Review Article Recent Advances in Understanding Magnetic...

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2013 Article ID 752973 17 pageshttpdxdoiorg1011552013752973

Review ArticleRecent Advances in Understanding MagneticNanoparticles in AC Magnetic Fields and Optimal Design forTargeted Hyperthermia

Hiroaki Mamiya

National Institute for Materials Science Tsukuba 305-0047 Japan

Correspondence should be addressed to Hiroaki Mamiya mamiyahiroakinimsgojp

Received 19 April 2013 Accepted 17 June 2013

Academic Editor Oleg Petracic

Copyright copy 2013 Hiroaki Mamiya 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

Targeted hyperthermia treatment using magnetic nanoparticles is a promising cancer therapy that enables selective heating ofhidden microcancer tissues In this review I outline the present status of chemical synthesis of such magnetic nanoparticles Thenthe latest progress in understanding their heat dissipationmechanisms under largemagnetic fields is overviewedThis review coversthe recently predicted novel phenomena magnetic hysteresis loops of superparamagnetic states and steady orientations of easy axesat the directions parallel perpendicular or oblique to the AC magnetic field Finally this review ends with future prospects fromthe viewpoint of optimal design for efficacy with a low side-effect profile

1 Introduction

Hippocrates said ldquothose diseases which drugs cannot curethe knife cures those which the knife cannot cure fire curesthose which fire does not cure must be considered incurablerdquoIn one respect medicine has not changed over time eventoday several treatments are used in combination to treatillnesses that have no established effective treatment protocolthe most prominent example being cancer The current stan-dard treatments for cancer include surgery chemotherapyand radiotherapy Beyond these treatments much researchis being undertaken to create several new treatment optionssuch as immunotherapy and the modern equivalent of Hip-pocratesrsquo ldquofirerdquo thermotherapyThermotherapy is a treatmentmethod that exploits the lowered heat resistance of canceroustissues compared with that of normal tissues Canceroustissues undergo cell death even at temperatures within therange of 42 to 43∘C thus rendering thermotherapy as apromising option to reduce the disease burden in a patient[1]

To reduce damage to normal tissues using standard treat-ments endo-laparoscopic surgical techniques have beendeveloped as amodern equivalent ofHippocratesrsquo ldquokniferdquo Forchemotherapy much effort has focused on drug delivery to

selectively transport anticancer agents to tumor tissues usingantigen-antibody reactions Such drug delivery systems arealso used to concentrate boron compounds in tumor tissuesIn the treatment known as boron neutron capture therapythe patient is radiated with epithermal neutrons whichselectively induce 120572-decay of boron nuclei concentratedinside hidden tumors thus specifically destroying cancercells Along these lines could thermotherapy also utilizedrug delivery technology to specifically deliver thermalseeds to cancer cells at unknown locations For exampleif it were possible to develop miniature induction-heatingcooking pans and selectively send these to hidden tumorsthen would this result in selective heating of the tumortissues in a human body on an induction-heating cookerIn contrast to a microwave oven we know that placing onersquoshand over the induction-heating cooker will not immediatelyresult in a burn (Usually we cannot confirm this featureusing the commercial IH cooker because it automaticallystops working when we take off a metal pan from it) Thisexperience shows that a radio-waveband AC magnetic fieldgenerated in the cooker can easily penetrate deep into tissueswhere a tumor may be embedded Therefore we believethat the simple idea mentioned here targeted hyperthermia

2 Journal of Nanomaterials

using magnetic nanoparticles has the potential to selectivelydestroy cancer cells hidden deep in the body [2ndash8]

Of course there are many questions about this con-cept that need to be addressed Is it safe to put magneticnanoparticles in the body Canmagnetic nanoparticles reallybe concentrated in hidden tumor tissues Can magneticnanoparticles be heated within the body To answer thesequestions DeNardo et al injected iron oxide nanoparticlesconjugated with monoclonal antibodies into mouse tails andfound that they accumulated at a concentration 119888 of approxi-mately 03 kgm3 (03mgcm3) in tumors [9] (The side effectsof iron oxide nanoparticles as anMRI contrast agent had beenpreviously studied and approved for intravenous injection)Wust et al showed that injecting high concentrations (119888 sim

10 kgm3) of magnetic nanoparticles directly into tumors atknown locations and irradiating them with an AC magneticfield caused the temperature of the tumors to increase to43∘C [10]Therefore it is known that iron oxide nanoparticlesare safe can be selectively accumulated in hidden tumors tosome degree and can be adequately heated when existingat high concentrations Nevertheless problems with targetedhyperthermia using magnetic nanoparticles are still evidentDoes the density of magnetic nanoparticles delivered totumors increase as it does when they are directly injectedinto tumors If not is it possible to compensate for a lowerdensity of magnetic nanoparticles by maximizing their heatdissipationThe first problem is primarily a biochemical onesomaterials researchers have primarily focused on improvingthe heating performance of magnetic nanoparticles [11ndash30] Consequent advances in chemical synthesis technologyhave resulted in the fabrication of magnetic nanoparticlesof engineered size shape and structure With respect tophysical heating mechanisms the nature of the nonlinearresponse and nonequilibrium dissipation in AC magneticfields of magnetic nanoparticles which are in contrast to theproperties of cooking pans have been uncoveredThis reviewaddresses this progress as follows In Section 2 conventionalmodels that are the basis of the traditional design of hyper-thermia treatments are introduced In Section 3 advances inthe synthesis of magnetic nanoparticles are described andlimitations in the conventional models when the monodis-perse nanoparticles are used in actual thermotherapy areconsidered In Section 4 recent advances in the knowledgeof heating mechanisms provided by numerical simulationsare explained Finally we summarize the optimal designof magnetic nanoparticles for hyperthermia treatment anddiscuss their potential as an effective and safe version ofHippocratesrsquo ldquofirerdquo in Section 5

2 Conventional Models for Magnetic Responseto AC Magnetic Fields [31ndash34]

The main advantage of hyperthermia treatment using mag-netic nanoparticles is that the nanoparticles can reachthe cancer tissue directly by travelling through the sub-micrometer spaces between blood cell walls Therefore forpractical use the nanoparticles should not form long chainsor large clusters Even though the many-body effects caused

by dipole-dipole coupling 119869dd are not fully understood [35ndash38] it is known that a dispersion becomes unstable if 119869ddbetween the closest nanoparticles is more than five times thethermal energy [35 38] Under these conditions the min-imum allowable distance between iron nanoparticles withdiameter119889 of 12 nm is roughly 27 nm and that between ferritenanoparticles with 119889 of 25 nm is almost 40 nm In contrastsmall-angle neutron scattering experiments have indicatedthat the thickness of an absorbed layer is normally severalnanometers [39] Therefore the upper limit of 119889 is estimatedto be roughly 12 nm for iron and 25 nm for ferrite Thesevalues would be references for considering the criteria foreasy delivery of the nanoparticles although agglomerationaggregation or flocculation may occur depending on thesurface charge of biofunctionalized nanoparticle or on theinteraction between tumor-targeting ligands Note that thesevalues are smaller than the critical diameters for the transitioninto a single-domain configuration and for the coherentreversal of all spins [40] Therefore it has been consideredthat a magnetic nanoparticle used in hyperthermia treatmenthas only one magnetic moment 120583 = M

119904119881 where M

119904

is the spontaneous magnetization and 119881 = 1205876 sdot 1198893 is

the volume of the magnetic core of the nanoparticle Suchmagnetic nanoparticles have been conventionally classifiedas ldquoferromagneticrdquo or ldquosuperparamagneticrdquo depending onwhether the direction of 120583 thermally fluctuates or not

Firstly a ferromagnetic nanoparticle with uniaxial mag-netic anisotropy anisotropy constant119870 is considered where119881 is large enough that its magnetic anisotropy barrier with aheight of119870119881 blocks the thermal fluctuations accordingly theremanent state appears to be permanent [41] If a magneticfield H is applied in the direction antiparallel to 120583 thestate becomes metastable as depicted in Figure 1(a) Then 120583reverses when the barrier disappears at the anisotropy field119867119870

= 2119870(1205830119872119904) consequently the Zeeman energy falls

from 1205830120583119867119870to minus1205830120583119867119870and the energy corresponding to the

difference dissipates where 1205830is permeability of vacuum In

this case the work done in one cycle of the ACmagnetic field119867ac sin(2120587119891 sdot 119905) is 0 for119867ac lt 119867

119870and 4120583

0120583119867119870for119867ac gt 119867

119870

This kind of heat dissipation has been termed ldquohysteresislossrdquo Briefly the heat dissipation from nanoparticles withunit weight during unit time also called specific loss power119875119867 abruptly increases from zero to 4120583

0120583119867119870sdot 119891 sdot 119908

minus1

(=

41205830119872119904119867119870sdot119891 sdot120588minus1

)when119867ac becomes higher than119867119870 where

119908 and 120588minus1 are the weight and density of the magnetic core

of the nanoparticles respectively Then 119875119867flattens out even

if 119867ac is strengthened further According to this argumentthe guiding principle for maximizing 119875

119867of ferromagnetic

nanoparticles is that 119867ac is adjusted to 119867119870and the number

of cycles 119891 is maximizedNext we move to smaller superparamagnetic nanoparti-

cles with thermally fluctuating reversal of 120583 [42]The reversalprobability in a zero magnetic field is expressed as

120591minus1

119873

= 1198910sdot exp(minus119870119881

119896119861119879

) (1)

where 120591119873

is the Neel relaxation time 1198910is the attempt

frequency of 109 sminus1 119896119861is the Boltzmann constant and 119879 is

Journal of Nanomaterials 3

minus2 minus1 0 1

1

1

2

21205830120583HK

21205830120583HK

HHK

MM

s

(a) Hysteresis loss

Susc

eptib

ility

Out-of-phase

In-phase

Brownian relaxationNeel relaxation

0

05

1

10minus3 100 103

2120587fmiddot120591120591minus1 = 120591minus1N + 120591minus1B

(b) Relaxation loss

Figure 1 Schematic diagrams of conventional models for magnetic loss (a) Hysteresis loss equivalent to the area of 119872-119867 loop and thepotential energy inmagnetic fields (b) Relaxation loss given by the out-of-phase component of AC susceptibility As illustrated in the sketchesin Neel relaxation the magnetic moment shown by the yellow arrow reverses (the particle does not rotate) while in Brownian relaxation themagnetic core (the red sphere) rotates with absorbed molecules (the green chains) as a whole (the magnetic moment does not reverse)

the temperature We must also consider Brownian rotationof the nanoparticles if they are dispersed in a liquid phaseIn this case the characteristic time of the rotation Brownianrelaxation time in a zero magnetic field is given by

120591119861=

3120578119881119867

(119896119861119879)

(2)

where 120578 is the viscosity of the liquid phase and 119881119867

is thehydrodynamic volume of the nanoparticles including surfacemodification layers If reversal and rotation occur in parallelthe characteristic time of relaxation 120591 could be expressed asthe following equation

120591minus1

= 120591minus1

119873

+ 120591minus1

119861

(3)

For very small superparamagnetic nanoparticles 120591 is deter-mined only by 120591

119873because 120591minus1

119873

increases exponentially withdecreasing 119881 while the increase of 120591minus1

119861

is inversely propor-tional to 119881

119867

If a linear response of the thermodynamic equilib-rium state of such nanoparticles is assumed for small 119867ac

the average out-of-phase component of AC susceptibility 12059410158401015840contributed from each nanoparticle is given by

12059410158401015840

=

12058301205832

(3119896119861119879)

sdot

2120587119891 sdot 120591

[1 + (2120587119891 sdot 120591)2

]

