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DOE OSTI2022

Observation of an Orientational Glass in a Superlattice of Elliptically-Faceted CdSe Nanocrystals

Extensive prior work has shown that colloidal inorganic nanocrystals coated with organic ligand shells can behave as artificial atoms and, as such, form superlattices with different crystal structures and packing densities. Although ordered superlattices present a high degree of long-range positional order, the relative crystallographic orientation of the inorganic nanocrystals with respect to each other tends to be random. Recent works have shown that superlattices can achieve orientational alignment through combinations of nanocrystal faceting and ligand modification, as well as selective metal particle attachment to particular facets. These studies have focused on the assembly of high-symmetry nanocrystals, such as cubes and cuboctahedra. In this work, we study the assembly of elliptically faceted CdSe/CdS core/shell nanocrystals with one distinctive crystallographic orientation along the major elliptical axis. We show that the nanocrystals form an unexpectedly well-ordered translational superlattice, with a degree of order comparable to that achieved with higher-symmetry nanocrystals. Additionally, we show that, due to the particles' faceted shape, the superlattice is characterized by an orientational glass phase in which only certain orientations are possible due to entropically frustrated crystallization. In this phase, the nanocrystals do not exhibit a local orientational ordering but rather have distinct orientations that emerge at different locations within the same domain. The distinct orientations are a result of a facet-to-facet lock-in mechanism that occurs during the self-assembly process. These facet-to-facet alignments force the nanocrystals to tilt on different lattice planes forming different projections that we termed apparent polydispersity. Our experimental realization of an orientational glass phase for multifaceted semiconducting nanocrystals can be used to investigate how this phase is formed and how it can be utilized for potential optical, electrical, and thermal transport applications.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

Elucidating facet dependent electronic and electrochemical properties of Cu 2 O nanocrystals using AFM/SCEM and DFT

Cuprous oxide (Cu 2 O) has extensively been studied owing to its excellent optical, magnetic, and catalytic properties. Many of these properties are facet-dependent and have not been well elucidated. This work synthesized cubic, cuboctahedral, octahedral, and rhombic dodecahedral shaped Cu 2 O nanocrystals of ~300 nm in size to evaluate the facet-dependent electrochemical activities. Here, cyclic voltammetry (CV) and electrochemical impedance spectroscopy (EIS) were firstly used to reveal the average electrochemical activities at the ensemble level. Atomic force microscopy-scanning electrochemical microscopy (AFM-SECM) was further used to assess the electrochemical activities of different Cu 2 O nanocrystals at the facet level. Hexaammineruthenium (III) chloride ({Ru(NH 3 ) 6 }Cl 3 ) was employed as the probe molecules that reacted with four different Cu 2 O nanocrystals under —400mV and yielded ~300 pA current between the probing tip and the nanocrystal surface. The tip-current mapping results indicate that rhombic dodecahedral Cu 2 O exhibits higher electrocatalytic activity than other shaped Cu 2 O, due to the presence of dominant exposed facet of {110} as indicated by the relatively high tip current. Density-functional theory (DFT) calculations confirmed the facet dependence of local surface energy and electronic structure of Cu 2 O nanocrystals. Besides electrochemical activity, the surface work function and adsorptive properties were both observed to vary with the shape and dominant exposed facets of Cu 2 O. This study presented a unique experimental and computational chemistry approach to analyze surface electrochemical properties of Cu 2 O crystals at a crystalline facet level.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Facet Preferencing by Chemical Substitution Controls Semi-Hydrogenation Selectivity in Ternary Pyrite-Type Intermetallic Compounds

Intermetallic compounds serve as model catalysts for selective hydrogenation reactions, offering precise control over the active site composition(s), geometric and electronic structure. The addition of a third element to form a ternary intermetallic alters the exposed crystal facet(s), demonstrating a strategy to impart improved catalytic behavior in intermetallic catalysts. The site-specific substitution of a small fraction of Pd atoms with Au in pyrite-type PdSb 2 results in the preferential exposure of the (100) facet over the (111) facet. Electron back scattered diffraction and density functional theory calculations confirm the facet change upon the substitution of Pd with Au to form the ternary Pd 1−x Au x Sb 2 (0.075 ≤ x ≤ 0.25). The (100) facet demonstrates higher net alkene selectivity due to significantly weaker alkene binding compared to the (111) facet. Distinct from our prior work on chemical substitution to directly alter the active site composition, this work demonstrates the indirect modification of active sites via preferential facet exposure.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗

