Elasticity of a real polymer chain for different modes of motion

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Abstract
A theory is given to evaluate the elastic constant of the force acting on the general atom of an (–A–)N polymer chain for any normal mode of motion allowed by skeletal rotations; recourse to the classical description in terms of beads and springs should thus be rendered unnecessary. The results are obtained after evaluation of the configurational free energy stored in the chain by the action of a general set of constant forces; according to the usual assumptions, the average displacements produced by the forces are assumed not to be too large compared with the statistical fluctuations. An algorithm to calculate the general force constant on the basis of the classical configurational statistics of polymer chains is given, and numerical results are reported for high molecular weight polyethylene. All other parameters being the same, the short relaxation times in the vicinity of the limit of the classical bead–spring theory (i.e., a sinusoidal wave comprising about 20 methylene units within each wavelength) are an order of magnitude smaller than calculated according to that theory. The whole spectrum of both the force constants and of the relaxation times (neglecting at present the internal viscosity as well as the hydrodynamic interactions) down to the shortest wavelength comprising two skeletal atoms is given.