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

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.

Effective one-component model of binary mixture: molecular arrest induced by the spatially correlated stochastic dynamics

Spatially correlated noise (SCN), i.e. the thermal noise that affects neighbouring particles in a similar manner, is ubiquitous in soft matter systems. In this work, we apply the over-damped SCN-driven Langevin equations as an effective, one-component model of the dynamics in dense binary mixtures. We derive the thermodynamically consistent fluctuation-dissipation relation for SCN to show that it predicts the molecular arrest resembling the glass transition, i.e. the critical slow-down of dynamics in the disordered phases. We show that the mechanism of singular dissipation is embedded in the dissipation matrix, accompanying SCN. We are also able to identify the characteristic length of collective dissipation, which diverges at critical packing. This novel physical quantity conveniently describes the difference between the ergodic and non-ergodic dynamics. The model is fully analytically solvable, one-dimensional and admits arbitrary interactions between the particles. It qualitatively reproduces several different modes of arrested disorder encountered in binary mixtures, including e.g. the re-entrant arrest. The model can be effectively compared to the mode coupling theory.

Refractive index matched, nearly hard polymer colloids

Refractive index matched particles serve as essential model systems for colloid scientists, providing nearly hard spheres to explore structure and dynamics. The poly(methyl methacrylate) latexes typically used are often refractive index matched by dispersing them in binary solvent mixtures, but this can lead to undesirable changes, such as particle charging or swelling. To avoid these shortcomings, we have synthesized refractive index matched colloids using polymerization-induced self-assembly (PISA) rather than as polymer latexes. The crucial difference is that these diblock copolymer nanoparticles consist of a single core-forming polymer in a single non-ionizable solvent. The diblock copolymer chosen was poly(stearyl methacrylate)-poly(2,2,2-trifluoroethyl methacrylate) (PSMA-PTFEMA), which self-assembles to form PTFEMA core spheres in n-alkanes. By monitoring scattered light intensity, n-tetradecane was found to be the optimal solvent for matching the refractive index of such nanoparticles. As expected for PISA syntheses, the diameter of the colloids can be controlled by varying the PTFEMA degree of polymerization. Concentrated dispersions were prepared, and the diffusion of the PSMA-PTFEMA nanoparticles as a function of volume fraction was measured. These diblock copolymer nanoparticles are a promising new system of transparent spheres for future colloidal studies.

Dramatic slowing of compositional relaxations in the approach to the glass transition for a bimodal colloidal suspension

Molecular dynamics simulation was used to study a model colloidal suspension with two species of slightly different sized colloidal particles in an explicit solvent. In this work we calculated the four interdiffusion coefficients for the ternary system, which were then used to calculate the decay coefficients D_{±} of the two independent diffusive modes. We found that the slower D_{-} decay mode, which is associated with the system's ability to undergo compositional changes, was responsible for the long-time decay in the intermediate scattering function. We also found that a decrease in D_{-} to negligible values at a packing fraction of Φ_{g}=0.592 resulted in an extreme slow-down in the long-time decay of the intermediate scattering function often associated with the glass transition. Above Φ_{g}, the system formed a long-lived metastable state that did not relax to its equilibrium crystal state within the simulation time window. We concluded that the inhibition of crystallization was caused by the inability of the quenched fluid to undergo the compositional changes needed for the formation of the equilibrium crystal.

Phase behavior of binary and polydisperse suspensions of compressible microgels controlled by selective particle deswelling

We investigate the phase behavior of suspensions of poly(N-isopropylacrylamide) (pNIPAM) microgels with either bimodal or polydisperse size distribution. We observe a shift of the fluid-crystal transition to higher concentrations depending on the polydispersity or the fraction of large particles in suspension. Crystallization is observed up to polydispersities as high as 18.5%, and up to a number fraction of large particles of 29% in bidisperse suspensions. The crystal structure is random hexagonal close-packed as in monodisperse pNIPAM microgel suspensions. We explain our experimental results by considering the effect of bound counterions. Above a critical particle concentration, these cause deswelling of the largest microgels, which are the softest, changing the size distribution of the suspension and enabling crystal formation in conditions where incompressible particles would not crystallize.

