Abstract/Summary

When considering the possibility of storing information in the sequence of monomer residues within an AB-type copolymer chain, it is constructive to model that sequence as a string of ones and zeros. The intramolecular environment around any given digit (say a "1") can then be represented by another string of integers - a code - obtained by summing pairs of digits at equivalent positions, in both directions, from that digit. The code can include only integers 0, 1 and 2, and can represent a number in any base b higher than 2. In base b = 3 the resulting set of codes includes all numbers (because only digits 0, 1 and 2 occur in ternary expansions), but in any base b > 3 the codes define a limited set of numbers comprising a fractal we term a Smith-Cantor set. The 1H NMR spectrum of a random, AB-type co(polyester-imide) shows, on complexation with pyrene, a pattern of complexation shifts approximating very closely to the Smith-Cantor set for which b = 4. Other co(polyimide) complexes show a 1H NMR pattern corresponding to a specific sub-set of this fractal. The sub-set arises from a “stop-at-zero” limitation, whereby digits in the initial string are set to zero for code-generating purposes if they occur beyond a zero, as viewed from the central "1". The limitation arises in copolymers where pyrene binds by intercalation between pairs of adjacent diimide residues. This numerical approach provides a complete, unifying theory to account for the emergence of fractal character in the 1H NMR spectra of AB-type copolymer complexes.