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*Symmetry*
**2012**,
*4*(3),
336-343;
doi:10.3390/sym4030336

## Abstract

**:**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.

## 1. Introduction

Small crystalline precipitates often form in alloys and have near-polyhedral shapes with round edges. Figure 1 is a transmission electron micrograph showing an example of this where the dark regions, which have shapes between a circle and a square, are Co-Cr alloy particles precipitated in a Cu matrix [1,2]. Why such precipitate shapes form has been explained by the anisotropies of physical properties of metals and alloys originating from the crystal structures [2,3]. Both the Co-Cr alloy particles and Cu matrix have cubic structures. The three-dimensional shapes of the particles shown in Figure 1 are intermediate between a sphere and a cube composed of crystallographic planes {100} as indicated by the Miller indices.

Even if the alloy system such as the Co-Cr alloy particles in the Cu matrix is fixed, the precipitate shapes change as a function of the precipitate size [1,2]. In the case of the Co-Cr alloy precipitates, the spherical to cubical shape transition occurs as the precipitate size increases [2,3]. 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 shapes [2,3,4,5]. In the present study, we discuss a simple equation that gives shapes intermediate between a sphere and various polyhedra.

## 2. Cubic Superspheres

The solid figure described by

(1)

expresses a sphere with radius R when p = 2 and a cube with edges 2R as p → ∞ [2,3,4]. It is reported in [6] that the 19th century French mathematician Gabriel Lamé first presented this equation. Intermediate shapes between these two limits can be represented by choosing the appropriate value of p > 2. In [2,3,4], such shapes are called superspheres, and Figure 2 shows the shapes given by (1) for (a) p = 2, (b) p = 4 and (c) p = 20. The parameter R determines the size and p determines the polyhedrality, i.e., the degree to which the supersphere is polyhedron. If |x| > |y| and |x| > |z|, |x/R|^{p} + |y/R|^{p} + |z/R|^{p}= 1 as p → ∞ means |x/R| = 1. This describes the limit for (1) as p → ∞ which gives a cube surrounded by three sets of parallel planes, x = ± R, y = ± R and z = ± R.

## 3. {111} Regular-Octahedral and {110} Rhombic-Dodecahedral Superspheres

Equation (1) can be rewritten as

(2)

This expression has been extended to describe other convex polyhedra [7]. Although the original superspheres discussed in [2,3,4] 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. We use this notation in the present study. The cube given by (2) as p → ∞ is the {100} cube composed of six {100} faces. Assuming crystals with cubic structures, the regular octahedron is the {111} octahedron and the rhombic dodecahedron is the {110} dodecahedron [7].

The {111} octahedral superspheres are given by the following equation:

(3a)

where

(3b)

The shapes given by (3) are shown in Figure 3.

**Figure 3.**Shapes of the {111} regular-octahedral superspheres given by (3); (

**a**) p = 4 and (

**b**) p = 40.

On the other hand, the {110} dodecahedral superspheres are given by

(4a)

where

(4b)

The shapes given by (4) are shown in Figure 4. Equations (2–4) expressed by the spherical coordinate system are shown in [7].

**Figure 4.**Shapes of the {110} rhombic-dodecahedral superspheres given by (4); (

**a**) p = 6 and (

**b**) p = 40.

## 4. {100}-{111}-{110} Polyhedral Superspheres

Combined superspheres can be expressed by combining the equations of each supersphere. Combining (2), (3) and (4), we get

(5)

The parameters a > 0 and b > 0 are those for determining the ratios of the {100}, {110} and {111} surfaces. The shapes of the supersphere given by (5) are shown in Figure 5 when , for two values of p.

**Figure 5.**Shapes of the {100}-{111}-{110} polyhedral superspheres given by (5); (

**a**) p = 20 and (

**b**) p = 100.

The a and b dependences of the shapes given by (5) are understood by examining the polyhedral shapes as p → ∞. Among the three polyhedra given by [h_{cube} (x,y,z)]^{1/p} = R, [h_{octa} (x,y,z)]^{1/p} = aR and [h_{dodeca} (x,y,z)]^{1/p} = bR, the innermost surfaces of the polyhedra are retained to form the combined polyhedron. Figure 6 shows the effect of a and b on the shapes given by (5) as p → ∞.The shape is determined by their location in the quadrilateral surrounded by the points P (a,b) = (3,2), Q (2,2), R (1,1) and S (3/2,1). Various shapes in and around the quadrilateral are shown by the insets in Figure 6 can be summarized as follows:

Three basic polyhedra

(a) {100} cube at point P.

