Consider a polymer brush made of chains with the degree of polymerization

$N=5000,$ repeat unit projection length $l=0\mathrm{.}3nm,$ Kuhn length $b_K=2\mathrm{.}0nm$ and excluded

volume

$v=l^3(1-\frac{}{T})$

in a solvent with $-$temperature of $20C.$ The chains are grafted to the substrate with a

grafting density ${10}^{-2}n{m}^{-2}.$

$($i$)$ Use a Flory approach to estimate thickness of the brush layer at $40C.$

$($ii$)$ Estimate a grafting density corresponding to a crossover to a mushroom regime of

nonoverlapping chains.

$($iii$)$ Use a blob description of the brush layer to estimate brush thickness.

$($iv$)$ How many repeat units are in a blob?