Growth competition between columnar dendritic grains – The role of microstructural length scales

Isotropic finite-difference approximations for phase-field simulations of polycrystalline alloy solidification

“…The grain competition at a relatively small was studied by 2D PF simulations with the conventional finite-difference scheme  1,0 [7,16]. Although  1,0 is generally sufficient for those studies, it becomes inadequate for quantitatively modeling grain competition in the well-developed dendritic regime at a relatively large [17]. In order to demonstrate the effects of finite-difference schemes in the latter scenario, we simply consider the PF simulations of directional solidification of a single crystal with different 0 .…”

“…To date, quantitative phase-field simulations of polycrystalline alloy solidification [7][8][9]16] have been typically carried out using standard finite-difference schemes where, with the exception of Laplacian terms, the discretization of spatial derivative terms are not rotationally invariant at order ℎ 2 of the lattice spacing ℎ. However, in a recent study of the growth competition of columnar grains focusing on a fully developed dendritic regime [17], where the entire dendrite tip region grows under isothermal conditions, we found that those schemes are insufficient to accurately resolve the dendrite growth orientation and the dendrite tip operating state for grains that have a large misorientation with respect to the principal axes of the lattice. The lattice anisotropy causes a significant deviation of the dendrite growth orientation from the principal crystal axes and also affects the tip selection parameter * = 2 * 0 ∕ 2 determined by microscopic solvability theory [12][13][14] (where and are the tip velocity and radius, respectively, is the solute diffusivity, and * 0 is the capillary length).…”

“…This new scheme was used to carry out the study of grain competition reported in Ref. [17]. The present paper provides a detailed exposition of the theoretical development of this scheme and numerical tests for isothermal and directional solidification in 2D and 3D.…”

“…Those include the formulation employed to study grain competition in Ref. [17], which makes use of an indicator field [7][8][9]16] to distinguish different grains but does not describe grain-grain interactions that occur at the late stages of solidification. They also include various formulations that use an orientation field [7,[20][21][22][23][24][25], multiple order parameters [26][27][28][29][30][31], or a vector field [32] to model those interactions.…”

“…Since grain boundaries tend to be more anisotropic than the solid-interface energy, it may not be necessary to use higher-order rotationally-invariant discretizations to quantitatively resolve their behavior. The present paper focuses primarily on suppressing the effect of the lattice anisotropy on the solidification dynamics of individual grains of arbitrary orientation for modeling the formation of well-developed dendritic alloy microstructures [17]. Grain coalescence at the late stages of solidification is not considered.…”