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Finding Critical Nuclei in Phase Transformations by Shrinking Dimer Dynamics and its Variants

Published online by Cambridge University Press:  03 June 2015

Lei Zhang ,
Jingyan Zhang  and
Qiang Du
Lei Zhang*
Affiliation:
Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
Jingyan Zhang*
Affiliation:
Department of Mathematics, Pennsylvania State University, PA 16802, USA
Qiang Du*
Affiliation:
Department of Mathematics, Pennsylvania State University, PA 16802, USA Beijing Computational Science Research Center, Beijing, 100084, China
*
Email:zhangl@math.pku.edu.cn
Email:j_zhang@math.psu.edu
Corresponding author.Email:qdu@math.psu.edu
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Abstract

We investigate the critical nuclei morphology in phase transformation by combining two effective ingredients, with the first being the phase field modeling of the relevant energetics which has been a popular approach for phase transitions and the second being shrinking dimer dynamics and its variants for computing saddle points and transition states. In particular, the newly formulated generalized shrinking dimer dynamics is proposed by adopting the Cahn-Hilliard dynamics for the generalized gradient system. As illustrations, a couple of typical cases are considered, including a generic system modeling heterogeneous nucleation and a specific material system modeling the precipitate nucleation in FeCr alloys. While the standard shrinking dimer dynamics can be applied to study the non-conserved case of generic heterogeneous nucleation directly, the generalized shrinking dimer dynamics is efficient to compute precipitate nucleation in FeCr alloys due to the conservation of concentration. Numerical simulations are provided to demonstrate both the complex morphology associated with nucleation events and the effectiveness of generalized shrinking dimer dynamics based on phase field models.

Keywords

37M0549K3537N3034K2865P99Critical nucleusphase field modelssaddle pointshrinking dimer dynamicsheterogeneous nucleationprecipitate nucleationphase transformation

