Glassy dynamics in a simple model of a one-dimensional homogeneous polymer

We investigate, using numerical simulations and analytical arguments, a simple one-dimensional model for the swelling or collapse of a closed polymer chain of size N, representing the dynamical evolution of a polymer in a Θ-solvent that is rapidly changed into a good solvent (swelling) or a bad solvent (collapse). In the case of swelling, the density profile for intermediate times is parabolic and expands in space as t^1/3, as predicted by a Flory-like continuum theory. The dynamics slows down after a time ∝N² when the chain becomes stretched and the polymer gets stuck in metastable `zig-zag' configurations, from which it escapes through thermal activation. The size of the polymer in the final stages is found to grow as (ln t)<sup>1/2</sup>. In the case of collapse, the chain very quickly (after a time of order unity) breaks up into clusters of monomers (`pearls'). The evolution of the chain then proceeds through a slow growth of the size of these metastable clusters, again evolving as the logarithm of time. We enumerate the total number of metastable states as a function of the extension of the chain, and deduce from this computation that the radius of the chain should decrease as 1/ln (ln t). We compute the total number of metastable states with a given value of the energy, and find that the complexity is non-zero for arbitrary low energies. We also obtain the distribution of cluster sizes, which we compare with simple `cut-in-two' coalescence models. Finally, we determine the aging properties of the dynamical structure. The sub-aging behaviour that we find is attributed to the tail of the distribution at small cluster sizes, corresponding to anomalously `fast' clusters (as compared with the average). We argue that this mechanism for sub-aging might hold in other slowly coarsening systems.