### 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 |
---|---|

Pages (from-to) | 216-225 |

Number of pages | 10 |

Journal | Computational Materials Science |

Volume | 81 |

DOIs | |

Publication status | Published - 2014 Jan 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Materials Science(all)
- Chemistry(all)
- Computer Science(all)
- Physics and Astronomy(all)
- Computational Mathematics
- Mechanics of Materials

### Cite this

*Computational Materials Science*,

*81*, 216-225. https://doi.org/10.1016/j.commatsci.2013.08.027

**Physical, mathematical, and numerical derivations of the cahn-hilliard equation.** / Lee, Dongsun; Huh, Joo Youl; Jeong, Darae; Shin, Jaemin; Yun, Ana; Kim, Junseok.

Research output: Contribution to journal › Article

*Computational Materials Science*, vol. 81, pp. 216-225. https://doi.org/10.1016/j.commatsci.2013.08.027

}

TY - JOUR

T1 - Physical, mathematical, and numerical derivations of the cahn-hilliard equation

AU - Lee, Dongsun

AU - Huh, Joo Youl

AU - Jeong, Darae

AU - Shin, Jaemin

AU - Yun, Ana

AU - Kim, Junseok

PY - 2014/1/1

Y1 - 2014/1/1

N2 - 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)+0.5â̂Š2c2]dx, where NV, 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.

AB - 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)+0.5â̂Š2c2]dx, where NV, 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.

KW - Cahn-Hilliard

KW - Chemical processes

KW - Mathematical modeling

KW - Numerical analysis

KW - Phase change

KW - Pseudospectral method

UR - http://www.scopus.com/inward/record.url?scp=84888377656&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84888377656&partnerID=8YFLogxK

U2 - 10.1016/j.commatsci.2013.08.027

DO - 10.1016/j.commatsci.2013.08.027

M3 - Article

VL - 81

SP - 216

EP - 225

JO - Computational Materials Science

JF - Computational Materials Science

SN - 0927-0256

ER -