IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 1, JANUARY 2014 4001604

Magnetic Hysteresis Loop in a Superparamagnetic StateHiroaki Mamiya and Balachandran Jeyadevan

National Institute for Materials Science, Tsukuba 305-0047, JapanThe University of Shiga Prefecture, Hikone 522-8533, Japan

Magnetization curves of superparamagnetic nanoparticles in ac magnetic field are numerically studied by Brownian dynamics sim-ulation with considering a more precisely estimated magnetic torque. Consequently, we confirmed that rotatable nanoparticles withmagnetic easy axes show magnetic hysteresis loops without remanence, even when the thermal fluctuations of the magnetizations aremuch faster than the oscillation of the magnetic fields. The origin is that magnetic alignments of the easy axes by the magnetic torquein the peak period of the oscillations of field alternate with their randomizations by Brownian relaxation in the period when the fieldbecomes almost zero. Because the equilibriummagnetization curve of the aligned nanoparticles is steeper than that of randomly orientedones, a new kind of hysteresis appears when both the Brownian relaxation time and the characteristic time of the field-driven rotationare comparable to the alternation period of magnetic field. This finding sheds a new light on the widely accepted association betweensuperparamagnetism and the anhysteretic magnetization curves. Furthermore, this knowledge will help reconsideration of the designof biomedical application using magnetic nanoparticles, since it has been optimized assuming magnetic response described by Langevinfunction.

Index Terms—Biomagnetics, magnetic hysteresis, magnetic liquids, magnetic nanoparticles.

I. INTRODUCTION

W ITH the recent progress of biomedical applicationsusing magnetic nanoparticles, such as tomographic

imaging and hyperthermia treatment, magnetic properties ofthe nanoparticles have received considerable scientific attentionin order to clarify their optimal design [1]–[3]. From the view-point of dispersion stability, the small magnetic nanoparticleshave been used; consequently, their diameters are smaller thanthe critical diameters for the transition into a single-domainconfiguration and for the coherent reversal of all spins [3]. Inother words, each nanoparticle has been considered to haveonly one large magnetic moment, , also called “superspin.”The following two properties have been well known as thefundamental features of superspin.In a static magnetic field , the energy potential is simply

given by Zeeman energy

(1)

where is permeability of vacuum. Therefore, magnetizationis expressed as , where is the particlenumber density, is the Langevin function, is the Boltzmannconstant, and is the temperature. The magnetization curvebecomes nonlinear at a relatively low magnetic field because ofthe large magnitude of . A new tomographic imaging, termedmagnetic particle imaging, utilizes this feature [4].From the view point of dynamics, the fluctuations of the di-

rection of have been considered on the following energy po-tential:

(2)

Manuscript received April 28, 2013; revised July 02, 2013; accepted July15, 2013. Date of current version December 23, 2013. Corresponding author:H. Mamiya (e-mail: MAMIYA.Hiroaki@nims.go.jp).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMAG.2013.2274072

where is the uniaxial anisotropy constant, is the volumeof the nanoparticle, is the unit vector along the easy axis, andis the unit vector along . In this case, energy barrier at hard

plane, , blocks the reversal of between the directions par-allel to the easy axes, and the Néel relaxation time is givenby , where is the attempt frequency of10 to 10 s . If the Néel relaxation is delayed to the alterna-tion of the ac magnetic field, relaxation loss dissipates from thenanoparticles. A new hyperthermia treatment, termed magneticfluid hyperthermia, utilizes this feature [5].These properties of the superspin have been called “super-

paramagnetism” because of the analogy to the properties of thetotal magnetic moment in each atom in the paramagnetic state.At the moment, one may, however, notice that the energy po-tentials, which are the basis of the understanding on superpara-magnetism, are completely different between (1) and (2). Forthis reason, advanced models have been studied; consequently,deviation from the value given by the Langevin function hasbeen clarified for the static magnetization curve [6]–[8], and anevident acceleration of in magnetic field has been reported[9], [10]. Furthermore, a numerical simulation predicted that amagnetic hysteresis exists in a superparamagnetic state, if thenanoparticles are rotatable [11]. This finding has provoked agreat deal of controversy, since it is incompatible with the com-monly accepted association between superparamagnetism andthe anhysteretic magnetization curves. It was pointed out thatthe direction of actually explores all over the potential wellswith precession [12], although it was assumed in [11] that istrapped at the lowest point of one of the wells and stochasti-cally jumps to another well by thermally activated process. Inother words, the assumption, the two-level approximation [8],[11], may be oversimplified. For this reason, we simulated themagnetic response for superparamagnetic nanoparticles rotat-able in a fluid using a more strict assumption, and verify theprediction that magnetic hysteresis loops in an ideal superpara-magnetic state.

