Concentration blob

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.