Green's function analysis of the vibrational spectra of polymer chains. I. Several approaches to the problem

Abstract

SynopsisVarious methods of solving the dynamical equation for defect polymers are discussed. It is shown that the Green's function (GF) can be conveniently used to reduce the order of the characteristic equation to the number of the degrees of freedom of a repeat unit in the case when the perturbation caused by each of the defects is localized within it. In the general case, the order of the reduced determinant depends on the range of the interaction forces between the defect unit and the regular chain.For a polymer chain containing one or two defect units, the Green's function method allows us to obtain the exact solution of the dynamical equation, while for a high concentration of randomly distributed defect units various approximation methods can be used. As a possible approach to the solution of the latter problem we suggest using either one of the versions of the perturbation theory (the dilute limit, the coherent potential approximation) or the cluster approximation.

Molecular Vibrations of Polymers

“…The Green's function method of Lipshitz, which has been extensively used to describe defects in inorganic materials, has been investigated by Schmid & Holzl (141,153,154) and by Kozyrenko et al (142,155) for application to chemical and conformational defects in polymers. The numerical method has been extensively used by Zerbi and co-workers (140,(156)(157)(158)(159).…”

“…Other techniques include the Green's function method (141,142) and a numerical approach using the negative eigenvalue theorem (NET) (140,151,152). The Green's function method of Lipshitz, which has been extensively used to describe defects in inorganic materials, has been investigated by Schmid & Holzl (141,153,154) and by Kozyrenko et al (142,155) for application to chemical and conformational defects in polymers.…”

“…The GF method has been applied to simple models of regular polymer chains and defect-containing chains (154,155). Schmid & Holzi (154) have solved the GF for Kirkwood's model (160) of the planar zig-zag chain as well as for the chain containing mass defects.…”