The collapse of chains with different architectures

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Abstract
The equilibrium properties and the Rouse–Zimm dynamics of polymer molecules with any architecture at temperatures T≤Θ, are treated using a bead-and-spring coarse-grained description. The collapsed globule model is adopted, whereby essentially all atoms are at the same mean-square distance 〈S2〉 from the center of mass; accordingly, at a given temperature the interatomic free energy is a single-valued function of 〈S2〉 and the self-consistent free-energy minimization is greatly simplified, in the Gaussian approximation. We prove that the connectivity matrix B and the bond-vector product matrix M=[〈li⋅lj〉] possess the same eigenvectors; these are the normal modes of the chain conformation. Furthermore, we show that 〈S2〉=N−1at∑kl2α2k/λk, where Nat is the total number of atoms, λk is the general nonzero eigenvalue of B, and l2α2k is the corresponding eigenvalue of M—the expansion ratio of the normal mode. Finally, we prove that in the free-draining limit the normal mode relaxation times are proportional to l2α2k/λk. Defining αS=√〈S2〉/〈S2〉ph as the overall strain ratio with respect to the phantom state, the plots of αS vs the reduced temperature τ=(T−Θ)/T≤0 indicate that polymers with more compact architectures display a prompter contraction for small ‖τ‖’s, although tending to larger αS’s at strong undercoolings, where the average density (∝Nat⋅〈S2〉−3/2) is about the same for all architectures. Concerning the dynamical behavior, at sufficiently large ‖τ‖’s the longest relaxation times reach a typical plateau, as already found for the linear chain.