The stability of a crystal with diamond structure for patchy particles with tetrahedral symmetry

Interplay between the Formation of Colloidal Clathrate and Cubic Diamond Crystals

Controlling the valency of directional interactions of patchy particles is insufficient for the selective formation of target crystalline structures due to the competition between phases of similar free energy. Examples of such are stacking hybrids of interwoven hexagonal and cubic diamonds with (i) its liquid phase, (ii) arrested glasses, or (iii) clathrates, all depending on the relative patch size, despite being within the one-bond-per-patch regime. Herein, using molecular dynamics simulations, we demonstrate that although tetrahedral patchy particles with narrow patches can assemble into clathrates or stacking hybrids in the bulk, this behavior can be suppressed by the application of external surface potential. Depending on its strength, the selective growth of either cubic diamond crystals or empty sII clathrate cages can be achieved. The formation of a given ordered network depends on the structure of the first adlayer, which is commensurate with the emerging network.

Self-Assembly Pathways of Triblock Janus Particles into 3D Open Lattices

The self-assembly of triblock Janus particles is simulated from a fluid to 3D open lattices: pyrochlore, perovskite, and diamond. The coarse-grained model explicitly takes into account the chemical details of the Janus particles (attractive patches at the poles and repulsion around the equator) and it contains explicit solvent particles. Hydrodynamic interactions are accounted for by dissipative particle dynamics. The relative stability of the crystals depends on the patch width. Narrow, intermediate, and wide patches stabilize the pyrochlore-, the perovskite-, and the diamond-lattice, respectively. The nucleation of all three lattices follows a two-step mechanism: the particles first agglomerate into a compact and disordered liquid cluster, which does not crystallize until it has grown to a threshold size. Second, the particles reorient inside this cluster to form crystalline nuclei. The free-energy barriers for the nucleation of pyrochlore and perovskite are ≈10 kBT, which are close to the nucleation barriers of previously studied 2D kagome lattices. The barrier height for the nucleation of diamond, however, is much larger (>20 kBT), as the symmetry of the triblock Janus particles is not perfect for a diamond structure. The large barrier is associated with the reorientation of particles, i.e., the second step of the nucleation mechanism.

Pursuing colloidal diamonds

The endeavor to selectively fabricate a cubic diamond is challenging due to the formation of competing phases such as its hexagonal polymorph or others possessing similar free energy. The necessity to achieve this is of paramount importance since the cubic diamond is the only polymorph exhibiting a complete photonic bandgap, making it a promising candidate in view of photonic applications. Herein, we demonstrate that due to the presence of an external field and delicate manipulation of its strength we can attain selectivity in the formation of a cubic diamond in a one-component system comprised of designer tetrahedral patchy particles. The driving force of such a phenomenon is the structure of the first adlayer which is commensurate with the (110) face of the cubic diamond. Moreover, after a successful nucleation event, once the external field is turned off, the structure remains stable, paving an avenue for further post-synthetic treatment.

Phases of surface-confined trivalent colloidal particles

Patchy colloids promise the design and modelling of complex materials, but the realization of equilibrium patchy particle structures remains challenging. Here, we assemble pseudo-trivalent particles and elucidate their phase behaviour when confined to a plane. We observe the honeycomb phase, as well as more complex amorphous network and triangular phases. Structural analysis performed on the three condensed phases reveals their shared structural motifs. Using a combined experimental and simulation approach, we elucidate the energetics of these phases and construct the phase diagram of this system, using order parameters to determine the phase coexistence lines. Our results reveal the rich phase behaviour that a relatively simple patchy particle system can display, and open the door to a larger joined simulation and experimental exploration of the full patchy-particle phase space.

Vectorial crystallization problems and collective behavior

We propose and analyze a class of vectorial crystallization problems, with applications to crystallization of anisotropic molecules and collective behavior such as birds flocking and fish schooling. We focus on two-dimensional systems of "oriented" particles: Admissible configurations are represented by vectorial empirical measures with density in [Formula: see text]. We endow such configurations with a graph structure, where the bonds represent the "convenient" interactions between particles, and the proposed variational principle consists in maximizing their number. The class of bonds is determined by hard sphere type pairwise potentials, depending both on the distance between the particles and on the angles between the segment joining two particles and their orientations, through threshold criteria. Different ground states emerge by tuning the angular dependence in the potential, mimicking ducklings swimming in a row formation and predicting as well, for some specific values of the angular parameter, the so-called diamond formation in fish schooling.

