Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

Curvature-guided depletion stabilizes Kagome superlattices of nanocrystals

Shape-anisotropic nanocrystals and patchy particles have been explored to construct complex superstructures, but most studies have focused on convex shapes. We report that nonconvex, dumbbell-shaped nanocrystals (nanodumbbells) exhibit globally interlocking self-assembly behaviors governed by curvature-guided depletion interactions. By tailoring the local curvature of nanodumbbells, we can precisely and flexibly adjust particle bonding directionality, a level of control rarely achievable with conventional convex building blocks. These nanodumbbells can undergo long-range ordered assembly into various intricate two-dimensional superlattices, including the chiral Kagome lattice. Theoretical calculations reveal that the Kagome lattice is a thermodynamically stable phase, with depletion interactions playing a crucial role in stabilizing these non-close-packed structures. The emergence of Kagome lattices and other unusual structures highlights the vast potential of nonconvex nanocrystals for creating sophisticated architectures.

Nanocrystal Assemblies: Current Advances and Open Problems

We explore the potential of nanocrystals (a term used equivalently to nanoparticles) as building blocks for nanomaterials, and the current advances and open challenges for fundamental science developments and applications. Nanocrystal assemblies are inherently multiscale, and the generation of revolutionary material properties requires a precise understanding of the relationship between structure and function, the former being determined by classical effects and the latter often by quantum effects. With an emphasis on theory and computation, we discuss challenges that hamper current assembly strategies and to what extent nanocrystal assemblies represent thermodynamic equilibrium or kinetically trapped metastable states. We also examine dynamic effects and optimization of assembly protocols. Finally, we discuss promising material functions and examples of their realization with nanocrystal assemblies.

Depletion-induced crystallization of anisotropic triblock colloids

The intricate interplay between colloidal particle shape and precisely engineered interaction potentials has paved the way for the discovery of unprecedented crystal structures in both two and three dimensions. Here, we make use of anisotropic triblock colloidal particles composed of two distinct materials. The resulting surface charge heterogeneity can be exploited to generate regioselective depletion interactions and directional bonding. Using extensive molecular dynamics simulations and a dimensionality reduction analysis approach, we map out state diagrams for the self-assembly of such colloids as a function of their aspect ratio and for varying depletant features in a quasi two-dimensional set-up. We observe the formation of a wide variety of crystal structures such as a herringbone, brick-wall, tilted brick-wall, and (tilted) ladder-like structures. More specifically, we determine the optimal parameters to enhance crystallization, and investigate the nucleation process. Additionally, we explore the potential of using crystalline monolayers as templates for deposition, thereby creating complex three-dimensional structures that hold promise for future applications.

Shape-induced crystallization of binary DNA-functionalized nanocubes

Leveraging the anisotropic shape of DNA-functionalized nanoparticles holds potential for shape-directed crystallization of a wide collection of superlattice structures. Using coarse-grained molecular dynamics simulations, we study the self-assembly of a binary mixture of cubic gold nanoparticles, which are functionalized by complementary DNA strands. We observe the spontaneous self-assembly of simple cubic (SC), plastic body-centered tetragonal (pBCT), and compositionally disordered plastic body-centered tetragonal (d-pBCT) phases due to hybridization of the DNA strands. We systematically investigate the effect of length, grafting density, as well as rigidity of the DNA strands on the self-assembly behavior of cubic nanoparticles. We measure the potential of mean force between DNA-functionalized nanocubes for varying rigidity of the DNA strands and DNA lengths. Using free-energy calculations, we find that longer and flexible DNA strands can lead to a phase transformation from SC to the pBCT phase due to a gain in entropy arising from the orientational degrees of freedom of the nanocubes in the pBCT phase. Our results may serve as a guide for self-assembly experiments on DNA-functionalized cubic nanoparticles.

