physical properties of aqueous suspensions of goethite nanorods

Depending on volume fraction, aqueous suspensions of goethite α-FeOOH nanorods form a liquid-crystalline nematic phase above 8.5% or an isotropic liquid phase below 5.5%.. After a brief

DOI 10.1140/epje/i2003-10078-6

Physical properties of aqueous suspensions of goethite

Part I: In the isotropic phase

B.J Lemaire1, P Davidson1,a, J Ferr´e1, J.P Jamet1, D Petermann1, P Panine2, I Dozov3, and J.P Jolivet4 1

Laboratoire de Physique des Solides, UMR CNRS 8502, Bˆatiment 510, Universit´e Paris-Sud, 91405 Orsay, France

2

European Synchrotron Radiation Facility, B.P 220, 38043 Grenoble, France

3

Nemoptic, 1 rue Guynemer, 78114 Magny-les-Hameaux, France

4

Laboratoire de Chimie de la Mati`ere Condens´ee, UMR CNRS 7574, Universit´e Paris 6, 4 Place Jussieu, 75252 Paris, France

Received 22 July 2003 and Received in final form 1 December 2003 /

Published online: 21 April 2004 – c
EDP Sciences / Societ`a Italiana di Fisica / Springer-Verlag 2004

Abstract Depending on volume fraction, aqueous suspensions of goethite (α-FeOOH) nanorods form a

liquid-crystalline nematic phase (above 8.5%) or an isotropic liquid phase (below 5.5%) In this article, we

investigate by small-angle X-ray scattering, magneto-optics, and magnetometry the influence of a magnetic

field on the isotropic phase After a brief description of the synthesis and characterisation of the goethite

nanorod suspensions, we show that the disordered phase becomes very anisotropic under a magnetic field

that aligns the particles Moreover, we observe that the nanorods align parallel to a small field (< 350 mT),

but realign perpendicular to a large enough field (> 350 mT) This phenomenon is interpreted as due to the

competition between the influence of the nanorod permanent magnetic moment and a negative anisotropy

of magnetic susceptibility Our interpretation is supported by the behaviour of the suspensions in an

alternating magnetic field Finally, we propose a model that explains all experimental observations in a

consistent way

PACS 61.30.-v Liquid crystals – 64.70.Md Transitions in liquid crystals – 75.50.Ee Antiferromagnetics

– 82.70.Kj Emulsions and suspensions

1 Introduction

Liquid-crystalline suspensions of mineral particles have

re-cently been the subject of renewed interest because they

may combine the fluidity and anisotropy of liquid

crys-tals with the specific magnetic and transport properties of

mineral compounds [1] Since their discovery by H Zocher

in 1925 [2], mineral liquid crystals (MLCs) have also

pro-vided convenient experimental systems to test

statistical-physics theories of phase transitions such as the Onsager

model [3, 4] Moreover, mineral low-dimensionality

pounds can provide very original building blocks

com-pared to the ones produced by usual organic synthesis [5]

In some cases, MLCs, such as laponite clay or V2O5

aqueous suspensions are already employed in the

indus-try [6–8]

As an illustration of these general ideas, we recently

reported in a letter the very unusual magnetic

proper-ties of the nematic suspensions of goethite (α-FeOOH)

nanorods [9] Goethite is one of the most common and

sta-ble iron oxides It was already used by the cavemen to

pro-duce the ochre colour for their wall paintings and it is still

a

e-mail: davidson@lps.u-psud.fr

being used nowadays in the paint industry [10] Goethite can be produced as nanorods through “chimie douce” techniques (i.e low-temperature solution chemistry) and dispersed in water in reasonable amounts to form stable suspensions [11] We have shown that these suspensions form a nematic phase that aligns in very weak magnetic fields Moreover, both in the nematic and isotropic phases, the nanorods orient parallel to the field at low field intensi-ties but reorient perpendicularly in higher fields It should

be noted that some of the magneto-optical effects in the isotropic phase reported here were already studied long ago by Majorana and by Cotton and Mouton at the be-ginning of the 20th century [12] However, it seems that they worked on suspensions of mixed iron oxides and they did not discuss their observations in the general context

of liquid crystals that were then very recently discovered

We have described and discussed these unexpected phe-nomena in detail in two papers The first and present one

is mostly devoted to the study of the magnetic proper-ties of the isotropic phase of the suspensions The second one deals with the isotropic/nematic phase transition and with the effects of applying a magnetic field or an electric field to nematic suspensions

L a ≈ 25 nm

L b ≈ 150 nm

L c ≈ 10 nm

Fig 1 Dimensions and crystallographic structure of goethite

nanorods The structure is made up of chains of oxygen

octa-hedra with an iron atom at the centre of each octahedron and

a hydrogen atom bonded to it

The outline of this article is as follows In the next

section, we describe the chemical synthesis and colloidal

stability of the suspensions of goethite nanorods We also

recall the crystallographic structure, the optical and

mag-netic properties of goethite, and we determine the nanorod

size and polydispersity distributions Section 3 describes

the various setups used in this study Our experimental

results about the magnetic properties of the isotropic

sus-pensions are detailed and briefly discussed in Section 4

A simple model is then given in Section 5 to account for

these quite unusual observations

2 Synthesis and characterisation of

suspensions of goethite nanorods

2.1 Synthesis

Goethite suspensions were synthesised by following an

al-ready described procedure [11] A molar solution of NaOH

is added, under stirring, to 400 ml of a decimolar solution

of Fe(NO3)3 until pH ≈ 11 is reached An ochre

ferrihy-drite precipitate instantly forms The suspension is left

for ten days at room temperature It is then centrifuged

(10000 rpm for 10 minutes), the precipitate is recovered

and redispersed in distilled water This operation, which

aims at removing unnecessary ions, is repeated twice The

suspension is again centrifuged and the precipitate is

re-dispersed in 3 M HNO3in order to electrostatically charge

the surface of the particles by proton adsorption Finally,

the suspension is rinsed three times to bring the pH b ack

to about 3 The final volume of the suspension is adjusted

so that the concentration is large enough to reach the

ne-matic/isotropic phase equilibrium (i.e a volume fraction

between 6 and 11% for different syntheses) The

suspen-sion demixes within a day into typically 5 ml of each phase;

