Fluctuation X-ray scattering from nanorods in solution reveals weak temperature-dependent orientational ordering.

Higher-order statistical analysis of X-ray scattering from dilute solutions of polydisperse goethite nanorods was performed and revealed structural information which is inaccessible by conventional small-angle scattering. For instance, a pronounced temperature dependence of the correlated scattering from suspension was observed. The higher-order scattering terms deviate from those expected for a perfectly isotropic distribution of particle orientations, demonstrating that the method can reveal faint orientational order in apparently disordered systems. The observation of correlated scattering from polydisperse particle solutions is also encouraging for future free-electron laser experiments aimed at extracting high-resolution structural information from systems with low particle heterogeneity.

Introduction

Non-crystalline materials, such as glasses, liquids and solutions, can accommodate various structural features which are intrinsically forbidden in systems with translational symmetry. Traditionally, X-ray scattering studies of disordered matter rely on small-angle X-ray scattering (SAXS) and pair-distribution function (PDF) analysis (Als-Nielsen & McMorrow, 2011 ▸; Warren, 1990 ▸). These widely used approaches, however, provide only limited structural information and are usually insufficient for the unambiguous derivation of the 3D structure that is crucial for a complete understanding of a material’s properties. Exploring higher-order statistics of the scattered intensity can provide additional information about disordered systems beyond that accessible by conventional analyses. It has been suggested that, by calculating angular cross-correlation functions (CCFs) of the scattered intensity images, it may be possible to extract higher-order scattering terms preserved in the measured intensity fluctuations beyond the isotropic averages (Kam, 1977 ▸). By performing so-called fluctuation X-ray scattering (FXS) experiments, one could, for example, facilitate biological structure determination from solution scattering (Kam, 1977 ▸; Kam et al., 1981 ▸) or detect local orientational order and hidden symmetries in amorphous materials (Clark et al., 1983 ▸; Ackerson et al., 1985 ▸).

Practical implementation of these techniques became possible only recently, mostly due to advances in X-ray instrumentation (Kurta et al., 2016 ▸). X-ray cross-correlation analysis (XCCA) based on CCFs allowed details to be revealed of the structural arrangements in partially ordered soft-matter systems such as colloids (Wochner et al., 2009 ▸; Lehmkühler et al., 2014 ▸; Schroer et al., 2015 ▸, 2016 ▸), liquid crystals (Kurta et al., 2013c ▸; Zaluzhnyy et al., 2015 ▸, 2017b ▸, 2018 ▸), polymer blends (Kurta et al., 2015 ▸) and mesostructures (Zaluzhnyy et al., 2017a ▸; Mancini et al., 2016 ▸; Lhermitte et al., 2017 ▸). Correlations have also been explored in electron scattering (Treacy et al., 2005 ▸, 2007 ▸; Treacy & Borisenko, 2012 ▸; Liu et al., 2013a ▸, 2015 ▸) where the requirement for a small scattering volume can be conveniently achieved to observe the intensity fluctuations associated with atomic scale structures (Clark et al., 1983 ▸). The revival of the field by new experimental X-ray capabilities has been accompanied by novel developments in the theory of correlated scattering and advanced data analysis (Altarelli et al., 2010 ▸; Kirian et al., 2011 ▸; Malmerberg et al., 2015 ▸; Liu et al., 2016 ▸; Martin, 2017 ▸).

