Problem points$).$ Phase transformation and microstructural evolutions: Suppression of

melting point in nanoparticles.

Consider first melting in the bulk phase.

$($a$)$ Sketch the molar free energy lines for the solid and liquid phases as a function of temperature, near

the melting point, $T_M.$ Pick a temperature below $T_M$ and indicate on your diagram the driving force for

solidification.

$($b$)$ Use your diagram to derive the expression for that driving force as a function of the enthalpy of

melting, the melting temperature, and the actual temperature.

Consider now the melting of spherical nanoprecipitates. Recall that capillary effects in a pure element

are equivalent to raising the free energy of the solid phase with respect to that of the liquid phase by $2\frac{}{r}$

$($per unit volume$),$ where $$ is the solid liquid interface energy and $r$ the radius of interfaces.

$($c$)$ Re$-$plot and modify your free energy diagram to account for capillary effects.

$($d$)$ Derive the expression for the melting temperature of nanoparticles as a function of their radius $r.$

Lastly consider the case of faceted nanoparticles. In this case, the capillary effects take the expression of

$2\frac{{}_{(hkl)}}{d_{(hkl)}}$ where $($hkl$)$ refers to a facet crystallographic orientation and $d(hkl)$ is the half$-$distance between

$2$ opposing $($hkl$)$ facets. The temperature is raised, starting from low temperature where the nanoparticles

are in a solid state.

$($e$)$ Should all the facets of a faceted nanoparticle melt at the same temperature?