(b) {111} octahedron at point R.

(c) {110} dodecahedron at point S.

Combination of two basic polyhedra

(a) {100}-{111} polyhedra changing from the {100} cube to the {111} octahedron along the line from P to R via Q, by truncating the eight vertices of the cube (The shape at point Q is {100}-{111} cuboctahedron).

(b) {111}-{110} polyhedra changing from the {111} octahedron to the {110} dodecahedron along the line from R to S, by chamfering the 12 edges of the octahedron.

(c) {110}-{100} polyhedra changing from the {110} dodecahedron to the {100} cube along the line from S to P, by truncating six of the 14 vertices of the dodecahedron.

Combinations of all three basic polyhedra

(a) {100}-{111}-{110} polyhedra with mutually non-connected {110} surfaces in Region 1 (R-1).

(b) {100}-{111}-{110} polyhedra with mutually connected {110} surfaces in Region 2 (R-2).

**Figure 6.**Diagram showing the variation in the shapes of the {100}-{111}-{110} polyhedral superspheres given by (5) as p → ∞.

The boundary between Regions 1 and 2, expressed by the line from P to R, is written as:

(6)

Figure 6 is essentially the same as Figure 3 in [7,8] where the parameters α = 1/a and β = 1/b are used instead of a and b. In the appendix, the volume and surface area of the polyhedra shown in Figure 6 are written as a function of a and b. The use of the parameters a and b gives a more intuitive diagram (Figure 6), compared with the diagram given by α and β.

## 5. Discussion

#### 5.1. Shape Transitions of Superspheres from a Sphere to Various Polyhedra

Shape transitions of superspheres from a sphere to a polyhedron are characterized by the change in the normalized surface area N = S/V^{2/3}, where S is the surface area and V the volume of the supersphere. For a sphere, N = 6^{2/3}π^{1/3} ≈ 4.84. Figure 7 shows the variations in N as a function of p for the following the superspheres as indicated by the insets:

(i) the {100} cube type given by (2),

(ii) the {111} regular-octahedral type given by (3),

(iii) the {110} rhombic-dodecahedral type given by (4) and

(iv) the {100}-{111}-{110} polyhedral type given by (5) with and .

**Figure 7.**Dependence of the normalized surface area N =S/V

^{2/3}on p, where S is the surface area and V the volume for various superspheres: (i) the {100} cube type given by (2); (ii) the {111} octahedral type given by (3); (iii) the {110} dodecahedral type given by (4) and (iv) the {100}-{111}-{110} polyhedral type given by (5) with and .

The broken lines at the right show the values of N for the polyhedra as p → ∞.

As shown in Figure 7, the change in N with increasing p becomes smaller as the number of faces of polyhedra increases from the {100} cube with 6 to the {100}-{111}-{110} polyhedron with 26. Among the various polyhedra shown in Figure 3, the polyhedron given by and in Region 1 with N = S/V^{2/3} ≈ 5.05 has the minimum total surface area for the same V [8,10]. The a and b dependence of N can be calculated easily using the results shown in the appendix.

#### 5.2. Shape of Small Metal Particles

The shapes of small metal particles observed in previous studies have been discussed previously using the superspherical approximation [8]. Menon and Martin reported the production of ultrafine Ni particles by vapor condensation in an inert gas plasma reactor [11]. They have also reported the crystallographic characterization of these particles by transmission electron microscopy [11]. Near-polyhedral shapes of nanoparticles have been observed to discuss their properties [12,13,14,15]. The superspherical approximation is a useful geometrical tool to describe the near-polyhedral shapes.

## Acknowledgment

This research was supported by a Grand-in-Aid for Scientific Research C (22560657) by the Japan Society for the Promotion of Science.

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## Appendix

The volume and surface area of the polyhedra shown in Figure 3.

The volume and the , and surface area, S_{100}, S_{111} and S_{110} of the polyhedra shown in Figure 6 are written as a function of a and b. In Region 1, these are given by

and

In Region 2, these are

and

when a = 1 and b = 1, the shape given by (5) as p → ∞ is the {111} regular-octahedron as shown by Figure 6. Since the {111} regular-octahedron belongs to both Regions 1 and 2, from both (A1) to (A4) and (A5) to (A8), we get V = (4/3)R^{3}, S_{100} = 0, S_{111} = 4√3R^{2} and S_{110} = 0 as it should be.

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