APCOM’07 in conjunction with EPMESC XI, December 3-6, 2007, Kyoto, JAPANDevelopment of Phase-Field Model of Boronizing Process on
SteelsR.D. Ramdan1*, T. Takaki2, and Y. Tomita3
1

Graduate School of Science and Technology, Kobe University, 1-1, Rokkodai-chou, Nada, Kobe,
657-8501, Japan
2
Graduate School of Science and Technology, Kyoto Institute of Technology, Matsugasaki, Sakyo, Kyoto,606-8585, Japan
3
Graduate School of Engineering, Kobe University, 1-1, Rokkodai-chou, Nada, Kobe, 657-8501, Japane-mail:ramdan@solid.mech.kobe-u.ac.jp, takaki@maritime.kobe-u.ac.jp, tomita@mech.kobe-u.ac.jp
Abstract Boronizing is a thermo-chemical treatment that has been widely used to improve surface
characteristics of metal. However limited study can be found on the detail behavior of phasetransformation during this process. Therefore the present research tries to develop phase-field model tosimulate the boride phase growth on the iron steels. The phase-field equations were derived from
Ginzburg-Landau free-energy functional using Allen-Cahn and Cahn-Hilliard equations. Pre-exponential
factor for interface mobility, M0, and temperature of process, T, were used as the variable parameter. It
was observed, even though the present variations of M0 does not influence the phase transformation rate,
an optimization of this parameter is required in order to meet a good agreement between the phasetransformation rate and boron concentration profile, since a slight difference of boron concentration
profile was found for the variations of M0 parameter. On the other hand, it was also observed that as thetemperature of process increases from 1000K up to 1300K there is an increasing of phase transformation
rate, which is also confirmed by the higher diffusion of boron. These observations agree with theprevious finding that higher temperature is required in order to create more Fe2B phase from austenite
phase.
Key words:

Phase-field method, Numerical simulation, Boronizing process, Iron steels

INTRODUCTION
Boronizing is a type of thermo-chemical treatment where boron atom is diffused into the metal substrateduring high temperature process to form hard boride structure. FeB and Fe2B are two common types ofboride structure that is resulted after the process [1, 2]. The former is considered to be harder than thelater but it is more brittle [1, 2]. Normally it is preferable to have single layer of Fe2B on the substraterather than having these two carbide phases in the coating layer. The presence of single or mixed layer ofboride coating mainly depends on the condition of boronizing process. In most cases FeB phase isconsidered to form firstly before Fe2B phase especially for the low temperature process [3]. However, forthe case of high temperature process, 950-1100oC, and with the longer boronizing time single Fe2B
boride phase is possible to exist [2].
Beside having higher hardness compared to other types of thermo-chemical treatment products,
boronizing product is also believed to offer a better strength to friction wear and abrasion [4-6].
Furthermore, its application in industry has been well accepted for long time such as EKabor boronizing
agents that have proved to be very effective in industrial practice for more than 3 decades [7]. Howeverless study on this process can be found in the literature compared to other types of thermo-chemicaltreatment. Therefore the present study made an effort to study the boronizing process by simulation work,
especially on the kinetics of phase transformation and the boron concentration diffusion. Experimentally,
this study requires a complex characterization work including the complex of sample preparation and thecharacterization technique it self. And it is expected, by conducting one dimensional phase-fieldsimulation, the simpler and easier way to study the kinetics of phase transformation of boride fromaustenite can be done and therefore the better understanding of this matter will also be obtained.
To the authors knowledge there are only few simulation works on boronizing process can be found in theliterature. Among them are simulation works on boronizing process conducted by Brakman et al. [8] andKeddam [1]. Brakman et al. assume a parabolic growth of boron diffusion during boronizing process andtake into account the specific volume difference between FeB and Fe2B. On the other hand, Keddam
works on the simulation of the growth kinetic of Fe2B, taking into account the thermodynamic propertiesof the Fe-B phase diagram and adopting diffusivity of boron by Campos et al. [9]. In the present study,
phase-field method was used to simulate the growth of single Fe2B phase from austenite phase. Thismethod has been proved to simulate many kinds of phase transformations, such as austenite-ferrite
transformation in steel [10]. One important point from this method is the unnecessary of calculating thecomplex condition of the interface, which might bring difficult problem for the simulation works. In thefollowing section, the construction of phase-field model for simulating the growth of Fe2B from austenite
phase will be presented.
PHASE-FIELD MODEL
Phase-field model for the evolution of boride phase during boronizing process was derived from
Ginzburg-Landau free energy functional that take into account the chemical free energy and gradientenergy as⎡ε22⎤∇φ ⎥dV ,G = ∫ ⎢ g (φ , X B , T ) +
2
⎦
V ⎣

