simulation study on the motion of magnetic particles in silicone … · 2019. 6. 5. · of...

Research ArticleSimulation Study on the Motion of Magnetic Particles inSilicone Rubber-Based Magnetorheological Elastomers

Zhiqiang Xu 12 HengWu1 QiuliangWang1 Liyin Yi1 and JunWang1

1School of Mechanical Engineering Xiangtan University Xiangtan China2Engineering Research Center of Complex Tracks Processing Technology and Equipment of Ministry of EducationXiangtan University Xiangtan China

Correspondence should be addressed to Zhiqiang Xu 201721541849smailxtueducn

Received 5 June 2019 Accepted 27 June 2019 Published 18 July 2019

Guest Editor Xizhong An

Copyright copy 2019 Zhiqiang Xu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Magnetorheological elastomer (MRE) is an intelligent composite material and has been widely used in various fields such asvibration reduction and sensing MRE has an excellent magnetorheological effect through the chaining of its internal magneticparticles Current studies on MREs mainly focus on the preparation of materials and characterization of mechanical propertiesHowever very few studies have been conducted on the mechanism of magnetic particle motion during MRE curing Based onthe silicone rubber-based MRE the motion mechanism of magnetic particles during curing was explored through numericalsimulation First we analyzed the magnetic force and viscous force of magnetic particles in MRE and discussed the equationsof motion of magnetic particles under applied magnetic field Further we established a uniform magnetic field model through thefinite elementmethod and simulated themotion of twomagnetic particles under themagnetic field Finally we discussed the effectsof particle distribution angles particle radii appliedmagnetic field strength and distance between particles on particle velocity anddisplacement The results show that the distance between particles has the greatest influence on the motion of magnetic particlesand the size of the distance between particles will affect the contact time of the particles thus affecting the chain formation ofmagnetic particles in the MRE

1 Introduction

Magnetorheological elastomer (MRE) is a smart compositematerial that consists of a polymeric matrix with embeddedmicron-sized magnetic particles then solidifying to obtainan elastic material with variable stiffness under an appliedmagnetic field [1 2] MRE overcomes the shortcomingsof magnetorheological fluid (MRF) such as leakage easysedimentation and instability [3ndash5] MRE has a wide rangeof engineering applications because its variable stiffnesscharacteristics such as tuning dampers dampers sensorsand magnetorheological elastic polishing bodies [6ndash9] Theexcellent mechanical properties of MRE are determined bythe internal magnetic particles which will form a magneticparticle chain parallel to the direction of the magnetic fieldunder applied magnetic field [10] The anisotropic MREwith different mechanical properties was finally obtained bycontrolling the strength of the appliedmagnetic field and the

better the arrangement of particles the better the mechanicalproperties of MRE [11] Therefore in order to improve themechanical properties of MRE andmake it more widely usedin engineering applications further study on its motion ofmagnetic particles is essential

At present the research on magnetic particles in MREmainly focuses on the types of magnetic particles particlesize concentration of particles and the influence of theapplied magnetic field on the magnetic particle chain [12ndash14] However there are few studies on the chain formationof magnetic particles in MRE under magnetic field theexisting research generally analyzes the formation of chainby magnetic dipole theory [15 16] or study on the finalarrangement of magnetic particle chains by changing theinfluence of different external conditions [17] and there isless quantitative analysis of magnetic particle motion duringMRE curing In addition themagnetic particlemotion underthe applied magnetic field was difficult to observe because of

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8182651 11 pageshttpsdoiorg10115520198182651

2 Mathematical Problems in Engineering

(a)

B

(b)

Figure 1 Schematic diagramof chain formation undermagnetic field in the curing stage (a) before curing and (b) after curing undermagneticfield

the limitation of particle size so the motion of the particlewas often described by numerical simulation [18ndash21]

Based on the existing research the equation of motionof magnetic particles in silicone rubber has been establishedThe motion of particles under magnetic field was simulatedby the finite element method The formation of magneticparticle chains was explained by the change of velocityand displacement during particle motion and the effectsof two magnetic particles at different distribution anglesparticle radius applied magnetic field and different distancebetween the particles on motion process were discussedMeanwhile the contact time of magnetic particles underdifferent conditions was analyzed and the results show thatchanging the distance between two magnetic particles underthe same ratio significantly increases the efficiency of particlechain formation Moreover the larger the distance betweenthe particles the longer the contact time which providesan effective theoretical basis for preparing a silicone rubber-based MRE with excellent performance

2 Theoretical Model

21 Movement Equation In the preparation process of MREanisotropic MRE and isotropic MRE are obtained by con-trolling the presence or absence of a magnetic field in thecuring stage [22]Themagnetic particles in the isotropicMREare uniformly distributed in the matrix and the magneticparticles inside the anisotropic MRE form a special chainstructure under the magnetic field Figure 1 is a schematicdiagram showing the distribution of magnetic particles of ananisotropic MRE during the curing stage

Analysis of the force of the magnetic particles in theMRE during the curing stage mainly includes the magneticforce by the applied magnetic field the viscous force bythe silicone rubber matrix buoyancy gravity the interactionforce between the particles and the Brownian force [23]The magnetic particles produce motion under the combinedaction of these forces so the following equation is obtainedaccording to Newtons formula

