Hierarchy of topological transitions in a network liquid.

The representation of complex systems as networks has become a critical tool across many fields of science. In the context of physical networks, such as biological neural networks, vascular networks, or network liquids where the nodes and edges occupy volume in three-dimensional space, the question of how they become densely packed is of special importance. Here, we investigate a model network liquid, which is known to densify via two successive liquid-liquid phase transitions (LLPTs). We elucidate the importance of rings-cyclic paths formed by bonded particles in the networks-and their spatial disposition in understanding the structural changes that underpin the increase in density across the LLPTs. Our analyses demonstrate that the densification of these networks is primarily driven by the formation of linked rings, and the LLPTs correspond to a hierarchy of topological transitions where rings form the fundamental building blocks. We envisage entanglement to emerge as a general mechanism for densification, with wide implications for the embedding of physical networks, especially in confined spaces.

Facile self-assembly of colloidal diamond from tetrahedral patchy particles via ring selection.

Diamond-structured crystals, particularly those with cubic symmetry, have long been attractive targets for the programmed self-assembly of colloidal particles, due to their applications as photonic crystals that can control the flow of visible light. While spherical particles decorated with four patches in a tetrahedral arrangement-tetrahedral patchy particles-should be an ideal building block for this endeavor, their self-assembly into colloidal diamond has proved elusive. The kinetics of self-assembly pose a major challenge, with competition from an amorphous glassy phase, as well as clathrate crystals, leaving a narrow widow of patch widths where tetrahedral patchy particles can self-assemble into diamond crystals. Here we demonstrate that a two-component system of tetrahedral patchy particles, where bonding is allowed only between particles of different types to select even-member rings, undergoes crystallization into diamond crystals over a significantly wider range of patch widths conducive for experimental fabrication. We show that the crystallization in the two-component system is both thermodynamically and kinetically enhanced, as compared to the one-component system. Although our bottom-up route does not lead to the selection of the cubic polytype exclusively, we find that the cubicity of the self-assembled crystals increases with increasing patch width. Our designer system not only promises a scalable bottom-up route for colloidal diamond but also offers fundamental insight into crystallization into open lattices.

Investigating the trade-off between color saturation and angle-independence in photonic glasses.

Photonic glasses-isotropic structures with short-range correlations-can produce structural colors with little angle-dependence, making them an alternative to dyes in applications such as cosmetics, coatings, and displays. However, the low angle-dependence is often accompanied by low color saturation. To investigate how the short-range correlations affect the trade-off between saturation and angle-independence, we vary the structure factor and use a Monte Carlo model of multiple scattering to investigate the resulting optical properties. We use structure factors derived from analytical models and calculated from simulations of disordered sphere packings. We show that the trade-off is controlled by the first peak of the structure factor. It is possible to break the trade-off by tuning the width of this peak and controlling the sample thickness. Practically, this result shows that the protocol used to pack particles into a photonic glass is important to the optical properties.

Supracolloidal reconfigurable polyhedra via hierarchical self-assembly.

Enclosed three-dimensional structures with hollow interiors have been attractive targets for the self-assembly of building blocks across different length scales. Colloidal self-assembly, in particular, has enormous potential as a bottom-up means of structure fabrication exploiting a priori designed building blocks because of the scope for tuning interparticle interactions. Here we use computer simulation study to demonstrate the self-assembly of designer charge-stabilised colloidal magnetic particles into a series of supracolloidal polyhedra, each displaying a remarkable two-level structural hierarchy. The parameter space for design supports thermodynamically stable polyhedra of very different morphologies, namely tubular and hollow spheroidal structures, involving the formation of subunits of four-fold and three-fold rotational symmetry, respectively. The spheroidal polyhedra are chiral, despite having a high degree of rotational symmetry. The dominant pathways for self-assembly into these polyhedra reveal two distinct mechanisms - a growth mechanism via sequential attachment of the subunits for a tubular structure and a staged or hierarchical pathway for a spheroidal polyhedron. These supracolloidal architectures open up in response to an external magnetic field. Our results suggest design rules for synthetic reconfigurable containers at the microscale exploiting a hierarchical self-assembly scheme.

Exploring energy landscapes: from molecular to mesoscopic systems.

