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

∆G =πr^2∆G, + 2myr + A - B Inr, where A - B Inr represents the dislocation energy within the cluster. 0).

(a) Sketch ∆G vs r (note at r= 0, G

(b) Determine the value of r*.

(c) Show that when ∆G, B/ry >+, ∆G monotonically decreases with r,

but when ∆G, B/ay² < 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) $\frac{\Delta Gv B}{\pi r^2} < \frac{1}{2}$