A Brownian quasi-crystal of pre-assembled colloidal Penrose tiles

Heptatic liquid quasi-crystals by colloidal lithographic pre-assembly

HYPOTHESIS

We hypothesize that pre-assembled lithographic Brownian seven-fold quasi-crystals (QCs) of colloidal tiles at high densities can exhibit a heptatic liquid quasi-crystal (LQC) phase upon release; such heptatic LQCs can undergo heterogeneous dynamics at different length scales, reflecting the underlying symmetry, corrugation, and hierarchy of local sets of tiles.

EXPERIMENTS

We design, fabricate, and release a seven-fold QC composed of three differently shaped rhombic tiles using the method of lithographically pre-assembled monolayers (litho-PAMs). High resolution optical microscopy enables spatio-temporal particle tracking of Brownian fluctuations of many tiles in a large area over a long time. We develop an edge-proximity tessellation method for analyzing nearest neighboring particles that can be applied to assemblies and dense systems of complex shapes.

FINDINGS

A fluctuating heptatic LQC phase is identified at high tile area fractions. Heterogenous dynamics and order at different length scales indicate diverse, hierarchical motif structures. We show that certain motifs can collectively rotate without any cage breaking, leading to alterations of the local tile-structure reminiscent of phason-flips in atomic QCs; this rotation causes a slow decline in the system's spatial order. We anticipate that edge-proximity tessellation will help elucidate phase transitions of other systems made of diverse building blocks having significant geometrical complexity at multiple length scales.

Self-Assembled Ring-Based Complex Colloidal Particles by Lock-And-Key Interaction and Their Self-Assembly into Unusual Colloidal Crystals

Creating hierarchical crystalline materials using simple colloids or nanoparticles is very challenging, as it is usually impossible to achieve hierarchical structures without nonhierarchical colloidal interactions. Here, we present a hierarchical self-assembly (SA) route that employs colloidal rings and anisotropic colloidal particles to form complex colloids and uses them as building blocks to form unusual colloidal columnar liquid crystals or crystals. This route is realized by designing hierarchical SA driving forces that is controlled by the colloidal shape and shape-dependent depletion attraction. Depletion-induced lock-and-key interaction is the first driving force, which ensures a high efficiency (>90%) to load colloidal particles of other shapes such as spheres, spherocylinders, and oblate ellipsoids into rings, providing high-quality building blocks. Their SA into ordered superstructures has to require a second driving force such as higher volume fraction and/or stronger depletion attraction. As a result, unusual hierarchical colloidal (liquid) crystals, which have previously been difficult to fabricate by simple binary assembly, can be achieved. This work presents a significant advancement in the field of hierarchical SA, demonstrating a promising strategy for constructing many unprecedented crystalline materials by the SA route.

A columnar liquid quasicrystal with a honeycomb structure that consists of triangular, square and trapezoidal cells

Quasicrystals are intriguing structures that have long-range positional correlations but no periodicity in real space, and typically with rotational symmetries that are 'forbidden' in conventional periodic crystals. Here, we present a two-dimensional columnar liquid quasicrystal with dodecagonal symmetry. Unlike previous dodecagonal quasicrystals based on random tiling, a honeycomb structure based on a strictly quasiperiodic tessellation of tiles is observed. The structure consists of dodecagonal clusters made up of triangular, square and trapezoidal cells that are optimal for local packing. To maximize the presence of such dodecagonal clusters, the system abandons periodicity but adopts a quasiperiodic structure that follows strict packing rules. The stability of random-tiling dodecagonal quasicrystals is often attributed to the entropy of disordering when strict tiling rules are broken, at the sacrifice of the long-range positional order. However, our results demonstrate that quasicrystal stability may rest on energy minimization alone, or with only minimal entropic intervention.

Hydrodynamic Enhancement of p-atic Defect Dynamics

We investigate numerically and analytically the effects of hydrodynamics on the dynamics of topological defects in p-atic liquid crystals, i.e., two-dimensional liquid crystals with p-fold rotational symmetry. Importantly, we find that hydrodynamics fuels a generic passive self-propulsion mechanism for defects of winding number s=(p-1)/p and arbitrary p. Strikingly, we discover that hydrodynamics always accelerates the annihilation dynamics of pairs of ±1/p defects and that, contrary to expectations, this effect increases with p. Our Letter paves the way toward understanding cell intercalation and other remodeling events in epithelial layers.

