School of Mathematical Sciences, University of Science and Technology of China; Wu Wen-Tsun Key Laboratory of Mathematics, Chinese Academy of Sciences, Hefei 230026, China
Teng Huang is an Associate Professor at the School of Mathematical Sciences, University of Science and Technology of China (USTC). He received his Ph.D. degree in Mathematics from USTC in 2016. His research mainly focuses on mathematical physics and differential geometry
In this note, we study the Yang–Mills bar connection A, i.e., the curvature of A obeys ˉ∂∗AF0,2A=0, on a principal G-bundle P over a compact complex manifold X. According to the Koszul–Malgrange criterion, any holomorphic structure on P can be seen as a solution to this equation. Suppose that G=SU(2) or SO(3) and X is a complex surface with H1(X,Z2)=0. We then prove that the (0,2)-part curvature of an irreducible Yang–Mills bar connection vanishes, i.e., (P,ˉ∂A) is holomorphic.
Graphical Abstract
Yang–Mills bar connection and holomorphic structure.
Abstract
In this note, we study the Yang–Mills bar connection A, i.e., the curvature of A obeys ˉ∂∗AF0,2A=0, on a principal G-bundle P over a compact complex manifold X. According to the Koszul–Malgrange criterion, any holomorphic structure on P can be seen as a solution to this equation. Suppose that G=SU(2) or SO(3) and X is a complex surface with H1(X,Z2)=0. We then prove that the (0,2)-part curvature of an irreducible Yang–Mills bar connection vanishes, i.e., (P,ˉ∂A) is holomorphic.
Public Summary
A connection A is called Yang–Mills bar connection if the curvature of the connection A satisfies ˉ∂∗AF0,2A=0.
When the structure group G=SU(2) or SO(3), we show that rank(F0,2A+F2,0A)≤1.
Suppose that H1(X,Z2)=0, following an idea from Donaldson, we prove that F0,2A=0.
Proton exchange membrane fuel cells (PEMFCs) have been regarded as the most promising low-carbon energy conversion technology because of their amazing energy density, high efficiency, and environmental protection characteristics[1-3]. However, the sluggish oxygen reduction reaction largely impedes its wide application in PEMFC cathodes[4-5]. Noble metals with high activity, such as platinum, account for a high proportion of the cost of fuel cells, and this problem will become increasingly obvious with large-scale industrialization. Therefore, the development of nonprecious metal catalyst substitutes has become an important issue. Among many substitutes, atomically FeN4 catalysts are favored by researchers due to their activity with promotion potential and low cost[6-8]. However, further requirements for selectivity and activity per unit are still driving the design and development of FeN4 with new configurations.
FeN4 with a conventional planar configuration has been studied as a potential cathode catalyst for PEMFCs for many years[9-11]. Nevertheless, this planar four-coordinate structure is easily attacked by protons or free radicals in acidic environments, causing leaching of iron ions in the FeN4 site and oxidization of the surrounding carbon structure[12-13]. In addition, exposed Fe active centers are easily poisoned in the electrolyte solution, resulting in a decline in both stability and selectivity[14]. Therefore, effectively regulating the local coordination structure and electronic structure to obtain a new FeN4 configuration arouses the extensive interest of researchers. For instance, most current reported regulation strategies are focused on heteroatom doping carbon substrates[15-16], fabricating carbon structure disorder[17], and introducing multiple active sites[18]. In these excellent works, it is worth noting that high-temperature thermal activation plays an important role in enhancing catalytic activity and stability. However, because high temperature greatly accelerates the diffusion of ions, the original structure of sites is often changed after thermal activation. Therefore, it is difficult to customize FeN4 sites with a specific structure on a large scale[19-21].
Herein, by using confinement space and specific adsorption of NO to hemin, we developed an NO (nitrogen oxide) group axial-modified central electron-enriched FeN4 structure (denoted as NO-FeN4) with excellent ORR mass activity and selectivity for the first time. This specific FeN4 site with a modified electronic structure shows complete 4-electron reaction selectivity and high mass activity, which paves the way for designing advanced ORR electrocatalysts for PEMFCs.
