The coil-globule transition of confined polymers

We study long polymer chains in a poor solvent, confined to the space between two parallel hard walls. The walls are energetically neutral and pose only a geometric constraint which changes the properties of the coil-globule (or "$\theta$-") transition. We find that the $\theta$ temperature increases monotonically with the width $D$ between the walls, in contrast to recent claims in the literature. Put in a wider context, the problem can be seen as a dimensional cross over in a tricritical point of a $\phi^4$ model. We roughly verify the main scaling properties expected for such a phenomenon, but we find also somewhat unexpected very long transients before the asymptotic scaling regions are reached. In particular, instead of the expected scaling $R\sim N^{4/7}$ exactly at the ($D$-dependent) theta point we found that $R$ increases less fast than $N^{1/2}$, even for extremely long chains.