Rheological properties and particle behaviors of a non-dilute colloidal dispersion composed of ferromagnetic spherocylinder particles subjected to a simple shear flow (analysis by means of mean-field approximation)

We have investigated the rheological properties and the orientational distributions of particles of a non-dilute colloidal dispersion, which is composed of ferromagnetic spherocylinder particles, subject to a simple shear flow. The mean-field approximation is applied to take into account the interactions between spherocylinder particles. The basic equation of the orientational distribution function has been derived from the balance of the torques (including the term due to the mean field approximation) acting on the particle in an applied magnetic field; this is an integrodifferential equation. Then, the governing equation has been solved by means of the method of successive approximation and Galerkin's method. The results obtained here are summarized as follows. For the case of strong magnetic interactions between particles, the particle exhibits a sharp peak of the orientational distribution even for a weak applied magnetic field. In this case, the mean magnetic moment of the particle becomes large, which leads to strong interactions between the applied magnetic field and the particle. Thus, the particle tends to point to the magnetic filed direction under these situations. Also, in this case, a large increase in viscosity is obtained due to such a restriction concerning the particle orientation.     

Rheological properties and particle behaviors of a nondilute colloidal dispersion composed of ferromagnetic spherocylinder particles subjected to a simple shear flow: analysis by means of mean-field approximation for the two typical external magnetic field directions

We have analyzed the orientational distributions and rheological properties of a nondilute colloidal dispersion composed of ferromagnetic spherocylinder particles subjected to a simple shear flow. In order to understand the effects of the magnetic interactions between the particles, we have applied the mean-field theory to a nondilute colloidal dispersion for the two typical external magnetic field directions, that is, the direction parallel to the shear flow and the direction parallel to the angular velocity vector of the shear flow. The main results are summarized as follows. The particle-particle interactions suppress the Brownian motion of the particles and, therefore, make the particles incline toward the same direction. For the magnetic direction parallel to the shear flow, the influence of the particle-particle interactions makes the peak of the orientational distribution sharper and higher. The viscosity generally increases as the interactions between particles become stronger in the case where the effects of the shear flow and magnetic field are relatively small. For the magnetic direction parallel to the angular velocity vector of the shear flow, the influence of the particle-particle interactions on the orientational distribution appears significantly, when the influences of the shear flow and magnetic field are not so strong that the particles can be aligned sufficiently to form stable chainlike clusters in a certain direction.     

Two-dimensional Monte Carlo simulations of a colloidal dispersion composed of polydisperse ferromagnetic particles in an applied magnetic field

We have investigated aggregation phenomena in a polydisperse colloidal dispersion of ferromagnetic particles simulated by employing the cluster-moving Monte Carlo method in an applied magnetic field. The influence of both particle-particle and particle-field interactions on the aggregate structures is analyzed in terms of a pair correlation function. The results obtained in this study are summarized as follows: Under a strong magnetic field, chainlike clusters are formed along the magnetic field direction, and they become thickly clustered with an increase in the strength of the external magnetic field. Moreover, the thickly clustered chains are formed for a polydisperse system that has a large standard deviation of particle diameters. In contrast, for a very weak magnetic field, the strong interaction between the larger particles gives rise to the formation of various shapes in the chainlike clusters, including bending, looping, and branching. With an increase in the external magnetic field, these structures reorganize to form straight chainlike clusters. Furthermore, the thickness of the chainlike clusters for the polydisperse system is found to depend on the standard deviation of the particle-size distribution but is found to be independent of the magnetic field strength.     

Two-dimensional Monte Carlo simulations of a polydisperse colloidal dispersion composed of ferromagnetic particles for the case of no external magnetic field

We have investigated the aggregation phenomena in a polydisperse colloidal dispersion composed of ferromagnetic particles by means of the cluster-moving Monte Carlo method. The results have been compared with those for a monodisperse system. The internal structures of aggregates have been analyzed in terms of the radial distribution function in order to clarify the quantitative differences in the internal structures of clusters. In addition, the cluster size distribution and angular distribution function have been investigated. The results obtained in the present study are summarized as follows. In a monodisperse system, open necklacelike clusters are formed and they extend with increasing strength of the magnetic particle-particle interaction. In a polydisperse system with a small standard deviation in the particle size distribution, sigma=0.2, larger necklacelike clusters are formed and some looplike clusters can also be observed. In a polydisperse system with a larger standard deviation, sigma=0.35, clumplike clusters are formed for a weak magnetic particle-particle interaction. For a stronger magnetic interaction, larger size clusters that exhibit a complicated network structure are formed. These complicated cluster formations found in a polydisperse system are mainly due to the effect of the presence of larger particles.     

