= Semidilute osmotic pressure of a two-dimensional polymer solution
{title2=$\Pi\sim(k_BT/\ell^2)\phi^3$}
In a planar good-solvent solution, a correlation blob of $g$ monomers has size $\xi\sim\ell g^{3/4}$. Its area fraction is $\phi\sim g\ell^2/\xi^2\sim g^{-1/2}$, giving $g\sim\phi^{-2}$ and $\xi\sim\ell\phi^{-3/2}$. Each blob contributes order $k_BT$ to the osmotic free energy, so the surface <osmotic pressure> is $\Pi\sim k_BT/\xi^2\sim(k_BT/\ell^2)\phi^3$. 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 $\Pi\sim k_BT\phi/(N\ell^2)$. The two expressions match at the <two-dimensional polymer overlap fraction> $\phi^*\sim N^{-1/2}$.
Back to article page
