Crystallization kinetics of polydisperse hard-sphere-like microgel colloids: Ripening dominated crystal growth above melting

Stress overshoot, hysteresis, and the Bauschinger effect in sheared dense colloidal suspensions

The mechanical nonlinear response of dense Brownian suspensions of polymer gel particles is studied experimentally and by means of numerical simulations. It is shown that the response to the application of a constant shear rate depends on the previous history of the suspension. When the flow starts from a suspension at rest, it exhibits an elastic response followed by a stress overshoot and then a plastic flow regime. Conversely, after flow reversal, the stress overshoot does not occur, and the apparent elastic modulus is reduced while numerical simulations reveal that the anisotropy of the local microstructure is delayed relative to the macroscopic stress.

Hard Sphere Crystal Nucleation Rates: Reconciliation of Simulation and Experiment

Over the past two decades, a large number of studies addressed the topic of crystal nucleation in suspensions of hard spheres. The shared result of all these efforts is that, at low supersaturations, experimentally observed nucleation rates and numerically computed ones differ by more than 10 orders of magnitude. We present precise simulation results of crystal nucleation rate densities in the metastable hard sphere liquid. To compare these rate densities to experimentally measured ones, we propose an interpretation of the experimental data as a combination of nucleation and crystal growth processes (rather than purely the nucleation process). This interpretation may resolve the long-standing dispute about the differing rates.

Role of diffusion in crystallization of hard-sphere colloids

Vital for a variety of industries, colloids also serve as an excellent model to probe phase transitions at the individual particle level. Despite extensive studies, origins of the glass transition in hard-sphere colloids discovered about 30 y ago remain elusive. Results of our numerical simulations and asymptotic analysis suggest that cessation of long-time particle diffusivity does not suppress crystallization of a metastable liquid-phase hard-sphere colloid. Once a crystallite forms, its growth is then controlled by the particle diffusion in the depletion zone surrounding the crystallite. Using simulations, we evaluate the solid-liquid interface mobility from data on colloidal crystallization in terrestrial and microgravity experiments and demonstrate that there is no drastic difference between the respective mobility values. The insight into the effect of vanishing particle mobility and particle sedimentation on crystallization of colloids will help engineer colloidal materials with controllable structure.

Self-Avoiding Random Walks as a Model to Study Athermal Linear Polymers under Extreme Plate Confinement

Monte Carlo (MC) simulations, built around chain-connectivity-altering moves and a wall-displacement algorithm, allow us to simulate freely-jointed chains of tangent hard spheres of uniform size under extreme confinement. The latter is realized through the presence of two impenetrable, flat, and parallel plates. Extreme conditions correspond to the case where the distance between the plates approaches the monomer size. An analysis of the local structure, based on the characteristic crystallographic element (CCE) norm, detects crystal nucleation and growth at packing densities well below the ones observed in bulk analogs. In a second step, we map the confined polymer chains into self-avoiding random walks (SAWs) on restricted lattices. We study all realizations of the cubic crystal system: simple, body centered, and face centered cubic crystals. For a given chain size (SAW length), lattice type, origin of SAW, and level of confinement, we enumerate all possible SAWs (equivalently all chain conformations) and calculate the size distribution. Results for intermediate SAW lengths are used to predict the behavior of long, fully entangled chains through growth formulas. The SAW analysis will allow us to determine the corresponding configurational entropy, as it is the driving force for the observed phase transition and the determining factor for the thermodynamic stability of the corresponding crystal morphologies.

Formation of Laves phases in buoyancy matched hard sphere suspensions

Colloidal Laves phases (LPs) are promising precursors for photonic materials. Laves phases have not yet been observed to form in experiments on colloidal suspensions of hard spheres (HS), even though they have been reported in computer simulations. LP formation so far has been achieved only for binary mixtures of colloidal charged spheres or ligand-stabilized nano-particles after drying. Using static light scattering, we monitored LP formation and annealing in a binary mixture of buoyant hard sphere approximants (size ratio Γ = 0.77, number or molar fraction of small spheres xS = 0.76) for volume fractions in the fluid-crystal coexistence regions. All samples spontaneously formed MgZn2 type LPs on the time scale of weeks to months via bulk nucleation and growth. Irrespective of the initial suspension volume fractions, the LP volume fraction at coexistence is ΦCOEX = 0.59 which is significantly below the close packing limit ΦMAX = 0.615 and remarkably close to the expectation from simulation. At low volume fractions, crystals anneal to high quality during coarsening which is in line with recent theoretical expectations for the thermodynamic stability of different LP types. At large volume fractions, however, the diffractograms evolve towards a more MgCu2-like appearance which we attribute to the formation of randomly stacked LPs. Such structures are not known from atomic systems.

