Magnetic properties of micro-particles with different shapes and postures in the high precision particles detection

Powder Technology 356 (2019) 628–639

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Magnetic properties of micro-particles with different shapes and postures in the high precision particles detection Ran Jia a, Biao Ma a, Changsong Zheng a,⁎, Xin Ba a,b, Liyong Wang c, Qiu Du a a

School of Mechanical Engineering, Beijing Institute of Technology, No. 5 South Zhongguancun Street HaiDian District, Beijing 100081, PR China Faculty of Engineering and Information, University of Technology Sydney, Ultimo, NSW 2007, Australia The Ministry of Education Key Laboratory of Modern Measurement and Control Technology, Beijing Information Science and Technology University, No. 12 Xiaoying East Street HaiDian District, Beijing 100192, China

b c

a r t i c l e

i n f o

Article history: Received 30 March 2019 Received in revised form 13 August 2019 Accepted 16 August 2019 Available online 03 September 2019 Keywords: Magnetic property Particle shape Particle posture Magnetic perturbation

a b s t r a c t Although some established electromagnetic particle detectors have been used to detect and distinguish microparticles in different fields, the adopted particle equivalent model leads to large detection error because that ignores the difference of the magnetic perturbations induced by particles with different shapes and postures. To improve the development of high-precision electromagnetic micro-particle detectors, the magnetic models of particles with different shapes and postures are established. The distribution of magnetic flux density and the magnetic energy change caused by particles are calculated. The theoretical and experimental results illustrate that the change of magnetic energy caused by spherical particles is much lower than that caused by ellipsoidal and flaky particles of the same volume, meanwhile, with the increase of the rotation angle of ellipsoidal and flaky particles, the variation of magnetic energy decreases significantly. Based on the results, precise particle equivalent functions considering the magnetic properties of particles are proposed. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Micro-particle detection is a promising technology with a wide range of applications in different fields including medicine, biology, semiconductor, precision machinery, environment [1]. To meet the detection requirements for the particles of different scales in a variety of fields, the particle detection sensors, based on the principles of optics [2], ultrasonic [3,4], capacitance [5,6], and electromagnetics [7,8], were proposed. In recent years, a variety of novel electromagnetic particle detection sensors have been widely investigated because of the rich detectable objects, better detection effect, and high reliability [9,10]. This kind of sensors can estimate the size of the particle and assist in determining the particle type through detecting the disturbance on the magnetic field. The structure of the electromagnetic particle detection sensor greatly affects the sensitivity, the minimum detectable particle size, and the consistency of test results. Flanagan et al. [11] and Li et al. [12,13] proposed a single-coil particle detection sensor. The sensor coil is incorporated into a marginal oscillator circuit. The composition and size of the particles can be reflected by the perturbation of the resonance frequency of the sensor. The experiment results show that this sensor can detect ferromagnetic particles with a diameter larger than ⁎ Corresponding author. E-mail address: [email protected] (C. Zheng).

https://doi.org/10.1016/j.powtec.2019.08.044 0032-5910/© 2019 Elsevier B.V. All rights reserved.

100 μm. To improve the anti-interference ability and the detectability for smaller particles, the particle detection sensors with double-coil [14] and parallel three-coil [15–17] were presented. The magnetic perturbation of the sensors caused by particles makes the induction coil output the induced electromagnetic force whose phase and amplitude indicate the magnetization characteristic and the size of the particle, respectively. However, the non-uniform magnetic distribution in the radial direction of the above sensors leads to the poor consistency of the test results, and then the planar spiral-coil particle detection sensors [18–20] were proposed. Because the flaky detection area is adopted in this type of sensors, the uniformity of the magnetic field of the sensor and the consistency of the test results are largely enhanced. The sensitivity of the electromagnetic particle detection sensor is strongly associated with the diameter of the sensor aperture. With the reduction of the aperture diameter, the sensitivity of the sensor is sharply increased. In recent years, the requirements of high accuracy detection for micro solid particles and bio-particles rise greatly. Therefore, some novel electromagnetic micro-particle detection sensors with microfluidic technology were studied. Wu et al. [21] designed a microfluidic chip-based inductive sensor with double coils. In this chip, the coil is a part of the flow channel, which minimizes the distance between the target particles and the embedded coil to zero, and leads to a great improvement in the sensitivity of the particle detection. The experiment results illustrate that this sensor can detect magnetic particles with a diameter of 5–10 μm. Besides that, the microfluidic particle