Information

Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 3 , September 2014 , pp. 781 - 798
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Cahn, J. and Hilliard, J., Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid, J. Chem. Phys., 31, 688–699, 1959.CrossRefGoogle Scholar
[2]Schlegel, H., Exploring potential energy surfaces for chemical reactions: an overview of some practical methods, J. Comput. Chem, 24,1514–1527, 2003.Google Scholar
[3]Wales, D., Energy landscapes: calculating pathways and rates, International Reviews in Physical Chemistry, 25, 237–282, 2006.Google Scholar
[4]Zhang, L., Chen, L.-Q. and Du, Q., Morphology of critical nuclei in solid state phase transformations, Phys. Rev. Lett., 98,265703, 2007.Google Scholar
[5]Cheng, X.-Y., Lin, L., E, W., Zhang, P.-W., and Shi, A.-C., Nucleation of Ordered Phases in Block Copolymers, Phys. Rev. Lett., 104,148301, 2010.Google Scholar
[6]Zhang, L., Chen, L.-Q. and Du, Q., Simultaneous prediction of morphologies of a critical nucleus and an equilibrium precipitate in solids, Comm. Comp. Phys., 7,674–682, 2010.Google Scholar
[7]Henkelman, G. and Jonsson, H., Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points, J. Chem. Phys., 113,9978, 2000.Google Scholar
[8]RenW. E, W. W. E, W. and Vanden-Eijnden, E., String method for the study of rare events, Phys. Rev. B, 66,052301, 2002.Google Scholar
[9]RenW. E, W. W. E, W. and Vanden-Eijnden, E, Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys., 126,164103, 2007.Google Scholar
[10]Crippen, G. and Scheraga, H., Minimization of polypeptide energy XI. The method of gentlest ascent, Archives of biochemistry and biophysics, 144,462–466, 1971.Google Scholar
[11]E, W. and Zhou, X., The gentlest ascent dynamics, Nonlinearity, 24,1831–1842, 2011.Google Scholar
[12]Henkelman, G. and Jonsson, H., A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives, J. Chem. Phys., 111,7010, 1999.Google Scholar
[13]Zhang, J.Y. and Du, Q., Shrinking dimer dynamics and its applications to saddle point search, SIAM J. Numer. Anal., 50 (2012), 1899–1921.Google Scholar
[14]Henkelman, G., Johannesson, G. and Jonsson, H., Methods for finding saddle points and minimum energy paths. In Progress in Theoretical Chemistry and Physics, ed. by Lipscomb, W., Prigogine, I. and Schwartz, S., Springer, Dordrecht, 269–302, 2002.Google Scholar
[15]Du, Q. and Zhang, L., A constrained string method and its numerical analysis, Comm. Math. Sci., 7,1039–1051, 2009.Google Scholar
[16]Zhang, J.Y. and Du, Q., Constrained shrinking dimer dynamics for saddle point search with constraints, J. Computational Phys., 231,4745–4758, 2012.Google Scholar
[17]Zhang, L., Chen, L.-Q., and Du, Q., Diffuse-interface description of strain-dominated morphology of critical nuclei in phase transformations, Acta Materialia, 56,3568–3576, 2008.Google Scholar
[18]Zhang, L., Chen, L.Q., and Du, Q., Mathematical and Numerical Aspects of Phase-field Approach to Critical Morphology in Solids, Journal of Scientific Computing, 37,89–102, 2008.Google Scholar
[19]Zhang, L., Chen, L.-Q. and Du, Q., Diffuse-Interface Approach to Predicting Morphologies of Critical Nucleus and Equilibrium Structure for Cubic to Tetragonal Transformations, J. of Comp. Phys., 229,65746584, 2010.CrossRefGoogle Scholar
[20]Heo, T., Zhang, L., Du, Q. and Chen, L.-Q., Incorporating diffuse-interface nuclei in phase-field simulations, Scripta Mater., 63,8–11, 2010.Google Scholar
[21]Backofen, R. and Voigt, A., A phase-field-crystal approach to critical nuclei, J. Phys.: Condens. Matter, 22,364104, 2010Google Scholar
[22]Castro, M., Phase-field approach to heterogeneous nucleation, Phys. Rev. B, 67,035412, 2003.CrossRefGoogle Scholar
[23]Granasy, L., Pusztai, T., Saylor, D. and Warren, J., Phase Field Theory of Heterogeneous Crystal Nucleation, Phys. Rev. Lett., 98,035703, 2007.CrossRefGoogle Scholar
[24]Khachaturyan, A.G., Theory of Structural Transformations in Solids, Wiley, New York, 1983.Google Scholar
[25]He, Y., Liu, Y., and Tang, T., On large time-stepping methods for the CahnHilliard equation, Applied Numerical Mathematics, 57, 616–628, (2007).Google Scholar
[26]Shen, J. and Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations, Discrete and Continuous Dynamical Systems, A, 28,1669–1691, 2010.Google Scholar
[27]Hu, S. Y. and Chen, L. Q., Solute Segregation and Coherent Nucleation and Growth near a Dislocation - a Phase-Field Model Integrating Defect and Phase Microstructures, Acta Materialia, 49(3), 463–472,2001Google Scholar
[28]Heo, T.-W., Phase-field Modeling of Microstructure Evolution in Elastically Inhomogeneous Polycrystalline Materials. PhD diss., The Pennsylvania State University, 2012.Google Scholar
[29]Yu, P., Hu, S.Y., Chen, L.Q., and Du, Q., An iterative perturbation schemes for treating inhomo- geneous elasticity in phase field models, with J. Computational Physics, 208, 34–50, 2005Google Scholar
[30]Grobner, P. J., The 885 F (475 C) Embrittlement of Ferritic Stainless Steels, Metall. Trans., 4,251–260, 1973.Google Scholar
[31]A., Caro, M., Caro, P., Klaver, B., Sadigh, E. M., Lopasso, S. G., Srinivasan, The computational modeling of alloys at the atomic scale: From ab initio and thermodynamics to radiation-induced heterogeneous precipitation. JOM, 59(4), 52–57, 2007.Google Scholar
[32]Li, Yulan, Hu, Shenyang, Zhang, Lei, Sun, Xin, Non-classical nuclei and growth kinetics of Cr precipitates in FeCr alloys during aging, Modelling Simul. Mater. Sci. Eng., 22,025002, 2014.Google Scholar
[33]Bronsard, L., and Stoth, B., Volume-preserving mean curvature flow as a limit of a nonlocal Ginzburg-Landau equation, SIAM Journal on Mathematical Analysis, 28, 769–807, 1997.CrossRefGoogle Scholar