{'fraction_non_alphanumeric': 0.025906468891635764, 'fraction_numerical': 0.014266584260403791, 'mean_word_length': 5.283818770226537, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Structural Study of a Self-Assembled Gold Mesocrystal Grain by Coherent X-ray Diffraction Imaging

Mesocrystals are nanostructured materials consisting of individual nanocrystals having a preferred crystallographic orientation. On mesoscopic length scales, the properties of mesocrystals are strongly affected by structural heterogeneity. Here, we report the detailed structural characterization of a faceted mesocrystal grain self-assembled from 60 nm sized gold nanocubes. Using coherent X-ray diffraction imaging, we determined the structure of the mesocrystal with the resolution sufficient to resolve each gold nanoparticle. The reconstructed electron density of the gold mesocrystal reveals its intrinsic structural heterogeneity, including local deviations of lattice parameters, and the presence of internal defects. The strain distribution shows that the average superlattice obtained by angular X-ray cross-correlation analysis and the real, multidomain structure of a mesocrystal are very close to each other, with a deviation less than 10 percent. These results will provide an important impact to understanding of the fundamental principles of structuring and self-assembly including ensuing properties of mesocrystals.

Assembling of individual nanocrystals (NCs) in an organized superstructure offers the possibility of combining the physical properties of the individual NCs, like surface plasmon resonances or superparamagnetism, with other physical properties provided by the superstructure, such as mechanical stiffness or geometric coordination. 1,2 NCs can be self-assembled into a superstructure, which is also known as a mesocrystal, where the separate NCs share a common crystallographic orientation. [3][4][5] Among suitable building blocks for mesocrystals, gold nanoparticles have attracted special attention due to the prospect of utilizing their surface plasmon resonance properties. For example, a cancer-selective amplification of chemoradiation with plasmonic nanobubbles has been reported for clusters of gold nanoparticles. 6,7 Similarly, gold mesocrystals show high potential in the detection of traces of chemical species using surfaceenhanced Raman scattering (SERS). [8][9][10][11][12][13] In these applications, both the shape and size of the NCs as well as their mutual arrangement are highly influential on the plasmonic properties. 14 In particular, the crystallographic structure of the assemblies (mesocrystals) determines the coupling strength and coherent superposition of the NCs plasmon polariton modes, which leads to the measured enhancement of the electromagnetic field 13,15 and the emergence of hybridized plasmon bands. 16 Moreover, when anisotropic NCs are arranged in a mesocrystalline structure, coupling and amplification of directional physical properties can be expected. 4 While the plasmonic properties of individual nanoparticles or two-dimensional (2D) assemblies can be easily characterized and related to the particle shape and size, the evaluation of optical responses from three-dimensional (3D) superstructures is not straight forward and requires exact knowledge not only of the internal symmetry, translational, and orientational order of the building blocks within the superlattice, but also structural heterogeneity (e.g. presence of defects). This knowledge is also important for the fundamental understanding of the phase behavior of nanocubes during the self-assembly process as well as the stability of superstructures. [17][18][19] Previously, we presented a detailed structural characterization of PbS and iron oxide mesocrystalline superstructures using the combination of different transmission electron microscopy (TEM) techniques (including imaging, electron diffraction, high resolution TEM, electron tomography and electron holography). [20][21][22][23][24][25] TEM is capable to resolve very small structures, but quickly reaches an application limit for multilayered superstructures and larger mesocrystals, in particular if composed of more than 20 nm sized gold particles due to their strong electron scattering and absorption. In the few micrometers thick 2D layers of mesocrystals, the structural relationship between the atomic lattice and superlattice of nanocrystals can be studied by X-ray nanodiffraction and angular X-ray cross-correlation analysis (AXCCA) techniques. [26][27][28][29][30] However, to fully understand the mesocrystal formation process and to get insight into the fundamental principles of structure -property relationship of such complex material, detailed structural characterization in 3D is crucial. To study the structure of these materials in a nondestructive way and resolve potential defects, we use here coherent X-ray diffraction imaging (CXDI). 31,32 The CXDI is a lens-less imaging technique that makes use of the coherence properties of the Xray beam. When a finite object is illuminated by a coherent X-ray beam, interferences between the incoming wavefront and the scattered beams generate a diffraction pattern, which can be recorded in the far-field by a 2D detector with proper sampling. 33 By stepwise rotating the sample and recording 2D diffraction patterns, one can measure a full 3D diffraction pattern and then use iterative algorithms to determine the 3D electron density of a sample. 34,35 A complex amplitude object is reconstructed, whose modulus is directly related to the object's electron density in the forward scattering geometry. 33 This technique has already been applied successfully to a colloidal grain of silica nanospheres (diameter 230 nm), where the accuracy of the determined positions of each colloidal sphere in the lattice was ~9 nm. 36 Here, we push forward this approach to solve the structure of a gold mesocrystal grain self-assembled from 60 nm sized gold nanocubes.

