Introduction

We report the phase behaviour of aqueous dispersionsof nanometer-sized magnetic spherical particles (ferro-¯uids) [1]. Considering the solvent as a continuousmedium, an analogy between such suspensions andatomic systems is usually made. Indeed the shape of theinteraction potential between objects is the same in bothsystems [2]. Fluid, liquid, gas and solid phases are thusobserved in colloids, but, in contrast to atomic systems,the interparticle interactions in colloidal suspensions canbe tuned using several experimental parameters, such assalinity, temperature and, in the case of ferro¯uids, themagnetic ®eld. Because of the variety of the potentialsspeci®c behaviours, predicted by numerous theoreticalstudies [3], are induced. The polydispersity in the

colloidal suspensions also strongly in¯uences their phasebehaviour [1, 4].

Few experimental studies concern aqueous suspen-sions of nanoparticles. Recent studies have reported thestability of aqueous magnetic ¯uids in the low-particle-volume-fraction regime and have pointed out theastonishing stability of these suspensions ± Derjaguin±Landau±Verwey±Overbeek (DLVO) theory fails todescribe such systems [5, 6]. In this low-volume-fractionregime and for the particle size under consideration(d £ 12 nm), the dipolar interactions are su�cientlyweak compared to electrostatic repulsions so the mag-netic ¯uids can be considered as isotropic dispersions.An increase in ionic strength lowers the strength ofrepulsions and turns the ¯uid phase into a biphasic``gas±liquid'' suspension where droplets of a dense phase

Progr Colloid Polym Sci (2000) 115 : 77±83Ó Springer-Verlag 2000 POLYMER COLLOID AND SOLID PARTICLES

F. CousinV. Cabuil

Fluid±solid transitions in aqueous ferro¯uids

F. Cousin á V. Cabuil (&)Laboratoire des Liquides Ioniqueset Interfaces Charge esEquipe ColloõÈ des Magne tiquesUniversite Paris 6, Case 6375252 Paris Cedex 5Francee-mail: cabuil@ccr.jussieu.frTel.: +33-1-44273174Fax: +33-1-44273675

F. CousinCentre de Recherche sur la MatieÁ re Divise eCentre National de la RechercheScienti®que, 45071 Orle ans Cedex 2France

Abstract We report the phase be-haviour of aqueous dispersions ofmagnetic nanoparticles in the high-volume-fraction regime. Osmoticcompression experiments are used toobtain high particle volume fractionsin the range of low ionic strength.The suspensions exhibit a reversiblerheological liquid±solid transitionfor a given particle volume fraction.The threshold of the transition isshifted towards low volume frac-tions by a decrease in the ionicstrength. The structure factor of thesuspensions obtained from small-angle neutron scattering measure-ments shows that the sum of theinteractions in the system is domi-nated by strong electrostatic repul-sions and that the suspensionsexhibit a liquidlike structure even in

the solid phase. The polydispersity ofthe system does not allow the crystalphase to be reached; the solid phaseis thus a Wigner glass. The thresholdof the transition is consistent with thetheoretical threshold of a solid tran-sition for a system of hard spheres,taking for the radius of the particlean e�ective radius: the sum of theradius and the Debye length. Repul-sions are so e�cient that the sum ofthe interactions remains isotropicunder the appliance of a magnetic®eld. It is nevertheless possible toinduce a solid±liquid transition bythe appliance of a magnetic ®eld.

Key words Magnetic ¯uids áOsmotic compression á Structurefactor á Phase transition áWigner glass

spontaneously nucleate in a dilute one. Such gas±liquid-like transitions have been described elsewhere [6±8].The e�ect of temperature has always been found to benegligible compared to the e�ect of ionic strength,except near the phase-transition line.

In the present work, we investigate the regime of highparticle volume fractions and low ionic strength, i.e. theregime of strong repulsions. We show that in this casea ¯uid±solid transition is observed and we discuss thenature of this transition.

Experimental

Materials

The spherical nanoparticles under consideration are made of amagnetic ferric oxide, maghemite (c-Fe2O3). They were synthesizedby condensation of metallic salts in an alkaline medium accordingto Massart's method [9] and can be dispersed either in acidic or inalkaline media when they are uncoated. Nevertheless, the e�ectivecharge at the surface of the particles depends on the pH [5, 10]. Inorder to get particles with a constant surface charge density for allpH ³ 3, the particles are coated with citrate species and dispersed atpH 7. They bear negative charges due to the acidic-basic behaviourof the citrate species. Above a concentration of citrate in solutionof 2 ´ 10)3 mol/l, the plateau of adsorption is reached [6]. Theionic strength in such suspensions is thus ensured by theunadsorbed electrolyte (3:1), i.e. trisodium citrate.

