Analytical Study On The Stability Of Rod Eutectic Growth In Directional Solidification

The study of interfacial instability and mode formation in crystal growth is a hot spot and focus in a wide range of fields of nonlinear science.These phenomena occur in various dynamic systems far from equilibrium.In many practical and very important physical systems,the solid-liquid interface or the two-liquid interface always shows some thought-provoking growth patterns.The two archetypes of these phenomena are dendritic growth in solidification and sticky fingers in Hele-Shaw cells.These two phenomena occur in completely different scientific fields,but they are both described by using the nonlinear free boundary problem of similar partial differential equation systems;In both cases,the boundary condition on the interface contains a curvature operator involving surface tension,which is nonlinear.In addition,both of these cases raise the same challenging theoretical problems,such as the interface instability mechanism and the choice of mode,and it is found that these problems can be solved by the same analysis method.Therefore,these two phenomena are considered as special examples of a kind of nonlinear pattern formation phenomenon in nature,and they are outstanding topics in the new interdisciplinary field of nonlinear science.The stability and selectivity of eutectic growth during solidification are always very important frontier topics in the fields of materials science,condensed matter physics and nonlinear science.The breakthrough of relevant theories about revealing eutectic growth mechanism will promote the mutual penetration of various disciplines and open up broad prospects for the preparation and development of eutectic and other in-situ composite materials.In the growth system of eutectic,rod-like eutectic has many morphological characteristics of stable growth（cylindrical,oval,peanut,etc.）;All kinds of rod-like eutectic have stable growth regions.In addition to the stable morphological characteristics,the growth morphology of rod eutectic also has oscillation modes such as single-period oscillation growth mode and double-period growth mode.Studying the stability of rod eutectic growth is to reveal the growth mechanism of rod eutectic with various morphological characteristics and determine the specific stable growth area of each mode.The study of rod eutectic growth selectivity is to analyze the transformation law of different eutectic growth modes and determine the critical transition zone or critical point of each growth mode.In order to further study the minimum critical point of stable growth of rod-like eutectic,this paper analyzes the stability of rod-like eutectic growth on the basis of uniformly effective asymptotic solution.The main contents and conclusions of this paper are as follows:（1）In this paper,the influence of time t on the growth of rod-like eutectic is considered,and based on the experimental system of rod-like eutectic growth,combined with its solidification thermodynamics and kinetics,the mathematical model of unsteady growth of rod-like eutectic is established,and the disturbance model of rod-like eutectic is obtained.（2）In this paper,the asymptotic analysis method is used to find the analytical solution of the perturbation model,and the stability of the analytical solution is analyzed by using the nonlinear stability analysis theory,and the corresponding dispersion relation equation of the interface wave is determined,and the perturbation interface solutions and quantization conditions of four modes（AA-,AS-,SA-,SS-,A stands for symmetric mode and S stands for asymmetric mode）are obtained.（3）Under the condition ofσ₀=0,it is found that a kind of global instability mechanism is involved in the rod-shaped growth system:the"stability exchange"caused by the non-oscillatory unstable mode,that is,the asymmetric-asymmetric（AA）mode.In this paper,the global mode solution and quantization conditions of this model are derived,and the critical small parameter_ST⁰ of rod-like eutectic growth is obtained underσ＿0=0.When≤_ST⁰,rod-like eutectic growth is stable,while>_ST⁰,rod-like eutectic growth is unstable,and the stable growth region of rod-like eutectic is quantitatively determined.