Multiphase Coexistence in Binary Hard Colloidal Mixtures: Predictions from a Simple Algebraic Theory

A general theoretical framework is proposed to quantify the thermodynamic properties of multicomponent hard colloidal mixtures. This framework is used to predict the phase behavior of mixtures of rods with spheres and rods with plates taking into account (liquid) crystal phases of both components. We demonstrate a rich and complex range of phase behaviors featuring a large variety of different multiphase coexistence regions, including two five-phase coexistence regions for hard rod/sphere mixtures, and even a six-phase equilibrium for hard rod/plate dispersions. The various multiphase coexistences featured in a particular mixture are in line with a recently proposed generalized phase rule and can be tuned through subtle variations of the particle shape and size ratio. Our approach qualitatively accounts for certain multiphase equilibria observed in rod/plate mixtures of clay colloids and will be a useful guide in tuning the phase behavior of shape-disperse mixtures in general.

Unusual Mathematical Approaches Untangle Nervous Dynamics

The massive amount of available neurodata suggests the existence of a mathematical backbone underlying neuronal oscillatory activities. For example, geometric constraints are powerful enough to define cellular distribution and drive the embryonal development of the central nervous system. We aim to elucidate whether underrated notions from geometry, topology, group theory and category theory can assess neuronal issues and provide experimentally testable hypotheses. The Monge’s theorem might contribute to our visual ability of depth perception and the brain connectome can be tackled in terms of tunnelling nanotubes. The multisynaptic ascending fibers connecting the peripheral receptors to the neocortical areas can be assessed in terms of knot theory/braid groups. Presheaves from category theory permit the tackling of nervous phase spaces in terms of the theory of infinity categories, highlighting an approach based on equivalence rather than equality. Further, the physical concepts of soft-matter polymers and nematic colloids might shed new light on neurulation in mammalian embryos. Hidden, unexpected multidisciplinary relationships can be found when mathematics copes with neural phenomena, leading to novel answers for everlasting neuroscientific questions. For instance, our framework leads to the conjecture that the development of the nervous system might be correlated with the occurrence of local thermal changes in embryo–fetal tissues.

Hydrodynamic theory of p-atic liquid crystals

We formulate a comprehensive hydrodynamic theory of two-dimensional liquid crystals with generic p-fold rotational symmetry, also known as p-atics, of which nematics (p=2) and hexatics (p=6) are the two best known examples. Previous hydrodynamic theories of p-atics are characterized by continuous O(2) rotational symmetry, which is higher than the discrete rotational symmetry of p-atic phases. By contrast, here we demonstrate that the discrete rotational symmetry allows the inclusion of additional terms in the hydrodynamic equations, which, in turn, lead to novel phenomena, such as the possibility of flow alignment at high shear rates, even for p>2. Furthermore, we show that any finite imposed shear will induce long-ranged orientational order in any p-atic liquid crystal, in contrast to the quasi-long-ranged order that occurs in the absence of shear. The induced order parameter scales like a nonuniversal power of the applied shear rate at small shear rates.

Fractional defect charges in liquid crystals with p-fold rotational symmetry on cones

Conical surfaces, with a δ function of Gaussian curvature at the apex, are perhaps the simplest example of geometric frustration. We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones. For free boundary conditions at the base, we find both the ground state(s) and a discrete ladder of metastable states as a function of both the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a set of fractional defect charges at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions for the ground-state energies numerically for a set of commensurate cone angles (corresponding to a set of commensurate Gaussian curvatures concentrated at the cone apex), whose surfaces can be polygonized as a perfect triangular or square mesh, and find excellent agreement with our theoretical predictions.

2D Colloids: Size- and Shape-Controlled 2D Materials at Fluid-Fluid Interfaces

Advances in synthesis of model 3D colloidal particles with exotic shapes and physical properties have enabled discovery of new 3D colloidal phases not observed in atomic systems, and simulations and quasi-2D studies suggest 2D colloidal systems have an even richer phase behavior. However, a model 2D (one-atom-thick) colloidal system has yet to be experimentally realized because of limitations in solution-phase exfoliation of 2D materials and other 2D particle fabrication technologies. Herein, we use a photolithography-based methodology to fabricate size- and shape-controlled monolayer graphene particles, and then transfer the particles to an air-water interface to study their dynamics and self-assembly in real-time using interference reflection microscopy. Results suggest the graphene particles behave as "hard" 2D colloidal particles, with entropy influencing the self-assembled structures. Additional evidence suggests the stability of the self-assembled structures manifests from the edge-to-edge van der Waals force between 2D particles. We also show graphene discs with diameters up to 50 μm exhibit significant Brownian motion under optical microscopy due to their low mass. This work establishes a facile methodology for creating model experimental systems of colloidal 2D materials, which will have a significant impact on our understanding of fundamental 2D physics. Finally, our results advance our understanding of how physical particle properties affect the interparticle interactions between monolayer 2D materials at fluid-fluid interfaces. This information can be used to guide the development of scalable synthesis techniques (e.g., solution-phase processing) to produce bulk suspensions of 2D materials with desired physical particle properties that can be used as building blocks for creating thin films with desired structures and properties via interfacial film assembly.

