zbMATH — the first resource for mathematics

Basic principles and practical applications of the Cahn-Hilliard equation. (English) Zbl 1400.35157
Summary: The celebrated Cahn-Hilliard (CH) equation was proposed to model the process of phase separation in binary alloys by Cahn and Hilliard. Since then the equation has been extended to a variety of chemical, physical, biological, and other engineering fields such as spinodal decomposition, diblock copolymer, image inpainting, multiphase fluid flows, microstructures with elastic inhomogeneity, tumor growth simulation, and topology optimization. Therefore, it is important to understand the basic mechanism of the CH equation in each modeling type. In this paper, we review the applications of the CH equation and describe the basic mechanism of each modeling type with helpful references and computational simulation results.

35K35 Initial-boundary value problems for higher-order parabolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
74N20 Dynamics of phase boundaries in solids
76T99 Multiphase and multicomponent flows
80A22 Stefan problems, phase changes, etc.
92B25 Biological rhythms and synchronization
Full Text: DOI
[1] Cahn, J. W., On spinodal decomposition, Acta Metallurgica, 9, 9, 795-801, (1961)
[2] Cahn, J. W.; Hilliard, J. E., Free energy of a nonuniform system. I. Interfacial free energy, The Journal of Chemical Physics, 28, 2, 258-267, (1958)
[3] Maraldi, M.; Molari, L.; Grandi, D., A unified thermodynamic framework for the modelling of diffusive and displacive phase transitions, International Journal of Engineering Science, 50, 1, 31-45, (2012) · Zbl 1423.80023
[4] Choksi, R.; Peletier, M. A.; Williams, J. F., On the phase diagram for microphase separation of diblock copolymers: an approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69, 6, 1712-1738, (2009) · Zbl 1400.74089
[5] Jeong, D.; Lee, S.; Choi, Y.; Kim, J., Energy-minimizing wavelengths of equilibrium states for diblock copolymers in the hex-cylinder phase, Current Applied Physics, 15, 7, 799-804, (2015)
[6] Jeong, D.; Shin, J.; Li, Y.; Choi, Y.; Jung, J.-H.; Lee, S.; Kim, J., Numerical analysis of energy-minimizing wavelengths of equilibrium states for diblock copolymers, Current Applied Physics, 14, 9, 1263-1272, (2014)
[7] Bertozzi, A. L.; Esedoglu, S.; Gillette, A., Inpainting of binary images using the Cahn-Hilliard equation, IEEE Transactions on Image Processing, 16, 1, 285-291, (2007) · Zbl 1279.94008
[8] Badalassi, V. E.; Ceniceros, H. D.; Banerjee, S., Computation of multiphase systems with phase field models, Journal of Computational Physics, 190, 2, 371-397, (2003) · Zbl 1076.76517
[9] Heida, M., On the derivation of thermodynamically consistent boundary conditions for the Cahn–Hilliard–Navier–Stokes system, International Journal of Engineering Science, 62, 126-156, (2013)
[10] Kotschote, M.; Zacher, R., Strong solutions in the dynamical theory of compressible fluid mixtures, Mathematical Models and Methods in Applied Sciences, 25, 7, 1217-1256, (2015) · Zbl 1329.76060
[11] Asle Zaeem, M.; El Kadiri, H.; Horstemeyer, M. F.; Khafizov, M.; Utegulov, Z., Effects of internal stresses and intermediate phases on the coarsening of coherent precipitates: a phase-field study, Current Applied Physics, 12, 2, 570-580, (2012)
[12] Zhu, J.; Chen, L.-Q.; Shen, J., Morphological evolution during phase separation and coarsening with strong inhomogeneous elasticity, Modelling and Simulation in Materials Science and Engineering, 9, 6, 499-511, (2001)