(1)

where φ is the phase field variable which varies smoothly across the interface from φ = 0 in the austenite
phase to φ = 1 in the boride (Fe2B) phase, XB is the mole fraction of boron, T is the absolute temperatureand ε is the gradient energy coefficient. The chemical free energy density g(φ, uB, T) is expressed asBBg (φ , u B , T ) = p(φ )GmFe 2 B + (1 − p(φ ))Gmγ + Wq(φ ) ,(2)

where GmFe 2 B and Gmγ are the chemical free energies of pure Fe2B and austenite phases, respectively.
p(φ) is an interpolating function expressed as p(φ) = φ 3(10-15 φ +6φ2) and q(φ) is a double well function
with q(φ) = φ2(1- φ) 2. W is the potential height defined as a function of interface energy (σ) and interface
thickness (δ) as in the following equation
W =

6σb

δ

,

(3)

where b is the constant which is defined as 2 tanh −1 (1 − 2λ ) and λ is constant that is set at 0.1. Sincephase field is a non-conserved order parameter, its time evolution equation can be derived from theabove functional with the assumption that the total free energy decreases monotonically with time
following the Allen-Cahn equation as follows,

⎧
∂φ
1 ⎞⎫
δG
⎛ 15
= −M φ= M φ ⎨ε 2∇ 2φ + 4Wφ (1 − φ )⎜ −φ (1 − φ )Δg + φ − ⎟⎬ ,
∂t
2 ⎠⎭
δφ
⎝ 2W
⎩

(4)

where Mφ is the kinetic parameter related to the mobility of the austenite-boride interface M as
Mφ =

2W
M.
6ε

(5)

On the other hand the interface mobility value, M, that will be used in the present research will follow
the similar approach that is used by Mecozzi et al. [11] and Krielaart [12], where they defined interface
mobility has a simple exponential dependence on temperature,

M = M 0 exp(− E / RT ) ,(6)where the activation energy, E, was taken from the experimental data conducted by Hasan et al. [13].
Pre-exponential factor, M0, will be used as the variable or fitting parameter as also performed by
Mecozzi et al. and Krielaart. Vooijs et al. [14] proposed that the expected order of magnitude of M0 canbe predicted using the following equation
M0 =

kd
,
hR

(7)

where d is the real interface thickness, h, R and k is the Planck’s, gas and Boltzmann’s constants,
respectively. However the reliability of this approach requires further investigation, therefore Krielaartused a wide variation value in order to optimize this parameter, even though their prediction on this value
is around 0.8 m/Js as calculated by Eq.7 using real interface thickness of 3 angstrom. For the present
simulation work the optimization of M0 values was done and it was varied at 0.8, 1,1.2, 1.4, 1.6 and 1.8
mol m/Js.
In Eq. 4, Δg is the chemical driving force and is defined as Δg = GmFe2 B − Gmγ . Where GmFe2 B and G mγ
are the molar Gibbs free energy for boride and austenite respectively, and is also used as the value forchemical free energy of boride and austenite in the Eq.2. The molar Gibbs free energy in austenite phase,Gmγ , was derived using CALPHAD method for two-sublattice model aso γγGmγ = yFe yB oGFe
: B + y Fe y◊ GFe:◊ + RTyB ln y B + RTy◊ ln y◊ ,where y Fe and yB is the site fraction for iron and boron, respectively.

(8)
o

γ
GFe
:B is the reference energyγterm for boron austenite and oGFe:◊ is the reference energy state for vacancy in austenite. In stead ofusing site fraction, molar fraction (XFe and XB, molar fraction for iron and boron respectively) is used in
the present research. The site fraction is defined by molar fraction as
B

yB =

a XB
,
c (1 − X B )(9)where a is equal to c since we consider the fcc austenite Fe. On the other hand yFe is set as 1 since there isno other alloying element in the matrix.
o
o

γ
GFe
: B is obtained from Fe-B thermodynamic optimization value by Van Rompaey et al. [15] as follow

γ
ser
ser
G Fe
:B = H B + H Fe + 45496 − 77 .534 T ,

(10)

ser
where H Fe
and H Bser is the enthalpy at reference state for iron and boron respectively and is obtained
from SGTE database for pure elements [16]. On the other hand, the reference energy state for vacancy inγaustenite, oGFe:◊ , in Eq.8 is obtained from Per Gustafson [17] and defined as

o

γ
SER
2
−8 3
GFe
:◊ = H Fe − 237.57 + 132.416T − 24.6643T ln T − 0.00375752T − 5.89269 E T+ 77358.5 / T(11)In order to allow a solubility range in the Fe2B phase, the chemical free energy of Fe2B is consideredhaving a parabolic function of composition, XB, following the similar approach by Huh et al. [18]B

