Hard-sphere Colloidal Suspensions Research Articles

Two-point microrheology and diffusing wave spectroscopy

The dynamics of charged and hard-sphere silica colloidal suspensions are measured using diffusing wave spectroscopy and are interpreted by the Generalized Stokes–Einstein Relation (GSER). At high concentration, the resulting moduli are in good qualitative agreement with bulk rheology, including the frequency response, but are higher by a quantitative factor. We show that the “two-point” GSER, which derives from the correlated motion of the colloidal particles, provides a better quantitative agreement between bulk and microrheology. The two-point GSER applies generally when the average scattering vector sampled by the multiple scattering events corresponds to a length scale greater than the scatterer diameter. We discuss applications for characterizing colloid interactions by measuring the high-frequency moduli of suspensions and extend the two-point analysis to tracer particle microrheology measurements of a semiflexible biopolymer network.

Coupling effect of solvent–colloid interaction on diffusivity and viscosity of colloidal hard-sphere suspensions: Statistical mechanical free-volume approach

Apart from describing the transport behaviors to a reliable accuracy, it remains as yet far from satisfactory to comprehend the explicit role of the solvent–colloid interaction on the diffusivity and viscosity of colloidal hard-sphere suspensions. Here, we exploit the free volume effect on the diffusivity and viscosity by means of statistical mechanics, which enables to express them in terms of the solvent–solvent, colloid–colloid, and solvent–colloid pair correlation functions as a function of the colloidal concentration. The agreement between the theory and experiment is found to be very good over the entire colloidal volume fraction ϕ investigated by experiment, i.e., up to ϕ∼0.60. The present theory predicts a crossover in the solvent effect on the diffusivity: Above ϕc=0.38, the solvent–colloid interaction enhances the diffusivity of the colloidal particles suspended therein, whereas below ϕc, it serves as a diffusion barrier against the particles.

Microscopic activated dynamics theory of the shear rheology and stress overshoot in ultradense glass-forming fluids and colloidal suspensions

We formulate a particle and force level, activated dynamics-based statistical mechanical theory for the continuous startup nonlinear shear rheology of ultradense glass-forming hard sphere fluids and colloidal suspensions in the context of the elastically collective nonlinear Langevin equation approach and a generalized Maxwell model constitutive equation. Activated structural relaxation is described as a coupled local-nonlocal event involving caging and longer range collective elasticity which controls the characteristic stress relaxation time. Theoretical predictions for the deformation-induced enhancement of mobility, the onset of relaxation acceleration at remarkably low values of stress, strain, or shear rate, apparent power law thinning of the steady-state structural relaxation time and viscosity, a nonvanishing activation barrier in the shear thinning regime, an apparent Herschel–Buckley form of the shear rate dependence of the steady-state shear stress, exponential growth of different measures of a yield or flow stress with packing fraction, and reduced fragility and dynamic heterogeneity under deformation were previously shown to be in good agreement with experiments. The central new question we address here is the defining feature of the transient response—the stress overshoot. In contrast to the steady-state flow regime, understanding the transient response requires an explicit treatment of the coupled nonequilibrium evolution of structure, elastic modulus, and stress relaxation time. We formulate a new quantitative model for this aspect in a physically motivated and computationally tractable manner. Theoretical predictions for the stress overshoot are shown to be in good agreement with experimental observations in the metastable ultradense regime of hard sphere colloidal suspensions as a function of shear rate and packing fraction, and accounting for deformation-assisted activated motion appears to be crucial for both the transient and steady-state responses.

Microscopic theory for the pair correlation function of liquidlike colloidal suspensions under shear flow.

We present a theoretical framework to investigate the microscopic structure of concentrated hard-sphere colloidal suspensions under strong shear flows by fully taking into account the boundary-layer structure of convective diffusion. We solve the pair Smoluchowski equationwith shear separately in the compressing and extensional sectors of the solid angle, by means of matched asymptotics. A proper, albeit approximate, treatment of the hydrodynamic interactions in the different sectors allows us to construct a potential of mean force containing the effect of the flow field on pair correlations. We insert the obtained pair potential in the Percus-Yevick relation and use the latter as a closure to solve the Ornstein-Zernike integral equation. For a wide range of either the packing fraction η and the Péclet (Pe) number, we compute the pair correlation function and extract scaling laws for its value at contact. For all the considered values of Pe, we observe a very good agreement between theoretical findings and numerical results from the literature, up to rather large values of η. The theory predicts a consistent enhancement of the structure factor S(k) at k→0, upon increasing the Pe number. We argue this behavior may signal the onset of a phase transition from the isotropic phase to a nonuniform one, induced by the external shear flow.

