Grain growth in two dimensions

The (N — 6) rule is obeyed for all grains except those intersecting the edge of the domain. Calculation using Surface Evolver [22]. Courtesy of Ellen J. Siem. 

Topology of Two-Dimensional Polycrystals. In two dimensions, three (and only three) grains meet at every grain boundary vertex, as in Fig. 15.10. Vertices where 

An important geometrical growth parameter is the average number of sides per grain, (N), in the ensemble, which can be determined with Euler s theorem, which states that 

Each boundary segment provides a side for two grains, therefore (N) = 2NB/NG. Since NG, NB, and Nv in Eq. 15.25 are all much greater than unity, NG + Nv = Nb to a good approximation, and, using Eq. 15.26, NB = 3NG. Therefore, 

In the relationships above, 8 is the angle that the boundary normal makes with a fixed direction in the plane of the specimen. Because the curvature is the rate of change of the boundary normal as the line integral is carried out, k = dO/ds. Also, 6 varies between 0 and 27r in the integration, because the normal rotates by 2/T as the boundary is traversed. Therefore, independent of the shape of the grain, Eq. 15.30 becomes 

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Figure 15.10 Simulation of isotropic and uniform grain growth in two dimensions.

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