Binary Hard-sphere Mixtures Research Articles

Phase behavior of binary hard-sphere mixtures: Free volume theory including reservoir hard-core interactions.

Comprehensive calculations were performed to predict the phase behavior of large spherical colloids mixed with small spherical colloids that act as a depletant. To this end, the free volume theory (FVT) of Lekkerkerker et al. [Europhys. Lett. 20, 559 (1992)] is used as a basis and is extended to explicitly include the hard-sphere character of colloidal depletants into the expression for the free volume fraction. Taking the excluded volume of the depletants into account in both the system and the reservoir provides a relation between the depletant concentration in the reservoir and that in the system that accurately matches with computer simulation results of Dijkstra et al. [Phys. Rev. E 59, 5744 (1999)]. Moreover, the phase diagrams for highly asymmetric mixtures with size ratios q ≲ 0.2 obtained by using this new approach corroborate simulation results significantly better than earlier FVT applications to binary hard-sphere mixtures. The phase diagram of a binary hard-sphere mixture with a size ratio of q = 0.4, where a binary interstitial solid solution is formed at high densities, is investigated using a numerical free volume approach. At this size ratio, the obtained phase diagram is qualitatively different from previous FVT approaches for hard-sphere and penetrable depletants but again compares well with simulation predictions.

Navier–Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture

The Navier–Stokes transport coefficients for a model of a confined quasi-two-dimensional granular binary mixture of inelastic hard spheres are determined from the Boltzmann kinetic equation. A normal or hydrodynamic solution to the Boltzmann equation is obtained via the Chapman–Enskog method for states near the local version of the homogeneous time-dependent state. The mass, momentum, and heat fluxes are determined to first order in the spatial gradients of the hydrodynamic fields, and the associated transport coefficients are identified. They are given in terms of the solutions of a set of coupled linear integral equations. In addition, in contrast to the previous results obtained for low-density granular mixtures, there are also nonzero contributions to the first-order approximations to the partial temperatures Ti(1) and the cooling rate ζ(1). Explicit forms for the diffusion transport coefficients, the shear viscosity coefficient, and the quantities Ti(1) and ζ(1) are obtained by assuming steady state conditions and by considering the leading terms in a Sonine polynomial expansion. The above transport coefficients are given in terms of the coefficients of restitution, concentration, and the masses and diameters of the components of the mixture. The results apply, in principle, for arbitrary degree of inelasticity and are not limited to specific values of concentration, mass, and/or size ratios. As a simple application of these results, the violation of the Onsager reciprocal relations for a confined granular mixture is quantified in terms of the parameter space of the problem.

Size-sieving separation of hard-sphere gases at low concentrations through cylindrically porous membranes.

Membranes are compelling devices for many industrial separation processes, which are all subject to the intrinsic permeability-selectivity tradeoff. A general strategy to enhance separation performance is to reduce the pore size distribution and, ideally, make the membrane isoporous. In this study, we focus on a minimal model for regularly porous membranes, which consists of hard spheres moving through cylindrical pores. The collision dynamics is solved exactly and implemented in nonequilibrium event-driven molecular dynamics simulations. For such size-sieving porous membranes, we show that the permeability P of hard spheres of size σ through cylindrical pores of size d follows the hindered diffusion mechanism due to size exclusion as P ∝ (1 - σ/d)2. According to this law, the separation of binary mixtures of large and small particles exhibits a linear relationship between α-1/2 and P-1/2, where α and P are the selectivity and permeability of the smaller particle, respectively. The mean permeability through polydisperse pores is the average of the permeabilities of individual pores, weighted by the fraction of the single pore area over the total pore area.

Correlation functions in mixtures with energetically favoured nearest neighbours of different kind: a size-asymmetric case

Binary mixtures of hard spheres with different diameters and square-well attraction between different particles are studied by theory and Monte Carlo simulations. In our mesoscopic theory, local fluctuations of the volume fraction of the two components are taken into account. Semi-quantitative agreement between the simulation and theoretical results is obtained, except from very small distances. The correlation functions exhibit exponentially damped oscillations, with the period determined by the interaction potential, and both the amplitude and the correlation length increasing significantly with increasing diameter ratio. Increasing size asymmetry leads also to decreasing fluctuations of the number of the smaller particles in the attractive shell of the bigger ones. For small size asymmetry, the strongest correlations occur for comparable volume fraction of the two components. When the size ratio increases, the maximum of the structure factor moves to a larger volume fraction of the bigger particles, and for the size ratio as large as 4, the maximum goes beyond the accessible range of volume fractions. Our results show that when the neighbourhood of different particles is energetically favoured, the particles are much more uniformly distributed than in the random distribution even at relatively high temperature, especially for large size asymmetry.

