On the influence of the hydrodynamic interactions on the aggregation rate of magnetic spheres in a dilute suspension

doi:10.1016/j.jmmm.2010.08.033

Journal of Magnetism and Magnetic Materials

Volume 323, Issue 1, January 2011, Pages 77-82

https://doi.org/10.1016/j.jmmm.2010.08.033 Get rights and content

Abstract

Magnetostatic attraction may lead to formation of aggregates in stable colloidal magnetic suspensions and magneto-rheological suspensions. The aggregation problem of magnetic composites under differential sedimentation is a key problem in the control of the instability of non-Brownian suspensions. Against these attractive forces are the electrostatic repulsion and the hydrodynamic interactions acting as stabilizing effects to the suspension. This work concerns an investigation of the pairwise interaction of magnetic particles in a dilute sedimenting suspension. We focus attention on suspensions where the Péclet number is large (negligible Brownian motion) and where the Reynolds number (negligible inertia) is small. The suspension is composed of magnetic micro-spheres of different radius and density immersed in a Newtonian fluid moving under the action of gravity. The theoretical calculations are based on direct computations of the hydrodynamic and the magnetic interactions among the rigid spheres in the regime of low particle Reynolds number. From the limiting trajectory in which aggregation occurs, we calculate the collision efficiency, representing the dimensionless rate at which aggregates are formed. The numerical results show clear evidence that the hydrodynamic interactions are of fundamental relevance in the process of magnetic particle aggregation. We compare the stabilizing effects between electrostatic repulsion and hydrodynamic interactions.

Introduction

Technological control of the stability against aggregate formation in colloidal magnetic suspensions and rheological magnetic suspensions (MRS) is important in a number of industrial processes. Aggregation may be induced to increase the settling rate, modifying the rheological properties of suspensions. In the treatment of waste water, suspended material must be separated so that the water can be recycled or so that the pollution of streams and rivers is reduced [1]. The phenomena examined for a gravitationally driven motion of magnetic particles made of homogeneous suspensions, a complex shape evolution, including aggregates formation, are fundamental to the mechanics of multiphase flows, and have direct and indirect applications in many chemical process such as instability of MRS. For colloidal suspensions, particles must first be brought close together by Brownian motion. They then experience an attractive force which may be sufficiently strong to overcome any repulsive forces and the fluid—mechanical lubrication to relative motion. At a critical size, however, their sedimentation speed becomes significant and thereafter the aggregates grow by the larger and faster aggregates sweeping up the smaller ones. So, even in an unstable colloidal magnetic suspension, after a critical size of the micro-aggregates, differential sedimentation becomes the key mechanism to the particle aggregate rather than Brownian motion. Fig. 1 shows clearly the aggregates formation in a suspension of micro-spherical magnetic composites of about $100 μ m$ . Owing to particle polydispersity differential sedimentation might bring (in the absence of Brownian motion) particles close together and attractive magnetic force between the particles lead to particle aggregate. The near field hydrodynamic interactions (lubrication regime between the composites) on the other hand appear to decrease the flocculation efficiency. This new effect is explored in this paper. The motivation is to understand how can a MRS become unstable. Aggregates form and they precipitate as shown in the experiments carried out in this work about sedimentation of magnetic composites, see Fig. 1, in which we can see typical aggregates that form in a dilute suspension of magnetic non-Brownian particles, initially statistically homogeneous.

The central issue in these systems is to understand and to predict the macroscopic behavior of such suspension from their microstructure. When particle aggregation occurs in a dilute suspension, the quantity of interest is the rate of doublet formation. The pioneer studies to estimate the rate of aggregation [2] considered that the particles are submitted only to a sticking force on contact, i.e. without any hydrodynamic interaction (HI) or interparticle forces. After this, the effects of hydrodynamic interaction and interparticle force were included [3] to compute the rate of aggregation of a dispersion of equal spheres subject to simple shear and to uniaxial extensional flow. For gravity-induced aggregation of rigid spheres, Davis [4] developed a theoretical model to investigate the influence of van der Waals and Maxwell slip on the collision efficiency for rapid aggregation regimes. Moreover, the collision rates of spherical and deformable drops in a shear flow were calculated [5], [6], [7]. While the subject of particle aggregation is generally well known, it must be appreciated that the present study is the first one based on a hydrodynamically interacting magnetic suspensions of non-Brownian particles. Several works have examined by numerical simulations the aggregation process of magnetic spheres in suspensions studying the formation of clusters and chain-like structures in the suspension [8], [9], [10]. Surprisingly, they have neglected the influence of the hydrodynamic interactions in their simulations considering free-energy models and Monte Carlo simulations only. In a study related to the present one [11], [12] included the effects of hydrodynamic interaction and interparticle force, and computed the relative trajectories and the hydrodynamic diffusivities for two spherical particles interacting in a simple shearing flow and in sedimentation, respectively.

