Arrested states in colloidal fluids with competing interactions: A static replica study.

We present the first systematic application of the integral equation implementation of the replica method to the study of arrested states in fluids with microscopic competing interactions (short-range attractive and long-range repulsive, SALR), as exemplified by the prototype Lennard-Jones-Yukawa model. Using a wide set of potential parameters, we provide as many as 11 different phase diagrams on the density (ρ)-temperature (T) plane, embodying both the cluster-phase boundary, TC(ρ), and the locus below which arrest takes place, TD(ρ). We describe how the interplay between TC and TD-with the former falling on top of the other, or the other way around, depending on thermodynamic conditions and potential parameters-gives rise to a rich variety of non-ergodic states interspersed with ergodic ones, of which both the building blocks are clusters or single particles. In a few cases, we find that the TD locus does not extend all over the density range subtended by the TC envelope; under these conditions, the λ-line is within reach of the cluster fluid, with the ensuing possibility to develop ordered microphases. Whenever a comparison is possible, our predictions favorably agree with previous numerical results. Thereby, we demonstrate the reliability and effectiveness of our scheme to provide a unified theoretical framework for the study of arrested states in SALR fluids, irrespective of their nature.

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.

Microphase versus macrophase separation in the square-well-linear fluid: A theoretical and computational study.

Due to the presence of competing interactions, the square-well-linear fluid can exhibit either liquid-vapor equilibrium (macrophase separation) or clustering (microphase separation). Here we address the issue of determining the boundary between these two regimes, i.e., the Lifshitz point, expressed in terms of a relationship between the parameters of the model. To this aim, we carry out Monte Carlo simulations to compute the structure factor of the fluid, whose behavior at low wave vectors accurately captures the tendency of the fluid to form aggregates or, alternatively, to phase separate. Specifically, for a number of different combinations of attraction and repulsion ranges, we make the system go across the Lifshitz point by increasing the strength of the repulsion. We use simulation results to benchmark the performance of two theories of fluids, namely, the hypernetted chain (HNC) equation and the analytically solvable random phase approximation (RPA); in particular, the RPA theory is applied with two different prescriptions as for the direct correlation function inside the core. Overall, the HNC theory proves to be an appropriate tool to characterize the fluid structure and the low-wave-vector behavior of the structure factor is consistent with the threshold between microphase and macrophase separation established through simulation. The structural predictions of the RPA theory turn out to be less accurate, but this theory offers the advantage of providing an analytical expression of the Lifshitz point. Compared to simulation, both RPA schemes predict a Lifshitz point that falls within the macrophase-separation region of parameters: in the best case, barriers roughly twice higher than predicted are required to attain clustering conditions.

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.

Competition between clustering and phase separation in binary mixtures containing SALR particles.

We investigate by Monte Carlo simulations a mixture of particles with competing interactions (hard-sphere two-Yukawa, HSTY) and hard spheres (HS), with same diameters σ and a square-well (SW) cross attraction. In a recent study [G. Munaò et al., J. Phys. Chem. B, 2022, 126, 2027-2039], we have analysed situations-in terms of relative concentration and attraction strength-where HS promote the formation of clusters involving particles of both species under thermodynamic conditions that would not allow for clustering of the pure HSTY fluid. Here, we focus on the role played by the range of cross attraction in determining the equilibrium structure of the mixture, starting from a homogeneous low-density state. When the width of the well exceeds approximately σ, clustering takes place in the system, with aggregates characterised by various sizes and shapes. Only for low HSTY concentrations (less than 10%) a single big cluster appears, anticipating the behaviour observed for a wider well, around 1.2σ. In the latter case, a spherical cluster encompassing almost all particles is the stable structure at equilibrium. We interpret this outcome as a macrophase, liquid-vapour separation where the spherical cluster is just the form taken at low density by the liquid phase inside the vapour phase: indeed, when the density takes larger values, periodic boundary conditions select liquid-vapour interfaces with other non-spherical shapes, similarly as found for a finite sample of simple fluid going through the liquid-vapour coexistence region. For still higher densities we document the existence of a solid phase characterized by the alternation of bilayers filled with particles of one species and bilayers of the other species, giving the solid a peculiar wafer structure.

Clustering in Mixtures of SALR Particles and Hard Spheres with Cross Attraction.

