Periodic ordering of clusters and stripes in a two-dimensional lattice model. II. Results of Monte Carlo simulation

Lattice Model Results for Pattern Formation in a Mixture with Competing Interactions

A monolayer consisting of two types of particles, with energetically favored alternating stripes of the two components, is studied by Monte Carlo simulations and within a mesoscopic theory. We consider a triangular lattice model and assume short-range attraction and long-range repulsion between particles of the same kind, as well as short-range repulsion and long-range attraction for the cross-interaction. The structural evolution of the model upon increasing temperature is studied for equal chemical potentials of the two species. We determine the structure factor, the chemical potential-density isotherms, the specific heat, and the compressibility, and show how these thermodynamic functions are associated with the spontaneous formation of stripes with varying degrees of order.

Spontaneous pattern formation in monolayers of binary mixtures with competing interactions

A model for a monolayer of two types of particles spontaneously forming ordered patterns is studied using a mesoscopic theory and MC simulations. We assume hard-cores of the same size a for both components. For r > a, like particles attract and repel each other at short and large distances, respectively, with the same potential u(r) for both species, and the cross-interaction is -u(r). The model is inspired by oppositely charged particles or macromolecules with preferential solubility in different components of a solvent that is close to a miscibility critical point, in particular by inclusions in biological membranes. We obtain the phase diagram in the chemical potentials and temperature variables as well as in the concentration, density and temperature variables, using the mean-field one-shell approximation. We find that the presence of the second component significantly extends the temperature range of stability of the ordered phases. We obtain three stable phases with periodic concentration: the lamellar L phase with alternating stripes of the two components for similar chemical potentials, and a hexagonal arrangement of the clusters of the minority component in the liquid of the majority component. The latter two phases, however, are stable only at relatively high temperatures. At lower temperatures, the L phase coexists with a disordered one-component fluid or with very dilute gas with mixed components. At still lower temperatures, the one-component phase coexisting with the L phase can be disordered or ordered, depending on the chemical potentials. The theoretical results are confirmed by MC simulations for selected thermodynamic states.

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.

Pattern Formation in Two-Component Monolayers of Particles with Competing Interactions

Competing interactions between charged inclusions in membranes of living organisms or charged nanoparticles in near-critical mixtures can lead to self-assembly into various patterns. Motivated by these systems, we developed a simple triangular lattice model for binary mixtures of oppositely charged particles with additional short-range attraction or repulsion between like or different particles, respectively. We determined the ground state for the system in contact with a reservoir of the particles for the whole chemical potentials plane, and the structure of self-assembled conglomerates for fixed numbers of particles. Stability of the low-temperature ordered patterns was verified by Monte Carlo simulations. In addition, we performed molecular dynamics simulations for a continuous model with interactions having similar features, but a larger range and lower strength than in the lattice model. Interactions with and without symmetry between different components were assumed. We investigated both the conglomerate formed in the center of a thin slit with repulsive walls, and the structure of a monolayer adsorbed at an attractive substrate. Both models give the same patterns for large chemical potentials or densities. For low densities, more patterns occur in the lattice model. Different phases coexist with dilute gas on the lattice and in the continuum, leading to different patterns in self-assembled conglomerates ('rafts').

Hydroxide films on mica form charge-stabilized microphases that circumvent nucleation barriers

Crystal nucleation is facilitated by transient, nanoscale fluctuations that are extraordinarily difficult to observe. Here, we use high-speed atomic force microscopy to directly observe the growth of an aluminum hydroxide film from an aqueous solution and characterize the dynamically fluctuating nanostructures that precede its formation. Nanoscale cluster distributions and fluctuation dynamics show many similarities to the predictions of classical nucleation theory, but the cluster energy landscape deviates from classical expectations. Kinetic Monte Carlo simulations show that these deviations can arise from electrostatic interactions between the clusters and the underlying substrate, which drive microphase separation to create a nanostructured surface phase. This phase can evolve seamlessly from a low-coverage state of fluctuating clusters into a high-coverage nanostructured network, allowing the film to grow without having to overcome classical nucleation barriers.

