Probability model theory of chain‐end activated polymer degradation. I. First‐order radical termination

Abstract

AbstractSkellam and Shenton have recently described a theoretical approach to the treatment, in terms of probability models, of distributions associated with random walk and recurrent events. Within this framework, analytical solutions are here obtained for a number of problems in the kinetics of chain‐end activated depolymerization. The model employs the process of random drawing of chains from urns containing the three distributions which define the problem: the initial polymer chain distribution, the zip‐number distribution which is a function of time, and the zip‐length distribution. The validity and ease of the method is demonstrated by deriving, in somewhat improved form, the solution obtained by Simha, Wall, and Blatz for a special case, viz., a homodisperse polymer undergoing chain‐end activated degradation without chain transfer to polymer. As was recently shown by Gordon, the assumption of an initially exponential polymer chain distribution greatly simplifies the solution, and also makes the model more realistic for most synthetic polymers. The simplification is shown to arise in the present method, because the exponential distribution conveniently furnishes the kernel of a Laplace transform. Solutions are also obtained here for the polymer chains obeying λ‐fold self convolution of the exponential distribution. The case λ = 1, which is directly applicable to polymers formed under a regime of radical combination, yields a solution which, though new, is quite simple. The case λ = ∞ can be shown to revert to the case of a homodisperse polymer already mentioned. It is readily shown, and explained intuitively, that for all finite λ the convoluted exponential chain distributions revert asymptotically to the unconvoluted exponential distribution at high conversion (i.e., “zipping undoes convolution”). Accordingly, the kinetics approach asymptotically to a first order reaction. This is not true, however, of the monodisperse polymer case, in which λ is infinite. The linear superposition of polymer chain distributions with different values of λ is finally shown to be amenable to the present treatment, and yields analytic solutions for the pure chain‐end activated degradation of polymers under unrestrictive conditions: the initial polymer chain distribution requires to be expandable in the form Gram‐Charlier Type A, but the zip‐number and zip‐length distributions can be chosen at will. In the present paper, initiation is taken to be first order in the total chain‐end concentration, and termination is regarded as first order with respect to the radical concentration. Solutions appertaining to radical disproportionation, and to variously reactive chain‐ends, are reserved for a further paper.