Numerical Simulation Of Dendritic Growth In Pure Material's Solidification Process

The dynamics and morphology of the growing dendrites have a direct impact on the bulk material properties of the final solidified material or product, such as mechanical, electrical, and chemical properties. So it's very important to understand and simulate the dendritic growth process. Unfortunately, because it is belonged to the moving boundary problem and related to the nonlinear system, the simulation of a dendrite is an extremely complicated problem. As a starting point, a better understanding to the growth of a single free dendrite is an important step towards a more complete understanding of the characteristics of dendritic solidification. This is the primary concern of this paper.This paper begins with an overview introduction to physical backgrounds, theoretic progresses and experimental method related to the dendritic growth. Among them, three mostly used methods are specifically discussed, they are sharp-interface model, diffusive-interface model and stochastic model. Been an important factor for the dendritic growth, the anisotrppy is also introduced.Take advantages of the sharp-interface model and the diffusive interface model, the simulation work of this paper is focused on pure materials. Including the undercooling, latent heat and anisotropy, the complicated dendritic microstructure is simulated by the finite difference method. The result is discussed according to the curvature, solidified fraction and tip velocity.Combined with the following modified Stefan equations, a front-tracking method is developed for the implementation of sharp-interface model:According to above model, the interface is not of finite thickness but is tracked as a discontinuity and explicitly tracked. Lagrange polynomial is built to calculate the curvature, normal and tangent. With function similar to Peskin's method, the interface heat source is solved and redistributed to the grid. Harmonic averaging method is used to deal with the discontinuity in the vicinity of the interface. An indicator matrix is constructed to represent the phase discontinuity.A finite difference algorithm is applied to conduct the numerical simulation. Simulation results can reproduce experimentally observed free dendrite, tip splitting as well as side branch.About the diffusive interface model, the following phase-field model is developed by entropy function in a thermodynamically consistent way:As the theoretical background of the phase-field model, the physical meaning of the order parameter is discussed, combined with some concepts of condensed physics. In order to relate model parameters with physical properties of material, asymptotic analysis is conducted. The result shows that as the interface width approach to zero, the phase-field model will converge to the modified Stefan equations. Take advantages of the Matlab, A finite-difference algorithm is developed to simulate the dendritic solidification process.A conclusion remark is given out at the end of the paper; further research directions are also discussed.