Crystallization of hard spheres revisited. I. Extracting kinetics and free energy landscape from forward flux sampling

We investigate the kinetics and the free energy landscape of the crystallization of hard spheres from a supersaturated metastable liquid though direct simulations and forward flux sampling. In this first paper, we describe and test two different ways to reconstruct the free energy barriers from the sampled steady state probability distribution of cluster sizes without sampling the equilibrium distribution. The first method is based on mean first passage times, and the second method is based on splitting probabilities. We verify both methods for a single particle moving in a double-well potential. For the nucleation of hard spheres, these methods allow us to probe a wide range of supersaturations and to reconstruct the kinetics and the free energy landscape from the same simulation. Results are consistent with the scaling predicted by classical nucleation theory although a quantitative fit requires a rather large effective interfacial tension.

Electronic crystal growth

Interacting atoms or molecules condense into liquid, and, when cooled further, they form a crystal. The time evolution of the atomic or molecular ordering has been widely studied as a nonequilibrium emergence of order from a supercooled liquid or a glass. Interacting electrons in a variety of correlated electron systems also form crystals, but observing the time evolution of electronic crystallization has been experimentally challenging. Here, working with an organic conductor exhibiting a supercooled charge liquid or charge glass as a metastable state, we observed electronic crystal growth through resistivity and nuclear magnetic resonance measurements. The temperature profile of the crystal growth is similar to those observed in classical systems and reveals two distinct regimes for the mechanism of electronic crystallization.

On tuning microgel character and softness of cross-linked polystyrene particles

Polystyrene (PS) microgel colloids have often been used successfully to model hard sphere behaviour even though the term "gel" invokes inevitably the notion of a more or less soft, deformable object. Here we systematically study the effect of reducing the cross-link density from 1 : 10 (1 cross-link per 10 monomers) to 1 : 100 on particle interactions and "softness". We report on the synthesis and purification of 1 : 10, 1 : 25, 1 : 50, 1 : 75 and 1 : 100 cross-linked PS particles and their characterization in terms of single particle properties, as well as the behaviour of concentrated dispersions. We are able to tune particle softness in the range between soft PNiPAM-microgels and hard PMMA particles while still allowing the mapping of the microgels onto hard sphere behavior with respect to phase diagram and static structure factors. This is mainly due to a rather homogeneous radial distribution of cross-links in contrast to PNiPAM microgels where the cross-link density decreases radially. We find that up to a cross-link density of 1 : 50 particle form factors are perfectly described by a homogeneous sphere model whereas 1 : 75 and 1 : 100 cross-linked spheres are slightly better described as fuzzy spheres. However the fuzziness is rather small compared to typical PNiPAM microgels so that a hard sphere mapping still holds even for these low cross-link densities. Finally, by varying the reaction conditions - changing from batch to semibatch emulsion polymerization and varying the feed rate or by adjusting the monomer to initiator ratio we can tune the fuzziness or significantly alter the inner structure to a more open, star-like architecture.

Atomic-Level Mechanisms of Nucleation of Pure Liquid Metals during Rapid Cooling

To obtain a material with the desired performance, the atomic-level mechanisms of nucleation from the liquid to solid phase must be understood. Although this transition has been investigated experimentally and theoretically, its atomic-level mechanisms remain debatable. In this work, the nucleation mechanisms of pure Fe under rapid cooling conditions are investigated. The local atomic packing stability and liquid-to-solid transition-energy pathways of Fe are studied using molecular dynamics simulations and first-principle calculations. The results are expressed as functions of cluster size in units of amorphous clusters (ACs) and body-centered cubic crystalline clusters (BCC-CCs). We found the prototypes of ACs in supercooled liquids and successfully divided these ACs to three categories according to their transition-energy pathways. The information obtained in this study could contribute to our current understanding of the crystallization of metallic melts during rapid cooling.

