Uniform dispersion of fine particles in a magnetic fluid and its evaluation

Uniform and

its

dispersion evaluation

MASANORI

HORIZOE,

of

fine

particles

in

RYUZO ITOH and KEISHI

a magnetic

fluid

GOTOH

Department of EnergyEngineering, ToyohashiUniversityof Technology,Tempaku-cho,Toyohashi441, Japan Published in JPTJ Vol. 31 No. 3 (1994);English Version for APT received27 January 1995 Abstract-Particles confinedin a thin horizontallayer of magneticfluid wereuniformlydispersedby applying a vertical magnetic field. The particle arrangementswere observedboth in experimentsand in twodimensionalcomputer simulations.The regularityof particle dispersionwas definedby the mean value of areasof the Voronoipolygonsand its variance,R = 2/σ2,for the evaluationof the particlearrangements. The clustersof particleswerevirtuallyproduced by uniformlyswellingall of the particles.The distribution of the clustersize can be utilizedfor detailedevaluationof the particlearrangement.The regularityof the particledispersiondependson the repulsiveforce acting betweenthe particles.The uniformity of particle dispersionincreaseswith the intensityof the magneticfield and the initial concentrationof the particles. NOMENCLATURE (a) average area of Voronoi polygons [-] c coverageor bulk-mean area fraction of particles [-] d particle diameter [m] e, unit vector in the radial direction [-] et unit vector in the circumferentialdirection [-] F. adhesion force between particle and wall [N] Fa F. divided by Fc [-] Fd magnetic repulsion force of particle i [N] F) Fd divided by Fc [-] Ff drag force of particle i [N] H magnetic field [Alm] k average size of clusters [-] m diamagneticmoment [wbm] mp mass of single particle [kg] N number of particles [-] Np demagnetizingfactor for particles [-] N, demagnetizingfactor for magnetic fluid [-] n, number of clusters of sizei [-] R regularity [-] position vector of particle i [m] r, SF swellingfactor [-] s distance betweentwo particle centers [m] t time [s] Ud magnetic interaction energy of particle i [J] Vp volume of singleparticle [m3] ?I viscosityof magnetic fluid [Pa s] 0 angle between magnetic moment of particle and applied magnetic field [rad] uf permeabilityof magnetic fluid (HIm] u2 variance of areas of Voronoi polygons [-] X susceptibilityof magnetic fluid [-] susceptibilityof magnetic fluid [-] effective

140 1. INTRODUCTION Particles confined in a thin layer of magnetic fluid are uniformly dispersed or form chain-like clusters depending on the direction of the applied magnetic field [1, 2]. fluids [3, 4]. This can be a model for Similar phenomena occur in electro-rheological studying the phase change from solid to liquid as well as a visible model of colloidal systems. It is applicable to the filters for electromagnetic and sound waves, diffractive lattices, polarizing plates, etc. [5, 6]. The chain-like clusters were studied in detail as one of a series of experiments on the packing operation of fine particles and structure measurement [7]. When the magnetic force is applied in the vertical direction, particles confined in a thin layer of magnetic fluid disperse uniformly due to the repulsion force. Davies et al. [8] evaluated the dispersed state of particles by introducing the spatial distribution function. Skjeltrop [9] used the r- and 0-directional correlation functions for this purpose. These evaluation methods require two parameters to express the dispersed state of particles. In this paper, Voronoi tessellation is conducted and the square of the mean area divided by the variance is used as the measure of regularity [10]. First particles are confined in a thin layer of magnetic fluid and the magnetic field is applied in the vertical direction. Resulting particle movements are investigated both from experiments and from computer simulations. Effects of the magnetic field and the bulk-mean particle concentration on the regularity of the particle dispersion are examined. Furthermore, a new method is proposed for evaluating the local structure of the particle dispersion, where all particles are supposed to be equally swelled and the size distribution of the resulting clusters is discussed. 2. EXPERIMENTAL The experimental apparatus is the same as explained in Inagaki et al. [7]. Figure 1(a) is the test sample, of which the thickness is adjusted by a 5 um spacer. The space was filled with the magnetic fluid, including prescribed amounts of Si02 particles (3,um

Figure 1. Main section of experimentalapparatus (a) Test sample; (b) Test section.