With the emergence of X-ray free-electron lasers (XFELs), the FXS approach has been extensively revised for biological structure determination from solution scattering (Saldin et al., 2009 ▸, 2011 ▸; Poon & Saldin, 2011 ▸; Kurta et al., 2012 ▸, 2013b ▸; Liu et al., 2012 ▸; Chen et al., 2013 ▸; Malmerberg et al., 2015 ▸; Kurta, 2016 ▸). Experimental demonstrations for disordered ensembles of various engineered nanostructures like nano-rice (Liu et al., 2013b ▸), polymer dumb-bells (Chen et al., 2012 ▸; Starodub et al., 2012 ▸) and three-bladed nano-propellers (Pedrini et al., 2013 ▸) indicated the general feasibility of the FXS approach. In particular, FXS measurements on crystalline nanoparticles in solution demonstrated the possibility of measuring atomic scale correlated scattering (Mendez et al., 2014 ▸, 2016 ▸). Recently, the FXS approach has successfully been applied to data taken at the Linac Coherent Light Source (LCLS) in the USA for biological structure determination. In combination with a novel iterative phasing algorithm (MTIP; Donatelli et al., 2015 ▸), FXS allowed the reconstruction of aerosolized single virus particles (Kurta et al., 2017 ▸) and multiple virus particles in solution (Pande et al., 2018 ▸) with nanometre precision.

In studies of FXS from solution, a uniform distribution of particle orientations is often assumed since it is a necessary requirement for a successful 3D reconstruction. The question is whether such a requirement is strictly fulfilled in real experiments and how it affects the resolution. Clearly, interparticle interaction may be responsible for the appearance of orientational particle correlations in concentrated solutions. In a generic SAXS experiment, the thermodynamic argument can be neglected when using dilute particle solutions, where vanishingly small particle–particle interactions result in a structure factor value close to unity. The effect of orientational order, however, has not been explored in FXS experiments, where the high sensitivity to orientational inhomogeneities may lead to the manifestation of subtle thermodynamic effects in the FXS data. On the other hand, particle motion during X-ray exposure, particularly rotational diffusion for elongated particles, can blur the contrast of the FXS data. In this work we investigate how the orientational distribution and rotational diffusion of particles in solution affect experimental FXS data. Measurements were performed on aqueous suspensions of polydisperse goethite nanorods at different volume fractions and temperatures. Our results show a pronounced temperature dependence of the correlated scattering which, to a great extent, can be associated with orientational particle correlations.

The analysis reveals that the higher-order scattering terms have larger values than expected for an isotropic distribution of particle orientations, indicative of weak orientational (nematic) ordering. This demonstrates that FXS can also be used as a high-sensitivity probe of orientational order in apparently disordered systems.

Theoretical background

We consider X-ray scattering from a dilute polydisperse mixture of N particles. The ensemble-averaged SAXS intensity for such a system can be specified as (Als-Nielsen & McMorrow, 2011 ▸)where A is an experimental normalization factor (see Appendix A ), I (q) is the scattered intensity at a momentum transfer of magnitude q for a particle of size s and D(s) is the normalized particle size distribution function, so = 1. The integration is performed over all particle sizes s, and denotes statistical averaging. The SAXS intensity, equation (1), commonly used to characterize polydisperse systems, can be interpreted as a zeroth-order term in the context of our work.

Here we introduce higher-order scattering terms for a dilute polydisperse system of particles as (see Appendix A for more details)where 〈C (q 1, q 2)〉 is the ensemble-averaged nth order Fourier component (FC) of the angular cross-correlation function (CCF) C(q 1, q 2, Δ) [see e.g. Kam (1977 ▸) and Kurta et al. (2016 ▸)], defined here for a polydisperse system of particles, and are the ensemble-averaged FCs defined for a specific particle size s. In equation (2) it is assumed that the particle orientations are uniformly distributed and is in fact a single-particle quantity.

Equation (2) defines higher-order scattering terms which are unavoidably lost in the isotropic SAXS intensity equation (1) and which we seek to extract here. For practical applications, the FCs 〈C (q 1, q 2)〉 can be experimentally approximated by the so-called difference FCs (see Appendix B ) which help in reducing the effect of various errors in the experimental data analysis (Kurta et al., 2017 ▸).