119898119894 119889997888rarrV 119894119889119905 = 997888rarr119865119898119894 + 997888rarr119865 V119894 + 997888rarr119865 119903119894 + 997888rarr119865119892119894 + 997888rarr119865 119887119894 + 997888rarr119865119861119894 (1)

where119898119894 is themass ofmagnetic particle i 997888rarrV 119894 denotes theparticle velocity 997888rarr119865119898119894 is the applied magnetic force 997888rarr119865 V119894 is theviscous force 997888rarr119865 119903119894 is the interaction force between particles

and 997888rarr119865119892119894 997888rarr119865 119887119894 and 997888rarr119865119861119894 represent the gravity buoyancy andBrownian forces respectively

The magnetic particles used in this study are micron-sized particles thus gravity and buoyancy can be consideredto be balanced thus they are not considered Further themagnetic and viscous forces of micron-sized particles inincompressible Newtonian liquids are 200 times that of otherforces [18] Therefore to facilitate calculation for this thelast four terms in (1) are omitted and only the magneticand viscous forces of particles moving in the silicone rubbermatrix are considered and the equation of movement ofparticles can be simplified to

119898119894 119889997888rarrV 119894119889119905 = 997888rarr119865119898119894 + 997888rarr119865 V119894 (2)

In this case the ordinary differential equation of displace-ment can be expressed as follows

rarr119906 =997888rarr119865119898119894 minus 997888rarr119865 V119894

119898119894 (3)

The ordinary differential equation of velocity can also beexpressed as follows

997888rarrV 119894 =997888rarr119865119898119894 minus 997888rarr119865 V119894

119898119894 (4)

997888rarr119906 119894 = 997888rarrV 119894 (5)

where 997888rarr119906 119894 and 997888rarrV 119894 represent the velocity and displacement ofparticle movement respectively

22 Magnetic Force The magnetic force on the magneticparticles can be divided into two aspects one is the forceproduced by the magnetization of particles under the appliedmagnetic field and the other is the force between themagneticdipoles Assuming that themagnetic particles in themagneticfield are spherical and uniform in size the magnetic momentof the magnetic particle can be expressed as follows

997888rarr119898 = 119881997888rarr119872 = 431205871198773120594997888rarr119867 (6)

where V is the volume of a single particleV=(4pilowast119877and3)3 119877 is the radius 997888rarr119872 represents magnetization

Mathematical Problems in Engineering 3

N

XR

Y

H

xj

Fmj

Fmij

rij

ij

mi

mj

Fmi

S

rij yj

Fmij

ij

Fi

i

j

Fj

Figure 2 Force diagram of particles i and j in a 2D plane under applied magnetic field

997888rarr119872 = 120594997888rarr119867 120594 is the magnetic susceptibility and 997888rarr119867 is theapplied magnetic field strength

The magnetic force between the magnetic particles i andj is as follows

997888rarr119865119898119894119895 = 997888rarr119898 sdot nabla997888rarr119861 (7)

= sum119895 =119894

1205830997888rarr119898119894997888rarr11989811989541205871199033119894119895 [997888rarr119889 119894 sdot 997888rarr119889 119895 minus 3 (997888rarr119889 119894 sdot 119903119894119895) (997888rarr119889 119895 sdot 119903119894119895)] (8)

The magnetic force acting on particle i in the appliedmagnetic field is as follows

997888rarr119865119898119894 = 1205830997888rarr119898119894997888rarr119867(997888rarr119889 119894 sdot ℎ) (9)

Based on the above equation it is concluded that thecomprehensive magnetic force of particle i is as follows

997888rarr119865 119894119898 = 997888rarr119865119898119894119895 + 997888rarr119865119898119894 (10)

997888rarr119865 119894119898 = 31205830997888rarr119898119894997888rarr11989811989541205871199034119894119895 sum

119895 =119894

[(997888rarr119889 119894 sdot 997888rarr119889 119895) 119903119894119895 + (997888rarr119889 119894 sdot 119903119894119895)997888rarr119889 119895

+ (997888rarr119889 119895 sdot 119903119894119895)997888rarr119889 119894 minus 5 (997888rarr119889 119894 sdot 119903119894119895) (997888rarr119889 119895 sdot 119903119894119895) 119903119894119895]+ 1205830997888rarr119898119894997888rarr119867(997888rarr119889 119894 sdot ℎ)

(11)

When only twomagnetic particles act the motion of the par-ticles is a two-dimensional plane motion so (11) is rewritten

997888rarr119865 119894119898 = 4120587120583012059421198672119877631199034119894119895 [(1 minus 5 cos2120579119894119895) 119903119894119895 + 2 cos 120579119894119895ℎ] (12)

Then the magnetic field force 997888rarr119865 119894119898 is decomposed in the x andy directions and obtained its matrix expression

[[[

997888rarr119865 119894119898x997888rarr119865 119894119898119910

]]]= 41205871205830120594211986721198776

31199034119894119895[[(1 minus 5 cos2120579119894119895) sin 120579119894119895(3 minus 5 cos2120579119894119895) cos 120579119894119895

]]

(13)

Since the direction of the magnetic force of the particleis related to the angle in the equation (13) so let (1 minus5 cos2120579119894119895) sin 120579119894119895 = 0 and get 120579119894119895 = 1205872 or 120579119894119895 = arccos(radic155)Therefore if the particles are to form a particle chain parallelto the direction of the magnetic field the angle between thetwo particles 120579119894119895 lt arccos(radic55) is required to be satisfiedat this time and particles appear as close to each If 120579119894119895 gtarccos(radic55) the magnetic particles repel each other

As shown in Figure 2 1205830 is the vacuum permeability 997888rarr119898119894and997888rarr119898119895 are the particle magnetic moments

