On the Spectra of 3-D Lamellar Solutions of the Diblock Copolymer Problem ∗ Xiaofeng Ren Department of Mathematics and Statistics Utah State University Logan, UT 84322-3900, USA Juncheng Wei † Department of Mathematics Chinese University of Hong Kong Shatin, Hong Kong December 23, 2002 Abstract One-dimensional free energy local minimizers are viewed as three-dimensional lamellar type critical points in a box. To determine whether they model the lamellar phase of diblock copoly- mers in the strong segregation region, we analyze their spectra. We obtain the asymptotic expansions of their eigenvalues and eigenfunctions. Consequently we find that they are stable, i.e. are local minimizers in space, only if they have sufficiently many interfaces. Interestingly the 1-D global minimizer is near the borderline of 3-D stability. Key words. spectrum, 3-D stability, lamellar solution, diblock copolymer 2000 Mathematics Subject Classification. 35J55, 34D15, 45J05, 82D60 1 Introduction In a di-block copolymer melt a molecule is a linear-chain consisting of two sub-chains grafted cova- lently to each other. The first sub-chain has N A type A monomer units and the second sub-chain has N B type B monomer units. In polymer systems even a weak repulsion between unlike monomers A and B induces a strong repulsion between sub-chains. With many chain molecules in a polymer melt the different type sub-chains tend to segregate below some critical temperature, but as they are chemically bonded in chain molecules, even a complete segregation of sub-chains cannot lead to a macroscopic phase separation. Only a local micro-phase separation occurs: micro-domains rich in A and B are formed. These micro-domains form morphological patterns/phases in a larger scale. The commonly observed phases include the spherical, cylindrical and lamellar, depicted in Figure 1. We consider a scenario that a diblock copolymer melt is placed in a domain D and maintained at fixed temperature. D is scaled to have unit volume in space. Let a = N A /(N A + N B ) ∈ (0, 1) be the relative number of the A monomers in a chain molecule. Similarly b = N B /(N A + N B ), so a + b = 1. The relative A monomer density field u is an order parameter. u ≈ 1 stands for high concentration * Abbreviated title. Spectra of Lamellar Solutions † Supported in part by a Direct Grant from CUHK and an Earmarked Grant of RGC of Hong Kong. 1