Issue 
Metall. Res. Technol.
Volume 117, Number 2, 2020



Article Number  202  
Number of page(s)  9  
DOI  https://doi.org/10.1051/metal/2020011  
Published online  23 March 2020 
Regular Article
Application of integral method for investigating the boriding kinetics of AISI 316 steel
Application de la méthode intégrale pour l’étude de la cinétique de boruration de l’acier AISI 316
Laboratoire de Technologie des Matériaux, Faculté de Génie Mécanique et Génie des Procédés, USTHB,
B.P No. 32,
16111,
ElAlia, BabEzzouar,
Algiers, Algeria
^{*} email: keddam@yahoo.fr
Received:
27
July
2019
Accepted:
12
February
2020
The present work is dealing with the modelling of boriding kinetics of AISI 316 steel in the temperature range 1123–1273 K. A diffusion model based on the integral method was used in order to investigate the kinetics of formation of FeB and Fe_{2}B layers and that of diffusion zone formed on AISI 316 steel by considering the presence of boride incubation times. By using a particular solution of the resulting differential algebraic system, the diffusion coefficients in FeB, Fe_{2}B and diffusion zone (DZ) were estimated as well as the corresponding values of activation energies. Finally, this present diffusion model has been experimentally validated for two additional boriding conditions (1243 K for 3 and 5 h of treatment). A good concordance was observed between the experimental and the simulated results in terms of layers’ thicknesses.
Résumé
Ce présent travail est consacré à la modélisation de la cinétique de boruration de l’acier AISI 316 dans le domaine de température 1123–1273 K. Un modèle de diffusion basé sur la méthode intégrale a été utilisé dans le but d’étudier la cinétique de formation des couches FeB, Fe_{2}B et celle de la zone de diffusion formées sur l’acier AISI 316 en considérant la présence des temps d’incubation des borures. Les coefficients de diffusion du bore dans FeB, Fe_{2}B et dans la zone de diffusion (ZD) ainsi que les valeurs correspondantes des énergies d’activation ont été estimés en utilisant une solution particulière du système algébrodifférentiel résultant. Finalement, ce présent diffusion modèle a été expérimentalement validé pour deux conditions de boruration additionnelles (1243 K pour 3 et 5 h de traitement). Une bonne concordance a été observée entre les résultats expérimentaux et simulés en termes d’épaisseurs des couches.
Key words: boriding / iron borides / incubation times / integral method / activation energies / DAE system / kinetics / diffusion coefficient / Arrhenius Law / thermodiffusion
Mots clés : boruration / borures de fer / temps d’incubation / méthode intégrale / énergies d’activation / système EAD / cinétique / coefficient de diffusion / loi d’Arrhenius / thermodiffusion
© EDP Sciences, 2020
1 Introduction
The boriding process is widely used to produce hard boride layers with outstanding properties [1] such as: a high surface hardness of about 2000 HV, a good tribological properties as well as resistance against corrosion in different media [2]. It is a method of enrichment of metallic surface by boron atoms via a thermodiffusion process from a source rich in boron element in the temperature range 800–1050°C between 0.5 and 10 h. Two types of iron borides could be observed when boriding the ferrous alloys such as steels and cast irons. The FeB phase crystallizes in an orthorhombic structure while the Fe_{2}B phase has a body central tetragonal structure with narrow composition ranges for both iron borides.
It is known that the functional properties in service of borided steels are strongly influenced by the boriding parameters (the time and the temperature), the type of boriding process as well as the nature of boron source.
For these reasons, the modeling of boriding kinetics is crucial for optimizing the desired boride layer thickness that matches the practical utilization of borided steels at industrial scale. To reach this objective, several studies were devoted in the literature for modeling the boriding kinetics of ferrous alloys: (Armco iron [3–6], steels [7–24] and cast irons [25–28]) by using different approaches. Mebarek et al. [4] have proposed an approach based on the Least Squares Support Vector Machines (LSSVM) for modelling the growth kinetics of FeB and Fe_{2}B layers on Armco iron. Nait Abdellah et al. [6] have investigated the kinetics of formation of Fe_{2}B layers on Armco iron by using two different approaches (a diffusion model including the effect of boride incubation times and dimensional analysis).
