On the formation of stripe, sigma, and honeycomb phases in a core-corona system

Fingerprints of ordered self-assembled structures in the liquid phase of a hard-core, square-shoulder system

We investigate the phase ordering (pattern formation) of systems of two-dimensional core-shell particles using Monte Carlo (MC) computer simulations and classical density functional theory (DFT). The particles interact via a pair potential having a hard core and a repulsive square shoulder. Our simulations show that on cooling, the liquid state structure becomes increasingly characterized by long wavelength density modulations and on further cooling forms a variety of other phases, including clustered, striped, and other patterned phases. In DFT, the hard core part of the potential is treated using either fundamental measure theory or a simple local density approximation, whereas the soft shoulder is treated using the random phase approximation. The different DFTs are benchmarked using large-scale grand-canonical-MC and Gibbs-ensemble-MC simulations, demonstrating their predictive capabilities and shortcomings. We find that having the liquid state static structure factor S(k) for wavenumber k is sufficient to identify the Fourier modes governing both the liquid and solid phases. This allows us to identify from easier-to-obtain liquid state data the wavenumbers relevant to the periodic phases and to predict roughly where in the phase diagram these patterned phases arise.

A density functional theory and simulation study of stripe phases in symmetric colloidal mixtures

In a binary mixture, stripes refer to a one-dimensional periodicity of the composition, namely, a regular alternation of layers filled with particles of mostly one species. We have recently introduced [Munaò et al., Phys. Chem. Chem. Phys. 25, 16227 (2023)] a model that possibly provides the simplest binary mixture endowed with stripe order. The model consists of two species of identical hard spheres with equal concentration, which mutually interact through a square-well potential. In that paper, we have numerically shown that stripes are present in both liquid and solid phases when the attraction range is rather long. Here, we study the phase behavior of the model in terms of a density functional theory capable to account for the existence of stripes in the dense mixture. Our theory is accurate in reproducing the phases of the model, at least insofar as the composition inhomogeneities occur on length scales quite larger than the particle size. Then, using Monte Carlo simulations, we prove the existence of solid stripes even when the square well is much thinner than the particle diameter, making our model more similar to a real colloidal mixture. Finally, when the width of the attractive well is equal to the particle diameter, we observe a different and more complex form of compositional order in the solid, where each species of particle forms a regular porous matrix holding in its holes the other species, witnessing a surprising variety of emergent behaviors for a very basic model of interaction.

Structures Formed by Particles with Shoulderlike Repulsive Interaction in Thin Systems

When particles are constructed in thin systems between two parallel flat walls, structures that are not observed in bulk systems are created and the created structures change, depending on the width between the walls. In this study, the structures formed by particles constructed in thin systems were investigated through performing isothermal-isobaric Monte Carlo simulations, where the interaction between the particles is given by the hard-core square shoulder potential. By controlling the width of the shoulder-like repulsive interaction and the system width, several novel structures such as the connection of rhombuses and the square lattice of the (100) face of the body-centered cubic lattice were created.

Like aggregation from unlike attraction: stripes in symmetric mixtures of cross-attracting hard spheres

Self-assembly of colloidal particles into striped phases is at once a process of relevant technological interest-just think about the possibility to realise photonic crystals with a dielectric structure modulated along a specific direction-and a challenging task, since striped patterns emerge in a variety of conditions, suggesting that the connection between the onset of stripes and the shape of the intermolecular potential is yet to be fully unravelled. Hereby, we devise an elementary mechanism for the formation of stripes in a basic model consisting of a symmetric binary mixture of hard spheres that interact via a square-well cross attraction. Such a model would mimic a colloid in which the interspecies affinity is of longer range and significantly stronger than the intraspecies interaction. For attraction ranges shorter enough than the particle size the mixture behaves like a compositionally-disordered simple fluid. Instead, for wider square-wells, we document by numerical simulations the existence of striped patterns in the solid phase, where layers of particles of one species are interspersed with layers of the other species; increasing the attraction range stabilises the stripes further, in that they also appear in the bulk liquid and become thicker in the crystal. Our results lead to the counterintuitive conclusion that a flat and sufficiently long-ranged unlike attraction promotes the aggregation of like particles into stripes. This finding opens a novel way for the synthesis of colloidal particles with interactions tailored at the development of stripe-modulated structures.

