A variational model for patterns in polymer melts

In nature one encounters patterns everywhere, from the stripes on zebras and the sand patterns in a desert, to the beating of your heart and the fingerprints on your fingers. A pattern forming system that has received extensive attention from experimentalists in recent decades is the diblock copolymer melt. Two types of polymer molecules are chemically bonded together to form a diblock copolymer. These molecules mutually repel each other, but due to the chemical bond they are restricted in their movement away from each other. These competing influences lead to pattern formation on a length scale between that of the system and that of the molecules. In recent years also mathematicians began studying models for diblock copolymer melts.

We are investigating a model for the simplest extension of this system, a blend of diblock copolymers and a third type of polymer, called homopolymer, which is not bonded to the copolymer molecules. This blend-model, which is variational in nature, is now well-understood in one dimension. We know how minimisers of the energy look and under which conditions they are non-unique. In higher dimensions we have bounds on the energy and we investigated the stability of some specific morphologies in two dimensions. Our study shows an interesting dependence of stability in these cases on the strength of the mutual repulsion between the different types of polymer molecules.