Mixture Of Hard Spheres Research Articles

Density functional theory for responsive hard-sphere fluids

We investigate the equilibrium properties of responsive hard-sphere fluids, where the particle size is a coarse-grained property that fluctuates within an internal free-energy landscape, responding to changes in concentration or the application of external fields. For this purpose, we employ a generalised density functional theory for responsive colloids based on the fundamental measure theory for polydisperse hard-sphere mixtures. We find that increasing particle softness yields a substantial reduction in mean particle size, volume fraction and pressure. We also examine the density profiles and size segregation of responsive hard-sphere fluids subjected to three representative external potentials: a planar hard wall, gravitational and Archimedes buoyant fields. We observe that increasing stiffness or particle concentration enhances density oscillations, converging to the behaviour of a monodisperse fluid. In a gravitational field, size segregation occurs, with smaller particles accumulating at the bottom due to sedimentation. This effect is more pronounced when the Archimedes force is included, causing larger colloids to accumulate at the top, leading to density oscillations in this region that intensify with particle softness, an effect not observed in monodisperse systems. Finally, we demonstrate that responsiveness leads to distinct behaviour compared to conventional polydisperse non-responsive fluids, in which particle compression is not allowed.

Short-time self-diffusion in binary colloidal suspensions

The Brownian dynamics of a colloidal particle is consistently modified by the presence in the solvent of other particles of comparable size, whose effects on the particle diffusion coefficient cannot be attributed to a change of the effective solvent viscosity. So far, despite their impact on subjects ranging from microrheology to phoretic transport in crowded environments, a detailed experimental survey of these effects is still lacking. By exploiting the peculiar properties of fluorinated colloidal particle, we have performed an extensive dynamic light scattering (DLS) investigation of short-time self-diffusion in binary colloidal mixtures, focusing on systems where one of the two species (the “tracer” particles) is very diluted compared to the other one (the “host” particles). From the dependence on the host concentration of the DLS correlation function, we have obtained the first-order correction hs1s to the tracer single-particle diffusion coefficient, varying the tracer-to-host size ratio q in the range 0.2 ≤ q ≤ 2. Our results support the functional relation of hs1s on q proposed to account for the theoretical and numerical results for hard-sphere mixtures. However, hs1s seems to have a weaker dependence on the size ratio than theoretically predicted, possibly because of an imperfect matching of the suspensions we used for an ideal hard-sphere mixture. This may be due to the presence of a stabilizing surfactant layer on the particle surface that, although very thin, has significant effects on hydrodynamic lubrication forces.

Calculation of liquidus in eutectic alkali halide mixtures using thermodynamic perturbation theory

A statistical-thermodynamic model that makes it possible to describe with good accuracy phase equilibria between melt and crystal in binary alkali halide mixtures with common cations and anions is proposed. This model is based on the thermodynamic perturbation theory, which takes into account the charge-dipole contribution to the interionic interaction of binary molten salts using a multicomponent mixture of charged hard spheres as a reference system. Specific expressions for chemical potentials, taking into account the charge-induced dipole interaction in the molten phase, are presented within the developed approach. Considering binary eutectic mixtures NaF–NaCl, CsF–CsCl, NaF–CsF, and NaCl–CsCl as an example, the liquidus curves obtained in two series of calculations are shown: with the equation of state and without it. The discrepancies between the calculated and experimental data on the position of eutectic equilibrium for most of the considered mixtures do not exceed about 10 percent on the temperature scale and 5 percent on the composition when using only the tabulated values of ionic radii and polarizabilities without any adjustment.

Dynamic neighbours: a proposal of a tool to characterize phase transitions

For molecular dynamics simulations of hard particles, we define dynamic neighbours as the distinct particles that collide with a given reference one during a specific time interval. This definition allows us to determine the distribution of the number of dynamic neighbours, its average, and its standard deviation. We will show that regardless of the time window used to identify dynamic neighbours, their distribution is correlated with diffusion coefficients, structure, and configurational entropy. Thus, it is likely that the distribution of the number of dynamic neighbours may be employed as another tool to gain insights into the dynamic behaviour of hard-core systems. We tested this approach on 2D and 3D systems consisting of monodisperse and binary mixtures of hard disks and spheres. Results show that implementing dynamic neighbours to define order parameters can sharpen the signals where transitions take place.

Electron scattering and transport in simple liquid mixtures

The theory for electron transport in simple liquids developed by Cohen and Lekner (1967 Phys. Rev. 158 305), is extended to simple liquid mixtures. The focus is on developing benchmark models for binary mixtures of hard-spheres, using the Percus-Yevick model (Lebowitz 1964 Phys. Rev. A 133 895, Hiroike 1969 J. Phys. Soc. Japan 27 1415) to represent the density structure effects. A multi-term solution of Boltzmann’s equation is employed to investigate the effect of the binary mixture structure on hard-sphere electron scattering cross-sections and transport properties, including the drift velocity, mean energy, longitudinal and transverse diffusion coefficients. Benchmark calculations are established for electrons driven out of equilibrium by a range of reduced electric field strengths ( 0.1−100 Td).