We review a comprehensive computational framework to survey the potential energy landscape for systems composed of rigid or partially rigid molecules. Illustrative case studies relevant to a wide range of molecular clusters and soft and condensed matter systems are discussed.

Effect of coat-protein concentration on the self-assembly of bacteriophage MS2 capsids around RNA.

Self-assembly is a vital part of the life cycle of certain icosahedral RNA viruses. Furthermore, the assembly process can be harnessed to make icosahedral virus-like particles (VLPs) from coat protein and RNA in vitro. Although much previous work has explored the effects of RNA-protein interactions on the assembly products, relatively little research has explored the effects of coat-protein concentration. We mix coat protein and RNA from bacteriophage MS2, and we use a combination of gel electrophoresis, dynamic light scattering, and transmission electron microscopy to investigate the assembly products. We show that with increasing coat-protein concentration, the products transition from well-formed MS2 VLPs to ``monster'' particles consisting of multiple partial capsids to RNA-protein condensates consisting of large networks of RNA and partially assembled capsids. We argue that the transition from well-formed to monster particles arises because the assembly follows a nucleation-and-growth pathway in which the nucleation rate depends sensitively on the coat-protein concentration, such that at high protein concentrations, multiple nuclei can form on each RNA strand. To understand the formation of the condensates, which occurs at even higher coat-protein concentrations, we use Monte Carlo simulations with coarse-grained models of capsomers and RNA. These simulations suggest that the the formation of condensates occurs by the adsorption of protein to the RNA followed by the assembly of capsids. Multiple RNA molecules can become trapped when a capsid grows from capsomers attached to two different RNA molecules or when excess protein bridges together growing capsids on different RNA molecules. Our results provide insight into an important biophysical process and could inform design rules for making VLPs for various applications.

Effect of coat-protein concentration on the self-assembly of bacteriophage MS2 capsids around RNA.

Self-assembly is a vital part of the life cycle of certain icosahedral RNA viruses. Furthermore, the assembly process can be harnessed to make icosahedral virus-like particles (VLPs) from coat protein and RNA in vitro. Although much previous work has explored the effects of RNA-protein interactions on the assembly products, relatively little research has explored the effects of coat-protein concentration. We mix coat protein and RNA from bacteriophage MS2, and we use a combination of gel electrophoresis, dynamic light scattering, and transmission electron microscopy to investigate the assembly products. We show that with increasing coat-protein concentration, the products transition from well-formed MS2 VLPs to "monster" particles consisting of multiple partial capsids to RNA-protein condensates consisting of large networks of RNA and partially assembled capsids. We argue that the transition from well-formed to monster particles arises because the assembly follows a nucleation-and-growth pathway in which the nucleation rate depends sensitively on the coat-protein concentration, such that at high protein concentrations, multiple nuclei can form on each RNA strand. To understand the formation of the condensates, which occurs at even higher coat-protein concentrations, we use Monte Carlo simulations with coarse-grained models of capsomers and RNA. These simulations suggest that the formation of condensates occurs by the adsorption of protein to the RNA followed by the assembly of capsids. Multiple RNA molecules can become trapped when a capsid grows from capsomers attached to two different RNA molecules or when excess protein bridges together growing capsids on different RNA molecules. Our results provide insight into an important biophysical process and could inform design rules for making VLPs for various applications.

Programmed Self-Assembly of Single Colloidal Gyroids for Chiral Photonic Crystals.

Gyroid structures are of extensive interest because they provide a rich platform for chiroptics as well as topological photonics. While the double-gyroid morphology as a bicontinuous structure is not uncommon in self-assembled soft materials, direct self-assembly of single-network gyroids has proven elusive. Here, an enantiomorphic pair of single-gyroid crystals comprising colloidal spheres is presented, and two distinct routes are demonstrated for programmed self-assembly of each single colloidal gyroid enantiomorph from rationally designed patchy spheres. The designer colloidal patchy spheres, which closely hew to their synthetic feasibility, are chiral, having either two staggered rectangular patches at opposite poles or four circular patches arranged in a well-defined geometry. The single colloidal gyroid, as well as its inverse structure, is shown to support a wide complete photonic bandgap in addition to exhibiting rich chiroptical properties, making them attractive chiral photonic crystals. The versatility of this single colloidal gyroid, the bottom-up routes devised here in silico, and the robustness of the design space for the chiral colloidal patchy spheres together make a strong case for single colloidal gyroids to supersede colloidal diamond, as a target for programmed self-assembly, in the quest for photonic crystals operating at optical frequencies.