Zonal Estimators for Quasiperiodic Bosonic Many-Body Phases

In this work, we explore the relevant methodology for the investigation of interacting systems with contact interactions, and we introduce a class of zonal estimators for path-integral Monte Carlo methods, designed to provide physical information about limited regions of inhomogeneous systems. We demonstrate the usefulness of zonal estimators by their application to a system of trapped bosons in a quasiperiodic potential in two dimensions, focusing on finite temperature properties across a wide range of values of the potential. Finally, we comment on the generalization of such estimators to local fluctuations of the particle numbers and to magnetic ordering in multi-component systems, spin systems, and systems with nonlocal interactions.

Inverse design of soft materials via a deep learning-based evolutionary strategy

Colloidal self-assembly—the spontaneous organization of colloids into ordered structures—has been considered key to produce next-generation materials. However, the present-day staggering variety of colloidal building blocks and the limitless number of thermodynamic conditions make a systematic exploration intractable. The true challenge in this field is to turn this logic around and to develop a robust, versatile algorithm to inverse design colloids that self-assemble into a target structure. Here, we introduce a generic inverse design method to efficiently reverse-engineer crystals, quasicrystals, and liquid crystals by targeting their diffraction patterns. Our algorithm relies on the synergetic use of an evolutionary strategy for parameter optimization, and a convolutional neural network as an order parameter, and provides a way forward for the inverse design of experimentally feasible colloidal interactions, specifically optimized to stabilize the desired structure.

Free energy cost to assemble superlattices of polymer-grafted nanoparticles

Mesoparticles consisting of a hard core and a soft corona like polymer-grafted nanoparticles (PGNPs) can assemble into various superlattice structures, in which each mesoparticle assumes the shape of the corresponding Wigner-Seitz (or Voronoi) cell. Conventional wisdom often perceives the stability of these superlattices in a mean-field view of surface area minimization or corona entropy maximization, which lacks molecular interpretation. We develop a simulation method to calculate the free energy cost to deform spherical PGNPs into Wigner-Seitz polyhedra, which are then relaxed in a certain crystalline superlattice. With this method, we successfully quantify the free energy differences between model BCC, FCC and A15 systems of PGNPs and identify BCC as the most stable structure in most cases. Analysis of polymer configurations in the corona, whose boundary is blurred by chain interpenetration, shows that the radial distribution of grafted chains and the corresponding entropy are almost identical between BCC and FCC, suggesting that the higher stability of the BCC structure cannot be explained by the mean-field description of the corona shape.

Multiple 2D crystal structures in bilayered lamellae from direct self-assembly of 3D systems of soft Janus particles

Numerous crystals and Frank-Kasper phases in two-dimensional (2D) systems of soft particles have been presented by theoretical investigations. How to realize 2D crystals or Frank-kasper phases by direct self-assembly of...

Defined core-shell particles as the key to complex interfacial self-assembly

The two-dimensional self-assembly of colloidal particles serves as a model system for fundamental studies of structure formation and as a powerful tool to fabricate functional materials and surfaces. However, the prevalence of hexagonal symmetries in such self-assembling systems limits its structural versatility. More than two decades ago, Jagla demonstrated that core-shell particles with two interaction length scales can form complex, nonhexagonal minimum energy configurations. Based on such Jagla potentials, a wide variety of phases including cluster lattices, chains, and quasicrystals have been theoretically discovered. Despite the elegance of this approach, its experimental realization has remained largely elusive. Here, we capitalize on the distinct interfacial morphology of soft particles to design two-dimensional assemblies with structural complexity. We find that core-shell particles consisting of a silica core surface functionalized with a noncrosslinked polymer shell efficiently spread at a liquid interface to form a two-dimensional polymer corona surrounding the core. We controllably grow such shells by iniferter-type controlled radical polymerization. Upon interfacial compression, the resulting core-shell particles arrange in well-defined dimer, trimer, and tetramer lattices before transitioning into complex chain and cluster phases. The experimental phase behavior is accurately reproduced by Monte Carlo simulations and minimum energy calculations, suggesting that the interfacial assembly interacts via a pairwise-additive Jagla-type potential. By comparing theory, simulation, and experiment, we narrow the Jagla g-parameter of the system to between 0.9 and 2. The possibility to control the interaction potential via the interfacial morphology provides a framework to realize structural features with unprecedented complexity from a simple, one-component system.

