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This chapter presents an introduction to some of the key dimensionality reduction methods currently used in data science. The intended audience is pure mathematicians. The chapter puts special emphasis on connecting the fundamental ideas in dimensionality reduction to a diverse set of mathematical domains, including geometry, dynamical systems, representation theory, and category theory.
The use of nanoporous metals as catalysts has attracted significant interest in recent years. Their high-curvature, nanoscale ligaments provide not only high surface area but also a high density of undercoordinated step edge and kink sites. However, their long-term stability, especially at higher temperatures, is often limited by thermal coarsening and the associated loss of surface area. Herein, it is demonstrated that the nanoscale morphology of nanoporous Cu can be regenerated by applying oxidation/reduction cycles at 250 °C. Specifically, the morphological evolution and H 2 dissociation activity of hierarchical nanoporous Cu catalysts doped with Ti during structural rearrangement triggered by oxidative and reductive atmospheres at elevated temperatures are studied. In addition to coarsening of the structure at elevated temperatures, oxidation at 400 °C causes an expansion of the ligaments. Further, subsequent reduction at 400 °C leads to the formation of particles and a drop in the H 2 dissociation activity compared the fresh catalyst. However, performing the redox cycle at 250 °C reverses coarsening and boosts the H 2 dissociation activity for the hydrogen–deuterium (H 2 –D 2 ) reaction. Herein, the possibility to reverse coarsening is demonstrated, thereby mitigating the loss of activity frequently observed in nanoporous catalysts.
Platinum group metal (PGM)-free catalysts for oxygen reduction reaction have shown high oxygen reduction reaction activity in alkaline media. In order to further increase the power density of anion-exchange membrane fuel cells (AEMFCs), PGM-free catalysts need to have a high site density to reach high current densities. In this work, synthesis, characterization, and utilization of heat-treated iron porphyrin aerogels are reported as cathode catalysts in AEMFCs. The heat treatment effect is thoroughly studied and characterized using several techniques, and the best performing aerogel is studied in AEMFC, showing excellent performance, reaching a peak power density of 580 mW cm –2 and a limiting current density of as high as 2.0 A cm –2 , which can be considered the state-of-the-art for PGM-free based AEMFCs.
Abstract Atomically dispersed 3D transitional metal active sites with nitrogen coordination anchored on carbon support have emerged as a kind of promising electrocatalyst toward oxygen reduction reaction (ORR) in the field of fuel cells and metal–air cells. However, it is still a challenge to accurately modulate the coordination structure of single‐atom metal sites, especially first‐shell coordination, as well as identify the relationship between the geometric/electronic structure and ORR performance. Herein, a carbon‐supported single‐atom nickel catalyst is fabricated with boron and nitrogen dual coordination (denoted as Ni‐B/N‐C). The hard X‐ray absorption spectrum result reveals that atomically dispersed Ni active sites are coordinated with one B atom and three N atoms in the first shell (denoted as Ni‐B 1 N 3 ). The Ni‐B/N‐C catalyst exhibits a half‐wave potential ( E 1/2 ) of 0.87 V versus RHE, along with a distinguished long‐term durability in alkaline media, which is superior to commercial Pt/C. Density functional theory calculations indicate that the Ni‐B 1 N 3 active sites are more favorable for the adsorption of ORR intermediates relative to Ni‐N 4 , leading to the reduction of thermodynamic barrier and the acceleration of reaction kinetics, which accounts for the increased intrinsic activity.
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Structural engineering of nanomaterials offers a promising way for developing high-performance catalysts toward catalysis. However, the delicate modulation of thermodynamically unfavorable nanostructures with unconventional phases still remains a challenge. Here, in this study, the synthesis of hierarchical AuCu nanostructures is reported with hexagonal close-packed (2H-type)/face-centered cubic (fcc) heterophase, high-index facets, planar defects (e.g., stacking faults, twin boundaries, and grain boundaries), and tunable Cu content. The obtained 2H/fcc Au 99 Cu 1 hierarchical nanosheets exhibit excellent performance for the electrocatalytic CO 2 reduction to produce CO, outperforming the 2H/fcc Au 91 Cu 9 and fcc Au 99 Cu 1 . The experimental results, especially those obtained by in-situ differential electrochemical mass spectroscopy and attenuated total reflection Fourier-transform infrared spectroscopy, suggest that the enhanced catalytic performance of 2H/fcc Au 99 Cu 1 arises from the unconventional 2H/fcc heterophase, high-index facets, planar defects, and appropriate alloying of Cu. Impressively, the 2H/fcc Au 99 Cu 1 shows CO Faradaic efficiencies of 96.6% and 92.6% at industrial current densities of 300 and 500 mA cm -2 , respectively, as well as good durability, placing it among the best CO 2 reduction electrocatalysts for CO production. The atomically structural regulation based on phase engineering of nanomaterials (PEN) provides an avenue for the rational design and preparation of high-performance electrocatalysts for various catalytic applications.
