Dynamics of a polymer test chain in a glass forming matrix: The Hartree approximation

We consider the Langevin dynamics of a Gaussian test polymer chain coupled with a surrounding matrix which can undergo the glass transition. The Martin-Siggia-Rose generating functional method and the nonpertubative Hartree approximation are used to derive the generalized Rouse equation for the test chain. It is shown that the interaction of the test chain with the surrounding matrix renormalizes the bare friction and the spring constants of the test chain in such a way that the memory function as well as the bending dependent elastic modulus appear. We find that below the glass transition temperature T G of the matrix the Rouse modes of the test chain can be frozen and moreover the freezing temperatures (or the ergodicity-nonergodicity transition temperature) T c (p) depends from the Rouse mode index p.