Freezing Of Hard Spheres Research Articles

Finding the differences: Classical nucleation perspective on homogeneous melting and freezing of hard spheres.

By employing brute-force molecular dynamics, umbrella sampling, and seeding simulations, we investigate homogeneous nucleation during melting and freezing of hard spheres. We provide insights into these opposing phase transitions from the standpoint of classical nucleation theory. We observe that melting has both a lower driving force and a lower interfacial tension than freezing. The lower driving force arises from the vicinity of a spinodal instability in the solid and from a strain energy. The lower interfacial tension implies that the Tolman lengths associated with melting and freezing have opposite signs, a phenomenon that we interpret with Turnbull's rule. Despite these asymmetries, the nucleation rates for freezing and melting are found to be comparable.

Ising model for the freezing transition.

We introduce a three-state Ising model with entropy-volume coupling suitably incorporating a packing mechanism into a lattice gas with no attractive interactions. On working in a great grand canonical ensemble in which the energy, volume, and number of particles are all allowed to fluctuate simultaneously, the model's mean-field solutions illuminate a strictly first-order transition akin to hard-sphere freezing while describing the thermodynamics of solid and fluid phases. Further implementation of attractive interactions in a natural way allows every aspect of the phase diagram of a simple substance to be reproduced, thereby accomplishing the van der Waals picture of the states of matter from first principles of statistical mechanics. This fairly accurate qualitative description plausibly renders mean-field theory a reasonable approach for freezing in three dimensions. At the same time, our mean-field treatment itself suggests freezing to persist in infinitely many dimensions, as advanced from recent simulations [Charbonneau et al., Eur. Phys. J. E 44, 101 (2021)10.1140/epje/s10189-021-00104-y].

Superlattice formation in colloidal nanocrystal suspensions: Hard-sphere freezing and depletion effects

Superlattice formation in dried suspensions of colloidal silver nanocrystals (AgNCs) is investigated through a combination of experiment, theory, and simulation. Using microscopic and spectroscopic techniques, we explore the phase behavior of dried AgNC suspensions, and we model the system using Monte Carlo simulations of a coarse-grained model with an effective pair potential that accounts for the composition of the nanocrystal core, the stabilizing ligand shell, and entropic effects associated with unbound ligand in solution. In the absence of free ligands, the effective potential at ligand contact is purely repulsive, and we find superlattice formation at an effective AgNC volume fraction close to that anticipated for hard-sphere freezing. In the presence of free ligands, the effective potential becomes attractive, and crystallization is accompanied by phase separation and multiphase coexistence, as anticipated for colloid-polymer mixtures.

Nonmonotonic pressure as a function of the density in a fluid without attractive forces

A simple result for the pressure of a hard sphere fluid that was developed many years ago by Rennert is extended in a straightforward manner by adding the terms that are of the same form as the Rennert's formula. The resulting expression is moderately accurate but its accuracy does not necessarily improve as additional terms are included. This expression has the interesting consequence that the pressure can have a maximum, as the density increases, which is consistent with the freezing of hard spheres. This occurs solely as a consequence of repulsive interactions. Only the Born-Green-Yvon and Kirkwood theories show such a behavior for hard spheres and they require a numerical solution of an integral equation. The procedure outlined here is ad hoc but is, perhaps, useful just as the popular Carnahan-Starling equation for the hard sphere pressure is also ad hoc but useful.

Precise simulation of the freezing transition of supercritical Lennard-Jones

The fluid-solid transition of the Lennard-Jones model is analyzed along a supercritical isotherm. The analysis is implemented via a simulation method which is based on a modification of the constrained cell model of Hoover and Ree. In the context of hard-sphere freezing, Hoover and Ree simulated the solid phase using a constrained cell model in which each particle is confined within its own Wigner-Seitz cell. Hoover and Ree also proposed a modified cell model by considering the effect of an external field of variable strength. High-field values favor configurations with a single particle per Wigner-Seitz cell and thus stabilize the solid phase. In previous work, a simulation method for freezing transitions, based on constant-pressure simulations of the modified cell model, was developed and tested on a system of hard spheres. In the present work, this method is used to determine the freezing transition of a Lennard-Jones model system on a supercritical isotherm at a reduced temperature of 2. As in the case of hard spheres, constant-pressure simulations of the fully occupied constrained cell model of a system of Lennard-Jones particles indicate a point of mechanical instability at a density which is approximately 70% of the density at close packing. Furthermore, constant-pressure simulations of the modified cell model indicate that as the strength of the field is reduced, the transition from the solid to the fluid is continuous below the mechanical instability point and discontinuous above. The fluid-solid transition of the Lennard-Jones system is obtained by analyzing the field-induced fluid-solid transition of the modified cell model in the high-pressure, zero-field limit. The simulations are implemented under constant pressure using tempering and histogram reweighting techniques. The coexistence pressure and densities are determined through finite-size scaling techniques for first-order phase transitions which are based on analyzing the size-dependent behavior of susceptibilities and dimensionless moment ratios of the order parameter.

Closure-based perturbative density-functional theory of hard-sphere freezing: Properties of the bridge functional

The study of freezing using perturbative classical density-functional theory is revisited, using a bridge functional approach to resum all terms beyond second order in the free energy expansion. More precisely, the first-order direct correlation function of the solid phase is written as a functional expansion about the homogeneous liquid phase, and the sum of all higher-order terms is represented as a functional of the second-order term. Information about the shape and uniqueness of this bridge functional for the case of hard spheres is obtained via an inversion procedure that employs Monte Carlo fluid-solid coexistence data from the literature. The parametric plots obtained from the inversion procedure show very little scatter in certain regions, suggesting a unique functional dependence, but large scatter in other regions. The scatter is related to the anisotropy of the solid lattice at the particle scale. Interestingly, the thermodynamic properties of the phase transition are quite insensitive to the regions where the scatter is large, and several simple closures (i.e., analytical forms of the bridge function) reproduce exactly the liquid-solid coexistence densities and Lindemann parameter from simulation. The form of these closures is significantly different from the usual closures employed in liquid-state integral equation theory.