2.
Experiments
2.1
The synthesis of NO-FeN4 product
All of the chemicals were of analytical grade and used without further purification. In a typical synthesis, 1 mmol hemin and 1 mmol Zn(NO3)2 were dissolved in 25 mL dimethylformamide under vigorous stirring for 30 min, and then 600 mg commercial activated carbon black (Ketjenblack EC600JD) and 25 mL dimethylformamide were snoicated for 1 h and dipped into the above solution. The mixed solution was continuously stirred for 4 h at 10 °C for adsorption until most of the hemin was adsorbed onto the carbon black. Then, the carbon black with adsorbed hemin/Zn(NO3)2 was separated from the mixture solution by washing with dimethylformamide several times, filtrating and freeze drying for 48 h. The obtained composite was directly pyrolyzed (at 650 °C for 2 h, heating rate of 10 °C·min−1, in an Ar atmosphere). Then, the pyrolyzed product was leached in 0.5 mol·L−1 H2SO4 at 80 °C for 8 h to remove inactive iron and zinc species. The leached sample was washed to neutral with water and alcohol several times and dried in vacuum at 60 °C overnight to generate the carbon-supported NO-FeN4 catalyst. For comparison, carbon black with only adsorbed hemin was also prepared under similar conditions, and the resultant sample was denoted as carbon black-supported FeN4.
2.2
Fabrication of the working electrode for ORR catalytic activity testing
To prepare the working electrode, 4 mg catalysts mixed with 40 μL Nafion solution (Sigma Aldrich, 5 wt%) were dispersed into 1 mL isopropanol and water mixture solution (volume ratio 3∶1) and sonicated for at least 60 min to form a homogeneous catalyst ink. A certain volume of catalyst ink was then drop-cast onto the glassy carbon electrode with a 0.6 mg·cm−2 loading for all samples.
2.3
Fabrication of single PEMFC battery
The catalyst inks were prepared by using catalyst, isopropanol, deionized water and Nafion solution (Sigma Aldrich, 5 wt%) with a weight ratio of 1/90/30/11. The catalyst inks were ultrasonicated for 1 h and then brushed on carbon paper with an effective area of 5 cm2 until the loading reached 4 mg·cm−2. With a similar preparation process, commercial Pt/C (20 wt%) ink was dispersed on carbon paper with a loading of 0.2 mg Pt cm−2 as the anode. Using a thermocompressor, the prepared cathode and anode were then pressed onto the two sides of a Nafion 211 membrane (DuPont) at 130 °C and 5 MPa for 5 min to fabricate membrane electrode assemblies (MEAs). The MEA was measured in a single-cell and condition-controlled fuel cell test station (Scribner 850e, Scribner Associates). The flow rates of H2 and O2 were both 400 mL·min−1, and the relative humidity was 100% during the PEFMC tests. During the test, the cell and input fuel temperature were maintained at 80 °C, and the back pressure was set at 0.2 MPa.
3.
