Extending the Peierls-Nabarro model using machine learning

The Peierls-Nabarro (PN) model is a well known approximation that removes the singularity of the linear elastic stress field of a dislocation by modeling the dislocation core structure as a competition between elastic and surface energies. Despite its many limitations, the PN model provides valuable insight into the dislocation core structure and is significantly more efficient than a full atomistic simulation. A variety of enhancements of the PN model have been proposed in the past to improve upon the various assumptions inherent in the model. In particular, it is standard practice to use atomistically informed models using the gamma surface, introduced originally by Vitek. In this work, we revisit the assumptions involved in using the gamma surface and study non-local extensions of the surface energy on the structure of the dislocation core. We achieve this by constructing a non-local version of the gamma surface, using data from molecular dynamics simulations, by employing both kernel ridge and Gaussian process regression. The resulting enhanced surface energy is used in conjunction with the PN model to study the dislocation core structure. The dynamic changes to the core structure as the dislocation crosses a Peierls barrier are also analyzed using the machine-learned PN model. Comparison with fully atomistic calculations will be provided for special cases involving edge and screw dislocations in metallic and covalently bonded crystalline solids to evaluate the accuracy and efficiency of the proposed extension to the PN model. A comparison of the predictions of the proposed model with other well known regularization techniques to model the dislocation core singularity, and planned future extensions of this work will also be discussed.