Experiment

The SAXS experiment was performed on beamline ID10 at the European Synchrotron Radiation Facility (ESRF, France) with 10 keV photon energy at about 1.3% bandwidth (pink beam). Using a double-mirror system and a set of slits, the X-ray beam was focused and collimated to a size of about 20 × 20 µm with about 1013 photons s−1 hitting the sample. The scattered intensity was recorded by a Maxipix detector (consisting of 256 × 256 square pixels, 55 µm in size) situated 515 mm downstream from the sample. A 2 mm round beamstop of Pb was placed in front of the detector to protect it from the direct beam transmitted through the sample.

The image acquisition time was 1 ms to minimize rotational motion of the nanoparticles during exposure, and a fast shutter before the sample protected it from the X-ray beam during the 0.1 s of waiting time between successive exposures. Scattering measurements were performed on dilute aqueous solutions (80 wt% propane-1,3-diol in water) of goethite (α-FeOOH) nanorods contained in glass capillaries of diameter 0.7 mm. About 104 diffraction patterns were acquired for the correlation analysis to accumulate sufficient statistics for each volume fraction of goethite particles (φg = 0.05% and 0.5%) at the different temperatures.

The powerful X-ray beam is potentially able to damage the sample, for instance resulting in gas bubbles that would lead to very strong and unwanted stray scattering. Hence, a procedure was established where every spot of the capillary only received an exposure of 10–20 ms, after which a new spot was illuminated. The waiting time between exposures ensures that diffusion creates a new spatial arrangement of nanoparticles inside the scattering volume, which is important for correct ensemble averaging.

The structure and dynamics of goethite solutes have previously been studied by X-ray scattering (Lemaire et al., 2004 ▸; Poulos et al., 2010 ▸). For the suspensions used here it has been established in structure factor studies that there is no orientational (nematic) ordering below about 4% volume concentration of particles. The exposure time of 1 ms was chosen as a compromise between the need for a strong scattering signal and the requirement that the particles remain in quasi-fixed positions during exposure. The latter can be ensured by cooling the sample, since the solvent (80 wt% propane-1,3-diol in water) increases in viscosity at low T and hence slows down the rotational diffusion over 1 ms, from ∼10° at room temperature (297 K) to ∼1° at 229 K and ∼0.1° at 209 K.

Results and discussion

SAXS images were corrected for background scattering and normalized by the average intensity per pixel. Saturated pixels, dead pixels and pixels shadowed by the beamstop were masked in the analysis, and a flat-field correction was applied to the detector. For the correlation analysis we employed the difference spectra (see Appendix B ), calculated in the range of scattering vectors from q = 0.13 to 0.43 nm−1. Fig. 1 ▸ shows a set of 2D correlation maps illustrating the amplitudes of six FCs , n = 1–6, determined at φg = 0.05% and T = 229 K. Dotted and dashed lines in Fig. 1 ▸(d) define four different sections through the 2D maps. These sections are shown in detail in Fig. 2 ▸ for n = 1–12.

Figure 1

Amplitudes (log scale, arbitrary units) of the difference FCs for n = , calculated from X-ray scattering images for the sample with a volume fraction φg = 0.05% of goethite nanorods at T = 229 K. The colour map and the axes specified in panel (d) are the same for all maps. The dashed line at q 1 = q 2 = q, and the dotted lines at q 2 = 0.15, 0.25 and 0.40 nm−1 in panel (d) indicate sections through the 2D maps (maps for n > 6 are not shown here). These sections are shown in Fig. 2 ▸.

Figure 2

Amplitudes (log scale) of the difference FCs for n = , determined at (a) q 1 = q 2 = q, (b) q 2 = 0.15 nm−1, (c) q 2 = 0.25 nm−1 and (d) q 2 = 0.40 nm−1 as a function of q 1 (see Fig. 1 ▸). Three dominant FCs of the orders n = 2 (red), n = 4 (blue), and n = 6 (green) clearly stand out from the background level [proximately indicated with dashed lines] formed by the degenerate FCs of other orders.

