Many-body Hydrodynamic Interactions Research Articles

Nature of the divergence in low shear viscosity of colloidal hard-sphere dispersions.

Measurements of the low-shear viscosity eta(o) with a Zimm-Crothers viscometer for dispersions of colloidal hard spheres are reported as a function of volume fraction phi up to 0.56. Nonequilibrium theories based on solutions to the two-particle Smoluchoski equation or ideal mode coupling approximations do not capture the divergence. However, the nonhydrodynamic contribution to the relative viscosity Deltaeta(o) is correlated over a wide range of volume fractions by the Doolittle and Adam-Gibbs equations, indicating an exponential divergence at phi(m)=0.625+/-0.015. The data extend the previously proposed master curve, providing a test for improved theories for the many-body thermodynamic and hydrodynamic interactions that determine the viscosity of hard-sphere dispersions.

Stability of colloidal clusters in shear flow near a wall: Stokesian dynamics simulation studies

An earlier algorithm for Stokesian dynamics simulation of colloid particles in a fluid bounded by a hard wall is extended to the case when a linear shear flow is applied. The algorithm includes many-body hydrodynamic interactions arising from the shear flow with lubrication corrections for pairs of close particles. The extended algorithm is used to simulate small clusters of particles which may interact with each other and with the bounding wall through model potentials with repulsive cores and attractive tails. The pair problem is studied first, in bounded and unbounded fluid, both with and without pair and wall interactions. The critical shear rate necessary to break a bound pair is determined for a range of initial configurations. Whereas in unbounded fluid a bound pair rotates with the vorticity of the shear flow, near a wall the hydrodynamic forces introduce a new breakup mechanism in which the pair tilts through a finite angle relative to the wall before separating. This tilt mechanism requires significantly stronger shear fields than are needed to separate a pair in unbounded fluid. Linear arrays of particles as well as examples of 2- and 3-D bound clusters are studied next to show how the wall and the tilt mechanism modify the shear-induced breakup. With a wall potential included, significant changes are seen in the final distribution of particles resulting from cluster breakup.

Fluid fields due to many-body hydrodynamic interactions

We consider the characteristics of the unsteady flows, generated in the incompressible, viscous fluid by the hydrodynamic interactions between a finite number of solid spheres. The non-stationary properties of the velocity fields, due to the local inertia of the fluid, are discussed. To take into account the local acceleration effects, we apply the time-dependent Green's tensor. The four different types of the time-dependent velocity profiles, generated at different régimes of the low Reynolds number hydrodynamic interactions, are regarded.

The Effect of Many-Body Interactions on the Sedimentation of Monodisperse Particle Dispersions

An experimental investigation was made of the sedimentation rate of low-charged monodisperse silica and polystyrene latex particle dispersions as a function of the particle volume fraction. It was found that the normalized sedimentation velocityU/U0, corrected for the effect of the two-body hydrodynamic interaction, increases with the particle volume fraction, which indicates that the degree of particle aggregation inside the dispersions increases with the particle volume fraction. This phenomenon results from attractive many-body hydrodynamic interactions between colloidal particles. It is reported for the first time that the many-body hydrodynamic interaction becomes important at the particle concentration of 6.5 vol% in monodisperse dispersions, and the many-body thermodynamic interaction is negligible at a low particle concentration, i.e., less than 15 vol%. The effect of many-body hydrodynamic interaction on the particle microstructure was also experimentally examined by using a nondestructive Kossel diffraction technique based on the principle of back-light scattering. It was found that the particle packing structure inside the dispersion initially becomes more ordered with the increase of the particle volume fraction. However, there is less increase in the particle ordering structure after 6 vol%. Furthermore, after the particle concentration reaches 10 vol%, the particle packing structure decreases to a value lower than that of 6 vol% due to the increased particle aggregation, as found in the sedimentation experiments. Predictions of a statistical thermodynamic model were compared with the experimental data on structure factors. It is found that particle dimerization occurs around 10 vol%, which agrees with the sedimentation results.

Hydrodynamic interactions between a particle and two rigid walls: Effect of surface roughness and many-body hydrodynamic interactions

The sedimentation of a particle trapped between two rigid walls allow us to demonstrate the effect of many-body hydrodynamic effects arising in a confined geometry. A theoretical treatment of this problem is presented together with an experimental investigation of the sedimentation of a macroscopic particle by image analysis which allows us to directly obtain the trajectory of the particle. Theoretical drag force in the presence of the two planes is shown to be in good agreement with the experimental results. The determination of the sedimentation of a particle, initially applied on the upper plate by a magnetic field gradient, is shown to provide a relevant information on the average surface roughness.