Softness-Enhanced Self-Assembly of Pyrochlore- and Perovskite-like Colloidal Photonic Crystals from Triblock Janus Particles

It remains extremely challenging to build three-dimensional photonic crystals with complete photonic bandgaps by simple and experimentally realizable colloidal building blocks. Here, we demonstrate that particle softness can enhance both the self-assembly of pyrochlore- and perovskite-like lattice structures from simple deformable triblock Janus colloids and their photonic bandgap performances. Dynamics simulation results show that the region of stability of pyrochlore lattices can be greatly expanded by appropriately increasing softness, and the perovskite lattices are unexpectedly obtained at enough high softness. Photonic calculations show that the direct pyrochlore lattices formed from overlapping soft triblock Janus particles exhibit even larger photonic bandgaps than the ideal nonoverlapping pyrochlore lattice, and proper overlap arising from softness can also dramatically improve the photonic properties of the inverse pyrochlore and perovskite lattices. Our study offers a new and feasible self-assembly path toward three-dimensional photonic crystals with large and robust photonic bandgaps.

Growth transitions and critical behavior in the non-equilibrium aggregation of short, patchy nanorods

We have carried out Monte Carlo simulations to study the non-equilibrium aggregation of short patchy nanorods in two dimensions. Below a critical value of patch size ([Formula: see text]), the aggregates have finite sizes with small radii of gyration, [Formula: see text]. At [Formula: see text], the average radius of gyration shows a power law increase with time such that [Formula: see text], where [Formula: see text]. Above, [Formula: see text], the aggregates are fractal in nature and their fractal dimension depends on the value of patch size. These morphological differences are due to the fact that below the critical value of patch size ([Formula: see text]), the growth of the clusters is suppressed and the system reaches an 'absorbed state.' Above [Formula: see text], the system reaches an 'active state,' in which the cluster size keeps growing with a fixed rate at long times. Thus, the system encounters a non-equilibrium phase transition. Close to the transition, the growth rate scales as [Formula: see text], where [Formula: see text]. The long-time growth rate varies as [Formula: see text] where [Formula: see text]. These scaling exponents indicate that the transition belongs to the directed percolation universality class. The patchy nanorods also display a threshold patch size ([Formula: see text]), beyond which the long-time growth rate remains constant. We present geometric arguments for the existence of [Formula: see text]. The fractal dimension of the aggregates increases from 1.75, at [Formula: see text], to 1.81, at [Formula: see text]. It remains constant beyond [Formula: see text].

From colloidal particles to photonic crystals: advances in self-assembly and their emerging applications

Over the last three decades, photonic crystals (PhCs) have attracted intense interests thanks to their broad potential applications in optics and photonics. Generally, these structures can be fabricated via either "top-down" lithographic or "bottom-up" self-assembly approaches. The self-assembly approaches have attracted particular attention due to their low cost, simple fabrication processes, relative convenience of scaling up, and the ease of creating complex structures with nanometer precision. The self-assembled colloidal crystals (CCs), which are good candidates for PhCs, have offered unprecedented opportunities for photonics, optics, optoelectronics, sensing, energy harvesting, environmental remediation, pigments, and many other applications. The creation of high-quality CCs and their mass fabrication over large areas are the critical limiting factors for real-world applications. This paper reviews the state-of-the-art techniques in the self-assembly of colloidal particles for the fabrication of large-area high-quality CCs and CCs with unique symmetries. The first part of this review summarizes the types of defects commonly encountered in the fabrication process and their effects on the optical properties of the resultant CCs. Next, the mechanisms of the formation of cracks/defects are discussed, and a range of versatile fabrication methods to create large-area crack/defect-free two-dimensional and three-dimensional CCs are described. Meanwhile, we also shed light on both the advantages and limitations of these advanced approaches developed to fabricate high-quality CCs. The self-assembly routes and achievements in the fabrication of CCs with the ability to open a complete photonic bandgap, such as cubic diamond and pyrochlore structure CCs, are discussed as well. Then emerging applications of large-area high-quality CCs and unique photonic structures enabled by the advanced self-assembly methods are illustrated. At the end of this review, we outlook the future approaches in the fabrication of perfect CCs and highlight their novel real-world applications.

Phase behavior and percolation in mixed patchy colloids

Patchy colloids can be modeled as hard spheres with directional conical association sites. A variety of physical phenomena have been discovered in the patchy colloid system due to its short range and directional interactions. In this work, we combined a cluster distribution theory with generalized Flory and Stockmayer percolation theory to investigate the interplay between phase behavior and percolation for a binary patchy colloid system. The binary patchy colloid system consists of solute molecules with spherically symmetric bonding sites and solvents with two singly bondable sites. Wertheim's first order thermodynamic perturbation theory (TPT1) has been widely applied to the patchy colloids system and it has been combined with percolation theory to study the percolation threshold. However, due to assumptions behind TPT1, it will lose accuracy for a system in which particles have multiple association sites or multiply bondable sites. A recently proposed cluster distribution theory accurately models association at sites that can form multiple bonds. In this work, we investigate the comparison among cluster distribution theory, TPT1, and Monte Carlo simulation for the bonding states of this binary system in which cluster distribution theory shows excellent agreement with Monte Carlo simulation, while TPT1 has a large deviation with the simulation. Cluster distribution theory was further combined with the Flory and Stockmayer percolation theory to investigate the interplay between phase behavior and percolation threshold. We found that the reduced density and the relative bonding strength of solvent-solvent association and solute-solvent association are key factors for the phase behavior and percolation. Percolation can form at low density and low temperature in the vapor phase of this binary system, where the star-like molecules with 12 long branches formed.

