Precursor-mediated crystallization process in suspensions of hard spheres

Mode expansion for the density profiles of crystal-fluid interfaces: hard spheres as a test case

We present a technique for analyzing the full three-dimensional density profiles of planar crystal-fluid interfaces in terms of density modes. These density modes can also be related to crystallinity order parameter profiles which are used in coarse-grained, phase field type models of the statics and dynamics of crystal-fluid interfaces and are an alternative to crystallinity order parameters extracted from simulations using local crystallinity criteria. We illustrate our results for the hard sphere system using finely resolved, three-dimensional density profiles from a density functional theory of fundamental measure type.

Nucleation in short-range attractive colloids: ordering and symmetry of clusters

Results from extensive Brownian dynamics simulations are presented for nucleation in a system of colloidal particles interacting via a short-range attractive potential. Our analysis shows that, even though the system is not in equilibrium, structures of small size clusters compare well with the theoretically predicted and experimentally observed ground state structures for short-range colloidal systems. In addition, the distribution of the symmetric structures in nucleation is comparable to the distribution seen in equilibrium. We also investigate how the shape and structure of fluctuating clusters in the prenucleation regime affect the formation of a stable nucleating cluster.

Bond orientational order in liquids: Towards a unified description of water-like anomalies, liquid-liquid transition, glass transition, and crystallization: Bond orientational order in liquids

There are at least three fundamental states of matter, depending upon temperature and pressure: gas, liquid, and solid (crystal). These states are separated by first-order phase transitions between them. In both gas and liquid phases a complete translational and rotational symmetry exist, whereas in a solid phase both symmetries are broken. In intermediate phases between liquid and solid, which include liquid crystal and plastic crystal phases, only one of the two symmetries is preserved. Among the fundamental states of matter, the liquid state is the most poorly understood. We argue that it is crucial for a better understanding of liquids to recognize that a liquid generally has the tendency to have a local structural order and its presence is intrinsic and universal to any liquid. Such structural ordering is a consequence of many-body correlations, more specifically, bond angle correlations, which we believe are crucial for the description of the liquid state. We show that this physical picture may naturally explain difficult unsolved problems associated with the liquid state, such as anomalies of water-type liquids (water, Si, Ge, ...), liquid-liquid transition, liquid-glass transition, crystallization and quasicrystal formation, in a unified manner. In other words, we need a new order parameter representing a low local free-energy configuration, which is a bond orientational order parameter in many cases, in addition to a density order parameter for the physical description of these phenomena. Here we review our two-order-parameter model of liquid and consider how transient local structural ordering is linked to all of the above-mentioned phenomena. The relationship between these phenomena is also discussed.

The microscopic pathway to crystallization in supercooled liquids

Despite its fundamental and technological importance, a microscopic understanding of the crystallization process is still elusive. By computer simulations of the hard-sphere model we reveal the mechanism by which thermal fluctuations drive the transition from the supercooled liquid state to the crystal state. In particular we show that fluctuations in bond orientational order trigger the nucleation process, contrary to the common belief that the transition is initiated by density fluctuations. Moreover, the analysis of bond orientational fluctuations shows that these not only act as seeds of the nucleation process, but also i) determine the particular polymorph which is to be nucleated from them and ii) at high density favour the formation of fivefold structures which can frustrate the formation of crystals. These results can shed new light on our understanding of the relationship between crystallization and vitrification.

Structure of finite sphere packings via exact enumeration: implications for colloidal crystal nucleation

We analyze the geometric structure and mechanical stability of a complete set of isostatic and hyperstatic sphere packings obtained via exact enumeration. The number of nonisomorphic isostatic packings grows exponentially with the number of spheres N, and their diversity of structure and symmetry increases with increasing N and decreases with increasing hyperstaticity H≡N_{c}-N_{ISO}, where N_{c} is the number of pair contacts and N_{ISO}=3N-6. Maximally contacting packings are in general neither the densest nor the most symmetric. Analyses of local structure show that the fraction f of nuclei with order compatible with the bulk (rhcp) crystal decreases sharply with increasing N due to a high propensity for stacking faults, five- and near-fivefold symmetric structures, and other motifs that preclude rhcp order. While f increases with increasing H, a significant fraction of hyperstatic nuclei for N as small as 11 retain non-rhcp structure. Classical theories of nucleation that consider only spherical nuclei, or only nuclei with the same ordering as the bulk crystal, cannot capture such effects. Our results provide an explanation for the failure of classical nucleation theory for hard-sphere systems of N≲10 particles; we argue that in this size regime, it is essential to consider nuclei of unconstrained geometry. Our results are also applicable to understanding kinetic arrest and jamming in systems that interact via hard-core-like repulsive and short-ranged attractive interactions.

