Characterization of microscopic structural order and in particular medium range order (MRO) in amorphous materials is challenging. A new technique is demonstrated that allows analysis of MRO using X-rays. Diffraction data were collected from a sample consisting of densely packed polystyrene-latex micro-spheres. Ptychography is used to reconstruct the sample transmission function and fluctuation microscopy applied to characterize structural order producing a detailed `fluctuation map' allowing analysis of the sample at two distinct length scales. Independent verification is provided via X-ray diffractometry. Simulations of dense random packing of spheres have also been used to explore the origin of the structural order measured.
© 2013 Optical Society of America
Determination of the structure of amorphous materials is an area of significant interest due to the large number of such materials that exist and their wide ranging applications. However, characterization of these materials, particularly at medium range length scales, has proven to be a challenging problem. Investigations have shown that the properties of a material may be critically dependent on the presence or absence of medium range order (MRO). Here we present a proof-of-principle demonstration of a new X-ray based approach to characterizing MRO in materials which may be approximated as ‘amorphous’ at long-range, but which exhibit structure at shorter length scales.
Although truly amorphous materials contain no structural order, many materials may appear to be amorphous at long length scales whilst still containing order at shorter length scales. For such materials the order length scales can be divided into two categories, short range order (SRO), corresponding to 2-body correlations within a material, and medium range order (MRO), corresponding to 4-body or pair-pair correlation. The length scales defined as MRO are not well established in the literature and several definitions have been proposed (see for example [1–3]). In purely practical terms however, it can be considered as order which occurs at length scales too long to appear in the pair distribution function but too short to cause peaks in Bragg diffraction .
Methods that measure the short and medium range structure of disordered materials include measurement of the first sharp diffraction peak , Raman spectroscopy , the use of X-ray diffraction microscopy to measure magnetic domains  and X-ray cross-correlation analysis . The high resolution achievable with electron and X-ray microscopy methods would appear to offer the opportunity of directly examining MRO. It has been found however, that it is difficult to reliably separate the signal due to MRO from the signal due to random structural correlations within the material , it is therefore necessary to use statistical analysis to positively identify MRO.
Fluctuation microscopy measures MRO in disordered materials by analysing statistical fluctuations in the intensity coherently scattered by the material. Originally fluctuation microscopy was developed as an electron microscopy technique  where it has been used successfully to analyse the structure of amorphous semiconductors showing the presence of strong MRO within these materials . Fluctuation microscopy studies indicate that MRO can be affected by processes such as annealing , light soaking , ion implantation  and hydrogenation .
To enable the technique to be applied to thicker materials and at different length scales fluctuation microscopy has more recently been performed using X-rays  instead of electrons. The key differences in Fluctuation X-ray microscopy (FXM) compared to the electron technique are the longer wavelength which means the characteristic length scales examined will be correspondingly longer, ranging from tens of nanometres to micrometres. Additionally, the longer scattering length of X-rays means that bulk materials can be examined, such that the method is not limited to thin films. It is therefore possible to examine subsurface or buried features using FXM and to characterize the structure of a sample through the bulk, not just at the surface. Finally, radiation sensitive materials may be more safely probed using X-rays rather than electrons to minimize beam damage.
Examples where the properties of a material may be altered by the MRO include; the optical properties of biological nano-structures, such as butterfly wings  or bird feather barbs , which depend on the ordering within the photonic crystal structure. Understanding the structure of these naturally occurring nano-structures can then be extended to the construction of artificial photonic band gap materials . Another example involves the formation of nano-composite materials such as amorphous aluminum alloys with embedded copper nano-particles . These materials have a wide range of potential applications, including use in the aircraft and automotive industries, with properties  that depend on the elements used to create the alloy.
We here present a new X-ray based technique, ptychographic fluctuation microscopy (PFM), that combines FXM with the phase retrieval method of ptychography  to characterize MRO in disordered materials. Ptychography uses diffraction data obtained from adjacent, overlapping areas of the sample to recover the phase of the diffracted wavefield and thus obtain the complex sample transmission function. The transmission function can then be analysed via a FXM algorithm to determine the MRO within the sample. This allows characterization of MRO in samples which are amorphous at long-range using one single, efficient, experimental scan representing significant improvement in terms of the amount of data required in comparison to a standard FXM experiment where multiple scans are needed at several different probe radii. In addition, the current method requires no changes to the basic experimental set up and can probe a continuous range of MRO length scales.
