The imaginary part of susceptibility, chi , in frozen ferrofluids was measured as functions of temperature, frequency, and concentration of magnetic particles. In a very dilute region, where magnetic interaction is negligible, the relaxation time is mainly determined by Neel\u27s relaxation in isolated particles and the frequency-dependent peak temperature in chi (T), T-p, obeys the Arrhenius law. In the moderate concentration range, T-p has a Vogel-Fulcher (VF) relaxation with the measuring frequency. However, for concentration larger than a certain value, deviation from VF law occurs. Simulations show that the VF relation could be attributed to Ising spin-glass-like random and frustrated interaction between magnetic moments of particles. We suggest that particle configuration before freezing is critical to determine the distribution of pair exchange parameters. For high concentration, there are more ferromagnetic than antiferromagnetic bonds. The deviation from VF law at high concentration could be due to short range correlation among spins

Duan, Xiaodong

Luo, Weili

Zhang, Jinlong

English

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University of Central Florida STARS Faculty Bibliography 2000s Faculty Bibliography 1-1-2000 Crossover behavior in dynamics of frozen ferrofluids Xiaodong Duan University of Central Florida Jinlong Zhang University of Central Florida Weili Luo University of Central Florida Find similar works at: https://stars.library.ucf.edu/facultybib2000 University of Central Florida Libraries http://library.ucf.edu This Article; Proceedings Paper is brought to you for free and open access by the Faculty Bibliography at STARS. It has been accepted for inclusion in Faculty Bibliography 2000s by an authorized administrator of STARS. For more information, please contact STARS@ucf.edu. Recommended Citation Duan, Xiaodong; Zhang, Jinlong; and Luo, Weili, "Crossover behavior in dynamics of frozen ferrofluids" (2000). Faculty Bibliography 2000s. 2509. https://stars.library.ucf.edu/facultybib2000/2509 Journal of Applied Physics 87, 6935 (2000); https://doi.org/10.1063/1.372891 87, 6935© 2000 American Institute of Physics.Crossover behavior in dynamics of frozenferrofluidsCite as: Journal of Applied Physics 87, 6935 (2000); https://doi.org/10.1063/1.372891Published Online: 28 April 2000Xiaodong Duan, Jinlong Zhang, and Weili LuoCrossover behavior in dynamics of frozen ferrofluidsXiaodong Duan, Jinlong Zhang, and Weili Luoa)Department of Physics, University of Central Florida, Orlando, Florida 32816The imaginary part of susceptibility, x9, in frozen ferrofluids was measured as functions oftemperature, frequency, and concentration of magnetic particles. In a very dilute region, wheremagnetic interaction is negligible, the relaxation time is mainly determined by Neel’s relaxation inisolated particles and the frequency-dependent peak temperature in x9(T), Tp , obeys theArrhenius law. In the moderate concentration range, Tp has a Vogel–Fulcher ~VF! relaxation withthe measuring frequency. However, for concentration larger than a certain value, deviation from VFlaw occurs. Simulations show that the VF relation could be attributed to Ising spin-glass-likerandom and frustrated interaction between magnetic moments of particles. We suggest that particleconfiguration before freezing is critical to determine the distribution of pair exchange parameters.For high concentration, there are more ferromagnetic than antiferromagnetic bonds. The deviationfrom VF law at high concentration could be due to short range correlation among spins. © 2000American Institute of Physics. @S0021-8979~00!24908-6#The relaxation behavior in frozen ferrofluids has beenrecently investigated as a disordered system with competinginteractions.1–8 Recently, some results indicate the existenceof a spin-glass-like behavior in frozen ferrofluids4,6 Thesepapers have shown a critical slowing down for the averagerelaxation time and a divergence of the nonlinear suscepti-bility at finite temperature Tg .4,5 It is also found that thefrequency-dependent peak temperatures in the imaginarysusceptibility in ferrofluids obey Vogel–Fulcher law.3 Theaging