Question #163094

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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.)

Expert's answer

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}"

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