PHASE-FIELD SIMULATION OF ISOTHERMAL
TRANSFORMATION IN Fe-C ALLOY
Akinori Yamanaka*, Tomohiro Takaki** and Yoshihiro Tomita****, ***

Graduate School of Science and Technology, Kobe University
657-8501, 1-1, Rokkodai-chou, Nada, Kobe, Hyogo, Japan.
*yamanaka@solid.mech.kobe-u.ac.jp, ***tomita@mech.kobe-u.ac.jp**Department of Maritime Sciences, Kobe University,658-0022, 5-1-1, Fukae minami-chou, Higashinada, Kobe, Hyogo, Japan.
takaki@maritime.kobe-u.ac.jp
ABSTRACT: Phase-field simulations of the isothermal austenite-to-ferrite
transformation and the formation of Widmanstätten ferrite in Fe-C alloys are carried out.
The effects of the strong anisotropy of interfacial properties on the formation and growthof Widmanstätten ferrite are studied by the regularized gradient coefficient method. The
simulation results clarified that Widmanstätten ferrite plates develop due tomorphological instability in the early stage of formation and that strong interface
anisotropy is required for the formation of Widmanstätten ferrite from allotriomorph
ferrite.
INTRODUCTION: The mechanical properties of steels are closely related to themorphology and composition of microstructures produced during the austenite to ferrite
transformation. Therefore, it is essential to understand the transformation kinetics and theevolution of microstructures. In particular, the formation of plate or lath like ferrite byhigh supercooling, such as Widmanstätten ferrite, pearlite and bainite, has been thesubject of many studies. However, since the transformation involves a complex
competition between interface migration and fast solute diffusion, the formation
processes of the microstructures are difficult to investigate only by experiments. In thisstudy, we employ the phase-field model of the transformation in Fe-C alloys and theregularized gradient coefficient method in order to introduce strong interface anisotropy.
We clarify the effects of interfacial properties on the formation and growth mechanism ofWidmanstätten ferrite through phase-field simulations.
PHASE-FIELD METHOD: The total free energy of Fe-C alloy is defined by the Gibbs
free energy functional [Loginova, I. et al, 2001]⎧ε (θ )22⎫(1)
∇φ ⎬dVG = ∫ ⎨ g (φ , u C ,T ) +V2⎩⎭
where φ is the phase field taking a value of φ =1 in ferrite and φ = 0 in austenite.
Temperature T is assumed to be constant due to isothermal transformation and u C is thecarbon concentration related to the sub lattice carbon concentration y C as y C = (a c )u C .
Here, a and c are the site numbers of sub-lattice position per Fe atom position defined asa = c = 1 in austenite and a = 1, c = 3 in ferrite, respectively. The chemical free energy
density of Fe-C alloy g (φ , u C ,T ) is postulated to be a function of the phase field, carbon
concentration and temperature.
(2)g (φ , u C , T ) = p (φ )g α (u C , T ) + {1 − p (φ )}g γ (u C , T ) + Wq (θ )Here, g α (φ , u C , T ) and g γ (φ , u C , T ) are the chemical free energy density of pure ferriteand pure austenite phases, respectively. p(φ ) and q(φ ) are the energy density function
and double-well potential function, respectively. W is the potential height related to theinterfacial energy σ and interface thickness δ by W =6σ b /δ. We assume that theinterface region is λ≦φ ≦1−λ and b = 2tanh-1(1-2λ) [Takaki, T. et al, 2005].
We assume the interface anisotropy is caused by anisotropic interfacial energy σand adopt the gradient energy coefficient ε (θ ) by the following equation:3σδ{1 + γ cos k (θ − θ 0 )}≡ ε (θ )(3)bwhere γ is the strength of anisotropy, k is the mode number of anisotropy, and θ isdefined as the angle between the interface normal and the x-axis. When we introduce
strong anisotropy such as γ ≥ 1 k 2 − 1 , the equilibrium shape cannot be provided in therange of 2πi / k − θ m ≤ θ − θ 0 ≤ 2πi / k + θ m , where θ m is the first missing orientation. Inorder to consider such strong anisotropy and provide the equilibrium shape, we employ
the following regularized gradient coefficient for missing orientations [Takaki, T. et al,2006]
ε (θ m + θ 0 )ε (θ ) =cos(θ − θ 0 )(4)cosθ mε (θ ) =()Considering that the total Gibbs free energy G decreases monotonically with time,
the time evolutions of the phase-field φ and carbon concentration uc are governed by theAllen-Cahn equation and the Cahn-Hilliard equation, respectively:⎧∂ε (θ ) ∂φ ⎞⎫∂ε (θ ) ∂φ ⎞ ∂ ⎛∂g ∂ ⎛∂φ2(5)− ⎜ ε (θ )= M φ ⎨∇ ⋅ ε (θ ) ∇φ −⎟ + ⎜ ε (θ )⎟⎬∂θ ∂x ⎠⎭∂θ ∂y ⎠ ∂y ⎝∂φ ∂x ⎝∂t⎩()⎧⎪⎛ ∂2g⎞⎫⎪∂u C∂2g⎟⎬
= ∇ ⋅ ⎨ DC (φ , u C ,T )⎜⎜∇+∇φuC2⎟⎪∂t∂∂φu⎪⎩∂uC⎝ C⎠⎭(6)where M φ is the mobility of the phase field related to the mobility of the α / γ interface

M and the interface thickness δ as M φ = Mb 3δ . DC (φ , u C ,T ) is the diffusion

coefficient of carbon atoms in the Fe-C alloy and is related to the carbon concentration,
vacant sub lattice concentration y v and mobility of carbon atoms in Fe-C alloy Mc, as
DC = u C y v M C . In numerical simulations, Eqns.(5) and (6) are solved by the CrankNicolson centered finite difference method for time and the adaptive finite element
method for space [Takaki, T. et al, 2005].

COMPUTATIONAL MODEL AND RESULTS: Figure 1 shows the computationalmodel and initial conditions. Allotriomorph ferrite is set at the bottom of thecomputational domain and the interface is disturbed by a combination of sinusoidal
curves. Temperature is set to be constant at T =1000K. In order to simulate the formationand growth processes of Widmanstätten ferrite, the mode number of anisotropy k = 2 andthe growth direction θ 0 = 60 degrees are adopted. Figure 2 and Figure 3 show the timeevolutions of ferrite and the distribution of carbon concentration with interfacial energyσ =1.0 J/m2 for γ =0.3 and γ =0.4, respectively. Although the ferrite phase grows into theaustenite matrix with carbon diffusion in both cases, Widmanstätten ferrite is not formedin the case of γ =0.3 due to the low strength of anisotropy. On the other hand, the growth
process of Widmanstätten ferrite from allotriomorph ferrite is observed for γ =0.4. At theearly stage of the growth, initial tips of plate like ferrite are formed on the α / γ interface.
Widmanstätten ferrite plates develop competitively after the coalescence of plate tips.
These results clarified that in the early stage of formation, the Widmanstätten ferrite isformed from allotriomorph ferrite due to morphological instability and strong interface
anisotropy is required for this growth process.

Fig.2. Growth of Allotriomorph Ferrite with γ = 0.3.

Fig.1. Computational Modeland Initial Conditions.

Fig.3. Formation of Widmanstätten Ferrite Plates with γ = 0.4.
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