Phase transitions in highly asymmetric binary hard-sphere fluids: Fluid-fluid binodal from a two-component mixture theory

Entropy-Driven Crystallization of Hard Colloidal Mixtures of Polymers and Monomers

The most trivial example of self-assembly is the entropy-driven crystallization of hard spheres. Past works have established the similarities and differences in the phase behavior of monomers and chains made of hard spheres. Inspired by the difference in the melting points of the pure components, we study, through Monte Carlo simulations, the phase behavior of athermal mixtures composed of fully flexible polymers and individual monomers of uniform size. We analyze how the relative number fraction and the packing density affect crystallization and the established ordered morphologies. As a first result, a more precise determination of the melting point for freely jointed chains of tangent hard spheres is extracted. A synergetic effect is observed in the crystallization leading to synchronous crystallization of the two species. Structural analysis of the resulting ordered morphologies shows perfect mixing and thus no phase separation. Due to the constraints imposed by chain connectivity, the local environment of the individual spheres, as quantified by the Voronoi polyhedron, is systematically more spherical and more symmetric compared to that of spheres belonging to chains. In turn, the local environment of the ordered phase is more symmetric and more spherical compared to that of the initial random packing, demonstrating the entropic origins of the phase transition. In general, increasing the polymer content reduces the degree of crystallinity and increases the melting point to higher volume fractions. According to the present findings, relative concentration is another determining factor in controlling the phase behavior of hard colloidal mixtures based on polymers.

Structure ordering and glass transition in size-asymmetric ternary mixtures of hard spheres: Variation from fragile to strong glasses

We investigate the structure and activated dynamics of a binary mixture of colloidal particles dispersed in a solvent of much smaller-sized particles. The solvent degrees of freedom are traced out from the grand partition function of the colloid-solvent mixture which reduces the system from ternary to effective binary mixture of colloidal particles. In the effective binary mixture colloidal particles interact via effective potential that consists of bare potential plus the solvent-induced interaction. Expressions for the effective potentials and pair correlation functions are derived. We used the result of pair correlation functions to determine the number of particles in a cooperatively reorganizing cluster (CRC) in which localized particles form "long-lived" nonchemical bonds with the central particle. For an event of relaxation to take place these bonds have to reorganize irreversibly, the energy involved in the processes is the effective activation energy of relaxation. Results are reported for hard sphere colloidal particles dispersed in a solvent of hard sphere particles. Our results show that the concentration of solvent can be used as a control parameter to fine-tune the microscopic structural ordering and the size of CRC that governs the glassy dynamics. We show that a small variation in the concentration of solvent creates a bigger change in the kinetic fragility which highlights a wide variation in behavior, ranging from fragile to strong glasses. We conclude that the CRC which is determined from the static pair correlation function and the fluctuations embedded in the system is probably the sole player in the physics of glass transition.

Critical point for demixing of binary hard spheres

We use a two-level simulation method to analyze the critical point associated with demixing of binary hard-sphere mixtures. The method exploits an accurate coarse-grained model with two- and three-body effective interactions. Using this model within the two-level methodology allows computation of properties of the full (fine-grained) mixture. The critical point is located by computing the probability distribution for the number of large particles in the grand canonical ensemble and matching to the universal form for the 3D Ising universality class. The results have a strong and unexpected dependence on the size ratio between large and small particles, which is related to three-body effective interactions and the geometry of the underlying hard-sphere packings.

Structural and thermodynamic properties of hard-sphere fluids

This Perspective article provides an overview of some of our analytical approaches to the computation of the structural and thermodynamic properties of single-component and multicomponent hard-sphere fluids. For the structural properties, they yield a thermodynamically consistent formulation, thus improving and extending the known analytical results of the Percus-Yevick theory. Approximate expressions linking the equation of state of the single-component fluid to the one of the multicomponent mixtures are also discussed.

