In this paper we derive a new model for diblock copolymer melts with a dominant phase that is simple enough to be amenable not only to numerics but also to analysis, yet sophisticated enough to reproduce the hexagonally packed cylinder patterns observed in experiments.
Starting from a sharp-interface continuum model, a nonlocal energy functional involving a Wasserstein cost, we derive the new model using Gamma-convergence in a limit where the volume fraction of one phase tends to zero. The limit energy is defined on atomic measures; in three dimensions the atoms represent small spherical blobs of the minority phase, in two dimensions they represent thin cylinders of the minority phase.
We then study minimisers of the limit energy. Numerical minimisation is performed in two dimensions by recasting the problem as a computational geometry problem involving power diagrams. The numerical results suggest that the small particles of the minority phase tend to arrange themselves on a triangular lattice as the number of particles goes to infinity. This is proved in the companion paper [BPT] and agrees with patterns observed in experiments. This is a rare example of a nonlocal energy-driven pattern formation problem in two dimensions where it can be proved that the optimal pattern is periodic, and the first time it has been proved that minimisers of a diblock copolymer energy are periodic.
|Number of pages||18|
|Publication status||Published - 2013|