The effective diffusion coefficient of boron in the Fe 2 B layers formed on the iron substrate

In this current work, the boron diffusion coefficient in Fe 2B was firstly evaluated using a diffusion model. It considers the effect of the incubation times required to form the Fe 2B layers by the paste-boriding process on the iron substrate. This model solves the mass balance equation at the (Fe 2B/substrate) interface under certain assumptions. Afterwards, the effective boron diffusion coefficient in Fe 2B was evaluated through an application of a simple equation. As a result, the estimated value of boron activation energy in the presence of chemical stresses was found to be equal to 146.5 kJ mol -1 on the basis of experimental data taken from the literature.


Introduction
The boriding is a diffusion-related surface treatment with the purpose of improving the tribological properties, the fatigue endurance and the corrosion resistance of ferrous and non-ferrous alloys.This thermochemical treatment can be carried out between 1123 and 1323 K with the exposure times varying from 0.5 to 10 h [1].The diffused boron atoms react with the substrate surface to form two types of borides FeB and Fe 2 B. The suggested model takes into the effect of chemical stresses through the determination of effective diffusion coefficient of boron in Fe 2 B grown on the iron substrate.The objective of this work was to evaluate the diffusivity of boron in Fe 2 B (with and without the presence of chemical stresses) in the temperature range of 1223-1323 K by applying a recent diffusion model [2,3] for the Fe 2 B layer development.

Diffusion model
The diffusion model considers the growth of Fe 2 B layer on a saturated substrate with boron atoms.A schematic non-linear concentration profile of boron through the Fe 2 B layer is illustrated in Figure 1.The boundary conditions of the diffusion problem are given by Equations ( 2) and ( 3): (3) for u x ≤ ≤ 0 .The Fick's second law of diffusion [5], relating the change in boron concentration through the Fe 2 B layer with time t and location ) (t

The constant
where The continuity equation at the (Fe 2 B/substrate) interface is expressed by Equation (6): The Fe 2 B layer thickness u can be expressed by Equation ( 7): where k represents the parabolic growth constant at the (Fe 2 B/substrate) interface.As stated by Brakman et al. [6], the boride incubation time is decreased with an increase of the boriding temperature.The ) (T β parameter can be approached by a linear relationship [2] (Equation ( 8)) on the basis of experimental data taken from [7]: It is possible to numerically evaluate the diffusion coefficient of boron in Fe 2 B B Fe B D 2 using the Newton-Raphson method [8].For this purpose, a computer program was written in Matlab (version 6.5) to find the roots of Equation (6).By ignoring the pressure effect on the diffusion, the effective diffusion coefficient of boron in Fe 2 B can be evaluated from Equation (9) as follows [3]  3)   are the Young's modulus and Poisson's ratio of the Fe 2 B layer, respectively [10,11].

Simulation results
The experimental results available in reference [7] were firstly used to evaluate the diffusion coefficient of boron in Fe 2 B using Equation ( 6), and secondly to determine the effective diffusion coefficient of boron in Fe 2 B based on Equation ( 9).The past-boriding process was carried out on the iron substrate under an argon atmosphere in a conventional furnace at 4 temperatures (1223, 1253, 1273 and 1323 K) and for variable times (2, 4, 6 and 8 h).Due to the saw-tooth morphology of the (boride layer/substrate) interface, twenty five measurements were done on different cross-sections of the borided samples to estimate the Fe 2 B layer thickness.In Figure 2 is plotted the variation of the squared value of experimental boride layer thickness [7] as a function of the treatment time.Variation of the squared value of the experimental boride layer thickness [7] versus the boriding time at increasing temperatures.

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The model uses as input data the following parameters: the time, the temperature, the upper and lower boron concentrations in the Fe 2 B iron boride and the experimental values of the parabolic growth constants at the (Fe 2 B/substrate) interface.6) and ( 9) for where R is the universal gas constant (= 8.314 J.mol -1 .K - 1 ), and T represents the boriding temperature in Kelvin.

Table 2. Determination of the diffusion coefficient of boron in
The obtained value of boron activation energy (=157.4kJ.mol -1 ), without the chemical stresses, is close to that determined by Campos et al. [12] (i.e.151 kJ mol - 1 ).This value represents the required energy to stimulate the boron diffusion in the preferential crystallographic direction [001] (i.e. in the direction perpendicular to the sample substrate).In this direction, it is reported that the boron element diffuses more speedily with a minimum resistance during the preferential growth of boride crystals.
In the direction parallel to the growth direction, the active flux of boron atoms accelerates the diffusion rate leading to a decrease in activation energy due to the chemical stresses effect.The deduced value of activation energy of boron (= 146.5 kJ.mol -1 ) under the chemical stresses is lower than that of 157.4 kJ.mol -1 due to enhancement of the boron diffusion.

Conclusion
In the present work, the diffusivity of boron in the Fe 2 B layers grown on the iron substrate was firstly evaluated without the presence of chemical stresses through an application of a kinetic approach.This approach was based on solving the mass balance equation at the (Fe 2 B/substrate) interface under certain assumptions.For the case of incorporating chemical stress effects, the boron effective diffusion coefficient in Fe 2 B was secondly evaluated by applying a simple equation.
A lower value of activation energy for the effective diffusivity (= 146.5 kJ.mol -1 ) was obtained for an upper boron content in Fe 2 B equal to

C 2 Figure 1 .
Figure 1.A schematic non-linear concentration profile of boron through the Fe 2 B layer.The initial condition of the diffusion problem is expressed by Equation (1):

Fe 2 B
free of chemical stresses and ) concentration through the Fe 2 B layer.The distribution of boron concentration as a function of the depth satisfies Equation (5):) the diffusion coefficients of boron in Fe 2 B with and without the presence of chemical stresses, respectively.The partial molar volume V is taken equal to ( -1 )[9].E (=290 GPa) and ν (=0.

Figure 2 .
Figure 2. Variation of the squared value of the experimental boride layer thickness[7] versus the boriding time at increasing temperatures.

Figure 3 .
Figure 3. Temperature dependence of the diffusivity of boron in Fe 2 B.
mol.m -3 .This result is a consequence of the chemical stresses enhancing the boron diffusion through the Fe 2 B layer.01012-p.3

Table 1 .
Experimental parabolic growth constants exp

Table 2 ,
are gathered the computed values of