Rayane and Allaoui [7] have employed the neural network model to predict the boride layer thickness on XC38 steel with a backpropagation algorithm. Keddam and Kulka [8] have suggested an alternative approach based on the integral method to investigate the boriding kinetics of AISI D2 steel. A particular solution of the resulting differential algebraic equations was obtained to estimate the boron diffusion coefficients in the FeB and Fe_{2}B layers on AISI D2 steel. A numerical treatment of this diffusion problem was also performed to validate experimentally this kinetic model. CamposSilva et al. [11] have formulated a diffusion model based on the mass balance equations at the two interfaces (FeB/Fe_{2}B) and (Fe_{2}B/substrate) in order to estimate the boron diffusion coefficients in FeB and Fe_{2}B on AISI M2 steel by taking into account the presence of boride incubation times. Campos et al. [13] have applied two techniques (neural network model and the least square method) to simulate the kinetics of formation of Fe_{2}B layers on AISI 1045 steel by changing the boron paste thickness. The simulation results showed a mean error of 5.31% for the neural network and 3.42% for the least square method for the Fe_{2}B layer’s thicknesses. CamposSilva et al. [17] have applied a diffusion model for the (FeB/Fe_{2}B) bilayer and the diffusion zone in the case of packboriding of AISI 316 steel. This model assumes a linear boron concentration through the FeB and Fe_{2}B layers with the inclusion of boride incubation times. It did take into account the precipitation of metallic borides inside the boride layers. Keddam et al. [20] have proposed a diffusion model based on the integral method for the boriding kinetics of AISI 12L14 steel. It is found that the boron diffusion coefficient in Fe_{2}B is proportional to the square of parabolic growth constant at the (Fe_{2}B/substrate) interface. The model has been validated experimentally for four additional boriding conditions.
In the present work, a diffusion model based on the integral model was proposed to estimate the values of boron diffusion coefficients in the FeB and Fe_{2}B layers as well as in the diffusion zone for AISI 316 steel. This kinetic approach is an extension of a diffusion model previously published in the reference work [8]. It does not consider the precipitation phenomenon of metallic borides during the diffusion process of boron atoms into the substrate of AISI 316 steel. This diffusion model assumes that the boron concentration profiles are non linear inside the FeB and Fe_{2}B layers and in the diffusion zone with the occurrence of boride incubation times. Finally, the present model was experimentally verified for two additional boriding conditions (1243 K for 3 and 5 h) by comparing the predicted results with the experimental values.
2 The diffusion model based on the integral method
The diffusion of boron atoms into the AISI 316 steel substrate was treated as a one dimensional problem. The present model, based on the integral model, investigates the kinetics of formation of FeB and Fe_{2}B layers and that of diffusion zone in a saturated matrix by boron atoms. The boronconcentration profiles inside the FeB, Fe_{2}B layers and in the diffusion zone are shown in Figure 1.
The terms (= 16.40 wt.%B) and (= 16.23 wt.%B) indicate respectively the upper and lower boron concentrations in the FeB phase while (= 9.00 wt.%B) and (= 8.83 wt.%B) are respectively the values of upper and lower boron concentrations in the Fe_{2}B phase [10,22,24]. (= 35 × 10^{−4} wt.%B) and (= 0.00 wt.%B) represent the upper and lower boron concentrations in the diffusion zone [23]. C_{ads} denotes the adsorbed boron concentration at the material surface [16]. The variables u, v and w represent respectively the layers’ thicknesses of FeB, (FeB + Fe_{2}B) and (FeB + Fe_{2}B + diffusion zone). During the formulation of this diffusion model the following assumptions are considered:

the mass flux of boron atoms is perpendicular to the material surface;

the FeB and Fe_{2}B layers and the diffusion zone are formed after specified incubation times;

boron concentrations do not change during the boriding treatment.

