3.1.1 MOCs and their assemblies

Metal oxide clusters (MOCs) can be considered as soluble metal oxides and they are polynuclear species with multiple metal centers, linked and stabilized by H₂O, OH⁻, and O²⁻ ligands. These include classic early transition metal polyoxometalates (POMs), uranyl peroxide capsules, polyoxocations, and organically ligated metal-oxo clusters. Unlike metastable colloidal suspensions, ionic charge and discrete and small sizes (from sub-nm to several nms) lead to stabilizations.^{15, 38} Though MOCs have been studied for over a century, they are still fertile for research discovery. The heavy metals of MOCs yield SAXS patterns with well-resolved and characteristic features from which the size and shape can be derived.³⁹ Moreover, the typical q range of SWAXS (small and wide angles combined) lies between 0.01 to 2.5 Å⁻¹, corresponding to hundreds to sub-nm spatial coverage, which is appropriate to describe the structure of discrete MOCs⁴⁰ and higher order assemblies.⁴¹

To determine MOC topologies, model fitting should be conducted, and the development of the scattering functions of classical geometrical models is critical.⁴² Generally available shapes are limited, including spheroids, discs, and cylinders. {Mo₁₅₄}, ([Mo₁₅₄(NO)₁₄O₄₄₈(H₂O)₇₀H₂₈]¹⁴⁻), a donut/torus-shaped cluster has been documented in prior studies⁴³ (Figure 2a) to undergo structural transformation to derivatives such as {Mo₁₅₀} and {Mo₁₂₈Eu₄}, or binding with rare earth ions, if present. To better describe the structures of {Mo₁₅₄} and its derivatives, Yin developed a custom carved ellipsoid model and the scattering function was systematically derived (Figure 2b).⁴⁴ This work provided a new geometric model for the scattering community and facilitates the structural studies of other toroidal clusters including [P₈W₄₈O₁₈₄]⁴³⁻,⁴⁵ M^IV₇₀ rings (M=U, Ce, Zr),⁴⁶ Mn-oxo rings,⁴⁷ and palladate rings.⁴⁸

In some cases, structural information of MOCs can be read directly from the scattering curves without complex model fitting. Tools such as SolX⁴⁹ can very accurately simulate scattering data from crystal structures. For example, Yin et al. studied solution behavior of a series of POMs with diverse architectures (Figure 2c). The theoretical curves derived from crystal structure models match very well with the experimental SAXS, implying that their solid-state structures remain intact upon redissolution (Figure 2d).⁵⁰ This analytical process is valuable when determining stability conditions, extracting information from polydisperse solutions and understanding assembly pathways. However, the necessary perquisite is a single crystal X-ray data for the cluster of interest. Oftentimes, supramolecular assembly or change in speciation can yield scattering data that is not identical to simulated data of the isolated cluster.⁵¹ Nonetheless, the structure is always a good starting point to understand complex and dynamic speciation.

Ion pairing is common for POMs and orchestrates assembly and function. However, studies of cations-POM assembly are usually based on indirect methods that are blind to structure. Antonio first used SAXS to directly observe the ion pairing of hexaniobate [Nb₆O₁₉]⁸⁻ in aqueous solution.^39b The value of radius of gyration (R_g) determined by Guinier analysis was used to detect ion-pairing. The R_g of [Nb₆O₁₉]⁸⁻ associated with alkali metal ions is 0.3–0.8 Å larger than that of the bare anionic cluster and is consistent with the theoretical R_g calculated from SCXRD, revealing remarkable similarity between solution and solid association of alkalis and POMs. However, for large POMs, i.e. {Mo₇₂Fe₃₀}, the change of R_g due to ion association is negligible. ASAXS also accesses ion-pairing information by resolving element-specific contributions to the X-ray scattering intensity. By varying the incident X-ray energy around the Rb⁺ absorption edge, the contribution of Rb⁺ to the SAXS of {Mo₇₂V₃₀} varies and indicated replacement of K⁺ with Rb⁺ in solution (Figure 3a). Based on ASAXS, Liu et al. also determined the accurate distribution of Rb⁺ and Sr²⁺ around {Mo₁₃₂}.⁵² The resonant term of ASAXS data is only related to the spatial distribution of counterions and is fitted with multi-layered spherical shell model (Figure 3b) to calculate the radial concentration profiles of Rb⁺ and Sr²⁺ (Figure 3c). It was found that most Rb⁺ ions were in the Stern layer of {Mo₁₃₂} while more Sr²⁺ ions loosely associated with {Mo₁₃₂} in the diffuse layer. This result explained the lower critical coagulation concentration of {Mo₁₃₂} with Rb⁺ compared with Sr²⁺ (Figure 3d).⁵³

For X-ray and neutron scattering, the structure factor $$ describes the contribution of interactions among scatters and it can be used to characterize the interaction potential among POMs as well as the formation of supramolecular assembly (Figure 4a). SAXS is exploited by Antonio et al. to study the solution behaviors of Keggin type POMs and the structure factor $$ is used to accurately identify the interaction potential.⁵⁴ Two correlation peaks (P₁ and P₂) can be observed from the $$ plot of P-HPA ([PW₁₂O₄₀]³⁻), while only P₁ peak appears for Si-HPA ([SiW₁₂O₄₀]⁴⁻), and Al-HPA ([AlW₁₂O₄₀]⁵⁻) (Figure 4b). The P₁ peak of Si-HPA and Al-HPA solutions corresponds to a correlation distance of 14.6 Å, which is larger than the sizes of Keggin type POMs. This suggests they remain as discrete clusters without aggregation. As for the P₂ peak of P-HPA, a correlation distance (8.6 Å) close to the POM's diameter can be quantified, demonstrating the aggregation of P-HPAs via direct contact. Meanwhile, its P₁ peak corresponding to 20.2 Å manifests the existence of long-range interactions among the P-HPA units. Further study shows that the temperature and concentration are important factors in governing the interplays among POMs in solutions (Figure 4c and 4d). These experimental observations are well consistent with simulation data. Also, the pH-regulated interactions of POMs are systematically investigated by Bera et al.⁵⁵ The $$ at pH=0 exhibits a shoulder peak near q=0 due to the strong attractive forces, whereas the repulsive interactions dominate and the peak at low q disappears. MSA model fitting on $$ data provides comprehensive structure information, verifing that the POM charge changes from −3 to −7 at pH=4.7 (Figure 4e and 4f). Bauduin et al. applied the same protocol to study the effects of ionic strength in regulating the inter-POM interactions.⁵⁶ Moreover, SAXS methodology offers deeper insights into the crystallization process for Keggin POMs.⁵⁷ The structure factor data directly reflects that the crystallization is proceed via two-step processes. The first step involves the formation of liquid-like correlated phases, followed by the crystal growth inside the as-formed dense liquid phase in the next step.

