Phase field study on crystal orientation effects in eutectoid phase transformation
Phase field study on crystal orientation effects in eutectoid phase transformation
Document code: A 0
0 Dong-qiao Zhang , Ya-jun Yin
1 Jian-xin Zhou, and Zhi-xin Tu State Key Laboratory of Materials Processing and Die & Mould Technology, Huazhong University of Science and Technology , Wuhan 430074 , China
In this present work, a multi-phase field model was used to simulate the eutectoid transformation process, and on the basis of the nucleation model that was previously proposed by our research team, anisotropic and orientation relationship models were introduced to study the growth mechanism of the pearlite lamellae with anisotropy. It was found that the growth direction of the pearlite lamellae is related to its orientation and spacing. In the process of lamellar growth, deflection growth of pearlite will appear along with the adjustment of lamellar spacing, and the deflection angle is equal to the orientation difference between the austenite and the pearlite. Comparison between experimental and numerical results indicates a good consistency in pearlite morphology.
simulation; phase field; lamellar growth; pearlite morphology; growth direction
Aon the isotropic eutectoid growth process, with
t present, most of the literature focused only
less research on the eutectoid nucleation process and
anisotropy. In the previous study, Zhang, et al. 
proposed a cooperative nucleation model, but did not
consider anisotropy for the eutectoid growth. However,
in studies by Zhi  and Miyamoto , it was found that
eutectic and eutectoid growth process does include
As early as the 1950s, Smith  studied the orientation
of ferrite nuclei at the austenite grain boundary, and
it was observed that the ferrite nuclei have a certain
relationship with the orientation of austenite grains.
Guo et al  studied the kinetics and crystallography
of the complex sedimentary surface nucleation of
the intergranular pearlite in Fe-Mn-C, and found that
the incoherent MnS in the austenite cannot be strong
nucleation position of pearlite, unless the transition time
is extended. There is no special orientation relationship
between lamellar ferrite and austenite matrix, but there
Male, born in 1975, Ph.D. and Professor. His research interests mainly focus
on the computer applications in foundry industry, especially on casting process
simulation and intelligent manufacturing for foundry enterprises.
is a Pitsch-Petch orientation relationship between the
ferrite and cementite of pearlite. Furuhara et al.  added
a small amount of S,V and N to Fe-2Mn-(0.13, 0.2) C,
and analyzed the effect of intergranular inclusions or
precipitates on the kinetics of ferrite transformation at
temperatures between 973-823 K and the orientation
relationship between the inclusion and austenite,
ferrite was obtained by similar experiments . The
phase concentration, dislocation density of austenite
and ferrite/bainite/ martensite, lattice distortion and
interstitial carbon atoms in austenite were investigated
by Wixess et al.  using X-ray diffraction techniques.
The lattice distortion and the gap between the carbon
atoms will affect the anisotropy of the phase. The
phenomenon of ferrite nucleation on non-metallic
inclusions was observed by Cheng et al. . When the
ferrite is at an elevated temperature, it presents an
isolated island. The discoid ferrite is derived from the
nucleation of the ferrite on the inclusions. It associated
with the larger degree of subcooling, and with the parent
austenite into a fixed orientation of growth. In 2010,
Cheng et al.  observed the nucleation phenomenon of
ferrite in the surface, edge and corner of the austenite
grain boundary, and analyzed the mechanism of ferrite
nucleation under three-dimensional morphology.
When ferrite nucleated on the boundary surface of
austenite grains, most of the ferrite nuclei were flat
ellipsoid, some of which are plate-like or
pyramidlike. When they nucleated on the edge of the grains, the
morphology presented triangular pyramid or triangular double
cone. When the nucleation position is at the corner of the grain
boundary, it shows irregular shape. In 2013, the nucleation
of ferrite and the three-dimensional kinetics of ferrite from
austenite in low-carbon micro-alloyed high-strength steels
were studied systematically by high temperature confocal laser
microscopy, hot-pressing and electron backscatter diffraction
experiments. Then Cheng et al.  used continuous interface and
computer aided technology to reconstruct the three-dimensional
morphology of ferrite, and found that the anisotropy of ferrite
can be obtained from its 3D morphology. Wang et al.  studied
the interfacial structure with Bagaryatsky orientation of ferrite/
cementite by approximating the repositioned lattice simulation.
When the ferrite/cementite crystal orientation presents a specific
relationship, the structural steps and mismatched dislocations
can be formed on the interface, but this step is not an intrinsic
structure and has no specific size,so the two-phase interface
can be fluctuated within a certain range. This causes the ferrite/
cementite to have no macroscopic habitat. This phenomenon is
conducive to bending growth of pearlite lamellar structure.
