Special Issue Article

Advances in Mechanical Engineering2015, Vol. 7(7) 1–11� The Author(s) 2015DOI: 10.1177/1687814015589686aime.sagepub.com

Dynamics of ferromagnetic nanowiresin a rotating magnetic field

Lixin Yang1, Nan Zhao1 and Dong Liu2

AbstractManipulating nanowires with external magnetic fields has emerged as a powerful tool in various engineering applications,which prompts an urgent need to better understand the dynamics of nanowire rotation under different control condi-tions. In this article, the motion of ferromagnetic nickel (Ni) nanowires under a rotating magnetic field was investigatedboth theoretically and experimentally. The synchronous and asynchronous rotations were characterized in detail.Analytical models were developed for the major modes of motion by solving the governing equations of rotation.Particularly, a selection of theoretical formula for fluid viscous torque on nanowires of large aspect ratios was madebased on the computational fluid dynamics simulation results. The comparisons of the theoretical prediction and theexperimental data showed very good agreement. The effects of various system variables, such as the strength and rotat-ing frequency of the magnetic field and the nanowire aspect ratio, were examined. Hence, the insights gained from thiswork can be applied to future exploration of magnetic manipulation of nanowires.

KeywordsFerromagnetic nanowire, rotation magnetic field, synchronous and asynchronous rotation, viscous torque, analystresolution

Date received: 25 December 2014; accepted: 12 May 2015

Academic Editor: Hyung Hee Cho

Introduction

Manipulating rod-like nanoparticles, such as nano-wires, with external magnetic fields has attracted exten-sive interests in a myriad of engineering applications inmicro/nanofluidics and biomedical engineering.1–15 Forinstance, in a study of nanowire alignment in solidify-ing films where the fluid viscosity increased with time,it was found that a large portion of the nanowires can-not properly align due to the viscous force. Hence, byanalyzing the relaxation time of a single nanowireunder different viscosities, the total time required forthe whole field to align can be extracted.1 To measurethe protein generation rate, proteins were adhered tothe surface of nanowires. The nanowire’s geometry wasmodified by the protein coating and their rotation tra-jectories were changed. Thus, the average generationrate of protein can be determined from the

measurement of the angle variety of single nanowire indifferent position.3 In an application involving laserbeam transmission, the beam transmission was accu-rately controlled with manipulating the movement ofnanorods.9 Other applications requiring specific nano-wire motion include nanomotor,12 drug delivery,14 and

1School of Mechanical, Electronic and Control Engineering, Beijing

Jiaotong University, Beijing, China2Department of Mechanical Engineering, University of Houston,

Houston, TX, USA

Corresponding authors:

Lixin Yang, School of Mechanical, Electronic and Control Engineering,

Beijing Jiaotong University, Beijing 100044, China.

Email: lxyang1@bjtu.edu.cn

Dong Liu, Department of Mechanical Engineering, University of Houston,

Houston, TX 77204, USA.

Email: dongliu@uh.edu

Creative Commons CC-BY: This article is distributed under the terms of the Creative Commons Attribution 3.0 License

(http://www.creativecommons.org/licenses/by/3.0/) which permits any use, reproduction and distribution of the work without

further permission provided the original work is attributed as specified on the SAGE and Open Access pages (http://www.uk.sagepub.com/aboutus/

openaccess.htm).

nanosensors.15 Therefore, a good understanding of thedynamics of nanowire motion is imperative to ensurethe success and future advancement of these newtechnologies.

The study of magnetic rotation of small particleswas initially conducted with a pair of bounded spheri-cal particles or single ellipsoidal particle. Depending onthe angular frequency of the magnetic field, either syn-chronous rotation, in which the lag angle between themagnetic field and the long axis of the ellipsoidal nano-particle (or the centerline connecting the pair of spheri-cal nanoparticles) remains constant, or asynchronousrotation, in which the lag angle exhibits periodic oscil-lation, can be observed.16,17 Analytical models havebeen successfully developed to account for the mechan-isms of both modes of particle motion.18,19 However,new challenges arise when these models are extended todescribe the magnetic rotation of nanowires, which aretypically characterized by extremely large aspect ratios(defined as the ratio of nanowire length to its dia-meter). The anisotropic shape of the nanowires can nolonger be approximated by ellipsoidal geometries,thereby rendering the existing theoretical models of theviscous torque inaccurate.18 Some efforts have beentaken to rectify this issue. For example, Edwards andDoi20 treated the slim nanowire as a string of spheresin ‘‘shish-kebab’’ model and calculated the viscous tor-que as a summation of those experienced by the indi-vidual constituent spheres. Tirado and De La Torre21

