Concentrated Hard-sphere Suspensions Research Articles

Theory of tracer diffusion in concentrated hard-sphere suspensions

A phenomenological theory of diffusion and cross-diffusion of tracer particles in concentrated hard-sphere suspensions is developed. Expressions for the diffusion coefficients as functions of the host particle volume fraction are obtained up to the close-packing limit. In concentrated systems the tracer diffusivity decreases because of the reduced pore space available for diffusion. The tracer diffusivity can be modelled by a Stokes–Einstein equation with an effective viscosity that depends on the pore size. Tracer diffusion and segregation during sedimentation cease at a critical trapping volume fraction corresponding to a tracer glass transition. The tracer cross-diffusion coefficient, however, increases near the glass transition and diverges in the close-packed limit.

Near-wall dynamics of concentrated hard-sphere suspensions: comparison of evanescent wave DLS experiments, virial approximation and simulations.

In this article we report on a study of the near-wall dynamics of suspended colloidal hard spheres over a broad range of volume fractions. We present a thorough comparison of experimental data with predictions based on a virial approximation and simulation results. We find that the virial approach describes the experimental data reasonably well up to a volume fraction of ϕ≈ 0.25 which provides us with a fast and non-costly tool for the analysis and prediction of evanescent wave DLS data. Based on this we propose a new method to assess the near-wall self-diffusion at elevated density. Here, we qualitatively confirm earlier results [Michailidou, et al., Phys. Rev. Lett., 2009, 102, 068302], which indicate that many-particle hydrodynamic interactions are diminished by the presence of the wall at increasing volume fractions as compared to bulk dynamics. Beyond this finding we show that this diminishment is different for the particle motion normal and parallel to the wall.

Effect of repulsive interactions on structure and rheology of sheared colloidal dispersions

A previously developed Smoluchowski theory for concentrated hard-sphere suspensions in shear flow is extended to study structure and rheology of colloidal suspensions with soft repulsive interactions. Accelerated Stokesian Dynamics simulations are carried out to provide insight and to enable direct comparison with theoretical predictions. The effect of extended range repulsive interactions is studied by considering repulsive interactions with different steepness, using identical potentials in simulation and theory, for varying shear rates characterized in dimensionless form as 0.1 ≤ Pe ≤ 100; here, Pe = 6πηa3/kbT is the ratio of hydrodynamic to Brownian forces and η is the fluid viscosity, is the shear rate, a is the particle radius and kbT is the thermal energy. Examples of predicted microstructures and the equivalent simulated results for hard-sphere suspensions at ϕ = 0.40 are also presented for comparison. The predicted pair distribution function is in good agreement with simulations before the onset of a shear-induced ordering transition in simulations of the soft colloids. The calculations of shear viscosity based on the predicted microstructure were also in general agreement with simulation results. The role of hydrodynamic interactions on flow-induced structures is discussed in the context of the proposed theory.

Slow dynamics of supercooled colloidal fluids: spatial heterogeneities and nonequilibrium density fluctuations

The coupled diffusion equations recently proposed by Tokuyama for concentrated hard-sphere suspensions are numerically solved, starting from nonequilibrium initial configurations. The most important feature of those equations is that the self-diffusion coefficient D S ( Φ) becomes zero at the glass transition volume fraction φ g as D S ( Φ)∼ D 0|1− Φ( x , t)/ φ g | γ with γ=2 where Φ( x , t) is the local volume fraction of colloids, D 0 the single-particle diffusion constant, and φ g=( 4 3 ) 3/(7 ln 3−8 ln 2+2) . This dynamic anomaly results from the many-body correlations due to the long-range hydrodynamic interactions. Then, it is shown how small initial disturbances can be enhanced by this anomaly near φ g , leading to long-lived, spatial heterogeneities. Those heterogeneities are responsible for the slow relaxation of nonequilibrium density fluctuations. In fact, the self-intermediate scattering function is shown to obey a two-step relaxation around the β-relaxation time t β ∼|1− φ/ φ g | −1, and also to be well approximated by the Kohlrausch–Williams–Watts function with an exponent β around the α-relaxation time t α ∼|1− φ/ φ g | − η , where η= γ/ β, and φ is the particle volume fraction. Thus, the nonexponential α relaxation is shown to be explained by the existence of long-lived, spatial heterogeneities.