Blunt-End Driven Re-entrant Ordering in Quasi Two-Dimensional Dispersions of Spherical DNA Brushes

We investigate the effects of crowding on the conformations and assembly of confined, highly charged, and thick polyelectrolyte brushes in the osmotic regime. Particle tracking experiments on increasingly dense suspensions of colloids coated with ultralong double-stranded DNA (dsDNA) fragments reveal nonmonotonic particle shrinking, aggregation, and re-entrant ordering. Theory and simulations show that aggregation and re-entrant ordering arise from the combined effect of shrinking, which is induced by the osmotic pressure exerted by the counterions absorbed in neighbor brushes and of a short-range attractive interaction competing with electrostatic repulsion. An unconventional mechanism gives origin to the short-range attraction: blunt-end interactions between stretched dsDNA fragments of neighboring brushes, which become sufficiently intense for dense and packed brushes. The attraction can be tuned by inducing free-end backfolding through the addition of monovalent salt. Our results show that base stacking is a mode parallel to hybridization to steer colloidal assembly in which attractions can be fine-tuned through salinity and, potentially, grafting density and temperature.

Adsorption time scales of cluster-forming systems

A microscopic model of adsorption in cluster forming systems with competing interaction is considered. The adsorption process is described by the master equation and modelled by a kinetic Monte Carlo method. The evolution of the particle concentration and interaction energy during the adsorption of particles on a plane triangular lattice is investigated. The simulation results show a diverse behavior of the system time evolution depending on the temperature and chemical potential and finally on the formation of clusters in the system. The characteristic relaxation times of adsorption vary in several orders of magnitude depending on the thermodynamic parameters of the final equilibrium state of the adsorbate. A very fast adsorption of particles is observed for highly ordered adsorbate equilibrium states.

Structural and Thermodynamic Peculiarities of Core-Shell Particles at Fluid Interfaces from Triangular Lattice Models

A triangular lattice model for pattern formation by core-shell particles at fluid interfaces is introduced and studied for the particle to core diameter ratio equal to 3. Repulsion for overlapping shells and attraction at larger distances due to capillary forces are assumed. Ground states and thermodynamic properties are determined analytically and by Monte Carlo simulations for soft outer- and stiffer inner shells, with different decay rates of the interparticle repulsion. We find that thermodynamic properties are qualitatively the same for slow and for fast decay of the repulsive potential, but the ordered phases are stable for temperature ranges, depending strongly on the shape of the repulsive potential. More importantly, there are two types of patterns formed for fixed chemical potential-one for a slow and another one for a fast decay of the repulsion at small distances. In the first case, two different patterns-for example clusters or stripes-occur with the same probability for some range of the chemical potential. For a fixed concentration, an interface is formed between two ordered phases with the closest concentration, and the surface tension takes the same value for all stable interfaces. In the case of degeneracy, a stable interface cannot be formed for one out of four combinations of the coexisting phases, because of a larger surface tension. Our results show that by tuning the architecture of a thick polymeric shell, many different patterns can be obtained for a sufficiently low temperature.

Statics and dynamics of a finite two-dimensional colloidal system with competing attractive critical Casimir and repulsive magnetic dipole interactions

We continue our theoretical study of a recently proposed two-dimensional colloidal system with attractive critical Casimir and repulsive magnetic dipole forces that can be tuned easily and independently from each other via the temperature and the strength of an external magnetic field, respectively [K. Marolt, M. Zimmermann, and R. Roth, Phys. Rev. E 100, 052602 (2019)2470-004510.1103/PhysRevE.100.052602]. Using this freedom, it is possible to construct a competing interaction potential that causes microphase separation featuring spatially inhomogeneous cluster, stripe, and bubble phases in the bulk, i.e., in an infinite system without an external potential. In the present work, we demonstrate by means of density functional theory that microphase separation can also occur in finite geometries. In a square cell with a side length of 20 or 30 colloid diameters, we observe the emergence of highly structured cluster and ring phases at intermediate bulk densities in addition to almost uniform fluid phases for lower and higher bulk densities. We then employ dynamic density functional theory to determine how the system reacts when the temperature and the magnetic field are altered over time, and we show how to induce a transition from the liquid to the cluster/ring phase and also from the cluster directly to the ring phase. We find that often a slowly varying and nontrivial path in parameter space is required to reach a stable state, whereas abrupt changes are prone to lead to metastable configurations.

