Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A4 / d

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.

For the planar good-solvent coil, $R\sim \ell N^{3/4}$. The two-dimensional polymer overlap fraction is thereforeThis is a scaling estimate; the monomer geometry and choice of coil radius set order-one numerical factors.