### Abstract

We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small balls of one component. Mean-field models for the evolution of a large ensemble of such spheres have been formally derived in Glasner and Choksi (Physica D, 238:1241-1255, 2009), Helmers et al. (Netw Heterog Media, 3(3):615-632, 2008). It turns out that on a time scale of the order of the average volume of the spheres, the evolution is dominated by coarsening and subsequent stabilization of the radii of the spheres, whereas migration becomes only relevant on a larger time scale. Starting from the free boundary problem restricted to balls we rigorously derive the mean-field equations in the early time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow.

Original language | English |
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Pages (from-to) | 273-305 |

Number of pages | 33 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 39 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2010 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

**A rigorous derivation of mean-field models for diblock copolymer melts.** / Niethammer, Barbara; Oshita, Yoshihito.

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 39, no. 3, pp. 273-305. https://doi.org/10.1007/s00526-010-0310-x

}

TY - JOUR

T1 - A rigorous derivation of mean-field models for diblock copolymer melts

AU - Niethammer, Barbara

AU - Oshita, Yoshihito

PY - 2010

Y1 - 2010

N2 - We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small balls of one component. Mean-field models for the evolution of a large ensemble of such spheres have been formally derived in Glasner and Choksi (Physica D, 238:1241-1255, 2009), Helmers et al. (Netw Heterog Media, 3(3):615-632, 2008). It turns out that on a time scale of the order of the average volume of the spheres, the evolution is dominated by coarsening and subsequent stabilization of the radii of the spheres, whereas migration becomes only relevant on a larger time scale. Starting from the free boundary problem restricted to balls we rigorously derive the mean-field equations in the early time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow.

AB - We study the free boundary problem describing the micro phase separation of diblock copolymer melts in the regime that one component has small volume fraction such that micro phase separation results in an ensemble of small balls of one component. Mean-field models for the evolution of a large ensemble of such spheres have been formally derived in Glasner and Choksi (Physica D, 238:1241-1255, 2009), Helmers et al. (Netw Heterog Media, 3(3):615-632, 2008). It turns out that on a time scale of the order of the average volume of the spheres, the evolution is dominated by coarsening and subsequent stabilization of the radii of the spheres, whereas migration becomes only relevant on a larger time scale. Starting from the free boundary problem restricted to balls we rigorously derive the mean-field equations in the early time regime. Our analysis is based on passing to the homogenization limit in the variational framework of a gradient flow.

UR - http://www.scopus.com/inward/record.url?scp=77957982353&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957982353&partnerID=8YFLogxK

U2 - 10.1007/s00526-010-0310-x

DO - 10.1007/s00526-010-0310-x

M3 - Article

AN - SCOPUS:77957982353

VL - 39

SP - 273

EP - 305

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 3

ER -