Correlation between three-dimensional and cross-sectional characteristics of ideal grain growth: large-scale phase-field simulation study
Grain growth is one of the most fundamental phenomena affecting the microstructure of polycrystalline materials. In experimental studies, three-dimensional (3D) grain growth is usually investigated by examining two-dimensional (2D) cross sections. However, the extent to which the 3D microstructural characteristics can be obtained from cross-sectional observations remains unclear. Additionally, there is some disagreement as to whether a cross-sectional view of 3D grain growth can be fully approximated by 2D growth. In this study, by employing the multi-phase-field method and parallel graphics processing unit computing on a supercomputer, we perform large-scale simulations of 3D and 2D ideal grain growth with approximately three million initial grains. This computational scale supports the detailed comparison of 3D, cross-sectional, and 2D grain structures with good statistical reliability. Our simulations reveal that grain growth behavior in a cross section is very different from those in 3D and fully 2D spaces, in terms of the average and distribution of the grain sizes, as well as the growth kinetics of individual grains. On the other hand, we find that the average grain size in 3D can be estimated as being around 1.2 times that observed in a cross section, which is in good agreement with classical theory in the stereology. Furthermore, based on the Saltykov–Schwartz method, we propose a predictive model that can estimate the 3D grain size distribution from the cross-sectional size distribution.
This research was supported by Grant-in-Aid for Scientific Research (B) (No. 16H04490) and for JSPS Fellows (No. 17J06356) from the Japan Society for the Promotion of Science (JSPS), the Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures, and the High Performance Computing Infrastructure in Japan (Project ID: jh170018-NAH), and MEXT as a social and scientific priority issue (Creation of new functional devices and high-performance materials to support next-generation industries) to be tackled using the post-K computer.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
- 1.Humphreys FJ, Hatherly M (2004) Recrystallisation and related annealing phenomena, 2nd edn. Elsevier Ltd., OxfordGoogle Scholar
- 15.Doherty RD (1984) Stability of the grain structure in metals. J Mater Educ 6:845–883Google Scholar
- 26.Suwa Y (2013) Phase-field simulation of grain growth. Nippon Steel Tech Rep 102:19–24Google Scholar
- 46.Shimokawabe T, Takaki T, Endo T, Yamanaka A, Maruyama N, Aoki T, Nukada A, Matsuoka S (2011) Peta-scale phase-field simulation for dendritic solidification on the TSUBAME 2.0 supercomputer. In: Proceedings of 2011 international conference for high performance computing, networking, storage and analysis. ACM, Seattle, pp 1–11Google Scholar
- 68.von Neumann J (1952) Discussion—shape of metal grains. In: Herring C (ed) Metal Interfaces. American Society for Metals, Cleveland, pp 108–110Google Scholar
- 70.Fullman RL (1953) Measurement of particle sizes in opaque bodies. Trans AIME 197:447–452Google Scholar
- 72.Schwartz HA (1934) The metallographic determination of the size distribution of temper carbon nodules. Met Alloy 5:139–140Google Scholar
- 78.Hull FC, Houk WJ (1953) Statistical grain structure studies: plane distribution curves of regular polyhedrons. Trans AIME 197:565–572Google Scholar
- 85.Pabst W, Treza U (2017) A generalized class of transformation matrices for the reconstruction of sphere size distributions from section circle size distributions. Ceram-Silik 61:147–157Google Scholar