Stokesian Dynamics Simulations Research Articles

Particle dispersion through porous media with heterogeneous attractions.

Porous media used in many practical applications contain natural spatial variations in composition and surface charge that lead to heterogeneous physicochemical attractions between the media and transported particles. We performed Stokesian dynamics (SD) simulations to examine the effects of heterogeneous attractions on quiescent diffusion and hydrodynamic dispersion of particles within geometrically ordered arrays of nanoposts. We find that transport under quiescent conditions occurs by two mechanisms, diffusion through the void space and intermittent hopping between the attractive wells of different nanoposts. As the attraction heterogeneity increases, the latter mechanism becomes dominant, resulting in an increase in the particle trajectory tortuosity, deviations from Gaussian behavior in the particle displacement distributions, and a decrease in the long-time particle diffusivity. Similarly, under flow conditions corresponding to low Péclet number (Pe), increased attraction heterogeneity leads to transient localization near the nanoposts, resulting in a broadening of the particle distribution and enhanced longitudinal dispersion in the direction of flow. At high Pe where advection strongly dominates, however, the longitudinal dispersion coefficient is insensitive to attraction heterogeneity and exhibits Taylor-Aris dispersion behavior. Our findings provide insight into how heterogeneous interactions may influence particle transport in complex 3-D porous media.

Effects of Interfacial Shear on Particle Aggregation at an Oil/Water Interface.

Using a Stokesian dynamics simulation, the microstructure of particle aggregates at an oil/water interface with an applied Couette flow is studied. The results of the aggregation are consistent with previously published experimental work demonstrating multiple regimes of behavior based on the relative strength of shear and capillary forces. In previous work, densification of aggregates at low shear rates was theorized to occur due to short time scale fragmentation/reaggregation of aggregates with rigid particle bonds. In simulations, densification is observed at low shear rates but occurs due to local reorganization of particles due to capillary torques over long time scales. Moderate shear rates create mobile bonds between particles at shorter time scales, allowing aggregates to fragment without reaggregation into smaller isolated clusters, consistent with prior experimental work. At the highest shear rates, aggregation is inhibited completely.

Rheology and microstructure of discontinuous shear thickening suspensions

We report experimental studies on rheology and microstructure in discontinuous shear thickening of fumed silica suspensions. Formation of particle clusters was observed after the critical shear rate, and their size increases during shear thickening. At higher shear rates, these clusters were found to break down due to strong shear forces, and a continuous decrease in viscosity was observed. The suspension viscosity and the first normal stress difference variation with the shear rate showed similar dependence. The sign of first normal stress difference was negative during shear thickening, which is consistent with the hydrodynamic model of cluster formation. A linear variation of the first normal stress difference with shear rate during shear thickening further indicates its predominant hydrodynamic origin and supports the recent Stokesian dynamics simulation studies on discontinuous shear thickening based on the hydrodynamic model of nonsmooth colloids by Wang et al. [J. Rheol. 64, 379–394 (2020)]. On the other hand, a nonlinear decrease in the first normal stress difference with shear rate in the second shear-thinning region is primarily due to breaking of large clusters into smaller ones and with a possibility of frictional contacts within these hydroclusters. The oscillatory shear measurements were also performed and the samples displayed strain thickening similar to shear thickening. The similarity between the steady and dynamic shear rheology at high strain amplitudes was observed using the modified Cox–Merz rule.

Nanoparticle dispersion in porous media: Effects of attractive particle-media interactions.

We investigate the effects of physicochemical attractions on the transport of finite-sized particles in three-dimensional ordered nanopost arrays using Stokesian dynamics simulations. We find that weak particle-nanopost attractions negligibly affect diffusion due to the dominance of Brownian fluctuations. Strong attractions, however, significantly hinder particle diffusion due to localization of particles around the nanoposts. Conversely, under flow, attractions significantly enhance longitudinal dispersion at low to moderate Péclet number (Pe). At high Pe, by contrast, advection becomes dominant and attractions weakly enhance dispersion. Moreover, attractions frustrate directional locking at moderate flow rates, and shift the onset of this behavior to higher Pe.