The suspensions are usually polydisperse systems. The particlesize distribution is described by a log±normal law [1]:

P�d� � 1������2pp

rdexp ÿ 1

2r2ln

dd0

� �2" #

;

where d0 is the mean diameter and r the standard deviation. Theparticles obtained have a mean diameter ranging between 5 and12 nm according to the experimental conditions used for theirsynthesis. It is possible to reduce the polydispersity by a size-sortingprocess [11] but it is not possible to get suspensions with a r inferiorto 0.1. The samples under consideration here have a mean diameterof 8 nm and a standard deviation of 0.35 (d and r are deduced fromthe analysis of the shape of the magnetization curve [12]). The meanvolume, VW, is estimated by taking into account the polydispersitythrough r3

� � r30 �

expÿ32r2

2

�. It allows an average diameter, dmean

of 9.6 nm to be deduced for the samples described here.

Methods

Osmotic compression experiment

The ionic strength and the osmotic pressure are imposed byosmotic compression [13], which is a suitable way to getsuspensions with a high volume fraction of particles: the suspen-sions after synthesis are dialysed against a bath, called a reservoir,which has the required ionic strength. A dextran polymer(Mw = 110000 g/mol) is added in the bath and imposes its ownosmotic pressure, which is neither dependent on temperature nordependent on ionic strength or pH. The phenomenological lawfollowed by the osmotic pressure as a function of the concentrationof polymer has been established elsewhere [14]: log10 (Pdyn/cm2) =1.826 + 1.715w0.297 (100w is the mass fraction of polymer insolution). The reservoir is replaced as many times as necessary toreach equilibrium, which usually takes about 3 weeks.

The volume fraction of the suspensions after the osmoticcompression is determined either by chemical titration [15] or fromthe value of the saturation magnetization [12].

Small-angle neutron scattering experiments

We investigated the structure of the concentrated suspensions bysmall-angle neutrons scattering (SANS) experiments. The mea-surements were performed at LLB (CEA-Saclay) on the two-dimensional PAXY spectrometer. The neutron wavelength was10 AÊ and two sample±multidetector distances were used: 3.20 and1.05 m. The scattering vector ranged from 8 ´ 10)3 to 7 ´ 10)2 AÊ )1

in the ®rst con®guration and from 5 ´ 10)2 to 4 ´ 10)1 AÊ )1 in thesecond one.

In order to evaluate the e�ect of the appliance of a magnetic®eld on the suspensions, we performed SANS measurements undera constant magnetic ®eld. The magnetic ®eld was parallel to theplane of the multidetector. The magnetic ®eld had a constant valueof 0.03 T. The sample±multidetector distance was 3.20 m and theneutron wavelength was 10 AÊ (8 ´ 10)3 AÊ )1 < q < 7 ´ 10)2 AÊ )1).

Results

Establishment of the rheological phase diagramof the concentrated suspensions

Osmotic compression of liquid samples allows solidsamples to be obtained. The ``liquid'' and ``solid'' phasesare here rheological de®nitions: a sample is consideredto be a liquid if it ¯ows and to be a solid if it does not.The osmotic decompression of the solid sample allowsthe liquid to be recovered: the liquid±solid transition isreversible.

The threshold of the ``liquid±solid'' transition isshifted towards the low volume of particles when theionic strength decreases.

The experimental phase diagram at room tempera-ture of the liquid±solid transition is presented in Fig. 1as a function of the ionic strength. All the samples wereobtained by osmotic compression.

Equation of state of the system

The equation of state of the system of concentratedsuspensions for a citrate ionic strength of 2.5 ´10)3 mol/l is presented in Fig. 2. The regime of thelow volume fraction is compared in Fig. 2a to previousresults reported in Ref. [5]. The high-volume-fractionregime is investigated in Fig. 2b. It was obtained usingthe osmotic compression experiment. It has revealedthat it is impossible to determine the volume fraction ofsamples for which the imposed osmotic pressure is about18000 Pa.

Structure of the suspensions

The structure of the suspensions was determined usingSANS. The scattered intensity of a colloidal suspension

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in a SANS experiment is I�q� � n��Dq�2�P �q��S�q�,where q is the wave factor, n the density of the system,(Dq)2 the contrast, P(q) the form factor of the particlesand S(q) the structure factor of the suspension.