Ternary Mixtures of Hard Spheres and Their Multiple Separated Phases

We study the liquid phase behavior of ternary mixtures of monodisperse hard spheres in solution. The interactions are modeled in terms of the second virial coefficient and can be additive hard sphere (HS) or non-additive hard sphere (NAHS) interactions. We give the set of equations that defines the phase diagram for mixtures of three components. We calculate the theoretical liquid–liquid phase separation boundary for two-phase separation (the binodal) and, if applicable, the three-phase boundary, as well as the plait points and the spinodal. The sizes of the three components are fixed. The first component (A) is the smallest one, the second component (B) is four times the size of the smallest component, and the third (C) component is three times the size of the smallest one. The interaction between the first two components is fixed, and this AB sub-mixture shows phase separation. The interactions of component C with the other two components are varied. Component C can be compatible or incompatible with components A and B. Depending on the compatibility of the components, the phase diagram is altered. The addition of the third component has an influence on the phase boundary, plait points, stability region, fractionation, and volume ratio between the different phases. When all sub-mixtures (AB, AC, and BC) show phase separation, a three-phase system becomes possible when the incompatibility among all components is high enough. The position and size of the three-phase region is dependent on the interactions between the different sub-mixtures. We study the fractionation off all components depending on specific parent concentrations.

Rheology of a dilute binary mixture of inertial suspension under simple shear flow

Abstract The rheology of a dilute binary mixture of inertial suspension under simple shear flow is analyzed in the context of the Boltzmann kinetic equation. The effect of the surrounding viscous gas on the solid particles is accounted for by means of a deterministic viscous drag force plus a stochastic Langevin-like term defined in terms of the environmental temperature Tenv. Grad’s moment method is employed to determine the temperature ratio and the pressure tensor in terms of the coefficients of restitution, concentration, the masses and diameters of the components of the mixture, and the environmental temperature. Analytical results are compared against event-driven Langevin simulations for mixtures of hard spheres with the same mass density m1/m2 = (σ(1)/σ(2))3, mi and σ(1) being the mass and diameter, respectively, of the species i. It is confirmed that the theoretical predictions agree with simulations of various size ratios σ(1)/σ(2) and for elastic and inelastic collisions in a wide range of parameter space. It is remarkable that the temperature ratio T1/T2 and the viscosity ratio η1/η2 (ηi being the partial contribution of the species i to the total shear viscosity η = η1 + η2) discontinuously change at a certain shear rate as the size ratio increases; this feature (which is expected to occur in the thermodynamic limit) cannot be completely captured by simulations due to the small system size. In addition, a Bhatnagar–Gross–Krook (BGK)-type kinetic model adapted to mixtures of inelastic hard spheres is exactly solved when Tenv is much smaller than the kinetic temperature T. A comparison between the velocity distribution functions obtained from Grad’s method, the BGK model, and simulations is carried out.

Single-file transport of binary hard-sphere mixtures through periodic potentials.

Single-file transport occurs in various scientific fields, including diffusion through nanopores, nanofluidic devices, and cellular processes. We here investigate the impact of polydispersity on particle currents for single-file Brownian motion of hard spheres when they are driven through periodic potentials by a constant drag force. Through theoretical analysis and extensive Brownian dynamics simulations, we unveil the behavior of particle currents for random binary mixtures. The particle currents show a recurring pattern in dependence of the hard-sphere diameters and mixing ratio. We explain this recurrent behavior by showing that a basic unit cell exists in the space of the two hard-sphere diameters. Once the behavior of an observable inside the unit cell is determined, it can be inferred for any diameter. The overall variation of particle currents with the mixing ratio and hard-sphere diameters is reflected by their variation in the limit where the system is fully covered by hard spheres. In this limit, the currents can be predicted analytically. Our analysis explains the occurrence of pronounced maxima and minima of the currents by changes in the effective potential barrier for the center-of-mass motion.

