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
The surface diffusion of Pt dimers on Wulff polyhedral clusters with 586-12934 atoms was studied by the embedded atom method. The minimum energy diffusion path and the corresponding energy barrier for dimer diffusion ...The surface diffusion of Pt dimers on Wulff polyhedral clusters with 586-12934 atoms was studied by the embedded atom method. The minimum energy diffusion path and the corresponding energy barrier for dimer diffusion on cluster surfaces were determined through a combination of the quenched Molecular Dynamics and the Nudged Elastic Band method. It was found that the diffusion of a dimer across the step-edge by dissociation and consecutive single-atom exchange with the edge atoms, rather than diffusion over the edge as a unit. Therefore, the step can enhance the dissociation of the dimer. For small Wulff polyhedral clusters containing up to 2000 atoms, the energy barrier for dimer diffusion from the {111} to the {100} facet is almost equal to that for the single adatom diffusion on the flat {111} surface. From the calculated cluster size dependence of the energy barrier, we conclude that the Wulff polyhedral cluster is stable, as the cluster has more than about 2000 atoms.展开更多
We establish the mean width inequalities for symmetric Wulff shapes by a direct approach.We also yield the dual inequality along with the equality conditions.These new inequalities have Barthe’s mean width inequaliti...We establish the mean width inequalities for symmetric Wulff shapes by a direct approach.We also yield the dual inequality along with the equality conditions.These new inequalities have Barthe’s mean width inequalities for even isotropic measures and its dual form as special cases.展开更多
As cathode materials for alkali-ion batteries,sodium manganese oxides have been receiving considerable and continuous attention in recent decades.In this work,the structure and environment-dependent stability of NaMn_...As cathode materials for alkali-ion batteries,sodium manganese oxides have been receiving considerable and continuous attention in recent decades.In this work,the structure and environment-dependent stability of NaMn_(2)O_(4) surface were studied based on the first principles calculations.The surface stability diagram of NaMn_(2)O_(4) involving various different terminations of(100),(110)and(111)surfaces was constructed,and the stability of these different terminations could be compared as a function of chemical environment.It is found that the(100)-MnO and(111)-ONa terminations are two more stable terminations under the investigated chemical conditions.And the surface energies of(110)surfaces are negative under the investigated chemical potential,hence,(110)surfaces are unstable.The surface energy of NaMn_(2)O_(4) as a function of O chemical potential is also investigated under constant Na chemical potential.The structure relaxation indicates that the surface rumpling and surface reconstruction can affect the electronic structure of the surface,thereby reducing surface energy and stabilizing the surface.Furthermore,the Wulff shape of NaMn_(2)O_(4) was also constructed based on Gibbs-Wulff theorem.展开更多
A numerical simulation based on a regularized phase field model is developed to describe faceted dendrite growth morphology. The effects of mesh grid, anisotropy, supersaturation and fold symmetry on dendrite growth m...A numerical simulation based on a regularized phase field model is developed to describe faceted dendrite growth morphology. The effects of mesh grid, anisotropy, supersaturation and fold symmetry on dendrite growth morphology were investigated, respectively. These results indicate that the nucleus grows into a hexagonal symmetry faceted dendrite. When the mesh grid is above 640×640, the size has no much effect on the shape. With the increase in the anisotropy value, the tip velocities of faceted dendrite increase and reach a balance value, and then decrease gradually. With the increase in the supersaturation value, crystal evolves from circle to the developed faceted dendrite morphology. Based on the Wulff theory and faceted symmetry morphology diagram, the proposed model was proved to be effective, and it can be generalized to arbitrary crystal symmetries.展开更多
In this study we performed density functional theory calculations of the adsorption of oxygen on theε-Fe_(2)surfaces to investigate the Wulff construction evolution at different oxygen coverage(θ_(O))and potential(...In this study we performed density functional theory calculations of the adsorption of oxygen on theε-Fe_(2)surfaces to investigate the Wulff construction evolution at different oxygen coverage(θ_(O))and potential(μ_(O))as well as the mechanism of oxygen removal.At low oxygen coverage(θ_(O)=0.25)and potential(μ_(O)<-7.1 eV),the(■)facets with high activity remain the high exposure(55%to 90%)while the inert(101)facets contribute less than 25%.With increasing theμ_(O)andθ_(O),the(■)and(100)become the main exposing facets.Considering the reaction condition(300℃,1000 kPa,V(CO)/V(H_(2))=1),theμo is determined as-7.5 eV.Thus,theε-Fe_(2)exposes mainly the(■)surfaces indicating its antioxidation ability.Oxygen removal mechanisms indicate that the adsorbed oxygen is easily removed on(■)surface with a barrier of 1.33 eV in good agreement with the simulated Wulff constructions.展开更多
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
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.