Two-dimensional Monte Carlo simulations of a colloidal dispersion composed of rod-like ferromagnetic particles in the absence of an applied magnetic field

Influences of the magnetic interaction between particles and the aspect ratio of particles on aggregate structures in a colloidal dispersion composed of rod-like ferromagnetic particles were investigated by means of the cluster-moving Monte Carlo method. The internal structures of the aggregates obtained in simulations were analyzed in terms of the number density distribution of the clusters and radial distribution functions. The results show that as the magnetic interaction between particles increases, many small clusters such as anti-parallel particle pairs, raft-like clusters, and small loop-like clusters are formed; these gather to form larger aggregates. In the case of a relatively strong magnetic interaction between particles, solid-like rectangular clusters are formed when the aspect ratio is approximately 2, since the suitable distance between magnetic charges enables particles to form a fundamental structure of two normal anti-parallel particle pairs. As the aspect ratio increases beyond 2, many more stable raft-like clusters are formed, since the increase in distance between magnetic charges makes the two normal anti-parallel particle pair structures unstable. For a significantly larger aspect ratio, large network microstructures are produced by the formation of many chain-like and loop-like structures.     

Rheology and orientational distributions of rodlike particles with magnetic moment normal to the particle axis for semi-dense dispersions (analysis by means of mean field approximation)

We have considered a semi-dense dispersion composed of ferromagnetic rodlike particles with a magnetic moment normal to the particle axis to investigate the rheological properties and particle orientational distribution in a simple shear flow as well as an external magnetic field. We have adopted the mean field approximation to take into account magnetic particle-particle interactions. The basic equation of the orientational distribution function has been derived from the balance of the torques and solved numerically. The results obtained here are summarized as follows. For a very strong magnetic field, the magnetic moment of the rodlike particle is strongly restricted in the field direction, so that the particle points to directions normal to the flow direction (and also to the magnetic field direction). This characteristic of the particle orientational distribution is also valid for the case of a strong particle-particle interaction, as in the strong magnetic field case. To the contrary, for a weak interaction among particles, the particle orientational distribution is governed by a shear flow as well as an applied magnetic field. When the magnetic particle-particle interaction is strong under circumstances of an applied magnetic field, the magnetic moment has a tendency to incline to the magnetic field direction more strongly. This leads to the characteristic that the viscosity decreases with decreasing the distance between particles, and this tendency becomes more significant for a stronger particle-particle interaction. These characteristics concerning the viscosity are quite different from those for a semi-dense dispersion composed of rodlike particles with a magnetic moment along the particle direction.     

Rheological properties and orientational distributions of dilute ferromagnetic spherocylinder particle dispersions. Part II. Analysis for the two typical magnetic field directions

We have investigated the orientational distributions and rheological properties of dilute colloidal dispersions, which consist of ferromagnetic spherocylinder particles. First, the governing equation of the orientational distribution function has been derived for the typical two cases of magnetic field directions: the direction parallel to the shear flow and the direction parallel to the angular velocity vector of the shear flow. The equation has been solved approximately by Galerkin's method. With these numerical solutions we have obtained the results of the orientational distribution and viscosity. The results obtained for the magnetic field in the shear flow direction are summarized as follows. In the case of a weak magnetic field, the particle tends to orient nearly toward the shear flow direction and its opposite direction. As the magnetic field increases, the orientation of the particle is restricted and the viscosity increases significantly. As the influence of the magnetic field becomes dominant, an overshoot in the viscosity curve appears. This is due to the fact that there is a maximum deviation of the averaged particle direction from the magnetic field direction. When the strength of the magnetic field increases significantly, the particle inclines close to the magnetic field direction and the viscosity converges to a constant value. Particles with a larger aspect ratio give rise to a larger increment in the viscosity since such elongated particles induce larger resistance in a flow field. We also have obtained results for the case of the magnetic field in the direction parallel to the angular velocity vector of the shear flow. When the flow field is dominant over both the rotational Brownian motion and the magnetic interaction, the particle rotates in the plane nearly perpendicular to the magnetic field direction. As the magnetic field increases, the particle inclines toward the magnetic direction. For this direction of field, the viscosity is independent of the magnetic field and is always zero.     