Long-time correlations in a binary mixture: analysis of the nonlinearities of fluctuating-hydrodynamic equations

The equations of fluctuating nonlinear hydrodynamics (FNH) following the proper conservation laws are considered for a binary mixture. We focus on the density (ρ) correlations’ renormalization due to the FNH equations’ nonlinearities. The consequence of density nonlinearities treated in simplest approximations gives rise to the well-studied form of the mode coupling theory (MCT) for a two-component system. The MCT predicts a sharp ergodicity–nonergodicity transition similar to the one-component fluid in this idealized form. In the first part of the present paper, we compare the predictions of the idealized MCT model with the computer simulation results for a hard sphere mixture. We show that there is clear disagreement in long-time dynamic behaviour. Next, we consider the full set of nonlinearities in the FNH equations using a Martin–Siggia–Rose field theory. From the time reversal properties of the correlation and response function of the associated field theory, a set of fluctuation–dissipation relations (FDR) are obtained. These FDRs impose constraints on the long-time behaviour of the correlation functions. Our non-perturbative analysis considers the viability of freezing the time correlations for the two-component fluid over the longest time scale. Due to the FDR constraints arising from the nonlinearity in the FNH equations, a sharp ergodicity–nonergodicity transition for the binary mixture is not supported. If the are replaced as in terms of the average density ρ 0, ad hoc, while the density-nonlinearities in the pressure term of the corresponding FNH equations are kept, the ideal transition model of the simplified MCT is recovered.

An Excess Chemical Potential for Binary Hard-Sphere Mixtures from Integral Equation Theory

We solve the site-site Ornstein-Zernike equation using the Percus-Yevick closure for binary hard-sphere mixture. We calculate an excess chemical potential for the mixture’s diameter ratios of 0.3, 0.5, 0.6 and 0.9, and at packing fraction of 0.49 using the analytical expression. Our numerical results are in good agreement with those in the literature.

Reformulation of the Ornstein-Zernike relation for a homogeneous isotropic fluid of spherical symmetry.

A reformulation of the Ornstein-Zernike equation for a homogeneous isotropic fluid composed of m species, with spherical symmetry, is formally derived. Based on a factorization of matrices of composed functions, this reformulation provides an interesting new set of functions. As a test to this reformulation, the resulting equations are solved for a binary mixture of hard spheres and compared to those obtained from the standard solution of the Ornstein-Zernike equation and with molecular dynamics simulations.

Assessment of kinetic theories for moderately dense granular binary mixtures: Shear viscosity coefficient

Two different kinetic theories [J. Solsvik and E. Manger (SM), Phys. Fluids 33, 043321 (2021) and Garzó et al. (GDH), Phys. Rev. E 76, 031303 (2007)] are considered to determine the shear viscosity η for a moderately dense granular binary mixture of smooth hard spheres. The mixture is subjected to a simple shear flow and heated by the action of an external driving force (Gaussian thermostat) that exactly compensates the energy dissipated in collisions. The set of Enskog kinetic equations is the starting point to obtain the dependence of η on the control parameters of the mixture: solid fraction, concentration, mass and diameter ratios, and coefficients of normal restitution. While the expression of η found in the SM-theory is based on the assumption of Maxwellian distributions for the velocity distribution functions of each species, the GDH-theory solves the Enskog equation by means of the Chapman–Enskog method to first order in the shear rate. To assess the accuracy of both kinetic theories, the Enskog equation is numerically solved by means of the direct simulation Monte Carlo method. The simulation is carried out for a mixture under simple shear flow, using the thermostat to control the cooling effects. Given that the SM-theory predicts a vanishing kinetic contribution to the shear viscosity, the comparison between theory and simulations is essentially made at the level of the collisional contribution ηc to the shear viscosity. The results clearly show that the GDH-theory compares with simulations much better than the SM-theory over a wide range of values of the coefficients of restitution, the volume fraction, and the parameters of the mixture (masses, diameters, and concentration).

Estimating random close packing in polydisperse and bidisperse hard spheres via an equilibrium model of crowding.

We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in d = 3 dimensions and, thus, obtain an estimate of the random close packing volume fraction, ϕRCP, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributions. For binary systems, we show agreement between our predictions and simulations using both our own results and results reported in previous studies, as well as agreement with recent experiments from the literature. We then apply our approach to systems with continuous polydispersity using three different particle size distributions, namely, the log-normal, Gamma, and truncated power-law distributions. In all cases, we observe agreement between our theoretical findings and numerical results up to rather large polydispersities for all particle size distributions when using as reference our own simulations and results from the literature. In particular, we find ϕRCP to increase monotonically with the relative standard deviation, sσ, of the distribution and to saturate at a value that always remains below 1. A perturbative expansion yields a closed-form expression for ϕRCP that quantitatively captures a distribution-independent regime for sσ < 0.5. Beyond that regime, we show that the gradual loss in agreement is tied to the growth of the skewness of size distributions.

A heuristic approach for the densest packing fraction of hard-sphere mixtures

In a previous work, a simple approach to derive the jamming packing fraction of a hard-sphere mixture from the knowledge of the random close-packing fraction of the monocomponent system was proposed. Now, an extension of that approach is applied to provide an approximate formula for the densest packing fraction of a given hard-sphere mixture in terms of the fcc close-packing fraction of a monocomponent hard-sphere system and of a single parameter encapsulating the dependence on the size ratios and the number of spheres in the unit cell. Comparison with recent results for such densest packing fraction of binary and ternary systems is performed and reasonable agreement is obtained.

