Howard M. Colquhoun

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.

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.

"Polycondensation of an aromatic ether-ketone AB2 monomer in the presence of bis(4-fluorobenzoyl)-terminated PEEK affords semi-crystalline triblock copolymers in which the central block is a readily-sulfonatable PEEK chain and the outer blocks are non- sulfonatable hyperbranched moieties. Post-polymerisation sulfonation of each these copolymers in 96% sulfuric acid, after extraction of a cyclic oligomer fraction originating from the AB2 monomer, yields the corresponding ionomer in which sulfonation has occurred exclusively and quantitatively on the central block. The ion-exchange capacities of these ionomers were in the range 1.4 to 1.8 mmol g-1, and their inherent viscosities were 3.9 to 4.8 dl g-1. The ionomers were successfully solution-cast into tough, transparent, thin-film membranes (ca. 40 μm in thickness) which were evaluated for direct-methanol fuel cell (DMFC) performance in terms of proton conductivity, methanol diffusion coefficient, limiting current density and maximum power density. The triblock ionomers showed substantially higher DMFC performance than a control membrane based on the industry-standard fluorocarbon ionomer Nafion® 115, and their methanol diffusion coefficients were two to three times lower."

" Linear polycondensation of activated aromatic dihalides including 4,4'-difluoro-diphenylsulfone, 1,3-bis(4-fluorobenzoyl)benzene and 3,3'-bis(4-fluorobenzoyl)biphenyl with a combination of the two bisphenol monomers 4,4'-dihydroxybenzophenone and 2,8- dihydroxydibenzofuran affords a novel series of high molecular weight aromatic polyethers. Post-polymerisation sulfonation of these copolymers in 96% sulfuric acid yields a corresponding series of ionomers in which sulfonation has occurred at the 3- and 7-positions of the dibenzofuran rings. The ion-exchange capacities of these ionomers were in the range 1.3 to 1.8 mmol g-1, and their molecular weights (Mn) by GPC were between 39 and 56 kD with dispersities (Đ) between 2 and 3. All the ionomers were successfully solution-cast into tough, transparent, thin-film membranes (40 - 90 μm) which were evaluated for direct- methanol fuel cell (DMFC) performance in terms of proton conductivity, methanol diffusion coefficient, limiting current density and maximum power density. A dibenzofuran-based ionomer derived specifically from 3,3'-bis(4-fluorobenzoyl)biphenyl showed DMFC performance equivalent to, or even slightly better than, a control membrane produced from the industry-standard fluorocarbon ionomer Nafion 115."

Catalytic ester-interchange reactions, analogous to mutation and recombination, allow new sequence information to be written statistically into poly(ester-imide) chains based on NDI (1,4,5,8-naphthalenetetracarboxylic diimide) units. Thus, both the insertion of the cyclic ester cyclopentadecanolide (“exaltolide”) into an NDI-based homopolymer and quantitative sequence exchange between two different homopoly(ester-imide)s are catalyzed by di-n-butyl tin(IV) oxide. Emerging sequences are identified at the triplet and quintet levels using supramolecular complexation of pyrene-d₁₀ at the NDI residues to amplify the separation of ¹H NMR resonances associated with different sequences. In such systems, pyrene is able to act as a “reader molecule” by generating different levels of ring-current shielding from the different patterns of supramolecular binding to all NDI-centered sequences of a given length.

Nine ¹H NMR resonances assignable to specific copoly(ester-imide) sequences identified from a fractal model result from 1 : 1 supramolecular binding of pyrene to NDI residues.