II. PROCEDURE

In numerical simulation, magnetic nanoparticles are consid-ered to be superparamagnetic. In other words, the distribution of

0018-9464 © 2013 IEEE

4001604 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 1, JANUARY 2014

the direction of , was assumed immediately equilibratedin the energy potential

(3)

at each step of Brownian dynamics simulation, where is thesolid angle of . Because an ideal superparamagnetic state isconsidered, we disregarded dipole–dipole interactions betweenthe nanoparticles, although intriguing collective properties havebeen discussed [13], [14]. On the other hand, the energy poten-tial as to the direction of is . Thus,mean torque on the easy axis at the step is given by

(4)

Nanoparticles are easily rotatable if they are dispersed in anaqueous phase mimicking the cytoplasm [15]. As well, in situX-ray diffractometry on a single nanoparticle showed that thenanoparticle anchored to a substrate by an antigen-antibodycombination can slowly rotate on the time scale of millisecondsto seconds [16]. Therefore, we consider magnetic nanoparticlesrotatable in a Newtonian fluid for simplicity, although actualmicro-viscoelasticity of the local environment in living cellsis still unknown. In this case, the inertia of nanoparticles canbe neglected and the frictional torque balances with magnetictorque and Brownian random torque are as follows:

(5)

(6)

(7)

where is the viscosity of the surrounding media, is the hy-drodynamic volume of the nanoparticles including surface mod-ification layers, is the angular velocity of rotation, and indi-cate the Cartesian vector components, is the Kronecker deltaand is the Dirac delta function.Using these equations, the simulation was carried out as

follows. At the beginning, an assembly of nanoparticles withrandomly oriented was generated at zero magneticfield, where their number ensures an optimal compromisebetween calculation time and precision. Then, the time evo-lution of was computed by the following steps: 1)was calculated for with the latest and

, where and are theamplitude and frequency of the ac magnetic field; 2) wascomputed by substituting into (4); 3) was calculatedby substituting into (5); and 4) was finally replaced by

, where the time stepwas s. This calculation was repeated until transientfactors depending on the initial conditions disappeared.In this paper, we report, as a typical example, the results com-

puted for spherical magnetic nanoparticles with a core diameterof nm, a spontaneous magnetization of 450 kA/m, andof kJ/m [5]. At body temperature of K, of the

nanoparticles in zero magnetic field is s for of 10 sand s for of s . For this reason, can be con-sidered to be equilibrated within of 1 ms at Hz. Thehydrodynamic diameter was assumed to be . For consis-tency with the observed slow rotation of the anchored nanopar-ticle, was set at Pa . This environment may be, however,

Fig. 1. Magnetic torque computed under the previous assumptions and thenew one, as a function of the angle between and . Broken curves indi-cate for fixed at the direction parallel to the easy axis, the solid curvesexhibit for trapped at a local minimum of the energy potential , opensymbols show computed for that thermally fluctuates between the localminima as described in [11], and the solid symbols show calculated forthat distributes in using (4). Right axis indicates the corresponding angularvelocity. Dashed-dotted line demonstrates the thermal energy at 310 K.Inset shows a schematic diagram of .

closer to glucose syrup rather than the actual surrounding mediain living cells.

III. RESULTS AND DISCUSSION

First of all, we begin by discussing effects of the improved as-sumption on the magnetic torque . Fig. 1 shows the values ofcomputed under the previous assumptions and the new one,

as a function of the angle between and . The broken curvesindicate for fixed at the direction parallel to the easy axis,the solid curves exhibit for trapped at a local minimum ofthe energy potential , the open symbols show computedfor that thermally fluctuates between the local minima as de-scribed in [11], and the solid symbols show calculated forthat distributes in using (4). We can find that the solid curveis lower than the broken curve at kA/m. This differencecan be attributed to canting of , which is a shift of the directionof the local minimum of toward that of due tothe large Zeeman energy at kA/m (see the black solidarrow in the inset of Fig. 1). On the other hand, at a relativelysmall of 4 kA/m, there are evidently differences between thesolid curve and the open symbols. The origin of the decrease isthat the normal torque when stays the potential well at the di-rection more parallel to is offset by the counter-torque whenit is more anti-parallel in the other well. At higher , this offsetis insignificant because the length of stay in the shallower wellbecomes very short. The exception is the case that the depths ofthe two wells are almost the same with one another at .In contrast to these distinct variations, the differences betweenthe open and solid symbols are not remarkable at any . In brief,distribution of the direction of in each potential well does notseriously affect the magnitude of in this example. Furthersystematic calculations are required to discuss the universalityof this result. In any case, a more precise magnitude of magnetictorque can be now obtained by using (4). Therefore, we shall

