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ISSN: 1600-5767

Application of small-angle neutron scattering to the study of forces between magnetically chained monodisperse ferrofluid emulsion droplets

aSchool of Chemistry, University of Sydney, Sydney, New South Wales, 2006, Australia, bInstitute of Materials Research and Engineering, Agency for Science Technology and Research, 3 Research Link, 117602, Singapore, and cNeutron Instrument and Source Design Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
*Correspondence e-mail:

(Received 20 May 2013; accepted 2 November 2013; online 25 December 2013)

The optical magnetic chaining technique (MCT) developed by Leal-Calderon, Stora, Mondain-Monval, Poulin & Bibette [Phys. Rev. Lett. (1994), 72, 2959–2962] allows precise measurements of force profiles between droplets in monodisperse ferrofluid emulsions. However, the method lacks an in situ determination of droplet size and, therefore, requires a combination of separately acquired measurements of droplet chain periodicity versus an applied magnetic field from optical Bragg scattering and droplet diameter inferred from dynamic light scattering (DLS) to recover surface force–distance profiles between the colloidal particles. Compound refractive lens (CRL) focused small-angle neutron scattering (SANS) MCT should result in more consistent measurements of droplet size (form factor measurements in the absence of field) and droplet chaining period (from structure factor peaks when the magnetic field is applied), and, with access to shorter length scales, extend force measurements to closer approaches than possible by optical measurements. This article reports on CRL-SANS measurements of monodisperse ferrofluid emulsion droplets aligned in straight chains by an applied field perpendicular to the incident beam direction. Analysis of the scattering from the closely spaced droplets required algorithms that carefully treated resolution and its effect on mean scattering vector magnitudes in order to determine droplet size and chain periods to sufficient accuracy. At lower applied fields, scattering patterns indicate structural correlations transverse to the magnetic field direction owing to the formation of intermediate structures in early chain growth.

1. Introduction

Ferrofluids are colloidal dispersions of magnetic nanoparticles, usually of magnetite or cobalt, in a continuous phase, usually an organic solvent or water. Owing to a high magnetic sus­ceptibility, ferrofluids have been termed superpara­mag­netic and exhibit strong structural and rheological responses to applied magnetic fields, which have been used to control these liquids in a great many technological applications [for examples see Odenbach (2009[Odenbach, S. (2009). Editor. Colloidal Magnetic Fluids: Basics, Development and Applications of Ferrofluids. Heidelberg: Springer.])]. Ferrofluid emulsions contain droplets of ferrofluid hundreds of times larger than the constituent nanoparticles as the dispersed phase, and as in conventional emulsions, the individual drops are stabilized by a surfactant dissolved primarily in the continuous phase. Both direct (oil-based ferrofluid-in-water) and inverse (water-based ferrofluid-in-oil) emulsions can be prepared depending on the continuous medium of the ferrofluid. Comparative studies on a direct ferrofluid emulsion and the ferrofluid itself (Montagne et al., 2002[Montagne, F, Mondain-Monval, O., Pichot, C., Mozzanega, H. & Elaissari, A. (2002). J. Magn. Magn. Mater. 250, 302-312.]) demonstrated that superparamagnetic behavior is retained by the emulsion and that the colloidal stability of the ferrofluid is preserved inside the emulsion droplets. As a result, the ferrofluid emulsion droplets may align to form chains and other structural transitions induced by an external magnetic field. A striking consequence of this behavior was reported in a study by Bibette (1993[Bibette, J. (1993). J. Magn. Magn. Mater. 122, 37-41.]) on a concentrated mono­disperse ferrofluid emulsion: when a homogenous exter­nal magnetic field was applied droplet chain alignment resulted in strong Bragg scattering in the visible range. The initially brown emulsion changed in color to red, and then through yellow to green to blue as the applied field was increased.

On the basis of these magnetic and optical properties, Leal-Calderon et al. (1994[Leal-Calderon, F., Stora, T., Mondain-Monval, O., Poulin, P. & Bibette, J. (1994). Phys. Rev. Lett. 72, 2959-2962.]) developed the optical magnetic chaining technique (MCT), enabling the direct measurement of forces between monodisperse ferrofluid emulsion droplets. The attractive forces between magnetic dipoles within the emulsion droplets result in the formation of linear chains parallel to the applied field, from which, knowing the magnetic susceptibility of the ferrofluid droplets, the exact force between them can be directly calculated for a given field (Zhang & Widom, 1995[Zhang, H. & Widom, M. (1995). Phys. Rev. E, 51, 2099-2103.]). In a monodisperse emulsion, measuring the strong scattering of visible light in the backward direction allows the determination of the spatial periodicity of the droplet chains. Combined with an independent measurement of the droplet hydrodynamic radius by dynamic light scattering, this allows determination of the droplet separation distance down to separations of about 5 nm (Dreyfus et al., 2009[Dreyfus, R., Lacoste, D., Bibette, J. & Baudry, J. (2009). Eur. Phys. J. E, 28, 113-123.]). The distance between the droplets reflects the attractive magnetic forces between the ferrofluid within the emulsion droplet and the repulsive forces induced by the stabilizing surfactant at the droplet surfaces. By controlled variation of an external applied magnetic field, the MCT allows direct measurement of the force–distance profile between these thin liquid films at scales of the order of pico- to nano-newtons. Further, while atomic force microscopy and surface forces apparatus micropipette techniques are limited to measuring the interactions between single pairs of emulsion droplets or interfaces, the MCT provides information representative of true emulsion systems, averages the forces across many inter-droplet contacts and enables the complete statistical system to be studied.

One limitation of the optical MCT is that it lacks a method for simultaneously monitoring of particle size and shape with chain formation. This limits the accuracy of determinations of interfacial separation and possible distortion of the droplets as they approach contact, factors which are obviously important for the study of phenomena such as droplet coalescence and its effect on emulsion stability or molecular forces and transfer in possible biomimetic applications (Montagne et al., 2006[Montagne, F., Braconnot, S., Eläissari, A., Pichot, C., Daniel, J. C., Mandrand, B. & Mondain-Monval, O. (2006). J. Nanosci. Nanotechnol. 6, 2312-2319.]). This limitation also becomes critical in emulsions where the stabilizing forces are very short ranged, exemplified by inverse emulsions with small surfactant stabilizers. Recent developments in small-angle neutron scattering (SANS) techniques employing compound refractive lens (CRL) systems have greatly increased neutron current in measurements on the length scales applicable to the MCT, making it feasible to apply small-angle scattering measurements to the technique (Eskildsen et al., 1998[Eskildsen, M. R., Gammel, P. L., Isaacs, E. D., Detlefs, C., Mortensen, K. & Bishop, D. J. (1998). Nature, 391, 563-566.]). Since SANS is sensitive to both the form of emulsion particles and their structural organization, CRL-SANS offers the possibility of more consistent determination of both of these aspects of emulsion chains and details of the process of chain formation. Also, with access to shorter length scales using neutron scattering, CRL-SANS offers the possibility of extending the MCT to closer droplet approach distances.