Configurational entropy of colloidal particles in a confined space

We calculate the configurational entropy of colloidal particles in a confined geometry interacting as hard disks using Monte Carlo integration. In particular, we consider systems with three kinds of boundary conditions: hard, periodic, and spherical. For small to moderate packing fraction ϕ values, we find the entropies per particle for systems with the periodic and the spherical boundary conditions tend to reach the same value with the increase of the particle number N, while that for the system with the hard boundary conditions still has obvious differences compared to them within the studied N range. Surprisingly, despite the small system sizes, the estimated entropies per particle at infinite system size from extrapolations in the periodic and spherical systems are in reasonable agreement with that calculated using thermodynamic integration. Besides, as N increases we find the pair correlation function begins to exhibit similar features as that of a thermally equilibrated hard-disk fluid at the same packing fraction. Our findings may contribute to a better understanding of how the configurational entropy changes with the system size and the influence of boundary conditions, and provide insights relevant to engineering particles in confined spaces.

Double-Lattice Packing of Pentagonal Gold Bipyramids in Supercrystals with Triclinic Symmetry

Pentagonal packing is a long-standing issue and a rich mathematical topic, brought to the fore by recent progress in nanoparticle design. Gold pentagonal bipyramids combine fivefold symmetry and anisotropy and their section varies along the length. In this work, colloidal supercrystals of pentagonal gold bipyramids are obtained in a compact arrangement that generalizes the optimal packing of regular pentagons in the plane. Multimodal investigations reveal a two-particle unit cell with triclinic symmetry, a lower symmetry than that of the building blocks. Monte Carlo computer simulations show that this lattice achieves the densest possible packing. Going beyond pentagons, further simulations show an odd-even effect of the number of sides on the packing: odd-sided bipyramids are non-centrosymmetric and require the double-lattice arrangement to recover inversion symmetry. The supercrystals display a facet-dependent optical response that is promising for sensing, metamaterials applications, and for fundamental studies of self-assembly processes.

Mesoscopic Modeling of a Highly-Ordered Sanidic Polymer Mesophase and Comparison With Experimental Data

Board-shaped polymers form sanidic mesophases: assemblies of parallel lamellae of stacked polymer backbones separated by disordered side chains. Sanidics vary significantly with respect to polymer order inside their lamellae, making them “stepping stones” toward the crystalline state. Therefore, they are potentially interesting for studying crystallization and technological applications. Building on earlier mesoscopic models of the most disordered sanidics Σ_d, we focus on the other extreme, near-crystalline order, and develop a generic model that captures a highly ordered Σ_r mesophase. Polymers are described by generic hindered-rotation chains. Anisotropic nonbonded potentials, with strengths comparable to the thermal energy, mimic board-like monomer shapes. Lamellae equilibrated with Monte Carlo simulations, for a broad range of model parameters, have intralamellar order typical for Σ_r mesophases: periodically stacked polymers that are mutually registered along their backbones. Our mesophase shows registration on both monomer and chain levels. We calculate scattering patterns and compare with data published for highly ordered sanidic mesophases of two different polymers: polyesters and polypeptoids. Most of the generic structural features that were identified in these experiments are present in our model. However, our mesophase has correlations between chains located in different lamellae and is therefore closer to the crystalline state than the experimental samples.

Re-entrant transition as a bridge of broken ergodicity in confined monolayers of hexagonal prisms and cylinders

The entropy-driven monolayer assembly of hexagonal prisms and cylinders was studied under hard slit confinement. At the conditions investigated, the particles have two distinct and dynamically disconnected rotational states: unflipped and flipped, depending on whether their circular/hexagonal face is parallel or perpendicular to the wall plane. Importantly, these two rotational states cast distinct projection areas over the wall plane that favor either hexagonal or tetragonal packing. Monte Carlo simulations revealed a re-entrant melting transition where an intervening disordered Flipped-Unflipped (FUN) phase is sandwiched between a fourfold tetratic phase at high concentrations and a sixfold triangular solid at intermediate concentrations. The FUN phase contains a mixture of flipped and unflipped particles and is translationally and orientationally disordered. Complementary experiments were conducted with photolithographically fabricated cylindrical microparticles confined in a wedge cell. Both simulations and experiments show the formation of phases with comparable fraction of flipped particles and structure, i.e., the FUN phase, triangular solid, and tetratic phase, indicating that both approaches sample analogous basins of particle-orientation phase-space. The phase behavior of hexagonal prisms in a soft-repulsive wall model was also investigated to exemplify how tunable particle-wall interactions can provide an experimentally viable strategy to dynamically bridge the flipped and unflipped states.

