The size of a polymer in random media

Stimulated by the recent numerical simulation results of Baumgartner and Muthukumar, an analytic derivation is given of the size of a polymer in random media. It is shown that, for long Gaussian chains in three dimensions, the mean square end-to-end distance 〈∼(R2)〉 is proportional to (uρ0)−2, where uρ0 is the measure of the scattering power of the medium. For example, ρ0 is the number density of the scatterers and u is the strength of the pseudopotential between the chain segments. A simple extrapolation formula has been obtained using the replica theory for 〈∼(R2)〉 at intermediate values of (uρ0)2L, 〈∼(R2)〉 =(Ll/z) [1−exp(−z)], where L is the chain length, l is the Kuhn step length, and z=εu2ρ20Ll5 with ε being a numerical coefficient. This theory agrees well with the simulation results. The effects of space dimensionality and the replica breaking symmetry are also addressed.