A one$-$component material has three possible structures: $,,$ and $.$ At high

temperatures, the system is $.$ If it is cooled slowly, it transforms to $$ at some

temperature below the transformation temperature, i$.$e$.,GTxy|a|xyT{T}_{}{T}_{}.$

$a.(3$ points$)$ Suppose that both the $$ and $$ transformations

occur $by$ nucleation and growth. Sketch plausible forms $of$

the free energy curves $G$ versus temperature $T$ for the three

phases that may allow for the above behavior.

Hint: you should have one curve for each phase.

$G$

$b.(4$ points$)$ Are the kinetics $of$ the nucleated transformations more likely

$tobe$ governed $by$ Figure $xory$ below? Explain.

$|a|$

$c.(4$ points$)$

$d.(6$ points$)$

Now consider the case $in$ which $is$ a liquid, phase $isa$

crystalline solid, and phase $is$ a glass. $In$ this alternate

scenario, are the kinetics $of$ the transformations more likely

$tobe$ governed $by$ Figure $xory$ above? Explain.

While $itis$ possible $to$ suppress the $$ transformation $by$

rapid cooling, $itis$ not ordinarily possible $to$ suppress the

reverse $$ transformation $by$ rapid heating. Why?

Hint: consider how the nucleation rate varies with

temperature $T$ increasing from $T_{}.T.Ifitis$ cooled

quickly, $$ transforms $toat$ some temperature $T_{}.$

$a.(3$ points$)$ Suppose that both the $$ and $$ transformations

occur $by$ nucleation and growth. Sketch plausible forms $of$

the free energy curves $G$ versus temperature $T$ for the three

phases that may allow for the above behavior.

Hint: you should have one curve for each phase.

$G$

$b.(4$ points$)$ Are the kinetics $of$ the nucleated transformations more likely

$tobe$ governed $by$ Figure $xory$ below? Explain.

$|a|$

$c.(4$ points$)$

$d.(6$ points$)$

Now consider the case $in$ which $is$ a liquid, phase $isa$

crystalline solid, and phase $is$ a glass. $In$ this alternate

scenario, are the kinetics $of$ the transformations more likely

$tobe$ governed $by$ Figure $xory$ above? Explain.

While $itis$ possible $to$ suppress the $$ transformation $by$

rapid cooling, $itis$ not ordinarily possible $to$ suppress the

reverse $$ transformation $by$ rapid heating. Why?

Hint: consider how the nucleation rate varies with

temperature $T$ increasing from $T_{}.$