Table Of ContentDynamics of magnetic nano particles in a viscous fluid driven by rotating magnetic
fields
Klaus D. Usadel
Theoretische Physik, Universita¨t Duisburg-Essen, 47048 Duisburg, Germany
(Dated: February 16, 2017)
The rotational dynamics of magnetic nano particles in rotating magnetic fields in the presence
of thermal noise is studied both theoretically and by performing numerical calculations. Kinetic
equationsforthedynamicsofparticleswithuniaxialmagneticanisotropyarestudiedandthephase
lag between the rotating magnetic moment and the driving field is obtained. It is shown that for
large enough anisotropy energy the magnetic moment is locked to the anisotropy axis so that the
particle behaves like a rotating magnetic dipole. The corresponding rigid dipole model is analyzed
both numerically by solving the appropriate Fokker-Planck equation and analytically by applying
an effective field method. In the special case of a rotating magnetic field applied analytic results
are obtained in perfect agreement with numerical results based on the Fokker-Planck equation.
Theanalyticformulasderivedarenotrestrictedtosmallmagneticfieldsorlowfrequenciesandare
thereforeimportantforapplications. Theillustrativenumericalcalculationspresentedareperformed
7
for magnetic parameters typical for iron oxide.
1
0
PACSnumbers: 75.40.Gb,75.40.Mg,75.75.Jn,75.60.Jk
2
n
a I. INTRODUCTION originated from coupling of the spins to the surrounding
J
medium (the heat bath). Additionally, for mobile par-
6 ticles Brownian relaxation has to be taken into account
Recently, magnetic nanoparticles (MNPs) have been
1
[5, 9–16].
studied for many biomedical applications such as mag-
] netic particle imaging, separation of biological targets, Recently the dynamical properties of mobile particles
n immunoassays, drug delivery, and hyperthermia treat- have been studied in some detail in connection with bio-
y
ment [1–3]. In these applications the magnetic moment logical applications, i. e. the realization of homogeneous
d
- mayrotatewithintheparticlewithrespecttosomecrys- biosensors [17–20] based on the response of a suspension
u tal axis - i.e. Ne´el relaxation - and move along with the ofMNPstoanappliedac magneticfield. Inthesebiolog-
fl particle with respect to the liquid, i.e. Brownian relax- ical applications a dilute suspension of MNPs is present
s. ation. Hyperthermia, for instance, is based on the fact thephysicalpropertiesofwhicharedeterminedtoalarge
c that power is absorbed locally by the MNP when placed extend by the magnetic and mechanical dynamics of in-
si in an applied oscillating magnetic field. A high absorp- dependent MNPs.
y tionrateisachievedifbothNe´elrelaxationandBrownian Quite recently biosensors were proposed [21–23] which
h
relaxation contribute [4–6]. are based on the response of MNPs with Brownian dy-
p
[ On the other hand if the anisotropy energy is large namics to rotating magnetic fields, i.e. to differences in
comparedtothethermalenergyNe´elrelaxationbecomes thephaselagbetweentherotatingmagneticmomentand
1 unimportant. InthislimittheMNPcanbeconsideredas the driving field in order to determine the changes in
v a rigid body (RB) having a constant magnetic moment hydrodynamic volume caused by analytes bound to the
3
firmly attached to it. In this rigid dipole model (RDM) surface of the MNPs. For this method to be successful,
7
6 theexternaldrivingfieldwillcreateatorquetothemag- it is necessary to quantitatively clarify the dynamics of
4 netic moment which is transferred to the RB. In a rotat- MNPsinrotatingfields. Theoreticalstudiesonthisprob-
0 ing magnetic field, for instance, the MNP will rotate as lem reported in Ref. [24] are based on an effective field
2. well. TheoreticallythedynamicsofMNPshasbeenstud- method (EFM) [25, 26] for the dynamics of an ensem-
0 ied in recent years in very many papers for the case that ble of MNPs treated as rigid dipoles placed in a viscous
7 the particles are fixed in space. The dynamics is then medium. The EFM was derived from a Fokker-Planck
1 reduced to that of magnetic moments in external fields equation (FPE) describing the rotational dynamics at fi-
v: for which the stochastic Landau-Lifshitz-Gilbert (LLG) nite temperatures.