Mikto-Arm Stars as Soft-Patchy Particles: From Building Blocks to Mesoscopic Structures

We present an atomistic molecular dynamics study of self-assembled mikto-arm stars, which resemble patchy-like particles. By increasing the number of stars in the system, we propose a systematic way of examining the mutual orientation of these fully penetrable patchy-like objects. The individual stars maintain their patchy-like morphology when creating a mesoscopic (macromolecular) self-assembled object of more than three stars. The self-assembly of mikto-arm stars does not lead to a deformation of the stars, and their shape remains spherical. We identified characteristic sub-units in the self-assembled structure, differing by the mutual orientation of the nearest neighbor stars. The current work aims to elucidate the possible arrangements of the realistic, fully penetrable patchy particles in polymer matrix and to serve as a model system for further studies of nanostructured materials or all-polymer nanocomposites using the mikto-arm stars as building blocks.

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.

A microgel-Pickering emulsion route to colloidal molecules with temperature-tunable interaction sites

A simple Pickering emulsion route has been developed for the assembly of temperature-responsive poly(N-isopropylacrylamide) (PNIPAM) microgel particles into colloidal molecules comprising a small number of discrete microgel interaction sites on a central oil emulsion droplet. Here, the surface activity of the microgels serves to drive their assembly through adsorption to growing polydimethylsiloxane (PDMS) emulsion oil droplets of high monodispersity, prepared in situ via ammonia-catalysed hydrolysis and condensation of dimethyldiethoxysilane (DMDES). A dialysis step is employed in order to limit further growth once the target assembly size has been reached, thus yielding narrowly size-distributed, colloidal molecule-like microgel-Pickering emulsion oil droplets with well-defined microgel interaction sites. The temperature-responsiveness of the PNIPAM interaction sites will allow for the directional interactions to be tuned in a facile manner with temperature, all the way from soft repulsive to short-range attractive as the their volume phase transition temperature (VPTT) is crossed. Finally, the microgel-Pickering emulsion approach is extended to a mixture of PNIPAM and poly(N-isopropylmethacrylamide) (PNIPMAM) microgels that differ with respect to their VPTT, this in order to prepare patchy colloidal molecules where the directional interactions will be more readily resolved.

Crystal structures in binary hard-sphere colloid-droplet mixtures with patchy cross interactions

A binary mixture of droplets and patchy colloids, where patches are arranged in tetrahedral symmetry, is studied with Metropolis Monte Carlo simulations. The colloidal patches attract droplets, while both the colloid-colloid and the droplet-droplet interactions are hard sphere like. We find stable crystal structures with atomic analogs ZnS, CaF_{2}, and fcc or hcp (face centered cubic or hexagonal close packed) of the droplets coexisting with a dispersed fluid of the colloids. The simulated crystal structures agree well with those predicted by close-packing calculations for an intermediate range of droplet-colloid size ratios. A discrepancy between the simulations and theoretical predictions occurs at low and high size ratios. The results of the simulations for mixtures with anisotropic colloid-droplet interactions reveal a richer phase diagram, with ZnS-gas and ZnS-fluid coexistence, as compared to the isotropic case. For the example of a square planar patch arrangement, we find a particular crystal structure, consisting of two interpenetrating fcc or hcp lattices with right bond angles. Such a structure has no known atomic analog. Our study of generic models of anisotropic colloid-droplet mixtures could provide a promising way towards the fabrication of novel and complex colloidal structures.

Inverse design of self-assembling colloidal crystals with omnidirectional photonic bandgaps

Open colloidal lattices possessing omnidirectional photonic bandgaps in the visible or near-visible regime are attractive optical materials the realization of which has remained elusive. We report the use of an inverse design strategy termed landscape engineering that rationally sculpts the free energy self-assembly landscape using evolutionary algorithms to discover anisotropic patchy colloids capable of spontaneously assembling pyrochlore and cubic diamond lattices possessing complete photonic bandgaps. We validate the designs in computer simulations to demonstrate the defect-free formation of these lattices via a two-stage hierarchical assembly mechanism. Our approach demonstrates a principled strategy for the inverse design of self-assembling colloids for the bottom-up fabrication of desired crystal lattices.