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detection sensors are also widely used in the detection of bio-particles [22–24] based on the magnetic permeability of specific cells or the magnetic bead. Liu et al. [25,26] proposed a novel cell detection device based on the microfluidic magnetic bead cell assay and analyzed the effect factors on the signal output of the sensor. The results indicate that the pulse magnitude change of the sensor is proportional to the cell volume, and this sensor can be used to detect the target cell and estimate the size of the cell. Lee et al. [27] proposed a cell detection device based on the giant-magneto resistance (GMR) effect. In the sensor, a highfrequency magnetic field is produced around the detection area by a coil. The sensor estimates the size and the type of the cell by detecting the impedance change of the sensing element under an alternating magnetic field. Meanwhile, the detection methods of different proteins based on the magnetic beads were studied [28]. Because the magnetic beads are generally ferromagnetic or superparamagnetic, the bioparticles marked with magnetic beads can be equivalent to magnetic micro-particles. Though the electromagnetic micro-particle detection sensors with different structures for various fields were studied, the researches on the mechanism of the magnetic disturbance caused by micro-particles are severely insufficient. Fan et al. [29] established the static magnetization model of a spherical particle to approximately estimate the magnetic perturbation of the sensor. The research results illustrate that the distribution of the magnetic field in the spherical particle is uniform and the change of the sensor magnetic field is determined by the demagnetization factor. Zhang et al. [30] and Fan et al. [31] studied the magnetic distribution around the spherical micro-particles under alternating magnetic field using the finite element method (FEM). The results show that with the increase of the field frequency, the magnetic distribution in the particle becomes severe uneven, which changes the variation of magnetic energy caused by micro-particles. Nabaei et al. [32] built the magnetization model of a bio-particle and simulated the change of the magnetic field caused by bio-particles through the FEM. However, during the studies of the particle detection, the hysteresis loss and eddy current loss in particles have always been ignored. Meanwhile, the particles with various shapes have been simply equivalent to the sphere based on the equivalent volume method, and the difference of the magnetic properties of particles with various shapes and postures has not been considered. However, the uncertainty and variability of the precise particle shape and posture lead to indelible detection error. Therefore, during the detection process of common particles, the shape distribution and the posture distribution of particles (generally satisfying the Gaussian distribution) should be considered for more accuracy detection result. The development of the high-precision electromagnetic microparticle detection technology asks urgently for the precise magnetic model of micro-particles. Therefore, the magnetic models of particles with different shapes and postures under the alternating magnetic field are established. The magnetic distribution around the particle and the energy change of the local magnetic field caused by the particle are studied based on the models. The results show that the change of the magnetic energy caused by particles is considerably sensitive to the

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Fig. 2. The dynamic hysteresis loops under the alternating magnetic field with different frequencies.

particle shape and particle posture. The research on the magnetic properties of particles with different shapes and postures contributes to improve the correctness of the particle detection results and lay the foundation of the high-precision particle detection technologies. 2. Magnetic model of the particle The alternating magnetic field is generally adopted in the electromagnetic particle detection sensor, and the magnetic perturbation of the sensor is sensitive to the particle shape and posture. According to the statistics of the classification of particles, large numbers of real particles can be approximately equivalent to three typical shapes including sphere-like, ellipsoid-like and flake-like [33]. Therefore, to study the nature and mechanism of the magnetic disturbance of the sensor caused by micro-particles, the magnetic models of spherical, ellipsoidal and flaky particle, under alternating magnetic field, are established as shown in Fig. 1a-c. These models are made to contain micro particles and surrounding air because the variation of the magnetic distribution in and around the particle leads to the energy change of local magnetic field together. A uniform alternating magnetic field oscillating along the x axis is exerted on these particles. The background magnetic flux density Bb can be described as: Bb ¼ Bp sinðωt Þi

ð1Þ

where Bp is the peak value of background magnetic flux density, ω = 2πf0 is the angle frequency, f0 is the frequency of the magnetic field, i is the unit vector of the x axis. Based on the constitutive equation B = ∇ × A, the background vector magnetic potential Ab is given by: 2

3 2 3 0 Ax 5½i; j; k Ab ¼ 4 Ay 5½i; j; k ¼ cosðωt Þ4 0 Bp  y Az

ð2Þ

where Ax, Ay, and Az are the components of the vector magnetic potential along the x, y and z axis respectively. i, j, and k are the unit vector of the x, y and z axis. Table 1 The magnetic parameters of the particle material.

Fig. 1. The magnetic model of the particles under the alternating magnetic field.

Field frequency (kHz)

Bm (mT)

Hm (Am)

δ (°)

0.5 3 10 20 40 60 80 100 200

1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01 1.01

13.4 33.0 63.4 80.9 107.6 128.5 143.6 164.5 197.6

49.57 53.44 33.75 41.385 34.603 24.951 20.9 16.584 41.003

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Fig. 3. The magnetic distribution of the spherical and ellipsoidal particles under the different magnetic fields.

