Reaction Kinetics in Polymer Melts

We study the reaction kinetics of end-functionalized polymer chains dispersed in an unreactive polymer melt. Starting from an infinite hierarchy of coupled equations for many-chain correlation functions, a closed equation is derived for the 2nd order rate constant $k$ after postulating simple physical bounds. Our results generalize previous 2-chain treatments (valid in dilute reactants limit) by Doi, de Gennes, and Friedman and O’Shaughnessy, to arbitrary initial reactive group density $n_0$ and local chemical reactivity $Q$. Simple mean field (MF) kinetics apply at short times, $k\sim Q$. For high $Q$, a transition occurs to diffusion-controlled (DC) kinetics with $k \approx x_t^3/t$(where$x_t$isrmsmonomerdisplacementintime$t$) leading to a density decay $n_t\approx n_0-n_0^2{x}_t^3$. If $n_0$ exceeds the chain overlap threshold, this behavior is followed by a regime where $n_t \approx 1/x_t^3$duringwhich$k$hasthesamepowerlawdependenceintime,$k \approx x_t^3/t$, but possibly different numerical coefficient. For unentangled melts this gives $n_t\sim t^{-3\mathrm{/}4}$ while for entangled cases one or more of the successive regimes $n_t\sim t^{-3\mathrm{/}4}$, $t^{-3\mathrm{/}8}$ and $t^{-3\mathrm{/}4}$ may be realized depending on the magnitudes of $Q$ and $n_0$. Kinetics at times longer than the longest polymer relaxation time $\tau$ are always MF. If a DC regime has developed before $\tau$ then the long time rate constant is $k \approx R^3/\tau$where$R$ is the coil radius. We propose measuring the above kinetics in a model experiment where radical end groups are generated by photolysis.