Mesoscopic modeling of spacing and grain selection in columnar dendritic solidification: Envelope versus phase-field model

Abstract

We investigate and assess the capability of the mesoscopic envelope model of dendritic solidification to represent the growth of columnar dendritic structures. This is done by quantitative comparisons to phase-field simulations in two dimensions. While the phase-field model resolves the detailed growth morphology at the microscale, the mesoscopic envelope model describes a dendritic grain by its envelope. The envelope growth velocities are calculated by an analytical dendrite-tip model and matched to the numerical solution of the solute concentration field in the vicinity of the envelope. The simplified representation of the dendrites drastically reduces the calculation time compared to phase field. Larger ensembles of grains can therefore be simulated. We show that the mesoscopic envelope model accurately reproduces the evolution of the primary branch structure, the undercooling of the dendrite tips, and the solidification path in the columnar mushy zone. We further show that it can also correctly describe the transient adjustments of primary spacing, both by spacing increase due to elimination of primary branches and by spacing reduction due to tertiary rebranching. Elimination and tertiary rebranching are also critical phenomena for the evolution of grain boundaries between columnar grains that have a different crystallographic orientation with respect to the temperature gradient. We show that the mesoscopic model can reproduce the macroscopic evolution of such grain boundaries for small and moderate misorientation angles, i.e., up to 30 •. It is therefore suitable for predicting the texture of polycrystalline columnar structures. We also provide guidelines for the calibration of the main parameters of the mesoscopic model, required to obtain reliable predictions.

Quantitative 3D mesoscopic modeling of grain interactions during equiaxed dendritic solidification in a thin sample

“…For our simulations we use the in-house computer code CrystalFOAM, which employs the finite-volume library OpenFOAM [27]. The complete description of CrystalFOAM is available in [15,16].…”

“…Several mesoscopic models exist, such as the dendrite needle network (DNN) model, already applied for equiaxed [8] and columnar dendritic grain growth [9], the cellular-automaton finite-element (CAFE) model [10,11] and the dendrite envelope model. In the envelope model, the complex dendritic structure of a crystal grain is described by means of an envelope, a smooth surface that wraps the tips of the actively growing branches of the dendrite [12,13,14,15,16]. The time-evolution of the envelope is computed by means of an analytical dendrite tip growth model coupled to the numerical solution of the solute concentration in the liquid surrounding the grain.…”

“…The time-evolution of the envelope is computed by means of an analytical dendrite tip growth model coupled to the numerical solution of the solute concentration in the liquid surrounding the grain. This method can accurately predict the morphology of the actual dendritic grains for both columnar [14,17,16] and equiaxed morphologies [12,13,15]. The mesoscopic envelope model can be used to simulate solidification phenomena for which interactions at the scale of a group of grains (up to 100 grains) are important, such as grain competition in columnar growth [16], or interactions in equiaxed growth [15].…”

Simulation and Experimental Studies on Grain Selection and Structure Design of the Spiral Selector for Casting Single Crystal Ni-Based Superalloy

“…They proposed that the coupling effect of the heat flow direction, the preferred growth direction and the geometrical restriction of the spiral wall makes a contribution to the crystal selection in the spiral passage. There are other simulation methods used to model competitive grain growth, such as envelope method and phase-field model [ 23 ], which still attract researchers’ attentions.…”

Evolution of a mushy zone in a static temperature gradient using a volume average approach

“…An additional contribution that lowers the solid fraction is the segregation of solute that transports solute by diffusion through the mushy zone towards the tips. In steady-state directional solidification, this segregation results in an increased average solute concentration (C > C 0 ) behind the tips and thus delays solidification [21,2]. The difference near the base of the mushy zone is due to the fact that the phase field simulation includes diffusion in the solid, while the Scheil equation does not [2].…”