Lattice Transformations in 3D Crystals

  • Oliver Kastner
Part of the Springer Series in Materials Science book series (SSMATERIALS, volume 163)


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.


Martensite Variant Transformation Zone Twin Interface Corner Atom Potential Energy Field 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Oliver Kastner
    • 1
  1. 1.Faculty of Mechanical Engineering, Institute for MaterialsRuhr University BochumBochumGermany

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