Question

.

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.)

Answer 1

a) $\frac{d\Delta G}{dr} =0$

b) $r^* = \frac{-\pi r +\sqrt (\pi^2 r^2+2B\pi \Delta Gv)}{2\pi \Delta Gy}$

c) $\frac{\Delta Gv B}{\pi r^2} < \frac{1}{2}$