DOE OSTI2023

High-Resolution Thermal Profiling of a High-Power Diode Laser Facet During Aging

Here, we study the facet temperature distribution of a high-power diode laser over its lifetime using a noncontact, high spatial resolution CCD-based thermoreflectance technique. Based on the known correlation between non-radiative defects and heating, thermal maps can provide valuable information regarding the formation and evolution of small point defects that are at or near the facet during aging. In the laser under study in this work we measure the appearance of local hot spots on the facet, including concentrated hot spots that appear just before or just after COD and are correlated with loss of local light emission. The locations of these hot spots do not exhibit morphology changes in high-resolution SEM imaging of the facet, indicating that the related defects are too small to be observable in SEM or are located at some depth under the facet. Prior to COD, we measure a gradual facet temperature increase accompanied by a gradual optical power decrease and gradual facet optical absorption increase, indicating gradual degradation of the laser.

47 OTHER INSTRUMENTATION↗

DOE OSTI2024

Theoretical Investigation of the Adsorbate and Potential–Induced Stability of Cu Facets During Electrochemical CO 2 and CO Reduction

The activity and product selectivity of electrocatalysts for reactions like the carbon dioxide reduction reaction (CO 2 RR) are intimately dependent on the catalyst's structure and composition. While engineering catalytic surfaces can improve performance, discovering the key sets of rational design principles remains challenging due to limitations in modeling catalyst stability under operating conditions. Herein, we perform first-principles density functional calculations adopting implicit solvation methods with potential control to study the influence of adsorbates and applied potential on the stability of different facets of model Cu electrocatalysts. Using coverage dependencies extracted from microkinetic models, we describe an approach for calculating potential and adsorbate-dependent contributions to surface energies under reaction conditions, where Wulff constructions are used to understand the morphological evolution of Cu electrocatalysts under CO 2 RR conditions. Here we identify that CO*, a key reaction intermediate, exhibits higher kinetically and thermodynamically accessible coverages on (100) relative to (111) facets, which can translate into an increased relative stabilization of the (100) facet during CO 2 RR. Our results support the known tendency for increased (111) faceting of Cu nanoparticles under more reducing conditions and that the relative increase in (100) faceting observed under CO 2 RR conditions is likely attributed to differences in CO* coverage between these facets.

30 DIRECT ENERGY CONVERSION↗

DOE OSTI2025

Facet-Dependent Adsorption of Pb(II) on Hematite (001), (116), and (104) Surfaces

Hematite’s common (001) and (012) facets are frequently used in model studies of lead (Pb) adsorption behavior, but there is a lack of research on the high-energy facets, e.g. (104), present in nature. Also, few studies have attempted to connect the molecular details of facet-specific Pb adsorption to macroscopic uptake behavior. Here, to address these knowledge gaps, we investigated Pb(II) adsorption behaviors on facet-engineered hematite nanoparticles dominated by (001), (104), and (116). Adsorption experiments revealed significant variations in Pb(II) uptake among the three samples, with (001) demonstrating the highest capacity and (116) showing the best adsorption efficiency when normalized to specific surface area. Adsorption kinetics followed the pseudo-second-order model, indicating the adsorption process is governed mostly by chemisorption. Adsorption isotherms were well fitted by the Langmuir model, indicating uptake proceeds until roughly monolayer adsorption. Detailed characterization revealed Pb(II) was adsorbed as single atoms with complex inner-sphere binding modes that varied across different facets, indicating adsorption is both structurally and energetically facet-dependent. Co-adsorption experiments further demonstrated Cu 2+ , Zn 2+ , and humic acid significantly promoted Pb(II) adsorption. This study advances the understanding of hematite surface reactivity in controlling macroscopic wet adsorption behaviors, providing valuable insights into the environmental fate of Pb(II).

