Interaction between two magnetic dipoles in a uniform magnetic field , J G Ku , X Y Liu, H H Chen, R D Deng, and Q X Yan Citation: AIP Advances 6, 025004 (2016); doi: 10.1063/1.4941750 View online: http://dx.doi.org/10.1063/1.4941750 View Table of Contents: http://aip.scitation.org/toc/adv/6/2 Published by the American Institute of Physics AIP ADVANCES 6, 025004 (2016) Interaction between two magnetic dipoles in a uniform magnetic field J G Ku,1,a X Y Liu,1 H H Chen,2 R D Deng,1 and Q X Yan1 College of Zijin Mining, Fuzhou University, Fuzhou 350116, Fujian, China School of Chemical Engineering, The University of Queensland, Brisbane, QLD 4067, Australia (Received 17 December 2015; accepted 27 January 2016; published online February 2016) A new formula for the interaction force between two magnetic dipoles in a uniform magnetic field is derived taking their mutual magnetic interaction into consideration and used to simulate their relative motion Results show that when the angle β between the direction of external magnetic field and the centerline of two magnetic dipoles is ◦ or 90 ◦, magnetic dipoles approach each other or move away from each other in a straight line, respectively And the time required for them to contact each other from the initial position is related to the specific susceptibility and the diameter of magnetic particles, medium viscosity and magnetic field strength When β is between ◦ and 90 ◦, magnetic dipole pair performs approximate elliptical motion, and the motion trajectory is affected by the specific susceptibility, diameter and medium viscosity but not magnetic field strength However, time required for magnetic dipoles to complete the same motion trajectory is shorter when adopting stronger magnetic field Moreover, the subsequent motion trajectory of magnetic dipoles is ascertained once the initial position is set in a predetermined motion trajectory Additionally, magnetic potential energy of magnetic dipole pairs is transformed into kinetic energy and friction energy during the motion C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/) [http://dx.doi.org/10.1063/1.4941750] I INTRODUCTION During the last half century, with the perfection and development of magnetic dipole theory, it has been widely used in diverse fields, such as nondestructive testing, magnetic localization, geophysical prospecting, induction log, magnetic fluid seal, magnetic fluid damper, magnetic-fluid separation and magnetic separation The theory of magnetic dipoles, based on a closed circulation of electric current, can be employed in theoretical calculation with high accuracy when the distance between magnetic dipoles is relatively far away (l > 10R).1,2 Whereas, the calculation error would increase rapidly with the decreasing distance and even wrong conclusion might be drawn when the distance and particle diameter are in the same order of magnitude.3 Finite element analysis is a relatively accurate method in the near field, whereas, the calculation quantity is quite huge.4,5 Hence, modified magnetic dipole theory was proposed to address this issue Research shows that the calculation accuracy of time’s simple divided magnetic dipole is relatively high (less than 1%), which facilitates the application of magnetic dipole theory in the near field.2 Magnetic chain formation of magnetic particles due to the existence of magnetic dipole–dipole attraction widely exists in magnetic fluid seal, magnetic fluid damper, magnetic-fluid separation, magnetic separation, etc At present, it is generally studied via combined photography and Monte Carlo simulation/magnetic dipole theory based simulation.6–9 Current research mainly focuses on the formed chains and the system response time, and overlooks the microcosmic formation process a Corresponding author: kkcc22@163.com (J G Ku) 2158-3226/2016/6(2)/025004/9 6, 025004-1 © Author(s) 2016 025004-2 Ku et al AIP Advances 6, 025004 (2016) of magnetic chains, especially for the interaction between two adjacent magnetic particles-magnetic dipole pair, which constitutes the fundamental research of magnetic chain formation mechanism Research work has been carried out to study the interaction of magnetic dipoles The magnetic interaction of two magnetic particles was investigated and magnetic force was proposed.10–14 Besides, the contact time of ferromagnetism