2. Experimental

2.1. Monodisperse ferrofluid emulsion system

The ferrofluid synthesized for the initial CRL-SANS MCT measurements was a 60 wt% (∼24 vol.%) solution of maghe­mite (Fe2O3) nanoparticles suspended in D2O, emulsion drop­lets of which were stabilized by the surfactant sorbitan monooleate (Span 80) in a continuous phase of tetradecane. To produce a sufficiently monodisperse emulsion sample from this polydisperse starting emulsion, we employed the method of repeated magnetic fractionation (Bibette, 1991[Bibette, J. (1991). J. Colloid Interf. Sci. 147, 474-478.]). A 10 wt% ferrofluid solution of the crude emulsion was magnetically sedimented and washed with a 2 wt% Span 80 in tetradecane solution. Placed in the magnetic field of a solenoid, large droplets having greater dipole interactions relative to the thermal energy will tend to chain more readily than smaller ones; groups of chains once formed will tend to bundle and sediment more rapidly under gravity than single droplets or chains. At a given field, sampling from the bottom solution selects against smaller droplets while sampling near the top selects against larger droplets. Thus, by decantation, separation at a carefully chosen series of field strengths (with staged dilutions with 2 wt% Span 80 in tetradecane solution), a ferrofluid emulsion was produced with a markedly narrower size distribution in the required `optical' size range as indicated by the signature iridescence of these solutions under the applied fields. The final concentration of ferrofluid droplets was somewhat less than 0.01 wt%, suitable for SANS measurements without the risk of significant multiple coherent scattering. The stability of our emulsion sample was checked by dynamic light scattering measurements on a Malvern high-performance particle sizer at 298 K. Over a 12 h period, there was no indication of Ostwald ripening increasing the droplet diameter from the initially measured value of ∼170 nm.

The nominal neutron scattering length density calculated for the D2O–ferrofluid emulsion droplets is 6.8 × 10−6 Å−2 against −0.4 × 10−6 Å−2 for the Span 80–tetradecane phase, giving an overall scattering length density contrast between the droplets and the continuous phase Δβ of ∼7.2 × 10−6 Å−2. Obviously a similar neutron contrast could have been achieved for a light water ferrofluid emulsified in deuterated tetradecane. However, this was not done as the multiple washing and dilution steps with the Span 80–tetradecane supporting solution in the synthesis would make it much more expensive to produce a sufficiently monodisperse ferrofluid emulsion sample from the initial concentrated polydisperse emulsion using d-tetradecane. This economy in the preparation of these initial samples comes at some cost in neutron transmission and a slightly higher detector background owing to incoherent hydrogen scattering.

2.2. CRL-SANS instrument and sample magnet

Measurements on the Span 80 ferrofluid emulsion were performed on the NG3 SANS instrument at the NIST Center for Neutron Research (Glinka et al., 1998[Glinka, C. J., Barker, J. G., Hammouda, B., Krueger, S., Moyer, J. J. & Orts, W. J. (1998). J. Appl. Cryst. 31, 430-445.]) utilizing one of that instrument's CRL sets: comprising 30 biconcave MgF2 lenses positioned just upstream of the sample position, which focus neutrons at a wavelength of 8.4 Å from the 2.54 cm-diameter entrance aperture of the 16 m pre-sample flight path to an image of that aperture at a detector position 13.2 m beyond the nominal sample position [Choi et al., 2000[Choi, S.-M., Barker, J. G., Glinka, C. J., Cheng, Y. T. & Gammel, P. L. (2000). J. Appl. Cryst. 33, 793-796.]; NIST Center for Neutron Research (2002[NIST Center for Neutron Research (2002). NG3 and NG7 30-meter SANS Instruments Data Acquisition Manual. National Institute of Standards and Technology Center for Neutron Research, Gaithersburg, MD, USA.])]. The precision Helmholtz electromagnet (NCNR HM1) that was necessary to perform our MCT measurements on the ferrofluid emulsion sample was installed on a stage nominally 0.55 m upstream of the instrument's standard design sample position, increasing the sample-to-detector distance to about 13.7 m. Note that since this is a focusing system it did not significantly change the direct-beam image size as it would for the normal pinhole SANS camera configuration. Since the entrance aperture, lens system and detector remain placed at their design positions, where the magnification of the lens setup is a little less than unity (the ratio of a lens-to-detector distance of about 14 m to the entrance-aperture-to-lens distance of about 15.5 m), the focused direct beam can be stopped by a 2.54 cm beamstop. In this configuration the nominal minimum scattering vector magnitude (Q) measurable on the instrument's two-dimensional 3He detector (64 × 64 cm active area, 0.5 cm resolution; Ordela Inc., Oak Ridge, TN, USA) is about 0.0010 Å−1. A 1.27 cm-diameter sample aperture defined the beam on the sample solution in a standard quartz `banjo' cell with a sample path length of 1 mm. At a neutron wavelength of 8.4 Å (velocity selector triangular FWHM 14.3%), the measured sample transmission was 43.5%; as noted previously, this is due mainly to incoherent hydrogen scattering in the sample.

For our SANS measurements the HM1 electromagnet provided a vertical applied field, and thus droplet chain direction, perpendicular to the horizontally incident neutron beam. Bragg scattering due to magnetic chaining along this field direction then appears perpendicular to it as horizontal bands of intensity across the two-dimensional detector of the SANS camera. This differs from the configuration of the optical MCT in which spectroscopic measurements are made in a backscattering configuration with the sample in a solenoid and the incident and Bragg scattered beams travel along the axis of the solenoid, parallel to the droplet chains. After initial measurements without magnetic field (0 G) of the ferrofluid emulsion, empty cell and a `background' sample of the Span 80 in tetradecane continuous phase solution, CRL-SANS MCT measurements were taken: the first at an applied field of 205 G, followed by a series reducing the field in stages to 50 G, and then another back to 205 G (1 G = 10−4 T). This range was chosen on the basis of previous optical MCT results on a similar sample and its double crossing was interspersed with short checking measurements at 0 G, to verify reversibility of magnetic chaining in the emulsion and its stability against droplet coalescence during the data acquisition period. All primary scattering runs were acquired for 180 min; the zero field checks for 30 min each. A final shorter data set was also acquired at a stronger applied field of 307 G. The experiments were performed at ambient neutron guide hall temperature (∼295 K).

3. Analysis

3.1. Scattering from ferrofluid droplet chains under an applied magnetic field

Fig. 1[link](a) shows two-dimensional CRL-SANS data collected at a representative range of applied magnetic fields from zero to 205 G after subtraction of transmission-normalized empty cell data. The initial 0 G data clearly show the expected isotropic form factor scattering, P(Q), from the ferrofluid droplets. Since this modulates the overall scattering, limiting the signal above the incoherent background to a scattering vector magnitude of ∼0.014 Å−1, we show a little less than one-quarter of the area of the SANS detector – a square section ∼0.02 Å−1 on a side. As the applied field is increased and the dipole–dipole interaction between droplets begins to form them into chains, the expected structure factor modulation of the developing chains is clearly visible as bands of Bragg scattering across the detector. In Fig. 1[link](b), we make the development of this structure factor modulation more evident by subtracting a continuous phase background and dividing the applied field scattering data by the 0 G form factor data. The 66 G data show typical scattering from the onset of magnetic ordering: the zeroth-order Bragg band is narrow and the first-order Bragg bands are very weak, indicating that the average chain length is small. The relatively uniform strong scattering within the circle defined by the form factor indicates that many of the droplets are still unchained – thermodynamically free of the magnetic interaction. At 88 G, with increasing incorporation of droplets into stable chains, the scattering from this unchained fraction is much weaker, the zeroth-order Bragg band wider and the first-order Bragg scattering clearly visible. As the applied field is increased, this trend to stronger chain scattering continues and the chain population essentially saturates above ∼100 G. The 205 G data show this strong ordering; the scattering at this field is dominated by anisotropic Bragg bands with little or no isotropic free droplet scattering evident. While the final 307 G data set appeared very similar to the 205 G scattering sets and within error showed the same chain period, its Bragg intensity was lower, indicating some reduction in the number of emulsion droplets in the beam. This suggests that the applied magnetic field had exceeded a threshold beyond which transverse interactions between long chains coming into contact were no longer disrupted by thermal effects, allowing the formation and precipitation of larger aggregate structures.