phenomenon was found at low temperature and it con-firmed the existence of glass or spin glass phase.6,7 There-fore, it is now accepted that there is cooperative behavior infrozen ferrofluids. However, the relaxation behavior dependson magnetic particle concentration and so far most of workmentioned only study one or a small range of concentration.In this article, we experimentally study the dynamic suscep-tibility for concentration from extreme dilute limit~0.0045%! to concentrated regime ~13%!. We found that thepeak temperature versus measuring frequency can be de-scribed by Arrhenius law in a very dilute limit, crossing overto Vogel–Fulcher law at more concentrated regime, and fi-nally deviating from VF relation at highest concentration.Comparing with the simulation results suggests that thesecrossover behaviors are associated with single moment relax-ation, cooperated relaxation from random, frustrated interac-tion, to the one originated from nonfrustrated interaction.Since the frozen ferrofluid could be used as a model systemto study the effects of dipolar interaction on relaxation, acomplete study of such a system is necessary and could sheda light on other disordered systems such as glass-formingliquids,9 spin-glasses,10 polymers,11 dielectric relaxors,12 andothers.Experiments were performed on ferrofluid samples con-sisting of magnetic particles dispersed in alkylnaphthalenes,abbreviated as ABF. Each particle has a mean diameter of 7nm. Particles are coated with surfactant to avoid agglomera-tion. Particles interact with each other through the long rangedipole force. The sample was sealed inside a small quartztube. The ac susceptibility measurements were performedwith a SQUID ac susceptometer. The samples with very di-lute concentration ~0.0045%, named ABFL!, moderate con-centration ~10.4%, named ABFM!, and high concentration~12.9%, named ABFH! are used in the experiments. Eachsample was first rapidly cooled down from room temperatureto 4.5 K in zero field. The cooling rate was approximately100 K/min. Then, the susceptibility x9 was measured at tem-perature T in intervals of 4 K from 6 to 300 K for fivedifferent frequencies f50.1, 1, 10, 100, and 1000 Hz, respec-tively. The field amplitude was 1 Oe except for sampleABFL, where the amplitude was 5 Oe because of weak sig-nal. The linear relation between the magnetization and thefield was confirmed in this field region. In Fig. 1~a!, theimaginary part of ac susceptibility, x9, is plotted as functionof temperature, T, at different frequencies for ABFM. In Fig.1~b!, the peak temperature in x9(T) were plotted as a func-tion of 1/(ln f2ln f0), where 1/f 0 is 1029 s for all concentra-tion ~ABFL, ABFM, and ABFH!. The straight line representsthe Vogel–Fulcher law: f 5 f 0 exp@A/k(T2T0)#. In order tounderstand the experiment results, we perform simulation inthe following paragraph.In a very dilute region, we can neglect dipole–dipoleinteraction between particles. The relaxation originates fromNeel mechanism, following activated Arrhenius law:t5t0 exp(Kv/kBT), where K is the anisotropy energy coeffi-cient, v the volume of a particle. The general expression ofx9 can be obtained from fluctuation and dissipation theory,13x9~v ,T !5M s2v3kBTE0‘g~t ,T !vt11~vt!2 dt , ~1!M s is the saturation magnetization. The 1/3 factor comesfrom averaging the random angle between a dipole and thefield directions. g(t ,T) is the distribution function for relax-ation time. For noninteracting system, g(t ,T) could be duea!Electronic mial: luo@pegasus.cc.ucf.eduJOURNAL OF APPLIED PHYSICS VOLUME 87, NUMBER 9 1 MAY 200069350021-8979/2000/87(9)/6935/3/$17.00 © 2000 American Institute of Physicsto the size distribution of particles. If all the particles havethe same volume, Eq. ~1! can be simplified asx9~v ,T !5M s2v3kBTvt0 exp~Kv/kBT !@vt0exp~Kv/kBT !#211. ~2!Considering the size distribution effect, one can assume thatM s is proportional to the particle volume, then x9 can bewritten asx9~v ,T !5M¯ s23kBTv¯E0‘ vt0 exp~Kv/kBT !@vt0 exp~Kv/kBT !