Influence of perpendicular external magnetic field on microstructures of monolayer composed of ferromagnetic particles: analysis by means of quasi-two-dimensional Monte Carlo simulation

We investigated the influences of the magnetic field strength and particle areal density on the microstructure of a quasi-two-dimensional monolayer composed of ferromagnetic particles by means of a Monte Carlo simulation. The magnetic field was applied along a direction perpendicular to the plane of the monolayer. Microstructures of the monolayer obtained in the simulations were analyzed in terms of radial distribution and orientational distribution functions. Formation of the microstructures is discussed from the perspective of particle-particle interaction energy and the perpendicular magnetic susceptibility of the monolayer was calculated from simulated magnetization curves. The obtained results are summarized as follows. For small areal density of particles, formation of chain-like structures is prevented by the repulsive magnetic interaction between particles due to orientations of the magnetic moments in the particles along the magnetic field direction. For intermediate areal density of particles, the chain-like structures remain even when a relatively strong magnetic field is applied, because contributions of the attractive magnetic interactions increase. For large areal density of particles, mixtures of chain-like and locally ordered structures appear due to the anisotropic attractive magnetic interactions in the absence of the magnetic field. However, when a sufficiently strong magnetic field is applied, the magnetic interactions between particles change to isotropic repulsive interactions, which results in the short-range repulsive steric interactions between particles becoming dominant with the appearance of hexagonal close packed structures.     

Transport coefficients and orientational distributions of rodlike particles with magnetic moment normal to the particle axis under circumstances of a simple shear flow

We have investigated the influences of the magnetic field strength, shear rate, and random forces on transport coefficients such as viscosity and diffusion coefficient, and also on the orientational distributions of rodlike particles of a dilute colloidal dispersion. This dispersion is composed of ferromagnetic spheroidal particles with a magnetic moment normal to the particle axis. In the present analysis, these spheroidal particles are assumed to conduct the rotational Brownian motion in a simple shear flow as well as an external magnetic field. The basic equation of the orientational distribution function has been derived from the balance of the torques and solved numerically. The results obtained here are summarized as follows. For a very strong magnetic field, the rodlike particle is significantly restricted in the field direction, so that the particle points to a direction normal to the flow direction (and also to the magnetic field direction). However, the present particle does not exhibit a strong directional characteristic, which is one of the typical properties for the previous particle with a magnetic moment parallel to the particle axis. That is, the particle can rotate around the axis of the magnetic moment, although the magnetic moment nearly points to the field direction. The viscosity significantly increases with the field strength, as in the previous particle model. The particle of a larger aspect ratio leads to the larger increase in the viscosity, since such elongated particles induce larger resistance in a flow field. The diffusion coefficient under circumstances of an applied magnetic field is in reasonable agreement between theoretical and experimental results.     

Three-dimensional Monte Carlo simulations of internal aggregate structures in a colloidal dispersion composed of rod-like particles with magnetic moment normal to the particle axis

We have treated a suspension composed of ferromagnetic rod-like particles with a magnetic moment normal to the particle axis in order to investigate aggregation phenomena of such a suspension by means of cluster-moving Monte Carlo simulations. In the present study, we have considered a three-dimensional mono-dispersed model system composed of such rod-like particles. Internal structures of self-assembled clusters have been discussed quantitatively in terms of radial distribution, pair correlation, orientational pair correlation functions, number distributions of clusters, and order functions. The main results obtained here are summarized as follows. Rod-like particles tend to aggregate to form raft-like clusters along the magnetic moment direction more significantly with magnetic particle-particle interactions. In such raft-like clusters, the direction of each particle axis has a tendency to incline in parallel formation, but is not so parallel as in a two-dimensional dispersion. As the volumetric fraction increases, longer raft-like clusters are formed, but such raft-like clusters do not aggregate further to form thicker clusters, which is in significantly contrast with a dispersion of spherical particles, where thicker chain-like clusters are observed under certain conditions. For the case of strong magnetic particle-particle interactions, sufficiently long raft-like clusters are formed along the magnetic field direction, even if the influence of an external magnetic field is of the same order of that of the thermal energy. However, rod-like particles in such clusters do not necessarily incline in significantly parallel formation along a certain direction. Self-assembled tube-like clusters are formed when magnetic particle-particle interactions are much more dominant than the rotational Brownian motion under circumstances of rod-like particles inclining in a certain direction.     

