored. Here, we report simulation results of the mechanical deformation of colloidal crystalline shells that illustrate the subtle role played by geometrically necessary topological defects in controlling plastic yielding and failure. We observe plastic deformation attributable to the migration and reorientation of grain boundary scars, a collective process assisted by the intermittent proliferation of disclination pairs or abrupt structural failure induced by crack nucleating at defects. Our results provide general guiding principles to optimize the structural and mechanical stability of curved colloidal crystals.The morphology of crystals becomes peculiar when self-assembled on curved shells. For example, the Gaussian curvature of a sphere demands the presence of geometrically necessary rotational defects (disclinations) such as the 12 pentagons in a soccer ball. Disclinations can be found in thin shell structures at different length scales: from the world of carbon allotropes (1) [as in fullerenes, nanotubes, and graphene (2)] to biological systems [such as in lipid membranes (3), solid domains on vesicles (4, 5), or in viral capsids (6–8)], and in synthetic structures such as colloidosomes, colloidal particle shells lying at the interface between two fluids (9–12). Thin-shell structures are often conceived for encapsulation purposes at various scales (i.e., as delivery vehicles of different kinds of cargo, from drugs to flavors and cosmetics) and arise naturally in biological systems. Examples include crystalline and glassy colloidosomes, capsules of Janus and patchy particles, nematic vesicles, and viral capsids.Theoretical considerations indicate that arranging a colloidal crystal into a curved geometry involves elastic deformation and the presence of geometrically necessary disclinations showed by simulations to be attached to extended grain boundary scars (13–15), as also confirmed in several experiments (5, 9, 16, 17). Thus, grain boundary scars are different from standard grain boundaries in that they have a nonzero disclination charge, which makes them more costly. Such a complex topological structure is bound to interfere with the mechanical response of the shell in a way that is still unclear. Understanding this point, however, is of utmost importance to control the deformation of many functionalized self-assembled materials (9, 18, 19). Numerical and theoretical approaches to date are typically based on solving the elasticity field between grain-boundary scars and deriving equilibrium particle configurations from the effective free energy of the interacting defects (15, 20–23). Dynamic models have also been considered (24) to describe the experimentally observed dislocation gliding within the grain-boundary scars (5), or to precisely relate the crystallization dynamics to the surface curvature (25), but no studies so far have inspected the stress–strain relationships or the microstructural reorganizations occurring in response to different protocols of deformation of curved crystals under load.Here, we study the mechanical response of crystalline colloidal shells by molecular dynamics simulations. To this end, we consider a crystal made by colloidal particles confined to the surface of a sphere and then analyze its response to geometrical changes of this surface. We first simulate an isotropic inflation of the sphere, which induces tensile stresses in the crystal, leading eventually to its failure. Geometrically necessary scars act in this case as weak spots where fracture is nucleated. Next, we consider shape deformations that modify the local curvature, such as the squeezing of the sphere. In particular, we discuss shape deformations that preserve the shell surface area, so that stretching is not relevant. In this case, the crystal deforms irreversibly and intermittently by reorganizing and reorienting its grain boundary scar structure. Nevertheless, disclinations and/or scars do not seem to glide easily through the crystal. We observe that these defects only move through the reaction with new dislocations that are nucleated in the crystal along the deformation process. To better understand the motion of disclinations and scars, we also study the response of the crystal to a localized deformation attributable to indentation that leads to the formation of a hole in the crystal. The newly created hole changes the crystal topological characteristics and provokes the reorganization of its existing scar structure. We corroborate that the basic microscopic mechanism undergoing scar motion always requires the assistance of new dislocations. All of these peculiar microscopic processes trigger a heterogeneous response in the form of scale-free plastic avalanches (26, 27). The scaling properties of plastic deformation in curved geometries, however, are observed to deviate from the statistical behavior experimentally and numerically observed for similar phenomena in flat geometry.     

Shape multistability in flexible tubular crystals through interactions of mobile dislocations

We study avenues to shape multistability and shape morphing in flexible crystalline membranes of cylindrical topology, enabled by glide mobility of dislocations. Using computational modeling, we obtain states of mechanical equilibrium presenting a wide variety of tubular crystal deformation geometries, due to an interplay of effective defect interactions with out-of-tangent-plane deformations that reorient the tube axis. Importantly, this interplay often stabilizes defect configurations quite distinct from those predicted for a two-dimensional crystal confined to the surface of a rigid cylinder. We find that relative and absolute stability of competing states depend strongly on control parameters such as bending rigidity, applied stress, and spontaneous curvature. Using stable dislocation pair arrangements as building blocks, we demonstrate that targeted macroscopic three-dimensional conformations of thin crystalline tubes can be programmed by imposing certain sparse patterns of defects. Our findings reveal a broad design space for controllable and reconfigurable colloidal tube geometries, with potential relevance also to architected carbon nanotubes and microtubules.