[Figure 1]
Figure 1
(a) Ferrofluid emulsion scattering patterns for applied magnetic fields shown relative to schematics of the sample cell droplet ordering without and with applied magnetic field in the z direction. Empty cell scattering has been subtracted. Note that the beamstop shadow size in these data is roughly indicative of the angular resolution of these CRL-focused measurements. (b) Subtraction of the continuous phase background and normalization against the single droplet isotropic form factor (0 G data) reveals the general behavior of the structure factor of droplet ordering for the same range of applied field. As indicated in this analysis of the CRL-SANS data we designate the magnetic field/chain direction as z vertical in the data set, the transverse scattering direction as x, with the neutron beam incident in the y direction, into the page. These axes are shown superimposed on direct-beam image data.

3.2. Scattering by an aligned chain of identical spheres

To analyze scattering by an aligned chain of identical spheres with the CRL-SANS MCT we used a modified form of the scattering analysis for a straight chain of identical spheres given by Kawaguchi (2001[Kawaguchi, T. (2001). J. Appl. Cryst. 34, 771-772.]). The scattering from individual droplets was simply modeled with the familiar form factor for a sphere of diameter d as function of the scattering vector magnitude Q:

[P(Q) = \{ 3[\sin(dQ/2)- (dQ/2)\cos(dQ/2)]/ (dQ/2)^{3} \}^{2}. \eqno(1)]

No attempt was made to model details of the surfactant structure at the droplet interface. The Span 80 head group is smaller than the expected resolution of our measurements and there will be negligible neutron contrast between the hydrogenous 2 nm tail of the Span 80 and the tetradecane continuous phase.

The modulating structure factor for a straight chain of uniformly spaced identical spheres can be modeled as the Fourier transform of a Dirac comb of delta functions δ(zna), where z is the chain (magnetic field) direction, a is the chain period and na is then the distance of the center of the nth sphere on one side from the center of a sphere at the middle of the chain. Since our low magnetic field data indicate a lesser degree of order and a more transient occupation, we introduced a correlation length l to model this behavior. This results in a scaled structure factor dependent only on the z component of the scattering vector:

[S(Q) = S(Q_{z}) = {{| 1+2\textstyle\sum\limits_{n = 1}^{\infty}\exp[-(na)^{2}/2l^{2}]\cos(naQ_{z})|^{2}} \over {1+2\textstyle\sum\limits_{n = 1}^{\infty} \exp[-(na)^{2}/2l^{2}]}}. \eqno(2)]

For nonzero correlation length fits, the summation was performed numerically using four correlation lengths on either side of the central sphere of the chain. In the event, this had little effect on the data analysis. Essentially equivalent fits were also obtained assuming a uniform chain of the same normalized occupancy, i.e. having ∼(2π)1/2l/a members.

To account for scattering from free droplets in solution, we modeled the overall scattering intensity in terms of contributions from a chained fraction c of the droplets and a free fraction (1 − c) contributing as independent scatterers. Thus

[I(Q) = A_0[c S(Q_{z}) P(Q)- (1-c)P(Q)] + B_{\rm inc}, \eqno(3)]

where A0 is a normalization parameter (for c = 0 this will be I0, the total scattering power per unit volume of the free droplets in solution) and Binc is a flat background due primarily to incoherent hydrogen scattering in the sample.

3.3. Data reduction

From inspection of the scattering patterns shown in Fig. 1[link], it is immediately clear that careful treatment of resolution will be quite important in analysis of this data. The CRL `top hat' direct-beam image of the entrance aperture on the detector is about 2 cm wide; just a little smaller than the beamstop shadow evident in the scattering patterns and about 15% of the distance to the first Bragg band peak. The resultant 15% smearing of the Q value of the Bragg signal can be expected to limit the resolution of the correlation length determinations to something just beyond about six droplet diameters, or a uniform chain that comprises more than 20 members.

The anisotropic nature of the data and the scattering function derived above also make clear the dependence of the important parameters for the MCT on the component of the scattering vector parallel to the magnetic field alignment direction z. Therefore, the most accurate determination of the chain periodicity can be obtained with binning in cuts along this direction, carrying the component's resolution correctly. At lower fields, binning along not yet fully formed Bragg bands in the x direction should give some information about transverse correlations established as the droplet chains grow.

Detailed treatments of the resolution and of the imaging performance of the NIST CRL-SANS instrument have been presented by Mildner et al. (2005[Mildner, D. F. R., Hammouda, B. & Kline, S. R. (2005). J. Appl. Cryst. 38, 979-987.]) and Hammouda & Mildner (2007[Hammouda, B. & Mildner, D. F. R. (2007). J. Appl. Cryst. 40, 250-259.]). Building on these treatments, algorithms were developed to reduce and bin the scattering patterns more generally with respect to the angular resolution of the scattering vector and its components. Briefly, our routine takes advantage of the (attenuated) direct-beam calibration scan used to determine the nominal center of the scattering pattern, but considers this image rather as a measurement of the distribution of possible centers of scattering – essentially treating the measured SANS data set as an overlapping set of scattering patterns centered on each pixel of the direct-beam measurement and weighted by the measured direct-beam intensity in that pixel. Before binning data cuts against a detector scattering parameter, an initial map of the mean and standard deviation (resolution) of that parameter for each detector pixel is created over the distribution represented by the beam image. These values are then applied to the binning of that parameter in a statistically valid manner to obtain the one-dimensional data presented here. The phrase `detector scattering parameter' is used to emphasize that this procedure not only is applicable to determinations of the mean and variance of a pixel's displacement across the detector relative to the direct-beam-center distribution, which determines the usual isotropic mean scattering angle and its resolution, but equally can be applied to the components of that displacement, leading, in this work, to a natural extension to reliable determinations of the mean and variance of Qz and Qx, or generally to any geometrical scattering pattern parameter. The most important improvement achieved by this reduction method is the reliable determination of scattering vector values near the beamstop. In the present case, since these lower Q values contribute strongly to droplet size determinations in the form factor measurement, we found that this treatment was essential to achieve sufficient accuracy for consistent MCT analysis at closest approach. Some further details and justification of the reduction routine are outlined in Appendix A[link].

3.4. CRL-SANS data reduction and analysis

The primary structural parameters required by the MCT are the droplet separation and the droplet diameter, which determine the interfacial separation of droplet surfaces. Fig. 2[link](a) shows the simple isotropic spherical form factor fit [setting c [\equiv] 0 in equation (3)[link]] to the reduced summed data of the ten scattering patterns recorded at zero applied field throughout the measurement series. (The total acquisition time is 7.5 h, comprising the initial 180 min run and nine subsequent 30 min zero field check runs.) The good visibility of the first interference minimum indicates a polydispersity at least better than a few percent, the upper limit set by the resolution of the SANS measurements (the fit shown assumes monodispersity and was not improved by the introduction of a polydispersity parameter). The diameter d of the ferrofluid droplets derived from this fit was 172.0 (3) nm. Fits to the shorter 30 min individual scattering patterns in the series varied randomly within ∼0.5 nm of this value, indicating that the ferrofluid emulsion remained stable over the duration of the SANS experiment. The extrapolated absolute intensity at Q = 0 for this fit, I0, is 157 cm−1. Dividing this value by the nominal integrated scattering strength of an individual ferrofluid droplet, (Δβπd3/6)2 ≃ 3.4 × 108 Å2, gives a particle number density of 4.7 × 109 cm−3, corresponding to a suitably low initial droplet volume fraction of 0.0013%.

[Figure 2]
Figure 2
(a) Isotropic form factor fit to absolutely normalized CRL-SANS ferrofluid emulsion data for zero applied field. Empty cell scattering has been subtracted. (b) Reduced data cuts in the Qz direction for 〈Qx〉 ≃ 0 at 0 G [the same data set sum as in Fig. 2[link](a)], and for applied magnetic fields of 77 and 205 G. The inset indicates the six-pixel-wide cut region marked on 205 G data. To ensure the best Qz resolution for peak determinations, each plotted point comprises a bin of six horizontal pixels. The fits are discussed in the main text. Shading indicates scattering regions compromised by partial shading of pixels by the beamstop.

