Triblock Copolymer Theory: Ordered ABC Lamellar Phase

The ABC lamellar phase of a triblock copolymer in the strong segregation region is studied on periodic and bounded intervals. In the periodic case we find a family of local minimizers of the free energy functional all with a fine lamellar structure. Among these local minimizers we identify the one most favored by the free energy, and hence determine the thickness of lamellar microdomains. In the bounded interval case we show that perfect lamellar structure does not exist due to the boundary effect. We view the strong segregation limit as a Γ-limit of the free energy by a proper choice of the material sample size. The key step is the spectral analysis of a large matrix resulted from the second derivative of the Γ-limit.