Diverse biological and synthetic systems at a range of scales are self-organized in ordered two-dimensional (2D) assemblies of cylindrical topology, including single-walled carbon nanotubes (SWCNTs) (1), filamentous viral capsids (2), microtubules (MTs) (3), and colloidal systems (4). Such tubular crystals frequently have circumferences of order only 10 times the interparticle spacing. This has the important consequence of restricting the orientations of the crystal axes, which trace out helical paths called parastichies, to a discrete set of possible angles with the tube axis. The number of distinct parastichies defines a pair of integer parastichy numbers, which index the possible crystalline tessellations of the cylinder, and which, for SWCNTs, determine the nanotube’s electrical conductivity (5). The “parastichy” terminology arises from an intriguing connection with the botanical study of phyllotaxis, which examines plant structures with repeating patterns that follow parallel helices (or spirals); examples include the arrangements of seeds on a pine cone, scales on a pineapple, or leaves on a stem (6, 7).Along with the importance of tubular crystals to molecular biology and the study of 2D solids, there is a growing interest in tubular crystals among soft matter physicists, due to the potential for exploiting phyllotaxis as a self-organization principle for colloidal particles or nanoparticles, and thus for creating assemblies of controllable helical pitch and chirality (4, 8–11). Higher-scale organization can occur through the coexistence of distinct phyllotactic tessellations on the same tube (12). Topological defects are central to this higher-scale organization: A change in parastichy numbers requires one or more dislocation defects at the boundary between domains (13). In SWCNTs, the analogous Stone–Wales defects in the honeycomb lattice of graphene are of great interest for their influence on plastic deformations (14) and electrical conductivity (5). Similarly, the observation of MTs with varying protofilament number (i.e., circumference) along their length implies the presence of dislocations in the rhombic packing of tubulin proteins (15).Much of the recent work on frustrated phyllotactic self-organization has focused on particles constrained to lie in or adjacent to a fixed cylindrical surface (16–21). That version of the tubular crystal is realized in recent experiments on colloids confined in capillaries (4, 8, 22) or assembling on a cylindrical substrate (21), as well as a macroscopic “magnetic cactus” model system (23, 24). Tubular crystals on fixed cylinders exhibit a rich variety of phenomena such as oblique (rhombic) lattices (17), helical faults known as line slips (16), and individual dislocations behaving analogously to infinite grain boundaries (25).However, in order to design colloidal analogs of MTs and SWCNTs, we must examine a different version of the problem, namely, the freestanding tubular crystal. Here, the particles are not constrained to any fixed surface; instead, the tubular surface emerges as the set of mechanical equilibrium positions of particles whose bond network has the topology of a tube (26). The ability of the tube shape to adapt locally and dynamically removes the source of frustration, specifically, the fixed circumference, that underlies the rich defect phenomenology of the fixed-cylinder crystals. On the other hand, this geometrical adaptability offers potential routes to stabilizing nontrivial tube shapes in the presence of defects, a possibility that has remained mostly unexplored.In this work, we demonstrate, numerically, that freestanding tubular crystals possess controllable and composable mechanically multistable geometries, enabled by rearrangements of dislocations through glide mobility. This effect, which we refer to as “dislocation-mediated shape multistability,” operates through a costabilization of kinks in the tube axis with defect configurations that would be unstable on the fixed cylinder or the plane. While such kinks can also be formed with isolated disclinations (27, 28), our focus on dislocation pairs emphasizes mechanical shape reconfigurability through dislocation glide moves, which are purely local disruptions in the lattice. Recent experiments demonstrate that dislocations can be precisely generated and moved in 2D colloidal crystals using optical tweezers (29, 30) or local melting by a focused laser beam (31).Using a minimal model of a tubular crystalline membrane, we show not only that a tubular crystal may possess distinct, mechanically stable geometries but also that it is possible to controllably switch between these competing states. Such switching can be achieved through quantitative changes in material properties, which, in principle, might be accomplished in situ by varying the temperature, or through application of external bending forces. Furthermore, by repeating certain stable dislocation motifs along the length of a tubular crystal, we show how multiple-kink structures approximating arbitrary curves can be targeted as equilibrium geometries, offering proofs of principle for large-scale shape manipulations of tubes into bent and helical structures.The important distinction between freestanding and fixed-surface crystals is well known for the spherical topology, providing the difference between the scars of “spherical crystallography” (32) on rigid spheres and buckled crystalline shells resembling viral capsids (33) in flexible membranes. More generally, ordered soft matter confined to rigid, curved surfaces tends to relax the stresses imposed by Gaussian curvature through pair nucleation of topological defects (28, 34, 35); conversely, freestanding crystalline membranes can spontaneously adopt buckled geometries in the presence of defects such as dislocations and disclinations (36, 37). An analogous effect in nematic elastomer sheets allows for targeted shape transitions in the vicinity of a topological defect (38, 39), where nonzero Gaussian curvature arises to relax the elastic stresses.At the same time, it is worth emphasizing key differences between tubular and spherical crystals. For spheres, topology demands a net excess disclination charge of +2, whereas tubes (even if closed as a torus) have no topologically required defects. In addition, the Gaussian curvature of surfaces such as spheres and tori can promote lines of dislocations known as scars (32) and stabilization of excess unbound disclinations (40). In contrast, the Gaussian curvature in a perfect cylinder is everywhere zero. Rather, defects in a tubular crystal are intimately related to the discretization of crystal axes orientations: A change of parastichy numbers, whether occurring spontaneously or to mediate plastic deformation, necessarily requires an intervening dislocation.We study the emergent interplay of dislocation interactions and surface deformation geometries by modeling the crystalline membrane as a triangular-lattice network of harmonic spring bonds with a bending rigidity, and with the overall topology of a tube. We assume that dislocations glide freely into energy-minimizing configurations, with a rapid relaxation of the crystal’s elastic energy between glide steps, and with a small, finite temperature able to overcome the Peierls barrier, which we ignore (26, 41). We prohibit climb motion for simplicity, with the justification that climb requires an exchange of mass with the surrounding medium along with breaking and forming of multiple bonds, a process typically much slower than glide relaxation (41). By this means, we calculate effective energy landscapes for dislocations interacting on a tubular crystal whose surface deformations respond strongly to changes in defect position, creating multistable energy landscapes in both defect configuration and tube shape.Our approach extends the methodology widely used to create soft elastic actuators by encoding locally preferred membrane geometries in the in-surface order to develop target three-dimensional (3D) morphologies (42–44). Viewed as plastically deformed actuators, our shape-morphing tubular crystals represent potential routes to creating carbon nanotube actuators (45): metamaterials with exceptional mechanical properties (46), electronic behavior (47, 48), and biomedical potential (49, 50). Further, the metastable tubular crystals can be used as building blocks for larger-scale architected structures of different dimensionality, for example, yarns (51), ribbons (52, 53), or 3D networks (scaffolds) of connected tubules (54, 55). Our findings offer a potential mechanism for explaining and exploiting large deformations in tubular crystals at the nanoscale (56, 57). Defects in the tubulin lattice can significantly alter the mesoscale shape of an MT, and can provide a means of plastic deformation to accommodate external stress without breaking (58) or folding (59), and can even reinforce the structure (60).     

Knots and nonorientable surfaces in chiral nematics

Knots and knotted fields enrich physical phenomena ranging from DNA and molecular chemistry to the vortices of fluid flows and textures of ordered media. Liquid crystals provide an ideal setting for exploring such topological phenomena through control of their characteristic defects. The use of colloids in generating defects and knotted configurations in liquid crystals has been demonstrated for spherical and toroidal particles and shows promise for the development of novel photonic devices. Extending this existing work, we describe the full topological implications of colloids representing nonorientable surfaces and use it to construct torus knots and links of type (p,2) around multiply twisted Möbius strips.     

Optically driven translational and rotational motions of microrod particles in a nematic liquid crystal

