Long Polymers Near Wedges and Cones

We perform a Monte Carlo study of $N$-step self-avoiding walks, attached to the corner of an impenetrable wedge in two dimensions ($d=2$), or the tip of an impenetrable cone in $d=3$, of sizes ranging up to $N=10^6$ steps. We find that the critical exponent $\gamma_{\alpha}$ which determines the dependence of the number of available conformations on $N$ for a cone/wedge with opening angle $\alpha$, is in good agreement with the theory for $d=2$. We study the end-point distribution of the walks in the allowed space and find similarities to the known behavior of random walks (ideal polymers) in the same geometry. For example the ratio between the mean square end-to-end distances of a polymer near the wedge and a polymer in free space depends linearly on $\gamma_{\alpha}$ as is known for ideal polymers. We show that the end-point distribution of polymers attached to a wedge does not separate into a product of angular and radial functions, as it does for ideal polymers in the same geometry. The angular dependence of the end-position of polymers near the wedge differs from theoretical predictions.