## Introduction

The boriding process is a thermochemical treatment involving a chemical modification of surfaces of treated parts with the purpose of increasing their tribological properties and corrosion resistance [1]. This thermochemical process, used to form iron borides, is carried out in the temperature range 700–1000 °C depending on the boriding media used [2, 3, 4, 5, 6, 7]. This process results in the formation of a wear-resistant iron boride layer on the surface of steel parts by thermodiffusion. The boride layer is either a single-phase layer (Fe_{2}B) or a double-phase layer with FeB and Fe_{2}B. For industrial applications, the monolayer configuration (Fe_{2}B) is preferred to the bilayer configuration (FeB and Fe_{2}B) since FeB, is harder and brittle than Fe_{2}B, and having a large coefficient of thermal expansion. As a consequence, cracks can form and propagate along the (FeB/Fe_{2}B) interface because of the high hardness of FeB. Many boriding methods exist for producing wear-resistant boride layers including different media (solid, liquid, gas, and plasma). However, the powder-pack-boriding presents some important advantages such as: easy handling, the possibility of modifying the composition of the boriding agent, minimal equipment, and cost-effectiveness.

From a kinetic viewpoint, several models dealing with the growth kinetics of boride layers on different substrates were reported in many research papers [8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

For example, the authors VillaVelázquez-Mendoza et al. [8]. have used the response surface methodology for studying the evolution of boride layers produced by paste-boriding on the AISI 1018 steel in the temperature range 1073–1273 K as a function of temperature, time and substrate roughness. Based on the ANOVA analysis, the boriding temperature had the highest contribution on the boride layer thickness.

The authors Campos et al. [9] have employed the neural network and the least square models in order to investigate the boriding kinetics of AISI 1045 steel. In this research paper, a single boride phase (Fe_{2}B) layer was produced by the paste-boriding process in the temperature range 1193–1273 K, by modifying the boron paste thickness. These two approaches have been experimentally validated for the samples borided at 1223 K for 5 h of treatment by varying the value of boron paste thickness of 2–5 mm. The technique of fuzzy logic was also employed by Campos et al. [10]. to investigate the growth kinetics of Fe_{2}B layers at the surface of AISI 1045 steel by the paste-boriding treatment. The experimental results in terms of boride layer thickness were compared with those provided by the technique of fuzzy logic and a satisfactory agreement was observed between these two set of data. It is concluded that the utilization of fuzzy logic approach constitutes an alternative for the modelling the growth kinetics of boride layers. The phase-field model has been applied by Ramdan et al. [11]. to simulate the boron-concentration profile through the Fe_{2}B layers formed on Armco iron based on the experimental data taken from the reference work [10]. This kinetic approach was derived from Ginzburg-Landau free energy functional that uses the thermodynamic data of Fe–B phase diagram and the following physical parameters of the material (interface energy and interface thickness). The Fe_{2}B phase was assumed to be stoichiometric during the formulation of phase-field model.

Campos et al. [12]. have used a simple diffusion model based on the kinetic model developed by Brakman et al. [13]. The model considered the principle of mass conservation at the (Fe_{2}B/substrate) interface during the formation of Fe_{2}B layers on Armco-iron by ignoring the effect of boride incubation times and taking into account the difference in molar volume between the Fe_{2}B phase and the substrate. A linear boron-concentration profile through the Fe_{2}B layer was assumed for estimating the diffusion coefficient of boron in Fe_{2}B in the temperature range 1223–1323 K.

Recently, Gómez-Vargas et al. [14]. have suggested a mathematical model for investigating the growth kinetics of Fe_{2}B layers on AISI 1025 steel in the temperature range 1123–1273 K. This kinetic model was based on the principle of mass conservation at the (Fe_{2}B/substrate) interface. A non-linear boron-concentration profile that satisfies the solution of Second Fick’s law was considered. For solving this diffusion problem, a non dimensional kinetic parameter was introduced where the boride incubation time for the formation of Fe_{2}B layers was independent of the boriding temperature.

Mebarek et al. [15]. have used a diffusion model for predicting the boride incubation times during the formation Fe_{2}B layers on XC 38 steel based on solving the mass balance equation at the (Fe_{2}B/substrate) interface and by using the concept of surface boron concentration instead of upper and lower limits for boron concentration in Fe_{2}B.

These different approaches can be used to select the optimum boride layers’ thicknesses according to the industrial applications of the borided materials.

The aim of this present work was to characterize the Fe_{2}B layers formed on AISI 1518 steel and to investigate its boriding kinetics.