A small amount of azo-dendrimer molecules dissolved in a liquid crystal enables translational and rotational motions of microrods in a liquid crystal matrix under unpolarized UV light irradiation. This motion is initiated by a light-induced trans-to-cis conformational change of the dendrimer adsorbed at the rod surface and the associated director reorientation. The bending direction of the cis conformers is not random but is selectively chosen due to the curved local director field in the vicinity of the dendrimer-coated surface. Different types of director distortions occur around the rods, depending on their orientations with respect to the nematic director field. This leads to different types of motions driven by the torques exerted on the particles by the director reorientations.Liquid crystals (LCs) are self-organized mesomorphic materials that exhibit various symmetries and structures (1). They are widely used in flat panel displays for their exceptional electrooptical properties and a combination of orientational elasticity and fluidity. For example, nematic LCs (NLCs) are distinguished by their long-range orientational order, which favors alignment of the molecules (mesogens) in a preferred direction denoted as the director n. An exceptional feature of NLCs is that, despite their fluidity, they exhibit anisotropic optical and mechanical properties, and thus can transmit mechanical torque because of directional elasticity (1). Such torque occurs in response to deformations away from a uniform equilibrium state.Such unique features of LCs can be exploited for designing smart multifunctional materials. Among these materials, colloidal dispersions of microparticles and nanoparticles in LCs have been actively studied in research on soft-matter physics (2–7). Tunable anisotropic interactions between microparticles dispersed in LCs give rise to self-assembled 1D and 2D colloidal structures (6, 8, 9). Such colloidal dispersions are interesting not only from a fundamental point of view but also from a technological one. A wide range of self-assembled structures of particles and topological defects stabilized by LC-mediated interactions find numerous applications in designing metamaterials (10), photonic devices (5, 11), sensors (12), and microrheology (5, 10–12). Here, we demonstrate a phenomenon that can be used in intelligent devices using colloidal dispersions: controlled light-driven translational and rotational motions of microrods in a NLC matrix.The orientation of LC molecules at an interface is governed by anchoring conditions, i.e., whether the director is perpendicular (homeotropic) or parallel (planar) to the interface. The orientation of the director at surfaces can be controlled through interfacial energy, anisotropy of the surface tension, and surface topography, by the pretreatment of the surfaces using surface agents, such as polymers and surfactants, together with mechanical or optical treatments. In most of the previous experiments, the solid interface was fixed. Here we use a so-called command surface (13) for LC alignment and its real-time control; the director manipulation is achieved by manipulating the anchoring condition through a light-induced isomerization of a photoactive azo-dendrimer adsorbed at the surface of microrods. As the light-induced isomerization takes place, the anchoring conditions provided by the cis isomer are different from those of the trans isomer. The initial equilibrium state of the director is lost. An important point here is that the bending direction in the cis form is not random but determined by the distorted local director near the surfaces. The torque on the particle exerted by the liquid crystal director reorientation results in specific particle motions toward a new equilibrium state. Such molecular-assisted manipulation of particles provides a tool for studying interfacial effects in their interplay with the topology of the nematic director field, which is a key concept for smart microdevices. Development of such devices requires better understanding of the photoisomerization. Some studies of azobenzenes bound to a dendrimer core have been conducted and reported in the literature (14, 15). The photoisomerization mechanism, the dependence of the quantum yield on the phenyl ring substituents, the solvent properties, and the irradiation wavelength, are still not fully understood (16). The molecular groups attached to the dendrimer core with highly regular branching do not entangle as in the case of conventional macromolecules and seem to express similar properties to the original azobenzene chromophores. Thus, this provides another way to study isomerization and photochemical properties of azobenzene molecules.Active steering of particles in LC hosts is usually achieved by convection mechanisms, such as electrophoresis, or using high-power laser radiation in optical tweezers (17). Manipulation of the mesogen orientations is another way to control colloidal particles in an LC host. This approach mimics molecular motors, which use conformational changes of molecules (18). Several types of artificial molecular motors and actuators were designed whereby the energy of light is transformed into the mechanical energy (19–21). In the case of LCs, a change of the anchoring conditions can be achieved by photosensitive functionalizations of the surface with mesogen-like moieties connected via light-sensitive azo linkages. Yamamoto et al. used a photosensitive surfactant and succeeded in manipulating the anchoring condition of colloidal particles in NLCs (22). There is, of course, a large number of papers dealing with light-driven motion and deformation in (soft) solids, e.g., liquid crystal elastomers that make use of cis−trans isomerization (for instance, refs. 23–29).Yonetake et al. synthesized a poly (propyleneimine) liquid crystalline dendrimer, which spontaneously adsorbs at LC−glass interfaces and favors homeotropic alignment of NLC molecules (30). This material was used to fabricate in-plane-switching-mode LC displays without pretreatment of substrate surfaces (31). Li et al. recently synthesized a photosensitive dendrimer with azo linkages (azo-dendrimer) in the mesogenic end chains (Fig. S1) and demonstrated the adsorption of the dendrimer at a glass interface (32). By UV irradiation, an orientational change of the mesogenic moieties occurs associated with the trans−cis photoisomerization, as illustrated in Fig. 1 A and B. Such asymmetric adsorption of dendrimers at surfaces has already been proved with a surface second-harmonic generation experiment (33). Hence, the director field was distorted under suitable illumination. The azo-dendrimer has already been used for controlling ordering transitions in mobile LC microdroplets in a polymer matrix (34) and defect structures in microparticles in LCs (35). In contrast, the situation is different in our study: Because the embedded particles are mobile and anisometric (rod-shaped), light irradiation brings about a dynamic motion of the enclosed colloids. We exploit the spontaneous adsorption of the photoactive dendrimer on various interfaces to functionalize the surfaces of rod-shaped microparticles suspended in a nematic host. As a result, we not only can control the molecular orientation at the immobile rods by light but can also mechanically rotate and translate them.Open in a separate windowFig. 1.Conformations of the photoactive dendrimer: (A) trans-conformation favoring homeotropic alignment of the mesogens and (B) cis-conformation favoring planar alignment obtained under UV light irradiation.Now we describe our present experimental results: All of the experiments were performed using 4′-n-pentyl-4-cyanobiphenyl liquid crystal (5CB) mixed with 0.1 wt% azo-dendrimer and glass rods of 10- to 20-µm length and 1.5-µm diameter. For details, the reader is referred to Experimental Procedures. First, we describe the photo-induced director field change around immobile microrods. The changes of the director configuration can be easily studied when the rods are immobile, i.e., attached to the glass substrate. The case of cells with homeotropic anchoring is described in Fig. S2. The situation is more complicated in cells with a planar anchoring condition. Several configurations are possible: rods aligned parallel (i), perpendicular (ii), and diagonal (iii) to the rubbing direction (Fig. S3). Experimentally, many rods are initially aligned at an angle Θ of 60°–70° to the director n, although normal anchoring conditions seem to favor Θ = 90°. This discrepancy may be attributed to an asymmetry of the director field at the ends of the rod and in the vertical dimension, and the different energies of the associated defect lines. Two kinds of director field configurations were found: dipolar with a single hyperbolic defect (Fig. 2A) in case i and quadrupolar with a disclination loop surrounding the rod (Fig. 2C) in case iii. These configurations have been established earlier in thin 2-μm cells by Tkalec et al. (6). In our case, to study mobile rods, thicker cells are preferred, which, however, makes the optical characterization difficult. Without UV irradiation, the director is normal to the surface of the rod (Fig. 2 A and C). This corresponds to a radial hedgehog configuration of the director field with the effective topological strength +1. To comply with the homogeneous far-field director outside the rod, a disclination loop is expected to encircle the rod in case iii. Evidence of that is shown in Fig. 2 E and F, and the structure of the disclination loop is schematically sketched in Fig. 2C. The loop is attracted to the diagonal edges of the rod to relive the mechanical strain on the director field.Open in a separate windowFig. 