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Journal of Magnetism and Magnetic Materials 321 (2009) 2985–2991

Contents lists available at ScienceDirect

Journal of Magnetism and Magnetic Materials

0304-88

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/jmmm

Rotational Brownian dynamics simulations of non-interacting magnetizedellipsoidal particles in d.c. and a.c. magnetic fields

Jorge H. Sanchez a,b, Carlos Rinaldi a,�

a Department of Chemical Engineering, University of Puerto Rico, Mayaguez campus, P.O. Box 9046, Mayaguez, PR 00681, Puerto Ricob Facultad de Ingenierıa Quımica, Universidad Pontificia Bolivariana, Medellın, Colombia

a r t i c l e i n f o

Article history:

Received 28 March 2008

Received in revised form

23 December 2008Available online 6 May 2009

Keywords:

Rotational Brownian motion

Ellipsoidal particle

Magnetic susceptibility

Relaxation time

Strong probe magnetic field

53/$ - see front matter & 2009 Elsevier B.V. A

016/j.jmmm.2009.04.066

esponding author.

ail address: crinaldi@uprm.edu (C. Rinaldi).

a b s t r a c t

The rotational Brownian motion of magnetized tri-axial ellipsoidal particles (orthotropic particles)

suspended in a Newtonian fluid, in the dilute suspension limit, under applied d.c. and a.c. magnetic

fields was studied using rotational Brownian dynamics simulations. The algorithm describing the

change in the suspension magnetization was obtained from the stochastic angular momentum equation

using the fluctuation-dissipation theorem and a quaternion formulation of orientation space.

Simulation results are in agreement with the Langevin function for equilibrium magnetization and

with single-exponential relaxation from equilibrium at small fields using Perrin’s effective relaxation

time. Dynamic susceptibilities for ellipsoidal particles of different aspect ratios were obtained from the

response to oscillating magnetic fields of different frequencies and described by Debye’s model for

the complex susceptibility using Perrin’s effective relaxation time. Simulations at high equilibrium and

probe fields indicate that Perrin’s effective relaxation time continues to describe relaxation from

equilibrium and response to oscillating fields even beyond the small field limit.

& 2009 Elsevier B.V. All rights reserved.

1. Introduction

Colloidal suspensions of spherical magnetic nanoparticlesare commonly referred to as magnetic fluids. Typically, the fluidwhere the particles are dispersed is Newtonian, such as water oran organic solvent. The characteristic diameter of each particleis of the order of 5–15 nm and each particle typically consistsof a single magnetic domain [1–4]. Under certain conditionsthe nanoparticles are considered to be permanently magnetizedand the direction of the magnetic dipole moment is parallel to themagnetization easy axis. In a fluid medium, there exist twomechanisms by which the particle’s magnetic dipole may relaxin order to align with an external applied magnetic field [1]:Neel relaxation and Brownian relaxation. In Neel relaxationthe magnetic moment rotates without physical rotation of theparticle. Brownian relaxation implies mechanical rotation of theparticle. Experimental techniques for measuring the frequency-dependent complex susceptibility, wðOÞ ¼ w0ðOÞ � jw00ðOÞ, havebeen used to determine the Brownian relaxation time tB andhence the particle size of magnetic fluids composed of sphericalparticles [5–7]. Due to the capability to modify the physicaland chemical properties of the suspending fluid in responseto impressed magnetic fields, there exists a great interest of

ll rights reserved.

magnetic fluids in novel applications such as bio-sensors [8–10].Sensing of the target molecule is based on the change ofhydrodynamic radius due to its binding to the magneticnanoparticles. This increase in the hydrodynamic radius leads toan increase in the Brownian relaxation time and therefore to achange in the magnetic response to an alternating magnetic field.Thus, an accurate knowledge of the rotational dynamics of theparticles is necessary to develop such sensing schemes.