Are strongly confined colloids good models for two dimensional liquids?

Quasi-two-dimensional (quasi-2D) colloidal hard-sphere suspensions confined in a slit geometry are widely used as two-dimensional (2D) model systems in experiments that probe the glassy relaxation dynamics of 2D systems. However, the question to what extent these quasi-2D systems indeed represent 2D systems is rarely brought up. Here, we use computer simulations that take into account hydrodynamic interactions to show that dense quasi-2D colloidal bi-disperse hard-sphere suspensions exhibit much more rapid diffusion and relaxation than their 2D counterparts at the same area fraction. This difference is induced by the additional vertical space in the quasi-2D samples in which the small colloids can move out of the 2D plane, therefore allowing overlap between particles in the projected trajectories. Surprisingly, this difference in the dynamics can be accounted for if, instead of using the surface density, one characterizes the systems by means of a suitable structural quantity related to the radial distribution function. This implies that in the two geometries, the relevant physics for glass formation is essentially identical. Our results provide not only practical implications on 2D colloidal experiments but also interesting insights into the 3D-to-2D crossover in glass-forming systems.

Fickian Non-Gaussian Diffusion in Glass-Forming Liquids.

Fickian yet non-Gaussian diffusion (FnGD), a most intriguing open issue in soft matter, is generically associated with some dynamical and/or structural heterogeneity of the environment. Here we investigate the features of FnGD in glass-forming liquids, the epitome of dynamical heterogeneity, drawing on experiments on hard-sphere colloidal suspensions and simulations of a simple model of molecular liquid. We demonstrate that FnGD strengthens on approaching the glass transition, by identifying distinct timescales for Fickianity, τ_{F}, and for restoring of Gaussianity, τ_{G}>τ_{F}, as well as their associated length scales, ξ_{F} and ξ_{G}. We find τ_{G}∝τ_{F}^{γ} with γ≃1.8 for both systems. In the deep FnGD regime, the displacement distributions display exponential tails. We show that, in simulations, the time-dependent decay lengths l(t) at different temperatures all collapse onto a power-law master curve [l(t)/(ξ_{G})]∝(t/τ_{G})^{α}, with α=0.33. A similar collapse, if less sharp, is also found in experiments, seemingly with the same exponent α. We further discuss the connections of the timescales and length scales characterizing FnGD with structural relaxation and dynamic heterogeneity.

Machine learning of lubrication correction based on GPR for the coupled DPD-DEM simulation of colloidal suspensions.

Hydrodynamic interactions have a major impact on the suspension properties, but they are absent in atomic and molecular fluids due to a lack of intervening medium at close range. To reproduce the correct hydrodynamic interactions, lubrication correction is essential to compensate the missing short-range hydrodynamics from the fluids. However, lubrication correction requires many simulations in particle-based simulations of colloidal suspensions. To address the problem, we employ an active learning strategy based on Gaussian process regression (GPR) for normal and tangential lubrication corrections to significantly reduce the number of necessary simulations and apply the correction to the coupled multiscale simulation of monodisperse hard-sphere colloidal suspensions. In particular, a single-particle dissipative particle dynamics (DPD) model with parameter correction is used to describe the solvent-solvent and colloid-solvent interactions, and a discrete element method (DEM) model to depict the colloid-colloid frictional contacts. The lubrication correction results demonstrate that only six and four independent simulations (observation points for GPR training) are required to achieve accurate normal and tangential lubrication corrections, respectively. To validate the machine learning of lubrication correction based on GPR, we investigate the self-diffusion coefficients of colloids, suspension rheology and microstructure using the coupled DPD-DEM model with GPR lubrication correction. Our simulation results show that the machine learning of lubrication correction based on GPR is effective and the lubrication corrected DPD-DEM model is indeed capable of accurately capturing hydrodynamic interactions and correctly reproducing dynamical and rheological properties of colloidal suspensions. Moreover, the machine learning of lubrication correction based on GPR is not limited to the coupled DPD-DEM simulation of colloidal suspensions presented here, but can be easily applied to other particle-based simulations of particulate suspensions.

