Stokesian Dynamics Research Articles

Charge and force on a conductive sphere between two parallel electrodes: A Stokesian dynamics approach

We present an accurate and efficient method to compute the electrostatic charge and force on a conductive sphere between two parallel electrodes. The method relies on a Stokesian dynamics-like approach, in which the capacitance tensor is divided into two contributions: (1) a far field contribution that captures the long range, many body interactions between the sphere and the two electrodes and (2) a near field contribution that captures the pairwise interactions between nearly contacting surfaces. The accuracy of this approach is confirmed by comparison to “exact” numerical results obtained by finite element modeling. From the capacitance tensor, we derive the charge and dipole moment on the sphere, the electrostatic free energy of the system, and the electrostatic force on the sphere. These quantities are used to describe the dynamics of micron-scale particles oscillating within a viscous dielectric liquid between two parallel electrodes subject to constant voltage. Simulated particle trajectories agree quantitatively with those captured experimentally by high speed imaging.

Behavior of Particles in the Process of Magnetic Compound Fluid Polishing of Inner Surface of Micro-Tube with Axial Flow

Behavior of both suspended micrometer-size magnetic particles and micrometer-size nonmagnetic abrasive particles in a micro-tube filled with a magnetic compound fluid is investigated by using discrete particles method based on the simplified Stokesian dynamics in order to reveal the polishing process of inner surface of the tube with an axial flow. Giving the axial flow of the appropriate velocity is effective on performing uniform polishing of the surface in the magnetic compound fluid polishing.

A parallel implementation of the Stokesian Dynamics method applied to the simulation of colloidal suspensions

We present a strategy for the numerical simulation of polydisperse colloidal suspensions on supercomputers using distributed memory. In order to simulate large colloidal systems we have parallelized the Stokesian Dynamics method and combined it with DLVO theory. Through an efficient parallelization using the message passing system MPI we are able to reduce the computational costs of the problem from O(N2) to O(Pnp,max2), where N is the number of particles in the system, P the number of used processes and np,max=⌈NP⌉, respectively. This allows the simulation of large colloidal systems, as the computational time now not only depends on the number of particles, but also on the number of cores, through which the CPU time can be reduced dramatically.

Material properties of the shear-thickened state in concentrated near hard-sphere colloidal dispersions

Reversible shear thickening is common in concentrated dispersions of Brownian hard-spheres at high-shear rates. We confirm the existence of a well-defined colloidal shear-thickened state through experimental measurements of the shear stress and the first and second normal stress differences in the shear-thickened state as a function of the particle volume fraction for a model dispersion of near hard-spheres. The shear stress and normal stress differences are observed to grow linearly with the shear rate in the shear-thickened state and both normal stress differences are observed to be negative. Our experimental results show the shear-thickened state of colloidal dispersions can be described by three material properties—the shear viscosity and first and second normal stress difference coefficients—that are a function of the volume fraction. All three material properties are found to diverge with a power law scaling as (1−ϕϕmax)−2 close to maximum packing, ϕmax, which is found to be 0.54 ± 0.01. We find ηr,sts > ϒ2,sts > ϒ1,sts. These results are consistent with theoretical predictions for shear thickening by hydrocluster formation and quantitatively comparable to Stokesian Dynamics simulations. We further postulate and show that these material properties are consistent with those measured for non-Brownian suspensions.

Particle dynamics modeling methods for colloid suspensions

We present a review and critique of several methods for the simulation of the dynamics of colloidal suspensions at the mesoscale. We focus particularly on simulation techniques for hydrodynamic interactions, including implicit solvents (Fast Lubrication Dynamics, an approximation to Stokesian Dynamics) and explicit/particle-based solvents (Multi-Particle Collision Dynamics and Dissipative Particle Dynamics). Several variants of each method are compared quantitatively for the canonical system of monodisperse hard spheres, with a particular focus on diffusion characteristics, as well as shear rheology and microstructure. In all cases, we attempt to match the relevant properties of a well-characterized solvent, which turns out to be challenging for the explicit solvent models. Reasonable quantitative agreement is observed among all methods, but overall the Fast Lubrication Dynamics technique shows the best accuracy and performance. We also devote significant discussion to the extension of these methods to more complex situations of interest in industrial applications, including models for non-Newtonian solvent rheology, non-spherical particles, drying and curing of solvent and flows in complex geometries. This work identifies research challenges and motivates future efforts to develop techniques for quantitative, predictive simulations of industrially relevant colloidal suspension processes.

