2.1 The kinetic model

The model takes into account the growth of layer on a saturated substrate with boron atoms as illustrated in Figure 1.

Figure 1 A schematic boron- concentration profile through the Fe₂B layer. 

The function represents the boron distribution in the ferritic matrix before the nucleation of phase. corresponds to the incubation time required to form the phase when the matrix reaches a saturation state with boron atoms. represents the upper limit of boron content in () ^{Kulka et al. 2013} , 197-199), is the lower limit of boron content in () and the point represents the layer thickness, see Figure 1, ^{Brakman et al. 1989} , 1357-1359).

The term is the effective adsorbed boron concentration during the boriding process ^{Yu et al. 2005} , 2364-2365). From Figure 1, defines the homogeneity range of the layer, is the miscibility gap and is the boron solubility in the matrix. This diffusion zone in the substrate underneath the compound layer can be ignored () ^{Okamoto 2004} , 297). The following assumptions are considered for the diffusion model:

The initial and boundary conditions for the diffusion problem are represented as:

Boundary conditions:

is a thin layer with a thickness of nm formed during the nucleation stage (Dybkpv 2010, 7). Thus, when compared to the thickness of layer (). The mass balance equation at the (/substrate) interface can be formulated by Equation (4) as follows:

Where is defined as the unit area and represents the boron concentration in the matrix. The flux and are obtained from the Fick's First law as:

and

The term is null since the boron solubility in the matrix is very low ( mol m^-3 )^{Okamoto 2004} , 297).

Thus, Eq. (4) can be written as:

If the boron concentration profile in is constant for the treatment time, Fick's Second law is reduced to an ordinary second-order differential equation as follows:

By solving Eq. (8), and applying the boundary conditions proposed in Eqs. (2) and (3), the boron concentration profile in is expressed by Eq. (9) if the boron diffusion coefficient in is constant for a particular temperature :

By substituting Eq. (9) into Eq. (7), equation (10) is obtained:

for .

Substituting the expression of the parabolic growth law: () into Eq. (10), Eq. (11) is deduced:

The normalized growth parameter () for the (/substrate) interface can be estimated numerically by the Newton-Raphson method. It is assumed that expressions, , and , do not depend significantly on temperature (in the considered temperature range) (^{Brakman et al. 1989}, 1357-1359).

A schematic representation of the square of the layer thickness against linear time () is depicted in Figure 2. is the effective growth time of the layer and is the treatment time.

Figure 2 A Schematic representation of the square of the layer thickness against treatment time. 

2.2. Experimental procedure

2.2.1. The boriding process

The material to be borided is AISI 1026 steel. It has a nominal chemical composition of 0.22-0.28 % C, 0.60-0.90% Mn, 0.07-0.6% Si, 0.050% S, and 0.040% P. The samples have a cubic shape with dimensions of Prior to the boriding process, the specimens were polished, ultrasonically cleaned in an alcohol solution and deionized water for 15 min at room temperature, and dried and stored under clean-room conditions.

The samples were packed along with a Durborid fresh powder mixture in a closed cylindrical case (AISI 304L) as shown in Figure 3. This powder mixture has an average size of 30 μm as illustrated on Figure 4.

Figure 3 Schematic view of the stainless steel AISI 304L container for the pack-powder boriding treatment (1: lid; 2: powder boriding medium (B₄C + KBF₄ + SiC); 3: sample; 4: container). 

Figure 4 The powder boriding medium (B₄C + KBF₄ + SiC). 

The powder-pack boriding process was carried out in a conventional furnace under a pure argon atmosphere in the temperature range of 1123 -1273 K. Four treatment times (2, 4, 6 and 8 h) were selected for each temperature. After the completion of boriding treatment, the container was removed from the furnace and slowly cooled to room temperature. As a result of preliminary experiments it was estimated that boriding started approximately after transferring the specimen to the furnace; after that, the so-called boride incubation time sets in.

2.2.2. Microscopical observations of boride layers

The borided samples were cross-sectioned for metallographic examinations using a LECO VC-50 cutting precision machine. The cross-sectional morphology of the boride layers was observed with the Olympus GX51 optical microscope in a clear field. Figure 5 shows the cress-sectional view of optical images of the layers of AISI 1026 steel formed at a temperature of 1173 K for different process durations. The resultant microstructure of layers appears to be very dense and homogenous, exhibiting 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 crystals to make contact with adjacent crystals and forced them to retain an acicular shape ^{Palombarini - Carbucicchio 1987} , 417). It is seen that the thickness of layer increased with an increase of the boriding time (Figure 5) since the boriding kinetics is influenced by the treatment time ^{Campos-Silva et al. 2013} , 415). For a kinetic study, the boride layer thickness was automatically measured with the aid of MSQ PLUS software (Ortíz-Dominguez, 2013). To ensure the reproducibility of the measured layers thicknesses, fifty measurements were collected in different sections of the borided AISI 1026 steel samples to estimate the layer thickness; defined as an average value of the long boride teeth ^{Campos-Silva 2010} , 410). All thickness measurements were taken from a fixed reference on the surface of the borided AISI 1026 steel, as illustrated in Figure 6.

