Three-dimensional dendritic needle network model for alloy solidification

Abstract

We present a three-dimensional (3D) formulation of the multiscale Dendritic Needle Network (DNN) model for dendritic microstructure growth. This approach is aimed at simulating quantitatively the solidification dynamics of complex hierarchical networks in spatially extended dendritic arrays, hence bridging the scale gap between phase-field simulations at the scale of a few dendrites and coarse-grained simulations on the larger scale of entire polycrystalline structures. In the DNN model, the dendritic network is represented by a network of branches that interact through the solutal diffusion field. The tip velocity V(t) and tip radius ρ(t) of each needle is determined by combining a standard solvability condition that fixes the product ρ2V and a solute flux conservation condition that fixes the product ρV2 in 2D and ρV in 3D as a function of a solutal flux intensity factor F(t). The latter measures the intensity of the solute flux in the dendrite tip region and can be calculated by contour (2D) or surface (3D) integration around the tip of each needle. We first present an extended formulation of the 2D DNN model where needles have a finite thickness and parabolic tips. This formulation remains valid for a larger range of tip Péclet number than the original thin needle formulation and is readily extended to 3D needles with paraboloidal tips. The 3D DNN model based on this thick-needle formulation is developed for both isothermal and directional solidification. Model predictions are validated by comparisons with known analytical solutions that describe the early transient and steady-state growth regimes. We exploit the power of the DNN model to characterize the competitive growth of well-developed secondary branches in 3D on the scale of the diffusion length. The results show that the length of active secondary branches increases as a power law of distance behind the tip with an exponent in good quantitative agreement with experimental measurements. Finally, we apply the model to simulate the three-dimensional directional solidification of an Al-7wt% Si alloy, which we directly compare to observed microstructures from microgravity experiments onboard the International Space Station. The predictions of selected microstructural features, such as dendrite arm spacings, show a good agreement with experiments. The computationally-efficient DNN model opens new avenues for investigating the dynamics of large dendritic arrays at length and time scales relevant to solidification experiments and processes.