Glass transition and reversible gelation in asymmetric binary mixtures: A study by mode coupling theory and molecular dynamics

The glass transition and the binodals of asymmetric binary mixtures are investigated from the effective fluid approach in the mode coupling theory and by molecular dynamics. Motivated by previous theoretical predictions, the hard-sphere mixture and the Asakura-Oosawa models are used to analyze experimental results from the literature, relative to polystyrene spheres mixed either with linear polymers or with dense microgel particles. In agreement with the experimental observations, the specificity of the depletant particles is shown to favor lower density gels. It further favors equilibrium gelation by reducing also the tendency of the system to phase separate. These results are confirmed by a phenomenological modification of the mode coupling theory in which the vertex functions are computed at an effective density lower than the actual one. A model effective potential in asymmetric mixtures of hard particles is used to further check this phenomenological modification against molecular dynamics simulation.

Pseudo hard-sphere potential for use in continuous molecular-dynamics simulation of spherical and chain molecules

We present a continuous pseudo-hard-sphere potential based on a cut-and-shifted Mie (generalized Lennard-Jones) potential with exponents (50, 49). Using this potential one can mimic the volumetric, structural, and dynamic properties of the discontinuous hard-sphere potential over the whole fluid range. The continuous pseudo potential has the advantage that it may be incorporated directly into off-the-shelf molecular-dynamics code, allowing the user to capitalise on existing hardware and software advances. Simulation results for the compressibility factor of the fluid and solid phases of our pseudo hard spheres are presented and compared both to the Carnahan-Starling equation of state of the fluid and published data, the differences being indistinguishable within simulation uncertainty. The specific form of the potential is employed to simulate flexible chains formed from these pseudo hard spheres at contact (pearl-necklace model) for m(c) = 4, 5, 7, 8, 16, 20, 100, 201, and 500 monomer segments. The compressibility factor of the chains per unit of monomer, m(c), approaches a limiting value at reasonably small values, m(c) < 50, as predicted by Wertheim's first order thermodynamic perturbation theory. Simulation results are also presented for highly asymmetric mixtures of pseudo hard spheres, with diameter ratios of 3:1, 5:1, 20:1 over the whole composition range.

Non-ergodicity transition and multiple glasses in binary mixtures: on the accuracy of the input static structure in the mode coupling theory

We examine the question of the accuracy of the static correlation functions used as input in the mode coupling theory (MCT) of non-ergodic states in binary mixtures. We first consider hard-sphere mixtures and compute the static pair structure from the Ornstein-Zernike equations with the Percus-Yevick closure and more accurate ones that use bridge functions deduced from Rosenfeld's fundamental measures functional. The corresponding MCT predictions for the non-ergodicity lines and the transitions between multiple glassy states are determined from the long-time limit of the density autocorrelation functions. We find that while the non-ergodicity transition line is not very sensitive to the input static structure, up to diameter ratios D(2)/D(1) = 10, quantitative differences exist for the transitions between different glasses. The discrepancies with the more accurate closures become even qualitative for sufficiently asymmetric mixtures. They are correlated with the incorrect behavior of the PY structure at high size asymmetry. From the example of ultra-soft potential it is argued that this issue is of general relevance beyond the hard-sphere model.

Depletion potentials in highly size-asymmetric binary hard-sphere mixtures: comparison of simulation results with theory

We report a detailed study, using state-of-the-art simulation and theoretical methods, of the effective (depletion) potential between a pair of big hard spheres immersed in a reservoir of much smaller hard spheres, the size disparity being measured by the ratio of diameters q ≡ σ(s)/σ(b). Small particles are treated grand canonically, their influence being parameterized in terms of their packing fraction in the reservoir η(s)(r). Two Monte Carlo simulation schemes--the geometrical cluster algorithm, and staged particle insertion--are deployed to obtain accurate depletion potentials for a number of combinations of q ≤ 0.1 and η(s)(r). After applying corrections for simulation finite-size effects, the depletion potentials are compared with the prediction of new density functional theory (DFT) calculations based on the insertion trick using the Rosenfeld functional and several subsequent modifications. While agreement between the DFT and simulation is generally good, significant discrepancies are evident at the largest reservoir packing fraction accessible to our simulation methods, namely, η(s)(r) = 0.35. These discrepancies are, however, small compared to those between simulation and the much poorer predictions of the Derjaguin approximation at this η(s)(r). The recently proposed morphometric approximation performs better than Derjaguin but is somewhat poorer than DFT for the size ratios and small-sphere packing fractions that we consider. The effective potentials from simulation, DFT, and the morphometric approximation were used to compute the second virial coefficient B(2) as a function of η(s)(r). Comparison of the results enables an assessment of the extent to which DFT can be expected to correctly predict the propensity toward fluid-fluid phase separation in additive binary hard-sphere mixtures with q ≤ 0.1. In all, the new simulation results provide a fully quantitative benchmark for assessing the relative accuracy of theoretical approaches for calculating depletion potentials in highly size-asymmetric mixtures.