シアシックニングと接触摩擦

In a recent article (R. Seto, R. Mari, J. F. Morris, and M. M. Denn, Phys. Rev. Lett., 111:218301, 2013) we found frictional contact forces to be essential for reproducing the shear thickening behavior of non-Brownian suspensions. Although the introduction of frictional contact to a Stokesian Dynamics simulation is speculative and requires the existence of particle-particle contact despite fluid lubrication, the simulation results are in good agreement with experimental data. The model also provides physical insight into the relation between shear thickening and a jamming transition. This article describes the transition from a fluid mechanics perspective to a granular physics perspective.

Anisotropic diffusion of concentrated hard-sphere colloids near a hard wall studied by evanescent wave dynamic light scattering

Evanescent wave dynamic light scattering and Stokesian dynamics simulations were employed to study the dynamics of hard-sphere colloidal particles near a hard wall in concentrated suspensions. The evanescent wave averaged short-time diffusion coefficients were determined from experimental correlation functions over a range of scattering wave vectors and penetration depths. Stokesian dynamics simulations performed for similar conditions allow a direct comparison of both the short-time self- and collective diffusivity. As seen earlier [V. N. Michailidou, G. Petekidis, J. W. Swan, and J. F. Brady, Phys. Rev. Lett. 102, 068302 (2009)] while the near wall dynamics in the dilute regime slow down compared to the free bulk diffusion, the reduction is negligible at higher volume fractions due to an interplay between the particle-wall and particle-particle hydrodynamic interactions. Here, we provide a comprehensive comparison between experiments and simulations and discuss the interplay of particle-wall and particle-particle hydrodynamics in the self- and cooperative dynamics determined at different scattering wave vectors and penetration depths.

A fictitious domain approach for the simulation of dense suspensions

Low Reynolds number concentrated suspensions do exhibit an intricate physics which can be partly unraveled by the use of numerical simulation. To this end, a Lagrange multiplier-free fictitious domain approach is described in this work. Unlike some methods recently proposed, the present approach is fully Eulerian and therefore does not need any transfer between the Eulerian background grid and some Lagrangian nodes attached to particles. Lubrication forces between particles play an important role in the suspension rheology and have been properly accounted for in the model. A robust and effective lubrication scheme is outlined which consists in transposing the classical approach used in Stokesian Dynamics to our present direct numerical simulation. This lubrication model has also been adapted to account for solid boundaries such as walls. Contact forces between particles are modeled using a classical Discrete Element Method (DEM), a widely used method in granular matter physics. Comprehensive validations are presented on various one-particle, two-particle or three-particle configurations in a linear shear flow as well as some O(103) and O(104) particle simulations.

Dynamics of suspensions of spherical doublets in simple shear and pressure driven flow

Suspensions of non-spherical particles are commonly encountered in the flow procedures used in material processing industries and biological applications. The rheological properties of these suspensions are influenced by both the dynamics and the orientation of the suspended particles, which is a characteristic that can be affected by hydrodynamic interactions. The rheology and dynamics of a suspension of spherical doublets in bounded flow between plane parallel walls were examined using the Stokesian Dynamics simulation method. The doublets consisted of two rigid spheres connected by inter-particle forces. These doublets are commonly observed in the agglomeration of particles during the processing of suspensions. The simulations were performed for particles undergoing both simple shear and pressure driven flow between two parallel walls. For all concentrations, the viscosity of the suspension of doublets was observed to be higher than the values determined for spherical particles. The wall normal stresses were similar for both types of suspensions. In addition to the shear and normal stresses, the velocity and concentration profiles in simple shear and pressure driven flow in the channel were also studied. In pressure driven flow, the shear induced migration observed with the doublets was similar to the behaviour observed with the spherical particles. Compared with the suspension of spherical particles in pressure driven flow, the maximum centreline concentration for the doublets always exhibited a lower value. An analysis of the orientation parameter indicated that the doublets near the wall were aligned in the flow direction, with the particles in the centre of the channels exhibiting random orientations.

Dynamic simulation of concentrated macromolecular solutions with screened long-range hydrodynamic interactions: Algorithm and limitations

Hydrodynamic interactions exert a critical effect on the dynamics of macromolecules. As the concentration of macromolecules increases, by analogy to the behavior of semidilute polymer solutions or the flow in porous media, one might expect hydrodynamic screening to occur. Hydrodynamic screening would have implications both for the understanding of macromolecular dynamics as well as practical implications for the simulation of concentrated macromolecular solutions, e.g., in cells. Stokesian dynamics (SD) is one of the most accurate methods for simulating the motions of N particles suspended in a viscous fluid at low Reynolds number, in that it considers both far-field and near-field hydrodynamic interactions. This algorithm traditionally involves an O(N(3)) operation to compute Brownian forces at each time step, although asymptotically faster but more complex SD methods are now available. Motivated by the idea of hydrodynamic screening, the far-field part of the hydrodynamic matrix in SD may be approximated by a diagonal matrix, which is equivalent to assuming that long range hydrodynamic interactions are completely screened. This approximation allows sparse matrix methods to be used, which can reduce the apparent computational scaling to O(N). Previously there were several simulation studies using this approximation for monodisperse suspensions. Here, we employ newly designed preconditioned iterative methods for both the computation of Brownian forces and the solution of linear systems, and consider the validity of this approximation in polydisperse suspensions. We evaluate the accuracy of the diagonal approximation method using an intracellular-like suspension. The diffusivities of particles obtained with this approximation are close to those with the original method. However, this approximation underestimates intermolecular correlated motions, which is a trade-off between accuracy and computing efficiency. The new method makes it possible to perform large-scale and long-time simulation with an approximate accounting of hydrodynamic interactions.

