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 $\nzero$ 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 $k \approx \xt^3/t$ (where $\xt$ is rms monomer displacement in time $t$) leading to a density decay $\nt \approx \nzero - \nzero^2 \xt^3$. If $\nzero$ is above the chain overlap threshold, this regime is followed by $\nt \approx 1/\xt^3$ during which $k$ has the same power law dependence, $k \approx \xt^3/t$, but presumably different numerical coefficient. For unentangled melts this leads to $\nt \sim t^{-3/4}$ while for entangled cases successive regimes $\nt \sim t^{-3/4}$, $t^{-3/8}$ and $t^{-3/4}$ may occur depending on $Q,\nzero$. 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.