Oberdisse et al. proposed a general approach based on the reverse Monte Carlo (RMC) computation to identify the percolation.⁵⁸ The experimental SAXS profile is reproduced using RMC in a simulation box with length $$ . The information of NP aggregations was extracted from the configurations by employing an aggregation recognition algorithm (Figure 5a and 5b).⁵⁸ The slice images of the RMC simulation box at different volume fractions of NPs are consistent with the transmission electron microscopy (TEM) observations, which proves the validity of this protocol (Figure 5c). This method can be universally applied to diverse chemical systems.⁵⁹ Moreover, the application of this method doesn't require any experience in analyzing SAXS data, which further promotes its versatility. The combination of scattering and RMC techniques provides a powerful toolbox for quantitative studies on the complicated solution behaviors of nanostructures, in addition to correlating non-intuitive SAXS with intuitive imaging.

Capsule-like POMs are ideal models for studying the nanoconfined water.⁶⁰ Yin used contrast variation small angle neutron scattering (CV-SANS) to investigate the distributions of water molecules inside capsule POMs (Figure 5d).⁶¹ The scattering contrasts are regulated by varying the ratios of H₂O and D₂O. A global-fit strategy assuming the water-filled POM as a core-shell structure is adopted to analyze the experimental data at various contrasts, and the number of water molecules inside the cavity was quantified (Figure 5e). Also, the density of confined water was quantified by excluding the volume of internally bound ligands (alkyl chain ligands with different lengths), and it was found that the water density decreases with the increasing length of ligands. Interestingly, the POM cavity exhibited complete dewetting with a core dimension of ≈11 Å, which is much larger than water molecules. This critical dewetting dimension is consistent with the length scale of a hydrogen bonded network of water, yet, a hydrogen bond network is geometrically impossible in this condition. The observed unique dewetting behavior confirms the existence of crossover of hydrophobicity at the nanometer scale.

3.1.2 Hybrid nanostructures and supramolecular assemblies

Through efforts of Cadot and Falaise,⁶³ there has been considerable success in using cyclodextrin (CD) to create supra-structures of POMs and metal-metal bonded clusters. Hydrogen bonding between the CD and the anionic clusters is the driving force for assembly. In one example, they assembled a cluster inside two CD rings, inside a molybdate ring, formulated [Re₆Te₈(CN)₆]@2(γ-CD))@[Mo₁₅₂O₄₅₇H₁₄ (H₂O)₆₈] or Re₆Te₈@CD@Mo₁₅₂ for short (Figure 6a).⁶² They used both SAXS and SCXRD to benchmark the assembly process. SAXS comparing {Mo₁₅₂} alone and {Mo₁₅₂} plus CD and Re₆Te₈ shows differences, which are readily explained by the pair distance distribution function (PDDF) analysis (Figure 6b). Specifically, while the maximum linear extent remains identical between the two solutions, commensurate with the diameter of {Mo₁₅₂} (≈35 Å), there is a distinct difference in intensity of the maxima centered at ≈15 Å. The PDDF of {Mo₁₅₂} alone has the characteristic shape of a core-shell profile with a lower shoulder on the left side, due to diminished scattering from the empty center. The maximum increases in intensity above that of the peak at ≈30 Å (from the {Mo₁₅₂}shell) with introduction of the Re₆Te₈ cluster plus the CD, which is due to the higher electron density of the core when it contains this heavy element cluster. This SAXS study clearly shows pre-assembly of Re₆Te₈@CD@Mo₁₅₂ prior to crystallization.

Metal organic polyhedra (MOP) skeletons provide robust molecular templates for the fabrication of hybrid nanostructures.⁶⁴ Enriched functional groups including alkyl chains,⁶⁵ polymers,⁶⁶ and clusters,⁶⁷ can be appended to the surface of MOPs via coordination driven self-assembly. Scattering methods offer overwhelming advantages in characterizing this type of hybrid structures, due to the intrinsic paramagnetic features and complicated hierarchical structures.⁶⁸ Taking PS-MOPs (polystyrene functionalized Cu-MOPs) as an example, scattering methods are used to characterize their structures.⁶⁹ Regarding the organic-inorganic structural features of PS-MOP, SAXS and SANS cooperatively reveal their core-shell structures with a high electron density core and hydrogen-rich shell from the geometric model fitting, which agrees well with the molecular design (Figure 7a and 7b). The surface-grafted polymer layer significantly altered the solvation structure of PS-MOP, as determined by SANS. In the deuterated THF solution of PS-MOPs, the chemical composition of the solvent inside the porous MOP core, which is indexed by the scattering length density (SLD), remains that of the original solvent, rather than the deuterated solvent. This suggests the presence of the rigid and thick PS layer severely prohibits mass exchange between the MOP core and external environment.