The interest here is to examine the influence of the hydrodynamic interaction on the dynamics of aggregate formation in a dilute magnetic suspension of non-Brownian micro-particles. From the computation of the relative trajectories of pairwise interactions between magnetic spherical composites at low particle Reynolds number we investigate the influence of particle magnetization on the rate of particle aggregation for different conditions of the physical parameter which considers the relative importance between interparticle and gravity forces. We calculate theoretically the rate at which aggregates are formed (collision efficiency) in a dilute suspension of magnetic composites.

Section snippets

Formulation of the problem

Consider a dilute suspension composed of two species of rigid smooth magnetic spheres of radius a₁ and a₂, densities $ρ_{1}$ and $ρ_{2}$ and magnetizations $M_{1} = M_{1} {\hat{d}}_{1}$ and $M_{2} = M_{2} {\hat{d}}_{2}$ suspended in a Newtonian fluid of density $ρ$ and viscosity $μ$ . Here, the inertial effects are neglected, as we have small particle Reynolds numbers, so that the creeping flow equations can be applied in the scale of the particle motion. The suspension is submitted to a settling motion so that the uniform gravitational force per

Trajectory equations

Making the balance between the hydrodynamic force and the applied force $F_{i}$ , one obtains the governing equation for the relative motion, i.e. the vector $s$ time evolution, given by the differential equation $d s / dt = U_{12} \equiv U_{2} - U_{1}$ . By substituting the expression for $F_{i}$ (11) into (3) the mobility relations in Section 2.1, one obtains an expression for the relative velocity $U_{12}$ in terms of dimensionless quantities, namely $U_{12} = e_{2} \cdot M^{g} - (Q_{M} \nabla φ_{M}^{d} + Q_{E} \nabla φ_{E}^{d}) \cdot M^{i},$ with $M^{g} = [\frac{ss}{s^{2}} L (s) + (I - \frac{ss}{s^{2}}) M (s)]$ and $M^{i} = [\frac{ss}{s^{2}} G (s) + (I - \frac{ss}{s^{2}}) H (s)],$ where

Numerical results

In order to perform the integration of the system of differential equations that govern the relative motion of the particles, Eq. (14), a fourth-order Runge–Kutta scheme is used, i.e. a predictor–corrector method with four steps of evaluations of the velocity function $U_{12}$ at each time-step. The asymptotic forms of the mobility functions for widely separated spheres (7), (8), (9), (10) were used for $s > 2.3$ . Otherwise, the near field mobilities given by (5), (6) are considered. Furthermore, in

Acknowledgements

The authors are grateful to the Brazilian research funding agencies CNPq and CAPES for their generous support of this work. We thank Prof. Yuri Dumaresq Sobral (MAT-UnB) for helpful comments on this work.

References (16)

L.L. Castro et al.
J. Magn. Magn. Mater.
(2005)
M. Aoshima et al.
J. Colloid Interface Sci.
(2005)
J.P. Huang et al.
J. Magn. Magn. Mater.
(2005)
E.J. Hinch
Disorder and Mixing
(1988)
M. von Smoluchowski
Z. Phys. Chem.
(1917)
G.R. Zeichner et al.
AIChE J.
(1977)
R.H. Davis
J. Fluid Mech.
(1984)
A.Z. Zinchenko et al.
Phys. Fluids
(1994)

There are more references available in the full text version of this article.