the (boride layer + diffusion zone) is thin in comparison with the sample thickness;

planar morphology is assumed for the growing interfaces.
The boundary conditions: (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
The time dependence of FeB layer thickness u is given by equation (13), is the parabolic growth constant of FeB layer and the associated boride incubation time. (13)
The time dependence of (FeB + Fe_{2}B) layer thickness v is described by equation (14), where denotes the parabolic growth constant at the (Fe_{2}B/DZ) interface with the boride incubation time of (FeB + Fe_{2}B) layer. (14)
The time evolution of total thickness of (FeB + Fe_{2}B + DZ) w is given by equation (15), where k_{DZ} is the parabolic growth constant of diffusion zone with the corresponding boride incubation time. (15)
Mathematically, equation (13) can be rewritten as equation (16) by taking the boride incubation time of (FeB + Fe_{2}B + DZ) with a new value of parabolic growth constant k_{1} at the (FeB/Fe_{2}B) interface: (16)
In the same way, the (FeB + Fe_{2}B) layer thickness can be expressed by equation (17) with respect to the boride incubatim time of (FeB + Fe_{2}B + DZ) as follows: (17) where k_{2} is the new value of parabolic growth constant at the (Fe_{2}B/DZ) interface. The boronconcentration profiles in the FeB and Fe_{2}B layers as well as in the diffusion zone are parabolic as suggested by Goodman [29]. Their expressions are given by equations (18)–(20) as follows: (18) (19) (20)
It is noted that the boron concentration profiles inside the FeB and Fe_{2}B layers as well as in the diffusion zone depend on the timedependent parameters a_{1}(t), b_{1}(t), a_{2}(t), b_{2}(t), a_{3}(t) and b_{3}(t). These parameters are subjected to the boundary conditions imposed by the integral method. A particular solution of DAE (Differential Algebraic Equations) was found in order to calculate the values of boron diffusion coefficients in iron borides and in the diffusion zone. The present diffusion problem requires a resolution of resulting DAE system formed by six algebraic constraints (Eqs. (21)–(26)) and three ordinary differential equations (27)–(29): (21) (22) (23) (24) (25) (26) (27) (28) (29) with
Therefore, the following new variable changes given by equations (30)–(35) were used to find the expressions of boron diffusivities in iron borides and in the diffusion zone: (30) (31) (32) (33) (34) (35) where the constants (αi and βi with i = 1 to 3) are defined positive due to the decreasing nature of boronconcentration profiles.
After derivation of equations (15)–(17) and (30)–(35) with respect to the time and substitution, the DAE system becomes a system of non linear equations solved by the NewtonRaphson routine [30]. The expressions of boron diffusion coefficients in iron borides and in the diffusion zone are respectively given by: (36) (37) and (38)
Fig. 1 Schematic representation of the boronconcentration profile along the borided layer and the diffusion zone. 
3 Results and discussions
The kinetics data obtained by CamposSilva et al. [17] were used for estimating the boron diffusivities in the iron borides (FeB and Fe_{2}B) and in the diffusion zone for AISI 316 steel by means of a kinetic model based on the integral method. The nominal chemical composition of AISI 316 steel was the following: 0.08 wt.%C max, 2.0 wt.%Mn max, 1.0 wt.%Si max, 10 to 14 wt.% Ni, 16 to 18 wt.%Cr, 2 to 3 wt.%Mo, 0.045 wt.%P max and 0.030 wt.%S max. In their experiments [17], the powderpack boriding was carried out in the temperature range 1123–1273 K for a treatment time between 2 and 10 h. The boriding agent was Durborid fresh powder mixture containing boron carbide (B_{4}C). The samples were placed in a closed container made of 304 AISI steel, packed into this boriding agent and heat treated in the furnace at 1123, 1173, 1223 and 1273 K. To guarantee the accuracy of thicknesses’ measurements, fifty measurements were made on different locations of crosssections of borided samples to estimate the layers’ thicknesses of FeB, (FeB + Fe_{2}B) and (FeB + Fe_{2}B + diffusion zone). By fitting the experimental data [17] with equations (13)–(15) the parabolic growth constants at the interfaces along with the corresponding boride incubation times could be obtained as shown in Table 1. The values of parabolic growth constants represent the slopes of straight lines and the boride incubation times are deduced from the intercepts with time axis for a null value of thickness.
