1. Introduction

Surface hardening of steel can be achieved, mainly through two procedures: modifying the chemical composition of the surface by diffusion of some chemical element (carbon, nitrogen, boron, sulfur, etc.) in which case it is known as thermochemical treatment (Table 1) or modifying only the microstructure of the surface by thermal treatment, then known as surface treatment. The current technological demands highlight the need to have metallic materials with high performance under critical service conditions, consequently, the increase in the wear resistance, preserving its ductility and the toughness of the core.

According to Table 1, there are three methods of surface hardening:


The current technological requirements highlight the need to have metallic materials with specific characteristics, for increasingly critical service conditions. For example, the metal dies used in the metallurgical processes of cold working and hot metals need a high toughness and surface hardness, especially at high temperature. Surface hardening of steel can be achieved, basically, by two processes: modifying the chemical composition of the surface by diffusion of some chemical elements (carbon, nitrogen, sulfur, boron, aluminum, zinc, chromium, and so on). Only boriding process for surface hardening is briefly reviewed in this chapter, boriding is a thermochemical treatment in which boron atoms are diffused into the surface of a workpiece and form borides with the base metal. Apart from constructional materials, which meet these high demands, processes have been developed which have a positive effect on the tribological applications including abrasive, adhesive, fatigue and corrosion wear of the component surface [1–3]. Boride layers are of particular benefit when the components have to withstand abrasive wear. The fundamental advantage of the borided layers (FeB and Fe2B) is that they can reach high hardness near the surface (1800 HV0.1 and 2000 HV0.1), maintained at high temperatures [4–8]. In this chapter, the growth kinetics of single phase layer (Fe2B) on the ferrous substrate was studied during the iron powder-pack boriding (steady state and non-steady state). The parabolic growth law for the borided layers was


#### Table 1. Engineering methods for surface hardening of steels.

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

mathematically estimated. Likewise, a mass balance equation was proposed at the Fe2B/substrate (AISI 12L14) interface. Moreover, the boron diffusion coefficients (DFe2<sup>B</sup>) in the Fe2B layers were determined considering two mathematical models for mass transfer. The Fe2B layers formed on the alloy surface is controlled by the diffusion of boron atoms, and the presence of the Fe2B layers was checked by the XRD technique. Finally, the distribution of the alloy elements in AISI 12L14 borided steel was verified by chemical microanalysis technique (EDS) used in conjunction with SEM.

#### 1.1 The diffusion models

One of the most important parameters that characterizes the Fe2B layers is the thickness, since the properties of the coating depend on it, such as: resistance to wear, fatigue, hardness, and dynamic loads, as well as to a large extent determining the grip with the substrate. Having an expression that allow estimating the layer thickness during the boriding process, facilitates the appropriate selection of the technological parameters, in order to guarantee the desired properties. The layer thickness exhibits a time dependence such that:

$$\text{layer thickness } \mathbf{v} \approx t^{1/2}, \tag{1}$$

### 1.1.1 Derivation of the parabolic growth law

In diffusion processes, parabolic kinetics occurs when the mass gain on a sample is proportional to the square root of time. In general, parabolic kinetics indicates that diffusion of reactants (such as boron) through a growing layer is ratedetermining. If the diffusion of B atoms is rate-determining, the layer rate is proportional to the flux through the substrate:

$$\frac{d\boldsymbol{\mathfrak{x}}}{dt} \approx \boldsymbol{J}\_{\text{Fe}\_2\text{B}}(\boldsymbol{\mathfrak{x}}, t). \tag{2}$$