Alkyl chain-coated MOPs (C_n-MOPs, n denotes carbon atom number of the alkyl chain) have demonstrated potential applications as selective ion channels, yet the underlying molecular mechanisms remain unclear.⁷⁰ The neutral ‘hairy’ MOPs remain as discrete particles in solution, which enables topology studies. Scattering data confirms the typical core-shell architectures of the C_n-MOPs (Figure 7c).⁷¹ CV-SANS experimental results demonstrate that the solvent molecules can penetrate the surface alkyl chains and reside inside the MOP cores, which means the mass exchange between the C_n-MOPs and external environment is possible. LB thin films of C_n-MOPs were fabricated at the water/air interface. Neutron reflectometry measurements were conducted to probe the monolayer structures normal to the surface of hairy MOPs under compression (Figure 7d). The neutron reflective data suggests the MOP cores are in the middle while alkyl chains are mainly distributed upward and downward. The bottom alkyl layer is thinner with higher packing density due to hydrophobic interactions.

SoINs can be chemically integrated into gel networks.⁷² Johnson et al. prepared a set of hybrid gels, in which the MOP cores are served as junctions for stabilizing the gel networks.⁷³ SANS well describes the hierarchical structures of these gels (Figure 8a and 8b).⁷³ A summed model fits the experimental scattering data well and the derivate structure parameters conform to the expected values, proving that the gel structures are as designed. Photo-reversible molecular units are further employed for the fabrication of smart hybrid gels.⁷⁴ After UV irradiation, the SAXS curve exhibits a well-resolved peak at low q region and five new peaks arise in the high-q regime, corresponding to the form factor of Pd₂₄L₄₈ cage with a 2.9 nm radius (Figure 8c). SAXS verifies the structural transitions of MOP junctions within the gels (Figure 8d), which provides the molecular origin of the observable changes of their mechanical properties. Recently, Richtering et al. harnessed the superchaotropic effect of silicotungstic acid (HSiW) to crosslink the hydroxypropylcellulose (HPC), and thereby facilitating the gelation process.⁷⁵ Notably, upon on the addition of 1 mM HSiW, the correlation peak at low q can be observed as the structure factor for the repulsive interaction among charged NPs. Cylinder model fitting at high q region affords a characteristic dimension, which shows monotonic increase with increasing HSiW loadings. This suggests that nanoscaled HSiW induces the aggregation of HPC chains (Figure 8f). Due to the electrostatic screening effects of NaCl, the repulsions among HPC chains are dramatically suppressed, supported by the disappearance of correlation peaks in the SANS plot (Figure 8e).

3.2.1 Reaction kinetics and formation mechanisms of metal oxide clusters

Many metal oxide clusters are first discovered by crystallization and structural analysis, however dedicating efforts and time are required for crystallization. Solution scattering characterization provides new insight into kinetics and mechanisms of cluster formation, and then the SCXRD data becomes supporting information via simulated scattering data. The structure of ferrihydrite, poorly crystalline iron oxyhydroxide that comprises rust, soil, and the core of ferritin, has been the subject of debate for decades. Central to the debate is the presence of tetrahedral iron in the structure. Michel and Parise⁷⁶ proposed a structural model that contains the Fe₁₃ Keggin cluster as the core building unit, via PDF from X-ray total scattering. The Fe₁₃ Keggin cluster contains tetrahedral iron surrounded by twelve octahedral irons, akin to POM Keggin ions discussed throughout this review. Sadeghi applied SAXS to study the formation and manipulation of Fe₁₃ in solution (Figure 9),^51a informed by crystallizing the compound with capping Bi³⁺-countercations. This highly reactive Fe₁₃-core can be stabilized in water via capping with six Bi³⁺ plus ligands, Bi₆[FeO₄Fe₁₂O₁₂(OH)₁₂(O₂C(CCl₃)₁₂]⁺¹ (Bi₆Fe₁₃, Figure 9a, blue box inset). Bi₆Fe₁₃ redissolved in acetone yields scattering consistent with the complete Bi₆Fe₁₃ cluster (Figure 9a, blue). The scattering of the synthesis solution (Figure 9a, green), on the other hand is consistent with the Bi₆Fe₁₃ cluster without the ligands, suggesting the Bi³⁺ is important in stabilizing the Fe₁₃ core, but the ligands just assist crystallization. In contrast, SAXS of the synthesis solution without Bi³⁺ shows higher scattering intensity at low q, representing polydispersity and formation of larger aggregates (Figure 9a, red). A log-normal particle size distribution analysis informs a mode radius of 11.2 Å (Figure 9a, inset), approximately 8× larger in volume than the isolated Fe₁₃. This highlights the significance of Bi³⁺ in stabilizing the small Fe₁₃ cluster without aggregation. Removing the ligands and Bi³⁺ leads to supramolecular assembly of the cluster into chain-like nanostructures with a primary particle radius of 11 Å (Figure 9b), consistent with the linear arrangements of natural ferrihydrite NPs. TEM images provide intuitive evidence for the chosen form factors for data modeling, and selected area electron diffraction indicates incipient formation of ferrihydrite (Figure 9b). The SAXS analysis supports that Fe₁₃ is a key pre-nucleation cluster for the Michel-Parise model for ferrihydrite formation.⁷⁶ Shaw and colleagues later executed a chemically simpler experiment by exploiting pseudo in situ SAXS measurement (Figure 9c).⁷⁷ They simply pH-adjusted a ≈7 mM solution of Fe³⁺ from 0.1 to 4 with NaOH. The reaction solution was continuously stirred and pumped through the sample chamber for analysis, and data was collected as a function of increasing pH. This scattering data similarly showed the formation of Fe₁₃, prior to assembly of larger linear aggregates.