#211v2 f ~v !dv ,~3!where M¯ s , v¯ are average values. f (v) is the distributionfunction for particle volume. f (v) is often represented by thelog-normal function.14 In moderate concentration region, theparticles are still randomly distributed. However, the dipole-dipole interaction becomes important. Because of the barrierdue to the anisotropy energy, the magnetic moment tends toalign with the easy axis within the particle and can be treatedas a point dipole. The interaction between two dipole mo-ments m1 and m2 isJˆ i j523cos w1 cos w22cos~w12w2!r123 m1m2 , ~4!where r12 is the distance between m1 and m2, w1 and w2 arethe angles between m1, m2, and the vector from m1 to m2,respectively. In order to make a clear discussion, the magni-tude and direction of mi are separated by introducing a quan-tity SI . When the projection of dipole to the external field ispositive, SI takes 11, otherwise, SI takes 21. An effectiveexchange parameter Ji j is introduced. So the model Hamil-tonian is Hmod 52( iÞ jJ i jSiSi . The effective exchange pa-rameter Ji j is determined by the random direction and thedistance of magnetic dipoles, the distribution of Ji j could beapproximated by a Gaussian P(Ji j)}exp@2Jij2/2(DJ)2# . Thisis the standard model or Ising spin glass. The imaginary partof the susceptibility can be calculated by Eq. ~2!, where therelaxation time distribution function g(t ,T) was found byMonte Carlo ~MC! simulation, using the above modelHamiltonian.8 For a polydispersed system, both dipole inter-action and the size distribution contribute to g(t ,T). Wenoticed that the intrinsic relaxation time is different for par-ticles with different size. Generally, two magnetic particleswith different intrinsic time clock could not synchronize ifthe dipole–dipole interaction energy is smaller than the en-ergy barrier. In that case, the interaction for long time aver-age will be zero for two particles with different sizes. So wecan approximately write down the relaxation time distribu-tion function as g(t ,T)5gi(t ,T)gv(t ,T), where gi is thedistribution function for the dipole interaction and gv for theparticle size distribution. The imaginary susceptibility forpolydispersed frozen ferrofluid can be expressed asx9~v ,T !5M¯ s23kBTv¯E0‘v2 f ~v !dv3E0‘vt exp~Kv/kBT !gI~t ,T !11@vt exp~Kv/kBT !#2dt . ~5!gi could be obtained from MC simulation.8 In a high con-centration region, the dipole–dipole interaction becomeseven stronger and can affect the particle configuration in liq-uid state. When the solvent is frozen, the configuration isfrozen as well. We compare two situations shown in theinsert of Fig. 2. When the easy magnetization axes of twoparticles are in alignment @Fig. 2, inset ~a!#, the exchangeparameter Ji j is greater than zero, representing ferromagneticbond. While for parallel easy axes that are not in alignment@Fig. 2, inset ~b!#, Ji j is less than zero, representing the an-tiferromagnetic bond. Due to the nature of dipole–dipole in-teraction, the configuration in Fig. 2 @inset ~a!# has lower freeFIG. 1. ~a! Imaginary part of susceptibility, x9, for ABFM vs temperaturefor following measurement frequencies: 0.1 Hz ~s!, 1 Hz ~d!, 10 Hz ~j!,100 Hz ~h!, and 1000 Hz ~n!. ~b! Experimental results for peak tempera-tures, Tp in x9 ~T! vs 1/(ln f2ln f0) for ABFL ~m!, ABFM ~d!, and ABFH~s!.FIG. 2. Results from MC simulation: peak temperatures in x9~T! vs the1/(ln f2ln f0) for the dilute ~n!, moderate ~d!, and concentrated ~s and m!samples. The temperature is normalized by the distribution width DJ in ~8!.In the simulation for concentrated samples, we assumed that J050.5DJ ~s!and J05DJ ~m! in the drift Gaussian distribution. For simulation of diluteconcentration, the energy barrier is assumed E52DJ . Insert: the easy mag-netization axis of two particles are in alignment. Ji j.0, suggesting ferro-magnetic bond in model Hamiltonian. The attractive energy is E f522m0M s2/d2, where d is the particle diameter. ~b! The easy magnetizationaxes of two particles are in parallel. Ji j,0 indicating antiferromagneticbond in the model Hamiltonian. The attractive energy is Ea f52m0M s2/d2, uEa f u,uE f u.6936 J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Duan, Zhang, and Luoenergy than the one in Fig. 2 @inset ~b!