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Scaling laws in microphase separation of diblock copolymers. (English) Zbl 1023.82015
Summary: We study a nonlocal variational problem modeling microphase separation of diblock copolymers. We apply certain new tools developed in [R. Choksi, R. V. Kohn and F. Otto, Commun. Math. Phys. 201, 61-79 (1999; Zbl 1023.82011)] to determine the principal part of the asymptotic expansion of the minimum free energy. That is, we prove a scaling law for the minimum energy and confirm that it is attained by a simple periodic lamellar structure. A previous result of I. Ohnishi, Y. Nishiura, M. Imai and Y. Matsushita [Chaos 9, 329-341 (1999; Zbl 0970.35151)] was for one space dimension. Here, we obtain a similar result for the full three-dimensional problem.

MSC:
82D60 Statistical mechanical studies of polymers
49N90 Applications of optimal control and differential games
49J10 Existence theories for free problems in two or more independent variables
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