Amorphous nucleation precursor in highly nonequilibrium fluids

Dynamical density-functional simulations reveal structural aspects of crystal nucleation in undercooled liquids: The first appearing solid is amorphous, which promotes the nucleation of bcc crystals but suppresses the appearance of the fcc and hcp phases. These findings are associated with features of the effective interaction potential deduced from the amorphous structure.

Nucleation barriers for the liquid-to-crystal transition in Ni: experiment and simulation

Nucleation in undercooled Ni is investigated by a combination of differential scanning calorimetry (DSC) experiments and Monte Carlo (MC) simulation. By systematically varying the sample size in the DSC experiments, nucleation rates J over a range of 8 orders of magnitude are obtained. Evidence is given that these rates correspond to homogeneous nucleation. Free energy barriers ΔG*, as extracted from the measured J, are in very good agreement with those from the MC simulation. The MC simulation indicates a nonspherical geometry of crystalline clusters, fluctuating between prolate and oblate shape at a given size. Nevertheless, the temperature dependence of ΔG* is well described by classical nucleation theory.

Roles of bond orientational ordering in glass transition and crystallization

It is widely believed that crystallization in three dimensions is primarily controlled by positional ordering, and not by bond orientational ordering. In other words, bond orientational ordering is usually considered to be merely a consequence of positional ordering and thus has often been ignored. This one-order-parameter (density) description may be reasonable when we consider an equilibrium liquid-solid transition, but may not be enough to describe a metastable state and the kinetics of the transition. Here we propose that bond orientational ordering can play a key role in (i) crystallization, (ii) the ordering to quasi-crystal and (iii) vitrification, which occurs under rather weak frustration against crystallization. In a metastable supercooled state before crystallization, a system generally tends to have bond orientational order at least locally as a result of a constraint of dense packing. For a system interacting with hard-core repulsions, the constraint is intrinsically of geometrical origin and thus the basic physics is the same as nematic ordering of rod-like particles upon densification. Furthermore, positional ordering is easily destroyed even by weak frustration such as polydispersity and anisotropic interactions which favour a symmetry not consistent with that of the equilibrium crystal. Thus we may say that vitrification can be achieved by disturbing and prohibiting long-range positional ordering. Even in such a situation, bond orientational ordering still survives, accompanying its critical-like fluctuations, which are the origin of dynamic heterogeneity for this case. This scenario naturally explains both the absence of positional order and the development of bond orientational order upon cooling in a supercooled state. Although our argument is speculative in nature, we emphasize that this physical picture can coherently explain crystallization, vitrification, quasi-crystallization and their relationship in a natural manner. For a strongly frustrated system, even bond orientational order can be destroyed. Even in such a case there may still appear a structural signature of dense packing, which is linked to slow dynamics.

Fivefold symmetry as an inhibitor to hard-sphere crystallization

Through molecular simulations we investigate the dynamics of crystallization of hard spheres of uniform size from dense amorphous states and the role that hidden structures in an otherwise disordered medium might have on it. It is shown that short-range order in the form of sites with fivefold symmetry acts as a powerful inhibitor to crystal growth. Fivefold sites not only retard crystallization, but can self-assemble into organized structures that arrest crystallization at high densities or lead to the formation of defects in a crystal. The latter effect can be understood in terms of a random polyhedral model.

Crystallization mechanism of hard sphere glasses

In supercooled liquids, vitrification generally suppresses crystallization. Yet some glasses can still crystallize despite the arrest of diffusive motion. This ill-understood process may limit the stability of glasses, but its microscopic mechanism is not yet known. Here we present extensive computer simulations addressing the crystallization of monodisperse hard-sphere glasses at constant volume (as in a colloid experiment). Multiple crystalline patches appear without particles having to diffuse more than one diameter. As these patches grow, the mobility in neighboring areas is enhanced, creating dynamic heterogeneity with positive feedback. The future crystallization pattern cannot be predicted from the coordinates alone: Crystallization proceeds by a sequence of stochastic micronucleation events, correlated in space by emergent dynamic heterogeneity.

