NUMERICAL SIMULATION FOR TRANSFORMAION IN

Fe-Mn-C SYSTEM BY PHASE-FIELD MODEL

Dong-Hee Yeon, Pil-Ryung Cha, Jong-Kyu Yoon

School of Materials Science and Engineering, Seoul National University,

Seoul 151-742, Korea

ABSTRACT

In Fe-Mn-C system, there is a large difference in mobility of solute Mn and C. Substitutional element Mn diffuses much slower than interstitial element C in Fe. By definition, paraequilibrium means that only the mobile elements are equilibrated while the sluggish ones behave as a single element. The  transformation may be under local equilibrium, paraequilibrium, or lying between these two limits. The evolution mode could be determined by the diffusion velocity of solute elements in the matrix in front of the moving interface and the interface migration. Phase-field model could be applicable to simulate these phenomena without any constraints or boundary conditions at the interface. The objective of this study is to find out evolution mode for paraequilibrium or full thermodynamic equilibrium by the phase-field model of ternary system.

1. INTRODUCTION

Many researches have been carried out for the  transformation of Fe-Mn-C system both theoretically and experimentally [1-5]. In  transformation of Fe-Mn-C system, there is a regime that can be characterized by the absence of partitioning of the substitutional alloying element Mn. This type of transformation is called paraequilibrium. Under paraequilibrium, carbon diffuses at an appreciable rate but the alloying element Mn is almost immobile relative to iron, the growing phase inherits the alloy content form the parent phase. While the mobile element carbon is in the local equilibrium at the phase interface, the chemical potentials of alloying substitutional element have no physical meaning and thus these substituional elements behave thermodynamically as if they were only a single element [4]. Many works have been done mainly about the limit in which the transformation under paraequilibrium can be possible thermodynamically and about the transition from fully local equilibrium to paraequilibrium. But, both fully local equilibrium and paraequilibrium states are thermodynamically unique states and the reaction of course may choose any one of infinite numbers of undefined nonequilibrium states lying between the equilibrium and paraequilibrium limits. Whether nature in fact attains such states between two limits is a very subtle problem in kinetics. So the model is needed that can select the state with considering the thermodynamics and kinetics of the system correctly.

The Phase field model can be recommended as the unified model that is applied to any systems under fully local equilibrium, paraequilibrium and nonequilibrium states lying between two limits. It has been reported that phase field model is able to describe successfully the interface kinetics such as solute trapping effect when it is applied to solidification in binary system [6,7]. In order to describe the paraequilbrium state, the phase field model for multicomponent alloy is needed and it can distinguish the interstitial alloy elements from the substitutional alloy elements because the large difference in atomic mobility is induced from the different diffusion mechanism. However, the existing phase field models for alloys are restricted to binary alloys. And, although these models are applicable to only binary system that contains only substitutional alloy element, they are often applied to the system that has interstitial alloy element such as carbon. [8,9] In this study, the  transformation in Fe-Mn-C system is simulated by the phase field model for multicomponent alloys. The result is compared with the simulation by the model that does not distinguish between intestitial and substituiuional alloy elements. The phase field model is certified by the comparison with the results of the system that is composed of only substituional alloy elements, so the difference of atomic mobility is small.

2. THE PHASE FIELD MODEL FOR MULTICOMPONENT ALLOYS

When the phase field model is applied to the multicomponent alloy system, the phase field evolution and diffusion equation could be established from a Landau-Ginzberg free energy functional F(c1,c2,). For this purpose, (x,t) is postulated to characterize the phase of the system. The phase field  is defined as a continuous variable between =1 at one phase and =0 at the other phase. In the interfacial region that has the finite width, (x,t) has the value between 0 and 1 and varies steeply but smoothly. In multicompnent systems, the free energy functional F(c1,c2,) is the functional of the phase field variable with solute concentration. [10]

(1)

Here, f(c1,c2,) is the free energy density (free energy per unit volume) and it may be written in the form,

,(2)

whereand are the free energy densities of each phase, respectively and functions of solute concentration ci. It is assumed that the phases in the interfacial region are the mixture of phases that are composed of same concentration (WBM Model) [6]. In equation (2), we choose h(), g() in the following forms.

(3)

(4)

where g() is the double-well potential associated with phase change.

The evolution equation of phase field and diffusion equations are driven from gradients of the above functional.

(5)

(6)

In the equations (5) and (6), M is the phase field mobility and Mki is the mobility matrix. From equations (5) and (6), it is possible to get governing equations.

(7)

(8)

In the above equations, f and fCi mean , respectively. Since equation (8) should be the same form with traditional diffusion equation in one phase region, we can determine Mki using . The parameter  in equation (7) and (8) should be related to the interfacial properties, the interfacial energy  and interface thickness 2 respectively. The parameters are determined at the sharp interface limit condition, , when we define interfacial regionas the region where changes from 0.1 to 0.9 at - to . The phase field mobility can be determined at the sharp interface limit as , whereis the driving force per unit volume and V is the interface velocity. The phase field mobility can be calculated with the relationship between driving force and interface velocity where Miis interface mobility. [6,10]

3. CONSIDERATION OF INTERSTITIAL ALLOY ELEMENT

The existing phase field models cannot be applied to the system that contains interstitial alloy element because these models use the Cahn-Hilliard type diffusion equation and they usually use atomic fraction of solute atoms as the concentration variables. These models cannot afford to distinguish the different diffusion mechanisms of substitutional alloy elements and interstitial alloy elements. The diffusion of substitutional solute atoms always accompanies with the diffusion of solvent atoms while the diffusion of interstitial solute atoms does not. If one uses the atomic fraction of solute atoms as the concentration variables, the increase of concentration of interstitial alloy elements means the decrease of the concentration of solvent atoms. In order to consider the interstitial alloy element in the diffusional transformation, the site fraction is used as the independent variables [3,11]. However, the definition of the site fraction in the mixture phases is a delicate problem. So, we use moles per unit volume as the concentration variables.