Preparation of colloidal molecules with temperature-tunable interactions from oppositely charged microgel spheres

The self-assembly of small colloidal clusters, so-called colloidal molecules, into crystalline materials has proven extremely challenging, the outcome often being glassy, amorphous states where positions and orientations are locked. In this paper, a new type of colloidal molecule is therefore prepared, assembled from poly(N-isopropylacrylamide) (PNIPAM)-based microgels that due to their well documented softness and temperature-response allow for greater defect tolerance compared to hard spheres and for convenient in situ tuning of size, volume fraction and inter-particle interactions with temperature. The microgels (B) are assembled by electrostatic adsorption onto oppositely charged, smaller-sized microgels (A), where the relative size of the two determines the valency (n) of the resulting core-satellite ABn-type colloidal molecules. Following assembly, a microfluidic deterministic lateral displacement (DLD) device is used to effectively isolate AB4-type colloidal molecules of tetrahedral geometry that possess a repulsive-to-attractive transition on crossing the microgels' volume phase transition temperature (VPTT). These soft, temperature-responsive colloidal molecules constitute highly promising building blocks for the preparation of new materials with emergent properties, and their optical wavelength-size makes them especially interesting for optical applications.

Assembly of clathrates from tetrahedral patchy colloids with narrow patches

Here, we revisit the assembly of colloidal tetrahedral patchy particles. Previous studies have shown that the crystallization of diamond from the fluid phase depends more critically on patch width than on the interaction range: particles with patches narrower than 40° crystallize readily and those with wide patches form disordered glass states. We find that the crystalline structure formed from the fluid also depends on the patch width. Whereas particles with intermediate patches assemble into diamond (random stacking of cubic and hexagonal diamond layers), particles with narrow patches (with width ≈20° or less) crystallize frequently into clathrates. Free energy calculations show that clathrates are never (in the pressure-temperature plane) thermodynamically more stable than diamond. The assembly of clathrate structures is thus attributed to kinetic factors that originate from the thermodynamic stabilization of pentagonal rings with respect to hexagonal ones as patches become more directional. These pentagonal rings present in the fluid phase assemble into sII clathrate or into large clusters containing 100 particles and exhibiting icosahedral symmetry. These clusters then grow by interpenetration. Still, the organization of these clusters into extended ordered structures was never observed in the simulations.

Mobile obstacles accelerate and inhibit the bundle formation in two-patch colloidal particle

Aggregation of protein into bundles is responsible for many neurodegenerative diseases. In this work, we show how two-patch colloidal particles self-assemble into chains and a sudden transition to bundles takes place by tuning the patch size and solvent condition. We study the kinetics of formation of chains, bundles, and networklike structures using patchy Brownian cluster dynamics. We also analyze the ways to inhibit and accelerate the formation of these bundles. We show that in the presence of inert immobile obstacles, the kinetics of formation of bundles slows down. However, in the presence of mobile aggregating particles, which exhibit interspecies hard sphere repulsion and intraspecies attraction, the kinetics of bundle formation accelerates slightly. We also show that if we introduce mobile obstacles, which exhibit interspecies attraction and intraspecies hard sphere repulsion, the kinetics of formation of bundles is inhibited. This is similar to the inhibitory effect of peptide P4 on the formation of insulin fibers. We are providing a model of mobile obstacles undergoing directional interactions to inhibit the formation of bundles.

Breakdown of the law of rectilinear diameter and related surprises in the liquid-vapor coexistence in systems of patchy particles

The phase diagram of molecular or colloidal systems depends strongly on the range and angular dependence of the interactions between the constituent particles. For instance, it is well known that the critical density of particles with "patchy" interactions shifts to lower values as the number of patches is decreased [see Bianchi et al. Phys. Rev. Lett. 97, 168301 (2006)]. Here, we present simulations that show that the phase behavior of patchy particles is even more interesting than had been appreciated. In particular, we find that, upon cooling below the critical point, the width of the liquid-vapor coexistence region of a system of particles with tetrahedrally arranged patches first increases, then decreases, and finally increases again. In other words, this system exhibits a doubly re-entrant liquid-vapor transition. As a consequence, the system exhibits a very large deviation from the law of rectilinear diameter, which assumes that the critical density can be obtained by linear extrapolation of the averages of the densities of the coexisting liquid and vapor phases. We argue that the unusual behavior of this system has the same origin as the density maximum in liquid water and is not captured by the Wertheim theory. The Wertheim theory also cannot account for our observation that the phase diagram of particles with three patches depends strongly on the geometrical distribution of the patches and on the degree to which their position on the particle surface is rigidly constrained. However, the phase diagram is less sensitive to small angular spreads in the patch locations. We argue that the phase behavior reported in this paper should be observable in experiments on patchy colloids and may be relevant for the liquid-liquid equilibrium in solutions of properly functionalized dendrimers.