Results and discussion
3.1
Coordination structure analysis of central electron-enriched NO-FeN4 sites
In this work, central electron-enriched NO-FeN4 sites with optimized electronic configurations were prepared by adsorption of hemin and zinc nitrate into the mesopores of carbon black (Ketjenblack EC600JD) and rapid heat treatment. Through in situ mass spectrometry, we found that nitrate can rapidly generate a large number of nitric oxide groups (NO) with 650 °C fast pyrolysis treatment, and the generated NO groups tend to adsorb onto the central atom of porphyrin iron[22] (Supporting information Fig. S1a). In contrast, in Fig. S1b, nitric oxide cannot be detected at 30 m/z for FeN4 precursors without adding nitrate (denoted as FeN4). The detailed structural features of the as-prepared NO-FeN4 electrocatalysts were investigated by X-ray diffraction (XRD) and Raman spectroscopy. As shown in Figs. S2 and S3, both NO-FeN4 and FeN4 show only two distinct peaks at approximately 26.2° and 43.2°, which could be ascribed to the characteristic (002) and (100) planes of graphitic carbon[23]. Meanwhile, the Raman spectra of the NO-FeN4 and FeN4 samples (Fig. S4) also display similar D-band (1326 cm−1) for lattice defects and G-band (1585 cm−1) for sp2 hybrid carbon atoms. Additionally, the IG/ID ratios of NO-FeN4 and FeN4 were 0.81 and 0.83, respectively, indicating the similar graphitization degrees of the two catalysts[24]. These results suggest that no obvious Fe-based compounds exist in either the NO-FeN4 or FeN4 electrocatalysts. Moreover, the morphology and microstructure were also investigated by transmission and scanning electron microscopy. As shown in Figs. 1b and S5, the NO-FeN4 electrocatalyst displays a spherical porous carbon structure, and no obvious nanoparticles could be found, indicating that the Fe atoms might be confined in carbon pores in atomically dispersed forms. Further investigation by high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM) in Fig. 1c showed that many bright spots were homogeneously distributed in the porous carbon framework, confirming the atomically dispersed FeN4 sites existing in the NO-FeN4 sample. The corresponding energy-dispersive spectroscopy (EDS) mapping (Fig. 1d) shows that only Fe, N, and C are uniformly dispersed in the carbon framework, and no Zn can be detected, indicating that Zn has little residue after the wash and heat treatment processes. This conclusion can also be confirmed by X-ray photoelectron spectroscopy (XPS) analysis in Fig. S6. Similarly, FeN4 samples prepared without confined NO treatment, which are denoted as FeN4, are also presented in Figs. S1a, S3–4, S6, and S8–9. After systematic characterizations, the traditional FeN4 sample seems to have similar features to NO-FeN4 in terms of phase composition, morphology structure, and element distributions. Thus, it can be easily concluded that confined small molecule synthesis has no obvious effect on morphology and element distributions and that Fe atoms might form as single FeN4 sites or clusters in the NO-FeN4 sample. The detailed carbon structure was also investigated to exclude its influence on catalyst performance. Brunauer-Emmett-Teller (BET) analysis was also presented to analyze the BET surface area and pore size distribution of NO-FeN4 and its contrast. In Fig. 1e, NO-FeN4 exhibits a surface area of 760.7 m2·g−1, which is comparable to that of the FeN4 sample (794.5 m2·g−1). In addition, the isotherm hysteresis loops of the NO-FeN4 and FeN4 samples exhibit obvious type-IV isotherms, indicating the same mesoporous carbon configuration of the two catalysts[25]. Moreover, the corresponding pore size distribution also confirms that these two catalysts both have a large number of mesopores with a diameter of approximately 3.2 nm (Fig. 1e inset). Comprehensive analysis of the results above, it can be concluded that NO-FeN4 catalysts have no significant difference in phase, element composition, morphology or detailed carbon structure properties. Therefore, the specific modified FeN4 sites might play a key role in enhancing ORR activities.
Figure
1.
(a) Schematic representation of the synthesis for NO-FeN4. (b) HRTEM images of NO-FeN4 material. (c) HAADF-STEM images of NO-FeN4. (d) Elemental mapping images for the NO-FeN4. (e) N2 adsorption/desorption isotherms and corresponding pore size distribution curves (inset) of as-prepared NO-FeN4 product.
