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The graph-like state of matter. VII. The glass transition of polymers and Hamiltonian walks

1976; Institute of Physics; Volume: 9; Issue: 5 Linguagem: Inglês

10.1088/0305-4470/9/5/011

1361-6447

M Gorodn, P Kapadia, A. Malakis,

Phase Equilibria and Thermodynamics

The sensitivity of the Gibbs-DiMarzio theory for the glass transition of polymers to its basic assumptions is analysed. The underlying model, and all the problems it raises, are graph-theoretical in nature. It is shown that the value of the flexing energy parameter epsilon calculated from a measurement of the glass transition temperature Tg, is dominated by the result appropriate to the limiting case when the concentration of holes is zero, the number of chains unity, and the chain length goes to infinity. Accordingly, the problem is dominated by the classical Hamiltonian-walk problem on a lattice graph. The nature of the lattice graph, its coordination number, and the boundary conditions, are examined. The dimensionality of the embedding space (e.g. the distinction between 'two-dimensional' and 'three-dimensional' lattices) is discarded in favour of the parameter actually relevant, called the r-degree of the lattice graph. Asymptotic results on the enumeration of Hamiltonian walks are presented for the unoriented honeycomb and for the oriented square and other lattices, including the covering lattices of certain orientations of the diamond and cubic lattices.