[13] Hilhorst, D.; Kampmann, J.; Nguyen, T.; Van Der Zee, K. G., Formal asymptotic limit of a diffuse-interface tumor-growth model, Mathematical Models and Methods in Applied Sciences, 25, 6, 1011-1043, (2015) · Zbl 1317.35123
[14] Wise, S. M.; Lowengrub, J. S.; Frieboes, H. B.; Cristini, V., Three-dimensional multispecies nonlinear tumor growth? I: model and numerical method, Journal of Theoretical Biology, 253, 3, 524-543, (2008)
[15] Farshbaf-Shaker, M. H.; Heinemann, C., A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media, Mathematical Models and Methods in Applied Sciences, 25, 14, 2749-2793, (2015) · Zbl 1325.35212
[16] Zhou, S.; Wang, M. Y., Multimaterial structural topology optimization with a generalized Cahn–Hilliard model of multiphase transition, Structural and Multidisciplinary Optimization, 33, 2, 89-111, (2007) · Zbl 1245.74077
[17] Wise, S. M.; Lowengrub, J. S.; Cristini, V., An adaptive multigrid algorithm for simulating solid tumor growth using mixture models, Mathematical and Computer Modelling, 53, 1-2, 1-20, (2011) · Zbl 1211.65123
[18] Lee, D.; Huh, J.-Y.; Jeong, D.; Shin, J.; Yun, A.; Kim, J., Physical, mathematical, and numerical derivations of the Cahn-Hilliard equation, Computational Materials Science, 81, 216-225, (2014)
[19] Yang, S.-D.; Lee, H. G.; Kim, J., A phase-field approach for minimizing the area of triply periodic surfaces with volume constraint, Computer Physics Communications, 181, 6, 1037-1046, (2010) · Zbl 1216.65021
[20] Cherfils, L.; Petcu, M., A numerical analysis of the Cahn-Hilliard equation with non-permeable walls, Numerische Mathematik, 128, 3, 517-549, (2014) · Zbl 1325.65133
[21] Christlieb, A.; Promislow, K.; Xu, Z., On the unconditionally gradient stable scheme for the Cahn-Hilliard equation and its implementation with Fourier method, Communications in Mathematical Sciences, 11, 2, 345-360, (2013) · Zbl 1305.65250
[22] Fernandino, M.; Dorao, C. A., The least squares spectral element method for the Cahn-Hilliard equation, Applied Mathematical Modelling, 35, 2, 797-806, (2011) · Zbl 1205.76158
[23] Hintermüller, M.; Hinze, M.; Tber, M. H., An adaptive finite-element Moreau-Yosida-based solver for a non-smooth Cahn-Hilliard problem, Optimization Methods and Software, 26, 4-5, 777-811, (2011) · Zbl 1366.74070
[24] Lecoq, N.; Zapolsky, H.; Galenko, P., Numerical approximation of the Cahn–Hilliard equation with memory effects in the dynamics of phase separation, Discrete and Continuous Dynamical Systems A, 31, 953-962, (2011) · Zbl 1306.65266
[25] Lezama-Alvarez, S.; Avila-Davila, E. O.; Lopez-Hirata, V. M.; Gonzalez-Velazquez, J. L., Numerical analysis of phase decomposition in A-B binary alloys using Cahn-Hilliard equations, Materials Research, 16, 5, 975-981, (2013)
[26] Tierra, G.; Guillén-González, F., Numerical methods for solving the Cahn-Hilliard equation and its applicability to related energy-based models, Archives of Computational Methods in Engineering, 22, 2, 269-289, (2015) · Zbl 1348.82080
[27] Wodo, O.; Ganapathysubramanian, B., Computationally efficient solution to the Cahn-Hilliard equation: adaptive implicit time schemes, mesh sensitivity analysis and the 3D isoperimetric problem, Journal of Computational Physics, 230, 15, 6037-6060, (2011) · Zbl 1416.65364
[28] Ohta, T.; Kawasaki, K., Equilibrium morphology of block copolymer melts, Macromolecules, 19, 10, 2621-2632, (1986)
