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.

Phase diagram of SALR fluids on spherical surfaces.

We investigate the phase diagram of a fluid of hard-core disks confined to the surface of a sphere and whose interaction potential contains a short-range attraction followed by a long-range repulsive tail (SALR). Based on previous work in the bulk we derive a stability criterion for the homogeneous phase of the fluid, and locate a region of instability linked to the presence of a negative minimum in the spherical harmonics expansion of the interaction potential. The inhomogeneous phases contained within this region are characterized using a mean-field density functional theory. We find several inhomogeneous patterns that can be separated into three broad classes: cluster crystals, stripes, and bubble crystals, each containing topological defects. Interestingly, while the periodicity of inhomogeneous phases at large densities is mainly determined by the position of the negative minimum of the potential expansion, the finite size of the system induces a richer behavior at smaller densities.

On the degeneracy of ordered ground state configurations of the aspherical Gaussian core model.

We provide rigorous evidence that the ordered ground state configurations of a system of parallel oriented, ellipsoidal particles, interacting via a Gaussian potential (termed in the literature as Gaussian core nematics), must be infinitely degenerate; we have demonstrated that these configurations originate from the related ground state configuration of the corresponding symmetric Gaussian core system via a suitable stretching operation of this lattice in combination with an arbitrary rotation. These findings explain related observations in former investigations, which then remained unexplained. Our conclusions have far reaching consequences for the search of ground state configurations of other nematic particles.

Formation of cluster crystals in an ultra-soft potential model on a spherical surface.

We investigate the formation of cluster crystals with multiply occupied lattice sites on a spherical surface in systems of ultra-soft particles interacting via repulsive, bounded pair potentials. Not all interactions of this kind lead to clustering: we generalize the criterion devised in C. N. Likos et al., Phys. Rev. E, 2001, 63, 031206 to spherical systems in order to distinguish between cluster-forming systems and fluids which display reentrant melting. We use both DFT and Monte Carlo simulations to characterize the behavior of the system, and obtain semi-quantitative agreement between the two. We find that the number of clusters is determined by the ratio between the size σ of the ultra-soft particles and the radius R of the sphere in such a way that each stable configuration spans a certain interval of σ/R. Furthermore, we study the effect of topological frustration on the system due to the sphere curvature by comparing the properties of disclinations, i.e., clusters with fewer than six neighbors, and non-defective clusters. Disclinations are shown to be less stable, contain fewer particles, and be closer to their neighbors than other lattice points: these properties are explained on the basis of geometric and energetic considerations.

Some general features of mesophase formation in hard-core plus tail potentials.

The formation of mesophases in fluids with hard-core plus tail interactions is investigated and compared with the occurrence of cluster crystals in ultra-soft repulsive potentials by using a simple variational expression for the Helmholtz free energy. The purpose of this study is mostly qualitative, i.e., to explain the origin of the different behavior of these systems, and the reason why, in the hard-core case, interactions which are apparently quite different display a common pattern for the phase diagram, featuring spheres, cylinders, lamellae, inverted cylinders, and inverted spheres as the density is increased. In the limit of zero temperature, our approach also yields some simple predictions for the densities at which the transitions between different mesophases are expected to take place, as well as for the size of their clusters at the transitions. We find that these results compare favorably with those obtained in a former study of a model fluid with competing attractive and repulsive interactions by density-functional theory with numerical minimization.

Pattern formation and self-assembly driven by competing interactions.

Colloidal fluids interacting via effective potentials which are attractive at the short range and repulsive at the long range have long been raising considerable attention because such an instance provides a simple mechanism leading to pattern formation even for isotropic interactions. If the competition between attraction and repulsion is strong enough, the gas-liquid phase transition is suppressed, and replaced by the formation of mesophases, i.e., inhomogeneous phases displaying periodic density modulations whose length, although being larger than the particle size, cannot nevertheless be considered macroscopic. We describe a fully numerical implementation of density-functional theory in three dimensions, tailored to periodic phases. The results for the equilibrium phase diagram of the model are compared with those already obtained in previous investigations for the present system as well as for other systems which form mesophases. The phase diagram which we find shows a strong similarity with that of block copolymer melts, in which self-assembly also results from frustration of a macroscopic phase separation. As the inhomogeneous region is swept by increasing the density from the low-density side, one encounters clusters, bars, lamellae, inverted bars, and inverted clusters. Moreover, a bicontinuous gyroid phase consisting of two intertwined percolating networks is predicted in a narrow domain between the bar and lamellar phases.