Cited by (15)

On the magnetization of a dilute suspension in a uniform magnetic field: Influence of dipolar and hydrodynamic particle interactions
2020, Journal of Magnetism and Magnetic Materials
Citation Excerpt :
Relaxation phenomena in dipolar suspensions have been also investigated by other works [10–12]. Current interest has focused on Brownian or Langevin numerical simulations of the behavior of magnetic suspensions have provided a valuable tool for computing microstructure transitions and macroscopic properties such as the magnetization, in the presence of particle interactions, sedimentation, shear flows and applied magnetic field [13–18]. While these simulations have a strong physical basis, they neglect the details of hydrodynamic interactions under condition of strong flows and are most applicable to sedimentation or low-to-moderate shear flows in the absence of hydrodynamic interactions (i.e. low and moderate Péclet numbers).
In this work, the magnetization of a dilute magnetic suspension undergoing a shear flow and a uniform field in the presence of dipolar and hydrodynamic interactions is investigated. Firstly, the problem of a single magnetic particle in a simple shear flow in the presence of an external uniform magnetic filed is examined. From this solution, we study the influence of non-equilibrium effects on the magnetization at condition of strong flows, including the spinning behavior of the particles. Next, we use a non-renormalized cluster expansion in order to derive the magnetization $O (ϕ^{2})$ in a sheared suspension of non-Brownian magnetic from a solution of a creeping flow problem of two magnetic spherical particles interacting magnetically and hydrodynamically in the presence of a uniform applied magnetic field, where $ϕ$ is the particle volume fraction. The numerical results suggest that under condition of strong flows the aggregative nature of the dipolar interactions may lead to a decrease in the magnetization component in the field direction. In contrast, the dispersive character of the viscous hydrodynamic interactions produces a substantial increase in the same component of magnetization even under the condition of strong flow. This result is attributed to the misalignment produced by the hydrodynamic disturbances on the particle dipoles orientated with the flow direction.
Rheology of a very dilute magnetic suspension with micro-structures of nanoparticles
2016, Journal of Magnetism and Magnetic Materials
The main objective of this present paper is to measure the apparent viscosity of a magnetic suspension in the presence of particle agglomerates of different sizes for several applied magnetic fields, shear rates and particle volume fractions. A secondary goal is to investigate suspension microstructure transition, when subjected to a magnetic field. We show that an employed like virial expansion of two empirical coefficients based only on the experimental data gives a good quantitative description of the magnetorheological suspension effective viscosity up to particle volume fraction less than 0.01. The observed shear rate dependence viscosity is a direct consequence of the stretching, breaking particle structures of different sizes and shapes formed by the action of magnetic attractive force between the polarized particles as observed previously in the context of dense ferrofluids. We have identified even in the limit of a very small particle volume fraction a strong non-linear behavior of the examined suspension due to formation of suspended blobs-like aggregates of different sizes and anisotropic chains of particles. These structures are induced by the presence of an external magnetic field and particle–particle magnetic interactions. A histogram of the structure size distribution is also examined. The results of this paper are important to those who are interested on the magnetorheological suspensions.
Dynamic numerical simulations of magnetically interacting suspensions in creeping flow
2015, Powder Technology
Citation Excerpt :
The effect of particle rotation motion and particle inertia on these interactions was explored. Particle trajectories, particle aggregation and particle self-diffusion induced by particle–particle hydrodynamic and magnetic interactions have been also investigated [20,21,29–32]. Numerical codes performing many interacting bodies simulation are usually based on Langevin dynamics (LD) or non-equilibrium molecular dynamics (NEMD), static Monte-Carlo (MC) simulations and on a more theoretical approach describing a probability density function of a suspension quantity (e.g. dipole orientation) which requires the solution of a Fokker–Planck nonlinear differential equation as part of the problem [17].