Among the POM families, polyoxoniobates (PONbs) are unique with high negative charges that lead to strong interactions with alkali-countercations. Beyond the role in charge-balancing, Sures showed that alkalis alone can drive speciation changes in PONbs (Figure 10a).⁷⁸ Specifically, addition of ACl (A=Li, Na, K, Rb, Cs) to aqueous decaniobate (Nb₁₀O₂₈⁶⁻, Nb₁₀), as observed by SAXS, leads to formation of much larger clusters. Moreover, the rate of the reaction and the resultant cluster size increase with alkali size. The increasing size can be seen with increasing scattering intensity and shift of the Guinier region to lower-q, Figure 10a. At the time of publication,⁷⁸ Nyman was never able to grow crystals to prove what was present in solution. In a separate study, they employed the same strategy using alkali carbonates, and the increased basicity from the carbonate yielded [Nb₂₄O₇₂]²⁴⁻ (Nb₂₄) in solution that are not linked by the alkalis.⁷⁹ Again, this was determined primarily by SAXS analysis. PDDF analysis to obtain a real-space shape representation of these cluster solutions (Figure 10b) produced a profile representative of a core-shell type structure with a size similar to Nb₂₄ (Figure 10c), reported prior.⁸⁰ With this hypothesis as a starting point, they simulated alkali-linked Nb₂₄ units and determined that increase size of species detected by SAXS with increased alkali size is due to increased alkali-linking of Nb₂₄ (i.e. dimers for Li and Na, tetramers for Rb and Cs).⁷⁸ They characterized both Nb₁₀ and Nb₂₄ by benchtop PDF in solution, strengthening the argument for Nb₁₀ to Nb₂₄ conversion.⁷⁸ Moreover, they hypothesized the challenge to crystallization was the lability of the three H₂O−Nb=O units (Figure 10c) that link the three Nb₇ units (polyhedral representation, Figure 10c) into the ring structure. In follow-on attempts to crystallize Nb₂₄, they added uranyl to the reaction, with a more stable O=U=O unit. Indeed two structures were obtained that feature the Nb₂₄ unit with substitution of the uranyl on the H₂O−Nb=O site,⁸¹ confirming assignment of structure from only the SAXS data was correct.

POMs are stable in solution and can be detected via SAXS in reaction solutions to optimize conditions for crystallization. Falaise⁸² demonstrated that the capsule-shaped uranyl peroxide polyoxometalate U₂₈ ([(UO₂) (O₂)_1.5]₂₈²⁸⁻), forms instantaneously upon dissolving studtite (UO₂(O₂)(H₂O)₂ ⋅ 2 H₂O) in LiOH. The SAXS of these uranyl peroxide capsules (Figure 11a) are very distinct due to the high electron density contrast between core (containing water and alkali-cations) and shell (composed of uranyl peroxide), including the presence of strong oscillations. The solution was aged for 60 days, which led to a slight shift in both the oscillations and the Guinier elbow to higher q, indicating evolution to a slightly smaller capsule. Falaise presumed this indicated evolution to U₂₄ ([UO₂(O₂) (OH)]₂₄²⁴⁻), with decomposition of the peroxides and replacement by dihydroxides. Indeed, U₂₄ was crystallized from solution via vapor diffusion of methanol, proving this hypothesis correct.

On the other hand, in multiple instances, reaction solutions of polyoxocations reveal polydispersity and smaller cluster fragments via SAXS. This is due to: 1) the lability of these clusters that feature water and hydroxyl ligands instead of the multiply bound oxo-ligands that stabilize the surface of transition metal and uranyl polyoxometalates,⁸³ or 2) extreme synthesis conditions that promote equilibrium between clusters and fragments. The latter phenomenon was observed in assembly and crystallization of Zr₂₅-peroxide ([Zr₂₅O₁₀(OH)₅₀(O₂)₅(H₂O)₄₀]²⁰⁺, Figure 11b, bottom, inset).⁸⁴ Zr₂₅ was obtained simply by room temperature evaporation of a 1 : 1 peroxide:Zr solution. Over four days of evaporation prior to crystallization, the SAXS remained invariant, suggesting a mixture of cluster fragments much smaller than the fully-formed pentagonal shaped Zr₂₅ (Figure 11b, top). Isolated Zr₂₅, can be redissolved in methanol and DMF, in addition to water. Redissolved in water, the SAXS of Zr₂₅ again indicates fragmentation. However, dissolution of Zr₂₅ crystals in the organic solvents show intact Zr₂₅; again, by comparison to simulated scattering (Figure 11b, bottom).

Aluminium Keggin ions, i.e. Al₁₃ (AlO₄Al₁₂(OH)₂₄(H₂O)₁₂⁷⁺) are amongst the most studied polyoxocations for developing aqueous reactivity and speciation diagrams, building blocks for materials, and commercial products in water treatment and personal care.⁸⁵ Given the similarities between aqueous Al³⁺ and Cr³⁺, there were early attempts to isolate a Cr₁₃ analogue.⁸⁶ Wang⁸⁷ and Amiri⁸⁸ accomplished this goal, and a central tetrahedral Zn²⁺ was one key to isolate both a mixed Zn(Cr,Al)₁₂ Keggin ion and a ZnCr₁₂ Keggin ion; the former confirms the similarity of Al³⁺ and Cr³⁺. However, this was not the only key, and X-ray scattering providing insight into unusual reaction pathways. SAXS and mass spectroscopy indicated that both Zn(Cr,Al)₁₂⁸⁷ and ZnCr₁₂⁸⁸ never fully assembled in the reaction solution, rather fragment formulae were detected (Figure 11c, Zn(Cr,Al)₁₂). To complement SAXS and ESI-MS, the ZnCr₁₂ assembly was studied by benchtop PDF (Figure 11d), and this also evidenced reaction solutions only contained cluster fragments. In this combined PDF/SAXS study, reaction time points differing by the amount of nitric acid resulted in growth of peaks a and b (Figure 11d), which represents Cr−Cr and Zn−Cr pair correlations indicating assembly of larger cluster fragments. On the other hand, SAXS and PDF of redissolved crystals indicated their stability in water, again by comparison to simulated scattering (Figure 11b and 11c). The solution conditions for both studies (high acidity and high nitrate) supress hydrolysis, which only permits formation of smaller clusters. Yet synthesis conditions that could support the full Keggin ion (lower ionic strength) only yielded amorphous precipitates.