Soft Particles at Liquid Interfaces: From Molecular Particle Architecture to Collective Phase Behavior

Soft particles such as microgels can undergo significant and anisotropic deformations when adsorbed to a liquid interface. This, in turn, leads to a complex phase behavior upon compression. To date, experimental efforts have predominantly provided phenomenological links between microgel structure and resulting interfacial behavior, while simulations have not been entirely successful in reproducing experiments or predicting the minimal requirements for the desired phase behavior. Here, we develop a multiscale framework to link the molecular particle architecture to the resulting interfacial morphology and, ultimately, to the collective interfacial phase behavior. To this end, we investigate interfacial morphologies of different poly(N-isopropylacrylamide) particle systems using phase-contrast atomic force microscopy and correlate the distinct interfacial morphology with their bulk molecular architecture. We subsequently introduce a new coarse-grained simulation method that uses augmented potentials to translate this interfacial morphology into the resulting phase behavior upon compression. The main novelty of this method is the possibility to efficiently encode multibody interactions, the effects of which are key to distinguishing between heterostructural (anisotropic collapse) and isostructural (isotropic collapse) phase transitions. Our approach allows us to qualitatively resolve existing discrepancies between experiments and simulations. Notably, we demonstrate the first in silico account of the two-dimensional isostructural transition, which is frequently found in experiments but elusive in simulations. In addition, we provide the first experimental demonstration of a heterostructural transition to a chain phase in a single-component system, which has been theoretically predicted decades ago. Overall, our multiscale framework provides a phenomenological bridge between physicochemical soft-particle characteristics at the molecular scale and nanoscale and the collective self-assembly phenomenology at the macroscale, serving as a stepping stone toward an ultimately more quantitative and predictive design approach.

Self-assembly of polymer-tethered nanoparticles with uniform and Janus surfaces in nanotubes

In this study, a coarse-grained molecular simulation was performed to investigate the morphologies and phase diagrams of self-assembled polymer-tethered nanoparticles (NPs) confined in nanotubes (NTs). Unlike ordinary NPs, polymer-tethered NPs have two distinct characteristic lengths, which are key factors that determine their self-assembly. Herein, two distinct types of NT walls and three types of polymer-tethered NPs were considered: hydrophilic and hydrophobic walls, and hydrophilic, hydrophobic, and Janus surfaces. First, the qualitative phase diagrams of the axial pressure, Pz, versus the ratio of the NT radius to the NP radius, L, were derived. The results revealed that diverse self-assembled morphologies, which are not formed in non-tethered NPs, were observed in the polymer-tethered NPs. For example, three types of ordered structures with different structural characteristic lengths, depending on Pz, were obtained. In addition, the effect of the chemical nature of the polymer-tethered NP surface on the self-assembled morphology confined in NTs was investigated. Clusters of water molecules were formed, particularly in the hydrophobic polymer-tethered NPs, and these clusters caused the structural distortion of the NP. Moreover, in the polymer-tethered NPs with the Janus amphiphilic surface, the hydrophobic and hydrophilic polymer tethered NPs assembled in the axial direction to form an ordered structure, and a double-helix structure was formed at L = 3.0 in the hydrophobic NT. The results of these simulations indicate that the self-assembly behaviours of polymer-tethered NPs can be qualitatively predicted based on the chemical nature of the NT walls and the surface design of the polymer-tethered NP.