Two-stage crystallization of charged colloids under low supersaturation conditions

We report simulations on the homogeneous liquid-fcc nucleation of charged colloids for both low and high contact energy values. As a precursor for crystal formation, we observe increased local order at the position where the crystal will form, but no correlations with the local density. Thus, the nucleation is driven by order fluctuations rather than density fluctuations. Our results also show that the transition involves two stages in both cases, first a transition of liquid → bcc, followed by a bcc → hcp/fcc transition. Both transitions have to overcome free energy barriers, so that a spherical bcc-like cluster is formed first, in which the final fcc structure is nucleated mainly at the surface of the crystallite. This means that the second stage bcc-fcc phase transition is a heterogeneous nucleation in the partially grown solid phase, even though we start from a homogeneous bulk liquid. The height of the bcc → hcp/fcc free energy barrier strongly depends on the contact energies of the colloids. For low contact energy this barrier is low, so that the bcc → hcp/fcc transition occurs spontaneously. For the higher contact energy, the second barrier is too high to be crossed spontaneously by the colloidal system. However, it was possible to ratchet the system over the second barrier and to transform the bcc nuclei into the stable hcp/fcc phase. The transitions are dominated by the first liquid-bcc transition and can be described by classical nucleation theory using an effective surface tension.

Crystallization kinetics of colloidal model suspensions: recent achievements and new perspectives

Colloidal model systems allow studying crystallization kinetics under fairly ideal conditions, with rather well-characterized pair interactions and minimized external influences. In complementary approaches experiment, analytic theory and simulation have been employed to study colloidal solidification in great detail. These studies were based on advanced optical methods, careful system characterization and sophisticated numerical methods. Over the last decade, both the effects of the type, strength and range of the pair-interaction between the colloidal particles and those of the colloid-specific polydispersity have been addressed in a quantitative way. Key parameters of crystallization have been derived and compared to those of metal systems. These systematic investigations significantly contributed to an enhanced understanding of the crystallization processes in general. Further, new fundamental questions have arisen and (partially) been solved over the last decade: including, for example, a two-step nucleation mechanism in homogeneous nucleation, choice of the crystallization pathway, or the subtle interplay of boundary conditions in heterogeneous nucleation. On the other hand, via the application of both gradients and external fields the competition between different nucleation and growth modes can be controlled and the resulting microstructure be influenced. The present review attempts to cover the interesting developments that have occurred since the turn of the millennium and to identify important novel trends, with particular focus on experimental aspects.

Solidification of a colloidal hard sphere like model system approaching and crossing the glass transition

We investigated the process of vitrification and crystallization in a model system of colloidal hard spheres. The kinetics of the solidification process was measured using time resolved static light scattering, while the time evolution of the dynamic properties was determined using time resolved dynamic light scattering. By performing further analysis we confirm that solidification of hard sphere colloids is mediated by precursors. Analyzing the dynamic properties we can show that the long time dynamics and thus the shear rigidity of the metastable melt is highly correlated with the number density of solid clusters (precursors) nucleated. In crystallization these objects convert into highly ordered crystals whereas in the case of vitrification this conversion is blocked and the system is (temporarily) locked in the metastable precursor state. From the early stages of solidification one cannot clearly conclude whether the melt will crystallize or vitrify. Furthermore our data suggests that colloidal hard sphere glasses can crystallize via homogeneous nucleation.

The crystallization process of liquid vanadium studied by ab initio molecular dynamics

We present a study of the crystallization process in liquid vanadium over a temperature range from 3000 K down to 1500 K using ab initio molecular dynamics simulations. Short-range order evolution during solidification is studied using various structural analysis methods. We show that the icosahedral-like short-range order is detected in the stable liquid phase and grows upon supercooling. The system undergoes a first-order phase transition (from a liquid to a solid state) at a temperature of about 1600 K. The crystal nucleation process is further studied using the time-temperature transformation mechanism by annealing the system at 1650 K. The nucleation is examined using bond-orientational order and density fluctuation analysis. Our finding is that various precursors appear in the region of high bond-orientational order with the majority having body-centered cubic (bcc)-like symmetry. This bcc-like region grows on annealing via thermal fluctuations. Our results reveal that the bond-orientational order precedes the density fluctuation, and is the main driving factor for nucleation.