One can see from Figs. 1 ▸ and 2 ▸ that within the analysed portion of reciprocal space, a dominant contribution to the difference spectrum originates from FCs of low even orders, i.e. n = 2, 4 and 6. Apart from a slightly increased value of the n = 1 FC, most probably due to imperfect centring of the SAXS patterns and/or absorption effects, all other FCs have vanishing values.

Fig. 3 ▸ illustrates the importance of utilizing the difference CCF instead of the commonly used CCF 〈C (q 1, q 2, Δ)〉. As one can see from Fig. 3 ▸, both and [see equation (17)] have a complex structure with similar magnitudes of even- and odd-order FCs, indicating a strong direct correlation of diffraction patterns which we attribute to an uncompensated and structured background. In contrast, shows a smooth variation of the even-order FCs of interest, as expected for this small-angle scattering experiment.

Figure 3

Experimental FXS data calculated for φg = 0.05% and T = 229 K, showing the amplitudes of the FCs of (a) the intra-image CCF, , (b) the inter-image CCF, , and (c) the difference, [see equation (17)], determined at q 1 = q 2 = q (autocorrelation part of the Fourier spectrum) for n = .

It is noteworthy that the Fourier spectrum of the autocorrelation [Fig. 2 ▸(a)] differs substantially from the cross-correlation terms1 [Figs. 2 ▸(b)–2 ▸(d)]. This is to be expected due to the contribution of self-correlation of pixels and coherent speckle patterns (Altarelli et al., 2010 ▸; Kurta et al., 2012 ▸, 2013a ▸) that are enhanced by autocorrelation (note the bright diagonal streak on all the 2D maps in Fig. 1 ▸). As a result, the contrast of higher-order FCs of the autocorrelation function diminishes rapidly as a function of q. In the present case, the FC of the autocorrelation of order n = 6 cannot be detected above the background and the FC of the autocorrelation of order n = 4 diminishes rapidly, see Fig. 2 ▸(a). Yet, the FCs of the CCF of orders n = 2, 4 and 6 are clearly dominant across all measured q values, see Figs. 2 ▸(b)–2 ▸(d). Considering a smooth q dependence of the amplitudes (see Fig. 1 ▸), it is sufficient to analyse the FCs of the CCF determined at one particular q 2 value. Therefore, all further analyses will exclusively involve the cross-correlation terms .

The temperature dependence of the FXS data is illustrated in Fig. 4 ▸. SAXS intensities [Figs. 4 ▸(a) and 4 ▸(c)] and the dominant FCs [Figs. 4 ▸(b) and 4 ▸(d)] are determined for two samples with different volume fractions of goethite particles, φg = 0.05% [Figs. 4 ▸(a) and 4 ▸(b)] and φg = 0.5% [Figs. 4 ▸(c) and 4 ▸(d)], at different temperatures. For each volume fraction, the data measured at different temperatures were scaled according to equations (1) and (2), assuming the same average number of particles N in the beam; see Appendix C for details of the scaling procedure. The analysis reveals substantial variation in the magnitudes of as a function of temperature, while the q dependence remains almost unchanged for a particular φg. Indeed, by increasing the solution temperature from T = 229 to 297 K (φg = 0.05%), the dominant FC of order n = 2 decreases by almost two orders of magnitude, as indicated by the arrow in Fig. 4 ▸(b), while the higher-order FCs vanish completely. Similarly, by increasing the solution temperature in a sample with a higher volume fraction (φg = 0.5%), the dominant FCs (n = 2 and 4) become approximately one order of magnitude smaller, see arrow in Fig. 4 ▸(d).

Figure 4

The temperature dependence of the FXS data (log scale). (a) and (c) The SAXS intensities, and (b) and (d) the amplitudes of the FCs for n = 2, 4 and 6 and q 2 = 0.25 nm−1, determined at temperatures T = 229 and 297 K for the sample with a volume fraction of goethite nanorods φg = 0.05% [panels (a) and (b)], and at temperatures T = 209 and 229 K for the sample with φg = 0.5% [panels (c)–(d)]. The black arrows in panels (b) and (d) indicate the observed temperature drop of .