Self-assembling complex fluids are often modeled as particles with effective competing isotropic interactions, combining a short-range attraction (SA) followed by a longer-range repulsion (LR). For moderately low temperatures and densities, SALR particles form clusters in equilibrium, at least provided that the potential parameters are appropriate. Here we inquire into the possibility that cluster formation in SALR fluids might be pushed by a foreign species even under thermodynamic conditions that would not allow for clusterization of the pure system. To this aim, we study by Monte Carlo simulations a mixture of hard-sphere two-Yukawa particles and hard spheres, with a cross interaction modeled by a square-well attraction, and we investigate the conditions of clustering in terms of strength of attraction and relative concentration of the two species. We find that clusters can occur in the mixture for the same temperature and density where the pure SALR fluid is almost structureless. In particular, we single out a cross attraction such that clusters are formed with a SALR concentration as low as 5%. We also find a situation where nearly pure droplets of hard spheres are held together by a shell of SALR particles. Conversely, we show that clustering can be undermined in the mixture under conditions for which this process takes place in the parent SALR fluid. Using a simple criterion, based on the second virial coefficients of the attractive part of interaction potentials (the so-called "reference attractive fluids"), we are able to predict accurately whether clustering is favored (or hindered) in the mixture, as compared to the pure SALR fluid.

Early stages of aggregation in fluid mixtures of dimers and spheres: a theoretical and simulation study.

We use Monte Carlo simulation and the Reference Interaction Site Model (RISM) theory of molecular fluids to investigate a simple model of colloidal mixture consisting of dimers, made up of two tangent hard monomers of different size, and hard spheres. In addition to steric repulsion, the two species interact via a square-well attraction only between small monomers and spheres. Recently, we have characterized the low-temperature regime of this mixture by Monte Carlo, reporting on the spontaneous formation of a wide spectrum of supramolecular aggregates [Prestipino et al., J. Phys. Chem. B, 2019, 123, 9272]. Here we focus on a regime of temperatures where, on cooling, the appearance of local inhomogeneties first, and the early stages of aggregation thereafter, are observed. In particular, we find signatures of aggregation in the onset of a low-wavevector peak in the structure factors of the mixture, as computed by both theory and simulation. Then, we link the structural information to the microscopic arrangement through a detailed cluster analysis of Monte Carlo configurations. In this regard, we devise a novel method to compute the maximum distance for which two spheres can be regarded as bonded together, a crucial issue in the proper identification of fluid aggregates. The RISM theory provides relatively accurate structural and thermodynamic predictions in comparison with Monte Carlo, but with slightly degrading performances as the fluid progresses inside the locally inhomogeneous phase. Our study certifies the efficacy of the RISM approach as a useful complement to numerical simulation for a reasoned analysis of aggregation properties in colloidal mixtures.

Efficient Hybrid Particle-Field Coarse-Grained Model of Polymer Filler Interactions: Multiscale Hierarchical Structure of Carbon Black Particles in Contact with Polyethylene.

In the present study, we propose, validate, and give first applications for large-scale systems of coarse-grained models suitable for filler/polymer interfaces based on carbon black (CB) and polyethylene (PE). The computational efficiency of the proposed approach, based on hybrid particle-field models (hPF), allows large-scale simulations of CB primary particles of realistic size (∼20 nm) embedded in PE melts. The molecular detailed models, here introduced, allow a microscopic description of the bound layer, through the analysis of the conformational behavior of PE chains adsorbed on different surface sites of CB primary particles, where the conformational behavior of adsorbed chains is different from models based on flat infinite surfaces. On the basis of the features of the systems, an optimized version of OCCAM code for large-scale (up to more than 8 million of beads) parallel runs is proposed and benchmarked. The computational efficiency of the proposed approach opens the possibility of a computational screening of the bound layer, involving the optimal combination of surface chemistry, size, and shape of CB aggregates and the molecular weight distribution of the polymers achieving an important tool to address the polymer/fillers interface and interphase engineering in the polymer industry.

Effect of Trapped Solvent on the Interface between PS-b-PMMA Thin Films and P(S-r-MMA) Brush Layers.

