Colloidal Hard Spheres Research Articles

BINARY HARD-SPHERE COLLOID-DROPLET MIXTURES WITH THE PYRITE-TYPE STRUCTURE

We investigate the assembly of a binary mixture of patchy colloids and droplets into crystal structures by use of the space-filling principle and Metropolis Monte Carlo simulations. Here, colloids with six patches in an octahedral symmetry attract droplets, whereas the colloid-colloid and droplet-droplet pairwise interactions are purely hard-core repulsions. Within parameter space regions that allow for stable binary crystals, we find the formation of different structure types, which are colloidal analogs of the NaCl phase and pyrite (FeS2) phase, as a function of the droplet-to-colloid sphere diameter ratio. This finding is consistent with theoretical predictions. Notably, while the NaCl structure type is a commonly known structure of hard-sphere colloids and nanoparticles, the formation of the FeS2 structure type from colloidal dispersion has not been explored. Our approach suggests a potential route to obtain colloidal crystals with more complex structures.

Multiscale heterogeneous dynamics in two-dimensional glassy colloids.

On approaching the glass transition, a dense colloid exhibits a dramatic slowdown with minute structural changes. Most microscopy experiments directly follow the motion of individual particles in real space, whereas scattering experiments typically probe the collective dynamics in reciprocal space at variable wavevector q. Multiscale studies of glassy dynamics are experimentally demanding and, thus, seldom performed. By using two-dimensional hard-sphere colloids at various area fractions ϕ, we show here that Differential Dynamic Microscopy (DDM) can be effectively used to measure the collective dynamics of a glassy colloid in a range of q within a single experiment. As ϕ is increased, the single decay of the intermediate scattering functions is progressively replaced by a more complex relaxation that we fit to a sum of two stretched-exponential decays. The slowest process, corresponding to the long-time particle escapes from caging, has a characteristic time τs = 1/(DLq2) with diffusion coefficient DL∼(ϕc-ϕ)2.8, and ϕc ≃ 0.81. The fast process exhibits, instead, a non-Brownian scaling of the characteristic time τf(q) and a relative amplitude a(q) that monotonically increases with q. Despite the non-Brownian nature of τf(q), we succeed in estimating the short-time diffusion coefficient Dcage, whose ϕ-dependence is practically negligible compared to the one of DL. Finally, we extend DDM to measure the q-dependent dynamical susceptibility χ4(q, t), a powerful yet hard-to-access multiscale indicator of dynamical heterogeneities. Our results show that DDM is a convenient tool to study the dynamics of colloidal glasses over a broad range of time and length scales.

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.

Phase stability of colloidal spheres mixed with semiflexible supramolecular polymers

HypothesisMixtures of colloids and supramolecular polymers may exhibit stimuli-responsive phase behaviour. However, in theoretical descriptions of such systems, the polymers are commonly described either as flexible chains or as rigid rods, while in experimental systems supramolecular polymers usually fall in between these two limits. We expect the flexibility of the polymers to have a profound effect on the stimuli-responsive phase behaviour. TheoryWe propose a general approach to predict the phase behaviour of colloidal hard spheres mixed with covalent or supramolecular polymers of arbitrary persistence length using free volume theory and an interpolation between flexible and rigid chains. FindingsThe binodals are predicted to shift to lower monomer concentrations as the persistence length is increased, making the polymers more efficient depletants. The persistence length is therefore an additional degree of freedom for manipulating the phase behaviour of colloid–polymer mixtures. We show that by manipulating the persistence length of temperature responsive supramolecular polymers, a wide range of phase diagrams with various topologies can be obtained. For example, we find phase diagrams with a critical point but no triple point or displaying two triple points for temperature-sensitive supramolecular polymers mixed with hard spheres.

Influence of solvent quality on depletion potentials in colloid-polymer mixtures.

