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![Surface-energy contours showing the values of c h /c 210 given by Eq. 6 based on the broken-bond model for face-centered cubic crystals [12]](profile/Susumu-Onaka-2/publication/235185808/figure/fig1/AS:668938430054436@1536498758937/Surface-energy-contours-showing-the-values-of-c-h-c-210-given-by-Eq-6-based-on-the.png)
Surface-energy contours showing the values of c h /c 210 given by Eq. 6 based on the broken-bond model for face-centered cubic crystals [12]
Source publication
Small crystalline particles or precipitates are often formed comprising near polyhedral shapes with round edges. Using the anisotropy of the surface energy given by a simple broken-bond model for fcc crystals, a geometrical analysis is performed to consider the particle-shape dependence of surface energy. Polyhedral and nearly polyhedral particles...
Contexts in source publication
Context 1
... is the maximum surface- energy density for the {210} plane, and v 210 is the unit normal vector of the {210} plane. To calculate the value of c h , the unit vector v 210 should be selected so that it is on the edge of the stereographic triangle that contains v h . The contours of c h /c 210 and the values for {100}, {110}, and {111} are shown in Fig. 7. As shown in Fig. 7, c h for {111}, c 111 is the lowest, while c 100 is less than c 110 . The ratio c {111 /c {100 given by Eq. 6 is ffiffi ffi 3 p ...
Context 2
... energy density for the {210} plane, and v 210 is the unit normal vector of the {210} plane. To calculate the value of c h , the unit vector v 210 should be selected so that it is on the edge of the stereographic triangle that contains v h . The contours of c h /c 210 and the values for {100}, {110}, and {111} are shown in Fig. 7. As shown in Fig. 7, c h for {111}, c 111 is the lowest, while c 100 is less than c 110 . The ratio c {111 /c {100 given by Eq. 6 is ffiffi ffi 3 p ...
Citations
... The superspherical-shape approximation is useful to discuss various near polyhedral shapes of crystalline nanomaterials [1,8,10,[12][13][14][15]. The extended superspheres are also treated in mechanics as possible shapes of inclusions and pores in materials [16][17][18][19][20][21]. ...
Crystalline nanoparticles or nanoprecipitates with a cubic structure often have near polyhedral shapes composed of low-fndex planes with {100}, {611} and {110}. To consider such near polyhedral shapes, algebraic fomadas of extended superspheres that can expeess intermediate shapes betwaen spheres and various polyhedra have been presented. Four extended superspheres, (i) {100} regular-hexahedral; (ii) {111} regular-octahedral (iii) {610} rhombic-dodecahedral and (iv) {100}-{111}-{110} rhombicuboctahedral superspheres are treated in this study. A measure Π to indicate the degree of polyhedrality is presented to discuss shape transitions of the extended superspheres. As an application of Π superspherical coherent precipitate is shown.
... The size dependence of the precipitate's equilibrium shape determines the shape transitions [2,3]. When we discuss such physical phenomenon, it is convenient to use simple equations that can approximate the precipitate shapes2345. In the present study, we discuss a simple equation that gives shapes intermediate between a sphere and various polyhedra.Figure 1. Transmission electron micrograph showing the Co-Cr alloy precipitates in a Cu matrix [1,2]. ...
... Although the original superspheres discussed in234 are intermediate shapes between a sphere and a cube, now the superspheres can refer to shapes intermediate between various convex polyhedra and a sphere [8]. Superspheres have been used to discuss the shapes of small crystalline particles and precipitates [2,3,5,8,9]. The planes of crystal facets are indicated by their Miller indices. ...
Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, where {100}, {111} and {110} are the Miller indices of crystals with cubic structures. The three parameters p, a and b are included to describe the {100}-{111}-{110} near-polyhedral shapes, where p describes the degree to which the shape is a polyhedron and a and b determine the ratios of the {100}, {111} and {110} surfaces.
... Onaka et al. used an equation for superspheres to describe intermediate shapes6789. Onaka recently extended this equation for superspheres, and derived basic equations to describe shapes intermediate of various convex polyhedra and spheres [10,11]. In the present paper, T. Miyazawa · M. Aratake · S. Onaka (B) Department of Materials Science and Engineering, Tokyo Institute of Technology, 4259-J2-63 Nagatsuta, Yokohama 226-8502, Japan e-mail: onaka.s.aa@iem.titech.ac.jp superspheres refer to intermediate shapes described by these basic equations [10]. ...
... Crystallographic indices of cubic crystals are used in the present study. A distinct characteristic of superspheres is that the combined shapes of polyhedra and superspheres can be described by combining the equations of each polyhedron [10,11].Figure 1 shows a {100} – {111} – {110} polyhedron formed by the combination of a {100} cube, {111} octahedron and {110} rhombic dodecahedron, in which the innermost surfaces of the three polyhedra are retained to form the combined polyhedron. Equations describing the shapes of superspheres have been derived from the spherical coordinates [10]. ...