From predictive modelling to machine learning and reverse engineering of colloidal self-assembly

An overwhelming diversity of colloidal building blocks with distinct sizes, materials and tunable interaction potentials are now available for colloidal self-assembly. The application space for materials composed of these building blocks is vast. To make progress in the rational design of new self-assembled materials, it is desirable to guide the experimental synthesis efforts by computational modelling. Here, we discuss computer simulation methods and strategies used for the design of soft materials created through bottom-up self-assembly of colloids and nanoparticles. We describe simulation techniques for investigating the self-assembly behaviour of colloidal suspensions, including crystal structure prediction methods, phase diagram calculations and enhanced sampling techniques, as well as their limitations. We also discuss the recent surge of interest in machine learning and reverse-engineering methods. Although their implementation in the colloidal realm is still in its infancy, we anticipate that these data-science tools offer new paradigms in understanding, predicting and (inverse) design of novel colloidal materials.

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.

Classifying Crystals of Rounded Tetrahedra and Determining Their Order Parameters Using Dimensionality Reduction

Using simulations we study the phase behavior of a family of hard spherotetrahedra, a shape that interpolates between tetrahedra and spheres. We identify 13 close-packed structures, some with packings that are significantly denser than previously reported. Twelve of these are crystals with unit cells of N = 2 or N = 4 particles, but in the shape regime of slightly rounded tetrahedra we find that the densest structure is a quasicrystal approximant with a unit cell of N = 82 particles. All 13 structures are also stable below close packing, together with an additional 14th plastic crystal phase at the sphere side of the phase diagram, and upon sufficient dilution to packing fractions below 50-60% all structures melt. Interestingly, however, upon compressing the fluid phase, self-assembly takes place spontaneously only at the tetrahedron and the sphere side of the family but not in an intermediate regime of tetrahedra with rounded edges. We describe the local environment of each particle by a set of l-fold bond orientational order parameters q̅l, which we use in an extensive principal component analysis. We find that the total packing fraction as well as several particular linear combinations of q̅l rather than individual q̅l's are optimally distinctive, specifically the differences q̅4 - q̅6 for separating tetragonal from hexagonal structures and q̅4-q̅8 for distinguishing tetragonal structures. We argue that these characteristic combinations are also useful as reliable order parameters in nucleation studies, enhanced sampling techniques, or inverse-design methods involving odd-shaped particles in general.

Effect of Patch Area and Interaction Length on Clusters and Structures Formed by One-Patch Particles in Thin Systems

Assuming that the interaction between particles is given by the Kern-Frenkel potential, Monte Carlo simulations are performed to study the clusters and structures formed by one-patch particles in a thin space between two parallel walls. In isothermal-isochoric systems with a short interaction length, tetrahedral tetramers, octahedral hexamers, and pentagonal dipyramidal heptamers are created with increasing patch area. In isothermal-isobaric systems, the double layers of a triangular lattice, which is the (111) face of the face-centered cubic (fcc) lattice, form when the pressure is high. For a long interaction length, a different type of cluster, trigonal prismatic hexamers, is created. The structures in the double layers also changed as follows: a simple hexagonal lattice or square lattice, which is the (100) face of the fcc structure, is created in isothermal-isobaric systems.

Solvent Effects on Catechol Crystal Habits and Aspect Ratios: A Combination of Experiments and Molecular Dynamics Simulation Study

This work could help to better understand the solvent effects on crystal habits and aspect ratio changes at the molecular level, which provide some guidance for solvent selection in industrial crystallization processes. With the catechol crystal habits acquired using both experimental and simulation methods in isopropanol, methyl acetate and ethyl acetate, solvent effects on crystal morphology were explored based on the modified attachment energy model. Firstly, morphologically dominant crystal faces were obtained with the predicted crystal habit in vacuum. Then, modified attachment energies were calculated by the molecular dynamics simulation to modify the crystal shapes in a real solvent environment, and the simulation results were in agreement with the experimental ones. Meanwhile, the surface properties such as roughness and the diffusion coefficient were introduced to analyze the solvent adsorption behaviors and the radial distribution function curves were generated to distinguish diverse types of interactions like hydrogen bonds and van der Waals forces. Results show that the catechol crystal habits were affected by the combination of the attachment energy, surface structures and molecular interaction types. Moreover, the changing aspect ratios of catechol crystals are closely related to the existence of hydrogen bonds which contribute to growth inhibition on specific faces.