As first explained by the classic Asakura-Oosawa (AO) model, effective attractive forces between colloidal particles induced by depletion of nonadsorbing polymers can drive demixing of colloid-polymer mixtures into colloid-rich and colloid-poor phases, with practical relevance for purification of water, stability of foods and pharmaceuticals, and macromolecular crowding in biological cells. By idealizing polymer coils as effective penetrable spheres, the AO model qualitatively captures the influence of polymer depletion on thermodynamic phase behavior of colloidal suspensions. In previous work, we extended the AO model to incorporate aspherical polymer conformations and showed that fluctuating shapes of random-walk coils can significantly modify depletion potentials [W. K. Lim and A. R. Denton, Soft Matter 12, 2247 (2016); J. Chem. Phys. 144, 024904 (2016)]. We further demonstrated that the shapes of polymers in crowded environments sensitively depend on solvent quality [W. J. Davis and A. R. Denton, J. Chem. Phys. 149, 124901 (2018)]. Here, we apply Monte Carlo simulation to analyze the influence of solvent quality on depletion potentials in mixtures of hard-sphere colloids and nonadsorbing polymer coils, modeled as ellipsoids whose principal radii fluctuate according to random-walk statistics. We consider both self-avoiding and non-self-avoiding random walks, corresponding to polymers in good and theta solvents, respectively. Our simulation results demonstrate that depletion of polymers of equal molecular weight induces much stronger attraction between colloids in good solvents than in theta solvents and confirm that depletion interactions are significantly influenced by aspherical polymer conformations.

Universal Nucleation Behavior of Sheared Systems.

Using molecular simulations and a modified classical nucleation theory, we study the nucleation, under flow, of a variety of liquids: different water models, Lennard-Jones, and hard sphere colloids. Our approach enables us to analyze a wide range of shear rates inaccessible to brute-force simulations. Our results reveal that the variation of the nucleation rate with shear is universal. A simplified version of the theory successfully captures the nonmonotonic temperature dependence of the nucleation behavior, which is shown to originate from the violation of the Stokes-Einstein relation.

Oscillatory shear-induced bcc-fcc martensitic transformation in a colloidal suspension with long-range repulsive interactions.

We perform non-equilibrium Brownian dynamics simulations to investigate the out-of-equilibrium phase behavior of a suspension of charged colloids under external oscillatory shear. We independently vary the frequency f and the maximum strain amplitude γmax of the oscillations and map out an out-of-equilibrium phase diagram in the f-γmax plane. Similar to what has been observed in earlier studies on colloidal hard spheres, we find the formation of a twinned face-centered-cubic phase in a specific range of γmax, which displays a martensitic transition to a body-centered-cubic crystal within half of the oscillation cycle. We provide a comprehensive analysis of these structures and show how the system transforms from one to the other. We also report evidence of a sliding layer phase and a string phase.

Dynamical Density Functional Theory for the Drying and Stratification of Binary Colloidal Dispersions.

We develop a dynamical density functional theory based model for the drying of colloidal films on planar surfaces. We consider mixtures of two different sizes of hard-sphere colloids. Depending on the solvent evaporation rate and the initial concentrations of the two species, we observe varying degrees of stratification in the final dried films. Our model predicts the various structures described in the literature previously from experiments and computer simulations, in particular the small-on-top stratified films. Our model also includes the influence of adsorption of particles to the interfaces.

Fcc vs. hcp competition in colloidal hard-sphere nucleation: on their relative stability, interfacial free energy and nucleation rate

In this work, we provide a detailed computational characterization of the polymorphic crystallization scenario between fcc, hcp and a random stacking mixture of both crystal phases in colloidal hard-spheres.

The effect of morphology and particle-wall interaction on colloidal near-wall dynamics.

We investigated the near-wall Brownian dynamics of different types of colloidal particles with a typical size in the 100 nm range using evanescent wave dynamic light scattering (EWDLS). In detail we studied dilute suspensions of silica spheres and shells with a smooth surface and silica particles with controlled surface roughness. While the near wall dynamics of the particle with a smooth surface differ only slightly from the theoretical prediction for hard sphere colloids, the rough particles diffuse significantly slower. We analysed the experimental data by comparison with model calculations and suggest that the deviating dynamics of the rough particles are not due to increased hydrodynamic interaction with the wall. Rather, the particle roughness significantly changes their DLVO interaction with the wall, which in turn affects their diffusion.

