Brownian dynamics simulation of polydisperse hard spheres

Standard algorithms used for the numerical integration of the Langevin equation require that interactions should slowly vary during the integration time-step. This in not the case for hard-body systems, where there is no clear-cut between the correlation time of the noise and the time-scale of the interactions. Starting with a short time approximation of the Smoluchowski equation, we introduce an algorithm for the simulation of the over-damped Brownian dynamics of polydisperse hard-spheres in absence of hydrodynamic interactions and briefly discuss the extension to the case of external drifts.     

Equilibrium phase behavior of polydisperse hard spheres

We calculate the phase behavior of hard spheres with size polydispersity, using accurate free energies for the fluid and solid phases. Cloud and shadow curves are found exactly by the moment free energy method, but we also compute the complete phase diagram, taking full account of fractionation. In contrast to earlier, simplified treatments we find no point of equal concentration between fluid and solid or reentrant melting at higher densities. Rather, the fluid cloud curve continues to the largest polydispersity that we study (14%); from the equilibrium phase behavior a terminal polydispersity can thus be defined only for the solid, where we find it to be around 7%. At sufficiently large polydispersity, fractionation into several solid phases can occur, consistent with previous approximate calculations; we find, in addition, that coexistence of several solids with a fluid phase is also possible.     

Dense random packings of hard and compressible spheres

The sublattices of the tetrahedrally co-ordinated random network of Connell and Temkin are related to the dense random packing of equal spheres. Thus, by relaxation of the atomic co-ordinates of the former, the dependence of the pair distribution function and packing fraction of the latter on sphere compressibility can be investigated. The results are compared with experimental data on NiP alloys.     

Geometrical properties of a random packing of hard spheres

Some geometrical properties of a random packing of identical hard spheres generated by a ballistic deposition model with complete restructuring are investigated. The length distribution of chords in the space between spheres is numerically calculated and is shown to have an exponential form (up to chord lengths of about five diameters) as conjectured by Dixmier. The anisotropic properties of the packing are numerically investigated and are shown to modify the Dixmier relation between the packing fraction and the average coordination number.     

Brownian dynamics of polydisperse colloidal hard spheres: Equilibrium structures and random close packings

Recently we presented a new technique for numerical simulations of colloidal hard-sphere systems and showed its high efficiency. Here, we extend our calculations to the treatment of both 2- and 3-dimensional monodisperse and 3-dimensional polydisperse systems (with sampled finite Gaussian size distribution of particle radii), focusing on equilibrium pair distribution functions and structure factors as well as volume fractions of random close packing (RCP). The latter were determined using in principle the same technique as Woodcock or Stillinger had used. Results for the monodisperse 3-dimensional system show very good agreement compared to both pair distribution and structure factor predicted by the Percus-Yevick approximation for the fluid state (volume fractions up to 0.50). We were not able to find crystalline 3d systems at volume fractions 0.50–0.58 as shown by former simulations of Reeet al. or experiments of Pusey and van Megen, due to the fact that we used random start configurations and no constraints of particle positions as in the cell model of Hoover and Ree, and effects of the overall entropy of the system, responsible for the melting and freezing phase transitions, are neglected in our calculations. Nevertheless, we obtained reasonable results concerning concentration-dependent long-time selfdiffusion coefficients (as shown before) and equilibrium structure of samples in the fluid state, and the determination of the volume fraction of random close packing (RCP, glassy state). As expected, polydispersity increases the respective volume fraction of RCP due to the decrease in free volume by the fraction of the smaller spheres which fill gaps between the larger particles.     

A first-order phase transition defines the random close packing of hard spheres

Randomly packing spheres of equal size into a container consistently results in a static configuration with a density of ∼64%. The ubiquity of random close packing (RCP) rather than the optimal crystalline array at 74% begs the question of the physical law behind this empirically deduced state. Indeed, there is no signature of any macroscopic quantity with a discontinuity associated with the observed packing limit. Here we show that RCP can be interpreted as a manifestation of a thermodynamic singularity, which defines it as the “freezing point” in a first-order phase transition between ordered and disordered packing phases. Despite the athermal nature of granular matter, we show the thermodynamic character of the transition in that it is accompanied by sharp discontinuities in volume and entropy. This occurs at a critical compactivity, which is the intensive variable that plays the role of temperature in granular matter. Our results predict the experimental conditions necessary for the formation of a jammed crystal by calculating an analogue of the “entropy of fusion”. This approach is useful since it maps out-of-equilibrium problems in complex systems onto simpler established frameworks in statistical mechanics.     