FCC ↔ BCC Phase Transitions in Convex and Concave Hard Particle Systems

Solid-solid transitions are ubiquitous in nature and are important for technology. Understanding and exploiting transitions are complicated by the fact that multiple transition pathways can exist between small unit cell structures such as face-centered cubic (FCC) and body-centered cubic (BCC). By symmetry, FCC ↔ BCC transitions can occur via a pair of continuous transitions or via a discontinuous, first-order transition. However, how to, or whether it is possible to, select between pathways is unclear. Here, we use particle shape change to induce FCC ↔ BCC transitions in systems where particle valence is malleable. Though some particle shapes can eliminate metastable HCP stacking faults, we find that for both convex and concave particles, transitions are first-order.

Using symmetry to elucidate the importance of stoichiometry in colloidal crystal assembly

We demonstrate a method based on symmetry to predict the structure of self-assembling, multicomponent colloidal mixtures. This method allows us to feasibly enumerate candidate structures from all symmetry groups and is many orders of magnitude more computationally efficient than combinatorial enumeration of these candidates. In turn, this permits us to compute ground-state phase diagrams for multicomponent systems. While tuning the interparticle potentials to produce potentially complex interactions represents the conventional route to designing exotic lattices, we use this scheme to demonstrate that simple potentials can also give rise to such structures which are thermodynamically stable at moderate to low temperatures. Furthermore, for a model two-dimensional colloidal system, we illustrate that lattices forming a complete set of 2-, 3-, 4-, and 6-fold rotational symmetries can be rationally designed from certain systems by tuning the mixture composition alone, demonstrating that stoichiometric control can be a tool as powerful as directly tuning the interparticle potentials themselves.

Generic Model for Lamellar Self-Assembly in Conjugated Polymers: Linking Mesoscopic Morphology and Charge Transport in P3HT

We develop a generic coarse-grained model of soluble conjugated polymers, capable of describing their self-assembly into a lamellar mesophase. Polymer chains are described by a hindered-rotation model, where interaction centers represent entire repeat units, including side chains. We introduce soft anisotropic nonbonded interactions to mimic the potential of mean force between atomistic repeat units. The functional form of this potential reflects the symmetry of the molecular order in a lamellar mesophase. The model can generate both nematic and lamellar (sanidic smectic) molecular arrangements. We parametrize this model for a soluble conjugated polymer poly(3-hexylthiophene) (P3HT) and demonstrate that the simulated lamellar mesophase matches morphologies of low molecular weight P3HT, experimentally observed at elevated temperatures. A qualitative charge-transport model allows us to link local chain conformations and mesoscale order to charge transport. In particular, it shows how coarsening of lamellar domains and chain extension increase the charge carrier mobility. By modeling large systems and long chains, we can capture transport between lamellar layers, which is due to rare, but thermodynamically allowed, backbone bridges between neighboring layers.

A simulation study on the phase behavior of hard rhombic platelets

Using Monte Carlo simulations, we investigate the phase behavior of hard rhombic platelets as a function of the thickness of the platelets, T. The phase diagram displays a columnar phase and a crystal phase in which the platelets are stacked in columns that are arranged in a two-dimensional lattice. We find that the shape of the platelets determines the symmetry of the two-dimensional lattice, i.e., rhombic platelets form an oblique columnar phase and a simple monoclinic crystal phase. For sufficiently thick platelets, i.e., for a thickness-to-length ratio T/L>0.17, we find only an isotropic fluid, an oblique columnar phase, and a monoclinic crystal phase. Surprisingly, for an intermediate plate thickness, 0.083<T/L<0.17, we also find a region in between the isotropic (or nematic) phase and the columnar phase, where the smectic phase is stable. For sufficiently thin platelets, T/L<0.13, the phase diagram displays a nematic phase. With the exception of the smectic phase, our results resemble the phase behavior of discotic particles. Our results may guide the synthesis and future experiments on rhombic nanoplatelets.