Fig. 2[link](b) shows fits using equation (3)[link] to narrow reduced data cuts in the Qz direction for 〈Qx〉 ≃ 0, as indicated in the two-dimensional data inset for applied fields from 0, 77 and 205 G. The cut region was six pixels (3 cm) wide on the detector covering about the width of the direct-beam image. The fits versus Qz to the data shown were averaged across Qx to account for this width in addition to integration across the indicated Qz resolution. Care was taken to constrain the chain periodicity parameter a to ensure that the fit over the Bragg peak was not degraded by statistical improvements in less critical regions. The 205 G data clearly show the saturated Bragg peak position achieved for fields above ∼100 G at Qz = 0.00364 Å−1. A Δχ2 = 1 parameter estimate in the Bragg peak region of this fit gives an error of ∼0.00002 Å−1. We verified the reliability of this error by applying a rough error in the mean estimate to the peak position. The resolution-limited peak's r.m.s. width is ∼0.00033 Å−1, very close to the 0.00032 Å−1 Qz resolution calculated by the reduction routine. A summation in the two 3 h raw 205 G data sets finds about 700 raw neutron counts contributing to the peak region of the data cut. An N−1/2 error in the mean estimate from the peak width gives a slightly lower value of the same order, ∼0.00013 Å−1, which is the value we would expect to increase when the background and free particle contributions are accounted for. This first-order Bragg scattering peak at Qz = 0.00364 (2) Å−1 corresponds to a chain periodicity a = 172.6 (10) nm. In conjunction with the droplet diameter d = 172.0 (3) nm from the previous form factor fit, we can infer a chained droplet surface separation h = ad of 0.6 (11) nm, essentially consistent with direct contact at this applied field.

3.5. CRL-SANS MCT determinations

The results of the CRL-SANS data analysis are summarized in Fig. 3[link](a), which shows the variation of surface separation of the chained droplets, h   ad, as a function of the applied field. At the lower applied fields, droplets were separated by more than 10 nm and were brought into contact at an applied field just above 100 G. The lowest field for which satisfactory fits indicating significant chaining could be obtained was 67 G, where the droplet surface separation was about 12 nm. At 50 and 58 G, magnetic chains were either too short lived or misaligned owing to thermal effects for their scattering to be observed against a much stronger isotropic scattering from free droplets. Over a ∼77–125 G range of applied field, we observed a very rapid increase from ∼25% to over 90% in the chain fraction c parameter from our fits, as indicated by the rapid drop in isotropic form factor scattering. As we might expect, the chain correlation length parameter l/a determined by the fitting increased as well: from about one at 67 G consistent with the formation of dimers or trimers to about seven, equivalent to a uniform chain with ∼18 members. Above 125 G, the parameter saturates because of the resolution limit discussed above, although the fact that the Bragg intensity continued to rise between 125 and 205 G does suggest the formation of longer aggregates. From these data alone we cannot infer that significantly longer straight chains are forming at the higher fields. Optical micrographs of previous MCT test emulsion samples did show aggregations with aspect ratios consistent with longer chains, but not at resolution sufficient to confirm continuous coherent alignments.

[Figure 3]
Figure 3
(a) Droplet surface separation h (with errors) versus applied magnetic field. Inset shows chain fraction c and chain correlation parameter l/a (in period units) from fits to CRL-SANS data. (b) Log–lin plot of MCT-derived surface repulsion F versus droplet surface separation h.

The attractive force between a pair of identical aligned magnetic dipoles varies as the square of their magnetization and the inverse fourth power of their center-to-center distance. Along a chain this force is increased by about 20% owing to forces between pairs along the chain. The magnetization of an isolated ferrofluid droplet in an external magnetic field is proportional to the applied field and to the volume of the droplet multiplied by the magnetic susceptibility of the ferrofluid, which for maghemite ferrofluids is roughly constant over the range of fields applied in these measurements. In a chain of magnetically aligned superparamagnetic ferrofluid droplets the field due to neighboring droplets will further increase the magnetization. In the present case, the increase is about 30%, which is also nearly constant for these emulsion droplets in relatively close proximity (hd). Together these chaining effects amount to a doubling of the attractive force along the chain. (Details are given in Appendix B[link].) Knowing the susceptibility of the ferrofluid, the emulsion droplet volume and the droplet chain periodicity therefore allows us to determine the form and magnitude of the droplet chain force that must be opposed by repulsive forces between the droplet surfaces in a stable chain. Fig. 3[link](b) shows the repulsive force between droplets derived from these CRL-SANS measurements of the droplet chain structure as a function of their interfacial separation.

The droplets are primarily stabilized by a short-range repulsion with a range of approximately 2 nm, and a much weaker and longer range `tail'. The short-range component resembles the soft unstructured repulsions which have been observed between mica surfaces with an adsorbed surfactant monolayer immersed in various alkanes, and can be attributed to steric interactions between the hydrophobic tails of the adsorbed Span 80 (Gee & Israelachvili, 1990[Gee, M. L. & Israelachvili, J. N. (1990). J. Chem. Soc. Faraday Trans. 86, 4049.]). We see no evidence in either the force curve or the observed behavior of the ferrofluid emulsions to indicate a van der Waals attractive minimum. However, the MCT is not amenable to measuring attractions so a shallow minimum might not be evident.

The hydrophobic oleyl tails of Span 80 are estimated to be 1.9 nm long, which is consistent with a 1.7–1.8 nm-thick monolayer of oleate anions measured previously on colloidal quantum dots (Abel et al., 2012[Abel, K. A., FitzGerald, P. A., Wang, T., Regier, T. Z., Raudsepp, M., Ringer, S. P., Warr, G. G. & van Veggel, F. C. J. M. (2012). J. Phys. Chem. C, 116, 3968-3978.]). The near-zero separation at closest approach thus suggests that these ferrofluid droplets do not have a densely packed coating of emulsifier, so that at contact the chains intercalate or `lie down' at the interface. The results are also consistent with very stable emulsion droplets that neither coalesce easily nor deform significantly, even when forced into contact with their neighbors.

3.6. Chain morphology at lower applied magnetic fields

Ferrofluid droplet chains begin to form when the strength of the magnetic interaction overcomes the thermal energy. At the lower fields, the concept of a correlation length becomes somewhat notional – dimers and trimers may be aligned, but initially the alignment is weak and such structures can hardly be called chains. In the rapid chain growth region above about ∼70 G, we see a dramatic reduction in the scattering from free droplets, which obviously implies that shorter chains are combining to form longer chains rather than chains absorbing new members from a dispersal of new droplets. Initially we may expect that thermal effects will still be relatively strong, that the mooring of one chain to another will be more haphazard, and that the combined structures will be more irregular and transient than will be the case when the magnetic interactions are strong and very long chains dominate the population.

These observations arise from our attempts to analyze Qz data sets cut at an offset in Qx in order to sample scattering intensity from the zeroth-order Bragg band at the same time as the first-order peaks. In principle such scans might be expected to give us a better estimate of chain correlations and length from the intensity ratio between orders, at the cost of some reduction in the accuracy of chain periodicity determination. This reduction in accuracy would be due to a reduced first-order Bragg signal, which in turn would be due to a falloff in the form factor, since this Qz sampling would be at a slightly higher overall Q than for the 〈Qx〉 = 0 cuts shown in Fig. 2[link]. In practice this was not quite the case. While the general trends were the same and the chain periodicity agreed within errors at some lower fields, the fits apparently indicated noticeably slightly greater correlation lengths and even more first-order Bragg intensity than for the on-axis cuts.