it will be called “synthesis batch” in the following

The synthesis product was characterised by powder

X-ray diffraction The solid part of a sample of the synthesis

batch was recovered by centrifugation, then dried under

nitrogen atmosphere, and powdered Its diffractogram was

recorded and identified to that of pure goethite In

par-ticular, no traces of haematite (α-Fe2O3, a very stable

and common iron oxide) were found The crystallographic structure of goethite can be represented in thePnma

or-thorhombic space group (Fig 1) The unit-cell parameters

are a = 0.995 nm, b = 0.302 nm, c = 0.460 nm

(Actu-ally, this structure is also sometimes represented in the

Pbnm space group, with a = 0.460 nm, b = 0.995 nm,

c = 0.302 nm.) Oxygen atoms form a hexagonal compact

lattice along thec-direction and the Fe3+ cations occupy half of the octahedric sites The structure can also be de-scribed as the stacking of double chains of oxygen octahe-dra occupied by Fe3+cations, oriented in theb-direction.

These double chains are linked to adjacent ones by corner-sharing and hydrogen bonds Electron microscopy images andin situ electron diffraction show that the goethite

par-ticles obtained under these synthesis conditions are

rect-angular parallelepipeds with length L b , width L a, and thickness L c, respectively oriented along the b, a, and c

crystallographic axes

2.2 Colloidal stability

The electrostatic repulsion between particles must be as large as possible to ensure the stability of goethite suspen-sions against flocculation This is achieved when the sur-face electric charge is large and the ionic strength as low as

possible These two parameters, that depend on pH, must

be known to reach a reasonable description of the thermo-dynamic properties of the suspensions The surface charge

of goethite nanorods was measured as a function of pH

ac-cording to an already described method [11,13,14] At low

pH, hydroxo –OH groups adsorbprotons to form –OH+2

aquo groups, whereas, at high pH, other –OH groups lose

protons to form oxo –O− groups Therefore, the particles

are globally positively charged at low pH and negatively charged at high pH The isoelectric point was observed around pH ≈ 9 The pH of the suspensions was adjusted

around 3 where the measurement of the surface charge

density gave σ ≈ 0.2 C · m −2 The pH should not be

de-creased below 3, because goethite particles would dissolve

in too acidic conditions For our experiments, samples of various concentrations were prepared by dilution from the

synthesis batch with solutions of nitric acid at pH = 3 The

ionic strength of the synthesis batch is essentially due to the NO−3 ions, the molarity of which was also measured

and found to be (4.5 ± 0.5) × 10 −2M in the nematic phase

and (3.0±0.5)×10 −2M in the isotropic phase In order to detect any particle aggregation due to the high concentra-tion of the synthesis batch, samples were prepared in flat

glass capillaries of 50 µm thickness and observed by

opti-cal microscopy The suspension looked homogeneous and

no aggregates were observed This was further confirmed

by transmission electronic microscopy

All samples were stored in glass vessels, tightly capped and wrapped in teflon tape Most samples could be kept for more than a year, but we have noted slight changes

on the timescale of months For instance, we noticed that both volume fractions of the nematic and isotropic phases

at coexistence slowly drifted with time, from initial values

of 8.5% and 5.5% respectively, to 12.5% and 8.5% after a

year This suggests that sideways aggregation of the

parti-cles may take place to some extent Moreover, we actually

performed several syntheses of goethite suspensions that

led to materials with similar properties However, the

de-tailed examination of these properties showed that there

are subtle differences between the various batches For

in-stance, the colour of some batches turned progressively

dark red after months, which could be due to the

progres-sive transformation of goethite into haematite, another

thermodynamically more favoured iron oxide Therefore,

all experiments described in this work were performed

starting from the same synthesis batch, within a year

2.3 Determination of the particle dimensions

The dimensions of the particles and their polydispersity

are of course very important parameters to understand

the properties of the colloidal suspensions A single

ex-perimental technique can hardly give all this information

and we therefore had to combine various X-ray scattering

and electron microscopy techniques X-ray scattering

tech-niques are particularly useful because they perform

com-plete ensemble averages of the particles and do not require

any particular sample treatment The electron microscopy

techniques allow one to appreciate, in direct space, the

morphology and crystallographic quality of the particles,

their polydispersity, and their possible aggregation

The powder X-ray diffraction lines, used above to

iden-tify goethite as the only reaction product, are broadened

by the small size of the particles [15] Thanks to the

Scher-rer formula, the line broadening of an (hkl) reflection is

related to the size of the particles in the [hkl] direction:

∆(2θ) cos θ hkl , where λ = 0.1542 nm is the X-ray wavelength, ∆(2θ) is

the full width at half-maximum of the (hkl) diffraction

line corrected for experimental resolution, and 2θ hklis the

diffraction angle This reasoning is based on the

assump-tion that goethite particles are actually single crystals,

which is confirmed, for most particles in a batch, by

elec-tron microscopy The (200) reflection directly gives the

apparent mean width, L a ≈ 29 nm However, the (00l)

lines being too weak, one must consider the (10l) lines,

us-ing the approximate formula: L c ≈ L 10lcos

arctanlc

a



,

which gives L c ≈ 12 nm Unfortunately, the particle length

could not be evaluated in the same way because the (0l0)

lines are either too weak or superimposed onto other lines

Particle dimensions can also be obtained by

small-angle X-ray scattering techniques [15] In a very dilute

suspension, inter-particle interferences can be neglected;

in other words, the structure factor S(q) is equal to 1 (q is

the scattering vector modulus, q = 4π sin θ λ ) Moreover, the

particles are isotropically distributed and the scattered

in-tensity reads

I(q) = N |F (q )|2ρeg− ρe

0

2

Ie ,

Fig 2 Transmission electron microscopy (TEM) image of a

diluted goethite suspension

where N is the number of particles, ρeg and ρe0 are the

electron densities of goethite and water respectively, Ie is the intensity scattered by an electron,   represents the

ensemble average over all the possible particle

orienta-tions, and F (q ) is the Fourier transform of the particle

volume (form factor) The form factor of parallelepipedic

particles is well known [16] and, according to the q-range

probed, the various particle dimensions can in principle

be measured SAXS experiments on the ID2 beamline of the European Synchrotron Radiation Facility have been performed on diluted samples The samples were diluted enough that their scattering curves superimpose after cor-rection by a multiplicative factor that only accounts for the dilution This demonstrates that particle inter-ferences are indeed negligible The SAXS curves could be

fitted by the theoretical form factor in the whole q-range

probed However, the fit is not very sensitive to the parti-cle length and thickness, so that it only provides the width,

L a ≈ 22 ± 10 nm, in a reliable way Moreover,

polydisper-sity effects prevented the observation of a minimum in the form factor that could have given a precise measure of the particle thickness

Another type of SAXS experiment gave us, by chance,

an idea of the particle length In the course of the study of nematic samples aligned in a magnetic field (see, Part II, this issue p 309), we observed that the SAXS patterns

of a few samples displayed very weak but sharp diffrac-tion peaks at very small angles These peaks arise from

a very small proportion of smectic domains in these sam-ples Had the suspensions of goethite particles been quite monodisperse, they would probably have shown a smec-tic phase, as observed for monodisperse suspensions of viruses [17, 18] In these smectic domains, the particles

stack in layers with a period close to their length, L b At the ionic strength mentioned above (I = 4.5 × 10 −2 M), the Debye length is rather small (≈ 2 nm) and negligible compared to L b, within our experimental accuracy The smectic period then gives us L b ≈ 160 ± 10 nm.

Figure 2 shows an example of transmission electron microscopy image of a drop of diluted goethite suspen-sion left to dry on a microscopy grid When examined

0 100 200 300

0.0

0.1

0.2

0.3

0 10 20 30 40 50 0.0

0.1 0.2 0.3

b) a)

Width L

Fig 3 Size distributions of goethite nanorods obtained from

TEM images Solid lines are fits to a truncated Gaussian

statis-tics of standard deviation ∆ν = 0.4 (Eq (1)) a) Nanorod

length; b) nanorod width

carefully, it appears that most of the particles are

sin-gle crystals They lie on their largest face of (001) indices,

which allowed us to measure their length and width

distri-butions shown in Figure 3 (400 particles were measured)

Both distributions are Gaussian, with averages of 105 and

18 nm, respectively and standard deviations ∆L/L ≈ 0.4.

Another measurement gave averages of 118 and 24 nm,

respectively The standard deviation defines the size

poly-dispersity of particles which is very large here This

fea-ture must be considered in order to account quantitatively

for most experimental results, as will be shown in the

next sections (Note that log-normal distributions are

of-ten found for suspensions of nanoparticles, and it is likely

that it was so, right after the synthesis of goethite

How-ever, the subsequent centrifugations and dispersions may

have removed some of the smaller particles together with

the supernatants.) Finally, scanning electron microscopy

images yielded an average length of 150 nm and an average

width of 27 nm

In summary, considering the large experimental errors

and polydispersities affecting the particle dimensions, we

shall use in the following a mean length L b= 150±25 nm,

width L a = 25± 10 nm, and thickness L c = 10± 5 nm.

The particles will be modelled as homothetic rectangular

parallelepipeds scaled by a factor ν that obeys a Gaussian

statistics of standard deviation ∆ν = 0.4, truncated at

ν = 0:

−(ν−1)2/(2∆ν2 )

∞



0 dνe −(ν−1)2/(2∆ν2)

, for ν > 0 ,

This means that we assume the same distributions for

the particle length, width, and thickness

With the truncated Gaussian statistics, the average particle surface can be calculated:

∞

 0

dνP (ν)ν2,

where s0 = 2(L a L b + L a L c + L b L c ) = 1.1 × 10 −14m2 is

the surface of a particle of average dimensions (ν = 1).