Tracer diffusion in crowded solutions of sticky polymers

Macromolecular diffusion in strongly confined geometries and crowded environments is still to a large extent an open subject in soft matter physics and biology. In this paper, we employ large-scale Langevin dynamics simulations to investigate how the diffusion of a tracer is influenced by the combined action of excluded-volume and weak attractive crowder-tracer interactions. We consider two species of tracers, standard hard-core particles described by the Weeks-Chandler-Andersen (WCA) repulsive potential and core-softened (CS) particles, which model, e.g., globular proteins, charged colloids, and nanoparticles covered by polymeric brushes. These systems are characterized by the presence of two length scales in the interaction and can show waterlike anomalies in their diffusion, stemming from the inherent competition between different length scales. Here we report a comprehensive study of both diffusion and structure of these two tracer species in an environment crowded by quenched configurations of polymers at increasing density. We analyze in detail how the tracer-polymer affinity and the system density affect transport as compared to the emergence of specific static spatial correlations. In particular, we find that, while hardly any differences emerge in the diffusion properties of WCA and CS particles, the propensity to develop structural order for large crowding is strongly frustrated for CS particles. Surprisingly, for large enough affinity for the crowding matrix, the diffusion coefficient of WCA tracers display a nonmonotonic trend as their density is increased when compared to the zero affinity scenario. This waterlike anomaly turns out to be even larger than what observed for CS particle and appears to be rooted in a similar competition between excluded-volume and affinity effects.

Adsorption anomalies in a two-dimensional model of cluster-forming systems

Adsorption on a boundary line confining a monolayer of particles self-assembling into clusters is studied by Monte Carlo simulations. We focus on a system of particles interacting via competing interaction potential in which effectively short-range attraction is followed by long-range repulsion. For the chemical potential values below the order-disorder phase transition the adsorption isotherms were shown to undergo nonstandard behavior, i.e., the adsorption exhibits a maximum on structural transition between structureless and disordered cluster fluid. In particular, we have found that the adsorption decreases for increasing chemical potential when (i) clusters dominate over monomers in the bulk, (ii) the density profile in the direction perpendicular to the confining line exhibits an oscillatory decay, and (iii) the correlation function in the layer near the adsorption wall exhibits an oscillatory decay in the direction parallel to this wall. Our report indicates striking differences between simple and complex fluid adsorption processes.

Microphase separation in a two-dimensional colloidal system with competing attractive critical Casimir and repulsive magnetic dipole interactions

We propose and study theoretically a colloidal system in two dimensions with attractive critical Casimir and repulsive magnetic dipole forces, wherein the strength of attraction and repulsion can be easily and independently tuned by adjusting the temperature and an external magnetic field, respectively. We expect this setup to be experimentally accessible and are confident that it can serve to deepen our understanding of the mechanisms behind microphase separation due to competing interactions. We develop a density functional theory for our model and present first results of our calculations in the form of a phase diagram for fixed temperature, but varying magnetic fields and bulk densities. For certain values of these parameters, we are able to confirm the existence of thermodynamically stable inhomogeneous density profiles in the bulk, such as parallel lamellar stripes, as well as clusters and voids on a hexagonal lattice.

Self-assembly of spiral patterns in confined systems with competing interactions

Colloidal particles in polymer solutions and functionalized nanoparticles often exhibit short-range attraction coupled with long-range repulsion (SALR) leading to the spontaneous formation of symmetric patterns. Chiral nanostructures formed by thin films of SALR particles have not been reported yet. In this study, we observe striking topological transitions from a symmetric pattern of concentric rings to a chiral structure of a spiral shape, when the system is in hexagonal confinement. We find that the spiral formation can be induced either by breaking the system symmetry with a wedge, or by melting of the rings. In the former case, the chirality of the spiral is determined by the orientation of the wedge and thus can be controlled. In the latter, the spiral arises due to thermally induced defects and is absent in the average particle distribution, which forms highly regular hexagonal patterns in the central part of the system. These hexagonal patterns can be explained by interference of planar density waves. Thermodynamic considerations indicate that equilibrium spirals can appear spontaneously in any stripe-forming system confined in a hexagon with a small wedge, provided that certain conditions are satisfied by a set of phenomenological parameters.