Depletion torques between anisotropic colloidal particles

We explore how the entropic notion of depletion forces between spheres, introduced by Asakura and Oosawa, can be extended to depletion torques that affect the orientations of colloidal particles having complex shapes. In prior experimental work, systems of microscale plate-like particles in the presence of a nanoscale depletion agent have been shown to form polymer-like columnar chains; restoring depletion torques act to align lithographically-structured platelets within a chain orientationally about the chain's axis. We consider depletion torques corresponding to parallel, face-to-face, near-contact pair interactions for complex-shaped, plate-like, prismatic lithographic particles in colloidal dispersions containing a spherical nanoscale depletion agent. We calculate depletion torques for a wide variety of such particles, including rotationally symmetric, asymmetric, achiral, chiral, and elongated particles. Moreover, we determine depletion torques between two non-parallel proximate square platelets connected by a lossless hinge along a common edge. Our investigations show that depletion torques can be tailored through lithographic or synthetic design of specific geometrical features in the shapes of particles.

Brownian lithographic polymers of steric lock-and-key colloidal linkages

We design and lithographically fabricate two-dimensional preassembled colloidal linkages of custom-shaped, discrete, mobile microscale tiles that are sterically coupled together by lock-and-key sub-tile features, yielding hinge-like bonds between separate tiles. These mobile colloidal linkages, which we call polylithomers, provide top-down, preconfigured, morphologically controllable analogs of fluctuating molecular polymers. We illustrate the versatility of this approach by fabricating and studying curvilinear, branched, bridged-spiral, dendritic, and mesh-like polylithomers having controllable preassembled dimensions, topologies, configurations, intrinsic local curvatures, persistence lengths, and bond extensibilities. By advancing anisotropic particle tracking routines to handle lock-and-key tiles, we measure the dynamic conformational changes of polylithomers caused by Brownian excitations to the monomer scale, revealing markedly large bond extensibilities. Beyond modeling fluctuating semiflexible molecular polymers, polylithomers provide access to unusual polymer morphologies and bonding potentials that have not yet been synthesized through other kinds of assembly methods using either molecular or colloidal monomers.

Friction-directed self-assembly of Janus lithographic microgels into anisotropic 2D structures

We present a method for creating ordered 2D structures with material anisotropy from self-assembling micro-sized hydrogel particles (microgels). Microgel platelets of polygonal shapes (hexagon, square, and rhombus), obtained by a continuous scalable lithographic process, are suspended in an aqueous environment and sediment on an inclined plane. As a consequence of gravitational pull, they slide over the plane. Each half of the microgel is composed of a different type of hydrogel [poly(N-isopropylacrylamide) (PNIPAM), and poly(ethylene glycol) diacrylate (PEGDA), respectively] which exhibit different frictional coefficients when sheared over a substrate. Hence the microgels self-orientate as they slide, and the side with the lower frictional coefficient positions in the direction of sliding. The self-oriented microgels concentrate at the bottom of the tilted plane. Here they form densely packed structures with translational as well as orientational order.

Spontaneous organization of supracolloids into three-dimensional structured materials

Periodic nano- or microscale structures are used to control light, energy and mass transportation. Colloidal organization is the most versatile method used to control nano- and microscale order, and employs either the enthalpy-driven self-assembly of particles at a low concentration or the entropy-driven packing of particles at a high concentration. Nonetheless, it cannot yet provide the spontaneous three-dimensional organization of multicomponent particles at a high concentration. Here we combined these two concepts into a single strategy to achieve hierarchical multicomponent materials. We tuned the electrostatic attraction between polymer and silica nanoparticles to create dynamic supracolloids whose components, on drying, reorganize by entropy into three-dimensional structured materials. Cryogenic electron tomography reveals the kinetic pathways, whereas Monte Carlo simulations combined with a kinetic model provide design rules to form the supracolloids and control the kinetic pathways. This approach may be useful to fabricate hierarchical hybrid materials for distinct technological applications.