2.Immobilized microrods attached to a surface in a planarly aligned LC cell. The rubbing direction is the horizontal direction of the images. (A) A rod lying along the rubbing direction without UV irradiation. The director map with dipolar point defects is shown, together with a microscope image in the Inset. With the inserted λ wave plate diagonal to the crossed polarizers, different birefringence colors are seen at both sides along the rod, and they are consistent with the director map shown. (B) The same rod under UV irradiation. In the vicinity of the particle, each of the director fields in A and B is mirror symmetric in the cell plane and, in first approximation, axially symmetric about the rod axis. (C) A rod lying approximately diagonal to the rubbing direction without UV irradiation. The director map with quadrupolar point defects is shown, together with a microscope image in the Inset. (D) The same rod under UV irradiation. Both director fields in C and D have C₂ symmetry about the cell normal in the cell midplane. The disclination loop shown in C, that is present at normal anchoring, vanishes under UV irradiation when the director anchors tangentially, as shown in D. It leaves two point defects. The director orientation and its change by UV irradiation are clearly visible in the images with a wave plate (see Insets). (E) Image of a rod between crossed polarizers without UV irradiation. (F) A bright field image of the same rod without polarizers. The arrows indicate the position of topological defects at the ends of the rod. The cell thickness is 10 μm.Under UV irradiation, the director configuration changes: The normal configuration of the director transforms into a tangential one. This is evident from the changes of the interference colors to the complementary ones observed with a wave plate (see schematics in Fig. 2 A−D, and microscope images in Fig. 2 A−D, Insets). The loop collapses and transforms into two separated surface boojums with1/2 topological charge on the surface of the rod to satisfy planar anchoring conditions. This motion results in a pair of defects attached to the corners (see Fig. 2F). Disclination loops (Saturn ring) were confirmed by Tkalec et al. (6) for rods aligned perpendicular to the rubbing direction (case ii). Those loops were also tilted with respect to the rods axes.Free rods exhibit opto-mechanical responses: (i) rotation of rods in the cell plane, (ii) rotation in the vertical plane, and (iii) translation in the cell plane. In the first case, the rods initially appear at an angle of ∼70° to the nematic director (Fig. 3 A and B). Under applied UV irradiation, the rods align nearly along the director (Fig. 3 A and C). In the second case (Fig. S4), rotation occurs about an axis perpendicular to the rubbing direction and the cell normal. In both cases, the rotations are fully reversible. The rods return to the original state when UV irradiation is removed (Fig. 3A). The angular variation and the switching rate depend on the intensity of the UV irradiation (Fig. 3F). The rotational motion involves several stages. In the first stage, the reorientation of the dendrimer moieties takes place resulting in a change of the anchoring condition from nearly orthogonal to nearly planar. The next (fast, characteristic time scale is 0.01–0.05 s) stage is accompanied by a rearrangement of the topological defects and a continuous reorientation of the director field. This triggers the last (slow, characteristic time scale is 0.1–2 s) stage: motion of the rod. The time dependence of the rod reorientation is shown in Fig. 3F. The solid curve is a best fit to one of the experimental data based on theoretical consideration (SI Text). Even at low light intensities, the rotation occurs, but the rotation angle is reduced (Fig. 3G). Both angular variations and the switching rates show a saturation-type dependence on the UV intensity (Fig. 3 G and H), which is also explained theoretically (Fig. S5B).Open in a separate windowFig. 3.Dynamic motion of rods under the action of UV irradiation. (A) Time dependence of the angle between the rod and the rubbing direction. (B) Image of the initial state of a rod without UV, and (C) the final state under UV irradiation. The length of the rod is 20 μm. The rubbing direction is marked by a white arrow (parallel to n). B and C reveal the rotational motion. (D) Image of a rod without UV irradiation. (E) Image of a rod under UV irradiation at t_exp = 43 s. D and E reveal the translational motion. A white arrow marks the rubbing direction. A dashed line indicates the initial position of the rod. Images D and E were taken between polarizers parallel to the sides of the frame. The length of the rods in D and E was 15 μm. (F) Time dependence of the rod inclination at various intensities of the UV light. A solid line is the theoretical fit of the experimental data (see SI Text for details). (G and H) UV intensity dependences of the deflection angle and the switching rates, respectively. The saturation behaviors are theoretically explained (compare H and Fig. S5B).Translational motion was observed in some rods, and the direction was more or less parallel to the long axis of the rods. Fig. 3 D and E shows the initial and intermediate stages during UV irradiation. The translational movement was extended to displacements of the order of one rod length. We could observe a slight backlash motion when the UV irradiation was stopped.Since UV irradiation affects the conformational state of the dendrimer adsorbed at the rod surface, it is reasonable to consider that the director reorientation direction is directly related to the cis conformation. Straight (trans) to bent (cis) conformational change occurs upon photoisomerization and results in a change of the anchoring condition. The rotational motion of the rods originates from the torques exerted by the director. How do those torques develop? The initial state depicted in Fig. 3B is stabilized by the director deformation and the configuration of the topological defects around the rod as well as the anchoring on the rod surface. The torque by the director in the volume is counteracted by the (anchoring) torque at the rod/LC boundary. Under UV light irradiation, the orthogonal anchoring condition changes to the planar one and the torque in the volume is not in balance with the boundary any more. This leads to the rotation of the rod to the new equilibrium state. In the first approximation, one may describe this anchoring by a potential as W_asin²(θ), where θ is the polar anchoring angle with respect to the surface, and W_a is the strength of the anchoring energy (1). The anchoring energy is determined by the ratio of cis and trans isomers at a given light intensity and spectral composition. Planar anchoring corresponds to positive W_a, while the orthogonal anchoring is achieved with W_a < 0. Strong anchoring is given if the penetration length ξ = |K/W_a| is short respective to typical geometrical sizes of the experiment; for weak anchoring, ξ is comparable or larger than system sizes. Obviously, strong illumination (large fraction of cis molecules) will correspond to large positive W_a, while no illumination (mainly trans molecules) corresponds to large negative W_a. The experiments suggest a monotonous functional form of the transition between both states with increasing/decreasing illumination intensity. One can demonstrate that intermediate stationary W_a?situations can be reached at certain moderate UV intensities. Then, the director field is not influenced by the rod; the microscopy image has a uniform color around the inclusions. This corresponds to an infinite ξ. These arguments enable us to roughly estimate the switching rates and qualitatively describe their dependence on the light intensity, as described in SI Text. At strong anchoring, the torque acting on the director is proportional to KL, where L is a characteristic length (of the order of magnitude of the rod length) and K is the mean Frank elastic constant. The viscous torque is proportional to ηL³, where η is a mean viscosity of the liquid crystal. This results in a switching time τ = ηL²/K ≈ 0.5? s. In case of weak anchoring, the switching rate Γ depends on the light intensity through the penetration length?ξ : Γ = K/[ηL(L + ξ)]. Since short ξ correspond to high UV intensities, whereas low UV intensities yield long?ξ, the switching rate increases with UV intensity and is expected to reach some saturation Γ = 1/τ. A qualitative agreement of the estimated switching rate with the experiment can be found by comparing Fig. 3H and Fig. S5B.How is the reorientation direction chosen? For rods with a surrounding director field similar to that shown in Fig. 2A, the director has an opposite sense of bending on two sides of the rod, as recognized by different birefringence colors under a wave plate. UV light irradiation under this condition induces the opposite director rotation at both sides of the rod (Fig. 4A), leading to lateral translational motion. In contrast, for rods similar to those shown in Fig. 2C, the director bending direction is the same at both sides (Fig. 4B), as shown by the same birefringence color. Hence, the same director rotation at both sides of the rod upon UV light irradiation exerts torques with the same sign, causing the rod to reorient (Fig. 4).Open in a separate windowFig. 4.Schematic illustrations of the microrod motions and the acting torques. (A) Translational and (B) rotational motions. Azo-dendrimers and their bending direction are marked by brown lines and arrows, respectively. Red arrows show the director rotation from 1 (normal to the rod) to 4 (parallel to the rod) and the torque direction. Green arrows designate the rod motion.Interactions between microparticles dispersed in an LC matrix lead to a formation of complex ordered structures. Such agglomerates of particles may have a form of chains or even more intricate structures. In this case, optomechanical effect manifests in the motion of either the whole agglomerate or its separate parts. Different examples of such structures are shown in Fig. 5.Open in a separate windowFig. 5.Switchable agglomerates of particles stabilized by the director-mediated interactions: doublets of rods (crossed polarizers with a wave-plate) without UV (A) and under UV irradiation (B). There is no rotation of the doublet since the mechanical torques on two arms cancel out. The switching of the director is clearly seen from the interference colors. A dumbbell of a rod with two beads without UV (C) and under UV irradiation (D) in unpolarized light. The director orients along the vertical direction in the images.In conclusion, we demonstrated the optomechanical effect of light-induced rotation and translation of micrometer-sized rod particles in a nematic host. This system represents an optically driven molecular microactuator, which exploits molecular reorientation on a particle surface and transforms it into a mechanical torque.     