Water-like anomalies as a function of tetrahedrality

Water is the most common and yet least understood material on Earth. Despite its simplicity, water tends to form tetrahedral order locally by directional hydrogen bonding. This structuring is known to be responsible for a vast array of unusual properties, e.g., the density maximum at 4 °C, which play a fundamental role in countless natural and technological processes, with the Earth’s climate being one of the most important examples. By systematically tuning the degree of tetrahedrality, we succeed in continuously interpolating between water-like behavior and simple liquid-like behavior. Our approach reveals what physical factors make water so anomalous and special even compared with other tetrahedral liquids.

Tetrahedral interactions describe the behavior of the most abundant and technologically important materials on Earth, such as water, silicon, carbon, germanium, and countless others. Despite their differences, these materials share unique common physical behaviors, such as liquid anomalies, open crystalline structures, and extremely poor glass-forming ability at ambient pressure. To reveal the physical origin of these anomalies and their link to the shape of the phase diagram, we systematically study the properties of the Stillinger–Weber potential as a function of the strength of the tetrahedral interaction λ. We uncover a unique transition to a reentrant spinodal line at low values of λ, accompanied with a change in the dynamical behavior, from non-Arrhenius to Arrhenius. We then show that a two-state model can provide a comprehensive understanding on how the thermodynamic and dynamic anomalies of this important class of materials depend on the strength of the tetrahedral interaction. Our work establishes a deep link between the shape of the phase diagram and the thermodynamic and dynamic properties through local structural ordering in liquids and hints at why water is so special among all substances.

Low-Temperature Crystal Structures of the Hard Core Square Shoulder Model

In many cases, the stability of complex structures in colloidal systems is enhanced by a competition between different length scales. Inspired by recent experiments on nanoparticles coated with polymers, we use Monte Carlo simulations to explore the types of crystal structures that can form in a simple hard-core square shoulder model that explicitly incorporates two favored distances between the particles. To this end, we combine Monte Carlo-based crystal structure finding algorithms with free energies obtained using a mean-field cell theory approach, and draw phase diagrams for two different values of the square shoulder width as a function of the density and temperature. Moreover, we map out the zero-temperature phase diagram for a broad range of shoulder widths. Our results show the stability of a rich variety of crystal phases, such as body-centered orthogonal (BCO) lattices not previously considered for the square shoulder model.

Phase behaviour and gravity-directed self assembly of hard convex spherical caps

We investigate the phase behaviour and self-assembly of convex spherical caps using Monte Carlo simulations. This model is used to represent the main features observed in experimental colloidal particles with mushroom-cap shape [Riley et al., Langmuir, 2010, 26, 1648]. The geometry of this non-centrosymmetric convex model is fully characterized by the aspect ratio χ* defined as the spherical cap height to diameter ratio. We use NPT Monte Carlo simulations combined with free energy calculations to determine the most stable crystal structures and the phase behaviour of convex spherical caps with different aspect ratios. We find a variety of crystal structures at each aspect ratio, including plastic and dimer-based crystals; small differences in chemical potential between the structures with similar morphology suggest that convex spherical caps have the tendency to form polycrystalline phases rather than crystallising into a single uniform structure. With the exception of plastic crystals observed at large aspect ratios (χ* > 0.75), crystallisation kinetics seem to be too slow, hindering the spontaneous formation of ordered structures. As an alternative, we also present a study of directing the self-assembly of convex spherical caps via sedimentation onto solid substrates. This study contributes to show how small changes to particle shape can significantly alter the self-assembly of crystal structures, and how a simple gravity field and a template can substantially enhance the process.

The stabilization of tubular crystals in mixtures of spherical particles

Novel crystal structures in binary atomic mixtures arise when the attractive well is wide enough to allow double occupancy by small particles. The resulting crystals consist of ordered packings of self assembled linear structures comprised of a cylindrical tube of large particles enclosing a close packed core of small particles that corresponds to a stacking of overlapping icosahedra. We show that the stability of these structures depends on two essential features of the spherically symmetric pairwise interactions: (i) a radius ratio between 0.414 and 0.588, and (ii) a width w of the attractive well in the interaction between unlike particles that satisfies w > σ_SS where σ_SS is the diameter of the small particle.