MAMIYA AND JEYADEVAN: MAGNETIC HYSTERESIS LOOP IN SUPERPARAMAGNETIC STATE 4001604

Fig. 2. Evolution in the mean orientation of the easy axes of the nanoparticles,, in one cycle. Insets show the orientation distribution of the easy axesof nanoparticles at typical points. Solid lines in the insets show random

distribution: .

verify whether hysteresis loops appear in magnetization curvesof the superparamagnetic nanoparticles using the precise torque,next.Fig. 2 shows the computed evolution of mean orientation of

the easy axis, , driven by the above-mentioned torque.The conditions are of kA/m and of 0.2 Hz. It is note-worthy that the directions of the easy axes have butterfly-shapedhysteresis in each cycle for 5.0 s. In the period at low , theoccupation probabilities of in the two stable wells with thedirections parallel/antiparallel to the easy axis are equalized asstated above; consequently, Brownian random torque becomespredominant over the mean magnetic torque in the period (seeFig. 1). The characteristic time of the randomization of thedirections of is the Brownian relaxation time , which is0.125 s in this case. As increases, the occupation probabilityin the more stabilized well immediately increases on a timescale of . This leads to a steep increase of the magnetictorque, consequently, the easy axes are turned toward thedirection of . The angular velocity of this rotation is approx-imately and becomes several radians per secondin the peak period of the oscillations of (see Fig. 1 again).Therefore, an aligned state of the easy axes is formed, as shownin the upper inset. Subsequently, decreases to zero and theBrownian torque takes the place of the magnetic torque. As aresult, the orientations of the easy axes are randomized within, as shown in the lower inset. As described here, competition

between the magnetic and Brownian torques within one cyclecauses the butterfly-shaped hysteresis of .At this stage, let us remind the reader of the well-known

fact that the static magnetization curve of superparamagneticnanoparticles with magnetic anisotropy cannot be expressed byusing Langevin function [6]–[8]. In other words, the equilibriummagnetization highly depends on the orientations of the easyaxes, as shown in Fig. 3. The reason is that is bound in the di-rections parallel or antiparallel to the easy axis. Therefore, the

Fig. 3. Magnetization as a function of applied field . Dashed-dotted lineshows calculated by Langevin function, the solid lines indicate the static- curves for the nanoparticles with variously aligned easy axes, broken

line indicates the static - curve for the nanoparticles with randomly ori-ented easy axes, and open circles demonstrate the dynamic - curves for thenanoparticles with rotatable easy axes.

Fig. 4. Area of the dynamic hysteresis loop for the nanoparticles with rotatableeasy axes, , where it is normalized by the upper limit of the area .Broken line shows the Brownian relaxation time , and solid line in-dicate the typical angular velocity at .

alternation of randomly oriented and aligned states of the easyaxes, mentioned above, causes a hysteresis loop without rema-nence in the dynamic magnetization curve of the superparamag-netic nanoparticles (see Fig. 3). As demonstrated here, we arecertain the existence of magnetic hysteresis computed by theimproved simulation for a superparamagnetic system.Finally, we shall discuss the condition in which this in-

triguing phenomenon occurs. Fig. 4 shows a contour plot ofthe area of the dynamic hysteresis loop, , computed for thenanoparticles in the ac magnetic field with various and .A single peak can be found when both the Brownian relaxationtime and the characteristic time of the field-drivenrotation are comparable to the alternation period

4001604 IEEE TRANSACTIONS ON MAGNETICS, VOL. 50, NO. 1, JANUARY 2014

of magnetic field . This is reasonable because the hys-teresis comes from the competition between the magnetic andBrownian torques within one cycle.