G mFe2 B = 2. × 10 5 ( X B − 0.33) + o G Fe2 B ,2(12)where oG Fe2 B is the reference energy term for the stoichimetric composition of Fe2B and is obtained
from optimization value by Van Rompaey et al. [15] as followoser
G Fe2 B = H Bser + 2 H Fe
− 93363 + 481.992T − 79.04999T ln T − 0.007071T 2 + 731991T −1 ,(13)

From Ginzburg-Landau free energy functional in Eq. 1, the nonlinear diffusion equation can also bederived by following the Cahn-Hilliard equation as follow

⎧⎪
⎛ ∂2g
⎞⎫⎪
∂X B
∂2g
⎟⎟⎬ ,
∇
= ∇ ⎨M B (φ , X B , T )⎜⎜
∇
+
φ
X
B
2
∂
∂
∂t
X
φ
∂
X
⎪⎩
B
⎝ B
⎠⎪⎭

(14)

where M B is diffusion coefficient of boron, and is defined as a function of diffusion coefficient of boron
in austenite, M Bγ , and boride, M BFe 2 B .

(M B (φ , X B , T ) = M BFe 2 B) ( ) (M )( ( )) .
pφ

γ1− p φ(15)

B

Both M Bγ and M BFe 2 B has a temperature dependence function as

M Bi = K exp (− E / RT ) ,(16)

where M Bi is the diffusion coefficient of boron in austenite or boride (i = γ or Fe2B), and K is thepre-exponential parameter of diffusivity. K values for boron in austenite and boride are set at 4.4x10-8
m2/s and 1.13x10-6 m2/s from Ref. [1].
COMPUTATIONAL MODEL
We performed one dimensional finite difference simulations of Eqs. 4 and 14. The computational domain
is divided into 100 finite difference grids. Initial boron concentration was set at XB(i) = 0.5-0.001i, for iless than 10, where i is the grid point number from left end, and at XB(i) = 0.3-0.0018i for the subsequent
grids. On the other hand, initial phase-field variable was set as one dimensional equilibrium profile withorigin at i = 10. Both of initial phase-field and boron concentration profile are illustrated in the Fig. 1.
The size of computational domain is 100 μm and Zero Neumann boundary condition is applied at bothends of computational domain. The interface thickness δ is set at δ =12Δx = 12 μm where Δx is grid size,
whereas the interface energy, σ = 1.0 J/m2.
B

B

Fig. 1 Initial profiles for boron concentration (a) and phase field (b)SIMULATION RESULTS

Figure 2 shows the typical boride layer that is growth on the steel substrate after boronizing process.
Before this boride phases forms on the steel substrate, certain amount of boron has been diffused into
certain depth of thickness inside the steel substrate as also explain by Xu et al. [2]. Here, the initial boron
concentration profile is assumed to have a linier function on the depth of thickness and is set as XB(i) =0.5-0.001i, for i less than 10, and at XB(i) = 0.3-0.0018i for the subsequent grids, as has been explained in
the previous section. The grid points here, i, is a representation of the depth of profile of boron inside thesteel substrate.
BBborideSteel substrate

Fig. 2 Typical boride growth on steel substrate [19]Figure 3 shows the effects of pre-exponential factor of interface mobility, M0, on the phase-fieldevolution, where the temperature is set constant at 1300K. In each figure of Fig.3, the growth of phasefield is from the solid line (100 steps), followed with the long-dashed line (1000 steps) and ended withthe dot line (2000 steps). From these figures, it can be seen that there is no significant differences
between one figure to other, which give a suggestion that the present variations of M0 have no effect on
the kinetics of phase transformation. It is quite difficult to explain the reason why there is no effect of

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3 The effect of pre-exponential factor of interface mobility on the kinetic of phase transformation(a) 0.8 mol m/Js, (b) 1.0 mol m/Js, (c) 1.2 m/Js, (d) 1.4 m/Js, (e) 1.6 m/Js and (f) 1.8 mol m/Js

this parameter on the phase transformations. However two explanations may be can be applied on thisfact. The first is that the prediction on the order of magnitude of this parameter based on Eq. 7 is quite
good and it requires higher order of magnitude of this parameter to change the kinetic of phasetransformation. The second explanation is that other setting parameters for this simulation works might
suppress the effect of this parameter on the kinetics of phase transformation.