A possible explanation for this effect is outlined in Fig. 4[link], which shows data from several horizontal Qx cuts through scattering data across the region of the first-order Bragg band at Qz = 0.00364 Å−1. The cuts are from the same data sets at applied fields of 0, 77 and 205 G as shown in Fig. 2[link](b). The data for 0 and 205 G agree with the prediction of equation (3)[link], with the strong Bragg scattering of the 205 G data generally following the form factor behavior of the zero field data, decreasing from a maximum Qx = 0: although again a higher intensity indicates that extra ferrofluid droplets are being drawn up into the incident neutron beam path at these stronger fields. The 77 G data (dashed line), however, show a distinct minimum at smaller Qx, peaking at an offset to the axis. Dividing by the form factor data to estimate the position of the peak in the structure factor, we find that this is a maximum for |Qz| ≃ 0.0022 Å−1. This is presumably from four equivalent peaks at (Qx, Qy) ≃ (±0.0022, ±0.00364 Å−1), so at very nearly 30° to the magnetic field axis. Obviously the structure labeled (1) with two short chains packed side by side exhibits correlations which would give increased scattering at these scattering vectors, as would a single droplet as shown initially attracted to an intermediate position on the chain before being incorporated within it. The population of such structures at the field strength is in all probability somewhat higher than the observed extra intensity indicates: note that if the two chains are not in a plane perpendicular to the incident beam direction, i.e. the same structure rotated 90° about the magnetic field axis marked (2), no extra contribution to the scattering in this region would be observed. We note that similar structures have been observed in X-ray studies of magnetic field induced crystallization of monodisperse sus­pen­sions of silica/magnetite nanoparticles (Malik et al., 2012[Malik, V., Petukhov, A. V., He, L., Yin, Y. & Schmidt, M. (2012). Langmuir, 28, 14777-14783.]) and are also posited to form preferentially and sediment out in the magnetic fractionation process owing to the much stronger interactions between the largest ferrofluid droplets (Dreyfus et al., 2009[Dreyfus, R., Lacoste, D., Bibette, J. & Baudry, J. (2009). Eur. Phys. J. E, 28, 113-123.]; Ivey et al., 2000[Ivey, M., Liu, J., Zhu, Y. & Cutillas, S. (2000). Phys. Rev. E, 63, 011403.]). The weaker Bragg scattering observed in the final 307 G data is also probably due to the precipitation of some larger three-dimensional aggregations of this type from the sample solution.

[Figure 4]
Figure 4
Horizontal Qx cuts through scattering data across the region of the first-order Bragg band at Qz = 0.00364 Å−1 at applied fields of 0, 77 and 205 G. These cuts are from the same data sets as those displayed in Fig. 2[link](b) and the data now indicated by solid lines follow the same legend.

The impact of the presence of these, perhaps transient, structures on the MCT analysis is difficult to assess. Although we did not observe any statistically significant difference in determinations of the chain periodicity, the nearby presence of off-axis dipoles must to some extent perturb the chain forces and needs further consideration. Since this occurs only a little above the fields at which reasonably well aligned structures that can be called chains form, it seems that the most reliable measures of chain spacing should be derived for the 〈Qx〉 = 0 scan cuts we have made directly along the field direction. It is also possible that most couplings of this kind are of nearly end-to-end matches since the field lines near the center of a chain will condense closely around the chain, and make chains `invisible' to each other laterally, with field lines only available to attract another chain as they emerge from its extremities. In this case the major body of the chains and presumably the forces and consequent periodicities will not be much affected.

4. Conclusions

We have applied CRL-SANS to the magnetic chaining technique to derive inter-surface forces between 60 wt% maghemite ferrofluid in D2O droplets of diameter 172 nm stabilized by sorbitan monooleate (Span 80) in a continuous phase of tetradecane. The derived forces varied from about 0.4 pN at an applied field of 66 G at a separation of 12 nm up to 13 pN at 307 G, and the force versus separation curve is consistent with steric repulsions between monolayers of adsorbed emulsifier on adjacent droplets. Below 66 G, the strength of the magnetic interaction was not sufficient to form aligned chains in sufficient number with long enough lifetimes for their scattering (if any) to be observed against the isotropic scattering from a majority of free droplets. Above this field, chain growth proceeded rapidly. At early stages in chain growth, scattering correlations transverse to the field direction indicate transient structures – either adjacent chains interlacing prior to combination or perhaps individual droplets coming into or out of the chain.

To the accuracy achieved in our measurements, approximately 1 nm, effective contact between the Span 80 interfaces was achieved between the applied fields of 100 and 125 G at an inter-droplet force of about 1 pN. At these fields, the chain correlation length reached at least the resolution limit of these CRL-SANS measurements at about eight chain periods, corresponding to a uniform chain of about 20 droplets. Forces at higher fields and a constant chain period at contact indicate the droplets' resistance to deformation by forces in the range 1–13 pN and are consistent with a monodispersity better than 0.5%.

To achieve the 1 nm accuracy of these results required careful attention to the resolution of the CRL-SANS measurements and its effect on observed scattering patterns (see Appendix A[link]). The range of interfacial spacing accessible to this technique, ∼1–10 nm, is suitable for the study of short-ranged close approach forces and contact interactions between emulsion droplets. The combination of structural and force information over this range could be used to study details of interactions between stabilizing surfactants that extend further into the continuous phase, chemical modification of the interactions due to the addition of co-surfactants, alcohols or salts, and more complicated interfacial films chemically grafted onto the ferrofluid emulsion droplets (Montagne et al., 2006[Montagne, F, Mondain-Monval, O., Pichot, C., Mozzanega, H. & Elaissari, A. (2002). J. Magn. Magn. Mater. 250, 302-312.]). At higher fields, the ability of SANS to monitor droplet deformation by tracking transverse scattering along the zeroth-order Bragg band could provide information distinguishing the effects of surface rupture and emulsion coalescence from inter-chain crystallization effects.


SANS data reduction – parameter mapping over the direct-beam distribution

As noted in the main text we consider a SANS direct-beam measurement over the set of pixels in the direct-beam region [(xoi, zoi)] of the relative strength of the scattering contribution [foi(xoi, zoi)] centered on each pixel. The acquired scattering pattern may be considered as the sum of an overlapping set of the scattering patterns about each of these center pixels weighted by measured foi and further by the corresponding wavelength distribution over that pixel. In our reduction scheme, we have assumed the wavelength distribution over the direct beam to be essentially constant. Two conditions that might violate this assumption are any wavelength variation in the guides feeding the beam and, for sufficiently long wavelength neutrons, gravity, which will cause longer-wavelength neutrons and their scattering patterns to fall further and appear lower on the detector as they pass from sample to detector than those of shorter-wavelength neutrons. In principle, both these effects could be measured and accounted for in an averaging process. However, for the NG3 CRL system operating at 8.4 Å neither of these effects is very large. At the collimation of our measurement, for example, approximately 15.5 m separates the 2.54 cm entrance aperture from the 1.27 cm sample aperture, so the incident beam is sampled over a divergence of only about 2 mrad, which is only about 10% of the reflection divergence for 8.4 Å neutrons in an Ni-coated neutron guide. As for gravity, an 8.4 Å neutron travels at 470 m s−1 and will travel 13.7 m in 29 ms, in which time it would will fall 0.41 cm. This is the average displacement of the entire scattering pattern and the direct-beam image; the resolution effects will depend on the spread about this value. Neutrons at either extreme of the velocity selector's triangular wavelength distribution, i.e. 14.3% slower or faster than the average, will fall over a spread of ∼±30% or about 0.25 cm overall, while the majority of the neutrons will be spread more narrowly still: 75% of neutrons within the FWHM of the velocity selector fall over about 0.12 cm, just one-twentieth of the beam image diameter. Since this adds in quadrature to other resolution contributions, we may expect the resolution effects to be somewhat less than those of collecting the data in 0.5 × 0.5 cm pixels on the detector. Indeed, the fact that the r.m.s. width and height of the image in the direct-beam data set agreed to about 1% (σxo = σzo = 0.83 pixel width or 0.42 cm) indicates that this is not a significant effect for our measurements.

