Computer simulations of the phase transformation in real alloy systems based on the phase field method

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Abstract

The kinetic simulation based on the non-linear diffusion equation becomes a very powerful method in fundamental understanding of the dynamics of phase transformation with recent remarkable development of the computer. In the present study, we calculate the dynamics of microstructure changes in real alloy systems, i.e. Fe–Mo, Al–Zn and Fe–Al–Co based on the phase field method. The composition dependencies of atomic interchange energy are taken into account so as to be applicable for the phase diagram of the real alloy systems. The elasticity and mobility of atoms are assumed to depend on the local order parameters such as the composition, the degree of order, etc. Time dependent morphological changes of the microstructure such as formation of modulated structure by spinodal decomposition, strain induced morphological changes of precipitates, the order–disorder phase transition with phase decomposition will be demonstrated. The results simulated are quantitatively in good agreement with the experimental results in the real alloy systems.

Introduction

Many researchers have performed theoretical investigations on the diffusion-controlled phase transformation. Since Cahn and Hilliard [1], [2], [3], [4] proposed the nonlinear diffusion equation in the 1960's, many researchers have attempted the theoretical analysis of phase decomposition on the basis of that equation [5], [6], [7]. However, various assumptions and omissions were made in their calculations, because it was extremely difficult to get the analytical solution of the nonlinear term in the differential equation. Since the recent remarkable developments in computers have made the numerical analysis of the nonlinear diffusion equation possible, computer simulations have become very useful in understanding the dynamics of phase transformations in materials, i.e. not only in metallic alloys but also in ceramics and polymers. It is considered that the calculation of time-dependent phase transformation is essentially based upon one of the following four methods; Cahn–Hilliard nonlinear diffusion equation [8], [9], [10], TDGL (Time-Dependent Ginzburge-Landau) Model [11], [12], [13], [14], [15], Khachaturyan's diffusion equation [16], [17], [18], [19], [20] Phase field method [21], [22], [23], [24], [25]. These methods are considered to be very useful for the basic understanding of the phase decomposition process, but it is undeniable that the calculations based on these methods have hitherto been carried for the hypothetical phase diagrams and have given only qualitative information on the phase decomposition. Such qualitative results are insufficient for quantitative understanding of the phase decomposition of real alloy system. The composition and the temperature dependencies of the atomic interchange energy should be taken into account, because such dependencies are usually found in real alloy systems. In the present study, a new calculation method of the phase decomposition process through the phase equilibrium is proposed on the basis of ‘the phase field method’ of discrete type non-linear diffusion equation. The composition dependencies of atomic interchange energy are taken into account so as to be applicable for the microstructure formation in the real alloy systems. The computer simulations are performed for the phase decompositions of the Fe–Mo, Al–Zn alloys and Fe–Co–Al ordering alloys. The time-development of phase decomposition is quantitatively consistent with the experimental one in the real alloys. The new calculation method proposed here is considered to be very useful for the basic understanding of the whole thermodynamic process of phase decomposition through the phase equilibrium, since various types of ordering parameters such as the degree of order s and the crystallographic tetragonality η can be used in the equation.

Section snippets

Phase field method

The total free energy of microstructure, Gsystem, is expressed by plural parameters such as composition c1, c2, c3  and ordering parameter s1, s2, s3  in the phase field method. Therefore, Gsystem is given by a sum of the chemical free energy Gc, the interfacial energy Esurf and the elastic strain energy Estr, all of which are functions of order parameters c and s, as represented in the Eq. (1). The number of the ordering parameter ci and si is not restrictive to be one but plural.Gsystem=rGc{cp(

Fe–Mo alloy system

The computer simulations of phase decomposition of the Fe–Mo binary alloys are represented. Fig. 1 shows the phase diagram of Fe–Mo binary alloy system [31], where the dotted line and the chain line are the metastable coherent binodal and spinodal lines, respectively. The lines are biasymmetric against composition, because of proportional increment of elastic constants with Mo content. The chemical free energy of α-phase is given by Eq. (22) [31].GcjΩjTcjc2+RTclnc+1−cln1−cΩ0T=−36490.7j/mol,ΩjT

Conclusions

On the basis of the discrete type phase field method, the phase transformation process and morphological change of microstructure are theoretically simulated for Fe–Mo, Al–Zn and Fe–Al–Co ternary ordering alloy systems by using the thermodynamic data related to the equilibrium phase diagram. The composition dependencies of atomic interchange energy are taken into account so as to be applicable for the microstructure formation in the real alloy systems. The time-developments of phase

Acknowledgements

The authors are grateful to Drs S. Takagishi, M. Fukaya and Pi Zhi Zhao who were all students in the graduate school of Nagoya Institute of Technology for their experiments in part. The present research was financially supported, in part, by a Grant-in-Aid for Scientific Research on the Priority Area of Phase Transformations (1997–1999) from the Ministry of Education, Science and Culture of Japan, also by the research funds from the Research Promotion Association for Light Metals, Japan.

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