Geometrical analysis of near polyhedral shapes with round edges in small crystalline particles or precipitates

Abstract

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 composed of {100} and {111} planes are treated as examples. The effect of round edges on the variation of surface energy of the nearly polyhedral particles is discussed.

Appendix

The surface area A and volume V of the {100}−{111} polyhedra are given by Eq. 4 with $p\rightarrow \infty.$

For the {100}−{111} polyhedra given by Eq. 4 with $p\rightarrow \infty$ , the surface area A = A ₁₀₀ + A ₁₁₁ and the volume V are given by as follows, where A ₁₀₀ and A ₁₁₁ correspond to the areas of the {100} and {111} surfaces, respectively. The variation of shapes of the {100}−{111} polyhedra as a function of α is shown in the insets in Fig. 5.

• (1) When 0 ≤ α ≤ 1/3, Eq. 4 gives the {100} cube with edges 2R:

  $$ A_{100} =24R^{2}, \quad A_{111} =0\hbox{ and }V=8R^{3}. $$

• (2) When 1/3 ≤ α ≤1/2, Eq. 4 gives a polyhedron with triangular {111} surfaces, which is inscribed in the cube with edges 2R:

  $$ \begin{array}{l} A_{100} =24\left\{ {1-\frac{1}{2}\left({3-\frac{1}{\alpha }} \right)^{2}} \right\}R^{2}, \quad A_{111} = 4\sqrt{3}\left({3-\frac{1}{\alpha }} \right)^{2}R^{2}\,\,\hbox{and}\\ V=8\left\{ {1-\frac{1}{6}\left({3-\frac{1}{\alpha}}\right)^{3}} \right\}R^{3}. \end{array} $$

• (3) When 1/2 ≤ α ≤ 1, Eq. 4 gives a polyhedron with square {100} surfaces, which is inscribed in the cube with edges 2R:

  $$ \begin{array}{l} A_{100} =\frac{12}{\alpha ^{2}}\left({1-\alpha} \right)^{2}R^{2}, \quad A_{111} =\frac{4\sqrt{3}}{\alpha ^{2}}\left\{ {1-3\left({1-\alpha} \right)^{2}} \right\}R^{2}\,\,\hbox{and}\\ V=\frac{4}{3\alpha ^{3}}\left\{ {1-3\left({1-\alpha} \right)^{3}} \right\}R^{3}. \end{array} $$

• (4) When 1 ≤ α, Eq. 4 gives the {111} octahedron with edges $\sqrt{2}R/\alpha$:

  $$ \begin{array}{l} A_{100} =0, \quad A_{111} =\frac{4\sqrt{3}}{\alpha ^{2}}R^{2}\,\,\hbox{and}\,\,V=\frac{4}{3\alpha ^{3}}R^{3}. \end{array} $$