(4)

Consequently ldquorelaxation lossrdquo occurs and its heat dissipation119875119867is expressed as

119875119867= 120587120583012059410158401015840

sdot 1198672

ac sdot 119891 sdot 119908minus1

=

1

2

[1205832

0

1198722

s119881 (3119896B119879120591120588)] sdot 1198672

ac sdot (2120587119891 sdot 120591)2

[1 + (2120587119891 sdot 120591)2

]

(5)

Equation (5) indicates that 119875119867increases in proportion to 119891

2

in the low frequency range 2120587119891 ≪ 120591 whereas it flattensout at 12[1205832

0

1198722

119904

119881(3119896119861119879120591120588)] sdot 119867

2

ac even if 119891 is increasedfurther in the high frequency range 2120587119891 ≫ 120591 Accordingto this argument the guiding principle for maximizing 119875

119867of

superparamagnetic nanoparticles is that 119891 is adjusted to 120591minus1

and119867ac is maximized


3 Progress in Synthesis ofMagnetic Nanoparticle and Their Usein Thermotherapy

31 Size- Shape- and Composite-Controlled Synthesis of Mag-netic Nanoparticles As discussed above to improve hystere-sis loss ferromagnetic nanoparticles with an anisotropy field(119867119870

= 2119870(1205830119872119904)) matching the amplitude 119867ac of the AC

magnetic field generated in the oscillator of realistic medicalequipment need to be synthesized In contrast increasingrelaxation loss involves the synthesis of superparamagneticnanoparticles that have 120591

119873matching 119891 of the AC magnetic

field For these reasons a large number of studies havefocused on controlling the size shape or composite structureof nanoparticles to optimize119867

119870and 120591119873

The history of colloids (magnetic fluids) stably dispersingmagnetic nanoparticles in solution goes back to the 1960swhen magnetic suspensions were prepared by pulverizingbulk iron oxide and used for fuel delivery in a weightlessenvironment [43] such as those involvingNASA expeditionsElsewhere Sato of TohokuUniversity createdmagnetic fluidsfrom minute iron oxide particles using chemical methods[44]There have also been severalmajor subsequent advancesin magnetic fluid development such as the monodisperseiron nitride-based magnetic fluids developed by Nakataniet al [45] however because the industrial applications ofmagnetic fluids at that time did not require precise con-trol of size shape or structure more extensive researchwas not conducted in this field However in 2000 Sun etal from IBM described an ordered self-assembled film ofmonodisperse iron-platinum nanoparticles that could serveas an ultrahigh-tech magnetic recording medium [46] Sincethen researchers have focused on developing methods tosynthesize well-controlled nanoparticles which have beenreviewed extensively [47 48] Next we briefly summarizethese methods

Generally formation of nanoparticles starts with nucle-ation in a supersaturated melt solution or vapor Particlegrowth continues until the concentration of solute atoms fallsbelow the saturation solubility If nucleation and growth pro-ceed in parallel nanoparticles formed initially have alreadygrown when the last nanoparticle is formed thus resultingin nanoparticles of variable size Furthermore processessuch as coarsening and aggregation simultaneously occur inmany cases One way to obtain monodisperse nanoparticlesis the two-stage growth method in the first stage rapidheating causes fast supersaturated-burst nucleation and inthe second stage the gradual precipitation of solute atoms at atemperature below the critical point of supersaturation allowsonly the existing nanoparticles to grow slowly In this processsurfactants are often introduced to the solution to preventcoarsening and aggregation Because all nanoparticles followthe same growth process in this method their size aftergrowth should in principle be uniform In practice differ-ent groups have developed particular methods to producenanoparticles of specific composition and size

With respect to controlling the shape of a nanoparticlegrowth kinetics is essential in addition to thermodynamic

stability to minimize surface free energy For example if thegrowth rate for cubic 111 surfaces is slower than for 100surfaces the surface area of 100 facets will decrease withgrowth and the particles finally become octahedrons of 111facets only Similarly if the growth of 001 surfaces in ahexagonal crystal system is fast rods or conversely plates canbe formed For this reason the adsorption of surfactants onparticular surfaces has been intensively studied to fabricatea desired shape by controlling the growth rate of eachsurface Figure 2 shows examples of regular octahedral andcubic nanoparticles [49 50] With regard to compositingnanoparticles dispersed in solution are regularly conjugatedby substances such as surfactants to lower their surface energyor prevent aggregation forming a kind of compositematerialAdvanced compositing techniques have been developed toprotect easily oxidizedmetal cores or to enable the simultane-ous expression ofmultiple functions For example dumbbell-shaped junctions in different kinds of nanoparticles [51]and core-shell structures [27] have been produced recently(Figure 3)

32 Magnetic Nanoparticles to Maximize Heat DissipationUsing these advanced synthesis techniques researchers havefabricated magnetic nanoparticles to maximize heat dissi-pation based on the guiding principles described above Asan example I highlight the recent report by Lee et al [27]who fabricated novel superparamagnetic nanoparticles witha uniform diameter 119889 of 15 nm (see Figure 3 again) Oneof the reasons why they chose such a size may be to avoidaggregation In addition the oscillator of their equipment cangenerate an AC magnetic field of frequency 119891 = 500 kHz Asdiscussed above superparamagnetic nanoparticles that havea Neel relaxation time (120591

119873) that matches 119891 are required to

maximize relaxation loss Briefly 120591119873

should be (2120587119891)minus1

=

318 ns (Overall 120591 needs to be set to 400 ns when alsoconsidering the Brownian relaxation time 120591

119861= 16 120583s)

Substituting 120591119873

= 318 ns in (1) the required energy barrierheight (119870119881) is calculated to be 24 times 10

minus20 J This valuecorresponds to a uniaxial anisotropic particle with 119889 = 15 nmand119870 = 14times10

4 Jm3 However examination of parameterssuch as bulk crystalline magnetic anisotropy [40] revealedthat no suitable candidate substances had been reported Forsubstances with cubic symmetry the magnitude of 119870

1and

the barrier height minus(112)1198701119881 for negative 119870

1or (14)119870

1119881

for positive 1198701 calculated using 119889 = 15 nm are as follows

minus12 times 104 Jm3 and 018 times 10minus20 J (Fe3O4) minus046 times 104 Jm3

and 008 times 10minus20 J (120574-Fe2O3) minus025 times 104 Jm3 and 004 times

10minus20 J (MnFe2O4) and 18 times 104 Jm3 and 80 times 10minus20 J

(CoFe2O4) As a result shape control which affects shape

and surface magnetic anisotropy or composition control orcomposite structure control which influences the crystallinemagnetic anisotropy are therefore required From amongthe possibilities mentioned Lee et al selected core-shellstructures of cobalt and manganese ferrites and used a core-and-shell exchange coupling to control the magnitude ofeffective magnetic anisotropy As a result they obtainedthe core-shell structure shown in Figure 3 with a measuredmagnetic anisotropy constant 119870 of 17 times 104 Jm3 (Table 1)


(a)

(c)

(b)

50 nm

Figure 2 Transmission electron micrographs of shape-controlled magnetic nanoparticles with different projection shapes (a) hexagonaloutlines of octahedron-shaped Fe

3

O4

nanoparticles (zone axis ⟨111⟩) and (b) parallelogram outlines of the same Fe3

O4

nanoparticles as in(a) (zone axis ⟨110⟩) lowastReproduced from Li et al [49] with permission (Copyright 2010 American Chemical Society) (c) Hexagonal outlinesof cube-shaped Ni-Pt nanoparticles (zone axis ⟨111⟩) Private communication (Copyright 2011 B Jeyadevan)

Figure 3 Electron energy-loss spectroscopy (EELS) mappinganalysis of CoFe

2

O4

MnFe2

O4

nanoparticles where Co Fe andMn atoms are indicated as green red and blue respectivelylowastReproduced from Lee et al [27] with permission (Copyright 2011Nature)

When these core-shell nanoparticles were irradiated with anAC magnetic field of frequency 119891 = 500 kHz and amplitude119867ac = 373 kAm the heat dissipation (119875

119867) per unit weight

reached 3MWkg (3 kWg) which was significantly higherthan that using nanoparticles of cobalt ferrite (04MWkg)

Table 1 Size saturation magnetization (119872119904

) anisotropy constant(119870) and heat dissipation rate per unit weight 119875

119867

(at 119867ac =

373 kAm 119891 = 500 kHz) of ferrite nanoparticles experimentallydetermined in [27]

Sample Size(nm)

119872119904

(kAm)119870

(kJm3)119875119867

(MWkg)CoFe2O4 12 510 200 04MnFe2O4 18 700 3 02MnFe2O4CoFe2O4 15 570 17 30

or manganese ferrite (02MWkg) The heat generation ofthese core-shell nanoparticles is unprecedented so they havereceived widespread attention

This example suggests optimized design of nanoparticlesynthesis has succeeded in producing nanoparticles thatgenerate large amounts of heat However further consider-ation revealed two notable points First the actual amplitudeof 119867ac reached 373 kAm or 80 that of the anisotropicmagnetic field 119867K = 2119870119872

119904= 473 kAm This is large

enough for the energy barrier to magnetization reversal todisappear because of the Zeeman energy in cases where thedirection of the AC magnetic field is not completely parallelto the easy axis of nanoparticles Thus these conditionsdo not permit the application of the guiding principles


given in (2)ndash(5) because these assume a linear response forsuperparamagnetic nanoparticles in zeromagnetic field limitThis raises the question of whether irradiation with an ACmagnetic field with 119891 of 500 kHz and 119867ac of 373 kAm forcore-shell structured nanoparticles with 119889 of 15 nm and 119870 of17 times 104 Jm3 are really the optimum conditions Howeverit is difficult to apply the other guiding principle to maxi-mize hysteresis loss of ferromagnetic nanoparticles becausethermally assisted reversals of 120583 occur stochastically beforethe barrier disappears at119867

119870 Recalling that the characteristic

time of thermal fluctuationwas estimated to be a fewhundrednanoseconds even in a zero magnetic field the conditionsused by Lee et al are outside the scope of applicabilityof conventional models for ferromagnetic nanoparticles ata temperature of absolute zero and for superparamagneticnanoparticles in a zero magnetic field Consequently newguiding principles to maximize heat dissipation 119875

119867are

required The second point is that 119867ac = 373 kAm ismuch larger than the exposure restriction for this waveband[52] This point is examined further in Section 5 The nextsection will present results of recent numerical studies on thebehavior of nanoparticles under conditions outside the scopeof applicability of conventional models This knowledge willbe useful to establish sophisticated guiding principles that areadapted to advanced technologies that control the size shapeand composite structure of nanoparticles

4 Recent Numerical Simulations for NovelResponses to AC Magnetic Fields

To further improve the guiding principles for the designof magnetic nanoparticles we must clarify the behavior ofnanoparticles under conditions outside the scope of appli-cability of conventional models However it is difficult todiscuss nonlinear nonequilibrium responses algebraically asan alternative numerical simulation has been performedextensively because of recent advances in computing speedNoteworthy results obtained from these studies will beintroduced in this section To fully discuss their features fromthe viewpoint of efficiency the results are shown as the ratioof the simulated value of 119875

119867to the theoretical upper limit of

119875119867119875119867Max where 119875119867Max is expressed as 4120583

0119872119904119867ac sdot 119891 sdot 120588

minus1

for irradiating ACmagnetic field119867ac sin(2120587119891 sdot 119905) because theloss dissipated in one cycle is the area of the hysteresis loop