The equations governing the motion of N magnetic particles suspended in a viscous fluid at low Reynolds and finite Stokes numbers are solved by direct numerical simulations for different Péclet numbers. The Langevin dynamics simulations include all dipole–dipole magnetic interactions for force and torque. An external applied magnetic field and near field interactions represented by contact and repulsion forces are also considered. Repulsive forces are modeled using a variation of the screened-Coulomb type potential. The initial particle distribution is an ergodic ensemble in which each member consists of N mutually impenetrable spheres whose centers are randomly distributed in a prismatic cell of volume V with wall boundaries. The stability of the proposed numerical method and its convergence in calculating some relevant macroscopic properties of the magnetic suspension are carefully examined. The simulations are used to investigate structure transition from an isotropic random distribution of particles to other structures in the presence of an external magnetic field and magnetic particle–particle interactions. The simulations show dimmers and short chain formation in the suspension space at low volume fraction. When the volume fraction is increased long chains and thin anisotropic structures may be observed along the magnetic field direction. The numerical method is also used to calculate the steady state equilibrium magnetization, and accurate results are obtained for different particle volume fractions ϕ in agreement with $O (ϕ^{3})$ asymptotic theories. The method presented is able to consider up to 3000 particles with accuracy and computational efficiency. Typical configurations and particle trajectories are also shown and discussed from a physical point of view.
Symmetry breaking of particle trajectories due to magnetic interactions in a dilute suspension
2013, Journal of Magnetism and Magnetic Materials
Citation Excerpt :
In order to show this, we plotted in Fig. 8 reversibility diagrams for the case in which the particles have infinite rotational inertia, that is, they do not rotate while moving and the orientation of the dipoles remain constant throughout time evolution. This is carried out in the simulations by not solving the angular momentum of the particles [6,7]. We can see from these plots that configuration 1 gives a more regular and isotropic shape, and configuration 2 conserves the dumbbell shape with an anisotropic ratio I=1.5.
This work presents a numerical study of the relative trajectories of two magnetic particles interacting in a dilute suspension. The suspension is composed of magnetic spherical particles of different radius and density immersed in a Newtonian fluid. The particles settle relative to one another under the action of gravity and, when in close proximity, exert on each other magnetic force and torque due to their permanent magnetization. The equations of motion for both translation and rotation of the particles are solved and particle inertia is included in the calculation. The numerical simulations are based on the direct computations of the hydrodynamic and of the magnetic interactions between the rigid particles in the regime of non-zero Stokes number. A detailed study of the relative trajectories of two magnetic particles in a dilute suspension allows us to explore irreversible interactions that lead to particle aggregation and particle migration induced by the breaking of the time reversibility of the creeping flow due to magnetic effects. The calculation shows that the rotation of the particles produced by magnetic interactions change significantly the dynamics of collisions of magnetic particle.
Particles nanomanipulation by the enhanced evanescent field through a near-field scanning optical microscopy probe
2011, Sensors and Actuators, A: Physical
Citation Excerpt :
The Brownian motion is due to the thermal fluctuations. The fluctuation–dissipation theorem of Einstein states that the Brownian motion force is equal to 12πaηkbT, where η is the viscosity of water, kb is the Boltzmann constant, and T is the temperature of water [19]. At room temperature, the Brownian motion force is 1.0 × 10−13 N, which is of the same magnitude as the trapping force.
A near-field scanning optical microscopy (NSOM) probe and a polarized semiconductor laser (808 nm, cw) were applied to push the trapping resolution down to 120 nm on near-field optical manipulation. A multi-circular shape with a minimum size of 400 nm consisting of 120 nm polystyrene particles can be obtained. They are at a resolution of d (d: NSOM probe tip diameter) and λ/7 (λ: laser wavelength), respectively. It is proved that sample concentration and laser power can affect feature size of trapping patterns. In this paper, the effect of trapping forces acted on a nanoparticle along three axis directions on trapping positions is studied, and different trapping positions are generated: the aperture edge in polarization direction and center surface of the probe tip. The result indicates that the single mode NSOM fiber probe is able to trap nanoparticles in a circular shape with lower laser intensity than that required by conventional optical tweezers. The simulated trapping positions around the probe tip based on the conservation law of momentum are found to agree well with experimental results.
The influence of hydrodynamic effects on the complex susceptibility response of magnetic fluids undergoing oscillatory fields: New insights for magnetic hyperthermia
2020, Physics of Fluids

View all citing articles on Scopus

View full text

On the influence of the hydrodynamic interactions on the aggregation rate of magnetic spheres in a dilute suspension

Abstract

Introduction

Section snippets

Formulation of the problem

Trajectory equations

Numerical results

Acknowledgements

J. Magn. Magn. Mater.

J. Colloid Interface Sci.

J. Magn. Magn. Mater.

Disorder and Mixing

Z. Phys. Chem.

AIChE J.

J. Fluid Mech.

Phys. Fluids