physical properties of aqueous suspensions of goethite nanorods

DOI 10.1140/epje/i2003-10078-6 Eur. Phys. J. E 13, 291–308 (2004) THE EUROPEAN PHYSICAL JOURNAL E Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods Part I: In the isotropic phase B.J. Lemaire 1 ,P.Davidson 1,a ,J.Ferr´e 1 , J.P. Jamet 1 , D. Petermann 1 , P. Panine 2 ,I.Dozov 3 , and J.P. Jolivet 4 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 com- 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 V 2 O 5 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. 292 The European Physical Journal E 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(NO 3 ) 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 HNO 3 in order to electrostatically charge the surface of the particles by proton adsorption. Finally, the suspension is rinsed three times to bring the pHback 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-Fe 2 O 3 , a very stable and common iron oxide) were found. The crystallographic structure of goethite can be represented in the Pnma 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 the c-direction and the Fe 3+ 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 Fe 3+ cations, oriented in the b-direction. These double chains are linked to adjacent ones by corner- sharing and hydrogen bonds. Electron microscopy images and in 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,andc 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 pHac- cording to an already described method [11,13,14]. At low pH, hydroxo –OH groups adsorb protons 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 ≈ 0.2C·m −2 .ThepH 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 −2 M in the nematic phase and (3.0±0.5)×10 −2 M 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 B.J. Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) 293 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: L hkl = 0.9λ ∆(2θ)cosθ 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 hkl is 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 10l cos  arctan  lc 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)isequalto1(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 )| 2   ρ e g − ρ e 0  2 I e , Fig. 2. Transmission electron microscopy (TEM) image of a diluted goethite suspension. where N is the number of particles, ρ e g and ρ e 0 are the electron densities of goethite and water respectively, I e 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 inter-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 294 The European Physical Journal E 0 100 200 300 0.0 0.1 0.2 0.3 fraction (%) Length L b (nm) 0 1020304050 0.0 0.1 0.2 0.3 b) a) Width L a (nm) fraction (%) 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± 5nm. 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: P (ν)= e −(ν−1) 2 /(2∆ν 2 ) ∞  0 dνe −(ν−1) 2 /(2∆ν 2 ) , for ν>0 , P (ν)=0, for ν<0 . (1) 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: s m = s 0 ∞  0 dνP(ν)ν 2 , where s 0 =2(L a L b + L a L c + L b L c )=1.1 × 10 −14 m 2 is the surface of a particle of average dimensions (ν = 1). We find s m =1.17s 0 =1.3 ×10 −14 m 2 . The average particle volume V m is obtained in a similar way: V m = V ∞  0 dνP(ν)ν 3 , where V = L a L b L c =3.7 × 10 −23 m 3 is the volume of a particle of average dimensions (ν = 1). We find V m = 1.5V =5.6 × 10 −23 m 3 . 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 T N [20, 21], above which the material becomes paramagnetic, varies between 325 and 400 K. For the particles considered in this work, T N ≈ 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 T N . 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. B.J. Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) 295 Table 1. Crystalline goethite Density ρ g 4370 kg · m −3 Molar mass M g 88.85 g · mol −1 Optical indices (at 632.8 nm) n a 2.24 n b = n c 2.38 Goethite nanorods Dimensions Length L b 150 ± 25 nm Width L a 25 ± 10 nm Thickness L c 10 ± 5nm Polydispersity distribution P (ν) Gaussian Standard deviation ∆L i /L i 0.4 Average volume V m 5.6 × 10 −23 m 3 Average surface s m 1.3 × 10 −14 m 2 Magnetic susceptibility χ (295 K) (1.7 ± 0.2) × 10 −3 Goethite suspensions pH3 Ionic strength Isotropic phase (3.0 ± 0.5) × 10 −2 M Nematic phase (4.5 ± 0.5) × 10 −2 M Electrical surface charge σ 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 directions 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 ∆n int =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 φ N and φ I , at thermodynamic equilibrium. φ N is the smallest volume fraction that can be observed in the nematic phase, whereas φ I 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 φ N =8.5 ± 0.5% and φ I =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 296 The European Physical Journal E divergence and a typical camera length of 10 m. Beam size at the sample position was about 0.1×0.1mm 2 . 