We find sm = 1.17s0 = 1.3 × 10 −14m2

The average particle volume Vmis obtained in a similar way:

∞

 0

dνP (ν)ν3,

where V = L a L b L c = 3.7 × 10 −23m3 is the volume of

a particle of average dimensions (ν = 1) We find Vm =

1.5V = 5.6 × 10 −23m3

2.4 Magnetic properties of goethite nanorods

Since the aim of this work was to investigate the very pecu-liar magnetic properties of goethite nanorod suspensions,

it was first necessary to examine the magnetic structure of bulk goethite, a typical antiferromagnetic material This structure was determined by performing neutron diffrac-tion experiments on natural single crystals [19] The two main sub-lattices are oriented along the b-axis (i.e the

nanorod length) which is the antiferromagnetic axis The structure of goethite is based on double chains of octa-hedra occupied by iron atoms Their spins are parallel within a chain but there is an antiferromagnetic coupling between neighbouring chains The magnetic properties of goethite particles depend on their size For instance, the N´eel temperature TN [20, 21], above which the material becomes paramagnetic, varies between 325 and 400 K

For the particles considered in this work, TN ≈ 350 K,

but the size polydispersity should also result in a dis-persion of the N´eel temperature The anisotropy energy

is very large so that the so-called “spin-flop” transition only occurs at a field intensity of 20 T at 4.2 K [19] At room temperature, this threshold field should be some-what smaller but still not smaller than several teslas Indeed, we checked that the magnetisation depends lin-early on the field in the whole range explored (0–1.5 T)

In natural goethite, the parallel and perpendicular

sus-ceptibilities, χ  and χ ⊥, show the classical behaviour

ex-pected for an antiferromagnetic material The

magnetic-susceptibility anisotropy, ∆χ = χ  − χ ⊥, is negative; it decreases with temperature until it vanishes at TN.

Previous studies of natural and synthetic particles also report that small goethite nanorods bear a weak ferromag-netic moment oriented along the b-axis [19, 22] A likely

explanation of this behaviour is that the moment arises from non-compensated surface spins [23] In the follow-ing, this experimental observation will prove very impor-tant for the interpretation of the magnetic behaviour of goethite suspensions

Table 1.

Crystalline goethite

Molar massMg 88.85 g· mol −1

Optical indices (at 632.8 nm)

Goethite nanorods Dimensions

Polydispersity distributionP (ν) Gaussian

Standard deviation∆L i/L i 0.4

Average volumeVm 5.6 × 10 −23m3

Average surfacesm 1.3 × 10 −14m2

Magnetic susceptibility

χ (295 K) (1.7 ± 0.2) × 10 −3

Goethite suspensions

Ionic strength

Isotropic phase (3.0 ± 0.5) × 10 −2 M

Nematic phase (4.5 ± 0.5) × 10 −2 M

Electrical surface chargeσ 0.2 C· m −2

We performed additional measurements of the

mag-netic susceptibility of goethite powder at room

temper-ature with a SQUID magnetometer The sample was

prepared by drying a drop of the synthesis batch at

180◦C We found that the magnetisation depends

lin-early on the field, giving a (dimensionless) susceptibility

χ = (1.7 ± 0.2) × 10 −3 and no remanent magnetisation

was detected This latter point is due to the fact that

the particles in the suspension adopt random

magnetisa-tion direcmagnetisa-tions in zero-field, as ferrofluids do Upon

dry-ing, the macroscopic magnetisation of the sample would

remain null, in agreement with the random orientation

of nanorods, in dried samples, observed by electron

mi-croscopy

2.5 Optical properties of goethite nanorods

A large part of our studies is devoted to the

magneto-optical properties of goethite suspensions First, we briefly

summarise here the optical properties of goethite crystals

reported in the literature Since, to our knowledge, there

is no measurement of the refractive indices performed at

the wavelength (632.8 nm) of our He-Ne laser (see next

section), we interpolated between the values reported at

589 nm and 671 nm [24,25], assuming a monotonous

varia-tion In this wavelength region, the nanorods have a

uniax-ial negative birefringence with n a = 2.24 along their width

and n b = n c = 2.38 along their length and thickness The

intrinsic birefringence is then ∆nint = 0.14 Note that the

optical anisotropy is uniaxial with symmetry axis oriented

along the width rather than the length of the particles

Finally, Table 1 summarises the chemical and physical properties of the suspensions of goethite nanorods men-tioned in this section

3 Experimental section

3.1 Sample preparation

Most samples were kept in optical flat glass capillaries (VitroCom, NJ, USA) of inner thickness 20, 30, 50 or

100 µm The thickness of each glass wall is equal to that of

the sample The width of the capillaries is about ten times their thickness They are filled by capillarity except for the most concentrated samples that had to be sucked in with

a small vacuum pump Then, the capillaries are flame-sealed at each end Such samples can usually be kept for years They are particularly well suited for observations by

optical microscopy We found that the 50 µm thick

cap-illaries were also suitable for SAXS experiments in spite

of the glass absorption Lindemann glass cylindrical cap-illaries were not adapted to X-ray diffraction because of

their minimum diameter of at least 200 µm, which results

in a much too strong absorption due to the large iron con-centration of the suspensions However, Lindemann glass capillaries of 1.5 mm diameter were used for SQUID mag-netisation measurements at different temperatures The volume fractions of the suspensions were deter-mined by measuring the weight loss of samples heated at

180◦C for two hours (At higher temperatures, goethite dehydrates into haematite.) Dilutions were performed by

adding weighted amounts of solutions of nitric acid at pH

= 3 Concentrated samples were obtained by drying in an oven until they reached the desired weight

For a given synthesis batch, if polydispersity is negli-gible and if there is no temporal evolution, a very prac-tical way to be sure of the sample concentrations is to start from biphasic suspensions because their nematic and isotropic phases always have the same volume fractions,

respectively φN and φI, at thermodynamic equilibrium.