For this purpose, an alternative diffusion model based on the integral method was suggested [16, 17] to investigate the kinetics of formation of Fe_{2}B layers on AISI 1518 steel. The present model assumes a non-linear boron-concentration profile through the Fe_{2}B layer with an occurrence of boride incubation time. As an advantage compared to other models, a simple equation was obtained that relates the diffusion coefficient of boron in Fe_{2}B to the square of parabolic growth constant at the (Fe_{2}B/substrate) interface. By using this equation, the boron diffusion coefficients in Fe_{2}B and the value of activation energy for boron diffusion in AISI 1518 steel were estimated in the temperature range 1123–1273 K. Finally, an experimental validation of the present diffusion model was made by using an extra boriding condition (1253 K for 2 h).

## The diffusion model

The diffusion model considers the growth of a single boride layer (Fe_{2}B) over a saturated substrate with boron atoms of AISI 1518 steel Schematic boron concentration – profile through the Fe_{2}B layer is displayed in Figure 1. The *f*(*x,t*) function gives the distribution of boron concentration in the substrate before the nucleation of Fe_{2}B phase. _{2}B. _{2}B (=9 wt.%) while _{2}B (=8.83 wt.%). The point *x*(*t*)=u is the Fe_{2}B layer thickness or the position of (Fe_{2}B/substrate) interface. A small homogeneity range of about 1 at. % was reported for the Fe_{2}B layer [13]. The term

The following assumptions are considered during the formulation of the diffusion model:

- –The growth kinetics is controlled by the boron diffusion in the Fe
_{2}B layer. - –The Fe
_{2}B phase nucleates after a specific incubation time. - –The boride layer grows because of the boron diffusion perpendicular to the specimen surface.
- –Boron concentrations remain constant in the boride layer during the treatment.
- –The boride layer is thin compared to the sample thickness.
- –A uniform temperature is assumed throughout the sample.
- –Planar morphology is assumed for the phase interface.

The initial and boundary conditions for the diffusion problem are given by

_{Boundary conditions:}

The Second Fick’s law that describes the evolution of boron concentration in Fe_{2}B as a function of diffusion distance x(t) and time t is expressed by eq. (4):

where the boron diffusion coefficient is only dependent on the boriding temperature. The expression of boron-concentration profile through the Fe_{2}B layer was adopted from the Goodman’s method [21].

The three time-dependent unknowns *a*(*t*), *b*(*t*), and *u*(*t*) must satisfy the boundary conditions given by eqs. (2) and (3). It is noticed that the two parameters *a*(*t*) and *b*(*t*) must be positive because of a decreasing nature of the concentration profile of boron element. By applying the boundary condition on the surface, eq. (6) was obtained:

By integrating eq. (4) between 0 and *u*(*t*) and applying the Leibniz rule, the ordinary differential equation (ODE) given by eq. (7) was obtained:

The mass balance equation at the (Fe_{2}B/substrate) interface is given by eq. (8):

with

At the (Fe_{2}B/substrate) interface, the boron concentration remains constant and eq. (8) can be rewritten as follows:

Substituting eq. (4) into eq. (9) and after derivation with respect to the diffusion distance x(t), eq. (10) was obtained:

Equations (6), (7) and (10) form 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 determine the expression of boron diffusion coefficient in the Fe_{2}B layers, an analytic solution exists for this diffusion problem by setting:

and

where *u*(*t*) is the Fe_{2}B layer thickness, *k* the parabolic growth constant at the (Fe_{2}B/substrate) interface. It is noticed that the use of eq. (11) is acceptable from a practical point of view since it has been observed in many experiments. The two unknowns *α* and *β* have to be identified for solving this diffusion problem. After substitution of eqs. (11), (12) and (13) into the DAEs system and derivation, the expression of boron diffusion coefficient was obtained as follows:

with

along with the expressions of a(t) and b(t) given by eqs. (15) and (16):

with

and

## Experimental details

### The material and the boriding treatment

The material to be pack- borided was AISI 1518 steel. The chemical composition of AISI 1518 steel is given (in weight percent) in Table 1. The samples had a cubic shape with dimensions of _{4}C, 10 % KBF_{4} and 70 % SiC. The powder-pack boriding process was carried out in a conventional furnace under a pure argon atmosphere in the temperature range 1123–1223 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.

The chemical composition of AISI 1518 steel (in weight percent).