Results and Discussion
The gold NCs stabilized by cetyltrimethylammonium chloride (CTAC) were synthesized in a three-step seed-mediated method, 21 purified by centrifugation and assembled to faceted mesocrystals (slightly distorted tetragonal prisms, see Figure 1) using depletion forces (see Methods for details). To perform a detailed structural characterization of the superstructure by CXDI, a rectangular cuboid with dimensions about 1.25 μm × 1.25 μm × 1.5 μm was cut from the central part of one of the grown mesocrystals using a focused ion beam (FIB) and was mounted on a tungsten micromanipulator tip (see Figure 2).
The CXDI measurements were performed at the Coherence Applications Beamline P10, at the PETRA III (DESY, Germany) storage ring. A sketch of the experimental setup is shown in Figure 2 (see Methods for details). The stack of 2D diffraction patterns measured in our experiment was interpolated into an orthonormal frame, resulting in a 3D reciprocal space map of our grain.
An isosurface of this 3D diffraction pattern is shown in Figure 3a. One can distinguish streaks with interference fringes at low scattering angles in Figure 3b, which is due to interference of coherent X-rays on the opposite facets of the mesocrystal. In addition, several orders of superlattice reflections at larger wave vector transfers values can be observed. Between the Bragg peaks, we measured a complex speckle pattern, which encodes the local information about the relative positions of scatterers in the mesocrystal. The direct beam was masked, corresponding to the white area in the center of diffraction pattern as shown in Figure 3b. In Figure 3c, an angular average of the diffraction intensities is shown. Although we are dealing with a single-domain mesocrystal, Bragg peaks are broadened due to the highly defective structure of the ensemble. Near qz~ ±0.1 nm -1 , in Figure 3b, Bragg peaks are split, which suggests the presence of defects in the superlattice. 37 The first evaluation of the angular averaged X-ray diffraction profile shown in Figure 3c suggests a simple cubic symmetry (space group 3 � ) of the superlattice with a lattice parameter a ~ 62 nm.
We further analyzed the 3D reciprocal space data by means of the AXCCA technique, 26-30 which can provide more information about the superlattice structure. This method was specifically modified to perform analysis in 3D reciprocal space. We calculated the cross-correlation functions (CCFs) C(q1,q2,Δ) between the most prominent peaks in the reciprocal space radial profile of The 3D diffraction pattern was inverted to a real space 3D electron density map using iterative phase retrieval reconstruction algorithms, 31,32 providing a full overview of the internal structure of the mesocrystal (see for details Methods). An isosurface of the reconstructed mesocrystal together with 2D slices parallel to (100), (010) and (001) planes of the superlattice is shown in Figure 5.
The first striking observation is that CXDI can fully resolve individual nanocubes. The voxel size of the reconstruction is 9.4 nm × 9.4 nm × 9.4 nm, while the resolution estimated by the phaseretrieval transfer function 38 (PRTF) is ~21 nm, which is smaller than the 60 nm size of the individual NCs. The reconstructed structure is tilted in the vertical direction, in agreement with the unit cell parameters of the superlattice obtained by the AXCCA.
The 2D slices of the electron density of the mesocrystalline grain (see Figure 5b-d) show the presence of several type of defects including point defects, lattice bending, crack, and voids. One could argue that the structure was damaged during sample preparation by the FIB. Indeed, it has been shown that FIB milling can induce a dislocation network in the outer layers of gold NCs. 39 However, this effect is expected to be limited to the first few tens of nanometers at the surface of the object, which does not explain the high inhomogeneity of the mesocrystal structure.
Consequently, the defects likely formed during the crystal growth or the post-growth deformation processes (including the crystal contraction during the drying processes and solvent evaporation).
Defects of both origins are important since application of mesocrystals often requires the material in a dry state.
It is also evident from the reconstructed electron density (see Figure 5), that the packing order of nanoparticles within (001) planes increases substantially in the first three layers from the substrate.
We would like to note, that these (001) planes are parallel to the substrate and basal plane of mesocrystals and are perpendicular to the growth direction. In general, the ordering within (001) planes is also much higher than within (100) and (010) planes (see Figure 5). These findings are consistent with the layer growth mechanism of mesocrystals, proceeding by particle-by-particle attachment to the facet grown parallel to the substrate. In addition, during the drying process (e.g. solvent evaporation) the nonhomogeneous contraction of the mesocrystal occurs mainly perpendicular to the basal plane and additional shear stress induces the formation of additional cracks and voids, which mainly propagate perpendicular to substrate (see Figure 5). The most prominent here is a large crack through the entire mesocrystal, visible on the right side of It is worth to note, that the averaged AXCCA lattice can be seen as a distorted primitive cubic arrangement of the particles where each particle has 6 nearest neighbours and additionally 6 next-  (Figure 4d). This allows to achieve a more efficient space-filling arrangement in accordance with the so-called "bump-to-hollow" packing principle known for molecular crystals 43 and also reported for other mesocrystalline selfassembled structures. 18,22 In addition, we found, that this average primitive cell can actually be identified to have a higher symmetry and corresponds to a primitive cell of a centred monoclinic lattice that has the following parameters in the conventional setting I2/m: a = 62.2 nm, b = 89.38 nm, c = 88.25 nm, β = 109.52°. For convenience, however, we prefer to use the original primitive setting (see Figure 4d) in the whole manuscript. We also calculated the pair distribution function (PDF) based on the detected positions of individual nanocrystals within the superlattice (see Figure 6d). The PDF confirmed that the short-range order of the nanoparticles is close to an arrangement with the average cell, while the long-range order (at higher distances r) is significantly disturbed due to the presence of defects and lattice deformations in the real structure of the mesocrystal. The first peak of the PDF curve was fitted by a Gaussian function (see Figure 6d inset), revealing that the average distance to the neighboring particles of the first coordination shell is around 62.7 nm and that the precision of the detected particle positions (given by the standard deviation) is better than 5.9 nm, since the spread includes also the particle displacements due to lattice deformations.
Although the displacement map already gives a good understanding of the large lattice deformation across the whole mesocrystal volume, we additionally extracted a complete superlattice strain map. As a tool to describe the deformation of the superlattice, we defined the superlattice strain tensor as the deviation of the actually observed structure of a mesocrystal compared to the defect free lattice model (determined by the AXCCA) in analogy to the linear elasticity theory for atomic lattices 44 . This linear model, which neglects higher order derivatives, is of course only valid for small deformations. Although formally analogous to the strain as defined in linear elasticity theory (Hooke's law), herein it is merely used as a tool to describe the deviation of the observed experimental structure from the average model. Here, we are not aiming to distinguish an elastic and plastic deformation, nor can we say anything about the stress of the system. In addition, the superlattice dilatation (that is a sum of diagonal elements of the strain tensor) can easily be retrieved, giving a direct visualization of local lattice contractions and expansions (see Figure 7). Figure 7a,b illustrates the highly mosaic structure of the mesocrystal.
Even though there are many local fluctuations in the crystal structure, there certainly seems to be a tendency of positive dilatation (lattice expansion) in the vicinity of the huge crack.
The individual strain εij and rotation ωij components can be found in Figure 7b. The most obvious feature is the "bipolar" structure, for example, in εxy and ωyz, which is consistent with the lattice bending, and can be observed in the slices in Figure 5. The calculated strain tensor and rotation components are quantitatively summarized in the histograms (Figure 7c). The measured strain in all components is in a reasonable range of ±10%. The sharpest distribution is shown by the εyy tensor component, which is closest to the a3 growth direction of the mesocrystal. In order to verify the fidelity of our calculated superlattice strain, we reapplied the strain to an ideal crystal lattice (i.e. averaged superlattice determined by the AXCCA) and were able to reproduce most structural features of the experimentally measured mesocrystal.

