Coarsening and Self-organization in Dilute Diblock Copolymer Melts and Mixtures

This paper explores the evolution of a sharp interface model for phase separation of copolymers in the limit of low volume fraction. Particles both exchange material as in usual Ostwald ripening, and migrate because of an effectively repulsive nonlocal energetic term. Coarsening via mass diffusion only occurs while particle radii are small, and they eventually approach a finite equilibrium size. Migration, on the other hand, is responsible for producing self-organized patterns. We construct approximations based upon an ansatz of spherical particles similar to the classical LSW theory to derive finite dimensional dynamics for particle positions and radii. For large systems, kinetic-type equations which describe the evolution of a probability density are constructed. For systems larger than the screening length, we obtain an analog of the homogenization result of Niethammer & Otto (Calc. Var. and PDE, Vol. 13 (2001)). A separation of timescales between particle growth and migration allows for a variational characterization of spatially inhomogeneous quasi-equilibrium states. Ostwald ripening, a coarsening process described by the exchange of material between particles of the minority phase in binary mixtures (cf. [30, 17]), has been the topic of extensive study over the last 20 years. Much of the analytical work is based upon the Cahn-Hilliard equation which was conceived to model phase separation in binary alloys (cf [3]). This equation describes spinodal decomposition of a fine grained mixture and nucleation of phases rich in each component of the mixture. This is followed by interface coarsening via the Mullins-Sekerka law, which produces coarsening behavior with a wellestablished rate of t, i.e. the length scale of increases as a power law with exponent 1/3. In the dilute regime, the minority phase forms particles which interact through mass diffusion, and the resulting process is termed Ostwald ripening. One can derive both finitedimensional dynamics of individual particles, and in the limit of large particle number kinetic-type statistical descriptions ([16, 31, 30, 17, 19] and references therein). These descriptions provide a counterpart to rigorous upper bounds (e.g. [15, 24]) which predict and confirm temporal scaling laws. The Cahn-Hilliard approach applies equally well to phase separation of mixtures or melts of homogeneous polymers which are thermodynamically incompatible [9]. Block copolymers, on the other hand, are inhomogeneous chain molecules composed of two or more monomer types (see Figure 1 top). Below a critical temperature, weak local repulsion of different species induces phase separation. Complete separation does not occur, however, because of molecular attachment of distinct monomer chains. Instead, a remarkable array