Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T10:06:29.395Z Has data issue: false hasContentIssue false

3 - Fundamentals of Mesoscale Simulation Methods

Published online by Cambridge University Press:  29 June 2023

Yong Du ,
Rainer Schmid-Fetzer ,
Jincheng Wang ,
Shuhong Liu ,
Jianchuan Wang  and
Zhanpeng Jin
Yong Du
Affiliation:
Central South University, China
Rainer Schmid-Fetzer
Affiliation:
Clausthal University of Technology, Germany
Jincheng Wang
Affiliation:
Northwestern Polytechnical University, China
Shuhong Liu
Affiliation:
Central South University, China
Jianchuan Wang
Affiliation:
Central South University, China
Zhanpeng Jin
Affiliation:
Central South University, China
Get access

Summary

In Chapter 3, we mainly focus on the fundamentals of typical mesoscale simulation methods, which can provide a bridge between atomistic structures and macroscopic properties of materials. Among many mesoscale simulation methods, the phase-field and cellular automaton methods are extremely popular and powerful for simulating microstructure evolution. Consequently, we first give a detailed introduction on the fundamentals of the two methods, briefly describing some other mesoscale simulation methods, such as level set and front tracking. After that, application examples using individual mesoscale simulation methods and integrations of the phase-field method with other simulation methods such as atomistic simulation, crystal plasticity, CALPHAD, and machine learning are described in detail. Finally, a case study for design of high-energy-density polymer nanocomposites using the phase-field method is very briefly presented.

Keywords

Mesoscale simulationphase-field methodphase-field crystal methodcellular automation methodintegrating phase-field and other simulation methods
Type
Chapter
Information
Computational Design of Engineering Materials
Fundamentals and Case Studies
 , pp. 46 - 94
Publisher: Cambridge University Press
Print publication year: 2023

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Allan, H. (1974) Regular algebra and finite machines by J.H. Conway. Mathematical Gazette, 58(405), 243–244.Google Scholar
Allen, S. M., and Cahn, J. W. (1979) A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metallurgica, 27(6), 1085–1095.CrossRefGoogle Scholar
Aspray, W., and Burks, A. (1987) Papers of John von Neumann on Computing and Computer Theory. Los Angeles: MIT Press.Google Scholar
Beltran-Sanchez, L., and Stefanescu, D. M. (2004) A quantitative dendrite growth model and analysis of stability concepts. Metallurgical and Materials Transactions A, 35(8), 2471–2485.CrossRefGoogle Scholar
Bhattacharyya, S., Sahara, R., and Ohno, K. (2019) A first-principles phase field method for quantitatively predicting multi-composition phase separation without thermodynamic empirical parameter. Nature Communications, 10(1), 3451.CrossRefGoogle ScholarPubMed
Bishop, C. M., and Carter, W. C. (2002) Relating atomistic grain boundary simulation results to the phase-field model. Computational Materials Science, 25(3), 378–386.CrossRefGoogle Scholar
Boettinger, W. J., Warren, J. A., Beckermann, C., and Karma, A. (2002) Phase-field simulation of solidification. Annual Review of Materials Research, 32(1), 163–194.CrossRefGoogle Scholar
Böttger, B., Eiken, J., and Apel, M. (2009) Phase-field simulation of microstructure formation in technical castings – a self-consistent homoenthalpic approach to the micro-macro problem. Journal of Computational Physics, 228(18), 6784–6795.CrossRefGoogle Scholar
Böttger, B., Eiken, J., and Steinbach, I. (2006) Phase field simulation of equiaxed solidification in technical alloys. Acta Materialia, 54(10), 2697–2704.CrossRefGoogle Scholar
