Mixture Of Hard Spheres Research Articles

Importance of feature construction in machine learning for phase transitions.

Machine learning is an important tool in the study of the phase behavior from molecular simulations. In this work, we use un-supervised machine learning methods to study the phase behavior of two off-lattice models, a binary Lennard-Jones (LJ) mixture and the Widom-Rowlinson (WR) non-additive hard-sphere mixture. The majority of previous work has focused on lattice models, such as the 2D Ising model, where the values of the spins are used as the feature vector that is input into the machine learning algorithm, with considerable success. For these two off-lattice models, we find that the choice of the feature vector is crucial to the ability of the algorithm to predict a phase transition, and this depends on the particular model system being studied. We consider two feature vectors, one where the elements are distances of the particles of a given species from a probe (distance-based feature) and one where the elements are +1 if there is an excess of particles of the same species within a cut-off distance and -1 otherwise (affinity-based feature). We use principal component analysis and t-distributed stochastic neighbor embedding to investigate the phase behavior at a critical composition. We find that the choice of the feature vector is the key to the success of the unsupervised machine learning algorithm in predicting the phase behavior, and the sophistication of the machine learning algorithm is of secondary importance. In the case of the LJ mixture, both feature vectors are adequate to accurately predict the critical point, but in the case of the WR mixture, the affinity-based feature vector provides accurate estimates of the critical point, but the distance-based feature vector does not provide a clear signature of the phase transition. The study suggests that physical insight into the choice of input features is an important aspect for implementing machine learning methods.

Coarse-grained Hamiltonian and effective one component theory of colloidal suspensions

We develop a theory to trace out the solvent degrees of freedom from the grand partition function of colloid-solvent mixtures. Our approach to coarse-graining is based on density functional formalism of density profile and the grand thermodynamic potential of solvent. The solvent-induced interaction which is many-body in character is expressed in terms of two functionals; one that couples the solvent to the colloidal density distribution and the second represents the density–density correlation function of the solvent. The nature, strength, and range of the potential depend on these functionals and therefore on the thermodynamic state of the solvent. The solvent-induced contribution to free energy functional is also derived. A self-consistent procedure is developed to calculate the effective potential between colloidal particles, colloid-solvent, and colloid-colloid correlation functions. The theory is used to investigate both additive and nonadditive binary hard-sphere mixtures. Results are reported for the two systems for several values of packing fractions ηb and ηs and particles diameter ratio q=σsσb where symbols b and s refer to colloid and solvent, respectively. Several interesting features are found: The short-range attractive part of the potential shows non-monotonic dependence on ηb; when ηb is increased from zero, initially the potential becomes more attractive but beyond a certain value of ηb that depends on q, the attraction starts weakening. The repulsive peaks formed at R∼(1+12nq) where R is a distance between centers of colloidal particles expressed in units of σb and n is an integer, become stronger on increasing ηb. These results show that many-body contribution to the effective potential depends in a subtle way on packing fractions ηb,ηs, size ratio q, and on nature of the interaction model and makes a non-negligible contribution to the coarse-grained Hamiltonian.

Enhancement in the diffusivity of Brownian spheroids in the presence of spheres.

In the present paper, we have extended the simulation technique Brownian cluster dynamics (BCD) to analyze the dynamics of the binary mixture of hard ellipsoids and spheres. The shape dependent diffusional properties have been incorporated into BCD using Perrin's factor and compared with analytical results of a one-component ellipsoidal system. We have investigated pathways to enhance the diffusivity of spheroids in the binary mixture by manipulating the phase behavior of the system through varying the fraction of spheres in the binary mixture. We show that at low volume fraction the spherical particles have a higher diffusion coefficient than the ellipsoids due to the higher friction coefficient. However, at a higher volume fraction, we show that the diffusion coefficient of the ellipsoids increases irrespective of the aspect ratio due to the anisotropic shape.

Kinetic Theory of Polydisperse Granular Mixtures: Influence of the Partial Temperatures on Transport Properties-A Review.

