Multi-Phase-Field Study for Pearlite Transformation
with Grain Boundary Diffusion
Akinori Yamanaka1, Takashi Yamamoto2, Tomohiro Takaki3, Yoshihiro Tomita2
1

Graduate School of Science and Technology, Kobe University,
1-1, Rokkodai, Nada, Kobe, Hyogo, Japan, yamanaka@solid.mech.kobe-u.ac.jp;
2
Graduate School of Engineering, Kobe University,
1-1, Rokkodai, Nada, Kobe, Hyogo, Japan;
3
Graduate School of Science and Technology, Kyoto Institute of Technology,
Matsugasaki, Sakyo, Kyoto, Japan

ABSTRACT

In order to investigate the effect of grain boundary diffusion (GBD) of carbon on growth velocityof pearlitic microstructure, pearlitic transformation accompanying with the GBD in Fe-C-Mn
alloy is simulated by multi-phase-field (MPF) method. The results show that the growth velocityof pearlite decreases with increasing lamellar spacing of pearlite, as suggested by numericalstudies using the Zener-Hillert model and experimental studies. And, the present MPF simulation
also reveals that the GBD considerably enhances carbon diffusion from ferrite (α) phase to
cementite (θ ) and assists cooperative growth of α phase and θ. From these results, it can bementioned that the rate of the GBD largely governs the growth velocity of pearlite.

1. Introduction
Microstructure in steels consists of α phase, pearlite, bainite and martensite phase. In particular,pearlite exhibits lamellar structure of α phase and θ, and possesses good balance between
strength and toughness. Therefore, pearlite is widely used as constituent phase in many steels
such as ferrite-pearlite (FP) steel. Recently, authors have developed the integrated simulation
model for predicting microstructure formation and mechanical properties of steel using thephase-field (PF) method and the homogenization method [1]. Using this model, the mechanicalproperties of the FP steel depending on the FP microstructure was clarified. However, in order topredict the mechanical properties of the FP steel precisely, morphology of pearlite is needed to
be predicted in the PF simulation. For this issue, Steinbach et al. [2] and Nakajima et al. [3]proposed MPF models for the pearlitic transformation and simulated the formation of pearlitic
microstructure based on the volume diffusion-controlled mechanism. However, many studies
suggest that the GBD of carbon is an essential phenomenon controlling the pearlitictransformation [4]. Thus, the effects of the GBD should be considered in MPF simulation of thepearlitic transformation. Therefore, in this study, the pearlitic transformation is simulated by theMPF method considering both volume diffusion and the GBD of carbon. Furthermore, the role ofthe GBD on the growth of pearlite is investigated.

2. Multi-Phase-Field Model
In order to simulate the pearlitic transformation in Fe-C-Mn alloy, the generalized multi-phasefield (GMPF) method proposed by Steinbach et al. [5] is used. In the GMPF method, the system
of N grains can be considered. Thus, we use N phase field variables, φi ( i = 1, 2, … , N). φi
describes volume fraction of the i th grain and varies smoothly across an interface from φi = 1 in
the i th grain to φi = 0 in other grains. Hereafter, we consider a α + γ + θ three-grain (phase)system for simple description. By using these phase field variables, the total Gibbs free energy ofthe system, G, is defined by the Ginzburg-Landau type Gibbs free energy functional,G=⎧⎪ 3 3 ⎛ a 2⎫⎞⎜ − ik ∇φ i ⋅ ∇φ k + Wik φ i φ k ⎟ + g e ⎪⎬dV ,⎨⎜ 2⎟V⎪⎪⎭⎠⎩ i =1 k =i +1 ⎝∫ ∑∑(1)

where aij and Wij are the gradient coefficient and potential height, respectively. The evolution
equation of the phase field variable is derived by assuming that the total free energy decreases
monotonically with time.
φn 2M
∂φ i
1 2
⎡n ⎧⎤.
⎫ 82
2= −∑⎢∑ ⎨⎩(Wik −W jk )φ k + 2 (a ik − a jk )∇ φ k ⎬⎭ − π φ i φ j ∆G ij ⎥∂tn⎣ k =1
⎦j =1(2)Here, n is the number of phase fields in the arbitrary point and is given by n = ∑3i =1ξi . ξi is a stepfunction, which is expressed as ξi = 1 in a region 0 < φi < 1 and ξi = 0 in other region. The third
term on the right-hand side of Eqn. (2) is the phenomenological thermodynamic driving forceand ∆Gij is its magnitude. To simulate the carbon diffusion during the pearlite transformation in a
multiphase system, the total carbon concentration C is defined as a linear function of the localcarbon concentration Cλi and the phase-field variables φi. Therefore, when φ1, φ2 and φ3correspond to α, γ and θ phases, respectively, the total carbon concentration C is written as
C = φ1Cα1 + φ 2 Cγ2 + φ3Cθ3 . The evolution equation of the total carbon concentration C is expressedby the sum of diffusion fluxes of carbon, Ji, in individual grains as [6],∂C= ∇ ⋅ (φ1 J 1 + φ 2 J 2 + φ3 J 3 ) = ∇ ⋅{φ1 Dα1 ∇Cα1 + φ 2 Dγ2 ∇Cγ2 + φ3 Dθ3∇Cθ3 }.
∂t(3)Here, Dλi denotes diffusion coefficient of carbon atoms in the i th grain (λ phase). The local
concentration Cλi is related to the partition coefficient of carbon atoms kjr. The partition
coefficient kjr and the driving force ∆Gij are calculated with a linearized phase diagram [7].

