Isolated polymer molecule in a random environment

The shape of a polymer molecule in the presence of randomly distributed stationary impurities is investigated using the path-integral formulation. The polymer molecule is assumed to be described by the Edwards minimal model. Calculation of the mean-square end-to-end displacement 〈$R^2$〉 shows that at a critical density of the scatterers the polymer molecule makes a transition from a swollen state (chain with excluded-volume interactions) to a disordered, random coiled state (a Gaussian chain). At this density there is a crossover in the value of the exponent ν (in three dimensions), characterizing the scaling of 〈$R^2$〉, with the length of the polymer from about 0.6 (corresponding to the excluded-volume regime) to 0.5 (corresponding to the coiled state). For densities below the critical density, the exponent ν is well described by the Flory value.