Diffusive growth of polydisperse hard-sphere crystals

Nanoparticle Superlattices with Nonequilibrium Crystal Shapes

Nanoparticle assembly is a material synthesis strategy that enables precise control of nanoscale structural features. Concepts from traditional crystal growth research have been tremendously useful in predicting and programming the unit cell symmetries of these assemblies, as their thermodynamically favored structures are often identical to atomic crystal analogues. However, these analogies have not yielded similar levels of influence in programming crystallite shapes, which are a consequence of both the thermodynamics and kinetics of crystal growth. Here, we demonstrate kinetic control of the colloidal crystal shape using nanoparticle building blocks that rapidly assemble over a broad range of concentrations, thereby producing well-defined crystal habits with symmetrically oriented dendritic protrusions and providing insight into the crystals' morphological evolution. Counterintuitively, these nonequilibrium crystal shapes actually become more common for colloidal crystals synthesized closer to equilibrium growth conditions. This deviation from typical crystal growth processes observed in atomic or molecular crystals is shown to be a function of the drastically different time scales of atomic and colloidal mass transport. Moreover, the particles are spherical with isotropic ligand grafts, and these kinetic crystal habits are achieved without the need for specifically shaped particle building blocks or external templating or shape-directing agents. Thus, this work provides generalizable design principles to expand the morphological diversity of nanoparticle superlattice crystal habits beyond the anhedral or equilibrium polyhedral shapes synthesized to date. Finally, we use this insight to synthesize crystallite shapes that have never before been observed, demonstrating the ability to both predict and program kinetically controlled superlattice morphologies.

Microstructural diversity, nucleation paths, and phase behavior in binary mixtures of charged colloidal spheres

We study low-salt, binary aqueous suspensions of charged colloidal spheres of size ratio Γ = 0.57, number densities below the eutectic number density n_E, and number fractions of p = 1.00-0.40. The typical phase obtained by solidification from a homogeneous shear-melt is a substitutional alloy with a body centered cubic structure. In strictly gas-tight vials, the polycrystalline solid is stable against melting and further phase transformation for extended times. For comparison, we also prepare the same samples by slow, mechanically undisturbed deionization in commercial slit cells. These cells feature a complex but well reproducible sequence of global and local gradients in salt concentration, number density, and composition as induced by successive deionization, phoretic transport, and differential settling of the components, respectively. Moreover, they provide an extended bottom surface suitable for heterogeneous nucleation of the β-phase. We give a detailed qualitative characterization of the crystallization processes using imaging and optical microscopy. By contrast to the bulk samples, the initial alloy formation is not volume-filling, and we now observe also α- and β-phases with low solubility of the odd component. In addition to the initial homogeneous nucleation route, the interplay of gradients opens various further crystallization and transformation pathways leading to a great diversity of microstructures. Upon a subsequent increase in salt concentration, the crystals melt again. Wall-based, pebble-shaped β-phase crystals and facetted α-crystals melt last. Our observations suggest that the substitutional alloys formed in bulk experiments by homogeneous nucleation and subsequent growth are mechanically stable in the absence of solid-fluid interfaces but thermodynamically metastable.

Phase separation dynamics of polydisperse colloids: a mean-field lattice-gas theory

Strong theoretical evidence shows that dense colloidal mixtures phase-separate in two stages and the denser phase contains long-lived composition heterogeneities.

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.

Spinodal fractionation in a polydisperse square-well fluid

Using kinetic Monte Carlo simulation, we model gas-liquid spinodal decomposition in a size-polydisperse square well fluid, representing a "near-monodisperse" colloidal dispersion. We find that fractionation (demixing) of particle sizes between the phases begins asserting itself shortly after the onset of phase ordering. Strikingly, the direction of size fractionation can be reversed by a seemingly trivial choice between two interparticle potentials which, in the monodisperse case, are identical--we rationalize this in terms of a perturbative, equilibrium theory of polydispersity. Furthermore, our quantitative results show that kinetic Monte Carlo simulation can provide detailed insight into the role of fractionation in real colloidal systems.