The orientation of block copolymer (BCP) features in thin films can be obtained by spin-coating a BCP solution on a substrate surface functionalized by a polymer brush layer of the appropriate random copolymer (RCP). Although this approach is well established, little work reporting the amount and distribution of residual solvent in the polymer film after the spin-coating process is available. Moreover, no information can be found on the effect of trapped solvent on the interface between the BCP film and RCP brush. In this work, systems consisting of poly(styrene)-b-poly(methyl methacrylate) thin films deposited on poly(styrene-r-methyl methacrylate) brush layers are investigated by combining neutron reflectivity (NR) experiments with simulation techniques. An increase in the amount of trapped solvent is observed by NR as the BCP film thickness increases accompanied by a significant decrease of the interpenetration length between the BCP and RCP, thus suggesting that the interpenetration between grafted chains and block copolymer chains is hampered by the solvent. Hybrid particle-field molecular dynamics simulations of the analyzed system confirm the experimental observations and demonstrate a clear correlation between the interpenetration length and the amount of trapped solvent.

Simulation of self-heating process on the nanoscale: a multiscale approach for molecular models of nanocomposite materials.

A theoretical-computational protocol to model the Joule heating process in nanocomposite materials is presented. The proposed modeling strategy is based on post processing of trajectories obtained from large scale molecular simulations. This protocol, based on molecular models, is the first one to be applied to organic nanocomposites based on carbon nanotubes (CNT). This strategy allows to keep a microscopic explicit picture of the systems, to directly catch the molecular structure underlying the process under study and, at the same time, to include macroscopic boundary conditions fixed in the experiments. As validation and first application of the proposed strategy, a detailed investigation on CNT based organic composites is reported. The effect of CNT morphologies, concentration and working conditions on Joule heating has been modelled and compared with available experiments. Further experiments are performed also in this work to increase the number of comparisons especially in specific voltage ranges where available references from literature were missing. Simulations are in both qualitative and quantitative agreement with several experiments and trends reported in the recent literature, as well as with experiments performed in this work. The proposed approach combined with large scale hybrid particle-field molecular simulations can give insights and opens to way to a rational design of self-heating nanocomposites.

Evidence of Structural Inhomogeneities in Hard-Soft Dimeric Particles without Attractive Interactions.

We perform Monte Carlo simulations of a simple hard-soft dimeric model constituted by two tangent spheres experiencing different interactions. Specifically, two hard spheres belonging to different dimers interact via a bare hard-core repulsion, whereas two soft spheres experience a softly repulsive Hertzian interaction. The cross correlations are soft as well. By exploring a wide range of temperatures and densities we investigate the capability of this model to document the existence of structural inhomogeneities indicating the possible onset of aggregates, even if no attraction is set. The fluid phase behavior is studied by analyzing structural and thermodynamical properties of the observed structures, in particular by computing radial distribution functions, structure factors and cluster size distributions. The numerical results are supported by integral equation theories of molecular liquids which allow for a finer and faster spanning of the temperature-density diagram. Our results may serve as a framework for a more systematic investigation of self-assembled structures of functionalized hard-soft dimers able to aggregate in a variety of structures widely oberved in colloidal dispersion.

Complex Self-Assembly from Simple Interaction Rules in Model Colloidal Mixtures.

Building structures with hierarchical order through the self-assembly of smaller blocks is not only a prerogative of nature, but also a strategy to design artificial materials with tailored functions. We explore in simulation the spontaneous assembly of colloidal particles into extended structures, using spheres and size-asymmetric dimers as solute particles, while treating the solvent implicitly. Besides rigid cores for all particles, we assume an effective short-range attraction between spheres and small monomers to promote, through elementary rules, dimer-mediated aggregation of spheres. Starting from a completely disordered configuration, we follow the evolution of the system at low temperature and density, as a function of the relative concentration of the two species. When spheres and large monomers are of same size, we observe the onset of elongated aggregates of spheres, either disconnected or cross-linked, and a crystalline bilayer. As spheres grow bigger, the self-assembling scenario changes, getting richer overall, with the addition of flexible membrane sheets with crystalline order and monolayer vesicles. With this wide assortment of structures, our model can serve as a viable template to achieve a better control of self-assembly in dilute suspensions of microsized particles.

Monte Carlo simulation and integral equation study of Hertzian spheres in the low-temperature regime.

We investigate the behavior of Hertzian spheres in the fluid phase and in proximity of the freezing threshold by using Monte Carlo (MC) simulations and integral equation theories, based on the Ornstein-Zernike (OZ) approach. The study is motivated by the importance of the Hertzian model in representing a large class of systems interacting via soft interactions, such as star polymers or microgels. Radial distribution functions, structure factors, and excess entropy clearly show the reentrant behavior typical of the Hertzian fluid, well caught by both MC simulations and OZ theory. Then, we make use of some phenomenological one-phase criteria for testing their reliability in detecting the freezing threshold. All criteria provide evidence of the fluid-solid transition with different degrees of accuracy. This suggests the possibility to adopt these empirical rules to provide a quick and reasonable estimate of the freezing transition in model potentials of wide interest for soft matter systems.

