 Solution

Here the osmotic pressure is a two-dimensional surface pressure, with units of energy per area. In the dilute regime, independent whole chains provide the translational entropy, soIn the semidilute polymer solution, use a correlation blob containing $g$ monomers. Within it the chain still follows the planar good-solvent law $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, so $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. There is about one independent thermal contribution per blob area. The semidilute osmotic pressure of a two-dimensional polymer solution is thusIt is independent of total chain length once each chain contains many blobs. The two estimates agree at ${\varphi }^{\ast }\sim N^{-1/2}$, where each gives $(k_{B}T/{\ell }^2)N^{-3/2}$.