magnetic nanoparticle size constraints in ferronematic liquid crystals

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Journal of Magnetism and Magnetic Materials 292 (2005) 310–316

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Magnetic nanoparticle size constraints in ferronematicliquid crystals

Derek Waltona,�, Suhaila M. Shiblib, M.L. Vegab, E.A. Oliveirab

aPhysics Department, McMaster University, Hamilton, Ontario, Canada L8S 4M1bInstituto de Fisica, Universidade de S. Paulo, S.Paulo, Brazil

Received 26 May 2004; accepted 4 November 2004

Available online 30 November 2004

Abstract

Thanks to an induced diamagnetic moment, liquid crystals can be oriented in fields on the order of 1T. The addition

of a few percent of a ferrofluid (FF) whose grain size is 13 nm to a liquid crystal (LC) produces a ferronematic (FN)

which can be re-oriented in fields �.001T with the longest axis of the micelles becoming parallel to the field direction (J.

Phys. 31 (1971) 691, Phys. Rev. A 34 (1986) 3483, J. Magn. Magn. Mater. 122 ( 1993) 53), but one whose grain size is

3 nm cannot, although the 3 nm FF results in a ferromagnetic moment orders of magnitude larger than the diamagnetic

moment of the liquid crystal. We will show that the FN made with the 13 nm particles contain aggregates that are

blocked at room temperature, and that these aggregates are responsible for the realignment. Our data suggests that the

aggregates mainly consist of two grains.

r 2004 Published by Elsevier B.V.

PACS: 75.50.Tt; 75.50.Ck; 75.50Mm

Keywords: Ferronematics; Blocking; Colloidal aggregation

1. Introduction

The moments of the individual magnetic nano-particles in the ferrofluid (FF) can reverse on amuch shorter time scale than the experimentaltime, and are superparamagnetic. But the addition

- see front matter r 2004 Published by Elsevier B.V.

/j.jmmm.2004.11.146

onding author. Tel.: +9055 259 140 24635; fax:

252.

ddress: [email protected] (D. Walton).

of a small quantity of FF to the liquid crystals(LC) results in an equilibrium paramagneticsusceptibility whose magnitude is much higherthan that of the diamagnetic susceptibility of theundoped liquid crystal. Since a field on the orderof 1T can produce a preferred orientation ofthe LC that is stable for many hours it wouldbe plausible to suppose [4,5] that, provided themagnetic particles are coupled to the micelles, theferronematic (FN) can be oriented in proportionately

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D. Walton et al. / Journal of Magnetism and Magnetic Materials 292 (2005) 310–316 311

lower fields. Berejnov et al. [4] have, assuming thatindividual superparamagnetic grains were stronglyanchored to the micelles, argued that the fractionalmagnetization of an equilibrium distribution ofsuperparamagnetic grains, being roughly 3 ordersof magnitude larger than that of the liquid crystal,is able to orient their ferrodiscotics in fields 3orders of magnitude smaller. In a recent analysis,Morozov [5] argued that individual superpara-magnetic grains are ineffective and dimers arenecessary. However, he states that the magneticrelaxation time need only be longer than thatfor the grain aggregates to rotate in the fluid foralignment to be achieved. But the relaxation timeof the dimers considered in Ref. [5] is still shortcompared with the experimental time; so onthe experimental scale they are still superpar-amagnetic.These arguments appear to be incorrect:The experiment we are considering here is the

following: liquid crystals are birefringent; so thealignment of the micelles affects the transmissionof polarized light, and this provides a commontechnique for monitoring the orientation process.In a typical experiment the FN is completelyaligned, with the director field and the grainmagnetizations parallel to the field direction, in ahigh field �1T (a field of this order is capable ofaligning the LC which then maintains alignmentfor many hours after the field is removed). Thefield is then lowered to �0.01T and the samplerotated by 451. The realignment of the micelles canthen be followed by the change in opticaltransmission.A typical result is shown in Fig. 1. In particular

it shows that the FF with 3 nm grains is incapableof promoting any realignment. In fact this cannotbe achieved even if the realignment field is raisedto 0.05 T, or by using a FF with larger (4.5 nm)grains.The contribution to the magnetic free energy of

the coupled micelles and magnetic grains is [6]Um ¼ Aðn � MÞ

2 where n is a unit vector parallel tothe director field, and A is a negative couplingconstant that is a function of state variables suchas temperature, pressure, etc. [6]. Um is of roughlythe same magnitude in the LC in a 1T field and theFN in .01T field; so what is wrong? This question

can be answered if the time for realignment isconsidered.The time required for realignment depends on

the torque exerted on the micelles by the magneticparticles. This torque is M�H, where H is themagnetic field, and M is the magnetization; so,although the values of M may be comparable inthe liquid crystal and FN, if the field is less thetime required is longer. We will show below that itis not possible for a superparamagneticM to alignthe liquid crystal because the time would simplybe too long, and that the magnetic nanoparticlesmust be blocked (thereby increasing M) for themto be effective.

