conference papers
Reordering of magnetic colloid structures in external magnetic fields
^{a}HahnMeitnerInstitut Berlin, Glienicker Strasse 100, D14109 Berlin, Germany
^{*}Correspondence email: heinemann@hmi.de
Fieldinduced local ordering within a cobaltbased magnetic colloid has been studied with polarized smallangle scattering by turning the direction of the external magnetic field. By analysing the nuclear–magnetic cross term we were able to separate the scattering contributions from dipolar chains and domains of pseudocrystalline hexagonal arrangements. The magnetic moments were found to be aligned along the external magnetic field and to follow the change of the field direction. The observed variation of the particle moments must give rise to a rearrangement of the local hexagonal structure with respect to the new field direction to explain the scattering pattern. We confirm this by twodimensional simultaneous leastsquares fits for different Q ranges [where , λ is the Xray wavelength and 2θ is the scattering angle] with a model containing a core–shell particle form factor and a structure factor composed of Gaussian peaks and contributions from polydisperse cylinders.
Keywords: smallangle neutron scattering; polarized neutrons; magnetic scattering; magnetic liquid structure; ferrofluids.
1. Motivation and experimental setup
Smallangle neutron scattering with polarized neutrons (SANSPOL) is an excellent tool to obtain insights into the inherent and magnetic fieldinduced nanostructure of magnetic liquids (Wiedenmann et al., 2003). In some concentrated Cobased ferrofluids, well defined peaks appeared in the twodimensional scattering pattern which have been assigned to a local pseudocrystalline hexagonal ordering (see Fig. 1). In Wiedenmann et al. (2003), the observed SANS pattern is interpreted as an arrangement of nanometresized core–shell particles in hexagonal planes where the magnetic moments are aligned along the external field direction. Furthermore, in other samples (Wiedenmann & Heinemann, 2005), the possible coexistence of hexagonal ordered domains with segments of dipolar chains can explain the observed scattering patterns. When the strength of the external magnetic field was increased, this model applied to the experimental data leads to the conclusion of a continuous transition from chainlike aggregates to layered structures. This interpretation, based on common SANS and SAXS experiments, is not unique. We have tried to eliminate some of the ambiguities by a magnetic fielddirection variation study. It was not clear whether the local particle ordering is conserved while the magnetic moments follow the change in the external field, or if reordering of the particles occurs in a new structure defined by the new field direction.
A horizontal magnet was installed on a turntable to vary the angle between the incoming neutron beam and the external field , both spanning a plane perpendicular to the detector. By turning the magnet ( rotation), both the field and the sample change their orientations with respect to the neutron beam. The sample, a Cobased magnetic liquid, MFT3 (the same as used in Wiedenmann & Heinemann, 2005), with mean particle core radius R = 4.4 nm coated by a layer (D = 1.9 nm) of oleoylsarkosine as surfactant and dispersed in an oil, L9, could be rotated independently ( rotation). We performed three combinations of these rotational degrees of freedom and analysed the twodimensional scattering pattern, taking into account all influences of the varying orientation between the vector [, with the Xray wavelength λ and the scattering angle 2θ] and .
2. Twodimensional SANS and SANSPOL analysis
In this type of experiment, the change in the external magnetic field , which likewise defines the neutron spin direction, has different influences on the scattering pattern. Generally, only the magnetic contrast perpendicular to the vector contributes to the coherent scattering. A change in the external direction may generate a change in the local magnetization direction and so in the magnetic contribution to the total scattering. For a dilute system of particles with neither permanent nor induced interparticle correlations, this can be incorporated using an adapted law. Anisotropic magnetic structures with a local magnetization direction and symmetry axis or axis of preference lead to a more complicated dependence of the scattering cross section from the interplay of this axis with , and . In the following, we use the definitions shown in Fig. 2. For instance, in order to observe contributions from Bragglike scattering of magnetic correlations, it is not only necessary that the correlation direction is within the detector plane, but also that it must have a nonvanishing component perpendicular to the respective vector represented by a detector cell. When polarized neutrons are used to investigate anisotropic magnetic structures, the situation is even more complex. Here, a possible divergence of the local magnetization and the external magnetic field can lead to additional variations in the SANSPOL intensities.
