# Influence of coherency strain on the stability of carbide precipitates in bcc metals

_{x}and predict the critical thickness of the carbide as shown in the figure below. This carbide is reported in an experimental study [1] as a metastable precipitate forming a semi-coherent interface at the grain boundary. The precipitation growth can be coherent, semi-coherent or incoherent with the original matrix of the metal. For modelling microstructure evolution or growth kinetics, the critical thickness of the precipitate before the interface undergoes a transition from coherent to semi-coherent state is required. In this project, we use an interface model to predict this transition that is completely based on the results of density functional theory calculations. Our approach starts from a coherent interface that is modelled in a standard supercell approach. A computational study [2] showed that there is a significant strain contribution due to the lattice misfit to the interface energy.

The aim of this work is to quantify this elastic and chemical contributions to the total energy of the system including the interface. This elastic energy as a function of strain is determined from single-crystal calculations, which allow us to decompose the total energy into a cohesive bulk energy, the strain energy, and the chemical binding energy at the interface. The strain energy that makes the interface coherent is size dependent, i.e. it increases with the thickness of the precipitate until the strain is released to form misfit dislocations. The challenge is to have a realistic description of a semi-coherent interface, which is usually out of the scope of DFT calculations. In our work, we use the so-called square-lattice-model [3]} as shown in Fig. 2 (see further information) for the computation of the increase in interface energy due to the lattice misfit dislocations. The energy distribution in this model is obtained from γ-surface calculations (Fig. 3, see further information), which in addition yield possible Burgers vectors for the misfit dislocations. The interface model presented here allows for a variation of dislocation spacing as well as dislocation core width. Taking these parameters into account, we predict a critical thickness of one to two unit cells of the carbide phase, which is in good agreement with experimental observations.

[1] J. M. Penisson, M. Bacia, and M. Biscondi. Philos. Mag., A(73):859, 1996.

[2] R. Janisch and C. Elsässer. Phys. Rev. B, 77:094118, 2008.

[3] J. M. Albina, M.Mrovec, B.Meyer, and C. Elsässer. volume 18-22 of
3rd International Conference Multiscale Materials Modeling, pages 819-822.
Fraunhofer IRB Verlag Stuttgart, 2006.