φN is the smallest volume fraction that can be observed

in the nematic phase, whereas φI is the largest volume fraction an isotropic phase can reach In the case of very polydisperse samples, such as the present ones, this rea-soning fails because fractionation effects occur However, all experiments described here were performed, using alto-gether only a very small amount of the suspension Under

these conditions, the volume fractions φN = 8.5 ± 0.5% and φI = 5.5 ± 0.5% seem to be reproducible within

ex-perimental accuracy

3.2 SAXS experiments

3.2.1 Experimental description The SAXS measurements were performed on the High-Brilliance beamline (ID2) of the European Synchrotron Radiation Facility located in Grenoble, France The scat-tering setup consists in a pinhole geometry, with a very low

divergence and a typical camera length of 10 m Beam size

at the sample position was about 0.1×0.1 mm2 The highly

monochromatic incident wavelength was λ = 0.0995 nm.

The scattered photons were recorded on a 2-dimensional

detector composed of a FreLoN CCD camera optically

coupled to a Thomson X-ray image intensifier having an

active diameter of 20 cm This combination provided a

useful range of scattering vector modulus, 0.02
q

0.6 nm −1 The standard procedure for data acquisition,

treatment and correction is described elsewhere [26]

A strong, highly homogeneous, and stable magnetic

field could be applied at the sample position The field

was generated by a set of two stacks of five NdFeB

per-manent magnets of size 5× 8 × 1 cm3 each This allowed

us to easily vary the field strength from 0.001 T up to

1.7 T by adjusting the distance between the two stacks

The magnetic-field intensity is a function of the gap

be-tween the two stacks of magnets and was measured using

a 1 mm2calibrated Hall effect probe The field

homogene-ity in the sample region has been explored and the field

lines distortions exhibit a standard deviation less than 1%

over a 1 cm3 volume at 1 T The magnetic field could be

applied in a direction either perpendicular or parallel to

the X-ray beam The combination of the two

magnetic-field orientations has allowed us to completely explore the

reciprocal space of the goethite suspensions

3.2.2 Interpretation of the SAXS patterns

The X-ray intensity scattered at small angles by

ly-otropic nematic suspensions of rod-like particles very often

arises from interferences among particles perpendicularly

to their main axis A diffuse peak is then observed that

corresponds to the liquid-like positional order of the

par-ticles in the plane perpendicular to the director (n) The

position of this peak usually scales as φ 1/2 and gives the

average distance between particles More generally, the

distribution of scattered intensity in the reciprocal plane

perpendicular ton is directly related to the Fourier

trans-form of the pair distribution function of the rods The

isotropic phase of goethite suspensions shows the same

qualitative features (apart from the anisotropy) in the

vicinity of the nematic phase, but the positional order has

a smaller range In this SAXS study, we are mostly

inter-ested in the orientation of the particles with respect to the

field, which is directly inferred from the orientation of the

scattering, and in S2, the nematic order parameter S2 is

the second moment of f (θ), the orientational distribution

function (ODF) of the particles (θ is the angle between a

rod and the nematic director) The n-th moment of f (θ) is

S n =



dΩ f (θ) P n (cos θ) , (2)

where Ω = (θ, ψ) is the solid angle and P n the n-th

or-der Legendre polynomial (P0 = 1, P1(X) = X, P2(X) =

3X2−1

2 , etc.) We will mainly use S1 and S2.

Assuming locally-well-aligned rod-like particles

scat-tering in an equatorial torus and neglecting finite-size

ef-fects, Leadbetter et al obtained S2 from an azimuthal

scan I(ψ) of the scattered intensity via the inversion of

the following relation, in a now classical way [27, 28]:

I (ψ) = C (q)

π/2



ψ

cos2ψ

tan2θ − tan2ψ , (3) where C(q) describes the contributions of the positions

and the form factor of the particles In the case of a

dipo-lar symmetry, f (θ) must be replaced by [f (θ)+f (π−θ)]/2.

In order to invert the previous relation, one can assume the

Maier-Saupe form [29, 30] for the ODF:f (θ) = Z1e m cos2θ,

where Z is the orientational partition function and m is a

fit parameter directly related to S2 m can take positive

values, in the case of an usual nematic phase, or negative values in the case of an “antinematic” phase of negative

S2(i.e a phase in which the rods tend to align

perpendic-ular to a given direction [30]) m = 0 corresponds to the

isotropic phase We obtain

I (ψ) = C (q) i erf (

√

m cos ψ) 4π erf (i √

m) cos ψ e

m cos2ψ (m > 0) ,

(4)

I (ψ) = C (q)

erf



i

|m| cos ψ 4π i erf

|m| cos ψ e

m cos2ψ (m < 0) ,

(4) where erf is the error function Fitting the azimuthal scan

of the scattered intensity by these expressions yields m and then S2by using the following relations:

S2= 3

4m

2i √ m

√ πerf (i √

m) e

m − 1

−1

2 (m > 0) ,

(5)

S2= 3

4m





|m|

√ πerf

|m| e m − 1



 −1

2 (m < 0)

(5) Another approach [31], suggested by Deutsch, still re-lies on Leadbetter’s relation but does not involve any

as-sumption of the ODF; it directly relates S2 to the az-imuthal scan through the following relation:

S2= 1−

3

π/2

0 dψI (ψ)

 sin2ψ + sin ψ cos2ψ ln



1+sin ψ cos ψ



2

π/2

0 dψI (ψ)

.