Multiscale nature of electric-field-induced structural formations in non-colloidal suspensions.

Non-colloidal suspensions undergoing dipolar interactions in an electric field have been extensively studied and are also known as smart materials as they share similarities with electrorheological (ER) fluids. Although the macroscopic responses are well-documented, the multiscale nature of such suspensions is still lacking. In this study, a large-scale Stokesian dynamics simulation is used to investigate the structural formation of such suspensions in an electric field up to highly concentrated regimes across different length scales: from particle-level (microscale) to particle cluster-level (mesoscale) and stress response-level (macroscale). It is observed that at a volume fraction of ϕ ≈ 30%, the steady-state structures are the most isotropic at the microscale, but at the macroscale, their normal stress fields are the most anisotropic. Interestingly, these structures are also the most heterogeneous at both the microscale and mesoscale. Furthermore, the effects of confinement on the multiscale responses are explored, revealing that there could be a strong link between the mesoscale and macroscale. This multiscale nature can offer the potential for precisely controlling or designing ER fluids in practical applications.

The confined Generalized Stokes-Einstein relation and its consequence on intracellular two-point microrheology

Two-point microrheology (TPM) is used to infer material properties of complex fluids from the correlated motion of hydrodynamically interacting probes embedded in the medium. The mechanistic connection between probe motion and material properties is propagation of disturbance flows, encoded in current TPM theory for unconfined materials. However, confined media e.g. biological cells and particle-laden droplets, require theory that encodes confinement into the flow propagator (Green’s function). To test this idea, we use Confined Stokesian Dynamics simulations to explicitly represent many-body hydrodynamic couplings between colloids and with the enclosing cavity at arbitrary concentration and cavity size. We find that previous TPM theory breaks down in confinement, and we identify and replace the underlying key elements. We put forth a Confined Generalized Stokes-Einstein Relation and report the viscoelastic spectrum. We find that confinement alters particle dynamics and increases viscosity, owing to hydrodynamic and entropic coupling with the cavity. The new theory produces a master curve for all cavity sizes and concentrations and reveals that for colloids larger than 0.005 times the enclosure size, the new model is required.

A hierarchical matrix approach for computing hydrodynamic interactions

For simulations of large numbers of small, spherical particles in a Stokes flow, the long-range hydrodynamic interactions approximated by the Rotne–Prager–Yamakawa (RPY) kernel can be summed rapidly using, for example, the fast multipole method (FMM) or the particle-mesh Ewald (PME) method. In this paper, we develop new fast methods for computing these sums using the H2 hierarchical matrix representation, for open and for periodic boundary conditions. To the best of our knowledge, the method for infinite periodic sums using the H2 hierarchical matrix representation is the first such method developed. We also consider a more general RPY kernel that handles polydisperse particle radii, and show analytically and experimentally that the proxy surface method for efficiently constructing the H2 hierarchical matrix representation remains effective in this case. Numerical tests demonstrate the well-controlled accuracy of the H2 summation methods and their linear-scaling computation and storage cost. We find that the H2 matrix approach has lower cost for computing the summations compared to FMM and PME, but higher precomputation cost (required for each particle configuration). This precomputation cost can be amortized over several summations when computing Brownian displacements or forces in Brownian and Stokesian dynamics simulations with very large numbers of particles.

Cyclic force driven colloidal self-assembly near a solid surface

HypothesisSelf-assembled colloidal mobility out of a non-equilibrium system can depend on many external and interparticle forces including hydrodynamic forces. While the driving forces guiding colloidal suspension, translation and self-assembly are different and unique, hydrodynamic forces are always present and can significantly influence particle motion. Unfortunately, these interparticle hydrodynamic interactions are typically overlooked. ExperimentsHere, we studied the collective behavior of colloidal particles (4.0 µm PMMA), located near the solid surface in a fluid medium confined in a cylindrical cell (3.0 mm diameter, 0.25 mm height) which was rotated vertically at a low rotational speed (20 rpm). The observed colloidal behavior was then validated through a Stokesian dynamics simulation where the concept of hydrodynamic contact force or lubrication interactions are avoided which is not physically intuitive and mathematically cumbersome. Rather, we adopted hard-sphere like colloidal collision or mobility model, while adopting other useful simplification and approximations. FindingsUpon particles settling in a circular orbit, they hydrodynamically interact with each other and evolve in different structures depending on the pattern of gravity forces. Their agglomeration is a function of the applied rotation scheme, either forming colloidal clusters or lanes. While evolving into dynamic structures, colloids also laterally migrate away from the surface.