To the best of our knowledge, this is the first reported observation of temperature effects in fluctuation X-ray scattering. It was pointed out by Kam (1977 ▸) that temperature-dependent rotational diffusion of particles during X-ray exposure can smear the intensity fluctuations of diffraction patterns, thus reducing the contrast of the angular CCFs. By reducing the temperature, the viscosity of the solvent increases and rotational diffusion slows down, leading to an increase in contrast. At first glance, the observed temperature behaviour of is in agreement with this physical picture. However, as will be shown below, rotational diffusion dynamics is not the decisive factor responsible for the temperature variation in the contrast of observed in our experiment.

Experimental results for two samples with different volume fractions of goethite particles (φg = 0.5% and 0.05%) measured at the same temperature (T = 229 K) are compared in Fig. 5 ▸. The results were scaled assuming that the number of scattering particles N is ten times larger in the sample with the highest volume fraction (φg = 0.5% compared with φg = 0.05%). Ideally, in the absence of additional concentration effects the FXS data should overlap after such a rescaling, which is apparently not the case. In the q range shown in Fig. 5 ▸, both SAXS intensities and FCs have similar q dependencies at different concentrations, but the magnitudes of the FCs are notably different. To understand the origin of this effect we performed simulations of FXS for various model systems (see Appendix D ). We identified a model of the goethite solution that adequately reproduces the SAXS intensity at φg = 0.05% as well as the q dependence of the FCs (Fig. 6 ▸). The model takes particle polydispersity into account and assumes a uniform distribution of orientations, but the simulated FCs have considerably lower values (about three orders of magnitude) than the experimental results [see Fig. 6 ▸(b)]. Notably, not only is the entire simulated spectrum shifted down in magnitude, but the relative scaling of simulated FCs at different orders n is also different, for instance is much larger in the experiment than retrieved in the simulation.

Figure 5

The concentration dependence of the FXS data at T = 229 K (log scale). (a) The SAXS intensities and (b) the amplitudes of the FCs for n = 2, 4 and 6 and q 2 = 0.25 nm−1, determined for two samples with different volume fractions of goethite nanorods, φg = 0.5% and 0.05%.

Figure 6

The results of simulations for a polydisperse system of lath-shaped particles [with average dimensions a = 23 nm, b = 9.2 nm and c = 400 nm (see text)] scaled to the experimental FXS data measured at φg = 0.05% and T = 229 K. (a) The SAXS intensities and (b) the amplitudes for n = 2, 4 and 6 at q 2 = 0.25 nm−1. A Gaussian distribution of particle sizes and a uniform distribution of particle orientations were applied in the simulations.

Additional simulations have indicated that a nonuniform distribution of particle orientations may be responsible for the observed effects (see Figs. 7 ▸ and 9 ▸). Our results show that the FCs of the CCF have substantially higher values in the case of a Gaussian distribution of orientations around a mean direction, and for arbitrarily large numbers of particles they can be several orders of magnitude larger than for a perfectly uniform distribution. Importantly, in the case of a nonuniform distribution of particle orientations, the relative scaling of FCs of different orders n is also affected, closely resembling what is observed in the experiment (see Fig. 8 ▸). Similar effects can be observed for models where only a fraction of the particles obey a nonuniform distribution of orientations while the others exhibit truly random orientations (see Fig. 9 ▸).

Figure 7

The results of simulations for various types of nonuniform distribution of particle orientations. (a), (c) and (e) The SAXS intensities, and (b), (d) and (f) the amplitudes of the FCs for n = 2, 4 and 6 at q 2 = 0.25 nm−1. The results for a Gaussian distribution of particle orientations about the y axis (GaussY) with σ = 1.0 [panels (a) and (b)] and σ = 2.0 [panels (c) and (d)], as well as a Gaussian distribution of orientations about the xz plane (GaussXZ) with σ = 2.0 [panels (e) and (f)], are plotted together with the results for a uniform distribution (Unif) of orientations. The nonuniform distributions of orientations are shown schematically as insets in panels (a), (c) and (e) (see Appendix D for details).