For low concentrations of colloids, a suspensionbehaves as a perfect gas: there are no interactionsbetween particles and the structure factor S(q) is 1 forevery q. As the contrast and P(q) are similar forconcentrated and dilute suspensions, the structure factorof the concentrated suspensions was calculated asfollows: S(q) = [Iconc(q)/Fconc]/[Idil(q)/Fdil].

The curves obtained for a dilute suspension(F = 1%) and for three concentrated suspensions, twoliquid samples (A and B) and a solid sample (C) arepresented in Fig. 3a. The citrate ionic strength was set to10)2 mol/l. Despite anisotropic dipolar interactions, the2D spectra are isotropic and are thus presented in anintegrated form. As it is very di�cult to measure thevolume fraction of maghemite in concentrated suspen-sions, the volume fraction was deduced from thecalculation of I/I1%, which is linear to S�q��U. As S(q)® 1 at high q, F can be determined. We found thefollowing A: F = 17.6%; B: F = 18.8%; C: F =29.5% (Fig. 3b).

The structure factor of suspensions A, B and C ispresented in Fig. 4. It is obtained by dividing the curvesin Fig. 3b by F.

E�ect of the appliance of a magnetic ®eldon the samples

Macroscopically, a magnetic ®eld has no e�ect on thesolid samples, except for the ones which are just abovethe liquid±solid transition threshold. For these samplesthe appliance of a magnetic ®eld turns the solid phaseinto a liquid one.

Evolution of the structure of the suspensions underthe appliance of a magnetic ®eld

The scattered intensity of a suspension with F = 17.6%(citrate ionic strength of 10)2 mol/l) in the parallel andperpendicular directions to the magnetic ®eld is repre-sented in Fig. 5. The two spectra are similar, indicatingthat the scattered intensity remains globally isotropic.

As the experimental spectrum of a concentratedsuspension remains isotropic under the appliance of amagnetic ®eld, it is possible to integrate the measuredsignal over the surface of the detector in order to get the

Fig. 1 Experimental phase diagram of suspensions as a function ofthe citrate concentration. The line represents the theoretical value of aliquid±solid transition for a system of hard spheres supposed to occurfor a renormalized volume fraction Fe� = 50%: renormalization isperformed for each ionic strength taking into account for the particlevolume the range of the repulsions

Fig. 2a, b Experimental equation of state of a suspension of maghe-mite particles for a citrate concentration of 2.5 ´ 10)3 mol/l. a Low-volume-fraction regime: experimental results are compared withresults from Ref. [5]. b High-volume-fraction regime: open symbolsare related to liquid samples and ®lled symbols to solid samples

79

isotropic structure factor of the suspensions submittedto a magnetic ®eld. The structure factor of the suspen-sion with F = 17.6% (citrate ionic strength of10)2 mol/l) in zero magnetic ®eld is compared with thatin a ®eld of 0.03 T.

Discussion

Interactions in the suspensions

The interparticle interactions in the aqueous dispersionsunder consideration here are van der Waals attractions,magnetic dipolar interactions and electrostatic repul-sions.

For a dispersion of spherical particles of diameter d0,the reduced potential due to van der Waals attractionscan be written

UvdW�s�kT

� ÿ A6kT

2

s2 ÿ 4� 2

s2� ln

s2

s2 ÿ 4

� �;

where A is the Hamaker constant and s = 2r/d0, forparticles with r� d0, r being the distance between thecentres of the two particles (A = 10)19 J for themaghemite particles) [16].

In the dispersions under consideration, each particleis a magnetic monodomain, i.e. each particle has apermanent magnetic moment, the intensity, l, of whichdepends on the speci®c magnetization, ms, of thematerial and on the particle diameter, d0, l � mspd3

0=6(ms = 3.1 ´ 105 A/m for these maghemite particles)[12]. If ~li is the dipolar moment of particle i and~r is thevector joining the centres of the particles, the anisotropicpotential relative to the dipolar interactions between twodipoles in zero magnetic ®eld is

Udip�r�kT

� c4p�2 cos h1 cos h2 ÿ sin h1 sin h2 cosu� ;

where hi is the angle between ~li and ~r and u is theazimuthal angle between both dipoles.