Symmetries in hard polygon systems determine plastic colloidal crystal mesophases in two dimensions

Orientational ordering is a necessary step in the crystallization of molecules and anisotropic colloids. Plastic crystals, which are possible mesophases between the fluid and fully ordered crystal, are translationally ordered but exhibit no long range orientational order. Here, we study the two-dimensional phase behavior of hard regular polygons with edge number n = 3-12. This family of particles provides a model system to isolate the effect of shape and symmetry on the existence of plastic crystal phases. We show that the symmetry group of the particle, G, and the symmetry group of the local environment in the crystal, H, together determine plastic colloidal crystal phase behavior in two dimensions. If G contains completely the symmetry elements of H, then a plastic crystal phase is absent. If G and H share some but not all nontrivial symmetry elements, then a plastic crystal phase exists with preferred particle orientations that recover the absent symmetry elements of the crystal; we call this phase the discrete plastic crystal phase. If G and H share no nontrivial symmetry elements, then a plastic crystal phase exists without preferred orientations, which we call an indiscrete plastic crystal.

How to simulate patchy particles⋆

Patchy particles is the name given to a large class of systems of mesoscopic particles characterized by a repulsive core and a discrete number of short-range and highly directional interaction sites. Numerical simulations have contributed significantly to our understanding of the behaviour of patchy particles, but, although simple in principle, advanced simulation techniques are often required to sample the low temperatures and long time-scales associated with their self-assembly behaviour. In this work we review the most popular simulation techniques that have been used to study patchy particles, with a special focus on Monte Carlo methods. We cover many of the tools required to simulate patchy systems, from interaction potentials to biased moves, cluster moves, and free-energy methods. The review is complemented by an educationally oriented Monte Carlo computer code that implements all the techniques described in the text to simulate a well-known tetrahedral patchy particle model.

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.

Complex patchy colloids shaped from deformable seed particles through capillary interactions

We investigate the mechanisms underlying the reconfiguration of random aggregates of spheres through capillary interactions, the so-called "colloidal recycling" method, to fabricate a wide variety of patchy particles. We explore the influence of capillary forces on clusters of deformable seed particles by systematically varying the crosslink density of the spherical seeds. Spheres with a poorly crosslinked polymer network strongly deform due to capillary forces and merge into large spheres. With increasing crosslink density and therefore rigidity, the shape of the spheres is increasingly preserved during reconfiguration, yielding patchy particles of well-defined shape for up to five spheres. In particular, we find that the aspect ratio between the length and width of dumbbells, L/W, increases with the crosslink density (cd) as L/W = B - A·exp(-cd/C). For clusters consisting of more than five spheres, the particle deformability furthermore determines the patch arrangement of the resulting particles. The reconfiguration pathway of clusters of six densely or poorly crosslinked seeds leads to octahedral and polytetrahedral shaped patchy particles, respectively. For seven particles several geometries were obtained with a preference for pentagonal dipyramids by the rigid spheres, while the soft spheres do rarely arrive in these structures. Even larger clusters of over 15 particles form non-uniform often aspherical shapes. We discuss that the reconfiguration pathway is largely influenced by confinement and geometric constraints. The key factor which dominates during reconfiguration depends on the deformability of the spherical seed particles.

Novel Pyrochlorelike Crystal with a Photonic Band Gap Self-Assembled Using Colloids with a Simple Interaction Potential

Using computer simulations, we investigate the phase behavior of a system of particles interacting with a remarkably simple repulsive square-shoulder pair potential and report the formation of a novel (and stable) pyrochlorelike crystal phase. The lattice structure of the pyrochlorelike phase formed in our simulations possesses two inherent length scales corresponding to the inter- and intratetrahedral neighbors. We show that it can be used to fabricate a photonic crystal which displays complete photonic band gaps in both the direct and inverted dielectric structures.

Assembly of trivalent particles under confinement: from an exotic solid phase to a liquid phase at low temperature

Using computer simulations, we study the phase diagram of a two-dimensional system of disk particles with three patches distributed symmetrically along the particle equator. The geometry of the particles is compatible with a honey-comb lattice at moderately low temperature and pressure, whereas it is expected that the system forms a close-packed triangular lattice at high temperature and pressure. The effect of patch size within the single bond per patch regime was investigated, and it was found that the topology of the phase diagram changes drastically with patch size. Interestingly, in particles with small patches (with a half opening angle of 10°), the fluid transforms upon increasing the pressure into a rather exotic phase that can be understood as a honey-comb lattice whose voids are filled continuously with additional particles that remain, on average, unbound. Eventually, all the voids are occupied so that particles are located at the positions of a triangular lattice, but only two thirds of the particles are orientationally ordered whereas the remaining one third can rotate almost freely as in a plastic crystal. At moderately low temperature, the fluid transforms into a nearly empty honey-comb lattice, whereas at high temperature it transforms directly into the almost filled lattice. Interestingly, for particles with big patches (with a half opening angle of 20°), the honey-comb and triangular lattices are separated by a liquid phase that remains stable down to fairly low temperatures. Less surprisingly, only particles with big patches exhibit an equilibrium gas-liquid separation.