i equation as introduced by Brown [7] is often used. This Results obtained for the case that the applied field ro-
X
approach has gained increased interest recently because tates were compared with measurements of the phase
r ofitsapplicationtomagneticsingle-domainparticles. In lag [23, 27] revealing a reasonable agreement between
a
these MNPs the magnetic moments within the particles the measured and calculated values in the low frequency
are firmly tied together making it possible to describe regime and for small field amplitudes. However, signifi-
the magnetic moment of the particle as one macro spin cantdiscrepancieswereobservedoutsidethisregime. Nu-
ofconstantlength[8]. Itsdynamicsisexpectedtobewell merical calculations based directly on the FPE for the
describedbyaclassicalapproach. Thedynamicsofthese rigid dipole model [28] confirmed these findings so that
macrospinsisgreatlyinfluencedbythermalfluctuations a need for further theoretical work persists.
2
In the present paper we therefore study the dynamics can be written in compact form as
of mobile MNPs driven by a rotating magnetic field in
de γ (cid:16) 1
order to contribute to an understanding of the physics =ω×e− e×(B − ω)
dt 1+α2 e γ
underlying the functionality of the class of biosensors
mentioned [21–23]. The ensemble of MNPs is described 1 (cid:17)
+αe×(e×(B − ω)) (2)
by a set of kinetic equations proposed earlier [5] for the e γ
dynamicsofmobileparticlestreatingBrownianandNe´el
and
dynamics on equal footing.
In the first part of the paper we discuss briefly our Θdω = µsde +µ e×(B+ζ)−ξω+(cid:15). (3)
kinetic equation [5] and present numerical results for dt γ dt s
the dynamics of MNPs with finite uniaxial anisotropy
The effective field B entering Eq.(2) consists of the ex-
driven by rotating magnetic fields. Evidence is given to e
ternal driving field B which may be time dependent,
the expected blocking of the magnetic moment to the
a term proportional to the anisotropy energy D and a
anisotropyaxisforlargeenoughanisotropyenergies. The
stochastic field ζ relevant at elevated temperatures,
relation between our kinetic equations and the RDM is
discussed and it is shown how the RDM emerges from
2D
our kinetic equations. Be =B+ µ (e·n)n+ζ. (4)
s
Inthesecondpartofthepaperwediscussindetailthe
RDM and sketch the derivation of the basic equations The quantity α appearing in Eq.(2) is the dimen-
of the EFM [25, 26]. We show that for the special case sionless damping parameter usually used in the litera-
of an applied rotating magnetic field these equations can ture [29–31], ξ denotes the friction coefficient usually ex-
be solved without further assumptions. Analytic formu- pressed as ξ = 6ηV where η is the dynamical viscosity
d
las are obtained for the phase lag and for the magnetic while V with radius r denotes the hydrodynamic or to-
d d
moment and it is shown that these results are in nearly tal volume of the particle, i.e. it is assumed that the
perfect agreement with results which follow from numer- particle consists of a magnetic core region of volume V
m
ical solutions of the FPE. The analytic formulas derived withradiusr eventuallycoveredbyanonmagneticsur-
m
are not restricted to small magnetic fields and/or fre- factant layer.
quencies and are therefore important for applications. For the thermal fluctuations introduced above it is as-
sumed as usual that they are Gaussian distributed with
zero mean. Their correlators are chosen in such a way
that in equilibrium Boltzmann statistics is recovered.
II. DYNAMICS OF MAGNETIC NANO
This leads to
PARTICLES WITH FINITE ANISOTROPY
(cid:104)ζ (0)ζ (t)(cid:105) =δ δ(t)2αk T/(µ γ). (5)
l m ζ l,m B s
A. Kinetic equations
for the magnetic field fluctuations [7] and
Ananoparticleconsideredinthepresentpaperismod-
(cid:104)(cid:15) (0)(cid:15) (t)(cid:105) =δ δ(t)2ξk T. (6)
eledasauniformlymagnetizedsphericalrigidbody(RB) l m (cid:15) l,m B
with uniaxial magnetic anisotropy. The particle can ro-
for the Brownian rotation [11, 32]. The angular brackets
tate in a viscous medium. Its orientation in space is
denote averages over ζ and (cid:15), respectively, and l and m
described by a time dependent unit vector n(t) parallel
label cartesian components of the fluctuating fields.