and Tirado et al.22 followed a similar approach in con-sidering the nanowire as a cluster of spheres, whichrefined the accuracy of solution. Keshoju et al.6 mod-eled the nanowire as several segments of short cylindersand estimated the overall viscous torque as the accumu-lation of each of them. Despite the improvement by thenew models, the applicability of these models is limitedto a specific range of nanowire aspect ratios.Additionally, the available studies in the open literaturefocus on either theoretical analysis or experimentalcharacterization of the nanowire motion in a magneticfield. Much essential information of the underlyingrotation dynamics, especially that which is related tothe experimental corroboration of theoretical models,is scattered or even missing. Therefore, it is the goals ofthis work to perform an in-depth analysis of thedynamics of magnetic field–driven nanowire rotation,to conduct comprehensive experimental measurementsof the key parameters, and, eventually, to compare thetheoretical predictions with the measured data.

In this article, the motion of ferromagnetic nickel(Ni) nanowires under a rotating magnetic field wasinvestigated both theoretically and experimentally. Thesynchronous and asynchronous rotations were charac-terized in detail. Analytical models were developed forthe major modes of motion by solving the governingequations of rotation. Particularly, a selection of fluid

viscous torque was made according to computationalfluid dynamics (CFD) simulations for nanowires oflarge aspect ratios. The comparisons of the theoreticalprediction and the experimental data show very goodagreement. The effects of various system variables, suchas the strength and rotating frequency of the magneticfield and the nanowire aspect ratio, were examined.Hence, the insights gained from this work can beapplied to future exploration of magnetic manipulationof nanowires.

Experimental apparatus and method

Ni nanowires were synthesized using electrochemicaldeposition in porous anodic aluminum oxide (AAO)templates.23 The average nanowire radius r was 300nmand the average length l was in the range of 10–50 mm,as determined by scanning electron microscopy (SEM).Dry nanowire powders were dispersed in ethylene gly-col (EG) to formulate the sample solution, and theresulting volume fraction was less than 0.1 vol%. Atthis concentration, the nanowire can be deemed asfreely rotating without interfering with its nearestneighbors.24 The dilute solution was then pre-magnetized by a piece of Nd-Fe-B magnet with 1Tmagnetic intensity and solicited for 30 min to ensureuniform dispersion. Prior to the magnetic rotationexperiments, 1.5mL sample solution was collected andtransferred into a polystyrene container, which mea-sures 20mm in diameter and 5mm in height. A singlenanowire suspended at the middle height of the con-tainer was selected as the test wire for observation,although all the nanowires were experiencing similarrotation motion. The experiments were conducted atan ambient temperature of 15 �C, at which point theviscosity of the EG solution is 26mPa s. The relativehigh viscosity helps to alleviate the precipitation ofnanowires and improve the solution stability.

A pair of Helmholtz coils was mounted on a rotat-ing disk driven by a brushed direct current (DC) motorto create the magnetic field with controlled angular fre-quency (as shown in Figure 1). The field strength canbe varied from 0 to 20Gs, and the rotating frequencyof the field was between 1 and 3Hz. Dynamics of nano-wire rotation was visualized using a biological micro-scope (Eclipse Ti-U; Nikon). A 503 objective lens (MPlan APO; Mitutoyo) was used to achieve sufficientmagnification and long working distance. A high-speedcharge-coupled device (CCD) camera (X-MOTION;AOS Technology) was employed to record the real-time nanowire rotation with a frame rate of up to 1000frames per second (fps) and a maximum resolution of1280 3 600 pixels. Static images were extracted fromthe recorded videos and analyzed with an in-houseMATLAB code to yield information of the location

2 Advances in Mechanical Engineering

and phase angle of the nanowire at desired timeinstants.

