Kinetics of Anchoring of Polymer Chains on Substrates with Chemically Active Sites

We consider dynamics of an isolated polymer chain with a chemically active end-bead on a 2D solid substrate containing immobile, randomly placed chemically active sites (traps). For a particular situation when the end-bead can be irreversibly trapped by any of these sites, which results in a complete anchoring of the whole chain, we calculate the time evolution of the probability $P_{ch}(t)$ that the initially non-anchored chain remains mobile until time $t$. We find that for relatively short chains $P_{ch}(t)$ follows at intermediate times a standard-form 2D Smoluchowski-type decay law $ln P_{ch}(t) \sim - t/ln(t)$, which crosses over at very large times to the fluctuation-induced dependence $ln{P}_{ch}(t)\sim -t^{1\mathrm{/}2}$, associated with fluctuations in the spatial distribution of traps. We show next that for long chains the kinetic behavior is quite different; here the intermediate-time decay is of the form $ln{P}_{ch}(t)\sim -t^{1\mathrm{/}2}$, which is the Smoluchowski-type law associated with subdiffusive motion of the end-bead, while the long-time fluctuation-induced decay is described by the dependence $ln{P}_{ch}(t)\sim -t^{1\mathrm{/}4}$, stemming out of the interplay between fluctuations in traps distribution and internal relaxations of the chain.