Abstract The activity and durability of chemical/electrochemical catalysts are significantly influenced by their surface environments, highlighting the importance of thoroughly examining the catalyst surface. Here, Cu‐substituted La 0.6 Sr 0.4 Co 0.2 Fe 0.8 O 3‐δ is selected, a state‐of‐the‐art material for oxygen reduction reaction (ORR), to explore the real‐time evolution of surface morphology and chemistry under a reducing atmosphere at elevated temperatures. Remarkably, in a pioneering observation, it is discovered that the perovskite surface starts to amorphize at an unusually low temperature of approximately 100 °C and multicomponent metal nanocatalysts additionally form on the amorphous surface as the temperature raises to 400 °C. Moreover, this investigation into the stability of the resulting amorphous layer under oxidizing conditions reveals that the amorphous structure can withstand a high‐temperature oxidizing atmosphere (≥650 °C) only when it has undergone sufficient reduction for an extended period. Therefore, the coexistence of the active nanocatalysts and defective amorphous surface leads to a nearly 100% enhancement in the electrode resistance for the ORR over 200 h without significant degradation. These observations provide a new catalytic design strategy for using redox‐dynamic perovskite oxide host materials.
Abstract CO 2 reduction is a highly attractive route to transform CO 2 into useful feedstocks, of which C 2 products are more desired than C 1 , yet face high kinetic barriers of C−C electrocoupling. Here, the engineering of pore‐enabled local confinement reaction environments is reported for tuning the enrichment of surface‐adsorbed oxygen‐relevant species and the establishment of their pronounced benefits in promoting C−C coupling over oxide‐derived Cu‐based catalysts. A new approach of utilizing the microphase separation of a block copolymer is developed to fabricate bicontinuous mesoporous CuO nanofibers (CuO‐BPNF). The enhanced confinement from long‐range mesochannels enables the adsorption of OH ad /O ad on the Cu surface at a wide negative potential range of −0.7 – −1.3 V in CO 2 reduction, which cannot be achieved over conventional deficient and short‐range pores. Constant‐potential DFT calculations reveal that the surface‐bound oxygen species weakens *CO affinity with the Cu (111) surface and lowers the kinetic barriers for both *CO−CO dimerization and *CO hydrogenation to enable *CO−CHO coupling. Accordingly, a CO 2 ‐to‐C 2 Faradaic efficiency of 74.7% over CuO‐BPNF is shown, significantly larger than counterparts with conventional pores. This work offers a general design principle of confinement engineering to manage the adsorption of reactive species for steering reaction pathways in interfacial catalysis.
Electroreduction of CO 2 to value-added products is a promising strategy for CO 2 reuse, where copper has a unique ability to produce oxygenates and C 2+ products. Unfortunately, the electronic factors making copper so unique are unknown, which limits the ability to design improved catalysts. By combining in situ surface-sensitive X-ray resonant photoelectron spectroscopy with density functional theory calculations, the complex electronic structure of copper is uncovered during the electrocatalytic reduction of CO 2 . It is found that the pseudocapacitive charging of copper, produced by the incorporation of protons and electrons into the subsurface, facilitates the activation of CO 2 , while simultaneously increasing the barrier for H-H coupling. The net result is that cathodic pseudocapacitive charge suppresses the hydrogen evolution reaction and promotes the production of hydrocarbons and oxygenated products on copper. These results represent a new paradigm in the understanding of CO 2 reduction, highlighting the key role of pseudocapacitive charge in the reaction.
Abstract Aerogels are fascinating materials that can be used for a wide range of applications, one of which is electrocatalysis of the important oxygen reduction reaction. In their inorganic form, aerogels can have ultrahigh catalytic site density, high surface area, and tunable physical properties and chemical structures—important features in heterogeneous catalysis. Herein, we report on the synthesis and electrocatalytic properties of an iron–porphyrin aerogel. 5,10,15,20‐(Tetra‐4‐aminophenyl)porphyrin (H 2 TAPP) and Fe II were used as building blocks of the aerogel, which was later heat‐treated at 600 °C to enhance electronic conductivity and catalytic activity, while preserving its macrostructure. The resulting material has a very high concentration of atomically dispersed catalytic sites (9.7×10 20 sites g −1 ) capable of catalyzing the oxygen reduction reaction in alkaline solution ( E onset =0.92 V vs. RHE, TOF=0.25 e − site −1 s −1 at 0.80 V vs. RHE).