3. Computational Model
The growth of a single pearlite lamellar during isothermal pearlitic transformation in Fe-C-Mn
alloy is simulated in two-dimensions. Because the diffusion of manganese atoms is much slowerthan that of carbon atoms, it is not considered. In this study, the effects of the lamellar spacing λand the GBD on the growth of pearlite are investigated. When the effect of λ is studied, we use a
rectangular computational domain. The size of computational domain along x direction DX,
which corresponds to λ, is changed from 0.24 µm to 0.48 µm. And, when the effect of the GBD
is studied, λ is set to be constant at 0.24 µm and a DX × DY = 0.24 × 0.48 µm2 rectangularcomputational domain is used. The initial α phase and θ is set at the bottom of the computational
domain. Temperature is set to be constant at T0 = 973 K. The interfacial energy of all interfaces
is assumed to be constant at 0.5 J/m2. The initial carbon concentrations of both α phase and θ arein the equilibrium values. The governing equations, Eqns. (2) and (3), are solved by the finite
difference method with periodic boundary condition for x direction and zero Neumann boundary
condition for y direction. And, the GBD is described by considering k (k = 1, 2, 3, 4) times larger
diffusion coefficient than that of bulk phase in the grain boundary region defined as 0 < φi < 1.

4. Results and Discussions
Figure 1 shows the evolution of carbon concentration field during the growth of pearlite fordifferent λ. From these results, it can be confirmed that the growth velocity of pearlite V
decreases with increasing λ, because the increase of λ promotes long diffusion distance ofcarbon atom from α phase to θ. This dependency of V upon λ is qualitatively identical with thenumerical results obtained by the Zener-Hillert model and experiments [7].
Figure 2 shows the distribution of carbon concentration during the pearlitic transformationaccompanying with the GBD. It is found that as the carbon partitioning from α phase into θ
becomes fast by the GBD, V increases. In the case of k = 4, V equals to 3.0 µm/s. Furthermore,according to Ridley [8], λ and V satisfies the following relationship: λV m = constant. Here, m isa constant taking the value of m = 2 ~ 3 in the range of the obtained V. Figure 3 shows therelationship between λ and V obtained by the present MFP simulation. It is clarified that therelationship, λV m = constant, is also satisfied in the present study and the value of m becomes
1.2 ~ 1.4. Although m is smaller than the conventional values (m = 2 ~ 3), m tends to increase

Figure 1. Distribution of carbon concentration
during growth of pearlite for different λ.

Figure 2. Distribution of carbon concentration
during growth of pearlite with GBD.

Figure 3. Relationship between lamellar spacing λ and growth velocity of pearlite V.
with increasing the rate of the GBD (the value of k). Therefore, it can be concluded that thegrowth of pearlite observed in real system can be simulated by considering quite fast GBD.

Acknowledgements
The authors acknowledge the financial support by the Japan Society for the Promotion ofScience for Young Scientists and by a Grant-in-Aid for Scientific Research from the Ministry ofEducation, Culture, Sports, Science and Technology of Japan.
References[1] A. Yamanaka, T. Takaki and Y. Tomita, “Coupled Simulation of Microstructure Formation
and Deformation Behavior of Ferrite-Pearlite Steel by Phase-Field Method andHomogenization Method”, Materials Science and Engineering A, 480, 244-252 (2008).
[2] I. Steinbach and M. Apel, “The Influence of Lattice Strain on Pearlite Formation”, Acta
Materialia, 55, 4817-4822 (2007).
[3] K. Nakajima, M. Apel and I. Steinbach, “The Role of Carbon Diffusion in Ferrite on theKinetics of Cooperative Growth of Pearlite”, Acta Materialia, 54, 3665-3672 (2006).
[4] K. Hashiguchi and J. S. Kirkaldy, “Pearlite Growth by Combined Volume and Phase
Boundary Diffusion“, Scandinavian Journal of Metallurgy, 13, 240-248 (1984).
[5] I. Steinbach and F. Pezzolla, “A Generalized Field Method for Multiphase Transformations
Using Interface Fields”, Physica D, 34, 385-393 (1999).
[6] J. Tiaden, B. Nestler, H. J. Diepers and I. Steinbach, ”The Multiphase-field Model with an
Integrated Concept for Modeling Solute Diffusion”, Physica D, 115, 73-86 (1998).
[7] C. Zener, “Kinetics of the Decomposition of Austenite”, Transaction of AIME, 167, 550-595(1947).
[8] N. Ridley, “A Review of the Data on the Interlamellar Spacing of Pearlite”, MetallurgicalTransactions, 15A, 1019-1036 (1984).