Commonly, magnetic fluids are composed of spherical mag-netic nanoparticles and analytical models have been developedin order to describe the magnetization of dilute suspensionsof this type of particles. In the case of concentrated systems,interparticle interactions become relevant and a numericalapproach is typically required. Indeed, numerical solutions ofthe Langevin equations have been obtained and several methodsfor simulating the Brownian dynamics of spherical particles havebeen developed, such as the work by Ermak and McCammon [11],who considered hydrodynamic interactions between sphericalparticles, and Dickinson [12], who incorporated the effects ofrotation–translation coupling. Their results show values of thetranslational diffusion coefficient and the rotational decay ratethat are in good agreement with the theoretical values from Oseenand Rotne–Prager diffusion tensors. More recently, Meriguet et al.[13] simulated the influence of an effective pair-interactionpotential between spherical particles on the magnetizationrelaxation of ferrofluids through integration of the stochastictranslational and rotational Langevin equations. They found that

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J.H. Sanchez, C. Rinaldi / Journal of Magnetism and Magnetic Materials 321 (2009) 2985–29912986

the relaxation time increases as the volume fraction of particlesincreases.

Other types of particles, such as fibers, rods, spheroids, plates,and their aggregation clusters, have been studied due to theirapplicability in many different fields of modern technology[14]. The paint, ceramic, and pharmaceutical industries usesuspensions of anisotropic particles, the dynamics of which differsignificantly from systems composed of spherical particles [15].The macroscopic properties of such suspensions—thermal con-ductivity, viscosity, magnetization, etc.—strongly depend onthe orientation of the particles in the medium. Thus, for example,spheroidal and rod-like particles are used to develop high-qualitymagnetic recording materials where the orientational distributionof the particles is carefully controlled by applied magnetic andflow fields [16].

The Brownian motion of ellipsoids in suspension withoutexternal forces has been recently studied experimentally by Hanet al. [17]. The experiments were restricted to two dimensions(2D) and the suspensions were made very dilute in order to avoidinteractions between particles. In their work it was possible toverify that a particle with a given initial angle will diffuse morerapidly along its long axis than along its short axis if the diffusiontime is lower than the rotational relaxation time, ty (anisotropicdiffusion). As time progresses, memory of the initial orientationis lost, and diffusion becomes isotropic. More recently, directvisualization of dynamics of colloidal rods was carried out byMukhija and Solomon [15]. They reported a particle trackingconfocal microscopy method to characterize the translational andorientational dynamics of anisotropic particles undergoing Brow-nian motion in three dimensions. Their results show thattranslational and rotational diffusion coefficients agree well withtheory for prolate shaped particles in a dilute suspension.

Nonspherical particles have been commonly modeled asaxisymmetric particles with two characteristic relaxation times,corresponding to rotation about an axis parallel and perpendicularto the symmetry axis, making feasible analytical solutionsor numerical approximations, even in the case of interactingparticles. In the case of axisymmetric particles, some theoreticalmodels have been obtained by solving the Fokker-Planck equationfor dilute suspensions [18–20]. For semi-concentrated suspen-sions interactions between particles make necessary numericalsolution for the probability distribution function, which describesthe average orientations of particles in the bulk of the suspension,or for the stochastic Langevin equation [11–13,21]. Simulationshave also been used to study the effects of an external forcefield on suspensions of nonspherical particles. An approach formodeling the rotational dynamics of axially symmetric magneticparticles in fluid suspensions, in the dilute limit, was developed byScherer and Matuttis [22]. Based on a generalized Lagrangianformulation for the equation of motion, they showed the effect ofparticle inertia on the susceptibility, suggesting that the peak ofthe imaginary part of the complex susceptibility shifts to higherfrequencies as the moment of inertia increases.

All of the work discussed above has considered suspensionsof solids of revolution, such as spheres, spheroids and rods. Theseparticles are characterized by a symmetry axis and therefore twointrinsic relaxation times. On the other hand, the nanoparticlesobtained in magnetic fluid synthesis are not perfectly sphericalor axisymmetric. Moreover, as mentioned above, in the use offerrofluids as bio-sensors once the particles attach to the analytethe particle loses its symmetry and the models developed are notable to describe the behavior of these suspensions and may notbe suitable to interpret experimental observations. Finally, litho-graphic methods may provide an alternative as a method toproduce magnetic particles suitable in sensors, in which case theparticles may be orthotropic by design.

