The molecular weight distribution problem and reptation mixing rules

Mixing rules model how the physical properties of a polymer, such as its re laxation modulus G(t), depend on the distribution w(m) of its molecular wei ghts m. They are of practical importance because, among other things, they allow estimates of the molecular weight distribution (MWD) w(m) of a polyme r to be determined from measurements of its physical properties including t he relaxation modulus. The two most common mixing rules are "single" and "d ouble" reptation. Various derivations for these rules have been published. In this paper, a conditional probability formulation is given which identif ies that the fundamental essence of ''double" reptation is the discrete bin ary nature of the "entanglements", which are assumed to occur in the corres ponding topological model of the underlying polymer dynamics. In addition, various methods for determining the MWD are reviewed, and the computation o f linear functionals of the MWD motivated and briefly examined.