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

Darae Jeong, Jaemin Shin, Yibao Li, Yongho Choi, Jae Hun Jung, Seunggyu Lee, Junseok Kim

Research output: Contribution to journalArticlepeer-review

24 Citations (Scopus)


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 languageEnglish
Pages (from-to)1263-1272
Number of pages10
JournalCurrent Applied Physics
Issue number9
Publication statusPublished - 2014 Sept

Bibliographical note

Funding Information:
The first author (D. Jeong) was supported by a Korea University Grant . The corresponding author (J.S. Kim) acknowledges the support of the MSIP (Ministry of Science, Ict & future Planning) .

Copyright 2014 Elsevier B.V., All rights reserved.


  • Diblock copolymer
  • Lamellar phase
  • Nonlocal Cahn-Hilliard equation
  • Phase separation
  • Wavelength

ASJC Scopus subject areas

  • General Materials Science
  • General Physics and Astronomy


Dive into the research topics of 'Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers'. Together they form a unique fingerprint.

Cite this