Multiscale dendritic needle network model of alloy solidification

Abstract We present a novel dendritic needle network (DNN) model for simulating quantitatively the solidification of dendritic alloys. This approach is intended to reliably bridge the gap between phase-field simulations on the scale of dendrite tip radius ρ and cellular-automaton simulations on the several orders of magnitude larger scale of an entire dendritic grain. In the DNN model, the dendritic network of primary, secondary and higher other branches is represented by a network of sharp needles that interact through the solutal diffusion field. The tip velocity V of each needle is determined by combining a standard solvability condition that fixes the product ρ2V and an additional solutal flux balance condition that fixes the product ρ V 2 ∼ F 2 , where F measures the intensity of the solutal flux in the dendrite tip region. This solutal flux intensity factor F can be accurately computed by contour integral methods commonly used to compute stress intensity factors in fracture mechanics. This formulation provides an asymptotically exact description of the dendritic network dynamics in the limit of small Peclet number ρV/D ≪ 1, where D is the solute diffusivity. The DNN model is developed and implemented for both isothermal and directional solidification in two dimensions and is validated by comparison with analytical solutions for both early-stage and steady-state equiaxed growth as well as phase-field simulations. The latter comparison shows that DNN simulations are roughly four orders of magnitude faster than phase-field simulations while remaining reasonably accurate. DNN simulations of directional solidification demonstrate that the approach can be used to efficiently investigate the stability and dynamics of spatially extended dendritic arrays. This is illustrated for an Al–7 wt.% Si alloy by computing the stable range of primary array spacing and the history-dependent dynamic selection of this spacing following an abrupt change of solidification rate or of sample cross-section. We also compare DNN model predictions to microgravity experiments for the same alloy.