Characterization of storage modulus of starch suspensions during the initial stages of pasting using Stokesian dynamics simulations

This study performs Stokesian dynamics simulations of suspensions of rigid spheres to determine the conditions under which swollen starch granules can be considered rigid for rheology measurements. The first part of the study benchmarks the numerical simulations to experimental measurements of storage modulus (G′) for monodisperse polystyrene suspensions. The second part of the study compares the simulations to experimental measurements of G′ for waxy maize and normal rice starch suspensions swollen to volume fractions 0.4 < φ < 0.6. These starch granules were fractionated before swelling to create a narrow range of swollen granule sizes. For granules with a narrow size distribution, the Stokesian dynamics simulations quantitatively match the experimental G′ for volume fractions φ ≤ 0.5, but deviate significantly when φ > 0.5. These results indicate that granule deformability plays an important role when φ > 0.5, and indicates the range where one can apply theories of rigid suspension rheology.

Nanoparticle dispersion in porous media: Effects of array geometry and flow orientation.

We investigate the effects of array geometry and flow orientation on transport of finite-sized particles in ordered arrays using Stokesian dynamics simulations. We find that quiescent diffusion is independent of array geometry over the range of volume fraction of the nanoposts examined. Longitudinal dispersion under flow depends on the direction of incident flow relative to the array lattice vectors. Taylor-Aris behavior is recovered for flow along the lattice directions, whereas a nonmonotonic dependence of the dispersion coefficient on the Péclet number is obtained for flow orientations slightly perturbed from certain lattice vectors, owing to a competition between directional locking and spatial velocity variations.

Stokesian dynamics simulations of a magnetotactic bacterium

The swimming of bacteria provides insight into propulsion and steering under the conditions of low-Reynolds number hydrodynamics. Here we address the magnetically steered swimming of magnetotactic bacteria. We use Stokesian dynamics simulations to study the swimming of single-flagellated magnetotactic bacteria (MTB) in an external magnetic field. Our model MTB consists of a spherical cell body equipped with a magnetic dipole moment and a helical flagellum rotated by a rotary motor. The elasticity of the flagellum as well as magnetic and hydrodynamic interactions is taken into account in this model. We characterized how the swimming velocity is dependent on parameters of the model. We then studied the U-turn motion after a field reversal and found two regimes for weak and strong fields and, correspondingly, two characteristic time scales. In the two regimes, the U-turn time is dominated by the turning of the cell body and its magnetic moment or the turning of the flagellum, respectively. In the regime for weak fields, where turning is dominated by the magnetic relaxation, the U-turn time is approximately in agreement with a theoretical model based on torque balance. In the strong-field regime, strong deformations of the flagellum are observed. We further simulated the swimming of a bacterium with a magnetic moment that is inclined relative to the flagellar axis. This scenario leads to intriguing double helical trajectories that we characterize as functions of the magnetic moment inclination and the magnetic field. For small inclination angles (lesssim {20^{circ }}) and typical field strengths, the inclination of the magnetic moment has only a minor effect on the swimming of MTB in an external magnetic field. Large inclination angles result in a strong reduction in the velocity in direction of the magnetic field, consistent with recent observations that bacteria with large inclination angles use a different propulsion mechanism.Graphic abstract

Predictions of microstructure and stress in planar extensional flows of a dense viscous suspension

Abstract

Nanoparticle dispersion in porous media: Effects of hydrodynamic interactions and dimensionality

AbstractWe investigate the effect of steric and hydrodynamic interactions (HI) on quiescent diffusion and flow‐driven transport of finite‐sized nanoparticles through periodic 2D (two‐dimensional) and 3D (three‐dimensional) nanopost arrays using Stokesian dynamics simulations. We find that steric and HI hinder particle diffusivity under quiescent conditions and enhance longitudinal dispersion under flow. Moreover, the presence of HI leads to a power‐law increase in the longitudinal dispersion coefficient with Pe due to spatial variations in the fluid velocity. Lastly, our simulations reveal that longitudinal particle dispersion coefficients behave similarly in 2D and 3D when HI are included.

