= Polymer overlap concentration
{title2=$c^*\sim M/(N_AV_{\mathrm{coil}})$}
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 $M/(N_AV_{\mathrm{coil}})$. 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>, $R\sim bN^{1/2}$ and a site-volume estimate $v_0\sim b^3$ gives overlap <volume fraction> $\phi^*\sim N^{-1/2}$. A <good solvent> chain with the <Flory approximation> $R\sim bN^{3/5}$ instead gives $\phi^*\sim N^{-4/5}$.
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