Frictional active Brownian particles

Frictional forces affect the rheology of hard-sphere colloids, at high shear rate. Here we demonstrate, via numerical simulations, that they also affect the dynamics of active Brownian particles and their motility-induced phase separation. Frictional forces increase the angular diffusivity of the particles, in the dilute phase, and prevent colliding particles from resolving their collision by sliding one past to the other. This leads to qualitatively changes of motility-induced phase diagram in the volume-fraction motility plane. While frictionless systems become unstable towards phase separation as the motility increases only if their volume fraction overcomes a threshold, frictional systems become unstable regardless of their volume fraction. These results suggest the possibility of controlling the motility-induced phase diagram by tuning the roughness of the particles.

Shear-induced crystallisation in binary colloidal suspensions investigated using confocal microscopy

Hard sphere colloids crystallise at large volume fractions. However, crystallisation is suppressed in the presence of large polydispersities. We investigate the effect of polydispersity on shear-induced crystallisation using binary suspensions of hard spheres. Depending on the size ratio, the samples fully crystallise, partially crystallise or remain amorphous. Using confocal microscopy, the structural changes are characterised on a single particle level. This allows us to determine the local bond order parameter, the number of nearest neighbours, the Voronoi volume, the local volume fraction and other quantities. Upon the application of shear, only minor changes of these quantities are detected in regions that remain amorphous whereas noticeable changes are observed in regions where shear-induced crystallisation occurs. These changes mainly reduce the effects of the different particle sizes on the particle arrangement. This allows substitutionally disordered crystals to form. However, with increasing size disparity the volume fraction in substitutionally disordered crystals decreases while the volume fraction in amorphous regions slightly increases. Beyond a size ratio of 1.18, the volume fraction of the crystals is smaller than the volume fraction of the amorphous regions. At the same size ratio shear-induced crystallisation ceases. The terminal size ratio of 1.18 is close to the value given by the empirical Hume-Rothery rule for metallic alloys and the corresponding polydispersity (about 0.1) is consistent with the polydispersity beyond which crystallisation is found to be suppressed.

Interaction between nearly hard colloidal spheres at an oil-water interface

We show that the interaction potential between sterically stabilized, nearly hard-sphere [poly(methylmethacrylate)-poly(lauryl methacrylate) (PMMA-PLMA)] colloids at a water-oil interface has a negligible unscreened-dipole contribution, suggesting that models previously developed for charged particles at liquid interfaces are not necessarily applicable to sterically stabilized particles. Interparticle potentials, $U(r)$, are extracted from radial distribution functions [$g(r)$, measured by fluorescence microscopy] via Ornstein-Zernike inversion and via a reverse Monte Carlo scheme. The results are then validated by particle tracking in a blinking optical trap. Using a Bayesian model comparison, we find that our PMMA-PLMA data is better described by a screened monopole only rather than a functional form having a screened monopole plus an unscreened dipole term. We postulate that the long range repulsion we observe arises mainly through interactions between neutral holes on a charged interface, i.e., the charge of the liquid interface cannot, in general, be ignored. In agreement with this interpretation, we find that the interaction can be tuned by varying salt concentration in the aqueous phase. Inspired by recent theoretical work on point charges at dielectric interfaces, which we explain is relevant here, we show that a screened $\frac{1}{r^2}$ term can also be used to fit our data. Finally, we present measurements for poly(methyl methacrylate)-poly(12-hydroxystearic acid) (PMMA-PHSA) particles at a water-oil interface. These suggest that, for PMMA-PHSA particles, there is an additional contribution to the interaction potential. This is in line with our optical-tweezer measurements for PMMA-PHSA colloids in bulk oil, which indicate that they are slightly charged.