The indepth Zn(Cr,Al)₁₂⁸⁷ study concluded that the fully-formed clusters must assemble at the air-liquid interface where nano-environments promote lower acidity and higher metal concentration, forcing hydrolysis and nucleation, without uncontrolled bulk precipitation. Bera et al.⁸⁹ showed exactly this by in situ grazing incidence XAS, comparing the bulk (monomers) to the surface (clusters) of an erbium chloride solution. To summarize, controlled growth and especially crystallization of aqueous polyoxocations (examples reviewed here include Zr-peroxide, Al and Cr) require conditions that do not support persistence of the fully-formed clusters, and X-ray scattering was key to providing this information. However, much information is still missing. To access this information, advanced synchrotron instrumentation that allows spatiotemporal studies (i.e. distinguishing bulk from surface) such as that demonstrated by Bera is extremely important.

Recently, Sommers et al.⁹⁰ used crystallization and SAXS to describe the atomic level details of the Zr−Hf industrial solvent extraction separation process, crucial to the nuclear industry since the 1950s.⁹¹ Hf is extracted selectively into an organic phase from crude Zr ore (containing a few % Hf) in an aqueous HCl-ammonium thiocyanate solution. SAXS informed every step of the extraction. First, oxo-centered tetrahedral clusters OM₄ (M=Hf, Zr, [M₄O(OH)₆(NCS)₁₂](NH₄)_4, Figure 12a inset) were crystallized from both aqueous Zr-ammonium thiocyanate and Hf-ammonium thiocyanate solutions, prepared separately to determine speciation differences between Zr and Hf. This result did not explain the difference between Zr and Hf, but SAXS of these aqueous phases began to provide an answer (Figure 12a). First, simulated SAXS of OHf₄ matches very well with the aqueous Hf solution, while the aqueous Zr solution shows the presence of larger species (≈10 % by volume larger species, determined by size distribution analysis). The larger clusters were coaxed out of solution with addition of Cs-salt, and SCXRD revealed a cluster with a nuclearity of 48, Zr₄₈. Simulation of Zr₄₈ scattering (Figure 12a) confirmed the Zr₄₈ was responsible for the shifted Guinier region and increased intensity, compared to the analogous Hf solution. SAXS of the organic phases, post Hf and Zr extraction, respectively show extraction of OHf₄, and minimal extraction of any Zr-species (Figure 12b). While the simulated OHf₄ matches well with the organic Hf-extractant, the organic Zr-extractant exhibits a shift to higher-q compared to the simulated scattering. Aside from comparison to the simulated scattering data, there are other indications of poor extraction. The intensity at the lowest observed q-value of the Hf scattering is an order of magnitude higher than that of the Zr scattering. Second, comparison of solvent scattering (q>1 Å⁻¹) to molecule scattering (q<1 Å⁻¹) shows that the Hf-organic phase contains considerably higher concentration of extracted species than the Zr-organic phase.

All of these SAXS and crystallization experiments were performed using a 3.3 : 1 NH₄SCN : M (M=Hf, Zr) ratio, identical to that used in the industrial process, and only 10 % excess SCN compared to stoichiometry of OM₄. By doubling the ratio to 6.6 : 1, formation of Zr₄₈ is supressed and both the Hf and Zr solutions clearly show pure phase OM₄ (Figure 12c). In this experiment, the power of X-ray scattering, specifically sensitivity to electron density, is revealed. Although isostructural OZr₄ and OHf₄ have the exact same size, OHf₄ appears smaller in both the experimental and simulated scattering, corresponding with a shift to higher-q in the Guinier region, compared to OZr₄. This is due to higher electron density contrast between core Hf and peripheral S, which decreases the magnitude of scattering vectors through the cluster.

Small clusters and counterions are usually considered as the transient molecular templates in the formation pathway of protein-sized MOCs, which refers to the well-accepted templated synthesis mechanism. Yin et al. used time-resolved SAXS to reveal the formation mechanism of [Mo^VI₇₂Mo^V₆₀O₃₇₂(CH₃COO)₃₀(H₂O)₇₂]⁴²⁻ ({Mo₁₃₂}).⁹² A theoretical scattering curve of {Mo₁₃₂} provides a good match with experimental SAXS, confirming it's stability in solution. The original molybdenum precursors deliver featureless scattering plot. With time flies, characteristic oscillations appear at high q and become more distinct, indicative of {Mo₁₃₂} assembling (Figure 13a). More importantly, the intensity of the first oscillation at q≈0.4 Å⁻¹ can be used to quantitatively analyze the growth kinetics. The plot of peak intensity versus time gives a sigmoid curve, which is consistent with the kinetics data obtained by NMR. Combined with UV/Vis data, multi-step formation mechanisms, in which {MoV₂(CH₃COO)} is the transient template is proposed. In a separate study, SAXS was employed to uncover the formation processes for core-shell shaped Keg@{Mo₇₂Fe₃₀}.^60b The operando SAXS data demonstrates formation of Keg@{Mo₇₂Fe₃₀} in solution with time (Figure 13b). Finally, a three-stage formation mechanism can be rationally deduced based on the in situ SAXS data with high spatiotemporal resolution.