All of the above studies, both in terms of nucleation
mechanism and orientation relationship, relate to the anisotropy
of ferrite and cementite. Considering the anisotropy of the
lamellar phase can lead to better understanding of the orientation
relationship between austenite, ferrite, cementite and pearlite.
In this work, with T8 steel as the research material, based on
the nucleation model of proposed by our research team  and
the introduction of anisotropy and orientation relationship, the
growth evolution process of pearlite lamellae was simulated.
Then the nucleation sites of pearlite lamellae were discussed.
1 Model and parameter conditions
1.1 Phase field model
In this work, the multi-phase field model was used to simulate
the eutectoid transformation process. The total free energy
function is shown in equation (
). Equations (
) and (
are interface energy density and chemical energy density,
where and are field variables of phases and , which
are required to satisfy the equation. They further satisfy the
constraints of equation (
). is the interface thickness, is the
interface energy between and , is the volume free
where CAnis is anisotropy coefficient, kin , and st are
constants, is the initial interface energy.
In the process of austenite transformation into pearlite,
two kids of orientation relationships are involved: one for the
energy density of phase , is the Lagrange multiplier, is
the concentration of carbon in phase , and N is the number of
The phase field calculation function is shown in the equation.
Where the variables , and are calculated from equations
) and (
where is the interface field between phase and ,
is the transformation entropy, is the reference temperature,
is slope of a corresponding line in phase diagram, is the
carbon concentration at the reference temperature , and is
) is the diffusion equation, and the concentration
field can be solved by combining equation (
) and equation (
N δ F N
c = ∇ ∑ M ∇ = ∇ ⋅ ∑φα Dα ∇cα
α =1 δ c α =1
c = ∑φα cα
cα = kαβ cβ
where represents the ratio of the concentration between
and . M is the chemical mobility and is the diffusion
coefficient of carbon in phase .
1.2 Anisotropy and orientation models
The anisotropic and orientation relationship models are
introduced on the basis of the early nucleation model.
Combining the growth patterns of ferrite and cementite,
two different anisotropy models are used, respectively. The
anisotropy model of ferrite is a common 4-symmetric model,
as shown in Equation (
). The cementite is characterized by
faceted anisotropy, its model is shown by equations (
) and (
which was constructed by Steinbach . The relevant parameters
can find from references[13, 14].
0 s2t ( ks2t cos2 θ + sin2 θ )−1.5
σ = σ k
orientation relationship between austenite and ferrite/cementite,
and the other for the orientation relationship between ferrite and
The relationship between austenite and ferrite is generally
divided into: K-S relationship and Non K-S relationship. The
K-S relationship is more refined by Goro Miyamoto  into:
K-S relationship and Near K-S relationship. The definition of
the interval is shown Table 1, where the K-S relationship is
( , ).
For the K-S orientation relationship, the interface energy can
be processed by the Read Shockley equation [15, 16]:
As the present work focused on the growth behavior of
pearlite, it was assumed that ferrite and cementite have no
orientation relationship, and were therefore not specifically
2 Results and discussion
The initial phase field distribution is shown in Fig. 1, and the upper
and lower parts are two different austenite grains. The middle
region is the interface of two austenite grains and . The
initial position of pearlite nucleus is set at the center of the whole
region. The critical nucleation concentrations of the ferrite and
cementite are 3.2 mol% and 3.72 mol% , respectively.
In the initial conditions, 200 × 200 mesh was used for the
simulation. The grid size is 4 nm and the isothermal temperature
is 970 K. The simulation parameters are shown in Table 3. The
boundary conditions are set to be adiabatic.
where is one of ferrite, cementite or austenite phase, is
interfacial energy of same phase between different grains. is
interfacial energy between ferrite and cementite.
There are three main relationships between ferrite and
cementite, namely Bagaryatsky, Isaichev, and Pitsch-Petch, as
shown in Table 2.
2.1 Effect of anisotropy on pearlite
The anisotropy of ferrite and cementite is represented by a
4-symmetric model. The orientations of two adjacent austenite
grains are set as 0° and 45°, respectively. The nucleus orientation
of pearlite varies from 0° to 22.5°.