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.

Easy-use and low-cost fiber-based two-color dynamic light-scattering apparatus

We present a small and compact two-color cross-correlation light-scattering setup designed to study the structure and dynamics of colloidal suspensions in the regime of considerable turbidity. Using a homemade concentric four-arm goniometer, combined with fiber-optical illumination and detection devices, allows an easy and long time stable alignment-even under a temperature variation of ∼5 °C. Tests with a nearly multiple scattering free suspension of small Rayleigh scatterers show intercepts in cross correlation near 0.4 over a wide range of scattering angles from 20° to 150°. Measuring slightly turbid samples in cross-correlation mode multiple scattering is sufficiently suppressed allowing the determination of multiple scattering free dynamic structure factors as well as particle form factors.

Kinetics study of crystallization with the disorder-bcc-fcc phase transition of charged colloidal dispersions

Structure transformation (disorder-bcc-fcc) in charged colloidal dispersions, as a manifestation of the Ostwald's step rule, was confirmed by means of reflection spectrum (RS) measurements in our previous study. By taking advantage of a reflection spectrum containing plenty of information about the crystallization behaviors, time-dependent changes of parameters associated with the crystal structure and composition during the disorder-bcc-fcc transition are reported by treating the data from RS in this article. In addition, Avrami's model is adopted to analyze the transition process and investigate the transition rate. On the basis of the above investigations, associated kinetic features of crystallization with the disorder-bcc-fcc transition are described.

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.

Seed- and wall-induced heterogeneous nucleation in charged colloidal model systems under microgravity

Understanding the process that drives an undercooled fluid to the crystal state is still a challenging issue for condensed matter physics and plays a key role in designing new materials. The crystallization kinetics and the resulting polycrystalline morphology are given by a complex interplay of crystal nucleation, growth, and ripening. A great deal of progress has been made in recent years using colloidal suspensions as model systems in the study of crystallization. Close analogies to atomic systems are observed which can be exploited to address questions not accessible in atomic solidification. Here we present systematic measurements of the crystallization kinetics of a charged colloidal model system adding small amounts of seeds using time resolved scattering techniques. Large seeds show strong sedimentation under gravity even on the time scale of the crystallization process. To avoid this problem we performed our measurements under microgravity during parabolic flights. We report how the seed size and structure affect crystal nucleation and growth as functions of metastability giving the possibility to modify the crystallization process and the resulting microstructure of the polycrystal.

Colloidal crystal growth monitored by Bragg diffraction interference fringes

We monitored the crystal growth kinetics of crystallization of a shear melted crystalline colloidal array (CCA). The fcc CCA heterogeneously nucleates at the flow cell wall surface. We examined the evolution of the (1 1 1) Bragg diffraction peak, and, for the first time, quantitatively monitored growth by measuring the temporal evolution of the Bragg diffraction interference fringes. Modeling of the evolution of the fringe patterns exposes the time dependence of the increasing crystal thickness. The initial diffusion-driven linear growth is followed by ripening-driven growth. Between 80 and 90 microM NaCl concentrations the fcc crystals first linearly grow at rates between 1.9 and 4.2 microm/s until they contact homogeneously nucleated crystals in the bulk. At lower salt concentrations interference fringes are not visible because the strong electrostatic interactions between particles result in high activation barriers, preventing defect annealing and leading to a lower crystal quality. The fcc crystals melt to a liquid phase at >90 microM NaCl concentrations. Increasing NaCl concentrations slow the fcc CCA growth rate consistent with the expectation of the classical Wilson-Frenkel growth theory. The final thickness of wall-nucleated CCA, that is determined by the competition between growth of heterogeneously and homogenously nucleated CCA, increases with higher NaCl concentrations.