The sensitivity of both FXM and PFM to MRO arises from the statistical analysis of the diffracted intensity collected from many scattering regions of the sample. MRO length scales can be determined by examining the variation in diffraction signal as a function of probe diameter [22, 23]. Maximum variation in the diffracted intensity from many spatial regions of a sample is observed when the probe beam has a diameter comparable to the characteristic MRO length scale.
In FXM the probing beam is defined by placing an aperture close to the sample plane. The conventional way of altering the probe diameter is to change the size of the aperture that defines the incident illumination. A full set of fluctuation microscopy data is then collected using apertures of several different sizes  chosen such that they span the length scale range of MRO. This approach thus assumes some a priori knowledge of the MRO in the sample. The use of multiple apertures can be tedious, time-consuming and requires a significant radiation dose be applied to the sample. In addition, experimental constraints may limit the number and range of aperture diameters that can be used to characterize MRO.
It is clear that the collection of data for more probe beam sizes would greatly improve the generality of the technique. In addition, if the corresponding data collection time could be substantially reduced FXM would become much more accessible. An obvious solution to this is to use an alternative method for defining the probing beam . It has previously been suggested that the diverging beam past the focus of a Fresnel zone plate could be used as a suitable probe. In such an experiment the focus-to-sample distance can be easily altered to provide access to a wide range of probe sizes. Unfortunately, the phase curvature of the beam inherent in the system means that it is not possible to perform a conventional FXM analysis of such a data set ; PFM however, provides an alternative approach. PFM is able to address the problems associated with conventional FXM by using ptychography to reconstruct the sample transmission function from which MRO may be extracted, via fluctuation microscopy analysis. This eliminates the need for multiple scans over one sample which arises due to the normal FXM requirement for multiple beam diameters.
2. Theory and Methods
Ptychography is used to reconstruct a sub-section of an extended sample using measured diffraction data. In ptychography, diffraction patterns are collected from multiple overlapping, regions on a sample which is translated in the plane perpendicular to the beam direction. The diffraction data are combined and inverted, using knowledge of the amount of overlap, to recover the sample transmission function. Ptychography was originally proposed for electron microscopy  and was first demonstrated using the direct inversion methods of phase space deconvolution ; iterative approaches have since been developed [21, 28]. More recently the ptychographic method has been adapted to allow illumination of the sample by curved beam illumination, rather than plane wave illumination as required previously. In this case, the sample is illuminated with the diverging beam past the focus of a Fresnel zone plate, which is raster scanned over the sample such that multiple diffraction patterns are collected from overlapping regions. This divergent illuminating beam introduces phase curvature across the sample exit surface wave and a sharp intensity fall off, aiding the image reconstruction algorithms and providing a real space support constraint . Full details of the ptychographic Fresnel coherent diffractive imaging (CDI) reconstruction algorithm are provided elsewhere [30, 31].
Once reconstructed, the sample transmission function contains all necessary information about structure in the sample to the resolution limit of the reconstructed image (typically 20-60 nm) and MRO can be analysed. In the X-ray regime this means that it is possible to analyse order length scales from tens of nanometres to tens of microns. The recovered sample transmission function is then used as the input for the next stage of PFM. Both PFM and FXM characterize MRO through calculation of the normalized variance.32], but the mean square intensity, , is dependent on higher order correlations, resulting in being sensitive to MRO.
Analysis of the sample transmission function proceeds as follows: a set of probe diameters are defined. For each probe diameter N diffraction patterns are collected from different scattering regions of the sample and the normalized variance V(r) (as in Eq. (1)) is calculated. The probe function P(r) is defined to be a circle of radius 0.5r convolved with a Gaussian function of FWHM of 1 pixel, this convolution acts to blur the boundary of the circle and ameliorate any numerical effects due to the sharp edge. For each probe diameter, a set of N random positions is chosen on the sample function, T(x, y); N must be large enough to ensure adequate statistics. In more detail:
- 1. For , define P(r).