El flux, JFe2<sup>B</sup>ðx; tÞ, can be written as

$$J\_{Fe\_2B}(\mathbf{x}, t) = C\_{Fe\_2B}(\mathbf{x}, t)(d\mathbf{x}/dt),\tag{3}$$

where CFe2<sup>B</sup>ðx; <sup>t</sup><sup>Þ</sup> is the boron concentration profile in mol/m3 and is the velocity dx=dt of Fe2B layer in m/s, JFe2<sup>B</sup>ðx; <sup>t</sup><sup>Þ</sup> giving units of mol/m2 s. The velocity of a particle is proportional to the force, F, on the particle:

$$d\mathfrak{x}/dt = B\_{Fe\_2B} F\_\bullet \tag{4}$$

where BFe2<sup>B</sup> is the mobility of the boron. Writing the chemical potential as μFe2<sup>B</sup>, this force is written as

$$F = -\partial \mu\_{Fe\_2B} / \partial \mathfrak{x},\tag{5}$$

for a Fe2B layer with thickness x. Combining Eq. (4) and (5) yields

$$J\_{Fe2}(\mathbf{x}, t) = -\mathcal{C}\_{Fe2}(\mathbf{x}, t) B\_{Fe2} \partial \mu\_{Fe2} / \partial \mathbf{x},\tag{6}$$

from the relationship

Heat and Mass Transfer - Advances in Science and Technology Applications

$$
\mu\_{Fe\_2B} = \mu\_{Fe\_2B}^{\rho} + k\_B T \ln a\_{Fe\_2B} \tag{7}
$$

where kB is the Boltzmann's constant, we can write

$$\frac{\partial \mu\_{Fe\_2B}}{\partial \mathbf{x}} = \frac{\partial \left(\mu\_{Fe\_2B}^o + k\_B T \ln \mathcal{a}\_{Fe\_2B}\right)}{\partial \mathbf{x}} = k\_B T \frac{\partial \ln \mathcal{a}\_{Fe\_2B}}{\partial \mathbf{x}}.\tag{8}$$

In an ideal system, the concentration, CFe2Bðx; tÞ, is equivalent to activity, aFe2Bðx; tÞ. Substituting the Eq. (8) into Eq. (6), we get

$$J\_{Fe\_2B}(\mathbf{x},t) = -C\_{Fe\_2B}(\mathbf{x},t)B\_{Fe\_2B}k\_BT\frac{\partial \ln C\_{Fe\_2B}(\mathbf{x},t)}{\partial \mathbf{x}} = -B\_{Fe\_2B}k\_BT\frac{\partial C\_{Fe\_2B}(\mathbf{x},t)}{\partial \mathbf{x}}.\tag{9}$$

As shown in Eq. (2),

$$\frac{d\mathbf{x}}{dt} = (\mathbf{constant}) f\_{Fe\_2B}(\mathbf{x}, t), \tag{10}$$

so that a combination of Eqs. (2) and (9) gives

$$\frac{d\mathbf{x}}{dt} = -(\text{constant})B\_{F\epsilon\_2 B}k\_B T \frac{\partial C\_{F\epsilon\_2 B}(\mathbf{x}, t)}{\partial \mathbf{x}}.\tag{11}$$

If we assume that the potential is fixed at each boundary of the Fe2B layer, we can replace ∂CFe2<sup>B</sup>ðx; <sup>t</sup>Þ=∂<sup>x</sup> in Eq. (11) with the slope (<sup>¼</sup> <sup>Δ</sup>CFe2<sup>B</sup>=x). We then introduce the parabolic growth constant kFe2<sup>B</sup>, and set:

$$k\_{Fe\_2B} = -(\mathbf{constant})B\_{Fe\_2B}k\_BT\Delta C\_{Fe\_2B}.\tag{12}$$

Combining Eqs. (11) and (12) then gives

$$\frac{d\infty}{dt} = \frac{k\_{Fe\_2B}}{\infty}.\tag{13}$$

Eq. (13) can be rewritten as

$$
\pi d \mathfrak{x} = k\_{Fe\_2B} dt.\tag{14}
$$

Upon integration of Eq. (14),

$$\int\_{x=0}^{x=v} \varkappa d\infty = k\_{Fe2B} \int\_{t=0}^{t=t} dt. \tag{15}$$