In most of the simulations it was assumed that mag-netic nanoparticles were individually delivered to tumortissues and accumulate randomly inside them apart fromthe present status of this treatment [53] Because the actualconcentration of nanoparticles in tumors 119888 does not exceed10 kgm3 (10mgcm3) as stated above effects caused bydipole-dipole interactions 119869dd between the accumulatednanoparticles were considered insignificant at room tem-perature For example at the mean distance ⟨119903⟩ asymp 119889 sdot

12058813

sdot 119888minus13

119869dd119896119861 asymp 12058301205832

(⟨119903⟩3

119896119861) is estimated to be

25K for magnetite nanoparticles with 119889 = 15 nm 119872119904=

450 kAm and 119888 = 10 kgm3 Thus the nanoparticlesin this hyperthermia treatment simulation were consideredmagnetically isolated from each other

41 Neel Relaxation in Magnetic Fields In a magneticallyisolated nanoparticle the potential energy 119880 with respectto the direction of 120583 is simply given by the sum of magneticanisotropy energy and Zeeman energy As a first approxima-tion uniaxial magnetic anisotropy has usually been assumedfor the former term although it contains contributionsfrom various kinds of magnetic anisotropy such as shapecrystalline and surface anisotropy In this case 119880 can beexpressed as

119880 (120601 120595) = 119870119881sin2120601 minus 1205830120583119867ac sin (2120587119891 sdot 119905) cos120595 (6)

where 120601 is the angle between the easy axis and 120583 and 120595

is the angle between 120583 and H The detailed trajectories of120583 in this potential can be precisely simulated by solvingthe stochastic Landau-Lifshitz-Gilbert equations [53ndash57]However we are only interested in the reversal of120583 once everymicrosecond because the frequency used for hyperthermiatreatment is limited Carrey et al calculated the behavior of120583 using a well-known coarse-grained approach or ldquotwo-levelapproximationrdquo [58 59] which considers thermally activatedreversals between the metastable directions via the midwaysaddle point in the energy barrier In this calculation easyaxes of the accumulated nanoparticles were assumed to befixed This assumption seems valid when the nanoparticlesare strongly anchored to structures resembling organelles

Figures 4(a) 5(a) and 6(a) show contour plots of 119875119867

119875119867Max calculated for cobalt ferrite manganese ferrite and

their core-shell nanoparticles introduced above respectivelywhere the time evolution of the occupation probabilitiesof the directions parallel to the randomly oriented easyaxes are simulated in the same way as Carrey et al usingthe parameters given in Table 1 At low 119867ac of 1 kAm119875119867119875119867Max of the core-shell nanoparticles increases with 119891

and a single maximum is found at a peak frequency 119891119901 of

110 kHz (Figure 6(a)) This behavior is consistent with theabove prediction that 119875

119867is maximized when 119891

minus1 is adjustedto the Neel relaxation time It is notable that 119891

119901shifts to

higher frequency as 119867ac increases This acceleration of Neelrelaxation can be attributed to lowering of the energy barrierby the Zeeman energy As indicated by the dashed line inFigure 6(a) the shift of 119891

119901can be explained by 120591

119873(119867ac)

calculated using the conventional Brownrsquos equation as follows[60]

[120591119873(119867ac)]

minus1

= 1198910sdot (1 minus ℎ

2

)

times (1 + ℎ) exp [(minus119870119881

119896119861119879

) (1 + ℎ)2

]

+ (1 minus ℎ) exp [(minus119870119881

119896119861119879

) (1 minus ℎ)2

]

(7)

where ℎ is 119867119867119870 In Figure 6(a) 119891

119901at 119867ac = 20 kAm a

typical 119867ac for hyperthermia treatment is about 40 timesfaster than that in a zero magnetic fieldThis fact clearly indi-cates thatmaximumheat dissipation cannot be obtained if weprepare nanoparticles according to the conventional guidingprinciples expressed in (1)ndash(5)This problembecomes serious


30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)

Figure 4 Calculated efficiency of heat dissipation byCoFe2

O4

nanoparticles that are (a) nonrotatable and (b) rotatableDashed lines representthe Neel relaxation time (2120587120591

119873

)minus1 and the solid line indicates 119891

119901

which was calculated using (11) Diamonds denote the conditions used inthe experiment

30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)

Figure 5 Calculated efficiency of heat dissipation by MnFe2

O4

nanoparticles that are (a) non-rotatable and (b) rotatable Dashed linesrepresent the Neel relaxation time (2120587120591

119873


119901

which was calculated using (11) Diamonds denote the conditionsused in the experiment

when monodisperse nanoparticles are synthesized althoughwe barely noticed the problem because we used polydispersenanoparticles with a broad distribution of 120591

119873

It is very important that these calculated results arecompared with experimental data even under only oneset of conditions with 119891 = 500 kHz and 119867ac = 373 kAm

In Figure 6(a) 40 of 119875119867Max that is 14MWkg is expected

for the core-shell nanoparticles at 119891 = 500 kHz and119867ac = 373 kAm (diamonds) whereas a larger value of30MWkgwas actually observed In Figure 4(a) almost zerodissipation was calculated for the cobalt ferrite nanoparticlesunder the same conditions because these nanoparticles


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)

Figure 6 Calculated efficiency of heat dissipation by core-shell nanoparticles that are (a) non-rotatable and (b) rotatable Dashed linesrepresent the Neel relaxation time (2120587120591

119873

)minus1 the solid line indicates 119891

119901

which was calculated using (11) and the dashed-dotted line shows thevalue calculated using (12) Diamonds denote the conditions used in the experiment

are ferromagnetic so no hysteresis loss is dissipated when119867ac = 373 kAm because it is sufficiently lower than119867119870

= 630 kAm In contrast considerable dissipation of04MWkg was experimentally reported for the cobalt ferritenanoparticles In Figure 5(a) a small amount of dissipationis expected for the manganese ferrite nanoparticles under thesame conditions because these nanoparticles are typicallysuperparamagnetic and little relaxation loss dissipates at119891 = 500 kHz that is sufficiently lower than [2120587120591

119873(119867ac)]

minus1

of several tens of megahertz However a considerabledissipation of 02MWkg was experimentally reportedfor the manganese ferrite nanoparticles Some of theseinconsistenciesmay be attributed to the fact that themagneticnanoparticles were easily rotatable in a low viscous liquid oftoluene Hence Brownian rotations would be described next

42 Brownian Relaxation in Magnetic Fields In this sub-section ferromagnetic nanoparticles in Newtonian fluidsare considered because toluene is a typical Newtonian fluid(120578 = 055mPasdots) although the actual microviscoelasticityof the local environment in cancer cells is still unknownIn this case the inertia of nanoparticles with a typical sizeof 10 nm can be neglected in considering their rotation byBrownian dynamics simulation [61 62] In the inertia-lesslimit frictional torque for the rotation of a sphere balanceswith magnetic torque 120583(119905) times 119867(119905) and Brownian torque 120582(119905)as follows

6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)

where 120596(119905) is the angular velocity of rotation for the unitvector e(119905) along the easy axis given by 119889119890119889119905 = 120596(119905) times e(119905)and 120575(119905

1minus1199052) is the Dirac delta function Yoshida and Enpuku

[63] simulated the rotation of ferromagnetic nanoparticlesusing the Fokker-Planck equation equivalent to the aboverelationships they assumed that 120583(119905) was permanently fixedat the direction parallel to e(119905) as long as 119867ac lt 119867

119870 As a

result they confirmed that at zero magnetic field limit thefrequency-dependence of heat dissipation exhibits a singlemaximumat119891

119901= (2120587120591

119861)minus1 as predicted by (2)ndash(5)They also

found that 119891119901increases with119867ac according to the equation

2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)

This equation indicates that the driving force of the rotationchanges from Brownian random torque to magnetic torqueas119867ac increases

As an example this equation is applied to the cobaltferrite nanoparticles discussed above The solid curve inFigure 4(b) shows the values of 2120587119891

119901calculated using (11)

with the parameters in Table 1 The obtained line is closeto the position of the diamond located at 119891 = 500 kHzand 119867ac = 373 kAm In other words the magnetic torquefrom the magnetic field at 373 kAm happened to satisfythe conditions of rotating the cobalt ferrite nanoparticles


with an appropriate delay to the alternation at 500 kHzconsequently a considerable amount of heat 37MWkgdissipates Apart from the magnitude this is the reason why119875119867= 04MWkg was experimentally observed for the cobalt

ferrite nanoparticles despite the conventional prediction ofno hysteresis loss under the experimental conditions Asexemplified here delayed rotations are caused by magnetictorque (not Brownian torque) even at 119867ac much lower than119867119870 resulting in significant heat dissipationResearchers are also interested in the magnetic response

when119867ac becomes comparable to119867119870 In this case the above-

mentioned assumption that 120583(119905) is permanently fixed at thedirection parallel to e(119905) is invalid because 120583(119905) is cantedfrom the easy axis by the Zeeman energy Furthermore 120583(119905)stochastically reverses by thermal fluctuations even in ferro-magnetic nanoparticles because the Zeeman energy lowersthe barrier height sufficiently Therefore I simultaneouslycomputed the rotations of the nanoparticles using (8)ndash(10)with the thermally activated reversals of 120583(119905) on the potentialgiven by (6) [64] Note that (8) is valid within the two-levelapproximation [65] The results calculated for these cobaltferrite nanoparticles are shown as the contour lines (andcolor difference) in Figure 4(b) Firstly we are certain thatat 119867ac ≪ 119867

119870asymp 630 kAm the location of the ridge in

the contour plot of 119875119867119875119867Max is consistent with the solid

line given by (11) This result indicates that ferromagneticnanoparticles are rotated by the magnetic torque before thereversal of 120583(119905) occurs within it However the ridge seemsturn to the position extrapolated from the dashed curve givenby (7) when119867ac becomes comparable to119867

119870 In other words

120583(119905) is promptly reversed before the rotation because theNeel relaxation is accelerated enough in this119867ac rangeTheserelationships can be written as

2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)

This equation is an extended relationship of 120591minus1 = 120591minus1

119873

+ 120591minus1

119861

((3)) for a large AC magnetic field It is noteworthy thatthe first term 120591

119873(119867ac) usually becomes extremely small for

ferromagnetic nanoparticles at 119867ac asymp 119867119870in an aligned case

(eH) or at 119867ac asymp 1198671198702 in tilted cases while the second

term is approximately expressed as 05(1205830120583119867ac6120578119881119867) when

1205830120583119867ac ≫ 119896

119861119879 Therefore the changeover from rotation to

reversal occurs at 2120587119891 asymp 05(12058301205831198671198706120578119881119867) = 119870119881(6120578119881

119867)

or 119870119881(12120578119881119867) for aligned and tilted cases respectively For

example this changeover frequency corresponds to 4MHzfor the aligned cobalt ferrite nanoparticles with 119889 = 12 nm119881119881119867

= 063 119870 = 200 kJm3 and 120578 = 055mPasdots Impor-tantly the changeover frequency is independent of the size ofnanoparticles as long as the ratio 119881119881

119867is constant In other

words rotations predominate over the magnetic responseat 1MHz even for much larger cobalt ferrite nanoparti-cles (119889 = 120 nm (2120587120591

119861)minus1

= 200Hz) We must keep inmind that even when ferromagnetic nanoparticles are largeenough for their Brownian relaxation to be negligible mag-netic torque can easily rotate such nanoparticles at a timescale of microseconds if they are in a liquid phase This

knowledge is helpful when considering the optimal frequencyfor hyperthermia treatment even if it is for a simplifiedsystem

