#### Abstract

An indispensable tool to choose the suitable process parameters for obtaining boride layer of an adequate thickness is the modeling of the boriding kinetics. In this work, two mathematical approaches were used in order to determine the value of activation energy in the Fe_{2}B layers on ASTM A36 steel during the iron powder-pack boriding in the temperature range of 1123–1273 K for treatment times between 2 and 8 h. The first approach was based on the mass balance equation at the interface (Fe_{2}B/substrate) and the solution of Fick’s second law under steady state (without time dependent). The second approach was based on the same mathematical principles as the first approach for one-dimensional analysis under non-steady-state condition. The measurements of the thickness (Fe_{2}B), for different temperatures of boriding, were used for calculations. As a result, the boron activation energy for the ASTM A36 steel was estimated as 161 kJ·mol^{−1}. This value of energy was compared between both models and with other literature data. The Fe_{2}B layers grown on ASTM A36 steel were characterized by use of the following experimental techniques: X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy dispersive X-ray Spectroscopy (EDS). Finally, the experimental value of Fe_{2}B layer’s thickness obtained at 1123 K with an exposure time of 2.5 h was compared with the predicted thicknesses by using these two approaches. A good concordance was achieved between the experimental data and the simulated results.

#### 1. Introduction

Nowadays and due to the increasing technological development, it is necessary to have metallic materials with specific features that must be maintained in critical service conditions: for example, the metal dies are used in the different hot and cold working metallurgical processes, which given the working conditions require high toughness and high surface hardness. The thermochemical treatments applied to steel are those in which the composition of the surface of the workpiece is altered by the addition of carbon, nitrogen, sulphur, boron, aluminium, zinc, chromium, or other elements. The most common treatments in the industry are carburization, nitriding, carbonitriding, and boriding. Boriding is a thermochemical treatment controlled by the diffusion of boron atoms, which modifies the properties of the material generating hard surfaces. Likewise, the boriding process has a positive effect on the tribological applications: abrasive, adhesive, fatigue, and corrosion wear in acid and alkaline media. The process involves heating a ferrous or nonferrous alloy for a temperature range of 700°C to 1000°C with a treatment time of 1 to 12 hours for powder-pack methods. When boron diffuses in a substrate, a monolayer (Fe_{2}B) or a double layer (FeB–Fe_{2}B) can be formed, depending on the chemical composition of the substrate and the chemical potential of boron. These phases consist of orthorhombic and tetragonal lattices (body-centred), respectively. One basic advantage of boride layers is that they can reach high hardness values (between 1800 and 2000 HV), kept at high temperatures [1–4]. Abrasion and adhesion wear are forms of wear by contact between a particle and solid material, being the characteristic result of almost all types of mechanical stress. Borided steels are resistant to abrasion because of their extreme hardness on the surface; this characteristic makes them suitable to be applied in pneumatic conveying systems; dies for stamping; components of plastic processing machines, such as extrusion screws; bearings for oil extraction pumps; ball valves; plungers for use in manufacturing glass; and components in textile machinery. In practice, there are many techniques of surface modification by boriding, such as powder-pack boriding [5], paste boriding [6], gaseous boriding [7], plasma boriding [8], plasma paste boriding [9], and laser boriding [10]. However, the most frequently used method in industry is the powder-pack boriding, which demands a low investment cost of equipment and an easy handling. From a kinetic point of view, several approaches [3, 5–7, 11–33] were developed in the objective of optimizing the thicknesses of borided layers in order to meet the functional requirements during industrial use of borided steels. Some of these models that estimate the thickness of the monolayer (Fe_{2}B) or a double layer (FeB–Fe_{2}B) are based on the solution of Fick’s second law without time dependent ( ⟶ steady state) [3, 6, 7, 16–18, 20–22, 24–26, 30] and some others on the solution of Fick’s second law with time dependent ( ⟶ non-steady state) [5, 24, 27, 28, 31, 33].