To evaluate the local atomic structure around the NO-FeN4 catalyst, X-ray absorption structure (XAS) measurements were performed. As shown in Fig. 2a and b, the XANES spectra of both the NO-FeN4 and FeN4 samples present similar preedge peaks at 7113.6 eV compared with the standard hemin precursor, suggesting their similar main structure of a typical square-planar Fe-N4 structure (D4h symmetry)[26]. However, the negative energy shifting of the absorption edge (Fig. 2c), as well as the lower white-line intensity (Fig. 2b) of NO-FeN4, suggests that its 3d orbital electron density is higher than that of the FeN4 sample, which means that the enrichment of d-electrons in the Fe center of NO-FeN4 catalysts occurs[27]. Moreover, an in-depth investigation of NO-FeN4 was also performed, as shown in Fig. 2d. From Fourier transform (FT) plots of the Fe K-edge results in R-space, NO-FeN4 presented one main peak at 1.51 Å, which was considered to be an elongated Fe-N shell compared with the FeN4 sample at 1.48 Å. To further reveal the local coordination structure of Fe atoms, Fe K-edge EXAFS spectra of NO-FeN4 and FeN4 catalysts were both fitted. As shown in Fig. S10, NO-FeN4 exhibited an Fe–N/O bond length of 2.03 Å ± 0.03 Å with a N/O coordination number of 5.6 ± 0.6, which is different from that of the FeN4 sample (2.00 Å ± 0.03 Å, 6.4 ± 0.6), demonstrating that increased Fe–N/O bond length has an obvious effect on the coordination number of Fe center atoms, which might change the adsorption/desorption energy of reactants. Based on the XAFS results, NO-FeN4 has a larger bond length than FeN4, which provides the opportunity to discuss the structure-function relationship between the bond length and the catalytic activity. As is well known, the metal iron center can act as an oxygen molecule adsorption center during the ORR process[26]. Therefore, the intrinsic ORR catalytic activity of the FeN4 materials is highly dependent on the nature of the Fe center. In fact, due to the stretched Fe–N bond length, NO-FeN4 with an electron-rich Fe atom center could suppress the Fe-to-ligand back-donation between the Fe atom center and its adsorbed oxygen species, causing enhancement of the Fe2+/3+ redox potential and reducing site-blocking effects on the ORR [28]. As far as we know, the scaled customization of the FeN4 structure in the high-temperature pyrolysis process is still a great challenge. As shown in Fig. S11, the NO-FeN4 site can be customized on a large scale during the pyrolysis process owing to the confinement space and specific adsorption of NO to hemin.
Figure
2.
(a) Schematic illustration of electronic structure for NO-FeN4. (b) and (c) Fe K-edge X-ray absorption near edge structures (XANES). (d) The Fe K-edge extended XAFS oscillation function of NO-FeN4 and FeN4 samples.
3.2
Electron structure analysis of central electron-enriched NO-FeN4 sites
It has been widely recognized that the NO group has a strong electron absorption effect[29], while the electronic structure and valence state around the Fe center have been systematically studied by soft X-ray absorption near-edge spectroscopy (XANES) and X-ray photoelectron spectroscopic (XPS) analysis. As shown in Fig. 3a, the normalized XANES spectrum of NO-FeN4 at the Fe L3 edge has a lower peak intensity and a negative energy shift compared to that of the FeN4 electrocatalyst, indicating that more electrons occupy the Fe 3d orbitals of NO-FeN4, which is consistent with the Fe K-edge result above. Moreover, in Fig. 3b, the C K-edge XANES spectra of the NO-FeN4 and FeN4 products exhibit a δ*C–C transition centered at ~303.5 eV and a π*C=C transition at ~295.6 eV, which represent typical sp2-hybridized carbon[30]. The increased π*C=C intensity for NO-FeN4 suggests that it has fewer electrons in the C 2p orbitals than FeN4. This phenomenon hints that more electrons might flow into the Fe center of NO-FeN4, demonstrating a stronger charge transfer effect. In addition, N K-edge XANES and XPS spectra can also verify the findings discovered above. As shown in Fig. 3c, the N K-edge of NO-FeN4 represents two typical spectroscopic features: the 1s→π* transition in the region of 393.0–401.0 eV and the 1s→σ* transition at 401.0–410.0 eV. Specifically, peak a can be ascribed to the π* transition of Fe-bonded pyrrole-type N at 398.5 eV, and b can be assigned to the π* transition of graphitic-type N-groups at 401.5 eV[31]. Notably, peak a of NO-FeN4 exhibits lower absorption intensity in the two resonances. This result suggests that the formation of the Fe−N−C chemical bond in NO-FeN4 has a stronger charge transfer effect. Furthermore, in Fig. 3d, the X-ray photoelectron spectroscopy (XPS) spectra of N 1s also reveal that the Fe-pyrrolic N peak in NO-FeN4 shifts to a higher binding energy (399.1 eV) compared with FeN4 materials (398.8 eV)[32]. In Fig. S12, the core-level scan spectrum of Fe 2p shows that the Fe 2p3/2 peak of NO-FeN4 has a slight negative shift compared with that of FeN4. These results confirm that a faster electron transfer pathway has formed between the Fe center and its substrate due to the strong electron absorption effect of NO. This suggests establishing a fast electron transport pathway between the substrate and the active sites.