... where γ 210 and ν 210 are the maximum surface-energy density and unit normal vector for the {210} plane, respectively. The Wulff shape for the anisotropy of surface-energy density given by Eq. (3) is a {100} – {111} polyhedron with planar faces and sharp edges [5,11]. However, the shapes of small fcc metal particles usually have round edges, as is demonstrated in Figs. 4 and 5. ...
Small crystalline particles are often formed comprising near-polyhedral shapes with round edges. When near-polyhedral shapes are analyzed and discussed, it is convenient if these shapes can be expressed by equations with simple parameters. Superspheres are solids expressing various shapes between those of polyhedra and spheres. The superspherical-shape approximation is used in this study to consider the morphology of cubic crystal structure particles. Various near-polyhedral shapes composed of {100}, {111} and {110} planes are described using a simple equation with three shape-related parameters. It is shown that the superspherical-shape approximation is a useful geometrical tool for evaluating the morphology of small crystalline particles.
We present an improved algorithm for
quasi-properly
learning convex polyhedra in the realizable PAC setting from data with a margin. Our learning algorithm constructs a consistent polyhedron as an intersection of about
$t \log t$
halfspaces with constant-size margins in time polynomial in
$t$
(where
$t$
is the number of halfspaces forming an optimal polyhedron). We also identify distinct generalizations of the notion of margin from hyperplanes to polyhedra and investigate how they relate geometrically; this result may have ramifications beyond the learning setting.
In this paper, a new method to analytically carry out the exterior elastic fields of a class of non-elliptical inclusions, i.e., those characterized by Laurent polynomials, is developed. Two complex variable fields, which exactly characterize the Eshelby's tensor, are explicitly achieved for the hypocycloidal and the quasi-parallelogram inclusions. Numerical examples show that the exterior fields near the inclusion are dominated by the boundary shape, but the fields far away from the inclusion tend to be convergent and can be well approximated by those of its equivalent circular/elliptical inclusion. These solutions are firstly reported, and largely make up for the deficiency in the list of the analytical results of non-elliptical inclusions in 2D isotropic elasticity.
Monocrystalline silicon nanoparticles with a mean diameter of between 30 and 40 nm have been synthesised by hot wire thermal catalytic and spark pyrolysis at a pressure of 40 and 80 mbar respectively. For the production a mixture of the precursor gases, silane and diborane or silane and phosphine were used. While hot wire pyrolysis always results in multifaceted particles, those produced by spark pyrolysis are spherical. Electrical resistance measurements of compressed powders showed that boron doped silicon powders have a much higher conductivity than those doped with phosphorus. TEM and XPS analysis reveals that the difference in electrical resistivity between boron an phosphorus doped particles can be attributed to phosphorus dopants being located at the surface of the particles where an oxide layer is also observed. In contrast, boron doped particles are far less oxidised and the dopant atoms can be found in the core of the particle. The results demonstrate that hot wire and spark pyrolysis offer a new simple route to the production of monocrystalline doped silicon nanoparticles suitable for printed electrical devices.
Doped silicon nanoparticles have successfully been produced by hot wire thermal catalytic pyrolysis at 40 mbar and a filament temperature of 1800 °C, using a mixture of silane and diborane or phosphine. All particles are monocrystalline with shapes ranging from an octahedron to varying degrees of truncation of this basic shape, with an average diameter of 22 nm. To determine the doping activity, the resistivity of the nanopowders was measured at successive compression levels. While boron doped particles have clean surfaces and are electrically active, with compacted powder having a resistivity of the order of 103 Ω m, phosphorus doped particles are covered by an oxide layer whose thickness increases from 0.3 nm to 0.6 nm with higher phosphine concentrations. Furthermore, the phosphor atoms are localised at the interface to this surface layer, where they are electrically inactive. These powders have a resistivity in the order of 107 Ω m.
Using a single crystal of a Cu–6 mass% Ag alloy aged at 723K for various times, nano-sized Ag precipitates in a Cu matrix
were observed by conventional and ultra high-voltage transmission electron microscopes. The nano-sized Ag precipitates with
a radius of about 5nm had a nearly spherical shape. The shape change from the nearly spherical shape to the {111} octahedral
shape occurred with increasing the size of the Ag precipitates to 11nm. Elastic states of the Ag precipitates were evaluated
by observing moiré fringes formed between the Ag precipitates and the Cu matrix. The origin of the shape change of the nano-sized
Ag precipitates was discussed by considering the sum of elastic strain energy and interfacial energy.