We arrive at the parabolic growth law:

$$\mathbf{v}^2 = 2k\_{Fe2}\mathbf{t}.\tag{16}$$

#### 1.1.2 Steady state diffusion model

Steady state means that there will not be any change in the composition profile with time. A linear boron concentration profile is considered along the depth of the Fe2B layer as depicted in Figure 1. The f(x) represents to the boron distribution in the substrate before the nucleation of iron boride layers on AISI 12L14 steel. t Fe2<sup>B</sup> is <sup>0</sup> the boride incubation time indispensable to form the Fe2B phase. Moreover, CFe2B up

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 1.

A schematic linear concentration profile of boron through the Fe2B layer is used to describe the steady state diffusion model.

represents the boron concentration on the surface in Fe2B layer Fe2B (<sup>¼</sup> <sup>60</sup>� <sup>10</sup><sup>3</sup>molm‐<sup>3</sup>), <sup>C</sup> represents the boron concentration at the Fe2B/sub- low strate interface (<sup>¼</sup> <sup>59</sup>:8 � <sup>10</sup><sup>3</sup>molm‐<sup>3</sup> ) and x tð ¼ tÞ ¼ v is the layer thickness of the boride layer (m) [9, 10].

<sup>B</sup> The term Cads is the effective adsorbed boron concentration during the boriding Fe2B Fe2B process [11]. From Figure <sup>1</sup>, a1 <sup>¼</sup> <sup>C</sup> � <sup>C</sup> defines the homogeneity range of up low the Fe2B layer, a2 <sup>¼</sup> <sup>C</sup>Fe2B � <sup>C</sup>0 represents the range of miscibility and C0 is the low boron concentration in the substrate (AISI 12L14) assumed as null [10, 12, 13]. During the establishment of the steady-state diffusion model, a linear concentration-profile of boron along the Fe2B layer is considered. Likewise, the assumptions proposed by Campos-Silva et al. [8], are taken account.

v0 is the first boride layer formed on the surface of the substrate (ASI 12 L14) during the boride incubation time [14], its thickness is very small in magnitude compared to the thickness of the boride layer (v). Moreover, regarded the mass balance equation at the growth interface (Fe2B/substrate), which is described as follows [15–18]:

$$\frac{\left.\left(\mathbf{C}\_{\mathrm{up}}^{\mathrm{Fe},\mathrm{B}} + \mathbf{C}\_{\mathrm{low}}^{\mathrm{Fe},\mathrm{B}} - 2\mathbf{C}\_{0}\right)}{2} \left.\frac{d\mathbf{x}(t)}{dt}\right|\_{\mathbf{x}(t) = \mathbf{v}} = -\left.D\_{\mathrm{Fe};\mathrm{B}}\frac{\partial \mathbf{C}\_{\mathrm{Fe};\mathrm{B}}[\mathbf{x}(t = t), t = t]}{\partial \mathbf{x}}\right|\_{\mathbf{x}(t) = \mathbf{v}}.\tag{17}$$

When the concentration field is independent of time and DFe2B is independent of CFe2Bðx; tÞ, Fick's second law is reduced to Laplace's equation,

$$\nabla^2 \mathcal{C}\_{\text{Fe}\_2\text{B}}[\varkappa(t)] = \frac{d^2 \mathcal{C}\_{\text{Fe}\_2\text{B}}[\varkappa(t)]}{d\varkappa^2} = \mathbf{0}.\tag{18}$$

By solving Eq. (18), and applying the boundary conditions proposed in Figure 1, the distribution of boron concentration in Fe2B is expressed as:

$$\mathbf{C\_{Fe\_2B}}[\mathbf{x}(t)] = \frac{\mathbf{C\_{low}^{Fe\_2B}} - \mathbf{C\_{up}^{Fe\_2B}}}{\mathbf{v}} \mathbf{x} + \mathbf{C\_{up}^{Fe\_2B}}.\tag{19}$$