totheanisotropyaxisoftheRB.Thisvectorisfirmlyat-
For given fluctuating fields (cid:15)(t) and ζ(t) the quanti-
tached to the RB so that its equation of motion is given
tiesn(t)ande(t)assolutionsofthestochasticequations
by
are trajectories on the unit sphere because the equations
of motion conserve the length of these vectors. Physical
dn
=ω×n (1) quantities of interest describing properties of an ensem-
dt ble of identical particles are obtained as averages over
these trajectories. The reduced magnetic moment at fi-
where ω denotes the angular velocity of the RB. The
nite temperatures, for instance, is given by
differentiation in Eq. (1) is performed in a coordinate
system fixed in space, the laboratory frame.
m(t)=(cid:104)e(t)(cid:105) . (7)
ζ,(cid:15)
ThemagneticmomentµoftheNPwithconstantmag-
nitude µ can rotate in space along with the RB and Eqs. (1-6)constituteaclosedsetofkineticequations
s
can also rotate relative to it depending primarily on the for the quantities n(t), e(t) and ω(t) specifying the dy-
strengthoftheanisotropyenergyD oftheNP.Weintro- namics of the MNP. In deriving these equations we were
duce e, an unit vector in the direction of the magnetic guided by the requirement that for the isolated particle
moment, e=µ/µ . For e and for theangular velocity ω the total angular momentum L+S has to be conserved
s
of the RB kinetic equations were proposed in [5] which irrespectively of internal interactions within the MNP.
3
Here, L = Θω denotes the angular momentum and S
the spin momentum, S = −γ−1µ, where γ denotes the
gyromagnetic ratio. For more details we refer to [5].
1.5
The kinetic equations proposed treat the dynamics of
the magnetic moment and the rotational motion of the
NP on the same footing. In general, both processes are
1
coupled leading to a rather complex behavior. In the
limitofalargeanisotropyenergy,however,itisexpected f
that the magnetic moment is locked into a position par-
0.5
allel to the anisotropy axis of the RB. This limiting case
is the essence of the rigid dipole model.
0
0 0.5 1 1.5 2 2.5
K / K
B. Numerical analysis: dependence on anisotropy 1 10
energy
1
Using our kinetic equations we investigate numeri-
cally the dependence of phase lag and induced mag-
netic moment on the anisotropy energy considering the 0.8
anisotropy constant K as being adjustable. Other pa-
1
rametersarechosenastypicalforparticlesofironoxides m
(magnetite). The magnetic moment µ is expressed as 0.6
s
µ =M V with M =4·105A/m and the reduced vis-
s s m s
cosity η˜ is defined as η˜ = η/η with η = 10−3
water water
kg/ms. ResultsareexpressedasfunctionofK /K with 0.4
1 10
K =104J/m3 whichorrespondstotheanisotropycon-
10
0 0.5 1 1.5 2 2.5
stant of magnetite. The rotating magnetic field is given
K / K
by 1 10
B=B (cos(ω˜t)xˆ+sin(ω˜t)yˆ) (8) FIG.1. (coloronline)Phaselagφ(upperpanel)andin-plane
0
magnetization m ( lower panel ) versus reduced anisotropy
where ω˜ = 2πF denotes the frequency of the driving parameterK /K fordifferentvaluesofthemagneticradius
1 10
field. A cartesian coordinate system fixed in space is rm: rm =19 nm (circles, black), 14 nm (squares, blue), and
used spanned by unit vectors xˆ, yˆ and zˆ. 10 nm (diamonds, blue). α = 0.01, η˜ = 1.0, rd/rm = 2.5,
F = 4 kHz, B = 5 mT. The horizontal lines indicate the
The initial conditions necessary for the numerical in-
values obtained within the RDM.
tegration of the kinetic equations are specified as e(t =
0) = n(t = 0) and ω(t = 0) = 0 with randomly dis-
tributed e(t = 0). For the integration of the stochastic
nearlyparallelto(cid:104)e(t)(cid:105) (notshown),i.e. magneticmo-
equationsEqs.(1-3)methodswellknownbynowfromlit- ζ,(cid:15)
ment and anisotropy axis rotate unisono.
eratureareused[29–31]. Inthestationarystatewhichis
TheRDMisstudiedindetailinthenextsections. Nu-
reached after a time interval δt the ensemble averaged
mericalresultswillbeobtainedsomeofwhichareshown
quantities (cid:104)e(t)(cid:105) and (cid:104)n(t)(cid:105) are monitored. Note
ζ,(cid:15) ζ,(cid:15)
as dashed horizontal lines in Fig.(1). They represent
that δt depends very much on parameters.