CONCLUSION
In conclusion, we have used a combination of the angular X-ray cross-correlation analysis and coherent X-ray diffractive imaging, to study the 3D structure of a gold mesocrystal. The achieved resolution of 21 nm allowed imaging of individual nanocubes in the mesocrystal grain. Importantly, the precision of the detected particle positions was better than 6 nm. The results reveal a strongly defective structure of the mesocrystal with overall monoclinic symmetry. This technique shows a strong potential in systems that cannot be studied by classical optical or electron microscopy methods and contributes to a better understanding of structural features of self-assembled mesocrystals and especially intrinsic structural heterogeneity (including deviation of crystal symmetry, variation of lattice parameter, distribution of defects and strains). Importantly, the real mesocrystal structure was analyzed with unprecedented detail, calculating a displacement map and the entire strain tensor of the whole specimen.
The unexplored deformation behavior and related changes in structure and functional properties of nanoparticles superlattices limit their promising implementation into devices. 45 Therefore, the precise determination of their crystal structure and especially intrinsic structural heterogeneity is crucial for understanding this complex behavior. The proposed methodological approach could be used not only to perform such non-destructive structural characterization of nanostructured materials, but also to build more adequate structural models which in turn could serve as a basis for the precise prediction of the physical properties of nanoparticles self-assemblies using the computational methods.