(6)

A comparison between these two methods shows that

they actually give the same values of S2 (± 0.01

differ-ence), well within the error bars of± 0.05, as long as the

signal-to-noise ratio is good enough

3.3 Magnetic-field–induced birefringence experiments

The most sensitive method to measure linear

birefrin-gence, ∆n, of the samples as a function of the volume

a) b) c) d)

B

-200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 215

220

225

230

235

240

245

250

255

Azimuthal angle

60 80 100 120 140 160 180 200 220 240 260 280 300 0

50 100 150 200

Azimuthal angle

Fig 4 Small-angle X-ray scattering (SAXS) patterns of an isotropic suspension (φ = 5.5%), recorded with the magnetic

field perpendicular to the X-ray beam (a-d) and parallel to the beam (e-h) at different field intensities: a), e): B = 0 T;

b), f):B = 0.25 T; c), g): B = 0.4 T; d), h): B = 1.4 T i), j): Azimuthal scans (solid square symbols) of the scattered intensity

in b), d), respectively, and their fits by equations (3, 4) (see text) shown as examples

fraction and of magnetic-field intensity and frequency, uses

a setup with a photoelastic modulator, as already

de-scribed [32, 33] This setup consists of the following

ele-ments: a He-Ne laser source (λ = 633 nm), a vertical

po-lariser, a photoelastic birefringent modulator whose main

optical axis lies at 45◦from the vertical direction and

oscil-lating at a frequency f = 50 kHz, the sample of thickness

d immersed in a horizontal magnetic field applied

perpen-dicular to the light beam, an analyser rotated by 45◦, and

a photomultiplier A lock-in amplifier measures the

com-ponent of the photomultiplier signal If at the modulation

frequency, which is related to the birefringence by the

fol-lowing equation:

If = I0 sin2π∆nd

where I0 is kept constant and determined through

cali-bration of the experiment We have checked that the

in-fluence of the linear dichroism can be neglected in the

re-lation between the signal Ifand the birefringence, for field

intensities up to 800 mT at the highest volume fraction

(φ = 5.5%) (The linear dichroism was measured

sepa-rately with a very similar setup.) The magnetic field was produced by three different magnets We first used small coils in Helmholz configu-ration, which produced a constant field of about 6 mT

and a 20 µs characteristic switching time, to evaluate the

relaxation time of the angular distribution of the particles

We also used an electromagnet for producing an a.c field with a sawtooth-like time variation, at a frequency low enough (0.02 Hz) to consider that the orientational dis-tribution function was always at equilibrium Then, the birefringence evolved in phase with the field The birefrin-gence was plotted as a function of the magnetic-field inten-sity To measure the birefringence at a higher frequency,

we used a nitrogen-cooled coil that produced fields up to

27 mT at 1 kHz At high enough frequency, we observed that the birefringence saturates We then measured its value as a function of the rms field intensity

4 Results

The isotropic phase of goethite suspensions is easier to

study than the nematic phase for three main reasons:

i) the orientation and relaxation times are far shorter in

the isotropic phase, ii) the nematic texture needs to be

well defined by removing topological defects iii) the

ne-matic anchoring at the surfaces of the sample must be

controlled, at least to some extent In other words, the

isotropic phase is usually homogeneous and is much less

sensitive to surface effects than the nematic phase

4.1 Study by SAXS of the orientation reversal upon

magnetic-field increase

The SAXS patterns of a sample of an isotropic

suspen-sion, at different field intensities and for both parallel and

perpendicular configurations, are shown in Figure 4 We

consider here an isotropic suspension at phase coexistence

(φ = 5.5%) because it is the most concentrated one and

therefore it shows the largest effects All X-ray scattering

patterns recorded with the magnetic field parallel to the

X-ray beam are actually isotropic, demonstrating that the

phase keeps uniaxial symmetry around the magnetic-field

direction at all field intensities Let us now examine the

SAXS patterns recorded in the perpendicular

configura-tion As expected, in the absence of field, the scattering

pattern is also isotropic In contrast, at low field

inten-sity (250 mT), the scattering pattern becomes anisotropic

The diffuse ring due to the liquid-like positional order of

the nanorods concentrates in the vertical direction, i.e.

perpendicular to the field direction This shows that the

nanorods then tend to point along the field direction

Un-expectedly, upon a further field increase (around 400 mT),

the SAXS pattern becomes isotropic again Moreover, at

still higher field intensities (1400 mT), the diffuse ring is

very much aligned along the horizontal direction, which

proves that the nanorods strongly tend to orient

perpen-dicular to the field At this stage, the SAXS pattern looks

quite like that of a nematic phase and the optical texture

of the sample is still completely homogeneous To the best

of our knowledge, this is the first example of what is

some-times called an “antinematic” phase, but it is induced here

by the field Moreover, we do not observe the nucleation

of any other phase

The values of the nematic order parameter, S2, were

extracted from the SAXS patterns, as a function of field

intensity (Fig 5) S2 increases from zero in zero-field to

reach a maximum of about 0.05 at B ≈ 250 mT S2 then

decreases back to zero at B ≈ 400 mT, takes negative

val-ues beyond, and reaches about −0.35 at 1.4 T More

di-luted (down to φ ≈ 1%) isotropic suspensions display the

same qualitative behaviour but with smaller absolute

val-ues of S2 All these suspensions displayed isotropic SAXS

patterns for the same value (around 400 mT) of the

mag-netic field

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 -0.4

-0.3 -0.2 -0.1 0.0

lock-in setup compensator SAXS

B (T)

