## Abstract

We review physical, mathematical, and numerical derivations of the binary Cahn-Hilliard equation (after John W. Cahn and John E. Hilliard). The phase separation is described by the equation whereby a binary mixture spontaneously separates into two domains rich in individual components. First, we describe the physical derivation from the basic thermodynamics. The free energy of the volume Ω of an isotropic system is given by ^{NV}∫ _{Ω}[F(c)+0.5^{â̂Š2}c^{2}]dx, where N_{V}, c, F(c), â̂Š, and c represent the number of molecules per unit volume, composition, free energy per molecule of a homogenous system, gradient energy coefficient related to the interfacial energy, and composition gradient, respectively. We define the chemical potential as the variational derivative of the total energy, and its flux as the minus gradient of the potential. Using the usual continuity equation, we obtain the Cahn-Hilliard equation. Second, we outline the mathematical derivation of the Cahn-Hilliard equation. The approach originates from the free energy functional and its justification of the functional in the Hilbert space. After calculating the gradient, we obtain the Cahn-Hilliard equation as a gradient flow. Third, various aspects are introduced using numerical methods such as the finite difference, finite element, and spectral methods. We also provide a short MATLAB program code for the Cahn-Hilliard equation using a pseudospectral method.

Original language | English |
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Pages (from-to) | 216-225 |

Number of pages | 10 |

Journal | Computational Materials Science |

Volume | 81 |

DOIs | |

Publication status | Published - 2014 |

### Bibliographical note

Funding Information:The first author (D. Lee) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2013003181 ), and J.Y. Huh was supported by a grant from the Fundamental R&D Program for Core Technology of Materials funded by the Ministry of Knowledge Economy, Republic of Korea. The corresponding author (J.S. Kim) would like to thank Professor Kyungkeun Kang for helpful conversations on the theoretical part. The authors are grateful to the anonymous referees whose valuable suggestions and comments significantly improved the quality of this paper.

## Keywords

- Cahn-Hilliard
- Chemical processes
- Mathematical modeling
- Numerical analysis
- Phase change
- Pseudospectral method

## ASJC Scopus subject areas

- General Computer Science
- General Chemistry
- General Materials Science
- Mechanics of Materials
- General Physics and Astronomy
- Computational Mathematics