A survey of the potential energy surface for the (benzene)13 cluster.

We report on a survey of the potential energy surface for the 13-molecule benzene cluster, (C(6)H(6))(13), bound by an atom-atom intermolecular potential developed from first principles. The potential, which has an anisotropic repulsion term, is found to support distinct pairs of structures of C(3), C(i), and S(6) symmetry as low-lying minima, including a C(3) global minimum. The organisation of the low-lying region of the potential energy surface suggests that one of the S(6) structures is likely to act as a kinetic trap, hindering efficient relaxation to the global minimum, in agreement with the hypothesis that two isomers coexist in this system, suggested by spectroscopic experiments.

Self-Assembly of Patchy Colloidal Rods into Photonic Crystals Robust to Stacking Faults.

Diamond-structured colloidal photonic crystals are much sought-after for their applications in visible light management because of their ability to support a complete photonic band gap (PBG). However, their realization via self-assembly pathways is a long-standing challenge. This challenge is rooted in three fundamental problems: the design of building blocks that assemble into diamond-like structures, the sensitivity of the PBG to stacking faults, and ensuring that the PBG opens at an experimentally attainable refractive index. Here we address these problems simultaneously using a multipronged computational approach. We use reverse engineering to establish the design principles for the rod-connected diamond structure (RCD), the so-called "champion" photonic crystal. We devise two distinct self-assembly routes for designer triblock patchy colloidal rods, both proceeding via tetrahedral clusters to yield a mixed phase of cubic and hexagonal polymorphs closely related to RCD. We use Monte Carlo simulations to show how these routes avoid a metastable amorphous phase. Finally, we show that both the polymorphs support spectrally overlapping PBGs. Importantly, randomly stacked hybrids of these polymorphs also display PBGs, thus circumventing the requirement of polymorph selection in a scalable fabrication method.

Leveraging Hierarchical Self-Assembly Pathways for Realizing Colloidal Photonic Crystals.

Colloidal open crystals are attractive materials, especially for their photonic applications. Self-assembly appeals as a bottom-up route for structure fabrication, but self-assembly of colloidal open crystals has proven to be elusive for their mechanical instability due to being low-coordinated. For such a bottom-up route to yield a desired colloidal open crystal, the target structure is required to be thermodynamically favored for designer building blocks and also kinetically accessible via self-assembly pathways in preference to metastable structures. Additionally, the selection of a particular polymorph poses a challenge for certain much sought-after colloidal open crystals for their applications as photonic crystals. Here, we devise hierarchical self-assembly pathways, which, starting from designer triblock patchy particles, yield in a cascade of well-separated associations first tetrahedral clusters and then tetrastack crystals. The designed pathways avoid trapping into an amorphous phase. Our analysis reveals how such a two-stage self-assembly pathway via tetrahedral clusters promotes crystallization by suppressing five- and seven-membered rings that hinder the emergence of the ordered structure. We also find that slow annealing promotes a bias toward the cubic polymorph relative to the hexagonal counterpart. Finally, we calculate the photonic band structures, showing that the cubic polymorph exhibits a complete photonic band gap for the dielectric filling fraction directly realizable from the designer triblock patchy particles. Unexpectedly, we find that the hexagonal polymorph also supports a complete photonic band gap, albeit only for an increased filling fraction, which can be realized via postassembly processing.

Energy landscape of a model discotic liquid crystal.

The potential energy surface of a model discotic liquid crystal is investigated across mesophases as a function of temperature by characterizing the local potential energy minima. The average depth of the minima sampled by the system gradually grows as orientational order increases through the discotic-nematic phase upon cooling, while it remains fairly constant in the isotropic phase for isochoric temperature variation. The high-temperature Arrhenius behavior of the single-particle orientational correlation times is found to break down at a temperature that marks the onset of exploration of deeper potential energy minima. The structural features of the minima reveal an interplay between orientational and translational order, in particular, when the parent phase is discotic-nematic. The local minima then exhibit short-range columns that tend to have local hexagonal packing. The present study and recent work on calamitic liquid crystals [D. Chakrabarti and B. Bagchi, Proc. Natl. Acad. Sci. U.S.A. 103, 7217 (2006)] together reveal a striking similarity between thermotropic liquid crystals and supercooled liquids in the exploration of the energy landscape and the breakdown of Arrhenius behavior for relaxation times.

