Numerical representations of AB-type copolymer complexes: analysis of 1H NMR chemical shift patterns in terms of a Smith–Cantor set

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.