Self-assembly of two-dimensional binary quasicrystals: a possible route to a DNA quasicrystal

We use Monte Carlo simulations and free-energy techniques to show that binary solutions of penta- and hexavalent two-dimensional patchy particles can form thermodynamically stable quasicrystals even at very narrow patch widths, provided their patch interactions are chosen in an appropriate way. Such patchy particles can be thought of as a coarse-grained representation of DNA multi-arm 'star' motifs, which can be chosen to bond with one another very specifically by tuning the DNA sequences of the protruding arms. We explore several possible design strategies and conclude that DNA star tiles that are designed to interact with one another in a specific but not overly constrained way could potentially be used to construct soft quasicrystals in experiment. We verify that such star tiles can form stable dodecagonal motifs using oxDNA, a realistic coarse-grained model of DNA.

Self-Assembly of Colloidal Nanocrystals: From Intricate Structures to Functional Materials

Chemical methods developed over the past two decades enable preparation of colloidal nanocrystals with uniform size and shape. These Brownian objects readily order into superlattices. Recently, the range of accessible inorganic cores and tunable surface chemistries dramatically increased, expanding the set of nanocrystal arrangements experimentally attainable. In this review, we discuss efforts to create next-generation materials via bottom-up organization of nanocrystals with preprogrammed functionality and self-assembly instructions. This process is often driven by both interparticle interactions and the influence of the assembly environment. The introduction provides the reader with a practical overview of nanocrystal synthesis, self-assembly, and superlattice characterization. We then summarize the theory of nanocrystal interactions and examine fundamental principles governing nanocrystal self-assembly from hard and soft particle perspectives borrowed from the comparatively established fields of micrometer colloids and block copolymer assembly. We outline the extensive catalog of superlattices prepared to date using hydrocarbon-capped nanocrystals with spherical, polyhedral, rod, plate, and branched inorganic core shapes, as well as those obtained by mixing combinations thereof. We also provide an overview of structural defects in nanocrystal superlattices. We then explore the unique possibilities offered by leveraging nontraditional surface chemistries and assembly environments to control superlattice structure and produce nonbulk assemblies. We end with a discussion of the unique optical, magnetic, electronic, and catalytic properties of ordered nanocrystal superlattices, and the coming advances required to make use of this new class of solids.

Consensus on the solubility of NaCl in water from computer simulations using the chemical potential route

The solubility of NaCl in water is evaluated by using three force field models: Joung-Cheatham for NaCl dissolved in two different water models (SPC/E and TIP4P/2005) and Smith Dang NaCl model in SPC/E water. The methodology based on free-energy calculations [E. Sanz and C. Vega, J. Chem. Phys. 126, 014507 (2007)] and [J. L. Aragones et al., J. Chem. Phys. 136, 244508 (2012)] has been used, except, that all calculations for the NaCl in solution were obtained by using molecular dynamics simulations with the GROMACS package instead of homemade MC programs. We have explored new lower molalities and made longer runs to improve the accuracy of the calculations. Exploring the low molality region allowed us to obtain an analytical expression for the chemical potential of the ions in solution as a function of molality valid for a wider range of molalities, including the infinite dilute case. These new results are in better agreement with recent estimations of the solubility obtained with other methodologies. Besides, two empirical simple rules have been obtained to have a rough estimate of the solubility of a certain model, by analyzing the ionic pairs formation as a function of molality and/or by calculating the difference between the NaCl solid chemical potential and the standard chemical potential of the salt in solution.

DNA brick self-assembly with an off-lattice potential

We report Monte Carlo simulations of a simple off-lattice patchy-particle model for DNA 'bricks'. We relate the parameters that characterise this model with the binding free energy of pairs of single-stranded DNA molecules. We verify that an off-lattice potential parameterised in this way reproduces much of the behaviour seen with a simpler lattice model we introduced previously, although the relaxation of the geometric constraints leads to a more error-prone self-assembly pathway. We investigate the self-assembly process as a function of the strength of the non-specific interactions. We show that our off-lattice model for DNA bricks results in robust self-assembly into a variety of target structures.