During each experiment, a stationary magnetic fieldwas first established by actuating the Helmholtz coils.The initially randomly oriented nanowires quicklyaligned along the direction of the field. Then, the DCmotor was turned on to rotate the magnetic field at aspecified frequency, and the high-speed camera startedsimultaneously to record the nanowire rotation. It wasfound that a frame rate of 63 fps and a resolution of800 3 600 pixels were sufficient to yield an optimalimage quality for subsequent analysis. In a series ofexperiments, the rotating frequency was varied andboth synchronous and asynchronous motions of thesingle nanowire with respect to the magnetic field wereobserved.

Theoretical analysis

Magnetically driven rotation of a single nanowire isshown schematically in Figure 2. The external magneticfield with frequency vH and strength H was applied in

the horizontal plane (X–Y plane) at an instantaneousangle uH around the Z-axis (vH= duH/dt). The pre-magnetized nanowire tends to follow the rotating fieldwith the long axis laying in the same plane and aninstantaneous angle of uW. However, due to inertia andviscous drag from the surrounding medium, there isalways a lag angle uL between the nanowire and thefield directions, uL= uH2 uW. The nanowire rotatingfrequency vW (vW= duW/dt) also usually differs fromvH during the dynamic process.

The nanowire rotation is governed by the magnetictorque, the fluid viscous torque, and the Brownianmotion. The magnetic torque provides the driving forcefor the nanowire motion, whereas the viscous torqueacts as the resisting mechanism. Due to the extremesmall size of nanowire, the Brownian motion may havesome effect on the rotation dynamics. All these factorswill be discussed in this section.

Magnetic torque

The magnetic torque tm of the nanowire is given by25

Figure 1. Schematic of the experimental setup.

Figure 2. Schematic of a magnetically driven nanowire.

Yang et al. 3

tm =~m 3 ~H =Mspr2lH sin uL ð1Þ

where ~m is the magnetic moment, ~H is the applied mag-netic field, and Ms (Ms=4.85 3 105 A/m for nickelnanowires in this work) is the spontaneous magnetiza-tion. Equation (1) shows clearly that tm reaches themaximum when the nanowire’s orientation is perpendi-cular to the magnetic field (uL=p/2 and 3p/2) andvanishes when they are perfectly aligned (uL=0and p).

Fluid viscous torque

Due to its nanoscale size, the Reynolds number of theflow is far smaller than 1. So, the viscous force domi-nated the fluid field. Motion of the nanowire can betreated as the rotation of a long slender cylinder inStokes flow. Consequently, the viscous torque td canbe calculated as

td = gvw =1

3vwpml3C ð2Þ

where g is the fluid drag coefficient, m is the viscosityof fluid, and C is a geometric constant related to theaspect ratio of the nanowire. A few hydrodynamicmodels for estimating the C value can be found in theliterature (as summarized in Figure 4), where the cylin-der was approximated as a series of spherical or ellip-soidal segments, each having a finite length. Knowingthe drag force on the individual segments, the viscoustorque can be obtained by summating the product ofthe drag force and the distance from the rotatingaxis over all the segments. To validate the analyticalmodels, full-scale CFD models were developed usingthe multiple frames of reference (MFR) approach.26

The numerical results for viscous torque on nanowireswith various aspect ratios are plotted in Figure 3, where

the analytical predictions are also included. It is seenthat the Tirado model provides the best estimate over awide range of nanowire aspect ratios (20\ p\ 100)

C =1

lnp� 0:662ð3Þ

and, therefore, it was selected in this work to estimatethe fluid viscous torque in equation (2).

Brownian motion

Nanowires suspended in a liquid are subject to theeffects of Brownian motion. In particular, the rota-tional Brownian motion may compete with the mag-netic and viscous torques to disorient the nanowire.For a given time interval Dt, the Brownian-induced

Figure 3. Comparison of fluid viscous torque models.

Figure 4. Theoretical prediction of synchronous versusasynchronous rotation of nanowire.

4 Advances in Mechanical Engineering

rotation angle u�W can be estimated using the Langevinequation28

u�W =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

kT

gDt

sð4Þ

where k=1.38 3 10223 J/K is the Boltzmann constant,T is the absolute temperature, and Dt=1/63 s is thetime interval between consecutive video frames. In theexperiments, m=26mPa s, l=26.7 mm, r=300nm,T=288K, and the corresponding Brownian rotationspeed v�W = u�W=Dt = 0:055 rad=s, which is muchsmaller than the driving magnetic rotation speed13.5 rad/s. Therefore, the Brownian movement will notbe considered in the following analysis.