particles formed via magnetic agglomeration15 and magnetic particles in aqueous solutions16 was calculated Unfortunately, to the authors’ best knowledge, a systematic research on the motion trajectory, the speed and the energy conversion of magnetic dipole pairs during the motion in a uniform magnetic field has not been reported so far In this work, based on the model of time-division magnetic dipole and mutual magnetic interaction when magnetic particles approaching each other, a new magnetic interaction formula was derived and the 3D characteristic magnetic force distribution of magnetic particles was established The velocity-Verlet algorithm was then adopted to simulate the interaction of magnetic particles and the subsequent 2D dynamic process And effects of process parameters such as magnetic susceptibility and diameter of particles, magnetic field strength, and medium viscosity on the motion trajectory and motion time of magnetic dipole pairs were investigated and the energy conversion during the motion process was also calculated II MATHEMATICAL MODEL A Equations of interaction energy and interaction force of a magnetic dipole pair The 3D coordinate of a magnetic dipole pair can be seen in Fig The distance between two magnetic dipoles, the angle between their centerline and the Z-axis, and the angle between their centerline and the X-axis can be represented as l, θ, and ϕ, respectively The interaction energy of two magnetic dipoles with dipole moments m1 and m2 is expressed17 as: m1m2 − (m1 · er ) (m2 · er ) Em 1m = , (1) 4π µ |l|3 where µ is the medium permeability, l is the distance between two magnetic dipoles, and er is the unit vector of the center line of the two magnetic dipoles When the magnetic moments of the two magnetic dipoles are exactly the same (both in magnitude and direction), Eq (1) can be simplified as: Emm = |m|2 − 3cos2 θ 4π µ|l|3 , FIG Spatial position of a magnetic dipole pair (2) 025004-3 Ku et al AIP Advances 6, 025004 (2016) where m is the magnetic dipole moment, m = 4π µ βR3 H, θ is the angle between the direction of external magnetic field and the centerline of two magnetic dipoles The mutual magnetization of the magnetic dipole pair is negligible when the distance between them is far However, when the distance is relatively short, the mutual magnetization will affect the individual magnetic dipole’s magnetic moment Considering the unsaturated mutual magnetization, the magnetic moment of magnetic dipoles can then be represented as follows: 12π µxz β 2r H m ′x = (l + r 2)5/2 (1 − 4π µ βr 3) 12π µy z β 2r H ⃗ ′ = m ′y = m , + r 2)5/2 (1 − 4π µ βr 3) (l m z′ = 4π µ βr H (1 − 4π µ βr 3) (3) µ −µ where µ is the medium permeability, β = µ k+2µrr ; µk and µr correspond to the relative permeability k of the magnetic particles and the medium, respectively; r is the radius of the magnetic dipole, m; H is the intensity of the magnetic field, kA/m Substituting sin θ cos ϕ = x/l, sin θ sin ϕ = y/l, cos θ = z/l, and l = x + y + z into Eq (3), and taking the substituted equation’s partial derivatives with respect to x, y, and z, the relevant component forces of magnetic particles can be written as: Fx = Fy = Fz = m ′x 5cos2 θ − sin θ cos ϕ 4π µ l4 m ′y 5cos2 θ − sin θ sin ϕ 4π µ l4 m z′ 5cos2 θ − cos θ 4π µ l4 (4) Due to the symmetric distribution of forces in 3D space, the y − z plane (ϕ = π/2) is chosen to represent the force orientation The relevant forces of two magnetic particles can then be calculated using Eq (5): Fy = Fz = m ′y 5cos2 θ − sin θ 4π µ l4 ′ 2 m z 5cos θ − cos θ 4π µ l4 (5) B Orientation of interaction forces of a magnetic dipole pair The relative position and orientation of interaction forces of a magnetic dipole pair can be seen in Fig It can be seen from Fig 2(a) that the orientation of interaction forces acting on magnetic dipole m2 varies with the angle between the centerline of the magnetic dipole pair and Z-axis The relative values of interaction forces acting on magnetic dipole m2 also vary with the angle, as can be seen from Fig 2(b) Specifically, for interaction force in Y-axis acting on dipole m2, repulsive force decreases from the maximum to zero with the angel increasing from 0◦ to 28◦, and attractive force increases with the angel increasing from 28◦ to 59◦, followed by dropping to zero when the angle reaches 90◦ However, interaction force in