Abstract
In this paper we study a new model for patterns in two dimensions, inspired by diblock copolymer melts with a dominant phase. The model is simple enough to be amenable not only to numerics but also to analysis, yet sophisticated enough to reproduce hexagonally packed structures that resemble the cylinder patterns observed in block copolymer experiments. Starting from a sharpinterface continuum model, a nonlocal energy functional involving a Wasserstein cost, we derive the new model using Gammaconvergence 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, and in two dimensions they represent thin cylinders of the minority phase. We then study local minimizers of the limit energy. Numerical minimization 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 [D. P. Bourne, M. A. Peletier, and F. Theil, Comm. Math. Phys. , 329 (2014), 117140] and agrees with patterns observed in block copolymer experiments. This is a rare example of a nonlocal energydriven pattern formation problem in two dimensions where it can be proved that the optimal pattern is periodic.
Original language  English 

Pages (fromto)  13151337 
Number of pages  23 
Journal  SIAM Journal on Applied Mathematics 
Volume  74 
Issue number  5 
DOIs  
Publication status  Published  9 Sep 2014 
Keywords
 Crystallization
 Diblock copolymers
 Energydriven pattern formation
 Nonlocal energy
 Small volume fraction limit
 Voronoi diagrams
ASJC Scopus subject areas
 Applied Mathematics
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David Bourne
 School of Mathematical & Computer Sciences  Associate Professor
 School of Mathematical & Computer Sciences, Mathematics  Associate Professor
Person: Academic (Research & Teaching)