Binary non-additive hard sphere mixtures: fluid demixing, asymptotic decay of correlations and free fluid interfaces

Using a fundamental measure density functional theory we investigate both bulk and inhomogeneous systems of the binary non-additive hard sphere model. For sufficiently large (positive) non-additivity the mixture phase separates into two fluid phases with different compositions. We calculate bulk fluid-fluid coexistence curves for a range of size ratios and non-additivity parameters and find that they compare well to simulation results from the literature. Using the Ornstein-Zernike equation, we investigate the asymptotic, [Formula: see text], decay of the partial pair correlation functions, g(ij)(r). At low densities a structural crossover occurs in the asymptotic decay between two different damped oscillatory modes with different wavelengths corresponding to the two intra-species hard-core diameters. On approaching the fluid-fluid critical point there is a Fisher-Widom crossover from exponentially damped oscillatory to monotonic asymptotic decay. Using the density functional we calculate the density profiles for the planar free fluid-fluid interface between coexisting fluid phases. We show that the type of asymptotic decay of g(ij)(r) not only determines the asymptotic decay of the interface profiles, but is also relevant for intermediate and even short-ranged behaviour. We also determine the surface tension of the free fluid interface, finding that it increases with non-additivity, and that on approaching the critical point mean-field scaling holds.

Binary fluid with attractions near a planar wall

It is well known that a mixture of big and small hard spheres next to a planar wall will exhibit segregation based on their size difference. The larger spheres will tend to locate next to the substrate because the overall system entropy loss per unit area is less. In the present study we determine the role of attraction between the small particles and the wall to displace the larger particles. Both fluids density-functional theory and discontinuous molecular dynamics simulations demonstrate that at a certain attractive potential, which is on the order of the thermal energy, the large particles can indeed be dislodged from the surface layer so the small particles are now the major surface component. Exploration of a range of parameters, including relative sphere size and concentration, as well as attractions between the small spheres in the bulk, shows that the phenomenon is quite robust.

Generalization of Rosenfeld's functional to non-additive hard-spheres: pair structure and test-particle consistency

The accuracy of the structural data obtained from the recently proposed generalization to non-additive hard-spheres (Schmidt 2004 J. Phys.: Condens. Matter 16 L351) of Rosenfeld's functional is investigated. The radial distribution functions computed from the direct correlation functions generated by the functional, through the Ornstein-Zernike equations, are compared with those obtained from the density profile equations in the test-particle limit, without and with test-particle consistency. The differences between these routes and the role of the optimization of the parameters of the reference system when the functional is used to obtain the reference bridge functional are discussed in the case of symmetric binary mixtures of non-additive hard-spheres. The case of highly asymmetric mixtures is finally briefly discussed.

Gelation and phase coexistence in colloidal suspensions with short-range forces: generic behavior versus specificity

The interplay between physical gelation and equilibrium phase transitions in asymmetric binary mixtures is analyzed from the effective fluid approach, in which the big particles interact via a short-range effective attraction beyond the core due to the depletion mechanism. The question of the universality of the scenario for dynamical arrest is then addressed. The comparison of the phase diagrams of the hard-sphere mixture and the Asakura-Oosawa models at various size ratios shows that strong specificity is observed for nonideal depletants. In particular, equilibrium gelation, without the competition with fluid-fluid transition is possible in mixtures of hard-sphere colloids. This is interpreted from the specificities of the effective potential, such as its oscillatory behavior and its complex variation with the physical parameters. The consequences on the dynamical arrest and the fluid-fluid transition are then investigated by considering in particular the role of the well at contact and the first repulsive barrier. This is done for the actual effective potential in the hard-sphere mixture and for a square well and shoulder model, which allows a separate discussion of the role of the different parameters, in particular on the localization length and the escape time. This study is next extended to mixtures of "hard-sphere-like" colloids with residual interactions. It confirms the trends relative to equilibrium gelation and illustrates a diversity of the phase behavior well beyond the scenarios expected from simple models.

