Phase-field crystal modeling of lamellar precipitation reactions

In this paper, we employ the phase-field crystal model to examine the atomistic and diffusive behavior of lamellar precipitate colonies in binary alloys. First, we briefly review steady-state growth theories for lamellar precipitation, including the generic velocity-spacing relationships predicted for surface and volume diffusion. We next detail a novel formulation of the binary alloy structural phase-field crystal model. This includes new forms for the compositional dependence of the two-point density correlation function, which allows for robust control over the free energy and the introduction of several intermediate solid phases. We also augment the dynamics to allow for spatial variations in the mobility, allowing for control over the enhanced diffusion that occurs near grain boundaries. We perform two-dimensional numerical simulations using our model to examine lamellar precipitate growth, demonstrating agreement with steady-state theory in a time-averaged sense. We also interpret our results from a dynamic (non-steady-state) perspective by considering the roles of volume versus surface diffusion of the solute. For the case of rapid volume diffusion, we observe a large oscillatory lamellar colony velocity caused by regular, sequential buildup of coherency strain energy ahead of the grain boundary and consequent relaxation through dislocation nucleation. Conversely, for the case of surface-dominated diffusion, velocity fluctuations are more stochastic as they do not involve significant buildup of coherency strains. We end by discussing our results in the context of steady-state spacing theories and motivate the experimental relevance of our findings.