1. Introduction
The boriding process is a thermochemical treatment in which the boron atoms are diffused into the surface of a workpiece to form hard layers composed of iron borides and metallic boride in the case of high alloy steels^{1}. For carbon steels, two kinds of iron borides can be formed by boriding in the temperature range 8001050°C.
The iron borides are interesting phases because of their high hardness. Nevertheless, the Fe_{2}B phase is preferred to FeB, when the resistance to wear under impact was required, since the doublé boride layer (FeB and Fe_{2}B) is prone to cracking during service. Among the boriding processes, the powderpack boriding is widely used in the industry because of its easy handling and low cost^{2}. In this boriding method, a mixture of powders that consists of a boron yielding substance, an activator and a diluent is used. The samples to be borided are then packed in a stainless steel container and placed in the furnace.
In the literature, no kinetic study was reported on the boriding of AISI S1 steel. The modeling of boriding process can be used as a tool to optimize the boriding parameters to produce boride layers with sufficient thicknesses that meet the requirements during service life.
From a kinetic point of view, several approaches have beeen developed to study the kinetics of formation of Fe_{2}B layers on Armco iron and steels as substrates^{3}^{}^{12}.
All these diffusion models considered the principle of the mass balance equation at the (Fe_{2}B/substrate) interface under certain assumptions (with and without boride incubation times).For instance, OrtizDomínguez et al.^{5} have developed a kinetic model for studying the growth kinetics of Fe_{2}B layers on gray cast iron by introducing a kinetic parameter that depends on the values of upper and lower boron concentrations in Fe_{2}B and on the boride incubation time.EliasEspinosa et al.^{6} have modeled the growth kinetics of Fe_{2}B layers on AISI O1 steel by using a diffusion model that assumes a nonlinear boron concentration profile with the presence of a constant boride incubation time. They have introduced a non dimensional kinetic parameter to evaluate the boron diffusion coefficients in the Fe_{2}B layers in the temperature range 11231273 K. Similarly, Nait Abdellah et al.^{7} have also suggested a kinetic model based on the mass balance equation at the (Fe_{2}B/Fe) interface by assuming a nonlinear boron concentration profile through the Fe_{2}B layers on Armco substrate. They introduced the β(T) parameter that depends on the boriding temperature. FloresRentería et al.^{8} have modelled the kinetics of formation of Fe_{2}B layers on AISI 1026 steel by using a kinetic model. In their model, they introduced a kinetic parameter called ε which is independent on the boriding temperature with a linear boron concentration profile in the Fe_{2}B layer.
In the present study, a recent kinetic approach based on the integral method ^{3}^{,}^{4} has been suggested to investigate the boriding kinetics of AISI S1 steel by taking into account the presence of boride incubation time.
The aim of the present work was to investigate the growth kinetics of Fe_{2}B layers on AISI S1 steel based on the integral method in the temperature range 11231273 K.
This diffusion problem can be solved either analytically or numerically.An analytic solution for the integral method has been obtained in order to estimate the boron diffusion coefficients in Fe_{2}B. An experimental validation of the integral method and other used diffusion models was also made for an upper boron concentration of 9 wt.% in Fe_{2}B.Furthermore, the value of activation energy for boron diffusion in AISI S1 steel was estimated on the basis of the integral method and compared with that obtained from another diffusion model ^{6}. Finally, the estimated value of boron activation energy from the integral method was compared with the data available in the literature.
2. The Diffusion Model
The diffusion model deals with the growth kinetics of Fe_{2}B layer on a saturated matrix with boron atoms. The boron concentration profile along the Fe_{2}B layer is depicted in Figure 1.
The f(x,t) function shows the distribution of boron concentration within the substrate before the formation of Fe_{2}B phase
The assumptions taken into account during the formulation of integral model are given in the reference works^{3}^{,}^{4}:
The initial and boundary conditions for the diffusion problem are given by:
t=0,x>0, with
Boundary conditions:
The Second Fick’s law describing the change in the boron concentration within the Fe_{2}B layer is given by Equation (4):
The three timedependent unknowns a(t), b(t) and u(t) have to meet the boundary conditions given by Equations (2) and (3). By applying the boundary condition on the surface, Equation (6) was obtained:
By integrating Equation (4) between 0 and u(t) and applying the Leibniz rule, the ordinary differential equation (ODE) given by Equation (7) was derived:
The mass balance equation at the (Fe_{2}B/substrate) interface can be formulated by Equation (8):
With
At the (Fe_{2}B/substrate) interface, the boron concentration remains constant and Equation (8) can be rewritten as follows:
Substituting Equation (4) into Equation (9) and after derivation with respect to the diffusion distance x(t), Equation (10) was deduced:
Equations (6), (7) and (10) constitute a set of differential algebraic equations (DAE) in a(t), b(t) and u(t) subjected to the initial conditions of this diffusion problem. To obtain the expression of boron diffusion coefficient in the Fe_{2}B layers, an analytic solution is possible by setting:
and
where u(t) is the Fe_{2}B layer thickness and k the corresponding parabolic growth constant at the (Fe_{2}B/substrate) interface. The two unknowns α and β which are positive have to be searched for solving this diffusion problem. After substitution of Equations (11), (12) and (13) into the DAE (differential algebraic equations) system and derivation, the expression of boron diffusion coefficient was obtained as follows:
where η is a dimensionless parameter.
with
along with the expressions of a(t) and b(t) given by Equations (15) and (16):
With
and
3. Experimental Details
3.1 The material and the boriding treatment
The AISI S1 steel was used as substrate for the powderpack boriding. The chemical composition of AISI S1 steel is listed (in weight percent) in Table 1.
C  Mn  Si  Cr  Mo  W 
0.400.55  0.100.40  0.151.20  1.001.80  0.350.50  1.503.00 
V  Ni  Cu  P  S  Fe 
0.150.30  0.3  0.25  0.03  0.03  balance 
The samples had a cubic shape with nominal dimensions of 10 mm×10 mm×10 mm. Before the boriding treatment, the samples were cut and the crosssections were polished metallographically and then etched by Nital solution to reveal the microstructure. The powderpack boriding was carried out by embedding the samples in a closedcontainer containing a mixture of powders as shown in Figure 2. The used boriding agent was composed of 20% B_{4}C, 10% KBF_{4} and 70% SiC. Figure 3 gives an SEM image of the mixture of powders having an average size of 30 µm.
The container was placed in a conventional furnace under a pure argon atmosphere in the temperature range 11231223 K. Four treatment times (2, 4, 6 and 8 h) were selected for each temperature. Once the boriding treatment was finished the container was removed from the furnace and slowly cooled to room temperature.
3.2 Experimental techniques
The crosssections of formed boride layers were examined by SEM (JEOL JSM 6300 LV). The boride layer thickness was automatically measured by means of MSQ PLUS software. For the reproducibility of measurements, seventy tests were performed from a fixed reference on different sections of borided samples to estimate the Fe_{2}B layer thickness; defined as an average value of the long boride teeth. The presence of the iron boride formed at the surface of treated sample was verified by use of XRay Diffraction (XRD) equipment (Equinox 2000) with CoKα radiation of wavelength λ_{Co} = 0.179 nm.
4. Results and Discussions
4.1 SEM examinations of Fe _{2} B layers
Figure 4 shows the crosssections of borided samples at a temperature of 1173 K for different treatment times (2, 4, 6 and 8 h). It is seen the formation of a dense and compact Fe_{2}B layer with a peculiar morphology. The SEM pictures revealed the presence of a sawtooth morphology. Such morphology is typical for borided Armco iron and carbon steels where as for high alloy steels the obtained morphology is very different. In fact, when increasing the contents of alloying elements the (boride layer/substrate) interface tends to be flat as observed, for example, in the packborided AISI 316 L steel^{18}. Carbucicchio et al.^{19} explained the occurrence of a sawtooth morphology to the enhanced growth at the tips of boride needles. Consequently, the iron borides developed a textured growth along the preferred crystallographic direction [001] after Palombarini et al.^{20}.
The Fe_{2}B layer thickness increased with the treatment time at 1173 K. The value of Fe_{2}B layer thickness ranged from 41.93 ± 8.25 µm for 2 h to 95.48 ± 17.4 µm for 8 h at 1173 K.
Figure 5 gives the SEM micrographs of the boride layers formed on the AISI S1 steels at increasing temperatures and for an exposure time of 4 h. The (boride layer/substrate) interface exhibited a sawtooth morphology. The kinetics of formation of boride layers is a thermally activated phenomenon with a change in the layer thickness with increasing temperatures.
4.2 XRD analysis
Figure 6 shows the XRD patterns obtained at the surface of borided AISI S1 steels at 1123, 1173 and 1223 K for 4 h of treatment. The XRD patterns revealed the presence of diffracting peaks belonging to the Fe_{2}B phase.The formation of Fe_{2}B layer is related to the quantity of active boron present in the boriding agent. The observed difference in the diffracted intensities can be explained by the textured growth along the easiest crystallographic direction 001 that minimizes the growth stress^{20}.
4.3 Estimation of boron activation energy in AISI S1 steel
The experimental results are needed to evaluate the values of boron diffusion coefficients in Fe_{2}B in the temperature range 11231273 K by plotting the variation of the square of Fe_{2}B layer thickness as a function of treatment time. The intercept with the time axis yields the value of boride incubation time. Figure 7 gives the evolution of the square of Fe_{2}B layer thickness as a function of time for increasing values of boriding temperatures. The growth kinetics of Fe_{2}B layers is governed by the parabolic growth law. The slope of each straight line depicted in the Figure 7 represent the square of parabolic growth constant at each boriding temperature.
The experimental values of parabolic growth constants at the (Fe_{2}B/substrate) interface along with the corresponding boride incubation times are shown in Table 2.
T (K)  Experimental parabolic growth constant k (µm∙s^{0.5})  Boride incubation time