In principle then we can envisage fitting against two-dimensional scattering patterns calculated as weighted distributions over the direct-beam image and further convoluted if necessary with its position-dependent wavelength distribution (which could be measured by time-of-flight techniques). Fortunately, the latter is not necessary for this data analysis, but two-dimensional data fitting would nevertheless generally seem to be more costly in computing time than it is worth, particularly in cases like the present one where selected regions of the data set may be expected to better represent the information required by the analysis. The reduction of scattering data to a one-dimensional form suitable for graphical presentation is usually advantageous. To do this in our analysis, we consider each pixel on the detector (xj, zj) in relation to the beam image distribution of scattering centers and obtain a map over the detector of the mean and standard deviation of the scattering parameter required for that pixel relative to those centers. So

[\eqalign{\langle p\rangle_{j} &= \langle p[(x_j,z_j) x_{{\rm o}i},z_{{\rm o}i})]\rangle_{\rm o}\,\,\, {\rm and} \cr \sigma _{j}(p) &\equiv \left(\langle p_{j}^{2}\rangle_{\rm o}-\langle p_{j}\rangle_{\rm o}^{2}\right)^{1/2}, } \eqno(4)]

where the subscript o indicates numerical averages calculated over the beam image distribution [foi(xoi, zoi)]. As noted previously these averages could also be taken over the triangular wavelength distribution of the SANS. In the current study, sufficiently accurate scattering parameter mean values (〈Qj, 〈Qzj etc.) for the maps were simply (and more quickly) calculated for a nominal wavelength of 8.4 Å, while the equivalent standard deviation for the FWHM triangular distribution (14.3%/61/2 = 5.83%) was added in proportionate quadrature to the standard deviation values of the map values [σ(Q)j, σ(Qz)j etc.]. Small standard deviation contributions for widths of pixels in the beam image and target pixel were also made. No correction needs to be made for detector resolution (assuming it is constant across the detector) as that should already be accounted for in the consequent smearing of the direct-beam image data. On a pixel-by-pixel basis the means and standard deviations calculated in this way can be used as the centers and standard deviations of a Gaussian averaging procedure carried out over a calculated scattering function. The accuracy of the procedure is indicated by the Bragg peak measurement in the Qz cut of the 205 G scattering data discussed above, with the calculated pixel standard deviation differing by only a few percent from the r.m.s. width calculated for this resolution-limited peak measurement. For the combination of data from similar pixels to improve statistics – annular binning to constant |Q| is the obvious example – the individual pixel means and standard deviations are combined, weighted by scattering intensity and its estimated error, and with the mean and standard deviation over each binning region (Bevington & Robinson, 2002[Bevington, P. R. & Robinson, D. K. (2002). Data Reduction and Error Analysis for the Physical Sciences. New York: McGraw-Hill.]). Fig. 5[link](a) shows a schematic of the beam image averaging scheme superimposed over a direct-beam measurement and a normalized scattering pattern.

[Figure 5]
Figure 5
(a) Schematic of the numerical calculation of scattering parameter means and standard deviations for individual pixels weighted over the direct-beam intensity (left hand image) – in this work a region 8 × 8 pixel is shown. Scattering observed at a pixel in the scattering data (right hand image) will be a similarly weighted combination of scattering from each of these possible centers and therefore a Gaussian sampling of the scattering function around the same mean value of the scattering parameter and with an integration width (resolution) of its standard deviation may be used to account for the effects of finite incident-beam size. As shown φ is the angle for a given pixel from the x axis on a coordinate system (x, z) with origin at the mean center of the direct-beam image. (b) Direct-beam image beam profiles versus x and z pixels summed over a transverse band 8 pixels wide in linear and log–lin plots. The 8 × 8 pixel region can be seen to contain the direct-beam image down about ∼0.1% of the peak intensity. Some of our data were analyzed only down to the 1% level, i.e. using a 6 × 6 pixel beam region; this did not significantly change the values derived from analysis of the scattering data.

The MCT-SANS measurements above require consistent accuracy in determinations of the droplet diameter from the form factor measurement and the chain period which is dominated by the structure factor signal – to rather better than 1%. To explain why obtaining mean scattering vectors and their resolution by the method outlined above was necessary for this data reduction, we need to consider the various distances over the detector plane relevant to the problem of determining scattering angles in SANS. The nominal radial distance to a pixel at some position (x, z) on the detector from the nominal beam center at (〈xoo, 〈zoo) is

[\eqalign{R_{\rm o}(x,z) \equiv [(x - \langle x_{\rm o}\rangle _{\rm o})^{2} + (z-\langle z_{\rm o}\rangle _{\rm o}^{2}]^{1/2} \equiv (x^{2} + z^{2})^{1/2}}. \eqno(5)]

{Without loss of generality we have set the weighted beam center as the origin. Furthermore, for clarity we have dropped the subscript to pixel (x, z) in the two-dimensional mapping with the understanding that all parameter map averaging is done for the pixel position (x, z) with respect to a weighted mean over the set of pixels [(xoi, zoi)] in the chosen beam image region.} The weighted mean distance to this pixel from pixels in the direct-beam region is

[\eqalign{\langle R(x,z)\rangle \equiv \langle[(x-x_{{\rm o}i})^{2} + (z-z_{{\rm o}i})^{2}]^{1/2}\rangle_{\rm o}}. \eqno(6)]

At the beam center, now origin, Ro(0, 0) [\equiv] 0 and we find

[\eqalign{\langle R(0,0)\rangle \equiv \langle (x_{{\rm o}i}^{2}+z_{{\rm o}i}^{2})^{1/2}\rangle_{\rm o} = \langle r_{\rm o}\rangle}, \eqno(7)]

i.e. the mean radius of the beam image, which is obviously greater than Ro(0, 0) [\equiv] 0. The mean scattering deviation [\langle]R(x, z)[\rangle] is always greater than the nominal value Ro(x, z) – so using the nominal value will underestimate scattering angles, with a progressively greater deviation at smaller angles. For larger distances, we can show that [\langle]R(x, z)[\rangle] and Ro are related as

[\eqalign{\langle R(x,z)\rangle \cong R_{\rm o}\left[1+{{\sigma ^{2}(x_{\rm o}) \sin^{2} \varphi + \sigma^2(z_{\rm o})\cos^{2}\varphi} \over {2R_{\rm o}^2}} -(0^4)\right]}, \eqno(8)]

where σ(xo) and σ(zo) are the r.m.s. width and height of the beam image, respectively. As shown in Fig. 5[link](a), φ is the angle of the target pixel away from the x axis: tan−1(x/z). So to this order of approximation we see that the contribution to the difference (shift) between R and Ro is due to the width of the beam spot transverse to their direction.

To estimate the resolution, we combine the weighted mean square distance of the pixel from pixels in the beam image

[\eqalign{\langle R^{2}(x,z)\rangle \equiv R_{\rm o}^2 + \sigma ^2(x_{\rm o}) + \sigma^2(z_{\rm o})}, \eqno(9)]

with the estimated value of [\langle]R(x, z)[\rangle], equation (8[link]), to obtain an estimate of the standard deviation

[\eqalignno{\sigma [R(x,z)]\equiv &\big(\langle R^2\rangle - \langle R\rangle^{2}\big)^{1/2}\cr \cong &\big[\sigma^2(x_{\rm o})\cos^2\varphi +\sigma^2(z_{\rm o})\sin^2\varphi\big]^{1/2}. &(10)}]

So, to this order of approximation, the resolution of the distance R(x, z) is due to the width of the beam spot in line with the direction of the target pixel from the beam center.