Dynamical modes of sheared confined microscale matter.

Based on (overdamped) Stokesian dynamics simulations and video microscopy experiments, we study the non equilibrium dynamics of a sheared colloidal cluster, which is confined to a two-dimensional disk. The experimental system is composed of a mixture of paramagnetic and non magnetic polystyrene particles, which are held in the disk by time shared optical tweezers. The paramagnetic particles are located at the center of the disk and are actuated by an external, rotating magnetic field that induces a magnetic torque. We identify two different steady states by monitoring the mean angular velocities per ring. The first one is characterized by rare slip events, where the inner rings momentarily depin from the outer ring, which is kept static by the set of optical traps. For the second state, we find a bistability of the mean angular velocities, which can be understood from the analysis of the slip events in the particle trajectories. We calculate the particle waiting- and jumping time distributions and estimate a time scale between slips, which is also reflected by a plateau in the mean squared azimuthal displacement. The dynamical transition is further reflected by the components of the stress tensor, revealing a shear-thinning behavior as well as shear stress overshoots. Finally, we briefly discuss the observed transition in the context of stochastic thermodynamics and how it may open future directions in this field.

Stokesian Dynamics Simulation of Suspension Flow in Porous Media

Suspension flow through porous medium was studied using the Stokesian dynamics simulation method. Stokesian dynamics is an efficient tool to carry out numerical simulations for suspension of rigid particles interacting through hydrodynamic and non-hydrodynamic forces. After validating the simulation method for a single particle flowing through an array of fixed grain particles, we have analysed the suspension transport through porous medium. It was observed that the hydrodynamic interactions and the inter-particle non-hydrodynamic forces between the moving and fixed grain particles have a strong influence on the particle trajectories. This was apparent from the change in particle flux with the fractional channel width in the presence of non-hydrodynamic forces. Hydrodynamic interaction between the suspension and grain particles was also studied for large-scale porous system that was generated by a random arrangement of the particles in a periodic cell. It was found that the change in porosity leads to change in the average fluctuation velocity of the suspension. The fluctuation velocity was observed to vary linearly with the particle concentration and average suspension velocity. Finally, a comparative study was performed with suspension flow in a straight channel and it was observed that the shear-induced particle migration in porous medium is altered by the presence of grain particles.

Dynamics of groups of magnetically driven artificial microswimmers.

Magnetically driven artificial microswimmers have the potential to revolutionize many biomedical technologies, such as minimally invasive microsurgery, microparticle manipulation, and localized drug delivery. However, many of these applications will require the controlled dynamics of teams of these microrobots with minimal feedback. In this work, we study the motion and fluid dynamics produced by groups of artificial microswimmers driven by a torque induced through a uniform, rotating magnetic field. Through Stokesian dynamics simulations, we show that the swimmer motion produces a rotational velocity field in the plane orthogonal to the direction of the magnetic field's rotation, which causes two interacting swimmers to move in circular trajectories in this plane around a common center. The resulting overall motion is on a helical trajectory for the swimmers. We compare the highly rotational velocity field of the fluid to the velocity field generated by a rotlet, the point-torque singularity of Stokes flows, showing that this is a reasonable approximation on the time average. Finally, we study the motion of larger groups of swimmers, and we show that these groups tend to move coherently, especially when swimmer magnetizations are uniform. This coherence is achieved because the group center remains almost constant in the plane orthogonal to the net motion of the swimmers. The results in the paper will prove useful for controlling the ensemble dynamics of small collections of magnetic swimmers.

