A kinetic 5-vertex model is used to investigate hexagon-islands formation on growing single-walled carbon nanotubes (SWCNT). In the model, carbon atoms adsorption and migration processes on the SWCNT edge are consider...A kinetic 5-vertex model is used to investigate hexagon-islands formation on growing single-walled carbon nanotubes (SWCNT). In the model, carbon atoms adsorption and migration processes on the SWCNT edge are considered. These two dynamic processes are assumed to be mutually independent as well as mutually dependent as far as the whole growth of the nanotube is concerned. Key physical parameters of the model are the growth time t, the diffusion length Γ defined as the ratio of the diffusion rate D to the carbon atomic flux F and the SWCNT chiral angle. The kinetic equation that describes the nanotube edge dynamics is solved using kinetic Monte Carlo simulations with the Bortz, Kalos and Lebowitz update algorithm. The behaviors of islands density and size distribution are investigated within the growth parameters’ space. Our study revealed key mechanisms that enable the formation of a new ring of hexagons at the SWCNT edge. The growth occurs either by pre-existing steps propagation or by hexagon-islands growth and coalescence on terraces located between dislocation steps, depending on values of model parameters. This should offer a road map for edge design in nanotubes production. We also found that in appropriate growth conditions, the islands density follows Gaussian and generalized Wigner distributions whereas their size distribution at a given growth time shows a decreasing exponential trend.展开更多
An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 i...An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.展开更多
文摘A kinetic 5-vertex model is used to investigate hexagon-islands formation on growing single-walled carbon nanotubes (SWCNT). In the model, carbon atoms adsorption and migration processes on the SWCNT edge are considered. These two dynamic processes are assumed to be mutually independent as well as mutually dependent as far as the whole growth of the nanotube is concerned. Key physical parameters of the model are the growth time t, the diffusion length Γ defined as the ratio of the diffusion rate D to the carbon atomic flux F and the SWCNT chiral angle. The kinetic equation that describes the nanotube edge dynamics is solved using kinetic Monte Carlo simulations with the Bortz, Kalos and Lebowitz update algorithm. The behaviors of islands density and size distribution are investigated within the growth parameters’ space. Our study revealed key mechanisms that enable the formation of a new ring of hexagons at the SWCNT edge. The growth occurs either by pre-existing steps propagation or by hexagon-islands growth and coalescence on terraces located between dislocation steps, depending on values of model parameters. This should offer a road map for edge design in nanotubes production. We also found that in appropriate growth conditions, the islands density follows Gaussian and generalized Wigner distributions whereas their size distribution at a given growth time shows a decreasing exponential trend.
基金Supported by the National Natural Science Foundation of China (Grant No.10831001)the Science-TechnologyFoundation for Young Scientists of Fujian Province (Grant No.2007F3070)
文摘An edge e of a k-connected graph G is said to be a removable edge if G O e is still k-connected, where G e denotes the graph obtained from G by deleting e to get G - e, and for any end vertex of e with degree k - 1 in G- e, say x, delete x, and then add edges between any pair of non-adjacent vertices in NG-e （x）. The existence of removable edges of k-connected graphs and some properties of 3-connected and 4-connected graphs have been investigated [1, 11, 14, 15]. In the present paper, we investigate some properties of 5-connected graphs and study the distribution of removable edges on a cycle and a spanning tree in a 5- connected graph. Based on the properties, we proved that for a 5-connected graph G of order at least 10, if the edge-vertex-atom of G contains at least three vertices, then G has at least （3│G│ ＋ 2）/2 removable edges.