Governing equation

The equation of nanowire motion is obtained fromNewton’s second law for rotation

tm � td =Ia ð5Þ

where the moment of inertia is I=pr2rNil3=12, and the

angular acceleration is a= d2uW=dt2. Recalling thatuL= uH2 uW, vH=constant, equation (5) can berecast as

d2uL

dt2+

4mC

r2rNi

duL

dt+

12MsH

rNil2

sin uL =4mC

r2rNi

vH ð6Þ

A scaling analysis of equation (6) reveals that for nickelnanowires suspended in EG, (4mC=r2rNi);o(106) and(12MsH=rNil

2);o(108). Therefore, the inertial termd2uL=dt2 is neglected, and equation (6) becomes

duL

dt+vC sin uL =vH ð7Þ

and

vC =3MsH

mC

r

l

� �2

ð8Þ

Note that essentially, equation (7) is equivalent totm � td = 0.

Solutions

The solution to equation (7) can be obtained for differ-ent system conditions.

1. When a stationary magnetic field (vH=0) isapplied abruptly, equation (7) reduces to

duL

dt+vC sin uL = 0 ð9Þ

Depending on the initial lag angle uL,0, three solu-tions are possible:

(a) If uL,0=0, uL=0 (stable equilibrium).(b) If uL,0=p, uL=p (unstable equilibrium).(c) Otherwise

uL =p � 2 arctan exp(vCt+b�)½ � ð10Þ

where b�= ln½tan (p � uL, 0)=2�. This solutiondescribes the transient re-orientation of the nano-wire with respect to the field.

2. When the angular frequency of the magneticfield satisfies vH � vC, a steady-state solutionexists for equation (7)

uL =arcsinvH

vC

� �ð11Þ

In this case, the lag angle remains constant, that is, themotion is phase locked, and the nanowire rotates syn-chronously with the magnetic field. In particular, if vH

approaches the critical frequency vC, the nanowire willbe oriented perpendicularly to the magnetic field, thatis, uL=p/2, at which point the driving torque reachesthe maximum Mspr

2 Hl, according to equation (1).3. When the frequency of the magnetic field sur-

passes the threshold value, vH. vC, there is aperiodic solution for equation (7)

uL = 2 arctan

vC

vH

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� vC

vH

� �2s

tan(vH t+b)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� vC

vH

� �2r2

0BB@

1CCA

when 0� uL\p ð12Þ

uL = 2p + 2 arctan

vC

vH

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� vC

vH

� �2s

tan(vH t+b)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� vC

vH

� �2r2

0BB@

1CCA

when p\uL� 2p ð13Þ

where b is a constant determined by the initial condi-tion, and if the nanowire aligns perfectly with the mag-netic field at t=0s, b= � 2 arctan((vC=vH )=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� (vC=vH )2

q)=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� (vC=vH )

2q

. Mathematically, the

value of arctangent function is confined between 2p

and p. Since the lag angle varies from 0 to 2p in oneperiod, a value of 2p is added to uL when the arctan-gent function is negative to ensure the continuity of uL.Accordingly, uL exhibits time periodic oscillations,

Yang et al. 5

which correspond to a back-and-forth motion of thenanowire with respect to the magnetic field.

Results and discussion

Criterion for nanowire rotation trajectory

As shown in the previous section, the parameter vH=vC

plays an important role, mathematically, in determiningthe rotational dynamics of the nanowire. Its physicalsignificance can be found by re-arranging equation (7)as

vC sin uL = vH �duL

dt

� �ð14Þ

where the left-hand side (LHS) of the equation is asso-ciated with the magnetic driving torque, and the right-hand side (RHS) represents the viscous resistance tor-que. As the lag angle uL increases, the magnetic torquealso increases till uL reaches 90�. Clearly, the maximummagnetic torque occurs when uL=90� (the exact formis tm,max =Mspr2Hl according to equation (1)), whereasthe maximum viscous torque occurs when duL=dt= 0

(the exact form is td,max = 1=3vH pml3C according toequation (2)). Therefore, vH=vC represents the ratio oftd,max and tm,max

vH

vC

=td,max

tm,max

ð15Þ

Physically, when vH=vC � 1, the magnetic drivingtorque is able to balance the viscous resistance torque,and consequently, the nanowire can synchronize withthe magnetic field and maintain a stable motion with aconstant uL. In contrast, if vH=vC.1, the viscous tor-que may exceed the driving torque. Consequently, twotypes of nanowire rotation may be identified, as shownin Figure 4: synchronous rotation and asynchronousrotation. In the synchronous rotation, after very shortstartup time, uW= uH. In the asynchronous rotation,uW\ uH.