Cahn, J. W., and Hilliard, J. E. (1958) Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics, 28(2), 258–267.CrossRefGoogle Scholar
Chen, L. Q. (2002) Phase-field models for microstructure evolution. Annual Review of Materials Research, 32(1), 113–140.CrossRefGoogle Scholar
Chen, L. Q., and Wang, Y. (1996) The continuum field approach to modeling microstructural evolution. JOM, 48(12), 13–18.CrossRefGoogle Scholar
Chen, L. Q., and Yang, W. (1994) Computer simulation of the domain dynamics of a quenched system with a large number of nonconserved order parameters: the grain-growth kinetics. Physical Review B, 50(21), 15752–15756.CrossRefGoogle ScholarPubMed
Choudhury, A., Reuther, K., Wesner, E., August, A., Nestler, B., and Rettenmayr, M. (2012) Comparison of phase-field and cellular automaton models for dendritic solidification in Al–Cu alloy. Computational Materials Science, 55, 263–268.CrossRefGoogle Scholar
Cottura, M., Appolaire, B., Finel, A., and Le Bouar, Y. (2016) Coupling the phase field method for diffusive transformations with dislocation density-based crystal plasticity: application to Ni-based superalloys. Journal of the Mechanics and Physics of Solids, 94, 473–489.CrossRefGoogle Scholar
Dantzig, J. A., Di Napoli, P., Friedli, J., and Rappaz, M. (2013) Dendritic growth morphologies in Al-Zn alloys – Part II: phase-field computations. Metallurgical and Materials Transactions A, 44(12), 5532–5543.CrossRefGoogle Scholar
Du, Q., Chen, M., and Xie, J. (2021) Modelling grain growth with the generalized Kampmann–Wagner numerical model. Computational Materials Science, 186, 110066.CrossRefGoogle Scholar
Du, Q., Tang, K., Marioara, C. D., Andersen, S. J., Holmedal, B., and Holmestad, R. (2017) Modeling over-ageing in Al–Mg–Si alloys by a multi-phase CALPHAD-coupled Kampmann–Wagner numerical model. Acta Materialia, 122, 178–186.CrossRefGoogle Scholar
Elder, K. R., Provatas, N., Berry, J., Stefanovic, P., and Grant, M. (2007) Phase-field crystal modeling and classical density functional theory of freezing. Physical Review B, 75(6), 064107.CrossRefGoogle Scholar
Emmerich, H., Löwen, H., Wittkowski, R., et al. (2012) Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview. Advanced Physics, 61(6), 665–743.CrossRefGoogle Scholar
Gandin, C. A., and Rappaz, M. (1994) A coupled finite element-cellular automaton model for the prediction of dendritic grain structures in solidification processes. Acta Metallurgica et Materialia, 42(7), 2233–2246.CrossRefGoogle Scholar
Ginzburg, V. L., and Landau, L. D. (1950) On the theory of superconductivity. Journal of Experimental and Theoretical Physics, 20(1064).Google Scholar
Glicksman, M. E., Koss, M. B., Hahn, R. C., Rojas, A., Karthikeyan, M., and Winsa, E. A. (1993) The isothermal dendritic growth experiment: scientific status of a USMP – 2 space flight experiment. Advanced Space Research, 13(7), 209–213.CrossRefGoogle Scholar
Guo, C., Wang, J., Wang, Z., Li, J., Guo, Y., and Tang, S. (2015) Modified phase-field-crystal model for solid-liquid phase transitions. Physical Review E, 92(1), 013309.CrossRefGoogle ScholarPubMed
Hohenberg, P. C., and Halperin, B. I. (1977) Theory of dynamic critical phenomena. Reviews of Modern Physics, 49(3), 435–479.CrossRefGoogle Scholar
Hoyt, J. J., and Asta, M. (2002) Atomistic computation of liquid diffusivity, solid–liquid interfacial free energy, and kinetic coefficient in Au and Ag. Physical Review B, 65(21), 214106.CrossRefGoogle Scholar
Jiang, X., Zhang, R., Zhang, C., Yin, H., and Qu, X. (2019) Fast prediction of the quasi phase equilibrium in phase field model for multicomponent alloys based on machine learning method. CALPHAD, 66, 101644.CrossRefGoogle Scholar
