This problem concerns the JMAK model of concurrent nucleation and growth.

a$.$ Make a qualitative plot of the volume fraction $f$ transformed, from $0$ to $1,$ versus time $t,$ for a general transformation occurring via nucleation and growth. If you had this data, how would you find the temporal exponent $n?$

b$.$ Derive the JMAK equation for the volume fraction transformed as a function of time $t$ for $f1$ under the assumption that $($i$)$ the specimen

$1$

contains a number $p$ of effective point heterogeneities or "seeds" per unit volume, $($ii$)$ nucleation occurs preferentially at all of these seeds, $($iii$)$ the nuclei have a spherical shape, and $($iv$)$ the growth rate is isotropic and $($v$)$ constant in time. What would you expect the temporal exponent $n$ to be$?$

c$.$ Derive the JMAK equation for the volume fraction transformed as a function of time $t$ for $f1$ under all of the same assumptions as above except $($i$)$ and $($v$).$ Here, the nuclei do not form at time $t=0$ but rather form randomly throughout a specimen at a constant rate, which is $N_v^{}$ nuclei formed per unit volume per unit time of untransformed material. Furthermore, the growth rate is now limited by the bulk diffusion of solute. What would you expect the temporal exponent $n$ to be in this case?