Highly cooperative stress relaxation in two-dimensional soft colloidal crystals

Stress relaxation in crystalline solids is mediated by the formation and diffusion of defects. Although it is well established how externally generated stresses relax, through the proliferation and motion of dislocations in the lattice, it remains relatively unknown how crystals cope with internal stresses. We investigate, both experimentally and in simulations, how highly localized stresses relax in 2D soft colloidal crystals. When a single particle is actively excited, by means of optical tweezing, a rich variety of highly collective stress relaxation mechanisms results. These relaxation processes manifest in the form of open strings of cooperatively moving particles through the motion of dissociated vacancy-interstitial pairs, and closed loops of mobile particles, which either result from cooperative rotations in transiently generated circular grain boundaries or through the closure of an open string by annihilation of a vacancy-interstitial pair. Surprisingly, we find that the same collective events occur in crystals that are excited by thermal fluctuations alone; a large thermal agitation inside the crystal lattice can trigger the irreversible displacements of hundreds of particles. Our results illustrate how local stresses can induce large-scale cooperative dynamics in 2D soft colloidal crystals and shed light on the stabilization mechanisms in ultrasoft crystals.Stress relaxation in crystalline solids is governed by the formation and diffusion of defects in the crystal lattice. For small deformations, it is well known that relaxation occurs through the motion of sparse dislocations (1–5). However, it remains unclear how a crystalline solid copes with stresses that are generated well inside the crystal, either caused by external sources (6, 7) or by thermal excitations, which can become especially important in superheated states (8–12). Particle rearrangements that result from large internal perturbations must necessarily involve the motion of many of the constituent particles simultaneously. Often these collective dynamics are rare due to large activation barriers in the dense solid state. As a result, studying large-scale collective dynamics inside crystalline solids is challenging. One may expect that sufficiently large fluctuations, which could drive collective rearrangements, may only appear when the elastic energy associated with a fluctuation becomes on the order of the thermal energy. In crystals formed from colloidal particles that interact through long-range repulsive interactions, low-density and ultrasoft solid states are experimentally accessible in which large thermal excitations can be easily observed using optical microscopy (13). These very weak solids may exhibit fragility, the phenomenon that weakly stable solids display a nonlinear response to even very small external perturbations. Understanding the microscopic mechanisms of stress relaxation in these marginally stable materials is of fundamental importance to understand mechanical instabilities such as creep, yield, and fracture. Such colloidal systems, in which very weak solids can be formed, create the experimental possibility to manipulate the kinetic states of individual particles by means of optical tweezers, for example, to create vacancies and interstitials (14–18), or to manipulate many-particle defect reactions (19). However, the response of colloidal crystals to large thermal and external excitations of a single particle within the lattice is largely unexplored. As a result, the relationship between stress relaxation mechanisms, in response to internal perturbations, and the ultimate stability of the solid phase remains poorly understood.In this paper, we investigate how stresses relax in 2D soft colloidal crystals using a combination of experiments and computer simulations. When a single particle inside the crystal is actively driven out of equilibrium, a rich variety of collective stress relaxation mechanisms result, mainly in the form of open and closed strings of rearranging particles. Surprisingly, we find that these unusual collective rearrangements are not restricted to crystals that are actively perturbed but also appear in soft colloidal crystals excited through thermal fluctuations alone. A sufficiently large internal agitation inside the lattice can cause the irreversible rearrangement of hundreds of particles from their previous equilibrium positions. These results illustrate the complexity of internal stress relaxation through collective and activated modes, and shed light on the origins of stability and instability in marginally stable crystalline solids.     

Linked topological colloids in a nematic host

Geometric shape and topology of constituent particles can alter many colloidal properties such as Brownian motion, self-assembly, and phase behavior. Thus far, only single-component building blocks of colloids with connected surfaces have been studied, although topological colloids, with constituent particles shaped as freestanding knots and handlebodies of different genus, have been recently introduced. Here we develop a topological class of colloids shaped as multicomponent links. Using two-photon photopolymerization, we fabricate colloidal microparticle analogs of the classic examples of links studied in the field of topology, the Hopf and Solomon links, which we disperse in nematic fluids that possess orientational ordering of anisotropic rod-like molecules. The surfaces of these particles are treated to impose tangential or perpendicular boundary conditions for the alignment of liquid crystal molecules, so that they generate a host of topologically nontrivial field and defect structures in the dispersing nematic medium, resulting in an elastic coupling between the linked constituents. The interplay between the topologies of surfaces of linked colloids and the molecular alignment field of the nematic host reveals that linking of particle rings with perpendicular boundary conditions is commonly accompanied by linking of closed singular defect loops, laying the foundations for fabricating complex composite materials with interlinking-based structural organization.Interlocking closed loops in physical field lines (1–3), small molecules (4), DNA and synthetic polymer chains (5), and various vortices (6–8) can lead to new physical behavior, biological functionality, and material properties that largely stem from the underlying topology (1). For example, linking looped lines of the liquid crystal (LC) molecular alignment field n(r) (3), which describes spatial changes in local average orientations of constituent rod-like molecules (9), causes formation of topologically protected particle-like structures resembling mathematical Hopf and Seifert fibrations (1, 3, 10, 11). Similar field configurations with linked closed loops or linked torus knots of field lines are also predicted to exist in electromagnetic fields (2, 12, 13), in Bose-Einstein condensates (14, 15), and in magnetization of various ferromagnets (16–18), often defining novel types of physical behavior that arise from topological stabilization of such field configurations. However, the implications of topological linking on behavior of colloidal particles have not been considered thus far, neither experimentally nor theoretically, although many types of complex-shaped colloidal particles have been recently fabricated (8, 19, 20).In this work, we fabricate micrometer-sized colloidal particles with differently linked components shaped as closed solid polymeric rings with disconnected surfaces that, when dispersed in a fluid host like water or LC, undergo Brownian motion both relative to each other and as a whole. In a nematic LC host (8, 19–23), these particles induce a large variety of field configurations and looped and linked vortex lines that entangle the linked components of the colloidal particles, resulting in elastic coupling between them. Using a combination of 3D nonlinear optical imaging, videomicroscopy, and noncontact laser manipulation (8, 24, 25), we characterize the interplay between topologies of colloidal surfaces, n(r) configurations, and defects, as well as probe the strength of elastic coupling between the colloidal particle’s components. We supplement these experiments with a theoretical analysis based on numerical minimization of the bulk Landau-de Gennes and surface anchoring free energies (26–29) that yields configurations topologically homeomorphic to experimental counterparts. Finally, we discuss the prospects for interlocking-based assembly of composite materials and for experiment-driven fundamental explorations of topological interaction of physical links with nonpolar fields.     

Knot theory realizations in nematic colloids

Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.From the invention of ropes and textiles, up to the present day, knots have played a prominent role in everyday life, essential crafts, and artistic expression. Beyond the simple tying of strings, the intriguing irreducibility of knots has led to Kelvin’s vortex model of atoms, and, subsequently, a more systematic study of knots and links in the context of knot theory (1–3). As a branch of topology, knot theory is a developing field, with many unresolved questions, including the ongoing search for an algorithm that will provide an exact identification of arbitrary knots.As knots cannot be converted one into another without the crossing of the strands––a discrete singular event––knotting topologically stabilizes the structure. In physical fields, this coexistence of discrete and continuous phenomena leads to the stabilization of geometrically and topologically nontrivial high-energy excitations (4, 5). Examples of strand-like objects in physics that can be knotted include vortices in fluids (6–9), synthetic molecules (10, 11), DNA, polymer strands and proteins (12–14), electromagnetic field lines (15, 16), zero-intensity loci in optical interference patterns (17), wave functions in topological insulators (18), cosmic strings (19), and defects in a broad selection of ordered media (20–23).Nematic liquid crystals (NLC) are liquids with a local apolar orientational order of rod-like molecules. The director field, which describes the spatial variation of the local alignment axis, supports topological point and line defects, making it an interesting medium for the observation of topological phenomena (20, 24). Defect structures in NLC and their colloidal composites (25) have been extensively studied for their potential in self-assembly and light control (26), but also to further the theoretical understanding of topological phenomena in director fields (23, 27). Objects of interest include chiral solitons (28, 29), fields around knotted particles (30–35), and knotted defects in nematic colloids (36–42). Each of these cases is unique, as the rules of knot theory interact with the rules and restrictions of each underlying material and confinement. The investigation of knotted fields is thus a specialized topic where certain theoretical aspects of knot theory emerge in a physical context.In nematic colloids––dispersions of spherical particles confined in a twisted nematic (TN) cell––disclination lines entangle arrays of particles into “nematic braids,” which can be finely controlled by laser tweezers to form various linked and knotted structures (38, 39, 43). In this paper, we focus on the diverse realizations of knot theory in such nematic colloidal structures. We complement and extend the classification and analysis of knotted disclinations from refs. 32, 38, 40 with the direct application of graph and knot theory to polarized optical micrographs. We further analyze the nematic director with constructed Pontryagin–Thom surfaces and polynomial knot invariants, which enables a comprehensive topological characterization of the knotted nematic field based on experimental data and analytical tools. We use a λ-retardation plate to observe and distinguish differently twisted domains in the optical micrograph, which correspond to medial graphs of the represented knots and contribute to the Pontryagin–Thom construction of the nematic director. Finally, we explore the organization of the space of possible configurations on a selected rectangular particle array and discuss the observed hierarchy of entangled and knotted structures.     