Figure 5 Optical micrographs of the boride layers formed at the surface of AISI 1026 steel treated at 1173 K during a variable time: (a) 2, 4, 6 and (b) 8 h. 

Figure 6 Schematic diagram illustrating the procedure for estimation of boride layer thickness in AISI 1026 steel. 

The phases of the boride layers were investigated by an X-Ray Diffraction (XRD) equipment (Equinox 2000) using radiation of 0.179 nm wavelength. 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 Daimler-Benz Rocwell-C was performed to attain qualitative information on the adhesive strength of the boride layers to the substrate (Verein, 1991, 4). The well-known Rockwell-C indentation test is prescribed by the VDI 3198 norm, as a destructive quality test of coated compounds ^{Vidakis et al. 2003} , 483). The principle of this method was reported in the reference work ^{Taktak 2007} , 1837 - 1839). A load of 1471 N was applied to cause coating damage adjacent to the boundary of the indentation. Three indentations were conducted for each borided sample and scanning electron microscopy (SEM) was used to assess the adhesion test.

3.1. SEM observations and EDS analysis

The cross- sectional view of SEM micrograph obtained on the borided AISI 1026 steel at 1223 K for 8 h, is shown in Figure 7(a). The boride layer is grown on the substrate with a saw- toothed morphology. The needles of, with a difference in length, are visible on the SEM micrograph and penetrating into the substrate. This typical morphology is responsible of a good adhesion to the substrate. On this point, is important to know the behavior of the others alloy elements, as such Mn, Si, in the interphase and surface of the boride layer as explained ahead by an EDS analysis.

Figure 7 (a) SEM micrographs of the cross-sections of the borided AISI 1026 steel at 1223 K for 8 h, (b) EDS spectrum of borided sample at surface and (c) EDS spectrum of borided sample at interface. 

The cross section of SEM micrograph of the AISI 1026, borided at temperature of 1223 K for 8 h, is shown in Fig. 7(a). The EDS analysis obtained by SEM is shown in Fig. 7(b) and ^(c) . The amounts of manganese appear to be lower than that of iron in the boride layer because of lower solubility. Thus, the deficiency of Mn results in a negative effect on the boride layer in terms of both thickness and morphology.

Fig. 7(c) shows that the carbon and silicon do not dissolve significantly over the phase and they do not diffuse through the boride layer, being displaced to the diffusion zone, and forms together with boron, solid solutions like silicoborides ( and ) and boroncementite () [28].

3.2 X-ray diffraction analysis

Figure 8 shows the XRD pattern recorded on the surface of borided AISI 1026 steel at a temperature of 1273 K for a treatment time of 8 h. The diffraction peaks relative to the phase are easily identified.

Figure 8 XRD pattern obtained at the surface of the borided AISI 1026 steel treated at 1273 K for 8 h, where the mains Fe₂B peaks are showed. 

Crystals of the type orientate themselves with the z-axis perpendicular to the surface. Consequently, the peaks of the phase belonging to crystallographic planes, having a deviation from zero of the l index, showed increased intensities in the X-ray diffraction spectra ^{Badini y Mazza 1988} , 663-665). The growth of boride layers is a controlled diffusion process with a highly anisotropic nature. In case of the phase, the crystallographic direction [001] is the easiest path for the boron diffusion in , because of the tendency of boride crystals to grow along a direction of minimum resistance, perpendicular to the external surface. As the metal surface is covered, an increasing number of crystals comes in contact with adjacent crystals and are forced to grow inside the metal, retaining an acicular shape ^{Palombarini - Carbucicchio 1987} , 415). In the powder-pack boriding, the active boron is supplied by the powder mixture. To form the phase on any borided steel, a low boron potential is required as reported in the reference works ^{Vipin 2002} , 21-22) ( ^{Bindal y Ucisik 1999} , 209-211), where a high amount of active boron in the powder mixture gives rise to the bilayer configuration consisting of FeB and .

3.3 Rockwell-C adhesion test

An indenter hardness tester was used to assess the Daimler-Benz Rockwell-C adhesion, as a destructive quality test for examined layers, it was employed for determination of cohesion. The well-know adhesion test prescribed by the VDI 3198 norm was used (Verein, 1991, 4). The principle of this method was presented in Figure 9. A conical diamond indenter penetrated into the surface of an investigated layer, thus inducing massive plastic deformation to the substrate and fracture of the boride layer.