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.

Exclusion of impurity particles in charged colloidal crystals

Uniformly shaped, charged colloidal particles dispersed in water form ordered "crystal" structures when the interaction between the particles is sufficiently strong. Herein, we report the behavior of "impurity" particles, whose sizes and/or charge numbers are different from those of the bulk, on addition to the charged colloidal crystals. These impurities were excluded from the crystals during the homogeneous crystallization, crystal grain growth, and unidirectional crystallization processes. Such systems will be useful as models for studying the refinement of materials and crystal defects.

Heterogeneous nucleation and crystal growth on curved surfaces observed by real-space imaging

We present a real-space imaging study of homogeneous and heterogeneous crystal nucleation and growth in colloidal suspensions of slightly charged and polydisperse particles. Heterogeneous crystallization is observed close to curved surfaces with radii of curvature, R, in the range from 4 to 40 particle diameters, d. Close to a curved surface, we find crystal nucleation and growth to be suppressed for R approximately < 10d. For R approximately > 15d, fast crystal growth is observed similar to that on a flat wall (R = ∞). We use the purely topological method of shortest path rings to determine the orientation of the crystal on the length scale of the nearest neighbor distance. Crystal nuclei forming close to a curved surface are oriented analogous to crystal growth on a flat wall with hexagonal planes parallel to the wall. While the smallest nuclei appear to be unaffected by the surface, larger nuclei are found to be suppressed for radii of curvature R approximately < 10d. The critical nucleus size in the vicinity of a curved surface is found to be about the same as for homogeneous nucleation.

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.

Exclusion of impurity particles during grain growth in charged colloidal crystals

We examine the spatial distribution of fluorescent-labeled charged polystyrene (PS) particles (particle volume fraction ϕ = 0.0001 and 0.001, diameter d = 183 and 333 nm) added to colloidal crystals of charged silica particles (ϕ = ϕ(s) = 0.035-0.05, d = 118 nm). At ϕ(s) = 0.05, the PS particles were almost randomly distributed in the volume-filling polycrystal structures before the grain growth process. Time-resolved confocal laser scanning microscopy observations reveal that the PS particles are swept to the grain boundaries of the colloidal silica crystals owing to grain boundary migration. PS particles with d = 2420 nm are not excluded from the silica crystals. We also examine influences of the impurities on the grain growth laws, such as the power law growth, size distribution, and existence of a time-independent distribution function of the scaled grain size.

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.

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.

Crystallization in three- and two-dimensional colloidal suspensions

Despite progress in the understanding of crystal nucleation and crystal growth since the first theories for nucleation were developed, an exact quantitative prediction of the nucleation rates in most systems has remained an unsolved problem. Colloidal suspensions show a phase behavior that is analogous to atomic or molecular systems and serve accordingly as ideal model systems for studying crystal nucleation with an accuracy and depth on a microscopic scale that is hard to reach for atomic or molecular systems. Due to the mesoscopic size of colloidal particles they can be studied in detail on the single-particle level and their dynamics is strongly slowed down in comparison with atomic or molecular systems, such that the formation of a crystal nucleus can be followed in detail. In this review, recent progress in the study of homogeneous and heterogeneous crystal nucleation in colloids and the controlled growth of crystalline colloidal structures is reviewed. All this work has resulted in unprecedented insights into the early stage of nucleation and it is also relevant for a deeper understanding of soft matter materials in general as well as for possible applications based on colloidal suspensions.

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.

Crystallization of dense binary hard-sphere mixtures with marginal size ratio

Molecular dynamics simulations are performed for binary hard-sphere mixtures with a size ratio of gamma =0.9 and a volume fraction of phi=0.58 over a range of compositions. We show how, at this high volume fraction, crystallization depends sensitively on the composition. Evidence is presented that crystallization in these mixtures does not proceed by the standard nucleation and growth paradigm. Rather, some crystallite forms almost immediately and then an interplay between compositional fluctuations and crystal growth is able to dramatically extend the time scale on which further crystallization occurs. This can be seen as a form of geometric frustration.