Figure
3.
(a) Electron transfer diagram of N, C adjacent to Fe central of NO-FeN4. (b) Soft-XAS spectra of Fe L-edge, (c) C K-edge and (d) N K-edge of NO-FeN4 and FeN4. (e) XPS spectra of N 1s of NO-FeN4.
To verify the ORR activity of the NO-FeN4 and FeN4 electrocatalysts, rotating ring-disk electrode (RRDE) measurements were first performed in an acidic medium. As shown in Fig. 4a, the linear scan sweep voltammetry (LSV) curves exhibited a positive onset potential of 0.90 V and a half-wave potential of 0.82 V for NO-FeN4, which is much higher than that of the FeN4 electrocatalyst (0.75 V) and close to that of commercial Pt/C. To quantitatively evaluate the ORR activity of the as-prepared NO-FeN4 sample, Tafel slopes and electrochemical impedance spectroscopy (EIS) were also performed, as shown in Fig. 4b and Fig. S13. NO-FeN4 has a relatively small Tafel slope value (77 mV/dec) compared to that of FeN4 (90 mV/dec) and Pt/C (110 mV/dec). Moreover, it also shows the smallest semicircle, which represents the minimum interface charge transfer resistance, indicating the extremely high kinetic activity for the NO-FeN4 sample. In Fig. 4c, the mass activity at 0.85 V was also calculated to demonstrate the high catalytic activity of the NO-FeN4 sample. In Fig. 4d, NO-FeN4 exhibits the largest mass activity of 9.3 A·g−1. Moreover, the turnover frequency (TOF) of NO-FeN4 is 1.14 e·s−1·site−1. It is 1.375 times that of the FeN4 sample, indicating that the Fe–N bond length endows its ultrahigh intrinsic activity. Moreover, the peroxide (H2O2) generation percent and electron transfer number were also evaluated using the RRDE technique. As shown in Fig. 4d, NO-FeN4 exhibited a relatively low H2O2 yield ranging from 1.3% to 1.8%, and the corresponding electron transfer number exceeded 3.97 below 0.8 V, suggesting its high selectivity and a direct four-electron pathway. These results revealed that NO-FeN4 could suppress the generation of H2O2 and reduce the Fenton reaction during the ORR process. As shown in Fig. 4e, only a 20 mV loss can be detected after 7000 CV cycles for NO-FeN4 in O2-saturated 0.5 mol·L−1 H2SO4. However, the half-wave potential of the FeN4 sample decreased by 47 mV under the same conditions (Fig. S14). These results prove that NO-FeN4 exhibits significantly enhanced catalytic activity as well as stability and could be a promising cathode electrocatalyst for PEMFCs. The excellent ORR activity of the NO-FeN4 sample was also demonstrated by integrating it into membrane-electrode assemblies (MEAs) with a total catalyst loading of 4.0 mg·cm−2 for PEMFC testing. As shown in Fig. 4f, under the test condition of 1.5 bar H2/O2 back pressure, the NO-FeN4 exhibited an open-circuit potential of 1.01 V and generated a current density of 355 mA·cm−2 at 0.8 V and 1828 mA·cm−2 at 0.4 V, which surpasses the FeN4 catalyst. Moreover, the corresponding peak power density of NO-FeN4 reached 735 mW·cm−2, which was much higher than that of FeN4 (481 mW·cm−2), indicating its broad application prospects in PEMFCs.
Figure
4.