Under the alternating magnetic field, the eddy current effect is produced in particles and the magnetic distribution of particles satisfies the quasi-static equation as [34]:

σ

  ∂A þ ∇  μ 0 −1 μ r −1 ∇  A ¼ J s ∂t

ð3Þ

Where A is the vector magnetic potential, σ is conductivity of the medium, μr is the relative permeability of material, μ0 = 4π × 10−7 is the permeability of vacuum and Js is the source current density. This research only focuses on the magnetic properties of microparticles, therefore, the source current density Js = 0. According to the vector analysis theory, in Cartesian coordinates, the quasi-static

equation can be described as the equation set as: "

! 2 2 ∂ Ay ∂ Ax − 2 − ∂x∂y ∂y " 2 ! 2 1 ∂ Az ∂ Ay − jωσ Ay þ − μ 0 μ r ∂z∂y ∂z2 " 2 ! 2 1 ∂ Ax ∂ Az jωσ Az þ − − μ 0 μ r ∂z∂x ∂x2 jωσ Ax þ

1 μ0μr

!# 2 2 ∂ Ax ∂ Az − ¼0 ∂z2 ∂x∂z ! # 2 2 ∂ Ay ∂ Ax ¼0 − ∂x2 ∂x∂y !# 2 2 ∂ Az ∂ Ay − ¼0 ∂y2 ∂z∂y

ð4Þ

Under the alternating magnetic field, the hysteresis effect exists in ferromagnetic particles, which makes the permeability of the particle show the nonlinear characteristic and produces the hysteresis loss. To precisely calculate the change of magnetic energy caused by particles, the complex relative permeability of the particle is estimated from the

Fig. 4. The distribution of the magnetic flux density in and around the particles under various magnetic fields.

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Fig. 5. The distribution of magnetic flux density in and around the particle with different postures under various magnetic fields.

dynamic hysteresis loop and used for solving the equations. The real measurement result shows that the amplitude of the magnetic flux density along the axis of the adopted sensor is 1.01mT. In this situation, the dynamic hysteresis loop of the particle material is measured by the testing system for AC magnetic properties of soft magnetic material (TD8120, Tunkia Co.,Ltd., Changsha, China) and shown in Fig. 2. It can be seen that, under the weak alternating magnetic field, the dynamic hysteresis loops are approximately ellipse. The standard oblique elliptic function, as Eq. (5), is used to fit the dynamic hysteresis loops based on the least square principle.

The real part of the complex relative permeability represents the ability of causing magnetic energy variation by particles and the imaginary part of the relative permeability indicates the comprehensive energy loss from both the hysteresis effect and the eddy current effect. Based on the data from Fig. 2, the magnetic parameters of the particle material are listed in the Table 1. The distribution of the magnetic vector potential in and around the particle, when considering the hysteresis loss and the eddy current loss, can be calculated based on the above equations. Then the change of the magnetic flux density in and around the particle is given by:

ðH cosθ−B sinθÞ2 ðH sinθ þ B cosθÞ2 þ ¼1 2 a2 b

ΔB ¼ ∇  A − Bb

ð5Þ

where H is the magnetic field intensity, B is the magnetic flux density, θ is the oblique angle, a and b are the semi-major axis and the semi-minor axis of the hysteresis loop respectively. The result indicates that all the dynamic hysteresis loops can be characterized by the standard ellipse. In this case, the magnetic flux density B and the magnetic field intensity H develop with sinusoidal fluctuations and can be described as: H ¼ H m sinðωt Þ B ¼ Bm sinðωt−δÞ

ð8Þ

The change of the magnetic flux density leads to the energy variation of the local magnetic field which is the essential factor of the detectability for particles by the electromagnetic particle detection sensors. The total change of the energy can be calculated by: ΔW t ¼ ΔW p þ ΔW a ¼

1 2

Z

ΔB2 1 dV p þ 2 μ0μ0

Z

ΔB2 dV a μ0

ð9Þ

ð6Þ

where Hm = a cos θ and Bm = Bp = 1.01mT are the maximum value of the H and B respectively, and δ is the phase difference between the H and B. Then, the complex relative permeability of the particle can be obtained as: μ r ¼ μ 0 þ iμ 00 ¼

B Bm ¼ ð cosδ‐i sinδÞ μ 0 H μ 0 Hm

ð7Þ

where μ' and μ '' are the real part and the imaginary part of the relative permeability.

Fig. 6. The energy change caused by the ellipsoidal particles with different shape factors.

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Fig. 7. The energy change caused by the ellipsoidal particle (re = 3.0) with different rotation angles.

where ΔWt is the total change of the magnetic energy, ΔWp and ΔWa are the changes of magnetic energy in particle and surrounding air, Vp and Va are the volumes of particle and surrounding air. 3. Simulation and result 3.1. The magnetic properties of the ellipsoidal particle with different shape factors and postures To study the difference of the magnetic disturbances caused by particles with different shapes, the magnetic properties of the spherical and ellipsoidal particles of the same volume are studied in this section. The radius of the spherical particle is set to ra = 125 μm. For ellipsoidal particles, the shape factors re is defined as (10) to adjust the exact shape of the particle.

