Dielectric relaxation and dynamic susceptibility of a one-dimensional model for perpendicular-dipole polymers
The dielectric properties of a simple model are studied. The model consists of a one-dimensional lattice of interacting objects, called spins. Each spin is oriented in a plane perpendicular to the lattice axis and is free to rotate in that plane. The spins undergo harmonic interactions with each other and, if they are dipolar, a cosine interaction with external fields. At sufficiently low temperature and for sufficiently strong spin-spin coupling, the model is equivalent to one in which the spin-spin interactions are proportional to the cosine of the angular difference between spin positions. It is shown that solution of the rotational diffusion equation leads to an interaction-independent, nonexponential decay function that is quite similar to a decay function derived empirically from polymer data by Williams and his co-workers. Exact decay functions are derived for both strong and weak applied fields, for both periodic and open boundary conditions, and for both nearest-neighbor and longer-range spin-spin interactions. Results of numerical calculations of dynamic susceptibitlity are presented, and it is shown that the model's qualitative behavior is consistent with much experimental data, especially polymer data.