M ¼P

i mi where mi is the moment of particle i.For the experiments being considered here, afterthe high field alignment the moments of theindividual grains are parallel, and M ¼ nm islarge. When the field is reduced and the samplerotated, the individual moments start to relax to adistribution appropriate to that temperature andthe small field being applied, eventually reducingM to a fraction of its initial value. If the relaxationtime is longer than the experimental time, i.e. if thegrains are blocked, the change in M is small, M

remains large, and the grains are able to re-orientthe micelles. However, if the relaxation time isshort the grains soon relax to their equilibriumdistribution, and a much smaller value for M. Forgrains on the scale of 10 nm the thermal equili-brium value for M is about 3 orders of magnitudesmaller than nm; so that only a small number ofgrains need to become blocked, by aggregation orother means, to dominate the reorientation processin our experiments.It would appear to be desirable to determine

what fraction, if any, of the magnetic grains in theFN are blocked and, if so, if the same fraction isblocked in the pure FF.A simple way to determine whether the mag-

netic grains in a material are blocked is to measurethe susceptibility of the material to changes in themagnetic field on the time scale of interest as afunction of temperature: the contribution to thesusceptibility is much larger when the grains aresuperparamagnetic than when they are blocked,so a large change in susceptibility occurs at theblocking temperature. Unfortunately, the fluid

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0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000

Time (secs)

Tra

nsm

itte

d li

gh

t (a

rb. u

nit

s) 13 nm

3 nm

Fig. 1. Optical transmission results from two ferronematics initially aligned in a 0.3 T field, after having the field reduced to 0.01T and

rotated 451. The larger grains are able to reorient the director field, but the smaller, entirely paramagnetic grains, cannot.

D. Walton et al. / Journal of Magnetism and Magnetic Materials 292 (2005) 310–316312

medium makes this measurement impossible onthe time scales of interest, since the magneticgrains can rotate. However, if the fluids are frozen,rotation is prevented, and the magnetic suscept-ibility can be measured. Furthermore, those grainsthat are blocked will provide an increased con-tribution to the signal when the fluid melts, andthey are once more able to rotate. Thus theexperiment simply consists of freezing the FN(being careful to wait long enough, before freezing,for aggregation, which would require diffusion ofthe grains through the liquid crystal, to occur), andthen monitoring its magnetic susceptibility as it iswarmed through the freezing point. The blockedgrains cannot reorient in the frozen solid, and donot contribute appreciably to the susceptibility,but when the sample melts they are able to rotate,and the susceptibility should increase. From theincrease in susceptibility the fraction of theblocked grains can be determined, and makingthe obvious assumption that the larger grainsaggregate preferentially, size limits can be placedon the sizes of the grains that can aggregate. Ofcourse, on freezing the material ceases to be aferronematic, but this is if no consequence since weare only interested in determining what fraction ofthe FF grains are blocked.

2. Experimental

The FN studied here are lyotropic nematicliquid crystals doped with small amounts of FF.This material consists of an orientationallyordered aqueous solution of short rodlike micellesof a surfactant whose size is approximately 8 nm.The lyotropic liquid crystals employed are amixture of potassium laurate/1-decanol/water(KL/DeOH/H2O), with concentrations in weightpercent of 28.90/7.10/64.0, respectively.The FF added to this material was purchased

from Ferrotec Corporation, and consisted ofnanoparticles of magnetite (Fe3O4) suspended inwater. The concentration of FF added to the liquidcrystal was about 1013 magnetic particles/cm3. Thesize distribution of the magnetic particles, deter-mined magnetically, obeyed a log-normal distribu-tion with a mean size of 13 nm and a full-width athalf-maximum of 8 nm [7]. They are coated withdispersive agent (oleic acid) to prevent theiraggregation [2,3]. The thickness of the coating isabout 2 nm [7]. The solvent used is water, whichallows the FF to be miscible with the liquid crystal.Roughly, equal amounts of FF were placed in

two small Pyrex containers 0.5 cm, in diameter by1.5 cm long, with a 2 cm long neck that had a 2mm

