Diffusion model and characterisation of fe2 b layers on aisi 1018 steel

Int. J. Surface Science and Engineering, Vol. 9, No. 4, 2015 281
Copyright © 2015 Inderscience Enterprises Ltd.
Diffusion model and characterisation of Fe2B layers
on AISI 1018 steel
Martín Ortiz-Domínguez and Jorge Zuno-Silva
Universidad Autónoma del Estado de Hidalgo,
Campus Sahagún, Carretera Cd. Sahagún-Otumba s/n, Hidalgo, México
Email: mortizd@ipn.mx
Email: zunojorge@gmail.com
Mourad Keddam*
Laboratoire de Technologie des Matériaux,
Département de Sciences des Matériaux,
Faculté de Génie Mécanique et Génie des Procédés,
USTHB, B.P. No. 32, 16111 El-Alia,
Bab-Ezzouar, Algiers, Algeria
Email: Keddam@yahoo.fr
*Corresponding author
Omar Damián-Mejía
Instituto de Investigación en Materiales,
Universidad Nacional Autónoma de México-UNAM,
Circuito Exterior, s/n Ciudad Universitaria,
Coyoacán, CP 04510, México
Email: odamian@iim.unam
Milton Elias-Espinosa
Instituto Tecnológico y de Estudios Superiores de Monterrey-ITESM,
Campus Santa Fe, Av. Carlos Lazo No. 100,
Del. Álvaro Obregón, CP. 01389, México
Email: mielias@itesm.mx
Abstract: An approach was suggested to study the kinetics of formation of
Fe2B layers grown at the surface of AISI 1018 steel by the pack-boriding
process. The kinetic model considered the principle of mass conservation at the
(Fe2B/substrate) interface to evaluate the boron diffusion coefficient of Fe2B in
the temperature range of 1,123–1,273 K. Validation of the model was made by
comparing the experimental Fe2B layer thickness with the predicted value for
the borided sample at 1,253 K during 5 h. In addition, the generated boride
layers were analysed by optical microscopy, scanning electron microscopy
(SEM), energy dispersive X-ray spectroscopy (EDS) and X-ray diffraction
analysis. Finally, a contour diagram describing the evolution of Fe2B layer
thickness as a function of treatment time and temperature was also proposed.
Finally, the boron activation energy for AISI 1018 steel was estimated as
159.3 kJ mol–1
basing on our experimental results.

282 M. Ortiz-Domínguez et al.
Keywords: boriding; incubation time; diffusion model; growth kinetics;
activation energy.
Reference to this paper should be made as follows: Ortiz-Domínguez, M.,
Zuno-Silva, J., Keddam, M., Damián-Mejía, O. and Elias-Espinosa, M. (2015)
‘Diffusion model and characterisation of Fe2B layers on AISI 1018 steel’,
Int. J. Surface Science and Engineering, Vol. 9, No. 4, pp.281–297.
Biographical notes: Martín Ortiz-Domínguez is a Research Professor at
Escuela Superior de Ciudad Sahagún-UAEH. He received his PhD in
Mechanical Engineering from Escuela Superior de Ingeniería Mecánica y
Eléctrica-Zacatenco, Instituto Politécnico Nacional, México – 2013. He has
published over 35 publications in the field of boriding process and classical
electrodynamics. He is a member of the National System of Researchers of the
National Council of Science and Technology at México.
Jorge Zuno-Silva studied a PhD on Materials at the University of Sheffield,
UK in 2009. He works mainly with steels on heat treatments, mechanical
properties, processing, wear, microscopy characterisation, coatings. Currently,
he is in charge of the Research Group of the Escuela Superior de Ciudad
Sahagún-UAEH, developing projects research on advanced high strength steels
in collaboration with international automotive industries.
Mourad Keddam is a Full Professor in the Department of Materials Science
at USTHB, Algiers, Algeria. He received his PhD in Metallurgy from
Ecole Nationale Polytechnique, Algiers in 2004. He has published over
50 publications in the field of thermochemical treatments and modelling of the
mass transfer phenomena.
Omar Damián-Mejía studied PhD on Materials at UNAM, México in 2014. He
works mainly with steels on thermochemical treatments, mechanical properties,
processing, wear, microscopy characterisation and thin films. He has published
nine articles focused on mass transfer models.
Milton Elias-Espinosa received his PhD in Mechanical Engineering from
Escuela Superior de Ingeniería Mecánica y Eléctrica-Zacatenco, Instituto
Politécnico Nacional, México in 2013. His scientific interests include materials,
mechanical design and wear. Since 2001, he is a Full Professor in the Instituto
Tecnológico y de Estudios Superiores de Monterrey in México.
1 Introduction
Boriding is a thermochemical treatment where boron atoms diffuse into the substrates of
ferrous and non-ferrous alloys to form a hard layer consisting of metallic borides (Sinha,
1991). The boriding process provides a high surface hardness, wear properties in terms of
adhesion, abrasion and surface fatigue. The boriding treatment is carried out by heating
the substrates in the temperature range of 800–1,050°C for a period of time ranging from
0.5 to 12 h in. Boron is supplied at the material surface by a gaseous, liquid or solid
(termed as ‘pack’ and paste) medium (Kulka et al., 2013; Nait Abdellah, 2012a; Campos
et al., 2003; Kaouka et al., 2014).
The most frequently used method is pack-boriding because of its technical advantages
and cost-effectiveness (Meric et al., 2000; Vipin and Sundararajan, 2002). The