基金supported by the National Natural Science Foundation of China (Grant No. 50971058)the Hunan Provincial Natural Science Foundation of China (Grant No. 10JJ3050)
文摘The surface diffusion of Pt dimers on Wulff polyhedral clusters with 586-12934 atoms was studied by the embedded atom method. The minimum energy diffusion path and the corresponding energy barrier for dimer diffusion on cluster surfaces were determined through a combination of the quenched Molecular Dynamics and the Nudged Elastic Band method. It was found that the diffusion of a dimer across the step-edge by dissociation and consecutive single-atom exchange with the edge atoms, rather than diffusion over the edge as a unit. Therefore, the step can enhance the dissociation of the dimer. For small Wulff polyhedral clusters containing up to 2000 atoms, the energy barrier for dimer diffusion from the {111} to the {100} facet is almost equal to that for the single adatom diffusion on the flat {111} surface. From the calculated cluster size dependence of the energy barrier, we conclude that the Wulff polyhedral cluster is stable, as the cluster has more than about 2000 atoms.
基金supported by National Natural Science Foundation of China (Grant No. 11271244)Shanghai Leading Academic Discipline Project (Grant No. S30104)
文摘We establish the mean width inequalities for symmetric Wulff shapes by a direct approach.We also yield the dual inequality along with the equality conditions.These new inequalities have Barthe’s mean width inequalities for even isotropic measures and its dual form as special cases.
基金Project(BK20241969)supported by the Natural Science Foundation of Jiangsu Province,ChinaProject(51971249)supported by the National Natural Science Foundation of China。
文摘As cathode materials for alkali-ion batteries,sodium manganese oxides have been receiving considerable and continuous attention in recent decades.In this work,the structure and environment-dependent stability of NaMn_(2)O_(4) surface were studied based on the first principles calculations.The surface stability diagram of NaMn_(2)O_(4) involving various different terminations of(100),(110)and(111)surfaces was constructed,and the stability of these different terminations could be compared as a function of chemical environment.It is found that the(100)-MnO and(111)-ONa terminations are two more stable terminations under the investigated chemical conditions.And the surface energies of(110)surfaces are negative under the investigated chemical potential,hence,(110)surfaces are unstable.The surface energy of NaMn_(2)O_(4) as a function of O chemical potential is also investigated under constant Na chemical potential.The structure relaxation indicates that the surface rumpling and surface reconstruction can affect the electronic structure of the surface,thereby reducing surface energy and stabilizing the surface.Furthermore,the Wulff shape of NaMn_(2)O_(4) was also constructed based on Gibbs-Wulff theorem.
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
文摘A numerical simulation based on a regularized phase field model is developed to describe faceted dendrite growth morphology. The effects of mesh grid, anisotropy, supersaturation and fold symmetry on dendrite growth morphology were investigated, respectively. These results indicate that the nucleus grows into a hexagonal symmetry faceted dendrite. When the mesh grid is above 640×640, the size has no much effect on the shape. With the increase in the anisotropy value, the tip velocities of faceted dendrite increase and reach a balance value, and then decrease gradually. With the increase in the supersaturation value, crystal evolves from circle to the developed faceted dendrite morphology. Based on the Wulff theory and faceted symmetry morphology diagram, the proposed model was proved to be effective, and it can be generalized to arbitrary crystal symmetries.
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).
文摘In this study we performed density functional theory calculations of the adsorption of oxygen on theε-Fe_(2)surfaces to investigate the Wulff construction evolution at different oxygen coverage(θ_(O))and potential(μ_(O))as well as the mechanism of oxygen removal.At low oxygen coverage(θ_(O)=0.25)and potential(μ_(O)<-7.1 eV),the(■)facets with high activity remain the high exposure(55%to 90%)while the inert(101)facets contribute less than 25%.With increasing theμ_(O)andθ_(O),the(■)and(100)become the main exposing facets.Considering the reaction condition(300℃,1000 kPa,V(CO)/V(H_(2))=1),theμo is determined as-7.5 eV.Thus,theε-Fe_(2)exposes mainly the(■)surfaces indicating its antioxidation ability.Oxygen removal mechanisms indicate that the adsorbed oxygen is easily removed on(■)surface with a barrier of 1.33 eV in good agreement with the simulated Wulff constructions.
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.
