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Computer simulation of monolayer growth kinetics of Fe2B phase during the paste-boriding process: Influence of the paste thickness
Applied Surface Science 253 (2006) 757–761 www.elsevier.com/locate/apsusc
Computer simulation of monolayer growth kinetics of Fe2B phase during the paste-boriding process: Influence of the paste thickness M. Keddam * De´partement de Sciences des Mate´riaux, Faculte´ de Ge´nie Me´canique et Ge´nie des Proce´de´s, USTHB, B.P No. 32, 16111 El-Alia, Bab-Ezzouar, Algiers, Algeria Received 1 October 2005; received in revised form 3 January 2006; accepted 4 January 2006 Available online 3 March 2006
Abstract This paper deals with the effect of boron paste thickness on the study of the monolayer growth kinetics of Fe2B phase forming on AISI 1045 steel by the paste-boriding process. A mathematical diffusion model based on the Fick’s phenomenological equations was applied in order to estimate the growth rate constant at (Fe2B/g-Fe) interface, the layer thickness of iron boride as well as the associated mass gain depending on the boriding parameters such as time, temperature and surface boron concentration related to the boron paste thickness. The simulation results are found to be in a fairly good agreement with the experimental data derived from the literature. # 2006 Elsevier B.V. All rights reserved. Keywords: Computer simulation; Fe–B system; Paste-boriding; Fick’s laws; Growth kinetics
1. Introduction Boriding (or boronizing) is one of the thermochemical surface treatments widely used in the industry. This is a technique by which active boron atoms are penetrated by thermodiffusion followed by chemical reaction into the surface. Its use increases strongly the surface hardness (about 2000 Hv), wear resistance and anticorrosion properties of the borided layers [1]. This treatment is generally performed between 1123 and 1323 K and it can be applied in solid, liquid or gaseous media. The paste-boriding is simpler and more economical comparable to other boriding techniques. In this technique, a boron source as the boron carbide (B4C) along with an activator and a diluent are thoroughly mixed to form the paste of boron. By lowering the boron potential of the reactive medium, the formation of Fe2B phase can be achieved during this treatment. In practical applications, its formation is more desirable than a dual-phase consisting of FeB and Fe2B. This latter phase is less harder and more tougher than FeB iron boride [1].
* Tel.: +213 21 24 79 19; fax: +213 21 24 79 19. E-mail address: [email protected]. 0169-4332/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.01.011
Understanding the diffusion phenomenon is a key towards modeling the boriding process and its control in automatic installations, requires a knowledge of kinetic parameters. However, little is known in the literature about the modeling of boride layers growth [2–7]. This current work attempts to apply a diffusion model (see Ref. [6], for more details) for studying the layer growth kinetics on the paste-borided AISI 1045 steel between 1193 and 1273 K for variable boron paste thickness ranging from 2 to 5 mm. Depending on the process parameters (time, temperature and surface boron concentration), the model was capable of predicting the boride layer thickness grown on AISI 1045 steel during this thermochemical treatment. 2. Model The solid state diffusion of atomic boron into the g-Fe matrix is governed by Fick’s second law given by Eq. (1): @Ci @2 Ci ðx; tÞ ¼ DiB @x2 @t
(1)
where DiB refers to the effective diffusion coefficient of boron in the i phase, with i = (g-Fe, Fe2B) and Ci(x, t) is the boron concentration as a function of position x and time t.
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M. Keddam / Applied Surface Science 253 (2006) 757–761
Nomenclature ai, bi
constants depending on initial and boundary conditions Fe B=g-Fe CB 2 the upper limit of boron solubility at (Fe2B/gFe) interface S=Fe B CB 2 boron concentration at the surface g-Fe=Fe2 B CB the lower limit of boron solubility at (Fe2B/gFe) interface Ci(x, t) boron concentration in the i phase (i = Fe2B or gFe) DiB diffusion coefficient of boron in the i phase (i = Fe2B or g-Fe) G(t) mass gain per unit area as a function of time JBi mass flux of i phase (i = Fe2B or g-Fe) k growth rate constant at (Fe2B/g-Fe) interface x distance from the surface t time variable Greek letter l layer thickness of Fe2B phase
The diffusion problem involves the solution of Fick’s second law of diffusion, subject to a mass conservation condition at the moving phase interface, which accounts for discontinuities and equilibrium concentration values at the phase interface. In particular, the general solution of Eq. (1) in a semi-infinite couple for concentration-independent diffusion is: Ci ðx; tÞ ¼ ai þ bi erf
x pffiffiffiffiffiffiffiffi 2 DiB t
Fig. 1. Schematic illustration of boron concentration profile for a monophase configuration Fe2B into the material matrix.