141 diameter). The sample was fixed on the stage of the microscope as shown in Fig. 1 and the magnetic field was applied. Before and after applying the magnetic field, the test sample was videotaped and the coordinates of all particle centers were read from the pictures. 3. COMPUTER SIMULATION Consider a mono-layer of particles dispersed in the magnetic fluid. The thermal motion of particles is neglected. The particle movement is assumed to be very slow so that it does not affect the surrounding flow field. When the magnetic field is applied to the system, the diamagnetic moment m is induced in the particles in the opposite direction.

where Pf is the permeability of the magnetic fluid, H is the magnetic field and Vp is the volume of a single particle. x?m is the effective susceptibility of magnetic fluid expressed by

where x is the susceptibility of magnetic fluid, NS is the demagnetizing factor of the magnetic fluid determined by the shape of the test sample and Np is the demagnetizing factor of particles. NS = 1 in the present study and Np = 1/3 for spherical particles [11]. The energy Ua of magnetic interaction between two particles at s is expressed as follows. separation

where B is the angle of the line joining two particle centers from the direction of the applied magnetic field. 9 = nl2 in the present study. The magnetic repulsion force Fd between two particles becomes as follows:

with

where er and et are the unit vector, respectively, for the radial and angular directions, and d is the particle diameter. For particle movements in the magnetic fluid, the Stokes law is applicable to the drag force Ff.

where ?7is the viscosity of magnetic

fluid and ri is the position

vector of particle i.

142 system, it cannot be ideal and Although we are considering the two-dimensional the adhesion force always exists between particles and vessel walls. Under the above condition the equation of motion of particle i in the magnetic fluid becomes as follows.

where only two-body interactions are taken into account and mp is the mass of the particle. When the total interaction is smaller than the adhesion force, the particle i cannot move. Next the following dimensionless variables are introduced:

Hence (7) becomes:

with

When H = 40000 [A/m] for example, G = 3.8 x 10-5, Ft 2: 1.6 x 10-3 and Fa = 2.5 x 10-2. Since only equilibrium particle arrangement is necessary, the lefthand side of (8) is neglected, yielding

Equation (9) was solved numerically by Euler's method. In computer simulation, a prescribed number of particles were placed one by one in the square region of side 40 in the particle diameter under the non-overlapping and periodic boundaries conditions. The particle interaction was calculated for neighboring particles lying within five diameters from the central one and longer distant interactions were neglected. The time increment was set 0 tk = 0.005 [-]. iterations were required for the calculation to reach an Twenty thousand in The numerical constants used in the calculation are listed Table 1. Fa equilibrium. was obtained from measuring the critical magnetic field above which increasing H induces particle movement. Table 1. Numerical constants

143

Figure 2. Particle arrangement. c = 0.10 [-]. (a) Initial random configuration. (b) Final configuration. H = 25200 A/m. (c) Final configuration. H = 40000 A/m.

144

Figure 3. Particle arrangement. c = 0.30 [-]. (a) Initial random configuration. (b) Final configuration. H = 25 200A/m. (c) Final configuration. H = 40000 A/m.

145 4. RESULTS AND DISCUSSION Figures 2 and 3 show results of the computer simulation for c = 0.10 [-] (particle numbers N = 204) and c = 0.30 [-] (N = 611), respectively. Figures 2(a) and 3(a) for are initial random configurations; Figs. 2(b) and 3(b) are final configurations H = 25200 [A/m]; Figs 2(c) and 3(c) are final configurations for H = 40000 [A/m]. In the case of c = 0.10 [-], the repulsive force is small because of the large separation between particles. Hence the particle arrangement remains unchanged. In the case of c = 0.30 [-], however, regular arrangement is achieved. The Voronoi tessellation method is available to evaluate the dispersed state of particles, where the regularity of cell areas is expressed by the average area (a) and its variance as follows [10]:

Figure 4 depicts the regularity in relation to the coverage, i.e. the bulk-mean fractional area of particles. The final configurations for H = 25 200 [A/m] and H = 40 000 [A/m] were obtained both from experiments and computer simulations. The results of experiments agree well with computer simulations. Figure 5 shows the effect of the magnetic field H on the regularity R for the simulation data. The relation between R and c for initial random configurations agrees well with the previous result [10], as depicted by the solid curve. R remains almost unchanged in the range c < 0.10 [-]. In the range c >_ 0.1 [-], however, the repulsion force of particles becomes larger because of the shorter separation, so that particles are liable to disperse uniformly. Needless to say, R increases with the magnetic field. Although the regularity obtained from the Voronoi polygonal analysis can express the dispersed state of particles only by the single parameter, it is an average for the whole range under consideration. A new method is proposed in the following to obtain detailed information on the particle dispersion. First, all particles are

Figure 4. Comparison between simulation and experiment for H = 25 200 A/m. x Experiment, o Simulation.