Electrocatalytic performance. (a) ORR polarization curves and (b) corresponding Tafel plots of NO-FeN4 and FeN4 in O2-saturated 0.5 mol·L−1 H2SO4 and 20% Pt/C in 0.1 mol·L−1 HClO4 under a rotating rate of 1600 r·min−1. (c) Comparison of mass activity at 0.85 V and half-wave potential of NO-FeN4 and FeN4 catalysts. (d) Peroxide yield and electron transfer number of NO-FeN4 and FeN4 sample during ORR process. (e) LSV curves of NO-FeN4 catalysts before and after 7000 CV cycles. (f) Polarization and power density curves of NO-FeN4 and FeN4 based membrane electrode assemblies in PEMFCs. Cell temperature: 80 ℃; RH: 100%, H2/O2: 1.5 bar.
The specific reaction mechanism of NO-FeN4 was further investigated by operando XAFS tests. As shown in Fig. 5a, operando XAS tests were conducted on a three-electrode system by using a gas diffusion electrode composited with NO-FeN4 as a cathode in fluorescence mode. In Fig. 5b, the first derivative of the XANES spectra of NO-FeN4 at the open circuit exhibited obvious positive energy shifting compared to the dry powder sample, indicating that the adsorption of oxygen species in the electrolyte may lead to an increase in the oxidation state for central Fe atoms, while excessive impurity adsorption in the electrolyte may be harmful to the fast oxygen reduction reaction FeN4 sites. Additionally, Fig. 5c shows the normalized XANES spectra of the Fe K-edge for NO-FeN4, and the adsorption edge exhibited negative shifting along with the decrease in applied potential, indicating that the ORR is closely related to the Fe2+/Fe3+ redox potential. Notably, the adsorption edge of 1.0 V and 1.0 V after cycles coincide, which proves that NO-FeN4 has good cycle stability. As the adsorption of reactants at FeN4 sites plays a key role in the ORR process, a surface-sensitive delta-mu (Δμ) technique in the X-ray absorption near-edge spectra (XANES) region was used to study the adsorption state of reactants at the NO-FeN4 sites. For example, in 0.5 mol·L−1 H2SO4, the experimental Δμ spectrum is obtained by subtracting the region of the Fe K-edge at 0.1 and 1.0 V (or 0.9, 0.7, and 0.3 V). As shown in Fig. 5d, the corresponding Δμ-XANES also shows that NO-FeN4 exhibited a significant increase in |Δμ| between 0.7 and 0.9 V, which means that it could decompose adsorbed oxygen molecules at a relatively high potential, avoid the influence of adsorbed impurities, and improve the ORR activity[33]. Since most FeN4 catalysts are easily poisoned by impurities at the beginning of the reaction process, a low reaction potential is required to decompose impurities and conduct an oxygen reduction reaction. Therefore, the high reaction potential of NO-FeN4 indicates that it is not easily poisoned, leading to a higher reaction activity and stability. In conclusion, all the above results demonstrate that NO-FeN4 exhibits accelerated kinetics and relatively high stability for oxygen reduction under acidic conditions.
Figure
5.
(a) Schematic of operando XAS measurement device and three-electrode half-cell. (b) First derivative of XANES spectra of NO-FeN4 for dry powder and operando sample. (c) Operando XANES spectra of Fe K-edge for NO-FeN4. (d) Normalized difference spectra for Fe K-edge of NO-FeN4 at different potentials at operando condition.
The much-enhanced ORR performance of NO-FeN4 can be ascribed to the following features: (i) the highly dispersed FeN4 sites could significantly improve ORR catalytic activity. (ii) Δμ-XANES analysis shows that NO-FeN4 sites possess highly reversible Fe2+/Fe3+ redox potential and are less likely to be affected by adsorbed impurities. (iii) Rapid electron transfer occurs between the Fe center and its coordinated nitrogen and carbon atoms, which greatly expedites the reduction process of adsorption oxygen species. (iv) The porous and highly graphitization carbon configuration increases the utilization and robust ability of electrocatalysts. Benefiting from the combination of highly active sites and open exposure areas, NO-FeN4 exhibits superior ORR catalytic activity in an acidic medium as well as high power density when integrated into PEMFC systems.
