Polymer chains and vulcanisation

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Abstract
A discrete Hamiltonian to describe the vulcanisation which occurs when linear polymer chains are mixed with cross-linking units is proposed. Here the vulcanisation of the chains can occur via clusters of cross-linking units. The Hamiltonian is a simple combination of the n-vector model in the limit when n goes to zero, and the m-states Potts model when m goes to unity. The partition function is discussed. The Migdal renormalisation group shows that the chain behaviour is always controlled by the self-avoiding walk (SAW) fixed point. The vulcanisation is described by percolation exponents except in the vicinity of a higher-order critical point where it crosses over the SAW exponents.