Thermodynamic geometry for binary mixtures: A constrained approach

Thermodynamic geometry for binary mixtures is explored in a constrained version. In order to have an insight into the interpretation of the scalar curvature (R) for mixtures, we develop and discuss the thermodynamic geometry for binary mixtures of hard spheres and van der Waals fluids with a restriction in the total particle number. For the case of hard spheres we have an interaction which is exclusively repulsive and van der Waals fluids account for repulsive and attractive interactions, which is a more realistic model. From our analysis, it is seen that the physical interpretation of R given for monocomponent systems, is no longer satisfied for the case of mixture of ideal gases and hard spheres. For the constrained version of the mixture of van der Waals fluids, we have that the scalar curvature nicely reproduces the behavior observed for the monocomponent fluid, including the interpretation of R as an indicator of repulsive or attractive intermolecular interactions and phase transitions.

Temperature relaxation in binary hard-sphere mixture system: Molecular dynamics and kinetic theory study.

Computational schemes to describe the temperature relaxation in the binary hard-sphere mixture system are given on the basis of molecular dynamics (MD) simulation and renormalized kinetic theory. Event-driven MD simulations are carried out for three model systems in which the initial temperatures and the ratios of diameter and mass of two components are different to study the temporal evolution of each component temperature in nanoscale molecular conditions mimicking those in living cells. On the other hand, the temperature changes of the two components are also described in terms of a mean-field kinetic theory with the correlation functions calculated in the Percus-Yevick approximation. The calculated results by both the computational approaches have shown fair agreement with each other, whereas slight deviations have been found in the temporal range of femto- to picoseconds when the initial temperatures of the two components are significantly different, such as 300 K vs 1000 K. This discrepancy can be ascribed to the fast intra-component temperature relaxation assumed in the kinetic theory, and its violation in the MD simulations can be evaluated in terms of the Kullback-Leibler divergence between the equilibrated Maxwell-Boltzmann distribution at each temperature and the actual non-equilibrium velocity distribution realized in the MD. Thus, the present analysis provides a quantitative basis for addressing the temperature inhomogeneities experimentally observed in nanoscale crowding conditions.

Multiple character of non-monotonic size-dependence for relaxation dynamics in polymer-particle and binary mixtures

Adding plasticizers is a well-known procedure to reduce the glass transition temperature in polymers. It has been recently shown that this effect shows a non-monotonic dependence on the size of additive molecules (2019 J. Chem. Phys. 150 024903). In this work, we demonstrate that, as the size of the additive molecules is changed at fixed concentration, multiple extrema emerge in the dependence of the system’s relaxation time on the size ratio. The effect occurs on all relevant length scales including single monomer dynamics, decay of Rouse modes and relaxation of the chain’s end-to-end vector. A qualitatively similar trend is found within mode-coupling theoretical results for a binary hard-sphere mixture. An interpretation of the effect in terms of local packing efficiency and coupling between the dynamics of minority and majority species is provided.

Crystal structures in binary hard-sphere colloid-droplet mixtures with patchy cross interactions.

A binary mixture of droplets and patchy colloids, where patches are arranged in tetrahedral symmetry, is studied with Metropolis Monte Carlo simulations. The colloidal patches attract droplets, while both the colloid-colloid and the droplet-droplet interactions are hard sphere like. We find stable crystal structures with atomic analogs ZnS,CaF_{2}, and fcc or hcp (face centered cubic or hexagonal close packed) of the droplets coexisting with a dispersed fluid of the colloids. The simulated crystal structures agree well with those predicted by close-packing calculations for an intermediate range of droplet-colloid size ratios. A discrepancy between the simulations and theoretical predictions occurs at low and high size ratios. The results of the simulations for mixtures with anisotropic colloid-droplet interactions reveal a richer phase diagram, with ZnS-gas and ZnS-fluid coexistence, as compared to the isotropic case. For the example of a square planar patch arrangement, we find a particular crystal structure, consisting of two interpenetrating fcc or hcp lattices with right bond angles. Such a structure has no known atomic analog. Our study of generic models of anisotropic colloid-droplet mixtures could provide a promising way towards the fabrication of novel and complex colloidal structures.

