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 Euler-Lagrange equation by singular perturbation techniques. First, we consider a solution of a single sphere pattern that models a cell in the spherica...

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Bibliographic Details
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
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Summary: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.
ISSN:0036-1399
1095-712X
DOI:10.1137/040618771