Diffusion model and characterisation of Fe2B layers on AISI 1018 steel 283
morphology of the boride layer is influenced by the presence of alloying elements in the
matrix. Saw-tooth shaped layers are obtained in low-alloy steels or Armco iron whereas
in high-alloy steels, the interfaces are flat.
Boriding has received modelling attention. So, the modelling of boriding kinetics is
considered as a suitable tool to select the optimised parameters for obtaining a desired
boride layer of the treated material for its practical utilisation in the industry. For
instance, some recent diffusion models were reported in the literature for analysing the
growth of Fe2B layers grown on different substrates (Campos-Silva et al., 2009, 2010,
2011; Keddam and Chegroune, 2010; Nait Abdellah et al., 2013) by considering the
boride incubation times. In the work published by Campos-Silva et al. (2009), the boron
diffusivity in Fe2B was evaluated in the temperature range of 1,173–1,273 K by
considering the temperature dependence of boride incubation time. The growth kinetics
of (FeB/Fe2B) bilayer formed on the AISI 316 L steel’s substrate was also investigated by
Campos-Silva et al. (2010). In their work, the boron activation energy was evaluated for
each boride layer with the presence of a diffusion zone. The kinetics of formation of Fe2B
layers on the Armco iron substrate by the paste-boriding process was studied by
Campos-Silva et al. (2011) and boron activation energy was determined as 157 kJ mol–1
.
The effect of chemical stresses on the boron diffusivity in Fe2B was studied by Nait
Abdellah et al. (2013) when pack-boriding the AISI 4140 steel in the temperature range
of 1,123–1,273 K.
The aim of the present study was to propose a diffusion model for estimating the
boron diffusion coefficient in Fe2B during the pack-boriding of AISI 1018 steel. This
approach was based on the principle of mass conservation at the (Fe2B/substrate)
interface. The pack-borided AISI 1018 was characterised by means of the following
experimental techniques: (optical microscopy, scanning electron microscopy (SEM),
EDS and XRD analyses). Based on the experimental data, the boron activation energy
was also evaluated when pack-boriding the AISI 1018 steel in the temperature range of
1,123–1,273 K.
2 The kinetic model
The model considers the growth of Fe2B layer on a saturated substrate with boron atoms
as displayed schematically in Figure 1.
The f(x, t) function represents the boron distribution in the substrate before the
nucleation of Fe2B phase. 2Fe B
0t corresponds to the incubation time required to form the
Fe2B phase when the matrix reaches a saturation state with boron atoms. 2Fe B
upC denotes
the upper limit of boron content in Fe2B ( = 60 × 103
mol m–3
), 2Fe B
lowC is the lower limit
of boron content in Fe2B ( = 59.8 × 103
mol m3
) and the point x(t = t) = v represents the
Fe2B layer thickness (Brakman et al., 1989). The term B
adsC is the effective adsorbed
boron concentration during the boriding process (Yu et al., 2005). From Figure 1,
2 2Fe B Fe B
1 up lowa C C= − defines the homogeneity range of the Fe2B layer, 2Fe B
2 0lowa C C= − is
the miscibility gap and C0 is the boron solubility in the matrix. The diffusion zone
underneath the Fe2B layer can be ignored by setting (C0 ≈ 0 mol m–3
(Okamoto, 2004;
Nait Abdellah, 2012b). The following assumptions are considered during the formulation
of the diffusion model:

• the kinetics is dominated by the diffusion-controlled mechanism
• the growth of Fe2B layers is a consequence of the boron diffusion perpendicular to
the sample surface
• the iron boride nucleates after a certain incubation time
• the boride layer is thin in comparison to the sample thickness
• local equilibrium is attained at the moving interface
• planar morphology is assumed for the phase interface
• the volume change during the phase transformation is not considered
• the diffusion coefficient of boron in Fe2B is independent on the concentration and
follows an Arrhenius relationship
• the presence of porosity is neglected during the boron diffusion
• a uniform temperature is assumed throughout the sample
• the influence of the alloying elements on the layer growth kinetics is not taken into
account.
Figure 1 A schematic boron-concentration profile through the Fe2B layer (see online version
for colours)
The initial and boundary conditions for the diffusion problem are represented as:
2Fe B 00, 0, with: [ ( ), 0] 0.t x C x t t C= > = = ≈ (1)

Boundary conditions:
( )2 2
2
2
Fe B Fe B
Fe B 00 0
Fe B
up
B 3 3
ads
v ,
(the upper boron concentration is kept constant),
for 60 10 mol m ,
C x t t t t
C
C −
⎫⎡ ⎤= = =⎣ ⎦ ⎪⎪
⎬=
⎪
> × ⎪⎭
(2)
2
2
Fe B
Fe B
low
B 3 3
ads
[ ( ) v, ]
(the boron concentration at the interface is kept constant),
59.8 10 mol m ,
C x t t t t
C
C −
= = = ⎫
⎪
= ⎬
⎪< × ⎭
(3)
v0 is a thin layer with a thickness of ≈ 5 nm that formed during the nucleation stage
(Dybkov, 2010). Thus, v0 ( ≈ 0) when compared to the thickness of Fe2B layer (v). The
mass balance equation at the (Fe2B/substrate) interface can be formulated by equation (4)
as follows:
2 2
2
Fe B Fe B
up 0low
Fe B
Fe
2
( v) ( v, )( )
2
( v v, )( ),
C C C
A d J x t t A dt
J x d t t A dt
⎛ ⎞+ −
⋅ = = = ⋅⎜ ⎟
⎝ ⎠
− = + = ⋅
(4)
where A ( = 1·1) is defined as the unit area and C0 represents the boron concentration in
the matrix. The flux JFe2B and JFe are obtained from the Fick’s first law as:
2 2 2Fe B Fe B Fe B v[ ( ) v, ] [ ( ) v, ]/ ,{ }xJ x t t t t D C x t t t t x == = = = − ∂ = = = ∂ (5)
and
Fe Fe Fe v v[ ( ) v v, ] [ ( ) v v, ]/{ }x dJ x t t d t t D C x t t d t t x +== = + = = − ∂ = = + = ∂ (6)
The term JFe is null since the boron solubility in the matrix is very low ( ≈ 0 mol m–3
)
(Okamoto, 2004; Nait Abdellah, 2012b).
Thus, equation (4) can be written as:
2 2
2
2
Fe B Fe B
up 0 Fe Blow
Fe B
( ) v ( ) v
2 [ ( ), ]( )
.
2 x t x t
C C C C x t t t tdx t
D
dt x= =
⎛ ⎞+ − ∂ = =
= −⎜ ⎟
⎝ ⎠ ∂
(7)
If the boron concentration profile in Fe2B is constant for the treatment time, the Fick’s
second law is reduced to an ordinary second-order differential equation as follows:
2 2
2
2
Fe B Fe B
Fe B 2
[ ( ), ] [ ( ), ]
.
C x t t C x t t
D
t x
∂ ∂
=
∂ ∂
(8)
By solving equation (8) and applying the boundary conditions proposed in equations (2)
and (3), the boron concentration profile in Fe2B is expressed by equation (9) if the boron
diffusion coefficient in Fe2B is constant for a particular temperature:
2 2
2
2
2
2
Fe B Fe B
uplowFe B
Fe B up[ ( ), ] .
v 2
2
Fe B
Fe B
C C x
C x t t C erf
D terf
D t
− ⎛ ⎞
= + ⎜ ⎟⎛ ⎞ ⎝ ⎠
⎜ ⎟
⎝ ⎠
(9)