997888rarr119889 119894 and997888rarr119889 119895 are thedirection vectors of the magnetic moment of particles i andj 119903119894119895 is the direction vectors from particle i to particle j 997888rarr119867 isthe direction vector of the applied magnetic field intensity ℎis the direction vector of the applied magnetic field intensityrij denotes the distance between the centers of two particles iand j and satisfies the following requirement

119903119894119895 = 1003816100381610038161003816100381611990311989411989510038161003816100381610038161003816 (14)

23 Viscous Force In a static incompressible fluid the mag-netic particles are subjected to the viscous force 997888rarr119865 V of thematrix when they move in the polymer matrix The particlesize is less than 3 mm thus according to the Stokes [24]

997888rarr119865 V = minus6120587119877120578119889119903119889119905 (15)

119877 is the radius of the particle and 120578 is the dynamicviscosity The selected silicone rubber is purchased from


Start

A choose 2d dimension

B increase physical field MagneticFields- No Currents Moving Mesh

Global ODEs and DAEs

C select the transient in the presetsolver

D the geometric modeling forpermanent magnet and particles

E set the material

F add equation set up the parametersof the physical fields

G mesh the geometry

H calculate

I post-processing

End

Figure 3 Simulation flow chart

Dow Corning USA and the viscosity is 35 Pasdots at roomtemperature 119889119903119889119905 is the velocity of the particle motion

3 Numerical Simulation

Based on the force analysis of the magnetic particles in thesecond section it is shown that the magnetic particles moveunder magnetic force and viscous force In this section webring the derived motion equation into the finite elementsoftware to simulate the two magnetic particles under themagnetic field which links the magnetic particle motiontheory with the numerical simulation

First a two-dimensional plane is selected to establish anempty model and then the physical field interface is addedincludingmagnetic field no currentmovingmesh and globalordinary differential and differential algebraic equationsFinally transient research is added to three physics Themagnetic particles under the action of the field are simulatedThe magnetic field no current module generates the appliedmagnetic field required for the movement of the magneticparticles The moving mesh realizes the movement of themagnetic particles at different time steps The global ordi-nary differential and differential algebra equations provide

calculations for the velocity and displacement of the particlesThe specific flow chart of the simulation is shown in Figure 3

31 Establishment of Permanent Magnetic Field The appliedmagnetic field adopts a uniform magnetic field to ensurethe accuracy of the calculation result As shown in Figure 4the main magnetic field is generated by a pair of horizon-tally placed permanent magnets and the magnetic particlemoving region is located in the middle of the permanentmagnet Aiming to form a uniform magnetic field havingthe same magnetic induction intensity in the horizontal andvertical directions we used a yoke at the end of themagnet asshown in the red part of Figure 4 The magnetic field designparameters are shown in Table 1

In order to verify the uniformity of the two-dimensionalplane magnetic field the cut line 1 and the cut line 2 wererespectively plotted in the horizontal and vertical midpointpositions of the moving region of the particle and themagnetic flux density on the cut linewas calculatedAs shownin Figure 5 it can be found that after the application of theyoke the variation of the magnetic induction intensity in themoving direction of the particles in the horizontal directionand the vertical direction is remarkably reduced with respect


Cut line 2

Magnetic yoke

Magnetic

2

2 1

1

0 001 002 003 004 005 006 007 008 009 01

Magnetic

Movement field

Air domain

Cut line 1

s

d

dL

L s

Figure 4 Magnetic field generating device (the uniform magnetic field is generated by a pair of rectangular permanent magnets and yokesand the middle square area is the moving area of the magnetic particles)

28

Cut line 1

Length (um)

Cut line 2Cut line 1 fixedCut line 2 fixed

233

0

0015

0020

0025B (m

T)

0030

0035

0040

minus20 minus15 5 10 15 20minus10 minus5

Figure 5Themagnitude of the induced intensity on the horizontal and vertical lines (the solid line indicates the distribution of the magneticfield without the yoke and the dotted line indicates the distribution of the magnetic field with the yoke)

to the nonapplied yoke and thus can be regarded as a uniformmagnetic field

32 Simulation Results and Discussion Since the movementof magnetic particles in MRE directly affects its mechanicalproperties the single factor analysis method was used toanalyze the influence of four different factors on motionof particles in chain formation including the distributionangle between two magnetic particles particle radius thestrength of applied magnetic field and distance between theparticles In the progress of simulation the relative magneticpermeability of the particles is 3000 and the permanent

magnetic field described in Section 31 is used as the appliedmagnetic field of the particles

321 Influence of Distribution Angles on the Movement ofMagnetic Particles The different types of distribution anglesof magnetic particles in the matrix result in different move-ments of the particles under the magnetic field In thissection the relationship between the different distributionangles (120579) of twomagnetic particles and their motion param-eters was discussed The angle of distribution was selected tobe 0∘ 30∘ 60∘ and 90∘ the applied magnetic field strengthis 50 mT the radius of the magnetic particles is 10 120583m and


t=0s t=0001s t=0002s t=0003s t=00032s

t=0s t=0002s t=0004s t=0006s t=0008s t=001s t=0012s t=00136s

BS

L R

Nh

t=0 s t=0002s t=0004s t=0006s t=0008s t=001s t=0012s t=0014s

t=0s

(a)

(b)

(c)

