Application of Fractional Calculus to the Modeling of the

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Application of Fractional Calculus to the Modeling of theComplex Magnetic Susceptibility for Polymeric-MagneticNanocomposites Dispersed Into a Liquid Media

M. E. Reyes-Melo, 1,2 M. A. Garza-Navarro, 1 V. A. Gonza lez-Gonza lez, 1,2C. A. Guerrero-Salazar, 1,2 J. Mart nez-Vega, 3 U. Ortiz-Me ndez 1,2

1 Doctorado en Ingenier a de Materiales, Facultad de Ingenier a Meca nica y Ele ctrica, Universidad Auto noma de NuevoLeo n, Ave. Universidad S/N, Cd. Universitaria, San Nicola s de los Garza, NL 66450, Me xico2 Centro de Innovacio n, Investigacio n y Desarrollo en Ingenier a y Tecnolog a, Km. 10 de la Nueva Carretera al Aeropuerto Internacional de Monterrey, PIIT Monterrey, Apodaca, NL 66600, Me xico3 Laboratoire Plasma et Conversion dEnergie-UMR CNRS 5213, Universite de Toulouse, 118 route de Narbonne,31062 Toulouse, France

Received 21 July 2008; accepted 24 September 2008DOI 10.1002/app.29308Published online 12 February 2009 in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: The objective of the present work is to de-velop a fractional model which describe in a preciselymanner the real and imaginary parts of the complex mag-netic susceptibility for polymer-magnetic nanocompositesdispersed into a liquid media, using differential and/or in-tegral operators of fractional order (fractional calculus).The theoretical results show that the shape of the curvesof both real and imaginary parts is depending of the frac-tional order of this new model named Fractional Magnetic

Model (FMM). Moreover, the comparison between theoret-ical and experimental results show that the FMM describequite well the complex susceptibility response of the poly-meric systems containing magnetic nanoparticles at lowfrequencies (from 0.1 to 10 5 Hz). VVC 2009 Wiley Periodicals,Inc. J Appl Polym Sci 112: 19431948, 2009

Key words: fractional calculus; complex susceptibility;magnetic nanoparticles, polymeric systems

INTRODUCTION

The magnetic nanoparticles have specic propertiesdifferent from massive magnetic material. 1 Commonlymagnetic nanoparticles, suspended in liquid organicmediums, are used in the development and design of apparatus and devices. 24 However, these colloidalsuspensions are not stable, because there is coagulationof magnetic nanoparticles and the magnetic behaviorcould be modied. 5 To avoid this particular phenom-enon, an alternative is to stabilize magnetic nanopar-ticles in a polymer matrix. These nanocompositesmaterials are composed by polymer lled with mag-netic nanoparticles, which has many advantages com-pared with those systems of magnetic nanoparticlesdispersed in organic media. For example, in the caseof magnetite nanoparticles rmly xed in a polysty-rene matrix, there is no coagulation of particles. 6 Thepolymer matrix also acts to stabilize the magneticnanoparticles, preventing their oxidation and stabiliz-

ing other properties, such as superparamagnetic relax-

ation response, over long periods7

; these nanomaterialsare so-called polymer-magnetic nanocomposite. 8,9

From this polymer-magnetic nanocomposite, it is pos-sible to obtain a ferrouid when magnetic nanopar-ticles xed in a matrix polymer are dispersed intoliquid media. 10 Accordingly, these ferrouids arewidely used in technical applications that are eithermechanical (e.g., seals, bearings, and dampers) or elec-tromechanical (e.g., loudspeakers, stepper motors, andsensors) in nature. Besides those technical applications,ferrouids are gaining increasing interest for biologicaland medical applications (e.g., high-gradient magneticseparation techniques, magnetic drug targeting, mag-netic hyperthermia, and contrast agents for magneticresonance imaging). 713

Nevertheless, all of these applications require acomplete understanding of the magnetic relaxationphenomena of the mentioned composite materials.The complex magnetic susceptibility, v v0 iv00,as a function of the frequency of an applied ac mag-netic eld is a powerful tool for the characterizationof the dynamic properties of polymer-magneticnanocomposites dispersed into a liquid media.