Juric, D., and Tryggvason, G. (1996) A front-tracking method for dendritic solidification. Journal of Computational Physics, 123(1), 127–148.CrossRefGoogle Scholar
Karma, A. (2001) Phase-field formulation for quantitative modeling of alloy solidification. Physical Review Letters, 87(11), 115701.CrossRefGoogle ScholarPubMed
Kaufman, L., and Bernstein, H. (1970) Computer Calculation of Phase Diagrams with Special Reference to Refractory Metals. New York: Academic Press, 334.Google Scholar
Khachaturyan, A. G. (1983) Theory of Structural Transformations in Solids. New York: John Wiley & Sons.Google Scholar
Kim, S. G., Kim, D. I., Kim, W. T., and Park, Y. B. (2006) Computer simulations of two-dimensional and three-dimensional ideal grain growth. Physical Review E, 74(6), 061605.CrossRefGoogle ScholarPubMed
Kim, S. G., Kim, W. T., and Suzuki, T. (1999) Phase-field model for binary alloys. Physical Review E, 60(6), 7186–7197.CrossRefGoogle ScholarPubMed
Kitashima, T. (2008) Coupling of the phase-field and CALPHAD methods for predicting multicomponent, solid-state phase transformations. Philosophical Magazine, 88(11), 1615–1637.CrossRefGoogle Scholar
Kobayashi, R. (1993) Modeling and numerical simulations of dendritic crystal growth. Physica D, 63(3), 410–423.CrossRefGoogle Scholar
Kobayashi, R. (1994) A numerical approach to three-dimensional dendritic solidification. Experimental Mathematics, 3(1), 59–81.CrossRefGoogle Scholar
Kobayashi, R., Warren, J. A., and Carter, W. C. (1998) Vector-valued phase field model for crystallization and grain boundary formation. Physica D, 119(3), 415–423.CrossRefGoogle Scholar
Langer, J. S. (1986) Models of pattern formation in first-order phase transitions, in Grinstein, G. and Mazenko, G. (eds), Directions in Condensed Matter Physics. Series on Directions in Condensed Matter Physics. Singapore: World Scientific, 165–186.CrossRefGoogle Scholar
Li, C. Y., Garimella, S. V., and Simpson, J. E. (2003) Fixed-grid front-tracking algorithm for solidification properties, part I: Method and validation. Numerical Heat Transfer, Part B, 43(2), 117–141.CrossRefGoogle Scholar
Li, J., Wang, Z., Wang, Y., and Wang, J. (2012) Phase-field study of competitive dendritic growth of converging grains during directional solidification. Acta Materialia, 60(4), 1478–1493.CrossRefGoogle Scholar
Li, J. J., and Wang, J. C. (2021) Macro-micro coupled simulation of microstructure during laser additive manufacturing process, [Lecture].unpublished.Google Scholar
Lian, Y., Gan, Z., Yu, C., Kats, D., Liu, W. K., and Wagner, G. J. (2019) A cellular automaton finite volume method for microstructure evolution during additive manufacturing. Materials and Design, 169, 107672.CrossRefGoogle Scholar
Liu, H., and Nie, J. F. (2017) Phase field simulation of microstructures of Mg and Al alloys. Materials Science and Technology, 33(18), 2159–2172.CrossRefGoogle Scholar
Louchez, M. A., Thuinet, L., Besson, R., and Legris, A. (2017) Microscopic phase-field modeling of Hcp|Fcc interfaces. Computational Materials Science, 132, 62–73.CrossRefGoogle Scholar
Marek, M. (2013) Grid anisotropy reduction for simulation of growth processes with cellular automaton. Physica D, 253, 73–84.CrossRefGoogle Scholar
Miyoshi, E., Takaki, T., Shibuta, Y., and Ohno, M. (2018) Bridging molecular dynamics and phase-field methods for grain growth prediction. Computational Materials Science, 152, 118–124.CrossRefGoogle Scholar
Moelans, N., Blanpain, B., and Wollants, P. (2008) An introduction to phase-field modeling of microstructure evolution. CALPHAD, 32(2), 268–294.CrossRefGoogle Scholar