2.1. SANSPOL for uncorrelated systems
For a polarized neutron beam, let P denote the degree of polarization and the spinflipper efficiency. If the polarization of the scattered neutrons is not analysed, one can distinguish between I^{+} and I^{} scattering by switching a spin flipper in front of the sample on and off. Here (see e.g. Heinemann & Wiedenmann, 2003), I^{+} evaluates to:
The or spin direction is defined by the external field (F_{N} is the nuclear scattering amplitude and F_{M} is the magnetic scattering amplitude). Thus, it appears that the interference term in equation (1) depends on the relative orientations of , and , whereas the second term in equation (1) is identical to the magnetic scattering for nonpolarized neutrons and is affected only by the angle between and . We will focus here on the result that a deviation of the local magnetization direction from the external field direction is at least detectable via the interference term using polarized neutrons. For superparamagnetic particles, such as for the nanosized core–shell particles in magnetic fluids used in our experiment, equation (1) evaluates to:
I^{} is obtained from equation (2) for . is the Langevin function, with (k_{B} is Boltzmann's constant and T is temperature). Using the definitions in Fig. 2, the cross term (intensity difference) reads
In the special case where is aligned along , equation (3) is simplified to:
2.1.1. Interparticle correlations: the structure factor S(Q)
Interparticle correlations are commonly taken into account in SANS cross sections by a structure factor . This general approximation is based on the splitting of the total intensity into contributions from isolated scatterers and from interparticle interference using different types of approach (Pedersen, 1994). We compare the results from a local monodisperse model,
and a decoupling model based on
and find no significant differences in the data fitting procedure. Thus, we present the results from the simpler local monodisperse approximation. In equations (5) and (6), is an abbreviation for the squared amplitude average and the amplitude average itself over the particle size distribution N(R). For the form factor F(Q), we used a polydisperse core–shell model, which was very successful for parameter determination of dilute samples [see e.g. Heinemann & Wiedenmann (2003) and references therein].
The model for the structure factor is composed of two different contributions. The Bragglike peaks from the pseudocrystalline structure are described by a Gaussian profile of the form
where the define the peak positions and the related peak widths. After correction for instrumental resolution, the widths are correlated to the mean sizes of the ordered domains by the Debye–Scherrer equation . The second contribution describes the scattering at low Q values, which dominates the twodimensional images for the sample–detector (SD) distance at 12 m (low Q range) and which shows the typical scattering pattern of long but small objects aligned along the field direction. We model this contribution as a structure factor constructed of polydisperse cylinder scattering amplitudes (Guinier & Fournet, 1955) via
with
where dN_{L} and dN_{R} indicate integration over the normalized length and radius distributions. J_{1}(z) is the firstorder Bessel function, and and are the moduli of the scattering vector parallel and perpendicular to the orientation axis, respectively. This nonunique choice turned out to be the simplest model with the capability of reflecting the observed scattering pattern for small Q values. Twodimensional fitting with this double integral in equation (9) is very time consuming and not practicable. With the assumption of a gammatype size distribution (Heinemann et al., 2000) for the radius and an independent cylinderlength size distribution of the same type, we obtained an analytical result for in equation (9):
where H2F1 is an abbreviation for the hypergeometric function (see e.g. Weisstein, 1999), L is the mean cylinder length and its standard deviation, and R is the mean radius and its standard deviation. In this approach the distributions for the radii and lengths of the cylinders are totally decoupled. To exclude parameter correlations in the leastsquares fit as far as possible, we applied constraints to the parameters. For instance, the mean cylinder radius and the mean singleparticle radius are connected to vary in the same direction during the fit.
3. Experimental results
The following discussion will reveal possible traps during a straightforward interpretation of the scattering pattern obtained and shows ways out of these. As mentioned in §2.1, only the magnetic component perpendicular to leads to a scattering contribution. If we turn the magnetic field with respect to the fixed detector, identical detector cells {R_{a},R_{b},0} defining a special vector collect different magnetic scattering contributions even without any change in the sample structure. The strong influence of this effect is illustrated in Fig. 3. This seems to suggest that by turning the magnetic field the scattering intensity for detector cells at small angles with respect to the b axis increases, but the reason is that equal cells now collect different magnetic scattering contributions. The implication of this effect is clearly demonstrated by rescaling the intensities with the change of the magnetic contribution () and plotting the intensities versus , now assuming that the moment direction of the particles corresponds to the direction of the external field. If the particle arrangement were still the same at any angle , the scaling of the intensities in Fig. 3 by should lead to one single master curve for all values of . The rescaled intensities in Fig. 4 show that this is obviously not the case. While the central peak at 90° is not affected by the tilting, clear differences appear at = 30 and 150°, corresponding to the hexagonal peaks. It is now evident that the intensity is not only influenced by the direction of the magnetic moment but also by a reordering of the particles. Fig. 4 shows clearly that by turning the external magnetic field the detector image cannot provide us with complete information about the anisotropic angle dependence of the magnetic scattering. This kind of processing and presentation of the data gives good insights into the possible changes of a structure factor which follows directly the magnetic field direction, like the S_{cyl} in our model, while for S_{hkl} it does not. The reason is that the appearance of peaks in the detector is a result of fulfilling the Laue conditions for a special vector. This leads to the Q and dependence of P_{hkl} in equation (7), whereas F_{cyl}, which defines S_{cyl} by equation (8), depends on via .