Fig. 3 a Small-angle neutron scattering spectra obtained for the high-volume-fraction samples A, B and C and for a dilute suspension ofmaghemite particles (F = 1%). b Renormalization of the spectra forsamples A, B and C by the form factor P(q) and the contrast (Dq)2.Open symbols represent the data obtained from the ®rst con®gurationof the experimental setup and ®lled symbols represent the dataobtained from the second con®guration

Fig. 4 Structure factor of samples A (F = 17.6%), B (F = 18.8%)and C (F = 29.5%). Open symbols represent data obtained from the®rst con®guration of the experimental setup and ®lled symbolsrepresent data obtained from the second con®guration

Fig. 5 Scattered intensity of sample A (F = 17.6%) under a 0.03 Tmagnetic ®eld in parallel and perpendicular directions to the ®eld

80

A reduced parameter, c � l0m2s V 2

kTr3 (V is the volume ofa particle), allows the magnetic coupling between theparticles to be quanti®ed:

� If c=4p� 1, the dipoles rotate independently and themagnetic potential in zero ®eld, averaged over alldirections, may be simpli®ed to hUMi

kT � ÿ c2

48p2 (lowmagnetic coupling).

� If c=4p� 1, there are correlations between thedipoles and the potential becomes anisotropic. Nev-ertheless, it is possible to integrate over the space toobtain a mean value of the dipolar interaction:hUMi

kT � ÿ c2p (high magnetic coupling) [10].

When a magnetic ®eld is applied, the expansion of thepotential around the most probable con®guration leadsto the mean potential decreasing as 1/r3.

Concerning repulsions, such aqueous dispersions ofparticles with surface charges are usually described usingthe celebrated DLVO formalism [17], which takes intoaccount the van der Waals attractions and the Coulom-bic repulsions; however, this DLVO theory fails todescribe the astonishing stability of aqueous suspensionsof maghemite particles coated by citrate ions at highionic strength as reported in Ref. [6]. The repulsiveCoulombic potential is nevertheless well described by aYukawa potential:

UCoul

kT� K

d0rexp ÿ�r ÿ d0�

D

� �;

where K corresponds to the contact value and Dcharacterizes the interaction range, with r being thedistance from the interparticle centre and d0 the particlediameter. The value of K is much higher than the oneused in the DLVO potential that takes into account thee�ective surface charge. These strong repulsions atcontact can be due to the ®nite size of the trivalentcitrate ions which induce an additional steric repulsion

at the surface of the particles. The range of theinteractions, D, is of the order of the Debye length,

1/j, de®ned by j � ÿ e2e0ekT

Pi ciz2i

�1=2.

Structure of the suspensionsin the high-volume-fraction regime

The structure factors obtained from SANS for liquidsamples A and B clearly indicate that the structure of thedispersions of the particles is that of a liquid (or a ¯uid).The isotropy of the spectra and the value of themaximum of S(q) indicate that the sum of the interac-tions is dominated by the electrostatic repulsions despitedipolar interactions. The shape of the structure factor ofsolid sample C is similar to that of liquid samples A andB and no second Bragg peak is observed, indicating thatthe structure stays liquidlike.

The mean distance between particles, rmean, is calcu-lated from the abscissa qmax corresponding to themaximum of the structure factor by rmean = 2p/qmax.rmean is perfectly linear to (p/6F)1/3dmean (the values arereported in Table 1), proving that the particles arehomogeneously dispersed. The mean diameter dmean ofthe particles deduced from rmean (9.5 nm) is in goodaccordance with the mean diameter obtained from themagnetization curve (9.6 nm).

Nature of the transition

Figure 2a indicates that for the low ionic strength underconsideration in this study the regime of the low volumefractions is dominated by strong electrostatic repulsions.The suspensions remain ¯uid in those experimentalconditions, far from the regime of lower strengthrepulsions where the transition from a ¯uid phase to abiphasic liquid±gas system occurs.

According to Fig. 2b, it is very di�cult to conclude ifthe equation of state presents a plateau. Moreover theevidence of the biphasic samples has not been estab-lished.

As shown by the structure factor, the regime of thehigh volume fractions is also dominated by strongelectrostatic repulsions. The strength of these electro-static repulsions is so high that it governs the structureof the system, which remains isotropic. Nevertheless, in

Table 1 Mean interparticle distance, rmean, deduced from qmax as afunction of the volume fraction, F, of the samples

F (%) rmean (nm)

17.6 12.9318.8 12.5829.5 10.58

Fig. 6 Structure factor of sample A (F = 17.6%) in zero magnetic®eld (open symbols) and in a 0.03 T magnetic ®eld (®lled symbols)

81

contrast to the case of the low volume fractions, forwhich the dipolar interactions remain in the low-coupling regime (i.e. c=4p� 1), an intermediate mag-netic coupling regime is reached for the high volumefractions as the interparticle distance is strongly reduced(we get c=4p � 0:4 for the most concentrated suspensionunder consideration here). The dipolar interactions arethus anisotropic.