re ¼

m s

ð10Þ

where m and s are the semi-major axis and semi-minor axis of the ellipsoidal particle. The distributions of the magnetic flux density of the ellipsoidal particles with shape factors re equaling to 1.0 (sphere), 2.0 and 3.0 are simulated and shown in Fig. 3. The results demonstrate that the magnetic distributions in spherical and ellipsoidal particles, under the static magnetic field, are uniform as shown in Fig. 3 a-c. However, under the alternating magnetic field with the frequency of 200 kHz, the eddy current

effect makes the magnetic distribution in the particles become more uneven as shown in Fig. 3 d-f. The detailed distributions of the magnetic flux density of the spherical and ellipsoidal particles are displayed in the Fig. 4. Fig. 4a and b show the distribution of magnetic flux density in and around the particles under the static magnetic field (the background magnetic flux density is 1.01mT). The results illustrate that the magnetic distribution in all the particles, under the static magnetic field, is uniform and among the spherical and ellipsoidal particles with the same volume, the magnetic flux density in the spherical particle is the lowest (3.02mT). With the increase of the shape factor re, the magnetic flux density in the ellipsoidal particles largely rises and reaches to 7.49mT when re = 3.0. Besides that, for the distribution of the magnetic flux density in the air, along the x axis, it gradually declines to the background magnetic flux density (1.01 mT). However, in the direction of the y axis, the magnetic flux density reduces to zero sharply at the surface of the particle and then gradually increases to the background magnetic flux density (1.01 mT). Fig. 4c and d show the distribution of magnetic flux density in and around the particles under the alternating magnetic field with the frequency of 200 kHz. The results indicate that, in this case, the variation rule of the magnetic flux density in the air is similar with that under the static magnetic field, and the magnetic flux density on the particle surface is much larger than that in the center of the particle. For the ellipsoidal particle with re = 3.0 and the spherical particle, in the direction of x axis, the magnetic flux densities on the particle surface are 8.25 mT and 2.77 mT respectively, and in the direction of y axis, the magnetic flux densities on the particle surface reach to 14.7 mT and 7.51 mT, respectively. On the whole, under arbitrary magnetic fields, the overall magnetic flux density around particles rises with the increase of re, which means that the ellipsoidal particles lead to larger magnetic energy change than the spherical particle of the same volume. Specifically, the energy variation caused by an ellipsoidal particle with re = 3.0 is more than twice as that caused by a spherical particle of the same volume (The energy change caused by particles with different shape factors are detailedly analyzed later and the results are shown in Fig. 6). Therefore, during the particle detection process, ellipsoidal particles cause larger induced electromotive force than spherical particles of the same volume. For the non-spherical particle, the various postures of particles passing through the magnetic field may cause the difference of the change of magnetic energy. Based on the orientation analysis of particles in the fluid, the ellipsoidal particle generally rotates a certain angle along the z axis. Therefore, the rotation angle of the particle β is defined as shown in Fig. 1b. To study the difference of the magnetic perturbations

Fig. 8. The magnetic distribution of the flaky particles under the different magnetic field.

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Fig. 9. The distribution of magnetic flux density in and around the flaky particles under different magnetic fields.

Fig. 10. The distribution of the magnetic flux density along the geometric axis of the flaky particle with different postures.

