Drag and Torque on Clusters of N Arbitrary Spheres at Low Reynolds Number

Hydrodynamics of particle clusters suspended in viscous fluids is a subject of considerable theoretical and practical importance. Using a multipole expansion of the flow velocity in a series of spherical harmonics, Lamb's fundamental solution of the Stokes flow outside a single sphere is generalized in this work to the case of N nonoverlapping spheres of arbitrary size with slip boundary conditions. The expansion coefficients are found by transforming the boundary conditions to the Lamb form and by transforming the spherical coordinates and solid spherical harmonics centered at different spheres. The problem is reduced to the solution of the linear system of equations for the expansion coefficients, which is carried out numerically. Based on the developed theory, the relation between the hydrodynamic and gyration radius of fractal-like aggregates with different structure is established. In another application, an asymptotic slip-regime dependence of the aggregate hydrodynamic radius on the Knudsen number and the number of particles is found by performing calculations of drag forces acting on the gas-borne fractal-like and straight chain aggregates. A good agreement is shown in comparing predictions of the described theory with available experimental and theoretical results on motion of various small sphere clusters in viscous fluid. Copyright 2000 Academic Press.     

The diffusion coefficient of a swollen microgel particle

The drag on a permeable particle traversing through a Newtonian liquid is calculated. This is in terms of a single dimensionless group, Da, the Darcy number, which relates the particle permeability to the radius. For small values of the Darcy number the solution reverts to the well-known Stokes drag for smooth hard particles. For larger values of the Darcy number the drag is reduced. This drag allows calculation of the diffusion coefficient of such particles, again as a function of the Darcy number. The results are discussed in relation to microgel particles, which display swelling under certain conditions. The size of such particles is typically measured with dynamic light scattering, which measures the diffusion coefficient of particles and as such the analysis presented here shows the conditions under which dynamic light scattering is accurate. The Darcy number for microgel particles is estimated to be on the order of 10(-7).     

Deposition efficiency of fractal-like aggregates in fibrous filters calculated using Brownian dynamics method

Nonspherical particles, such as fractal-like aggregates emitted by diesel engines, are commonly met in the ambient air. Some of them are believed to be carcinogenic to humans, thus their efficient removal is of crucial practical importance. A fibrous filter is the device commonly used for aerosol purification but the literature lacks experimental data concerning aggregates filtration. Effect of aggregates' parameters (fractal dimension, primary particle radius) as well as fiber diameter and air velocity on the filtration efficiency is investigated theoretically using the modified Brownian dynamics method. Three different expressions for the friction coefficient evaluation for the aggregates were examined. The results obtained indicate that structure of an aggregate, filter structure and process conditions strongly influence the aggregates deposition efficiency, which significantly differs from the values determined for mass-equivalent spherical particles. The results determined using the Brownian dynamics approach were compared with the values calculated using classical single fiber theory and noticeable discrepancy was observed for the most penetrating particles, while both approaches agree for the limiting cases of small or large particles. Peclet number based on the mobility radius and the interception parameter based on the outer radius are the proper criteria to describe diffusional and deterministic deposition of aggregates.     

The drag of the row of parallel chains of spherical particles in the Stokes flow

The drag of thin-layered porous deposit consisting of dendrites of identical spherical particles with respect to the flow of viscous incompressible liquid is calculated. The deposit is approximated by a model system, a row of parallel chains of particles oriented perpendicular to a flow direction. The expression is derived for the dimensionless drag force acting on the unit chain length as a function of the ratio of a particle radius to a half-distance between chain axes, a/h. It is shown that, at a/h < 0.5, the hydrodynamic equivalent of the chains is the smooth cylinder whose radius is 1.16 times smaller than the particle radius that agrees with the experiment. It is also shown that, at a/h = 1, the drag force of a particle contacting with four adjacent particles in the layer with square packing is equal to F = 44F _St, where F _St is the Stokes drag force of a spherical particle. The pressure drop in this single layer is by 3.5% higher than in the layer of spherical particles with cubic packing. At a/h = 2/√3, drag force F of the particle contacting with six particles in a single layer with hexagonal packing is equal to 340F _St.     