3.2.2 Structural Dynamics and assembly pathways of hybrid nanostructures

Hybrid surfactant can self-organize into micelles, which further coagulates into nanobelts in water (Figure 14a). By applying time-resolved SAXS, the stepwise assembly processes were revealed by Liu et al.⁹³ SAXS of the freshly prepared hybrid surfactants solution shows typical scattering of core-shell structures. The total radius, core radius, and shell thickness can all be quantified from model fitting. With aging, the characteristic core-shell oscillations became weaker and completely disappeared after 680 min, while a sharp peak centered at q=0.202 Å⁻¹ rises and grows simultaneously (Figure 14b). This scattering feature corresponds to an interlayer distance of 3.11 nm, originating from the lamellar packing of the hybrids. Sigmoidal shaped feature was observed for the peak intensity plot as a function of time, suggesting the slow coalescence rate of the micelles at the initial time. More interestingly, Weinstock et al⁹⁴ realized the successful polymerization of a bifunctional Keggin cluster to afford a linear polymer with entirely inorganic backbones (Figure 14c). The flexible μ-oxo-linked backbone permits the as-formed polymer to coil, analogous to typical linear polymers (inset picture in Figure 14d). By fitting the SAXS data, it is confirmed that compact sphere with average radius of ca. 15 nm (green curve) and individual Keggin ions (red curve) both exist (Figure 14d), which is supported by dynamic light scattering and cryo-TEM. This work demonstrates the exciting potentials of inorganic nanostructures as buliding blocks for the fabrication of functional materials with higher ordered structures.

The study on the dynamics of coordination bonds is still in it's fancy. Yin et al. explored the dynamic behaviors of MOPs, that is, the coordination bonds can be dynamically broken and reformed at characteristic time scales in solutions.⁹⁵ The high sensitivity towards light elements of SANS techniques brings overwhelming advantages for uncovering the hierarchical inorganic-organic nanostructures as well as their dynamics.⁹⁶ The ligand exchange processes can be tracked by time-resolved SANS. The addition of excess PS ligand to the solution of alkyl chain-coated MOPs triggers the ligand exchange process. The free PS ligands adapted with Gaussian coil conformations are bonded together to form core-shell shaped nanostructures, supported by the SANS data (Figure 15a). In addition, ligand exchange among different MOP species is monitored during the reaction process. Initially, the SANS pattern implies a wider size distribution of mixed MOPs in the solution. Up to 48 h, the exchange reaction has reached equilibrium. The final product presents a narrower size distribution and relatively lower scattering contrast, which demonstrates the two cages are chemically fused yielding a stable hybrid MOP, resulting from dynamic ligand exchange (Figure 15b). This unique dynamic behavior can be further harnessed for the construction of MOP cross-linked vitrimer for gas separation.⁹⁷

POSSs are a family of cage like molecular entities comprised of alternative connected Si−O bonds.⁹⁸ Recently, norbornene was chemically-appended to the POSS surface.⁹⁹ The polymerization of this precursor leads to the generation of a millipede-shaped hybrid polymer. Unexpectedly, this polymer has excellent mechanical properties without cross-linking and strong supramolecular interactions. SAXS and contrast complemented SANS provided comprehensive structural information from the solution structure to the packing structure at multi-scales (Figure 15c). Operando SAXS (solid-state) enabled in situ monitoring of structural transition in response to an external force (Figure 15d). Typical orientation of polymer chains are clearly detected, evidenced by the in situ SAXS. Additionally, the polymer backbone to backbone distance decreases, characteristic of more condensed interpenetrated microstructures resulting from stretching. The directional packing of polymer backbones greatly enhances the friction between polymer chains, accounting for the observable strain-hardening behaviors (Figure 15e). Overall, clear physical images of complex structures and dynamics were obtained.

3.3.1 Coordination of metal ions in solution and cluster formation, M₃₈ (M=Ce^IV, Bi^III, Pu^IV)

Here we review related studies on a family of M₃₈ (M=Ce^IV, Bi^III, Pu^IV) oxocluster compounds, in which SAXS, PDF and XAS were employed in four separate studies. Notably, the Np₃₈¹⁰⁰ and U₃₈¹⁰¹ analogues have also been synthesized and characterized by SCXRD. Briefly, the M^IV₃₈ (M^IV=Pu, Ce, Np, U) all have isostructural cores that are similar to the CaF₂ structure, and Bi^III₃₈ is quite different, likely a direct consequence of the lone-pair effect that drives distortion. Each M₃₈ is centered around a M₆O₈ hexamer that is the inorganic node of the UiO-66 MOFs,¹⁰² but the Bi₃₈ has an addition μ₆-oxo in the center.

PDF from X-ray total scattering has been used by Soderholm, Skanthakumar and collaborators to characterize the coordination environment of metal ions in solution¹⁰³ and the oligomerization and hydrolysis processes leading to cluster formation.¹⁰⁴ Studying the hydrolysis and oligomerization of Pu^IV, they found monodisperse, nanometer size, surface stabilized (Cl⁻) clusters of plutonium dioxide. Crystals obtained from the solution revealed well-defined Pu₃₈ clusters ([Pu₃₈O₅₆Cl₅₄(H₂O)₈]¹⁴⁻)^104b (Figure 16a), and the match between the measured PDF and the model built from the crystallized Pu₃₈ supports proposed reaction pathways (Figure 16b). The absence of hydroxo-bridged Pu in the Pu₃₈ cluster suggests that the condensation reaction of Pu^IV may proceed by oxolation rather than olation, contrary to what previously reported,¹⁰⁵ and that is similar to the hydrolysis chemistry seen for POMs.

Anker et al. recently combined in situ SAXS and PDF from total X-ray scattering to investigate the formation of the Bi₃₈ oxo-nanoclusters (Figure 16c), and identify intermediate cluster nuclearities.¹⁰⁶ In their study, they started with dissolution of [Bi₆O₅(OH)₃(NO₃)₅] in DMSO, and followed the reaction, leading from the [Bi₆O₈] units to the [Bi₃₈O₄₅] final product. SAXS also provided information about the size, shape, and polydispersity of the intermediate products, while PDF from total scattering was used to determine the short-range atomic arrangement (Figure 16d). Contrary to the expectation, they identified [Bi₂₂O_y] as the main metastable intermediate. The results are solid based on the parallel refinement of PDF and SAXS data with the same structural models.¹⁰⁷

Amiri et al.¹⁰⁸ took this study a step further and used SAXS to benchmark the formation of coordination polymer networks that feature the Bi₃₈ node linked by DDBS (2,2′-[biphenyl-4,4′-diylchethane-2,1-diyl] dibenzenesulphonate) (Figure 16e). Using a benchtop instrument and in situ heating, not surprisingly, they also observed Bi₃₈ rather than Bi₆. With heating and addition of the DDBS linker, they were also able to track pre-organization in solution by modeling the structure factor, followed by crystallization of the framework, with the observation of diffraction peaks (Figure 16f). With the benefit of the Bi₃₈-DDBS framework crystal structure, they were able to correlate cluster-to-cluster distances in solution and in the solid state, as well as confirm identification of the compound that crystallized in the in situ study.

