A cylindrical pill-like cluster of radius r nucleates on a dislocation that emerges from the substrate. The free-energy change per unit thickness is given by
DG = pr2Gv + 2pyr + A - B ln r,
where A — B In r represents the dislocation energy within the cluster.
(a) Sketch DG vs r (note at r = 0, DG = µ
(b) Determine the value of r*.
(c) Show that when AGvJB/7iy2 > i, AG monotonically decreases with r, but when AGvB/7ry2 < i, there is a turnaround in the curve. (The latter case corresponds to a metastable state and associated energy barrier.)