Depletion forces in dense mixtures of spheres and rods.

We evaluate depletion forces in molecular dynamics simulation of a binary mixture of spheres (depleted particles) and rods (depletant particles) for a wide range of densities for both species. This evaluation was carried out using a recently proposed least squares fitting algorithm. We found that the restriction of the rods' rotational degrees of freedom, when the distance between two spheres is less than the rods length, creates a shallow, and apparently linear, attractive force ramp. For intersphere distances smaller than the rods' diameter, a much stronger attractive force is found, and a large repulsive barrier appears between these aforementioned regimes, roughly at the distance of the rods' thickness. The evaluated forces are validated via a comparison of the pairwise correlation functions obtained from molecular dynamics simulation of a mono-disperse sphere fluid, using the evaluated effective forces, against the original (full system) pairwise correlation functions. Agreement is excellent. We also record the angular pairwise correlation function, using the P2(x) Legendre polynomial, and find that for high densities of both species, a local nematic ordering starts to appear. This nematic order may be a factor in the small differences found between original and effective pairwise correlation functions at high densities of rods.

Dynamical arrest transition of a bidisperse two-patchy colloidal dispersion: A dynamic Monte Carlo study.

During the last few years, patchy colloidal dispersions have emerged as ideal candidates of glass-formers of systems composed of particles that interact with non-isotropic potentials. However, from the computational point of view, the characterization of their dynamical properties close to the glass transition via any kind of molecular dynamics simulation technique can be very difficult due to the slowing down of both the rotational and translational dynamics. Although a plethora of dynamical techniques have been developed to account for the dynamics of patchy colloids, new and complementary simulation techniques are required to explore, much faster and more efficiently, the dynamical arrest transition of patchy colloidal dispersions when computer simulation consists of a large number of particles and, due to the slow particle dynamics at the glass transition, an extended time window is explicitly required. Then, in this contribution, by means of the so-called dynamic-Monte Carlo method, we report on the dynamical arrest transition, both rotational and translational, of a bidisperse patchy colloidal dispersion, following three different paths along the density-temperature plane, including high densities and low temperatures. Although this method has not been extensively tested at extreme thermodynamic conditions, we show that even at the dynamical arrest transition, it allows us to extract good dynamical data from a complex system. Therefore, it turns out to be a promising technique to explore the onset of vitrification of anisotropic colloidal particles.

Determining depletion interactions by contracting forces.

Depletion forces are fundamental for determining the phase behavior of a vast number of materials and colloidal dispersions and have been used for the manipulation of in- and out-of-equilibrium thermodynamic states. The entropic nature of depletion forces is well understood; however, most theoretical approaches, and also molecular simulations, work quantitatively at moderate size ratios in much diluted systems since large size asymmetries and high particle concentrations are difficult to deal with. The existing approaches for integrating out the degrees of freedom of the depletant species may fail under these extreme physical conditions. Thus, the main goal of this contribution is to introduce a general physical formulation for obtaining the depletion forces even in those cases where the concentration of all species is relevant. We show that the contraction of the bare forces uniquely determines depletion interactions. Our formulation is tested by studying depletion forces in binary and ternary colloidal mixtures. We report here results for dense systems with total packing fractions of 45% and 55%. Our results open up the possibility of finding an efficient route to determine effective interactions at a finite concentration, even under non-equilibrium thermodynamic conditions.

Non-equilibrium relaxation and aging in the dynamics of a dipolar fluid quenched towards the glass transition.

The recently developed non-equilibrium self-consistent generalized Langevin equation theory of the dynamics of liquids of non-spherically interacting particles [2016J. Phys. Chem. B1207975] is applied to the description of the irreversible relaxation of a thermally and mechanically quenched dipolar fluid. Specifically, we consider a dipolar hard-sphere liquid quenched (attw= 0) from full equilibrium conditions towards different ergodic-non-ergodic transitions. Qualitatively different scenarios are predicted by the theory for the time evolution of the system after the quench (tw> 0), that depend on both the kind of transition approached and the specific features of the protocol of preparation. Each of these scenarios is characterized by the kinetics displayed by a set of structural correlations, and also by the development of two characteristic times describing the relaxation of the translational and rotational dynamics, allowing us to highlight the crossover from equilibration to aging in the system and leading to the prediction of different underlying mechanisms and relaxation laws for the dynamics at each of the glass transitions explored.

