## Abstract

We investigate the energy-minimizing wavelengths of equilibrium states for diblock copolymers in the hex-cylinder phase. The mathematical model is the Cahn-Hilliard equation with long-range interactions. The numerical scheme is based on a linearly gradient stable method and the resulting discrete system of equations is solved by a Fourier-spectral method. We solve the equations in non-square domains because the periodic unit is not a square. We choose the computational domains as rectangles of aspect ratio 3 (height/width). We run the computation until the system reaches a numerical equilibrium state. We repeat these calculations in domains of gradually increasing size and then find the wavelength that minimizes the domain-size-scaled total energy. We investigate the effect of the parameters on the energy-minimizing wavelength. We also propose a formula for a non-square domain that is close to a square domain and has an exact periodicity.

Original language | English |
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Pages (from-to) | 799-804 |

Number of pages | 6 |

Journal | Current Applied Physics |

Volume | 15 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2015 Apr 25 |

### Bibliographical note

Funding Information:The first author (D. Jeong) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( 2014R1A6A3A01009812 ). The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIP) ( NRF-2014R1A2A2A01003683 ). The authors are grateful to the reviewers whose valuable suggestions and comments significantly improved the quality of this paper.

Publisher Copyright:

© 2015 Elsevier B.V. All rights reserved.

## Keywords

- Cahn-Hilliard equation
- Diblock copolymer
- Fourier-spectral method
- Hex-cylinder phase
- Nonlocal

## ASJC Scopus subject areas

- General Materials Science
- General Physics and Astronomy