Semidilute polymer solution

A solution above the polymer overlap concentration but below the concentrated regime, with interpenetrating chains. Local polymer structure is characterized by a correlation length or concentration-blob size. Scaling formulas describe a broad semidilute regime most reliably when the concentration greatly exceeds overlap while remaining small compared with one in volume fraction.

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 ${\varphi }^{\ast }\sim N^{-1/2}$.

A region of size $\xi$ containing a typical correlated chain segment whose internal monomer concentration matches the bulk concentration. If it contains $g$ statistical units with isolated size law $\xi \sim b{g}^{\nu }$ and unit volume $v_0$, concentration matching gives $\varphi \sim g{v}_0/{\xi }^3$. Eliminating $g$ gives $\xi \sim b(\varphi b^3/v_0{)}^{-\nu /(3\nu -1)}$. Numerical prefactors and distinctions among scattering, osmotic and geometric correlation lengths require additional definitions.

In a three-dimensional semidilute good solvent, take $g$ statistical segments per blob with size $\xi \sim a{g}^{3/5}$ and concentration matching $\varphi \sim g{a}^3/{\xi }^3$. Eliminating $g$ gives the displayed relations. A developed blob regime requires $1\ll g\ll N$ and a volume fraction well below close packing. Within a blob the chain retains isolated swollen statistics; on larger scales interpenetrating chains screen excluded-volume effects. The exponents use $\nu =3/5$ and the site-volume assumption. For general $\nu$, $\xi \sim a{\varphi }^{-\nu /(3\nu -1)}$.

For an ideal chain inside a blob, $\nu =1/2$ and $\xi \sim b\sqrt{g}$. Together with $\varphi \sim g{v}_0/{\xi }^3$, this gives the displayed concentration-blob size. Under the site-volume approximation $v_0\sim b^3$, $\xi \sim b/\varphi$ and $g\sim {\varphi }^{-2}$. The estimate describes local overlap rather than the full chain size and requires $1\ll g\ll N$ for a well-developed semidilute regime.