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caused by the particle with various postures, the magnetic distribution of an ellipsoidal particle with re = 3.0 at different rotation angles is simulated and displayed in Fig. 5. Fig. 5a and b show the distribution of the magnetic flux density along the major-axis and minor-axis of the particle under the static magnetic field. It can be clearly seen that, the magnetic flux density in the particle with different postures is evenly distributed and with the increase of the rotation angle, the magnetic flux density in the particle reduces gradually. When the rotation angle β = 0° (the major-axis of the particle is parallel to the x axis), the magnetic flux density in the particle is 7.51mT, and when the rotation angle increases to 90° (the major-axis of the particle is vertical to the x axis), the magnetic flux density in the particle sharply decreases to 2.16mT. Under the alternating magnetic field with the frequency of 200 kHz, the distributions of magnetic flux density along the direction of major-axis and minor-axis of the particle are displayed in Fig. 5c and d, respectively. The results indicate that the distributions of the magnetic flux density in the particle with different postures show obvious ununiformity and the overall magnetic flux density in the particle with smaller ration angle is higher than that in the particle with larger ration angle. When the rotation angle β = 0°, the magnetic flux density at the surface of the particle along the major-axis and minor-axis of the particle can reach to 8.24 mT and 14.7 mT respectively, however, when the rotation angle β = 90°, the magnetic flux density at the surface of the particle along the major-axis and minor-axis of the particle are only 3.19 mT and 2.21 mT respectively. That the distribution of magnetic flux density in the particle weakens with the increase of the rotation angle means that the particle posture greatly influences on the magnetic perturbation. More specifically, the energy variation caused by a horizontal ellipsoidal particle (the major-axis of the particle is parallel to the x axis) is more than twice as that caused by the same vertical ellipsoidal particle (the major-axis of the particle is vertical to the x axis. The energy changes caused by particles with different rotation angles are detailedly analyzed later and the results are shown in Fig. 7). It can be seen from Figs. 4 and 5, the changes of the magnetic distribution in particles and the surrounding air jointly lead to the energy variation of local magnetic field. Considering the eddy current loss and the hysteresis loss of the particles under the alternating magnetic field, the total change of the magnetic energy caused by the ellipsoidal particles with different shape factors (β = 0) are calculated and shown in Fig. 6. The results indicate that with the increase of field frequency, the energy change caused by each particle declines gradually. This phenomenon may because that the hysteresis loss and eddy current loss of the particles increase with the rise of the field frequency, which weakens the magnetic perturbation caused by particles. Meanwhile, the energy change caused by the ellipsoidal particle is much larger than that caused by the spherical particle of the same volume. Under the magnetic field with low frequency, the energy change caused by the spherical particle with a radius of 125 μm is 7.23 × 10−12 J, and the variation of magnetic energy caused by the ellipsoidal particle of the same volume (re = 3.0) is 6.58 × 10−11 J which is about nine times as much as the energy change caused by the spherical particle. To study the difference of the magnetic disturbances caused by the particle with different postures, the energy change caused by the ellipsoidal particle (re = 3.0) with different rotation angles is simulated and displayed in Fig. 7. It can be seen that with the increase of the field frequency, the overall change of magnetic energy caused by particles gradually declines. Meanwhile, the change of the magnetic energy is sensitive to the particle posture. With the increase of the rotation angle, under the magnetic field with the frequency of 10 kHz, the change of the magnetic energy caused by the particle decreases from 4.16 × 10−11 J to 8.24 × 10−12 J; and under the magnetic field with the frequency of 200 kHz, the energy change caused by the particle reduces from 1.32 × 10−11 J to 4.52 × 10−12 J. Because the exact posture of particles, when passing through the magnetic field, has some uncertainty and variability to a certain extent, the difference of the energy

Fig. 11. The energy change caused by the flaky particles with different shape factors.

changes, caused by the variation of the particle posture, leads to a poor consistency of the particle detection result. But it can be seen that higher frequency of the magnetic field improves the consistency of the particle detection result at a certain degree, however, weakens the detectability for the particle. 3.2. The magnetic properties of the flaky particle with different shape factors and postures The flaky particle is one kind of typical particle. In the mechanical system, the flaky metal wear particle always indicates the severe erosion wear [35].. In the microwave field, some specific flaky magnetic metal particles are used as a wave-absorbing material [36,37]. Therefore, the detection of the flaky magnetic particle is of great significance. The previous study illustrates that the exact shape parameters of the particle greatly affect the magnetic perturbation of the sensor, and further analysis shows that the thickness of the flaky particle has the greatest effect on the magnetic perturbation. Hence, the side lengths of flaky particles are supposed to be the same. In this case, the shape factor of the flaky particle rf is defined as: rf ¼

h h ¼ w l

ð11Þ

where w and l are the side lengths of the flake particle; h is thickness of the flaky particle. The magnetic distributions of the flaky particles, which have the same volume with the above spherical particle, were simulated in this section. In this simulation, the shape factor rf was set to 0.1, 0.2, 0.5 and 1.0 (cubic), respectively. Fig. 8a-d show the distributions of magnetic flux density of the flaky particles under the static magnetic field. It can be seen that differing from the spherical and ellipsoidal particles, the magnetic distribution in flaky particles, under the static magnetic field, is uneven, and the magnetic flux density reaches its maximum in the center of the particles. Meanwhile, the overall distributions of the magnetic flux density in the thinner flaky particles are significantly

Fig. 12. The energy change caused by the flaky particle (rf = 0.2) with different rotation angles.

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635

Fig. 13. The diagram of the experiment system.

greater than that in the thicker flaky particles. Fig. 8e-h show the distributions of the magnetic flux density of the flaky particles under the alternating magnetic field with the frequency of 200 kHz. The results indicate that the eddy current effect greatly increases the ununiformity of the magnetic distribution in the particles and the magnetic flux densities on the top and bottom surface of particles are much larger than that in the center of particles. However, with the decrease of the particle thickness, the magnetic ununiformity in particles is improved dramatically. The detailed distribution of the magnetic flux density in and around the flaky particles, along the x axis, y axis and z axis, are shown in Fig. 9ac. The results illustrate that the overall distribution of magnetic flux density in particles declines sharply with the increase of the shape factor re. Under the static magnetic field, the magnetic flux densities in the center of the particles with shape factor rf equaling to 0.1, 0.2, 0.5 and 1.0 are 13.2mT, 8.40mT, 4.85mT and 3.37mT, respectively, which means that the thinner flaky particle has the larger overage magnetic flux density and surely leads the greater energy change of the local magnetic field. Under the alternating magnetic field, the eddy current effect enhances the ununiformity of the magnetic distribution in particles and weakens the magnetic flux density in the center of particles. For the flaky particle with rf equaling 1.0, the magnetic flux density in the center of the particle is only 1.52mT which is reduced by 54.9% from that under static magnetic field (3.37mT). However, for the flaky particle with rf equaling 0.1, the magnetic flux density in the center of the particle is 12.7mT which is only 3.56% below that under static magnetic field (13.2mT). This phenomenon indicates that the eddy current is suppressed to a certain extent in the thinner flaky particle, which decreases loss the magnetic energy in particles.