Programming hierarchical self-assembly of colloids: matching stability and accessibility.

Encoding hierarchical self-assembly in colloidal building blocks is a promising bottom-up route to high-level structural complexity often observed in biological materials. However, harnessing this promise faces the grand challenge of bridging hierarchies of multiple length- and time-scales, associated with structure and dynamics respectively along the self-assembly pathway. Here we report on a case study, which examines the kinetic accessibility of a series of hollow spherical structures with a two-level structural hierarchy self-assembled from charge-stabilized colloidal magnetic particles. By means of a variety of computational methods, we find that for a staged assembly pathway, the structure, which derives the strongest energetic stability from the first stage of assembly and the weakest from the second stage, is most kinetically accessible. Such a striking correspondence between energetics and kinetics for optimal design principles should have general implications for programming hierarchical self-assembly pathways for nano- and micro-particles, while matching stability and accessibility.

Correction: Hierarchical self-assembly of colloidal magnetic particles into reconfigurable spherical structures.

Correction for 'Hierarchical self-assembly of colloidal magnetic particles into reconfigurable spherical structures' by Daniel Morphew et al., Nanoscale, 2015, 7, 8343-8350.

Programming Hierarchical Self-Assembly of Patchy Particles into Colloidal Crystals via Colloidal Molecules.

Colloidal self-assembly is a promising bottom-up route to a wide variety of three-dimensional structures, from clusters to crystals. Programming hierarchical self-assembly of colloidal building blocks, which can give rise to structures ordered at multiple levels to rival biological complexity, poses a multiscale design problem. Here we explore a generic design principle that exploits a hierarchy of interaction strengths and employ this design principle in computer simulations to demonstrate the hierarchical self-assembly of triblock patchy colloidal particles into two distinct colloidal crystals. We obtain cubic diamond and body-centered cubic crystals via distinct clusters of uniform size and shape, namely, tetrahedra and octahedra, respectively. Such a conceptual design framework has the potential to reliably encode hierarchical self-assembly of colloidal particles into a high level of sophistication. Moreover, the design framework underpins a bottom-up route to cubic diamond colloidal crystals, which have remained elusive despite being much sought after for their attractive photonic applications.

Glassiness of thermotropic liquid crystals across the isotropic-nematic transition.

The orientational dynamics of thermotropic liquid crystals across the isotropic-nematic phase transition have traditionally been investigated at long times or low frequencies using frequency domain measurements. The situation has now changed significantly with the recent report of a series of interesting transient optical Kerr effect (OKE) experiments that probed orientational relaxation of a number of calamitic liquid crystals (which consist of rod-like molecules) directly in the time domain, over a wide time window ranging from subpicoseconds to tens of microseconds. The most intriguing revelation is that the decay of the OKE signal at short to intermediate times (from a few tens of picoseconds to several hundred nanoseconds) follows multiple temporal power laws. Another remarkable feature that has emerged from these OKE measurements is the similarity in the orientational relaxation behavior between the isotropic phase of calamitic liquid crystals near the isotropic-nematic transition and supercooled molecular liquids, notwithstanding their largely different macroscopic states. In this article, we present an overview of the understanding that has emerged from recent computational and theoretical studies of calamitic liquid crystals across the isotropic-nematic transition. Topics discussed include (a) single-particle as well as collective orientational dynamics at a short-to-intermediate time window, (b) heterogeneous dynamics in orientational degrees of freedom diagnosed by a non-Gaussian parameter, (c) fragility, and (d) temperature-dependent exploration of underlying energy landscapes as calamitic liquid crystals settle into increasingly ordered mesophases upon cooling from the high-temperature isotropic phase. A comparison of our results with those of supercooled molecular liquids reveals an array of analogous features in these two important classes of soft matter systems. We further find that the onset of growth of the orientational order in the parent nematic phase induces translational order, resulting in smectic-like layers in the potential energy minima of calamitic systems if the parent nematic phase is sandwiched between the high-temperature isotropic phase and the low-temperature smectic phase. We discuss implications of this startling observation. We also discuss recent results on the orientational dynamics of discotic liquid crystals that are found to be rather similar to those of calamitic liquid crystals.