Self-diffusion coefficients of charged particles: Prediction of nonlinear volume fraction dependence

We report on calculations of the translational and rotational short-time self-diffusion coefficients $D^t_s$ and $D^r_s$ for suspensions of charge-stabilized colloidal spheres. These diffusion coefficients are affected by electrostatic forces and many-body hydrodynamic interactions (HI). Our computations account for both two-body and three-body HI. For strongly charged particles, we predict interesting nonlinear scaling relations $D^t_s\propto 1-a_t\phi^{4/3}$ and $D^r_s\propto 1-a_r\phi^2$ depending on volume fraction $\phi$, with essentially charge-independent parameters $a_t$ and $a_r$. These scaling relations are strikingly different from the corresponding results for hard spheres. Our numerical results can be explained using a model of effective hard spheres. Moreover, we perceptibly improve the known result for $D^t_s$ of hard sphere suspensions.

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.

Hydrodynamic screening in sedimenting suspensions of non-Brownian spheres.

It has been suggested [D. L. Koch and E. S. G. Shaqfeh, J. Fluid Mech. 224, 275 (1991)] that the long-range hydrodynamic interactions in sedimenting suspensions of non-Brownian spheres are screened by changes in the pair correlation function. However, a large-scale numerical simulation, using more than 32 000 spheres, with full many-body hydrodynamic interactions, has found no evidence of the predicted changes in suspension microstructure. The absence of hydrodynamic screening in the simulations is shown to lead to divergent velocity fluctuations, in disagreement with recent experimental results [H. Nicolai and E. Guazzelli, Phys. Fluids 7, 3 (1995)].

Rheology of hydrodynamically interacting concentrated hard disks

We have developed a novel Lagrangian–Eulerian numerical scheme to model the dynamics of dense hard-disk suspensions with many-body hydrodynamic interactions in an incompressible Newtonian solvent. A simple derivation of analytical solution of lubrication force between disks is given. Reasonable agreement of viscosity calculated by lattice–gas cellular automata and our method is obtained, despite the lack of Brownian forces in our work.

Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation

A new and very general technique for simulating solid–fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in a companion paper (Part 2), extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite-Reynolds-number flows, are reported.

Short-time dynamics of charge-stabilized colloidal suspensions

We have calculated the combined effects of the electrostatic and hydrodynamic interactions on the short-time dynamics of charge-stabilized colloidal suspensions, which can be investigated experimentally from the first cumulant of the dynamic scattering function. The first cumulant is obtained using two methods, a pairwise-additivity approximation for the many-body hydrodynamic interaction combined with a far-field expansion of the mobility tensors, and a renormalized density fluctuation expansion due to Beenakker and Mazur. An effective hard-sphere model is employed to provide a simple physical interpretation of the results derived from both methods. It is shown that even for moderately charged colloidal particles the hydrodynamic interaction cannot be neglected, and that its effects are more pronounced for collective diffusion than for self-diffusion. Moreover, our calculations predict a non-analytic volume fraction dependence of the sedimentation velocity, with an exponent whose numerical value is sensitive to the amount of added electrolyte.

Short-time motion of colloidal particles: Numerical simulation via a fluctuating lattice-Boltzmann equation.

A new and general technique for simulating solid-fluid suspensions, which combines molecular dynamics for the solid particles with a lattice-Boltzmann model for the fluid, is described. The many-body hydrodynamic interactions are fully accounted for, both for small particle velocities and at higher Reynolds numbers. Brownian motion of the solid particles is included by adding a fluctuating component to the fluid stress tensor. Simulations of the dynamics of colloidal particles at short times compare favorably with recent diffusing-wave spectroscopy experiments.

Diffusion, dispersion, and settling of hard spheres.

Using a multiple light scattering technique and a stabilizing counterflow (i.e., a fluidized bed), we measured the average sedimentation velocity, its variance, and the short-time self-diffusion coefficient in a concentrated hard-sphere suspension. Many-body hydrodynamic interactions slow both the Brownian diffusion and mean sedimentation and provide a novel mechanism for long time dispersion.

Stokesian dynamics simulations of particle trajectories near a plane

The trajectories of spherical particles close to a plane in shear flow are computed, taking into account lubrication forces and many-body hydrodynamic interactions between spheres, and between spheres and the plane. It is shown that the relative motion of two particles may be reversed by the presence of the plane, and eddylike recirculation motions can occur. The resistance coefficient of a linear chain of N spheres is calculated as a function of the distance of the chain from the plane, and is shown to agree well with a simple model based on a screening coefficient derived from a single pair of spheres.