Soft self-assembled nanoparticles with temperature-dependent properties

The fabrication of versatile building blocks that reliably self-assemble into desired ordered and disordered phases is amongst the hottest topics in contemporary materials science. To this end, microscopic units of varying complexity, aimed at assembling the target phases, have been thought, designed, investigated and built. Such a path usually requires laborious fabrication techniques, especially when specific functionalisation of the building blocks is required. Telechelic star polymers, i.e., star polymers made of a number of f di-block copolymers consisting of solvophobic and solvophilic monomers grafted on a central anchoring point, spontaneously self-assemble into soft patchy particles featuring attractive spots (patches) on the surface. Here we show that the tunability of such a system can be widely extended by controlling the physical and chemical parameters of the solution. Indeed, under fixed external conditions the self-assembly behaviour depends only on the number of arms and on the ratio of solvophobic to solvophilic monomers. However, changes in temperature and/or solvent quality make it possible to reliably change the number and size of the attractive patches. This allows the steering of the mesoscopic self-assembly behaviour without modifying the microscopic constituents. Interestingly, we also demonstrate that diverse combinations of the parameters can generate stars with the same number of patches but different radial and angular stiffness. This mechanism could provide a neat way of further fine-tuning the elastic properties of the supramolecular network without changing its topology.

A versatile model for soft patchy particles with various patch arrangements

We propose a simple and general mesoscale soft patchy particle model, which can felicitously describe the deformable and surface-anisotropic characteristics of soft patchy particles. This model can be used in dynamics simulations to investigate the aggregation behavior and mechanism of various types of soft patchy particles with tunable number, size, direction, and geometrical arrangement of the patches. To improve the computational efficiency of this mesoscale model in dynamics simulations, we give the simulation algorithm that fits the compute unified device architecture (CUDA) framework of NVIDIA graphics processing units (GPUs). The validation of the model and the performance of the simulations using GPUs are demonstrated by simulating several benchmark systems of soft patchy particles with 1 to 4 patches in a regular geometrical arrangement. Because of its simplicity and computational efficiency, the soft patchy particle model will provide a powerful tool to investigate the aggregation behavior of soft patchy particles, such as patchy micelles, patchy microgels, and patchy dendrimers, over larger spatial and temporal scales.

Controllable Synthesis and Surface Wettability of Flower-Shaped Silver Nanocube-Organosilica Hybrid Colloidal Nanoparticles

Synthesis of hybrid colloidal particles with complex and hierarchical structures is attracting much interest theoretically and technically in recent years, but still remains a tremendous challenge. Here, we present a mild and controllable wet-chemical method for the synthesis of silver nanocube (Ag NC)-organosilica hybrid particles with finely tuned numbers (with one, two, three, four, five, or six) and sizes of organosilica petals, by simply controlling the affinity with Ag NC/nature, amount, and prehydrolysis process of alkoxysilanes. The morphologies of hybrid colloidal particles have an obvious influence on the surface wettability of the hybrid particle-based films. More and larger organosilica petals can increase the surface hydrophobicity of the hybrid particle-based films.

Crystal-Templated Colloidal Clusters Exhibit Directional DNA Interactions

Spherical colloids covered with grafted DNA have been used in the directed self-assembly of a number of distinct crystal and gel structures. Simulation suggests that the use of anisotropic building blocks greatly augments the variety of potential colloidal assemblies that can be formed. Here, we form five distinct symmetries of colloidal clusters from DNA-functionalized spheres using a single type of colloidal crystal as a template. The crystals are formed by simple sedimentation of a binary mixture containing a majority "host" species that forms close-packed crystals with the minority "impurity" species occupying substitutional or interstitial defect sites. After the DNA strands between the two species are hybridized and enzymatically ligated, the results are colloidal clusters, one for each impurity particle, with a symmetry determined by the nearest neighbors in the original crystal template. By adjusting the size ratio of the two spheres and the timing of the ligation, we are able to generate clusters having the symmetry of tetrahedra, octahedra, cuboctahedra, triangular orthobicupola, and icosahedra, which can be readily separated from defective clusters and leftover spheres by centrifugation. We further demonstrate that these clusters, which are uniformly covered in DNA strands, display directional binding with spheres bearing complementary DNA strands, acting in a manner similar to patchy particles or proteins having multiple binding sites. The scalable nature of the fabrication process, along with the reprogrammability and directional nature of their resulting DNA interactions, makes these clusters suitable building blocks for use in further rounds of directed self-assembly.