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  0.6nm −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 × 1cm 3 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 a1mm 2 calibrated 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% overa1cm 3 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 to n 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 S 2 , the nematic order parameter. S 2 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 =  dΩf(θ) P n (cos θ) , (2) where Ω =(θ, ψ) is the solid angle and P n the n-th or- der Legendre polynomial (P 0 =1,P 1 (X)=X, P 2 (X)= 3X 2 −1 2 , etc.). We will mainly use S 1 and S 2 . Assuming locally-well-aligned rod-like particles scat- tering in an equatorial torus and neglecting finite-size ef- fects, Leadbetter et al. obtained S 2 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 dθ sin θ f (θ) cos 2 ψ  tan 2 θ − tan 2 ψ , (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)= 1 Z e m cos 2 θ , where Z is the orientational partition function and m is a fit parameter directly related to S 2 . 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 S 2 (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)cosψ e m cos 2 ψ (m>0) , (4) I (ψ)=C (q) erf  i  |m|cos ψ  4πierf   |m|  cos ψ e m cos 2 ψ (m<0) , (4  ) where erf is the error function. Fitting the azimuthal scan of the scattered intensity by these expressions yields m and then S 2 by using the following relations: S 2 = 3 4m  2i √ m √ πerf (i √ m) e m − 1  − 1 2 (m>0) , (5) S 2 = 3 4m   2  |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 S 2 to the az- imuthal scan through the following relation: S 2 =1− 3 π/2  0 dψI (ψ)  sin 2 ψ +sinψ cos 2 ψ 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 S 2 (± 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 B.J. Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) 297 a) b) c) d) e) f) g) h) B B -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 20 215 220 225 230 235 240 245 250 255 i ) Scattered intensity (a.u.) Azimuthal angle 60 80 100 120 140 160 180 200 220 240 260 280 300 0 50 100 150 200 250 j ) Scattered Intensity (a.u.) 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 =0T; b), f): B =0.25 T; c), g): B =0.4T;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 I f at the modulation frequency, which is related to the birefringence by the fol- lowing equation: I f = I 0 sin 2π∆nd λ , where I 0 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 I f and 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. 298 The European Physical Journal E 4Results 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, S 2 , were extracted from the SAXS patterns, as a function of field intensity (Fig. 5). S 2 increases from zero in zero-field to reach a maximum of about 0.05 at B ≈ 250 mT. S 2 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 S 2 . 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 S 2 lock-in setup compensator SAXS B (T) Fig. 5. Evolution of the nematic order parameter S 2 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 ∆n = ∆n sat φS 2 , (7) where ∆n sat is the specific birefringence [34]. The specific birefringence can be estimated from the expression ∆n sat = n s 2  n 2 b − n 2 s n 2 s + N b (n 2 b − n 2 s ) − 1 2  n 2 a − n 2 s n 2 s + N a (n 2 a − n 2 s ) + n 2 c − n 2 s n 2 s + N c (n 2 c − n 2 s )   , (8) where n s is the refraction index of the solvent and N a , N b ,andN 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: ∆n sat =0.71 for ellipsoids and ∆n sat =0.64 for parallelepipeds. The specific birefringence was also estimated by mea- suring the optical path difference introduced by a nematic sample (φ = 8.5%) held in an optical flat glass capillary of thickness e =20µm, 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: S 2 =0.95± 0.05, which yields: ∆n sat =0.80 ±0.04 in reasonable agreement with the values predicted above. B.J. Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) 299 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 S 2 = −0.35 (S 2 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 B 2 , as usual. We find that the coeffi- cient ∆n/B 2 =0.03 T −2 in the case of goethite suspen- sions, compared to 1.5 × 10 −7 T −2 for suspensions of the Tobacco mosaic virus [36], 1.1 ×10 −7 T −2 for suspensions of the fd virus [37] and 6.5 ×10 −3 T −2 for suspensions of V 2 O 5 ribbons (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% ∆ n / max( ∆ n) 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. 