(d)

t=0001s t=0002s t=0003s t=0004s t=00042s

d

d

=90∘

=60∘

=30∘

=0∘

2

1

Figure 6 Variation of particle movement position with time in different particle distribution angles ((a) (b) (c) and (d) corresponding tothe distribution angle between particles 120579 = 0∘ 30∘ 60∘ and 90∘)

Table 1 Simulation parameters of uniform magnetic field

Parameter Value UnitMovement field (1198891 lowast 1198892) 40lowast40 umMagnetic size (1198711 lowast 1198712) 80lowast50 umWidth of magnetic yoke (s) 10 um

the initial distribution distance of the two particles is 60 120583mFigure 6 shows the movement of the particles at differenttimes the black circle in the figure represents the initialdistribution of the particles and the red arrow representsthe direction of movement of the particles From the motionstate of the particles it can be found that the two particleshave two effects of attraction and repulsionWhen 120579 =0∘ 30∘and 60∘ the particles become mutually attracted and when120579 =90∘ the particles repel each other this is because whenthe angle between the particles is 90∘ it is known from (13)that 120579 = 90∘ gt arccos(radic55) so the particles are mutuallyexclusive

The three distribution angles of the two particles attractedto each other in Figure 6 were selected and the changesof velocity and displacement during the movement of theparticles were calculated as shown in (a) (b) and (c) ofFigure 7 As shown in Figure 7 the contact time to attracteach other is different where in the corresponding particlecontact time is 120579= 0∘ 30∘ and 60∘ and t = 00032s t=00042s and t= 00136s and as 120579 increases the contact timeis longer As shown in Figures 7(a) 7(b) and 7(c) enlargedview comparing the velocities of the particles at t= 0002sthe average velocity v=(v1+v2)2 corresponding to 120579= 0∘ 30∘

and 60∘ gradually decreases indicating that the smaller 120579 isthe larger moving speed of the particles is in the same timethereby the contact time of the two particles is shorten At thesame time Figures 7(a) 7(b) and 7(c) show the displacementchange of the different distribution angles as 120579 increasesu1 x and u2 x also gradually increases This is because theparticles first move in the x direction to reduce the angle 120579and then attract each other until the particles contact Alsoit can be seen from the displacements in the y direction thatthe u1 y and u2 y increase during the movement indicatingthat the particles have a vertical movement tendency fromthe beginning of the movement and this trend was moreobvious when the particles are in contact Finally due to theparticularity of the grid calculation when the particlemotionstops the calculation still uses the contact velocity as the finalvelocity In fact the velocity of the particle is 0 at this timethat is when the velocity of the particle tends to be constantand it can be regarded as the particles come into contact andthe movement stops

322 Effect of Different Particle Radius on the Movement ofMagnetic Particles MRE usually use micron-sized magneticparticles so the radius of particles also is a significant effect


0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)

Figure 7 The velocity and displacement of two particles with time at different distribution angles (the two particles corresponding to (a)(b) and (c) are 0∘ 30∘ and 60∘ respectively)


minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y

R=5umR=10umR=15umR=20um

u1_xu1_yu2_xu2_yR=5umR=10umR=15umR=20um

0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)

Figure 8 Variations of velocity and displacement during the mutual attraction of four different magnetic particle radii

on the magnetorheological effect of MRE To analyze themovement of magnetic particles with different radii undermagnetic field 5um 10um 15um and 20um ofmagnetic par-ticles radius were simulated and the changes of velocity anddisplacement parameters during the motion were obtainedIn order to consider the influence of the magnetic field inthe horizontal direction and the vertical direction the angle120579 between the particle connection and the applied magneticfield strength was 45∘ The distance between the two particleswas 60 um

As shown in Figure 8 the contact time and displacementof the particles become smaller with the larger particlesradius it is because the distance between the particles isconstant with the particle radius increasing and the corre-sponding particle volume and mass increase relatively andthen the magnetic force of the particle increases so that thecontact takes less time which demonstrated that controllingthe size of the magnetic particles can effectively control theparticle contact time

323 Effect of Different Strength of the Applied Magnetic Fieldon Particle Motion The greater the applied magnetic fieldstrength the greater the magnetic force of the particles Inorder to reflect the influence of different applied magneticfield strength on the movement of magnetic particles fourkinds of uniform magnetic fields were selected in this studyThe magnetic field strengths were 50mT 100mT 150mT and200mT respectively the corresponding magnetic particleshave a radius of 5 um the distance between the particles is30 um and the angle between the particles is 45∘

It can be seen from Figure 9 that the velocity and dis-placement of the two particles change significantly with thestrength of the applied magnetic field increases the velocityof the particles increases slowly especially at the beginning ofthe motion but the particles rapidly increase upon contactat the same time the displacement of the two particles in

the x and y directions gradually increases indicating that theparticles move in parallel and perpendicular to the directionof the magnetic field When the particles are in contactingthe displacement in the y direction increases rapidly andthe contact of the particles forms a particle chain parallelto the direction of the applied magnetic field From thecontact time of the two particles the greater the magneticfield strength the shorter the contact time of the particlesThe particle contact time is 0003s in 50 mT while the timeis 00002s in 200mT That is the time for the two particlesto be chained is reduced by 93 Because the stronger themagnetic field strength the stronger the magnetic force ofthe particles under the same magnetic permeability It showsthat increasing the strength of the applied magnetic field cansignificantly improve the efficiency of particle contact into thechain

