Short polymer chain statistics and the relationship to end to end electronic excitation transport: random walks with variable-step lengths

Abstract

The problem of the distribution of distances between the ends of a polymer chain for short chains (a small number of statistical segments) is considered. A formalism developed by Rayleigh is utilized to calculate the exact end to end distribution functions for random walks with a small number of steps and variable step lengths. In particular, distribution functions for walks in which the first and last step differ in length from the intervening steps are obtained. These are used as models for low molecular weight polymers in which the first and last statistical segments are different from internal segments because of chain end effects or chromophores attached to the chain termini. The results are used to calculate the ensemble averaged time dependence of end to end electronic excitation transport. G8((t), the part of the transport Green function which yields the time-dependent fluorescence depolarization observable, is calculated with exact random walk distribution functions and approximate distribution functions. It is demonstrated that the results are similar but that experiments which examine several distance ranges have the capability of distinguishing the distribution functions.