"Polymer Physics" $5\mathrm{.}22.$ Consider a spherical micelle made of $f$ deblock copolymers. The

insoluble block consists of $N_B$ monomers of size $b_B$ and its Flory interaction parameter with

solvent is ${}_B.$

$($i$)$ Calculate the size $R_{\text{c}\text{o}\text{r}\text{e}}$ of the core of the micelle.

$($ii$)$ Calculate the interfacial energy of the core $($per chain$).$

$($iii$)$ Calculate the size of the micelle $R_{\text{m}\text{i}\text{c}\text{e}\text{l}\text{l}\text{e}}$ in an athermal solvent for the outer block

${}_A=0.$ Calculate the free energy of the corona $($per chain$).$

$($iv$)$ Optimize the aggregation number $f$ by balancing the interfacial and corona parts of

free energy per chain.