Influence of the spin Brownian motion on the negative magneto-rheological effect in a rod-like haematite particle suspension

We have shown the basic equation of the orientational distribution function of prolate spheroidal haematite particles with rotational Brownian motion in a simple shear flow under an applied magnetic field. The equation has been solved numerically in order to investigate the influence of the spin Brownian motion on the orientational distribution, the negative magneto-rheological effect and the characteristics of magnetisation. With respect to the orientational distribution, the present results are in good agreement with those of the theory without that motion, which shows that the spin Brownian motion does not significantly influence the orientational distribution. In contrast, the influence of the spin Brownian motion appears more significantly in the negative viscosity: the negative viscosity effect decreases in comparison with that of the previous theory without the spin Brownian motion. Moreover, a more significant negative magneto-rheological effect is obtained for a larger particle aspect ratio. Since the magnetisation has a strong relationship with the orientation of the magnetic moment, the effect of the spin Brownian motion appears more significantly in this characteristic.     

On the orientational characteristics and transport coefficients of an oblate spheroidal hematite particle in a simple shear flow

We have developed the basic equation of the orientational distribution function of oblate spheroidal hematite particles with rotational Brownian motion in a simple shear flow under an applied magnetic field. An oblate spheroidal hematite particle has an important characteristic in that it is magnetized in a direction normal to the particle axis. Since a dilute dispersion is addressed in the present study, we have taken into account only the friction force (torque) whilst neglecting the hydrodynamic interactions among the particles. This basic equation has been solved numerically in order that we may investigate the dependence of the orientational distribution on the magnetic field strength, shear rate and rotational Brownian motion and the relationship between the orientational distribution and the transport coefficients such as viscosity and diffusion coefficient. We found that if the effect of the magnetic field is more dominant, the particle inclines in such a way that the oblate surface aligns in the magnetic field direction. If the Peclet number increases and the effect of the shear flow becomes more dominant, the particle inclines such that the oblate surface tilts in the shear flow direction. The viscosity due to the magnetic torque is shown to increase as the magnetic field increases, since the magnetic torque due to the applied magnetic field becomes the more dominant effect. Moreover, the viscosity increase is shown to be more significant for a larger aspect ratio or for a more oblate hematite particle. We have applied the analysis to the problem of particle sedimentation under gravity in the presence of a magnetic field applied in the sedimentation direction. The particles are found to sediment with the oblate surface aligning more significantly in the sedimentation direction as the applied magnetic field strength increases.     

Application of the Brownian dynamics method to a rod-like hematite particle dispersion

We have investigated various problems that arise in applying the Brownian dynamics method to a suspension composed of rod-like hematite particles, which have a magnetic moment normal to the particle axis direction. The accuracy and the deviation of simulation results from theoretical solutions have been discussed by comparison with the theoretical solutions that were obtained by solving the basic equations of the orientational distribution function. The characteristics of the negative viscosity are not observed to be dependent on a time interval unless a sufficiently short time interval is used. The present simulation results can satisfactorily reproduce the qualitative characteristics of the negative magneto-rheological effect that was predicted by the previous theoretical investigation. Good quantitative agreement is obtained in the range of small-applied magnetic fields, but agreement is not significantly good in the large magnetic field region. The deviation of the negative viscosity from the theoretical prediction cannot be improved by using a more accurate numerical algorithm such as moving from Euler to second-order or fourth-order Runge–Kutta. The results of the orientational distribution can well reproduce the characteristic features that the distribution has a gradual shape with low linear-like peak, which is in significant contrast to the sharp single-peak distribution of a ferromagnetic rod-like particle dispersion. The present orientational distributions are in significantly good agreement with those of the theoretical prediction in regard to the position and the height of a peak and the general shape of the overall profile. Good agreement of the present magnetisation curves with the theoretical prediction verifies that the spin rotational Brownian motion is activated at a physically reasonable level in the present simulations.     

On aggregate structures in a rod-like haematite particle suspension by means of Brownian dynamics simulations

We have investigated aggregation phenomena in a suspension composed of rod-like haematite particles by means of Brownian dynamics simulations. The magnetic moment of the haematite particles lies normal to the particle axis direction and therefore the present Brownian dynamics method takes into account the spin rotational Brownian motion about the particle axis. We have investigated the influence of the magnetic particle–field and particle–particle interactions, the shear rate and the volumetric fraction of particles on the particle aggregation phenomena. Snapshots of aggregate structures are used for a qualitative discussion and the cluster size distribution, radial distribution function and the orientational correlation functions of the direction of particle axis and magnetic moment are the focus for a quantitative discussion. The significant formation of raft-like clusters is found to occur at a magnetic particle–particle interaction strength much larger than that required for a magnetic spherical particle suspension. This is because the rotational Brownian motion has a significant influence on the formation of clusters in a suspension of rod-like particles with a large aspect ratio. An applied magnetic field enhances the formation of raft-like clusters. A shear flow does not have a significant influence on the internal structure of the clusters, but influences the cluster size distribution of the raft-like clusters.     

