1.2.2 Characterization of boride layers

The borided samples were prepared metallographically for their characterization using GX51 Olympus equipment. As a result of preliminary experiments it was estimated that boriding started at approximately t Fe2B <sup>¼</sup> <sup>29</sup>:55 min after transfer- <sup>0</sup> ring the sample to the furnace; after that, the so-called boride incubation time sets in. The borided and etched samples were cross-sectioned, for microstructural investigations, to be observed by scanning electron microscope. The equipment used was the Quanta 3D FEG-FEI JSM7800-JOEL. Figure 4 shows the crosssections of boride layers formed on the surfaces of AISI 12L14 steel at different exposure times (2, 4, 6 and 8 h) and for 1173 K of boriding temperature.

The resultant microstructure of Fe2B layers appears to be very dense and homogenous, exhibiting a sawtooth morphology where the boride needles with different lengths penetrate into the substrate [19, 20]. These elements tend to concentrate in the tips of boride layers, reducing the boron flux in this zone. The Fe2B crystals preferably grow along the crystallographic direction [0 0 1], because it is the easiest path for the diffusion of boron in the body-centered tetragonal lattice of the Fe2B phase [19].

It is seen that the thickness of Fe2B layer increased with an increase of the boriding temperature (Figure 4) since the boriding kinetics is influenced by the

#### Figure 4.

SEM micrographs of the cross-sections of AISI 12L14 steel samples borided at 1173 K during different exposure times: (a) 2, (b) 4, (c) 6, and (d) 8 h.

Figure 5.

Schematic diagram illustrating the procedure for estimation of boride layer thickness in AISI 12L14 steel.

treatment time. To estimate the boride layer thickness, 50 measurements were made from the surface to the long boride teeth in different sections, as shown in Figure 5; the boride layer thickness was measured using specialized software [20–22].

The identification of phases was carried out on the top surface of borided sample by an X-ray diffraction (XRD) equipment (Equinox 2000) using CoK<sup>α</sup> radiation of 0.179 nm wavelength. In addition, the elemental distribution of the transition elements within the cross-section of boride layer was determined by electron dispersive spectroscopy (EDS) equipment (Quanta 3D FEG-FEI JSM7800-JOEL) from the surface.

#### 1.3 Results and discussions

#### 1.3.1 SEM observations and EDS analysis

The metallography of coating/substrate formed in AISI 12L14 borided steel at different exposure times (2, 4, 6 and 8 h) and for 1173 K of boriding temperature are shown in Figure 4. The EDS analysis obtained by SEM is shown in Figure 6(a) and (b).

The results show in Figure 6(a) that the sulfur can be dissolve in the Fe2B phase, in fact, the atomic radiuses of S (= 0.088 nm) is smaller than that of Fe (= 0.156 nm), and it can then be expected that S dissolved on the Fe sublattice of the borides. In Figure 6(b), the resulting EDS analyses spectrums revealed that the manganese, carbon and silicon do not dissolve significantly over the Fe2B phase and they do not diffuse through the boride layer, being displaced to the diffusion zone, and forms together with boron, solid solutions [10, 23, 24]. On boriding carbon is driven ahead of the boride layer and, together with boron, it forms borocementite, Fe3(B, C) as a separate layer between Fe2B and the matrix with about 4 mass% B corresponding to Fe3(B0.67C0.33) [10]. Thus, part of the boron supplied is used for the formation of borocementite. Likewise, silicon forming together with boron, solid solutions like silicoborides (FeSi0.4B0.6 and Fe5SiB2) [24].

#### 1.3.2 X-ray diffraction analysis

Figure 7 shows the XRD pattern recorded on the surface of borided AISI 12L14 steel at a temperatures of: 1123 K for a treatment time of 2 h, and 1273 K for a treatment time of 8 h. The patterns of X-ray diffraction (see Figure 7) show the

Comparison and Analysis of Diffusion Models for the Fe2B Layers Formed on the AISI 12L14… DOI: http://dx.doi.org/10.5772/intechopen.84846

#### Figure 6.

A SEM micrographs of the cross-sections of the borided AISI 12L14 steel micrograph image of microstructure of the AISI 12L14 boride layer obtained at 1173 K with exposure time of 8 h, (a) and (b) EDS spectrum of borided sample.

Figure 7. XRD patterns obtained at the surface of borided AISI 12L14 steels for two boriding conditions: (a) 1123 K for 2 h and (b) 1273 K for 8 h.

presence of Fe2B phase which is well compacted. Likewise, the patters show that there is a preferential orientation in the crystallographic plane (0 0 2) whose strength increases as the depth of the analysis increases. In a study by Martini et al. [18], the growth of the iron borides (Fe2B) near at the Fe2B/substrate interface only shows the diffraction peak of Fe2B in the crystallographic plane (002).