4.
Conclusions
In conclusion, the central electron-enriched NO-FeN4 structure has been successfully developed as a high-efficiency cathode electrocatalyst. Because the confined small molecule modification significantly increases the charge densities of central iron atoms, the as-prepared electrocatalyst exhibits much enhanced intrinsic catalytic activity, fast electron transfer ability and full four-electron reaction selectivity. Moreover, the as-prepared NO-FeN4 electrocatalyst shows high mass activity (1.1 A·g−1 at 0.85 V) and TOF activity (1.14 e·s−1·site−1), and its half-wave potential is also close to that of the commercial Pt/C sample. As a result, the PEMFC built by this electrocatalyst has a high power density of 725 mW·cm−2 at 80 °C, 1.5 bar back pressure, and 100% relative humidity, demonstrating its promising potential in practical applications. This work will pave new avenues for the design of advanced nonnoble electrocatalysts for PEMFCs.
Conflict of interest
The authors declare that they have no conflict of interest.
Conflict of Interest
The authors declare that they have no conflict of interest.
A connection A is called Yang–Mills bar connection if the curvature of the connection A satisfies ˉ∂∗AF0,2A=0.
When the structure group G=SU(2) or SO(3), we show that rank(F0,2A+F2,0A)≤1.
Suppose that H1(X,Z2)=0, following an idea from Donaldson, we prove that F0,2A=0.
Dai B, Guan R. Transversality for the full rank part of Vafa–Witten moduli spaces. Comm. Math. Phys., 2022, 389: 1047–1060. DOI: 10.1007/s00220-021-04176-x
[2]
Donaldson S K. Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 1985, 50 (1): 1–26. DOI: 10.1112/plms/s3-50.1.1
[3]
Donaldson S K, Kronheimer P B. The Geometry of Four-Manifolds. Oxford, UK: Oxford University Press, 1990 .
[4]
Huybrechts D. Complex Geometry: An Introduction. Berlin: Springer, 2005 .
[5]
Itoh M. Yang–Mills connections over a complex surface and harmonic curvature. Compositio Mathematica, 1987, 62: 95–106.
[6]
Le H V. Yang–Mills bar connections over compact Kähler manifolds. Archivum Mathematicum (Brno), 2010, 46: 47–69.
[7]
Mares B. Some analytic aspects of Vafa–Witten twisted N = 4 supersymmetric Yang–Mills theory. Thesis. Cambridge, USA: Massachusetts Institute of Technology, 2010 .
[8]
Koszul J L, Malgrange B. Sur certaines structures fibrées complexes. Archiv der Mathematik, 1958, 9: 102–109. DOI: 10.1007/BF02287068
[9]
Newlander A, Nirenberg L. Complex analytic coordinates in almost complex manifolds. Ann. Math., 1957, 65 (3): 391–404. DOI: 10.2307/1970051
[10]
Păunoiu A, Rivière T. Sobolev connections and holomorphic structures over Kähler surfaces. J. Func. Anal., 2021, 280 (12): 109003. DOI: 10.1016/j.jfa.2021.109003
[11]
Stern M. Geometry of minimal energy Yang–Mills connections. J. Differential Geom., 2010, 86 (1): 163–188. DOI: 10.4310/jdg/1299766686
[12]
Tanaka T. Some boundedness properties of solutions to the Vafa–Witten equations on closed 4-manifolds. The Quarterly Journal of Mathematics, 2017, 68 (4): 1203–1225. DOI: 10.1093/qmath/hax015
[13]
Uhlenbeck K, Yau S T. On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure and Appl. Math., 1986, 39 (S1): S257–S293. DOI: 10.1002/cpa.3160390714
Liu, X., Fan, K., Huang, X. et al. Recent advances in artificial intelligence boosting materials design for electrochemical energy storage. Chemical Engineering Journal, 2024.
DOI:10.1016/j.cej.2024.151625
2.
