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

• Published in: SIAM journal on applied mathematics 2006-01, Vol.66 (3), p.1080-1099
• Main Authors: Ren, Xiaofeng, Wei, Juncheng
• Format: Article
• Language: English
• Subjects: Applied mathematics; Approximation; Copolymers; Density functional theory; Eigenfunctions; Eigenvalues; Energy; Euler Lagrange equation; Free energy; Lagrange multiplier; Mathematics; Molecules; Monomers; Morphology; Optimal size; Spheres
• ISSN: 0036-1399; 1095-712X
• DOI: 10.1137/040618771

The relatively simple Ohta-Kawasaki density functional theory for diblock copolymer melts allows us to construct and analyze exact solutions to the Euler-Lagrange 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 a computer. We find two thresholds. Only when the size of the sample is larger than the first threshold value does a spherical lamellar pattern exist. This patten is stable only when the size 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 the first pattern. A similar bifurcation phenomenon also occurs in the single sphere pattern at its stability threshold.