Interfaces and Free Boundaries


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Volume 18, Issue 2, 2016, pp. 263–290
DOI: 10.4171/IFB/364

Published online: 2016-09-13

Sharp interface limit of an energy modelling nanoparticle-polymer blends

Stan Alama[1], Lia Bronsard[2] and Ihsan Topaloglu[3]

(1) McMaster University, Hamilton, Canada
(2) McMaster University, Hamilton, Canada
(3) McMaster University, Hamilton, Canada

We identify the  $\Gamma$-limit of a nanoparticle-polymer model as the number of particles goes to infinity and as the size of the particles and the phase transition thickness of the polymer phases approach zero. The limiting energy consists of two terms: the perimeter of the interface separating the phases and a penalization term related to the density distribution of the infinitely many small nanoparticles. We prove that local minimizers of the limiting energy admit regular phase boundaries and derive necessary conditions of local minimality via the first variation. Finally, we discuss possible critical and minimizing patterns in two dimensions and how these patterns vary from global minimizers of the purely local isoperimetric problem.

Keywords: Nanoparticles, isoperimetric problem, $\Gamma$-convergence, block copolymers, self-assembly, phase separation

Alama Stan, Bronsard Lia, Topaloglu Ihsan: Sharp interface limit of an energy modelling nanoparticle-polymer blends. Interfaces Free Bound. 18 (2016), 263-290. doi: 10.4171/IFB/364