Semidilute osmotic pressure of a two-dimensional polymer solution ($\Pi \sim (k_{B}T/{\ell }^2){\varphi }^3$)

In a planar good-solvent solution, a correlation blob of $g$ monomers has size $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, giving $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. Each blob contributes order $k_{B}T$ to the osmotic free energy, so the surface osmotic pressure is $\Pi \sim k_{B}T/{\xi }^2\sim (k_{B}T/{\ell }^2){\varphi }^3$. This regime requires many blobs per chain and area fraction below close packing. In the dilute regime entire chains are the independently translating objects and $\Pi \sim k_{B}T\varphi /(N{\ell }^2)$. The two expressions match at the two-dimensional polymer overlap fraction ${\varphi }^{\ast }\sim N^{-1/2}$.