An equilibrium model for chiral amplification in supramolecular polymers.

We describe a model that rationalizes amplification of chirality in cooperative supramolecular copolymerization. The model extends nucleation-elongation based equilibrium models for growth of supramolecular homopolymers to the case of two monomer and aggregate types. Using the principle of mass-balance for the two monomer types, we derive a set of two nonlinear equations, describing the thermodynamic equilibrium state of the system. These equations can be solved by numerical methods, but also analytical approximations are derived. The equilibrium model allows two-sided growth of the aggregates and can be applied to symmetric supramolecular copolymerizations, corresponding to the situation in which the monomers are enantiomerically related, as well as to the more general case of nonsymmetric supramolecular copolymerizations. In detail, so-called majority-rules phenomena in supramolecular systems with isodesmic as well as cooperative growth are analyzed. Comparison of model predictions with experimental data shows that the model gives a very good description of both titration and melting curves. When the system shows cooperative growth, the model leads to a phase diagram in which the presence of the various aggregate types is given as a function of composition and temperature.