Abstract Single‐atom catalysts (SACs) show great promise for electrochemical CO 2 reduction reaction (CRR), but the low density of active sites and the poor electrical conduction and mass transport of the single‐atom electrode greatly limit their performance. Herein, we prepared a nickel single‐atom electrode consisting of isolated, high‐density and low‐valent nickel(I) sites anchored on a self‐standing N‐doped carbon nanotube array with nickel–copper alloy encapsulation on a carbon‐fiber paper. The combination of single‐atom nickel(I) sites and self‐standing array structure gives rise to an excellent electrocatalytic CO 2 reduction performance. The introduction of copper tunes the d‐band electron configuration and enhances the adsorption of hydrogen, which impedes the hydrogen evolution reaction. The single‐nickel‐atom electrode exhibits a specific current density of −32.87 mA cm −2 and turnover frequency of 1962 h −1 at a mild overpotential of 620 mV for CO formation with 97 % Faradic efficiency.
Abstract Lanthanide triflates have been used to incorporate Nd III and Sm III ions into the 2.2.2‐cryptand ligand (crypt) to explore their reductive chemistry. The Ln(OTf) 3 complexes (Ln=Nd, Sm; OTf=SO 3 CF 3 ) react with crypt in THF to form the THF‐soluble complexes [Ln III (crypt)(OTf) 2 ][OTf] with two triflates bound to the metal encapsulated in the crypt. Reduction of these Ln III ‐in‐crypt complexes using KC 8 in THF forms the neutral Ln II ‐in‐crypt triflate complexes [Ln II (crypt)(OTf) 2 ]. DFT calculations on [Nd II (crypt)] 2+ ], the first Nd II cryptand complex, assign a 4f 4 electron configuration to this ion.
Abstract The electrocatalytic reduction of CO 2 is an appealing method for converting renewable energy sources into value‐added chemical feedstocks. We report a co‐electrocatalytic system for the reduction of CO 2 to CO comprised of a molecular Cr complex and dibenzothiophene‐5,5‐dioxide (DBTD) as a redox mediator, which achieves high activity (TOF=1.51–2.84×10 5 s −1 ) and quantitative selectivity. Under aprotic or protic conditions, DBTD produces a co‐electrocatalytic response with 1 by coordinating trans to the site of CO 2 binding and mediating electron transfer from the electrode with quantitative efficiency for CO. This assembly is reliant on through‐space electronic conjugation between the π frameworks of DBTD and the bpy fragment of the catalyst ligand, with contributions from dispersive interactions and weak sulfone coordination.
Abstract A new isolation protocol was recently reported for highly purified metallic Fullertubes D 5h ‐C 90 , D 3d ‐C 96 , and D 5d ‐C 100, which exhibit unique electronic features. Here, we report the oxygen reduction electrocatalytic behavior of C 60 , C 70 (spheroidal fullerenes), and C 90 , C 96 , and C 100 (tubular fullerenes) using a combination of experimental and theoretical approaches. C 96 (a metal‐free catalyst) displayed remarkable oxygen reduction reaction (ORR) activity, with an onset potential of 0.85 V and a halfway potential of 0.75 V, which are close to the state‐of‐the‐art Pt/C benchmark catalyst values. We achieved an excellent power density of 0.75 W cm −2 using C 96 as a modified cathode in a proton‐exchange membrane fuel cell, comparable to other recently reported efficient metal‐free catalysts. Combined band structure (experimentally calculated) and free‐energy (DFT) investigations show that both favorable energy‐level alignment active catalytic sites on the carbon cage are responsible for the superior activity of C 96 .