Contaminant removal↗

DOE OSTI2024

Anatase TiO 2 Surface Hydration: Infrared Vibrational Assignment and Facet-Specific Stability

Anatase TiO 2 nanoparticles support a subtle diversity of surface aqua and hydroxy coordination environments that are characteristic of distinct facets and defect sites. The disparate properties and stability of each molecularly or dissociatively absorbed water environment may provide unique reactivity for facet- and site-selective atomic layer deposition (ALD). Here we probe the temperature-dependent and facet-specific stretching frequency and stability of surface hydroxyls on faceted anatase TiO 2 nanoparticles via in situ diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS). Well-defined truncated octahedra, disks, and rods fabricated through hydrothermal synthesis present dominant (101), (001), and (010) facets, respectively. The selective dehydration of discrete local coordination environments as a function of temperature provides a potential pathway to site-selective and facet-selective ALD on metal oxide nanomaterials.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2026

Dynamic Nuclear Polarization Enhanced 113 Cd Solid-State Nuclear Magnetic Resonance Spectroscopy Reveals CdSe Nanocrystals with Triangular Two-Dimensional Projections are Terminated by {100} Facets

Surface structure plays an important role in particle growth and determining the chemical and photophysical properties of semiconductor nanocrystals (NCs). Therefore, there is a need for structural tools that can characterize and detect different surface facets. Here, we sought to investigate the structure of {111} CdSe facets by applying dynamic nuclear polarization (DNP) enhanced 113 Cd and 77 Se solid-state nuclear magnetic resonance (SSNMR) spectroscopy to zinc-blende CdSe NCs that exhibit triangular two-dimensional (2D) projections within transmission electron microscope (TEM) images. It was originally hypothesized in the literature that these CdSe NCs were tetrahedral in shape and terminated by facets from the {111} family of lattice planes of the zinc-blende structure. Surprisingly, we observe 113 Cd NMR spectra indicating that the primary facets are from the {100} family of lattice planes. We also obtained DNP-enhanced 113 Cd and 77 Se SSNMR spectra of recently reported right trigonal bipyramidal (rTriBP) CdSe NCs grown by seeded growth. The 113 Cd NMR spectra rTriBP CdSe NCs are consistent with {100} surface facets. TEM images show that rTriBP NCs may also exhibit triangular 2D projections that are similar to the so-called tetrahedral NCs. Based upon these results, we conclude that the so-called tetrahedral NCs are predominantly terminated by {100} surface facets and most likely have the same shape as rTriBP NCs. These results highlight the need for multiple complementary techniques when assigning the shape and surface termination of NCs.

Santhiran, Anuluxan [Ames Laboratory (AMES), Ames,↗

DOE OSTI2022

Synthesis of perovskite polyhedron nanocrystals with equivalent facets and the controlled growth of Pt nanoparticles with differing surface concentration of oxidized Pt 4+ /Pt 2+ species

Co-existence of different surfaces in traditional polycrystalline heterogeneous catalysts imposes a challenge to induce different kinds of catalytic species and average reactivity. Supported metal/oxide model catalyst designed with equivalent well-defined surfaces and active species provides an opportunity to control the uniformity, maintain efficient activity, and understand catalytic mechanisms at the atomic and molecular levels. Herein, sub-20 nm perovskite-structure BaTiO 3 polyhedron nanocuboids and Zr-substituted rhombic dodecahedron nanocrystals were synthesized and employed as the oxide support for the subsequent uniform growth of 1–2 nm Pt nanoparticles. These Pt nanoparticles were well dispersed on the 12 equivalent (110) facets of BaTi 0.75 Zr 0.25 O 3 rhombic dodecahedra and on the 6 equivalent (100) facets of BaTiO 3 nanocuboids. The main active centers on the Pt/BaTi 0.75 Zr 0.25 O 3 catalysts in the oxidized surface layer of Pt (111) facets are Pt 4+ versus Pt 2+ , which is in contrast to the oxidized surface layer of the Pt (100) facets on the Pt/BaTiO 3 cuboids where there are a greater number of Pt 2+ species. Compared with Pt/BaTiO 3 nanocuboids, the Pt/BaTi 0.75 Zr 0.25 O 3 catalysts demonstrated higher CO catalytic oxidation activity with a lower apparent Arrhenius activation energy. Finally, this work advances our ability to synthesize metal/oxide model catalyst that offer equivalent facets and highly uniform active centers with reasonable surface area in order to gain insight in reaction mechanisms.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2022

