摘要
Abstract
In this paper,we study the effects of anisotropic interface kinetics and surface tension on deep cellular crystal growth in directional solidification.The following assumptions are made:the process of solidification is viewed as a twodimensional problem;the minor species in this binary mixture system is considered as an impurity;the solute diffusion in the solid phase is negligible;the thermodynamic properties other than the diffusivities are the same for both solid and liquid phases;there is no convection in the system;the anisotropic interface kinetics and the anisotropic surface tension are a four-fold symmetry function each;neither the preferred directions of the anisotropic interface kinetics nor the anisotropic surface tensions are necessarily the same as their counterparts for the solid and liquid phases respectively;the angle between the preferred directions of the two anisotropies is θ0.By using the matched asymptotic expansion method and the multiple variable expansion method,we obtain the diagram of interface morphology for a deep cellular crystal in directional solidification.The results show that there exists a discrete set of the steady-state solutions subject to the quantization condition (35).The quantization condition yields the eigenvalue ξ* as a function of parameter ε and other parameters of the system,which determines the interface morphology of the cell.The results also show the variation of the minimum eigenvalue ξ*(0) with parameter ε.It is seen that when the preferred directions of the two anisotropies are the same,i.e.,θ0 =0,the minimum eigenvalue ξ*(0) reduces with the increase of anisotropic surface-tension coefficient α4,increases with the augment of parameter ε,and is unrelated to anisotropic interface kinetic coefficientβ4 in the low order;when the angle 0 ≤ θ0 ≤ π/4,as the θ0 increases,the minimum eigenvalue ξ*(0) increases;when the angle π/4 ≤ θ0 ≤ π/2,as the θo increases,the minimum eigenvalue ξ*(0) decreases.In addition,the results show the composite solution for the interface shape function ηB described on (X,Y) plane.It is seen that both of the anisotropy and the angle θ0 have a significant effect on the total length and the root of deep cellular crystal,however,have little influence on the other solid-liquid interface,such as the top of deep cellular crystal.When the angle θo is 0,as anisotropic coefficient increases,the total length of the finger increases,the curvature of the interface near the root increases or the curvature radius decreases.It is found that the influence of the anisotropic surface-tension coefficient on interface morphology is more remarkable than that of the anisotropic interface kinetics coefficient.when the angle 0 ≤ θ0 ≤ π/4,as the θ0 increases,the total length of the finger decreases,the curvature of the interface near the root decreases or the curvature radius increases;when the angle π/4 ≤ θ0 ≤ π/2,as θ0 increases,the total length of the finger increases,the curvature of the interface near the root increases or the curvature radius decreases.关键词
定向凝固/各向异性表面张力/各向异性界面动力学/深胞晶生长Key words
directional solidification/anisotropic surface tension/anisotropic interface kinetics/deep cellular crystal growth