Nonconventional Phases of Colloidal Nanorods with a Soft Corona

Using computer simulations, we investigate the phase behavior of hard-core spherocylinders with a length-to-diameter ratio L/σ=5 and coated by a soft deformable corona of length λ/σ=1.35. When quasi-two-dimensional layers are formed in smectic and solid phases at low temperatures, the competition between the two intrinsic length scales of the parallel aligned particles leads to the stabilization of different in-plane lattices of nonconventional symmetry, including low-density hexagonal, square, and high-density hexagonal crystals, as well as an intriguing dodecagonal quasicrystal. Our Letter opens up the opportunity to control the assembly of anisotropic nanoparticles into structures with preengineered symmetry-dependent physical properties.

Mechanisms of Nucleation and Solid-Solid-Phase Transitions in Triblock Janus Assemblies

A model, including the chemical details of core nanoparticles as well as explicit surface charges and hydrophobic patches, of triblock Janus particles is employed to simulate nucleation and solid-solid phase transitions in two-dimensional layers. An explicit solvent and a substrate are included in the model, and hydrodynamic and many-body interactions were taken into account within many-body dissipative particle dynamics simulation. In order not to impose a mechanism a priori, we performed free (unbiased) simulations, leaving the system the freedom to choose its own pathways. In agreement with the experiment and previous biased simulations, a two-step mechanism for the nucleation of a kagome lattice from solution was detected. However, a distinct feature of the present unbiased versus biased simulations is that multiple nuclei emerge from the solution; upon their growth, the aligned and misaligned facets at the grain boundaries are introduced into the system. The liquid-like particles trapped between the neighboring nuclei connect them together. A mismatch in the symmetry planes of neighboring nuclei hinders the growth of less stable (smaller) nuclei. Unification of such nuclei at the grain boundaries of misaligned facets obeys a two-step mechanism: melting of the smaller nuclei, followed by subsequent nucleation of liquid-like particles at the interface of bigger neighboring nuclei. Besides, multiple postcritical nuclei are formed in the simulation box; the growth of some of which stops due to introduction of a strain in the system. Such an incomplete nucleation/growth mechanism is in complete agreement with the recent experiments. The solid-solid (hexagonal-to-kagome) phase transition, at weak superheatings, obeys a two-step mechanism: a slower step (formation of a liquid droplet), followed by a faster step (nucleation of kagome from the liquid droplet).

Pattern detection in colloidal assembly: A mosaic of analysis techniques

Characterization of the morphology, identification of patterns and quantification of order encountered in colloidal assemblies is essential for several reasons. First of all, it is useful to compare different self-assembly methods and assess the influence of different process parameters on the final colloidal pattern. In addition, casting light on the structures formed by colloidal particles can help to get better insight into colloidal interactions and understand phase transitions. Finally, the growing interest in colloidal assemblies in materials science for practical applications going from optoelectronics to biosensing imposes a thorough characterization of the morphology of colloidal assemblies because of the intimate relationship between morphology and physical properties (e.g. optical and mechanical) of a material. Several image analysis techniques developed to investigate images (acquired via scanning electron microscopy, digital video microscopy and other imaging methods) provide variegated and complementary information on the colloidal structures under scrutiny. However, understanding how to use such image analysis tools to get information on the characteristics of the colloidal assemblies may represent a non-trivial task, because it requires the combination of approaches drawn from diverse disciplines such as image processing, computational geometry and computational topology and their application to a primarily physico-chemical process. Moreover, the lack of a systematic description of such analysis tools makes it difficult to select the ones more suitable for the features of the colloidal assembly under examination. In this review we provide a methodical and extensive description of real-space image analysis tools by explaining their principles and their application to the investigation of two-dimensional colloidal assemblies with different morphological characteristics.