It is assumed that the solvent atoms and the substitutional elements contribute to the total molar volume of the alloy and there is no contribution from the interstitial elements. Thereis no constriction on the movement of the interstitial atoms and total moles of substitutional atoms are conserved. The two constraints can be written as follows.

(9)

(10)

VS is the molar volume of substitutional alloy elements, cVa and ci is the mole numbers of vacancies per unit volume in the interstitial sites and S, I denote substitutional element and interstitial element respectively. It is assumed that all substitutional elements have the same molar volume and interstitial elements have no volume. We did not consider the vacancies in the substitutional sites.

From the above assumptions, we can evaluate correct driving force of solute diffusion. The free energy per unit volume can be written as the following,

(11)

From equation (8), if we do not consider the off diagonal term of atomic mobility matrix, the driving force of solute diffusion per unit volume is . For the case of substitutional elements, the driving force per unit volume is and for interstitial elements, .

4. NUMERICAL SIMULATION OF TRANSFORMAION

The isothermaltransformationin Fe-Mn-C system is simulated by numerical method. The overall compositions of each element are Mn 1.0% and C 1.0% and the temperature of the system is 873K. In this system, the carbon diffuses 107~109 times faster than Mn. So the ferrite grows under paraequilibrium. The simulation is performed that the ferrite, which occupies initially 10% of whole system with the same composition in austenite, grows into austenite phase. The result is shown in Fig. 1. Figure 2 is the same result in which the concentration is expressed as a function of the atomic fraction.

Fig. 1. Concentration Profile as a Function of Mole per Unit Volume

Fig. 2. Concentration Profile as a Function of Atomic Fraction

From the above results, we can see that the phase-field model describes the phase transformation under paraequilibrium. The result without considering interstitial element is shown in Fig. 3. Comparing with Fig. 2, there is no change of the atomic fraction of Mn and only concentration of carbon changes. This means that there is a change of the atomic fraction of Fe and only Fe atoms move without movement of Mn atoms. During the growth of ferrite, the difference of chemical potentials of carbon is very small as compared with that of Mn across the interface. Kirkaldy et. al said that under paraequilibrium as the mobile atom is fully local equilibrium state at the interface and the chemical potential of immobile atom is discrete [2]. The small difference in chemical potential of carbon might be kinetic effect.

Fig. 3. Concentration Profile as a Function of Atomic Fraction during  Transformation of Fe-Mn-C System Without Considering the Interstial Atom

In order to compare with the above calculation, another  Transformation in the Fe-Cr-Ni system, which contains only substitutional elements, is studied. The overall compositions of each element are Cr 23.0% and Ni 9.04% and the temperature of the system is 1373K. This system was verified thermodynamically and kinetically by the concept of local equilibrium [12,13]. In this system, Cr diffuses only 10~102 times faster than Ni. So Both Cr and Ni atoms are partitioned at the interface and the transformation is controlled by diffusion of Ni atoms because Ni moves slower. The simulation is performed that the austenite, which occupies initially 10% of whole system with the same composition with ferrite, grows into ferrite phase. Figure. 4 show that the transformation is governed by Ni diffusion.

Fig. 4. Concentration Profile as a Function of Atomic in Fe-Mn-C System

When the system reaches equilibrium state, the equilibrium volume fractions of ferrite and austenite are 20.11% and 79.89% respectively and equilibrium concentrations of Cr and Ni in ferrite are 29.26at% and 5.69at% respectively and those in austenite are 21.42at% and 9.88at% [13]. The result performed by the phase field model is that the equilibrium volume fractions of ferrite and austenite are 20.44% and 79.56% respectively and equilibrium concentrations of Cr and Ni in ferrite are 29.22at% and 5.70at% respectively and those in austenite are 21.43at% and 9.90at% respectively. From this result, the phase field model reflects thermodynamics and solute conservation properly. For Fe-Mn-C system, it takes too long to reach the equilibrium state. So, we calculate the equilibrium volume fraction of each phase and their compositions by increasing the diffusivities of Mn hypothetically in both phases. The result predicted by the thermodynamic calculation is that the ferrite (Fe-0.511at% Mn-0.077at%C) occupies 86.6% of total volume and the austenite (Fe-4.172at%Mn-6.984at%C) occupies 13.4% at equilibrium state. The result by phase field model is that the compositions of Mn and C in ferrite is Fe-0.507at% Mn-0.0737 at %C and that in austenite is Fe-4.169at%Mn-6.965at%C respectively and volume fractions of ferrite and austenite are 86.5% and 13.5% respectively.

From the results presented, we can say that the transformation mode between fully local equilibrium and paraequilibrium can be described properly using the phase field model. The selection of transformation mode depends on the relationship between the interface mobility and the diffusivities of the solute atoms.

5. CONCLUSION

Using the phase field model, the transformation of Fe-Mn-C system is simulated. It is shown that the phase field model can describe not only thermodynamics effects of the phase transformation but also the kinetics such as partitioning of solute atoms, so it is possible to simulate various modes of partitioning behavior at the interface without extra boundary condition for moving interface. By consideration of interstitial elements, we can simulate the transformation under paraequilibrium properly.

ACKNOWLEDGEMENTS

The authors appreciate Prof. Kyu-Hwan Oh in Seoul National University for his help in thermodynamic evaluation Authors are grateful to the financial support of Brain Korea 21 program supported by Ministry of Education, Korea.

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