It is well-recognized that granular media under rapid flow conditions can be modeled as a gas of hard spheres with inelastic collisions. At moderate densities, a fundamental basis for the determination of the granular hydrodynamics is provided by the Enskog kinetic equation conveniently adapted to account for inelastic collisions. A surprising result (compared to its molecular gas counterpart) for granular mixtures is the failure of the energy equipartition, even in homogeneous states. This means that the partial temperatures (measuring the mean kinetic energy of each species) are different to the (total) granular temperature T. The goal of this paper is to provide an overview on the effect of different partial temperatures on the transport properties of the mixture. Our analysis addresses first the impact of energy nonequipartition on transport which is only due to the inelastic character of collisions. This effect (which is absent for elastic collisions) is shown to be significant in important problems in granular mixtures such as thermal diffusion segregation. Then, an independent source of energy nonequipartition due to the existence of a divergence of the flow velocity is studied. This effect (which was already analyzed in several pioneering works on dense hard-sphere molecular mixtures) affects to the bulk viscosity coefficient. Analytical (approximate) results are compared against Monte Carlo and molecular dynamics simulations, showing the reliability of kinetic theory for describing granular flows.

Comparing machine learning techniques for predicting glassy dynamics.

In the quest to understand how structure and dynamics are connected in glasses, a number of machine learning based methods have been developed that predict dynamics in supercooled liquids. These methods include both increasingly complex machine learning techniques and increasingly sophisticated descriptors used to describe the environment around particles. In many cases, both the chosen machine learning technique and choice of structural descriptors are varied simultaneously, making it hard to quantitatively compare the performance of different machine learning approaches. Here, we use three different machine learning algorithms-linear regression, neural networks, and graph neural networks-to predict the dynamic propensity of a glassy binary hard-sphere mixture using as structural input a recursive set of order parameters recently introduced by Boattini et al. [Phys. Rev. Lett. 127, 088007 (2021)]. As we show, when these advanced descriptors are used, all three methods predict the dynamics with nearly equal accuracy. However, the linear regression is orders of magnitude faster to train, making it by far the method of choice.

Clustering in Mixtures of SALR Particles and Hard Spheres with Cross Attraction.

Self-assembling complex fluids are often modeled as particles with effective competing isotropic interactions, combining a short-range attraction (SA) followed by a longer-range repulsion (LR). For moderately low temperatures and densities, SALR particles form clusters in equilibrium, at least provided that the potential parameters are appropriate. Here we inquire into the possibility that cluster formation in SALR fluids might be pushed by a foreign species even under thermodynamic conditions that would not allow for clusterization of the pure system. To this aim, we study by Monte Carlo simulations a mixture of hard-sphere two-Yukawa particles and hard spheres, with a cross interaction modeled by a square-well attraction, and we investigate the conditions of clustering in terms of strength of attraction and relative concentration of the two species. We find that clusters can occur in the mixture for the same temperature and density where the pure SALR fluid is almost structureless. In particular, we single out a cross attraction such that clusters are formed with a SALR concentration as low as 5%. We also find a situation where nearly pure droplets of hard spheres are held together by a shell of SALR particles. Conversely, we show that clustering can be undermined in the mixture under conditions for which this process takes place in the parent SALR fluid. Using a simple criterion, based on the second virial coefficients of the attractive part of interaction potentials (the so-called “reference attractive fluids”), we are able to predict accurately whether clustering is favored (or hindered) in the mixture, as compared to the pure SALR fluid.

Excluded volume interactions and phase stability in mixtures of hard spheres and hard rods.

In this paper we study excluded volume interactions, the free volume fraction available, and the phase behaviour, in mixtures of hard spheres (HS) and hard rods, modeled as spherocylinders. We use free volume theory (FVT) to predict various physical properties and compare to Monte Carlo computer simulations. FVT is used at two levels. We use the original FVT approach in which it is assumed that the correlations of the HS are not affected by the rods. This is compared to a recent, more rigorous, FVT approach which includes excluded volume interactions between the different components at all levels. We find that the novel rigorous FVT approach agrees well with computer simulation results at the level of free volume available, as well as for the phase stability. The FVT predictions show significant quantitative and qualitative deviations with respect to the original FVT approach. The phase transition curves are systematically at higher rod concentrations than previously predicted. Furthermore, the calculations revealed that a certain asphericity is required to induce isostructural fluid-fluid coexistence and the stability region is highly dependent on the size ratio between the rods and the spheres.

