J. Phys. I France

**2**(1992) 581-598

## The random phase approximation for polymer melts with quenched degrees of freedom

**M. G. Brereton**

^{1}and T. A. Vilgis^{2}^{1}IRC in Polymer Science and Technology, The University of Leeds, Leeds LS2 9JT, G.B.

^{2}Max Planck Institut fur Polymerforschung, Postfach 3148, 6500 Mainz, Germany

(Received 1 December 1991, accepted 5 February 1992)

** Abstract **

There are many concentrated polymer systems of interest (e.g. crosslinked rubber networks) where the chains retain their configurational
freedom but the translational degree of freedom is absent. To investigate the effect of the loss of this degree of freedom
on the density fluctuations we have considered a simple case of a melt of polymer chains where one end of each chain is anchored
to a fixed point in space. The random phase approximation (RPA) used for melts cannot be immediately applied to this problem,
since the loss of translational freedom couples together density fluctuations belonging to different wavevectors. The quenched
chain end variables must first be accounted for using a "replica" calculation, then the RPA can be used for the annealed configurational
degrees of freedom. An explicit analytic result has been obtained, which shows that despite the loss of the translational
degree of freedom the density fluctuations in the quenched system remain similar to those in the true melt out to spatial
scales of the order of
*N*^{1/4}*R*, where
*R* is the size of a single chain coil (
) and
*N* is the degree of polymerisation.
We have also considered the concentration fluctuations a binary blend of chains with anchored chain ends. The situation is
shown to be very similar to a diblock polymer melt in so far as when the Flory interaction parameter exceeds a given value
microphase separation occurs. Again the loss of the translational freedom of the chains makes little difference to the stabilily
criterium.

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*Les Editions de Physique 1992*