Properties of Two-Dimensional Polymers

We have performed a series of Monte Carlo simulations for a two-dimensional polymer chain with monomers interacting via a Lennard-Jones potential. An analysis of a mean field theory, based on approximating the free energy as the sum of an elastic part and a fluid part, shows that in two dimensions there is a sharp collapse transition, at T = θ, but that there is no ideal or quasi-ideal behavior at the transition as there is in three dimensions. The simulations and mean field theory agree very well. While the simulations are not sensitive enough to extract precise values for the exponent v, it is clear that it is close to the Flory values: v = ¾, T > θ; v = ½, T < θ. The mean field theory also gives v = ⅔ at T = θ.