Two-dimensional polymer overlap fraction (${\varphi }^{\ast }\sim N{\ell }^2/R^2\sim N^{1-2\nu }$)

For polymers restricted to an interface, overlap means roughly one coil per coil area, $n_c{R}^2\sim 1$. A chain contains $N$ monomers with typical occupied area ${\ell }^2$, so its overlap area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$. With the planar good-solvent exponent $\nu =3/4$ from the two-dimensional Flory approximation, this gives ${\varphi }^{\ast }\sim N^{-1/2}$. The same exponent happens to occur for a three-dimensional ideal chain's volume fraction, but its geometric origin is different.