Z-axis acting on dipole m2 experiences a different changing trend As can be inferred from Fig 2(a) and 2(b), the closer the magnetic dipole m2 is to Y-axis, the larger repulsive force the magnetic dipole m2 will experience applied by magnetic dipole m1, while the closer to Z-axis, the larger attractive force the magnetic dipole m2 will experience applied by m1 Throughout the circumference, magnetic dipole forces push the magnetic dipole m2 toward the Z-axis, thus leading to magnetic particles oriented along the Z-axis 025004-4 Ku et al AIP Advances 6, 025004 (2016) FIG Relative values and orientation of interaction forces of a magnetic dipole pair (a) relative position and orientation of interaction forces acting on magnetic dipole m (b) Relative values of forces along different orientations acting on m C Frictional resistance FD The flow pattern of magnetic particles in the separation space of the magnetic separator is under turbulent flow conditions, and the Reynolds number (Re) range is widely distributed Thus, the frictional resistance FD is calculated using Eq (6) developed by Abraham:18 ( ) 2k FD = ψ t + √ ρd 2v 2, (6) Re where ψ t and k are the resistance coefficients related to the Reynolds number; Re is the Reynolds number; ρ is the density of the medium; d is the diameter of the magnetic particle; v is the velocity of the magnetic particle relative to the medium Modification to Eq (6) has been made by Concha and Almendra19 via combining boundary-layer theory and experimental data, where ψ t = 0.11 and k = 4.53; Eq (6) was developed in order to correlate the particle diameter at any value to Re D Kinetic equation The motion of magnetic particles is governed by Newton’s second law as shown in Eq (7) The involved forces include frictional resistance and magnetic dipole–dipole attraction m d 2s = Fmm + FD , dt (7) where m, s, and t are the mass of the particle, the spatial coordinate of the particle and time, respectively The position, velocity, and acceleration of magnetic particles remain synchronized during their motion In the numerical simulation, a certain time step was chosen to firstly calculate acceleration; the end velocity value and position were then obtained by using the velocity-Verlet algorithm.20 The initial positions of the magnetic particles were selected artificially and the initial velocity was set as zero Based on Newton’s second law, acceleration of magnetic particles at any time (t) can be obtained by:9 a (t) = Fmm (t) + FD (t) m (8) The velocity at time (t + ∆t) can be calculated using Eq (9) with a given time step (∆t): v (t + ∆t) = v (t) + [a (t) + a (t + ∆t)] ∆t (9) 025004-5 Ku et al AIP Advances 6, 025004 (2016) FIG Relationship between the relative speed of magnetic dipole pairs and relevant factors when β is ◦ (a) Relative positions of magnetic dipole pairs, solid and empty circles represent initial and final positions of magnetic dipole pairs, respectively (b) Effects of factors on the relative speed of magnetic dipole pairs in the Y-axis positive direction The position at time (t + ∆t) is: s (t + ∆t) = s (t) + v (t) ∆t + a (t) ∆t Computer simulation is performed by iterative calculation of Eqs (9) and (10) (10) III RESULTS AND DISCUSSION A Perpendicular orientation, β = ◦ When β is ◦, the relative position and speed of the magnetic dipole pair under the influence of different factors can be seen in Fig It is evident that magnetic dipoles run away from each other due to the mutual repulsion (Fig 3(a)) At the very early stage of the motion, the speed of magnetic dipoles increased rapidly, followed by the gradual decrease after the highest velocity was attained (Fig 3(b)) Moreover, the relative speed of magnetic dipole pair was positively related with magnetic susceptibility of particles and magnetic field strength, and negatively related with the density of the particle and medium viscosity, as evidenced by Fig 3(b) B Parallel orientation, β = 90 ◦ When β is 90 ◦, the relative position and speed of the magnetic dipole pair under the influence of different factors is shown in Fig It is obvious that the two magnetic dipoles move relative to FIG Relationship between the relative speed of magnetic dipole pairs and relevant factors when β is 90 ◦ (a) Relative positions of magnetic dipole pair, solid and empty circles represent initial and final positions of