Slow dynamics of nonequilibrium density fluctuations in concentrated hard-sphere suspensions

The coupled diffusion equations recently proposed for concentrated hard-sphere suspensions of interacting Brownian particles, the nonlinear deterministic diffusion equation with the self-diffusion coefficient DS(Φ(x, t)) for the average local volume fraction Φ(x, t), and the linear stochastic diffusion equation with DS(Φ(x, t)) for the density fluctuations δn(x, t) are numerically solved under a spatially inhomogeneous, nonequilibrium initial state. Thus, in a supercooled region where Φβ ≤ Φ < Φg, the slow evolution of the cluster-like glassy domains with Φ(x, t) ≥ Φg and the slow relaxation of the nonequilibrium density fluctuations are shown to be caused by the dynamic singularity of the self-diffusion coefficient, DS(Φ(x, t)) ∼ (1−Φ(x, t)/Φg)2, where Φ is a particle volume fraction, Φg = (4/3)3/(7 ln 3 − 8 ln 2 + 2) is the colloidal glass transition volume fraction, and Φβ is the crossover volume fraction.

Nonequilibrium effects on slow dynamics in concentrated colloidal suspensions.

An alternative stochastic diffusion equation is proposed to study the dynamics of nonequilibrium density fluctuations in concentrated hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions among particles. The singularity of the correlation effect of the many-body hydrodynamic interactions is shown to drastically influence the qualitative behavior of the relaxation of nonequilibrium density fluctuations, and thus to cause the two different slow relaxations whose time scales, ${t}_{\ensuremath{\beta}}$ and ${t}_{\ensuremath{\alpha}}$, diverge as the volume fraction of Brownian particles approaches the critical value ${\ensuremath{\varphi}}_{c}=\frac{{(\frac{4}{3})}^{3}}{(7\mathrm{ln}3\ensuremath{-}8\mathrm{ln}2+2)}$; ${t}_{\ensuremath{\beta}}\ensuremath{\sim}{(1\ensuremath{-}\frac{\ensuremath{\varphi}}{{\ensuremath{\varphi}}_{c}})}^{\ensuremath{-}1}$ and ${t}_{\ensuremath{\alpha}}\ensuremath{\sim}{(1\ensuremath{-}\frac{\ensuremath{\varphi}}{{\ensuremath{\varphi}}_{c}})}^{\ensuremath{-}2}$.

Slow dynamics of self-diffusion in metastable colloidal fluids

A simple theory for nonequilibrium density fluctuations in concentrated hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is proposed. The correlation effects due to the many-body hydrodynamic interactions among particles are shown to cause a structural arrest in the relaxation of nonequilibrium density fluctuations. A volume fraction dependence of slow relaxation process in concentrated colloidal suspensions is thus explored from a new unifying point of view.

On the theory of concentrated hard-sphere suspensions

A systematic theory for the dynamics of hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is presented. A generalized diffusion equation is derived for concentrated suspensions. The volume fraction (φ) dependence of the short- and long-time self-diffusion coefficients are thus explored from a unifying point of view. The long-range hydrodynamic interactions due to the Oseen tensor are shown to play a crucial role in both coefficients, while the short-range hydrodynamic interactions just lead to corrections. The importance of the correlation effects between particles due to the long-range hydrodynamic interactions is also stressed. The nonlocal correlation effect is an important factor, leading to the behavior of the long-time self-diffusion coefficient ( D S L) as D S L ∼ (1 − φ/ φ 0) 2 near the volume fraction of φ 0 = 0.5718. The direct interactions are also found to be drastically reduced by the short-range hydrodynamic interactions.

Dynamics of hard-sphere suspensions.

A systematic theory for the dynamics of concentrated hard-sphere suspensions of interacting Brownian particles with both hydrodynamic and direct interactions is presented. An alternative equation for the number density of particles is derived. The volume-fraction dependence of the short- and long-time self-diffusion coefficients is thus explored from a new point of view. Both the short-range and the long-range hydrodynamic interactions are shown to play an important role in both coefficients, while the direct interactions are reduced drastically by the hydrodynamic interactions.