324 Effect of Different Distance between Particles on ParticleMotion Due to the random distribution of magnetic parti-cles there are different distribution distances between the twoparticles In this section four different particle distributiondistances were selected which are 4lowastR 6lowastR 8lowastR and 10lowastRThemagnetic field is 50mT the radius of themagnetic particleR is 5um and the angle between the particle connection andthemagnetic field is 45∘ As shown in Figure 10 the longer thedistance between the particles the longer the contact time ofthe particles It can be seen from the displacement curve ofthe particles that the larger the distance between the particlesthat the larger the displacement in the x and y directions thelonger the contact time

325 Effect of Different Conditions on Contact Time of Mag-netic Particles Based on the analysis of the previous sectionsthe contact time t of the two particles at different angles (120579)particle radius (R) applied magnetic field strength (B) andthe distance (L) between the particles was considered and the


minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT

Figure 9 Variations in the velocity and displacement of magnetic particles with different applied magnetic field strengths

minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R

Figure 10 Variation of velocity and displacement of particles with time at different particle distribution distances

influence of the factor on themotion state of the particles wasjudged Select initial conditions 1205790 1198770 1198610 1198710 for 30∘ 10um100mT and 50um and the corresponding scale factor 120582=0510 15 and 20 under a single factor The contact time ofeach factor under the four scale factors was discussed Thecorresponding proportional conditions are shown in Table 2and the calculation results are shown in Table 3

According to Table 3 the change of particle contacttime under different conditions was obtained as shown inFigure 11 The height of the histogram reflects the contacttime t of the two particles and the red arrow reflects the

change of the contact time of the particles under differentratios As can be seen from Figure 11 changing the ratio ofthe four conditions has a significant effect on the contacttime of the two particles comparing the rate of change ofparticle contact time and the change rate of contact timecorresponding to the distance L between the two particles isthe largest reaching 1143 Secondly the particle radius Rthe magnetic field strength B and the distribution angle 120579and the corresponding contact time change rate are 345288 and 191 which means that if the particle chainformation efficiency is improved the particle contact time is


Table 2 Proportional condition setting

Factor 120582 = 05 120582 = 10 120582 = 15 120582 = 20120579 (∘) 05 lowast 1205790 1198770 1198610 1198710 1205790 1198770 1198610 1198710 15 lowast 1205790 1198770 1198610 1198710 2 lowast 1205790 1198770 1198610 1198710R (um) 1205790 05 lowast 1198770 1198610 1198710 1205790 1198770 1198610 1198710 1205790 15 lowast 1198770 1198610 1198710 1205790 2 lowast 1198770 1198610 1198710B (mT) 1205790 1198770 05 lowast 1198610 1198710 1205790 1198770 1198610 1198710 1205790 1198770 151198610 1198710 1205790 1198770 2 lowast 1198610 1198710L (um) 1205790 1198770 1198610 05 lowast 1198710 1205790 1198770 1198610 1198710 1205790 1198770 1198610 15 lowast 1198710 1205790 1198770 1198610 2 lowast 1198710

Table 3 Calculated contact time (t)

Factor 120582 = 05 120582 = 10 120582 = 15 120582 = 20120579 (∘) 432E-4 515E-4 760E-4 152E-3R (um) 290E-4 515E-4 103E-4 265E-5B (mT) 200E-3 515E-4 238E-4 140E-4L (um) 159E-5 515E-4 238E-3 642E-3

L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)

Figure 11 Change in contact time of two magnetic particles underdifferent conditions

shortened this is achieved by increasing the concentration ofthe particles to reduce the distance between the particles

For the magnetic particles in the silicone rubber-basedMRE this section discusses the effects of different distribu-tion angles particle radii applied magnetic field strengthand the distance between the particles on the velocity anddisplacement during mutual attraction and the results showthat these three different conditions have different effects onthe movement of the particles Secondly the contact time ofthe particles under different ratios was calculated and thedistance between the particles has the greatest influence onthe contact time and the influence of the particle distributionangle on the contact time of the particles is relatively small

4 Conclusions

In this paper through the theoretical analysis and simulationcalculation of magnetic particles in MRE the followingconclusions are obtained

(1) The force analysis of the magnetic particles in thesilicone rubber-based MRE was carried out and themotion model of the magnetic particles under theapplied magnetic field was established

(2) The motion of magnetic particles in the MRE weresimulated under the uniform magnetic field by thefinite element simulation method and the effects ofdifferent particle distribution angles different particlesizes different applied magnetic fields and differentparticle distances on the velocity and displacementduring particle motion were discussed the simulatedresult reveal that the distance between particles hasthe greatest influence on contact time

(3) The simulation on motion of two magnetic particlesprovided guiding for optimizing the chaining forma-tion conditions of MRE in the curing process

Data Availability

All data included in this study are available upon request bycontact with the corresponding author

Conflicts of Interest

The authors declared no potential conflicts of interest withrespect to the research authorship andor publication of thisarticle

Acknowledgments

This research was funded by the National Natural ScienceFoundation of China (grant nos 51605409 and 51605410) andthe Educational Committee Foundation of Hunan (grant no17C1531) The authors are grateful for the support

References

[1] M R Jolly J D Carlson B C Munoz and T A BullionsldquoThe magnetoviscoelastic response of elastomer composites


consisting of ferrous particles embedded in a polymer matrixrdquoJournal of Intelligent Material Systems and Structures vol 7 no6 pp 613ndash622 1996

[2] L Chen X-L GongW-Q Jiang J-J Yao H-X Deng andW-H Li ldquoInvestigation on magnetorheological elastomers basedon natural rubberrdquo Journal of Materials Science vol 42 no 14pp 5483ndash5489 2007