#. So the system willfavor the ferromagnetic bond at high concentration. A simplemodel for the high concentration frozen ferrofluid can beexpressed as Hmod el52( iÞ jJ i jSiSi and the distribution ofthe exchange parameter P(Ji j)}exp@2(Jij2J0)2/2(DJ)2#with J0.0. The imaginary susceptibility can be calculatedby Eq. ~1! or ~5! except that g(t ,T) is obtained from MCsimulation for the model Hamiltonian in which the distribu-tion of exchange interaction follows a shifted Gaussian.The simulation results for the peak temperatures, Tp , inthe imaginary part of the susceptibility, x9(T , f ), are shownin Fig. 2, where f is the measuring frequency of simulationand 1/f 0 is the t0, the unit spin-flip time used in MC simu-lation. In order to simplify the calculation, we only considerthe monodispersed system. As we can see from Fig. 2, thesimulation results qualitatively agreed with experiments verywell. For the dilute limit, the extrapolated line goes throughthe origin, suggesting T050, then Arrhenius law is recov-ered as expected since relaxation for single particle shouldexhibit activated behavior described by Neel’s relaxation. Atthe moderate concentration, VF law was found, the finitediverse temperature, T0, arose from the interaction betweenparticles. At high concentration, the behavior can no longerbe described by any simple relation. However, the simulationresults suggest that the deviation from VF law comes fromthe change of particle configuration. In moderate concentra-tion, the particle distribution is still completely random. Butin high concentration, the interaction energy between par-ticles in the liquid state can be comparable to or larger thanthe thermal energy and thus changes the particle distribution.As a result, there were more ferromagnetic than antiferro-magnetic type interactions when ABFH was frozen. We havemeasured the kerosene based ferrofluid ~KBF!2,8 and the de-viation from VF law occurred at concentration of 6%–8%compare to the 10%–12% of the ABF sample. The frozentemperature of kerosene is at 151 K which is lower than thatof alkylnaphthalenes, 263 K, it might be the reason of higherdeviation concentration in ABF.In conclusion, the dynamic susceptibility was studied forABF with different concentrations. We found that in a diluteregion, the relaxation time are mainly determined by Neel’srelaxation in the isolated particles. In the moderate and highconcentration ranges, the relaxation behavior not only de-pends on the strength of the pair interaction but also dependson the distribution of particles. Simulation showed that theVF relation could be attributed to Ising spin-glass-like inter-action where the distribution function for exchange param-eters is a Gaussian centered at zero. Shifting the center of theGaussian from zero to a nonzero value leads to the deviationfrom VF law at high concentration, suggesting correlation inthe system.The authors are grateful to Dr. S. Taketomi for providingsamples. This work is supported by the National ScienceFoundation Young Investigator Award ~W.L.!.1 W. Luo et al., Phys. Rev. Lett. 67, 2721 ~1991!.2 T. Jonson et al., Phys. Rev. Lett. 75, 4138 ~1995!.3 J. Zhang, C. Boyed, and W. Luo, Phys. Rev. Lett. 77, 390 ~1996!.4 C. Djurberg et al., Phys. Rev. Lett. 79, 5154 ~1997!.5 H. Mamiya, I. Nakatani, and T. Furubnayashi, Phys. Rev. Lett. 80, 177~1998!.6 T. Jonsson et al., Phys. Rev. Lett. 81, 3976 ~1998!.7 H. Mamiya, I. Nakatani, and T. Furubayashi, Phys. Rev. Lett. 82, 4332~1999!.8 X. Duan, J. Zhang, and W. Luo ~unpublished!.9 M. D. Ediger, C. A. Angell, and S. Nagel, J. Phys. Chem. 100, 13200~1996!.10 See, Spin Glasses and Random Fields, edited by A. P. Young ~WorldScientific, Singapore, 1998!.11 C. Angell, Science 267, 1924 ~1995!.12 D. Viehland, S. J. Jang, and L. E. Cross, J. Appl. Phys. 69, 414 ~1991!.13 S. Dattagupta, Relaxation Phenomena in Condensed Matter Physics ~Aca-demic, London, 1987!.14 T. Jonsson, J. Mattsson, P. Nordblad, and P. Svedlindh, J. Magn. Magn.Mater. 168, 269 ~1997!.6937J. Appl. Phys., Vol. 87, No. 9, 1 May 2000 Duan, Zhang, and Luo

Crossover behavior in dynamics of frozen ferrofluids

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Crossover behavior in dynamics of frozen ferrofluids

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