In the case of asymmetric particles, the rotational diffusionequation has 23 terms when expressed in terms of Euler angles[23], making its analytical solution difficult. Nevertheless, ananalytical model was obtained by Perrin [24] and Coffey et al. [25]for the dynamics of a collection of non-interacting asymmetric-top particles in the cases of relaxation from an equilibrium fieldand response to an oscillating field, both in the limit of smallapplied fields. Perrin’s solution was obtained by solving for theorientational distribution function. More recently, Kalmykov andTitov [26,27] have obtained analogous results by averaging thenon-inertial Langevin equation for rotational Brownian motion,considering the nonlinear transient response of asymmetric topsto arbitrary fields and susceptibility to small probe fields in thepresence of large bias fields.

Here we present a method to simulate the rotational Browniandynamics of non-interacting tri-axial ellipsoidal particles withan embedded magnetic dipole. The algorithm numerically solvesthe non-inertial Langevin equation for rotational Brownianmotion directly, obtaining the orientation evolution of a collectionof non-interacting particles. The method is not limited to smallequilibrium or probe fields, as are previous analytical solutions[24–27]. However, we assume for the present that the magneticdipole is rigidly locked to one of the ellipsoid axes, neglecting theeffect of Neel relaxation. Results from simulations with variousparticle geometries indicate that relaxation from an equilibriumfield is well described by a single exponential decay and Perrin’seffective relaxation time, even for large initial equilibrium fields.Simulations for oscillating fields show that for small probe fieldsthe response is well described by Debye’s model for the complexsusceptibility using Perrin’s effective relaxation time. At highprobe fields it is shown that Perrin’s effective relaxation time stilladequately collapses the response of particles of all the simulatedaspect ratios. In Section 2, we describe the particle model and thedifferent particle geometries investigated, we present the equa-tion of motion, introduce our numerical model for the orientationof the particle, and the dimensionless form of both magnetizationand complex susceptibility. In Section 3, the simulation para-meters are summarized. In Section 4, we discuss the resultsobtained from equilibrium and dynamic simulations. In Section 5,we include a concluding discussion.

2. Rotational Brownian dynamics of an ellipsoid

For a tri-axial ellipsoidal particle (orthotropic particle),physical and magnetic properties are most naturally writtenin a Cartesian coordinate system with axes aligned with theprincipal axes of the ellipsoid (primed axes) rather than relative tothe coordinate axes of the laboratory space (unprimed axes), bothillustrated in Fig. 1. An ellipsoid has three principal axes a1, a2, anda3, directed along the x0, y0, and z0-axis, respectively. In the presentwork we consider the following cases: spherical particlesa1 ¼ a2 ¼ a3; prolate ellipsoids ða1 ¼ a2Þoa3; oblate ellipsoidsa1oða2 ¼ a3Þ; and scalene ellipsoids a1aa2aa3.

We further consider that the magnetic dipole moment of theparticle, l0, is directed along one of the principal axes of theellipsoid, and the magnetic field, H, is assumed to be applied alongthe z-axis direction.

2.1. Rotational Brownian motion

Our Brownian dynamics simulations are based on the integra-tion of the stochastic angular momentum equation. There arethree kinds of torque acting on the particle: hydrodynamictorque Th, magnetic torque Tm, and Brownian torque TB. In theinertialess limit the angular momentum balance, relative to body

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z’

µµ ’

y’

z’

µ ’

x’

y

x

z

�

µ ’

Fig. 1. Particle model and system of coordinates.