- The hypothesis of pore formation on the iron surface is not considered. - The boriding potential does not vary with time. - The diffusion coefficient of boron in each phase is independent of concentration and follows an Arrhenius relationship. - The interface position as a function of time has a parabolic growth. - The temperature in the specimen is identical during the whole process.
The flux balance at (Fe2B/g-Fe) interface is expressed as:
(2)
dl ¼ ½JBFe2 B JBg-Fe x¼l dt
where ai and bi are constants that are defined by the boundary and initial conditions [6]. Fig. 1 illustrates the boron concentration distribution along the depth from the sample surface for a given temperature and under a boron potential which allows the formation of a monophase configuration Fe2B on the material substrate. Assumptions made in the formulation of the diffusion model are as follows:
W
- The diffusion process follows Fick’s second law in a semiinfinite medium, which states that the rate of compositional change is equal to the diffusivity multiplied by the rate of change in the concentration gradient. - The growth of borided layer occurs in a moving flat front parallel to the sample surface. - The iron boride Fe2B nucleates instantly on the material substrate at time zero. - Local equilibrium is attained at the phase interface. - The progress of the reaction front grows parabolically. - The borided layer is thin as compared to the thickness of the sample.
i.e. the difference in diffusional fluxes in and out of phase interface is equal to mass accumulation at the phase interface. The mass balance (Eq. (3)) satisfies all values of time t if the Fe2B phase exists. It is solved by Newton–Raphson’s numerical technique [8] in order to find a positive value of the growth rate constant k, i.e. a kinetically possible solution. Furthermore, the mass gain per unit surface area can be evaluated using Eq. (4). The calculation is based on the assumptions that the Fe2B layer forms instantaneously at t = 0 and immediately covers
(3)
and S=Fe2 B
W¼
CB
JBi ¼ DiB
Fe B=g-Fe
CB 2 2
@Ci @x
with
Fe B=g-Fe
þ ðCB 2
g-Fe=Fe2 B CB Þ
i ¼ ðFe2 B or g-FeÞ and
pffi l ¼ k t;
M. Keddam / Applied Surface Science 253 (2006) 757–761
759
Table 1 Diffusion and thermodynamic data required by the model with R = 8.32 J/mol K Diffusivities of B in Fe2B and g-Fe phases (m2/s) Fe B
DB 2
Fe B
DB 2
Fe B
DB 2
2B DFe B
3 ¼ exp 23:51 255:8410 RT 3 ¼ exp 21:92 232:9810 RT 3 ¼ exp 20:99 215:9210 RT 3 ¼ exp 20:05 202:5210 RT Fe B=g-Fe
(Fe2B/g-Fe) interface: CB 2
Fe B=g-Fe CB 2 qffiffiffiffiffiffiffiffiffiffiffi GðtÞ ¼ 2r 2B erfðk=ð2 DFe ÞÞ B S=Fe2 B
CB
2
13.30
3
13.38
4
13.43
5
13.50
(wt.% B)
g-Fe=Fe2 B ¼ 8:83 wt:%, (g-Fe/Fe2B) interface: CB ¼ 35 103 wt:%.
