The purpose of the research presented is to contribute to simulate directional solidification of a dilute SCN/acetone alloy by using quantitative phase-field model.
To aim at the morphological transition near the onset of instability, our calculations show a lot of progress on both phase-field modeling and the understanding of solidification physics.
For simulations of lower concentration, c0=0.01mol% is considered. We use the WBM model [54] derived from thermodynamics and a thin interface thickness δ=0.2µm is adopted. Under higher pulling, which is comparable to the value of DL/δ, we further add anti-trapping current (ATC) [64] to suppress the effect of solute trapping. The phase-field modeling shows good agreement with both MS [5] and WL [7] theories for planar solidification. The time to complete a morphological transition by increasing the solidification speed is up to 5DL/kV2, which could take days in a real experiment. With the increasing pulling speed, the transitions from planar to λc↔λc/2 bifurcation, Vλ2~constant, and then multiplet regions are also clearly illustrated. The transition phenomena, such as tip splitting, cell coarsening, amplitude overshooting, solute pinching-off, and swing asymmetric cells are consistent with the long-time scale experimental observations of higher concentrations [87].
Because of thermodynamic nature of the WBM model, a quantitative modeling is failed when much thicker δ (~1µm) is used. To release computational load of long time and length scale for realistic cases, we use the phase-field model derived from geometrical description [58], while the technique of simple interface model (SIM) is proposed to maintain quantitative modeling. Because this approach concerns local equilibrium only, the range for pulling and interface thickness is up to two orders.
Furthermore, the local Gibbs-Thompson condition is exactly obtained even at high speeds, and the effect of solute trapping is totally eliminated.
Therefore, for simulations of higher concentration (c0=0.04~0.1mol%), this method is adopted to study morphological transition near the onset of instability and formation of deep cells. In terms of the first one, we verify critical conditions under
different concentrations by decreasing speeds. The simulated critical velocities show good agreement with ones predicted from the MS theory, while the shallow cellular morphologies are all in the λc/2 branch. The λc branch can be obtained by introducing an initial λc perturbation. We found that the range of this branch is very small and is almost covered by the hysteresis and the λc /2 one.
For the first time, the morphological transition from shallow cells to deep cells is simulated. By using the phase-field model proposed here, the variation of wavelength, amplitude, and local tip radius with increasing of the pulling is illustrated.
Three different regions can be identified. They are λc/2 shallow cells, smaller wavelength finite-depth cells, and deep cells. Interestingly, the wavelength increases with the velocity during the transition from smaller wavelength finite-depth cells to deep cells, which is also observed in many experiments [17,93]. Furthermore, the time evolution of a deep cellular pattern starting from a planar interface in directional solidification simulation is performed. Our results show that the crossover wavelength λ0 decreases with increasing of the pull speed, while the overall trend shows nice agreement with the WL theory [7].
The investigations in this work lay the fundamentals for much exciting and fruitful future research. Some suggestions are described below:
1. Since the quantitative phase-field modeling of alloy solidification under higher concentrations is performed, the one-to-one comparison with experiments is promising. By further using this model, many interesting topics (e.g. cells to dendrite (CTD)) of alloy solidification can be studied quantitatively in the future.
2. Eutectic two-phase cells are commonly observed during the solidification of ternary alloys when the composition is close to the eutectic valley. The solute trapping still plays an important role in the quantitative phase-field modeling [96]. The simple interface model proposed here may gives some new idea on this topic; the interfacial kinetic is possible to be canceled out for approaching sharp-interface simulation.
3. Recently, phase field simulation is widely formulated to study many material processes for industrial manufacturing. A non-thoroughgoing search of the literature showed applications as diverse as island formation during epitaxial growth [97], modeling of electrochemistry [98], and excimer laser annealing process in Silicon [99]. To simulate these processes with local interface and global crystallization, the numerical solution based on efficient AMR scheme is necessary. These multi-scale problems will continue to challenge modeling research and the simulation tools developed.
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