On the behavior of an oblate spheroidal hematite particle in a simple shear flow under a uniform magnetic field applied in the flow direction

We discuss the orientational properties of an oblate spheroidal hematite particle and also its influence on the rheological characteristics of a dilute suspension of these magnetic particles, by means of an analytical approach based on the orientational distribution function. A hematite particle with oblate spheroidal shape has an important characteristic; that is, it is magnetized in a direction normal to the particle axis. From the balance of the torques acting on a particle, we have developed the basic equation of the orientational distribution function. This basic equation has been numerically solved in order to investigate the dependence of the orientational distribution on the various factors. If both the magnetic field and the shear flow are weak, the particle does not exhibit specific directional characteristics. If the magnetic field is more dominant, the particle inclines such that the oblate surface is parallel to the magnetic field direction. If the shear flow becomes more dominant, the particle shows a sharper peak of the orientational distribution in the shear flow direction. The viscosity due to the magnetic torque increases and finally converges to a constant value as the magnetic field increases. In a sedimentation process under the gravitational field, the translational diffusion coefficient decreases with increasing magnetic field strength in the present case of the magnetic field direction.     

Transport coefficients and orientational distributions of spheroidal particles with magnetic moment normal to the particle axis (Analysis for an applied magnetic field normal to the shear plane)

We have investigated the influence of the magnetic field strength, shear rate, and rotational Brownian motion on transport coefficients such as viscosity and diffusion coefficient, and also on the orientational distributions of rodlike particles of a dilute colloidal dispersion. The rodlike particle is modeled as a magnetic spheroidal particle which has a magnetic moment normal to the particle axis; such a particle may typically be a hematite particle. In the present study, an external magnetic field is applied in the direction normal to the shear plane of a simple shear flow. The basic equation of the orientational distribution function has been derived from the balance of torques and solved numerically. The results obtained here are summarized as follows. Although the orientational distribution function shows a sharp peak in the shear flow direction for a very strong magnetic field, such a peak is not restricted to the field direction alone, but continues in every direction of the shear plane. This is due to the characteristic particle motion that the particle can rotate around the axis of the magnetic moment in the shear plane, although the magnetic moment nearly points to the magnetic field direction. This particle motion in the shear plane causes negative values of the viscosity due to the magnetic field. The viscosity decreases, attains a minimum value, and then converges to zero as the field strength increases. Additionally, the diffusion coefficient is significantly influenced by such characteristic particle motion in the shear plane for a strong magnetic field.     

Alignment of plate-like particles in a colloidal dispersion under flow in a uniform pipe studied by high-energy X-ray diffraction

High-energy angle-dispersive X-ray diffraction has been used to study the alignment of colloidal suspension of kaolinite particles in water as they flow along a pipe. X-rays with energies above 25 keV have a major advantage, as they can penetrate through thick samples and walls of containers and permit investigation of samples under realistic flow conditions. As an example of the method, flow through a circular cross-section pipe with an internal diameter of 5 mm has been studied: this is typical of industrial applications. The angular distribution of intensities of peaks in the diffraction pattern as a function of the location of the pipe in the X-ray beam provides information about the alignment of particles under flow. Order parameters have been calculated to describe the alignment and direction of orientation. It is observed that the particles align in the direction of flow with their flat faces parallel to the flow. The experimental results are compared with the calculations of the local strain rate that help to explain the onset of alignment of the particles.     

Adsorption and desorption of colloidal particles on glass in a parallel plate flow chamber—Influence of ionic strength and shear rate

The adsorption and desorption rates of 736 nm diameter polystyrene particles on glass were studiedin situ using a parallel plate flow chamber and automated image analysis. Adsorption and desorption rates were measured simultaneously during deposition, enabling the determination of initial deposition rates, blocked areas per particle, desorption rate coefficients, and the number of adhering particles in the stationary state. Deposition experiments were done from suspensions with different potassium nitrate concentrations (1, 10 and 50 mM) and at varying shear rates (15 to 200 s^–1). The initial deposition rate, the desorption rate, the blocked area per particle and the number of adhering particles in the stationary state showed major variations with the shear rate and the ionic strength of the suspension. At low ionic strength, the number of adhering particles showed an oscillatory behavior in time, presumably due to a varying interaction between particle and collector surface. Blocked areas, determined from deposition kinetics, ranged 705 to 2374 cross-sections at low ionic strength, and from 10 to 564 at high ionic strength and corresponded well with those estimated from local pair distribution functions which were obtained from an analysis of the spatial arrangement of the adhering particles.