Quantum Critical Behavior of One-Dimensional Soft Bosons in the Continuum.

We consider a zero-temperature one-dimensional system of bosons interacting via the soft-shoulder potential in the continuum, typical of dressed Rydberg gases. We employ quantum Monte Carlo simulations, which allow for the exact calculation of imaginary-time correlations, and a stochastic analytic continuation method, to extract the dynamical structure factor. At finite densities, in the weakly interacting homogeneous regime, a rotonic spectrum marks the tendency to clustering. With strong interactions, we indeed observe cluster liquid phases emerging, characterized by the spectrum of a composite harmonic chain. Luttinger theory has to be adapted by changing the reference lattice density field. In both the liquid and cluster liquid phases, we find convincing evidence of a secondary mode, which becomes gapless only at the transition. In that region, we also measure the central charge and observe its increase towards c=3/2, as recently evaluated in a related extended Bose-Hubbard model, and we note a fast reduction of the Luttinger parameter. For two-particle clusters, we then interpret such observations in terms of the compresence of a Luttinger liquid and a critical transverse Ising model, related to the instability of the reference lattice density field towards coalescence of sites, typical of potentials which are flat at short distances. Even in the absence of a true lattice, we are able to evaluate the spatial correlation function of a suitable pseudospin operator, which manifests ferromagnetic order in the cluster liquid phase, exponential decay in the liquid phase, and algebraic order at criticality.

Corrigendum: Theory and computer simulation of hard-core Yukawa mixtures: thermodynamical, structural and phase coexistence properties.

A number of references in the bibliography were reported incorrectly.

Theory and computer simulation of hard-core Yukawa mixtures: thermodynamical, structural and phase coexistence properties.

We report extensive calculations, based on the modified hypernetted chain (MHNC) theory, on the hierarchical reference theory (HRT), and on Monte Carlo simulations, of thermodynamical, structural and phase coexistence properties of symmetric binary hard-core Yukawa mixtures (HCYM) with attractive interactions at equal species concentration. The obtained results are throughout compared with those available in the literature for the same systems. It turns out that the MHNC predictions for thermodynamic and structural quantities are quite accurate in comparison with the MC data. The HRT is equally accurate for thermodynamics, and slightly less accurate for structure. Liquid-vapor (LV) and liquid-liquid (LL) consolute coexistence conditions as emerging from simulations, are also highly satisfactorily reproduced by both the MHNC and HRT for relatively long ranged potentials. When the potential range reduces, the MHNC faces problems in determining the LV binodal line; however, the LL consolute line and the critical end point (CEP) temperature and density turn out to be still satisfactorily predicted within this theory. The HRT also predicts with good accuracy the CEP position. The possibility of employing liquid state theories HCYM for the purpose of reliably determining phase equilibria in multicomponent colloidal fluids of current technological interest, is discussed.

An unconstrained DFT approach to microphase formation and application to binary Gaussian mixtures.

The formation of microphases in systems of particles interacting by repulsive, bounded potentials is studied by means of density-functional theory (DFT) using a simple, mean-field-like form for the free energy which has already been proven accurate for this class of soft interactions. In an effort not to constrain the configurations available to the system, we do not make any assumption on the functional form of the density profile ρ(r), save for its being periodic. We sample ρ(r) at a large number of points in the unit cell and minimize the free energy with respect to both the values assumed by ρ(r) at these points and the lattice vectors which identify the Bravais lattice. After checking the accuracy of the method by applying it to a one-component generalized exponential model (GEM) fluid with pair potential ϵexp[ - (r/R)(4)], for which extensive DFT and simulation results are already available, we turn to a binary mixture of Gaussian particles which some time ago was shown to support microphase formation [A. J. Archer, C. N. Likos, and R. Evans, J. Phys.: Condens. Matter 16, L297 (2004)], but has not yet been investigated in detail. The phase diagram which we obtain, that supersedes the tentative one proposed by us in a former study [M. Carta, D. Pini, A. Parola, and L. Reatto, J. Phys.: Condens. Matter 24, 284106 (2012)], displays cluster, tubular, and bicontinuous phases similar to those observed in block copolymers or oil/water/surfactant mixtures. Remarkably, bicontinuous phases occupy a rather large portion of the phase diagram. We also find two non-cubic phases, in both of which one species is preferentially located inside the channels left available by the other, forming helices of alternating chirality. The features of cluster formation in this mixture and in GEM potentials are also compared.

Self-Consistent Ornstein-Zernike Approximation (SCOZA) and exact second virial coefficients and their relationship with critical temperature for colloidal or protein suspensions with short-ranged attractive interactions.