The ASTM A36 steel has a good machinability with an acceptable wear resistance. ASTM A36 steel has the following applications:(i)It is used in bolted, riveted, or welded construction of bridges, buildings, and oil rigs(ii)It is used in forming tanks, bins, bearing plates, fixtures, rings, templates, jigs, sprockets, cams, gears, base plates, forgings, ornamental works, stakes, brackets, automotive and agricultural equipment, frames, and machinery parts(iii)It is used for various parts obtained by flame cutting such as in parking garages, walkways, boat landing ramps, and trenches

In the present work, two different models were suggested for simulating the growth kinetics of monolayer (Fe_{2}B) on ASTM A36 steel in the range of boriding temperature 1123–1273 K. The parabolic growth constants of Fe_{2}B were determined. The boron diffusion coefficient in the boride layers was estimated from these two approaches based on the conditions of the boriding process in the Fe-B system. X-ray diffraction (XRD), scanning electron microscopy (SEM), and energy dispersive X-ray spectroscopy (EDS) were conducted on material borided to characterize the presence of the Fe_{2}B layer and the distribution of heavy elements in the surface of the ASTM A36 steel. Finally, the experimental value of Fe_{2}B layer’s thickness obtained at 1123 K with an exposure time of 2.5 h was compared with the predicted result from these two approaches.

#### 2. Mathematical Approaches

One basic parameter that characterizes the Fe_{2}B 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 allows 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 dependent such that

##### 2.1. Derivation of the Parabolic Growth Law

In diffusion in solid, 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 rate-determining. If the diffusion of B atoms is rate-determining, the layer rate is proportional to the flux through the substrate:

El flux, , can be written aswhere is the boron concentration profile in mol/m^{3} and is the velocity of Fe_{2}B layer in m/s, and giving units of mol/m^{2}·s. The velocity of a particle is proportional to the force, *F*, on the particle:where is the mobility of the boron atoms. Writing the chemical potential as , this force is written asfor a Fe_{2}B layer with thickness *x*. If combining equations (4) and (5), then equation (3) can be written asFrom the relationship,where is the Boltzmann’s constant, we can write

In an ideal system, the concentration, , is equivalent to activity, . Substituting equations (8) into (6), we get

As shown in equation (2),so that a combination of equations (2) and (9) gives

If we assume that the potential is fixed at each boundary of the Fe_{2}B layer, we can replace in equation (11) with the slope (). We then introduce the parabolic growth constant , and set

Combining equations (11) and (12) gives

Equation (13) can be rewritten as

Upon integration of equation (14),

We arrive at the parabolic growth law (with ):where represents the Fe_{2}B layer thickness.

##### 2.2. First Approach: Steady-State Diffusion Model

The first approach is based on the diffusion model proposed by Flores-Rentería et al. [3], where a mathematical model has been applied based on the mass balance equation at the (Fe_{2}B/substrate) interface by assuming a linear boron concentration profile through the Fe_{2}B layer ( ⟶ steady state). This approach was used to simulate the kinetics of formation of Fe_{2}B layer on ASTM A36 steel with the presence of boride incubation time. Steady state means that there will not be any change in the composition profile with time. and denote the upper and lower boron concentrations in the Fe_{2}B phase [3, 34, 35]. represents the boron solubility in the matrix and can be neglected [3, 34, 35]. The assumptions made during the mathematical formulation of the diffusion model are given in the reference work [3].

The mass balance equation [3], describing the evolution of displacement of growing interface with respect to the time, is given by

The linear boron concentration profile [3], through the Fe_{2}B layer, is given by the solution of Fick’s second law without time dependent ( ⟶ steady state ( (is called the Laplace operator or Laplacian in one dimension)) and is rewritten as

By substituting equations (18) and (16) into equation (17),

##### 2.3. Second Approach: Non-Steady-State Diffusion Model in One Dimension

The second approach [5] was applied to analyze the kinetics of formation of the monolayer (Fe_{2}B) generated at the surface of pack-borided AISI 1045 steel. In the present work, this mathematical model was adopted for studying the boriding kinetics of ASTM A36 steel. Likewise, the mass balance equation [5], describing the evolution of displacement of growing interface with respect to the time, is given by

The linear boron concentration profile [5], through the Fe_{2}B layer, is given by the solution of Fick’s second law with time dependent ( ⟶ non-steady state) and is deduced as follows:

By substituting equations (21) into (20), the following equation is obtained:

Substituting the expression of the parabolic growth law obtained from equation (16) () into equation (22), we have

The diffusion coefficient () can be estimated numerically from equation (23) by the Newton–Raphson method. An illustrative representation of the parabolic growth law of Fe_{2}B layer thickness () is represented in Figure 1. In addition, is the effective growth time of the Fe_{2}B layer and *t* is the boriding time [3, 5].

