Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Sundquist B.E.: A direct determination of the anisotropy of the surface free energy of solid gold, silver, copper, nickel, and alpha and gamma iron. Acta Metall. 12, 67–86 (1964)
Kimoto K., Nishida I.: The crystal habits of small particles of aluminium and silver prepared by evaporation in clean atmosphere of argon. Jpn. J. Appl. Phys. 16, 941–948 (1977)
Heyraud J.C., Metois J.J.: Equilibrium shape of gold crystallites on a graphite cleavage surface: surface energies and interfacial energy. Acta Metall. 28, 1789–1797 (1980)
Heyraud J.C., Metois J.J.: Equilibrium shape and temperature. Surf. Sci. 128, 334–342 (1983)
Menon S.K., Martin P.L.: Determination of the anisotropy of surface free energy of fine metal particles. Ultramicroscopy 20, 93–98 (1986)
Onaka S.: Averaged Eshelby tensor and elastic strain energy of a superspherical inclusion with uniform eigenstrains. Philos. Mag. Let. 81, 265–272 (2001)
Onaka S., Kobayashi N., Fujii T., Kato M.: Simplified energy analysis on the equilibrium shape of coherent γ′ precipitates in γ matrix with a superspherical shape approximation. Intermetallics 10, 343–346 (2002)
Onaka S., Kobayashi N., Fujii T., Kato M.: Energy analysis with a superspherical shape approximation on the spherical to cubical shape transitions of coherent precipitates in materials with cubic structures. Mater. Sci. Eng. A. A 347, 42–49 (2003)
Onaka S., Fujii T., Kato M.: Elastic strain energy due to misfit strains in a polyhedral precipitate composed of low-index planes. Acta Mater. 55, 669–673 (2007)
Onaka S.: Simple equations giving shapes of various convex polyhedra: the regular polyhedra and polyhedra composed of crystallographically low-index planes. Philos. Mag. Lett. 86, 175–183 (2006)
Onaka S.: Geometrical analysis of near polyhedral shapes with round edges in small crystalline particles or precipitates. J. Mater. Sci. 43, 2680–2685 (2008)
Suárez J., Gancedo E., Manuel Álvarez J., Morán A.: Truncating and chamfering diagrams of regular polyhedra. J. Math. Chem. 46, 155–163 (2009)
Herring C.: Some theorems on the free energies of crystal surfaces. Phys. Rev. 82, 87–93 (1951)
Cahn J.W., Handwerker C.A.: Equilibrium geometries of anisotropic surfaces and interfaces. Mater. Sci. Eng. A. A 162, 83–95 (1993)
Mackenzie J.K., Moore A.J.W., Nicholas J.F.: Bonds broken at atomically flat crystal surfaces-I. J. Phys. Chem. Solids 23, 185–196 (1962)
Suárez J., Gancedo E., Manuel Álvarez J., Morán A.: Optimum compactness structures derived from the regular octahedron. Eng. Struct. 30, 3396–3398 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Miyazawa, T., Aratake, M. & Onaka, S. Superspherical-shape approximation to describe the morphology of small crystalline particles having near-polyhedral shapes with round edges. J Math Chem 50, 249–260 (2012). https://doi.org/10.1007/s10910-011-9909-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-011-9909-1