Due to the anisotropy of ferrite and cementite, the growth
morphologies of ferrite and cementite are different. Figure 2
shows the growth state of the pearlite from the boundary into the
austenite grains under anisotropic conditions at 0.10 s, where the
orientation of austenite is 0°, and the orientation of austenite
is 45°. It can be clearly found that the growth direction of the
Note: α denotes ferrite, β denotes cementite, and γ denotes austenite.
pearlite into the austenite grains on both sides of the boundary is
different due to the different orientations of the austenite grains
on both sides. Figure 2(a) shows that the growth direction of the
pearlite in is perpendicular to the grain boundary, that is, the
angle between the growth direction of pearlite and the vertical
direction is 0°, which is equal to the orientation difference
between the austenite and the pearlite. However, the angle
between the growth direction of pearlite in and the grain
boundary of austenite is 45°, which is equal to the orientation
difference between the austenite and the pearlite. Figures
2(b), (c), (d), (e) and (f) have the same rules, as shown in Table
4. However, the growth direction of pearlite will deviate with
the growth of pearlite, as the second angle marked in Fig. 2(e)
and (f). When the cementite lamellae at the left side began to
grow, the growth direction of cementite lamellae at right side
deviated from the orientation angle. This results in a large
spacing between the two cementite lamellae. In order to reduce
the lamellar spacing, the growth direction of cementite lamellae
at the right side tends to shift toward the center of the two
cementite lamellae. This phenomenon indicates that the growth
direction of the pearlite lamellae is related to the orientation of
the pearlite lamellae and the lamellar spacing of pearlite.
2.2 Effect of orientation relation on pearlite
According to the experiments of Miyamoto , the orientation
difference between the pearlite lamellae and the austenite is
mainly concentrated between 0°-25°. Considering that the
orientation of two adjacent austenite grains is 0° and 25°, the
nucleus orientation of the pearlite lamellae is set to 5°, 10°,
15° and 20°, respectively. Due to the different orientations of
pearlite, the pearlite grows into different morphologies.
Figure 3 shows the growth morphology of pearlite in different
orientations. The grid region is the boundary between two
austenite grains. The interface energy between ferrite and
austenite would vary drastically with the orientation when the
orientation relationship is considered. When the orientation
difference is less than 5°, the lamellar ferrite does not grow
directly into the austenite grains, but grows in the form of
Widmanstatten pattern ferrite or boundary ferrite. When the
orientation difference is greater than 5°, it grows into the
austenite grains. The pearlite in the two austenite grains in
Fig. 3 (a) and (b) shows two different growth morphologies.
At the same time, the cementite in Fig. 3(a) and (d) is more
likely to grow along the austenite boundary. In the vicinity
of the austenite boundary, there are ferrite and cementite
lamellae parallel to the austenite boundary. The ferrite may be
firstly precipitated, then cementite would be precipitated from
the boundary between ferrite and austenite. The orientation
difference between the austenite and pearlite is 10° and 15° for
Fig. 3(b) and (c), so there was very little difference between the
growth rate of pearlite on both sides of the austenite boundary.
In particular, it is noted that the austenite grain boundary
varies with the different orientations of pearlite. At the
beginning, the austenite grain boundary is a horizontal straight
line. With the pearlite growing in the austenite grain, the grain
junction is changed from a two-phase junction to a three-phase
junction. The boundary evolution mechanism in Fig. 4 shows
that only exists at initial state and the interface of two
austenite grains will not bend. With the formation of the ferrite
nucleus, the interface [ and , as shown in Fig. 4(b)
and (e)] between austenite and ferrite appears. Since the two
austenite grains are in different orientations, and
have different values, and they vary with the normal values
of the interface at the junction point. At this time, if
still remains horizontal, the three interface energies would be
unbalanced at the junction. In view of < , and the
direction of interface energy between austenite and ferrite
being approximately horizontal, the vector sum of and
is as shown in Fig. 4(c) and (f). In order to ensure balance,
the direction of is the reverse of the vector sum. As the
growth continues, the interface will move toward the inside of
(a) Orientation is 5°
(b) Orientation is 10°
(c) Orientation is 15°
(d) Orientation is 20°
the grain , eventually showing such morphology as shown in
Fig. 3(a) and (d).
2.3 Effect of nucleation positions on pearlite
The nucleation site of the above pearlite growth process is at
the austenite grain boundaries, but the pro-eutectoid phase
will appear in the process of the actual growth of pearlite,
and preferentially nucleates and grows at the austenite grain
boundary. Therefore, the subsequent nucleation sites are mostly
at the interface of the pro-eutectoid phase and the austenite
phase. However, for the alloys of the different compositions,
the distribution of impurity particles inside the austenite grains
would be very different. The nucleation induced by the inclusion
in grains or boundaries is not considered herein.