Effect of small particles on the near-wall dynamics of a large particle in a highly bidisperse colloidal solution

We consider the hydrodynamic effect of small particles on the dynamics of a much larger particle moving normal to a planar wall in a highly bidisperse dilute colloidal suspension of spheres. The gap h(0) between the large particle and the wall is assumed to be comparable to the diameter 2a of the smaller particles so there is a length-scale separation between the gap width h(0) and the radius of the large particle b>h(0). We use this length-scale separation to develop a new lubrication theory which takes into account the presence of the smaller particles in the space between the larger particle and the wall. The hydrodynamic effect of the small particles on the motion of the large particle is characterized by the short time (or high frequency) resistance coefficient. We find that for small particle-wall separations h(0), the resistance coefficient tends to the asymptotic value corresponding to the large particle moving in a clear suspending fluid. For h(0)>a, the resistance coefficient approaches the lubrication value corresponding to a particle moving in a fluid with the effective viscosity given by the Einstein formula.     

Stability of 2-D colloidal particle aggregates held against flow stress in an ultrasound trap

The formation of a two-dimensional aggregate of 25 microm latex particles in a 1.5 MHz ultrasound standing wave (USW) field and its disintegration in a flow were studied. The aggregate was held in the pressure node plane, which allowed continuous microscope observation and video recording of the processes. The trajectories and velocities of the particles approaching the formation site were analyzed by particle image velocimetry (PIV). Since the direct radiation force on the particles dominated the drag due to acoustic streaming, the acoustic pressure profile in the vicinity of the aggregate was quantifiable. The drag coefficients D(coef) for 2- to 485-particle aggregates were estimated from the balance of the drag force FD and the buoyancy-corrected gravitational force during sedimentation on termination of the ultrasound when the long axis of the aggregate was in the vertical plane. D(coef) were calculated from FD as proportional to the aggregate velocity. Experiments on particle detachment by flow (in-plane velocity measured by PIV) from horizontal aggregates suspended in deionized water and CaCl2 solution of different concentrations showed that the mechanical strength of the aggregates depended on the acoustic pressure amplitude P0 and ionic strength of the solution. In deionized water the flow velocity required to detach the first single particle from an aggregate increased from 1 mm s-1 at P0 = 0.6 MPa to 4.2 mm s-1 at P0 = 1.4 MPa. The balance of forces acting on particles in a USW trap is discussed. The magnitude of the shear stress employed ( approximately 0.05 Pa) and separation forces suggests that this technique can be applied to studying the mechanical responses of cell aggregates to hydrodynamic flow, where cell-cell interaction can be separated from the effects of solid substrata.     

Hydrodynamic permeability of a membrane composed of porous spherical particles in the presence of uniform magnetic field

This work concerns the flow of an incompressible viscous fluid past a porous sphere in presence of transverse applied uniform magnetic field, using particle-in-cell method. The Brinkman equations are used in porous region and the Stokes equations for non-porous region. At the fluid-porous interface, the stress jump boundary condition for tangential stresses along with continuity of normal stress and velocity components are used. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). The hydrodynamic drag force experienced by a porous spherical particle in a cell and hydrodynamic permeability of membrane built up by porous spherical particles are evaluated. The patterns of streamlines are also obtained and discussed. The effect of stress jump coefficient, Hartmann number, dimensionless specific permeability of the porous particle and particle volume fraction on the hydrodynamic permeability and streamlines are discussed. Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified.     

Diffusion, sedimentation, and rheology of concentrated suspensions of core-shell particles

Short-time dynamic properties of concentrated suspensions of colloidal core-shell particles are studied using a precise force multipole method which accounts for many-particle hydrodynamic interactions. A core-shell particle is composed of a rigid, spherical dry core of radius a surrounded by a uniformly permeable shell of outer radius b and hydrodynamic penetration depth κ(-1). The solvent flow inside the permeable shell is described by the Brinkman-Debye-Bueche equation, and outside the particles by the Stokes equation. The particles are assumed to interact non-hydrodynamically by a hard-sphere no-overlap potential of radius b. Numerical results are presented for the high-frequency shear viscosity, η(∞), sedimentation coefficient, K, and the short-time translational and rotational self-diffusion coefficients, D(t) and D(r). The simulation results cover the full three-parametric fluid-phase space of the composite particle model, with the volume fraction extending up to 0.45, and the whole range of values for κb, and a/b. Many-particle hydrodynamic interaction effects on the transport properties are explored, and the hydrodynamic influence of the core in concentrated systems is discussed. Our simulation results show that for thin or hardly permeable shells, the core-shell systems can be approximated neither by no-shell nor by no-core models. However, one of our findings is that for κ(b - a) ? 5, the core is practically not sensed any more by the weakly penetrating fluid. This result is explained using an asymptotic analysis of the scattering coefficients entering into the multipole method of solving the Stokes equations. We show that in most cases, the influence of the core grows only weakly with increasing concentration.     

