Cobalt nanoparticles(NPs)catalysts are extensively used in heterogeneous catalytic reactions,and the addition of alkali metal promoters is a common method to modulate the catalytic performance because the catalyst'...Cobalt nanoparticles(NPs)catalysts are extensively used in heterogeneous catalytic reactions,and the addition of alkali metal promoters is a common method to modulate the catalytic performance because the catalyst's surface structures and morphologies are sensitive to the addition of promoters.However,the underlying modulation trend remains unclear.Herein,the adsorption of alkali metal promoters(Na and K)on the surfaces of face-centered-cubic(FCC)and hexagonal-closest packed(HCP)polymorphous cobalt was systematically investigated using density functional theory.Furthermore,the effect of alkali promoters on surface energies and nanoparticle morphologies was revealed on the basis of Wulff theory.For FCC-Co,the exposed area of the(111)facet in the nanoparticle increases with the adsorption coverage of alkali metal oxide.Meanwhile,the(311),(110),and(100)facets would disappear under the higher adsorption coverage of alkali metals.For HCPCo,the Wulff morphology is dominated by the(0001)and(1011)facets and is independent of the alkali metal adsorption coverage.This work provides insights into morphology modulation by alkali metal promoters for the rational design and synthesis of cobalt-based nanomaterials with desired facets and morphologies.展开更多
Given a positive function F on S^n which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in R^n+1 which is a generalization of the usual r-th mean curvature Hr. ...Given a positive function F on S^n which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in R^n+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in R^n+1. We give some new characterizations of the Wulff shape by the use of our integral formulas of Minkowski type, in case F=1 which reduces to some well-known results.展开更多
This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bou...This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.展开更多
基金financial support from the National Natural Science Foundation of China (Nos.21972157,21972160,and 22202224)the CAS Project for Young Scientists in Basic Research (No.YSBR-005)+2 种基金the Key Research Program of Frontier Sciences CAS (No.ZDBS-LY-7007)the CAS Project for Internet Security and Information Technology (No.CAS-WX2021SF0110)the funding support from Beijing Advanced Innovation Center for Materials Genome Engineering,Synfuels China,Co.Ltd.and Inner Mongolia University of Technology。
文摘Cobalt nanoparticles(NPs)catalysts are extensively used in heterogeneous catalytic reactions,and the addition of alkali metal promoters is a common method to modulate the catalytic performance because the catalyst's surface structures and morphologies are sensitive to the addition of promoters.However,the underlying modulation trend remains unclear.Herein,the adsorption of alkali metal promoters(Na and K)on the surfaces of face-centered-cubic(FCC)and hexagonal-closest packed(HCP)polymorphous cobalt was systematically investigated using density functional theory.Furthermore,the effect of alkali promoters on surface energies and nanoparticle morphologies was revealed on the basis of Wulff theory.For FCC-Co,the exposed area of the(111)facet in the nanoparticle increases with the adsorption coverage of alkali metal oxide.Meanwhile,the(311),(110),and(100)facets would disappear under the higher adsorption coverage of alkali metals.For HCPCo,the Wulff morphology is dominated by the(0001)and(1011)facets and is independent of the alkali metal adsorption coverage.This work provides insights into morphology modulation by alkali metal promoters for the rational design and synthesis of cobalt-based nanomaterials with desired facets and morphologies.
基金Projects(11102164,11304243)supported by the National Natural Science Foundation of ChinaProject(2014JQ1039)supported by the Natural Science Foundation of Shannxi Province,China+1 种基金Project(3102016ZY027)supported by the Fundamental Research Funds for the Central Universities of ChinaProject(13GH014602)supported by the Program of New Staff and Research Area Project of NWPU,China
基金The project was supported by the National Natural Science Foundation of China(21972170,22072184)the Fundamental Research Funds for the Central Universities,and South-Central Minzu University(CZY13005,CZT20010).
基金Tianyuan Fund for Mathematics of NSFC (Grant No.10526030)Grant No.10531090 of the NSFCDoctoral Program Foundation of the Ministry of Education of China (2006)
文摘Given a positive function F on S^n which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in R^n+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in R^n+1. We give some new characterizations of the Wulff shape by the use of our integral formulas of Minkowski type, in case F=1 which reduces to some well-known results.
文摘This paper presents the proof of several inequalities by using the technique introduced by Alexandroff, Bakelman, and Pucci to establish their ABP estimate. First,the author gives a new and simple proof of a lower bound of Berestycki, Nirenberg, and Varadhan concerning the principal eigenvalue of an elliptic operator with bounded measurable coefficients. The rest of the paper is a survey on the proofs of several isoperimetric and Sobolev inequalities using the ABP technique. This includes new proofs of the classical isoperimetric inequality, the Wulff isoperimetric inequality, and the Lions-Pacella isoperimetric inequality in convex cones. For this last inequality, the new proof was recently found by the author, Xavier Ros-Oton, and Joaquim Serra in a work where new Sobolev inequalities with weights came up by studying an open question raised by Haim Brezis.