Spontaneous Crystallization in Systems of Binary Hard Sphere Colloids.

Computer simulations of the fluid-to-solid phase transition in the hard sphere system were instrumental for our understanding of crystallization processes. But while colloid experiments and theory have been predicting the stability of several binary hard sphere crystals for many years, simulations were not successful to confirm this phenomenon. Here, we report the growth of binary hard sphere crystals isostructural to Laves phases, AlB_{2}, and NaZn_{13} in simulation directly from the fluid. We analyze particle kinetics during Laves phase growth using event-driven molecular dynamics simulations with and without swap moves that speed up diffusion. The crystallization process transitions from nucleation and growth to spinodal decomposition already deep within the fluid-solid coexistence regime. Finally, we present packing fraction-size ratio state diagrams in the vicinity of the stability regions of three binary crystals.

Long-time self-diffusion in quasi-two-dimensional colloidal fluids of paramagnetic particles.

The effect of hydrodynamic interactions (HI) on the long-time self-diffusion in quasi-two-dimensional fluids of paramagnetic colloidal particles is investigated using a combination of experiments and Brownian dynamics (BD) simulations. In the BD simulations, the direct interactions (DI) between the particles consist of a short-ranged repulsive part and a long-ranged part that is proportional to 1/r^{3}, with r the interparticle distance. By studying the equation of state, the simulations allow for the identification of the regime where the properties of the fluid are fully controlled by the long-ranged interactions, and the thermodynamic state solely depends on the dimensionless interaction strength Γ. In this regime, the radial distribution functions from the simulations are in quantitative agreement with those from the experiments for different fluid area fractions. This agreement confirms that the DI in the experiments and simulations are identical, which thus allows us to isolate the role of HI, as these are not taken into account in the BD simulations. Experiment and simulation fall onto a master curve with respect to the Γ dependence of D_{L}^{★}=D_{L}/(D_{0}Γ^{1/2}), with D_{0} the self-diffusion coefficient at infinite dilution and D_{L} the long-time self-diffusion coefficient. Our results thus show that, although HI affect the short-time self-diffusion, for a quasi-two-dimensional system with 1/r^{3} long-ranged DI, the reduced quantity D_{L}^{★} is effectively not affected by HI. Interestingly, this is in agreement with prior work on quasi-two-dimensional colloidal hard spheres.

The effect of hydrodynamics on the crystal nucleation of nearly hard spheres.

We investigate the effect of hydrodynamic interactions (HIs) on the crystal nucleation of hard-sphere colloids for varying supersaturations. We use molecular dynamics and stochastic rotation dynamics techniques to account for the HIs. For high supersaturation values, we perform brute force simulations and compute the nucleation rate, obtaining good agreement with previous studies where HIs were neglected. In order to access low supersaturation values, we use a seeding approach method and perform simulations with and without HIs. We compute the nucleation rates for the two cases and surprisingly find good agreement between them. The nucleation rate in both cases follows the trend of the previous numerical results, thereby corroborating the discrepancy between experiments and simulations. Furthermore, we investigate the amount of fivefold symmetric clusters (FSCs) in a supersaturated fluid under different physical conditions, following the idea that FSCs compete against nucleation. To this end, we explore the role of the softness of the pair interactions, different solvent viscosities, and different sedimentation rates in simulations that include HIs. We do not find significant variations in the amount of FSCs, which might reflect the irrelevance of these three features on the nucleation process.

High antisite defect concentrations in hard-sphere colloidal Laves phases.

Binary mixtures of hard spheres can spontaneously self-assemble into binary crystals. Computer simulations have been especially useful in mapping out the phase behaviour of these mixtures, under the assumption that the stoichiometry of the binary crystal is ideal. Here we show that for a size ratio of q = 0.82 this assumption is not valid near the coexistence region between the fluid and the stable binary crystal, the MgZn2 Laves phase. Instead we find a surprisingly high number of antisite defects: up to 2% of the large spheres are replaced by small spheres in equilibrium. We demonstrate that the defect concentration can be estimated using simple approximations, providing an easy way to identify systems where antisite defects play an important role. Our results shed new light on the self-assembly of colloidal Laves phases, and demonstrate the importance of antisite defects in binary crystals.