- (a) If iN then define a random position (xi, yi) on the sample T(x, y);
- (b) Multiply the probe and sample functions to get the exit surface wave:
ψESW (xi, yi) = P(r) × T(xi, yi).
where * denotes the complex conjugate.
- (e) Choose a new random position, i = i + 1.
and normalized variance,
- 3. Increment the probe diameter, r = r + δr
Full exploration of the variance space is achievable by plotting the variance against probe diameter and scattering vector to produce a fluctuation map. First the 2D variance images are radially averaged to obtain the variance as a function of q. A fluctuation map is constructed by plotting the radial averaged normalized variances for all. Any numerical artefacts due to the probe function, which remain in the calculated variances are removed using Fourier filtering.
Experiments were performed on a sample consisting of a disordered layer of polystyrene-latex spheres which was 7-12 µm thick (Fig. 1). Although this sample is ordered at very short length scales, at length scales much beyond the tenth coordination shell it can be considered to be disordered . This satisfies the sample requirement of short-range order and long-range disorder discussed in the introduction. An aqueous solution of 350 nm diameter Duke Scientific polystyrene-latex micro-spheres was first sonicated to ensure the even dispersal of the spheres. The solution was then deposited on a silicon nitride membrane which was gently heated to speed drying and encourage disorder in the resulting film.
The experiment was performed in vacuo at Sector 2-ID-B, at the Advanced Photon Source on the dedicated Fresnel imaging end-station [31, 33] using 1.83 keV X-rays. The sample was placed downstream of the focal point of a Fresnel zone plate with an outermost zone width of 60 nm; in this plane the beam had a diameter of 16 µm. An extended raster scan was performed with steps of 8 µm in x and y. A diffraction pattern was collected at each scan point with an exposure time of 0.2 s using a 2048 × 2048 pixel CCD with 13.5 µm square pixels placed 0.47 m from the sample plane. Images of the unscattered probe beam and dark field were taken before collection of sample data. Fresnel coherent diffractive imaging was used to recover the illumination function .
4. Results and Discussion
The reconstruction (Fig. 2) covers an 82 µm × 92 µm region of the sample using 90 positions with a pixel size of 24 nm. The coverage of this region allows the inclusion of all potential length scales of interest and allows sufficient statistics for fluctuation microscopy analysis.
A full fluctuation map was produced, as seen in Fig. 3, by plotting the radial averages against probe diameter and scattering vector. It is possible to perform the calculations for any desired number of probe diameters; in this analysis the probe diameters were varied between 0.04 µm and 16 µm, a range chosen to include suspected MRO length scales. The results indicate that N > 500 provides sufficient statistics; here N = 1000.
Examination of Fig. 2 shows a strong peak at q = (4.8 ± 0.2) × 10−4 Å−1 (indicated by an arrow in Fig. 2(a), which corresponds to a length scale in the material at 260 ± 9 nm. This signal is due to the close hexagonal packing of the spheres forming the sample. Additional weaker peaks may be seen in the fluctuation map at q = (7.3 ± 0.6) × 10−4 Å−1 and q = (9.3 ± 0.6) × 10−4 Å−1 corresponding to weak Bragg peaks due to the repeating units present at short length scales within the sample due to the hexagonal packing. The uncertainties are calculated with respect to the fit to the peak position in q.
MRO length scales can be extracted through analysis of the variance as a function of probe diameter. Analysis of the behaviour of the variance along q = (4.8 ± 0.2) × 10−4 Å−1 as a function of R exhibits an increase at a probe diameter of 4.94 ± 0.05 μm indicating the presence of additional order at this longer length scale. The variances along q = (7.3 ± 0.6) × 10−4 Å−1 and q = (9.3 ± 0.6) × 10−4 Å−1 also show an increase, identifying the presence of structure within the sample at 4.94 × 10−4 ± 0.05 μm, although the change in the variance is not as marked.