1123  0.3704  2038.8 
1173  0.5837  2039.4 
1223  0.8861  2040.2 
1273  1.3016  2038.4 
It is seen that the values of boride incubation times are nearly constant. The following reason can be provided for this experimental observation. According to the design of the thermochemical treatment, the container is always placed at ambient temperature in a conventional furnace under a pure argon atmosphere until the boriding temperature (1123 K ≤ T ≤ 1273 K), the boride incubation time
The value of boron diffusion coefficient in Fe_{2}B at each temperature was estimated from Equation (14) based on the integral method. This value can be easily obtained from the slope of straight line shown in Figure 8. The value of 199.16 kJmol^{1} indicates the amount of energy for the boron mobility in the easiest path corresponding to the crystallographic direction [001] along the Fe_{2}B layer. Therefore, the expression describing the evolution of boron diffusion coefficients in Fe_{2}B versus temperature is given by Equation (17) in the temperature range 11231273 K:
where R = 8.314 J mol^{1} K^{1} and T the absolute temperature in Kelvin.
Table 3 shows a comparison between the values of activation energy for boron diffusion in Armco iron and some ferrous alloys (steels and gray cast iron) and the estimated value of activation energy for boron diffusion in AISI S1 steel^{3}^{,}^{12}^{,}^{21}^{}^{28}.
Material  Boriding method  Activation energy for boron diffusion (kJ mol^{1})  References 

