Theory of chiral order in random copolymers.

Recent experiments have found that polyisocyanates composed of a mixture of opposite enantiomers follow a chiral “majority rule:” the chiral order of the copolymer, measured by optical activity, is dominated by whichever enantiomer is in the majority. We explain this majority rule theoretically by mapping the random copolymer onto the random-field Ising model. Using this model, we predict the chiral order as a function of enantiomer concentration, in quantitative agreement with the experiments, and show how the sharpness of the majority-rule curve can be controlled. PACS numbers: 61.41.+e, 05.50.+q, 78.20.Ek, 82.90.+j Typeset using REVTEX 1 Cooperative chiral order plays a vital role in the self-assembly of ordered supramolecular structures in liquid crystals [1], organic thin films [2–5], and lipid membranes [6–12]. One particularly simple and well-controlled example of cooperative chiral order is in random copolymers. Recent experiments have found that polyisocyanates formed from a mixture of opposite enantiomers follow a chiral “majority rule” [13]. The chiral order of the copolymer, measured by optical activity, responds sharply to slight differences in the concentrations of the enantiomers, and is dominated by whichever enantiomer is in the majority. In this paper, we show that the majority rule can be understood through a mapping of the random copolymer onto the random-field Ising model [14–16]. Using this model, we predict the chiral order as a function of enantiomer concentration, in quantitative agreement with the experiments, and show that the sharpness of the majority-rule curve is determined by two energy scales associated with the chiral packing of monomers. In a series of experiments, Green et al. have investigated chiral order in polyisocyanates [17]. This polymer consists of a carbon-nitrogen backbone with a pendant group attached to each monomer, as shown in Fig. 1. Although the backbone is nonchiral, steric constraints force the molecule to polymerize in a helical structure. If the pendant group is also nonchiral, the helix is randomly rightor left-handed. A long chain then consists of domains of fixed helicity, separated by occasional helix reversals. On average, there are equal rightand left-handed domains, leading to zero net optical activity. However, if the pendant group is chiral, there is a preference for one sense of the helix, which leads to a net optical activity. Because of the cooperative interaction between the monomers in a domain, even a very small chiral influence leads to a large optical activity [18]. Most recently, Green et al. have synthesized random copolymers with a mixture of rightand left-handed enantiomeric pendant groups, with concentrations p and 1 − p, respectively [13]. The resulting optical activity, shown in Fig. 2, has a surprisingly sharp dependence on p. A 56/44 mixture of enantiomers has almost the same optical activity as a pure 100/0 homopolymer, and even a 51/49 mixture has a third of that optical activity. To explain this cooperative chiral order theoretically, we map the random copolymer onto 2 the one-dimensional random-field Ising model, a standard model in the theory of random magnetic systems [14–16]. Although related models have been applied to other polymer systems [19–22], our theory gives a new, direct correspondence between the Ising order parameter and the optical activity. This correspondence provides a novel experimental test of predictions for the random-field Ising model. Let the Ising spin σi = ±1 represent the sense of the polymer helix at the monomer i. The energy of a polymer can then be written as