A quantitative dendrite growth model and analysis of stability concepts

Abstract

While a number of cellular automaton (CA) based models for dendrite growth have been proposed, none so far have been validated, casting doubt on their quantitative capabilities. All these models are mesh dependent and cannot correctly describe the influence of crystallographic orientation on growth morphology. In this work, we present an improved version of our previously developed CA based model for dendrite growth controlled by solutal effects in the low Péclet number regime. The model solves the solute and heat conservation equations subject to the boundary conditions at the interface, which is tracked with a new virtual front tracking method. It contains an expression equivalent to the stability constant required in analytical models, termed stability parameter, which is not a constant. The process determines its value, changing with time and angular position during dendrite formation. The article proposes solutions for the evaluation of local curvature, solid fraction, trapping rules, and anisotropy of the mesh, which eliminates the mesh dependency of calculations. Several tests were performed to demonstrate the mesh independence of the calculations using Fe-0.6 wt pct C and Al-4 wt pct Cu alloys. Computation results were validated in three ways. First, the simulated secondary dendrite arm spacing (SDAS) was compared with literature values for an Al-4.5 wt pct Cu alloy. Second, the predictions of the classic Lipton-Glicksman-Kurz (LGK) analytical model for steady-state tip variables, such as velocity, radius, and composition, were compared with simulated values as a function of melt undercooling for Al-4 wt pct Cu alloy. In this validation, it was found that the stability parameter approaches the experimentally and theoretically determined value of 0.02 of the stability constant. Finally, simulated results for succinonitrile-0.29 wt pct acetone (SCN-0.29 wt pct Ac) alloy are compared with experimental data. Model calculations were found to be in very good agreement with both the analytical model and the experimental data. The model is used to simulate equiaxed and columnar growth of Fe-0.6 wt pct C and Al-4 wt pct Cu alloys offering insight into microstructure formation under these conditions.

Simulation of dendritic growth of magnesium alloys with fluid flow

“…In the present paper, based on the CA method, a growth model was established in which the growth kinetics were calculated from the complete solution of the transport equations. By defining a special neighbourhood configuration with the square CA cell, and using a set of capturing rules which was proposed by BeltranSanchez and Stefanescu for the dendritic growth of cubic crystal metals during solidification [10] , modeling of dendritic growth of magnesium alloy with six-fold symmetry and different growth orientations was achieved. More details relating calculation of the growth kinetics (including the growth velocity, solid fraction increment and mean curvature of the interface cells, etc.)…”

“…Among those numerical modeling techniques, the CA method and PF (phase field) method have been extensively used as they both have the capability of predicting and depicting the evolution of complex solidification structures. For the dendritic growth of cubic metals, numerous simulation works have been reported [7][8][9][10][11] , whereas few have been done for the dendritic growth of magnesium alloys with hcp crystal structure. Using the PF method, Böttger et al [12,13] and Wang et al [14] successfully simulated the dendritic growth of magnesium alloys, reproducing most of the dendritic features observed experimentally.…”

Development of a CA-FVM Model with Weakened Mesh Anisotropy and Application to Fe–C Alloy

“…Therefore, the half length of the square diagonal Ldia is updated as follows [9,35,36]: The magnitude of V n is determined as follows [7]:…”

“…Therefore, before the dendritic growth reaches the steady state at ∆T = 5 K, it will be influenced by boundary conditions. Beltran-Sanchez and Stefanescu [7] proposed that the steady state can be determined as the concentration at the boundary contrary to the dendritic tip reaches 1.01 times the initial value. So, the domain size is adjusted according to the melt undercooling in the present work.…”

“…So, the domain size is adjusted according to the melt undercooling in the present work. Steady states at melt undercoolings of 5 K (n = 701) and 8 K (n = 301) are properly determined according to Beltran-Sanchez and Stefanescu [7], as shown in Figure 2. Figure 3 shows steady tip growth velocities of equiaxed dendrites predicted by CA-FVM model and the comparison with analytical results.…”