XC38 steel  Liquid boriding  207.8 (Fe_{2}B)  12 
IF steel  Electrochemical boriding  80.70100.16 (FeB + Fe_{2}B) depending on the current density  21 
AISI D2 steel  Saltbath  170.0 (FeB + Fe_{2}B)  22 
Q235 steel  Plasma electrolytic boriding  186.17(FeB + Fe_{2}B and Ni borides as precipitates)  23 
AISI 316  Plasma paste boriding  250.8 (FeB + Fe_{2}B)  24 
AISI 1018 Steel  Paste  159.3 (Fe_{2}B)  25 
Armco Iron  Powder  157.5 (Fe_{2}B)  26 
AISI 9840 steel  Powder  193.08 (Fe_{2}B)  27 
ENJL250 Gray cast iron  Powder  134.21 (FeB + Fe_{2}B)  28 
AISI P20 steel  Powder  200 (Fe_{2}B)  3 
AISI S1 steel  Powder  199.16 (Fe_{2}B) by the integral method 199.1 (Fe_{2}B) by the masse balance equation  This work 
It is observed that the obtained values of boron activation energy by different investigators are dependent on several factors such as: (the method of calculation, the boriding method, the nature of boriding agent, the chemical reactions involved and the chemical composition of the substrate.
For example, Şeşen et al.^{21} have borided an interstitial free (IF) steel by using the electrochemical boriding under different current densities. They produced a double boride layer (FeB and Fe_{2}B) where FeB was a dominant phase. A metastable iron boride was also identified by XRD analysis.
The calculated boron activation energies ranged between 80.70 and 100.16 kJ mol^{1}, depending on the value of current density in the range 0.1 0.4 A cm^{2}. It is noticed that these values are lower than those obtained from other reported works^{3}^{,}^{12}^{,}^{21}^{}^{28}. It should be attributed to the absence of carbon and nitrogen as interstitial atoms leading to the increase in the boron mobility within the material substrate. In addition, the estimated value of activation energy for boron diffusion in AISI S1 steel is very comparable to the values estimated for AISI 9840 and AISI P20 steels by using the same chemical composition of boriding agent^{3}^{,}^{27}.
4.4 Experimental validation of different diffusion models
To experimentally validate the diffusion model based on the integral method and four diffusion models, three extra boriding conditions were used for this purpose. Figure 9 gives the SEM images of the crosssections of Fe_{2}B layers formed at 1173 K for 3.5 h and 6.5 h and at 1223 K for 1.5 h, respectively.
For such boriding conditions, a compact single phase layer of Fe_{2}B was produced with a sawtooth morphology.On the basis of integral method, the expression of Fe_{2}B layer thickness depending on the boriding parameters (time and temperature) is given by Equation (18):
with
where Q is the activation energy for boron diffusion (kJmol^{1}), D_{0} represents a preexponential constant (m^{2} s^{1}) and T is the absolute temperature in Kelvin.
Table 4 provides the expressions of Fe_{2}B layer thickness as a function of boriding parameters derived for four diffusion models with Equation (18) valid for the integral method. It is seen that the estimated value of boron activation energy for AISI S1 steel by using the diffusion model^{6} is very close to that obtained from the integral method.
D0 (m^{2} s^{1} )  Activation energy Q (kJmol^{1})  Equations for evaluating the Fe_{2}B layer thickness  References 