(a)

(d)

(b)

(c)

(e)

(f)

Fig. 4 The effect of pre-exponential factor of interface mobility on the boron concentration profile(a) 0.8 mol m/Js, (b) 1.0 mol m/Js, (c) 1.2 m/Js, (d) 1.4 m/Js, (e) 1.6 m/Js and (f) 1.8 mol m/Js
On the other hand, Fig. 4 shows the effect of M0 on the boron concentration profile. The solid line,
long-dashed line and dot line in these figures indicate the boron concentration profile after 100, 1000 and2000 steps respectively. Generally, all figures show that there is a depletion of boron concentration at thegrid numbers less than 10 which may correspond with the near-surface of boron in the real boronizing
process. On the other hand, there is an increasing of boron concentration at the subsequent grid numbers.
This observations give a suggestion that the near-surface boron diffuse into the inner part of substrate toform the thicker layer of boride as also confirm in the Fig. 3. This finding is also in a flow with theobservation by Xu et al.[2] that the growth of boride structure is accompanied with the depletion of boron
concentration on the surface. In addition, by relating Figs. 3 and 4, it can be seen that generally, boride
phase in the inner part of substrate has a lower boron concentration compared with the boride phase in thenear surface depth of profile. This situation is also observed by Keddam [1], where his calculation on theboron concentration profile resulted a linier decreasing of boron concentration in the area of boride phaseas the depth of thickness increases.

(a)

(b)

(c)

(d)

Fig. 5 The effect of Temperature of process on the kinetics of phase transformation (a) T = 1000K, (b) T =1100K (c) T = 1200K and (d) T = 1300K
By comparing each figure in the Fig. 4, it can also be observed that a slight difference of boron
concentration profile exists with the present variations of M0. It can be seen that as the M0 value increases
from 0.8 up to 1.8, there is a decreasing of diffusion of boron which is represented by the slower
changing of boron profile from the black line up to the dot line. This condition indicates that higher valueof interface mobility might inhibit the role of diffusivity on the phase transformation and optimization ofthis value is important in order to get the correct relation between the phase transformation kinetics andthe diffusion of boron. Therefore it can be concluded that even though the present variations of M0 does
not have significant effect on the kinetics of phase transformation, however the determination of thecorrect value of this parameter is important in order to get the correct relation between the kinetics ofphase transformation and the boron concentration profile.
Figure 5 shows the effect of temperature of process on the kinetics of phase transformation where M0
value is set at 0.8 mol m/Js. It can be observed from this figure that there is an increasing of kinetics ofphase transformation as the temperature of process increases. This condition is considered due to theincreasing of kinetics of transformation as the temperature of process increases as also observed by
Keddam [1]. This observation is also in a flow with the observation by Xu et al. [2], that single phase of
Fe2B may be exist by higher temperature of boronizing process in the range of 950-1100oC. Thisobservation is also confirmed by the Fig. 6 that there is a slight difference of boron concentration profile

with the present variations of temperature. It can be seen from this figure that as the temperature ofprocess increases, a higher diffusion of boron is observed. One point to be noted here is, since the present
simulation works used similar value of pre-exponential diffusivity (K) for all temperature due to theunavailability data of K value of boron in boride for temperature other than 1300K, it is supposed that byimplementing precise value of K for each temperature, a better understanding on the effect of temperatureon the kinetic of phase transformation will be obtained.

(a)

(b)

(c)

(d)

Fig. 6 The effect of Temperature of process on the boron concentration profile, (a) T = 1000K, (b) T =1100K (c) T = 1200K and (d) T = 1300K

CONCLUSIONS

Phase-field model for the growth of Fe2B phase from austenite phase was developed. For this purpose,several important parameters were taken from several literatures while the pre-exponential factor forinterface mobility (M0) and temperature of process ( K ) were chosen as the variable parameters. It wasobserved that the present variations of M0 parameter does not influence the kinetics of phasetransformation, which might indicate that a higher order magnitude of this parameter may be required tochange the kinetics of phase transformation or the effect of this parameters might be suppressed by thesetting of other parameter in the present simulation works. However since a slight difference in the boron
concentration profiles were found with the present variations of M0 value, it indicates that determination
the optimum value of this parameter is required in order to get a good understanding between the kineticsof phase transformation and boron concentration profile. From the study on the effect of temperature ofprocess on the kinetics of phase transformation and boron concentration profile, it was observed that there
is a good agreement between the present simulation works and the previous study that higher temperatureis required in order to allow higher phase transformation from austenite to Fe2B. After all, it can beconcluded that one dimensional phase-field method can be a good option for the easier and quicker studyof the kinetics of phase transformation and boron concentration profile which might bring difficulty whenstudy this subject experimentally.
Acknowledgements

The authors would like to thank Ministry of Education, Culture, Sports, Science and Technology Japanfor the funding of the present studies.
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