The magnetic distributions of the flaky particle with different rotation angles are studied to estimate the change of the magnetic energy by the particle with various postures. Because of the symmetry of both the magnetic field in the sensor and the particle shape, the rotation angles along the x and y axes have no influence on the distribution of the magnetic flux density. Therefore, the rotation angle β of the particle along the z axis is defined as shown in Fig. 1c. The distributions of magnetic flux density on the geometric axes of the flaky particle with rf = 0.2, along its direction of the length, width, and thickness, are displayed in Fig. 10a-c. It can be seen that with the increase of the rotation angle, the overall distributions of magnetic flux density in the particle decrease gradually. Under the static magnetic field, the magnetic flux density in the center of the particle, when β = 0 and β = 90, are 8.87mT and 1.18mT respectively, which decreases by 86.7%. Under the highfrequency magnetic field, the eddy current effect in the particle is produced and becomes more significant with the rise of the rotation angle. When the rotation angle is 90°, the magnetic flux densities on the surfaces of the particle, along its length and thickness direction, rise greatly and reach to 5.94mT and 5.49mT respectively, which are much larger than that under the static magnetic field. Considering the hysteresis loss and eddy current loss of particles, the changes of the magnetic energy caused by the flaky particles (β = 0) and the spherical particle of the same volume are shown in Fig. 11. It can be clearly seen that with the increase of the frequency of the magnetic field, the change of the magnetic energy caused by each particle gradually declines, which weakens the magnetic perturbation. Meanwhile, the energy change caused by the flaky particles is larger than that caused by the spherical particle of the same volume. With the decrease of the shape factor rf (particles become thinner), the variation

Fig. 14. The adopted particles.

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Fig. 15. The measured induced electromotive force caused by the particles under various magnetic fields.

of the magnetic energy caused by the particle enhanced. The maximum energy changes caused by the spherical particle and the flaky particles with rf equaling to 1.0, 0.5, 0.2 and 0.1 are 7.23 × 10−12 J, 1.92 × 10−11 J, 2.83 × 10−11 J, 5.37 × 10−11 J, and 9.90 × 10−11 J, respectively, which shows that the variation of magnetic energy is closely related to the particle shape factors. The energy changes caused by a flaky particle with different rotation angles are calculated and shown in Fig. 12. The results illustrate that with the increase of the field frequency and the rotation angle, the change of magnetic energy caused by the particle gradually declines. Under the magnetic field with the frequency of 10 kHz, the energy change caused by the flaky particle with rf = 0.2 decreases from 5.21 × 10−11 J (β = 0) to 1.69 × 10−12 J (β = 90). With the increase of the magnetic field frequency to 200 kHz, the energy changes caused by the flaky particle (rf = 0.2), when β = 0 and β = 90, reduce to 1.43 × 10−11 J and 1.08 × 10−12 J, respectively, which reduces by 6.1% and 0.74% compared with the energy change under the magnetic field with the frequency of 100 kHz (1.52 × 10−11 J at β = 0 and 1.09 × 10−12 J at β = 90). The phenomenon indicates that the eddy current loss and the hysteresis loss in the particle increase sharply with the rise of the field frequency and stabilize gradually when the field frequency is larger than 100 kHz. Therefore, increasing the frequency of the magnetic field can improve the consistency of the detection results.

the spherical particle of the same volume, and the particle posture also influences significantly on the change of the magnetic energy. Therefore, the volume equivalent principle must cause a significant detection error. To improve the accuracy of the particle detection result, considering the difference of the magnetic properties of particles with various shapes and postures, the new equivalent functions he(f0, re, β) for the ellipsoidal particle and hf(f0, rf, β) for the flaky particle are proposed by fitting the data of Figs. 6–7 and 11–12. 0 he ðre ; f 0 ; β Þ ¼ @0:595 þ 

0

1 1:75  104   A 2 0:915 þ ðre −2:94Þ2  20:31 þ ð f 0 þ 59:34Þ

1 1:32−3:72  10‐3 f 0 −1:48  10‐2 β B þ7:08  10‐5 β2 þ 2:25  10‐5 f β C B C 0 B C 2 ‐4 @ 1 þ 0:031f −0:016β−1:31  10 f A 0 0 ‐4 2 ‐5 þ3:876  10 β −7:92  10 f 0 β

