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Using an

Small crystalline precipitates often form in alloys and have near-polyhedral shapes with round edges.

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 [

Transmission electron micrograph showing the Co-Cr alloy precipitates in a Cu matrix [

The solid figure described by

expresses a sphere with radius ^{p}^{p}^{p}

Shapes of the cubic superspheres given by (1); (

Equation (1) can be rewritten as

This expression has been extended to describe other convex polyhedra [

Superspheres have been used to discuss the shapes of small crystalline particles and precipitates [

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

where

The shapes given by (3) are shown in

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

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

where

The shapes given by (4) are shown in

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

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

The parameters

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

The _{cube} (^{1/p} = _{octa} (^{1/p} = _{dodeca} (^{1/p} =

Three basic polyhedra

(a) {100} cube at point

(b) {111} octahedron at point

(c) {110} dodecahedron at point

Combination of two basic polyhedra

(a) {100}-{111} polyhedra changing from the {100} cube to the {111} octahedron along the line from

(b) {111}-{110} polyhedra changing from the {111} octahedron to the {110} dodecahedron along the line from

(c) {110}-{100} polyhedra changing from the {110} dodecahedron to the {100} cube along the line from

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).

Diagram showing the variation in the shapes of the {100}-{111}-{110} polyhedral superspheres given by (5) as

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

Shape transitions of superspheres from a sphere to a polyhedron are characterized by the change in the normalized surface area ^{2/3}, where ^{2/3}^{1/3} ≈ 4.84.

(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

Dependence of the normalized surface area ^{2/3} on

The broken lines at the right show the values of

As shown in ^{2/3} ≈ 5.05 has the minimum total surface area

The shapes of small metal particles observed in previous studies have been discussed previously using the superspherical approximation [

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

The volume and surface area of the polyhedra shown in

The volume _{100}, _{111} and S_{110} of the polyhedra shown in

and

In Region 2, these are

and

when ^{3}, _{100} = 0, _{111} = 4√3^{2} and _{110} = 0 as it should be.