### Abstract

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 [B. Niethammer, F. Otto, Ostwald ripening: The screening length revisited, Calc. Var. Partial Differential Equations 13-1 (2001) 33-68]. A separation of timescales between particle growth and migration allows for a variational characterization of spatially inhomogeneous quasi-equilibrium states.

Original language | English (US) |
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Pages (from-to) | 1241-1255 |

Number of pages | 15 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 238 |

Issue number | 14 |

DOIs | |

State | Published - Jul 1 2009 |

### Keywords

- Coarsening
- Diblock copolymers
- Ostwald ripening
- Self-organization

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

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## Cite this

*Physica D: Nonlinear Phenomena*,

*238*(14), 1241-1255. https://doi.org/10.1016/j.physd.2009.04.006