Two typical magnetic relaxations are observed inpolymer-magnetic composites. At high frequencies ( f > 1GHz) it is observed the Ne el-relaxation which is

Journal ofAppliedPolymer Science,Vol.112, 19431948(2009)VVC 2009 Wiley Periodicals, Inc.

Correspondence to:M. E. Reyes-Melo ([email protected]).

Contract grant sponsor: CONACYT; contract grantnumber: 061313.


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characterized by the internal reorientation (insidenanoparticles) of the magnetization against an inter-nal energy barrier. And on the other hand, at lowfrequencies ( f < 1GHz) is observed the Brownianrelaxation due to the rotational diffusion of the

nanoparticles in the liquid.13,14

In polymer-magneticnanocomposite dispersed into a liquid media, therotational diffusion of the xed magnetic nanopar-ticles induces the motion of the polymeric matrix. 10

From experimental measurements of the v wecan study these phenomena, because we can sepa-rate into its real and imaginary components. Thereal part, v0, represents the component that is inphase with the applied ac magnetic eld, whereasthe imaginary part, v 00, is proportional to the p /2 outof phase or quadrature component of the magnetiza-tion. The imaginary component is related to theenergy dissipated by the sample from the ac eld,whereas real component is associated to partial stor-ing energy. For the interpretation of the experimen-tal spectra of both v0 and v 00 is necessary to use amathematical model because the dynamic magneticproperties for these materials are very complex,which makes them very difcult to handle analyti-cally. It is possible to nd models in the literaturewhich describe v0 and v 00, but most of them take intocount only one relaxation time, s . In this sense, theuse of differential and integral operators of fractionalorder (fractional calculus) is an alternative.

The goal of this work is the application of frac-tional calculus to model the complex magnetic sus-ceptibility. Using this new fractional model, we canassociate the molecular mobility to each relaxationphenomenon. It is important to remark that in thiswork we are interested only to modeling the low fre-quencies relaxation phenomenon, because this mag-netic relaxation is associated to the mobility of themagnetic nanoparticles xed in polymer matrix intoa liquid media.

FRACTIONAL CALCULUS AND THE NEWFRACTIONAL RESISTOR-INDUCTOR ELEMENT

Fractional calculus is the branch of mathematics thatdeals with the generalization of integrals and deriva-tives of all real orders. 15 In this work, for the modelingof the ac magnetization response, fractional calculus isused to obtain a new magnetic-fractional elementwhose perform an intermediary behavior between amagnetic-inductor and an electrical resistance. Wehave named to this new magnetic fractional element afractional resistor-inductor or FRI. The constitutiveequation, shown by eq. (1), of the FRI is based in a dif-ferential operator of fractional order a between 0 and1. From the FRI we obtain a resistance behavior if a 0, and a magnetic-inductor behavior if a 1 (see

Fig. 1). Therefore for 0 < a < 1 an intermediary behav-ior between a resistance and an inductance isobtained.

V t RLR

adaI t dta

Rs aDat I t with 0 a 1 (1)

In eq. (1), V is the circuit applied voltage, R and Lare the electric resistance and inductance magni-tudes, respectively, and s L/ R is the characteristicresponse time, called relaxation time, which can beassociated with the time required to the motion for acomplete reorientation of a given particle (with adipolar magnetic moment) to a new equilibriumstate. Finally, Dat I (t) is the fractional derivate of theath order of the electrical current in the FRI withrespect to time, 16,17 which can be dened by the Rie-mann-Liouville derivative:

Dat I t DZ t

01C 1 a I y d

st y a with a 2 0

; 1 (2)

where C is the Gamma function:

Cx Z 1

0euux1 du with x > 0 (3)

and y is a mathematical variable used in Riemann-Lioville derivative. It is important to mention thateq. (2) is obtained from a Riemann-Lioville integralwhich is represented as a fractional integral dened between 0 and t:

Dat I t DZ t

01C a I y dst y 1a with a 2 0

; 1 (4)

Figure 1 The new fractional resistor-inductor element(FRI), with 0 < a < 1.

1944 REYES-MELO ET AL.

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A fractional derivate [eq. (2)] represents a convolutionintegral in which the function I (t) is convolved withthe impulse-response function of ath order. In conse-

quence, eqs. (2) and (4) describe the state of the under-lying system inuenced by all states at early times. Onthe other hand, from the physical point of view, thefractional order of a fractional integral [eq. (4)] can beconsidered as an indication of the remaining energy inthe system, given from an applied magnetic eld sig-nal. In similar manner, the fractional order of a deri-vate reects the rate at which a portion of the energyhas been lost in the system.