magnetic dipole pairs, respectively (b) Effects of factors on the relative speed of magnetic dipole pairs in the Z-axis positive direction 025004-6 Ku et al AIP Advances 6, 025004 (2016) one another (Fig 4(a)) At the very early stage, the relative speed dramatically increased and then increased gradually after a certain speed was obtained in the Z-axes positive direction, which can be seen in the insert presented in Fig 4(b) The speed increased rapidly again when the distance of the two magnetic dipoles equals twice the diameter until collisions occurred (Fig 4(b)) It is worth noting from Fig 4(b) that the relative speed of the magnetic dipole pair was positively related with the magnetic susceptibility of the particle and magnetic field strength, and negatively related with the density of the particle and medium viscosity C Intermediate orientation, ◦ < β < 90 ◦ Magnetic dipoles share similar approximate elliptical motion trajectory regardless of the magnetic susceptibility and density of magnetic dipoles, magnetic field strength, and medium viscosity (Fig 5) What’s more, the angle between the centerline of two magnetic dipoles and Y axis was kept as ◦ when collisions happened The motion trajectory of the magnetic dipole pair was not significantly affected by the nature of magnetic dipoles (i.g., density and magnetic susceptibility) but strongly influenced by external factors (i.g., magnetic field strength and medium viscosity) To be specific, the motion trajectory is larger with higher magnetic field strength and smaller with higher medium viscosity The motion trajectory and relevant speed of magnetic dipole pairs in the same uniform magnetic field and medium can be seen in Fig and Fig 7, respectively It is shown that the motion trajectory of the magnetic dipole pair with a certain initial position is ascertained and overlaps perfectly if the initial position falls within a certain motion trajectory and the nature of magnetic dipoles and external conditions are kept constant (Fig 6) However, although the motion trajectory of magnetic dipole pairs with different initial positions overlaps given the same internal and external condition, their speeds at the early stage of motion are diverse To be specific, the speed increased rapidly at first, followed by a gradual decrease when the highest speed was achieved (Fig 7) When the speed was reduced to zero, the absolute value of speed once again increased gradually When β was more than 60 ◦, the absolute speed increased quickly until collisions happened Whereas, when FIG Motion trajectory of different magnetic dipole pairs at the same initial position: (a) with differing density; (b) with differing magnetic susceptibility; (c) under differing magnetic field strength; (d) in media with differing viscosity Solid and empty circles represent initial and final positions of magnetic dipole pairs, respectively 025004-7 Ku et al AIP Advances 6, 025004 (2016) FIG Motion trajectory of magnetic dipole pairs in the same uniform magnetic field under different initial positions: (a) β = ◦, (b) β = ◦, (c) β = 25 ◦, (d) β = 45 ◦, (e) β = 65 ◦ The initial positions of magnetic dipole pairs in Fig 6(b), Fig 6(c), Fig 6(d) and Fig 6(e) was chosen from the complete motion trajectory in Fig 6(a) Fig 6(f) is obtained by stacking Figures 6(a), 6(b), 6(c), 6(d) and 6(e) Solid and empty circles represent the initial and finial positions of magnetic dipole pairs, respectively β is different (i.g., 45 ◦), the speed increased fast at first, and then had the same speed compared with another magnetic dipole pair (whose initial β = ◦) when their positions (values of v x and v y ) were the same (Fig 7(c)) To understand how the magnetic field strength affects the motion of magnetic dipoles, photographs have been taken during the motion with respect to time Results are shown in Fig It can be seen that given the initial position, time required for magnetic dipole pairs to contact with each other increased with the decreasing magnetic field strength Magnetic potential energy, kinetic energy and frictional energy constitute the main energy forms of magnetic dipole pairs when they move in the magnetic field And the relevant energy conversion is shown in Fig The magnetic potential energy is highest in the initial position and decreases with time To be specific, the magnetic potential