Collapse or swelling dynamics of homopolymer rings: Self-consistent Hartree approach

We investigate by the use of the Martin–Siggia–Rose generating functional technique and the self-consistent Hartree approximation, the dynamics of the ring homopolymer collapse (swelling) following an instantaneous change into a poor (good) solvent condition. The equation of motion for the time-dependent monomer-to-monomer correlation function is systematically derived. It is argued that for describing the coarse-graining process (which neglects the capillary instability and the coalescence of “pearls”) the Rouse mode representation is very helpful, so that the resulting equations of motion can be simply solved numerically. In the case of the collapse there are two characteristic regimes. The earlier regime is analyzed in the framework of the hierarchically crumpled fractal picture, with crumples of successively growing scale along the chain. The presented numerical results are in line with the corresponding simple scaling argumentation which in particular shows that the characteristic collapse time of a segment of length $g$ scales at this earlier stage as $tcrump*∼ζ0g/τ$ (where $ζ0$ is a bare friction coefficient and τ is a depth of quench). The later regime is related with a rearrangement of a “fluid of thermal blobs” and can be described by de Gennes’ “sausage” model. In contrast to the collapse the globule swelling can be seen (in the case that topological effects are neglected) as a homogeneous expansion of the globule interior. The swelling of each Rouse mode as well as gyration radius $Rg$ is discussed.