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bore, using a graduated syringe. One of them wasthen filled with a liquid crystal. They were thensealed by melting the necks of the containers.The time scales of interest are many minutes,

which makes AC susceptibility measurementsdifficult; so the measurements were made byalternating a DC magnetic field in a SQUIDmagnetometer built by Quantum Design. Themeasurement procedure consisted in cooling thesample in zero magnetic field to 6K, and thenmeasuring the moment at increasing temperatureintervals of 5K after applying a 30Oe field for5min followed by a �30Oe field for an equal time.The susceptibility can be obtained from thedifference between the two readings.After the liquid crystal was mixed with the

ferrofluid the sample remained at room tempera-ture for about an hour before it was rapidly cooledin the cryostat, thereby allowing any aggregationto take place. Of course, on freezing, the materialceases to be a liquid crystal, but this is irrelevant tothe present investigation. The freezing process isnecessary to prevent rotation of the magneticgrains, and to prevent any separation of particlesthat have aggregated.

3. Results and discussion

The results for a pure FF sample, and a FN areshown in Fig. 2. Since the lower temperature dataare irrelevant for the present discussion, only theresults above 200K are plotted. The completedata, together with grain size distributions deter-mined from it, are being published elsewhere [8].It is clear from the data that none of the pure FF

is blocked on these time scales. The FN, on theother hand reveals an increase in susceptibility ofabout 9% when the sample melts, indicating thatabout 9% (by volume) of the grains are now freeto rotate. This fraction of the magnetic grains areblocked, and their moments do not reverse, butthey can follow the changing magnetic field byrotating when the sample melts. Electron micro-scopy revealed that the largest grains visible wereabout 20 nm in size, and grains larger than about17 nm occupied a volume fraction of 9%.Although they only comprise a fraction of the

total magnetic material, since these grains areblocked, they will make the major contribution tothe sample moment at room temperature after thesuperparamagnetic grains have relaxed to theirequilibrium distribution, and are therefore respon-sible for the reorientation of the director.The magnetic particles in a FF are single

domain ferromagnetic grains. Their ground stateis usually two-fold degenerate with the particlemagnetization parallel to the longer axis. Therate of moment reversal for a single domainparticle is [9]

1

t¼ ce�KV=kT ; (1)

where V is the volume of the particle, K theanisotropy constant, T the temperature, k Boltz-mann’s constant, and c an attempt frequency onthe order of the frequency of the spin-wave of zerowave-vector.The anisotropy of magnetite is mainly due to the

shape of the particle and, for an ellipsoid ofrevolution with semi axes a and b K ¼ 1

2ðDb �

DaÞJ2; where Db and Da are demagnetization

factors, and J is the magnetization density [9]. Itis extremely difficult to estimate demagnetizationfactors for ferrofluid grains that are not evenapproximately ellipsoids of revolution. However,there is general agreement that the size of grains inrocks and pottery blocked at room temperature(i.e. for which t is greater than �1000 s) is about35 nm. This is about 3 times the size of theferrofluid grains in FN. Since the grains in FF androcks and ceramics have similar shapes it is clearthat the individual grains are superparamagneticat room temperature, their relaxation rate isextremely fast, and they will have reached equili-brium in the experiments being considered here.However, the increase in volume and anisotropy

for dimers of large enough grains can lead toblocking: the magnetic moments of two touchingmagnetic grains are coupled, and they can betreated approximately as a single particle whoselength is the sum of the longest dimensions of thetwo grains. If the two touching particles aretreated as an ellipsoid with an axis ratio of 2:1,Db�Da ¼ 3.01, at room temperature, J ¼ 480gauss [10], and the minimum diameter of each of

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0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

200 220 240 260 280 300

Temperature, K

emu

Fig. 2. The difference between the sample moment in 30 and �30Oe as a function of temperature. The crosses are for the pure FF, and

the solid circles are for FN sample. The melting temperature of the FN was 260K. While an attempt was made to use equal amounts of

FF for both samples, this was not entirely successful, and the FN contained more magnetic material. The decrease in moment of the

FN just before melting can be explained by the formation of aggregates of 2 particles [8].