We focus on the second virial coefficient B2 of fluids with molecules interacting through hard-sphere potentials plus very short-ranged attractions, namely, with a range of attraction smaller than half hard-sphere diameter. This kind of interactions is found in colloidal or protein suspensions, while the interest in B2 stems from the relation between this quantity and some other properties of these fluid systems. Since the SCOZA (Self-Consistent Ornstein-Zernike Approximation) integral equation is known to yield accurate thermodynamic and structural predictions even near phase transitions and in the critical region, we investigate B2(SCOZA) and compare it with B2(exact), for some typical potential models. The aim of the paper is however twofold. First, by expanding in powers of density the condition of thermodynamic consistency included in the SCOZA integral equation, a general analytic expression for B2(SCOZA) is derived. For a given potential model, a comparison between B2(SCOZA) and B2(exact) may help to estimate the regimes where the SCOZA closure is reliable. Second, following the Vliegenthart-Lekkerkerker (VL) and Noro-Frenkel suggestions, the relationship between the critical B2 and the critical temperature Tc is discussed in detail for two prototype models: the square-well (SW) potential and the hard-sphere attractive Yukawa (HSY) one. The known simulation data for the SW model are revisited, while for the HSY model new SCOZA results have been generated. Although B2(HSY) at the critical temperature is found to be a slowly varying function of the range of Yukawa attraction ΔY over a wide interval of ΔY, it turns out to diverge as ΔY vanishes. For fluids with very short-ranged attractions, such a behavior contrasts with the VL assumption that B2 at the critical temperature should be nearly independent of the range of attraction. A very simple analytic representation is found for the available Monte Carlo data for Tc(HSY) and B2(HSY) as functions of the range of attraction, for ΔY smaller than half hard-sphere diameter.

Thermodynamic properties of short-range attractive Yukawa fluid: simulation and theory.

Coexistence properties of the hard-core attractive Yukawa potential with inverse-range parameter kappa=9, 10, 12, and 15 are calculated by applying canonical Monte Carlo simulation. As previously shown for longer ranges, we show that also for the ranges considered here the coexistence curves scaled by the critical density and temperature obey the law of corresponding states, and that a linear relationship between the critical density and the reciprocal of the critical temperature holds. The simulation results are compared to the predictions of the self-consistent Ornstein-Zernike approximation, and a good agreement is found for both the critical points and the coexistence curves, although some slight discrepancies are present.

Liquid-vapor transition from a microscopic theory: beyond the Maxwell construction.

A smooth cutoff formulation of the hierarchical reference theory (HRT) is developed and applied to a Yukawa fluid. The HRT equations are derived and numerically solved leading to the expected renormalization group structure in the critical region, nonclassical critical exponents and scaling laws, a convex free energy in the whole phase diagram (including the two-phase region), finite compressibility at coexistence, together with a fully satisfactory comparison with available numerical simulations. This theory, which also guarantees the correct short range behavior of two body correlations, represents a major improvement over the existing liquid-state theories.

Smooth cutoff formulation of hierarchical reference theory for a scalar phi4 field theory.

The phi4 scalar field theory in three dimensions, prototype for the study of phase transitions, is investigated by means of the hierarchical reference theory (HRT) in its smooth cutoff formulation. The critical behavior is described by scaling laws and critical exponents which compare favorably with the known values of the Ising universality class. The inverse susceptibility vanishes identically inside the coexistence curve, providing a first principle implementation of the Maxwell construction, and shows the expected discontinuity across the phase boundary, at variance with the usual sharp cutoff implementation of HRT. The correct description of first and second order phase transitions within a microscopic, nonperturbative approach is thus achieved in the smooth cutoff HRT.

Phase equilibria and glass transition in colloidal systems with short-ranged attractive interactions: application to protein crystallization.

We have studied a model of a complex fluid consisting of particles interacting through a hard-core and short-range attractive potential of both Yukawa and square-well form. Using a hybrid method, including a self-consistent and quite accurate approximation for the liquid integral equation in the case of the Yukawa fluid, perturbation theory to evaluate the crystal free energies, and mode-coupling theory of the glass transition, we determine both the equilibrium phase diagram of the system and the lines of equilibrium between the supercooled fluid and the glass phases. For these potentials, we study the phase diagrams for different values of the potential range, the ratio of the range of the interaction to the diameter of the repulsive core being the main control parameter. Our arguments are relevant to a variety of systems, from dense colloidal systems with depletion forces, through particle gels, nanoparticle aggregation, and globular protein crystallization.