Crystallization in suspensions of hard spheres: a Monte Carlo and molecular dynamics simulation study

The crystallization of a metastable melt is one of the most important non-equilibrium phenomena in condensed matter physics, and hard sphere colloidal model systems have been used for several decades to investigate this process by experimental observation and computer simulation. Nevertheless, there is still an unexplained discrepancy between the simulation data and experimental nucleation rate densities. In this paper we examine the nucleation process in hard spheres using molecular dynamics and Monte Carlo simulation. We show that the crystallization process is mediated by precursors of low orientational bond-order and that our simulation data fairly match the experimental data sets.

Role of the prestructured surface cloud in crystal nucleation

For the homogeneous crystal nucleation process in a soft-core colloid model, we identify optimal reaction coordinates from a set of novel order parameters based on the local structure within the nucleus, by employing transition path sampling techniques combined with a likelihood maximization of the committor function. We find that nucleation is governed by solid clusters that consist of an hcp core embedded within a cloud of surface particles that are highly correlated with their nearest neighbors but not ordered in a high-symmetry crystal structure. The results shed new light on the interpretation of the surface and volume terms in classical nucleation theory.

Densest local sphere-packing diversity. II. Application to three dimensions

The densest local packings of N three-dimensional identical nonoverlapping spheres within a radius R(min)(N) of a fixed central sphere of the same size are obtained for selected values of N up to N=1054. In the predecessor to this paper [A. B. Hopkins, F. H. Stillinger, and S. Torquato, Phys. Rev. E 81, 041305 (2010)], we described our method for finding the putative densest packings of N spheres in d-dimensional Euclidean space R(d) and presented those packings in R(2) for values of N up to N=348. Here we analyze the properties and characteristics of the densest local packings in R(3) and employ knowledge of the R(min)(N), using methods applicable in any d, to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In R(3), we find wide variability in the densest local packings, including a multitude of packing symmetries such as perfect tetrahedral and imperfect icosahedral symmetry. We compare the densest local packings of N spheres near a central sphere to minimal-energy configurations of N+1 points interacting with short-range repulsive and long-range attractive pair potentials, e.g., 12-6 Lennard-Jones, and find that they are in general completely different, a result that has possible implications for nucleation theory. We also compare the densest local packings to finite subsets of stacking variants of the densest infinite packings in R(3) (the Barlow packings) and find that the densest local packings are almost always most similar as measured by a similarity metric, to the subsets of Barlow packings with the smallest number of coordination shells measured about a single central sphere, e.g., a subset of the fcc Barlow packing. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance R(min)(N). In particular, we find two "maracas" packings at N=77 and N=93, each consisting of a few unjammed spheres free to rattle within a "husk" composed of the maximal number of spheres that can be packed with centers at respective R(min)(N).

Free energies, vacancy concentrations, and density distribution anisotropies in hard-sphere crystals: a combined density functional and simulation study

We perform a comparative study of the free energies and the density distributions in hard-sphere crystals using Monte Carlo simulations and density functional theory (employing Fundamental Measure functionals). Using a recently introduced technique [T. Schilling and F. Schmid, J. Chem. Phys. 131, 231102 (2009)] we obtain crystal free energies to a high precision. The free energies from fundamental measure theory are in good agreement with the simulation results and demonstrate the applicability of these functionals to the treatment of other problems involving crystallization. The agreement between fundamental measure theory and simulations on the level of the free energies is also reflected in the density distributions around single lattice sites. Overall, the peak widths and anisotropy signs for different lattice directions agree, however, it is found that fundamental measure theory gives slightly narrower peaks with more anisotropy than seen in the simulations. Among the three types of fundamental measure functionals studied, only the White Bear II functional [H. Hansen-Goos and R. Roth, J. Phys.: Condens. Matter 18, 8413 (2006)] exhibits sensible results for the equilibrium vacancy concentration and a physical behavior of the chemical potential in crystals constrained by a fixed vacancy concentration.