Motion of rigid aggregates under different flow conditions

The response of rigid aggregates to different flow fields is investigated theoretically using model clusters with realistic three-dimensional structure composed of identical spherical primary particles. The aim is to relate the main fluid dynamic properties of the system with the geometry and morphology of the aggregates. Our simulations are based on Stokesian dynamics. The dilute limit of a colloidal aggregate system was studied, where aggregates are very far from each other, and hence, mutual inter-aggregate interactions are negligible. The motion of aggregates is characterized in terms of translational mobility and angular velocity, and the ability of simple models, based on either the simplified aggregate geometry or the concept of permeability, to capture the main features of the motion is examined.

Comparison of three simulation methods for colloidal aggregates in Stokes flow: Finite elements, lattice Boltzmann and Stokesian dynamics

Dilute suspensions of colloidal aggregates of two different types are investigated numerically by three different methods; Stokesian dynamics (SD), the finite element method (FEM) and the lattice Boltzmann method (LBM). The Navier–Stokes equations are solved in the zero-Reynolds-number limit, and the fluid forces acting on particles within the aggregates as well as the aggregates’ influence on the fluid flow are evaluated by the discrete methods. Two fluid-flow profiles are considered; uniform plug flow and linear shear flow. The paper compares three methods in their own typical simulation environment. Thus, the results may provide practical guidelines on which methods to chose for certain accuracy requirements.

Pair-particle dynamics and microstructure in sheared colloidal suspensions:Simulation and Smoluchowski theory

The Smoluchowski equation (SE) approach reduced to pair level provides an accepted method for analysis of the pair microstructure, i.e., the pair distribution functiong(r), in sheared colloidal suspensions. Under dilute conditions, the resulting problem is well-defined, but for concentrated suspensions the coefficients of the pair SE are unclear. This work outlines a recently developed theoretical approach for analytical and numerical study of the pair SE for concentrated colloidal suspensions of spheres in shear flow, and then focuses upon evaluation of coefficients and related properties of the problem from Stokesian Dynamics simulation, over a wide range of particle volume fraction, ϕ, and Péclet number (ratio of shear to Brownian motion). The pair distribution functiondetermined from the SE theory is in generally good agreement with Stokesian Dynamics, as are the computed viscosity and normal stresses of the material. The primary focus of the work is to consider the pair relative velocity predicted by the theoryin comparison to Stokesian Dynamics simulations, as well as to evaluate quantities related to the hydrodynamic dispersion needed in the theoretical approach. The pair dynamics for moderate particle volume fraction, 0.20 ⩽ ϕ ⩽ 0.35, are found to be remarkably different from the form for an isolated pair of spheres, and at ϕ ⩾ 0.40 a qualitative change is again seen. Agreement of the theory and simulation on the primary features of the particle motion and structure is good, and discrepancies are clearly delineated.

Stokesian dynamics of pill-shaped Janus particles with stick and slip boundary conditions

We study the forces and torques experienced by pill-shaped Janus particles of different aspect ratios where half of the surface obeys the no-slip boundary condition and the other half obeys the Navier slip condition of varying slip lengths. Using a recently developed boundary integral formulation whereby the traditional singular behavior of this approach is removed analytically, we quantify the strength of the forces and torques experienced by such particles in a uniform flow field in the Stokes regime. Depending on the aspect ratio and the slip length, the force transverse to the flow direction can change sign. This is a novel property unique to the Janus nature of the particles.

Stokes flow past three spheres

In this paper we present a numerical method to calculate the dynamics of three spheres in a quiescent viscous fluid. The method is based on Lamb’s solution to Stokes flow and the Method of Reflections, and is arbitrarily accurate given sufficient computer memory and time. It is more accurate than multipole methods, but much less efficient. Although it is too numerically intensive to be suitable for more than three spheres, it can easily handle spheres of different sizes. We find no convergence difficulties provided we study mobility problems, rather than resistance problems.After validating against the existing literature, we make a direct comparison with Stokesian Dynamics (SD), and find that the largest errors in SD occur at a sphere separation around 0.1 radius. Finally, we present results for an example system having different-sized spheres.