When the nucleus location is set at the interface of the
austenite phase and the pro-eutectoid phase or austenite phase,
the K-S orientation relationship is not changed, as shown in
Fig. 5. Figure 5 (a) and (d) show unilateral austenite growth,
whereas (b) and (c) show lamellar growth. Compared with the
growth morphology of pearlite with different orientations in Fig.
3, it can be found that the pearlite at the nucleus location that
set at the interface of the austenite phase and the pro-eutectoid
phase or austenite phase can develop the bifurcation mechanism
(that is, a ferrite lamella is divided into two ferrite lamellae) of
lamellar growth, as shown in the virtual box of Fig. 5(b). The
growth rate of the lamellar region would decrease when the
lamellar spacing is large. The carbon atoms precipitated from
lamellar ferrite will be hoarded here until they are higher than
the critical nucleation concentration, then forming the nucleus
of cementite. Morphological characteristics are manifested as
In view of the actual situation, there will be granular pearlite
structure, but it cannot form a granular pearlite structure when
the nucleation site is at the interface of two austenite grains.
When the nucleus location is set at the interface of the austenite
phase and the pro-eutectoid phase or austenite phase, granular
pearlite can form in the simulation results, that is, a plurality of
granular cementite distribute in the ferrite matrix.
2.4 Experimental verification
It is difficult to obtain the mechanism of microstructure
evolution during the eutectoid phase transformation by
experiment. In the previous study on the formation of grain
boundary ferrite in eutectoid and hypereutectoid pearlitic steels
by Miyamoto et al , there are four microstructures, namely:
boundary ferrite, Widmanstatten ferrite, granular pearlite, and
lamellar pearlite when the carbon content is 0.75wt.% and
the transition temperature is 873 K, as shown as in Fig. 6. It
was found that the four microstructures are determined by the
orientation relationship between the austenite and pearlite. That
is, when the orientation difference is less than or equal to 5°,
Widmanstatten ferrite and boundary ferrite are liable to appear.
When the orientation difference is more than 5° and less than
or equal to 15°, discrete pearlite is liable to produce. When the
orientation difference is more than 15°, lamellar pearlite is liable
In the study, to verify the accuracy of the model indirectly,
the simulation initial conditions were set according to the actual
composition and temperature of the above experiment, with only
0.8 μm × 0.8 μm region being simulated due to the simulation
area limitation. Figure 7 shows the simulation results with the
same conditions as Fig. 6. The orientation of lamellar pearlite
is 4.5°, the orientation difference between austenite and the
lamellar pearlite is 4.5°, the orientation difference between
austenite and the lamellar pearlite is 15.5°.
Figures 6 and 7 show two kinds of microstructure with
different morphologies. The left side of Fig. 6 (a) is the pearlite,
the right side is the Widmanstatten ferrite and pearlite. The left
side of Fig. 6 (b) is the lamellar pearlite, the middle boundary
is boundary ferrite, and the right side is the martensite which
corresponds to the austenite of original microstructure. The
lower part of Fig. 7 is lamellar pearlite, the middle boundary
is the boundary ferrite, and the upper part is the austenite that
has not transformed. The reason for the formation of boundary
ferrite and Widmanstatten ferrite is that the orientation
difference between the austenite and the ferrite is less than 5º
[1.5 º for Fig. 6 (a), 3.6 º for Fig. 6 (b), and 4.5º for Fig. 7].
It can be found that the simulation results are in line with the
experimental conclusion. Boundary ferrite is easy to produce
when the orientation difference is less than 5º (austenite ),
here, as the characteristics of Widmanstatten ferrite were not set,
then Widmanstatten ferrite will not appear. Lamellar pearlite is
easy to produce when the orientation difference is greater than
15 º (austenite ).
By introducing anisotropy and orientation relationships, the
growth of the pearlite lamellae was studied by multi-phase
field method. It was found that the simulation results were
in agreement with the experimental results. Therefore the
following conclusions can be drawn:
) The growth direction of the pearlite lamellae is related to
the orientation and spacing of the pearlite lamellae. The growth
direction of the pearlite lamellae is deflected along with the
adjustment of lamellar spacing, and the deflection angle is equal
to the orientation difference between the pearlite lamellae and
) When the orientation difference between the pearlite
lamellae and austenite is less than 5 °, the pearlite lamellae does
not grow into the austenite grains, but grows in the form of
boundary ferrite. When the orientation difference between the
pearlite lamellae and austenite is greater than 5 °, the pearlite
lamellae grows into the austenite grains.