the particles for the dimer to have a relaxationtime greater than 600 s is 15 nm.The grain size distribution of our 13 nm ferro-

fluid is approximately log-normal with a width of8 nm [7]; so a significant fraction of the grains havesizes larger than 15 nm, and aggregation to formdimers and larger clusters can explain our results.If that is the case, the blocked aggregates willconstitute a significant fraction of the moment.The question of why grains aggregate in the

liquid crystal, but not in the FF will now beconsidered:Liquids cannot transmit torques, but liquid

crystals can. Thus a FF particle will displace thedirector, ~n; leading to a distortion free energy [1]

F ¼K

2

ZdV jr~nj2; (2)

where K is now a Frank elastic constant. If twoparticles are touching the volume integral iseffectively halved, thereby reducing the energy;so, since the strain fields are long range, this willlead to an effective interaction between theparticles, and a diffusive aggregation process.The magnitude of the interaction energy de-

pends on the particle size; so, if this energy is lessthan kT, no aggregation will occur. This conclu-sion is supported by a detailed analysis of the data

[8] from which the particle size distribution wasobtained. That analysis revealed that peak inthe size distribution had not shifted, and that thedistributions were similar up to about 15 nm whereit decreased in the ferronematic. Such a decreasewould occur if grains larger than �15 nm hadaggregated. On the other hand, the lack of changein the position of the peak in the distributionindicates that the smaller grains did not aggregate.A similar analysis can be undertaken to

compute the minimum grain size in triads andlarger aggregates. The constraints imposed by thenematic structure mean that the particles in liquidcrystals tend to aggregate in a linear fashion ratherthan forming clumps [11]. Following the procedureabove, it can be shown that the minimum averagesize of particles that would be blocked if 3 wereaggregated in a line is about 10 nm.The peak of thedistribution lies at 13 nm [8]; so if trimers had beenpresent in significant numbers the change in signalon melting would be larger than 50%. Thus, itmust be concluded that aggregates larger thanpairs do not make a significant contribution.Thermal activation can lead to aggregates

breaking apart, and the process of aggregationwill eventually reach a steady state. In order to testwhether or not a steady state had been reachedafter one hour, the same sample was measured a

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0.0E+00

2.0E-05

4.0E-05

6.0E-05

8.0E-05

1.0E-04

1.2E-04

1.4E-04

0.0E+00 5.0E-02 1.0E-01 1.5E-01 2.0E-01 2.5E-01

Field (T)

Rec

ipro

cal R

elax

atio

n t

ime

(1/s

ec)

Fig. 3. The change in relaxation time with applied field.


week later, and no change in the size of theincrease in susceptibility was observed, indicatingthat a steady state had already been establishedafter one hour.The torque exerted on the micelles by the

magnetic particles is M�H [9], where H is themagnetic field. Thus the time required for reor-ientation depends on the magnetic field, as shownin Fig. 3. The blocked grains make by far themajor contribution to the sample moment, andare responsible for these times. But, if noaggregation had occurred, and all the grains hadremained superparamagnetic, the value of M

would be two orders of magnitude less, and theapplied field would have to be 100 times larger inorder to re-orient the samples on a comparabletime scale. This suggests that superparamagneticgrains cannot be responsible for the phenomenareported in [4]. Of course the reason the pureliquid crystal can be oriented on a shorter timescale is that although the diamagnetic moment isof the same order, H is indeed about 100 timeslarger.

4. Conclusions

We have observed that the individual magneticparticles in a FF are superparamagnetic above

273K. The addition of a liquid crystal to the FFplaces the particles in an environment where eachparticle produces a strain field. The energy of thesystem can be reduced if the particles aggregate.The experimental results show that 9% of theresulting FN contains aggregates large enough tobe blocked at room temperature. The process ofaggregation is limited to pairs, since, if trimers orlarger clusters of smaller particles were present themuch larger numbers of the smaller particleswould lead to a much larger volume fractionblocked at room temperature. Therefore, weconclude that the decrease in energy is less thankT at room temperature for grains smaller thanabout 15 nm, and they do not aggregate in thisliquid crystal. It is possible that a differentliquid crystal, with a larger Frank elastic constant,would lead to aggregation of smaller particles.This possibility is being investigated. Finally, wesuggest that the data in [4] can only be explained ifblocked aggregates are present.

Acknowledgements

This research was supported by the NaturalSciences and Engineering Council of Canada(NSERC) and by the Fundacao de Amparo aPesquisa do Estado de S.Paulo (FAPESP).

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[10] C. Kittel, Introduction to Solid State Physics, Wiley,

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[11] H. Stark, Phys. Rep. 351 (2001) 387.

magnetic nanoparticle size constraints in ferronematic liquid crystals

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