In our case since σ(xo) = σ(zo), we can put equations (8[link]) and (10[link]) in terms of the nominal mean square radius of the beam spot [\langle r_{\rm o}^2 \rangle \equiv] σ2(xo) + σ2zo = 2σ2xo = 2σ2zo and further obtain the proportional shift and standard deviation of the distances from beam image pixels to (x, z). In proportional terms the difference between [\langle]R(x, z)[\rangle] and the nominal distance Ro(x, z) is

[\eqalign{{{\langle R(x,z)\rangle -R_{\rm o}} \over {R_{\rm o}}} \cong + {{\langle r_{\rm o}^{2}\rangle} \over {(2R_{\rm o})^{2}}}} \eqno(11)]

and to estimate the proportional resolution in our determination of distance to the pixel

[\eqalign{{{\sigma [R(x,z)]} \over {\langle R \rangle}} \cong {\sigma [R(x,z)] \over R_{\rm o}} \cong {{\big(\langle r_{\rm o}^2\rangle/2\big)^{1/2}} \over R_{\rm o}} }. \eqno(12)]

To convert these expressions to estimate the angular resolution contribution to the shift and resolution of the scattering vector from the nominal pixel value we define

[\eqalign{ \sigma (q_{\rm o}) \equiv 4\pi{ \sin\big\{{1\over2}\tan^{-1}\big[\big(\langle r_{\rm o}^2\rangle/2\big)^{1/2}/L\big]\big\} \over {\lambda_{\rm o}}} \cong {\pi \over \lambda_{\rm o} } {\big(2\langle r_{\rm o}^2\rangle\big)^{1/2} \over L} }, \eqno (13)]

where λo is the center of the incident wavelength distribution at 8.4 Å and L is the sample-to-detector distance 13.7 m. This estimates the angular uncertainty contribution to the resolution of the true scattering vector 〈Q〉. Now the estimated proportional shift due to averaging relative to the nominal scattering vector at a pixel can be written as

[\eqalign{{\Delta Q \over {\bf Q}_{\rm o}(x,z)} \simeq + { 1\over 2}\left[{{\sigma(q_{\rm o})} \over {{\bf Q}_{\rm o}}}\right]^{2} }. \eqno(14)]

From the direct-beam image data the r.m.s. width and height of the beam is 0.42 cm, which we recall has already been smeared by the detector resolution. To obtain a slightly better estimate of the angular uncertainty in this case we add in quadrature (as our averaging routine does on a pixel-by-pixel basis) the r.m.s. width contributions from 0.5 cm pixel binning in both image and target pixels, 0.5/(12)1/2 = 0.14 cm. Estimating the r.m.s. uncertainties in the horizontal and the vertical directions on the detector we find a resolution of 0.47 cm, and so 〈ro2〉 ≃ (0.66 cm)2 and the angular scattering vector contribution resolution is σ(qo) ≃ 0.00025 Å−1.

While a reliable determination of resolution contributions is obviously important for accurate determination of scattering parameters, in the present case determination of the relative shift in scattering vector proved to be critical for accurate determination of the droplet separation at higher fields. If we consider Fig. 2[link], the signal in the form factor measurements is stronger by about an order of magnitude in the Q range 0.001–0.002 Å−1 than it is in the region in which the first-order Bragg signal develops at QB = 0.00364 Å−1. From equation (14[link]) we can now estimate the resolution shift in Q at the Bragg peak to be +0.3%, while adding the angular contribution to resolution of ∼6.9% in quadrature to the 5.8% r.m.s. the contribution from the (14.3% FWHM triangular) wavelength distribution gives an overall Q resolution of ∼9.0% (0.00033 Å−1). In the center of the strong form factor region, Q ≃ 0.0015 Å−1, the difference between true and nominal scattering vector magnitudes will be +1.4%, rising to about +3% for pixels near the edge of the beamstop at Q ≃ 0.001 Å−1. Unless corrected for, the progressive differential shift between these regions gives an overestimate of the droplet diameter relative to the chain period, owing to the greater influence of stronger signals at lower Q and their greater underestimation of the true average scattering vector 〈Q〉 if the nominal value is used instead. This combined effect is a little larger than we might initially expect due to the fit resolution averaging at lower Q over relatively wider regions of more strongly underestimated values nearest the beamstop: we found the droplet diameter to be overestimated by about 4 nm (if the correction was omitted entirely) and underestimated somewhat if too large a correction were imposed. Even in 7.5 h of accumulated data the statistics at the less shifted first minimum and peak in the form factor are not good enough or these features are not well enough resolved to balance this effect if it is not corrected, as it amounts to only about 2% of the size determination overall. From inspection of Fig. 5[link](a), it is evident that a 4 nm error would limit the applied fields for which consistent droplet surface separations could be obtained in the present work to rather less than 100 G.

For the very small scattering angles in these measurements (the scattering angle of the saturated chain Bragg peak 2θB ≅ 0.28°), this resolution shift far exceeds other off-axis effects which can also give rise to systematic errors of angle determination. Parallax error in the detector at small scattering angles will shift apparent positions relative to the beam center by only the ratio of an 8.4 Å neutron's free path through the pressurized 3He gas in the two-dimensional Ordela detector, ∼1 cm (the full 3He depth is only 2.5 cm), to the sample-to-detector distance of 13.7 m, so less than 0.01% in this case. We might also consider that because of the curvature of the Ewald sphere for vertical alignment of the magnetic field the first orders of magnetic chain scattering, in three-dimensional reciprocal space broad discs, are misaligned by about their Bragg angles ±θB away from the true central maxima of their respective rocking curves. These first-order discs will then intersect the Ewald sphere in the vertical direction at observed scattering angles very slightly greater than the true first-order Bragg scattering angles, increased by a fraction [\sim \theta_{\rm B}^{2} /2]. We might note here that in traversing the sample-to-detector distance of 13.7 m at 470 m s−1 an 8.4 Å neutron will accelerate downward by 0.28 m s−1, effectively tilting the Ewald sphere at the detector plane `backward' by 0.035°. This effect and tilting of the field along the beam direction are similarly only second-order corrections to the observed scattering angles presented here. Over the range of wavelengths, distances and scattering angles covered by our measurements these effects are negligible in comparison to the resolution averaging shift.

We note that (to this second-order level of approximation) this shift does not affect the Qz component cuts shown in Fig. 2[link](b) since the equivalent to (6[link]) to derive the average deviation distance in that direction caused by the scattering is simply

[\eqalign{\langle z(x,z)\rangle \equiv \langle z-z_{{\rm o}i}\rangle_{\rm o} = z-\langle z_{\rm o}\rangle \equiv z}, \eqno(15)]

using the conventions applied above. This is generally true for one-dimensional scattering vector component cuts along any defined φ direction. However, the resolution as calculated by our reduction routine, since it depends on averages over the beam spot width along the line of the vector direction, will be the same. We also observe that these errors would probably not arise in size estimations obtained by an lnI versus Q2 Guinier gradient analysis, since as we may infer from equation (9[link]) the derived value of 〈Q2 would differ from the nominal only by a constant negative offset which would not affect the gradient determination. There would, however, be a slight underestimation of the extrapolated intensity Io at Q = 0, potentially leading to an underestimate of the total scattering power of the particles.