Competing interactions in the depletion forces of ternary colloidal mixtures.

Depletion interactions between colloidal particles surrounded by smaller depletants are typically characterized by a strong attraction at contact and a moderately repulsive barrier in front of it that extends at distances similar to the size of the depletants; the appearance and height of the barrier basically depend on the concentration and, therefore, the correlation between depletants. From a thermodynamic point of view, the former can drive the system to phase separation or toward non-equilibrium states, such as gel-like states, but its effects on both local and global properties may be controlled by the latter, which acts as a kind of entropic gate. However, the latter has not been entirely analyzed and understood within the context of colloidal mixtures mainly driven by entropy. In this contribution, we present a systematic study of depletion forces in ternary mixtures of hard spherical particles with two species of depletants, in two and three dimensions. We focus the discussion on how the composition of the depletants becomes the main physical parameter that drives the competition between the attractive well and the repulsive barrier. Our results are obtained by means of the integral equation theory of depletion forces and techniques of contraction of the description adapted to molecular dynamics computer simulations.

Spherical harmonic projections of the static structure factor of the dipolar hard sphere model: Theory vs simulations.

We investigate the static correlations of a dipolar fluid in terms of the irreducible coefficients of the spherical harmonic expansion of the static structure factor. To this end, we develop a theoretical framework based on a soft-core version of Wertheim's solution of the mean spherical approximation (MSA), which renders the analytical determination of such coefficients possible. The accuracy of this approximation is tested by a comparison against the results obtained with the assistance of extensive molecular dynamics simulations at different regimes of concentration and temperature. Crucial aspects for the comparison of the results provided by the two methods are carefully discussed, concerning the different reference frames used in theory and simulations to describe rotations and orientations, and leading to important differences in the behavior of correlation functions with the same combination of spherical harmonic indices. We find a remarkable agreement between the two approaches in the fluid regime, thus providing a first stringent comparison of the irreducible coefficients of the spherical harmonic expansion of the dipolar fluid's static structure factor, provided by the MSA theory and molecular dynamics simulations.

Arrested dynamics of the dipolar hard sphere model.

We report the combined results of molecular dynamics simulations and theoretical calculations concerning various dynamical arrest transitions in a model system representing a dipolar fluid, namely, N (soft core) rigid spheres interacting through a truncated dipole-dipole potential. By exploring different regimes of concentration and temperature, we find three distinct scenarios for the slowing down of the dynamics of the translational and orientational degrees of freedom: at low (η = 0.2) and intermediate (η = 0.4) volume fractions, both dynamics are strongly coupled and become simultaneously arrested upon cooling. At high concentrations (η≥ 0.6), the translational dynamics shows the features of an ordinary glass transition, either by compressing or cooling down the system, but with the orientations remaining ergodic, thus indicating the existence of partially arrested states. In this density regime, but at lower temperatures, the relaxation of the orientational dynamics also freezes. The physical scenario provided by the simulations is discussed and compared against results obtained with the self-consistent generalized Langevin equation theory, and both provide a consistent description of the dynamical arrest transitions in the system. Our results are summarized in an arrested states diagram which qualitatively organizes the simulation data and provides a generic picture of the glass transitions of a dipolar fluid.

Glassy dynamics in asymmetric binary mixtures of hard spheres.

We perform a systematic and detailed study of the glass transition in highly asymmetric binary mixtures of colloidal hard spheres, combining differential dynamic microscopy experiments, event-driven molecular dynamics simulations, and theoretical calculations, exploring the whole state diagram and determining the self-dynamics and collective dynamics of both species. Two distinct glassy states involving different dynamical arrest transitions are consistently described, namely, a double glass with the simultaneous arrest of the self-dynamics and collective dynamics of both species, and a single glass of large particles in which the self-dynamics of the small species remains ergodic. In the single-glass scenario, spatial modulations in the collective dynamics of both species occur due to the structure of the large spheres, a feature not observed in the double-glass domain. The theoretical results, obtained within the self-consistent generalized Langevin equation formalism, are in agreement with both simulations and experimental data, thus providing a stringent validation of this theoretical framework in the description of dynamical arrest in highly asymmetric mixtures. Our findings are summarized in a state diagram that classifies the various amorphous states of highly asymmetric mixtures by their dynamical arrest mechanisms.