Wang, X., Dong, C., Zhao, W. et al. Core-shell tin pyrophosphate-based composite membrane for fuel cell with durability enhancement at elevated temperatures. Electrochimica Acta, 2024.
DOI:10.1016/j.electacta.2023.143588
3.
Yu, S., Wang, Q., Wang, J. et al. Computational modeling guided design of metal-organic frameworks for photocatalysis - a mini review. Catalysis Science and Technology, 2023, 13(23): 6583-6603.
DOI:10.1039/d3cy00862b
4.
Xu, Y., Ge, J., Ju, C.-W. Machine learning in energy chemistry: introduction, challenges and perspectives. Energy Advances, 2023, 2(7): 896-921.
DOI:10.1039/d3ya00057e
Dai B, Guan R. Transversality for the full rank part of Vafa–Witten moduli spaces. Comm. Math. Phys., 2022, 389: 1047–1060. DOI: 10.1007/s00220-021-04176-x
[2]
Donaldson S K. Anti self-dual Yang–Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 1985, 50 (1): 1–26. DOI: 10.1112/plms/s3-50.1.1
[3]
Donaldson S K, Kronheimer P B. The Geometry of Four-Manifolds. Oxford, UK: Oxford University Press, 1990 .
[4]
Huybrechts D. Complex Geometry: An Introduction. Berlin: Springer, 2005 .
[5]
Itoh M. Yang–Mills connections over a complex surface and harmonic curvature. Compositio Mathematica, 1987, 62: 95–106.
[6]
Le H V. Yang–Mills bar connections over compact Kähler manifolds. Archivum Mathematicum (Brno), 2010, 46: 47–69.
[7]
Mares B. Some analytic aspects of Vafa–Witten twisted N = 4 supersymmetric Yang–Mills theory. Thesis. Cambridge, USA: Massachusetts Institute of Technology, 2010 .
[8]
Koszul J L, Malgrange B. Sur certaines structures fibrées complexes. Archiv der Mathematik, 1958, 9: 102–109. DOI: 10.1007/BF02287068
[9]
Newlander A, Nirenberg L. Complex analytic coordinates in almost complex manifolds. Ann. Math., 1957, 65 (3): 391–404. DOI: 10.2307/1970051
[10]
Păunoiu A, Rivière T. Sobolev connections and holomorphic structures over Kähler surfaces. J. Func. Anal., 2021, 280 (12): 109003. DOI: 10.1016/j.jfa.2021.109003
[11]
Stern M. Geometry of minimal energy Yang–Mills connections. J. Differential Geom., 2010, 86 (1): 163–188. DOI: 10.4310/jdg/1299766686
[12]
Tanaka T. Some boundedness properties of solutions to the Vafa–Witten equations on closed 4-manifolds. The Quarterly Journal of Mathematics, 2017, 68 (4): 1203–1225. DOI: 10.1093/qmath/hax015
[13]
Uhlenbeck K, Yau S T. On the existence of Hermitian–Yang–Mills connections in stable vector bundles. Comm. Pure and Appl. Math., 1986, 39 (S1): S257–S293. DOI: 10.1002/cpa.3160390714
Liu, X., Fan, K., Huang, X. et al. Recent advances in artificial intelligence boosting materials design for electrochemical energy storage. Chemical Engineering Journal, 2024.
DOI:10.1016/j.cej.2024.151625
2.
Wang, X., Dong, C., Zhao, W. et al. Core-shell tin pyrophosphate-based composite membrane for fuel cell with durability enhancement at elevated temperatures. Electrochimica Acta, 2024.
DOI:10.1016/j.electacta.2023.143588
3.
Yu, S., Wang, Q., Wang, J. et al. Computational modeling guided design of metal-organic frameworks for photocatalysis - a mini review. Catalysis Science and Technology, 2023, 13(23): 6583-6603.
DOI:10.1039/d3cy00862b
4.
Xu, Y., Ge, J., Ju, C.-W. Machine learning in energy chemistry: introduction, challenges and perspectives. Energy Advances, 2023, 2(7): 896-921.
DOI:10.1039/d3ya00057e