Rheology of colloidal and metallic glass formers

Colloidal hard-sphere suspensions are convenient experimental models to understand soft matter, and also by analogy the structural-relaxation behavior of atomic or small-molecular fluids. We discuss this analogy for the flow and deformation behavior close to the glass transition. Based on a mapping of temperature to effective hard-sphere packing, the stress–strain curves of typical bulk metallic glass formers can be quantitatively compared with those of hard-sphere suspensions. Experiments on colloids give access to the microscopic structure under deformation on a single-particle level, providing insight into the yielding mechanisms that are likely also relevant for metallic glasses. We discuss the influence of higher-order angular signals in connection with non-affine particle rearrangements close to yielding. The results are qualitatively explained on the basis of the mode-coupling theory. We further illustrate the analogy of pre-strain dependence of the linear-elastic moduli using data on PS-PNiPAM suspensions.

Dynamics of dense hard sphere colloidal systems: A numerical analysis.

The applicability to dense hard sphere colloidal suspensions of a general coarse-graining approach called Record Dynamics (RD) is tested by extensive molecular dynamics simulations. We reproduce known results as logarithmic diffusion and the logarithmic decay of the average potential energy per particle. We provide quantitative measures for the cage size and identify the displacements of single particles corresponding to intermittent cage breakings. We then partition the system into spatial domains and show that, within each domain, a subset of such intermittent events called quakes constitutes a log-Poisson process, as predicted by RD. Specifically, quakes are shown to be statistically independent and Poisson distributed with an average depending on the logarithm of time. Finally, we discuss the nature of the dynamical barriers surmounted by quakes and link RD to the phenomenology of aging hard sphere colloids.

Coupling of sedimentation and liquid structure: Influence on hard sphere nucleation.

The discrepancy in nucleation rate densities between simulated and experimental hard spheres remains staggering and unexplained. Suggestively, more strongly sedimenting colloidal suspensions of hard spheres nucleate much faster than weakly sedimenting systems. In this work, we consider first the effect of sedimentation on the structure of colloidal hard spheres by tuning the density mismatch between solvent and colloidal particles. In particular, we investigate the effect on the degree of the fivefold symmetry present. Second, we study the size of density fluctuations in these experimental systems in comparison to simulations. The density fluctuations are measured by assigning each particle a local density, which is related to the number of particles within a distance of 3.25 particle diameters. The standard deviation of these local densities gives an indication of the fluctuations present in the system. Fivefold symmetry is suppressed by a factor of two when sedimentation is induced in our system. Density fluctuations are increased by a factor of two in experiments compared to simulations. The change in fivefold symmetry makes a difference to the expected nucleation rates, but we demonstrate that it is ultimately too small to resolve the discrepancy between experiment and simulation, while the fluctuations are shown to be an artefact of 3d particle tracking.

A crucial role of long-range hydrodynamic interactions near the colloidal glass transition based on time-convolutionless mode-coupling theory

The time-convolutionless mode-coupling theory (TMCT) recently proposed for molecular systems is employed to derive a TMCT equation for the collective-intermediate scattering function in suspensions of hard-sphere colloids from the nonlinear Langevin equations with the hydrodynamic interactions between colloids. The nonlinear memory function contained in the TMCT equation consists of two types of interactions between colloids; a mechanical interaction and a hydrodynamic interaction. It is predicted from the TMCT equation how the hydrodynamic interactions can affect the ergodic to non-ergodic transition at a critical point. Then, such a hydrodynamic effect is tested by the experimental data for suspensions of hard-sphere colloids with size polydispersities. Thus, it is emphasized that the long-range hydrodynamic interactions are indispensable to explain the polydispersity dependence of the experimental critical points which is quite different from that of simulations where only the mechanical interactions are taken into.