IV. CONCLUDING REMARKS

We have adapted the method of estimating magnetic torqueto superparamagnetic states, where the distribution of the di-rection of is equilibrated in the energy potential at all times.Consequently, Brownian dynamics simulation with consideringa more precisely estimated magnetic torque indicates that themagnetization curve of superparamagnetic nanoparticles rotat-able in ac magnetic field has a hysteresis loop without rema-nence. The origin is that magnetic alignments of the easy axesby the magnetic torque in the peak period of the oscillations offield alternate with their randomizations by Brownian relaxationin the other period at lower magnetic fields. This finding castsa new light on the widely accepted association between super-paramagnetism and the anhysteretic magnetization curves, al-though it has not yet been be tested through experiments. In thiscontext, in situ magnetometry/diffractometry on monodispersemagnetic fluids in large ac magnetic fields are highly required toclarify the dynamics of magnetization and crystal orientation. Ifthe advanced model is established, it would be used as the basisof design of magnetic nanoparticles for biomedical applications.

ACKNOWLEDGMENT

This study was supported in part by a Grant-in-Aid for Sci-entific Research 24310071.

REFERENCES[1] Q. A. Pankhurst, N. K. T. Thanh, S. K. Jones, and J. Dobson, “Progress

in applications of magnetic nanoparticles in biomedicine,” J. Phys. D,vol. 42, no. 224001, 2009.

[2] B. Jeyadevan, “Present status and prospects of magnetite nanoparticles-based hyperthermia,” J. Ceram. Soc. Jpn, vol. 118, pp. 391–401, 2010.

[3] H. Mamiya, “Recent advances in understanding magnetic nanopar-ticles in ac magnetic fields and optimal design for targeted hyper-thermia,” J. Nanomater., vol. 2013, no. 752973, 2013.

[4] B. Gleich and J. Weizenecker, “Tomographic imaging using thenonlinear response of magnetic particles,” Nature, vol. 435, pp.1214–1217, 2005.

[5] R. E. Rosensweig, “Heating magnetic fluid with alternating magneticfield,” J. Magn. Magn. Mater., vol. 252, pp. 370–374, 2002.

[6] M. Hanson, C. Johansson, and S. Mørup, “The influence of magneticanisotropy on the magnetization of small ferromagnetic particles,” J.Phys.: Condens. Matter, vol. 5, pp. 725–732, 1993.

[7] H. Mamiya and I. Nakatani, “Magnetization curve for iron nitride fineparticle system with random anisotropy,” IEEE Trans. Magn., vol. 34,no. 3, pp. 1126–1128, Jun. 1998.

[8] J. Carrey, B. Mehdaoui, and M. Respaud, “Simple models for dynamichysteresis loop calculations of magnetic single-domain nanoparticles:Application to magnetic hyperthermia optimization,” J. Appl. Phys.,vol. 109, no. 083921, 2011.

[9] W. F. Brown, Jr., “Thermal fluctuations of a single-domain particle,”Phys. Rev., vol. 130, pp. 1677–1686, 1963.

[10] W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, andE. C. Kennedy, “Effect of an oblique magnetic field on the superparam-agnetic relaxation time. II. Influence of the gyromagnetic term,” Phys.Rev. B, vol. 58, pp. 3249–3266, 1998.

[11] H. Mamiya and B. Jeyadevan, “Hyperthermic effects of dissipativestructures of magnetic nanoparticles in large alternating magneticfields,” Sci. Rep., vol. l, no. 157, 2011.

[12] N. A. Usov and B. Ya. Liubimov, “Dynamics of magnetic nanopar-ticle in a viscous liquid: Application to magnetic nanoparticle hyper-thermia,” J. Appl. Phys, vol. 112, no. 023901, 2012.

[13] H. Mamiya, I. Nakatani, and T. Furubayashi, “Slow dynamics for spin-glass-like phase of a ferromagnetic fine particle system,” Phys. Rev.Lett., vol. 82, pp. 4332–4335, 1999.

[14] H. Mamiya, I. Nakatani, and T. Furubayashi, “Phase transitions of ironnitride magnetic fluids,” Phys. Rev. Lett., vol. 84, pp. 6106–6109, 2000.

[15] M. K. Kuimova et al., “Imaging intracellular viscosity of a single cellduring photoinduced cell death,” Nature Chem., vol. 1, pp. 69–73,2009.

[16] T. Sagawa, T. Azuma, and Y. C. Sasaki, “Dynamical regulations ofprotein- ligand bindings at single molecular level,” Biochem. Biophys.Res. Commun., vol. 355, pp. 770–775, 2007.