Time-nonlocal internal-variable LB-PF model for continuous nucleation during dendritic solidification

Continuous nucleation in nonequilibrium solidification is often treated as an instantaneous response controlled by the current undercooling. In welding, casting, and layer-wise metal additive manufacturing, however, liquid regions can undergo nonmonotonic thermal histories and reach similar undercooling states with different nucleation readiness. Here, an internal activation variable is incorporated into a continuous nucleation description and embedded in an LB-PF solver for dendritic solidification. The variable acts as an effective continuum state for the accumulation, retention, and weakening of prenucleation readiness. Constant cooling, piecewise cooling, low-temperature pretreatment, and high-temperature holding are used to examine the resulting path-dependent nucleation response. The simulations show that different thermal paths lead to different activation states, accepted-nucleus distributions, solid-fraction evolutions, and grain-scale features even under similar later undercooling conditions. The results suggest that time-nonlocal internal variables can help represent thermal-history-dependent nucleation in continuum phase-transformation simulations.