Brownian dynamics simulations of a cubic haematite particle suspension with a more effective treatment of steric layer interactions

We have proposed a new repulsive layer model for describing the interaction between steric layers of coated cubic particles. This approach is an effective technique applicable to particle-based simulations such as a Brownian dynamics simulation of a suspension composed of cubic particles. 3D Brownian dynamics simulations employing this repulsive interaction model have been performed in order to investigate the equilibrium aggregate structures of a suspension composed of cubic haematite particles. It has been verified that Brownian dynamics employing the present steric interaction model are in good agreement with Monte Carlo results with respect to particle aggregate structures and particle orientational characteristics. From the viewpoint of developing a surface modification technology, we have also investigated a regime change in the aggregate structure of cubic particle in a quasi-2D system by means of Brownian dynamics simulations. If the magnetic particle–particle interaction strength is relatively strong, in zero applied magnetic field the particles aggregate in an offset face-to-face configuration. As the magnetic field strength is increased, the offset face-to-face structure is transformed into a more direct face-to-face contact configuration that extends throughout the whole simulation region.     

Lattice Boltzmann simulation of the two-dimensional flow of a magnetic suspension between two parallel walls under a non-uniform magnetic field

We have developed a lattice Boltzmann method based on fluctuation hydrodynamics that is applicable to the flow problem of a particle suspension. In this method, we have introduced the viscosity-modifying method, rather than the velocity-scaling method, in which a modified viscosity is used for generating random forces in lattice Boltzmann simulations. The viscosity-modifying method is found to be applicable to the simulation of a magnetic particle suspension. We have applied this method to the two-dimensional Poiseuille flow of a magnetic suspension between two parallel walls in order to investigate the behavior of magnetic particles in a non-uniform applied magnetic field. From the results of the snapshots, the pair correlation function between the magnetic pole and the magnetic particles and the averaged local particle velocity and magnetization distributions, it was observed that the behavior of the magnetic particles changes significantly depending upon which factor dominates the phenomenon in the balance between the magnetic particle–particle interaction, the non-uniform applied magnetic field and the translational and rotational Brownian motion.     

Negative magneto-rheological effect of a dispersion composed of spindle-like hematite particles

We have experimentally investigated the negative magneto-rheological effect that is theoretically predicted to appear in a dispersion composed of spindle-like hematite (α-Fe₂O₃) particles. The spindle-like hematite particles were synthesized by aging a solution of FeCl₃ and KH₂PO₄ for 72?h at 373?K. The particle size distribution was determined by digital image analysis from electronic microscope observation of the particles. In the present study we considered a glycerol–water-based dispersion in order to clarify the influence of the shear rate and the magnetic field strength on the negative magneto-rheological effect. Measurement of the viscosity was carried out using a rotational-type rheometer in an external magnetic field generated by Helmholtz coils. The main results obtained can be summarized as follows. The viscosity of a hematite/glycerol–water dispersion relative to that with no applied magnetic field decreases with increasing magnetic field strength: that is, we observed the negative viscosity effect that has previously been predicted from theoretical considerations. Moreover, the negative magneto-rheological effect is observed to decrease with increasing shear rate, which also agrees qualitatively with the theoretical prediction.     

Influences of magnetic particle–particle interactions on orientational distributions and rheological properties for a colloidal dispersion composed of rod-like particle with a magnetic moment normal to the particle axis

We have investigated mainly the influences of magnetic particle–particle interactions on the orientational distribution and viscosity of a semi-dense dispersion, which is composed of rod-like particles with a magnetic moment magnetized normal to the particle axis. In addition, the influences of the magnetic field strength, shear rate, and random forces on the orientational distribution and rheological properties have been clarified. The mean field approximation has been applied to take into account magnetic interactions between rod-like particles. The basic equation of the orientational distribution function has been derived from the balance of torques and solved by the numerical analysis method. The results obtained here are summarized as follows. For a strong magnetic field, the rotational motion of the rod-like particle is restricted in a plane normal to the shearing plane since the magnetic moment of the particle is restricted in the magnetic field direction. Under circumstances of a very strong magnetic interaction between particles, the magnetic moment is strongly restricted in the magnetic field direction, so that the particle has a tendency to incline in the flow direction with the magnetic moment pointing to the magnetic field direction. For a strong shear flow, a directional characteristic of rod-like particles is enhanced, and this leads to a more significant one-peak-type distribution of the orientational distribution function. Magnetic interactions between particles do not contribute to the increase in the viscosity because the mean-field vector has only a component along the magnetic field direction.     

