Problem $1$

Flory theory of randomly branched polyelectrolytes

$($i$)$ Use Flory theory to calculate the size of randomly branched polyelectrolyte consisting of $N$ Kuhn

monomers of size $b$ with fraction $f$ of these Kuhn monomers carrying elementary charge $e.$

Hint: recall that the ideal Gaussian size of a randomly branched polymer is $R_0=b{N}^{\frac{1}{4}}.$

$($ii$)$ Note that the backbone of a randomly branched polymer contains $N^{\frac{1}{2}}$Kuhn segments $($so that

its ideal size $b(N^{\frac{\text{I}}{2}}{)}^{\frac{\text{I}}{2}}$ is the same as of the whole polymer $(R_0).$ Compare the answer of part

$($i$)$ with the length of the maximally extended backbone. At what value of the number of

Kuhn segments $N$ does the length of the maximally extended backbone becomes shorter than

the prediction of Flory theory? What is the size of randomly branched polyelectrolytes with

larger $N?$ What does it imply about the validity of the Flory theory of randomly branched

polyelectrolytes with large number of Kuhn monomers?