(111) Faceted Metal Oxides: A Review of Synthetic Methods

Material design and synthesis have made tremendous impacts in the scientific community by unleashing a material's true potential via enhanced properties and applications. Over the years, advanced synthetic strategies have emerged and have been expanded to not only control the size and shape of nanoparticles but also to control the preferential growth of surface facets, paving the way for new materials with facet-dependent properties. Metal oxide (111) facets as compared to their potentially more stable counterpart facets (e.g., (100), (110)) have recently exhibited enriched chemical properties owing to their unique surface arrangement. As a result, metal oxide (111) faceted surfaces have been used in applications such as catalysis, sorbents, batteries, etc. This work aims to provide a perspective on the synthetic processes utilized to expose (111) surfaces and the governing factors/synthetic parameters that expose them across various metal oxides of different crystal structures as well as some of their applications.

36 MATERIALS SCIENCE↗

DOE OSTI2020

Shuffling mode competition leads to directionally anisotropic mobility of faceted Σ11 boundaries in fcc metals

Faceted grain boundaries can migrate in interesting and unexpected ways. For example, faceted Σ11 <110> tilt grain boundaries were observed to exhibit mobility values that could be strongly dependent on the direction of migration. In order to understand whether this directionally-anisotropic mobility is a general phenomenon and to isolate mechanistic explanations for this behavior, molecular dynamics simulations of bicrystals evolved under an artificial driving force are used to study interface migration for a range of boundary plane inclination angles and temperatures in multiple face centered cubic metals (Al, Ni, and Cu). We find that directionally anisotropic mobility is active in a large fraction of these boundaries in Ni and Cu and should therefore impact the coarsening of polycrystalline materials. On the other hand, no such anisotropy is observed in any of the Al boundaries, showing that this behavior is material-dependent. Migration of the faceted boundaries is accomplished through transformation events at facet nodes and incommensurate boundary plane facets, which are termed shuffling modes. Three major shuffling modes have been identified, namely Shockley shuffling, slip plane shuffling, and disordered shuffling. A shift from the first two ordered modes to the third disordered mode is found to be responsible for reducing or removing directionally-anisotropic mobility, especially at the highest temperatures studied.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Facet-dependent electron transfer induces distinct arsenic reallocations on hematite

The interfacial electron transfer (ET) between electron shuttling compounds and iron (Fe) oxyhydroxides plays a crucial role in the reductive dissolution of Fe minerals and the fate of surface-bound arsenic (As). However, the impact of exposed facets of highly crystalline hematite on reductive dissolution and As immobilization is poorly understood. In this study, we systematically investigated the interfacial processes of the electron shuttling compound cysteine (Cys) on various facets of hematite and the reallocations of surface-bound As(III) or As(V) on the respective surfaces. Our results demonstrate that the ET process between Cys and hematite generates Fe(II) and leads to reductive dissolution, with more Fe(II) generated on {001} facets of exposed hematite nanoplates (HNPs). Reductive dissolution of hematite leads to significantly enhanced As(V) reallocations on hematite. Nevertheless, upon the addition of Cys, a raipd release of As(III) can be halted by its prompt re-adsorption, leaving the extent of As(III) immobilization on hematite unchanged throughout the course of reductive dissolution. This is due to that Fe(II) can form new precipitates with As(V), a process that is facet-dependent and influenced by water chemistry. Electrochemical analysis reveals that HNPs exhibit higher conductivity and ET ability, which is beneficial for reductive dissolution and As reallocations on hematite. In conclusion, these findings highlight the facet-dependent reallocations of As(III) and As(V) facilitated by electron shuttling compounds and have implications for the biogeochemical processes of As in soil and subsurface environments.