Evaluation of the translational and rotational diffusion coefficients of a cubic particle (for the application to Brownian dynamics simulations)

By estimating the force and torque acting on the cube for the two cases of a uniform flow field and a rotational flow field, we have discussed whether or not there is a coupling between the translational and the rotational motion. From the characteristics of the friction coefficients, we may understand that there is no coupling between the translation motion and the rotational motion in the situation of the Reynolds number being sufficiently smaller than unity. In contrast, in the case of a non-slow flow field with the Reynolds number larger than unity, the coupling characteristics of the motion of a cube is certainly recognised and therefore the interaction with the ambient fluid is characterised by a variety of friction coefficients including friction coefficients that relate the forces acting on the cube to the angular velocities of the rotational motion. Hence, the employment of these translational and rotational diffusion coefficients for a cube enables the implementation of Brownian dynamics simulations for a suspension composed of cubic particles in order to analyse the dynamic characteristics of a cubic particle suspension.

Highlights

1. We have considered a flow problem around a cube in order to numerically clarify the characteristics of the translational and rotational friction or diffusion coefficients.

2. In a slow flow field the motion of the cube need only to be characterised by two friction coefficients, i.e. the translational and rotational friction coefficients.

3. In the case of a non-slow flow field, the coupling characteristics between the translational motion and the rotational motion are recognised.

4. Employment of these diffusion coefficients enables the implementation of Brownian dynamics simulations for a suspension composed of cubic particles.

     

Rotational Brownian dynamics simulations of non-interacting magnetized ellipsoidal particles in d.c. and a.c. magnetic fields

The rotational Brownian motion of magnetized tri-axial ellipsoidal particles (orthotropic particles) suspended in a Newtonian fluid, in the dilute suspension limit, under applied d.c. and a.c. magnetic fields was studied using rotational Brownian dynamics simulations. The algorithm describing the change in the suspension magnetization was obtained from the stochastic angular momentum equation using the fluctuation-dissipation theorem and a quaternion formulation of orientation space. Simulation results are in agreement with the Langevin function for equilibrium magnetization and with single-exponential relaxation from equilibrium at small fields using Perrin's effective relaxation time. Dynamic susceptibilities for ellipsoidal particles of different aspect ratios were obtained from the response to oscillating magnetic fields of different frequencies and described by Debye's model for the complex susceptibility using Perrin's effective relaxation time. Simulations at high equilibrium and probe fields indicate that Perrin's effective relaxation time continues to describe relaxation from equilibrium and response to oscillating fields even beyond the small field limit.     

非球形颗粒布朗运动的涨落-格子Boltzmann数值研究

采用涨落-格子Boltzmann方法对非球形颗粒(二维)的布朗运动进行直接数值模拟.数值结果包括椭圆形、矩形颗粒的速度均方值及速度自相关函数等.研究发现,对于非球形颗粒,其方向性并没有影响能量均分原理的适用性,每个自由度的能量由其速度或角速度的均方值确定,而且计算的颗粒平动温度和转动温度一致.此外,颗粒的速度自相关函数在相对长的时间内以~ct^-1的规律衰减,其系数c与颗粒的形状无关.     

Correlation in the velocity of a Brownian particle induced by frictional anisotropy and magnetic field

We study the motion of charged Brownian particles in an external magnetic field. It is found that a correlation appears between the components of particle velocity in the case of anisotropic friction, approaching asymptotically zero in the stationary limit. If magnetic field is smaller compared to the critical value, determined by frictional anisotropy, the relaxation of the correlation is non-oscillating in time. However, in a larger magnetic field this relaxation becomes oscillating. The phenomenon is related to the statistical dependence of the components of transformed random force caused by the simultaneous influence of magnetic field and anisotropic dissipation.     