Stress distribution in two-dimensional silos.

Simulations of a polydispersed two-dimensional silo were performed using molecular dynamics, with different numbers of grains reaching up to 64 000, verifying numerically the model derived by Janssen and also the main assumption that the walls carry part of the weight due to the static friction between grains with themselves and those with the silo's walls. We vary the friction coefficient, the radii dispersity, the silo width, and the size of grains. We find that the Janssen's model becomes less relevant as the the silo width increases since the behavior of the stresses becomes more hydrostatic. Likewise, we get the normal and tangential stress distribution on the walls evidencing the existence of points of maximum stress. We also obtained the stress matrix with which we observe zones of concentration of load, located always at a height around two thirds of the granular columns. Finally, we observe that the size of the grains affects the distribution of stresses, increasing the weight on the bottom and reducing the normal stress on the walls, as the grains are made smaller (for the same total mass of the granulate), giving again a more hydrostatic and therefore less Janssen-type behavior for the weight of the column.

One-dimensional Gaussian-core fluid: ordering and crossover from normal diffusion to single-file dynamics.

The peculiarity of a bounded pair potential in combination with strong confinement brings some quite interesting new phenomenology in the structure and dynamics of one-dimensional colloidal systems. Such behaviour is atypical in comparison with colloidal systems interacting with potentials that diverge at the origin. In this contribution, by means of molecular dynamics simulations, a confined one-dimensional model of particles interacting via a Gaussian-core pair potential is studied. We explore the effects of confinement, density and temperature on the structural and dynamical correlation functions. Our findings indicate that the static and dynamic liquid-state anomalies already reported in open systems are also present in this 1D model system. Using the radial distribution function and the static structure factor to characterise the spatial ordering, it is observed that the system remains fluid at all densities. However, when the reduced temperature is above 0.03, it displays typical features of a liquid regime, i.e., there exist short-range spatial correlations among particles. In contrast, at lower temperatures and densities, where the particle-particle interaction dominates, the system behaves structurally and dynamically similar to a hard-core repulsive system. In such a region, interestingly, there is a crossover from a liquid to a solid-like regime. At any given temperature, the system undergoes a sort of reentrant structural behaviour as the density increases. At either high densities or temperatures, particle correlations vanish, thus, the system exhibits structural and dynamical properties similar to those of an ideal gas. To examine a possible correlation between the structural anomalies and the diffusive behaviour, the mean-square displacement and the self-intermediate scattering function are also computed. From these observables, we establish the thermodynamic phase-space points where the dynamical behaviour is non-monotonic. In conjunction with the observed anomalous diffusion, we have found a dynamical crossover from single-file diffusion, which is characteristic of one-dimensional systems with a well-defined hard-core, to the ordinary Fickian diffusion present in open systems.

Effective potentials in a bidimensional vibrated granular gas.

We present a numerical study of the spatial correlations of a quasi-two-dimensional granular fluid kept in a nonstatic steady state via vertical shaking. The simulations explore a wide range of vertical accelerations, restitution coefficients, and packing fractions, always staying below the crystallization limit. From the simulations we obtain the relevant pair distribution functions (PDFs), and effective potentials for the interparticle interaction are extracted from these PDFs via the Ornstein-Zernike equation with the Percus-Yevick closure. The correlations in the granular structures originating from these effective potentials are checked against the originating PDF using standard Monte Carlo simulations, and we find in general an excellent agreement. The resulting effective potentials show an increase of the spatial correlation at contact with the decreasing values of the restitution coefficient, and a tendency of the potentials to display deeper wells for more dissipative dynamics. A general exception to this trend appears for a range of values of the forcing, which depends on the restitution coefficient, but not on the density, where resonant bouncing increases correlations, resulting in deeper potential wells. The nature of these resonances is explored and shown to be the result of synchronization in the parabolic flights of the particles.

