Step growth of an AB2 monomer, with cycle formation

Abstract

A computer-based lattice model of the step growth reaction of an AB 2 monomer, the next elaborate system after an ABmonomer, has been devised that allows the simultaneous and explicit occurrence of inter- and intramolecular reactions of A and B groups of the flexible and moving molecules according to Monte Carlo selections of pairs adjacent on the lattice. Though cyclizations are infrequent in comparison to the reactions that develop the branched structures, they do occur, as they accumulate they consume a proportion of the A groups, and so they prevent the development of infinite branched molecules with fractalcharacteristics. Growth stops when each molecule contains a cycle. For the model explored, in which six lattice sites are used for each monomer, the limiting value of the number average degree of polymerization, 〈 x 〉 n,∞ , is 14.6(±0.3) (after infinite time). The occurrence within the system of rings of m residues ( m =1,2,3,…) is found to depend upon m and the extent of reaction of the A groups, p a , according to R m = C 0 p a m m −2.71 , the constant C 0 reflecting the structure of the lattice and the monomer, and being shown to determine the final degree of polymerization. The exponent of the integers m is apparently −e, so when p a =1 the total number of rings of the molecules is given by the product of C 0 and the Euler–Riemann ζ function, ζ(e), a finite number. C 0 is obtainable by experiment from 〈 x 〉 n,∞ at the end of the reaction of a real AB 2 monomer. Flory’s distribution functions for the numbers and weights of AB 2 polymers may be modified to allow for these cyclizations in a procedure which is useful during the early and middle courses of the reaction. However, at the end when cycle formation has come to dominate, the number and weight distribution functions for the first fifty molecules of size x also have the form of a power function [the terms of ζ(χ)] i.e., N x = N x,1 x −1.5 . As each molecule eventually contains one cycle, N x,1 = C 0 ζ(e)/ζ(1.5). Since for the weight distribution χ w =0.5, the total weight in the system N x,1 Σ x −χ w diverges, and so there is a limit to the size of x. We present a method for ordering the nodes within a particular structural isomer, priority going to the nodes that bear a loop, and then following its extent of reaction, and if necessary the extent of reaction of its neighbors (and so on). In this way the nodes in each structural isomer of the oligomers of a particular size may be characterized and identified. A mean extent of reaction vector, P x,p , may be obtained for the oligomers of size x at an extent of reaction, p a , to describe the mean extents of reaction of the ordered nodes and to convey the scope for further reactions at those nodes. From the data structures of the model we present information on the proportions of the different structural isomers of the smaller oligomers that are identified by this means, and provide mean connectivity or Kirchof matrices, K x,p , to describe the patterns of linking between the ordered nodes for examples of certain sizes of the species at selected stages of the reaction.