[29] Ohnishi, I.; Nishiura, Y.; Imai, M.; Matsushita, Y., Analytical solutions describing the phase separation driven by a free energy functional containing a long-range interaction term, Chaos, 9, 2, 329-341, (1999) · Zbl 0970.35151
[30] Jeong, D.; Li, Y.; Lee, H. G.; Kim, J., Fast and automatic inpainting of binary images using a phase-field model, Journal of the Korean Society for Industrial and Applied Mathematics, 13, 3, 225-236, (2009)
[31] Feng, X., Fully discrete finite element approximations of the Navier–Stokes–Cahn–Hilliard diffuse interface model for two-phase fluid flows, SIAM Journal on Numerical Analysis, 44, 3, 1049-1072, (2006) · Zbl 1344.76052
[32] Kim, J., A continuous surface tension force formulation for diffuse-interface models, Journal of Computational Physics, 204, 2, 784-804, (2005) · Zbl 1329.76103
[33] Chen, L.-Q.; Khachaturyan, A. G., Computer simulation of structural transformations during precipitation of an ordered intermetallic phase, Acta Metallurgica et Materialia, 39, 11, 2533-2551, (1991)
[34] Wang, Y.; Chen, L.-Q.; Khachaturyan, A. G., Kinetics of strain-induced morphological transformation in cubic alloys with a miscibility gap, Acta Metallurgica Et Materialia, 41, 1, 279-296, (1993)
[35] Khachaturyan, A. G., Microscopic theory of diffusion in crystalline solid solutions and the time evolution of the diffuse scattering of X rays and thermal neutrons, Soviet Physics Solid State, USSR, 9, 9, 2040-2046, (1968)
[36] Khachaturyan, A. G., Theory of Structural Transformations in Solids, (1985), New York, NY, USA: Wiley-Interscience, New York, NY, USA
[37] Moelans, N.; Blanpain, B.; Wollants, P., An introduction to phase-field modeling of microstructure evolution, Calphad, 32, 2, 268-294, (2008)
[38] Chen, L.-Q., Phase-field models for microstructure evolution, Annual Review of Materials Science, 32, 1, 113-140, (2002)
[39] Hu, S. Y.; Chen, L. Q., A phase-field model for evolving microstructures with strong elastic inhomogeneity, Acta Materialia, 49, 11, 1879-1890, (2001)
[40] Paris, O.; Langmyar, F.; Vogl, G.; Fratzl, P., Possible criterion for slowing down of precipitate coarsening due to elastic misfit interactions, Zeitschrift für Metallkunde, 86, 12, 860-863, (1995)
[41] Lee, J. K., A study on coherency strain and precipitate morphology via a discrete atom method, Metallurgical and Materials Transactions A: Physical Metallurgy and Materials Science, 27, 6, 1449-1459, (1996)
[42] Lee, J. K., Elastic stress and microstructural evolution, Materials Transactions, JIM, 39, 1, 114-132, (1998)
[43] Schmidt, I.; Mueller, R.; Gross, D., The effect of elastic inhomogeneity on equilibrium and stability of a two particle morphology, Mechanics of Materials, 30, 3, 181-196, (1998)
[44] Jeong, D.; Lee, S.; Kim, J., An efficient numerical method for evolving microstructures with strong elastic inhomogeneity, Modelling and Simulation in Materials Science and Engineering, 23, 4, (2015)
[45] Cristini, V.; Frieboes, H. B.; Li, X.; Lowengrub, J. S.; Macklin, P.; Sanga, S.; Wise, S. M.; Zheng, X., Nonlinear modeling and simulation of tumor growth, Selected Topics in Cancer Modeling, 113-181, (2008), Boston, Mass, USA: Birkhauser, Boston, Mass, USA
[46] Cristini, V.; Lowengrub, J., Multiscale Modeling of Cancer: An Integrated Experimental and Mathematical Modeling Approach, (2010), Cambridge University Press
[47] Travasso, R. D. M.; Castro, M.; Oliveira, J. C. R. E., The phase-field model in tumor growth, Philosophical Magazine, 91, 1, 183-206, (2011)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.