Structural properties of additive binary hard-sphere mixtures.

An approach to obtain the structural properties of additive binary hard-sphere mixtures is presented. Such an approach, which is a nontrivial generalization of the one recently used for monocomponent hard-sphere fluids [S. Pieprzyk, A. C. Brańka, and D. M. Heyes, Phys. Rev. E 95, 062104 (2017)2470-004510.1103/PhysRevE.95.062104], combines accurate molecular-dynamics simulation data, the pole structure representation of the total correlation functions, and the Ornstein-Zernike equation. A comparison of the direct correlation functions obtained with the present scheme with those derived from theoretical results stemming from the Percus-Yevick (PY) closure and the so-called rational-function approximation (RFA) is performed. The density dependence of the leading poles of the Fourier transforms of the total correlation functions and the decay of the pair correlation functions of the mixtures are also addressed and compared to the predictions of the two theoretical approximations. A very good overall agreement between the results of the present scheme and those of the RFA is found, thus suggesting that the latter (which is an improvement over the PY approximation) can safely be used to predict reasonably well the long-range behavior, including the structural crossover, of the correlation functions of additive binary hard-sphere mixtures.

High antisite defect concentrations in hard-sphere colloidal Laves phases.

Binary mixtures of hard spheres can spontaneously self-assemble into binary crystals. Computer simulations have been especially useful in mapping out the phase behaviour of these mixtures, under the assumption that the stoichiometry of the binary crystal is ideal. Here we show that for a size ratio of q = 0.82 this assumption is not valid near the coexistence region between the fluid and the stable binary crystal, the MgZn2 Laves phase. Instead we find a surprisingly high number of antisite defects: up to 2% of the large spheres are replaced by small spheres in equilibrium. We demonstrate that the defect concentration can be estimated using simple approximations, providing an easy way to identify systems where antisite defects play an important role. Our results shed new light on the self-assembly of colloidal Laves phases, and demonstrate the importance of antisite defects in binary crystals.

Active binary mixtures of fast and slow hard spheres.

We computationally studied the phase behavior and dynamics of binary mixtures of active particles, where each species had distinct activities leading to distinct velocities, fast and slow. We obtained phase diagrams demonstrating motility-induced phase separation (MIPS) upon varying the activity and concentration of each species, and extended current kinetic theory of active/passive mixtures to active/active mixtures. We discovered two regimes of behavior quantified through the participation of each species in the dense phase compared to their monodisperse counterparts. In regime I (active/passive and active/weakly-active), we found that the dense phase was segregated by particle type into domains of fast and slow particles. Moreover, fast particles were suppressed from entering the dense phase while slow particles were enhanced entering the dense phase, compared to monodisperse systems of all-fast or all-slow particles. These effects decayed asymptotically as the activity of the slow species increased, approaching the activity of the fast species until they were negligible (regime II). In regime II, the dense phase was homogeneously mixed and each species participated in the dense phase as if it were it a monodisperse system (i.e. not mixed at all). Finally, we showed that a weighted average of constituent particle activities, which we term the net activity, defines a binodal for the MIPS transition in active/active binary mixtures. We examined the critical point of the transition and found a critical exponent (β = 0.45) in agreement with similar studies on monodisperse systems, and distinct from equilibrium systems.

The role of inertia on a locally favoured structure in equimolar binary mass Hard Spheres

Molecular dynamics simulations have been carried out to investigate the effect of inertia on both dynamics and locally favoured structures (LFS) of a supercooled liquid toward glass transition. In this work, the equimolar supercooled liquid binary mixtures of Hard Spheres with the same diameter but different masses were studied. The mass ratio of the binary system was 2:1. The simulation was performed and sampled by using the microcanonical ensemble. Both dynamics and structure changing were analysed. A cluster structure analysis was detected by topological cluster classification (TCC) based on the results of local free energy minimisation. The slowing down dynamics were found to be related to cluster structure changing in both the binary mass Hard Sphere (BMHS) and Hard Sphere (HS). The inertia affected an increasing dynamical heterogeneity and had some effects on favouring some local structures before they get crystallisation.

Theoretical calculation of the buoyancy force on a particle in flowing granular mixtures.