) When the nucleus of ferrite lamellae or cementite lamellae
is located at the interface of austenite and ferrite, cementite or
austenite, the K-S orientation relationship is not changed. At the
same time, the nucleus at the interface of austenite and ferrite,
cementite or austenite can induce the bifurcation of thick ferrite.
 Zhang D , Yin Y , Zhou J , et al. Phase field study of microstructure evolution in eutectoid phase transformation - I nucleation . Archives of Foundry Engineering , 2017 , 17 ( 3 ): 155 - 162 .
 Zhi X , Liu J , Xing J , et al. Effect of cerium modification on microstructure and properties of hypereutectic high chromium cast iron . Materials Science and Engineering: A , 2014 , 603 : 98 - 103 .
 Miyamoto G , Karube Y , Furuhara T. Formation of grain boundary ferrite in eutectoid and hypereutectoid pearlitic steels . Acta Materialia , 2016 , 103 : 370 - 381 .
 Smith C S. Microstructure . Trans. ASM , 1953 , 45 : 533 .
 G u o Z , K i m u r a N , Ta g a s h i r a S , e t a l . K i n e t i c s a n d Crystallography of intragranular pearlite transformation nucleated at (MnS plus VC) complex precipitates in hypereutectoid FeMn-C alloys . ISIJ International, 2002 , 42 ( 9 ): 1033 - 1041 .
 Furuhara T , Yamaguchi J , Sugita N , et al. Nucleation of Proeutectoid Ferrite on Complex Precipitates in Austenite. ISIJ International , 2003 , 43 ( 10 ): 1630 - 1639 .
 Furuhara T , Shinyoshi T , Miyamoto G , et al. Multiphase crystallography in the nucleation of intragranular ferrite on MnS plus V(C, N) complex precipitate in austenite . ISIJ International , 2003 , 43 ( 12 ): 2028 - 2037 .
 Wiessner M , Angerer P , Prevedel P , et al. Advanced X-ray Diffraction Techniques for Quantitative Phase Content and Lattice Defect Characterization during Heat Treatment of High Speed Steels . BHM Berg- und Hüttenmännische Monatshefte , 2014 , 159 ( 9 ): 390 - 393 .
 Cheng L , Wu K M. New insights into intragranular ferrite in a low-carbon low-alloy steel . Acta Materialia , 2009 , 57 ( 13 ): 3754 - 3762 .
 Cheng L , Wan X L , Wu K M. Three-dimensional analysis of ferrite allotrimorphs nucleated on grain boundary faces, edges and corners . Materials Characterization , 2010 , 61 ( 5 ): 580 - 583 .
 Cheng L. The nucleation, three dimensional morphology and growth kinetics of ferrite in low carbon high strength microalloyed steels . Wuhan: Wuhan University of Science and Technology, 2013 . (In Chinese)
 Wang Z C , Li W , Guo Z H , et al. Simulation of Ferrite/Cementite Interface with Bagaryatsky Orientation Relationship . Journal of Shanghai Jiao Tong University, 2005 , 39 ( 1 ): 14 - 17 . (In Chinese)
 Steinbach I , Pezzolla F , Prieler R . Grain selection in faceted crystal growth using the phase field theory . Minerals , Metals and Materials Society , Warrendale, PA (United States) , 1995 .
 Eggleston J J , Mcfadden G B , Voorhees P W. A phase-field model for highly anisotropic interfacial energy . Physica D Nonlinear Phenomena , 2001 , 150 ( 1 ): 91 - 103 .
 Yamanaka A , Takaki T , Tomita Y. Multi-phase-field modeling of diffusive solid phase transition in carbon steel during continuous cooling transformation . Journal of Crystal Growth , 2008 , 310 ( 7- 9 ): 1337 - 1342 .
 Read W T , Shockley W. Dislocation Models of Crystal Grain Boundaries . Physical Review , 1950 , 78 ( 3 ): 275 - 289 .
 Mangan M A , Shiflet G J. The Pitsch- Petch orientation relationship in ferrous pearlite at small undercooling . Metallurgical and Materials Transactions A , 1999 , 30 ( 11 ): 2767 - 2781 .
 Nakajima K , Apel M , Steinbach I. The role of carbon diffusion in ferrite on the kinetics of cooperative growth of pearlite: A multiphase field study . Acta Mater. , 2006 , 54 ( 14 ): 3665 - 3672 .