43 Easy Axes Oriented to the Directions Parallel Perpendic-ular or Oblique to the AC Field As described above the fastreversals of 120583(119905) are predominant in the magnetic responseof ferromagnetic nanoparticles at frequencies higher than thechangeover frequency The simulations however revealedthat at the frequencies the rotation induces various kinds ofstationary orientations of the easy axes e(119905) which criticallyaffect the reversals [64 66] In this section we also examinethe results determined for cobalt ferrite nanoparticles with119889 = 12 nm 119881119881

119867= 063 119870 = 200 kJm3 and 120578 =

055mPasdots In the initial state before irradiation with the ACmagnetic field the easy axes are set to be randomly orientedin the fluid as shown in Figure 7(a) Therefore in the firstcycle themajor hysteresis loop obtained at119867ac = 640 kAmgt

119867119870is consistent with the magnetization curve predicted by

the Stoner-Wohlfarth model (see the inset) If the irradiationof the AC magnetic field at 119867ac = 640 kAm is continuedin the simulation the easy axes gradually turn toward thedirection parallel to H Note that in the case where the easyaxis is not parallel to H the direction of 120583 is not completelyparallel to H even though 120583 is already reversed at 119867 ge 119867

119870

Therefore a large magnetic torque proportional to sin 120595 canturn the easy axis even if the magnetization seems almostsaturated at 119867 asymp 119867

119870 For example sin 120595 is 043 when cos

120595 is 09 Consequently a longitudinally oriented structureof the easy axes is formed in the fluid (see Figure 7(d))The formation of this nonequilibrium structure makes thedynamic hysteresis loop squarer than the initial curve asshown in the inset of upper panel of Figure 7(d)

In contrast themagnetization curve at119867ac = 300 kAmlt

1198671198702 is a minor hysteresis loop as shown in Figure 7(b) In

this case the easy axis turns toward the direction perpen-dicular to H and they maintain planar orientations if theferromagnetic nanoparticles are continuously irradiated byan AC magnetic field at 119867ac = 300 kAm A question nowarises because we know that the longitudinal orientation ispreferred when the Zeeman energy is considered To clarifythe reason for this we consider an initial state in which ananoparticle with an easy axis at angle 120579 has a magneticmoment 120583 at a parallel direction 120595 = 120579 When a smallmagnetic field 119867 lt 119867

1198702 is applied to the nanoparticle

120583 immediately tilts to 120595 = 120579 minus 120601 without reversals (seeFigure 7(e)) because the position of the local minimum on119880(120601 120595) is changedThen themagnetic torqueminus120583

0120583119867 sin(120579minus

120601) rotates 120583 toward the longitudinal direction 120595 rarr 0Because120583drags the easy axis 120579 also decreases In otherwordsthe easy axis turns toward the direction parallel to H If H isreversed subsequently the direction of 120583 at this moment isalmost antiparallel to H at 120595 = 120579 + 120587 minus 120601 Then 120595 instantlychanges to 120579 + 120587 + 120601 because of the effect of variation of theminimum on 119880(120601 120595) (see Figure 7(e)) The magnetic torqueat this stage minus120583

0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583

to rotate toward the direction120595 = 2120587 via120595 = (32)120587 Because120583 is bound on the easy axis 120579 also increases In other words


01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640

Toluenea liquid phase

Magneticnanoparticles

120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04

minus1 minus05 0 05 1

0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis

(b)Hac = 300kAm (c) Hac = 340kAm (d)Hac = 640kAm

(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0

Figure 7 Calculated orientation distribution of the easy axes 120588(120579) of CoFe2

O4

nanoparticles in (a) thermal equilibrium at119867 = 0 and (b)ndash(d) nonequilibrium steady states under AC magnetic field at various 119867ac and 119891 = 30MHzThe inset shows the dynamic hysteresis loopsDiagrams of the magnetic torques in the AC field are depicted in (e) and (f) where the ellipsoid in each figure shows a nanoparticle andthe broken line open and closed arrows indicate the directions of the easy axis magnetic moment of the particle and the AC magnetic fieldrespectively The nonequilibrium structures under the high-frequency AC magnetic field are illustrated in the sketches in the lower column

the easy axis starts to turn toward the plane perpendicularto H If the direction of H alternates at a high frequencya planar orientation of the easy axis is formed on averagebecause 120583

0120583119867 sin(120579 + 120601) is larger than 120583

0120583119867 sin(120579 minus 120601) This

reduces the remanence of the hysteresis loop In contrast alongitudinal orientation is formed in a large AC magneticfield119867 ge 119867

119870as discussed above because120583 is always reversed

to the direction parallel toH immediately afterH is reversedOverall 120579 decreases toward 0 when the reversal of 120583 occurswith alternation of the direction of H whereas 120579 increasestoward 1205872 without reversal of 120583

This feature leads to formation of novel nonequilibriumstructures such as the obliquely oriented state found atan intermediate amplitude of 119867ac = 340 kAm Withoutconsidering thermal fluctuations the reversals should occurin the range of 120579 from 015120587 to 035120587 for Stoner-Wohlfarthnanoparticles with 119867

119870= 630 kAm while 120583 never reverses

in the other ranges If this feature simply applies 120579 shoulddecrease with time in the range between 015120587 and 035120587whereas it should increase both between 0 and 015120587 andbetween 035120587 and 1205872 These variations certainly lead toformation of a bimodal120588(120579)with doublemaxima at 120579 = 015120587

and 1205872 as found in Figure 7(c) Consequently the easy axesare oriented in both the planes perpendicular and oblique tothe magnetic field

Concisely in ferromagnetic nanoparticles in toluene oran aqueous phase longitudinal conical or planar orienta-tions are formed irrespective of the free energy as nonequi-librium structures under a high-frequency AC magneticfield As a result the major hysteresis loop becomes squarerand the minor loop becomes narrower compared with

the magnetization curve calculated for randomly orientednanoparticles These variations of the area of the loops causethe maximum of 119875

119867119875119867Max to shift towards higher 119867ac

from the optimal conditions predicted by the conventionalmodels in Section 2 This kind of averaging of the oscillatingrotations discussed using the cobalt ferrite nanoparticlesas an example should always occur as long as the alter-nation of the magnetic field is much more frequent thanthe characteristic time of rotation 05(120583

0120583119867ac6120578119881119867) For

this reason these nonequilibrium structures would form inthe radio-waveband used for hyperthermia treatment if theamplitude is somewhat smaller (sim10 kAm) or the viscosityis considerably higher (sim10mPasdots) Therefore we must keepin mind the important effects of nonequilibrium structureson heat dissipation when establishing the optimal design offerromagnetic nanoparticles for hyperthermia treatment

44 Magnetic Hysteresis of Superparamagnetic States Let usleave ferromagnetic nanoparticles and move on to super-paramagnetic manganese ferrite nanoparticles from whicha considerable amount of heat dissipation 02MWkg wasexperimentally reported at 119891 = 500 kHz The orientationof 120583 on these nanoparticles is easily equilibrated in themagnetic potential expressed in (6) within the scale of theNeel relaxation time 120591

119873(119867ac = 0) of 1 times 10minus8 s Therefore

little relaxation loss is expected using the conventionalmodelFor this reason I wish to examine this inconsistency from theviewpoint of the effects of slow rotations on the fast reversalsin superparamagnetic nanoparticles

The contour lines (and color difference) in Figure 5(b)show the results obtained from the simultaneous simulation


120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06

H (kAm)minus4 minus2 0 2 4

⟨cos120579⟩

(b) Mean orientation

Figure 8 Calculated magnetic response of MnFe2

O4

nanoparticles with an applied AC field with 119867ac = 4 kAm and 119891 = 100 kHz (a)Steady magnetization curves and (b) mean orientation of the easy axis of the nanoparticles ⟨cos 120579⟩ In the inset in (a) the ellipsoid shows ananoparticle and the broken line open and closed arrows indicate the directions of the easy axis magnetic moment of the particle and theAC magnetic field respectively The variation of easy axis orientations is illustrated in the sketches in (b)

of rotations and reversals for the manganese ferrite nanopar-ticles Firstly we find a secondary maximum of 119875

119867119875119867Max

around 119891 = 100 kHz in addition to the primary ridge of119875119867119875119867Max indicated by the dashed curve at frequencies of

several tens of megahertz which is explained by (7) for120591119873(119867ac) above To clarify the origin of the new kind of heat

dissipation the magnetization curve calculated under theconditions of119867ac = 4 kAm and 119891 = 100 kHz is presented inFigure 8(a) An S-shaped hysteresis loop without remanenceis observed In this cycle the directions of the easy axeshave butterfly-shaped hysteresis as shown in Figure 8(b)This behavior is explained by the following atypical magneticresponse in the period 119891

minus1 (10 120583s) Initially (at 119905 = 0)no magnetization exists because the occupation probabilitiesof 120583 in the two stable directions parallel to the easy axisare equalized in a zero magnetic field As 119867 increasesthe occupation probability in the more stabilized directionimmediately increases because of reversals on a time scaleof 120591119873(le10 ns) The reversed 120583 in the stabilized direction is

not completely parallel to H 120595 = 0 and the magnetic torque1205830120583119867 sin120595 turns the easy axis towards the direction of the

field The time constant of this process is approximatelyexpressed as [05(120583

0120583119867ac6120578119881119867)]

minus1 using the second term in(11) For the manganese ferrite nanoparticles it is 3 120583s when119867 is 4 kAm Therefore rotation is not negligible in the peakperiod of the oscillations of119867 Subsequently119867 decreases to

zero at 119905 = 05119891 = 5 120583s and the occupation probabilitiesare again equalized because reversal is rapid so the magnetictorque disappears Alternatively the Brownian torque ran-domizes the orientation of the easy axis on a time scale of 120591

119861

(= 2120583s) Therefore competition between the magnetic andBrownian torques can cause the butterfly-shaped hysteresisof ⟨cos 120579⟩ Because the equilibrium magnetization of thesuperparamagnetic nanoparticles with easy axes parallel to119867is higher than that of randomly oriented ones [58 67] themagnetization curve shows hysteresis without remanenceConsequently a secondary maximum appears even though120591119873≪ 120591119861if the nanoparticles are rotatable As discussed here

we should remove the stereotype of a single peak at a 2120587119891119901

value of 120591minus1(= 120591minus1

119873

+ 120591minus1

119861

)Needless to say there is still room for further study For

example 119875119867simulated at f = 500 kHz and 119867ac = 373 kAm

is 013MWkg which is inconsistent with the observed 119875119867of

02MWkg At present it is unclear whether the differencecan be attributed to the nontrivial polydisperse nature ofthe prepared sample or the accuracy of the simulationsbecause the experiment was performed under only one setof conditions with 119891 = 500 kHz and119867ac = 373 kAm Thusmeasurement of 119875

119867under various conditions will be helpful

to establish a model of heat dissipation in superparamagneticnanoparticles In addition it is certain that the protocols ofthese simulations are also improvable because it has been


assumed that the direction of 120583 is fixed at one of the localminima in the energy potential given by (6) although weknow 120583 stochastically explores all over the potential well [65]Briefly the magnetic torque is overestimated Recently morestrict calculations were carried out and they also show thesame kind of butterfly-shaped hysteresis [67] As describedhere much still remains to be done

45 Intermediate State between Ferromagnetic and Superpara-magnetic Nanoparticles Core-shell nanoparticles which cangenerate the largest amount of heat out of various nanopar-ticle structures fit neither ferromagnetic (120591

119873(119867ac = 0) ≫

119891minus1

) nor superparamagnetic (120591119873(119867ac = 0) ≪ 119891

minus1

) condi-tions This is because the value of the Neel relaxation time120591119873(119867ac = 0) calculated using the parameters in Table 1 is

1 120583s which is comparable with the alternation time of the ACmagnetic field used in hyperthermia treatment Furthermorethe Brownian relaxation time 120591