## 1.3.3 Estimation of boron activation energy with steady state model

The growth kinetics of Fe2B layers formed on the AISI 12L14 steel was used to estimate the boron diffusion coefficient through the Fe2B layers by applying the suggested steady state diffusion model. In Figure 8 is plotted the time dependence of the squared value of Fe2B layer thickness for different temperatures.

In Figure 8, the square of boride layer thicknesses were plotted vs. the treatment time, the slopes of each of the straight lines provide the values of the parabolic growth constants ð¼ 2kFe2BÞ. In addition, to determinate the boride incubation time, was necessary extrapolating the straight lines to a null boride layer thickness (see Figure 8). Table 2 provides the estimated value of growth constants in Fe2B at each temperature. The results, which are summarized in Table 2, reflect a diffusioncontrolled growth of the boride layers.

In Table 2, the boron diffusion coefficient in the Fe2B layers (DFe2<sup>B</sup>) was estimated for each boriding temperature. So, an Arrhenius equation relating the boron diffusion coefficient to the boriding temperature can be adopted.

As a consequence, the boron activation energy (QFe2<sup>B</sup>) and pre-exponential factor (D0) can be calculated from the slopes and intercepts of the straight line shown in coordinate system: lnDFe2<sup>B</sup> as a function of reciprocal boriding temperature, it is presented in Figure 9. The boron diffusion coefficient through Fe2B layers was deducted by steady state diffusion model as:

$$D\_{\rm Fe\_2B} = 2.444 \times 10^{-3} \exp\left(-165.0329 \, kJmol^{-1}/RT\right) \quad \left[\text{m}^2 \text{s}^{-1}\right]. \tag{27}$$

˛ ˝ where: <sup>R</sup> <sup>¼</sup> <sup>8</sup>:3144621 Jmol�<sup>1</sup> K�1 and T absolute temperature [K]. From the <sup>2</sup> Eq. (27), the pre-exponential factor ˜ <sup>D</sup>0 <sup>¼</sup> <sup>2</sup>:444 � <sup>10</sup>�<sup>3</sup><sup>m</sup> <sup>=</sup><sup>s</sup> ° and the activation


Comparison and Analysis of Diffusion Models for the Fe2B Layers Formed on the AISI 12L14… DOI: http://dx.doi.org/10.5772/intechopen.84846

Table 2.

The growth constants and boron diffusion coefficients as a function of boriding temperature.

#### Figure 9.

Arrhenius relationship for boron diffusion coefficient (DFe2<sup>B</sup>) through the Fe2B layer obtained with the steady state diffusion model.

˜ ° energy QFe2B <sup>¼</sup> <sup>165</sup>:0329 kJmol�1 values are affected by the contact surface between the boriding medium and the substrate, as well as the chemical composition of the substrate [9, 10, 23–28].

#### 1.3.4 Estimation of boron activation energy with non-steady state diffusion model

In Table 2 provides the growth constants (2kFe2B) at each temperature, and in Table 3 provides the boron diffusion coefficients (DFe2B), they were estimated numerically by the Newton-Raphson method from the Eq. (26).

˜ ° The boron activation energy QFe2B <sup>¼</sup> <sup>164</sup>:999 kJmol�1 and pre-exponential <sup>2</sup> factor ˜ <sup>D</sup>0 <sup>¼</sup> <sup>2</sup>:072 � <sup>10</sup>�<sup>3</sup> <sup>m</sup>=s ° can be calculated from the slopes and intercepts of the straight line shown in coordinate system: ln DFe2B as a function of reciprocal boriding temperature, it is presented in Figure 10, in the same way as above.

The boron diffusion coefficient through Fe2B layers was deducted by non-steady state diffusion model as:

$$D\_{\text{Fe}\,\text{B}}\,\,^{\circ} = 2.072 \times 10^{-3} \exp\left(-164.999 \,\text{kJ}\,\text{mol}^{-1}/\text{RT}\right) \,\,\,^{\circ} \,\text{[m}^{2}\,\text{s}^{-1}\text{]}.\tag{28}$$


Table 3.

The boron diffusion coefficients (DFe2B) as a function of boriding temperature.

#### Figure 10.

Arrhenius relationship for boron diffusion coefficient (DFe2<sup>B</sup>) through the Fe2B layer obtained with the nonsteady state diffusion model in on dimension.