5.9×10^{3}  199.15 

5 
5.9×10^{3}  199.15 

6 
5.9×10^{3}  199.15 

7 
5.9×10^{3}  199.15 

8 
3.4×10^{3}  199.16 

Present work 
Table 5 gives a comparison between the experimental values of Fe_{2}B layers’ thicknesses (obtained for three boriding conditions) and the predicted values using four different models and the integral method for an upper boron content in the Fe_{2}B phase equal to 9 wt.%.
Boriding conditions  Experimental value (µm)  Simulated value (µm)5  Simulated value (µm)6  Simulated value (µm)7  Simulated value (µm)8  Simulated value (µm) Equation (18) 

1173 K for 3.5 h  59.65±10.43  57.00  61.84  55.37  61.94  60.37 
1173 K for 6.5 h  75.14 ±13.76  77.68  84.28  75.45  84.41  85.86 
1223 K for 1.5 h  61.54±11.45  56.64  6146  55.02  61.55  51.71 
It is seen that the experimental values in terms of Fe_{2}B layers’ thicknesses coincide in a satisfactory way with the predicted results.
Equation (18) can be employed as a simple tool to predict the optimum value of Fe_{2}B layer thickness as a function of boriding parameters (the treatment time and the process temperature) as shown in Figure 10 to match the case depth that meets requirements for a practical use of AISI S1 steel in the industry.
5. Conclusions
In this current work, the AISI S1 steel was treated by the powderpack boriding in the temperature range 11231273 K with a variable treatment time (from 2 h to 8 h). The boriding agent was composed of 20% B_{4}C, 10% KBF_{4} and 70% SiC. The XRD analysis confirmed the presence of Fe_{2}B phase in the boride layer for all boriding conditions. For indication, the XRD patterns for the borided samples at 1123, 1173 and 1223 K for 4 h were only shown as experimental evidence. The SEM examinations revealed a saw tooth morphology for the Fe_{2}B layers formed on AISI S1 steel. The growth kinetics of Fe_{2}B layers on AISI S1 steel was described by the classical parabolic growth law with the occurrence of a constant boride incubation time. The value of activation energy for boron diffusion in AISI S1 steel was estimated as 199.16 kJmol^{1} on the basis of the integral method,and compared with that obtained from an alternative diffusion model. Furthermore, this value of boron activation energy was compared to the values found in the literature. The present kinetic approach based on the integral method and four diffusion models were experimentally validated by using three extra boriding conditions. As a consequence, a good agreement was noticed between the experimental values of Fe_{2}B layers’ thicknesses and the predicted thicknesses. Finally, an isothickness diagram was suggested to be used as a simple tool for selecting the optimized value of Fe_{2}B layer thickness for practical use of AISI S1 steel in the industry.