119861is also estimated to be

1 120583s Therefore it is worth discussing this intermediate casebefore concluding this section Figure 6(b) shows the resultsobtained by simultaneous simulation of rotation and reversalas contour lines (and color difference) In this figure weare certain that location of the ridge in the contour plotof 119875119867119875119867Max is consistent with neither the dashed curve

(7) nor the solid curve (11) but instead with the dashed-dotted curve given by (12) Furthermore the iso-heightcontour lines for example the boundary between yellowand light green shift toward lower frequency compared withthe randomly fixed case in Figure 6(a) Figure 9 shows themagnetization curve and variation of the directions of theeasy axes calculated for the core-shell nanoparticles underthe conditions of 119867ac = 373 kAm and 119891 = 500 kHz Weobserve eyeglass-shaped hysteresis in the variation of thedirections of the easy axes This behavior is attributed tocomplicated competition between normal rotations when 120583is parallel to H counter-rotations when 120583 is antiparallel toH and randomization at H asymp 0 The major point is that thefirst term seems to dominate the other terms because thebaseline of the eyeglass-shaped oscillations of the easy axesis considerably higher than the 05 expected for randomlyoriented nanoparticles This longitudinal orientation makesthe dynamic hysteresis loop squarer and leads to an increasein 119875119867(see Figure 9(a)) In addition to this effect on average

oscillation of the directions of the easy axes induced by thealternation of the counter-rotations and randomization fur-ther increases 119875

119867 Indeed we can observe that the hysteresis

loop of the rotatable nanoparticles in Figure 9(a) opens evenin the higher magnetic field where the loop of the non-rotatable nanoparticles in Figure 9(a) is closed after all 120583 arereversed Overall both the phenomena discussed for ferro-magnetic and superparamagnetic nanoparticles contribute toamplification of the hysteresis loop area in this intermediatestate as a result 119875

119867increases from 14MWkg for the non-

rotatable case to 24MWkg for the rotatable one We cansay that this value is fairly consistent with the observed 119875

119867

of 3MWkg in consideration that the simulation was carriedout for completely isolated monodisperse nanoparticles withuniform uniaxial anisotropy

minus40 minus20 0 20 40minus1

0

1

RotatableNonrotatable

0 20 4006

07

08

09

Rotatable

minus40 minus20H (kAm)

MM

s

⟨cos120579⟩

(a)

(b)

Figure 9 Calculated magnetic response of core-shell nanoparticleswith an applied AC field with119867ac = 373 kAm and119891 = 500 kHz (a)Steady magnetization curves and (b) mean orientation of the easyaxis of the nanoparticles ⟨cos 120579⟩

5 Optimized Design and Future Outlook

Magnetic nanoparticles for thermotherapy particularly rotat-able nanoparticles have been predicted to exhibit variousnovel responses to AC magnetic fields as described aboveExamples include magnetic hysteresis observed for super-paramagnetic states and nonequilibrium structures with easyaxes oriented to the directions parallel perpendicular oroblique to the magnetic field These nonlinear and nonequi-librium phenomena cannot be explained using conventionalmodels Further systematic simulations and their experimen-tal verification are required to establish sophisticated guidingprinciples for such magnetic nanoparticles However somefeel that the heat generation of 3MWkg achieved by Lee etal is sufficient for practical use in hyperthermia treatment somore sophisticated guidelines may not be necessary In thisfinal section we discuss this issue

Tumors less than 001m (= 1 cm) in size are consid-ered difficult to find with existing diagnostic methods sohere we examine whether or not the heat dissipation fromcurrent magnetic nanoparticles is enough to treat hiddentumors of such size According to Andra et al [68] raising


the temperature of a tumor by Δ119879 requires heat generationof approximately 3120582Δ119879119877

minus2 without considering blood flowwhere 120582 is thermal conductivity and 2119877 is the diameter ofa tumor If we assume 120582 = 06WKminus1mminus1 Δ119879 = 5Kand 2119877 = 0005 or 001m the required heat generationwould be 15 or 04MWm3 respectively The rate of bloodflow in tumor tissues is typically 1 per second by volume(60mLmin100 g) [69] thus when Δ119879 = 5K the heattransport caused by blood flow is estimated to be 02MWm3using a value of sim4MJ-mminus3 Kminus1 for the specific heat ofblood Therefore the total cooling power of hidden tumorsis between 06 and 2MWm3 for Δ119879 = 5K This assessmentindicates that the amount of heat dissipation 119875

119867required

to kill metastatic cancer cells is estimated to be within 03and 1MWkg if we can expect a nanoparticle concentrationwithin tumors of approximately 2 kgm3The developed core-shell magnetic nanoparticles thus clearly enable adequateheat dissipation However are they actually suitable for usein hyperthermia treatment

Note that Section 4 described how nanoparticles with119875119867of 3MWkg was obtained from irradiation using an AC

magnetic field of 119867ac = 373 kAm and 119891 = 500 kHzWhen this AC magnetic field is irradiated on a simple modelbody composed of a homogenous column with electricalconductivity 120590 = 02 Smminus1 and radius 119903 = 01m themaximum voltage generated on the outer circumference is119881 = 120587119903

2

2120587119891(1205830119867ac) = 4600V per revolution at which point

the eddy current loss 119875119890= 12120587

2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3

(5Wcm3) This heat generation is sufficient to raise the tem-perature of thermally insulated tissues by 10K or more in 10seconds For this reason we cannot ignore the side effects of119875119890onnormal tissues although themodel assuming a constant

120590 is oversimplified According to guidelines published bythe International Commission on Non-Ionizing RadiationProtection [52] the upper limit for work-related exposureof the torso is 10Wkg (corresponding to 001MWm3)However because this value is the upper limit for routineexposure it may not be indicative of the maximum exposurein medical treatment A slightly more specific value can becalculated as followsHeat generation only occurs in the outeredge of a human body if amagnetic field is irradiated over thewhole body therefore the heated region can be consideredas a cylinder that is a few centimeters thick This regioncan be cooled from the body surface area in medical treat-ment Its cooling power 12058212059721198791205971199032 sim 120582Δ119879(Δ119903)

minus2 is roughlyestimated to be 003MWm3 under the conditions of Δ119879 =

20K and Δ119903 = 002m Because blood vessels expandwhen temperature rises blood flow increases even in tissueswith little blood flow normally In subcutaneous tissuesfor example a blood flow rate of approximately 02 persecond by volume (12mLmin100 g) has been reported at42∘C [69 70] Under these conditions calculating the heattransport caused by blood flow using the samemethod yieldsa value of 003MWm3 when the temperature difference fromthe outside of the irradiated region is set at Δ119879 = 4KThese values are the result of rough calculations that donot consider the detailed structure of a human body [71]

but their sum which is about 006MWm3 can be used asan approximation of cooling ability When a magnetic fieldis applied to the model body mentioned above this valuecorresponds to 119875

119890for the condition119867ac119891 = 2 times 109 Amminus1 sminus1

Calculating the behavior of the above-mentioned core-shellnanoparticles (119889 = 15 nm) within this restriction (seeFigure 10) shows that 119875

119867does not reach the requirement of

03MWkg However Figure 10 indicates that if the size ofthe particle is increased slightly sufficient 119875

119867can be obtained

from the rotatable nanoparticles at higher119867ac (equivalent tolower 119891) even under this restriction and adequate heatingis expected inside hidden tumors with a diameter of 001mwithout serious side effects on normal tissues from 119875

119890

Our discussion up to this point applies to treatment usingcontinuous irradiation where heat balance holds Irradiationtime and interval can be controlled in medical treatment Forexample when tumors with a specific heat of 4MJmminus3Kminus1containing the above-mentioned core-shell nanoparticleswith a concentration 119888 of 2 kgm3 were irradiated with an ACmagnetic field of 119867ac = 373 kAm and 119891 = 500 kHz heatof approximately 119888119875

119867= 6MWm3 was generated Relative to

this value the quantity of heat diffused to the surroundingareas from 10mm tumors is negligible when Δ119879 lt 5K thusthe temperaturewill rise by 5K after approximately 3 secondsBecause the eddy current loss 119875

119890in this case is 5MWm3

it will take approximately 4 seconds for the temperatureof normal tissue to rise by 5K Stopping irradiation after3 seconds will thus enable selective heating of tumors by5K or more This is an extreme example however it doesindicate that there is another option apart from continuousirradiation The ratio of 119888119875

119867to 119875119890is important Although

obtaining robust values requires detailed protocol a factor of4-5 or so might be a criterion for 119888119875

119867119875119890 As an example we

calculated 119888119875119867119875119890for the core-shell nanoparticles and found

that this condition is satisfied for lower frequenciessmalleramplitudes than those indicated by the solid line in Figure 11[72] This finding reflects the fact that 119875

119867is the area of

the 119872-119867 curve times frequency which is proportional to 119867ac119891

at most whereas 119875119890increases in proportion to (119867ac119891)

2 aspreviously described Because it is impossible to attain a risein temperature of 5 K if 119888119875

119867is at least 06 (or 2)MWm3 irra-

diationmust therefore be conducted using a higher frequencyand larger amplitude to ensure that this condition is met (seedashed lines in Figure 12 [72]) Ultimately stronger fasterconditions are needed to destroy cancer cells and weakerslower conditions are needed to limit damage to normaltissue Using the core-shell nanoparticles of 119889 = 15 nma frequency of 119891 = 500 kHz is thus acceptable but 119867acneeds to be maintained at 9 kAm to resolve the conflictingrequirements

As discussed above the combination of the core-shellnanoparticles of 119889 = 15 nm and 119870 = 17 times 10

4 Jm3 withan AC magnetic field of 119891 = 500 kHz and 119867ac = 373 kAmmay not be optimal A narrow range of combinations of theseparameters will facilitate efficient hyperthermia treatmentand prevent side effects We have not yet optimized theconditions for hyperthermia treatment however establishingthe optimal combinations may be difficult particularly if


1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable

Figure 10 Calculated heat dissipation by core-shell nanoparticles that are (a) non-rotatable and (b) rotatable where 119867ac119891 is always 2 times

109 Amminus1 sminus1 (corresponding to the restriction that the eddy current loss 119875119890

is 006MWm3 in normal tissue) The size 119889 is changed in thesimulation but the other parameters were fixed at the values shown in Table 1

2 4 8 16 32 64 2 4 8 16 32 64

(a) Nonrotatable (b) Rotatable30000

3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe

Figure 11 Calculated selection ratio 119888119875119867

119875119890

for core-shell nanoparticles inACmagnetic fields with various119867ac and119891 Rotatable nanoparticlesare compared with randomly oriented ones The dashed lines show the isoplethic curves at 119875

119867

= 03 and 1MWkg (see Figure 12) while thesolid lines show the isoplethic curves at 119888119875

119867

119875119890

= 4 lowastReproduced fromMamiya [72] with permission (Copyright 2012 TIC)

a trial and error approach is used The routes used tosynthesize magnetic nanoparticles of controlled size shapeand composite structure have become increasingly advancedas described in this paper Dramatic advances in computingspeed have also promoted numerical simulation of non-linear nonequilibrium responses to AC magnetic fields If

we continue to improve material design on the bases ofsuch advanced nanotechnology and computer simulationsoptimal conditions will eventually be clarified Remarkableadvances are still continually being reported in clinicaltrials are being conducted even though the combination ofnanoparticles and oscillation of the equipment has not been


001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)

(a) Nonrotatable (b) Rotatable

Figure 12 Calculated magnetic loss 119875119867

for core-shell nanoparticles in AC magnetic fields for various119867ac and 119891 Rotatable nanoparticles arecompared with randomly oriented onesThe dashed lines show the isoplethic curves at 119875