146

Figure 5. Regularityin relation to coveragefor various magneticfieldsby numericalsimulation. 0 Initial random configuration, · H = 25 200 A/m,1 H = 40 000A/m, Solid curve is R = 3.6 é.4e supposed to be uniformly swelled, as illustrated in Fig. 6. The overlapping of i particles forms the cluster of size i. When the number of clusters of size i is denoted by n; , the total number of particles becomes:

The average size k of clusters is defined by

The average size of clusters is related to is the swelled diameter d, divided by the the result for the particle arrangement in be evaluated from the slope of the curve,

the swelling factor SF in Fig. 7, where SF average distance between particles. This is Fig. 3. The dispersed state of particles can from which the differential average size of

Figure 6. Schematicof uniformly expandedparticles.

147

Figure 7. Average-cluster-sizein relation to swelling factor. c = 0.30 [-], 0 Initial random configuration, 0 H = 25 200A/m, O H = 40 000A/m.

clusters was obtained as depicted in Fig. 8. The abscissa of Fig. 8 expresses the local average spacing of particles and the ordinate expresses the number fraction of the clusters. One can see that when the magnetic field is applied, the corresponding initial particle distribution curve is compressed from the left-hand side to become sharper than before. As the magnetic field increases, the peak of 0(klN) approaches unity at SF = 1 where the width of the curve becomes zero and the particle arrangement becomes regular. Comparing the curves of Fig. 8 with various distribution functions [12], the gamma distribution function was found to give the best fit, as shown by the solid curves in Fig. 8.

Figure 8. Differential average-cluster-sizein relation to swelling factor. c = 0.30 [-], ASF = 0.1, o Initial random configuration, 0 H = 25 200A/m, O H = 40 000A/m.

148 5. CONCLUSION Particles confined in a thin horizontal layer of magnetic fluid were uniformly dispersed by applying a vertical magnetic field and the state of particle dispersion was discussed both from experiments and from computer simulations. The particle dispersion depends on the replusive force between particles. As the magnetic field and the particle concentration increase, the particle arrangement becomes more regular. The local state of particle dispersion can be evaluated by the proposed method. All particles are supposed to be uniformly swelled so as to yield clusters of overlapping particles. The size distribution of the clusters provides detailed information on the local structure of the particle dispersion. REFERENCES 1. R. W. Chantrell, A. Bradbury, J. Popplewell,et al., J. Phys. D, 13, 123, 1980. 2. A. T. Skjeltrop, J. Appl. Phys., 57, 3285, 1985. 3. D. J. Klingengerg,F. van Swol and C. F. Zukoski, J. Chem. Phys., 91, 7888, 1989. 4. H. Tamura and M. Doi, J. Phys. Soc. Japan, 61, 3984, 1992. 5. A. T. Skjeltrop, J. Appl. Phys., 55, 2587, 1984. 6. A. T. Skjeltrop, Phys. Rev. Lett., 51, 2306, 1983. 7. Y. Inagaki, M. Furuuchi and K. Gotoh, Funtai kougaku gakkaishi, 30, 34, 1993. 8. P. Davies, J. Popplewell, G. Martin, et al. J. Phys. D: Appl. Phys., 19, 469, 1986. 9. A. T. Skjeltrop, J. Magn. Magn. Mater., 65, 195, 1987. 10. K. Gotoh, Phys. Rev. E, 47, 316, 1993. 11. S. Chikazumi,Kyoujiseitai no buturi(jyou). Tokyo: Syoukabou, 1987,p. 14. 12. M. Nakayama and T. Makabe, Kakuritu katei. Tokyo: Baifuukan, 1989,p. 41.