Chromatic patchy particles: Effects of specific interactions on liquid structure

We study the structural and thermodynamic properties of patchy particle liquids, with a special focus on the role of "color," i.e., specific interactions between individual patches. A possible experimental realization of such "chromatic" interactions is by decorating the particle patches with single-stranded DNA linkers. The complementarity of the linkers can promote selective bond formation between predetermined pairs of patches. By using MD simulations, we compare the local connectivity, the bond orientation order, and other structural properties of the aggregates formed by the "colored" and "colorless" systems. The analysis is done for spherical particles with two different patch arrangements (tetrahedral and cubic). It is found that the aggregated (liquid) phase of the "colorless" patchy particles is better connected, denser and typically has stronger local order than the corresponding "colored" one. This, in turn, makes the colored liquid less stable thermodynamically. Specifically, we predict that in a typical case the chromatic interactions should increase the relative stability of the crystalline phase with respect to the disordered liquid, thus expanding its region in the phase diagram.

Phase behaviour of inverse patchy colloids: effect of the model parameters

The phase behaviour of inverse patchy colloid systems composed of spherical particles with two oppositely charged patches at the poles is investigated by simulation-based thermodynamic integration schemes. The interaction between the particles is derived via a coarse-grained model characterized by three system parameters: the charge imbalance between the bare colloid and the patches, the patch surface extension and the particle interaction range. Starting from a set of parameters for which a stacking of parallel layers is thermodynamically stable, the effect of each of these three parameters on the phase diagram is studied. Our results show that the region of stability of the layered solid phase can be expanded by increasing the charge imbalance and/or by reducing the interaction range. A larger patch size, on the other hand, stabilizes the layered structure with respect to the competing face centered cubic solid at high pressures but destabilizes it with respect to the fluid phase at low pressures. The location of the liquid-vapour critical point in the temperature versus density plane is also investigated: while the charge imbalance and the patch size affect mainly the critical density, a change of the interaction range has a substantial impact also on the critical temperature.

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

Colloidal self-assembly has enormous potential as a bottom-up means of structure fabrication. Here we demonstrate hierarchical self-assembly of rationally designed charge-stabilised colloidal magnetic particles into ground state structures that are topologically equivalent to a snub cube and a snub dodecahedron, the only two chiral Archimedean solids, for size-selected clusters. These spherical structures open up in response to an external magnetic field and demonstrate controllable porosity. Such features are critical to their applications as functional materials.

Cooperative polymerization of one-patch colloids

We numerically investigate cooperative polymerization in an off-lattice model based on a pairwise additive potential using particles with a single attractive patch that covers 30% of the colloid surface. Upon cooling, these particles self-assemble into small clusters which, below a density-dependent temperature, spontaneously reorganize into long straight tubes. We evaluate the partition functions of clusters of all sizes to provide an accurate description of the chemical reaction constants governing this process. Our calculations show that, for intermediate sizes, the partition functions retain contributions from two different structures, differing in both energy and entropy. We illustrate the microscopic mechanism behind the complex polymerization process in this system and provide a detailed evaluation of its thermodynamics.

Soft-patchy nanoparticles: modeling and self-organization

We consider a novel class of patchy particles inspired by polymer-based complex units where the limited valence in bonding is accompanied by soft interactions and incessant fluctuations of the patch positions, possibly leading to reversible modifications of the patch number and size. We introduce a simple model that takes into account the aforementioned features and we focus on the role played by the patch flexibility on the self-organization of our patchy units in the bulk, with particular attention to the connectivity properties and the morphology of the aggregated networks.

"Crystal-clear" liquid-liquid transition in a tetrahedral fluid

For a model known to exhibit liquid-liquid transitions, we examine how varying the bond orientational flexibility affects the stability of the liquid-liquid transition relative to that of the crystal phases. For very rigidly oriented bonds, the crystal is favored over all amorphous phase transitions. We find that increasing the bond flexibility decreases both the critical temperature Tc for liquid-liquid phase separation and the melting temperature Tm. The effect of increasing flexibility is much stronger for melting, so that the distance between Tc and Tm progressively reduces and inverts sign. Under these conditions, a "naked" liquid-liquid critical point bulges out in the liquid phase and becomes accessible, without the possibility of crystallization. These results confirm that a crystal-clear, liquid-liquid transition can occur as a genuine, thermodynamically stable phenomenon for tetrahedral coordinated particles with flexible bond orientation, but that such a transition is hidden by crystallization when bonds are highly directional.

Phase diagram of inverse patchy colloids assembling into an equilibrium laminar phase

We numerically study the phase behavior of colloidal particles with two charged patches at the poles and an oppositely charged equatorial belt. Interactions between particles are described using the inverse patchy colloid model, where the term inverse emphasizes the difference with respect to conventional patchy particles: as a consequence of the heterogeneous charge distribution, the patches on the particle surface repel each other, whereas the patches and non-patch regions mutually attract. For the model parameters considered in this work, the system exhibits an unusual equilibrium phase diagram characterized by a broad region where a novel structure composed of parallel colloidal monolayers is stable.