energy decreased dramatically after the movement occurred and declined gradually when β is larger than 10 ◦, then it dropped to zero when β is around 35 ◦ and decreases quickly once again when β is larger than 70 ◦ The changing trend for friction energy whose initial value is zero is similar with that for magnetic potential, whereas, FIG (a) Speed of magnetic dipole pairs under different initial positions in the same uniform magnetic field v x and v y are the corresponding speed in the directions of X axis and Y axis, respectively (b) Selected zone of (a) (c) Selected zone of (b) with two different initial positions, β = ◦ and 45 ◦ 025004-8 Ku et al AIP Advances 6, 025004 (2016) FIG Photographs of magnetic dipole pairs in the same initial position and uniform magnetic field vs time with the magnetic field strength of (a) 860 Oe, (b) 690 Oe, (c) 470 Oe and (d) 370 Oe The initial β value was 45 ◦ kinetic energy demonstrates different changing trend throughout the motion Specifically, at the very initial stage of the motion, kinetic energy increased to a maximum ( β ≈ 1.1 ◦) and then dropped to a relatively small value with increasing β, followed by the rapid increase after β exceeds 80 ◦ until collisions happened FIG (a) Relationship between energy conversion and relative positions of magnetic dipole pairs in the uniform magnetic field, the value of horizontal coordinates is taken from Fig 6(a) (b) Selected zone of (a) 025004-9 Ku et al AIP Advances 6, 025004 (2016) IV CONCLUSIONS In this study, a new interaction force formula of magnetic dipoles was derived based on their mutual magnetization in a uniform magnetic field The motion trajectories of magnetic dipole pairs were simulated by adopting the derived formula and the velocity-Verlet algorithm to establish their 3D characteristic motion and to calculate relevant energy conversion during the motion Major results include that: (1) When β is ◦, magnetic dipoles run away from each other due to the mutual repulsion The speed of magnetic dipoles increases rapidly at the early stage, followed by a gradual decrease after the highest speed is attained; (2) When β is 90 ◦, magnetic dipoles move relative to one another due to the mutual magnetic attraction The speed increases very fast at the very beginning stage (the displacement distance is short) and then increases slowly when the maximum is achieved The speed would increase rapidly again when the distance of magnetic dipoles equals to twice the diameter before collisions occurs (3) When β is ◦ or 90 ◦, the relative speed of magnetic dipole pairs is positively related with the magnetic susceptibility of the particle and magnetic field strength, and negatively related with medium viscosity and the density of the particle (4) When β is between ◦ and 90 ◦, the magnetic dipoles share similar approximate elliptical motion trajectory regardless of the magnetic susceptibility and density of magnetic dipoles, magnetic field strength, and medium viscosity The motion trajectory is not significantly affected by the nature of magnetic dipoles (i.g., density and magnetic susceptibility) but significantly influenced by external factors (i.g., magnetic field strength and medium viscosity) To be specific, the motion trajectory is larger with stronger magnetic field strength and smaller with higher medium viscosity Besides, the motion trajectories are ascertained and overlap perfectly if the initial position is within a certain motion trajectory given the same magnetic dipoles and external conditions However, the initial position remarkably affects the speed of magnetic dipole pairs at the initial stage Moreover, magnetic dipole pairs with different initial positions move along the same motion trajectory after they pass the same position Additionally, magnetic potential energy of magnetic dipole pairs is transformed into kinetic energy and friction energy during the motion ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China (Grant No 51104048), for which the authors express their appreciation D J Griffiths, Introduction to Electrodynamics (Prentice Hall Press, Upper Saddle River, 1981) Z Z Tian and Y F Hou, Journal of China University of Mining and Technology 41, 299 (2012) (in Chinese) A J Petruska and J J Abbott, IEEE Trans Magn 49(2), 811–819 (2013) C W 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