Nanowire alignment when a stationary field isrotated abruptly

When a stationary magnetic field is rotated abruptly bya certain angle, the nanowire will re-orient and alignitself with the field. The relaxation of the lag angle canbe described by equation (10) as a function of time,which represents by line in the plot. In Figure 5, thetheoretical predictions are compared to the experimen-tal data measured under various test conditions, wherea good agreement is observed.

Figure 5. Lag angle versus t for various conditions of H, p, and initial angle: (a) Lag angle ; H; (b) Lag angle ; Initial angle; (c) Lagangle ; p.

6 Advances in Mechanical Engineering

Synchronous rotation

Figure 6(a) shows the instantaneous orientations of ananowire in a magnetic field (H=7.2Gs) that rotatesin the counterclockwise direction with a constant angu-lar frequency vH=13.5 rad/s. The dark arrow is thenanowire and the red arrow represents the direction ofmagnetic field. The nanowire under investigation isl=26.7 mm and r=0.3 mm. Thus, the geometric con-stant C can be estimated from equation (3) as C=0.31,and the critical frequency is obtained from equation (8)as vC=16.4 rad/s. Since vH/vC\ 1, the nanowire fallsin the synchronous rotation regime. Figure 6(b) illus-trates the time evolution of uH, uW, and uL, respectively.At t=0, the magnetic field and the nanowire arealigned. Once the magnetic field starts spinning, uHincreases linearly. The nanowire begins to follow thefield; however, the catching-up takes about 0.4 s beforethe wire can reach the same angular speed of the field.Afterward, uL remains unchanged. It is noted that theslow startup is not caused by the inertia of the nano-wire, which has been shown to be negligible in the dis-cussion of equation (6). Rather, it is because themagnetic driving torque (see equation (1)), being pro-portional to sin uL, is non-existent at uL=0 and onlypicks up as uL increases.

The startup and steady-state processes can be betterdiscerned from Figure 6(c) where the respective angularfrequencies are plotted as a function of time. It is easyto note that the measured steady-state lag angle (uL=1rad) is also predicted by equation (11) which yieldsuL = arcsin (vH=vC)= 1 rad. More experiments wereconducted for three nanowires rotating synchronously

in different magnetic fields. The measurements of thesteady-state lag angle uL are plotted in Figure 7 as afunction of vH/vC, which match the theoretical predic-tion from equation (11) very well.

Asynchronous rotation

When vH/vC . 1, the nanowire will rotate asynchro-nously with the magnetic field. The mathematical originof this oscillating behavior has been shown in equations(12) and (13). Physically, when the nanowire first startsmoving, it tries to follow the magnetic field frombehind, leading to the forward rotation. However,because the magnetic field spins too fast and the rota-tion period (tH=1/vH) is shorter than the time ofnanowire in one period rotation cost, the lag angle uL

Figure 6. Synchronous rotation of nanowire: (a) images of synchronous rotation of nanowire; (b) angle of nanowire, magnetic, andlag angle; and (c) angular velocity of nanowire, magnetic, and lag angle.

Figure 7. uL with different values of vH and p.

Yang et al. 7

keeps increasing, that is, vL = duL=dt.0, and willexceed p before the nanowire finishes one lap of rota-tion. Consequently, the magnetic moment ~m varies infront of the field ~H , and the nanowire will instantlydecelerate from its forward motion and eventuallymove backward in order to align with ~H .

Figure 8 shows the asynchronous rotation of a nano-wire (l=26.7 mm) over one period in a magnetic field(H=3.9Gs and vH=13.5 rad/s). The instantaneousorientations of the nanowire and the magnetic field aredepicted in Figure 8(a), and the forth-and-back motionof the nanowire can be identified more clearly fromFigure 8(b) and (c).

