Phase-field modeling of diffusion controlled phase ... - KTH Mechanics

LOGINOVA et al.: PHASE-FIELD SIMULATIONS OF BINARY ALLOY SOLIDIFICATION577when a0. The steady-state growth velocity of thedendrite tip was chosen as a reliable quantity to definetime-step. It was found that with t0.2 the numericalscheme gives an error of the tip speed ca 3%.Solute redistribution obtained for isothermal calculationsis given in Fig. 1.4. TEMPERATURE FIXED ON THE BOUNDARYTo complete the model describing cooling of a meltwith growing nuclei, thermal boundary conditions areto be specified. Different types of boundary conditionsdetermine different thermal regimes affectingthe temperature variation in space. In this sectionFig. 1. Concentration field, isothermal model. Time is 2.75 ms.Fig. 2. Concentration field given by the non-isothermal modelwith temperature fixed on the boundary. All the physical andphase-field parameters are the same as in isothermal calculations.Time is 2.75 ms. The color scheme uses yellow–redpalette in grey scale for low and high concentration values,respectively.equations (14), (18) and (26) are solved withT1574K as initial and boundary conditions for theheat equation. This is done in order to fit the nonisothermaltemperature regime to the isothermal case.The results of this simulation are shown in Figs 2and 3.As one would expect, the temperature does notvary much, neither in time nor in space, for example,the spatial temperature difference does not exceed1.6K all the time (Fig. 4). However, comparison ofthe solute patterns (Figs 1 and 2) shows that the compositionbehaviour is very sensitive to even smallchanges in the temperature field. Increased melt temperaturereduces sources of instability, which leads toless developed structure of the non-isothermal dendrite.The length of the primary arms is about 6% lessthan the corresponding one calculated under isothermalconditions. The solute diffusion length islarger compared to the isothermal growth, as well asspacing of interdendritic liquid pockets. The value ofthe composition at the solid/liquid interface isdecreased since the operating point in the phase diagramhas moved to the left due to the increased temperature.Figure 3 demonstrates spatial redistribution of thetemperature field with imposed interface location attime 2.75 ms. As predicted by the Gibbs–Thomsoncondition, the tips are the coldest parts of the dendritedue to the large curvature here. The location of thehottest point during the crystal growth varies and ingeneral corresponds to one of the tips of those sidebrancheswhich grow towards each other and form aclosed liquid pocket. Melt in this pocket is of highesttemperature due to release of latent heat by thesegrowing sidebranches. The less sharp the sidebranchtip, the higher temperature of the sidebranch.The maximal value of the system temperature as afunction of time is given in Fig. 4. The curve isslightly oscillating because the temperature is takenat different grid points wherever the maximum valueoccurs. The local peaks occur when two or more sidebranchesgrowing towards each other merge and stopgrowing and consequently to produce the latent heat.Then another sidebranch surrounded by hot meltstarts to release more latent heat than the others andbecomes the hottest place in the system. Fig. 5 showsthe area of the dendrite tip with imposed isotherms,which are refracted over the interface. This reflectsthe change of the temperature gradient owing to therelease of latent heat.The results presented in this section indicate clearlythat for a supersaturated binary alloy the developmentof dendritic patterns is governed mainly by solute diffusion,but the temperature variation alters significantlythe morphology of the microstructure.5. RECALESCENCE CALCULATIONSThis section represents the results obtained forsimulations of dendritic solidification in the thermal