Simulations and experiments of granular mixtures comprising two different sizes flowing over an inclined plane under the influence of gravity show segregation, where large particles rise to the free surface. The segregation results from differential forces acting on the particles. The buoyancy force experienced by the particles is an important component of the total force acting on the particles and, in this work, we theoretically calculate the buoyant force on intruder particles of different sizes in a flowing granular medium. The effective particle volume for buoyancy force calculation is obtained as the partial molar volume of the large particle from the equation of state for a binary mixture of hard spheres. The theoretical results are in good agreement with results obtained from discrete element method (DEM) simulations. Our calculations show that the buoyancy force on a particle is always smaller than the weight of the particle for larger particles. The ratio of the buoyancy force to the weight of the large particles decreases with increasing size ratio and increases with increasing concentration of the particles. This ratio approaches unity in the limiting cases of size ratio and large particle concentration approaching unity. These theoretical calculations establish a method for obtaining the buoyancy force experienced by particles in flowing mixtures.

Effect of size disparity on the structure and dynamics of the small component in concentrated binary colloidal mixtures.

We determined, using confocal microscopy, the structure and dynamics of the small component in concentrated binary colloidal mixtures with moderate and large size ratios and different compositions of Polymethyl methacrylate particles. We show that when increasing the content of small spheres at fixed total volume fraction, a transition in the local environment of the small particles is observed, from a mixed environment of other small and large particles to a local environment of only small particles. The transition is rather abrupt for moderate size ratios, while it becomes particularly broad for large size ratios. This can be associated with the improved ability of the small particles to pack in between the large particles for larger size ratios. The dynamics reflect the transition with an increase of the mobility observed at intermediate mixing. This increase becomes particularly pronounced for large size ratios, leading to diffusive dynamics of the small particles, in agreement with predictions of theories of the glass transition in binary hard-sphere mixtures. The composition at which the fastest dynamics are observed is apparently independent of the size ratio.

Glass-transition asymptotics in two theories of glassy dynamics: Self-consistent generalized Langevin equation and mode-coupling theory.

We contrast the generic features of structural relaxation close to the idealized glass transition that are predicted by the self-consistent generalized Langevin equation theory (SCGLE) against those that are predicted by the mode-coupling theory of the glass transition (MCT). We present an asymptotic solution close to conditions of kinetic arrest that is valid for both theories, despite the different starting points that are adopted in deriving them. This in particular provides the same level of understanding of the asymptotic dynamics in the SCGLE as was previously done only for MCT. We discuss similarities and different predictions of the two theories for kinetic arrest in standard glass-forming models, as exemplified through the hard-sphere system. Qualitative differences are found for models where a decoupling of relaxation modes is predicted, such as the generalized Gaussian core model, or binary hard-sphere mixtures of particles with very disparate sizes. These differences, which arise in the distinct treatment of the memory kernels associated to self- and collective motion of particles, lead to distinct scenarios that are predicted by each theory for partially arrested states and in the vicinity of higher-order glass-transition singularities.

Theory for the single-particle dynamics in glassy mixtures with particle size swaps

We present a theory for the single-particle dynamics in binary mixtures with particle size swaps. The general structure of the theory follows that of the theory for the collective dynamics in binary mixtures with particle size swaps, which we developed previously (Szamel 2018 Phys. Rev. E 98 050601). Particle size swaps open up an additional relaxation channel, which speeds up both the collective dynamics and the single-particle dynamics. To make explicit predictions, we resort to a factorization approximation similar to that employed in the mode-coupling theory of glassy dynamics. We show that, like in the standard mode-coupling theory, the single-particle motion becomes arrested at the dynamic glass transition predicted by the theory for the collective dynamics. We compare the non-ergodicity parameters predicted by our mode-coupling-like approach for an equimolar binary hard sphere mixture with particle size swaps with the non-ergodicity parameters predicted by the standard mode-coupling theory for the same system without swaps. Our theory predicts that the ‘cage size’ is bigger in the system with particle size swaps.

Thermodynamic behavior of binary mixtures of hard spheres: Semianalytical solutions on a Husimi lattice built with cubes.

