Single-chain statistics and the upper wave-vector cutoff in polymer blends.

Abstract

We derive the equation for the single-chain correlation function in polymer blends. The chains in the incompressible blend have a radius of gyration smaller than the radius of gyration for ideal chains. The chains shrink progressively as we approach the critical temperature ${\mathit{T}}_{\mathit{c}}$. The correction responsible for shrinking is proportional to 1/ \ensuremath{\surd}N , where N is the polymerization index. At T=${\mathit{T}}_{\mathit{c}}$ and for N=1000, the size of the chain has been estimated to be 10% smaller than the size of the ideal coil. The estimate relies on the appropriate cutoff. In the limit of N\ensuremath{\rightarrow}\ensuremath{\infty} the chains approach the random walk limit. Additionally, we propose in this paper a self-consistent determination of the radius of gyration and the upper wave-vector cutoff. Our model is free from any divergences such as were encountered in the previous mean-field studies; we make an estimate of the chain size at the true critical temperature and not the mean-field one.