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.

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.

Correlation between structures and atomic transport properties of compound forming liquid Cu-In alloys

The Lebowitz solution of the Percus-Yevick hard-sphere mixture is invoked with the square-well potential function to determine Ashcroft-Langreth type three partial correlation functions in binary Cu-In liquid alloys. These computed partial correlation functions have been employed to generate Bhatia-Thornton fluctuations in the considered melts. Using these structural parameters we investigate the temperature and composition-dependent diffusion coefficients of considered melts. The computed values of concentration-concentration fluctuations, in the long-wavelength limit, and the Warren-Cowley short-range order parameter, show the formation of chemical compounds between unlike atoms in the In-rich region of Cu-In melts. We find a very good agreement between theoretically formulated and computed data of with the corresponding experimental values. Computed results of and suggest that hetero coordination is favorable over homo coordination in the investigated melts. The validity of the Stokes-Einstein relation was observed over a wide range of temperatures and compositions in the investigated melts. Further, a new correlation between two body pair excess entropy and the Stoke-Einstein relation has been formulated for square-well binary liquid alloys. Surface tension and compressibility as a function of In composition have been computed through microscopic structural functions of the alloys. Computed surface tension and ratio of mutual to intrinsic diffusion (D m /D id ) are found comparable to available simulated data. The computed values of shear viscosity are in good agreement with the experimental data. Calculated results suggest that the combination of hard sphere potential with square-well tail under mean spherical model approximation is one of the good method for determining the structures and transport coefficients of compound forming binary liquid alloys.

Multilevel simulation of hard-sphere mixtures.

We present a multilevel Monte Carlo simulation method for analyzing multi-scale physical systems via a hierarchy of coarse-grained representations, to obtain numerically exact results, at the most detailed level. We apply the method to a mixture of size-asymmetric hard spheres, in the grand canonical ensemble. A three-level version of the method is compared with a previously studied two-level version. The extra level interpolates between the full mixture and a coarse-grained description where only the large particles are present-this is achieved by restricting the small particles to regions close to the large ones. The three-level method improves the performance of the estimator, at fixed computational cost. We analyze the asymptotic variance of the estimator and discuss the mechanisms for the improved performance.

Towards Predicting Partitioning of Enzymes between Macromolecular Phases: Effects of Polydispersity on the Phase Behavior of Nonadditive Hard Spheres in Solution.

The ability to separate enzymes, or cells or viruses, from a mixture is important and can be realized by the incorporation of the mixture into a macromolecular solution. This incorporation may lead to a spontaneous phase separation, with one phase containing the majority of one of the species of interest. Inspired by this phenomenon, we studied the theoretical phase behavior of a model system composed of an asymmetric binary mixture of hard spheres, of which the smaller component was monodisperse and the larger component was polydisperse. The interactions were modeled in terms of the second virial coefficient and could be additive hard sphere (HS) or nonadditive hard sphere (NAHS) interactions. The polydisperse component was subdivided into two subcomponents and had an average size ten or three times the size of the monodisperse component. We gave the set of equations that defined the phase diagram for mixtures with more than two components in a solvent. We calculated the theoretical liquid–liquid phase separation boundary for the two-phase separation (the binodal) and three-phase separation, the plait point, and the spinodal. We varied the distribution of the polydisperse component in skewness and polydispersity, and we varied the nonadditivity between the subcomponents as well as between the main components. We compared the phase behavior of the polydisperse mixtures with binary monodisperse mixtures for the same average size and binary monodisperse mixtures for the same effective interaction. We found that when the compatibility between the polydisperse subcomponents decreased, the three-phase separation became possible. The shape and position of the phase boundary was dependent on the nonadditivity between the subcomponents as well as their size distribution. We conclude that it is the phase enriched in the polydisperse component that demixes into an additional phase when the incompatibility between the subcomponents increases.

Calculation of interfacial free energy for binary hard sphere mixtures

In this work, we present a method to calculate interfacial free energy for a binary hard sphere fluid mixture in coexistence with disordered face centered cubic solid. Monte Carlo(MC) Simulations are performed at 0.97 size ratio of small to large particle diameters. Interfacial free energy is calculated across composition for (100),(110) and (111) orientations of the substitutionally disordered solid phase along the coexistence line. Our calculations are based on the concept of Gibbs equimolar dividing surface. First, we calculate the surface excess quantities using the thermodynamic properties obtained from MC simulations then we integrate rewritten equivalent form of Gibbs adsorption equation with respect to pressure using pure hard sphere system interfacial free energy as a reference to get binary hard sphere mixture interfacial free energy. In our simulations we also varied cross sectional areas of solid exposed to fluid mixtures by varying system sizes, the computed interfacial free energies are consistent within error limits and confirms to scaling laws.