119867

= 03 and 1MWkg while the solid lines show theisoplethic curves at the selection ratio 119888119875

119867

119875119890

= 4 (see Figure 11) lowastReproduced fromMamiya [72] with permission (Copyright 2012 TIC)

optimized yet [73 74] Once optimization improves localheating ability then thermotherapy should be established asa fourth or fifth standard cancer treatment method to reducethe disease burden of a patient

Acknowledgment

This work was partly supported by a Grant-in-Aid for Scien-tific Research (No 24310071)

References

[1] W C Dewey L E Hopwood S A Sapareto and L EGerweck ldquoCellular responses to combinations of hyperthermiaand radiationrdquo Radiology vol 123 no 2 pp 463ndash474 1977

[2] S Mornet S Vasseur F Grasset and E Duguet ldquoMagneticnanoparticle design for medical diagnosis and therapyrdquo Journalof Materials Chemistry vol 14 no 14 pp 2161ndash2175 2004

[3] P Tartaj M Del Puerto Morales S Veintemillas-Verdaguer TGonzalez-Carreno and C J Serna ldquoThe preparation of mag-netic nanoparticles for applications in biomedicinerdquo Journal ofPhysics D vol 36 no 13 pp R182ndashR197 2003

[4] T Neuberger B Schopf H Hofmann M Hofmann and B vonRechenberg ldquoSuperparamagnetic nanoparticles for biomedicalapplications possibilities and limitations of a new drug deliverysystemrdquo Journal of Magnetism and Magnetic Materials vol 293no 1 pp 483ndash496 2005

[5] Q A Pankhurst N K T Thanh S K Jones and J Dob-son ldquoProgress in applications of magnetic nanoparticles inbiomedicinerdquo Journal of Physics D vol 42 no 22 Article ID224001 2009

[6] K M Krishnan ldquoBiomedical nanomagnetics a spin throughpossibilities in imaging diagnostics and therapyrdquo IEEE Trans-actions on Magnetics vol 46 no 7 pp 2523ndash2558 2010

[7] B Jeyadevan ldquoPresent status and prospects of magnetitenanoparticles-based hyperthermiardquo Journal of the CeramicSociety of Japan vol 118 no 1378 pp 391ndash401 2010

[8] I Sharifi H Shokrollahi and S Amiri ldquoFerrite-basedmagneticnanofluids used in hyperthermia applicationsrdquo Journal of Mag-netism andMagneticMaterials vol 324 no 6 pp 903ndash915 2012

[9] S J DeNardo G L DeNardo A Natarajan et al ldquoThermaldosimetry predictive of efficacy of111In-ChL6 nanoparticleAMF-induced thermoablative therapy for human breast cancerinmicerdquo Journal of NuclearMedicine vol 48 no 3 pp 437ndash4442007

[10] P Wust U Gneveckow M Johannsen et al ldquoMagneticnanoparticles for interstitial thermotherapymdashfeasibility tol-erance and achieved temperaturesrdquo International Journal ofHyperthermia vol 22 no 8 pp 673ndash685 2006

[11] R Hergt R Hiergeist I Hilger et al ldquoMaghemite nanoparti-cles with very high AC-losses for application in RF-magnetichyperthermiardquo Journal of Magnetism and Magnetic Materialsvol 270 no 3 pp 345ndash357 2004

[12] M Ma Y Wu J Zhou Y Sun Y Zhang and N Gu ldquoSizedependence of specific power absorption of Fe

3

O4

particlesin AC magnetic fieldrdquo Journal of Magnetism and MagneticMaterials vol 268 no 1-2 pp 33ndash39 2004

[13] T N Brusentsova N A Brusentsov V D Kuznetsov and V NNikiforov ldquoSynthesis and investigation of magnetic propertiesof Gd-substituted Mn-Zn ferrite nanoparticles as a potentiallow-TC agent for magnetic fluid hyperthermiardquo Journal ofMagnetism and Magnetic Materials vol 293 no 1 pp 298ndash3022005

[14] G Glockl R Hergt M Zeisberger S Dutz S Nagel andW Weitschies ldquoThe effect of field parameters nanoparticle


properties and immobilization on the specific heating power inmagnetic particle hyperthermiardquo Journal of Physics vol 18 no38 pp S2935ndashS2949 2006

[15] J P Fortin C Wilhelm J Servais C Menager J-C Bacriand F Gazeau ldquoSize-sorted anionic iron oxide nanomagnets ascolloidal mediators for magnetic hyperthermiardquo Journal of theAmerican Chemical Society vol 129 no 9 pp 2628ndash2635 2007

[16] G Baldi D Bonacchi C Innocenti G Lorenzi and C Sangre-gorio ldquoCobalt ferrite nanoparticles the control of the particlesize and surface state and their effects on magnetic propertiesrdquoJournal of Magnetism and Magnetic Materials vol 311 no 1 pp10ndash16 2007

[17] L Y Zhang H-C Gu and X-M Wang ldquoMagnetite ferrofluidwith high specific absorption rate for application in hyperther-miardquo Journal of Magnetism and Magnetic Materials vol 311 no1 pp 228ndash233 2007

[18] D-H Kim D E Nikles D T Johnson and C S Brazel ldquoHeatgeneration of aqueously dispersed CoFe

2

O4

nanoparticles asheating agents for magnetically activated drug delivery andhyperthermiardquo Journal of Magnetism and Magnetic Materialsvol 320 no 19 pp 2390ndash2396 2008

[19] J-P Fortin F Gazeau and CWilhelm ldquoIntracellular heating ofliving cells through Neel relaxation of magnetic nanoparticlesrdquoEuropean Biophysics Journal vol 37 no 2 pp 223ndash228 2008

[20] L-M Lacroix R B Malaki J Carrey et al ldquoMagnetic hyper-thermia in single-domain monodisperse FeCo nanoparticlesevidences for Stoner-Wohlfarth behavior and large lossesrdquoJournal of Applied Physics vol 105 no 2 Article ID 023911 4pages 2009

[21] C L Dennis A J Jackson J A Borchers et al ldquoNearly com-plete regression of tumors via collective behavior of magneticnanoparticles in hyperthermiardquoNanotechnology vol 20 no 39Article ID 395103 2009

[22] M Gonzales-Weimuller M Zeisberger and K M KrishnanldquoSize-dependant heating rates of iron oxide nanoparticles formagnetic fluid hyperthermiardquo Journal of Magnetism and Mag-netic Materials vol 321 no 13 pp 1947ndash1950 2009

[23] R Sharma and C J Chen ldquoNewer nanoparticles in hyper-thermia treatment and thermometryrdquo Journal of NanoparticleResearch vol 11 no 3 pp 671ndash689 2009

[24] E Kita T Oda T Kayano et al ldquoFerromagnetic nanoparticlesfor magnetic hyperthermia and thermoablation therapyrdquo Jour-nal of Physics D vol 43 no 47 Article ID 474011 2010

[25] B Mehdaoui A Meffre L-M Lacroix et al ldquoLarge specificabsorption rates in the magnetic hyperthermia properties ofmetallic iron nanocubesrdquo Journal of Magnetism and MagneticMaterials vol 322 no 19 pp L49ndashL52 2010

[26] T Kikuchi R Kasuya S Endo et al ldquoPreparation of magnetiteaqueous dispersion for magnetic fluid hyperthermiardquo Journalof Magnetism and Magnetic Materials vol 323 no 10 pp 1216ndash1222 2011

[27] J-H Lee J-T Jang J-S Choi et al ldquoExchange-coupledmagnetic nanoparticles for efficient heat inductionrdquo NatureNanotechnology vol 6 no 7 pp 418ndash422 2011

[28] S-H Noh W Na J Jang et al ldquoNanoscale magnetism controlvia surface and exchange anisotropy for optimized ferrimag-netic hysteresisrdquoNano Letters vol 12 no 7 pp 3716ndash3721 2012

[29] KNakamura K Ueda A Tomitaka et al ldquoSelf-heating temper-ature and AC hysteresis of magnetic iron oxide nanoparticlesand their dependence on secondary particle sizerdquo IEEE Trans-actions on Magnetics vol 49 no 1 pp 240ndash243 2013

[30] CMartinez-Boubeta K Simeonidis AMakridis et al ldquoLearn-ing from nature to improve the heat generation of iron-oxide nanoparticles for magnetic hyperthermia applicationsrdquoScientific Reports vol 3 article 1652 2013

[31] J L Dormann D Fiorani and E Tronc ldquoMagnetic relaxationin fine-particle systemsrdquo Advances in Chemical Physics vol 98pp 283ndash494 1997

[32] X Batlle and A Labarta ldquoFinite-size effects in fine particlesmagnetic and transport propertiesrdquo Journal of Physics D vol35 no 6 pp R15ndashR42 2002

[33] H Mamiya Magnetic Properties of Nanoparticles YushodoTokyo Japan 2003

[34] P E Jonsson ldquoSuperparamagnetism and spin glass dynamicsof interacting magnetic nanoparticle systemsrdquo Advances inChemical Physics vol 128 pp 191ndash248 2004

[35] P C Scholten ldquoHowmagnetic can amagnetic fluid berdquo Journalof Magnetism and Magnetic Materials vol 39 no 1-2 pp 99ndash106 1983

[36] H Mamiya I Nakatani and T Furubayashi ldquoBlocking andfreezing of magnetic moments for iron nitride fine particlesystemsrdquoPhysical Review Letters vol 80 no 1 pp 177ndash180 1998

[37] H Mamiya I Nakatani and T Furubayashi ldquoSlow dynamicsfor spin-glass-like phase of a ferromagnetic fine particle systemrdquoPhysical Review Letters vol 82 no 21 pp 4332ndash4335 1999

[38] H Mamiya I Nakatani and T Furubayashi ldquoPhase transitionsof iron-nitride magnetic fluidsrdquo Physical Review Letters vol 84no 26 pp 6106ndash6109 2000

[39] A Wiedenmann M Kammel A Heinemann and U Keider-ling ldquoNanostructures and ordering phenomena in ferrofluidsinvestigated using polarized small angle neutron scatteringrdquoJournal of Physics vol 18 no 38 pp S2713ndashS2736 2006

[40] H Kronmuller and M Fahnle Micromagnetism and theMicrostructure of Ferromagnetic Solids Cambridge UniversityPress Cambridge UK 2003

[41] R Hergt S Dutz and M Roder ldquoEffects of size distribution onhysteresis losses of magnetic nanoparticles for hyperthermiardquoJournal of Physics vol 20 no 38 Article ID 385214 2008

[42] R E Rosensweig ldquoHeating magnetic fluid with alternatingmagnetic fieldrdquo Journal of Magnetism and Magnetic Materialsvol 252 pp 370ndash374 2002

[43] S S Papell US Patent No 3 215 1965[44] T Sato S Higuchi and J Shimoiizaka in Proceedings of the 19th

Annual Meeting of the ChemicalSociety of Japan 293 1966[45] I Nakatani M Hijikata and K Ozawa ldquoIron-nitride magnetic

fluids prepared by vapor-liquid reaction and their magneticpropertiesrdquo Journal of Magnetism and Magnetic Materials vol122 no 1ndash3 pp 10ndash14 1993

[46] S Sun C B Murray D Weller L Folks and A MoserldquoMonodisperse FePt nanoparticles and ferromagnetic FePtnanocrystal superlatticesrdquo Science vol 287 no 5460 pp 1989ndash1992 2000