Assessment of the micro-structure and depletion potentials in two-dimensional binary mixtures of additive hard-disks.

Depletion forces are a particular class of effective interactions that have been mainly investigated in binary mixtures of hard-spheres in bulk. Although there are a few contributions that point toward the effects of confinement on the depletion potential, little is known about such entropic potentials in two-dimensional colloidal systems. From theoretical point of view, the problem resides in the fact that there is no general formulation of depletion forces in arbitrary dimensions and, typically, any approach that works well in three dimensions has to be reformulated for lower dimensionality. However, we have proposed a theoretical framework, based on the formalism of contraction of the description within the integral equations theory of simple liquids, to account for effective interactions in colloidal liquids, whose main feature is that it does not need to be readapted to the problem under consideration. We have also shown that such an approach allows one to determine the depletion pair potential in three-dimensional colloidal mixtures even near to the demixing transition, provided the bridge functions are sufficiently accurate to correctly describe the spatial correlation between colloids [E. López-Sánchez et al., J. Chem. Phys. 139, 104908 (2013)]. We here report an extensive analysis of the structure and the entropic potentials in binary mixtures of additive hard-disks. In particular, we show that the same functional form of the modified-Verlet closure relation used in three dimensions can be straightforwardly employed to obtain an accurate solution for two-dimensional colloidal mixtures in a wide range of packing fractions, molar fractions, and size asymmetries. Our theoretical results are explicitly compared with the ones obtained by means of event-driven molecular dynamics simulations and recent experimental results. Furthermore, to assess the accuracy of our predictions, the depletion potentials are used in an effective one-component model to reproduce the structure of either the big or the small disks. This demonstrates the robustness of our theoretical scheme even in two dimensions.

Clustering-Induced Attraction in Granular Mixtures of Rods and Spheres.

Depletion-induced aggregation of rods enhanced by clustering is observed to produce a novel model of attractive pairs of rods separated by a line of spheres in a quasi-2D, vertically-shaken, granular gas of rods and spheres. We show that the stability of these peculiar granular aggregates increases as a function of shaking intensity. Velocity distributions of spheres inside and outside of a pair of rods trapping a line of spheres show a clear suppression of the momentum acquired by the trapped spheres. The condensed phase formed between the rods is caused by a clustering instability of the trapped spheres, enhanced by a vertical guidance produced by the confining rods. The liberated area corresponding to direct excluded-volume pairs and indirect depletion-aggregated pairs is measured as a function of time. The stability of rod pairs mediated by spheres reveals an attraction comparable in strength to the one purely induced by depletion forces.

Demixing transition, structure, and depletion forces in binary mixtures of hard-spheres: the role of bridge functions.

Asymmetric binary mixtures of hard-spheres exhibit several interesting thermodynamic phenomena, such as multiple kinds of glassy states. When the degrees of freedom of the small spheres are integrated out from the description, their effects are incorporated into an effective pair interaction between large spheres known as the depletion potential. The latter has been widely used to study both the phase behavior and dynamic arrest of the big particles. Depletion forces can be accounted for by a contraction of the description in the multicomponent Ornstein-Zernike equation [R. Castañeda-Priego, A. Rodríguez-López, and J. M. Méndez-Alcaraz, Phys. Rev. E 73, 051404 (2006)]. Within this theoretical scheme, an approximation for the difference between the effective and bare bridge functions is needed. In the limit of infinite dilution, this difference is irrelevant and the typical Asakura-Osawa depletion potential is recovered. At higher particle concentrations, however, this difference becomes important, especially where the shell of first neighbors is formed, and, as shown here, cannot be simply neglected. In this work, we use a variant of the Verlet expression for the bridge functions to highlight their importance in the calculation of the depletion potential at high densities and close to the spinodal decomposition. We demonstrate that the modified Verlet closure predicts demixing in binary mixtures of hard spheres for different size ratios and compare its predictions with both liquid state and density functional theories, computer simulations, and experiments. We also show that it provides accurate correlation functions even near the thermodynamic instability; this is explicitly corroborated with results of molecular dynamics simulations of the whole mixture. Particularly, our findings point toward a possible universal behavior of the depletion potential around the spinodal line.