Effect of aggregation on adsorption phenomena

Adsorption at an attractive surface in a system with particles self-assembling into small clusters is studied by molecular dynamics simulation. We assume Lennard-Jones plus repulsive Yukawa tail interactions and focus on small densities. The relative increase in the temperature at the critical cluster concentration near the attractive surface (CCCS) shows a power-law dependence on the strength of the wall-particle attraction. At temperatures below the CCCS, the adsorbed layer consists of undeformed clusters if the wall-particle attraction is not too strong. Above the CCCS or for strong attraction leading to flattening of the adsorbed aggregates, we obtain a monolayer that for strong or very strong attraction consists of flattened clusters or stripes, respectively. The accumulated repulsion from the particles adsorbed at the wall leads to a repulsive barrier that slows down the adsorption process, and the accession time grows rapidly with the strength of the wall-particle attraction. Beyond the adsorbed layer of particles, a depletion region of a thickness comparable with the range of the repulsive tail of interactions occurs, and the density in this region decreases with increasing strength of the wall-particle attraction. At larger separations, the exponentially damped oscillations of density agree with theoretical predictions for self-assembling systems. Structural and thermal properties of the bulk are also determined. In particular, a new structural crossover associated with the maximum of the specific heat and a double-peaked histogram of the cluster size distribution are observed.

3D structure of the electric double layer of ionic liquid-alcohol mixtures at the electrochemical interface

Mixtures of the ionic liquid 1-butyl-3-methylimidazolium tetrafluoroborate with amphiphilic cosolvents, such as methanol and ethanol, nanoconfined between graphene walls are studied by means of molecular dynamics simulations and the results are compared with those of the pure ionic liquid and its mixtures with water confined in the same conditions. We investigate the adsorption of cosolvent molecules at the graphene walls as well as their distribution across the system. The results show that, due to a higher affinity of the polar groups to be close to the anions in combination with the electrostatic and excluded volume interactions, there exists a high tendency of the OH groups to lie close to the anode, inducing small changes in the first cation layer. The orientation of cosolvent molecules is found to be closely related to the alignment of the molecular dipole moment. We also investigate the lateral ionic distribution in the layers close to the electrodes, which shows a structural transition from liquid-like lamellar ordering to solid-like hexagonal patterns as the size of the cosolvent molecules increases leading to smaller position fluctuations of the ions. The dependence of the specific patterns on the nature of the electrodes is also studied. This study strongly suggests that the ionic patterns formed in the first ionic layers next to the charged interfaces are universal since their existence does not crucially depend on the atomic composition of the interfacial material, but only on the net charge density of the considered ionic layer, which significantly changes the ionic mobility in this region.

Combined density functional and Brazovskii theories for systems with spontaneous inhomogeneities

The low-T part of the phase diagram in self-assembling systems is correctly predicted by known versions of density functional theory (DFT). The high-T part obtained in DFT, however, does not agree with simulations even on the qualitative level. In this work, a new version of DFT for systems with spontaneous inhomogeneities on a mesoscopic length scale is developed. The contribution to the grand thermodynamic potential associated with mesoscopic fluctuations is explicitly taken into account. The expression for this contribution is obtained by methods known from the Brazovskii field theory. Apart from developing the approximate expression for the grand thermodynamic potential that contains the fluctuation contribution and is ready for numerical minimization, we develop a simplified version of the theory valid for weakly ordered phases, i.e. for the high-T part of the phase diagram. The simplified theory is verified by comparison with the results of simulations for a particular version of the short-range attraction long-range repulsion (SALR) interaction potential. Except for the fact that in our theory the ordered phases are stable at lower T than in simulations, a good agreement for the high-T part of the phase diagram is obtained for the range of density that was considered in simulations. In addition, the equation of state and compressibility isotherms are presented. Finally, the physical interpretation of the fluctuation-contribution to the grand potential is discussed in detail.

Orientational ordering of lamellar structures on closed surfaces

Self-assembly of particles with short-range attraction and long-range repulsion interactions on a flat and on a spherical surface is compared. Molecular dynamics simulations are performed for the two systems having the same area and the density optimal for formation of stripes of particles. Structural characteristics, e.g., a cluster size distribution, a number of defects, and an orientational order parameter (OP), as well as the specific heat, are obtained for a range of temperatures. In both cases, the cluster size distribution becomes bimodal and elongated clusters appear at the temperature corresponding to the maximum of the specific heat. When the temperature decreases, orientational ordering of the stripes takes place and the number of particles per cluster or stripe increases in both cases. However, only on the flat surface, the specific heat has another maximum at the temperature corresponding to a rapid change of the OP. On the sphere, the crossover between the isotropic and anisotropic structures occur in a much broader temperature interval; the orientational order is weaker and occurs at significantly lower temperature. At low temperature, the stripes on the sphere form spirals and the defects resemble defects in the nematic phase of rods adsorbed at a sphere.