Mohanty, S., and Ross, R. B. (2008) Overview of multiscale simulation methods for materials, in Ross, R. B. and Mohanty, S. (eds), Multiscale Simulation Methods for Nanomaterials. Hoboken: John Wiley & Sons, Inc., 1–7.Google Scholar
Nomoto, S., Seguwa, M., and Wakameda, H. (2018) Non-equilibrium PF model using thermodynamics data estimated by machine learning for additive manufacturing solidification. In Solid Freeform Fabrication 2018: Proceedings of the 29th Annual International Solid Freeform Fabrication Symposium – an Additive Manufacturing Conference, 1975–1886.Google Scholar
Osher, S., and Sethian, J. A. (1988) Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. Journal of Computational Physics, 79(1), 12–49.CrossRefGoogle Scholar
Poduri, R., and Chen, L. Q. (1998) Computer simulation of morphological evolution and coarsening kinetics of δ′ (Al3Li) precipitates in Al–Li alloys. Acta Materialia, 46(11), 3915–3928.CrossRefGoogle Scholar
Provatas, N., Dantzig, J. A., Athreya, B., et al. (2007) Using the phase-field crystal method in the multi-scale modeling of microstructure evolution. JOM, 59(7), 83–90.CrossRefGoogle Scholar
Provatas, N., and Elder, K. (2010) Phase-Field Methods in Materials Science and Engineering. Weinheim: Wiley–VCH Verlag GmbH & Co. KGaA.CrossRefGoogle Scholar
Qiu, D., Zhao, P., Shen, C., et al. (2019) Predicting grain boundary structure and energy in Bcc metals by integrated atomistic and phase-field modeling. Acta Materialia, 164, 799–809.CrossRefGoogle Scholar
Raabe, D. (2002) Cellular automata in materials science with particular reference to recrystallization simulation. Annual Review of Material Research, 32(1), 53–76.CrossRefGoogle Scholar
Rahnama, A., Dashwood, R., and Sridhar, S. (2017) A phase-field method coupled with CALPHAD for the simulation of ordered -carbide precipitates in both disordered and phases in low density steel. Computational Materials Science, 126, 152–159.CrossRefGoogle Scholar
Rajkumar, V. B., Du, Y., Zeng, Y., and Tang, S. (2020) Phase-field simulation of solidification microstructure in Ni and Cu–Ni alloy using the Wheeler, Boettinger and McFadden model coupled with the CALPHAD data. CALPHAD, 68, 101691.CrossRefGoogle Scholar
Rappaz, M. (1993) Modelling of microstructure formation in solidification processes. Acta Metallurgica et Materialia, 34(1), 93–124.Google Scholar
Reuther, K., and Rettenmayr, M. (2014) Perspectives for cellular automata for the simulation of dendritic solidification – a review. Computational Materials Science, 95, 213–220.CrossRefGoogle Scholar
Shen, Z. H., Wang, J. J., Lin, Y., Nan, C. W., Chen, L. Q., and Shen, Y. (2018) High-throughput phase-field design of high-energy-density polymer nanocomposites. Advanced Materials, 30(2), 1704380.CrossRefGoogle ScholarPubMed
Shibuta, Y., Ohno, M., and Takaki, T. (2018) Advent of cross-scale modeling: high-performance computing of solidification and grain growth. Advanced Theory and Simulations, 1(9), 1800065.CrossRefGoogle Scholar
Steinbach, I. (2009) Phase-field models in materials science. Modelling and Simulation in Materials Science and Engineering, 17(7), 073001.CrossRefGoogle Scholar
Steinbach, I., Böttger, B., Eiken, J., Warnken, N., and Fries, S. G. (2007) CALPHAD and phase-field modeling: a successful liaison. Journal of Phase Equilibria and Diffusion, 28(1), 101–106.CrossRefGoogle Scholar
Steinbach, I., Pezzolla, F., Nestler, B., et al. (1996) A phase field concept for multiphase systems. Physica D, 94(3), 135–147.CrossRefGoogle Scholar
Steinmetz, P., Yabansu, Y. C., Hötzer, J., et al. (2016) Analytics for microstructure datasets produced by phase-field simulations. Acta Materialia, 103, 192–203.CrossRefGoogle Scholar
