Commun. Comput. Phys., 16 (2014), pp. 781-798.


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

Lei Zhang 1, Jingyan Zhang 2, Qiang Du 3*

1 Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China.
2 Department of Mathematics, Pennsylvania State University, PA 16802, USA.
3 Department of Mathematics, Pennsylvania State University, PA 16802, USA; Beijing Computational Science Research Center, Beijing, 100084, China.

Received 25 September 2013; Accepted (in revised version) 24 March 2014
Available online 4 July 2014
doi:10.4208/cicp.250913.240314a

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.

AMS subject classifications: 37M05, 49K35, 37N30, 34K28, 65P99

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Key words: Critical nucleus, phase field models, saddle point, shrinking dimer dynamics, heterogeneous nucleation, precipitate nucleation, phase transformation.

*Corresponding author.
Email: zhangl@math.pku.edu.cn (L. Zhang), j_zhang@math.psu.edu (J. Y. Zhang), qdu@math.psu.edu (Q. Du)
 

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