By substituting the derivative of Eq. (19) with respect of the distance x(t) into Eq. (17), we have

$$\frac{\mathbf{C}\_{\rm up}^{\rm Fe\_2B} + \mathbf{C}\_{\rm low}^{\rm Fe\_2B} - 2\mathbf{C}\_0^{\rm O}}{2} \Bigg/ \underbrace{\mathbf{d}\_{\rm V}}\_{\rm dt} = \boldsymbol{D}\_{\rm Fe\_2B} \frac{\mathbf{C}\_{\rm up}^{\rm Fe\_2B} - \mathbf{C}\_{\rm low}^{\rm Fe\_2B}}{\mathbf{v}},\tag{20}$$

for 0 ≤ x≤ v. By substituting Eq. (16) into Eq. (20)

$$D\_{\rm Fe\_2B} = \frac{1}{2} \quad \frac{C\_{\rm up}^{\rm Fe\_2B} + C\_{\rm low}^{\rm Fe\_2B} - 2C\_{\rm O}^{\rm O}}{C\_{\rm up}^{\rm Fe\_2B} - C\_{\rm low}^{\rm Fe\_2B}} \Bigg/ \Big(\_{\rm Fe\_2B} \tag{21}$$

#### 1.1.3 Non-steady state diffusion model in one dimension

The general diffusion equation for one-dimensional analysis under non-steady state condition is defined by Fick's second law. The growth of single phase layer (Fe2B) with one diffusing element (boron) is observed as illustrated in Figure 2.

The f xð ; tÞ function represents the boron distribution in the ferritic matrix before the nucleation of Fe2B phase as a function of time. Likewise, for analysis, the kinetic model is imposing the same restrictions as in the previous model, except the last one, it is replaced by:

• The concentration-profile of boron is the solution of the Fick's second law and depends on initial and boundary conditions through the Fe2B zone.

The mass balance equation at the (Fe2B/substrate) interface can be formulated by Eq. (22) as follows:

$$\begin{cases} \begin{aligned} \mathbf{C}\_{\text{up}}^{\text{Fe},\text{B}} + \mathbf{C}\_{\text{low}}^{\text{Fe},\text{B}} - \mathbf{2}\mathbf{C}\_{\text{0}} \quad \left. \frac{d\mathbf{x}(t)}{dt} \right|\_{\mathbf{x}(t)=\mathbf{v}} = - & \left. D\_{\text{Fe},\text{B}} \frac{\partial \mathbf{C}\_{\text{Fe},\text{B}}[\mathbf{x}(t=t), t=t]}{\partial \mathbf{x}} \right|\_{\mathbf{x}(t)=\mathbf{v}} \end{aligned} \end{cases} \end{cases} \tag{22}$$

Fick's second law, isotropic one-dimensional diffusion, DFe2B independent of concentration:

$$\frac{\partial C\_{\rm Fe\_2B}[\varkappa(t), t]}{\partial t} = D\_{\rm Fe\_2B} \frac{\partial^2 C\_{\rm Fe\_2B}[\varkappa(t), t]}{\partial \varkappa^2}. \tag{23}$$

By solving Eq. (23), and applying the boundary conditions proposed in Figure 2, the boron concentration profile in Fe2B is expressed by Eq. (24), if the boron diffusion coefficient (DFe2B) in Fe2B is constant for a particular temperature:

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

$$\mathbf{C\_{Fe2B}}[\mathbf{x}(t), t] = \mathbf{C\_{up}^{Fe\_2B}} + \frac{\mathbf{C\_{low}^{Fe\_2B}} - \mathbf{C\_{up}^{Fe\_2B}}}{\text{erf}\left(\frac{\mathbf{v}}{2\sqrt{D\_{Fe\_2B}t}}\right)} \text{erf}\left(\begin{array}{c} \mathbf{x} \\\\ \hline \end{array}\right) \cdot \begin{pmatrix} \\\\ \end{pmatrix} \tag{24}$$