By substituting the derivative of equation (9) with respect to the distance x(t) into
equation (7), equation (10) is obtained:
2 2 2 2
2
2
2
Fe B Fe B Fe B Fe B 2
up 0 upFe Blow low2 v v
exp ,
v2 4
2
Fe B
Fe B
C C C C CDd
dt πt D t
erf
D t
⎛ ⎞+ − − ⎛ ⎞
= −⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎛ ⎞ ⎝ ⎠
⎜ ⎟
⎝ ⎠
(10)
for 0 ≤ x ≤ v.
By substituting the derivative of the parabolic growth law: ( )2
1/2 1/2
Fe Bv 2εD t= with
respect to the time t into equation (10), equation (11) is deduced:
( )
2 2 2 2Fe B Fe B Fe B Fe B
up 0 uplow low 2
2 1
exp .
2 ( )
C C C C C
ε ε
π erf ε
⎛ ⎞+ − −
= −⎜ ⎟
⎝ ⎠
(11)
The normalised growth parameter (ε) for the (Fe2B/substrate) interface can be estimated
numerically by the Newton-Raphson method. It is assumed that expressions 2Fe B
up ,C
2Fe B
low ,C and C0, do not depend significantly on temperature (in the considered temperature
range) (Brakman et al., 1989). 2
2 2
Fe B2 2 2
Fe B Fe B v 0(v 4 4 ( ))ε D t ε D t t= = + is depicted in
Figure 2 where 2Fe B
v 0( )t t t= − is the effective growth time of the Fe2B layer and t is the
treatment time.
Figure 2 A schematic representation of the square of the layer thickness against treatment time

3 Experimental details
The samples to be pack-borided are from the AISI 1018 steel. This steel has a nominal
chemical composition ( in weight percent) of 0.15–0.20% C, 0.15–0.35% Si, 0.60–0.90%
Mn, 0.040% P, 0.050% S. The samples were cut in the dimension of 10 mm × 10 mm ×
10 mm, polished, ultrasonically cleaned in an alcohol solution and deionised water for
15 min at room temperature. Finally, the samples were dried and stored under clean-room
conditions. The powder-pack boriding process was carried out in a conventional furnace
under a pure argon atmosphere in the temperature range of 1,123–1,273 K. Four
treatment times (2, 4, 6 and 8 h) were selected for each boriding temperature. The
samples were then packed in a closed cylindrical case (AISI 304L) that contains a
Durborid fresh powder mixture having an average size of 30 μm. After this
thermochemical treatment, the samples were removed from the furnace and slowly
cooled to room temperature. The cross-sectional morphology of the boride layers was
observed with the Olympus GX51 optical microscope in a clear field and SEM (JEOL
JSM 6300 LV). The elemental distribution within the cross-section of boride layer was
determined by Electron Dispersive Spectroscopy (EDS) equipment (JEOL JSM 6300 LV)
from the surface. The boride layer thickness was automatically measured with the aid of
MSQ PLUS software.
The phases of the boride layers were investigated by an X-ray diffraction (XRD)
equipment (Equinox 2000) using CoKα radiation of 0.179 nm wavelength. To identify the
phases from the diffraction peaks, the data of the JCPDS database were used (JCPDS,
1999).
4 Results and discussion
4.1 Microstructure of boride layers
Figure 3 shows the cross-sectional view of optical images of the Fe2B layers formed on
AISI 1018 steel at 1,273 K for different treatment times. The obtained microstructure is a
single phase layer composed of Fe2B. The Fe2B layers are very dense and homogenous,
having a saw-toothed morphology. Since the growth of the saw-toothed boride layer is a
controlled diffusion process with a highly anisotropic nature, higher temperatures and/or
longer times encouraged the Fe2B crystals to make contact with adjacent crystals and
forced them to retain an acicular shape (Palombarini and Carbucicchio, 1987).
The diffusion kinetics is accelerated with prolonged process duration and the Fe2B
layer thickness increased with the boriding time (Figure 3). To ensure the reproducibility
of the measured layers thicknesses, 50 measurements were collected in different sections
of the borided AISI 1018 steel’s samples to estimate the Fe2B layer thickness; defined as
an average value of the long boride teeth (Campos-Silva et al., 2010, 2013). All thickness
measurements were taken from a fixed reference on the surface of the borided AISI 1018
steel, as illustrated in Figure 4.