Evidence of pre-micellar aggregates in aqueous solution of amphiphilic PDMS-PEO block copolymer.

The self-assembly process in a water solution of an amphiphilic polydimethylsiloxane-b-polyethyleneoxide (PDMS-PEO) diblock copolymer was investigated by means of small-angle X-ray scattering (SAXS) experiments in the concentration region below (and near) the critical micellar concentration (c.m.c. = 0.007 g cm-3). In the highly diluted region, at the copolymer concentration of c = 0.002 g cm-3, the early stage of the self-assembly process was characterized by the formation of small (primary) micellar units (with a radius of R = 2.7 nm) with core-shell morphology, which coexisted with larger supramolecular aggregates of entangled micelles (with an average radius of R = 9.5 nm). The increase in the copolymer concentration (to c = 0.005 and c = 0.01 g cm-3) caused increase in the sizes of both the small micelles and supra-micellar aggregates. Interestingly, at the concentration of c = 0.005 g cm-3, both the size and micelle aggregation number (Nagg) were found to increase on increasing the temperature in the range of 10 ≤ T ≤ 55 °C. This phenomenon was characterised by the dehydration process of the ethylene oxide (EO) segments, as evidenced by the calculation of excess water in the hydrophilic shell of the micelles. The more compact (less hydrated) structure of the hydrophilic PEO chains, which strongly influenced the spontaneous curvature of the amphiphile hydrophilic region, turned out to be the driving factor that favoured the increase in the micelle aggregation number with the increase in temperature. The obtained results evidence that the self-assembly process of PDMS-PEO copolymer amphiphiles is a gradual process that is already present at the very low concentration region (far below the macroscopically determined c.m.c.); moreover, it is characterised by a multi-stage organization process, where the primary building blocks self-assemble into more complex secondary structures that encompass multiple length scales.

Molecular structure and multi-body potential of mean force in silica-polystyrene nanocomposites.

We perform a systematic application of the hybrid particle-field molecular dynamics technique [Milano, et al., J. Chem. Phys., 2009, 130, 214106] to study interfacial properties and potential of mean force (PMF) for separating nanoparticles (NPs) in a melt. Specifically, we consider Silica NPs bare or grafted with Polystyrene chains, aiming to shed light on the interactions among free and grafted chains affecting the dispersion of NPs in the nanocomposite. The proposed hybrid models show good performances in catching the local structure of the chains, and in particular their density profiles, documenting the existence of the "wet-brush-to-dry-brush" transition. By using these models, the PMF between pairs of ungrafted and grafted NPs in Polystyrene matrix are calculated. Moreover, we estimate the three-particle contribution to the total PMF and its role in regulating the phase separation on the nanometer scale. In particular, the multi-particle contribution to the PMF is able to give an explanation of the complex experimental morphologies observed at low grafting densities. More in general, we propose this approach and the models utilized here for a molecular understanding of specific systems and the impact of the chemical nature of the systems on the composite final properties.

Molecular dynamics determination of liquid-vapor coexistence in molten alkali halides.

We show by extensive molecular dynamics simulations that accurate predictions of liquid-vapor coexistence in molten alkali halides can be achieved in terms of a rigid ion potential description in which temperature-dependent ionic diameters are employed. The new ionic sizes result from the fitting of the experimental isothermal compressibilities, a condition whose physical implications and consequences are illustrated. The same diameters also allow us to formulate confident predictions for the compressibilities of salts in cases where the experimental data are lacking. The extension of the present approach to molten alkali-halide mixtures and to other classes of molten salts is discussed.

On the calculation of the potential of mean force between atomistic nanoparticles.