Equilibrium route to colloidal gelation: mixtures of hard-sphere-like colloids

The binodals and the nonergodicity lines of a binary mixture of hard-sphere-like particles with a large size ratio are computed for studying the interplay between dynamic arrest and phase separation in depletion-driven colloidal mixtures. Contrary to the case of hard core plus short-range effective attraction, physical gelation without competition with the fluid-phase separation can occur in such mixtures. This behavior due to the oscillations in the depletion potential should concern all simple mixtures with a nonideal depletant, justifying further studies of their dynamic properties.

Surface-induced first-order transition in athermal polymer-nanoparticle blends

We investigate the phase behavior of athermal polymer-nanoparticle blends near a substrate. We apply a recent fluids density functional theory of Tripathi and Chapman to a simple model of the blend as a mixture of hard spheres and freely jointed hard chains, near a hard wall. We find that there is a first-order phase transition in which the nanoparticles expel the polymer from the surface to form a monolayer. The nanoparticle transition density depends on the length of the polymer and the overall bulk density of the system. The effect is due both to packing entropy effects related to size asymmetry between the components and to the polymer configurational entropy. The simplicity of the system allows us to understand the so-called "entropic-push" observed in experiments.

Lattice-switch Monte Carlo simulation for binary hard-sphere crystals

We show how to generalize the lattice-switch Monte Carlo method to calculate the phase diagram of a binary system. A global coordinate transformation is combined with a modification of particle diameters, enabling the multicomponent system in question to be explored and directly compared to a suitable reference state in a single Monte Carlo simulation. We use the method to evaluate the free energies of binary hard-sphere crystals. Calculations at moderate size ratios alpha=0.58 and 0.73 are in agreement with previous results, and confirm AB2 and AB13 as stable structures. We also find that the AB(CsCl) structure is not entropically stable at the size ratio and volume where it has been reported experimentally, and therefore that those observations cannot be explained by packing effects alone.

Equilibrium and glassy states of the Asakura-Oosawa and binary hard sphere mixtures: effective fluid approach

Motivated by recent experimental results on model binary colloidal mixtures, especially for the glass transition, we investigate the phase diagram of two models of asymmetric binary mixtures: the hard sphere and the Asakura-Oosawa mixtures. This includes the binodals and the glass transition line, computed in the effective one-component representation using the corresponding potentials of mean force at infinite dilution. The reference hypernetted chain approximation is used for computing the static properties and the glass transition line is computed in the mode coupling approximation. The similarities and the differences between the two models are discussed for different size ratios. It is shown that while both models follow a universal behavior at large asymmetry, the hard sphere mixture model leads to more original results at moderate size ratio. These results show that a modeling beyond generic effective potentials might be necessary for an appropriate description of the complete phase diagram.

Fundamental measure density functional theory for nonadditive hard-core mixtures: the one-dimensional case

We treat a one-dimensional binary mixture of hard-core particles that possess nonadditive diameters. For this model, a density functional theory is constructed following similar principles as an earlier extension of Rosenfeld's fundamental measure theory to three-dimensional nonadditive hard-sphere mixtures. The theory applies to arbitrary positive and moderate negative nonadditivity and reduces to Percus' exact functional in the additive case. Bulk direct correlation functions are obtained as functional derivatives of the excess free energy functional. Results for the partial pair correlation functions in bulk, as calculated via the Ornstein-Zernike route and using the direct correlation functions as input, show very good agreement with results from our Monte Carlo computer simulations of the mixture.