[3] M Schumann and S Odenbach ldquoIn-situ observation of theparticle microstructure of magnetorheological elastomers inpresence of mechanical strain and magnetic fieldsrdquo Journal ofMagnetism and Magnetic Materials vol 441 pp 88ndash92 2017

[4] N F Alias A G Muthalif K A Arpan and N D NordinldquoExperimental investigation of static properties of magnetorhe-ological elastomerrdquo Iranian Journal of Science and TechnologyTransaction A-Science pp 1ndash13 2018

[5] Y Li J Li W Li and H Du ldquoA state-of-the-art review onmagnetorheological elastomer devicesrdquo Smart Materials andStructures vol 23 no 12 Article ID 123001 2014

[6] S B Kumbhar S Chavan and S Gawade ldquoAdaptive tunedvibration absorber based on magnetorheological elastomer-shape memory alloy compositerdquoMechanical Systems and SignalProcessing vol 100 pp 208ndash223 2018

[7] A K Bastola and L Li ldquoA new type of vibration isolator basedon magnetorheological elastomerrdquoMaterials amp Design vol 157pp 431ndash436 2018

[8] I Bica ldquoMagnetoresistor sensor with magnetorheological elas-tomersrdquo Journal of Industrial and Engineering Chemistry vol 17no 1 pp 83ndash89 2011

[9] Z Xu QWang K Zhu S Jiang HWu and L Yi ldquoPreparationand characterization of magnetorheological elastic polishingcompositesrdquo Journal of Intelligent Material Systems and Struc-tures vol 30 no 10 pp 1481ndash1492 2019

[10] A Boczkowska and S Awietjan ldquoMicrostructure and proper-ties of magnetorheological elastomersrdquo Advanced Elastomers -Technology Properties and Applications vol 595 2012

[11] K Danas S Kankanala and N Triantafyllidis ldquoExperimentsand modeling of iron-particle-filled magnetorheological elas-tomersrdquo Journal of the Mechanics and Physics of Solids vol 60no 1 pp 120ndash138 2012

[12] I A Perales-Martınez L M Palacios-Pineda L M Lozano-Sanchez O Martınez-Romero J G Puente-Cordova and AElıas-Zuniga ldquoEnhancement of a magnetorheological PDMSelastomer with carbonyl iron particlesrdquo Polymer Testing vol 57pp 78ndash86 2017

[13] T Liu X Gong Y Xu S Xuan and W Jiang ldquoSimulationof magneto-induced rearrangeable microstructures of magne-torheological plastomersrdquo So Matter vol 9 no 42 pp 10069ndash10080 2013

[14] J Li X Gong Z Xu and W Jiang ldquoThe effect of pre-structureprocess on magnetorheological elastomer performancerdquo Inter-national Journal of Materials Research vol 99 no 12 pp 1358ndash1364 2008

[15] M R Jolly J D Carlson and B C Munoz ldquoA model of thebehaviour of magnetorheological materialsrdquo Smart Materialsand Structures vol 5 no 5 pp 607ndash614 1996

[16] A Tsutomu H Noriyuki and W Hitoshi ldquoNumerical simula-tion of chainlike cluster movement of feeble magnetic particlesby induced magnetic dipole moment under high magneticfieldsrdquo Science and Technology of Advanced Materials vol 10Article ID 014609 2009

[17] M Yu B X Ju J Fu X Liu and Q Yang ldquoInfluence ofcomposition of carbonyl iron particles on dynamic mechanicalproperties of magnetorheological elastomersrdquo Journal of Mag-netism and Magnetic Materials vol 324 no 13 pp 2147ndash21522012

[18] Q Cao Z Li Z Wang F Qi and X Han ldquoDisaggregation andseparation dynamics ofmagnetic particles in amicrofluidic flowunder an alternating gradient magnetic fieldrdquo Journal of PhysicsD Applied Physics vol 51 no 19 p 195002 2018

[19] A Sand J F Stener M O Toivakka J E Carlson and BI Palsson ldquoA Stokesian dynamics approach for simulation ofmagnetic particle suspensionsrdquo Minerals Engineering vol 90pp 70ndash76 2016

[20] R Soda K Tanaka K Takagi and K Ozaki ldquoSimulation-aided development of magnetic-aligned compaction processwith pulsed magnetic fieldrdquo Powder Technology vol 329 pp364ndash370 2018

[21] M Hashemi M Manzari and R Fatehi ldquoDirect numericalsimulation of magnetic particles suspended in a Newtonianfluid exhibiting finite inertia under SAOSrdquo Journal of Non-Newtonian Fluid Mechanics vol 256 pp 8ndash22 2018

[22] H S Jung S H Kwon H J Choi J H Jung and Y G KimldquoMagnetic carbonyl ironnatural rubber composite elastomerand its magnetorheologyrdquo Composite Structures vol 136 pp106ndash112 2016

[23] U Banerjee P Bit R Ganguly and S Hardt ldquoAggregationdynamics of particles in a microchannel due to an appliedmagnetic fieldrdquoMicrofluidics and Nanofluidics vol 13 no 4 pp565ndash577 2012

[24] S Krishnamurthy A Yadav P E Phelan et al ldquoDynamicsof rotating paramagnetic particle chains simulated by particledynamics Stokesian dynamics and lattice BoltzmannmethodsrdquoMicrofluidics and Nanofluidics vol 5 no 1 pp 33ndash41 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


(a)

B

(b)