Orientational relaxation in a discotic liquid crystal.

We investigate orientational relaxation of a model discotic liquid crystal, consisting of disclike molecules, by molecular dynamics simulations along two isobars starting from the high temperature isotropic phase. The two isobars have been so chosen that (a) the phase sequence isotropic- (I-) nematic- (N-) columnar (C) appears upon cooling along one of them and (b) the sequence isotropic- (I-) columnar- (C) along the other. While the orientational relaxation in the isotropic phase near the I-N phase transition in system (a) shows a power law decay at short to intermediate times, such power law relaxation is not observed in the isotropic phase near the I-C phase boundary in system (b). In order to understand this difference (the existence or the absence of the power law decay), we calculated the growth of the orientational pair distribution functions (OPDFs) near the I-N phase boundary and also near the I-C phase boundary. We find that the OPDF shows a marked growth in long range correlation as the I-N phase boundary is approached in the I-N-C system (a), but such a growth is absent in the I-C system, which appears to be consistent with the result that I-N phase transition in the former is weakly first order while the I-C phase transition in the latter is not weak. As the system settles into the nematic phase, the decay of the single-particle second-rank orientational time correlation function follows a pattern that is similar to what is observed with calamitic liquid crystals and supercooled molecular liquids.

Glassy orientational dynamics of rodlike molecules near the isotropic-nematic transition.

We investigate the single-particle orientational dynamics of rodlike molecules across the isotropic-nematic transition in computer simulations of a family of model systems of thermotropic liquid crystals. Several remarkable features of glassy dynamics are on display including nonexponential relaxation, dynamical heterogeneity, and non-Arrhenius temperature dependence of the orientational relaxation time. In order to obtain a quantitative measure of glassy dynamics in line with the established methods in supercooled liquids, we construct a relaxation time versus scaled inverse temperature plot and demonstrate that one can indeed define a "fragility index" for thermotropic liquid crystals that depends on density and aspect ratio. The values of the fragility parameter are surprisingly in the range observed for glass-forming liquids. A plausible correlation between the energy landscape features and the observed fragility is discussed.

Anisotropic translational diffusion in the nematic phase: dynamical signature of the coupling between orientational and translational order in the energy landscape.

We find in a model system of thermotropic liquid crystals that the translational diffusion coefficient parallel to the director D(parallel) first increases and then decreases as temperature drops through the nematic phase, and this reversal occurs where the smectic order parameter of the underlying inherent structures becomes significant. We argue, based on an energy landscape analysis, that the coupling between orientational and translational order can play a role in inducing the nonmonotonic temperature behavior of D(parallel). Such a view is likely to form the foundation of a theoretical framework to explain the anisotropic translation diffusion.

Power law relaxation and glassy dynamics in Lebwohl-Lasher model near the isotropic-nematic phase transition.

Orientational dynamics in a liquid crystalline system near the isotropic-nematic (IN) phase transition is studied using molecular dynamics simulations of the well-known Lebwohl-Lasher model. As the IN transition temperature is approached from the isotropic side, we find that the decay of the orientational time correlation functions (OTCF) slows down noticeably, giving rise to a power law decay at intermediate time scales. The angular velocity time correlation function also exhibits a rather pronounced power law decay near the IN boundary. In the mean squared angular displacement at comparable time scales, we observe the emergence of a subdiffusive regime which is followed by a superdiffusive regime before the onset of the long-time diffusive behavior. We observe signature of dynamical heterogeneity through pronounced non-Gaussian behavior in orientational motion particularly at lower temperatures. This behavior closely resembles what is usually observed in supercooled liquids. We obtain the free energy as a function of orientational order parameter by the use of the transition matrix Monte Carlo method. The free energy surface is flat for the system considered here and the barrier between isotropic and nematic phases is vanishingly small for this weakly first-order phase transition, hence allowing large scale, collective, and correlated orientational density fluctuations. This might be responsible for the observed power law decay of the OTCFs.