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.

Two-dimensional pattern formation in ionic liquids confined between graphene walls

We perform molecular dynamics simulations of ionic liquids confined between graphene walls under a large variety of conditions (pure ionic liquids, mixtures with water and alcohols, mixtures with lithium salts and defective graphene walls). Our results show that the formation of striped and hexagonal patterns in the Stern layer can be considered as a general feature of ionic liquids at electrochemical interfaces, the transition between patterns being controlled by the net balance of charge in the innermost layer of adsorbed molecules. This explains previously reported experimental and computational results and, for the first time, why these pattern changes are triggered by any perturbation of the charge density at the innermost layer of the electric double layer (voltage and composition changes, and vacancies at the electrode walls, among others), which may help tuning electrode-ionic liquid interfaces. Using Monte Carlo simulations we show that such structures can be reproduced by a simple two-dimensional lattice model with only nearest-neighbour interactions, governed by highly screened ionic interactions and short-range and excluded volume interactions. We also show that the results of our simulations are consistent with those inferred from the Landau-Brazovskii theory of pattern formation in self-assembling systems. The presence of these patterns at the ionic liquid graphene-electrode interfaces may have a strong impact on the process of ionic transfer from the bulk mixtures to the electrodes, on the differential capacitance of the electrode-electrolyte double layer or on the rates of redox reactions at the electrodes, among other physicochemical properties, and is therefore an effect of great technological interest.

Thermodynamic signatures and cluster properties of self-assembly in systems with competing interactions

Colloidal particles, amphiphiles and functionalized nanoparticles are examples of systems that frequently exhibit short-range attraction coupled with long-range repulsion. We vary the ratio of attraction and repulsion in a simple isotropic model with competing interactions, using molecular simulations, and observe significant differences in the properties of the self-assembled clusters that form. We report conditions that lead to the self-assembly of clusters of a preferred size, accompanied by a change in the slope of the pressure with respect to density, similar to micelles formed by amphiphilic molecules. We also report conditions where repulsion dominates, clusters of a preferred size form and the pressure vs. density slope is unaffected by self-assembly. We investigate cluster structure by calculating the size distributions, free colloid density, cluster shape and density profiles. The system dynamics are characterized by cluster life-times. We do not find qualitative differences in structure or dynamics of the clusters, regardless the pressure behavior. Therefore, thermodynamic and structural quantities are required to classify the different clustering characteristics that are observable in systems with competing interactions. Our results have implications in terms of development of design principles for stable cluster self-assembly.

Exactly solvable model for self-assembly of hard core-soft shell particles at interfaces

A generic model for self-assembly of a monolayer of hybrid core-shell particles at an interface is developed. We assume that for distances larger than the size of the incompressible core a soft repulsion appears, and the repulsion is followed by an attraction at larger separations. The model is solved exactly in a one-dimensional lattice version. One, two or three periodic structures and variety of shapes of the pressure-density isotherms may occur in different versions of the model. For strong interactions the isotherm consists of nearly vertical segments at densities optimal for the periodic structures that are connected by segments with a small slope. The range of order depends very strongly on the strength of attraction and on the density. Our results agree with experimental observations.

Interactions and design rules for assembly of porous colloidal mesophases

Porous mesophases, where well-defined particle-depleted 'void' spaces are present within a particle-rich background fluid, can be self-assembled from colloidal particles interacting via isotropic pair interactions with competing attractions and repulsions. While such structures could be of wide interest for technological applications (e.g., filtration, catalysis, absorption, etc.), relatively few studies have investigated the interactions that lead to these morphologies and how they compare to those that produce other micro-phase-separated structures, such as clusters. In this work, we use inverse methods of statistical mechanics to design model isotropic pair potentials that form porous mesophases. We characterize the resulting porous structures, correlating features of the pair potential with the targeted pore size and the particle packing fraction. The former is primarily encoded by the amplitude and range of the repulsive barrier of the designed pair potential and the latter by the attractive well depth. We observe a trade-off with respect to the packing fraction of the targeted morphology: greater values support more spherical and monodisperse pores that themselves organize into periodic structures, while lower values yield more mobile pores that do not assemble into ordered structures but remain stable over a larger range of packing fraction. We conclude by commenting on the limitations of targeting a specific pore diameter within the present inverse design approach as well as by describing future directions to overcome these limitations.