Preparation and characterization of particles with small differences in polydispersity

Colloidal particles are widely used both in fundamental research and in materials science. One important parameter influencing the physical properties of colloidal materials is the particle size distribution (polydispersity) of the colloidal particles. Recent work on colloidal crystallization has demonstrated that even subtle changes in polydispersity can have significant effects. In this study we present centrifugation techniques for subtly manipulating the width and the shape of the particle size distribution, for polydispersities less than 10%. We use scanning electron microscopy as well as dynamic and static light scattering to characterize the particle size distributions. We compare the results and highlight the difficulties associated with the determination of accurate particle size distributions.

Effect of polydispersity on the crystallization kinetics of suspensions of colloidal hard spheres when approaching the glass transition

We present a comprehensive study of the solidification scenario in suspensions of colloidal hard spheres for three polydispersities between 4.8% and 5.8%, over a range of volume fractions from near freezing to near the glass transition. From these results, we identify four stages in the crystallization process: (i) an induction stage where large numbers of precursor structures are observed, (ii) a conversion stage as precursors are converted to close packed structures, (iii) a nucleation stage, and (iv) a ripening stage. It is found that the behavior is qualitatively different for volume fractions below or above the melting volume fraction. The main effect of increasing polydispersity is to increase the duration of the induction stage, due to the requirement for local fractionation of particles of larger or smaller than average size. Near the glass transition, the nucleation process is entirely frustrated, and the sample is locked into a compressed crystal precursor structure. Interestingly, neither polydispersity nor volume fraction significantly influences the precursor stage, suggesting that the crystal precursors are present in all solidifying samples. We speculate that these precursors are related to the dynamical heterogeneities observed in a number of dynamical studies.

Small changes in particle-size distribution dramatically delay and enhance nucleation in hard sphere colloidal suspensions

We present hard-sphere crystallization kinetics for three samples with small differences in polydispersity. We show that an increase in polydispersity of 1% is sufficient to cause dramatic changes in the crystallization kinetics: crystallization is delayed by almost one decade in time and quantitative and qualitative changes in the crystallization scenario are observed. Surprisingly the nucleation rate density is enhanced by almost a factor of 10. We interpret these results in terms of polydispersity limited growth, where local fractionation processes lead to a delayed but faster nucleation.

Crystal nucleation in binary hard sphere mixtures: A Monte Carlo simulation study

We present calculations of the nucleation barrier during crystallization in binary hard sphere mixtures under moderate degrees of supercooling using Monte Carlo simulations in the isothermal-isobaric semigrand ensemble in conjunction with an umbrella sampling technique. We study both additive and negatively nonadditive binary hard sphere systems. The solid-fluid phase diagrams of such systems show a rich variety of behavior, ranging from simple spindle shapes to the appearance of azeotropes and eutectics to the appearance of substitutionally ordered solid phase compounds. We investigate the effect of these types of phase behavior upon the nucleation barrier and the structure of the critical nucleus. We find that the underlying phase diagram has a significant effect on the mechanism of crystal nucleation. Our calculations indicate that fractionation of the species upon crystallization increases the difficulty of crystallization of fluid mixtures and in the absence of fractionation (azeotropic conditions) the nucleation barrier is comparable to pure fluids. We also calculate the barrier to nucleation of a substitutionally ordered compound solid. In such systems, which also show solid-solid phase separation, we find that the phase that nucleates is the one whose equilibrium composition is closer to the composition of the fluid phase.

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.

Stability of the binary colloidal crystals AB2 and AB13

Suspensions of binary mixtures of hard-sphere poly-methylmethacrylate colloidal particles were studied at six different size ratios alpha. The main aim was to determine the range of size ratios over which the binary colloidal crystals AB2 and AB13 are stable. Combining these results with those of earlier work, we found stability of AB2 for 0.60 approximately > alpha approximately > 0.425, in good agreement with theoretical predictions by computer simulation and cell model methods. AB13 was observed for 0.62 approximately > alpha approximately > 0.485, the lower limit being significantly smaller than the theoretical prediction of about 0.525. Rough measurements of crystallization rates showed that AB2 tended to crystallize fastest at small size ratios, whereas the opposite was true for AB13. These findings should provide a guide to the optimum conditions for materials applications of these binary colloidal crystals.