Connecting Packing Efficiency of Binary Hard Sphere Systems to Their Intermediate Range Structure.

Using computed x-ray tomography we determine the three dimensional (3D) structure of binary hard sphere mixtures as a function of composition and size ratio of the particles q. Using a recently introduced four-point correlation function we reveal that this 3D structure has on intermediate and large length scales a surprisingly regular order, the symmetry of which depends on q. The related structural correlation length has a minimum at the composition at which the packing fraction is highest. At this composition also the number of different local particle arrangements has a maximum, indicating that efficient packing of particles is associated with a structure that is locally maximally disordered.

Structural properties of additive binary hard-sphere mixtures. III. Direct correlation functions.

An analysis of the direct correlation functions c_{ij}(r) of binary additive hard-sphere mixtures of diameters σ_{s} and σ_{b} (where the subscripts s and b refer to the "small" and "big" spheres, respectively), as obtained with the rational-function approximation method and the WM scheme introduced in previous work [S. Pieprzyk et al., Phys. Rev. E 101, 012117 (2020)2470-004510.1103/PhysRevE.101.012117], is performed. The results indicate that the functions c_{ss}(r<σ_{s}) and c_{bb}(r<σ_{b}) in both approaches are monotonic and can be well represented by a low-order polynomial, while the function c_{sb}(r<1/2(σ_{b}+σ_{s})) is not monotonic and exhibits a well-defined minimum near r=1/2(σ_{b}-σ_{s}), whose properties are studied in detail. Additionally, we show that the second derivative c_{sb}^{''}(r) presents a jump discontinuity at r=1/2(σ_{b}-σ_{s}) whose magnitude satisfies the same relationship with the contact values of the radial distribution function as in the Percus-Yevick theory.

Fluid-fluid phase behaviour in the explicit hard spherocylinder solvent ionic model confined in a disordered porous medium

We propose a theoretical approach that takes into account the effects of a disordered porous confinement on the liquid–liquid phase behavior of the ionic solution modelled by the restricted primitive model with an explicit consideration of the neutral hard spherocylinder solvent. The theory is based on a combination of two approaches: the scale particle theory for the description of the reference system represented by a mixture of hard spheres and hard spherocylinders confined in a disordered hard-sphere matrix and the associative mean-spherical approximation for taking into account Coulomb interactions between ions. Alternatively, the mean spherical approximation is applied for the description of the ionic subsystem for comparison. For the considered solvent explicit model in a matrix, analytical expressions for pressure, free energy and partial chemical potentials are derived for the first time. Using these expressions, the liquid–liquid phase transition is studied, where one phase is enriched with ions and another one – with solvent particles. In the solvent-rich phase, the isotropic-nematic phase transition is observed due to the orientational ordering of the spherocylinder solvent particles at different porosities of a matrix. The effects of the matrix confinement and of the solvent particle elongation on the liquid–liquid and isotropic-nematric phase transitions are studied at different pressures. The role of association phenomena appearing between positively and negatively charged ions in phase behavior of the considered ionic solutions under confinement is discussed.

Critical point for demixing of binary hard spheres.

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

Structural properties of additive binary hard-sphere mixtures. II. Asymptotic behavior and structural crossovers.

The structural properties of additive binary hard-sphere mixtures are addressed as a follow-up of a previous paper [S. Pieprzyk et al., Phys. Rev. E 101, 012117 (2020)]2470-004510.1103/PhysRevE.101.012117. The so-called rational-function approximation method and an approach combining accurate molecular dynamics simulation data, the pole structure representation of the total correlation functions, and the Ornstein-Zernike equation are considered. The density, composition, and size-ratio dependencies of the leading poles of the Fourier transforms of the total correlation functions h_{ij}(r) of such mixtures are presented, those poles accounting for the asymptotic decay of h_{ij}(r) for large r. Structural crossovers, in which the asymptotic wavelength of the oscillations of the total correlation functions changes discontinuously, are investigated. The behavior of the structural crossover lines as the size ratio and densities of the two species are changed is also discussed.