As the SANS spectra indicate that the liquid and thesolid phases obtained are both ¯uid phases (no crystal-line order is observed), the liquid±solid transitionobserved can be described as a vitreous transition. Asit is driven by long-range electrostatic repulsions, thesolid phase is a Wigner glass [18]. The solid transitionfor a system of hard spheres occurs theoretically for agiven particle volume fraction of 50%. In Fig. 1 wecompare this theoretical threshold to our experimentalresults for di�erent ionic strength and renormalize thevolume of the particles by taking into account the rangeof the electrostatic repulsions: the radius of the particleis replaced by an e�ective radius: the sum of theparticle's radius, the citrate shell (6 AÊ [19]) and theDebye length. The experimental data are in goodagreement with this theoretical value of the threshold.

The polydispersity of our suspensions (r = 0.35)explains why it is not possible to make ¯uid±crystaltransitions [4]. When a signi®cant volume fraction ofparticles has a diameter greater than the meaninterparticle distance, too many defects in the latticeare created and the formation of a crystal is notallowed. An e�ective polydispersity, inferior to the realone, can be evaluated in our experimental system bytaking into account the range of the repulsions, but itremains too important to allow the crystal phase to bereached.

Dynamic measurements usually allow the ergodicityof the system to be discussed, in order to determine if thenature of a transition is vitreous [20]. Unfortunately thelong time which is necessary to reach equilibrium duringan osmotic compression experiment does not allowdynamic measurements to be made on our suspensions.Our conclusion concerning the vitreous nature of thetransition is thus only based on static arguments: theliquidlike structure of the solid and the high polydisper-sity of the system.

E�ect of a magnetic ®eld

The structure factor of the suspensions for which amagnetic ®eld was applied shows that the interactions inthe system continue to be dominated by repulsions andthus stay globally isotropic. Nevertheless, the sum of the

interactions is less repulsive when a magnetic ®eld isapplied (the ®rst peak of the structure factor is lowered).This has to be related to the rheological solid±liquidtransition observed when a magnetic ®eld is applied tosolid samples. The sum of the interactions is lowered andthe change in the e�ective volume fraction of the spheresthrough the lowering of the range of the interactions canbe su�cient for the threshold of the solid transition to becrossed.

It is astonishing that at the same time the meandistance between the particles increases: the value ofqmax of the structure factor is shifted towards lower q,meaning that the density of the objects changes throughrmean = (p/6F)1/3dmean. The decrease in qmax corre-sponds to a decrease in the volume fraction from 17.6to 16.5%. This has to be related to the appearance of asmall peak in the structure factor at q = 0.0665 AÊ )1

(9.45 nm in real space). This distance is of the order oftwo particle radii, which means that some doublets ofparticles are formed under the appliance of the magnetic®eld. For a ®eld of 0.03 T, the decrease in the volumefraction corresponds to a proportion of particles in-volved in doublets of about 12%. As the polydispersityof the interactions is very important in the system(dipolar interactions are function of d3), it is reasonableto suppose that the particles involved in the creation ofthe doublets are the biggest. Nevertheless, the systemcontinues to be dominated by repulsions.

Conclusion

This experimental work concerning the phase behaviorof aqueous dispersions of magnetic nanoparticles in thehigh-volume-fraction regime completes previous studieson the same systems performed for low volume fractionsand illustrates the fact that magnetic ¯uids are a suitableexperimental system for the study of the phase behav-iour of suspensions of nanometric particles (it is possibleto reach the ¯uid, solid, gas and liquid phases). Thestructure of concentrated suspensions is governed bystrong electrostatic repulsions and the suspensionsexhibit a vitreous transition. The threshold of thetransition can be monitored by the ionic strength inthe suspension. The suspensions keep their vitreousstructure when a magnetic ®eld is applied, but the sumof the interactions is reduced and it appears possible toinduce solid±liquid transitions, which give speci®cmagnetorheological properties to the suspensions.

Acknowledgements We thank FrancË ois Boue for his help duringthe SANS experiments and Emmanuelle Dubois for helpfuldiscussions.

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