Abstract We present a supramolecular approach to catalyzing photochemical CO 2 reduction through second‐sphere porosity and charge effects. An iron porphyrin box ( PB ) bearing 24 cationic groups, FePB‐2(P) , was made via post‐synthetic modification of an alkyne‐functionalized supramolecular synthon. FePB‐2(P) promotes the photochemical CO 2 reduction reaction (CO 2 RR) with 97 % selectivity for CO product, achieving turnover numbers (TON) exceeding 7000 and initial turnover frequencies (TOF max ) reaching 1400 min −1 . The cooperativity between porosity and charge results in a 41‐fold increase in activity relative to the parent Fe tetraphenylporphyrin ( FeTPP ) catalyst, which is far greater than analogs that augment catalysis through porosity ( FePB‐3(N ), 4‐fold increase) or charge (Fe p ‐tetramethylanilinium porphyrin ( Fe‐ p ‐TMA ), 6‐fold increase) alone. This work establishes that synergistic pendants in the secondary coordination sphere can be leveraged as a design element to augment catalysis at primary active sites within confined spaces.
The electrocatalytic reduction of CO 2 is an appealing method for converting renewable energy sources into value-added chemical feedstocks. We report a co-electrocatalytic system for the reduction of CO 2 to CO comprised of a molecular Cr complex and dibenzothiophene-5,5-dioxide (DBTD) as a redox mediator, which achieves high activity (TOF=1.51–2.84×10 5 s –1 ) and quantitative selectivity. Under aprotic or protic conditions, DBTD produces a co-electrocatalytic response with 1 by coordinating trans to the site of CO 2 binding and mediating electron transfer from the electrode with quantitative efficiency for CO. Furthermore, this assembly is reliant on through-space electronic conjugation between the π frameworks of DBTD and the bpy fragment of the catalyst ligand, with contributions from dispersive interactions and weak sulfone coordination.
Many metal coordination compounds catalyze CO 2 electroreduction to CO, but cobalt phthalocyanine hybridized with conductive carbon such as carbon nanotubes is currently the only one that can generate methanol. The underlying structure–reactivity correlation and reaction mechanism desperately demand elucidation. Here, in this study, we report the first in situ X-ray absorption spectroscopy characterization, combined with ex situ spectroscopic and electrocatalytic measurements, to study CoPc-catalyzed CO 2 reduction to methanol. Molecular dispersion of CoPc on CNT surfaces, as evidenced by the observed electronic interaction between the two, is crucial to fast electron transfer to the active sites and multi-electron CO 2 reduction. CO, the key intermediate in the CO 2 -to-methanol pathway, is found to be labile on the active site, which necessitates a high local concentration in the microenvironment to compete with CO 2 for active sites and promote methanol production. A comparison of the electrocatalytic performance of structurally related porphyrins indicates that the bridging aza-N atoms of the Pc macrocycle are critical components of the CoPc active site that produces methanol. In situ X-ray absorption spectroscopy identifies the active site as Co(I) and supports an increasingly non-centrosymmetric Co coordination environment at negative applied potential, likely due to the formation of a Co−CO adduct during the catalysis.
The nitrido-ate complex [(PN) 2 Ti(N){μ 2 -K(OEt 2 )}] 2 (1) (PN − =(N-(2-P i Pr 2 -4-methylphenyl)-2,4,6-Me 3 C 6 H 2 ) reductively couples CO and isocyanides in the presence of DME or cryptand (Kryptofix222), to form rare, five-coordinate Ti II complexes having a linear cumulene motif, [K(L)][(PN) 2 Ti(NCE)] (E=O, L=Kryptofix222, (2); E=NAd, L=3 DME, (3); E=N t Bu, L=3 DME, (4); E=NAd, L=Kryptofix222, (5)). Oxidation of 2–5 with [Fc][OTf] afforded an isostructural Ti III center containing a neutral cumulene, [(PN) 2 Ti(NCE)] (E=O, (6); E=NAd (7), N t Bu (8)) and characterization by CW X-band EPR spectroscopy, revealed unpaired electron to be metal centric. Moreover, 1e − reduction of 6 and 7 in the presence of Kryptofix222cleanly reformed corresponding discrete Ti II complexes 2 and 5, which were further characterized by solution magnetization measurements and high-frequency and -field EPR (HFEPR) spectroscopy. Furthermore, oxidation of 7 with [Fc*][B(C 6 F 5 ) 4 ] resulted in a ligand disproportionated Ti IV complex having transoid carbodiimides, [(PN) 2 Ti(NCNAd) 2 ] (9). Comparison of spectroscopic, structural, and computational data for the divalent, trivalent, and tetravalent systems, including their 15 N enriched isotopomers demonstrate these cumulenes to decrease in order of backbonding as Ti II →Ti III →Ti IV and increasing order of π-donation as Ti II →Ti III →Ti IV , thus displaying more covalency in Ti III species. Lastly, we show a synthetic cycle whereby complex 1 can deliver an N-atom to CO and CNAd.