phaselagandmagneticmoment, respectively, calculated
In the stationary state (cid:104)e(t)(cid:105) rotates around zˆ with
ζ,(cid:15)
withintheRDM.ObviouslyforlargeK1/K10thevalues
thefrequencyofthedrivingfield. Aphaselagisobserved
obtained from the RDM are approached asymptotically.
betweenthedirectionoftherotatingfieldandthisrotat-
ingmagneticmoment. Thephaselagφisobtainedasthe A reduction of K1/K10, on the other hand, leads to a
angel between (cid:104)e(t)(cid:105) and the direction of the driving sharpdropinthephaselagaccompaniedbyasignificant
ζ,(cid:15)
field averaged over about 1000 particles in the ensemble increase of m. In this limit the moment adjusts nearly
followedbyanaverageoverabout100fieldcycles. Fig. 1 parallel to the driving field and rotates with the field
shows the averaged phase lag as function of K /K for while the RBs stop rotating in a coherent fashion. This
1 10
three different magnetic radii rm of the MNP, rm = 10 follows from the time variation of (cid:104)n(t)(cid:105)ζ,(cid:15) (not shown)
nm (circles, black),14 nm (diamonds, blue) and 19 nm consisting of only small random variations around zero.
(squares, red). In the course of this work we kept the temperature
For large anisotropy energies it is observed that the fixed at T = 300 K. A variation of T will effect in par-
phase lag becomes independent of the anisotropy energy ticulartheNe´elrelaxationtimewhichdependsonK1/T.
which indicates that the moment is locked to the RB, We therefore expect that the value of K1 at which the
i.e. it behaves like a rotating rigid dipole. This is di- locking of the moment sets in will depend on T.
rectlyconfirmedbytheobservationthat(cid:104)n(t)(cid:105) remains The increase of m when reducing the anisotropy en-
ζ,(cid:15)
4
ergy sets in for values of K /K for which the magnetic Eqs.(11-12) can be solved for de with algebraic manip-
1 10 dt
moment unlocks from the RB moving towards the di- ulations known from the LLG-equation. The resulting
rection of the field. In the extreme limit K = 0 the equation for de depends on the parameter κ,
1 dt
magnetic moment is only subjected to a tiny frictional
µ
torque resulting from the Gilbert - damping α. Only for κ= s (13)
ξγ
unrealistichighvaluesofαthistorqueislargeenoughto
overcome the viscous damping necessary for the RB to
which is extremely small. It is suffice therefore to keep
rotate.
only the leading terms with respect to κ resulting in the
well-known equation of motion for the dynamics of a
magnetic NP in the rigid dipole approximation,
C. Large anisotropy energy
de
=−ξ−1µ e×(e×B)−ξ−1e×(cid:15). (14)
The numerical results presented suggest that for large dt s
anisotropyenergiesourkineticequationspassovertothe
Note that the fluctuating field ζ entering Eq.(12) can be
RDM. This is supported by the following qualitative ar-
shown to be negligible in lowest order in κ.
guments. Consider a rotating frame firmly attached to
the RB in which the anisotropy axis is fixed. The time
derivative de can be split into two parts,
dt III. RIGID DIPOLE MODEL
de de
dt =ω×e+(dt)rf (9) The rigid dipole model became a matter of particular
interestquiterecentlybecauseofitspotentialforbiomed-
wherethesecondterminthisequationisthetimederiva- ical applications. The stochastic equation describing its
tiveintherotatingframe. ComparisonwithEq.(2)shows dynamics at finite temperature, Eq.(14), can be studied
that this time derivative is equal to the second term of numerically. An alternative is using the FPE which cor-
Eq.(2) so that responds to this stochastic process.
de γ (cid:16) 1
( ) =− e×(B − ω)
dt rf 1+α2 e γ A. Fokker-Planck equation for the RDM
1 (cid:17)
+αe×(e×(B − ω)) , (10)
e γ FortheRDMtheprobabilitydensityP(S,t)isdefined
as
describing the dynamics of a spin with a LLG-type
equation with effective field B − 1ω in the rotating
e γ P(S,t)=<δ(S−e(t))>(cid:15) (15)
frame. Inthisrotatingframetheanisotropyaxisisfixed.