#### 3. Materials and Methods

##### 3.1. Powder-Pack Boriding Process

ASTM A36 steel was used for investigation. It had a nominal chemical composition of 0.25–0.29% C, 0.20–0.28% Si, 0.85–1.35% Mn, 0.15–0.20% Cu, 0.035–0.040% P, and 0.050% S. The steel samples were sectioned into small cubes with the following dimensions: 10 mm × 10 mm × 10 mm. Prior to the boriding process, the steel samples were grinded with SiC abrasive paper up to grit 2500 and cleaned using a multistage ultrasonic bath with n-heptane and ethanol for 20 min. The mean hardness of the substrate was 170 HV. The ASTM A36 steel samples were immersed in a closed cylindrical case made of AISI 316L steel as shown in Figure 2, using Ekabor 2 as a boron-rich mixture.

The thermochemical process was carried out in a conventional furnace model Nabertherm N 250/85 HA (this type of furnace is gas-tight; it is equipped with direct heating depending on the temperature. It is excellent for maintaining an atmosphere defined by an inert gas), maintaining a pure argon atmosphere, to eliminate the oxidation of the boron released in the chemical reaction of the boriding medium [1]. The thermochemical treatment was carried out at boriding temperatures of 1123, 1173, 1223, and 1273 K for a variable time (2, 4, 6, and 8 h). The treatment temperatures were selected according to the Fe-B phase diagram.

##### 3.2. Microscopical Observations of Boride Layers

The hardened samples were sectioned and prepared metallographically (the samples were polished using a diamond suspension with a particle size of 6 *μ*m, finishing with a particle size of 3 *μ*m), using a GX51 Olympus equipment. Likewise, the borided samples were analyzed through scanning electron microscope. The equipment used was the Quanta 3D FEG-FEI JSM7800-JOEL. Figure 3 shows the cross sections of Fe_{2}B layers formed on the surfaces of ASTM A36 steel at different exposure times (2, 4, 6, and 8 h) and for 1173 K of boriding temperature. The mechanical properties of borided alloys depend on the composition and structure of the boride layer. The images obtained from the scanning electron microscope (Figure 3) present a sawtooth morphology; this characteristic is typical of ARMCO pure iron, and low and medium carbon steels [24, 36].

**(a)**

**(b)**

**(c)**

**(d)**

When the alloying elements and/or the carbon content of the steel increases, the layer thickness tends to favor the formation of iron borides with flat growth fronts. The alloying elements have obvious effects on the formation of the layer thickness, restricted diffusion of the boron atoms, thus forming a diffusion barrier. Because of the diffusion of boron atoms, there is a segregation of the alloying elements from the surface to the (Fe_{2}B/substrate) interface. Some alloying elements tend to form compounds with the boron atoms and others cannot interact with them. Boron has some weird and wonderful chemistry. The alloying elements cannot form compounds and tend to concentrate at the tips of boride columns, decreasing the layer thickness [36].

Figure 3 shows that the boride layer thickness increases with respect to the boriding time for a predetermined temperature. For simulating the growth kinetics of the Fe_{2}B layer grown on ASTM A36 steel, an average measurement of the boride layer thickness was made, where the longest tips of the sawtooth morphology were taken into account (see Figure 4), and the software used was MSQ Materials Analysis. Fifty measurements were collected from the boride surface to the longest tips of boride columns of the borided ASTM A36 steel, as plotted in Figure 4 [3, 5]. The identification of phases formed on the surface of the borided sample was conducted through the X-ray diffraction technique (XRD). The equipment used for the study was an INEL EQUINOX 2000 X-ray diffractometer, using CoK_{α} radiation of 0.179 nm wavelength, operated at 30 mA and 20 kV.

