### Abstract

We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn-Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L^{*}/m, where h is the space grid size, L^{*} is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.

Original language | English |
---|---|

Pages (from-to) | 1263-1272 |

Number of pages | 10 |

Journal | Current Applied Physics |

Volume | 14 |

Issue number | 9 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

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### ASJC Scopus subject areas

- Materials Science(all)
- Physics and Astronomy(all)

### Cite this

*Current Applied Physics*,

*14*(9), 1263-1272. https://doi.org/10.1016/j.cap.2014.06.016

**Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers.** / Jeong, Darae; Shin, Jaemin; Li, Yibao; Choi, Yongho; Jung, Jae Hun; Lee, Seunggyu; Kim, Junseok.

Research output: Contribution to journal › Article

*Current Applied Physics*, vol. 14, no. 9, pp. 1263-1272. https://doi.org/10.1016/j.cap.2014.06.016

}

TY - JOUR

T1 - Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers

AU - Jeong, Darae

AU - Shin, Jaemin

AU - Li, Yibao

AU - Choi, Yongho

AU - Jung, Jae Hun

AU - Lee, Seunggyu

AU - Kim, Junseok

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn-Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L*/m, where h is the space grid size, L* is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.

AB - We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths of equilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on the nonlocal Cahn-Hilliard equation. The model consists of local and nonlocal terms associated with short- and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, we use a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To find energy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain an equilibrium state from a long time simulation of the time-dependent partial differential equation with varying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve the ordinary differential equation for the steady state. The results from these two methods are identical, which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula: h = L*/m, where h is the space grid size, L* is the energy-minimizing wavelength, and m is the number of the numerical grid steps in one period of a wave. Two- and three-dimensional numerical results are presented validating the usefulness of the formula without trial and error or ad hoc processes.

KW - Diblock copolymer

KW - Lamellar phase

KW - Nonlocal Cahn-Hilliard equation

KW - Phase separation

KW - Wavelength

UR - http://www.scopus.com/inward/record.url?scp=84904695583&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84904695583&partnerID=8YFLogxK

U2 - 10.1016/j.cap.2014.06.016

DO - 10.1016/j.cap.2014.06.016

M3 - Article

AN - SCOPUS:84904695583

VL - 14

SP - 1263

EP - 1272

JO - Current Applied Physics

JF - Current Applied Physics

SN - 1567-1739

IS - 9

ER -