Polymer overlap concentration ($c^{\ast }\sim M/(N_A{V}_{\mathrm{coil}})$)

The overlap concentration is the concentration at which distinct dilute polymer coils first substantially interpenetrate. It corresponds to roughly one chain per coil volume. If a chain contains $N$ repeat units, has molar mass $M$ and occupies a coil volume $V_{\mathrm{coil}}$, then its chain number density is approximately $1/V_{\mathrm{coil}}$, its repeat-unit number density is $N/V_{\mathrm{coil}}$, and its mass concentration is $M/(N_A{V}_{\mathrm{coil}})$. A scaling convention uses $V_{\mathrm{coil}}\sim R^3$; a commonly chosen geometric convention uses $V_{\mathrm{coil}}=4\pi R_g^3/3$. These give different order-one prefactors because a coil has no sharp surface. For an ideal chain, $R\sim b{N}^{1/2}$ and a site-volume estimate $v_0\sim b^3$ gives overlap volume fraction ${\varphi }^{\ast }\sim N^{-1/2}$. A good solvent chain with the Flory approximation $R\sim b{N}^{3/5}$ instead gives ${\varphi }^{\ast }\sim N^{-4/5}$.

The polymer overlap volume fraction counts the segment volume of one chain divided by its coil volume. If the segment volume is $v_0$ and $R\sim b{N}^{\nu }$, then ${\varphi }^{\ast }\sim (v_0/b^3)N^{1-3\nu }$. The site-volume convention $v_0\sim b^3$ gives $N^{-4/5}$ for the three-dimensional Flory approximation and $N^{-1/2}$ for an ideal chain. A geometric coil volume such as $4\pi R_g^3/3$ changes a prefactor; segment length and segment volume are not generally interchangeable without a model convention.

For polymers restricted to an interface, overlap means roughly one coil per coil area, $n_c{R}^2\sim 1$. A chain contains $N$ monomers with typical occupied area ${\ell }^2$, so its overlap area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$. With the planar good-solvent exponent $\nu =3/4$ from the two-dimensional Flory approximation, this gives ${\varphi }^{\ast }\sim N^{-1/2}$. The same exponent happens to occur for a three-dimensional ideal chain's volume fraction, but its geometric origin is different.