Likewise, Match version 3.3 was the software for phase identification. In addition, the elemental distribution of the transition elements within the cross section of boride layer was determined by using scanning electron microscopy-energy dispersive X-ray spectroscopy (SEM-EDS). The equipment used for the study was a Quanta™ 3D FEG scanning electron microscope, with an accelerating voltage of 200 V–30 kV and a magnification of 30 X–1280 kX in “quad” mode.

#### 4. Results and Discussion

##### 4.1. SEM Observations and EDS Analysis

The metallography of coating/substrate formed in ASTM A36 borided steel at different exposure times (2, 4, 6, and 8 h) and for 1123 K of boriding temperature is shown in Figure 5.

**(a)**

**(b)**

The EDS analysis obtained by SEM is shown in Figures 5(a) and 5(b). The results show in Figure 5(a) that the manganese (Mn) negatively affects the boride layer thickness and morphology. Likewise, as can be seen in Figure 5(b), carbon (C) and silicon (Si) do not dissolve and diffusive through the boride layer. During the boriding process, carbon is transmitted from the boride surface to the matrix and forms, together with boron, borocementite Fe_{3}(B, C) (or, more appropriately, Fe_{3}(B_{0.67}C_{0.33})) [21, 32]. In addition to carbon, silicon is also not soluble in the boride layer. This element is expelled from the surface by boron atoms to the growth interface (Fe_{2}B/substrate), forming iron-silicoborates (FeSi_{0.4}B_{0.6} and Fe_{5}SiB_{2}) [37].

##### 4.2. X-Ray Diffraction Analysis

Figure 6 shows the diffractogram recorded on the surface of the borided ASTM A36 steel at a temperature of 1123 K for a treatment time of 8 h. The X-ray diffraction patters (see Figure 6) show the intensity of X-rays scattered at different angles by a borided sample, where the presence of the Fe_{2}B phase is confirmed.

##### 4.3. First Approach: Estimation of Boron Activation Energy with Steady-State Model

To study the growth kinetic of Fe_{2}B on ASTM A36 steel, a simple diffusion model inspired from the mass balance equation at the (Fe_{2}B/substrate) interface and the solution of Fick’s second law under steady-state condition for one-dimensional analysis was applied [3, 6, 7, 16–18, 20–22, 24–26, 30]. The Fe_{2}B layer thickness () obeys the parabolic growth law given by equation (16), with the presence of boride incubation time ( after transferring the sample to the furnace) associated with the formation of the Fe_{2}B layer. Figure 7 displays the time dependent of the squared value of Fe_{2}B layer thickness for different temperatures, the slopes of each of the straight lines supply the values of the parabolic growth constants ().

Table 1 summarizes the experimental values of parabolic growth constants at the (Fe_{2}B/substrate) interface in the temperature range of 1123–1273 K with the associated boride incubation time. Also, the plots presented in Figure 7 demonstrate a diffusion-controlled process (Brackman et al. [21]).

The value of activation energy (is the energy required for a reaction to proceed) for boron diffusion in Fe_{2}B ( = 161.00061 kJ·mol^{−1}) and preexponential factor ( = 1.361592 × 10^{−3} m^{2}/s) can be calculated after a linear fitting of these data according to Arrhenius relationship from the slope and interception of the straight line, respectively, shown in coordinate system (see equation (24)): vs , it is displayed in Figure 8:with *R* = 8.314 Jmol^{−1}K^{−1} and *T* the absolute temperature in Kelvin.

##### 4.4. Second Approach: Estimation of Boron Activation Energy with Non-Steady-State Diffusion Model

The values of the growth constants () at each temperature reported in Table 1 are also used in the second approach and in Table 2, the values of the boron diffusion coefficients () for Fe_{2}B are gathered, which were calculated by the Newton–Raphson method using equation (23).