Understanding the evolution of atomic structure during cluster assembly with increasing nuclearity can be complemented by parallel investigation of the evolution of the electronic structure. In a recent study, Estevenon et al.¹⁰⁹ investigated the atomic and electronic structure of Ce oxo-hydroxo clusters with PDF and Ce L₃ edge HERFD XAS. Ce oxo-hydroxo clusters represent the fundamental building blocks in CeO₂ nanoscale architectures, where molecular clusters can exhibit reactivity similar to CeO₂ NPs.¹¹⁰ To explore assembly processes, a series of Ce-oxo clusters with nuclearity (Ce₂ to Ce₃₈) were synthesized and compared with small CeO₂ NPs and bulk CeO₂.¹⁰⁹ The employed methods consistently revealed trends as the system size progressed from molecular Ce_n clusters (n=2, 6, 24, 38, and 40) to bulk CeO₂. From PDF, the increased nuclearity can be readily seen and good agreement is found between the data and the PDF calculated from available structures of Ce_n clusters (Figure 16g). The shortest correlation in PDF, corresponding to Ce−O bonds, showed a decreasing disorder when going towards larger nuclearity to match the final single Ce−O distance in CeO₂. HERFD XAS at the Ce L₃ edge concomitantly probed the evolution of the electronic structure upon size increase.

Analysis of the Ce L₃ pre-edge region, denoted as peak A (Figure 16h) and arising from transitions between the 2p and 4f valence orbitals, revealed that Ce clusters are only made of Ce^IV,¹¹¹ and no Ce^III species were found (inset in Figure 16h). Inspection of the post edge region of Ce L₃ edge HERFD XAS additionally showed that the shape of Ce 5d states gradually changed from that of Ce^IV in aqueous solution, characterized by two primary peaks, labeled B and C in Figure 16h, to that of bulk CeO₂, where B and C are further split in B₁–B₂ and C₁–C₂ by the Oh crystal field of CeO₂ fluorite structure. This behavior reflects the effect of the building-up of the cubic crystal field, which splits the Ce 5d density of states into eg and t2g bands, corresponding to B1 and B2.¹¹² With increased nuclearity, the density of states is progressively more ordered until it matches that of crystalline CeO₂.

The role of 5d orbitals in the bonding, and electronic and magnetic structure of Ce complexes have been discussed by Moreau and co-authors.¹¹³ They conducted a study involving the synthesis and characterization of two novel alkali metal-capped Ce oxo compounds. Through the use of HERFD XAS measurements at the Ce L₃ edge, it was determined that both the imido and oxo materials possess an intermediate valence ground state for Ce. The authors′ calculations further indicated a significant 4f/5d hybridization and a variation in the distribution of d-state DOS, depending on the specific bonding configuration of Ce.

3.3.2 Oxide nanoparticles and colloids

Plakhova and colleagues conducted a study on the structural modifications of small CeO₂ NPs ranging from 2 to 8 nm.¹¹⁴ By employing ab-initio calculations using FEFF, they demonstrated that the Ce L₃ HERFD XAS spectral shape changes due to the presence of OH groups on the NP surfaces, indicating the significant influence of surface composition on the reactivity and performance of CeO₂ NPs. Particularly notable changes in the spectral profile were observed when the oxygen surface groups of the 2 nm CeO₂ NPs were hydroxylated. As the particles increased in size, the effects of the OH groups on the spectra diminished, becoming less visible for 5 nm CeO₂ NPs and disappearing completely for particles as large as 8 nm.

The structure of PuO₂ NPs has been extensively studied by a combination of techniques including SAXS, SANS, EXAFS and HERFD XAS.¹¹⁵ In a recent review, Virot and collaborators provide a comprehensive summary of the collective efforts made to study the size, morphology, and crystallinity of PuO₂ NPs.¹¹⁶ The center of the discussion was the oxidation state and the bonding of Pu at the surface. Conradson and co-workers were the first to fit EXAFS with multiple Pu−O bond lengths and claimed the presence of Pu(V).¹¹⁷ In the investigation of intrinsic colloid formation, Rothe et al. found only Pu(IV) and explained the splitting of the Pu−O bonds in EXAFS by condensation of Pu(IV) phases with Pu-OH groups at the surface.¹¹⁸ Bonato et al. also reported the splitting of the Pu−O shell in PuO₂ NPs and explained it with disorder of the local structure at the surface.¹¹⁹ The same group found further evidence of surface disorder by combining SAXS and XAS on PuO₂ NPs obtained by different synthetic methods.¹²⁰ SAXS was used to measure the outer size of NPs and reveal the different morphologies obtained by different methods. XAS showed a PuO₂-like core, smaller than the size estimated by SAXS. Gerber et al. combined EXAFS, Pu L₃ and M₄ HERFD XAS and PDF on PuO₂ synthesized from different Pu precursor and they found only Pu(IV) in the end products.¹²¹ The EXAFS analyzed by simple fitting and by sophisticated Monte-Carlo simulations was well-reproduced by a single Pu−O bond length. HERFD XAS was also employed by Kvashnina et al. to determine the structure of an unknown intermediate phase, which was stabilized during PuO₂ NPs synthesis.^114b Pu M₄ edge HERFD XAS identified the unknown phase as a Pu(V) phase. The structure was then determined to be NH₄PuO₂CO₃ by comparing the Pu L₃ edge HERFD data with XAS calculations from different possible Pu(V) inorganic structures. The same authors investigated the formation of PuO₂ NPs at low pH^121b and determined that the predominant oxidation state for the particles was Pu(IV). However, when the PuO₂ NPs were formed at pH=1, the presence of multiple oxidation states was observed, including Pu(III), Pu(IV), and Pu(VI). It was concluded that the contribution of Pu(III) and Pu(VI) observed in the HERFD XAS spectrum of PuO₂ NPs at pH=1 originated from the solution rather than from the NPs themselves. This detection was possible due to the HERFD method‘s high sensitivity to the concentration of Pu under these conditions.

