The Ising model in swollen vs. compact polymers: Mean-field approach and computer simulations.

We study the properties of the classical Ising model with nearest-neighbor interaction for spins located at the monomers of long polymer chains in 2 and 3 dimensions. We compare results for two ensembles of polymers with very different single chain properties: 1) swollen, self-avoiding linear polymer chains in good solvent conditions and 2) compact, space-filling randomly branching polymers in melt. By employing a mean-field approach and Monte Carlo computer simulations, we show that swollen polymers cannot sustain an ordered phase. On the contrary, compact polymers may indeed produce an observable phase transition. Finally, we briefly consider the statistical properties of the ordered phase by comparing polymer chains within the same universality class but characterized by very different shapes.