Valley Chern numbers and boundary modes in gapped bilayer graphene

Electronic states at domain walls in bilayer graphene are studied by analyzing their four- and two-band continuum models, by performing numerical calculations on the lattice, and by using quantum geometric arguments. The continuum theories explain the distinct electronic properties of boundary modes localized near domain walls formed by interlayer electric field reversal, by interlayer stacking reversal, and by simultaneous reversal of both quantities. Boundary mode properties are related to topological transitions and gap closures, which occur in the bulk Hamiltonian parameter space. The important role played by intervalley coupling effects not directly captured by the continuum model is addressed using lattice calculations for specific domain wall structures.     

Structures and topological defects in pressure-driven lyotropic chromonic liquid crystals

Lyotropic chromonic liquid crystals are water-based materials composed of self-assembled cylindrical aggregates. Their behavior under flow is poorly understood, and quantitatively resolving the optical retardance of the flowing liquid crystal has so far been limited by the imaging speed of current polarization-resolved imaging techniques. Here, we employ a single-shot quantitative polarization imaging method, termed polarized shearing interference microscopy, to quantify the spatial distribution and the dynamics of the structures emerging in nematic disodium cromoglycate solutions in a microfluidic channel. We show that pure-twist disclination loops nucleate in the bulk flow over a range of shear rates. These loops are elongated in the flow direction and exhibit a constant aspect ratio that is governed by the nonnegligible splay-bend anisotropy at the loop boundary. The size of the loops is set by the balance between nucleation forces and annihilation forces acting on the disclination. The fluctuations of the pure-twist disclination loops reflect the tumbling character of nematic disodium cromoglycate. Our study, including experiment, simulation, and scaling analysis, provides a comprehensive understanding of the structure and dynamics of pressure-driven lyotropic chromonic liquid crystals and might open new routes for using these materials to control assembly and flow of biological systems or particles in microfluidic devices.

Lyotropic chromonic liquid crystals (LCLCs) are aqueous dispersions of organic disk-like molecules that self-assemble into cylindrical aggregates, which form nematic or columnar liquid crystal phases under appropriate conditions of concentration and temperature (1–6). These materials have gained increasing attention in both fundamental and applied research over the past decade, due to their distinct structural properties and biocompatibility (4, 7–14). Used as a replacement for isotropic fluids in microfluidic devices, nematic LCLCs have been employed to control the behavior of bacteria and colloids (13, 15–20).Nematic liquid crystals form topological defects under flow, which gives rise to complex dynamical structures that have been extensively studied in thermotropic liquid crystals (TLCs) and liquid crystal polymers (LCPs) (21–29). In contrast to lyotropic liquid crystals that are dispersed in a solvent and whose phase can be tuned by either concentration or temperature, TLCs do not need a solvent to possess a liquid-crystalline state and their phase depends only on temperature (30). Most TLCs are shear-aligned nematics, in which the director evolves toward an equilibrium out-of-plane polar angle. Defects nucleate beyond a critical Ericksen number due to the irreconcilable alignment of the directors from surface anchoring and shear alignment in the bulk flow (24, 31–33). With an increase in shear rate, the defect type can transition from π-walls (domain walls that separate regions whose director orientation differs by an angle of π) to ordered disclinations and to a disordered chaotic regime (34). Recent efforts have aimed to tune and control the defect structures by understanding the relation between the selection of topological defect types and the flow field in flowing TLCs. Strategies to do so include tuning the geometry of microfluidic channels, inducing defect nucleation through the introduction of isotropic phases or designing inhomogeneities in the surface anchoring (35–39). LCPs are typically tumbling nematics for which α2α3 < 0, where α2 and α3 are the Leslie viscosities. This leads to a nonzero viscous torque for any orientation of the director, which allows the director to rotate in the shear plane (22, 29, 30, 40). The tumbling character of LCPs facilitates the nucleation of singular topological defects (22, 40). Moreover, the molecular rotational relaxation times of LCPs are longer than those of TLCs, and they can exceed the timescales imposed by the shear rate. As a result, the rheological behavior of LCPs is governed not only by spatial gradients of the director field from the Frank elasticity, but also by changes in the molecular order parameter (25, 41–43). With increasing shear rate, topological defects in LCPs have been shown to transition from disclinations to rolling cells and to worm-like patterns (25, 26, 43).Topological defects occurring in the flow of nematic LCLCs have so far received much more limited attention (44, 45). At rest, LCLCs exhibit unique properties distinct from those of TLCs and LCPs (1, 2, 4–6, 44). In particular, LCLCs have significant elastic anisotropy compared to TLCs; the twist Frank elastic constant, K2, is much smaller than the splay and bend Frank elastic constants, K1 and K3. The resulting relative ease with which twist deformations can occur can lead to a spontaneous symmetry breaking and the emergence of chiral structures in static LCLCs under spatial confinement, despite the achiral nature of the molecules (4, 46–51). When driven out of equilibrium by an imposed flow, the average director field of LCLCs has been reported to align predominantly along the shear direction under strong shear but to reorient to an alignment perpendicular to the shear direction below a critical shear rate (52–54). A recent study has revealed a variety of complex textures that emerge in simple shear flow in the nematic LCLC disodium cromoglycate (DSCG) (44). The tumbling nature of this liquid crystal leads to enhanced sensitivity to shear rate. At shear rates γ˙<1 s−1, the director realigns perpendicular to the flow direction adapting a so-called log-rolling state characteristic of tumbling nematics. For 1 s−1<γ˙<10 s−1, polydomain textures form due to the nucleation of pure-twist disclination loops, for which the rotation vector is parallel to the loop normal, and mixed wedge-twist disclination loops, for which the rotation vector is perpendicular to the loop normal (44, 55). Above γ˙>10 s−1, the disclination loops gradually transform into periodic stripes in which the director aligns predominantly along the flow direction (44).Here, we report on the structure and dynamics of topological defects occurring in the pressure-driven flow of nematic DSCG. A quantitative evaluation of such dynamics has so far remained challenging, in particular for fast flow velocities, due to the slow image acquisition rate of current quantitative polarization-resolved imaging techniques. Quantitative polarization imaging traditionally relies on three commonly used techniques: fluorescence confocal polarization microscopy, polarizing optical microscopy, and LC-Polscope imaging. Fluorescence confocal polarization microscopy can provide accurate maps of birefringence and orientation angle, but the fluorescent labeling may perturb the flow properties (56). Polarizing optical microscopy requires a mechanical rotation of the polarizers and multiple measurements, which severely limits the imaging speed. LC-Polscope, an extension of conventional polarization optical microscopy, utilizes liquid crystal universal compensators to replace the compensator used in conventional polarization microscopes (57). This leads to an enhanced imaging speed and better compensation for polarization artifacts of the optical system. The need for multiple measurements to quantify retardance, however, still limits the acquisition rate of LC-Polscopes.We overcome these challenges by using a single-shot quantitative polarization microscopy technique, termed polarized shearing interference microscopy (PSIM). PSIM combines circular polarization light excitation with off-axis shearing interferometry detection. Using a custom polarization retrieval algorithm, we achieve single-shot mapping of the retardance, which allows us to reach imaging speeds that are limited only by the camera frame rate while preserving a large field-of-view and micrometer spatial resolution. We provide a brief discussion of the optical design of PSIM in Materials and Methods; further details of the measurement accuracy and imaging performance of PSIM are reported in ref. 58.Using a combination of experiments, numerical simulations and scaling analysis, we show that in the pressure-driven flow of nematic DSCG solutions in a microfluidic channel, pure-twist disclination loops emerge for a certain range of shear rates. These loops are elongated in the flow with a fixed aspect ratio. We demonstrate that the disclination loops nucleate at the boundary between regions where the director aligns predominantly along the flow direction close to the channel walls and regions where the director aligns predominantly perpendicular to the flow direction in the center of the channel. The large elastic stresses of the director gradient at the boundary are then released by the formation of disclination loops. We show that both the characteristic size and the fluctuations of the pure-twist disclination loops can be tuned by controlling the flow rate.     