Dense and nearly jammed random packings of freely jointed chains of tangent hard spheres

Dense packings of freely jointed chains of tangent hard spheres are produced by a novel Monte Carlo method. Within statistical uncertainty, chains reach a maximally random jammed (MRJ) state at the same volume fraction as packings of single hard spheres. A structural analysis shows that as the MRJ state is approached (i) the radial distribution function for chains remains distinct from but approaches that of single hard sphere packings quite closely, (ii) chains undergo progressive collapse, and (iii) a small but increasing fraction of sites possess highly ordered first coordination shells.     

Experimental observation of structural crossover in binary mixtures of colloidal hard spheres

Using confocal microscopy, we investigate the structure of binary mixtures of colloidal hard spheres with size ratio q=0.61. As a function of the packing fraction of the two particle species, we observe a marked change of the dominant wavelength in the pair-correlation function. This behavior is in excellent agreement with a recently predicted structural crossover in such mixtures. In addition, the repercussions of structural crossover on the real-space structure of a binary fluid are analyzed. We suggest a relation between crossover and the lateral extension of networks containing only equally-sized particles that are connected by nearest-neighbor bonds. This is supported by Monte Carlo simulations which are performed at different packing fractions and size ratios.     

Mixtures of hard spherocylinders and hard spheres

Monte Carlo simulations have been performed for equimolar mixtures of hard prolate spherocylinders of length: breadth ratio 2:1 and hard spheres, in the fluid region. Two systems have been studied. In the first the breadth of the spherocylinder was equal to the hard sphere diameter, and in the second system both components were of equal molecular volume.

The compressibility factor, PV/NkT, has been obtained for both mixtures at four reduced densities (packing fractions) from 0·20 to 0·45. The results have been compared with the predictions of several analytical equations appropriate to mixtures of hard convex molecules, and an equation due to Pavlicek et al. was found to be very accurate. The results have been used to calculate the excess volumes of mixing at constant pressure, in an attempt to establish the relative importance of the effects of differences in molecular volume and shape on the thermodynamic properties.

The structural properties of the mixtures have also been investigated by calculating pair distribution functions for the three types of pair interactions present in these mixtures.     

Analytical representation of the higher virial coefficients of binary mixtures of additive hard spheres

Approximate expressions for the fourth and fifth virial coefficients of binary hard-sphere fluid mixtures are derived. The procedure used to obtain these expressions is based on that previously proposed by Wheatley [J. chem. Phys., 111, 5455 (1999)], but slightly modified. Wheatley's procedure starts from a prescribed general analytical form of the virial coefficients, from which the particular expression for each virial coefficient is obtained by imposing to the general form a number of limiting conditions. Here, we propose an alternative general expression of the virial coefficients and derive one more condition. This condition is satisfied when the fourth and fifth virial coefficients are expressed in the form we propose, but not when they are expressed in Wheatley's form. The agreement of the proposed analytical expressions with exact numerical data is excellent. The procedure can be extended to higher virial coefficients, although the lack of exact numerical data prevents any comparison.     

Monte Carlo data of dilute solutions of large spheres in binary hard sphere mixtures

Monte Carlo (MC) simulation data for additive binary hard sphere mixtures are reported for dilute concentrations of the large sphere. Using a single occupancy linked cell method, binary hard sphere solutions with a size ratio of 5 are simulated at high reduced density and low concentration of the large sphere. Data for the solute-solvent pair distribution function show that at the lowest concentrations of the large sphere simulated, the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation underestimates the contact value; whereas the recently proposed Henderson-Chan (HC) equation gives a good prediction. For the solute-solute contact value at the colloidal limit, the MC data lie between the two predictions. The BMCSL equation underestimates, while the HC equation overestimates, the correct solute-solute contact value.     

Effective dielectric constant of dilute polydisperse suspensions of spheres

We study the effective dielectric constant of a dilute, polydisperse suspension of spheres embedded in a uniform background. We consider a frequency region where the dipole polarizability of individual spheres exhibits a resonance. We evaluate the effective dielectric constant to second order in the volume fraction employing the dipole approximation, which in previous work has been shown to be applicable in resonance. We apply our results to suspensions of spheres with identical and uniform dielectric constant, assuming a log-normal distribution of sphere radii.     

