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

N1 - 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.

PY - 2014

Y1 - 2014

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

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U2 - 10.1016/j.commatsci.2013.08.027

DO - 10.1016/j.commatsci.2013.08.027

M3 - Article

AN - SCOPUS:84888377656

VL - 81

SP - 216

EP - 225

JO - Computational Materials Science

JF - Computational Materials Science

SN - 0927-0256

ER -