ð12Þ 7353 þ 5231r f þ 16:33f 0 þ 0:141 f 0 þ 42:28r f f 0

1 þ 5355r f þ 36 f 0 −666:4r f 2 −1:93  10‐2 f 0 þ 114:6r f f 0 1 1:40−4:28  10‐3 f 0 −1:95  10‐2 β B þ5:10  10‐5 β2 þ 4:59  10‐5 f β C B C 0 B C @ 1 þ 0:04f −0:02β−1:72  10‐4 f 2 A 2

0

0

þ3:8  10‐4 β2 −7:17  10‐5 f 0 β

4. Discussion The development of the high-precision electromagnetic microparticle detection technology asks urgently for the precise magnetic model of micro-particles. In the conventional particle detection process, the volume equivalent principle is generally used to simplify the various particles to sphere for estimating the size of the particle. However, the above study results show that the change of magnetic energy caused by the ellipsoidal and the flaky particle is larger than that caused by

!

2

  h f r f ; f 0; β ¼

0

ð13Þ The change of the magnetic energy caused by the ellipsoidal and flaky particle can be expressed as: Ee ¼ Es ðr a ; μ Þhe ðre ; f 0 ; βÞ E f ¼ Es ðr a ; μ Þh f ðr f ; f 0 ; βÞ

Fig. 16. The measured induced electromotive force caused by the particles at different rotation angles.

ð14Þ

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Fig. 17. The comparison analysis between normalized induced electromotive force and the normalized energy change.

where Es(ra, μ) is the change of the magnetic energy caused by an equivalent spherical particle, under the static magnetic field, ra is the equivalent radius of the particle calculated by the volume equivalent principle, μ is the magnetic permeability of particle material. Therefore, when estimating the size of the target particle through the change of magnetic energy, the real volume of the target particle should be expressed as: V e ¼ V s ðr a Þ=he ðre ; f 0 ; βÞ V f ¼ V s ðr a Þ=h f ðr f ; f 0 ; βÞ

ð15Þ

Where Vs(ra) is the volume of the equivalent spherical particle. Though most of the particles are spheroids and flakes approximately, the exact shape of particles is varied and generally unpredictable. Therefore, in the particle detection process, for the detection of particles with specific shape (e.g. cells marked by magnetic bead), the equivalent functions can be estimated precisely. However, for the detection of common particles, the shape distribution and the posture distribution of particles (generally satisfying the Gaussian distribution) should be considered for more accuracy detection result. 5. The experiment The study of the magnetic properties of particles demonstrates that the change of the magnetic energy caused by particles is sensitive to the particle shape and posture. Meanwhile, the energy variation can be reflected by the induced electromotive force of the particle detection sensor. To verify the variation rule of the magnetic energy caused by particles with different shapes and postures, the induced electromotive forces, when particles pass through the sensor, are measured. The diagram of the experiment system is shown in Fig. 13. The adopted particle detection sensor is a triple-coil electromagnetic particle detection sensor with an inner diameter of 2 mm, which is composed of two exciting coils and an inductive coil. The turns of exciting coil and inductive coil

are 45 and 38, respectively. A sinusoidal excitation signal (the peak current is 95 mA) is feed into the two exciting coils. Two magnetic fields with opposite direction and same magnitude are induced by the exciting coils on either side. The magnetic perturbation of the unilateral excitation coil caused by the particle makes the induction coil generate the induced electromotive force which is used to estimate the size of the particle. During the verification experiment, it's difficult to manufacture standard ellipsoidal particles, and the theoretical analysis illustrates that the change of the magnetic energy caused by ellipsoidal particles can be approximately estimated by that caused by the cylindrical particle which has the same major axis and minor axis as ellipsoidal particles. Therefore, the cylindrical and flaky particles of the same volume are adopted in the experiment. Fig .14a-d show the cylindrical particles with shape factor re equaling to 3.48, 2.22, 1.52 and 1.04, respectively. Fig .14e-h illustrate the flaky particles with shape factor rf equaling to 0.12, 0.31, 0.58 and 0.96, respectively. To precisely control the posture of each particle, when it passes through the sensor, the Perspex bars with the inclined angles α equaling to 20°, 30°, 40°, 50°, 60°, 70°, 80°, and 90° are made. Each particle is stuck on the center of the section of the Perspex bar with different inclined angles in sequence and move through the sensor separately. In this case, the induced electromotive force of the sensor, under the frequency of magnetic field ranges from 0 to 200 kHz, is measured to indirectly reflect the change of magnetic energy caused by single particle with different postures. Furthermore, the previous study [34] has shown that in the real particle detection process, the flow regime affects the posture of particles passing through the detection area but generally doesn't influence the magnetic perturbation caused by particles. Therefore, in the experiment, to better control the posture of particles passing through the detection area, the experiment is conducted under a fluid-free state. The induced electromotive force of the sensor caused by the ellipsoidal and flaky particles with different shape factors (β = 0), under the alternating magnetic field with various frequencies, are measured and

Fig. 18. The comparison analysis between normalized induced electromotive force and the normalized energy change caused by particles at different rotation angles.

