Relaxation of entangled polymers at the classical gel point

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Abstract
We examine the relaxation behaviour of an entangled cross-linked polymer gel as it approaches the gel point in mean field (Flory-Stockmayer) percolation. The calculation is based on a tube model for the topological interactions in which stress is lost via hierarchical fluctuation of the primitive paths between cross-links. The decay time of a segment is calculated via a recursion relation which has an analytic solution near the gel point. The startling conclusion is that all clusters relax in a finite time T∞ giving a relaxation modulus G (t ) = Go γ-2[α -1 ln (T∞/t)]4, where α counts the number of entanglements between cross-links and γ is a constant of order unity. For timescales much shorter than T∞ this may resemble a weak power law. The unphysically rapid relaxation of the largest clusters is prevented by a transition to percolation statistics at long length scales. The timescale separating the two regimes is close to T∞

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