In Table 1, the boride incubation times were decreased with boriding temperature as evidenced in many studies published in the literature [8–12,17]. Since the diffusion of boron atoms was activated at high temperatures, therefore the incubation periods were reduced.
Experimental values of parabolic growth constants at the interfaces along with the corresponding boride incubation times.
3.1 Estimation of boron diffusion coefficients in iron borides and in the diffusion zone
To determine the values of boron diffusion coefficients in the FeB and Fe_{2}B layers as well as in the diffusion zone by using equations (36)–(38), it is necessary to determine the new values of the parabolic growth constants at the two interfaces (FeB/Fe_{2}B) and (Fe_{2}B/DZ)
with respect to the boride incubation time of (FeB + Fe_{2}B + diffusion zone). Table 2 gives the new values of experimental parabolic growth constants at the interfaces obtained from a linear fitting of equations (16) and (17).
In Table 3 are gathered the estimated values of boron diffusion coefficients in FeB, Fe_{2}B and in the diffusion zone from the integral method for an upper boron concentration of 16.40 wt.% in FeB.
Figure 2 shows the temperature dependence of boron diffusion coefficients in the FeB and Fe2B layers as well as in the diffusion zone. Therefore the expressions of boron diffusion coefficients in the iron borides and in the diffusion zone (in m^{2} s^{−1} ) are given by equations (39)–(41) according to the Arrhenius relationships. (39) (40) and (41)
with R = 8.314 J mol^{−1} K^{−1} and T the temperature in Kelvin.
From the slopes in Figure 2, the values of boron activation energies for AISI 316 steel were respectively estimated as equal to 210.26, 193.80 and 140.55 kJ mol^{−1} in FeB, Fe_{2}B and diffusion zone.
New experimental values of parabolic growth constants of the growing interfaces with respect to the boride incubation times .
Estimated values of boron diffusion coefficients in iron borides and in the diffusion zone by the integral method.
Fig. 2 Arrhenius relationships describing the temperature dependence of boron diffusion coefficients. (a) FeB; (b) Fe_{2}B; (c) diffusion zone (DZ). 
3.2 Determination of activation energies for boron diffusion in iron borides and in the diffusion zone
The values of boron activation energies in some alloyed steels taken from the literature [8,17,31–36] were compared to the estimated values from the integral method for AISI 316 steel as shown in Table 4. The reported values in terms of activation energies depended on different factors such as: the chemical composition of steels, the boriding method, the method of their determination and the chemical reactions involved during the boriding process. It is noted that the values of activation energies obtained by the powderpack boriding are higher than those from the plasmapaste boriding due to the difference in the diffusion mechanism of boron atoms.
The values of activation energies in the FeB and Fe_{2}B layers as well as in the diffusion zone are respectively 210.26, 193.80 and 140.55 kJ mol^{−1} for AISI 316 steel. It is concluded that the obtained results from the integral method are consistent with the literature data.
Comparison of boron activation energies obtained in this work with the values taken from the literature.