Ordering on different length scales in liquid and amorphous materials

The ordering on different real-space length scales is considered for a variety of glass-forming materials, ranging from densely packed amorphous metals and hard-sphere glassy colloids, to simple tetrahedral systems that include amorphous silicon and patchy colloids, to decorated tetrahedral systems that include amorphous ice and network-forming glasses with the AX2 stoichiometry (A = Si, Ge or Zn; X = O, S, Se or Cl). The ordering manifests itself as distinct peaks in the total structure factor , where k denotes the magnitude of the scattering vector, with positions ki (i = 1, 2 or 3) that scale with the nearest-neighbour distance. Different length scales emerge with complexity of the bonding scheme. A peak at k3 is a generic feature associated with nearest-neighbour contacts, and is therefore present in for all of the materials. A second longer-length scale emerges as a peak at if the bonding scheme assumes a directional character, leading to the formation of tetrahedral motifs in amorphous silicon and patchy colloids, or to Se–Se–Se chain segments in glassy selenium. A third still-longer-length scale appears for AX2 glasses as a first sharp diffraction peak at , where the scaled peak position depends on the character of the local network of A atoms. The geometrical origin of the peaks in and corresponding partial structure factors is considered, and equations are given for predicting the peak positions. The change in system fragility with the emergence of ordering on different length scales is discussed, along with the effect of pressure.

Competing factors in grain boundary loop shrinkage: Two-dimensional hard sphere colloidal crystals.

A grain boundary (GB) loop in a two-dimensional solid is the boundary of a domain or grain whose lattice orientation is mismatched with its uniform surroundings. Understanding the factors that influence the rate at which the interior of a GB loop relaxes to the orientation of its surroundings is an important step toward control and predictability of grain coarsening in general. Recent computational and experimental studies looking at the rate of GB loop shrinkage in two-dimensional colloidal hard sphere solids have uncovered contradictory trends: in experiments, GB loops with low misorientation angles shrank the fastest, while in simulations, they persisted the longest. In this study, the computational system's behavior is brought into qualitative agreement with the experimental results through increasing the lateral packing pressure, decreasing the domain size, and mimicking the experimental protocol used to form the GB loop. Small GB loops with the same misorientation, but displaying either a hexagonal or starlike grain shape depending on the orientation of their six dislocations, are shown to differ in their rates of shrinkage by two orders of magnitude. The evidence suggests that both the barrier to generating new dislocations as well as the pattern of dislocations formed by different GB loop preparation methods will determine which trend is observed.

Measurement of the stiffness of hard-sphere colloidal crystal-liquid interfaces

We investigated the stiffness of the interfaces between hard-sphere colloidal face-centered cubic crystals sedimented onto (100), (110), and (111) oriented templates and their equilibrium liquid with special attention to the in-plane anisotropy. The stiffness was determined from thermal fluctuations in the interface position imaged by confocal microscopy. The colloidal particles of diameter $\ensuremath{\sigma}$ are nearly density-matched with the suspending fluid so that the amount of liquid above the interface was much greater than in an earlier similar hard-sphere colloid experiment. We find stiffness values of $0.47\phantom{\rule{0.16em}{0ex}}{k}_{B}T/{\ensuremath{\sigma}}^{2}$ for the (100) interface $0.53\phantom{\rule{0.16em}{0ex}}{k}_{B}T/{\ensuremath{\sigma}}^{2}$ for the (110) interface and $0.41\phantom{\rule{4pt}{0ex}}{k}_{B}T/{\ensuremath{\sigma}}^{2}$ for the (111) interface. These values are generally closer to those found in computer simulations than to the earlier colloid results. No evidence was found however for the expected in-plane anisotropy of the (110) interface. This surprising finding confirms that of the earlier colloid work.