Therefore e will settle nearly instantaneously parallel to where the angular brackets denote an average over all
the anisotropy axis and will stay there providing the trajectories e(t) of Eq. (14).
anisotropy energy is much larger than the contributions TheprobabilitydensityP(S,t)definedisasolutionof
from the driving field and from ω and that the temper- the FPE obtained from Eq.(14). It reads
ature is such that thermal switching of e does not take
place. Thismeansthataftera(short)initialtime(ddet)rf ∂P(S,t) =∇·(cid:104)µsS×(S×B)
will go to zero and we are left with ∂t ξ
1 (cid:105)
de − S×(S×∇) P(S,t). (16)
=ω×e. (11) 2τB
dt
For an elegant derivation of the FPE see for instance
NotethatthisargumentrequiresafiniteGilbertdamping Ref.[33]. The quantity τ introduced in Eq.(16) denotes
B
α. Ifthedampingparameterisstrictlyzerotherewillbe the Brownian relaxation time,
norelaxationtowardstheanisotropyaxissothatarather
complex precessional motion results. 3ηV
τ = d. (17)
Thus for large anisotropy energy we are lead to Eqs. B k T
B
(3)and(11)whichconstitutethebasisoftheRDM.Fur-
thersimplificationsarepossible. Firstwenotethatinthe Because the stochastic process conserves the length of e
equation of motion for ω inertia effects can be ignored the probability density is of general form
because of the small particle size combined with realistic
values of ξ so that one obtains P =δ(|S|−1)Q(S,t). (18)
1 µ de The probability density P(S,t) contains all informa-
ω = ( s +µ e×(B+ζ)+(cid:15)). (12)
ξ γ dt s tion about fluctuation averaged physical quantities. The
5
averagedreducedmagneticmoment,forinstance,isgiven The effective field A is determined by the requirement
by that this equation is fulfilled for the probability density
P(S,t) defined in Eq.(20) leading to
(cid:90)
m(t)= d3SP(S,t)S, (19)
dm =−µs(cid:2)(1− 3m)(B·mˆ)mˆ +(m −1)B(cid:3)− 1 m.
dt ξ A A τ
the first moment of the probability density, which is B
(25)
equivalent to the average over thermal fluctuations,
mˆ denotes a unit vector in the direction of m. Eqs.
Eq.(7).
(22,23,25) are the basic equations of the EFM [25, 26].
It is easy to see that the Boltzmann distribution P ∼
0 Note that for a time independent magnetic field B =
exp(µ S·B /k T) is an equilibrium solution of Eq.(16).
s 0 B B the equilibrium solution dm = 0 is parallel to B so
In general, however, exact solutions of the FPE are not 0 dt 0
known so that one has to rely on numerical solutions of thatweobtainfromEq. (25)A= µsB0. Forthereduced
kBT
the FPE or on approximations. magnetic moment we therefore obtain with Eq. (23)
For a numerical solution of the FPE the construction
µ B
of a fast and robust algorithm has been described previ- m=L( s 0), (26)
ously. The important point is to discretize the equation kBT
of motion for Q, Eq.(18), in such a way that the normal-
i.e. the equilibrium moment. This result is of course
ization of P is preserved independent of the mesh size.
expected because the equilibrium probability density is
For details the reader is referred to [34].
of the same form as that assumed in Eq.(20).
AnapproximatesolutionoftheFPE,theeffectivefield
In the general case, however, because of the nonlinear
method, has been described in [25, 26] the basic steps of
relationbetweenm(t)andA(t)inEq.(22),analyticsolu-
which are outlined in the next section.
tionsarenotobvious. Anexceptionisthespecialcaseof
a rotating magnetic field, Eq.(8), for which a stationary
solution of Eq.(25) can be found.
B. Effective field method
Thestartingpointisanansatzfortheprobabilityden-
C. Rotating magnetic fields
sity as in the works before, [25, 26], assuming for P(S,t)
an expression of the form of an equilibrium density with
For a rotating magnetic field a stationary solution ex-
adjustable parameters, (cid:112)
ists with time independent amplitude m = (m·m)
P =N−1δ(|S|−1)exp(A(t)·S) (20) implying a time independent A, Eq.(22).