Phase behavior of a two-dimensional core-softened system: new physical insights

In this work, we report results of extensive computer simulations regarding the phase behavior of a core-softened system. By using structural and thermodynamic descriptors, as well as self-diffusion coefficients, we provide a comprehensive view of the rich phase behavior displayed by the particular instance of the model studied in here. Our calculations agree with previously published results focused on a smaller region in the temperature-density parameter space (Dudalov et al 2014 Soft Matter 10 4966). In this work, we explore a broader region in this parameter space, and uncover interesting fluid phases with low-symmetry local order, that were not reported by previous works. Solid phases were also found, and have been previously characterized in detail by (Kryuchkov et al 2018 Soft Matter 14 2152). Our results support previously reported findings, and provide new physical insights regarding the emergence of order as disordered phases transform into solids by providing radial distribution function maps and specific heat data. Our results are summarized in terms of a phase diagram.

Pattern formation in two-dimensional hard-core/soft-shell systems with variable soft shell profiles

Hard-core/soft shell (HCSS) particles have been shown to self-assemble into a remarkably rich variety of structures under compression due to the simple interplay between the hard-core and soft-shoulder length scales in their interactions. Most studies in this area model the soft shell interaction as a square shoulder potential. Although appealing from a theoretical point of view, the potential is physically unrealistic because there is no repulsive force in the soft shell regime, unlike in experimental HCSS systems. To make the model more realistic, here we consider HCSS particles with a range of soft shell potential profiles beyond the standard square shoulder form and study the model using both minimum energy calculations and Monte Carlo simulations. We find that by tuning density and the soft shell profile, HCSS particles in the thin shell regime (i.e., shell to core ratio ) can form a large range of structures, including hexagons, chains, squares, rhomboids and two distinct zig-zag structures. Furthermore, by tuning the density and r1/r0, we find that HCSS particles with experimentally realistic linear ramp soft shoulder repulsions can form honeycombs and quasicrystals with 10-fold and 12-fold symmetry. Our study therefore suggests the exciting possibility of fabricating these exotic 2D structures experimentally through colloidal self-assembly.

Temperature expansions in the square-shoulder fluid. II. Thermodynamics

In Paper I [O. Coquand and M. Sperl, J. Chem. Phys. 152, 124112 (2020)], we derived analytical expressions for the structure factor of the square-shoulder potential in a perturbative way around the high- and low-temperature regimes. Here, various physical properties of these solutions are derived. In particular, we investigate the large wave number sector and relate it to the contact values of the pair-correlation function. Then, the thermoelastic properties of the square-shoulder fluids are discussed.

Amphiphile-Induced Anisotropic Colloidal Self-Assembly

Spherical colloidal particles typically self-assemble into hexagonal lattices when adsorbed at liquid interfaces. More complex assembly structures, including particle chains and phases with square symmetry, were theoretically predicted almost two decades ago for spherical particles interacting via a soft repulsive shoulder. Here, we demonstrate that such complex assembly phases can be experimentally realized with spherical colloidal particles assembled at the air/water interface in the presence of molecular amphiphiles. We investigate the interfacial behavior of colloidal particles in the presence of different amphiphiles on a Langmuir trough. We transfer the structures formed at the interface onto a solid substrate while continuously compressing, which enables us to correlate the prevailing assembly phase as a function of the available interfacial area. We observe that block copolymers with similarities to the chemical nature of the colloidal particles, as well as the surface-active protein bovine serum albumin, direct the colloidal particles into complex assembly phases, including chains and square arrangements. The observed structures are reproduced by minimum energy calculations of hard core-soft shoulder particles with experimentally realistic interaction parameters. From the agreement between experiments and theory, we hypothesize that the presence of the amphiphiles manipulates the interaction potential of the colloidal particles. The assembly of spherical colloidal particles into complex assembly phases on solid substrates opens new possibilities for surface patterning by enriching the library of possible structures available for colloidal lithography.

