Time evolution of polymer distribution functions from moment equations and maximum-entropy methods

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Abstract
We use the maximum-entropy method to calculate the chain-length distribution as a function of time for cooperative polymerization models involving nucleation and growth. At least the first two moments of the distribution are required for the maximum-entropy method. To obtain the moments we use a generating function to give the moment rate equations which in general involves an infinite set of coupled differential equations which can be truncated to give a finite set by using various closure approximations. In particular we use the maximum-entropy method to treat the reversible growth of chains from a fixed concentration of initiators in which case the initial distribution is a sharp Poisson-type one that then evolves slowly to the very broad equilibrium distribution. For this model we find that there is a scaled time that reduces the time dependence of the moments to a universal set of asymptotic curves.