To show this we note that for a time independent m
with normalization factor N given by we have
sinh(A(t)) dm
N(t)=4π (21) m· =0. (27)
A(t) dt
(cid:112)
and A(t)= (A(t)·A(t)). Multiplying Eq.(25) with m we obtain
The reduced magnetic moment resulting from this
Amk T
probability density, i.e. the first moment of the prob- B(t)·m(t)= B . (28)
ability density, is given by µs
Therefore a time independent m (and A) requires a ro-
A(t) tating magnetic moment m with constant phase lag, i.e.
m(t)=L(A(t)) (22)
A(t) m must be of the form
where
1 1 m=m(cos(ω˜t−φ)xˆ+sin(ω˜t−φ)yˆ). (29)
L(A)= − (23)
tanh(A) A
ThisformindeedsolvesEq.(25)andonefindsaftersome
denotes the Langevin function. For the first moment of algebra for the phase lag
theprobabilitydensityweobtainfromEq.(16)afterpar-
2ω˜τ
tial integration φ=arctan( B ) (30)
A/m−1
(cid:90) d3SS∂P(S,t) =(cid:90) d3S(cid:104)µsS×(S×B) and for the amplitude of the effective field
∂t ξ
− 1 S×(S×∇)(cid:105)P(S,t). (24) A= µsB0(cid:112) (A/m−1) (31)
2τB kBT (2ω˜τB)2+(A/m−1)2
6
with were obtained [24]:
m=L(A). (32) 2L(ξ)
φ=arctan(ω˜τ ) (35)
Bξ−L(ξ)
Eqs.(31-32)determineimplicitlythequantitiesAandm.
Wenoteinpassingthatthelagangelφdeterminesthe with
energyabsorptionoftheMNPsbytherotatingfield. The
µ B
absorbed power can be obtained from [4, 5, 12] ξ = s 0 (36)
k T
B
dB
W =−µ (cid:104)e (cid:105) (33)
s dt and
so that we obtain from Eqs.(8,29) m=L(ξ)cos(φ). (37)
W =µ ω˜B msin(φ). (34) The effective field A, Eq.(31), is frequency dependent
s 0
in general. For ω →0 it becomes identical to ξ, Eq.(36),
In the next section we will show that our analytic re- sothatinthislimittheresultsobtainedforthephaselag
sults, Eq.(30-32), are in nearly perfect agreement with coincide. Differences are obvious for finite frequencies.
those we obtained from numerical solutions of the FPE. Finally we would like to mention that for small fields
Differentresultsforthecasethatarotatingfieldisap- the analytic results presented agree with each other and
plied have been reported previously [24]. Those results they also agree with results from linear response theory.
are based on expansions of the EFM around the equilib- The reason is simply that for small fields the quantity
rium solution [25, 26]. The following expressions for the Q(S,t), Eq.(18), can be expanded as
phase lag and for the effective field - called ξ for clarity -
Q(S,t)=1+q(t)·S+... (38)
whereqisassumedtobelinearinthefield. Insertingthis
1.5 expression into Eq.(16) and keeping only terms linear in
B we arrive at a differential equation for q(t) which
0
can be solved easily. It is important to note that this
is an exact solution of the FPE (to linear order in B )
1 0
describing the stationary state.
m
, We do not present details of this calculation here be-
f causetheresultscanalsobeobtainedfromanexpansion
0.5
of Eqs.(30-32) in linear order in B . The reason for this
0
is that for small effective fields P(S,t), Eq.(20), agrees
with Eq.(38) in linear order in B so that the results for
0
0
0 10 20 30 40 φ and m obtained must coincide.
F [kHz]
D. Numerical analysis of the RDM
1.5
ResultsobtainedfromanumericalsolutionoftheFPE
fortheRDMwillbecomparedinthefollowingwiththose
1 obtained from our analytic results, Eqs.(30-32) and also
m with those obtained in Ref. [24] (Eqs. (35-37)).
, From the numerical solution of the FPE in a rotat-
f
0.5 ing field the magnetic moment m(t) is calculated using
Eq.(19). After an initial time interval with length de-
pending again strongly on the parameters of the system
0 astationarystatewithtimeindependentphaselagisob-
0 5 10
tained. Results for the phase lag and the reduced mag-
F [kHz]
neticmomentinthestationarystatearediscussedinthe
FIG. 2. (color online) Phase lag φ and magnetic moment m following.
versus frequency. Fig.(2)showsthephaselagφ(blackcirclesandascend-
Parameters: η˜=1.0,T =300,rm =15.0,B0 =5 mT. ing curves) and the magnitude of the induced magnetic
Upper panel: rd/rm =1.5, lower panel rd/rm =4.5. moment,m,(redsquaresanddescendingcurves)asfunc-
Dots black and squares red: numerical solution of the FPE. tion of frequency for r /r = 1.5 (upper panel) and for
d m
Solid, blue: present paper, dashed-dotted, green: Ref.[24].
r /r =4.5 (lower panel). An increase of the dynamical
d m
7
2
1.5
1.5
1
m m
1
, ,
f f
0.5
0.5
0 0
0 2 4 6 8 2 4 6
F [kHz] r /r
d m
2 FIG. 4. (color online) Phase lag φ and magnetic moment m
versus r /r .
d m
Parameters: η˜=1.0,T =300,r =19.0,B =5.0 mT, F =4
1.5 m
kHz.