Topological Defects in Topological Insulators and Bound States at Topological Superconductor Vortices

The scattering of Dirac electrons by topological defects could be one of the most relevant sources of resistance in graphene and at the boundary surfaces of a three-dimensional topological insulator (3D TI). In the long wavelength, continuous limit of the Dirac equation, the topological defect can be described as a distortion of the metric in curved space, which can be accounted for by a rotation of the Gamma matrices and by a spin connection inherited with the curvature. These features modify the scattering properties of the carriers. We discuss the self-energy of defect formation with this approach and the electron cross-section for intra-valley scattering at an edge dislocation in graphene, including corrections coming from the local stress. The cross-section contribution to the resistivity, ρ, is derived within the Boltzmann theory of transport. On the same lines, we discuss the scattering of a screw dislocation in a two-band 3D TI, like Bi_1−xSb_x, and we present the analytical simplified form of the wavefunction for gapless helical states bound at the defect. When a 3D TI is sandwiched between two even-parity superconductors, Dirac boundary states acquire superconductive correlations by proximity. In the presence of a magnetic vortex piercing the heterostructure, two Majorana states are localized at the two interfaces and bound to the vortex core. They have a half integer total angular momentum each, to match with the unitary orbital angular momentum of the vortex charge.     

Splitting,linking, knotting,and solitonic escape of topological defects in nematic drops with handles

Topologically nontrivial field excitations, including solitonic, linked, and knotted structures, play important roles in physical systems ranging from classical fluids and liquid crystals, to electromagnetism, classic, and quantum field theories. These excitations can appear spontaneously during symmetry-breaking phase transitions. For example, in cosmological theories, cosmic strings may have formed knotted configurations influencing the Early Universe development, whereas in liquid crystals transient tangled defect lines were observed during isotropic–nematic transitions, eventually relaxing to defect-free states. Knotted and solitonic fields and defects were also obtained using optical manipulation, complex-shaped colloids, and frustrated cholesterics. Here we use confinement of nematic liquid crystal by closed surfaces with varied genus and perpendicular boundary conditions for a robust control of appearance and stability of such field excitations. Theoretical modeling and experiments reveal structure of defect lines as a function of the surface topology and material and geometric parameters, establishing a robust means of controlling solitonic, knotted, linked, and other field excitations.Since the origins of the mathematical knot theory, development of which was prompted by early models of elementary building blocks of matter (1), knotted fields and structures arise in proteins (2), light (3–5), fluids (6–8), liquid crystals (LCs) (9–14), classic and quantum field theories (15, 16), topological insulators (17), and other physical systems (18). Such topologically nontrivial field configurations can be predicted from solutions of nonlinear field equations, but are rarely accessible to direct experimental visualization. On the other hand, LCs offer complexity in degrees of freedom and symmetries that allow for probing topologically analogous phenomena (19, 20) on completely different scales, such as kinetics of cosmic strings in the Early Universe (21). In this work, we develop polymer-dispersed nematic drops with nontrivial surface topology and perpendicular boundary conditions that prompt stable configurations of defect lines in forms of unknots, knots, links, 2D skyrmions, and other singular and solitonic structures that can be selected by controlling geometric and material parameters. This enables a robust control of defects in nematic drops of nonzero genus by shaping topology and varying geometric parameters of confining surfaces as well through the use of laser-guided temperature quenching of isotropic–nematic transition within the drops.Our nematic drops in a polymer matrix have handlebody shapes with genus g varying from 1 to 5 and the corresponding Euler characteristics χ = 2(1 − g) from 0 to −8 (22). The closed confining surfaces impose strong homeotropic (normal) anchoring on the nematic director n(r) describing average local orientation of LC molecules, so that n(r) aligns along the inner normal to a bounding surface S. The surface topology and these boundary conditions dictate bulk defects of net topological hedgehog charge m = ±(1 − g) in the nematic domain, which, to assure the topological charge conservation, compensate the hedgehog charge of the field on the inner closed confining surface of the nematic drop given by the Gauss–Bonnet and Poincaré–Hopf theorems (23). One would therefore simply expect that the nematic interior of a single torus is topologically uncharged, g = 2 drop hosts a defect of m = ±1 topological charge, and so on, where the sign of m depends on the choice of vector field direction when decorating n(r) to determine the charge (24). However, the mathematical theorems prescribe no particular ways in which the topological constraints should be satisfied. Our study shows that this “flexibility” of satisfying topological constraints, combined with the nematic LCs nature and ability of hosting both half-integer line defects and point defects, leads to a large number of topologically nontrivial configurations that can be selected as stable and metastable structures by controlling material and geometric parameters. Importantly, some of these field and defect configurations, such as linked and knotted loops of half-integer defect lines, are topologically different from what topological theorems predict for vector fields under such confinement. Moreover, a combination of tuning topology (genus) and geometric parameters of confining surfaces as well as laser-guided spatially resolved isotropic–nematic temperature quench allow us to generate the precise desired defect and field configurations out of a host of topology-satisfying stable and metastable states.     

Effect of Sr Doping on Structural and Transport Properties of Bi2Te3

Search for doped superconducting topological insulators is of prime importance for new quantum technologies. We report on fabrication of Sr-doped Bi2Te3 single crystals. We found that Bridgman grown samples have p-type conductivity in the low 1019 cm−3, high mobility of 4000 cm2V−1s−1, crystal structure independent on nominal dopant content, and no signs of superconductivity. We also studied molecular beam epitaxy grown SrxBi2−xTe3 films on lattice matched (1 1 1) BaF2 polar surface. Contrary to the bulk crystals thin films have n-type conductivity. Carrier concentration, mobility and c-lattice constant demonstrate pronounced dependence on Sr concentration x. Variation of the parameters did not lead to superconductivity. We revealed, that transport and structural parameters are governed by Sr dopants incorporation in randomly inserted Bi bilayers into the parent matrix. Thus, our data shed light on the structural position of dopant in Bi2Te3 and should be helpful for further design of topological insulator-based superconductors.     

From the Cover: Emergence of dynamic vortex glasses in disordered polar active fluids

In equilibrium, disorder conspires with topological defects to redefine the ordered states of matter in systems as diverse as crystals, superconductors, and liquid crystals. Far from equilibrium, however, the consequences of quenched disorder on active condensed matter remain virtually uncharted. Here, we reveal a state of strongly disordered active matter with no counterparts in equilibrium: a dynamical vortex glass. Combining microfluidic experiments and theory, we show how colloidal flocks collectively cruise through disordered environments without relaxing the topological singularities of their flows. The resulting state is highly dynamical but the flow patterns, shaped by a finite density of frozen vortices, are stationary and exponentially degenerated. Quenched isotropic disorder acts as a random gauge field turning active liquids into dynamical vortex glasses. We argue that this robust mechanism should shape the collective dynamics of a broad class of disordered active matter, from synthetic active nematics to collections of living cells exploring heterogeneous media.