Polymorphism, crystal nucleation and growth in the phase-field crystal model in 2D and 3D

We apply a simple dynamical density functional theory, the phase-field crystal (PFC) model of overdamped conservative dynamics, to address polymorphism, crystal nucleation, and crystal growth in the diffusion-controlled limit. We refine the phase diagram for 3D, and determine the line free energy in 2D and the height of the nucleation barrier in 2D and 3D for homogeneous and heterogeneous nucleation by solving the respective Euler-Lagrange (EL) equations. We demonstrate that, in the PFC model, the body-centered cubic (bcc), the face-centered cubic (fcc), and the hexagonal close-packed structures (hcp) compete, while the simple cubic structure is unstable, and that phase preference can be tuned by changing the model parameters: close to the critical point the bcc structure is stable, while far from the critical point the fcc prevails, with an hcp stability domain in between. We note that with increasing distance from the critical point the equilibrium shapes vary from the sphere to specific faceted shapes: rhombic dodecahedron (bcc), truncated octahedron (fcc), and hexagonal prism (hcp). Solving the equation of motion of the PFC model supplied with conserved noise, solidification starts with the nucleation of an amorphous precursor phase, into which the stable crystalline phase nucleates. The growth rate is found to be time dependent and anisotropic; this anisotropy depends on the driving force. We show that due to the diffusion-controlled growth mechanism, which is especially relevant for crystal aggregation in colloidal systems, dendritic growth structures evolve in large-scale isothermal single-component PFC simulations. An oscillatory effective pair potential resembling those for model glass formers has been evaluated from structural data of the amorphous phase obtained by instantaneous quenching. Finally, we present results for eutectic solidification in a binary PFC model.

Precursor-mediated crystallization process in suspensions of hard spheres

We report on a large scale computer simulation study of crystal nucleation in hard spheres. Through a combined analysis of real- and reciprocal-space data, a picture of a two-step crystallization process is supported: First, dense, amorphous clusters form which then act as precursors for the nucleation of well-ordered crystallites. This kind of crystallization process has been previously observed in systems that interact via potentials that have an attractive as well as a repulsive part, most prominently in protein solutions. In this context the effect has been attributed to the presence of metastable fluid-fluid demixing. Our simulations, however, show that a purely repulsive system (that has no metastable fluid-fluid coexistence) crystallizes via the same mechanism.

Optical experiments on a crystallizing hard-sphere-polymer mixture at coexistence

We report on the crystallization kinetics in an entropically attractive colloidal system using a combination of time resolved scattering methods and microscopy. Hard sphere particles are polystyrene microgels swollen in a good solvent (radius a=380 nm, starting volume fraction 0.534) with the short ranged attractions induced by the presence of short polymer chains (radius of gyration r g=3 nm, starting volume fraction 0.0224). After crystallization, stacking faulted face centered cubic crystals coexist with about 5% of melt remaining in the grain boundaries. From the Bragg scattering signal we infer the amount of crystalline material, the average crystallite size and the number density of crystals as a function of time. This allows to discriminate an early stage of conversion, followed by an extended coarsening stage. The small angle scattering (SALS) appears only long after completed conversion and exhibits Furukawa scaling for all times. Additional microscopic experiments reveal that the grain boundaries have a reduced Bragg scattering power but possess an increased refractive index. Fits of the Furukawa function indicate that the dimensionality of the scatterers decreases from 2.25 at short times to 1.65 at late times and the characteristic length scale is slightly larger than the average crystallite size. Together this suggests the SALS signal is due scattering from a foam like grain boundary network as a whole.

Competition between heterogeneous and homogeneous nucleation near a flat wall

We studied the competition between heterogeneous and homogeneous nucleation of an aqueous suspension of charged colloidal spheres close to the container walls. Samples of equilibrium crystalline structure were shear-melted and the metastable melt left to solidify after the cessation of shear. The crystallization kinetics was monitored using time-resolved scattering techniques: at low particle number densities n we applied an improved static light scattering method while at large particle concentrations ultra-small-angle x-ray scattering was applied for the first time. Our results show some unexpected behavior: the heterogeneous nucleation at the container walls is delayed in comparison to the homogeneous bulk nucleation and its rate density appears surprisingly slightly smaller, demonstrating the complexity of the observed crystallization process.