Figure 1 Schematic diagramof chain formation undermagnetic field in the curing stage (a) before curing and (b) after curing undermagneticfield

the limitation of particle size so the motion of the particlewas often described by numerical simulation [18ndash21]

Based on the existing research the equation of motionof magnetic particles in silicone rubber has been establishedThe motion of particles under magnetic field was simulatedby the finite element method The formation of magneticparticle chains was explained by the change of velocityand displacement during particle motion and the effectsof two magnetic particles at different distribution anglesparticle radius applied magnetic field and different distancebetween the particles on motion process were discussedMeanwhile the contact time of magnetic particles underdifferent conditions was analyzed and the results show thatchanging the distance between two magnetic particles underthe same ratio significantly increases the efficiency of particlechain formation Moreover the larger the distance betweenthe particles the longer the contact time which providesan effective theoretical basis for preparing a silicone rubber-based MRE with excellent performance

2 Theoretical Model

21 Movement Equation In the preparation process of MREanisotropic MRE and isotropic MRE are obtained by con-trolling the presence or absence of a magnetic field in thecuring stage [22]Themagnetic particles in the isotropicMREare uniformly distributed in the matrix and the magneticparticles inside the anisotropic MRE form a special chainstructure under the magnetic field Figure 1 is a schematicdiagram showing the distribution of magnetic particles of ananisotropic MRE during the curing stage

Analysis of the force of the magnetic particles in theMRE during the curing stage mainly includes the magneticforce by the applied magnetic field the viscous force bythe silicone rubber matrix buoyancy gravity the interactionforce between the particles and the Brownian force [23]The magnetic particles produce motion under the combinedaction of these forces so the following equation is obtainedaccording to Newtons formula

119898119894 119889997888rarrV 119894119889119905 = 997888rarr119865119898119894 + 997888rarr119865 V119894 + 997888rarr119865 119903119894 + 997888rarr119865119892119894 + 997888rarr119865 119887119894 + 997888rarr119865119861119894 (1)

where119898119894 is themass ofmagnetic particle i 997888rarrV 119894 denotes theparticle velocity 997888rarr119865119898119894 is the applied magnetic force 997888rarr119865 V119894 is theviscous force 997888rarr119865 119903119894 is the interaction force between particles

and 997888rarr119865119892119894 997888rarr119865 119887119894 and 997888rarr119865119861119894 represent the gravity buoyancy andBrownian forces respectively

The magnetic particles used in this study are micron-sized particles thus gravity and buoyancy can be consideredto be balanced thus they are not considered Further themagnetic and viscous forces of micron-sized particles inincompressible Newtonian liquids are 200 times that of otherforces [18] Therefore to facilitate calculation for this thelast four terms in (1) are omitted and only the magneticand viscous forces of particles moving in the silicone rubbermatrix are considered and the equation of movement ofparticles can be simplified to

119898119894 119889997888rarrV 119894119889119905 = 997888rarr119865119898119894 + 997888rarr119865 V119894 (2)

In this case the ordinary differential equation of displace-ment can be expressed as follows

rarr119906 =997888rarr119865119898119894 minus 997888rarr119865 V119894

119898119894 (3)

The ordinary differential equation of velocity can also beexpressed as follows

997888rarrV 119894 =997888rarr119865119898119894 minus 997888rarr119865 V119894

119898119894 (4)

997888rarr119906 119894 = 997888rarrV 119894 (5)

where 997888rarr119906 119894 and 997888rarrV 119894 represent the velocity and displacement ofparticle movement respectively

22 Magnetic Force The magnetic force on the magneticparticles can be divided into two aspects one is the forceproduced by the magnetization of particles under the appliedmagnetic field and the other is the force between themagneticdipoles Assuming that themagnetic particles in themagneticfield are spherical and uniform in size the magnetic momentof the magnetic particle can be expressed as follows

997888rarr119898 = 119881997888rarr119872 = 431205871198773120594997888rarr119867 (6)

where V is the volume of a single particleV=(4pilowast119877and3)3 119877 is the radius 997888rarr119872 represents magnetization


N

XR

Y

H

xj

Fmj

Fmij

rij

ij

mi

mj

Fmi

S

rij yj

Fmij

ij

Fi

i

j

Fj





= sum119895 =119894





997888rarr119865 119894119898 = 997888rarr119865119898119894119895 + 997888rarr119865119898119894 (10)


119895 =119894



(11)




[[[

997888rarr119865 119894119898x997888rarr119865 119894119898119910

]]]= 41205871205830120594211986721198776


]]

(13)




119903119894119895 = 1003816100381610038161003816100381611990311989411989510038161003816100381610038161003816 (14)


997888rarr119865 V = minus6120587119877120578119889119903119889119905 (15)



Start






E set the material


G mesh the geometry

H calculate

I post-processing

End










Cut line 2

Magnetic yoke

Magnetic

2

2 1

1

0 001 002 003 004 005 006 007 008 009 01

Magnetic

Movement field

Air domain

Cut line 1

s

d

dL

L s


28

Cut line 1

Length (um)


233

0

0015

0020

0025B (m

T)

0030

0035

0040








t=0s t=0001s t=0002s t=0003s t=00032s


BS

L R

Nh


t=0s

(a)

(b)

(c)