J.H. Sanchez, C. Rinaldi / Journal of Magnetism and Magnetic Materials 321 (2009) 2985–2991 2987

fixed axes, is

T0h þ T0m þ T0B ¼ 0. (1)

The hydrodynamic torque is expressed in terms of the rotationalhydrodynamic resistance dyadic f0r and the angular velocitypseudovector x0 of the particle as

T0h ¼ �f0r �x0. (2)

The resistance dyadic is dependent only on the particlegeometry and for an ellipsoidal particle is expressed as [28]

f0r ¼ ZK0r , (3)

where Z is the dynamic viscosity of the carrier fluid and the dyadicK0r is given in terms of the principal axes of the ellipsoid by

K0r ¼16p

3

a22 þ a2

3

ða22�2 þ a2

3�3Þix0 ix0 þ

a23 þ a2

1

ða23�3 þ a2

1�1Þiy0 iy0

"

þa2

1 þ a22

ða21�1 þ a2

2�2Þiz0 iz0

#, (4)

with

�j ¼

Z 10

dlðaj þ lÞ½ða2

1 þ lÞða22 þ lÞða2

3 þ lÞ�1=2; j ¼ 1;2;3. (5)

The uniform applied magnetic field H exerts a torque on theparticle tending to align its magnetic dipole moment with thefield

T0m ¼ m0ðl0 �H0Þ, (6)

where m0 is the permeability of free space and H0 ¼ A �H isthe magnetic field transformed to the body-fixed axes through thetransformation matrix A. In order to free the algorithm fromsingularities the transformation matrix is expressed in terms ofthe quaternion parameters e0, e1, e2, and e3 as [29]

A ¼

�e22 þ e2

1 � e23 þ e2

0 2ðe3e0 � e2e1Þ 2ðe1e3 þ e2e0Þ

�2ðe2e1 þ e3e0Þ e22 � e2

1 � e23 þ e2

0 2ðe1e0 � e2e3Þ

2ðe1e3 � e2e0Þ �2ðe2e3 þ e1e0Þ �e22 � e2

1 þ e23 þ e2

0

264

375.

(7)

The quaternion parameters satisfy the relation e20 þ e2

1 þ e22þ

e23 ¼ 1.

Substituting Eqs. (2) and (6) into Eq. (1) we obtain thefollowing expression:

�f0r �x0 þ m0ðl

0 �H0Þ þ T0B ¼ 0. (8)

Solving (8) for x0 ¼ dU0=dt, where dU0 is the infinitesimalrotation vector, and integrating from time t to time t þDt using afirst-order forward Euler method, we obtain

U0tþDt �U0t ¼ ðf0

rÞ�1� ½m0ðl

0 �H0Þ�Dt þ ½ðf0rÞ�1� T0B�Dt. (9)

The last term on the right-hand side of (9) is a stochastic termarising from the random collisions between solvent molecules andthe particle. Applying the fluctuation-dissipation theorem [12],Eq. (9) becomes

U0tþDt �U0t ¼ ðf0

rÞ�1� ½m0ðl

0 �H0Þ�Dt þ ðB0 �w0Þ, (10)

where B0 � B0T ¼ kBTðf0rÞ�1, kB is Boltzmann’s constant, T the

absolute temperature, and the superscript T means transposition.Vector w is a random vector characterized by a Gaussiandistribution with mean hwi ¼ 0 and variance hw �wi ¼ 2Dt.

In order to reduce the number of variables and parameters,Eq. (10) is expressed in dimensionless form using the followingscaled variables:

~f0

r ¼f0r

ZKr;min; ~l0 ¼

l0

m; ~H ¼

H

H0,

~t ¼ tDr;max; ~B0¼

B0ffiffiffiffiffiffiffiffiffiffiffiffiffiDr;max

p ; ~w ¼ wffiffiffiffiffiffiffiffiffiffiffiffiffiDr;max

p. (11)

In (11) m and H0 are the magnitudes of l0 and H respectively,and Dr;max is the larger of the diagonal terms in the rotationaldiffusion tensor D0r written relative to particle-fixed axes. Therotational diffusion tensor is given by the generalized Stokes-Einstein equation D0r ¼ kBTðf0rÞ

�1. Thus (10) is written in dimen-sionless form as

DU0 ¼ U0~tþD~t �U0~t ¼ að~f0

rÞ�1� ð ~l0 � ~H

0ÞD~t þ ð ~B

0� ~w0Þ, (12)

where a ¼ ðm0mH0Þ=ðkBTÞ is the Langevin parameter.