Criticality of the phase transition on stage two in a lattice-gas model of a graphite anode in a lithium-ion battery

Herein, a Monte Carlo study within the canonical assembly has been applied to elucidate the lithium-ion phase transition order of a stage II lithium–graphite intercalation compound (LiC₁₂).

Self-consistent theory for systems with mesoscopic fluctuations

We have developed a theory for inhomogeneous systems that allows for the incorporation of the effects of mesoscopic fluctuations. A hierarchy of equations relating the correlation and direct correlation functions for the local excess [Formula: see text] of the volume fraction of particles ζ has been obtained, and an approximation leading to a closed set of equations for the two-point functions has been introduced for the disordered inhomogeneous phase. We have numerically solved the self-consistent equations for one-dimensional (1D) and three-dimensional (3D) models with short-range attraction and long-range repulsion. Predictions for all of the qualitative properties of the 1D model agree with the exact results, but only semi-quantitative agreement is obtained in the simplest version of the theory. The effects of fluctuations in the two 3D models considered are significantly different, despite the very similar properties of these models in the mean-field approximation. In both cases we obtain the sequence of large-small-large compressibility for increasing ζ. The very small compressibility is accompanied by the oscillatory decay of correlations with correlation lengths that are orders of magnitude larger than the size of particles. In one of the two models considered, the small compressibility becomes very small and the large compressibility becomes very large with decreasing temperature, and eventually van der Waals loops appear. Further studies are necessary in order to determine the nature of the strongly inhomogeneous phase present for intermediate volume fractions in 3D.

Effects of confinement on pattern formation in two dimensional systems with competing interactions

Template-assisted pattern formation in monolayers of particles with competing short-range attraction and long-range repulsion interactions (SALR) is studied by Monte Carlo simulations in a simple generic model [N. G. Almarza et al., J. Chem. Phys., 2014, 140, 164708]. We focus on densities corresponding to formation of parallel stripes of particles and on monolayers laterally confined between straight parallel walls. We analyze both the morphology of the developed structures and the thermodynamic functions for broad ranges of temperature T and the separation L2 between the walls. At low temperature stripes parallel to the boundaries appear, with some corrugation when the distance between the walls does not match the bulk periodicity of the striped structure. The stripes integrity, however, is rarely broken for any L2. This structural order is lost at T = TK(L2) depending on L2 according to a Kelvin-like equation. Above the Kelvin temperature TK(L2) many topological defects such as breaking or branching of the stripes appear, but a certain anisotropy in the orientation of the stripes persists. Finally, at high temperature and away from the walls, the system behaves as an isotropic fluid of elongated clusters of various lengths and with various numbers of branches. For L2 optimal for the stripe pattern the heat capacity as a function of temperature takes the maximum at T = TK(L2).

Density functional theory for systems with mesoscopic inhomogeneities

We study the effects of fluctuations on the mesoscopic length scale on systems with mesoscopic inhomogeneities. Equations for the correlation function and for the average volume fraction are derived in the self-consistent Gaussian approximation. The equations are further simplified by postulating the expression for the structure factor consistent with scattering experiments for self-assembling systems. Predictions of the approximate theory are verified by a comparison with the exact results obtained earlier for the one-dimensional lattice model with first-neighbor attraction and third-neighbor repulsion. We find qualitative agreement for the correlation function, the equation of state and the dependence of the chemical potential μ on the volume fraction ζ. Our results confirm also that strong inhomogeneities in the disordered phase are found only in the case of strong repulsion. The inhomogeneities are reflected in an oscillatory decay of the correlation function with a very large correlation length, three inflection points in the [Formula: see text] curve and a compressibility that for increasing ζ takes very large, very small and again very large values.