Averaging Local Structure to Predict the Dynamic Propensity in Supercooled Liquids.

Predicting the local dynamics of supercooled liquids based purely on local structure is a key challenge in our quest for understanding glassy materials. Recent years have seen an explosion of methods for making such a prediction, often via the application of increasingly complex machine learning techniques. The best predictions so far have involved so-called Graph Neural Networks (GNNs) whose accuracy comes at a cost of models that involve on the order of 10^{5} fit parameters. In this Letter, we propose that the key structural ingredient to the GNN method is its ability to consider not only the local structure around a central particle, but also averaged structural features centered around nearby particles. We demonstrate that this insight can be exploited to design a significantly more efficient model that provides essentially the same predictive power at a fraction of the computational complexity (approximately 1000 fit parameters), and demonstrate its success by fitting the dynamic propensity of Kob-Andersen and binary hard-sphere mixtures. We then use this to make predictions regarding the importance of radial and angular descriptors in the dynamics of both models.

Effective potentials induced by mixtures of patchy and hard co-solutes.

The addition of co-solutes to colloidal suspensions is often employed to induce tunable depletion interactions. In this work, we investigate effective colloidal interactions arising from binary co-solute mixtures of hard spheres and patchy particles. By changing the relative concentration of the two species, we show that the resulting effective potential Veff continuously changes from the one obtained for a single-component hard sphere co-solute to that mediated by the single-component patchy particle co-solute. Interestingly, we find that, independent of the relative concentration of the two components, the resulting Veff is additive, i.e., it is well-described by the linear combination of the effective interactions mediated by respective pure co-solutes. However, a breakdown of the additivity occurs when the co-solute mixture is close to the onset of a demixing transition. These results represent a step forward in understanding and predicting colloidal behavior in complex and crowded environments and for exploiting this knowledge to design targeted colloidal superstructures.

Modelling the Diffusion Coefficients of Dilute Gaseous Solutes in Hydrocarbon Liquids

In this work, we present a model, based on rough hard-sphere theory, for the tracer diffusion coefficients of gaseous solutes in non-polar liquids. This work extends an earlier model developed specifically for carbon dioxide in hydrocarbon liquids and establishes a general correlation for gaseous solutes in non-polar liquids. The solutes considered were light hydrocarbons, carbon dioxide, nitrogen and argon, while the solvents were all hydrocarbon liquids. Application of the model requires knowledge of the temperature-dependent molar core volumes of the solute and solvent, which can be determined from pure-component viscosity data, and a temperature-independent roughness factor which can be determined from a single diffusion coefficient measurement in the system of interest. The new model was found to correlate the experimental data with an average absolute relative deviation of 2.7 %. The model also successfully represents computer-simulation data for tracer diffusion coefficients of hard-sphere mixtures and reduces to the expected form for self-diffusion when the solute and solvent become identical.

Evaluation of Osmotic Virial Coefficients via Restricted Gibbs Ensemble Simulations, with Support from Gas-Phase Mixture Coefficients.

We present a method for computing osmotic virial coefficients in explicit solvent via simulation in a restricted Gibbs ensemble. Two equivalent phases are simulated at once, each in a separate box at constant volume and temperature and each in equilibrium with a solvent reservoir. For osmotic coefficient , a total of N solutes are individually exchanged back and forth between the boxes, and the average distribution of solute numbers between the boxes provides the key information needed to compute . Separately, expressions are developed for as a series in solvent reservoir density ρ1, with the coefficients of the series expressed in terms of the usual gas-phase mixture coefficients Bij. Normally, the Bij are defined for an infinite volume, but we suggest that the observed dependence of Bij on system size L can be used to estimate L dependence of the , allowing them to be computed accurately at L → ∞ while simulating much smaller system sizes than otherwise possible. The methods for N = 2 and 3 are demonstrated for two-component mixtures of size-asymmetric additive hard spheres. The proposed methods are demonstrated to have greater precision than established techniques, for a given amount of computational effort. The ρ1 series for when applied by itself is (for this noncondensing model) found to be the most efficient in computing accurate osmotic coefficients for the solvent densities considered here.