In Figure 9, the vs is plotted, the values of activation energy and pre exponential factor for boron diffusion in Fe_{2}B from the second approach were = 160. 9922 kJ·mol^{−1} and = 1.35914 × 10^{−3} m^{2}/s, respectively. Equation (25) was deducted from a fitting of data according to the Arrhenius relationship with the coefficient of determination close to unity.with *R* = 8.314 Jmol^{−1}K^{−1} and *T* the absolute temperature in Kelvin.

##### 4.5. The Two Diffusion Models

This section describes the differences between the two diffusion models that have been used to compute the growth kinetics of boride layers. It is noticed that the estimated values of boron activation energy for ASTM A36 steel from the first approach (see equation (24)) and the second approach (see equation (25)) is approximately the same value for both diffusion models. Likewise, in the estimated values of pre exponential factor from the first approach (*D*_{0} = 1.361592 × 10^{−3} m^{2}/s) and the second approach (*D*_{0} = 1.35914 × 10^{−3} m^{2}/s), there is a small variation. To find out how this similarity is possible in the diffusion coefficients obtained by two different models, we first focus our attention on equation (23). The error function is a monotonically increasing function of . Its Maclaurin series (for small ) is given by [38, 39]:

According to the numerical value of the , equation (26) can be rewritten as

Similarly for the real exponential function : ℝ ⟶ ℝ can be characterized in a variety of equivalent ways. Most commonly, it is defined by the following power series [38, 39]:

Thus, equation (28) can be modified in the following manner:

By substituting Equations (27) and (29) into equation (23), we havewhere

The result obtained by equation (31) is the same as that obtained in equation (19) estimated by a steady-state diffusion model. The result from equation (19) 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 as in this case.

Table 3 presents the values of activation energies for boron diffusion in some different alloys and Armco iron along with the estimated value of boron activation energy in ASTM A36 steel. From Table 3, it is seen that the estimated value of activation energy in ASTM A36 steel is consistent with the literature data [3, 5–7, 27, 31, 33, 40–43].

##### 4.6. Experimental Validation of the Kinetic Models

The suggested models were validated by comparing the experimental value of Fe_{2}B layers’ thickness with the theoretical result. By substituting equation (19) into (16), we have

Figure 10 shows the SEM micrographs of the cross sections of borided ASTM A36 steels at 1123 K for 2.5 h. Table 4, presents the values of the theoretical result of Fe_{2}B layers’ thickness with the experimental data. A good concordance was the observed value between the experimental result and the simulated value for the given boriding condition.

#### 5. Conclusions

In this work, the ASTM A36 steel was pack-borided in the temperature range of 1123–1273 K for a variable exposure time ranging between 2 and 8 h. The kinetics data on treated ASTM A36 steel by the powder-pack boriding were used to estimate the value of activation energy for boron diffusion in the boride layer (Fe_{2}B) by means of two different mathematical approaches. In the first approach, the mass balance equation was formulated by assuming a linear boron concentration profile in the boride layer () for an upper boron content in Fe_{2}B of 60 × 10^{3} mol·m^{−3}. The second approach was based on the same mathematical principles as the first approach for one-dimensional analysis under non-steady-state condition by assuming the solution of Fick’s second law of diffusion when the bulk concentration is greater than the surface concentration (), was successfully applied to the boriding kinetics of ASTM A36 steel by considering the effect of boride incubation time. As a main result, the estimated value of activation energy for boron diffusion in the Fe_{2}B layer by the two approaches was estimated as 161 kJ·mol^{−1}. In a future prospect, these diffusion approaches are in general not identical, both are equivalent models, and this fact can be used as a tool to select optimum values of layers’ thicknesses for practical utilization of any borided steels to produce boride layers with sufficient thicknesses that meet the requirements during service life.

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The work described in this paper was supported by a grant of National Council of Science and Technology (CONACyT) and PRODEP México. Also, the authors want to thank Dr. Jorge Zuno Silva, who is a research professor and principal at Escuela Superior de Ciudad Sahagún-UAEH. Likewise, the authors would like to thank Dr. José Solís Romero for his helpful advice on various technical issues examined in this manuscript and Dr. Adrian Leyland for his advice and comments.