Figure 9 Principle of the VDI 3198 indentation test (Verein, 1991, 4) 

The damage of the boride layer was compared with the adhesion strength quality maps HF1-HF6 (see Figure 9). In general, the adhesion strength HF1 to HF4 defined sufficient adhesion, whereas HF5 and HF6 represented insufficient adhesion (Verein, 1991, 4). Scanning Electron Microscope (SEM) images of the indentation craters for samples borided at 1273 K for 4 h are given in Figure 10. The indentation craters obtained on the surface of the borided AISI 1026 revealed that there were radial cracks at the perimeter of indentation craters. However, a small quantity of spots with flaking of delamination was visible and the adhesion strength quality of this boride layer fits the HF3 category. In this context, the authors S. ^{Taktak and S. Tasgetiren (2006}, 571-572) have used the Daimler-Benz Rockwell-C adhesion technique to study the adhesion of boride layers grown on AISI H13 and AISI 304 steels. They found that their adhesion decreased with an increase of boriding time and temperature. ^{Rodriguez et al. (2009}, 63) also reported the adhesive strength of the boride layers using Daimler-Benz method.

Figure 10 SEM micrograph showing the indentation of VDI adhesion test on AISI 1026. 

3.4 Estimation of boron activation energy

The growth kinetics of layers formed on the AISI 1026 steel was used to estimate the boron diffusion coefficient through the layers by applying the suggested diffusion model. The determination of epsilon parameter () is then required by solving the mass balance equation at the (/substrate) interface (see Eq. (11)) using the Newton-Raphson method. The time dependence of the squared value of layer thickness is shown in Figure 11. The slopes of the straight lines in this figure provide the values of growth constants for each boriding temperature. The value of boron diffusion coefficient in the layers can be determined by knowing the value of epsilon parameter. The boride incubation times for can also be deduced from the straight lines displayed in Figure 11 by extrapolating to a null boride layer thickness.

Figure 11 Averaged squares of the layers () vs. boriding time () at different temperature. 

Table 1 summarizes the estimated value of boron diffusion coefficient in at each temperature along with the squared normalized value of epsilon determined from Eq. (11).

Table 1 The squared value of normalized growth parameter and boron diffusion coefficients in Fe₂B as a function of boriding temperature. 

To estimate the boron activation energy for the AISI 1026 steel, it is necessary to plot the natural logarithm of boron diffusion coefficient in versus the reciprocal temperature following the Arrhenius equation (see Figure 12). A linear fitting was adopted to obtain the temperature dpendence of boron diffusion coefficient in with a correlation factor of 0.9782.

Figure 12 Boron diffusion coefficient () as a function of boriding temperatures. 

where: and absolute temperature .

Table 2 shows a comparison of the boron activation energies in case of some borided steels (^{Celik et al. 2008}, 33-336) (^{Gunes et al. 2013}, 569-570). The found value of boron activation energy (= 178.4 kJ mol -1) for AISI 1026 is slightly different from the reported value in (Gunes et al. 2013, 569-570) depending on the boriding conditions (using the liquid boriding method).

Table 2 A comparison of the boron activation energies for some borided steels. 

3.5 Validation of the diffusion model

The present model was validated by comparing the experimental value of layer thickness with the predicted result at a temperature of 1253 K for a treatment time of 7 h using Eq. (13):

In Figure 13 is shown the optical image of the boride layer formed at 1253 K for 7 h of treatment.

Figure 13 Optical micrographs of boride layer formed on AISI 1026 during powder-pack at the 1253 K of boriding temperature with an exposure time of 7 h. 

Table 3 gives a comparison between the experimental value of layer thickness and the predicted one basis of Eq. (13). A good agreement was then obtained between the experimental value of layer thickness and the predicted one for the borided AISI 1026 steel at 1253 K for 7 h.

Table 3 Predicted and estimated values of the Fe₂B layer thickness obtained at 1253 K for a treatment time of 7 h. 

3.5 Future exploitation of the simulation results

This kinetic approach can be used as a tool to determine the layer thickness as a function of boriding parameters (time and temperature) for AISI 1026 steel. Eq. (13) predicts the layer thickness for any temperature and boriding time. An iso-diagram thickness was plotted as a function of the temperature and exposure time as shown in Figure 14.

Figure 14 Iso-diagram describing the evolution of Fe₂B layer as a function of boriding parameters. 

The results of Figure 14 can serve as a powerful tool to select the optimum value of layer thickness in relation with the potential applications of the borided AISI 1026 in 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 and low alloy steels, the optimum boride layer thicknesses ranges from 50 to 250 μm, and for high- alloy steels, the optimum boride layer thicknesses ranges from 25 to 76 μm. Finally, this model can be extended to predict the growth kinetics of bilayer configuration (FeB + ) grown on any boride steel.