Freezing line of the Lennard-Jones fluid: A phase switch Monte Carlo study

We report a phase switch Monte Carlo (PSMC) method study of the freezing line of the Lennard-Jones (LJ) fluid. Our work generalizes to soft potentials the original application of the method to hard-sphere freezing and builds on a previous PSMC study of the LJ system by Errington [J. Chem. Phys. 120, 3130 (2004)]. The latter work is extended by tracing a large section of the Lennard-Jones freezing curve, the results of which we compare with a previous Gibbs-Duhem integration study. Additionally, we provide new background material regarding the statistical-mechanical basis of the PSMC method and extensive implementation details.

Freezing of hard spheres in confinement

The influence of confinement on the freezing transition of hard spheres is investigated. Two limiting cases are considered: (1) large systems, where walls weakly perturb the bulk system, and (2) small systems where the influence of geometry becomes important. In the first situation, the shift in coexisting densities is a linear function of the area to volume ratio in the system. This is a manifestation of the Kelvin equation, and the phenomenon is thermodynamically equivalent to capillary condensation. A claim (by others) of “prefreezing” of hard spheres at a smooth hard wall is quantitatively attributed to capillary crystallization. It is shown that the coexistence region narrows as a function of the area to volume ratio. In the second limit two different confined geometries are studied. In these limits, widening of the coexistence region is observed, pointing to an upper and lower critical point at intermediate values of the area to volume ratio, or no critical point at all. In a slit geometry buckling transitions interfere with the freezing transition. In a box geometry, at large values of the area to volume ratio, fluctuations become important. These fluctuations determine the fate of the freezing transition at intermediate values of the area to volume ratio.

Freezing of hard spheres within the modified weighted density approximation

The real space version of the modified weighted density approximation is investigated in order to determine the effective liquid density that is used to represent a face centered cubic hard-sphere solid. It is shown that above a threshold average density of the solid the theory is unable to predict the existence of the crystal phase.

Phase behaviour and structure of colloidal suspensions

We describe three sets of experiments on suspensions of 'hard-sphere' colloids. Suspensions of equal-sized particles exhibit the hard-sphere freezing and glass transitions. Mixtures of spheres of two different sizes can form the ordered binary crystals AB2 and AB13. The addition of non-adsorbing polymer to a one-component suspension leads, through the depletion mechanism, to a range of phase behaviour, which includes colloidal gas, liquid, crystal and gel.

Entropically driven surface phase separation in binary colloidal mixtures.

We investigate the phase diagram of mixtures of charge stabilized polystyrene spheres of two different sizes in the hard-sphere limit. We observe bulk phase separation into two disordered phases and find a new ordered phase located on the cell walls. The ordered phase occurs at volume fractions as low as 0.2, much less than the value of 0.49 for hard-sphere freezing in monodisperse suspensions, and is qualitatively explained with simple geometric arguments.

Hard-sphere and hard-disk freezing from the differential formulation of the generalized effective liquid approximation.

We apply the differential formulation of the generalized effective liquid approximation to the study of hard-sphere and hard-disk freezing. We show that the thermodynamic properties of the solid phase are rather insensitive to the compressibility factor of the fluid phase used to map the solid onto the effective liquid. The solid-fluid coexistence data instead are quite dependent on the equation of state describing the fluid phase. Very accurate results, as compared with the simulation data, are obtained for both the freezing of hard spheres and hard disks.

Mapping a solid onto an "effective liquid"

The most notable success of the various density-functional theories of freezing is achieved for hard spheres where the density change upon freezing is relatively large, while they more or less equally fail when applied to soft inverse-power interactions where the density change upon freezing is much smaller. It is argued that this relatively large density change makes the hard-sphere freezing results insensitive to physics details in the density-functional calculational setup. As an example, it is demonstrated that the equations of state for the solid near melting as obtained by all the leading density-functional theories of freezing, in their particularly successful application to the freezing of hard spheres and hard disks, are related by a trivial parabola-shift transformation from the liquid excess free energy onto that for the solid. Conclusions regarding the relative merit of the free-energy functionals, which are based only on their performance for the hard-sphere fluid-solid transition, can be misleading.

Equation of state for hard disks and spheres: The Kirkwood transition

A nonlinear differential equation for the pressure P of a hard-core system gives rise to a unique solution for P< P 0 and a separate unique solution for P> P 0. The former corresponds to the uniform liquid and the latter to the crystalline solid. The two distinct solutions of the differential equation are separated by the singular solution P = P 0 which represents the liquid-solid coexistence phase. The existence of a singular solution signals the translational symmetry breaking associated with the liquid-solid transition. The hard-disk and hard-sphere freezing and melting transition densities obtained compare well with computer simulation results.

Thermodynamic Consistency and Entropy Change in the Density-Wave Theory of Freezing

Abstract Thermodynamic consistency within the density-wave theory of freezing is briefly discussed. The non equivalence of different routes to calculate the change in a thermodynamic quantity upon freezing is related to the approximations involved in practical calculations. In particular, a new way of calculating entropy and volume changes is proposed. It is shown that this new route avoids an unphysical feature of the entropy formula previously proposed by other authors. The freezing of hard spheres is discussed as an illustration.

The freezing of hard spheres

The freezing of hard spheres