Study of dynamical heterogeneities in colloidal nanoclay suspensions approaching dynamical arrest

The dynamics of aqueous Laponite clay suspensions slow down with increasing sample waiting time (tw). This behavior, and the material fragility that results, closely resemble the dynamical slowdown in fragile supercooled liquids with decreasing temperature, and are typically ascribed to the increasing sizes of distinct dynamical heterogeneities in the sample. In this article, we characterize the dynamical heterogeneities in Laponite suspensions by invoking the three-point dynamic susceptibility formalism. The average time-dependent two-point intensity autocorrelation and its sensitivity to tw are probed in dynamic light scattering experiments. Distributions of relaxation time scales, deduced from the Kohlrausch-Williams-Watts equation, are seen to widen with increasing tw. The calculated three-point dynamic susceptibility of Laponite suspensions exhibits a peak, with the peak height increasing with evolving tw at fixed volume fraction or with increasing volume fraction at fixed tw, thereby signifying the slowdown of the sample dynamics. The number of dynamically correlated particles, calculated from the peak-height, is seen to initially increase rapidly with increasing tw, before eventually slowing down close to the non-ergodic transition point. This observation is in agreement with published reports on supercooled liquids and hard sphere colloidal suspensions and offers a unique insight into the colloidal glass transition of Laponite suspensions.

Unsteady shear flows of colloidal hard-sphere suspensions by dynamic simulation

The rheology during the start-up and cessation of simple shear flow has been investigated for near hard-sphere colloidal suspensions. Simulations augmented by theoretical analysis are used to determine how the non-Newtonian stress development and relaxation depend on the microstructure. Accelerated Stokesian dynamics (ASD) and Brownian dynamics (BD) simulations are used for 0.05 ≤ Pe ≤ 500 in concentrated freely flowing suspensions; the Peclet number defining the ratio of shear to thermal motion is P e = 3 π η γ a 3 / k T with η the suspending fluid viscosity, γ the shear rate, and kT the thermal energy. Theoretical predictions based on the Smoluchowski equation for dilute suspensions are made, and these are primarily used for comparison with results from BD simulations in which hydrodynamic interactions are neglected. For suspensions with hydrodynamics, simulations by ASD are used to probe start-up and flow cessation over a large range of Pe; these studies focus on solid volume fraction ϕ = 0.4, with...

Colloidal gels tuned by oscillatory shear.

We examine the microstructural and mechanical changes which occur during oscillatory shear flow and reformation after flow cessation of an intermediate volume fraction colloidal gel using rheometry and Brownian Dynamics (BD) simulations. A model depletion colloid-polymer mixture is used, comprising a hard sphere colloidal suspension with the addition of non-adsorbing linear polymer chains. The results reveal three distinct regimes depending on the strain amplitude of oscillatory shear. Large shear strain amplitudes fully break the structure which results in a more homogenous and stronger gel after flow cessation. Intermediate strain amplitudes densify the clusters and lead to highly heterogeneous and weak gels. Shearing the gel to even lower strain amplitudes creates a less heterogonous stronger solid. These three regimes of shearing are connected to the microscopic shear-induced structural heterogeneity. A comparison with steady shear flow reveals that the latter does not produce structural heterogeneities as large as oscillatory shear. Therefore oscillatory shear is a much more efficient way of tuning the mechanical properties of colloidal gels. Moreover, colloidal gels presheared at large strain amplitudes exhibit a distinct nonlinear response characterized largely by a single yielding process while in those presheared at lower rates a two-step yielding process is promoted due to the creation of highly heterogeneous structures.

Transport coefficients and mechanical response in hard-disk colloidal suspensions**Project supported by the National Basic Research Program of China (Grant No. 2012CB821500) and the National Natural Science Foundation of China (Grant Nos. 21374073 and, 21574096).