Crystalline phases of polydisperse spheres

We use specialized Monte Carlo simulation methods and moment free energy calculations to provide conclusive evidence that dense polydisperse spheres at equilibrium demix into coexisting fcc phases, with more phases appearing as the spread of diameters increases. We manage to track up to four coexisting phases. Each of these is fractionated: it contains a narrower distribution of particle sizes than is present in the system overall. We also demonstrate that, surprisingly, demixing transitions can be nearly continuous, accompanied by fluctuations in local particle size correlated over many lattice spacings.

Direct observation of size fractionation during colloidal crystallization

We present a confocal microscopy study of the quasi-two-dimensional crystallization of a binary mixture of spherical colloids coated with long DNA strands. Our experiments show that in the crystalline phase the two colloidal species are completely demixed. Analysis of the lattice spacings in the two types of colloidal crystal shows that the diameters of the two species of colloids differ by 10%. We argue that the demixing in the crystalline phase is due to size segregation during crystallization. This phenomenon had been predicted in several theoretical studies. To our knowledge, the present study provides the first 'real-space' experimental confirmation of this effect.

Hard spheres: crystallization and glass formation

Motivated by old experiments on colloidal suspensions, we report molecular dynamics simulations of assemblies of hard spheres, addressing crystallization and glass formation. The simulations cover wide ranges of polydispersity s (standard deviation of the particle size distribution divided by its mean) and particle concentration. No crystallization is observed for s>0.07. For 0.02<s<0.07, we find that increasing the polydispersity at a given concentration slows down crystal nucleation. The main effect here is that polydispersity reduces the supersaturation since it tends to stabilize the fluid but to destabilize the crystal. At a given polydispersity (<0.07), we find three regimes of nucleation: standard nucleation and growth at concentrations in and slightly above the coexistence region; 'spinodal nucleation', where the free-energy barrier to nucleation appears to be negligible, at intermediate concentrations; and, at the highest concentrations, a new mechanism, still to be fully understood, which only requires small rearrangement of the particle positions. The cross-over between the second and third regimes occurs at a concentration, approximately 58 per cent by volume, where the colloid experiments show a marked change in the nature of the crystals formed and the particle dynamics indicate an 'ideal' glass transition.

Ripening-dominated crystallization in polydisperse hard-sphere-like colloids

We report on the crystal growth scenario in gravity-matched, polydisperse hard-sphere-like colloids at increasing particle concentration. In the fluid-crystal coexistence region, the crystal size as a function of time shows two separate regimes corresponding to crystal growth and crystal ripening. At higher supersaturation the crystal size grows according to the same power law through the whole experimental window of a few days: crystal growth and ripening merge together. We show that our observations cannot be explained by considering the slowing down of single-particle dynamics due to increasing volume fraction. We suggest that size fractionation occurring at the crystal-fluid interface is the dominant mechanism.

Two-step crystallization kinetics in colloidal hard-sphere systems

The crystallization kinetics of colloidal hard spheres was studied using a special Bragg spectrometer with high sensitivity. In contrast with the classical scenario we observe a two-step nucleation process: the number of crystallites increases slowly at early times, followed by a dramatic reduction at intermediate times, prior to undergoing a rapid increase at late times. We explain these results in terms of a polydispersity limited growth of crystallites, where the crystallization at early times is governed by local fractionation processes, leading to a long delay prior to final crystallization.

Effects of colloid polydispersity on the phase behavior of colloid-polymer mixtures

We study theoretically the equilibrium phase behavior of a mixture of polydisperse hard-sphere colloids and monodisperse polymers, modeled using the Asakura-Oosawa model [S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255 (1954)] within the free volume approximation of H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren [Europhys. Lett. 20, 559 (1992)]. We compute full phase diagrams in the plane of colloid and polymer volume fractions, using the moment free energy method. The intricate features of phase separation in pure polydisperse colloids combine with the appearance of polymer-induced gas-liquid coexistence to give a rich variety of phase diagram topologies as the polymer-colloid size ratio xi and the colloid polydispersity delta are varied. Quantitatively, we find that polydispersity disfavors fluid-solid against gas-liquid separation, causing a substantial lowering of the threshold value xi(c) above which stable two-phase gas-liquid coexistence appears. Phase splits involving two or more solids can occur already at low colloid concentration, where they may be kinetically accessible. We also analyze the strength of colloidal size fractionation. When a solid phase separates from a fluid, its polydispersity is reduced most strongly if the phase separation takes place at low colloid concentration and high polymer concentration, in agreement with experimental observations. For fractionation in gas-liquid coexistence we likewise find good agreement with experiment, as well as with perturbative theories for near-monodisperse systems.

