Quantitative phase-field modeling for two-phase solidification process involving diffusion in the solid

A quantitative phase-field model for two-phase solidification processes is developed based on the antitrapping current approach with the free energy functional formulated to suppress the formation of an extra phase at the interface. This model appropriately recovers the free boundary problem for the motion of interface in the thin-interface limit and, importantly, it is applicable to the solidification process in binary alloy systems with arbitrary values of the solid diffusivities and interfacial energies. The performance of the present model is investigated for the peritectic reaction process in carbon steel. The present model exhibits excellent convergence behavior with respect to the interface thickness

Diffusion-controlled peritectic reaction process in carbon steel analyzed by quantitative phase-field simulation

The pentectic reaction process in carbon steel, L + δ → γ, has been analyzed by a quantitative phase-field simulation. The calculated moving velocities of the γ-L and γ-δ planar interfaces in the isothermal peritectic transformation precisely agree with the corresponding experimental data, which strongly supports the accuracy of the present simulation. The diffusion-controlled peritectic reaction rate or the growing velocity of γ phase along the δ-L interface obtained by the present simulation is fairly consistent with the experimentally measured values. This fact indicates that the recent experimental findings can be explained by a diffusion-controlled mechanism. This is in marked contrast to the claims made on the basis of the experimental data and an analytical model that the peritectic reaction is not controlled by the diffusion of carbon

Finite Difference Simulation of Peritectic Transformation in Iron-Carbon Alloy

The process of peritectic transformation during cooling of iron-carbon alloys has been simulated by a numerical analysis based on the direct finite difference method. During the peritectic transformation, austenite is produced from δ-ferrite by precipitation and from liquid by crystallization. The amount of precipitated austenite is approximately four times as large as that of the crystallized one. Both the amounts of the precipitated austenite and the crystallized one increase when the carbon content of the alloy increases from 0.09 to 0.17 mass%, but they decrease when it increases from 0.17 to 0.53 mass%. The finishing temperature of the peritectic transformation falls with the increase in carbon content from 0.09 to 0.17 mass%, but it rises with the further increase in carbon content to 0.53 mass%. The increase in cooling rate accelerates the transformation rate and reduces the finishing temperature

Analysis of the effect of grain size distribution on grain growth by computer simulation

The geometrical features of 12 kinds of polyhedrons were investigated, and the results were applied to the relationship between the grain shape and the size. Taking account of grain size and shape distribution, an expression for grain growth was proposed. The expression was used to investigate the effect of the grain size distribution on the growth rate. It was found that the growth rate increased with increasing variation coefficient and with decreasing kurtosis. It was also found that growth exponent n was 2.2 during the steady growth

Quantitative phase-field modeling for dilute alloy solidification involving diffusion in the solid

An antitrapping current scheme for quantitative phase-field model A. Karma, Phys. Rev. Lett. 87, 115701 2001 is extended to solidification process in a dilute binary alloy system involving diffusion in the solid. It is demonstrated in an asymptotic analysis that in the case of an arbitrary value of the solid diffusivity, five types of constraints exist between interpolating functions used in the phase-field model, which need to be satisfied simultaneously to eliminate all anomalous interface effects. Then, the authors present an appropriate form of the antitrapping current term for the two-sided case to remove all the spurious effects. The convergence test of the output with respect to the interface thickness was carried out for the isothermal dendrite growth process, which demonstrates an excellent performance of the present model

Refinement of As-cast Austenite Microstructure in S45C Steel by Titanium Addition

The effect of Ti addition on as-cast austenite (γ) structure of S45C steel has been investigated. The as-cast γ structure without the Ti addition consists of coarse columnar grains. The Ti addition leads to formation of equiaxed γ-grains and also grain refinement of the γ structure. Fully equiaxed and very fine γ-grain structure forms in a limited range of Ti addition of 0.13 to 0.17 wt%. The thermodynamic calculation of phase diagram showed that Ti carbonitride crystallizes as a primary phase in this composition range. The formation of the very fine equiaxed γ-grain structure originates from the columnar-to-equiaxed transition (CET) in δ-dendrite solidification induced by the primary Ti carbonitride particles. These particles act as the nucleation sites of the equiaxed δ-dendrite. The experimental results are suggestive of existence of finer Ti carbonitride particles retarding the grain growth of γ-phase after the peritectic transformation, which leads to the refinement of the as-cast γ structure

Motion and Morphology of Triple Junction in Peritectic Reaction Analyzed by Quantitative Phase-field Model

Motion and morphology of triple junction during peritectic reaction process is analyzed for a model alloy system based on a quantitative-phase-field simulation for two-phase solidification involving diffusion in the solid. It is demonstrated that the dominative process controlling the motion of the reaction front gradually changes from the solid–solid transformation to the secondary solid solidification as the moving velocity of solid–solid interface decreases. On the other hand, the local shape of the triple junction is mainly determined by the balance between the interfacial energies regardless of the difference in the moving velocity of solid–solid interface

Estimation of Three-dimensional Grain Size Distribution in Polycrystalline Material

A new method has been proposed for the estimation of the three-dimensional grain size distribution from the two-dimensional distribution measured on the cross section of polycrystalline material. In this method, twelve types of polyhedra were employed as the grain models. The distributions of the cross-sectional diameters of the individual polyhedra were expressed as probability density functions. On the basis of the functions for each polyhedron, the supposed grain size distribution on the cross section of the material was calculated, and it was compared with the measured one. The comparison was repeated until the agreement between the both distributions. By operating this two-dimensional distribution reversely, the three-dimensional grain size distribution was estimated. The distribution of the vertex number of polygon-shaped grains on the cross section and that of the face number of polyhedron-shaped grains in three dimensions were calculated from the obtained results and were compared with the measured distributions. There was a good agreement between these distributions

Numerical Analysis on Columnar-to-equiaxed Transition of δ-Ferrite Dendrite in Carbon Steel Induced by Titanium Carbonitride Particles

Dendrite growth simulations have been performed to analyze Columnar-to-Equiaxed Transition (CET) of δ-ferrite dendrite structure triggered by fine particles of a primary Ti(C, N) crystal. The CET position estimated by Hunt's criterion and the present simulation indicated that the existence of a large number of the Ti(C, N) particles gives rise to the CET of the dendrite structure even in the vicinity of the mold wall under the present casting condition. Furthermore, the capability of the Ti(C, N) leading to the CET was discussed in the light of the different number of Ti(C, N) particles and the different thickness of cast ingot. It was shown that a sufficient number of Ti(C, N) leads to the formation of fully equiaxed dendrite structure irrespective of the ingot thickness