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 local (intramolecular) environment around a given monomer residue within the chain can then be represented by another string of integers, which we denote as a code, obtained by summing pairs of digits at equivalent positions, in both directions, from that monomer residue’s position. The code can include only the integers 0, 1 and 2, and in principle it can represent a number in any base b, higher than 2 (i.e. it cannot be a binary number). Whereas in base b = 3 the resulting set of codes includes all numbers (because only digits 0, 1 and 2 are required for ternary representations), in any base b > 3 the codes define a limited set of numbers comprising a type of fractal we term a last-fraction Smith-Cantor set. Experimentally, the 1H NMR spectrum of a random, AB-type co(polyester-imide) shows, on complexation with the ring-current-shielding molecule pyrene, a pattern of chemical shifts approximating very closely to the fourth-quarter Smith-Cantor set (i.e. b = 4). It is demonstrated that this pattern results from the binding of pyrene to single diimide residues under conditions of fast-exchange on the NMR timescale. Other co(polyimide) complexes show a related 1H NMR pattern, but one now corresponding to a specific sub-set of the same fractal. We show that this sub-set corresponds to codes based on a “stop-at-zero” limitation, whereby digits in the initial string are disregarded (i.e. set to zero) for code-generating purposes if they occur beyond a zero, when viewed from the central “1”. This limitation is found to arise in copolymer systems where the shielding molecule binds by intercalation between pairs of adjacent diimide residues. The present 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.