 Solution

The polymer overlap concentration is reached when the number of chains per coil area is of order one. If $n_c$ is the chain number per unit interfacial area, then $n_c^{\ast }{R}^2\sim 1$. Equivalently, the monomer area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$, assuming an occupied monomer area of order ${\ell }^2$. Below this scale coils are mostly separate; above it they interpenetrate.