Figure 3 Optical micrographs of the boride layers formed at the surface of AISI 1018 steel
treated at 1,273 K during a variable time, (a) 2 h (b) 4 h (c) 6 h (d) 8 h
(a) (b)
(c) (d)
Figure 4 Schematic diagram illustrating the procedure for estimation of boride layer thickness in
AISI 1018 steel

4.2 SEM observations and EDS analysis
Figure 5(a) gives the cross sectional view by SEM of the boride layer formed on the
AISI 1,018 steel at 1,223 K for 6 h. The morphology of the boride layer is a
saw-tooth structure. This typical structure helps improving the mechanical adherence
at the (Fe2B/substrate) interface. The EDS analysis presented in Figure 5(b)
indicates the presence of two elements: iron and manganese. Figure 5(c) shows the
presence of carbon and silicon elements in the vicinity of the (Fe2B/substrate)
interface. During the boriding process; the boride layer expels carbon and silicon from
the surface into the substrate matrix. As a consequence, there is a formation of
silicoborides (FeSi0.4B0.6 and Fe5SiB2) and boroncementite (Fe3B0.67C0.33) (Dukarevich et
al., 1973).
Figure 5 (a) SEM micrographs of the cross-sections of the borided AISI 1018 steel at 1,223 K for
6 h (b) EDS spectrum of borided sample at surface (c) EDS spectrum of borided sample
at interface
(a)
(b)

Figure 5 (a) SEM micrographs of the cross-sections of the borided AISI 1018 steel at 1,223 K for
6 h (b) EDS spectrum of borided sample at surface (c) EDS spectrum of borided sample
at interface (continued)
(c)
4.3 XRD analysis
Figure 6 gives the XRD pattern obtained at the surface of borided AISI 1018 steel at
1,273 K for a treatment time of 8 h. To identity the Fe2B phase, the file (00–036–1332)
from the JCPDS database was used (JCPDS, 1999) and XRD analysis confirms the
presence of Fe2B phase.
It is seen that the intensities of diffraction peaks are different and depend on the
crystallographic orientations of Fe2B crystals. During the first stage, the crystals of Fe2B
as needles start to form in the contact zone between the metal surface and the powder
particles by covering the metal surface.
During the second stage, significant amounts of Fe2B crystals continue to grow in the
direction of the substrate. During the third stage, all the Fe2B crystals develop along a
preferred direction in order to minimise the mechanical resistance due to the competitive
growth of adjacent crystals (Palombarini and Carbucicchio, 1987). In fact, this textured
growth is attributed to the crystallographic nature of Fe2B phase which crystallises in a
body-centred tetragonal lattice. In the powder-pack boriding, the active boron is supplied
by the powder mixture. To form the Fe2B phase on any borided steel, a low boron
potential is required as reported in the reference work (Vipin and Sundararajan, 2002)
where a high amount of active boron in the powder mixture leads to the bilayer
configuration consisting of FeB and Fe2B.
4.4 Determination of the value of boron activation energy
To estimate the boron diffusion coefficient trough the Fe2B layers, it is necessary to use
the experimental data in terms of the boride layer thickness as a function of the boriding
time. The mass balance equation [see equation (11)], based on the principle of mass
conservation, was able to determine the value of boron diffusion coefficient in Fe2B at
each boriding temperature.