2.2. Equilibrium and dynamic magnetization

The magnetization m of a collection of n magnetic particleswith magnetic dipole moment l dispersed in a volume V iscalculated from

m ¼1

V

Xn

i¼1

li. (13)

The dimensionless form of the magnetization vector, i.e., themagnetization divided by its saturation value, ms ¼ nm=V , is thengiven by

~m ¼m

ms¼

1

n

Xn

i¼1

~li. (14)

In this study we are concerned with the magnetization in thedirection of the applied field, ~mz ¼ ~m � iz.

When the suspension is subjected to an external magnetic fieldthe magnetic dipole moments of the particles tend to align withthe field. The magnetic field exerts a torque turning the dipolestowards the applied field direction. Several fundamental modelshave been developed in order to describe the magnetizationof suspensions of magnetic particles; thus, if the suspension issubjected to a steady d.c. magnetic field it eventually achieves anequilibrium state and the magnetization is described by the

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Table 1Aspect ratios and rotational hydrodynamic resistance coefficients of the different

types of particles used in the simulations.

Type of particle a1 a2 a3 Kr;x0x0 Kr;y0y0 Kr;z0z0

Sphere 1.0 1.0 1.0 25.13 25.13 25.13

Prolate 0.1 0.1 1.0 3.33 3.33 0.17

Scalene-1 0.1 0.25 1.0 4.06 4.69 0.64

Scalene-2 0.1 0.5 1.0 5.76 6.77 2.44

Scalene-3 1.0 0.5 0.1 2.44 6.77 5.76

Oblate 0.1 1.0 1.0 12.04 10.80 10.80

J.H. Sanchez, C. Rinaldi / Journal of Magnetism and Magnetic Materials 321 (2009) 2985–29912988

Langevin function:

~mz ¼ cotha� 1

a . (15)

This result should be applicable regardless of particle shape as itstems from the equilibrium Boltzmann distribution.

In the particular case of isotropic particles, the dynamicresponse of the magnetization can be described using Debye’smodel, which can be obtained by solving the Smoluchowskiequation for the orientation distribution function f of themagnetic dipole moment [23]

zr

kBT

@f

@t¼

1

sin y@

@ysin y

@f

@y

� �þmðm0H0Þ

kBT

1

sin y@

@yðsin2 yf Þ, (16)

where zr is the rotational hydrodynamic resistance coefficient ofthe isotropic particle. Solving (16) for small values of the Langevinparameter a51, and using the expression for the mean dipoleof the system, hm cos yi ¼

R p0 m cos yf sinydy, the magnetization

of the suspension mzðtÞ ¼ hm cos yi=V is obtained. Two specialsituations are readily solved: (i) relaxation upon switching off thefield and (ii) response to an oscillating field.

For isotropic particles in equilibrium with an external d.c.magnetic field H ¼ H0iz, if the field is suppressed, the magnetiza-tion relaxes to a new equilibrium state following an exponentialdecay

~mz ¼13a expð�t=tÞ, (17)

where t ¼ 1=ð2DrÞ is the Debye relaxation time.In the case of an external oscillating magnetic field,

H ¼ H0 cosðOtÞiz, the dipole moment of the particles follows theoscillations of the magnetic field with a phase-lag between thefield and the particle. The z-component of the magnetization isthen

~mz ¼13a½ ~w

0 cosðOtÞ þ ~w00 sinðOtÞ�. (18)

The nondimensional in-phase and out-of-phase susceptibilities, ~w0

and ~w00, respectively, are frequency dependent, where for isotropicparticles are given as