C | Si | Mn | P | S | Fe |
---|---|---|---|---|---|

0.15–0.21 | 0.20–0.40 | 1.10–1.40 | 0.040 | 0.050 | Balance |

### Experimental techniques

The borided and etched samples were cross-sectioned for microstructural investigations using a LECO VC-50 cutting precision machine and the cross-sections of formed boride layers were observed by SEM (JEOL JSM 6300 LV). For a kinetic study, the boride layer thickness was automatically measured with the aid of MSQ PLUS *software*. To ensure the reproducibility of the measured layers, seventy measurements were taken from different sections of the borided samples to estimate the Fe_{2}B layer thickness; defined as an average value of the long boride teeth [22]. The boride formed on the surface of borided sample was identified by means of X-Ray Diffraction (XRD) equipment (Equinox 2000) using

## Results and discussion

### SEM observations and EDS analysis

Figure 2 gives the SEM micrographs of cross-sections of Fe_{2}B layers formed at the surfaces of AISI 1518 steel borided at 1173 K for increasing treatment times. The obtained boride layers look dense and compact exhibiting a saw-tooth morphology for all boriding conditions. This particular morphology promotes a good adhesion to the substrate [23]. The thickness of Fe_{2}B layer increased with the change in the boriding temperature because the diffusion phenomenon of boron atoms into the substrate is a thermally activated process. The value of Fe_{2}B layer thickness ranged from 55.8±10.5 µm for 2 h of treatment to 149.3±22.5 µm for 8 h at 1173 h.

The EDS analysis was carried out at the surface of borided sample and in the vicinity of the (boride layer/substrate) interface as shown in Figure 3. Figure 3**(a)** indicated the presence of iron element with boron element. At the surface of borided sample, the iron atoms combine with the boron atoms to form the Fe_{2}B phase by a mechanism of nucleation and growth of Fe_{2}B crystals. Figure 3**(b)** showed an EDS analysis in the vicinity of the (Fe_{2}B/substrate) interface where the following elements: iron, carbon, silicon and manganese are present. Carbon and silicon are diffused towards the diffusion zone to form together with boron, solid solutions as silicoborides (FeSi_{0.4}B_{0.6} and Fe_{5}SiB_{2}) and boroncementite (Fe_{3}B_{0.67}C_{0.33}) [24].

### X-ray diffraction analysis

Figure 4 gives the XRD pattern obtained at the surface of pack-borided AISI 1518 steel at 1273 K during 8 h. The XRD pattern revealed the existence of Fe_{2}B layer over the surface of AISI 1518 steel. The diffraction peaks showed a difference in intensities that depends on the crystallographic orientations of Fe_{2}B crystals. Furthermore, the growth of Fe_{2}B layer is if a highly anisotropic nature [25].

### Growth kinetics of Fe_{2}B layers

The diffusion model requires the kinetic data to estimate the values of boron diffusion coefficients in the Fe_{2}B layers in the temperature range 1123–1273 K by using eq. (14).

Figure 5 describes the evolution of the square of Fe_{2}B layer thickness versus the treatment time for different boriding temperatures.

Table 2 gives the experimental values of parabolic growth constants at the (Fe_{2}B/substrate) interface along with the corresponding boride incubation times deduced from Figure 5. These data were obtained by plotting the square of Fe_{2}B layer thickness versus time according to eq. (11). In addition, it is noticed that the boride incubation time is independent on the boriding temperature.

The experimental values of parabolic growth constants at the (Fe_{2}B/substrate) interface along with the corresponding boride incubation times.

T(K) | Experimental parabolic growth constant | Boride incubation time |
---|---|---|

1123 | 0.5000 | 2025 |

1173 | 0.7416 | 2025 |

1223 | 1.0909 | 2025 |

1273 | 1.3491 | 2025 |

Figure 6 gives the temperature dependence of boron diffusion coefficients through the Fe_{2}B layers according to Arrhenius relationship. The expression of boron diffusion coefficient in the Fe_{2}B layer can be readily obtained using a linear fitting in the temperature range 1123–1273 K:

where *R*=8.314 J mol^{−1} K^{−1} and

Table 3 compares the value of activation energy for boron diffusion in AISI 1518 steel with the values of activation energy reported in the literature data for some borided materials (Armco iron and steels) [7, 12, 26, 27, 28, 29, 30]. It is noticed that the published values of activation energy for boron diffusion depended on various factors such as: (the temperature range considered, the boriding method, the chemical composition of treated material, the method of calculation and mechanism of boron diffusion). The observed differences in the values of activation energies for boron diffusion in the treated materials indicate that the rate-determining steps in powder and paste-boriding deviate from that for plasma paste-boriding and that for gas boriding process [7, 12, 14, 26]. In the work carried out by Altinsoy et al [30].,The obtained value of activation energy for boron diffusion in AISI 1020 steel was very comparable with that found in this work (=160.45 ± 5.7 kJ mol^{−1}) for AISI 1518 steel. This value of activation energy for boron diffusion was interpreted as the amount of energy for the movement of boron atoms in the easier path for the boron diffusion in the body centred tetragonal lattice of Fe_{2}B that minimizes the growth stresses [25]. Regarding the nature of boride coatings, Palombarini et Carbucicchio [31] reported that the saw-tooth morphology of the (boride layer/substrate) interface in low alloy steels can be explained by enhanced growth at the tips of boride needles.