Figure 9

The results of simulations for a partially ordered versus completely disordered system of particles. The amplitudes of the FCs for n = 2, 4 and 6 at q 2 = 0.25 nm−1 are shown. In the model of a mixed system of particles, a fraction of the particles have a Gaussian distribution of orientations about the xz plane (GaussXZ) with σ = 2.0, while all other particles obey a uniform distribution (Unif), where the corresponding fractions are specified in the figure legend.

Figure 8

The ratios between the FC amplitudes of different orders n determined for the model systems shown in Figs. 7 ▸(a), 7 ▸(c) and 7 ▸(e). The FC ratios plotted in panels (a), (b) and (c) were determined between the FCs shown in Figs. 7 ▸(b), 7 ▸(d) and 7 ▸(f), respectively. For instance, the ratio for a model of N = 104 particles with a Gaussian distribution of particle orientations about the y axis (GaussY) with σ = 1.0 is labelled in panel (a) with red triangular markers.

While the details depend on the particular parameters of the nonuniform orientational distribution, the simulations generally indicate a nonlinear dependence of the FCs on the number of particles N in the system, meaning that equation (2) does not hold in this case. Such a nonlinear scaling of FCs was deduced earlier for a 2D disordered system of particles with a Gaussian distribution of orientations about a certain direction (Kurta et al., 2012 ▸). It has been shown that the N-dependent scaling factor for an FC of nth order is equal to , where σ is the standard deviation of the Gaussian distribution. In the limit of σ → ∞ this expression tends towards N, which is the exact result for a uniform distribution2. This means that for a nonuniform distribution, FCs of different orders have different scaling parameters which depend in a nonlinear fashion on N. Clearly, in the 3D case relevant for our experiment, a specific non­uniform distribution of orientations could further alter the q dependence and relative scaling of individual FCs. While it is not feasible to obtain a general analytical result for an arbitrary non-uniform distribution of particle orientations in 3D [similar to equation (2)], simulations can still provide valuable information for qualitative analyses.

Taking into account the devised model of nanoparticle solution studied in our experiment, which involves orientational nonuniformity of particles, we can finally discuss possible origins of the observed temperature variation of , particularly the role of rotational diffusion of particles. Our simulations show (see Appendix E and Fig. 10 ▸) that, in the case of orientational order of particles, the rotational diffusion dynamics has a rather minor effect on the contrast of . Therefore, temperature-dependent orientational correlations of particles are predominantly responsible for the observed FXS contrast variation. The data available from our experiment make it difficult to distinguish whether direct particle–particle interactions or particle–wall interactions are responsible for these correlations. Considering the very low sample volume fraction, however, we are inclined to interpret our observations as a result of inadequate equilibration (due to the high solvent viscosity at low temperature) after sample loading, or an alignment effect induced by the capillary walls. A more systematic study is required to identify uniquely the physical origin of the observed temperature-dependent orientational correlations.

Figure 10

The results of simulations of rotational diffusion effects in systems with and without orientational order. The amplitudes of the FCs for n = 2, 4 and 6 at q 2 = 0.25 nm−1 are shown. In the model of a mixed system of particles, a fraction of the particles have a Gaussian distribution of orientations about the xz plane (GaussXZ) with σ = 2.0, while all other particles obey a uniform distribution (Unif), where the corresponding fractions are specified in the figure legends. The results of simulations with rotational diffusion are compared with those without rotational diffusion [shown in Fig. 9 ▸(a)]. The difference between panels (a) and (b) is in the distinct number of X-ray snapshots, (a) ten versus (b) 20, averaged to form individual averaged diffraction patterns, when simulating rotational diffusion effects (see Appendix E ).