Analytic estimates for the shift in 〈Q〉 relative to the nominal Qo for standard pinhole SANS have been presented by Mildner & Carpenter (1984[Mildner, D. F. R. & Carpenter, J. M. (1984). J. Appl. Cryst. 17, 249-256.]) and Barker & Pedersen (1995[Barker, J. G. & Pedersen, J. S. (1995). J. Appl. Cryst. 28, 105-114.]). We note that the expressions derived in these treatments are directly applicable to CRL-SANS in the absence of lens aberration or absorption and gravitational effects in the limit of a sharp pinhole image, i.e. zero sample aperture diameter. For the standard `focusing' aperture SANS configuration meant to maximize incident intensity at a given resolution, the beam image intensity is slightly sharper than conical and the corresponding 〈ro2〉 of the image is only about one-half of the value for a top hat image of the same full width, leading to a smaller overall shift and resolution effects. The present case is obviously an extreme example: the small-angle crystallography of large highly monodisperse objects in close proximity requiring the use of CRL-SANS. This is not a situation that arises often, although similar effects might occur in stacked or crystalline phase lamellar systems or in the packing together of biological moieties, a high degree of monodispersity being common in such systems.

Fig. 5[link](a) also illustrates to some degree the limitations of the `detector parameter' beam image averaging method as it was implemented in this initial work. There and in the second log–lin plot panel of beam image cuts (Fig. 5[link]b) we see that the primary beam image is surrounded by a weak apron of signal: three to four orders of magnitude lower in intensity, but covering a rather larger area than the primary image. Signal in this region could arise from a combination of a number of effects: lens aberration, detector position address errors, parasitic scattering from aperture edges, stray reflections in the guide and very probably small-angle pre-scattering of the incident beam in the MgF2 lens system. In combination with the wings of the actual triangular wavelength distribution, the spatial breadth of these angular resolution effects gives rise to a noticeable, if low, intensity departure of the instrument resolution from the quasi-Gaussian behavior we have assumed to arise from the convolution of well behaved distributions. The effects are visible at the edges of the first-order Bragg peak in the 205 G data and the fit shown in Fig. 2[link](b), where although the Gaussian approximation works extremely well for the central region of the peak region there is some departure at its wings. A first-order improvement to the code is currently being implemented so as to determine only angular resolution parameters for the two-dimensional map. Digital averaging, convoluting a Gaussian angular resolution distribution calculated from these parameters with the nominal triangular wavelength contribution, will then be applied within the one-dimensional reduction and fitting routines. At some computational cost further improved versions of the averaging code could have better digital representations of the angular resolution averaging function (calculated to higher moments) or undertake numerical averages as convolutions over improved numerical representations of the pixel-to-pixel angular resolution distribution and measured wavelength distributions.

Despite these minor limitations to the current implementation, the numerical data reduction method outlined above has the advantage of versatility, being applicable beyond delivering mean and resolution values for the normal isotropic small-angle scattering vector and its vector components, for which reasonably adequate estimates are available for standard instrument configurations. Since it is determined relative to a measurement of a SANS direct-beam image, it not only gives better resolution estimates on a pixel-by-pixel basis than the approximations presented above, which are truncated to second order, particularly for those close to the direct beam, but also can be applied simply in cases where approximations of parameter pixel mean and resolution values are not available and calculation methods are not readily apparent: for instance, binning against angle in the detector plane from a given axis on the detector if high-resolution orientation information is sought. Reasonable estimates of scattering vector shifts and resolution would also be delivered in the case of odd beam shapes (e.g. slit and rectangular rather than circular apertures), when the direct beam is distorted to some extent by lens imperfections or aberration, or uneven incident flux patterns (`shadows' from neutron guide gaps or misalignments), which might be due to unsuspected source configuration changes upstream of the instrument and not necessarily obvious. Also, since the detector binning and resolution affect the actual beam measurements done at the start of a measurement series, drifts or tuning errors from those aspects of instrument performance could be to some extent automatically corrected.


MCT force determination

Following Zhang & Widom (1995[Zhang, H. & Widom, M. (1995). Phys. Rev. E, 51, 2099-2103.]) and Dreyfus et al. (2009[Dreyfus, R., Lacoste, D., Bibette, J. & Baudry, J. (2009). Eur. Phys. J. E, 28, 113-123.]), we write the magnetic dipolar force between two droplets of the same aligned magnetization m as

[\eqalign{F_{\rm pair} = {{3\mu _{0}m^{2}} \over {2\pi a^{4}}}}, \eqno(16)]

where a is the center-to-center distance, our chain periodicity, and μ0 is the permeability of free space. For an extended chain of particles we add forces between pairs of particles in the chain to obtain the total attractive force along the chain,

[\eqalign{F_{\rm chain} = \sum\limits_{n = 1}^{N} n{{3\mu _{0}m^{2}} \over {2\pi (na)^{4}}} \simeq 1.2 {3\mu _{0}m^{2} \over 2\pi a^{4}}}, \eqno(17)]

where because of the rapid convergence of the series the approximation is valid for chains with only a few members. The magnetization of the ferrofluid droplets is given by

[\eqalign{m \cong {\pi d^{3}\chi^{B} \over 6\mu_{0}} \Big/ \left(1-1.2 {d^{3}\chi \over 6a^{3}}\right)}, \eqno(18)]

where the term in the numerator is the expected magnetization for a droplet of diameter d and magnetic susceptibility χ subjected to an external field B. The correction term in the denominator is another chaining effect and arises from an increased magnetization induced in each droplet owing to the magnetic field of the other droplets in the chain – with the same factor of ∼1.2 arising from the summation of pair contributions along the chain.

The magnetic properties of the maghemite ferrofluid were measured in the solid state at room temperature by vibrating sample magnetometry. The saturation magnetization and magnetic susceptibilities were obtained by fitting the experimental data to the Langevin equation (Jain et al., 2010[Jain, N., Wang, Y., Jones, S. K., Hawkett, B. S. & Warr, G. G. (2010). Langmuir, 26, 4465-4472.]). To the accuracy required here over the range of fields applied in the SANS measurements, the susceptibility can be considered approximately constant: χ ≃ 1.4. For a very similar ferrofluid, Dreyfus et al. (2009[Dreyfus, R., Lacoste, D., Bibette, J. & Baudry, J. (2009). Eur. Phys. J. E, 28, 113-123.]) found the same value using SQUID (superconducting quantum interference device) magnetometry. Combining this value with the results of equations (17[link]) and (18[link]), we obtain the attractive magnetic chaining force that must be balanced by repulsion between the droplet surfaces in a stable chain presented in the main text.


This article will form part of a virtual special issue of the journal, presenting some highlights of the 15th International Small-Angle Scattering Conference (SAS2012). This special issue will be available in early 2014.


Oak Ridge National Laboratory is managed for the US Department of Energy by UT-Batelle LLC under contract DE-AC05-00OR22725. The authors acknowledge support from the Australian Research Council, Dyno-Nobel Asia-Pacific Ltd, Clariant (Australia) Pty Ltd and travel funding from the Australian Access to Major Research Facilities Program. We acknowledge the support of the National Institute of Standards and Technology, US Department of Commerce, in providing the neutron research facilities used in this work, which are supported in part by the National Science Foundation under agreement No. DMR-0944772. We gratefully acknowledge the help of NIST Center for Neutron Research scientific and technical staff in mounting this experiment, in particular, P. D. Butler, L. Porcar and S. R. Kline. The SANS data reduction and analysis routines described herein were implemented on the eponymous MIRROR data collection, reduction and analysis code (J. B. Hayter & W. A. Hamilton, 1992–2012). The data were exported to this code using the standard NIST SANS data reduction software package (Kline, 2006[Kline, S. R. (2006). J. Appl. Cryst. 39, 895-900.]) and its results were benchmarked as appropriate against that standard.


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