It is obvious that a onedimensional presentation of the data can hardly show all aspects of the influence of the magnetic field direction on the different anisotropic structure factors. For the same reason, onedimensional data fitting methods will show application limitations here. Besides the different angle dependence of the two structure factors, they have different strong influences on the total scattering for different Q ranges, i.e. different sample–detector distances. This forces us to apply a twodimensional datafitting method with simultaneous fitting of the SD = 4 m and SD = 12 m experimental results. By doing this we obtain not only parameter values for the single particles, but also very reasonable values for the structure factor parameters (see Table 2). Within the error bars, the parameters describing the single particles are the same as obtained by fitting experimental results from dilute samples of the same type (see Table 1) fitted using a different program. Figs. 5 and 6 show comparisons of the experimental data and the simultaneous fitting results of the complete set of parameters as presented in Tables 1 and 2. In Fig. 6, only the influence of the cylindrical structure factor is relevant to explain the data. We assume that this structure factor represents the influence of chainlike structures in the ferrofluid, giving rise to lowQ anisotropic scattering. In Fig. 5, we observe an interplay between the assumed hexagonal pseudocrystalline structure and the chainlike correlations. It was essential to apply a simultaneous fit to the two different Q ranges, otherwise the procedure runs into local minima for the SD = 4 m data which cannot explain the SD = 12 m data.


4. Influences of the H field direction
After obtaining the parameter values for the single particles and the structure factors for the = 0° case, we study the influence of the change of the magnetic field direction on the structure. As long as our model of chainlike structures represented by cylindrical structure factors is applicable and the chains follow a change of the magnetic field direction, the influence on S_{cyl} is well known by substituting in equation (10). Now we look at the contribution from the fieldinduced hexagonal pseudocrystalline structure, which was found to disappear in zero field. If we assume that the particle ordering itself follows the direction of the external magnetic field, the Laue conditions for the reciprocal pseudolattice vectors are no longer fulfilled for the Q vectors within the detector plane. For longrange ordered structures, like peaks would disappear instantaneously by turning , and hence the reciprocal lattice, out of the detector plane. For real peaks broadened by the finite size of the ordered domains, the observed intensity is the result of a convolution of the peak width with the Ewaldsphere conditions. The main effect of the convolution must result in a decrease of the peak height, since only part of the full twodimensional peak is cut off in the detector plane when is increased from 0°. Due to the large polydispersity, no change of the observed positions and widths of the peaks could be detected. The width of the peaks in and was fixed to the value obtained at = 0. Therefore, in a simultaneous fit of all data for the different directions, we fixed all structure parameters and fitted only the height of the 100, , 010 and hexagonal peaks. Because the change of the magnetic field direction will not affect the symmetry for the and peaks, this contribution is fixed too.
Fig. 7 shows the S_{hkl} part of the fits for different orientations of the magnetic field. A rotation = 10° of the magnetic field with respect to the detector plane leads to a significant drop in intensity at the peak positions mentioned above. The loss of intensity continues by turning the H field more and more in the neutron beam direction (see Table 3). After reaching = 60°, there is no longer any evidence of these peaks in the fit. This behaviour clearly shows that both the nuclear and magnetic ordering are induced by the magnetic field. The particle arrangement in hexagonal layers is fully determined by the direction of the external magnetic field, as well as by the moment direction of the particles.

5. Conclusions
By turning the direction of the magnetic field, we were able to separate consistently with our model the scattering contributions resulting from dipolar chains and those from domains of pseudocrystalline hexagonal arrangements. An arrangement of particles in chainlike structures is manifested by explaining the data with a cylindrical structure factor. We also obtained values for the length of these chains of about nm. The thickness fits well to the singleparticle properties, as shown in Table 1. The simultaneous fits of the twodimensional patterns are consistent with the picture of local hexagonal ordering of the nanoparticles induced by the external magnetic field. The results show unambiguously that the magnetic moments are aligned along the external magnetic field. This reorientation of the particle moments along the new direction of must give rise to a rearrangement of the local structure with respect to the new field direction. The correlation length of the ordered domains induced by the external magnetic field was derived from the actual peak widths (see Table 2) at about 105 nm. Further work has to be done to apply this model to other magnetic fluids and to exclude a variety of alternative hypotheses, e.g. scattering from ordered aggregates. Very recent results (Klokkenburg et al., 2006) with nonscattering methods support the present model.
Acknowledgements
This project was supported by the German Research Foundation (DFG), project No. Wi 1151/2.
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