~w0 ¼ 1

1þ ðOtÞ2; ~w00 ¼ ðOtÞ

1þ ðOtÞ2(19)

and

t ¼ 1=Omax. (20)

Omax is the frequency at which ~w00 is a maximum.According to Perrin’s analysis [24,25] for the case of an

ellipsoid with its magnetic dipole along one of its principal axes,say the z0-axis, Eqs. (17) and (19) continue to apply insofar as therelaxation time t is replaced by an effective relaxation time, thatin dimensionless form is given by

~teff ¼2Dr;max

Dr;x0x0 þ Dr;y0y0¼

2

Kr;min

Kr;x0x0Kr;y0y0

Kr;x0x0 þ Kr;y0y0

� �. (21)

As has been mentioned before, Perrin’s analysis is limited torelaxation from a small applied field or response to a smalloscillating probe field.

In the simulations that follow, we consider particles whosedipole moment is aligned with one of the ellipsoid’s principalaxes. For simulations with small equilibrium and oscillating fieldswe show that (17) and (19) are correct. We further show thatrelaxation from a large field also follows exponential decay andthat Perrin’s relaxation time still applies for large equilibriumfields, as shown analytically by Kalmykov and Titov [26], and forlarge oscillating probe fields.

3. Simulation parameters

We carried out rotational Brownian dynamics simulationsimplementing an algorithm which allows us to compute themagnetization of the suspension under specified conditions ofexternal magnetic field. The particles are considered permanentlymagnetized along the z0-axis, and different particle aspect ratioswere used in the simulations, as summarized in Table 1. In ourcase, to calculate the evolution with time of the magnetization bynumerical simulation it is necessary to transform the magneticdipole moment of each particle to the laboratory coordinates andfind the z-component. Using the transformation matrix A themagnetization of the suspension is written as

~mz;ð~tþD~tÞ ¼1

n

Xn

i¼1

ð�e22;ð~tþD~tÞ � e2

1;ð~tþD~tÞ þ e23;ð~tþD~tÞ þ e2

0;ð~tþD~tÞÞi, (22)

where the quaternion parameters are related to the infinitesimalrotation vector, Eq. (12), by [29]

DFx0

DFy0

DFz0

0

26664

37775 ¼ 2

�e3 e0 e2 �e1

�e0 �e3 e1 e2

e1 �e2 e0 �e3

e2 e1 e3 e0

266664

377775

De2

De1

De3

De0

266664

377775. (23)

Three types of simulation runs were performed: (i) equilibriumin a d.c. magnetic field, (ii) relaxation after switching off anequilibrium field, and (iii) response to an oscillating magneticfield. All runs were performed using 105 noninteracting particles,a time step D~t ¼ 0:01, and different values of the Langevinparameter ð0:1 � a � 30Þ. For the relaxation runs the system wasallowed to achieve equilibrium and then the magnetic field wassuppressed. For the oscillating magnetic field runs the suspensionwas subjected to an a.c. magnetic field H ¼ H0 cosðOtÞiz and runswere conducted at dimensionless frequencies in the range of10�2–102. Values for hydrodynamic resistance coefficients werecalculated from Eqs. (4) and (5) and summarized in Table 1.

4. Results and discussion

Equilibrium and dynamic simulations were performed startingfrom a random configuration. Fig. 2 shows the dimensionlessequilibrium magnetization as a function of the Langevinparameter for the different types of particles. Clearly, theequilibrium magnetization is well described by the Langevinfunction independent of the aspect ratio of the particles. This is tobe expected because the energy necessary to rotate the particleinto the field direction, which enters into the equilibriumBoltzmann distribution, is a function only of the zenithal angle y(see Fig. 1).

Fig. 3 shows relaxation of the magnetization after the externalfield is suppressed. The system is allowed to achieve equilibriumin 104 time steps, then the external field is set to zero. The

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Fig. 2. Dimensionless equilibrium magnetization as a function of Langevin

parameter.