54 ENVIRONMENTAL SCIENCES↗

DOE OSTI2025

Rapid Screening of Single-Atom Catalyst Synthesis Conditions Using ToF-SIMS and Facet-Dependent Single-Crystal Substrates

Single-atom catalysts (SACs) offer superior catalytic performance compared to traditional nanoparticle catalysts but are challenging to develop because of the need for extensive optimization and specialized characterization techniques. Here, this study presents a rapid and versatile method for detecting synthesis conditions and elucidating deposition mechanisms of SACs on various substrates. By depositing active elements (Au, Cu, Ni and Rh) on facet-specific single-crystalline substrates (CeO 2 , TiO 2 , MgO and Al 2 O 3 ) and employing time-of-flight secondary ion mass spectroscopy (ToF-SIMS), we assessed facet-dependent deposition behaviors and identified optimal conditions for solution-based SAC synthesis. On CeO 2 and TiO 2 , we confirmed facet-dependent deposition, primarily influenced by oxygen vacancy density and photocatalytic activity, respectively. MgO exhibited the formation of metal oxide/hydroxide clusters for all active elements, and the degree of clustering for Cu and Ni was correlated with the facet hydrolysis susceptibility. Notably, Au and Rh deposition on MgO was facet-independent, attributed to the formation of hydroxide species in solution. Al 2 O 3 , due to its chemical stability and lack of surface defects, did not show active element deposition. This study not only provides a time and cost-efficient method for prescreening SAC synthesis conditions, but it also provides valuable insights into the various deposition mechanisms governing SAC formation on different substrates, paving the way for the rational design of tailored SACs for various catalytic applications.

ToF-SIMS↗

DOE OSTI2025

Continuous Encodable Reshaping of Gold Nanocrystals through Facet Modulation

Shape control of nanocrystals (NCs) is crucial for tuning their assembly behavior and functional properties, yet the precise manipulation of facet composition remains challenging. Here, we present a nanocrystal reshaping strategy to control and modulate the facets of gold (Au) NCs. Our one-pot approach, conducted at room temperature, requires only initial Au NCs, Au 3+ ions, and surfactants, distinguishing it from conventional reduction-mediated “etching-and-regrowth” methods. Detailed structural studies using electron microscopy, small-angle X-ray scattering (SAXS), and UV−vis spectroscopy reveal the surfactant-encoded pathway for NC transformation from shaped particles to spheres and then into various polyhedral shapes while preserving the individual particles' volume. The proposed reshaping mechanism involves the dissolution of surface Au atoms into Au + complexes in the presence of Au 3+ and surfactant, followed by surfactant-guided redeposition and formation of facets with different atomic planes. Using the ethanol oxidation reaction (EOR) as a probe, we observe a quasi-linear decrease in onset potential and an increase in activity with increasing {100} facet exposure. This work broadens synthetic strategies by offering precise NC reshaping and facet control.

36 MATERIALS SCIENCE↗

DOE OSTI2024

Facets control charge separation during photoelectrochemical water oxidation with strontium titanate (SrTiO 3 ) single crystals

The photocatalytic overall water splitting reaction provides a pathway to hydrogen fuel from sunlight. Photocatalysts must achieve the reaction without the application of an external bias, which requires an effective charge separation mechanism. Photolabeling studies and electrostatic simulations for the well-known CoOOH/Al:SrTiO 3 /Rh/Cr 2 O 3 photocatalyst suggest that charge separation is driven by work function differences at the (100) and (110) facets of SrTiO 3 , which are electron and hole selective, respectively. Here we use hydrogen annealed SrTiO 3–x single crystals to obtain the first quantitative assessment of the charge separation ability of the (100), or (110), or (111) facets during oxygen evolution. Under UV illumination (60 mW cm –2 ), the crystals exhibit variable water oxidation photocurrents (0.34, 0.82, 1.36 mA cm –2 at 1.23 V versus RHE) and photovoltage values of 1.40, 1.52 and 1.52 V for (100), (110), and (111) SrTiO 3–x , respectively. A surface photovoltage increase in that same order (0.31 V < 0.57 V < 0.67 V) is confirmed independently with vibrating Kelvin probe surface photovoltage spectroscopy (VKP-SPV) under 375 nm (1.91 mW cm –2 ) illumination. Mott Schottky measurements in aqueous K 3/4 [Fe(CN) 6 ] reveal facet-dependent flatband positions of –0.58, –0.71, and –0.74 V RHE for the (100), (110), and (111) crystals respectively. This confirms that the photoelectrochemical water oxidation performance of SrTiO 3–x crystals is controlled by the work function of each facet, which determines the electron transfer barrier height of the respective solid–liquid junctions. After correcting for differences in electron donor concentrations, barriers are found to increase in the order (100) < (111) < (110) and differ by as much as 0.16 eV, similar to an earlier prediction. Altogehter, these results explain the charge separation mechanism in SrTiO 3 photocatalysts and highlight the need for faceted semiconductor crystals as light absorbers in particle-based photocatalysts.