#### 1.3.5 The two diffusion models

� � In this section we want to illustrate the differences between the two diffusion models have been used to describe the growth kinetics of boride layers. It is noticed that the estimated values of boron activation energy QFe2B <sup>¼</sup> <sup>165</sup>:0 kJmol�1 for AISI 12L14 steel by steady state (see Eq. (27)) and non-steady state (see Eq. (28)), is exactly the same value for both diffusion models. Likewise, the estimated values of <sup>2</sup> pre-exponential factor by steady state � D0 <sup>¼</sup> <sup>2</sup>:444 � <sup>10</sup>�<sup>3</sup> <sup>m</sup>=s � and non-steady <sup>2</sup> state � D0 <sup>¼</sup> <sup>2</sup>:072 � <sup>10</sup>�<sup>3</sup> <sup>m</sup>=s � , there is a small variation. To find out how this similarity is possible in the diffusion coefficients obtained by two different models, �pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� we first focus our attention on Eq. (23). The error function erf kFe2<sup>B</sup>=2DFe2<sup>B</sup> is a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi monotonically increasing odd function of kFe2<sup>B</sup>=2DFe2<sup>B</sup>. Its Maclaurin series (for pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi small kFe2<sup>B</sup>=2DFe2<sup>B</sup>) is given by [29]:

$$\text{erf}\left(\sqrt{k\_{\text{Fe},\text{B}}/2D\_{\text{Fe},\text{B}}}\right) \stackrel{\text{(}\begin{array}{c} 2\\ \text{ } \sqrt{\pi} \end{array}}{\sqrt{\pi}} \Big(\sqrt{k\_{\text{Fe},\text{B}}/2D\_{\text{Fe},\text{B}}} \Big(\text{-}\frac{\left(\sqrt{k\_{\text{Fe},\text{B}}/2D\_{\text{Fe},\text{B}}}\right)^{3}}{3\cdot 1!} + \frac{\left(\sqrt{k\_{\text{Fe},\text{B}}/2D\_{\text{Fe},\text{B}}}\right)^{5}}{5\cdot 2!} - \text{:}\Big) \Big(\text{-}\frac{\left(\sqrt{k\_{\text{Fe},\text{B}}/2D\_{\text{Fe},\text{B}}}\right)^{2}}{\sqrt{2\pi}} \Big) \tag{29}$$

Comparison and Analysis of Diffusion Models for the Fe2B Layers Formed on the AISI 12L14… DOI: http://dx.doi.org/10.5772/intechopen.84846

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi According to the numerical value of the kFe2B=2DFe2B, Eq. (29) can be rewritten as:

$$\text{erf}\left(\sqrt{k\_{\text{Fe}\_2\text{B}}/2D\_{\text{Fe}\_2\text{B}}}\right) \left(= -2\sqrt{k\_{\text{Fe}\_2\text{B}}/2D\_{\text{Fe}\_2\text{B}}}\pi.\right) \tag{30}$$

Similarly for the real exponential function exp ð�kFe2B=2DFe2BÞ : R ! R can be characterized in a variety of equivalent ways. Most commonly, it is defined by the following power series [30]:

$$\exp\left(-k\_{\rm Fe\_2B}/2D\_{\rm Fe\_2B}\right) = \left(1 - k\_{\rm Fe\_2B}/2D\_{\rm Fe\_2B} + \left(-k\_{\rm Fe\_2B}/2D\_{\rm Fe\_2B}\right)^2/2! + \left(-k\_{\rm Fe\_2B}/2D\_{\rm Fe\_2B}\right)^3/3! + \cdots\right) \tag{31}$$

Thus, Eq. (31) can be written as:

$$\exp\left(-k\_{\text{Fe}\,\text{B}}/2D\_{\text{Fe}\,\text{B}}\right) = \mathbf{1}.\tag{32}$$

By substituting the Eqs. (30) and (32) into Eq. (26), we have

$$\frac{\text{C}\_{\text{up}}^{\text{Fe},\text{B}} + \text{C}\_{\text{low}}^{\text{Fe},\text{B}} - 2\text{C}\_{\text{ol}}^{\text{O}}}{4} \left(\text{2}l\_{\text{Fe},\text{B}}\right)^{1/2} = \sqrt{\frac{D\_{\text{Fe},\text{B}}}{\pi}} \frac{\mathcal{C}\_{\text{up}}^{\text{Fe},\text{B}} - \mathcal{C}\_{\text{low}}^{\text{Fe},\text{B}}}{2\sqrt{\frac{k\_{\text{Fe},\text{B}}}{2D\_{\text{Fe},\text{B}}}}} (1), \tag{33}$$
 
$$D\_{\text{Fe},\text{B}} = \frac{1}{2} \quad \frac{\mathcal{C}\_{\text{up}}^{\text{Fe},\text{B}} + \mathcal{C}\_{\text{low}}^{\text{Fe},\text{B}} - 2\mathcal{C}\_{\text{ol}}^{\text{O}}}{\mathcal{C}\_{\text{up}}^{\text{Fe},\text{B}} - \mathcal{C}\_{\text{low}}^{\text{Fe},\text{B}}} \Bigg) \not{k}\_{\text{Fe},\text{B}}. \tag{34}$$

The result obtained by Eq. (34) is the same as that obtained in Eq. (21) estimated by steady state diffusion model. The result from the Eq. (21) would appear to imply that the non-steady state diffusion model is superior to the steady state diffusion model and so should always be used. However, in many interesting cases the models are equivalent.
