Thermodynamic and topological properties of copolymer rings with a segregation/mixing transition.

Two ring polymers will be segregated if there is a strong repulsion between monomers in the polymers, and will be in a mixed phase if there is a strong attraction instead. These phases are separated by a segregated-mixed critical point similar to the Θ-point for homopolymers in poor solvents. In this talk self-avoiding polygons are used to model the ring polymers. Some bounds are proven on the free energy of the model, in particular showing that there is a critical point in the phase diagram of the model. Numerical data were obtained using a Multiple Markov Chain implementation using a combination of pivot and Verdier-Stockmayer elementary moves on the polygons. Numerical results on the metric and topological properties of the model will be presented, and it is shown that these properties change when the system passes through the critical point between the segregated and mixed phases. For example, in the segregated phase the data show that the ring polymers are expanded with a low probability of forming topological links, whereas in the mixed phase the ring polymers interpenetrate and are in a collapse state with a higher probability of forming non-trivial links. There is a sharp increase in the linking between the components as the system passes through the critical point. Links between the polygon components are detected by computing the two variable Alexander polynomial and the linking number (which detects homological links). Compactness of the polygons were determined by tracking the number of nearest neighbour contacts, metric properties, and the shape parameters of the model, as a function of the strength of the interaction between the two rings. The model is also appropriate for modeling figure-8 block copolymers with each ring in the figure-8 a block in the copolymer.