[47] X-M Lin and A C S Samia ldquoSynthesis assembly and physicalproperties of magnetic nanoparticlesrdquo Journal of Magnetismand Magnetic Materials vol 305 no 1 pp 100ndash109 2006

[48] A H Lu E L Salabas and F Schuth ldquoMagnetic nanoparticlessynthesis protection functionalization and applicationrdquoAnge-wandte Chemie vol 46 no 8 pp 1222ndash1244 2007

[49] L Li Y Yang J Ding and J Xue ldquoSynthesis of magnetitenanooctahedra and their magnetic field-induced two-three-dimensional superstructurerdquoChemistry ofMaterials vol 22 no10 pp 3183ndash3191 2010


[50] J L C Huaman S Fukao K Shinoda and B Jeyadevan ldquoNovelstandingNi-Pt alloy nanocubesrdquoCrystEngComm vol 13 no 10pp 3364ndash3369 2011

[51] Y Li Q Zhang A V Nurmikko and S Sun ldquoEnhancedmagne-tooptical response in dumbbell-like Ag-CoFe

2

O4

nanoparticlepairsrdquo Nano Letters vol 5 no 9 pp 1689ndash1692 2005

[52] The International Commission onNon-IonizingRadiation Pro-tection ldquoGuide-lines for limiting exposure to time-varying elec-tric magnetic and electro-magnetic fields (up to 300GHz)rdquoHealth Physics vol 74 no 4 pp 494ndash522 1998

[53] E Lima Jr E de Biasi and M V Mansilla ldquoHeat generation inagglomerated ferrite nanoparticles in an alternating magneticfieldrdquo Journal of PhysicsD vol 46 no 4 Article ID045002 2013

[54] S M Morgan and R H Victora ldquoUse of square waves incidenton magnetic nanoparticles to induce magnetic hyperthermiafor therapeutic cancer treatmentrdquo Applied Physics Letters vol97 no 9 Article ID 093705 2010

[55] E L Verde G T Landi and M S Carriao ldquoField dependenttransition to the non-linear regime in magnetic hyperthermiaexperiments comparison between maghemite copper zincnickel and cobalt ferrite nanoparticles of similar sizesrdquo AIPAdvances vol 2 no 3 Article ID 032120 23 pages 2012

[56] G T Landi and A F Bakuzis ldquoOn the energy conversionefficiency in magnetic hyperthermia applications a new per-spective to analyze the departure from the linear regimerdquoJournal of Applied Physics vol 111 no 8 Article ID 083915 2012

[57] N A Usov S A Gudoshnikov and O N Serebryakova ldquoProp-erties of dense assemblies of magnetic nanoparticles promisingfor application in biomedicinerdquo Journal of Superconductivityand Novel Magnetism vol 26 no 4 pp 1079ndash1083 2013

[58] J Carrey B Mehdaoui and M Respaud ldquoSimple modelsfor dynamic hysteresis loop calculations of magnetic single-domain nanoparticles application to magnetic hyperthermiaoptimizationrdquo Journal of Applied Physics vol 109 no 8 ArticleID 083921 17 pages 2011

[59] Z P Mendoza G A Pasquevich and S J Stewart ldquoStructuraland magnetic study of zinc-doped magnetite nanoparticles andferrofluids for hyperthermia applicationsrdquo Journal of Physics Dvol 46 no 12 Article ID 125006 2013

[60] W F Brown Jr ldquoThermal fluctuations of a single-domainparticlerdquo Physical Review vol 130 no 5 pp 1677ndash1686 1963

[61] H Mamiya and B Jeyadevan ldquoOptimal design of nanomagnetsfor targeted hyperthermiardquo Journal of Magnetism and MagneticMaterials vol 323 no 10 pp 1417ndash1422 2011

[62] D B Reeves and J B Weaver ldquoSimulations of magneticnanoparticle Brownian motionrdquo Journal of Applied Physics vol112 no 12 Article ID 124311 6 pages 2012

[63] T Yoshida and K Enpuku ldquoSimulation and quantitative clarifi-cation of AC susceptibility of magnetic fluid in nonlinear Brow-nian relaxation regionrdquo Japanese Journal of Applied Physics vol48 Article ID 127002 7 pages 2009

[64] H Mamiya and B Jeyadevan ldquoHyperthermic effects of dissi-pative structures of magnetic nanoparticles in large alternatingmagnetic fieldsrdquo Scientific Reports vol 1 article 157 2011

[65] N A Usov and B Ya Liubimov ldquoDynamics of magnetic nano-particle in a viscous liquid application tomagnetic nanoparticlehyperthermiardquo Journal of Applied Physics vol 112 no 2 ArticleID 023901 11 pages 2012

[66] H Mamiya and B Jeyadevan ldquoFormation of non-equilibriummagnetic nanoparticle structures in a large alternatingmagneticfield and their influence on magnetic hyperthermia treatmentrdquo

IEEE Transactions on Magnetics vol 48 no 11 pp 3258ndash32622012

[67] H Mamiya and B Jeyadevan ldquoMagnetic hysteresis loop in asuperparamagneticstaterdquo in press IEEE Transactions on Mag-netics

[68] W Andra C G DrsquoAmbly R Hergt I Hilger and W A KaiserldquoTemperature distribution as function of time around a smallspherical heat source of local magnetic hyperthermiardquo Journalof Magnetism and Magnetic Materials vol 194 no 1 pp 197ndash203 1999

[69] C W Song ldquoEffect of local hyperthermia on blood flow andmicroenvironment a reviewrdquo Cancer Research vol 44 no 10supplement pp 4721sndash4730s 1984

[70] T Hasegawa R Kudaka K Saito et al Bulletin of SuzukaUniversity of Medical Science vol 11 pp 58ndash64 2004

[71] J Bohnert and O Dossel ldquoSimulations of temperature increasedue to time varying magnetic fields up to 100 kHzrdquo in Pro-ceedings of the 5th European Conference of the InternationalFederation for Medical and Biological Engineering vol 37 ofIFMBE Proceedings pp 303ndash306 2012

[72] H Mamiya ldquoMagnetic response of nanoparticles to AC mag-netic fields and targeted thermotherapyrdquo Materials Integrationvol 25 pp 11ndash23 2012

[73] T Kobayashi ldquoCancer hyperthermia using magnetic nanopar-ticlesrdquo Biotechnology Journal vol 6 no 11 pp 1342ndash1347 2011

[74] B Thiesen and A Jordan ldquoClinical applications of magneticnanoparticles for hyperthermiardquo International Journal of Hyper-thermia vol 24 no 6 pp 467ndash474 2008

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of



CeramicsJournal of


CompositesJournal of

NanoparticlesJournal of



International Journal of

Biomaterials


NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research



MetallurgyJournal of


BioMed Research International

MaterialsJournal of


Nano

materials


Journal ofNanomaterials


using magnetic nanoparticles has the potential to selectivelydestroy cancer cells hidden deep in the body [2ndash8]

Of course there are many questions about this con-cept that need to be addressed Is it safe to put magneticnanoparticles in the body Canmagnetic nanoparticles reallybe concentrated in hidden tumor tissues Can magneticnanoparticles be heated within the body To answer thesequestions DeNardo et al injected iron oxide nanoparticlesconjugated with monoclonal antibodies into mouse tails andfound that they accumulated at a concentration 119888 of approxi-mately 03 kgm3 (03mgcm3) in tumors [9] (The side effectsof iron oxide nanoparticles as anMRI contrast agent had beenpreviously studied and approved for intravenous injection)Wust et al showed that injecting high concentrations (119888 sim

10 kgm3) of magnetic nanoparticles directly into tumors atknown locations and irradiating them with an AC magneticfield caused the temperature of the tumors to increase to43∘C [10]Therefore it is known that iron oxide nanoparticlesare safe can be selectively accumulated in hidden tumors tosome degree and can be adequately heated when existingat high concentrations Nevertheless problems with targetedhyperthermia using magnetic nanoparticles are still evidentDoes the density of magnetic nanoparticles delivered totumors increase as it does when they are directly injectedinto tumors If not is it possible to compensate for a lowerdensity of magnetic nanoparticles by maximizing their heatdissipationThe first problem is primarily a biochemical onesomaterials researchers have primarily focused on improvingthe heating performance of magnetic nanoparticles [11ndash30] Consequent advances in chemical synthesis technologyhave resulted in the fabrication of magnetic nanoparticlesof engineered size shape and structure With respect tophysical heating mechanisms the nature of the nonlinearresponse and nonequilibrium dissipation in AC magneticfields of magnetic nanoparticles which are in contrast to theproperties of cooking pans have been uncoveredThis reviewaddresses this progress as follows In Section 2 conventionalmodels that are the basis of the traditional design of hyper-thermia treatments are introduced In Section 3 advances inthe synthesis of magnetic nanoparticles are described andlimitations in the conventional models when the monodis-perse nanoparticles are used in actual thermotherapy areconsidered In Section 4 recent advances in the knowledgeof heating mechanisms provided by numerical simulationsare explained Finally we summarize the optimal designof magnetic nanoparticles for hyperthermia treatment anddiscuss their potential as an effective and safe version ofHippocratesrsquo ldquofirerdquo in Section 5

2 Conventional Models for Magnetic Responseto AC Magnetic Fields [31ndash34]

The main advantage of hyperthermia treatment using mag-netic nanoparticles is that the nanoparticles can reachthe cancer tissue directly by travelling through the sub-micrometer spaces between blood cell walls Therefore forpractical use the nanoparticles should not form long chainsor large clusters Even though the many-body effects caused

by dipole-dipole coupling 119869dd are not fully understood [35ndash38] it is known that a dispersion becomes unstable if 119869ddbetween the closest nanoparticles is more than five times thethermal energy [35 38] Under these conditions the min-imum allowable distance between iron nanoparticles withdiameter119889 of 12 nm is roughly 27 nm and that between ferritenanoparticles with 119889 of 25 nm is almost 40 nm In contrastsmall-angle neutron scattering experiments have indicatedthat the thickness of an absorbed layer is normally severalnanometers [39] Therefore the upper limit of 119889 is estimatedto be roughly 12 nm for iron and 25 nm for ferrite Thesevalues would be references for considering the criteria foreasy delivery of the nanoparticles although agglomerationaggregation or flocculation may occur depending on thesurface charge of biofunctionalized nanoparticle or on theinteraction between tumor-targeting ligands Note that thesevalues are smaller than the critical diameters for the transitioninto a single-domain configuration and for the coherentreversal of all spins [40] Therefore it has been consideredthat a magnetic nanoparticle used in hyperthermia treatmenthas only one magnetic moment 120583 = M

119904119881 where M

119904

is the spontaneous magnetization and 119881 = 1205876 sdot 1198893 is

the volume of the magnetic core of the nanoparticle Suchmagnetic nanoparticles have been conventionally classifiedas ldquoferromagneticrdquo or ldquosuperparamagneticrdquo depending onwhether the direction of 120583 thermally fluctuates or not

Firstly a ferromagnetic nanoparticle with uniaxial mag-netic anisotropy anisotropy constant119870 is considered where119881 is large enough that its magnetic anisotropy barrier with aheight of119870119881 blocks the thermal fluctuations accordingly theremanent state appears to be permanent [41] If a magneticfield H is applied in the direction antiparallel to 120583 thestate becomes metastable as depicted in Figure 1(a) Then 120583reverses when the barrier disappears at the anisotropy field119867119870

= 2119870(1205830119872119904) consequently the Zeeman energy falls

from 1205830120583119867119870to minus1205830120583119867119870and the energy corresponding to the

difference dissipates where 1205830is permeability of vacuum In

this case the work done in one cycle of the ACmagnetic field119867ac sin(2120587119891 sdot 119905) is 0 for119867ac lt 119867