Assembly of nothing: equilibrium fluids with designed structured porosity

Controlled micro- to meso-scale porosity is a common materials design goal with possible applications ranging from molecular gas adsorption to particle size selective permeability or solubility. Here, we use inverse methods of statistical mechanics to design an isotropic pair interaction that, in the absence of an external field, assembles particles into an inhomogeneous fluid matrix surrounding pores of prescribed size ordered in a lattice morphology. The pore size can be tuned via modification of temperature or particle concentration. Moreover, modulating density reveals a rich series of microphase-separated morphologies including pore- or particle-based lattices, pore- or particle-based columns, and bicontinuous or lamellar structures. Sensitivity of pore assembly to the form of the designed interaction potential is explored.

Pattern formation in binary fluid mixtures induced by short-range competing interactions

Molecular dynamics simulations and integral equation calculations of a simple equimolar mixture of diatomic molecules and monomers interacting via attractive and repulsive short-range potentials show the existence of pattern formation (microheterogeneity), mostly due to depletion forces away from the demixing region. Effective site-site potentials extracted from the pair correlation functions using an inverse Monte Carlo approach and an integral equation inversion procedure exhibit the features characteristic of a short-range attractive and a long-range repulsive potential. When charges are incorporated into the model, this becomes a coarse grained representation of a room temperature ionic liquid, and as expected, intermediate range order becomes more pronounced and stable.

Effects of rigid or adaptive confinement on colloidal self-assembly. Fixed vs. fluctuating number of confined particles

The effects of confinement on colloidal self-assembly in the case of fixed number of confined particles are studied in the one dimensional lattice model solved exactly in the grand canonical ensemble (GCE) in Pȩkalski et al. [J. Chem. Phys. 142, 014903 (2015)]. The model considers a pair interaction defined by a short-range attraction plus a longer-range repulsion. We consider thermodynamic states corresponding to self-assembly into clusters. Both fixed and adaptive boundaries are studied. For fixed boundaries, there are particular states in which, for equal average densities, the number of clusters in the GCE is larger than in the canonical ensemble. The dependence of pressure on density has a different form when the system size changes with fixed number of particles and when the number of particles changes with fixed size of the system. In the former case, the pressure has a nonmonotonic dependence on the system size. The anomalous increase of pressure for expanding system is accompanied by formation of a larger number of smaller clusters. In the case of elastic confining surfaces, we observe a bistability, i.e., two significantly different system sizes occur with almost the same probability. The mechanism of the bistability in the closed system is different to that of the case of permeable walls, where the two equilibrium system sizes correspond to a different number of particles.

Nematic phase in the J(1)-J(2) square-lattice Ising model in an external field

The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.

Bistability in a self-assembling system confined by elastic walls: exact results in a one-dimensional lattice model

The impact of confinement on self-assembly of particles interacting with short-range attraction and long-range repulsion potential is studied for thermodynamic states corresponding to local ordering of clusters or layers in the bulk. Exact and asymptotic expressions for the local density and for the effective potential between the confining surfaces are obtained for a one-dimensional lattice model introduced by J. Pȩkalski et al. [J. Chem. Phys. 138, 144903 (2013)]. The simple asymptotic formulas are shown to be in good quantitative agreement with exact results for slits containing at least 5 layers. We observe that the incommensurability of the system size and the average distance between the clusters or layers in the bulk leads to structural deformations that are different for different values of the chemical potential μ. The change of the type of defects is reflected in the dependence of density on μ that has a shape characteristic for phase transitions. Our results may help to avoid misinterpretation of the change of the type of defects as a phase transition in simulations of inhomogeneous systems. Finally, we show that a system confined by soft elastic walls may exhibit bistability such that two system sizes that differ approximately by the average distance between the clusters or layers are almost equally probable. This may happen when the equilibrium separation between the soft boundaries of an empty slit corresponds to the largest stress in the confined self-assembling system.

Stripe systems with competing interactions on quasi-one dimensional periodic substrates

We numerically examine the two-dimensional ordering of a stripe forming system of particles with competing long-range repulsion and short-range attraction in the presence of a quasi-one-dimensional corrugated substrate. As a function of increasing substrate strength or period we show that a remarkable variety of distinct orderings can be realized, including modulated stripes, prolate clump phases, two dimensional ordered kink structures, crystalline void phases, and smectic phases. Additionally in some cases the stripes align perpendicular to the substrate troughs. Our results suggest that a new route to self assembly for systems with competing interactions can be achieved through the addition of a simple periodic modulated substrate.