Fig. 3. Relaxation of dimensionless normalized magnetization for a ¼ 0:1

(markers) and a ¼ 10 (dashed line).

Table 2Dimensionless effective relaxation time from simulations and theory for each type

of particle.

Type of particle ~teff ;a¼0:1 ~teff ;a¼10 ~teff ;Eq. (21)

Sphere 1:1� 0:03 1:0� 0:01 1:0

Prolate 20:0� 2:0 20:8� 0:2 20:2

Scalene-1 6:4� 0:6 6:9� 0:2 6:89

Scalene-2 2:9� 0:6 2:6� 0:04 2:57

Scalene-3 1:4� 0:02 1:5� 0:005 1:47

Oblate 1:0� 0:06 1:1� 0:006 1:06

Fig. 4. Real (a) and imaginary (b) components of the dimensionless complex

susceptibility as a function of dimensionless frequency. Symbols from Fig. 2.

J.H. Sanchez, C. Rinaldi / Journal of Magnetism and Magnetic Materials 321 (2009) 2985–2991 2989

relaxation curves are shifted to higher values of dimensionlesstime when the aspect ratio of the particle is lower. Hence, prolateshaped particles take longer times relative to the minimumcharacteristic relaxation time to relax to a new equilibrium state,compared to spherical and oblate particles. For a ¼ 0:1 themagnetization curves show good agreement with Debye’stheory, Eq. (17), with the particles having an effective relaxationtime given by teff ; thus, the relaxation of the magnetization indimensionless form can be written as

~mzðtÞ ¼ ~mzð0Þ expð�2~t=~teff Þ. (24)

Eq. (24) can be used to obtain ~teff from the slopes of the curves inFig. 3. Values for ~teff , shown in Table 2, agree with Perrin’seffective relaxation time. For relaxation from high fields ða ¼ 10Þ,Fig. 3 shows the particles still display single exponential decayand the relaxation time is given by teff .

In the case of an external oscillating magnetic field, thedimensionless magnetization for a suspension composed of tri-axial particles has periodic behavior and therefore it can berepresented by a Fourier series as

~mz ¼1

3aX1n¼1

½ ~w0n cosðn ~O~tÞ þ ~w00n sinðn ~O~tÞ�, (25)

here n ¼ 1 denotes the fundamental susceptibility, while n ¼

2;3;4; . . . are the higher order harmonics in ~w. For a ¼ 0:1 it is

possible to obtain the fundamental in-phase and out-of-phasedimensionless dynamic susceptibilities, ~w0 and ~w00, respectively,by simple integration. If we first multiply each side in Eq. (25) bycosð ~O~tÞ and then integrate termwise on the interval ½0;2p�, theresult is

~w0 ¼ 3

pa

Z 2p

0

~mz0 cosð ~O~tÞdð ~O~tÞ. (26)

If we multiply each side in Eq. (25) by sinð ~O~tÞ and then integratetermwise, we find in a similar way that

~w00 ¼ 3

pa

Z 2p

0

~mz0 sinð ~O~tÞ dð ~O~tÞ. (27)

Eqs. (26) and (27) can be evaluated numerically using simulationresults.

Fig. 4 shows dimensionless susceptibilities as a function ofdimensionless frequency. All curves have the same shape asDebye’s model for particles with only one relaxation time, butparticles with aspect ratios less than one shift to the left

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Fig. 6. A.C. magnetization of suspensions of ellipsoidal particles for a ¼ 10 and~O ¼ 1:0. (–) spheres, (– –) prolates, (– � –) scalene-1, (� � �) scalene-2, (–) oblates

(indistinguishable from spheres).

