The Avrami law for the fraction transformed, $f(t)=1-$exp$(-k{t}^n),$ can be $($sort of$)$ justified using

various different growth models.

$($a$)$ For small $t,$ show that $f(t)=k{t}^n.$

$($b$)$ Suppose that all the growing nuclei are spherical and form very early in the phase

transformation $($near $t=0),$ and that their radius grows linearly with time; $r=vt,$ where $v$ is the

growth velocity. Determine, at short times $t,$ how the total volume of the nuclei scales with $t,$

and thus how $f(t)$ scales with $t.$ What is the Avrami exponent $n$ in this case?

$($c$)$ Suppose that all the growing nuclei are spherical and form very early in the phase

transformation $($near $t=0),$ but that their volume grows linearly with time; $r=v{t}^{\frac{1}{3}}.$ What is

the Avrami exponent $n$ in this case?

$($d$)$ Suppose that the radius of a growing nucleus increases linearly with time $(r=vt,$ but now the

nuclei appear throughout the phase transformation at a constant rate of $N$ nuclei per unit time.

Calculate the Avrami exponent in this case. Hint: at a given time, nuclei will be all different

sizes, depending on when they first appeared. You need to integrate over this distribution.