Deposition of spherical particles onto cylindrical solid surfaces. I. Numerical simulations

The permeability of fractal porous aggregates with realistic three-dimensional structure is investigated theoretically using model aggregates composed of identical spherical primary particles. Synthetic aggregates are generated by several techniques, including a lattice-based method, simulation of aggregation by differential settling and turbulent shear, and the specification of simple cubic structures, resulting in aggregates characterized by the number of primary particles, solid fraction, characteristic radius, and fractal dimension. Stokesian dynamics is used to determine the total hydrodynamic force on and the distribution of velocity within an aggregate exposed to a uniform flow. The aggregate permeability is calculated by comparing these values with the total force and velocity distribution calculated from the Brinkman equation applied locally and to the entire aggregate using permeability expressions from the literature. The relationship between the aggregate permeability and solid fraction is found to be best predicted by permeability expressions based on cylindrical rather than spherical geometrical elements, the latter tending to underestimate the aggregate permeability significantly. The permeability expressions of Jackson and James or Davies provide good estimates of the force on and flow through porous aggregates of known structure. These relationships are used to identify a number of general characteristics of fractal aggregates.     

Precise multipole method for calculating many-body hydrodynamic interactions in a microchannel

We introduce a novel and precise method for computing many-body hydrodynamic interactions in a cylindrical microchannel. The method is generic in the sense that we can easily change the radius and the character of particles (hard spheres, droplets, permeable spheres, etc.). These features are not available in any of the existing methods. Comparison with the available results validates our method. In particular we obtain excellent agreement with the analytically known expression for the single particle friction coefficient. Additionally we observe negative hydrodynamic coupling for finite particles which are consistent with the recently reported effect for point particles. As an example we compute the velocities of polymeric chains of particles in parabolic flow and compare them to unbounded space. The method will be helpful in the understanding of physical and physicochemical processes in a wide range of bio-, geophysical, and microfluidic systems.     

Anisotropic and hindered diffusion of colloidal particles in a closed cylinder

Video microscopy and particle tracking were used to measure the spatial dependence of the diffusion coefficient (D(α)) of colloidal particles in a closed cylindrical cavity. Both the height and radius of the cylinder were equal to 9.0 particle diameters. The number of trapped particles was varied between 1 and 16, which produced similar results. In the center of the cavity, D(α) turned out to be 0.75D(0) measured in bulk liquid. On approaching the cylindrical wall, a transition region of about 3 particle diameters wide was found in which the radial and azimuthal components of D(α) decrease to respective values of 0.1D(0) and 0.4D(0), indicating asymmetrical diffusion. Hydrodynamic simulations of local drag coefficients for hard spheres produced very good agreement with experimental results. These findings indicate that the hydrodynamic particle-wall interactions are dominant and that the complete 3D geometry of the confinement needs to be taken into account to predict the spatial dependence of diffusion accurately.     

Study of colloidal particle Brownian aggregation by low-coherence fiber optic dynamic light scattering

The aggregation kinetics of particles in dense polystyrene latex suspensions is studied by low-coherence fiber optic dynamic light scattering. Low-coherence fiber optic dynamic light scattering is used to measure the hydrodynamic radius of the aggregates. The aggregation kinetics data obtained can be fitted into a single exponential function, which is the characteristic of slow aggregation. It is found that the aggregation rate of particles increased with higher electrolyte levels and with larger particle concentrations. The experimental results can be explained by use of the Derjaruin-Landau-Verwey-Overbeer (DLVO) theory.     

Characterization of globular protein solutions by dynamic light scattering, electrophoretic mobility, and viscosity measurements