We investigate the transport properties and mechanical response of glassy hard disks using nonlinear Langevin equation theory. We derive expressions for the elastic shear modulus and viscosity in two dimensions on the basis of thermal-activated barrier-hopping dynamics and mechanically accelerated motion. Dense hard disks exhibit phenomena such as softening elasticity, shear-thinning of viscosity, and yielding upon deformation, which are qualitatively similar to dense hard-sphere colloidal suspensions in three dimensions. These phenomena can be ascribed to stress-induced “landscape tilting”. Quantitative comparisons of these phenomena between hard disks and hard spheres are presented. Interestingly, we find that the density dependence of yield stress in hard disks is much more significant than in hard spheres. Our work provides a foundation for further generalizing the nonlinear Langevin equation theory to address slow dynamics and rheological behavior in binary or polydisperse mixtures of hard or soft disks.

Correlation between dynamical and structural heterogeneities in colloidal hard-sphere suspensions

Experiments combining dynamic and static light scattering have probed a colloidal hard-sphere system for the formation of dynamical and structural heterogeneities, which play a role in both forms of solidification: crystallization and vitrification.

Apparent wall-slip of colloidal hard-sphere suspensions in microchannel flow

We investigated the dependence of the apparent slip length (β) on the particle concentration for suspensions of colloidal hard spheres flowing through microchannels with smooth walls. Hydrostatic pressure was used to pump monodisperse suspensions of 1 μm silica spheres in water–glycerol mixture through a poly(dimethylsiloxane) microchannel with rectangular cross section. Confocal scanning laser microscopy allowed accurate measurement of velocity profiles via particle tracking velocimetry. Apparent slip lengths were measured by extrapolating the velocity profiles to zero. Systematic variations were made of the particle volume fraction Φ (0.03–0.42) and the Peclet number (2–50). In these regimes the apparent slip length shows a weak dependence on Pe and the slip velocity near the wall grows linearly with shear rate irrespective of Φ. However, β turns out to increase strongly with Φ, tending toward an asymptotic behavior for Φ ≫ 0.3. To explain these observations, we worked out a conceptually simple model, in which the apparent slip is caused by a local reduction in volume fraction due to excluded volume, and the local viscosity is calculated using the Krieger–Dougherty equation. The predictions by this model match the experimental results remarkably well.

Communication: Dynamical density functional theory for dense suspensions of colloidal hard spheres.

We study structural relaxation of colloidal hard spheres undergoing Brownian motion using dynamical density functional theory. Contrary to the partial linearization route [D. Stopper et al., Phys. Rev. E 92, 022151 (2015)] which amounts to using different free energy functionals for the self and distinct part of the van Hove function G(r, t), we put forward a unified description employing a single functional for both components. To this end, interactions within the self part are removed via the zero-dimensional limit of the functional with a quenched self component. In addition, we make use of a theoretical result for the long-time mobility in hard-sphere suspensions, which we adapt to the inhomogeneous fluid. Our results for G(r, t) are in excellent agreement with numerical simulations even in the dense liquid phase. In particular, our theory accurately yields the crossover from free diffusion at short times to the slower long-time diffusion in a crowded environment.

Constant Stress and Pressure Rheology of Colloidal Suspensions.

We study the constant stress and pressure rheology of dense hard-sphere colloidal suspensions using Brownian dynamics simulation. Expressing the flow behavior in terms of the friction coefficient-the ratio of shear to normal stress-reveals a shear arrest point from the collapse of the rheological data in the non-Brownian limit. The flow curves agree quantitatively (when scaled) with the experiments of Boyer et al. [Phys. Rev. Lett. 107, 188301 (2011)]. Near suspension arrest, both the shear and the incremental normal viscosities display a universal power law divergence. This work shows the important role of jamming on the arrest of colloidal suspensions and illustrates the care needed when conducting and analyzing experiments and simulations near the flow-arrest transition.

Modeling diffusion in colloidal suspensions by dynamical density functional theory using fundamental measure theory of hard spheres.

We study the dynamics of colloidal suspensions of hard spheres that are subject to Brownian motion in the overdamped limit. We obtain the time evolution of the self- and distinct parts of the van Hove function by means of dynamical density functional theory. The free-energy model for the hard-sphere fluid that we use is the very accurate White Bear II version of Rosenfeld's fundamental measure theory. However, in order to remove interactions within the self-part of the van Hove function, a nontrivial modification has to be applied to the free-energy functional. We compare our theoretical results with data that we obtain from dynamical Monte Carlo simulations, and we find that the latter are well described by our approach even for colloid packing fractions as large as 40%.