Conclusions and outlook

Fluctuation scattering based on higher-order statistics of the scattered intensities, combined with novel capabilities of X-ray instrumentation and advanced approaches to data analysis, opens up exciting new opportunities for materials research with X-rays. Our results indicate that comprehensive information about the structure and dynamics of disordered systems can be extracted by means of angular cross-correlation functions. We have shown that FXS from solutions of nanoparticles can provide unique structural information, which is challenging or impossible to obtain by conventional SAXS approaches.

The ‘classical’ aim of FXS formulated by Kam (1977 ▸) is to use the angular CCFs as additional constraints in the particle structure determination problem. Such studies rely on a uniform distribution of particle orientations in solution, which is a prerequisite of a successful 3D reconstruction. However, the experimentally measured CCFs can be affected by two major factors, namely rotational diffusion dynamics and inter-particle interactions, leading to quite distinct outcomes for the FXS data.

According to earlier theoretical predictions for isotropic solutions (Kam, 1977 ▸; Kam et al., 1981 ▸), temperature-dependent rotational diffusion of particles during X-ray exposure can smear the intensity fluctuations of the diffraction pattern, thus reducing the contrast of the angular CCFs. Therefore, particle dynamics prevents structural information being accessed by FXS because the higher-order scattering terms vanish. To solve the problem it was suggested to cool the solution down in order to slow down the dynamics due to the resulting increase in solvent viscosity (Kam, 1977 ▸; Kam et al., 1981 ▸). The effect of rotational diffusion is naturally diminished at novel X-ray sources like X-ray free-electron lasers (XFELs) and diffraction-limited storage rings, where extremely high numbers of photons can be delivered to the sample in ultrashort pulses, much shorter than the characteristic rotational diffusion times of materials. This makes such X-ray sources well suited to structural characterization by FXS. On the other hand, angular cross-correlation functions provide a new tool for studying rotational diffusion dynamics that is notoriously difficult to access experimentally.

Interparticle interactions can introduce distortions in the angular CCFs, which may prevent them being used for particle 3D structure recovery. In conventional SAXS structural studies, the role of interparticle interactions can be effectively diminished by reducing the sample concentration, resulting in SAXS intensity curves that can be used directly for structure refinement. The FXS data appear to be much more sensitive to nonuniformities in the orientational distribution of particles, which manifest themselves in the measured CCFs even at subtle deviations from the isotropic case, in contrast to SAXS. Other sources of orientational nonuniformities may be induced by external forces (optical excitations, magnetic fields etc), the sample container (e.g. particle–wall interaction) or solute flow alignment. Therefore, the possibility of unwanted orientational ordering should be carefully considered in all experiments aimed at single-particle structure recovery from solution scattering, because the resulting nonlinear and non-trivial scaling of higher-order scattering terms will distort the structural information obtained. On the other hand, FXS may be considered as a tool to study weak orientational order and correlations in solutions, where other methods cannot provide the desired sensitivity.

In our experimental study, by varying the temperature of the goethite nanorod solution, we observed substantial changes in the FXS contrast. We also revealed significant deviations of the correlated scattering from that expected for an isotropically oriented sample. Our simulations show that nonuniformities in the orientational distribution of goethite particles may be responsible for the observed features in FXS. These results demonstrate that FXS can also be used as a sensitive probe of orientational alignment in apparently disordered systems, which is an essential capability for nanoscale studies of inhomogeneities, cooperativity and early stages of nucleation in solutions. The sample model involving orientational correlations also suggests that rotational diffusion dynamics plays a minor role in the temperature-dependent variation of the FXS contrast. In the present case, the goethite solutions are believed to be isotropic in bulk at low concentrations (≤4%), so the weak anisotropy observed is probably an artifact from the sample loading or an alignment effect induced by the capillary walls. In either case, FXS demonstrates extreme sensitivity to weak nematic ordering of anisotropic particles.