Several works have used empirical models todescribe complex magnetic susceptibility response of magnetic materials. 1821 However, these classicalmodels are characterized by only one relaxation

time, s , and hence explain the experimental resultsonly as a rst approximation. In the next section wedescribe the application of the FRI element for thedevelopment of a fractional magnetic model (FMM)to describe the complex magnetic susceptibilityresponse at low frequencies for systems containingmagnetic nanoparticles. By introducing fractionalcalculus tool into the FRI, it is possible take intoaccount a distribution of relaxation times, associatedwith the system magnetic response. This approachhas been successfully used over the past few yearsin the case of the dielectric manifestation of the vis-coelasticity of polymer-dielectric materials. 16,17

THE FRACTIONAL MAGNETIC MODEL

The Figure 2 shows the proposed FMM and the con-stitutive equations of their electric elements. The elec-trical behavior of the FMM it is described by eq. (5):

Dat V t s1aDat V t L0 L1 D

a 1t I t

L1 Da 1t I t L1 sa 1D2t I t 0 5

Applying the Fourier transform to eq. (5), thecomplex inductance, L, of the circuit is calculated:

L L1 L0 L1 ixs a

ixs a ixs (6)

From the inductance-magnetic susceptibility relation,L k(1 v ), where k is a constant which involvesgeometrical variables of the inductance, the complexsusceptibility equation of the FMM is obtained:

v v 1 v 0 v 1 ixs

a

ixs a ixs (7)

where v 0 and v 1 represents the susceptibility at lowand high frequencies, respectively. From this equa-tion the mathematical expressions of v0 and v 00 areobtained; v 0 is calculated as:

v 0 v 1 v 0 v 1 xs

2a xs 1 asin ap2 h ixs 2a 2 xs 1 asin ap2 xs

2 (8)

and v 00is dened as:

v 00v 0 v1 xs

1 acos ap2 xs 2a 2 xs 1 asin ap2 xs 2 (9)

TESTING THE RESPONSE OF THE FMM

To verify the magnetic behavior of the FMM dened by eqs. (8) and (9), we proceeded to vary systemati-cally the fractional order or the parameter a of theFMM. This parameter can take values only between0 and 1. The Figure 3 shows the isothermal predic-tions of the real part, whereas Figure 4 display thepredictions for the imaginary part of v at differentvalues of a, having v 0 1.0 and v 1 0.1. In both

Figure 2 The proposed fractional magnetic model (FMM).

Figure 3 The frequency dependence of the real part of complex susceptibility predicted by the FMM, for the indi-cated values of a.

APPLICATION OF FRACTIONAL CALCULUS 1945



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cases, the shape of curves is depending of the valueof parameter a.

For values of a < 1, the curves of the real partshow that at low frequencies and high frequencies v 0is not dependent of the frequency. Instead, at lowfrequencies v0 % v 0 , and at high frequencies v 0 % v 1(see Fig. 3). However, at intermediary frequencies v 0is depending of the frequency, and there is a notice-able decrease of the real part as frequency increases;this behavior corresponds to a maximum at theimaginary part (see Fig. 4), and is related to a typicalrelaxation phenomenon.

By another hand, as Figure 3 depicts, when a 1there is not energy dissipation, whereas when a 0we obtain the typical curves analogous to Debyemodel. 4 Moreover, for values of 0 < a < 1, but closeto 0, the decrement of the curves of the real part dis-play a step-like feature, whereas for values of pa-rameter a in the same range but close to 1 the step-like feature is stretched. As a consequence of thislast behavior, the amplitude of the peak at the imag-inary part curves decreases (see Fig. 4). It is wellknown that for viscoelastic systems a decrease in theamplitude of the peak at the imaginary curve isrelated to a decrease of the energy dissipated for thesystem.

