Distribution of occupied sequences in one-dimensional arrays. Models for the reaction of polymer substituents

Abstract

An analysis has been carried out to give, as functions of time or extent of occupation, the number R_w of sequences of w occupied members for the random pairwise occupation of a linear array which might represent the reaction of substituents of a polymer molecule. Results are given as a function of time for cyclic (loop) and linear arrays initially consisting of m unreacted members. The model, referred to as (I), has been heretofore analysed to give the time dependence of (i) the number N_x of sequences of x unreacted members, (ii) the extent ξ and (iii) the rate ξ of reaction. The basic postulate is that the probability that a hitherto unreacted pair of adjacent members reacts in the interval between t and t+ dt is kdt. It is found, for example, that for m→∞, more reacted pairs are in sequences of six reacted members than in any other kind of sequence at the completion of reaction (t→∞). The sequences are separated by me^–2(≈13.5 %) isolated and therefore unreacted members.

Analysis and results are also given in the limit t→∞, for a second model (II) in which there are two rows of reactive groups where diagonal, yet still adjacent (not horizontal) reaction is possible, [graphic omitted] In this case a proportion, m/2e(≈18.4 %), of pairs of groups become isolated in the limits t→∞, m→∞. In this final state there are more reacted pairs in sequences of four pairs than in any other kind of sequence.