Semiconductor nanowires provide a useful illustration of

the importance of thermodynamics and kinetics in

modern materials synthesis, and on effects that emerge

when the relevant length scales become very small. An

illustration of the growth process is shown at right.

Growth occurs at the interface between a gold liquid

phase $($where the Si solubility is quite high$)$ and a solid

Si phase $($which has a negligible solubility for Au$).$ The

Si$-$Au phase diagram is obviously relevant to this

problem. However, the issue is that this phase diagram is for bulk materials, and

will somehow be affected by the fact that the length scales are very small.

If the interfacial tension for $Au$ is $0\mathrm{.}177\frac{J}{m^2},$ the molar volume of solid $Au$ is

$1\mathrm{.}14{10}^{-5}{m}^3,$ the molar heat of fusion is $1\mathrm{.}25{10}^4\frac{J}{m}ol,$ the equilibrium

melting point is $1064C,$ and the droplet radius is $5nm,$ calculate the melting

point depression for the Au droplet. For the sake of simplicity, you can assume

that the droplet is approximately spherical. Is capillarity a significant effect?