08 HYDROGEN↗

DOE OSTI2026

Development and Validation of Low-Cost, High-Reflectance Composite CSP Facets: SIPS Final Report

This work investigates the various challenges associated with developing heliostat structural composite facets using 1 mm glass mirrors. Such facets are desirable for Concentrating Solar Power because 1 mm glass mirrors provide an absolute increase in reflectivity of 2-3 % over the industry standard of 4 mm glass mirrors. Prototypes of paraboloid composite facets with 1 mm glass mirrors were constructed that have a root mean square slope error on the order of 2 mrad while achieving greater than 96% reflectivity. These facets were constructed using a low-quality aluminum mold and a bill of materials that show potential to achieve cost parity with existing 4 mm glass mirrors supported by structural steel. The facets produced were able to survive up to 50 mm hail ball impacts and were robust against accelerated environmental cycling designed to expose durability concerns. Further work is needed to develop a scalable and cost-effective manufacturing process with a similar bill of materials.

14 SOLAR ENERGY↗

DOE OSTI2021

Probing Acid–Base Properties of Anatase TiO 2 Nanoparticles with Dominant {001} and {101} Facets Using Methanol Chemisorption and Surface Reactions

In the present study, we investigate the surface acid-base properties of anatase TiO 2 nanomaterials with dominant {101} and {001} facets via methanol titrations. Two anatase nanoparticles, TiO 2 (101) and TiO 2 (001), with well-defined morphology are prepared. TiO 2 (101) is predominantly enclosed by the {101} facets (> 90%), and TiO 2 (001) contains ~46% {001} facets and ~54% {101} facets. Upon adsorption of methanol 423 K, DRIFTS measurements show that both molecular and dissociative adsorption occurs on TiO 2 (101), while dissociative adsorption dominates on TiO 2 (001). During methanol TPD, TiO 2 (001) mainly generates acid-base product dimethyl ether and thermal cracking products CO and H 2 , as anticipated. In contrast, substantial amounts of formaldehyde and methane also desorb from TiO 2 (101), suggesting strong participation of surface defects (e.g., oxygen vacancies).

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2020

The interplay between surface facet and reconstruction on isopropanol conversion over SrTiO 3 nanocrystals

Strontium titanate (SrTiO 3 ) is an extensively investigated perovskite for various applications due to its optical, electrical and chemical properties. To gain an in-depth understanding of the active sites involved in heterogeneous catalysis over the broadly used SrTiO 3 (STO), we studied a model reaction, isopropanol conversion, on three differently shape-controlled nanocrystals: cube, truncated cube and dodecahedra. SEM, XRD and XPS confirmed the morphology, phase and composition of STO shapes. Low energy ion scattering (LEIS) revealed the occurrence of surface reconstruction over STO shapes during O 2 pretreatment at different temperatures. Based on the catalytic activities, scanning transmission electron microscopy images and density functional theory calculations, the step sites on STO derived from surface reconstruction were proposed to be the active sites for isopropanol conversion. This was further confirmed by steady state isotopic kinetic analysis (SSITKA) which demonstrated similar intrinsic turnover frequencies (TOFs) for the differently reconstructed STO shapes. It is concluded that the crystal facets impose an indirect effect on the catalysis of STO via controlling the degrees of surface reconstruction: the less stable STO (1 1 0) facet (dodecahedra) leads to more step sites after reconstruction and hence higher overall reaction rate than the more stable (1 0 0) facet (cube). This work highlights the important interplay between the crystal facet and surface reconstruction in controlling the nature and density of active sties and thus catalysis over complex oxides.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