= Two-dimensional polymer overlap fraction
{title2=$\phi^*\sim N\ell^2/R^2\sim N^{1-2\nu}$}
For polymers restricted to an interface, overlap means roughly one coil per coil area, $n_cR^2\sim1$. A chain contains $N$ monomers with typical occupied area $\ell^2$, so its overlap area fraction is $\phi^*\sim N\ell^2/R^2$. With the planar good-solvent exponent $\nu=3/4$ from the <two-dimensional Flory approximation>, this gives $\phi^*\sim N^{-1/2}$. The same exponent happens to occur for a three-dimensional ideal chain's volume fraction, but its geometric origin is different.
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