Fig 5 Evolution of the nematic order parameter S2 of an isotropic suspension (φ = 5.5%) versus field intensity,

mea-sured with three different techniques: with SAXS, with a mi-croscope using a compensator, with the field-modulation tech-nique described in the text

4.2 Measurement of the field-induced birefringence

4.2.1 Determination of the specific birefringence

For a dilute suspension (φ < 1%) of particles, much

smaller than the wavelength of light, the birefringence is given by

where ∆nsatis the specific birefringence [34] The specific birefringence can be estimated from the expression

∆nsat=ns

2



n2b − n2 s

n2s+ N b (n2b − n2

s)

−1

2

n2a − n2 s

n2s+ N a (n2a − n2

s)+

n2c − n2 s

n2s+ N c (n2c − n2

s)



, (8)

where ns is the refraction index of the solvent and N a,

N b , and N c are the depolarising factors [35] that can be

calculated by considering the particles either as ellipsoids

or parallelepipeds The values obtained in both cases are

very similar: N a = 0.28, N b = 0.02 and N c = 0.70 for ellipsoids of dimensions 150, 25 and 10 nm; N a = 0.29,

N b = 0.05 and N c = 0.66 for parallelepipeds of the same

dimensions Therefore, comparable values are obtained for

the specific birefringence: ∆nsat = 0.71 for ellipsoids and

∆nsat = 0.64 for parallelepipeds.

The specific birefringence was also estimated by mea-suring the optical path difference introduced by anematic sample (φ = 8.5%) held in an optical flat glass capillary

of thickness e = 20 µm, submitted to a magnetic field of intensity B = 110 mT The measurement was performed

in white light (of average wavelength 550 nm), and gives

e∆n = 1286 ± 4 nm The nematic order parameter of this

sample was independently determined by X-ray

scatter-ing: S2 = 0.95± 0.05, which yields: ∆nsat = 0.80 ± 0.04 in

reasonable agreement with the values predicted above

4.2.2 Birefringence measurement with an optical

compensator

The main interest of this method is that the homogeneity

of the samples can be directly checked while performing

the measurement of the birefringence, using a Derek

com-pensator, under the microscope As expected, the

evolu-tion of the nematic order parameter follows that already

observed by SAXS (Fig 5) The birefringence measured

at B = 900 mT (∆n = −0.016±0.002) is actually huge for

an isotropic phase This is due not only to the large

spe-cific birefringence and volume fraction of the suspension,

but also to its large nematic order parameter S2=−0.35

(S2 saturates at−0.5, in this orientation).

Let us briefly discuss the order of magnitude of the

birefringence At low field, the field-induced birefringence

is proportional to B2, as usual We find that the

coeffi-cient ∆n/B2 = 0.03 T −2 in the case of goethite

suspen-sions, compared to 1.5 × 10 −7T−2 for suspensions of the

Tobacco mosaic virus [36], 1.1 × 10 −7T−2for suspensions

of the fd virus [37] and 6.5 × 10 −3T−2 for suspensions of

V2O5ribbons (unpublished data) The field-induced

bire-fringence is therefore some 5 orders of magnitude larger for

goethite suspensions than for usual suspensions of organic

rod-like particles

4.2.3 Orientation kinetics of the isotropic suspensions

Before using the magnetic-field modulation technique in

order to measure the sample birefringence, it is first

nec-essary to check that the orientation kinetics of the

suspen-sions are fast enough to follow the field sweeping rate

Per-manent magnets were used to apply a constant field giving

rise to a large birefringence A small superimposed

tran-sient field, created by Helmholtz coils, allowed us to

esti-mate birefringence decay times larger than 1 ms

Exper-iments were performed on samples in the isotropic phase

at coexistence (φ = 5.5%) which are the most viscous

Ex-ponential decays of the birefringence were recorded upon

a small drop of the magnetic-field intensity from 43 to

37 mT (data not shown) The time constant of the

sus-pension was measured to be τ = 10.2±0.3 ms This means

that, in order to perform field sweeps with about 500 data

points per cycle, the field frequency has to be much less

than 0.2 Hz for the system to remain in quasi-static

con-ditions

4.2.4 Birefringence measurements with the magnetic-field

modulation setup

Compared to the previous technique using an optical

com-pensator, this setup gives much more accurate

measure-ments of the optical path difference (e∆n, where e is the

sample thickness), allowing us to measure a birefringence

variation as small as ∆n = 10 −8 for a 100 µm thick

capil-lary The birefringence of the suspensions was measured as

a function of the magnetic-field intensity at various

con-centrations in the isotropic phase (0.001 < φ < 0.055).

-0.4 -0.2 0.0 0.2 0.4 -0.5

0.0 0.5

1.0

φ =0.61%

φ =2.0%

φ =5.5%

B (T) Fig 6 Birefringence curves (at various volume fractions)

mea-sured with the optical-modulation technique versus field

inten-sity, rescaled in order to show their superposition over a decade

of volume fraction

-200 0 200

400

0.02 Hz

4 Hz

40 Hz

400 Hz

t/T (fraction of period)