Figure 6 XRD pattern obtained at the surface of the borided AISI 1018 steel at 1,273 k for 8 h
Figure 7 Time dependence of the square of Fe2B layer thickness for different temperatures

Figure 7 depicts the time dependence of the square of Fe2B layer thickness in the
temperature range of 1,123–1,273 K, where the slopes of the straight curves correspond
to the values of growth constants 2
2
Fe B( 4 ).ε D= The boride incubation time for the Fe2B
phase, which is independent on the boriding temperature can be easily obtained from
Figure 7 for a null boride layer thickness.
Table 1 The squared value of normalised growth parameter and boron diffusion coefficients
through the Fe2B layer in the temperature range of 1,123–1,273 K
Temperature (K) ε2
(dimensionless) 2
2
Fe B4ε D (μm2
s–1
)
1,123 1.747141 × 10–3
2.91 × 10–1
1,173 6.02 × 10–1
1,223 12.9 × 10–1
1,273 20.9 × 10–1
Table 1 gives the estimated values of boron diffusion coefficients through the Fe2B layers
in the temperature range of 1,123–1,273 K after determining the value of ε parameter
from equation (11).
Figure 8 The temperature dependence of boron diffusion coefficient 2Fe B( )D
The temperature dependence of boron diffusion coefficient in Fe2B is plotted in Figure 8
and fitted according to the Arrhenius equation. The resultant relation is given by equation
(12) with a correlation factor of 0.9959.
2
3 1
Fe B 1.0 10 ( 159.3kJmol / ),D exp RT− −
= × − (12)

where R = 8.3144621 [Jmol–1
K–1
] and T absolute temperature [K].
The value of boron activation energy for the AISI 1018 steel can be easily deduced
from the slope of this linear relationship.
Table 2 A comparison of the boron activation energies for some borided steels
Material
Boriding
method
Boron activation
energy (kJ mol–1
)
References
AISI 1020 Powder 110.3 Celik et al. (2008)
AISI 1040 Powder 118.8 Celik et al. (2008)
AISI 1018 Paste 153 Ortiz-Dominguez et al. (2012)
AISI 1018 Paste 157 Ortiz-Dominguez et al. (2011)
AISI 8620 Plasma paste 99.7–108.8 Gunes et al. (2013a)
AISI 8620 Plasma paste 124.7–138.5 Gunes et al. (2013b)
AISI 1018 Powder 159.3 Present study
Table 2 shows a comparison of the boron activation energies in the case of some borided
steels (Celik et al., 2008; Gunes et al., 2013a, 2013b; Ortiz-Dominguez et al., 2011,
2012). The found value of boron activation energy (= 159.3 kJ mol–1
) for the AISI 1018
steel is very comparable to the values obtained by the powder and paste-boriding
processes (Ortiz-Dominguez et al., 2011, 2012).
The reported values of boron activation energy depend on different factors such as:
the boriding method, the chemical composition of steel substrate and mechanism of
boron diffusion. The value of boron activation obtained on the AISI 1018 steel, is
required to stimulate the boron diffusion according to the easiest diffusion path
corresponding to [001] direction due to the textured growth of Fe2B needles.
4.5 Experimental validation of the diffusion model
The present model was validated by comparing the experimental Fe2B layer thickness
with the predicted value using equation (13) for the borided sample at 1,253 K during a
treatment time of 5 h:
22
1/2 1/2
Fe BFe Bv 2 17 2,500./εD t D t= = (13)
Figure 9 Optical micrograph of boride layer formed on the AISI 1018 steel at 1,253 K for 5 h

