Mathematical modeling of the directed growth of dendrites of Ni-20% al in non-isotherhermal conditions

The paper presents a phase-field model of solidification that describes the kinetics of dendrite growth under non-isothermal conditions on the basis of the Gibbs' potentials of the liquid phase and the FCC of phase Ni-Al with the molar concentration of aluminum less than 20 %. For the purposes of simplicity, the mutual influence of dendrites on each other is taken into account only through redistribution of the solute in liquid phase. Dynamics of an solute in each phase is described by a separate diffusion equation. Each dendrite is mapped to a separate phase field. The set of equation consistently with Gibbs potentials for Ni-20%Al was obtained and solved for the 3d-problem of directed growth which is initiated by temperature gradient. The growth of the dendrite from the spherical nucleus under initial undercooling 80 $^{\circ}$С was obtained. Grains of the growing nucleus are oriented according to the grid, primary and secondary dendrite arms are clearly seen. Secondary dendrite arm spacing is 0.4 mkm. The resulting fields of concentrations in liquid and solid phases and temperature were obtained. Concentration in liquid phase differs from the concentration in solid phase, and according to the simulation results it is shown that the segregation goes into the liquid. Also the growth of 5 dendrites is presented. Initial undercooling was 120 $^{\circ}$С, needle-shaped dendrites were obtained using that condition, it was observed that grains of the growing nucleus are undirected. The growth direction of the general arm is oriented randomly upwards, and the initial temperature gradient is directed up along the z-axis, that forces dendrites to choose main growth direction along the same line. The initial position of the embryo and its orientation are determined at random. The direction of solidification is specified with an external temperature gradient. The effect of orientation of dendrites on the growth rate is determined by anisotropy of coefficients of the model. As result of simulation of the received equations on the calculation grid 200$\times$200$\times$200 cells simultaneous growth of several dendrites was studied what allows to compare qualitatively the present model with earlier made calculations.