Collective diffusion of charged spheres in the presence of hydrodynamic interaction

The short-time behavior of the scattering intensity, which can be measured by dynamic light scattering, is investigated theoretically. We consider a monodisperse system of charged hard spheres and study the combined effects of electrostatic repulsion and hydrodynamic interactions. The electrostatic repulsion determines the static properties, which are calculated from the thermodynamically consistent Rogers-Young scheme. For comparison, some results obtained within the rescaled mean spherical approximation (RMSA) are shown as well. Many-body hydrodynamic interaction is treated within the renormalization approach due to Beenakker. We compare results depending on the scattering vector to experimental data. Further, we show various trends for the collective diffusion coefficient, which is obtained as the long wavelength limit of the collective diffusion function, when volume fraction of spheres, salinity, and charge of the spheres are changed. The results are aimed at the interpretation of dynamic light scattering properties of systems of weakly charged spheres, which could be ionic micelies or macromolecules at moderate concentrations.

Hydrodynamic transport coefficients of random dispersions of hard spheres

Accurate values for the hydrodynamic transport properties of random dispersions of hard spheres have been determined by numerical simulation. The many-body hydrodynamic interactions are calculated from a multipole-moment expansion of the force density on the surface of the solid particles; the singular lubrication forces are included exactly for pairs of particles near contact. It has been possible to calculate the transport properties of small periodic systems, at all packing fractions, with uncertainties of less than 1%; but for larger systems we are limited computationally to lower order, and therefore less accurate, moment approximations to the induced force density. Nevertheless, since the higher-order moment contributions are short range they are essentially independent of system size and we can use small system data to correct our results for larger systems. Numerical calculations show that this is a reliable and accurate procedure. The ensemble-averaged mobility tensors are strongly dependent on system size, with deviations from the thermodynamic limit varying as N−1/3. We show that these deviations can be accounted for by a straightforward calculation based on the length of the unit cell and the suspension viscosity and mobility. The remaining number dependencies are small. Problems involved in implementing this method for larger numbers of particles are considered, and alternative methods are discussed.

Many-body hydrodynamic interactions in suspensions

We consider hydrodynamic interactions between N rigid bodies of arbitrary shape immersed in an incompressible fluid. We study the generalized mobility matrix relating the translational and rotational velocities and the symmetric force dipole moments to the forces, the torques and the strain of an incident flow field. We show that the elements of the mobility matrix may be obtained as matrix elements of an operator related to the friction kernel. This allows a multiple scattering expansion of the mobility matrix.

Mutual and self-diffusion in concentrated α-crystallin protein dispersions. A dynamic light scattering study

The theory of Brownian motion has not yet reached the point where the dynamic properties of concentrated colloidal dispersions can be computed. This is mainly due to the complicated nature of many-body hydrodynamic interactions and difficulties in considering memory effects. Here we present mutual and self-diffusion coefficients for concentrated alpha-crystallin protein dispersions measured by the photon correlation technique. These measurements are compared to theoretical predictions considering only 2-body hydrodynamic interactions or screened hydrodynamic interactions without memory effects. While the limit of applicability of these theories is restricted to low concentrations, it is seen that the (long-time limit) self-diffusion follows the dispersion inverse viscosity and that mutual diffusion increases almost linearly with concentration. Although the dynamic behaviour observed may result from an accidental coincidence, similar tendencies have been reported by other authors and they could be used (with caution) as simple empirical rules for estimating diffusion coefficients in the high-concentration range.

Convective motion and transfer of force by many-body hydrodynamic interaction

We consider hydrodynamic interactions between N rigid bodies of arbitrary shape immersed in an incompressible fluid. When the bodies are carried along by an incident flow without exerting forces or torques on the fluid then their translational and rotational velocities are linearly related to the incident flow velocity by convection kernels. In the absence of an incident flow, but with applied forces and torques, the force density acting on the fluid is linearly related to the forces and torques by transfer kernels. We show that the convection and transfer kernels are simply related by a symmetry relation. For freely moving bodies the force density exerted on the fluid is related to the incident flow by a convective friction kernel. We show that this kernel is symmetric.

Dynamic simulation of hydrodynamically interacting particles

A general method for computing the hydrodynamic interactions among N suspended particles, under the condition of vanishingly small particle Reynolds number, is presented. The method accounts for both near-field lubrication effects and the dominant many-body interactions. The many-body hydrodynamic interactions reproduce the screening characteristic of porous media and the ‘effective viscosity’ of free suspensions. The method is accurate and computationally efficient, permitting the dynamic simulation of arbitrarily configured many-particle systems. The hydrodynamic interactions calculated are shown to agree well with available exact calculations for small numbers of particles and to reproduce slender-body theory for linear chains of particles. The method can be used to determine static (i.e. configuration specific) and dynamic properties of suspended particles that interact through both hydrodynamic and non-hydrodynamic forces, where the latter may be any type of Brownian. colloidal, interparticle or external force. The method is also readily extended to dynamically simulate both unbounded and bounded suspensions.