Negative viscosity due to magnetic properties of a non-dilute suspension composed of ferromagnetic rod-like particles with magnetic moment normal to the particle axis

The negative viscosity of a colloidal dispersion composed of ferromagnetic rod-like particles, which have a magnetic moment normal to the particle axis, have been investigated. A simple shear flow problem has been treated to clarify the particle orientational distribution and rheological properties of such a semi-dense dispersion, under circumstances of an external magnetic field applied in the direction normal to the shear plane of a simple shear flow. The results obtained here are summarized as follows. For the cases of a very strong magnetic field and magnetic interactions between particles, the magnetic moment of the rod-like particles is significantly restricted in the magnetic field direction, so that the particle approximately aligns in the shear flow direction. Also, the particle can easily rotate around the axis of the cluster almost freely even in a simple shear flow. Characteristic orientational properties of the particle cause negative viscosity, as in the previous study for a dilute dispersion. However, magnetic particle-particle interactions have a function to make such negative viscosity decrease.     

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

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.     

Active Brownian motion tunable by light

Active Brownian particles are capable of taking up energy from their environment and converting it into directed motion; examples range from chemotactic cells and bacteria to artificial micro-swimmers. We have recently demonstrated that Janus particles, i.e.?gold-capped colloidal spheres, suspended in a critical binary liquid mixture perform active Brownian motion when illuminated by light. In this paper, we investigate in more detail their swimming mechanism, leading to active Brownian motion. We show that the illumination-borne heating induces a local asymmetric demixing of the binary mixture, generating a spatial chemical concentration gradient which is responsible for the particle's self-diffusiophoretic motion. We study this effect as a function of the functionalization of the gold cap, the particle size and the illumination intensity: the functionalization determines what component of the binary mixture is preferentially adsorbed at the cap and the swimming direction (towards or away from the cap); the particle size determines the rotational diffusion and, therefore, the random reorientation of the particle; and the intensity tunes the strength of the heating and, therefore, of the motion. Finally, we harness this dependence of the swimming strength on the illumination intensity to investigate the behavior of a micro-swimmer in a spatial light gradient, where its swimming properties are space-dependent.     

Bistability and "negative" viscosity for a suspension of insulating particles in an electric field

It is shown that a suspension of insulating particles in a liquid with low conductivity possesses bistability and has a "negative" effective viscosity effect in the electric field due to internal rotations. By Brownian dynamics simulation it has been found that thermal fluctuations of the angular velocity of particles in this bistable system can have a large effect on the viscosity of the suspension.     

Lattice Boltzmann simulation of the dispersion of aggregated Brownian particles under shear flows

The deformation and breakup processes of a particle-cluster aggregate under shear flows are investigated by the two-phase lattice Boltzmann method. In the simulation the particle is modeled by a hard droplet with large viscosity and strong surface tension. The van der Waals attraction force is taken into account for the interaction between the particles. Also, the Brownian motion is considered for nano-particles. Two important dimensionless parameters are introduced in order to classify calculated results. One is the ratio of fluid force to the maximum inter-particle force, Y, and the other is the Péclet number which is the ratio of the rate of diffusion by a shear flow to the rate of diffusion by Brownian motion. It is found that Y is the key factor in dispersion and that the Brownian motion retards the dispersion.     

Light scattering studies of Brownian motion in confined geometries

Summary Dynamic light scattering can be a useful tool to determine the confinement of Brownian particles whose motion is restricted to dimensions comparable to the wavelength of the light. The theoretical form of the correlation function of the electric field scattered from such trapped particles has been derived and compared with the signal obtained both in a simulated experiment and in a real experiment where the particles are trapped in a glass wedge. This new result can be of relevance for particles trapped in various media such as a porous (transparent) media, a gel, a suspension of lamellar phases or even a concentrated colloidal suspension where a particle is ?trapped? by its neighbours. Paper presented at the I International Conference on Scaling Concepts and Complex Fluids, Copanello, Italy, July 4–8, 1994.     

对利用动态光散射法测量颗粒粒径和液体黏度的改进

光散射技术通过测量悬浮液中布朗运动颗粒的平移扩散系数,得到颗粒流体力学直径或液体黏度.本文由单参数模型入手,建立了低颗粒浓度下,单颗粒平移扩散系数与颗粒集体平移扩散系数和颗粒浓度之间的线性依存关系并将其引入光散射法中,从而对现有的测量方法进行了改进.改进后的测量方法可实现纳米尺度球型颗粒标称直径的测量和液体黏度的绝对法测量.以聚苯乙烯颗粒+水和二氧化硅颗粒+乙醇两个分散系为参考样本,通过实验,验证了改进后方法的可行性.此外,还针对上述两个分散系,实验探讨了温度和颗粒浓度对颗粒集体平移扩散系数的影响规律,发现聚苯乙烯颗粒+水分散系中,颗粒间相互作用表现为引力;二氧化硅颗粒+乙醇分散系中,颗粒间相互作用表现为斥力.讨论了颗粒集体平移扩散系数随颗粒浓度变化规律与第二渗透维里系数的关系.