To understand the origins of the structural signals observed in the data, simulations of the dense random packing of spheres were performed using a modified Lubachevsky-Stillinger algorithm  in both 2 and 3 dimensions. In agreement with both the electron microscope images and the reconstructed transmission function, dense random packing simulations show that the sample is highly ordered at length scales up to 2 μm but is not ordered at longer length scales. The characteristic length scale of 260 ± 9 nm is due to the close hexagonal packing of the spheres.
At longer length scales, simulations indicate that the dense random packing of spheres will result in grouping or domains of hexagonal close packing, surrounded by dislocations . The longer length scale measured at 4.94 ± 0.05 μm corresponds to the average size of the domains formed through the dense packing of spheres, a supposition supported through close examination of the electron microscope images. Figure 4 shows the results of modelling of dense random packing of spheres in both 2D and 3D. The models are not fully representative of the sample structure but show considerable SRO forming correlated regions corresponding to MRO within the structure.
An independent diffractometry measurement was used to verify the length scales of structural order measured in this proof of principle experiment. X-ray diffraction patterns  were collected at the Photon Factory, Japan using 8.05 keV X-rays. The first oscillation of the X-ray diffraction data (indicated by the arrow in Fig. 5 located at q = 0.4 × 10−3 Å−1 indicates structure within the sample at 265 nm agreeing with the length scales measured using PFM.
In this proof of principle system, diffractometry has been used to confirm the success of the method in measuring ordering length scales within the material. As such, much of the information available in the fluctuation map is also present using diffractometry. There is, however, a wealth of information available in the diffraction map that cannot be seen in the diffractometry experiment, such as the measurement of structural order within the sample at 4.94 ± 0.05 μm. In addition, diffractometry is primarily a 1D technique; extension to 2D is possible, but complicated. Therefore, another strong advantage of PFM is the capability of 2D (and potentially 3D) analysis which can be an important feature when different areas or volumes of the sample contain significantly different amounts of disorder at long length scales. We note however, that as implied by our definition of MRO in the introduction that the sample should possess some degree of anisotropy at smaller length scales in order for higher order correlations to be significant .
In conclusion, we have presented a novel method for characterizing the medium range structure of disordered materials. This method, ptychographic fluctuation microscopy provides a compact procedure for extracting information about a sample using much less experimental data then is currently required for FXM. Strictly speaking, it is only necessary that the resolution be greater than any medium range order of interest. With this in mind, it is possible to apply this method to a range of samples with structure of interest at length scales ranging from tens of nanometres to micrometres. Since the concepts and techniques discussed here are fully complementary to electron microscopy it is envisaged that ptychographic fluctuation microscopy may be readily implemented using electrons. Providing a full complex sample transmission function can be obtained, this may allow atomic level investigation of MRO.
Another, potential application of this technique is in the analysis of “hyperuniform” disordered materials. Unlike most amorphous materials hyperuniform materials suppress infinite-wavelength density fluctuations in a way which is analogous to crystals. Disorder in hyperuniform structures therefore amounts to a ‘hidden order’ with the degree of hyperuniformity charaterised by the structure factor S(q→0) . Experimentally determining the structure factor at low values of the scattering vector is therefore critical in determining whether an amorphous system is hyperuniform. Since in the present case PFM can access the appropriate spatial frequencies it should provide a window into hyperuniformity in amorphous materials.
With the advent of ultrabright pulsed sources such as X-ray free electron lasers, potential extensions of this method could enable time-resolved measurements of medium range order in a dynamical process, through ptychographic fluctuation microscopy. X-ray free electron lasers have femtosecond repetition rates offering the opportunity to study the evolution of structural order during dynamical process such as phase change. In such an experiment a full ptychographic scan could be replaced with a single field of view of the sample, where the beam may be used as the real space constraint. The only limitation of such an experiment is that the field of view must be large enough to allow statistical analysis. The method does not require that the basic unit cell be resolved, only that the level of resolution is sufficient to reconstruct any medium range order present within the sample.
We acknowledge useful discussions with D. Paterson of the Australian Synchrotron. The authors would like to acknowledge funding received from the Australian Research Council through its Centres of Excellence and Federation Fellowship programs. We also acknowledge funding from the Australian Synchrotron. Work at the Advanced Photon Source and the Center for Nanoscale Materials was supported by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357.
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