J.H. Sanchez, C. Rinaldi / Journal of Magnetism and Magnetic Materials 321 (2009) 2985–29912990

indicating a slower relaxation process characterized by a highereffective relaxation time, ~teff . Note that when an oscillatingmagnetic field with dimensionless frequency ~O ¼ O=Dr;max isapplied, the magnetization of the suspension changes with timefollowing the field oscillation. There exists a phase-lag betweenthe magnetization and the magnetic field, which is dependent onthe aspect ratio of the particle and the frequency of the magneticfield. At low frequencies the magnetic field has a characteristictime that is longer than the ~teff of the particles, then themagnetization fit to magnetic field and is independent of theaspect ratio as is observed from ~w0 and ~w00 values close to one andzero, respectively. As the frequency increases, the characteristictime of the magnetic field is lower than ~teff and thus thoseparticles with higher ~teff show out-of-phase and lower valuesof the dimensionless magnetization. As expected for particlessuspended in a Newtonian fluid, in-phase and out-of-phasesusceptibilities vanish at high frequencies.

Now, according to these observations and Perrin’s analysis thein-phase and out-of-phase susceptibilities may be expressed as

~w0 ¼ 1

1þ ð ~O~teff =2Þ2; ~w00 ¼

ð ~O~teff =2Þ

1þ ð ~O~teff =2Þ2. (28)

This implies that the dynamic susceptibilities, ~w0 and ~w00, can berepresented by a unique curve when plotted against ~O~teff =2.Indeed, as shown in Fig. 5, the complex susceptibility fits thismodel independent of the aspect ratio of the particles.

For a ¼ 10, the dimensionless magnetization for a suspensioncomposed of tri-axial particles also has periodic behavior, asshown in Fig. 6, exhibiting a plateau at the maximum value of themagnetization, which is due to the fact that a saturation state isachieved at low frequencies. There exists a phase-lag between the

Fig. 5. Real (a) and imaginary (b) components of the dimensionless complex

susceptibility as a function of ~Oð~teff =2Þ. Symbols from Fig. 2.

Fig. 7. A.C. magnetization of suspensions of ellipsoidal particles as a function of~t=~teff for a ¼ 10 and ~O~teff ¼ 0:1, 1:0, and 10.

magnetization and the magnetic field, which is dependent on theaspect ratio of the particle and the frequency of the magnetic field.As the frequency increases those particles with higher ~teff show

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J.H. Sanchez, C. Rinaldi / Journal of Magnetism and Magnetic Materials 321 (2009) 2985–2991 2991

out-of-phase and lower values of the dimensionlessmagnetization. Fig. 7 shows that using the effective relaxationtime to define a new dimensionless frequency, ~O~teff , thenormalized magnetization for an oscillating magnetic field athigh values of the Langevin parameter fits the curve for asuspension of isotropic particles, independent of the aspect ratioof the particles. Thus it is seen that Perrin’s effective relaxationtime applies even for large probe fields.

5. Conclusions

The magnetization of suspensions of noninteracting tri-axialpermanently magnetized particles under d.c. and a.c. externalmagnetic fields was studied. The inertialess approximation of theLangevin equation was solved numerically and simulated in orderto obtain the dynamic magnetic behavior of the suspension whenit is subjected to an external magnetic field and the carrier fluid isa Newtonian liquid.

The results show that the equilibrium magnetization of thesuspension is independent of aspect ratio of the particle and isdescribed by the Langevin function. Furthermore, relaxation fromsmall and high fields shows the magnetization being character-ized by Perrin’s effective relaxation time, as shown by Coffey et al.[25] and Kalmykov and Titov [26] for small and large fields,respectively. Also, simulations demonstrate that in-phase and out-of-phase susceptibilities can be represented by a unique functionusing the effective relaxation time, ~teff , independent of the aspectratio of the particle, and that the normalized magnetization for astrong a.c. magnetic field collapses to a single curve, independentof the aspect ratio of the particles, with ~O~teff as the dimensionlessfrequency.

Acknowledgments

The authors thank Juan J. de Pablo and Juan Pablo Hernandezfrom the Department of Chemical and Biological Engineering,University of Wisconsin. The work was supported by the National

Science Foundation (NSF) CAREER Program under Grant no. CBET-0547150.

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