Complex crystalline structures in a two-dimensional core-softened system

A transition from a square to a hexagonal lattice is studied in a 2D system of particles interacting via a core-softened potential. Due to the presence of two length scales of repulsion, different local configurations with four, five, and six neighbors are possible, leading to the formation of complex crystals. The previously proposed interpolation method is generalized to calculate pair correlations in crystals whose unit cell consists of more than one particle. The high efficiency of the method is illustrated using a snub square lattice as a representative example. Molecular dynamics simulations show that the snub square lattice is broken upon heating, generating a high-density quasicrystalline phase with 12-fold symmetry (HD12 phase). A simple theoretical model is proposed to explain the physical mechanism responsible for this phenomenon: with an increase in the density (from square to hexagonal phases), the concentrations of different local configurations randomly realized through a plane tiling change, which minimizes the energy of the system. The calculated phase diagram in the intermediate density range justifies the existence of the HD12 phase and demonstrates a cascade of first-order transitions "square - HD12 - hexagonal" solid phases with increasing density. The results allow us to better understand the physical mechanisms responsible for the formation of quasicrystals, and, therefore, should be of interest for broad community in materials science and soft matter.

Waterlike anomalies in a two-dimensional core-softened potential

We investigate the structural, thermodynamic, and dynamic behavior of a two-dimensional (2D) core-corona system using Langevin dynamics simulations. The particles are modeled by employing a core-softened potential which exhibits waterlike anomalies in three dimensions. In previous studies in a quasi-2D system a new region in the pressure versus temperature phase diagram of structural anomalies was observed. Here we show that for the two-dimensional case two regions in the pressure versus temperature phase diagram with structural, density, and diffusion anomalies are observed. Our findings indicate that, while the anomalous region at lower densities is due the competition between the two length scales in the potential at higher densities, the anomalous region is related to the reentrance of the melting line.

Stable in Bulk and Aggregating at the Interface: Comparing Core-Shell Nanoparticles in Suspension and at Fluid Interfaces

Colloidal particles are extensively used to assemble materials from bulk suspensions or after adsorption and confinement at fluid interfaces (e.g., oil-water interfaces). Interestingly, and often underestimated, optimizing interactions for bulk assembly may not lead to the same behavior at fluid interfaces. In this work, we compare model composite nanoparticles with a silica core coated with a poly-N-isopropylacrylamide hydrogel shell in bulk aqueous suspensions and after adsorption at an oil-water interface. Bulk properties are analyzed by confocal differential dynamic microscopy, a recently developed technique that allows one to simultaneously obtain structural and dynamical information up to high volume fractions. The results demonstrate excellent colloidal stability and the absence of aggregation in all cases. The behavior at the interface, investigated by a range of complementary approaches, is instead different. The same hydrogel shells that stabilize the particles in the bulk deform at the interface and induce attractive capillary interactions that lead to aggregation even at very low area fractions (surface coverage). Upon further compression of a particle-laden interface, a structural transition is observed where closely packed particle aggregates form. These findings emphasize the manifestation of different, and possibly unexpected, responses for sterically stabilized nanoparticles in the bulk and upon interfacial confinement.

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.

Periodic layers of a dodecagonal quasicrystal and a floating hexagonal crystal in sedimentation-diffusion equilibria of colloids

We investigate the behaviour of a system of colloidal particles interacting with a hard-core and a repulsive square shoulder potential under the influence of a gravitational field using event-driven Brownian dynamics simulations. We use a fixed square shoulder diameter equal to 1.4 times the hard-core diameter of the colloids, for which we have previously calculated the equilibrium phase diagram considering two-dimensional disks [H. Pattabhiraman et al., J. Chem. Phys. 143, 164905 (2015) and H. Pattabhiraman and M. Dijkstra, J. Phys.: Condens. Matter 20, 094003 (2017)]. The parameters in the simulations are chosen such that the pressure at the bottom of the sediment facilitates the formation of phases in accordance with the calculated phase diagram of the two-dimensional system. It is surprising that we observe the formation of layers with dodecagonal, square, and hexagonal symmetries at the relevant pressures in the three-dimensional sedimentation column. In addition, we also observe a re-entrant behaviour exhibited by the colloidal fluid phase, engulfing a hexagonal crystal phase, in the sedimentation column. In other words, a floating crystal is formed between the colloidal fluid regions.