First Principles Modelling of Shape Memory Alloys pp 151-163 | Cite as

# Lattice Transformations in 3D Crystals

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## Abstract

We discuss two simulation examples concerning transformations in 3D lattices, intended as appendix to the investigations of the 2D case presented in the previous chapter. The two examples concern a 3D Lennard–Jones material and an EAM model for zirconium. We follow this work’s paradigm of not artificially constraining the model’s surfaces in order to allow the transformation to freely evolve; including the resulting surface effects. Compared to the 2D case, a 3D model geometrically augments the impact of these surfaces: Assuming square and cubic assemblies in 2D and 3D with \(N_{2\mathrm{ D} }\) and \(N_{3\mathrm{ D} }\) atoms respectively, the fraction of surface atoms is approximately \(4\,\sqrt{N_{2\mathrm{ D} }}/N_{2\mathrm{ D} }\) in the 2D and \(6\,\root 3 \of {N_{3\mathrm{ D} }}^2/N_{3\mathrm{ D} }\) in the 3D case. These fractions are equal for \(N_{3\mathrm{ D} }={27}/{8}N_{2\mathrm{ D} }^{{3}/{2}}\). Our 2D studies have revealed that with model sizes of \(10^5\) atoms, surfaces may influence the nucleation, but not the microstructure’s formation in the bulk material. Using the above calculation, this figure translates into a 3D model size with \(10^8\) atoms. A five million 3D crystal therefore is “small” in regard to the surface influence, while in 2D, a quarter million atom crystal is “large”. Computational resources limit our 3D models to sizes of a few million atoms, although the impact of the surface is still tangible with such sizes.

## Keywords

Martensite Variant Transformation Zone Twin Interface Corner Atom Potential Energy Field## References

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