Monte-Carlo studies of a polymer between planes, crossover between dimensionalities

We describe the results of Monte-Carlo calculations of polymer chains with excluded volume interactions which are confined within a slab of width D. We studied, in particular, R2(N, D), the mean square of the end-to-end distance of a chain with N links. For large N and fixed D, R2(N, D) ~ R 22 (N), the squared end-to-end distance of a chain constrained to a plane. We find R22(N) ≡ R 2(N, 0) ∝ N2v with 2v = 3/2 in agreement with the prediction of Flory. Letting R23(N)≡ R2(N, ∞), the end-to-end distance for an unconstrained three dimensional chain, we examine the crossover scaling of R2(N, D)/R2 3(N) as a function f(x) of x ≡ D/R3(N). For x ≤ 0.45, f(x) ∝ x-1/2, in agreement with predictions of Daoud and de Gennes. The behaviour of a chain without excluded volume interactions in the same constraining geometry is also discussed.