$[20$ pts$]$ A single atom 'solidifying' on to a fat solid surface will be unstable and tend to rejoin the melt. However, if a sufficiently large number of atoms can come together to form a disc$-$shaped layer as shown on the right, it is possible for the arrangement to become self$-$stabilized and continue to grow.

The problem of disc creation is the two$-$dimensional analog of cluster formation during homogeneous nucleation. In this case, the edges of the disc contribute a positive energy, which must be counterbalanced by the volume free energy released in the process. Associated with the two$-$dimensional nucleus, therefore, there is a critical radius $(r^{**}),$ which decreases with increasing interface undercooling. Once nucleated, the disc will spread rapidly over the surface, and the rate of growth normal to the interface will be governed by the surface nucleation rate. Suppose the edge of the cap nucleus is associated with an energy $(J{m}^{-1}).$ Use nucleation theory to derive quantitative expressions for

$($a$)$ the critical radius $r^{**}$ and $($b$)$ the growth rate of an atomically smooth interface as a function of undercooling $($supercooling$).$