Crystallization time scales for polydisperse hard-sphere fluids.

We study the evolution of crystallization in dense mono- and polydisperse hard sphere fluids, initially quenched to an amorphous configuration. We use as signatures of crystallization both the decay of the reduced pressure Z and the increase in the local and global orientational order parameters Q[over ¯](6). For a given realization of the crystallization process these parameters show sudden changes, both large and small, separated by long periods of quiescence. However, averaging over a large number of realizations, a well-defined scenario for their evolution appears. We find an initial fast relaxation to a disordered state, followed by a period of slow variation, associated to the presence of nucleation events, followed by a fast change, composed of the growth of a few crystals with different orientations, and a final and slow coarsening in a domain-growth process. No clear scaling for this whole process was found. We also find that the transition to an stable glassy fluid is quite sharp as the polydispersity is increased, showing a probable first-order phase transition behavior. A well-defined boundary between crystallizing and permanently amorphous fluids should exist, at least for a region in packing fractions. We looked for segregation at large values of polydispersity, but no effects of this type were found.

Bouncing, rolling, energy flows, and cluster formation in a two-dimensional vibrated granular gas.

We study the formation of crystalline clusters for a two-dimensional (2D) sinusoidally vibrated granular gas, with maximum vertical acceleration smaller than gravity, using fully 3D simulations. It is found that this phenomenon arises from the spontaneous segregation of the granulate into two dynamical modes: one of grains that bounce in synchrony with the motion of the sustaining plate ("bouncers") and another of grains that cease to bounce and simply rolls on the plate, without ever loosing contact with it ("rollers"). These two dynamical categories are quite robust with respect to perturbations. The populations for bouncers and rollers depend on the preparation of the granulate and can be made to take arbitrary values in all the range of accelerations where both dynamical modes are present. It is found that the dynamical mode with the largest population coalesces in clusters under the influence of the other mode, whose grains act as a higher pressure gas that compresses the clusters. In this way it is possible to produce clusters of rollers or clusters of bouncers. A gas made of grains from only one dynamical class shows only weak density fluctuations. When the occupation fractions for both modes are similar, one observes segregation and clusters of both types. The clustering of the gas is monitored using both the average coordination number and the local hexatic order parameter ψ(6). Energy flows in the plane are monitored, and it is shown that roller-bouncer collisions increase horizontal kinetic energy, while all other types of collisions reduce this energy. We find that friction with the substrate is the main sink of horizontal energy for these granular gases.

Equilibration of concentrated hard-sphere fluids.

We report a systematic molecular dynamics study of the isochoric equilibration of hard-sphere fluids in their metastable regime close to the glass transition. The thermalization process starts with the system prepared in a nonequilibrium state with the desired final volume fraction ϕ for which we can obtain a well-defined nonequilibrium static structure factor S(0)(k;ϕ). The evolution of the α-relaxation time τ(α)(k) and long-time self-diffusion coefficient D(L) as a function of the evolution time t(w) is then monitored for an array of volume fractions. For a given waiting time the plot of τ(α)(k;ϕ,t(w)) as a function of ϕ exhibits two regimes corresponding to samples that have fully equilibrated within this waiting time [ϕ≤ϕ(c)(t(w))] and to samples for which equilibration is not yet complete [ϕ≥ϕ(c)(t(w))]. The crossover volume fraction ϕ(c)(t(w)) increases with t(w) but seems to saturate to a value ϕ(a)≡ϕ(c)(t(w)→∞)≈0.582. We also find that the waiting time t(w)(eq)(ϕ) required to equilibrate a system grows faster than the corresponding equilibrium relaxation time, t(w)(eq)(ϕ)≈0.27[τ(α)(eq)(k;ϕ)](1.43), and that both characteristic times increase strongly as ϕ approaches ϕ(a), thus suggesting that the measurement of equilibrium properties at and above ϕ(a) is experimentally impossible.