Glassy effects in the swelling/collapse dynamics of homogeneous polymers

We investigate, using numerical simulations and analytical arguments, a simple one-dimensional model for the swelling or the collapse of a closed polymer chain of size $N$, representing the dynamical evolution of a polymer in a $\Theta$-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 $\propto N^2$ 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 $\sqrt{\ln t}$. 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, that we compare to simple "cut-in-two"coalescence models. Finally, we determine the aging properties of the dynamical structure. The subaging behaviour that we find is attributed to the tail of the distribution at small cluster sizes, corresponding to anomalously "fast"clusters (as compared to the average). We argue that this mechanism for subaging might hold in other slowly coarsening systems