In Situ Quantification of Interactions between Charged Nanorods in a Predefined Potential Energy Landscape

Quantitative understanding of nanoscale interactions is a prerequisite for harnessing the remarkable collective properties of nanoparticle systems. Here, we report the combined use of liquid-phase transmission electron microscopy and electron beam lithography to elucidate the interactions between charged nanorods in a predefined potential energy landscape. In situ site-selective lift-off of surface-functionalized lithographed gold nanorods is achieved by patterning them with adhesion layer materials that undergo etching at different rates. Analysis of the subsequent nanorod motion, which is two-dimensionally confined as a result of the particle-substrate attraction, allows quantification of interparticle interactions in a lithographically engineered environment. For lithographed nanorods patterned with the same adhesion layer material, their self-assembly behavior following lift-off is tuned by changing their starting spatial arrangement. Our approach facilitates investigation of interparticle interactions in designed nanoparticle systems and affords fundamental insights into the role of the potential energy landscape in determining the kinetic pathway for nanoparticle self-assembly.

Orientational ordering in a fluid of hard kites: A density-functional-theory study

Using density-functional theory we theoretically study the orientational properties of uniform phases of hard kites-two isosceles triangles joined by their common base. Two approximations are used: scaled particle theory and a new approach that better approximates third virial coefficients of two-dimensional hard particles. By varying some of their geometrical parameters, kites can be transformed into squares, rhombuses, triangles, and also very elongated particles, even reaching the hard-needle limit. Thus, a fluid of hard kites, depending on the particle shape, can stabilize isotropic, nematic, tetratic, and triatic phases. Different phase diagrams are calculated, including those of rhombuses, and kites with two of their equal interior angles fixed to 90^{∘}, 60^{∘}, and 75^{∘}. Kites with one of their unequal angles fixed to 72^{∘}, which have been recently studied via Monte Carlo simulations, are also considered. We find that rhombuses and kites with two equal right angles and not too large anisometry stabilize the tetratic phase but the latter stabilize it to a much higher degree. By contrast, kites with two equal interior angles fixed to 60^{∘} stabilize the triatic phase to some extent, although it is very sensitive to changes in particle geometry. Kites with the two equal interior angles fixed to 75^{∘} have a phase diagram with both tetratic and triatic phases, but we show the nonexistence of a particle shape for which both phases are stable at different densities. Finally, the success of the new theory in the description of orientational order in kites is shown by comparing with Monte Carlo simulations for the case where one of the unequal angles is fixed to 72^{∘}. These particles also present a phase diagram with stable tetratic and triatic phases.

Fabrication of magnetic colloidal bicones and their switchable self-assembly

The creation and fabrication of novel shapes are of paramount importance for colloids to self-assemble into desired structured materials and in order to use them as model systems for fundamental studies. Here, we demonstrate a simple yet effective method to fabricate bicone-shaped colloids by using SiO₂/PS core/shell particles through a thermal stretching route. The shape parameters are tunable. The aspect ratio is found to be predictable, and the experimental results are consistent with the model calculations. The yields are high enough to be used for self-assembly studies. As an illustration, we investigate the phase behavior of particles when interacting with depletion forces and show that the particles can form hexagonal or non-hexagonal crystal lattices in quasi-two-dimensional space. The method also allows functional nanoparticles to be integrated into the cores, resulting in responsive colloidal bicones. Moreover, the magnetically responsive self-assembly of particles is demonstrated.

Artificial colloidal liquid metacrystals by shearing microlithography

Meta-periodicity beyond intrinsic atomic and molecular order, such as metacrystalline and quasicrystalline lattices, exists in solids, but is usually elusive in lyotropic liquid crystals for its energetic instability. The stable meta-periodicity in lyotropic liquid crystals in the absence of external stimuli remains unexplored, and how to achieve it keeps a great challenge. Here we create lyotropic liquid crystals with stable meta-periodicity in a free state, coined as liquid metacrystals, in colloidal systems by an invented shearing microlithography. The meta-periodicity is dynamically stabilized by the giant molecular size and strong excluded volume repulsion. Liquid metacrystals are designed to completely cover a library of symmetries, including five Bravais and six quasicrystalline lattices. Liquid metacrystal promises an extended form of liquid crystals with rich meta-periodicity and the shearing microlithography emerges as a facile technology to fabricate liquid meta-structures and metamaterials, enabling the digital design of structures and functionalities of liquid crystalline materials.

Stable periodic structures can be difficult to obtain in a liquid crystal compared to a solid due to the energetic instability of the former. Here the authors present a technique to fabricate quasicrystalline structures of graphene oxide liquid crystals which have high stability.