A silicate composition is formulated to prepare a glass$-$ceramic. A heat treatment is designed to achieve the desired microstructure to meet optical and mechanical property requirements. The transformation proceeds by nucleation and crystal growth. According to the Johnson$-$Meh$1-$Avrami Equation: $x_1=1-$exp$(-\frac{}{3}{I}_v{u}^3{t}^n)$

where $n$ is the power of the time, which is $4$ for time$-$independent nucleation and $3$ for time$-$dependent nucleation.

Here are schematics of microstructures in the early stage of transforming by nucleation and crystal growth. Compare the sequences $($a$)$ and $($b$)$:

$($a$)$

Beginning

$$

Middle

$$

End

$($b$)$ Beginning

Middle

End

a$.$ Which one $(($a$)$ or $($b$))$ fits the description:

i$.$ A transformation from metastable $($matrix$)$ to stable phase $($circles$),$ where nucleation occurs at a constant rate, and nuclei grow.

ii$.$ A transformation from metastable to stable phase, where nucleation saturates at the beginning.

b$.$ If $x_t={10}^{-2},u=2\mathrm{.}5{10}^{-6}c\frac{m}{s}$ and $I_v=100$

$u$ cle$\frac{i}{c}{m}^{3}s,$ how long does it take to reach this volume$-$fraction$-$crystallized? Does this describe microstructure $($a$)$ or $($b$)?$

c$.$ If $x_t={10}^{-2},u=2\mathrm{.}5{10}^{-6}c\frac{m}{s}$ and $I_v=100$ nuclei $\frac{?}{c{m}^3},$ how long does it take to reach this volume$-$fraction$-$crystallized? Does this describe microstructure $($a$)$ or $($b$)?$

e sure that you explain your answers in detail.