Fig 7 Birefringence curves (φ = 5.5%) measured with the

field modulation technique, versus time, at different frequencies

(T is the period of the magnetic field), at constant Beff =

35 mT

Whatever the concentration was, all the curves looked sim-ilar (Fig 6) Moreover, considering the approximations in-volved in the derivation of equations (5–7), all three tech-niques,i.e magnetic-field modulation setup, optical

com-pensator and X-ray scattering show the same behaviour

of the nematic order parameter (Fig 5) (In Fig 5, the small discrepancies between curves obtained by different techniques might be due to the very different durations of the experiments: 1 h with the optical compensator, 15 s with the lock-in amplifier, 10 min for X-ray scattering,

and to the different sizes of the samples: 20 or 50 µm Also, the measures were made at λ = 633 nm with the

modulation setup and in white light with the optical com-pensator.) The curves can be rescaled by a factor that depends on the concentration and that strongly increases

at the isotropic/nematic phase transition At small fields,

the birefringence scales as B2, as expected for this class

of lyotropic nematic phases [36, 38]

We have also performed birefringence measurements

at various frequencies (0.02 Hz < f < 1000 Hz) at a small magnetic-field intensity, B = 35 mT Since the

0.0 0.2 0.4 0.6

0.0

0.1

0.2

B (T)

2 k g

Fig 8 Magnetisation (per kg of dried mass) versus field

in-tensity of a nematic suspension (φ = 8.5%) frozen either in a

1 T field (solid squares) or in a 0.1 T field (solid circles) as

measured with the SQUID magnetometer Straight lines are

linear fits to the data

birefringence rise and decay happen on a time scale of

about 10 ms, we expect that the particles will not have

time to follow the field at high frequencies, which should

induce a regime of constant birefringence Indeed, the

modulation amplitude of the birefringence decreases as

the frequency increases and becomes even negligible

be-yond 400 Hz (Fig 7) At the same time, the curves

un-dergo a phase shift upon increasing frequency (The

cusp-like shape of the birefringence curve at 0.02 Hz is due

to the quadratic dependence of the birefringence on B,

in quasi-static conditions, whereas the rounded shapes of

the other curves are due to the nanorod reorientation time

lag.) These attenuation and phase shift are typical of an

intrinsic dynamical phenomenon that cannot follow the

field variation Moreover, the continuous component of

the birefringence decreases with the frequency; it becomes

negative beyond 20 Hz and does not change any more

be-yond 400 Hz Therefore, the particles that were aligned

parallel to the field at low frequency (∆n > 0) reorient

perpendicularly to the field at high frequency (∆n < 0).

The birefringence was also measured as a function of the

field for various concentrations at 400 Hz It is now

nega-tive but it still scales as B2 at low fields and diverges at

the I/N transition.

4.3 Static magnetic measurements

4.3.1 Magnetic anisotropy

The origin of the puzzling behaviour described above

clearly lies in the individual magnetic properties of the

goethite nanorods, because this behaviour is observed in

the isotropic phase even at high dilutions The magnetic

properties of the nanorods were thus investigated with a

SQUID magnetometer in the nematic phase In order to

measure anisotropic properties and to prevent the

parti-cles from realigning in the field, the solvent (i.e water)

was frozen below 273 K in various field intensities

Fig-ure 8 shows the phase magnetisation versus field

inten-sity of a sample (m = 7.9 mg) of a nematic suspension (φ = 8.5%) frozen in a 1 T field At such a field intensity,

all the nanorods are oriented perpendicularly to the field The curve obtained is a straight line that extrapolates to the origin and its slope represents the perpendicular sus-ceptibility of the phase The curve recorded on the same suspension frozen in a 0.1 T field is also a straight line but its slope is smaller and it clearly does not extrapolate

to the origin Its slope roughly represents the parallel sus-ceptibility of the phase These linear behaviours show that the anisotropy energy is very large; the spin-flop transi-tion is known to occur at very high fields and was never reached in our samples Taking into account the values

of the nematic order parameters in both orientations and the dependence (assumed linear) on temperature, we es-timated the value of the anisotropy of magnetic

suscepti-bility: ∆χ ≈ −3 × 10 −4 It is very important to note that this quantity is negative

4.3.2 Remanent magnetic moment The remanent magnetisation that is measured in the par-allel orientation is in fact smaller than the sum of the re-manent moments of all the nanorods Indeed, even though

the moments are roughly all parallel (S2 ≈ 0.95), they

can-not all point in the same direction for entropic reasons In the classical Langevin description, it can be shown that

the remanent magnetisation M of the suspension is

pro-portional to 

µ2

B/kBT , where   is the average on the size distribution P (ν) We measured M ≈ 4 × 10 −8Am2, which leads to (µ2) 1/2 ≈ 1.43 × 10 −20Am2 ≈ 1500 µB, where µB is the Bohr magneton In this respect, the size polydispersity should also be considered We have seen in Section 2.3 that the nanorod size polydispersity can be modelled by a Gaussian distribution of standard

devia-tion ∆ν = 0.4 Moreover, following N´eel [23], we assume that the nanorod remanent moment, µ ν, is due to

non-compensated surface spins and that it therefore scales as

µ ν = µν2 Then, we find



µ2

=

∞

 0

dν P (ν) ν4µ2 = 2.05 µ2,

µm=µ =

∞

 0

dν P (ν) ν2µ = 1.17µ ,

where µ is the moment of a nanorod of average dimensions The average moment is then µm ≈ 1300 µB and the

mo-ment of a nanorod of average dimensions is µ ≈ 1100 µB.

Actually, these values are quite comparable to that of the magnetisation induced by a magnetic field of magnitude

B = 0.1 T, i.e χVmB/µ0≈ 800 µB.

4.3.3 First moment of the ODF

A major consequence of the existence of a nanorod re-manent moment is that both the nematic and isotropic