From a physicist’s perspective, flocks are ensembles of living or synthetic motile units collectively moving along a common emerging direction (1–4). They realize one of the most robust ordered states of matter observed over five orders of magnitude in scale and in systems as diverse as motility assays, self-propelled colloids, shaken grains, and actual flocks of birds (3, 5–10). The quiet flows of flocks are in stark contrast with the spatiotemporal chaos consistently reported and predicted in active nematic liquid crystals, another abundant form of ordered active matter realized in biological tissues, swimming cells, cellular extracts, and shaken rods (2, 11). Active nematics do not support any form of long-range order (4, 12). Their structure is continuously bent and destroyed by the proliferation and annihilation of singularities in their local orientation: topological defects (11, 13–15). Unlike in active nematics, topological defects in flocking matter are merely transient excitations which annihilate rapidly and allow uniaxial order to extend over system-spanning scales (4).This idyllic view of the ordered phases of active liquids is limited, however, to pure systems. Disorder is known to profoundly alter the stability of topological defects and the corresponding ordered states in equilibrium condensed matter (16–18), but its role in active fluids remains virtually uncharted territory. All previous studies (19–26), including our own early experiments (22), have been limited to weak disorder and smooth perturbations around topologically trivial states. Unlike in equilibrium, no available experiment, simulation, or theory has ever demonstrated or predicted disorder-induced topological excitations in active matter.In this paper we show how isotropic disorder generically challenges the extreme robustness of flocking matter to topological defects. We map the full phase behavior of colloidal flocks navigating through disordered lattice of obstacles and reveal an unanticipated state of active matter: a dynamical vortex glass. In dynamical vortex glasses, millions of self-propelled particles can steadily cruise through disorder, maintaining local orientational order and without relaxing the topological singularities of their flows. The associated flow patterns are exponentially degenerated and shaped by amorphous ensembles of frozen topological defects, yielding a dynamical state akin to the static vortex-glass phase of dirty superconductors and random-gauge magnets (27–29). Building a theory of flock hydrodynamics beyond the spin-wave approximation, we elucidate the emergence and stabilization of topological vortices by quenched disorder. Finally, we discuss the universality of the dynamical vortex glass phase beyond the specifics of polar active matter and colloidal flocks.     

Interplay of local hydrogen-bonding and long-ranged dipolar forces in simulations of confined water

Spherical truncations of Coulomb interactions in standard models for water permit efficient molecular simulations and can give remarkably accurate results for the structure of the uniform liquid. However, truncations are known to produce significant errors in nonuniform systems, particularly for electrostatic properties. Local molecular field (LMF) theory corrects such truncations by use of an effective or restructured electrostatic potential that accounts for effects of the remaining long-ranged interactions through a density-weighted mean field average and satisfies a modified Poisson's equation defined with a Gaussian-smoothed charge density. We apply LMF theory to 3 simple molecular systems that exhibit different aspects of the failure of a naïive application of spherical truncations—water confined between hydrophobic walls, water confined between atomically corrugated hydrophilic walls, and water confined between hydrophobic walls with an applied electric field. Spherical truncations of 1/r fail spectacularly for the final system, in particular, and LMF theory corrects the failings for all three. Further, LMF theory provides a more intuitive way to understand the balance between local hydrogen bonding and longer-ranged electrostatics in molecular simulations involving water.     

The transition between the niche and neutral regimes in ecology

An ongoing debate in ecology concerns the impacts of ecological drift and selection on community assembly. Here, we show that there is a transition in diverse ecological communities between a selection-dominated regime (the niche phase) and a drift-dominated regime (the neutral phase). Simulations and analytic arguments show that the niche phase is favored in communities with large population sizes and relatively constant environments, whereas the neutral phase is favored in communities with small population sizes and fluctuating environments. Our results demonstrate how apparently neutral populations may arise even in communities inhabited by species with varying traits.The success of the neutral theory of biodiversity and biogeography (1, 2) at explaining patterns in biodiversity has resulted in a vigorous debate on the processes underlying the assembly, dynamics, and structure of ecological communities (1, 3–12). Starting with the pioneering work of MacArthur (13–15), ecologists have emphasized the roles of interspecific competition and environmental interactions in community assembly and dynamics. These niche-based models emphasize ecological selection as the driving force of community assembly, whereas neutral models of biodiversity assume a functional equivalence between species and emphasize the role of ecological drift (i.e., stochasticity) in community dynamics (1, 2, 16, 17). The success of both types of models at explaining ecological data highlights the crucial need for understanding the impacts of ecological drift and selection in community ecology (18).     

Unexpected patterns of fisheries collapse in the world's oceans

Understanding which species are most vulnerable to human impacts is a prerequisite for designing effective conservation strategies. Surveys of terrestrial species have suggested that large-bodied species and top predators are the most at risk, and it is commonly assumed that such patterns also apply in the ocean. However, there has been no global test of this hypothesis in the sea. We analyzed two fisheries datasets (stock assessments and landings) to determine the life-history traits of species that have suffered dramatic population collapses. Contrary to expectations, our data suggest that up to twice as many fisheries for small, low trophic-level species have collapsed compared with those for large predators. These patterns contrast with those on land, suggesting fundamental differences in the ways that industrial fisheries and land conversion affect natural communities. Even temporary collapses of small, low trophic-level fishes can have ecosystem-wide impacts by reducing food supply to larger fish, seabirds, and marine mammals.     

The role of fluctuations and stress on the effective viscosity of cell aggregates

Cell aggregates are a tool for in vitro studies of morphogenesis, cancer invasion, and tissue engineering. They respond to mechanical forces as a complex rather than simple liquid. To change an aggregate''s shape, cells have to overcome energy barriers. If cell shape fluctuations are active enough, the aggregate spontaneously relaxes stresses (“fluctuation-induced flow”). If not, changing the aggregate''s shape requires a sufficiently large applied stress (“stress-induced flow”). To capture this distinction, we develop a mechanical model of aggregates based on their cellular structure. At stress lower than a characteristic stress τ*, the aggregate as a whole flows with an apparent viscosity η*, and at higher stress it is a shear-thinning fluid. An increasing cell–cell tension results in a higher η* (and thus a slower stress relaxation time t_c). Our constitutive equation fits experiments of aggregate shape relaxation after compression or decompression in which irreversibility can be measured; we find t_c of the order of 5 h for F9 cell lines. Predictions also match numerical simulations of cell geometry and fluctuations. We discuss the deviations from liquid behavior, the possible overestimation of surface tension in parallel-plate compression measurements, and the role of measurement duration.     

Numerical Simulation and Experimental Validation of Squeeze Casting of AlSi9Mg Aluminum Alloy Component with a Large Size

The squeeze casting process for an AlSi9Mg aluminum alloy flywheel housing component was numerically simulated using the ProCAST software, and orthogonal simulation tests were designed according to the L16 (4) ⁵ orthogonal test table to investigate the alloy melt flow rule under four factors and four levels each of the pouring temperature, mold temperature, pressure holding time and specific pressure, as well as the distributions of the temperature fields, stress fields and defects. The results showed that the flywheel housing castings in all 16 test groups were fully filled, and the thinner regions solidified more quickly than the thicker regions. Hot spots were predicted at the mounting ports and the convex platform, which could be relieved by adding a local loading device. Due to the different constraints on the cylinder surface and the lower end surface, the solidification was inconsistent, the equivalent stress at the corner junction was larger, and the castings with longer pressure holding time and lower mold temperature had larger average equivalent stress. Shrinkage cavities were mainly predicted at mounting ports, the cylindrical convex platform, the peripheral overflow groove and the corner junctions, and there was also a small defect region at the edge of the upper end face in some test groups.