We study binary mixtures of hard particles, which exclude up to their kth nearest neighbors (kNN) on the simple cubic lattice and have activities z_{k}. In the first model analyzed, point particles (0NN) are mixed with 1NN ones. The grand-canonical solution of this model on a Husimi lattice built with cubes unveils a phase diagram with a fluid and a solid phase separated by a continuous and a discontinuous transition line which meet at a tricritical point. A density anomaly, characterized by minima in isobaric curves of the total density of particles against z_{0} (or z_{1}), is also observed in this system. Overall, this scenario is identical to the one previously found for this model when defined on the square lattice. The second model investigated consists of the mixture of 1NN particles with 2NN ones. In this case, a very rich phase behavior is found in its Husimi lattice solution, with two solid phases-one associated with the ordering of 1NN particles (S1) and the other with the ordering of 2NN ones (S2)-beyond the fluid (F) phase. While the transitions between F-S2 and S1-S2 phases are always discontinuous, the F-S1 transition is continuous (discontinuous) for small (large) z_{2}. The critical and coexistence F-S1 lines meet at a tricritical point. Moreover, the coexistence F-S1,F-S2, and S1-S2 lines meet at a triple point. Density anomalies are absent in this case.

A kinetic-based hyperbolic two-fluid model for binary hard-sphere mixtures

Starting from coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures is derived with separate mean velocities and energies for each phase. In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients (e.g. lift). In the derivation, the particles in a given phase have identical mass and volume, and have no internal degrees of freedom (i.e. the particles are adiabatic). The ‘hard-sphere-fluid’ phase is obtained in the limit where the particle diameter in one phase tends to zero with fixed phase density so that the number of fluid particles tends to infinity. The moment system resulting from the two BE kinetic equations is closed at second order by invoking the anisotropic Gaussian closure. The resulting two-fluid model for a binary, hard-sphere mixture therefore consists (for each phase $\unicode[STIX]{x1D6FC}=1,2$) of transport equations for the mass $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}$, mean momentum $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ (where $\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ is the velocity) and a symmetric, second-order, kinetic energy tensor $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}}=\frac{1}{2}\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}\otimes \boldsymbol{u}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})$. The trace of the fluctuating energy tensor $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}}$ is $\text{tr}(\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})=3\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ where $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ is the phase temperature (or granular temperature). Thus, $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}E_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\text{tr}(\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}})$ is the total kinetic energy, the sum over $\unicode[STIX]{x1D6FC}$ of which is the total kinetic energy of the system, a conserved quantity. From the analysis, it is found that the BE finite-size correction leads to exact phase pressure (or stress) tensors that depend on the mean-slip velocity $\boldsymbol{u}_{12}=\boldsymbol{u}_{1}-\boldsymbol{u}_{2}$, as well as the phase temperatures for both phases. These pressure tensors also appear in the momentum-exchange terms in the mean momentum equations that produce the Archimedes force, as well as drag contributions due to fluid compressibility and a lift force due to mean fluid-velocity gradients. The closed BE energy flux tensors show a similar dependence on the mean-slip velocity. The characteristic polynomial of the flux matrix from the one-dimensional model is computed symbolically and depends on five parameters: the particle volume fractions $\unicode[STIX]{x1D711}_{1}$, $\unicode[STIX]{x1D711}_{2}$, the phase density ratio ${\mathcal{Z}}=\unicode[STIX]{x1D70C}_{f}/\unicode[STIX]{x1D70C}_{p}$, the phase temperature ratio $\unicode[STIX]{x1D6E9}_{r}=\unicode[STIX]{x1D6E9}_{2}/\unicode[STIX]{x1D6E9}_{1}$ and the mean-slip Mach number $Ma_{s}=\boldsymbol{u}_{12}/\sqrt{5\unicode[STIX]{x1D6E9}_{1}/3}$. By applying Sturm’s Theorem to the characteristic polynomial, it is demonstrated that the model is hyperbolic over a wide range of these parameters, in particular, for the physically most relevant values.

Effect of particle exchange on the glass transition of binary hard spheres

We investigate the replica theory of the liquid-glass transition for a binary mixture of large and small additive hard spheres. We consider two different ansätze for this problem: the frozen glass ansatz (FGA) in which the exchange of large and small particles in a glass state is prohibited, and the exchange glass ansatz (EGA), in which it is allowed. We calculate the dynamical and thermodynamical glass transition points with the two ansätze. We show that the dynamical transition density of the FGA is lower than that of the EGA, while the thermodynamical transition density of the FGA is higher than that of the EGA. We discuss the algorithmic implications of these results for the density-dependence of the relaxation time of supercooled liquids. We particularly emphasize the difference between the standard Monte Carlo and swap Monte Carlo algorithms. Furthermore, we discuss the importance of particle exchange for estimating the configurational entropy.

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.