Spatially Localized Structures in Diblock Copolymer Mixtures

Abstract

Abstract. Above the critical temperature for the order-disorder transition, diblock copolymer melts have been observed to exhibit localized structures that exist within the homogeneous mixture. This paper uses an Ohta-Kawasaki-type density functional to explore this regime. Spatially localized peak-shaped equilibria are studied in one, two, and three dimensions, corresponding to amphiphilic bilayers, cylindrical micelles, and spherical micelles, respectively. A combination of rigorous estimates, asymptotic analysis, and numerical computation is used to characterize solutions and the regime where they exist. The interaction of superpositions of these solutions is studied by a perturbation analysis and shows how steady multipeak configurations can be achieved. Evidence is found for a secondary bifurcation slightly below the spinodal instability threshold, beyond which self-replication phenomena are observed. Dynamics in two dimensions are also illustrated, suggesting other mechanisms for instability and growth.Key words. diblock copolymer, micelle, density functional theory, self-replication AMS subject classifications. 82D60, 35K55DOI. 10.1137/080743913 1. Introduction. Diblock copolymers are linear chain molecules composed of two distinct subchains. At low temperatures, repulsion between subchains can result in phase segregation, but macroscopic segregation cannot occur because of covalent bonding between subchains. This incomplete segregation can give rise to a variety of stable microstructures (see, e.g., [1,11,14,24]), which are typically characterized as spatially periodic patterns whose domain boundaries closely resemble minimal surfaces, such as spheres, cylinders, and gyroids [39].Outside of the regime where mixtures are unstable to spinodal decomposition, the situation for copolymers is more complicated than simple binary systems, such as those described by the Cahn-Hilliard theory [5]. In addition to a stable homogeneous state, there are other observed behaviors. Physical experiments have identified inhomogeneous states above the critical temperature, such as sparse arrangements of micellar structures [35,36,43] and anisotropic lamellar grains [15]. This paper seeks to investigate this regime and its dynamics using density functional theory and the corresponding gradient flow dynamics.Theoretical work on the disordered regime has focused on spherical micelles. Dormidontova and Lodge [9] use Semenov's strong segregation theory [37] to predict the concentration of micelle assemblies. Their theory predicts that the number of micelles increases rapidly below a "critical micelle temperature" (CMT), which is greater than the temperature at which a homogeneous mixture is spinodally unstable. Wang, Wang, and Yang [42] use self-consistent field theory calculations to find the density profile of individual spherical micelles and evaluate their excess free energy. They find that micelles exist (within the mean field approximation) for volume frac-

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