Gas-liquid phase transition in a binary mixture with an interaction that creates constant density profiles.

If, in a hard sphere fluid, a single (test) particle is fixed, the other particles display a density profile that possesses long-ranged oscillations. Surprisingly, one can show via classical density functional theory that it takes a simple, purely repulsive (external) potential with a finite range in addition to the fixed hard sphere that forces these oscillations to vanish completely. This can give rise to interesting phenomena; however, it gained little attention in the past. In this work, we use the potential in question as an inter-component interaction in a binary hard-sphere mixture, where it is shown that the effective interaction induced by one component resembles qualitatively the well-known Asakura-Oosawa-Vrij potential and can lead to a liquid-gas phase transition in the other component.

Alternative formulation of the induced surface and curvature tensions approach

We develop a novel method to analyze the excluded volume of the multicomponent mixtures of classical hard spheres in the grand canonical ensemble. The method is based on the Laplace–Fourier transform technique and allows one to account for the fluctuations of the particle number density for the induced surface and curvature tensions equation of state. As a result one can go beyond the Van der Waals (VdW) approximation by obtaining the suppression of the induced surface and curvature tensions coefficients at moderate and high packing fractions. In contrast to the standard induced surface and curvature tensions equation of state the suppression of these coefficients is not the exponential, but a power-like one. The obtained alternative equation of state is further generalized to account for higher virial coefficients. This result is straightforwardly generalized to the case of quantum statistics.

Effects of Polydispersity on the Phase Behavior of Additive Hard Spheres in Solution.

The ability to separate enzymes, nucleic acids, cells, and viruses is an important asset in life sciences. This can be realised by using their spontaneous asymmetric partitioning over two macromolecular aqueous phases in equilibrium with one another. Such phases can already form while mixing two different types of macromolecules in water. We investigate the effect of polydispersity of the macromolecules on the two-phase formation. We study theoretically the phase behavior of a model polydisperse system: an asymmetric binary mixture of hard spheres, of which the smaller component is monodisperse and the larger component is polydisperse. The interactions are modelled in terms of the second virial coefficient and are assumed to be additive hard sphere interactions. The polydisperse component is subdivided into sub-components and has an average size ten times the size of the monodisperse component. We calculate the theoretical liquid–liquid phase separation boundary (the binodal), the critical point, and the spinodal. We vary the distribution of the polydisperse component in terms of skewness, modality, polydispersity, and number of sub-components. We compare the phase behavior of the polydisperse mixtures with their concomittant monodisperse mixtures. We find that the largest species in the larger (polydisperse) component causes the largest shift in the position of the phase boundary, critical point, and spinodal compared to the binary monodisperse binary mixtures. The polydisperse component also shows fractionation. The smaller species of the polydisperse component favor the phase enriched in the smaller component. This phase also has a higher-volume fraction compared to the monodisperse mixture.

An Artificial Neural Network Reveals the Nucleation Mechanism of a Binary Colloidal AB13 Crystal.

Colloidal suspensions of two species have the ability to form binary crystals under certain conditions. The hunt for these functional materials and the countless investigations on their formation process are justified by the plethora of synergetic and collective properties these binary superlattices show. Among the many crystal structures observed over the past decades, the highly exotic colloidal icosahedral AB13 crystal was predicted to be stable in binary hard-sphere mixtures nearly 30 years ago, yet the kinetic pathway of how homogeneous nucleation occurs in this system is still unknown. Here we investigate binary nucleation of the AB13 crystal from a binary fluid phase of nearly hard spheres. We calculate the nucleation barrier and nucleation rate as a function of supersaturation and draw a comparison with nucleation of single-component and other binary crystals. To follow the nucleation process, we employ a neural network to identify the AB13 phase from the binary fluid phase and the competing fcc crystal with single-particle resolution and significant accuracy in the case of bulk phases. We show that AB13 crystal nucleation proceeds via a coassembly process where large spheres and icosahedral small-sphere clusters simultaneously attach to the nucleus. Our results lend strong support for a classical pathway that is well-described by classical nucleation theory, even though the binary fluid phase is highly structured and exhibits local regions of high bond orientational order.