Grain growth in three dimensions

However, Plateau s laws are not obeyed by the grain boundary structure in the special polycrystal in Fig. 15.15. To achieve local equilibrium at all grain junctions, so that Plateau s laws are obeyed, the faces of each grain must be curved [31]. A boundary structure in which all junctions obey Plateau s laws is presented in Fig. 15.16, which shows a polycrystal consisting of six grains that meet at four vertices. Each grain fully occupies one face of the polycrystal. As in Fig. 15.15, 

3These conditions are Young s equation, qi/sin i = 72/sind 2 = 73/sin fe, which requires all grain-boundary angles to be 27r/3 for the uniform boundary-energy case and the requirement that quad- and higher-order junctions are unstable in two dimensions [30]. 

Each boundary in the system provides two faces for grains. Therefore, the average number of faces per grain, (/)syst, is 

because four edges emanate from each vertex and every edge connects two vertices, 

It is readily verified by inspection that Eqs. 15.46 and 15.47 are obeyed for the grain structure in Fig. 15.15. 

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