The sensitivity of the HERFD XAS method in detecting oxidation states has also been demonstrated in another study on the formation of UO₂ NPs.¹²³ The authors revealed that although U(IV) was the dominant oxidation state of the freshly prepared UO₂ NPs, they oxidized to U₄O₉ over time and under the influence of the X-ray beam. Additionally, it was observed that the oxidation process of the NPs was accompanied by an increase in their size to 6 nm. The importance of small polynuclear metal-clusters in the assembly path of actinide dioxide NPs was recently revealed for ThO₂ and PuO₂. Amidani et al. combined HERFD XAS and PDF to ThO₂ synthesized by chemical precipitation.¹²⁴ In a first study, they found that for freshly precipitated ThO₂ NPs (2.5 nm size), a feature in the post-edge of Th L₃ edge HERFD XAS disappears. They performed electronic structure calculations, considering non-relaxed structural models, which successfully reproduced the trends observed in the experimental data. Through examination of the simulations involving Th atoms located in the core and on the surface of the NP, it was revealed that the presence of the first post-edge feature is highly sensitive to the reduction in the number of coordinating atoms, as shown in Figure 17a. The signature of very weak Th−Th coordination at the surface was confirmed by an independent study on ThO₂ and UO₂ NPs.¹²⁵

In a following investigation, the same authors found the first evidence of the presence of Th-hexamer clusters in the first stages of ThO₂ NPs synthesis by using PDF and HERFS XAS at the Th M₄ edge. PDF data from X-ray total scattering were collected for freshly precipitated, 150 °C dried ThO₂ NPs and bulk ThO₂. The size increase with annealing can be readily seen in PDF data. The authors then fitted the data using a structural model of an ensemble of NPs of different sizes, cut-out from the structure of ThO₂ bulk. The good agreement (Figure 17c) revealed that the freshly prepared sample contained up to 60 % of [Th₆O₃₂] hexameric units, very similar to Th₆ cluster reported in literature (Figure 17d and 17e).^{122, 126} The HERFD XAS at the M₄ edge of Th shows also relevant differences. In particularly, compared to the spectrum of bulk ThO₂, the peak C is missed for freshly precipitated NPs (Figure 17b). Knowing from PDF that 60 % of the sample is made of Th hexameric units, the authors used the structure of published Th₆ clusters to calculate the change in 5 f electronic structure and found agreement with the experimental data (Figure 17b). Insight into the calculation result elucidated that the absence of peak C was due to the 45° rotation of Th−O bonds at the surface of [Th₆O₃₂] clusters, showing the power of HERFD XAS to elucidate the structure at the surface for samples of few nm size.

The presence of Pu-hexamer clusters was also found by Cot-Auriol et al. for PuO₂ NPs by SAXS and XAS.^115a Moreau et al. investigated ultra-small ThO₂ and UO₂ NPs by combining SAXS, EXAFS and XANES.¹²⁵ They found that a bulk-like core is accompanied by strong disorder of the Th−O and U−O bonds at the surface. Romanchuk et al. recently discussed EXAFS analysis as a tool to estimate NP size and disorder.¹²⁷ They proposed an effective coordination number derived from a core-shell NPs model that perfectly fits the variations in a metal-metal coordination observed in EXAFS with decreasing NP size. Understanding the chemistry of actinides in the environment is fundamental to predict their mobility and small NPs are believed to be an important cause of environmental contamination due to their high mobility.¹²⁸ Several studies are therefore utilizing structural characterization to understand the fate of actinides during environmentally relevant chemical processes, like their interaction with minerals. EXAFS was used to investigate the form assumed by Pu interacting with ferrihydrite. It was found that Pu sorption on ferrihydrite followed by transformation into goethite produces PuO₂ and Pu is incorporated into the Fe mineral. Differently, co-precipitation of ferrihydrite and Pu only results in Pu incorporation into the structure of the Fe minerals.¹²⁹ In another study, the same group investigated the interaction of Pu with calcite with EXAFS. They found that Pu(VI) is incorporated as plutonyl into the calcite structure and undergoes partial reduction to Pu(V).¹³⁰

In a recent work, Neill and collaborators investigated the formation, composition, and structure of U(IV)-silicate colloids under the alkaline conditions relevant to spent nuclear fuel storage and disposal. By exploiting the complementarity of SAXS, PDF and EXAFS, they were able to determine the complex structures of the colloids formed.¹³¹ SAXS analysis provided an average particle size of 4.6–6.3 nm, in net contrast with the absence of PDF signal above 1.5 nm. The interatomic distances from PDF indicated the presence of U(IV)-silicate and UO₂ interatomic distances at short range, with only UO₂ interatomic distances surviving in the long-range up to 1.5 nm. For the EXAFS fit, on the other hand, good agreement was reached by considering only the local environment of U(IV)-silicate and no evidence of UO₂ was found. The apparent conflicting results were interpreted as the evidence of core-shell structure colloids, i.e. a small 1.5 nm crystalline UO₂ core surrounded by a thicker amorphous U(IV)-silicate shell, up to 4.6–6.3 nm. SAXS measured the outer size of the colloids, independently from the crystallinity. PDF detected only short distances of the amorphous layer and all distances involved in the crystalline core. EXAFS on the other hand was blind to the small UO₂ core, constituting only 2.6 % of the colloids, and measured local structure of the dominant U(IV)-silicate phase.