(a) Forward motion: from t=0–0.44 s, the nano-wire is always going along the same directionas the magnetic field, as evidenced by theascending uW in Figure 8(b). The trend of vW

is shown in Figure 8(c). As discussed above,vC is the highest velocity the nanowire couldreach. It first increases till it reaches the highestspeed of 8.5 rad/s at t=0.22 s, at whichmoment the magnetic field is perpendicular to

the nanowire (uL=p/2) and 8.5 rad/s equalsto vC. Then, the angular speed will decrease asthe magnetic torque passes the peak. Att=0.44 s, vW=0rad/s, and uL=p, the mag-netic momentum of the nanowire and the mag-netic field are aligned opposite, and thenanowire becomes static.

(b) Backward motion: from t=0.44–0.60 s, thenanowire is rotating backward, as shown by thedescending uW in Figure 8(b). Since the nano-wire and the field are moving opposite to eachother, the lag angle uL increases sharply to 2p att=0.605 s. The rotation speed of the nanowirevW remains negative in this regime. The maxi-mum vW in the backward motion is 28.5 rad/sat t=0.52 s when uL=3p/2 and the magneticfield is again perpendicular to the nanowire. Theentire rotation cycle ends at t=0.605 s whereuL=2p. At this moment, the nanowire and themagnetic field are perfectly aligned.

The polar representation of the rotation dynamics ofa nanowire (l=26.7 mm) under a myriad of magnetic

Figure 8. Nanowire asynchronous rotation in one period: (a) images of asynchronous rotation of nanowire; (b) angle of nanowire,magnetic, and lag angle; and (c) angular velocity of nanowire, magnetic, and lag angle.

8 Advances in Mechanical Engineering

fields is shown in Figure 9. In this figure, the theoreticalvalue of uW is calculated for each frame (Dt=1/63 s),and uL is accumulated over consecutive periods. It canbe found that the theoretical results match well with theexperimental data.

From the foregoing discussion, the forward motionof the nanowire starts at uL=0 and ends at uL=p.Subsequently, the backward motion starts at uL=p

and ends at uL=2p. The respective duration of thetwo regimes is given as follows

tW forth =pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2H � v2

C

q � b

vH

ð16Þ

tW back =pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2H � v2

C

q +b

vH

ð17Þ

tW total =2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

v2H � v2

C

q ð18Þ

where tW total denotes the total time of a completeperiod. Figure 10 depicts the variations in time andangle belonging to nanowire for the forward and

backward motion regimes, respectively. As shown inFigure 10(a), when vC/vH=0, tW forth = tWback=(tH=2), which means when magnetic torque is small, thebackward and forward motion periods last for thesame time. As vC/vH increases, tW forth reaches a bound-less value when nanowire synchronous rotation occurs.However, tW back continually decreases to tH=4. Thisindicates the forward motion stage always outruns thebackward motion stage.

With the angle relation uL= uH–uW, uW of the for-ward and backward motion can be derived from equa-tions (12) and (13)

uW forth =pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� vc

vH

� �2r � b� p ð19Þ

uW back =pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� vc

vH

� �2r +b� p ð20Þ

uW total =2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� vc

vH

� �2r � 2p ð21Þ

Figure 9. Nanowire asynchronous rotation under different H.

Yang et al. 9

where uW forth and uW back are the nanowire angles ofthese two stages, and uW total is the nanowire angle inone period. As shown in Figure 10(b), uW forth =uW back = 0 when vC/vH=0. This means that nanowirewill oscillate in its previous position.

Conclusion

The dynamics of ferromagnetic nickel (Ni) nanowiresrotating in a driving magnetic field was investigated.The major findings are summarized as follows.

1. The dynamics of nanowire rotation is governedprimarily by the magnetic torque and the fluidviscous torque, whereas the Brownian motionand inertia have negligible impact.

2. There exists a critical rotation speed for themagnetic field, vC. When vH/vC � 1, the nano-wire rotates in the synchronous mode, wherethe lag angle uL remains constant; otherwise,the nanowire rotates in the asynchronous mode,where uL fluctuates with time and the nanowireundergoes back-and-forth oscillation.

3. Analytical models were developed to describethe major modes of nanowire motion. The com-parisons of the theoretical prediction and theexperimental data show very good agreement.

Acknowledgements

We thank Dr Li Sun at the University of Houston for supply-ing the nanowire samples.

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

This work was supported by National Natural ScienceFoundation of China (Grant No. 51376022).

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