Takaki, T. (2014) Phase-field modeling and simulations of dendrite growth. ISIJ International, 54(2), 437–444.CrossRefGoogle Scholar
Tan, L., and Zabaras, N. (2007) A level set simulation of dendritic solidification of multi-component alloys. Journal of Computational Physics, 221(1), 9–40.CrossRefGoogle Scholar
Tang, K., Du, Q., and Li, Y. (2018) Modelling microstructure evolution during casting, homogenization and ageing heat treatment of Al–Mg–Si–Cu–Fe–Mn alloys. CALPHAD, 63, 164–184.CrossRefGoogle Scholar
Tegze, G., Bansel, G., Tóth, G. I., Pusztai, T., Fan, Z., and Gránásy, L. (2009) Advanced operator splitting-based semi-implicit spectral method to solve the binary phase-field crystal equations with variable coefficients. Journal of Computational Physics, 228(5), 1612–1623.CrossRefGoogle Scholar
Teichert, G. H., and Garikipati, K. (2019) Machine learning materials physics: surrogate optimization and multi-fidelity algorithms predict precipitate morphology in an alternative to phase field dynamics. Computer Methods in Applied Mechanics and Engineering, 344, 666–693.CrossRefGoogle Scholar
Tonks, M. R., and Aagesen, L. K. (2019) The phase field method: mesoscale simulation aiding material discovery. Annual Review of Materials Research, 49(1), 79–102.CrossRefGoogle Scholar
Vaithyanathan, V., Wolverton, C., and Chen, L. Q. (2002) Multiscale modeling of precipitate microstructure evolution. Physical Review Letters, 88(12), 125503.CrossRefGoogle ScholarPubMed
Wang, Y., and Li, J. (2010) Phase field modeling of defects and deformation. Acta Materialia, 58(4), 1212–1235.CrossRefGoogle Scholar
Warren, J. A., and Boettinger, W. J. (1995) Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method. Acta Metallurgica et Materialia, 43(2), 689–703.CrossRefGoogle Scholar
Warren, J. A., Kobayashi, R., Lobkovsky, A. E., and Carter, W. C. (2003) Extending phase field models of solidification to polycrystalline materials. Acta Materialia, 51(20), 6035–6058.CrossRefGoogle Scholar
Wei, L., Lin, X., Wang, M., and Huang, W. (2011) A cellular automaton model for the solidification of a pure substance. Applied Physics A, 103(1), 123–133.CrossRefGoogle Scholar
Wheeler, A. A., Boettinger, W. J., and McFadden, G. B. (1993) Phase-field model of solute trapping during solidification. Physical Review E, 47(3), 1893–1909.CrossRefGoogle ScholarPubMed
Wolfram, S. (1986) Theory and Applications of Cellular Automata (Including Selected Papers 1983–1986) 1. Singapore: World Scientific.Google Scholar
Yabansu, Y. C., Steinmetz, P., Hötzer, J., Kalidindi, S. R., and Nestler, B. (2017) Extraction of reduced-order process-structure linkages from phase-field simulations. Acta Materialia, 124, 182–194.CrossRefGoogle Scholar
Zaeem, M. A., Yin, H., and Felicelli, S. D. (2012) Comparison of cellular automaton and phase field models to simulate dendrite growth in hexagonal crystals. Journal of Materials Science and Technology, 28(2), 137–146.CrossRefGoogle Scholar
Zhang, X., Zhao, J., Jiang, H., and Zhu, M. (2012) A three-dimensional cellular automaton model for dendritic growth in multi-component alloys. Acta Materialia, 60(5), 2249–2257.CrossRefGoogle Scholar
Zhou, N., Shen, C., Mills, M., and Wang, Y. (2010) Large-scale three-dimensional phase field simulation of -rafting and creep deformation. Philosophical Magazine, 90(1–4), 405–436.CrossRefGoogle Scholar
Zhu, M. F., and Hong, C. P. (2001) A modified cellular automaton model for the simulation of dendritic growth in solidification of alloys. ISIJ International, 41(5), 436–445.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×