Synthesis of Gold Nanocubes
The synthesis of Gold Nanocubes was performed by a three-step seed-mediated growth method based on a previously described procedure. 21

Preparation of Gold Mesocrystals.
An aqueous solution having approximately 10 11 particles/mL and a CPC concentration of 0.02 M was prepared. 400 µL of this solution were filled in a 1 mL shell vial with a silicon wafer (7 × 5 mm 2 ). The solution was carefully overlayered with 400 µL of 25 wt.% cetyltrimethylammonium chloride (CTAC) in H2O. After 12 h, the solution was carefully removed, and the silicon wafer was washed with acetone to yield mesocrystals.

Iterative phasing and resolution estimate
Phase retrieval was carried out on the interpolated diffracted intensity data using PyNX package, 46 imposing at each iteration that the calculated Fourier intensity of the current object agrees with the measured data. The metric used to estimate the goodness of the fit during phasing was the free log-likelihood, 47 available in PyNX. Defective pixels for experimental data and gaps in the detector were not used for imposing the reciprocal space constraint mentioned above and thus were evolving freely during phasing.
The initial support was estimated from the autocorrelation function of the 3D diffraction intensity that included only the first superlattice Bragg peaks. For larger reciprocal space this support was correspondingly rescaled. While the iterative phase retrieval this support was evolved by application of the ''shrink-wrap algorithm''. 48  iterations were used. The phasing process included implementation of the Lucy-Richardson deconvolution that takes into account partial coherence effects. 50 The resulting point spread function is shown in Figure 8a-c of Supporting Information. To ensure the best reconstruction possible, we kept only the best 10 reconstructions from 1000 with random phase start and performed the mode decomposition. 47 The weight of the most prominent mode which was considered as a final result was 69%.
The resolution of the reconstruction was estimated using the normalized Phase-Retrieval Transfer Function 3 (PRTF) at a cutoff value of 1/e (see Figure 8d). The PRTF is a measure of how well the retrieved Fourier amplitudes match the square root of the measured diffraction intensity.
After calculating the ratio of the reconstructed and measured amplitudes, the obtained 3D PRTF was azimuthally averaged over shells of constant q and normalized to obtain the result shown in