We study the potential of mean force (PMF) between atomistic silica and gold nanoparticles in the vacuum by using molecular dynamics simulations. Such an investigation is devised in order to fully characterize the effective interactions between atomistic nanoparticles, a crucial step to describe the PMF in high-density coarse-grained polymer nanocomposites. In our study, we first investigate the behavior of silica nanoparticles, considering cases corresponding to different particle sizes and assessing results against an analytic theory developed by Hamaker for a system of Lennard-Jones interacting particles (H.C. Hamaker, Physica A 4, 1058 (1937)). Once validated the procedure, we calculate effective interactions between gold nanoparticles, which are considered both bare and coated with polyethylene chains, in order to investigate the effects of the grafting density [Formula: see text] on the PMF. Upon performing atomistic molecular dynamics simulations, it turns out that silica nanoparticles experience similar interactions regardless of the particle size, the most remarkable difference being a peak in the PMF due to surface interactions, clearly apparent for the larger size. As for bare gold nanoparticles, they are slightly interacting, the strength of the effective force increasing for the coated cases. The profile of the resulting PMF resembles a Lennard-Jones potential for intermediate [Formula: see text], becoming progressively more repulsive for high [Formula: see text] and low interparticle separations.

Two-dimensional mixture of amphiphilic dimers and spheres: Self-assembly behaviour.

The emergence of supramolecular aggregates from simple microscopic interaction rules is a fascinating feature of complex fluids which, besides its fundamental interest, has potential applications in many areas, from biological self-assembly to smart material design. We here investigate by Monte Carlo simulation the equilibrium structure of a two-dimensional mixture of asymmetric dimers and spheres (disks). Dimers and disks are hard particles, with an additional short-range attraction between a disk and the smaller monomer of a dimer. The model parameters and thermodynamic conditions probed are typical of colloidal fluid mixtures. In spite of the minimalistic character of the interaction, we observe-upon varying the relative concentration and size of the two colloidal species-a rich inventory of mesoscale structures at low temperature, such as clusters, lamellæ (i.e., polymer-like chains), and gel-like networks. For colloidal species of similar size and near equimolar concentrations, a dilute fluid of clusters gives way to floating lamellæ upon cooling; at higher densities, the lamellæ percolate through the simulation box, giving rise to an extended network. A crystal-vapour phase-separation may occur for a mixture of dimers and much larger disks. Finally, when the fluid is brought in contact with a planar wall, further structures are obtained at the interface, from layers to branched patterns, depending on the nature of wall-particle interactions.

Self-assembly behaviour of hetero-nuclear Janus dumbbells.

We investigate the fluid structure and self-assembly of a system of Janus dumbbells by means of aggregation-volume-bias Monte Carlo simulations and Simulated Annealing techniques. In our approach, Janus dumbbells model asymmetric colloidal particles constituted by two tangent (touching) spheres (labelled as h and s) of different sizes and interaction properties: specifically, the h spheres interact with all other spheres belonging to different dumbbells via hard-sphere potentials, whereas two s spheres interact via a square-well potential. By introducing a parameter α ∈ [0,2] that controls the size ratio between the h and s spheres, we are able to investigate the overall phase behaviour of Janus dumbbells as a function of α. In a previous paper (O'Toole et al., Soft Matter, 2017, 13, 803) we focused on the region where the s sphere is larger than the h sphere (α > 1), documenting the presence of a variety of phase behaviours. Here we investigate a different regime of size ratios, predominantly where the hard sphere is larger than (or comparable to) the attractive one. Under these conditions, we observe the onset of many different self-assembled super-structures. Depending on the specific value of α we document the presence of spherical clusters (micelles) progressively evolving into more exotic structures including platelets, filaments, networks and percolating fluids, sponge structures and lamellar phases. We find no evidence of a gas-liquid phase separation for α ≤ 1.1, since under these conditions it is pre-empted by the development of self-assembled phases.

Self-assembly in a model colloidal mixture of dimers and spherical particles.

We investigate the structure of a dilute mixture of amphiphilic dimers and spherical particles, a model relevant to the problem of encapsulating globular "guest" molecules in a dispersion. Dimers and spheres are taken to be hard particles, with an additional attraction between spheres and the smaller monomers in a dimer. Using the Monte Carlo simulation, we document the low-temperature formation of aggregates of guests (clusters) held together by dimers, whose typical size and shape depend on the guest concentration χ. For low χ (less than 10%), most guests are isolated and coated with a layer of dimers. As χ progressively increases, clusters grow in size becoming more and more elongated and polydisperse; after reaching a shallow maximum for χ≈50%, the size of clusters again reduces upon increasing χ further. In one case only (χ=50% and moderately low temperature) the mixture relaxed to a fluid of lamellae, suggesting that in this case clusters are metastable with respect to crystal-vapor separation. On heating, clusters shrink until eventually the system becomes homogeneous on all scales. On the other hand, as the mixture is made denser and denser at low temperature, clusters get increasingly larger until a percolating network is formed.