Problem 4. Given the diffusivity of vacancies in a metal Virginium at 500K is 3.1 x 10-12 cm/s, answer the following questions. a) If a 1.0 mm diameter polycrystalline wire specimen of Virginium at 900K is rapidly quenched from 900K to 500K such that the equilibrium concentration of vacancies at 900K is retained, estimate how long it will take for the vacancy concentration to reach 95% of the equilibrium concentration characteristic of 500K. Assume that the excess vacancy concentration decays by migration of vacancies from grain interiors to grain boundaries. Thus, the grain boundaries act as sinks for the excess vacancy concentration. Assume further that the grains can be considered to be spherical with an effective average grain diameter d = 100 um and that the vacancy concentration in the lattice in the immediate vicinity of the grain boundaries will decrease instantaneously to the new equilibrium value of the new temperature upon quenching. b) How long would it take if the grain size is reduced 50%? c) Consider the problem described in part a). How long will it take for the vacancy concentration to reach 95% of the new equilibrium concentration at 500K assuming that now the surface of the wire acts as the vacancy sink instead of the grain boundaries? Problem 4. Given the diffusivity of vacancies in a metal Virginium at 500K is 3.1 x 10-12 cm/s, answer the following questions. a) If a 1.0 mm diameter polycrystalline wire specimen of Virginium at 900K is rapidly quenched from 900K to 500K such that the equilibrium concentration of vacancies at 900K is retained, estimate how long it will take for the vacancy concentration to reach 95% of the equilibrium concentration characteristic of 500K. Assume that the excess vacancy concentration decays by migration of vacancies from grain interiors to grain boundaries. Thus, the grain boundaries act as sinks for the excess vacancy concentration. Assume further that the grains can be considered to be spherical with an effective average grain diameter d = 100 um and that the vacancy concentration in the lattice in the immediate vicinity of the grain boundaries will decrease instantaneously to the new equilibrium value of the new temperature upon quenching. b) How long would it take if the grain size is reduced 50%? c) Consider the problem described in part a). How long will it take for the vacancy concentration to reach 95% of the new equilibrium concentration at 500K assuming that now the surface of the wire acts as the vacancy sink instead of the grain boundaries