(d)

t=0001s t=0002s t=0003s t=0004s t=00042s

d

d

=90∘

=60∘

=30∘

=0∘

2

1









0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)



minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









N

XR

Y

H

xj

Fmj

Fmij

rij

ij

mi

mj

Fmi

S

rij yj

Fmij

ij

Fi

i

j

Fj





= sum119895 =119894





997888rarr119865 119894119898 = 997888rarr119865119898119894119895 + 997888rarr119865119898119894 (10)


119895 =119894



(11)




[[[

997888rarr119865 119894119898x997888rarr119865 119894119898119910

]]]= 41205871205830120594211986721198776


]]

(13)




119903119894119895 = 1003816100381610038161003816100381611990311989411989510038161003816100381610038161003816 (14)


997888rarr119865 V = minus6120587119877120578119889119903119889119905 (15)



Start






E set the material


G mesh the geometry

H calculate

I post-processing

End










Cut line 2

Magnetic yoke

Magnetic

2

2 1

1

0 001 002 003 004 005 006 007 008 009 01

Magnetic

Movement field

Air domain

Cut line 1

s

d

dL

L s


28

Cut line 1

Length (um)


233

0

0015

0020

0025B (m

T)

0030

0035

0040








t=0s t=0001s t=0002s t=0003s t=00032s


BS

L R

Nh


t=0s

(a)

(b)

(c)

(d)

t=0001s t=0002s t=0003s t=0004s t=00042s

d

d

=90∘

=60∘

=30∘

=0∘

2

1









0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)



minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









Start






E set the material


G mesh the geometry

H calculate

I post-processing

End










Cut line 2

Magnetic yoke

Magnetic

2

2 1

1

0 001 002 003 004 005 006 007 008 009 01

Magnetic

Movement field

Air domain

Cut line 1

s

d

dL

L s


28

Cut line 1

Length (um)


233

0

0015

0020

0025B (m

T)

0030

0035

0040








t=0s t=0001s t=0002s t=0003s t=00032s


BS

L R

Nh


t=0s

(a)

(b)

(c)

(d)

t=0001s t=0002s t=0003s t=0004s t=00042s

d

d

=90∘

=60∘

=30∘

=0∘

2

1









0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)



minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









Cut line 2

Magnetic yoke

Magnetic

2

2 1

1

0 001 002 003 004 005 006 007 008 009 01

Magnetic

Movement field

Air domain

Cut line 1

s

d

dL

L s


28

Cut line 1

Length (um)


233

0

0015

0020

0025B (m

T)

0030

0035

0040








t=0s t=0001s t=0002s t=0003s t=00032s


BS

L R

Nh


t=0s

(a)

(b)

(c)

(d)

t=0001s t=0002s t=0003s t=0004s t=00042s

d

d

=90∘

=60∘

=30∘

=0∘

2

1









0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)



minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









t=0s t=0001s t=0002s t=0003s t=00032s


BS

L R

Nh


t=0s

(a)

(b)

(c)

(d)

t=0001s t=0002s t=0003s t=0004s t=00042s

d

d

=90∘

=60∘

=30∘

=0∘

2

1









0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)



minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









0000

0000 0002

004002000

0004

v1=000488

v2=000576

00 0

024u

(um

)

10

20v

(ms

) 01

02

03

0005

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0010 0015 0020t (s)

0000

0000 0001 0002

0005 0010 0015 0020t (s)

minus01

minus002

minus004minus02

minus03 minus20

minus2minus4

minus10

(a)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

0000

0000

000

002

0002

v1=000379

v2=0003860004

00

01

v (m

s)

02

0005 0010 0015 0020t (s)

minus02minus002

minus01

0

5

10

15

minus10

minus5

minus15

(b)

u (u

m)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

0000 0005 0010 0015 0020t (s)

v (m

s)

0000 0005

0000

000001002003

0004

v2=000213

v1=000288

0008 0012

0010 0015 0020t (s)

000

01

02

10

20

30

minus10minus01

minus02minus003

minus002

minus001

minus20

minus30

(c)



minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









minus01 minus5

minus10

minus15

minus20

minus25

minus02

minus03

minus04

v1_xv1_yv2_xv2_y



0000

0001

v (m

s)

020304

0005 0010 0015 0020 0025 0030 0035t (s)

0000

0

u (u

m) 5

10152025

0005 0010 0015 0020 0025 0030 0035t (s)










minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018









minus16 minus12

minus8

minus4

minus12

minus08

minus04

v1_x

00000

00 0

u (u

m)

4

8

12

04

v (m

s)

08

12

16

00005

B=150mT

B=200mT

B=100mT B=50mT

00010 00015t (s)

00020 00025 00030 00000 00005 00010 00015t (s)

00020 00025 00030

v1_yv2_xv2_y

u1_xu1_yu2_xu2_y

50mT100mT150mT200mT


minus04

minus16

minus12

minus8

minus4

minus03

minus02

minus01

0000 0005 0010 0015 0020t (s)

0000

00 048

121620

v (m

s)

u (u

m)01

02

03

04

0005 0010 0015

L=10RL=4R

L=6R L=8R

0020t (s)

v1_xv1_yv2_xv2_y

u1_xu1_yu2_xu2_y

L=4RL=6RL=8RL=10R










L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018













L (um)B (mT)R (um)

1143

288345191

0006

0005

0004

0003

0002

0001

0000

t (s)

=05=10=15=20

(∘)




4 Conclusions





Data Availability




Acknowledgments


References


































Journal of












Journal of







Volume 2018






Volume 2018








































Journal of












Journal of







Volume 2018






Volume 2018








simulation study on the motion of magnetic particles in silicone … · 2019. 6. 5. · of...

Documents