Figure 9 gives the optical image of the boride layer formed on the borided AISI 1018
steel at 1,253 K during 5 h.
Table 3 Predicted and estimated values of the Fe2B layer thickness obtained at 1,253 K for a
treatment time of 5 h
Temperature (K) Type of layer
Boride layer
thickness (µm)
estimated by
equation (13)
Experimental boride
layer thickness (µm)
1,253 Fe2B 174.26 167.84 ± 14.29
Table 3 compares between the experimental value of Fe2B layer thickness and the
predicted value one the basis of equation (13). It is seen that the experimental Fe2B layer
thickness agrees with the predicted value for the borided AISI 1018 steel at 1,253 K
during 5 h.
Figure 10 Iso-thickness diagram describing the evolution of Fe2B layer as a function of boriding
parameters

The suggested model can serve as a simple tool to determine the Fe2B layer thickness as a
function of the boriding parameters (time and temperature) for the AISI 1018 steel. An
iso-thickness diagram was plotted as a function of the temperature and the exposure time
as shown in Figure 10. It was obtained by fixing the value of the boride layer thickness
and varying the boriding parameters (time and temperature).
Figure 10 allows selecting the optimum value of Fe2B layer thickness according to
the practical utilisation of the borided AISI 1018 steel at the industrial scale.
As a rule, thin layers (e.g., 15–20 μm) are used to protect against adhesive wear (such
as chipless shaping and metal stamping dies and tools), whereas thick layers are
recommended to combat abrasive wear (extrusion tooling for plastics with abrasive fillers
and pressing tools for the ceramic industry).
In the case of low carbon steels, the optimum boride layers thicknesses vary between
50 and 250 µm. Furthermore, the present model can be extended to be able to simulate
the growth kinetics of bilayer configuration (FeB + Fe2B) formed at the surface of
borided steels.
5 Conclusions
The AISI 1018 steel was pack-borided in the temperature range of 1,123–1,273 K and the
following concluding points can be drawn from the present work:
1 An approach was suggested to estimate the boron diffusion coefficient through the
Fe2B layers. The boron activation energy of the AISI 1018 steel was estimated as
159.3 kJ mol–1
. This value was necessary to stimulate the boron diffusion along the
[001] preferred direction since the Fe2B phase is of anisotropic nature.
2 Validation of the model was made by comparing the experimental Fe2B layer
thickness with the predicted value for the borided sample at 1,253 K for 5 h.
A good agreement was obtained between these two sets of data.
3 In addition, a contour diagram was proposed to be used as a tool to select the
optimum boride layer thickness according to the practical use of this kind of steel in
the industry.
Acknowledgements
The work described in this paper was supported by a grant of CONACyT and PROMEP
México. The authors wish to thank Ing. Martín Ortiz Granillo director de la Escuela
Superior Campus Ciudad Sahagún-Universidad Autónoma del Estado de Hidalgo for
their valuable collaboration for this study.

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List of symbols
v Represents the boride layer thickness (m)
tv Is the effective growth time of the Fe2B layer (s)
t Is the treatment time (s)
2Fe B
0t Is the boride incubation time (s)
2Fe B
upC Represents the upper limit of boron content in Fe2B
(= 60 × 103
mol m–3
)
2Fe B
lowC Is the lower limit of boron content in Fe2B (= 59.8 × 103
mol m–3
)
B
adsC Is the adsorbed boron concentration in the boride layer (mol m–3
)
22 Fe BFe B
1 up lowa C C= − Defines the homogeneity range of the Fe2B layer (mol m–3
)
2Fe B
2 0lowa C C= − Is the miscibility gap (mol m–3
)
C0 Is the terminal solubility of the interstitial solute (≈ 0 mol m–3
)
2Fe B[ ( )]C x t Is the boron concentration profile in the Fe2B layer (mol m–3
)
2
( 2 )/ Fe Berf x D t Is the error function (it has no physical dimensions)
v0 Indicates the initial Fe2B layer (m)
ε Is the normalised growth parameter for the (Fe2B/substrate) interface
(it has no physical dimension)
2Fe BD Denotes the diffusion coefficient of boron in the Fe2B phase (m2
s–1
)
Ji[x(t)]
(with i = Fe2B and Fe)
Are the fluxes of boron atoms in the (Fe2B/substrate) interface boundary
(mol m–2
s–1
)

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