A possible four-phase coexistence in a single-component system

For different phases to coexist in equilibrium at constant temperature T and pressure P, the condition of equal chemical potential μ must be satisfied. This condition dictates that, for a single-component system, the maximum number of phases that can coexist is three. Historically this is known as the Gibbs phase rule, and is one of the oldest and venerable rules of thermodynamics. Here we make use of the fact that, by varying model parameters, the Gibbs phase rule can be generalized so that four phases can coexist even in single-component systems. To systematically search for the quadruple point, we use a monoatomic system interacting with a Stillinger-Weber potential with variable tetrahedrality. Our study indicates that the quadruple point provides flexibility in controlling multiple equilibrium phases and may be realized in systems with tunable interactions, which are nowadays feasible in several soft matter systems such as patchy colloids.

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.

Tic-Tac-Toe Binary Lattices from the Interfacial Self-Assembly of Branched and Spherical Nanocrystals

The self-organization of nanocrystals has proven to be a versatile route to achieve increasingly sophisticated structures of materials, where the shape and properties of individual particles impact the final functionalities. Recent works have addressed this topic by combining various shapes to achieve more complex arrangements of particles than are possible in single-component samples. However, the ability to create intricate architectures over large regions by exploiting the shape of multiply branched nanocrystals to host a second component remains unexplored. Here, we show how the concave shape of a branched nanocrystal, the so-called octapod, is able to anchor a sphere. The two components self-assemble into a locally ordered monolayer consisting of an intercalated square lattice of octapods and spheres, which is reminiscent of the "tic-tac-toe" game. These tic-tac-toe domains form through an interfacial self-assembly that occurs by the dewetting of a hexane layer containing both particle types. By varying the experimental conditions and performing molecular dynamics simulations, we show that the ligands coating the octapods are crucial to the formation of this structure. We find that the tendency of an octapod to form an interlocking-type structure with a second octapod strongly depends on the ligand shell of the pods. Breaking this tendency by ligand exchange allows the octapods to assemble into a more relaxed configuration, which is able to form a lock-and-key-type structure with a sphere, when they have a suitable size ratio. Our findings provide an example of a more versatile use of branched nanocrystals in self-assembled functional materials.

Nested sampling of isobaric phase space for the direct evaluation of the isothermal-isobaric partition function of atomic systems

Nested Sampling (NS) is a powerful athermal statistical mechanical sampling technique that directly calculates the partition function, and hence gives access to all thermodynamic quantities in absolute terms, including absolute free energies and absolute entropies. NS has been used predominately to compute the canonical (NVT) partition function. Although NS has recently been used to obtain the isothermal-isobaric (NPT) partition function of the hard sphere model, a general approach to the computation of the NPT partition function has yet to be developed. Here, we describe an isobaric NS (IBNS) method which allows for the computation of the NPT partition function of any atomic system. We demonstrate IBNS on two finite Lennard-Jones systems and confirm the results through comparison to parallel tempering Monte Carlo. Temperature-entropy plots are constructed as well as a simple pressure-temperature phase diagram for each system. We further demonstrate IBNS by computing part of the pressure-temperature phase diagram of a Lennard-Jones system under periodic boundary conditions.

Explicit excluded volume of cylindrically symmetric convex bodies

We represent explicitly the excluded volume V(e){B(1),B(2)} of two generic cylindrically symmetric, convex rigid bodies, B(1) and B(2), in terms of a family of shape functionals evaluated separately on B(1) and B(2). We show that V(e){B(1),B(2)} fails systematically to feature a dipolar component, thus making illusory the assignment of any shape dipole to a tapered body in this class. The method proposed here is applied to cones and validated by a shape-reconstruction algorithm. It is further applied to spheroids (ellipsoids of revolution), for which it shows how some analytic estimates already regarded as classics should indeed be emended.

Glassy dynamics of convex polyhedra

Self-assembly of polyhedral-shaped particles has attracted huge interest with the advent of new synthesis methods that realize these faceted particles in the lab. Recent studies have shown that polyhedral-shaped particles exhibit a rich phase behavior by excluded volume interactions alone; some of these particles are even alleged to show a transition to a glass phase by quenching the liquid sufficiently fast beyond the glass transition (supercooling), such that the formation of structures with long-range order is suppressed. Despite the recent progress, no study has been made on the glass formation of polyhedral-shaped particles. Here, we study the glass behavior of polyhedral particles using advanced Monte Carlo methods. We investigate the formation of a glass of monodisperse hard polyhedral-shaped particles, namely, octahedra, tetrahedra, and triangular cupola, using simulations. Finally, the fragility of these particles is determined and compared to that of a polydisperse hard-sphere system.