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shown in Fig. 15a and b. The results illustrate that the induced electromotive force for all the ellipsoidal and flaky particles declines as the increase of the field frequency, which may because that the hysteresis loss and the eddy current loss in particles rapidly increase as the rise of the field frequency. Meanwhile, the signals of the ellipsoidal particles with re equaling to 3.48, 2.22, 1.51 and 1.09, under the magnetic field with the frequency of 4 kHz, are 336 mV, 296 mV, 202 mV and 170 mV, respectively, and in this situation, the signals of the flaky particles with rf equaling to 0.12, 0.31, 0.58 and 0.96 are 534 mV, 289 mV, 165 mV and 120 mV, respectively. The results indicate that longer ellipsoidal particles (larger re) and thinner flaky particles (smaller rf) lead to larger magnetic perturbation. Fig. 16 shows the induced electromotive force caused by the above ellipsoidal and flaky particles at different rotation angles. The results indicate that with the increase of the rotation angle, the induced electromotive force declines obviously. The signals of the horizontal ellipsoidal particle with re = 3.48 and the horizontal flaky particle with rf = 0.12 are 336 mV and 534 mV, respectively. However, when the rotation angle is 90°, the induced electromotive forces decrease to 100 mV and 63 mV which reduce by 70.2% and 88.2%. The experimental results demonstrate that the particle shape and posture indeed greatly influence the perturbation of the magnetic field. Therefore, that the various particles are simply equivalent to the sphere and ignoring the change of the particle shape and posture must lead to large detection error. To validate the correctness of the magnetic model of the particles and the variation rule of the magnetic energy caused by particles, taking the ellipsoidal particle with re = 3.48 and the flaky particle with rf = 0.119 as the examples, the measured induced electromotive force and the energy variation calculated by the theory are normalized. Fig. 17 and Fig. 18 show the comparison results between the induced electromotive force and the change of the magnetic energy both under the magnetic field with various frequencies and in the occasion of the particles rotate different angles. It can be seen that in all the situations the theoretical results and the measured results present on trends in consistency, which further indicates that the particle equivalent method considering the magnetic properties of particles can precisely estimate the magnetic perturbation caused by particles and greatly enhances the accuracy of the conventional particle equivalent model.

increase of the rotation angle, the change of the magnetic energy caused by particles gradually declines. (4) During the particle detection, that the particles with various shapes are simply equivalent to the sphere, based on the equivalent volume method, leads to a great detection error. Therefore, a new particle equivalent method considering the difference in the magnetic properties of the particles with different shapes and postures is proposed. Current advancements in magnetic property of particles with different shapes and postures will trigger research efforts to develop the high-precision magnetic particle detection sensors and make it possible to detect and distinguish different types of particles based on the difference in the magnetic properties of particles with various shapes. This research concerns the fields of semiconductor (metal contamination detection), precision machinery (micro wear particle detection) and the fields that currently use magnetic or magnetized particles as tracers in the research of chemical and biological processes (detection and recognition of bio-particles and macromolecular particles), but also some other field where magnetic particles have potential to be used. Author contributions R.J., B.M. and C.S.Z. conceived the project and planned the experiments. X.B presented the methodology. L.Y·W and Q.D performed the data analysis. R.J. and L.Y·W wrote the manuscript with the help of the all authors. Declaration of Competing Interest None. Acknowledgements This work was funded by National Natural Science Foundation of China (NSFC) (grant number:51475044) and Beijing finance found of science and technology planning project (grant number: KZ201611232032). References

6. Conclusions and future perspectives In conclusion, the magnetic models of ellipsoidal and flaky particles, under the alternating magnetic field, are built to study the magnetic properties of the particles and to precisely estimate the magnetic perturbation caused by particles. These models consider the hysteresis effect and the eddy current effect of the particles and study the difference of the magnetic disturbances caused by the particles with different shapes and postures. Based on this work, the following conclusions can be obtained: (1) With the increase of the frequency of the magnetic field, the hysteresis loss and eddy current loss in particles gradually raise, which weakens the magnetic disturbance of the sensor and reduces the induced electromotive force output by the sensor. (2) The magnetic disturbance of the sensor caused by ellipsoidal and flaky particles is larger than that caused by spherical particles of the same volume. Meanwhile, with the increase of the re (ellipsoidal particles become longer), the change of the magnetic energy caused by ellipsoidal particles boosts; and with the decrease of the rf (flaky particles become thinner), the change of the magnetic energy caused by flaky particles increases. (3) The posture of particles passing through the magnetic field has a great effect on the magnetic disturbance of the sensor. The horizontal ellipsoidal and flaky particles lead to the greatest change of the magnetic energy in the local magnetic field, and with the

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