Existence and Stability of Spherically Layered Solutions of the Diblock Copolymer Equation

The relatively simple Ohta-Kawasaki density functional theory for diblock copolymer melts allows us to construct and analyze exact solutions to the EulerLagrange equation by singular perturbation techniques. First we consider a solution of a single sphere pattern that models a cell in the spherical morphology. We show the existence of the sphere pattern and find a stability threshold, so that if the sphere is larger than the threshold value, the sphere pattern becomes unstable. Next we study a spherical lamellar pattern, which may be regarded as a defective lamellar pattern. We reduce the existence and the stability problems to some finite dimensional problems which are accurately solved with the help of computer. We find two thresholds. Only when the size of the sample is larger than the first threshold value, a spherical lamellar pattern exists. This patten is stable only when the seize of the sample is less than the second threshold value. As the stability of the spherical lamellar pattern changes at the second threshold, a bifurcating branch with a pattern of wriggled spherical interfaces appears. The free energy of the latter pattern is lower than that of Corresponding author. Phone: 1 435 797-0755, Fax 1 435 797-1822, E-mail: [email protected] Supported in part by NSF grant DMS-0509725. Phone: 852 2609-7967, Fax 852 2603-5154, E-mail: [email protected] Supported in part by an Earmarked Grant of RGC of Hong Kong (RGC Proj. 402304).