Dots black and squares red: numerical solution of the FPE.
m
1 Solid blue: present paper, dashed-dotted green: Ref.[24].
f,
0.5
ment with our analytic results obtained from the EFM
(solid lines, blue) in the entire frequency region shown.
0 Results obtained from [24] (dashed-dotted, green) devi-
0 2 4 6 8
F [kHz] atefromthenumericallyexactresultespeciallyforlarger
frequencies.
The horizontal lines (solid, red) show the equilibrium
magnetic moment for a field B , the amplitude of the
0
1.5 rotating field. Obviously this result shows that for very
small frequencies the system is in a state of quasi equi-
libriuminwhichthemagnitudeofthemagneticmoment
m 1 isclosetoitsequilibriumvaluerotatingslowly. Withde-
, creasing frequency the equilibrium value is approached
f
asymptotically.
0.5
The results shown are representative in the sense that
we always found an extremely good agreement between
the results obtained from the FPE and those obtained
0
0 1 2 3 4 from our analytic results based on the EFM.
F [kHz]
Fig.(3) shows results for particles with an increased
magnetic radius, r = 19 nm. An increase in the field
m
FIG. 3. (color online) Phase lag φ and magnetic moment m
strength from B =2.5 mT (upper panel) to B =5 mT
versus frequency. 0 0
(middle panel) leads to a reduction in the lag angel and
Parameters: r /r =2.5, r =19 nm.
d m m to a significant increase in the magnetic moment in the
Upper panel: B =2.5 mT, η˜=1.0,
low frequency region. This is explained by the increased
middle panel: B =5 mT. η˜=1.0,
lower panel: B =5 mT. η˜=8.0. driving torque exerted by the larger magnetic field. An
Dots black and squares red: numerical solution of the FPE. increased frictional torque, on the other hand, leads to
Solid blue: present paper, dashed-dotted green: Ref.[24]. an increase of the lag angel and a decrease of m as can
be seen in the lower panel of Fig.(3) where the viscosity
isenlargedbyafactorof8(noteagainthedifferentscale
radiusr increasesthefrictionaltorquewhichiscompen- in the abscissa of the figure in the lower panel).
d
satedbytheenormousincreaseofthelagangelobserved Finally, in Fig.(4) we show results for the dependence
at small frequencies (note the different scales in the ab- of the lag angel and the magnetic moment on the ratio
scissa of these two graphs). This is consistent with the of the hydrodynamic to the magnetic radius, r /r , im-
d m
observation that an increase of the phase lag leads to an portantforbioassayapplications[22]. Remarkableisthe
increase of the absorbed power according to Eq. (34). sharpincreaseofφaroundr /r =2.5. Forlargervalues
d m
Phase lag and magnetic moment calculated from the of r /r the phase lag is nearly constant, i.e. insensitive
d m
numerical solution of the FPE are in very good agree- to variations of r .
d
8
IV. CONCLUSIONS temperature with magnetic parameters typical for iron
oxides (magnetite) the moment can be considered as be-
ing locked if the magnetic radius r is larger than about
m
12 nm, c.f. Fig.(1). Particles with a larger radius can be
Our numerical calculations based on the kinetic equa- described within the RDM.
tions for MNPs dissolved in a viscous liquid show that The EFM which is based on the FPE for the RDM
under the influence of a rotating magnetic field a tran- has been reconsidered and it has been applied to the dy-
sition takes place from a state with magnetic moment namicsofMNPinrotatingmagneticfields. Theanalytic
locked to the anisotropy axis of the MNP to a state with results we obtain are shown to be in perfect agreement
freerotationofthemomentdependingontheanisotropy with numerical solutions of the FPE in the range of pa-
energy. This scenario is expected physically and our rameters studied. Therefore these results can be used
results support this picture quantitatively. From these withsuccessinapplicationsavoidinglengthycalculations
investigations we can conclude that for MNPs at room on the basis of the FPE.
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