By substituting Eq. (24) into Eq. (22), Eq. (25) is obtained:

$$\frac{\mathbf{C}\_{\text{up}}^{\text{Fe}\_2\text{B}} + \mathbf{C}\_{\text{low}}^{\text{Fe}\_2\text{B}} - 2\mathbf{C}\_0^\dagger}{2} \left(\frac{\mathbf{v}}{dt} = \sqrt{\frac{D\_{\text{Fe}\_2\text{B}}}{\pi t}} \frac{\mathbf{C}\_{\text{up}}^{\text{Fe}\_2\text{B}} - \mathbf{C}\_{\text{low}}^{\text{Fe}\_2\text{B}}}{\text{erf}\left(\frac{\mathbf{v}}{2\sqrt{D\_{\text{Fe}\_2\text{B}}t}}\right)} \exp\left(-\frac{\mathbf{v}^2}{4D\_{\text{Fe}\_2\text{B}}t}\right), \quad \text{(25)}$$
  $\mathbf{v} \to \mathbf{v} \text{ :}$ 

for 0 ≤ x≤ v.

Substituting the expression of the parabolic growth law obtained from Eq. (16) pffiffiffiffiffiffiffiffiffiffiffiffiffi (v <sup>¼</sup> <sup>2</sup>kFe2Bt) into Eq. (25), we have

$$\frac{\left(\mathbf{C}\_{\rm up}^{\rm Fe\_2B} + \mathbf{C}\_{\rm low}^{\rm Fe\_2B} - 2\mathbf{C}\_{\rm Q}^{\rm Q}\right)}{4} \bigg/ \left(2k\_{\rm Fe\_2B}\right)^{1/2} = \sqrt{\frac{D\_{\rm Fe\_2B}}{\pi}} \frac{\mathbf{C}\_{\rm up}^{\rm Fe\_2B} - \mathbf{C}\_{\rm low}^{\rm Fe\_2B}}{\rm pf} \exp\left(-\frac{k\_{\rm Fe\_2B}}{2D\_{\rm Fe\_2B}}\right). \tag{26}$$

#### Figure 2.

A schematic non-linear concentration profile of boron through the Fe2B layer is used to describe the non-steady state diffusion model in on dimension.

The diffusion coefficient (DFe2B) can be estimated numerically by the Newton– , CFe2B Raphson method. It is assumed that expressions CFe2B , and C0, do not depend up low significantly on temperature (in the considered temperature range) [10].

#### 1.2 Materials and methods

#### 1.2.1 Powder pack boriding process

AISI 12L14 steel was used for investigation. It had a nominal chemical composition of 0.10–0.15% C, 0.040–0.090% P, 0.15–0.35% Pb, 0.80–1.20% Mn, 0.25– 0.35% S, 0.10% Si. The typical applications are: brake hose ends, pulleys, disc brake pistons, wheel nuts and inserts, control linkages, gear box components (case hardened), domestic garbage bin axles, concrete anchors, padlock shackles, hydraulic fittings, vice jaws (case hardened). The samples were sectioned into cubes with dimensions of 10 mm ˜ 10 mm ˜ 10 mm. Prior to the boriding process, the samples were polished with SiC sandpaper up 2500 grade, ultrasonically cleaned in an alcohol solution and deionized water for 15 min at room temperature, and dried and stored under clean-room conditions. The mean hardness was 237 HV. The samples were embedded in a closed cylindrical case (AISI 316L) as shown in Figure 3, using Ekabor 2 as a boron-rich agent.

The powder-pack boriding process was performed in a conventional furnace under a pure argon atmosphere. It is important to note that oxygen-bearing compounds adversely affect the boriding process [1]. The thermochemical treatment

#### 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) (millimeter scale).

was performed at temperatures of 1123, 1173, 1223, and 1273 K with 2, 4, 6 and 8 h of exposure time. When the boriding process was concluded, the steel container was removed from the heating furnace and placed in a room temperature.