119870and 4120583

0120583119867119870for119867ac gt 119867

119870

This kind of heat dissipation has been termed ldquohysteresislossrdquo Briefly the heat dissipation from nanoparticles withunit weight during unit time also called specific loss power119875119867 abruptly increases from zero to 4120583

0120583119867119870sdot 119891 sdot 119908

minus1

(=

41205830119872119904119867119870sdot119891 sdot120588minus1

)when119867ac becomes higher than119867119870 where

119908 and 120588minus1 are the weight and density of the magnetic core

of the nanoparticles respectively Then 119875119867flattens out even

if 119867ac is strengthened further According to this argumentthe guiding principle for maximizing 119875

119867of ferromagnetic

nanoparticles is that 119867ac is adjusted to 119867119870and the number

of cycles 119891 is maximizedNext we move to smaller superparamagnetic nanoparti-

cles with thermally fluctuating reversal of 120583 [42]The reversalprobability in a zero magnetic field is expressed as

120591minus1

119873

= 1198910sdot exp(minus119870119881

119896119861119879

) (1)

where 120591119873

is the Neel relaxation time 1198910is the attempt

frequency of 109 sminus1 119896119861is the Boltzmann constant and 119879 is


minus2 minus1 0 1

1

1

2

21205830120583HK

21205830120583HK

HHK

MM

s

(a) Hysteresis loss

Susc

eptib

ility

Out-of-phase

In-phase


0

05

1

10minus3 100 103


(b) Relaxation loss



120591119861=

3120578119881119867

(119896119861119879)

(2)



120591minus1

= 120591minus1

119873

+ 120591minus1

119861

(3)



119873


119861


119867



12059410158401015840

=

12058301205832

(3119896119861119879)

sdot

2120587119891 sdot 120591

[1 + (2120587119891 sdot 120591)2

]

(4)


119875119867= 120587120583012059410158401015840

sdot 1198672


=

1

2

[1205832

0

1198722

s119881 (3119896B119879120591120588)] sdot 1198672

ac sdot (2120587119891 sdot 120591)2

[1 + (2120587119891 sdot 120591)2

]

(5)


2


0

1198722

119904

119881(3119896119861119879120591120588)] sdot 119867

2


119867of










119870and 120591119873









=


119861= 16 120583s)





1and


1or (14)119870

1119881








(a)

(c)

(b)

50 nm


3

O4


O4



2

O4

MnFe2

O4







119867

(at 119867ac =


Sample Size(nm)

119872119904

(kAm)119870

(kJm3)119875119867










119867are






0119872119904119867ac sdot 119891 sdot 120588

minus1



12058813

sdot 119888minus13

119869dd119896119861 asymp 12058301205832

(⟨119903⟩3















119901shifts to



119873(119867ac)


[120591119873(119867ac)]

minus1


2

)

times (1 + ℎ) exp [(minus119870119881

119896119861119879

) (1 + ℎ)2

]

+ (1 minus ℎ) exp [(minus119870119881

119896119861119879

) (1 minus ℎ)2

]

(7)


119901at 119867ac = 20 kAm a



30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)


O4


119873


119901


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)


O4


119873


119901



119873





30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




minus2 minus1 0 1

1

1

2

21205830120583HK

21205830120583HK

HHK

MM

s

(a) Hysteresis loss

Susc

eptib

ility

Out-of-phase

In-phase


0

05

1

10minus3 100 103


(b) Relaxation loss



120591119861=

3120578119881119867

(119896119861119879)

(2)



120591minus1

= 120591minus1

119873

+ 120591minus1

119861

(3)



119873


119861


119867



12059410158401015840

=

12058301205832

(3119896119861119879)

sdot

2120587119891 sdot 120591

[1 + (2120587119891 sdot 120591)2

]

(4)


119875119867= 120587120583012059410158401015840

sdot 1198672


=

1

2

[1205832

0

1198722

s119881 (3119896B119879120591120588)] sdot 1198672

ac sdot (2120587119891 sdot 120591)2

[1 + (2120587119891 sdot 120591)2

]

(5)


2


0

1198722

119904

119881(3119896119861119879120591120588)] sdot 119867

2


119867of










119870and 120591119873









=


119861= 16 120583s)





1and


1or (14)119870

1119881








(a)

(c)

(b)

50 nm


3

O4


O4



2

O4

MnFe2

O4







119867

(at 119867ac =


Sample Size(nm)

119872119904

(kAm)119870

(kJm3)119875119867










119867are






0119872119904119867ac sdot 119891 sdot 120588

minus1



12058813

sdot 119888minus13

119869dd119896119861 asymp 12058301205832

(⟨119903⟩3















119901shifts to



119873(119867ac)


[120591119873(119867ac)]

minus1


2

)

times (1 + ℎ) exp [(minus119870119881

119896119861119879

) (1 + ℎ)2

]

+ (1 minus ℎ) exp [(minus119870119881

119896119861119879

) (1 minus ℎ)2

]

(7)


119901at 119867ac = 20 kAm a



30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)


O4


119873


119901


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)


O4


119873


119901



119873





30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










119870and 120591119873









=


119861= 16 120583s)





1and


1or (14)119870

1119881








(a)

(c)

(b)

50 nm


3

O4


O4



2

O4

MnFe2

O4







119867

(at 119867ac =


Sample Size(nm)

119872119904

(kAm)119870

(kJm3)119875119867










119867are






0119872119904119867ac sdot 119891 sdot 120588

minus1



12058813

sdot 119888minus13

119869dd119896119861 asymp 12058301205832

(⟨119903⟩3















119901shifts to



119873(119867ac)


[120591119873(119867ac)]

minus1


2

)

times (1 + ℎ) exp [(minus119870119881

119896119861119879

) (1 + ℎ)2

]

+ (1 minus ℎ) exp [(minus119870119881

119896119861119879

) (1 minus ℎ)2

]

(7)


119901at 119867ac = 20 kAm a



30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)


O4


119873


119901


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)


O4


119873


119901



119873





30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




(a)

(c)

(b)

50 nm


3

O4


O4



2

O4

MnFe2

O4







119867

(at 119867ac =


Sample Size(nm)

119872119904

(kAm)119870

(kJm3)119875119867










119867are






0119872119904119867ac sdot 119891 sdot 120588

minus1



12058813

sdot 119888minus13

119869dd119896119861 asymp 12058301205832

(⟨119903⟩3















119901shifts to



119873(119867ac)


[120591119873(119867ac)]

minus1


2

)

times (1 + ℎ) exp [(minus119870119881

119896119861119879

) (1 + ℎ)2

]

+ (1 minus ℎ) exp [(minus119870119881

119896119861119879

) (1 minus ℎ)2

]

(7)


119901at 119867ac = 20 kAm a



30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)


O4


119873


119901


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)


O4


119873


119901



119873





30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials







119867are






0119872119904119867ac sdot 119891 sdot 120588

minus1



12058813

sdot 119888minus13

119869dd119896119861 asymp 12058301205832

(⟨119903⟩3















119901shifts to



119873(119867ac)


[120591119873(119867ac)]

minus1


2

)

times (1 + ℎ) exp [(minus119870119881

119896119861119879

) (1 + ℎ)2

]

+ (1 minus ℎ) exp [(minus119870119881

119896119861119879

) (1 minus ℎ)2

]

(7)


119901at 119867ac = 20 kAm a



30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)


O4


119873


119901


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)


O4


119873


119901



119873





30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




30000

3000

300

30

3

f(k

Hz)

1 16 128 2048Hac (kAm)

(a)

1 16 128 2048Hac (kAm)

001

005

01

05

1PHPHmax

(b)


O4


119873


119901


30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

1 4 16 64Hac (kAm)

03

01

003

001

0003

PHPHmax

(b)


O4


119873


119901



119873





30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




30000

3000

300

30

31 4 16 64

f(k

Hz)

Hac (kAm)

(a)

001

005

01

05

1

PHPHmax

1 4 16 64Hac (kAm)

(b)


119873


119901




119873(119867ac)]

minus1



6120578119881119867sdot 120596 (119905) = 120583

0120583 (119905) timesH (119905) + 120582 (119905) (8)

⟨120582119894(119905)⟩ = 0 (9)

⟨120582119894(1199051) 120582119894(1199052)⟩ = 2119896

119861119879 sdot (6120578119881

119867) sdot 120575 (119905

1minus 1199052) (10)




119870 As a


119901= (2120587120591



2120587119891119901asymp 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

asymp

120591minus1

119861

at 1205830120583119867ac ≪ 119896

119861119879

05 (

1205830120583119867ac

6120578119881119867

) at 1205830120583119867ac ≫ 119896

119861119879

(11)















2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials













2120587119891119901asymp [120591119873(119867ac)]

minus1

+ 120591minus1

119861

[1 + 007(

1205830120583119867ac119896119861119879

)

2

]

05

(12)


119873

+ 120591minus1

119861






1205830120583119867ac ≫ 119896



119867)






119861)minus1




119867= 063 119870 = 200 kJm3 and 120578 =




119870









0120583119867 sin(120579minus


0120583119867 sin(120579+120587+120601) = 120583

0120583119867 sin(120579+120601) forces 120583



01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

Torque

Torque

Easy axis

120579

120579

120583

120583

120601

120601

H

H

120595

120595

01

02

03

04

minus1

minus1

minus05 0 05 1

1

00 0 0 0

1205914 1205912120579

MM

s

300340

640



120588(120579

)

01

02

03

04

minus1

minus1

minus05 0 05 1

1

0 1205914 1205912120579

MM

s

120588(120579

)

01

02

03

04


0 1205914 1205912120579

120588(120579

)

1

minus1

MM

s

Easyaxis


(e)

(f)

HHKHHK HHK HHK

Hac (kAm)

(a) Hac = 0


O4




0120583119867 sin(120579 minus 120601) This












0120583119867ac6120578119881119867) For







120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




120579 120595

120583

H

Longaxis

H (kAm)

0 1 2 3 4

075

05

025

0

MM

s

(a) 119872-119867 curves

Reversal

Rotation

Randomization

045

05

055

06


⟨cos120579⟩



O4



119867119875119867Max







0120583119867ac6120578119881119867)]



119861




119873

+ 120591minus1

119861










119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






119873(119867ac = 0) ≫

119891minus1


minus1











119867



0

1


0 20 4006

07

08

09

Rotatable


MM

s

⟨cos120579⟩

(a)

(b)








119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






119867required




2



2

1205832

0

1205901199032

1198912

1198672

ac is 5MWm3












119890








119867119875119890 As an example we












1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




1 5 10 50 1000

02

04

06

08501005001000 2

131415

182228

2000

Hac (kAm)

d (nm)

PH

(MW

kg)

f (kHz)

(a) Nonrotatable

1 5 10 50 1000

02

04

06

08501005001000 22000

Hac (kAm)

PH

(MW

kg)

f (kHz)

131415

182228

d (nm)

(b) Rotatable




2 4 8 16 32 64 2 4 8 16 32 64


3000

300

30

3

f(k

Hz)

Hac (kAm) Hac (kAm)

001

01

1

10

100cPHPe


119875119890


119867


119867

119875119890





001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials




001

01

1

10

100

2 4 8 16 32 64 2 4 8 16 32 64Hac (kAm) Hac (kAm)

30000

3000

300

30

3

f(k

Hz)

PH (MWkg)




119867


119867

119875119890



Acknowledgment


References













3

O4










2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials









2

O4




































2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials






2

O4

































CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials










CeramicsJournal of







Biomaterials




Journal of


Journal of





CoatingsJournal of

Advances in








MaterialsJournal of


Nano

materials



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