Numerical simulation of particle migration in suspension flow through heterogeneous porous media

Particle migration in porous media is relevant to many applications in chemical and petroleum engineering. The hydrodynamic interactions between the suspension and grain particles affect the distribution and transport of particles during suspension flow through porous media. Continuum models based on Darcy’s law are widely used to investigate the macroscopic properties of the system. However, these models are unable to capture the pore scale dynamics and microstructure. We have carried out particle migration studies during transport of suspension in heterogeneous porous media using the Stokesian dynamics simulations. The velocity and concentration distributions were first studied in a large scale constant porosity system, but with different concentration of particles in the two adjacent porous layers. It was found that particle migration across the interface is higher when the difference of concentration is large. Simulations were also carried out for constant areal concentration of particles, where the adjacent layers had different porosity. A high rate of migration near the interface was observed for the larger difference in porosity. It was observed that the hydrodynamic interactions are strongly affected by the increase in the number of grain particles of porous media in comparison to that of suspended particles.

Focusing and splitting streams of soft particles in microflows via viscosity gradients.

Microflows are intensively used for investigating and controlling the dynamics of particles, including soft particles such as biological cells and capsules. A classic result is the tank-treading motion of elliptically deformed soft particles in linear shear flows, which do not migrate across straight streamlines in the bulk. However, soft particles migrate across straight streamlines in Poiseuille flows. In this work we describe a new mechanism of cross-streamline migration by using soft capsules with a spherical equilibrium shape. If the viscosity varies perpendicular to the streamlines then the soft particles migrate across streamlines towards regions of a lower viscosity, even in linear shear flows. An interplay with the repulsive particle-boundary interaction causes then focusing of particles in linear shear flows with the attractor streamline closer to the wall in the low viscosity region. Viscosity variations perpendicular to the streamlines in Poiseuille flows leads either to a shift of the particle attractor or even to a splitting of particle attractors, which may give rise to interesting applications for particle separation. The location of attracting streamlines depend on the particle properties, like their size and elasticity. The cross-stream migration induced by viscosity variations is explained by analytical considerations, Stokesian dynamics simulations with a generalized Oseen tensor and lattice-Boltzmann simulations.

Emerging Attractor in Wavy Poiseuille Flows Triggers Sorting of Biological Cells.

Microflows constitute an important instrument to control particle dynamics. A prominent example is the sorting of biological cells, which relies on the ability of deformable cells to move transversely to flow lines. A classic result is that soft microparticles migrate in flows through straight microchannels to an attractor at their center. Here, we show that flows through wavy channels fundamentally change the overall picture. They lead to the emergence of a second, coexisting attractor for soft particles. Its emergence and off-center location depends on the boundary modulation and the particle properties. The related cross-stream migration of soft particles is explained by analytical considerations, Stokesian dynamics simulations in unbounded flows, and Lattice-Boltzmann simulations in bounded flows. The novel off-center attractor can be used, for instance, in diagnostics, for separating cells of different size and elasticity, which is often an indicator of their health status.

Protein Short-Time Diffusion in a Naturally Crowded Environment.

The interior of living cells is a dense and polydisperse suspension of macromolecules. Such a complex system challenges an understanding in terms of colloidal suspensions. As a fundamental test we employ neutron spectroscopy to measure the diffusion of tracer proteins (immunoglobulins) in a cell-like environment (cell lysate) with explicit control over crowding conditions. In combination with Stokesian dynamics simulation, we address protein diffusion on nanosecond time scales where hydrodynamic interactions dominate over negligible protein collisions. We successfully link the experimental results on these complex, flexible molecules with coarse-grained simulations providing a consistent understanding by colloid theories. Both experiments and simulations show that tracers in polydisperse solutions close to the effective particle radius Reff = ⟨ Ri3⟩1/3 diffuse approximately as if the suspension was monodisperse. The simulations further show that macromolecules of sizes R > Reff ( R < Reff) are slowed more (less) effectively even at nanosecond time scales, which is highly relevant for a quantitative understanding of cellular processes.