Modeling the austenite-ferrite isothermal transformation in an Fe-C binary system and experimental verification
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I. INTRODUCTION
DURING recent years, with the intensive application of different kinds of materials, the modeling of microstructure formation in materials processing has markedly progressed, especially in the fields of recrystallized structures,[1–4] grainstructure formation in solidification processes,[5,6] and dendritic growth.[7] Modeling is able to provide an image which reproduces “actual” microstructures. The austenite-ferrite isothermal transformation has also been modeled by numerically solving the diffusion equation.[8,9] In this method, the criterion for the onset of transformation is related to the carbon concentration. This means that once the concentration of carbon in a cell reaches the equilibrium carbon concentration determined by the Fe-C phase diagram, the cell begins to transform. However, the diffusion equation could not deal effectively with some transformations in which the diffusion procedure is not the only factor that controls the transformation, such as massive transformations and deformationinduced transformations. Therefore, this model is very limited. In essence, the diffusional transformation is a process in which the system evolves to reach the equilibrium state with minimized free energy by the diffusion of solute atoms. Therefore, the key to modeling this transformation seems to be the free-energy minimization procedure. II. FREE-ENERGY MODEL In the present article, the Landau–Ginzburg free-energy model is adopted. The free-energy density is expressed as f (c, ) ⫽
A1 A2 A3 4 A4 6 ⫹ (c ⫺ c1)2 ⫹ (c ⫺ c2) 2 ⫺ 2 2 4 6 [1]
Where c is the concentration of carbon; A1, A2, A3, and A4 are four constants; and c2 is the equilibrium carbon concentration in the ␣ phase. The parameter c1 has a value near
MINGMING TONG, Ph.D. Student, DIANZHONG LI, Professor, and YIYI LI, Professor and Academician, are with the Institute of Metal Research, Chinese Academy of Sciences, Shenyang, 110016, People’s Republic of China. Contact e-mail: [email protected] JUN NI, Professor, is with the Molecular and Nano Science Laboratory of Education Ministry, Department of Physics, Tsinghua University, Beijing 100084, People’s Republic of China. Manuscript submitted November 6, 2001. METALLURGICAL AND MATERIALS TRANSACTIONS A
that of the equilibrium carbon concentration in the ␥ phase. The order parameter, , is defined as the difference between 1 and the axis ratio; it equals 0 in the ␥ phase and 0.293 in the ␣ phase. According to the common tangent rule, the following relations can be obtained: A2 2 2 ␣
[2]
A3 4 A4 ⫺ 6␣ 2 ␣ 3
[3]
A1 (c␥ ⫺ c␣) ⫽ A1 (c␥ ⫺ c␣)2 ⫽
2 ⫽
A3 ⫽ 0.085849 A4
[4]
As the concentration of carbon approaches zero, the system becomes pure iron. The following equation can be given: A2 A1 A3 4 A4 6 ⭈ c12 ⫺ ⭈ c␣ ⭈ 2 ⫺ ⫹ ⫽ Giron (T ) 2 2 4 6 [5] where the Giron(T ) is the Gibbs free energy of pure iron at the temperature T. At the temperature of 1080 K,[10,11] c␣ ⫽ 0.015 wt pct, c1 ⫽ 0.3 wt pct, c␥ ⫽ 0.35 wt pct, and Giron (1080 K) ⫽ ⫺1.75 J/mol were assumed. Using Eqs. [2] t
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