Steric self-assembly of laterally confined organic semiconductor molecule analogues

Self-assembly of planar molecules can be a critical route to control morphology in organic optoelectronic systems. In this study, Monte Carlo simulations were performed with polygonal disc analogues to planar semiconducting molecules under confinement. By examining statistically the molecular density and configurations of such analogues, we have observed that the symmetry of the confining medium can have a greater impact on the final densified particle configurations than the intramolecular interactions. Using the steric frustration imparted by confinement, novel self-assembled (partially) ordered phases are available. Our Monte Carlo simulations suggest new avenues to control ordering and morphology of planar molecules, which are critical for high-performance organic optoelectronic devices.

Shaping the synthesis and assembly of symmetrically stellated Au/Pd nanocrystals with aromatic additives

Au/Pd octopods were synthesized with enhanced sample homogeneity through the use of aromatic additives. This increase in sample monodispersity facilitates large-area periodic assembly of stellated metal nanostructures for the first time. The aromatic additives were also found to influence the structures of the stellated nanocrystals with subtle shape modifications observed that can alter the packing arrangement of the Au/Pd octopods. These results indicate the possibility of tailored assembly of stellated nanostructures, which would be useful for optical applications that require strong and predictable coupling between plasmonic building blocks.

Fundamental challenges in packing problems: from spherical to non-spherical particles

Random packings of objects of a particular shape are ubiquitous in science and engineering. However, such jammed matter states have eluded any systematic theoretical treatment due to the strong positional and orientational correlations involved. In recent years progress on a fundamental description of jammed matter could be made by starting from a constant volume ensemble in the spirit of conventional statistical mechanics. Recent work has shown that this approach, first introduced by S. F. Edwards more than two decades ago, can be cast into a predictive framework to calculate the packing fractions of both spherical and non-spherical particles.

Self-assembly of colloidal hexagonal bipyramid- and bifrustum-shaped ZnS nanocrystals into two-dimensional superstructures

We present a combined experimental, theoretical, and simulation study on the self-assembly of colloidal hexagonal bipyramid- and hexagonal bifrustum-shaped ZnS nanocrystals (NCs) into two-dimensional superlattices. The simulated NC superstructures are in good agreement with the experimental ones. This shows that the self-assembly process is primarily driven by minimization of the interfacial free-energies and maximization of the packing density. Our study shows that a small truncation of the hexagonal bipyramids is sufficient to change the symmetry of the resulting superlattice from hexagonal to tetragonal, highlighting the crucial importance of precise shape control in the fabrication of functional metamaterials by self-assembly of colloidal NCs.

Phase diagram of one-patch colloids forming tubes and lamellae

We numerically calculate the equilibrium phase diagram of one-patch particles with 30% patch coverage. It has been previously shown that in the fluid phase these particles organize into extremely long tubelike aggregates (G. Munaò et al. Soft Matter 2013, 9, 2652). Here, we demonstrate by means of free-energy calculations that such a disordered tube phase, despite forming spontaneously from the fluid phase below a density-dependent temperature, is always metastable against a lamellar crystal. We also show that a crystal of infinitely long packed tubes is thermodynamically stable, but only at high pressure. The full phase diagram of the model, beside the fluid phase, displays four different stable crystals. A gas-liquid critical point, and hence a liquid phase, is not detected.

Phase diagram and structural diversity of a family of truncated cubes: degenerate close-packed structures and vacancy-rich states

Using Monte Carlo simulations and free-energy calculations, we determine the phase diagram of a family of truncated hard cubes, where the shape evolves smoothly from a cube via a cuboctahedron to an octahedron. A remarkable diversity in crystal phases and close-packed structures is found, including a fully degenerate crystal structure, several plastic crystals, as well as vacancy-stabilized crystal phases, all depending sensitively on the precise particle shape. Our results illustrate the intricate relation between phase behavior and building-block shape, and can guide future experimental studies on polyhedral-shaped nanoparticles.