3.3 Experimental validation of integral method with two additional boriding conditions
To check the validity of the present diffusion model, two additional boriding conditions were considered. For this purpose, a computer simulation program was written to solve numerically the obtained system of differential algebraic equations by using the Petzold’s DAE solver DASPK [37] with the Interactive Thermodynamics software. This numerical resolution required a choice of consistent initial conditions of timedependent parameters (ai(t) and bi(t) with i = 1, 3) and the initial values of layers’ thicknesses of FeB and (FeB + Fe_{2}B) and (FeB + Fe_{2}B + diffusion zone) for which a_{10} = 0.3519, a_{20} = 0.3688, a_{30} = 0.005134, b_{10} = 0.001103, b_{20} = 0.007603, b_{30} = 0.005548, u_{0} = 0.48 μm, v_{0} = 0.94 μm and w_{0} = 1.39 μm at 1243 K for a boride incubation time of . This value was obtained from a linear fitting of boride incubation times listed in Table 2 as a function of the boriding temperature.
Table 5 shows the experimental and simulated thicknesses of FeB and Fe_{2}B layers and those of diffusion zone for two boriding conditions. It is seen that the experimental data agreed in an acceptable manner with the predicted results. However a slight discrepancy was observed between the experimental values and the simulation results regarding the thickness of diffusion zone due to the fact that the present model did not take into account the influence of carbon on the diffusion of boron atoms within the diffusion zone. Furthermore, the precipitation phenomenon of metallic borides inside the twoboride layer (FeB + Fe_{2}B) was also ignored by assuming a planar morphology for the (FeB/Fe_{2}B) and (Fe_{2}B/DZ) interfaces during the mathematical formulation of the integral diffusion model. In spite of its limitations, the diffusion model based on the integral method was able to reproduce the experimental data obtained at 1243 K for 3 and 5 h.
Experimental and simulated thicknesses of FeB and Fe_{2}B layers and those of diffusion zone for two boriding conditions.
4 Conclusions
In the present work, a simulation of boronizing kinetics of AISI 316 steel was done by means of a diffusion model based on the integral method. After an appropriate choice of variables change, a particular solution of the system of differential algebraic equations was obtained to estimate the values of boron diffusion coefficients in FeB and Fe_{2}B as well as in the diffusion zone (DZ) on AISI 316 steel in the temperature range 1123–1273 K. By adopting the Arrhenius relationships for boron diffusivities in FeB, Fe_{2}B and in the diffusion zone, the values of activation energies for boron diffusion in AISI 316 steel were respectively found to be 210.26, 193.80 and 140.55 kJ mol^{−1} by the integral method.
Furthermore, the present diffusion model has been validated experimentally for two additional boriding conditions (1243 K for 3 and 5 h). A fairly good agreement was achieved between the experiments and predicted values of FeB and Fe_{2}B layers’thicknesses and those of diffusion zone. However, this model presents some limitations since it does not take into account the effect of carbon and precipitation of chromium and nickel borides inside the boride layer during the diffusion of boron atoms into the AISI 316 steel substrate. For further studies, the integral method could be applied to any borided materials (ferrous and non ferrous alloys) to simulate the growth kinetics of boride layers.
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Cite this article as: Chaima Zouzou, Mourad Keddam, Application of integral method for investigating the boriding kinetics of AISI 316 steel, Metall. Res. Technol. 117, 202 (2020)
All Tables
Experimental values of parabolic growth constants at the interfaces along with the corresponding boride incubation times.
New experimental values of parabolic growth constants of the growing interfaces with respect to the boride incubation times .
Estimated values of boron diffusion coefficients in iron borides and in the diffusion zone by the integral method.
Comparison of boron activation energies obtained in this work with the values taken from the literature.
Experimental and simulated thicknesses of FeB and Fe_{2}B layers and those of diffusion zone for two boriding conditions.
All Figures
Fig. 1 Schematic representation of the boronconcentration profile along the borided layer and the diffusion zone. 

In the text 
Fig. 2 Arrhenius relationships describing the temperature dependence of boron diffusion coefficients. (a) FeB; (b) Fe_{2}B; (c) diffusion zone (DZ). 

In the text 
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