Crystallization kinetics of polydisperse colloidal hard spheres. II. Binary mixtures

In this paper we present measurements of the crystallization kinetics of binary mixtures of two different sized hard sphere particles. The growth of the Bragg reflections over time were analyzed to yield the crystallite scattering vector, the total amount of crystal, and the average linear crystal size. It was observed that a particle size distribution skewed to higher sized particles has a less detrimental effect on the crystal structure than a skew to smaller sized particles. In the latter case we observe that initial crystallite growth occurs at only a small number of sites, with further crystallization sites developing at later times. Based on these measurements we elaborate further on the previously proposed growth mechanism whereby crystallization occurs in conjunction with a local fractionation process in the fluid, which significantly affects the kinetic growth of crystallites in polydisperse systems.

Fractionation effects in phase equilibria of polydisperse hard-sphere colloids

The equilibrium phase behavior of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to nonequilibrium effects. We find no evidence of reentrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity, and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using principal component analysis in the space of density distributions. We conclude by comparing our predictions to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.

Equilibrium phase behavior of polydisperse hard spheres

We calculate the phase behavior of hard spheres with size polydispersity, using accurate free energies for the fluid and solid phases. Cloud and shadow curves are found exactly by the moment free energy method, but we also compute the complete phase diagram, taking full account of fractionation. In contrast to earlier, simplified treatments we find no point of equal concentration between fluid and solid or reentrant melting at higher densities. Rather, the fluid cloud curve continues to the largest polydispersity that we study (14%); from the equilibrium phase behavior a terminal polydispersity can thus be defined only for the solid, where we find it to be around 7%. At sufficiently large polydispersity, fractionation into several solid phases can occur, consistent with previous approximate calculations; we find, in addition, that coexistence of several solids with a fluid phase is also possible.

Observation of a smecticlike crystalline structure in polydisperse colloids

We present the results of crystallographic measurements on samples of two latexes: one with a relatively symmetric particle size distribution, and another with a highly skewed pseudobimodal distribution. For the skewed latex, crystallites are clearly visible, but they exhibit only a single Bragg reflection, indicating long-range order in only one direction. We propose a schematic model that explains this result in terms of stacks of planes, which are unregistered due to a high incidence of stacking faults caused by the incorporation of a large number of small particles.

Crystallization kinetics of polydisperse colloidal hard spheres: experimental evidence for local fractionation

We present the crystallization kinetics for two polydisperse hard-sphere particle stocks with differing particle size distributions. One of the latexes had a relatively symmetrical distribution, the other had a more polydisperse distribution, which was highly skewed to smaller sizes. The emerging Bragg reflections from the crystallizing samples were measured using a technique that provides improved statistical averaging over our previous methods. It was observed that, for the more polydisperse particles, the onset of nucleation was delayed by up to an order of magnitude in reduced time, and displayed qualitatively different growth behavior compared to the particles with the more symmetric size distribution. Based on these measurements and time lapse photographs, we propose a growth mechanism whereby crystallization occurs in conjunction with a local fractionation process near the crystal-fluid interface, which significantly alters the kinetics of crystallite nucleation and growth. This fractionation effect becomes more significant as polydispersity or skewness increases.

Kinetics of growth process controlled by convective fluctuations

A model of the spherical (compact) growth process controlled by a fluctuating local convective velocity field of the fluid particles is introduced. It is assumed that the particle velocity fluctuations are purely noisy, Gaussian, of zero mean, and of various correlations: Dirac delta, exponential, and algebraic (power law). It is shown that for a large class of the velocity fluctuations, the long-time asymptotics of the growth kinetics is universal (i.e., it does not depend on the details of the statistics of fluctuations) and displays the power-law time dependence with the classical exponent 1/2 resembling the diffusion limited growth. For very slow decay of algebraic correlations of fluctuations asymptotically like t(-gamma), gamma in (0,1]), kinetics is anomalous and depends strongly on the exponent gamma. For the averaged radius of the crystal <R(t)> approximately t(1-gamma/2) for 0<gamma<1 or <R(t)> approximately (t ln t)1/2 for gamma=1.