0.0 0.2 0.4 0.6 0.8 1.0 -200 0 200 400 0.02 Hz 4 Hz 40 Hz 400 Hz ∆ n (arbitrary unit) 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 B eff = 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 B 2 , 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 300 The European Physical Journal E 0.0 0.2 0.4 0.6 0.0 0.1 0.2 B (T) M (A.m 2 .kg -1 ) 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 B 2 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 (S 2 ≈ 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  µ 2  B/k B T , where  is the average on the size distribution P (ν). We measured M ≈ 4 × 10 −8 Am 2 , which leads to (µ 2 ) 1/2 ≈ 1.43 × 10 −20 Am 2 ≈ 1500 µ B , where µ B 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 = µν 2 . Then, we find  µ 2  = ∞  0 dνP(ν) ν 4 µ 2 =2.05 µ 2 , µ m = µ = ∞  0 dνP(ν) ν 2 µ =1.17µ, where µ is the moment of a nanorod of average dimensions. The average moment is then µ m ≈ 1300 µ B and the mo- ment of a nanorod of average dimensions is µ ≈ 1100 µ B . Actually, these values are quite comparable to that of the magnetisation induced by a magnetic field of magnitude B =0.1T,i.e. χV m B/µ 0 ≈ 800 µ B . 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 [...]... field of low intensity 5.6 Polydispersity As will be seen in the discussion of the results, the polydispersity of the samples must explicitly be taken into account in the model Following Bacri et al in their study of ferrofluid suspensions [34a,43], the birefringence and the magnetization of a suspension of goethite particles can be B.J Lemaire et al.: Physical properties of aqueous suspensions of goethite. ..B.J Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) Induced magnetisation χ⊥H⊥ H M θ 2 M (A.m /kg) 0.02 301 0.01 χ H µ 0.00 0 1 2 3 4 Fig 10 Schematic representation of the remanent and induced magnetic moments of a goethite nanorod B (T) Fig 9 Remanent magnetisation (per kg of dried mass) of an isotropic suspension (φ = 5.5%)... values of S2 induced in polar liquids are usually much smaller and strong fields (electric fields of ∼ 30 V/µm) are needed to induce the so-called paranematic phase [45], even close to the B.J Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) isotropic/nematic transition Secondly, goethite suspensions differ from other systems by the different symmetry of the... properties of the isotropic phase of goethite suspensions in a magnetic field The zero-field magnetisation and the birefringence being respectively proportional to the first and second moments of the orientational distribution function (Eqs (7) and (9)), the model consists in giving expressions of the magnetic energy and of the free energy of a suspension, in deriving the ODF, and finally the moments of. .. We shall describe in Part II the physical properties of the nematic phase and show that they can also be explained within the same set of assumptions Future developments should include the investigation of the magnetic properties of the suspensions around and above the N´el temperature, where the permanent mage netic moment should vanish Size fractionations of the suspensions should also be performed... The fit parameters obtained agree fairly well with the experimental measurements of the magnetic properties of goethite suspensions At this stage, let us try to summarise the most salient features of these very peculiar suspensions S1 , the first moment of the orientational distribution function, arises from the reorientation of the permanent magnetic moments in the external magnetic field This phenomenon... evolutions of the first and ◦ ◦ second moments of the ODF, S1 and S2 , of an isotropic suspension versus magnetic-field intensity depending on the sign of the anisotropy of magnetic susceptibility and the existence of a remanent magnetic moment (Eqs (19) and (20)) These equations give the expression of the birefringence and the zero-field magnetisation of a frozen dilute suspension as a function of the intensity... Conclusion Aqueous suspensions of goethite nanorods, depending on volume fraction, form stable isotropic and nematic phases The isotropic phase has very peculiar magnetic properties because goethite nanorods align parallel to a weak magnetic field but perpendicular to a strong field The magnitude of these effects is remarkable, leading to very large nematic order parameters around 1 T On the basis of previous... B.J Lemaire et al.: Physical properties of aqueous suspensions of goethite (α-FeOOH) nanorods (Part I) Finally, the orientational distribution function takes a Boltzmann-type expression: f (θ) = Eexc (f, Ω) + Em (θ) 1 exp − Z kB T E (f,Ω)+E (θ) − exc k T m B where Z = dΩe tition function, and , (15) ∆χ >0, µ>0 is the orientational par- 5.4.1 Dilute suspensions In the limit of a dilute isotropic suspension,... ν, magnetic moment µν and volume Vν Let us give the scaling properties of µν and Vν Since the remanent magnetism of goethite particles is most often described as a surface effect, we assume that the longitudinal magnetic moment is proportional to the surface area of the particles: µν = ν 2 µ, where µ is the magnetic moment of a particle of average dimensions (and volume V ) Obviously, Vν = ν 3 V . interpretation of the magnetic behaviour of goethite suspensions. B.J. Lemaire et al.: Physical properties of aqueous suspensions of goethite ( -FeOOH) nanorods. birefrin- gence, ∆n, of the samples as a function of the volume B.J. Lemaire et al.: Physical properties of aqueous suspensions of goethite ( -FeOOH) nanorods (Part