In this work, physicochemical properties of two globular proteinsbovine serum albumin (BSA) having a molecular weight of 67 kDa and human serum albumin (HSA) having a molecular weight of 69 kDawere characterized. The bulk characteristics of these proteins involved the diffusion coefficient (hydrodynamic radius), electrophoretic mobility, and dynamic viscosity as a function of protein solution concentration for various pH values. The hydrodynamic radius data suggested an association of protein molecules, most probably forming compact dimers. Using the hydrodynamic diameter and the electropheretic mobility data allowed the determination of the number of uncompensated (electrokinetic) charges on protein surfaces. The electrophoretic mobility data were converted to zeta potential values, which allowed one to determine the isoelectric point (iep) of these proteins. It was found to be at pH 5.1 for both proteins, in accordance with previous experimental data and theoretical estimations derived from amino acid composition and p K values. To determine further the stability of protein solutions, dynamic viscosity measurements were carried out as a function of their bulk volume concentration for various pH values. The intrinsic viscosity derived from these measurements was interpreted in terms of the Brenner model, which is applicable to hard spheroidal particles. It was found that the experimental values of the intrinsic viscosity of these proteins were in good agreement with this model when assuming protein dimensions of 9.5 x 5 x 5 nm3 (prolate spheroid). The possibility of forming linear aggregates of association degree higher than 2 was excluded by these measurements. It was concluded that the combination of dynamic viscosity and dynamic light scattering can be exploited as a convenient tool for detecting not only the onset of protein aggregation in suspensions but also the form and composition of these aggregates.     

Polarizability and complex conductivity of dilute suspensions of spherical colloidal particles with uncharged (neutral) polymer coatings

The polarizability of polymer-coated colloidal particles, as measured via dielectric relaxation spectroscopy, reflects on the degree to which convection, diffusion, and electromigration deform the equilibrium double layer. With a polymer coating, convection and electro-osmosis are resisted by hydrodynamic drag on the polymer segments. The electro-osmotic flow near the underlying bare surface is therefore diminished. Characteristics of the particles and the adsorbed polymer can, in principle, be inferred by measuring the frequency-dependent polarizability. In this work, "exact" numerical solutions of the electrokinetic equations are used to examine how adsorbed polymer changes the particle polarizability and, hence, the conductivity and dielectric constant increments of dilute suspensions. For neutral polymer coatings, the conductivity and dielectric constant increments are found to be very similar to those of the underlying bare particles, so the response depends mostly on the underlying bare particles. These observations suggest that dielectric spectroscopy is best used to determine the underlying surface charge, with characteristics of the coating inferred from the electrophoretic or dynamic mobility, together with the hydrodynamic radius obtained from sedimentation or dynamic light scattering. Addressed briefly are the effects of added counterions and nonspecific adsorption. The electrokinetic model explored in this work can be used to guide experiments (frequency and ionic strength, for example) to either minimize or maximize the sensitivity of the complex conductivity to the coating thickness or permeability.     

Hydrodynamics of an ideal aggregate with quadratically increasing permeability

In this study, we consider the ideal aggregate with quadratically increasing permeability kappa = k2r2 and derive the analytical expression of the stream function within the porous aggregate by incorporating the Brinkman and continuity equations. The hydrodynamic properties of the aggregate are investigated by taking account of the hydrodynamic radius, settling velocity, and fluid collection efficiency, which are found to be solely dependent on the permeability prefactor k2. The fractal dimension Df and prefactor k2 of the ideal aggregate are found to be 5/3 (=1.67) and 0.20, respectively, and well describe the hydrodynamics of aggregates formed in the diffusion-limited-cluster-aggregation (DLCA) regime. More important, hydrodynamic similarity between the ideal aggregate and impermeable solid sphere is discovered in terms of variations of the hydrodynamic radius, settling velocity, and fluid collection efficiency with respect to the aggregate radius.     

Effect of electrostatic, hydrodynamic, and Brownian forces on particle trajectories and sieving in normal flow filtration

Particle deposition and fouling are critical factors governing the performance of microfiltration and ultrafiltration systems. Particle trajectories were evaluated by numerical integration of the Langevin equation, accounting for the combined effects of electrostatic repulsion, enhanced hydrodynamic drag, and Brownian diffusion. In the absence of Brownian forces, particles are unable to enter the membrane pores unless the drag associated with the filtration velocity can overcome the electrostatic repulsion. Brownian forces significantly alter this behavior, allowing some particles to enter the pore even at low filtration velocities. The average particle transmission, evaluated from the probability of having a particle enter the pore, increases with increasing filtration velocity due to the greater hydrodynamic drag force on the particle. These results provide important insights into particle behavior in membrane systems.     

On hydrodynamic permeability of a membrane built up by porous deformed spheroidal particles

This paper concerns the slow viscous flow of an incompressible fluid past a swarm of identically oriented porous deformed spheroidal particles, using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid region in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in a small parameter characterizing the deformation. The flow through the porous oblate spheroid is considered as the particular case of the porous deformed spheroid. The hydrodynamic drag force experienced by a porous oblate spheroid and permeability of a membrane built up by porous oblate spheroids having parallel axis are evaluated. The dependence of the hydrodynamic drag force and the hydrodynamic permeability on particle volume fraction, deformation parameter and viscosity of porous fluid are also discussed. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and hydrodynamic permeability have been verified. The model suggested can be used for evaluation of changing hydrodynamic permeability of a membrane under applying unidirectional loading in pressure-driven processes (reverse osmosis, nano-, ultra- and microfiltration).     