In addition, an important tool to estimate the mag-nitude of the fractional exponent a from experimen-tal results is the Cole-Cole diagram. The Figure 5shows the Cole-Cole diagrams obtained from FMMfor the same values of the parameter a used at Fig-ures 3 and 4, and also it is showed how the frac-tional parameter a can be estimated from a Cole-Cole diagram; for a 0, Cole-Cole diagram displaysa semi-circle whose radius is equal to ( v 0 v 1 )/2. Itis important to remark here that the fractional expo-

nent a, could be considered as a relative measure of partial stored energy by the system due to theapplied magnetic eld. Moreover, the exponent acould be related to a distribution function of relaxa-tion times, 3,4 as it has been reported by Alcoutlabi etal. for polymer systems. 22

COMPARISON BETWEEN THEORETICAL ANDEXPERIMENTAL RESULTS

The Figure 6 shows the comparison between thetheoretical curves of the real and imaginary parts,performed by the FMM, and those experimentalcurves reported for a system of polymer-magnetic

Figure 5 Cole-Cole diagrams obtained from the FMM,for the indicated values of a.

Figure 6 Comparison between the theoretical curves pre-dicted by the FMM (solid line) and the experimental dataof the real (solid circles) and imaginary (open circles) partsof the complex susceptibility taken from a colloid disper-sion of polymer-magnetic microspheres.

Figure 4 The frequency dependence of the imaginarypart of complex susceptibility predicted by the FMM, forthe indicated values of a.




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microspheres of latex lled with cobalt-ferrite nano-particles; 10 the microspheres are dispersed into aliquid media. As this gure displays, the FMMdescribe quite well the experimental results of bothreal and imaginary parts of a colloidal dispersion of polymer-magnetic microspheres. Additionally, Fig-ure 7 displays the theoretical Cole-Cole diagram andthose experimental points extracted from the behav-ior of the real and imaginary part of the polymer-magnetic microspheres systems. Here it is observedthat theoretical results are also in good agreementwith the experimental data, conrming that expo-nent a was correctly estimated. The values of the pa-rameters introduced to the FMM are given in theTable I.

The performance of the FMM was also evaluatedcomparing its theoretical predictions with thereported experimental results of a ferrouid com-posed by stabilized magnetite nanoparticles dis-persed into an aqueous dissolution. 4 As Figure 8depicts, the FMM also describes in a precisely man-ner the behavior of the rotational motion of magneticnanoparticles into the aqueous media. Moreover,Figure 9 shows how the experimental Cole-Cole dia-gram is described by the FMM. In this case, the the-

oretical curve describe quite well the experimentaldata, indicating that the value of the fractional expo-nent a was correctly estimated. The values of the pa-rameters introduced to the FMM to describe thissystem are also given in the Table I.

CONCLUSIONS

Using fractional calculus is possible to describe thecomplex susceptibility of colloidal dispersions of polymer-magnetic microspheres and also stabilizednanometric-sized magnetic particles. The shape of the theoretical curves of the real and imaginary parts

Figure 7 Comparison between the theoretical Cole-Colediagram predicted by the FMM (solid line) and the experi-mental data (open circles) taken from a colloid dispersionof polymer-magnetic microspheres.

TABLE IValues of the Parameters Used to Evaluate the FMM

FMMparameters

Magnetic systems

Polymer magneticmicrospheres

Stabilizedmagnetite ferrouid

a 0.02 0.40v 0 2.30 10

2 0.89v 1 1.00 10 3 0.11s(s) 5.20 10 4 1.27 10 4

Figure 8 Comparison between the theoretical curves pre-dicted by the FMM (solid line) and the experimental dataof the real (solid circles) and imaginary (open circles) partsof the complex susceptibility taken from a magnetite aque-ous ferrouid.

Figure 9 Comparison between the theoretical Cole-Colediagram predicted by the FMM (solid line) and the experi-mental data (open circles) taken from a magnetite aqueousferrouid.

APPLICATION OF FRACTIONAL CALCULUS 1947



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can be modied changing the order of the fractionalderivate, between 0 and 1. The comparison betweenthe theoretical results obtained from the FMM andthose reported experimental data shows that theFMM is capable to describe rotational diffusion fea-

tures associated to both polymer-magnetic micro-spheres and magnetic nanoparticles dispersed into aliquid media. Moreover, from the denition of thisnew FMM we can establish that the order of thefractional derivate could be considered as a relativemeasure of the partial dissipated or storing magneticenergy by the system.

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Application of Fractional Calculus to the Modeling of the