Kinetic theory of hard spheres

Kinetic equations for the hard-sphere system are derived by diagrammatic techniques. A linear equation is obtained for the one-particle-one particle equilibrium time correlation function and a nonlinear equation for the one-particle distribution function in nonequilibrium. Both equations are nonlocal, noninstantaneous, and extremely complicated. They are valid for general density, since statistical correlations are taken into account systematically. This method derives several known and new results from a unified point of view. Simple approximations lead to the Boltzmann equation for low densities and to a modified form of the Enskog equation for higher densities.     

Colour conductivity of hard spheres

We present an analytic solution for the d-dimensional (d > 1) hard-sphere free flight trajectories in a thermostatted colour field. The solution shows that particles can only reach a finite distance in the direction perpendicular to the field in the absence of collisions. Using a numerical algorithm we designed to simulate many-body hard-sphere systems with curved trajectories, we study the onset of the instability leading to phase separation in the two-dimensional case for a range of field strengths and three densities. For the two fluid densities we find that phase separation occurs for sufficiently strong fields regardless of the initial configuration, and that the phase-separated state eventually becomes a collisionless, non-ergodic steady state. For solid densities the phase-separated configuration is stable and conducting, but is not an attractor for other charge distributions because of the impossibility of particle rearrangement.     

Approximate solution of the Percus-Yevick equation for a binary mixture of hard spheres with non-additive diameters

We present an approximate solution of the Percus-Yevick integral equation for a binary mixture of hard spheres with non-additive diameters. Defining R_ij the distance of closest approach between particles of species i and j by R ₁₂ = ½(R ₁₁ + R ₂₂) + α, we obtain a closed set of equations for the direct correlation functions c_ij (r) when 0 < α ? min [½(R ₂₂ - R ₁₁), ½R ₁₁]. Our expressions for c_ii (r), and for c ₁₂(r) in the range 0 < r ? ½[R ₂₂ - R ₁₁] - α, agree with those previously obtained by Lebowitz and Zomick.     

Dissipative dynamics for hard spheres

The dynamics for a system of hard spheres with dissipative collisions is described at the levels of statistical mechanics, kinetic theory, and simulation. The Liouville operator(s) and associated binary scattering operators are defined as the generators for time evolution in phase space. The BBGKY hierarchy for reduced distribution functions is given, and an approximate kinetic equation is obtained that extends the revised Enskog theory to dissipative dynamics. A Monte Carlo simulation method to solve this equation is described, extending the Bird method to the dense, dissipative hard-sphere system. A practical kinetic model for theoretical analysis of this equation also is proposed. As an illustration of these results, the kinetic theory and the Monte Carlo simulations are applied to the homogeneous cooling state of rapid granular flow.     

Local size segregation in polydisperse hard sphere fluids

The structure of polydisperse hard sphere fluids, in the presence of a wall, is studied by the Rosenfeld density functional theory. Within this approach, the local excess free energy depends on only four combinations of the full set of density fields. The case of continuous polydispersity thereby becomes tractable. We predict, generically, an oscillatory size segregation close to the wall, and connect this, by a perturbation theory for narrow distributions, with the reversible work for changing the size of one particle in a monodisperse reference fluid.     

Equilibrium fluid-solid coexistence of hard spheres

We present a tethered Monte Carlo simulation of the crystallization of hard spheres. Our method boosts the traditional umbrella sampling to the point of making practical the study of constrained Gibbs' free energies depending on several crystalline order parameters. We obtain high-accuracy estimates of the fluid-crystal coexistence pressure for up to 2916 particles (enough to accommodate fluid-solid interfaces). We are able to extrapolate to infinite volume the coexistence pressure [p(co)=11.5727(10)k(B)T/σ(3)] and the interfacial free energy [γ({100})=0.636(11)k(B)T/σ(2)].     

Shear viscosity for inelastic hard spheres

The hydrodynamic equations of the Enskog theory for inelastic hard spheres is considered as a model for rapid flow granular fluids at finite densities. A detailed analysis of the shear viscosity of the granular fluid has been done using homogenous cooling state (HCS) and uniform shear flow (USF) models. It is found that shear viscosity is sensitive to the coefficient of restitution α and pair correlation function at contact. The collisional part of the Newtonian shear viscosity is found to be dominant than its kinetic part.