Relation between ordering and shear thinning in colloidal suspensions

Colloidal suspensions exhibit shear thinning and shear thickening. The most common interpretation of these phenomena identifies layering of the fluid perpendicular to the shear gradient as the driver for the observed behavior. However, studies of the particle configurations associated with shear thinning and thickening cast doubt on that conclusion and leave unsettled whether these nonequilibrium phenomena are caused primarily by correlated particle motions or by changes in particle packing structure. We report the results of stokesian dynamics simulations of suspensions of hard spheres that illuminate the relation among the suspension viscosity, shear rate, and particle configuration. Using a recently introduced sampling technique for nonequilibrium systems, we show that shear thinning can be decoupled from layering, thereby eliminating layering as the driver for shear thinning. In contrast, we find that there is a strong correlation between shear thinning and a two-particle measure of the shear stress. Our results are consistent with a recent experimental study.

Microstructure in sheared non-Brownian concentrated suspensions

The shear-induced microstructure in non-Brownian suspensions is studied. The pair distribution function (PDF) in the shear plane is experimentally determined for particle volume fractions ranging from 0.05 to 0.56. Transparent suspensions made of polymethylmetacrylate particles (172 μm in diameter) dispersed in a fluorescent index matched Newtonian liquid are sheared in a wide-gap Couette rheometer. A thin laser sheet lights the shear plane. The particle positions are recorded and the PDF in the shear plane is computed. The PDF at contact is shown to be anisotropic, with a depleted area in the receding side of the reference particle. The angular position of the depleted zone, close to the velocity axis at low particle concentration, is tilted toward the dilatation axis as the volume fraction is increased. At high concentrations (larger than 0.45), the shape of the PDF changes qualitatively with a secondary depleted area in the compressional quadrant of the main flow and a probability peak in the velocity direction. These experimental results are in good agreement with numerical simulations in Stokesian dynamics where the interaction force between particles has been tuned to reproduce the particle roughness effects.

Sedimentation of non-Brownian suspensions

We describe the sedimentation of a non-Brownian suspension of hard spheres by Stokesian dynamics simulations. Attention is focused on the mean settling velocity and the structure of particles distribution as the suspension attains a non-equilibrium steady sedimentation state. Many-body hydrodynamic interactions between particles are evaluated using the precise multipole method, encoded in the new FAST-HYDROMULTIPOLE program. With the complete neglect of Brownian motion, the sedimentation velocity and the particles configurational distribution in the steady state differ measurably from their equilibrium counterparts. Simulations reproduce short-range particle correlations in good agreement with the statistical description of sedimentation formulated by Cichocki and Sadlej [Europhys. Lett. 72, 936 (2005)].

Fluid Permeability in Stratified Unconsolidated Particulate Bed

Fluid permeability of polydisperse particulate bed with finite thickness has been examined. On the assumption of creeping flow, the permeability of monodisperse particles with arbitrary arrangement is calculated by means of Stokesian dynamics approach in which the interaction between individual particles and interstitial fluid is described by multipole expansion of the Oseen tensor. We have extended such calculation method to polydisperse particulate systems which have not so dense structures (up to particle volume fraction $${\phi \sim}$$ 0.2). The particles are located infinitely in space and their interaction has been taken into account by Ewald summation technique. For the spatial distribution of polydisperse particles, we consider locally stratified particulate beds and define stratification degree as a parameter which apparently and mathematically represents the thickness of the mixing region of different-sized particles. The permeability profiles in the particulate beds with different stratification degree show the dependence of local permeability on the spatial and size distribution of particles. Consequently, the calculation results indicate that the permeability of non-uniform polydisperse particulate bed can be predicted by integrating the local permeation resistance which is determined by the local specific surface area.

Restructuring of colloidal aggregates in shear flow

A method to couple interparticle contact models with Stokesian dynamics (SD) is introduced to simulate colloidal aggregates under flow conditions. The contact model mimics both the elastic and plastic behavior of the cohesive connections between particles within clusters. Owing to this, clusters can maintain their structures under low stress while restructuring or even breakage may occur under sufficiently high stress conditions. SD is an efficient method to deal with the long-ranged and many-body nature of hydrodynamic interactions for low Reynolds number flows. By using such a coupled model, the restructuring of colloidal aggregates under shear flows with stepwise increasing shear rates was studied. Irreversible compaction occurs due to the increase of hydrodynamic stress on clusters. Results show that the greater part of the fractal clusters are compacted to rod-shaped packed structures, while the others show isotropic compaction.