Effect of NaCl on the self-aggregation of n-octyl beta-D-thioglucopyranoside in aqueous medium

This report investigates the effect of sodium chloride (NaCl) on the micellization, surface activity, and the evolution in the shape and size of n-octyl beta-D-thioglucopyranoside (OTG) aggregates. By using surface tension measurements, information was obtained on both changes in the critical micelle concentration and adsorption behavior in the air-liquid interface with the electrolyte concentration. These data were used to obtain the thermodynamic properties of micellization along with the corresponding adsorption parameters in the air-liquid interface. From extended static and dynamic light scattering measurements, the micelle molecular weight, the mean aggregation number, and the second virial coefficient, the apparent diffusion coefficient and the mean hydrodynamic radius of micelles in a range of NaCl concentrations were obtained. The light scattering data have shown that when the surfactant concentration is lower to 4.5 g/L, only spherical micelles are formed. However, an increase in the surfactant concentration induces an increase in micellar size, suggesting a rodlike growth of the micelles. This deviation of micelle geometry from spherical to rodlike is supported both by the ratio between the hydrodynamic radius and the radius of gyration and by the angular dependence of light scattering. On the other hand, the studies performed in the presence of high NaCl concentration (0.2 and 0.5 M) provide strong support for the view that the micelles may overlap together to form an entangled network above certain crossover concentration.     

Dynamic drag force based on iterative density mapping: A new numerical tool for three‐dimensional analysis of particle trajectories in a dielectrophoretic system

Dielectrophoresis is a widely used means of manipulating suspended particles within microfluidic systems. In order to efficiently design such systems for a desired application, various numerical methods exist that enable particle trajectory plotting in two or three dimensions based on the interplay of hydrodynamic and dielectrophoretic forces. While various models are described in the literature, few are capable of modeling interactions between particles as well as their surrounding environment as these interactions are complex, multifaceted, and computationally expensive to the point of being prohibitive when considering a large number of particles. In this paper, we present a numerical model designed to enable spatial analysis of the physical effects exerted upon particles within microfluidic systems employing dielectrophoresis. The model presents a means of approximating the effects of the presence of large numbers of particles through dynamically adjusting hydrodynamic drag force based on particle density, thereby introducing a measure of emulated particle–particle and particle–liquid interactions. This model is referred to as “dynamic drag force based on iterative density mapping.” The resultant numerical model is used to simulate and predict particle trajectory and velocity profiles within a microfluidic system incorporating curved dielectrophoretic microelectrodes. The simulated data are compared favorably with experimental data gathered using microparticle image velocimetry, and is contrasted against simulated data generated using traditional “effective moment Stokes‐drag method,” showing more accurate particle velocity profiles for areas of high particle density.     

Structure and dynamic properties of colloidal asphaltene aggregates

The abundant literature involving asphaltene often contrasts dynamic measurements of asphaltene solutions, highlighting the presence of small particle sizes between 1 and 3 nm, with static scattering measurements, revealing larger aggregates with a radius of gyration around 7 nm. This work demonstrates the complementary use of the two techniques: a homemade dynamic light scattering setup adapted to dark and fluorescent solutions, and small-angle X-ray and neutron scattering. Asphaltene solutions in toluene are prepared by a centrifugation separation to investigate asphaltene polydispersity. These experiments demonstrate that asphaltene solutions are made of Brownian colloidal aggregates. The hydrodynamic radii of asphaltene aggregates are between 5 and 10 nm, while their radii of gyration are roughly comparable, between 3.7 and 7.7 nm. A small fraction of asphaltenes with hydrodynamic and gyration radii around 40 nm is found in the pellet of the centrifugation tube. The fractal character of the largest clusters is observed from small angle scattering nearly on a decade length scale. Previous results on aggregation mechanisms are confirmed ( Eyssautier, J., et al. J. Phys. Chem. B 2011 , 115 , 6827 ): nanoaggregates of 3 nm radius, and with hydrodynamic properties also frequently illustrated in the literature, aggregate to form fractal clusters with a dispersity of aggregation number.