Interfaces and Free Boundaries
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Stability of monolayers and bilayers in a copolymer-homopolymer blend model
Yves van Gennip (1) and Mark A. Peletier (2)
(1) Department of Mathematics, Eindhoven University of Technology, P.O.Box 513, NL-5600 MB, EINDHOVEN, NETHERLANDS(2) Department of Mathematics, Eindhoven University of Technology, P.O.Box 513, NL-5600 MB, EINDHOVEN, NETHERLANDS
We study the stability of layered structures in a variational model for diblock copolymerhomopolymer blends with respect to perturbations of their interfaces. The main step consists of calculating the first and second derivatives of a sharp-interface Ohta–Kawasaki energy for straight mono- and bilayers and determining the sign of the latter. By developing the interface perturbations in a Fourier series we fully characterise the stability of the structures in terms of the energy parameters. Both for the monolayer and for the bilayer there exist parameter regions where these structures are unstable. For strong repulsive interaction between the monomer types in the diblock copolymer the bilayer is always stable with respect to interface perturbations, irrespective of the domain size. The monolayer is only stable for small domain size.
In the course of our computations we also give a Green’s function for the Laplacian on a twodimensional periodic strip.
Keywords: Block copolymers, copolymer-homopolymer blends, pattern formation, variational model, partial localisation, Green’s function for Laplacian on a strip