The size of a single chain in a melt of anchored chains

(Received 7 April 1992, accepted 28 August 1992)

Abstract
This paper considers concentrated polymer systems where the translational degree of freedom of all the chains to move has been quenched. In particular we consider the screening properties of this constrained environment and the subsequent effect on the size of a single chain. The model consists of many flexible chains, each anchored at one end to fixed points randomly distributed in space. The model has applications to network problems. The quenched chain end variables are handled by a replica calculation and the chain configurations by a random phase approximation in replica space. After integrating over the density fluctuations present in the constrained environment the problem is reduced to a single chain interacting with replicas of itself through an effective interaction potential. Unlike the case for a melt the effective interation is not completely screened. In particular the inter replica interaction, which arises specifically from the constraint, is attractive. A perturbation calculation confirms that, not withstanding the lack of screening, the Gaussian nature of the chain is not affected. The fixed chain ends only lead to a contraction of the chain of the order N^-1/2, where N is the number of segments in the chain.