Comparison of activation energy for boron diffusion in AISI 1518 steel with other borided materials (Armco iron and steels).

Material | Boriding method | Activation energy (kJ mol ^{−1}) | Temperature range (K) | References |
---|---|---|---|---|

Armco iron | Gaseous | 87.857 (FeB) 117.508 (Fe _{2}B) | 1073–1273 | [26] |

Armco iron | Paste | 151 (Fe_{2}B) | 1223–1323 | [12] |

AISI 1018 steel | Electrochemical | 172.75 ± 8.6 (FeB + Fe_{2}B) | 1123–1273 | [27] |

AISI 440C steel | Plasma paste-boriding | 134.62 (FeB + Fe _{2}B) | 973–1073 | [7] |

AISI 304 steel | Salt bath | 253 0.35 (FeB + Fe _{2}B) | 1073–1223 | [28] |

AISI P20 steel | Pack- powder | 200 (FeB + Fe _{2}B) | 1073–1223 | [29] |

AISI 1020 steel | Pack- powder | 164.356 (Fe _{2}B) | 1073–1223 | [30] |

AISI 1518 steel | Pack- powder | 160.4 5 ± 5.7 (Fe _{2}B) | 1123–1273 | This work |

In the present study, the SEM observations revealed that many of borided needles are of dentritic nature as reported by Ninham and Hutchings [32]. The columnar nature of this interface is resulting from the side arm growth similar to that observed during the solidification of metallic alloys or metals [33].

### Experimental verification of the diffusion model

The validity of this diffusion model was verified by a comparison of experimental value of Fe_{2}B layer thickness obtained at 1253 K during 2 h of treatment with the predicted value of Fe_{2}B layer thickness given by eq. (18).

with *η*=13.3175

Figure 7 shows the SEM micrograph of the cross-section of the sample borided at 1253 K for 2 h. Table 4 shows a comparison between the experimental value of Fe_{2}B layer thickness obtained at 1253 K for 2 h and the predicted value given by eq. (18) for an upper boron content in the Fe_{2}B phase equal to 9 wt.%. Therefore, the predicted value of Fe_{2}B layer thickness agree with the data obtained experimentally, From a practical point of view, and for this kind of steel, knowledge of the variables that control the boriding treatment is of great importance for obtaining the optimum value of Fe_{2}B layer thickness.

Comparison between the experimental value of Fe_{2}B layer thickness obtained at 1253 K for 2 h and the predicted value using the integral method for an upper boron content in the Fe_{2}B phase equal to 9 wt.%.

Boriding conditions | Experimental Fe_{2}B layer thickness (µm) | Simulated Fe_{2}B layer thickness (µm) by eq. (18) |
---|---|---|

1253 K for 2 h | 102.6 ± 12.2 | 89.6 |

## Conclusions

In the present work, the AISI 1518 steel was subjected to the pack-boriding process in the mixture of powders composed of (20 % B_{4}C, 10 % KBF_{4} and 70 % SiC) in the temperature range 1123–1223 K for a variable treatment between 2 and 8 h. A monolayer configuration (Fe_{2}B) was seen in all SEM micrographs exhibiting a saw-toothed morphology. The crystalline nature of this iron boride was confirmed by XRD analysis. The growth kinetics of Fe_{2}B layers followed a parabolic growth law. A particular solution of a system of DAEs has been obtained in order to estimate the boron diffusion coefficients in the Fe_{2}B layers in the temperature range 1123–1273 K. On the basis of our experimental data, the value of activation energy for boron diffusion was estimated as 160.45±5.7 kJ mol^{−1} for AISI 1518 steel. This value of energy is needed to overcome the energetic barrier for activating the boron diffusion along the preferred crystallographic direction [100]. In addition, the present diffusion model was also verified experimentally for the sample borided at 1253 K for 2 h. A good agreement was observed between the experimental result and the predicted value of Fe_{2}B layer thickness.

## List of symbols

u(t) | is the Fe |

a(t) | and b(t) are the time-dependent parameters |

k | is the parabolic growth constant of the Fe |

t | is the treatment time ( |

is the boride incubation time ( | |

represents the upper limit of boron content in Fe | |

is the lower limit of boron content in Fe | |

is the adsorbed boron concentration in the boride layer (wt..%). | |

is the boron solubility in the matrix ( | |

is the boron concentration profile in the layer ( | |

represents the diffusion coefficient of boron in the |

The work described in this paper was supported by a grant of PRDEP and CONACyT México. Likewise, FCS reconoce los fondos del Departamento de Física y Matemáticas y de la División de Investigación de la UIA. The authors wish to thank the Laboratorio de Microscopía de la UIA.

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