the sample surface yielding a parabolic growth law and one gets: CB
S=Fe2 B
Paste thickness (mm)
sffiffiffiffiffiffiffiffiffiffiffiffiffi 2B DFe t B p
(4)
where G(t) denotes mass gain per unit surface area (g/cm2) and r is the specific volume of pure iron (=7.86 g/cm3). 3. Results of simulation In accordance with the diffusion model presented in Ref. [6] the case structure of the borided layer contains only the Fe2B phase by considering a boron potential which gives a value of surface boron concentration ranging from 8.83 to 16.23 wt.% B (boron content in FeB phase). This boron composition results in appearance of a monolayer configuration (Fe2B) on the material substrate. In order to test the validity of the mathematical model [6], the results of the paste-boriding experiments on AISI 1045 steel [9] were used to validate the model. The diffusivity of boron in Fe2B phase depends upon the base material composition and the boron potential that gives a surface boron concentration sufficient to develop a monolayer configuration, i.e. Fe2B phase. Table 1 summarizes values of diffusion coefficients of boron in Fe2B and g-Fe phases, the equilibrium concentrations at (Fe2B/g-Fe) interface as well as the corresponding surface boron concentration related to the used paste thickness. Note that the diffusion coefficient of boron in the matrix (austenitic phase) was taken from Ref. [10] and the boride concentration ranges are read from the Fe–B phase diagram [11] where it can be seen that the boron solubility is extremely low in g-Fe phase, even in elevated temperatures approximately 35 104 wt.% B. A computer simulation program [6] was run by considering the experimental data on paste-borided AISI 1045 steel [9] between 1193 and 1273 K for a boron paste thickness ranging from 2 to 5 mm. Fig. 2a–d shows a comparison between the simulated and measured boride layer thickness against square root of time at different temperatures with variable boron paste thickness. A linear relationship is observed between the boride layer
thickness and the square root of time for specified parameters. The controlled diffusion process is more accelerated with increasing temperature due to a high activation energy of boron within the boride layer. Moreover, the simulated curves agree fairly with the experimental data [9]. It can be observed that the boron potential plays an essential role in determining the composition and the thickness of the boride layer. The variation of growth rate constant as a function of temperature is depicted in Fig. 3. The increase of boron paste thickness leads to activate the mass transfer through the paste in contact with the sample surface. Boron transfer to the solid produces the nucleation and growth of iron boride (Fe2B) from the surface when exceeding a value of 8.83 wt.% B. Consequently, growth and coalescence of this phase gives rise to a compact boride layer on the material surface. It can be concluded that the growth rate constant is a complex function of temperature and boron potential. It increases with a rise of temperature for a given surface boron concentration. Fig. 4 describes the evolution of mass gain associated to the formation of Fe2B phase on the material surface in dependence on temperature. It is noticed that the mass gain is increased with increasing temperature due to an important mobility of boron within Fe2B phase. So the choice of a paste thickness of 5 mm, when performing the boriding process at 1273 K, ensures an optimal mass transfer through the borided layer. In order to follow the layer growth kinetics of borided layers, the growth rate constant is depicted (Fig. 5) as a function of boron paste thickness for T = 1273 K under two different surface boron concentrations 13.0 and 13.5 wt.% B. This plot can be used as a basis for matching optimum value of paste thickness in order to achieve the paste-boriding process in a short time duration according to the specified conditions. 4. Discussions A kinetic approach of the paste-boriding process requires a knowledge of influence of each parameter (time, temperature and boron potential). A diffusional approach is thus needed to simulate the layer growth kinetics in case of AISI 1045 carbon steel at 0.4 wt.% C. The applied model presents a limitation since it does not consider the effect of carbon on the boriding kinetics. (It is noted that a mean activation energy value of Fe2B phase, close to 227 kJ/mol in AISI 1045 steel is higher than that of pure iron
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M. Keddam / Applied Surface Science 253 (2006) 757–761
Fig. 2. Layer thickness of Fe2B phase vs. square root of time for variable paste thickness at different temperatures. The symbols denote experimental data taken from Ref. [9].
Fig. 3. Variation of growth rate constant vs. temperature with variable boron paste thickness.
Fig. 4. Evolution of mass gain against temperature with variable boron paste thickness.
M. Keddam / Applied Surface Science 253 (2006) 757–761
Fig. 5. Variation of growth rate constant in dependence on boron paste thickness at T = 1273 K.
[4], i.e. 151 kJ/mol, during the paste-boriding process.) Since the diffusivity of boron in iron boride depends upon the process temperature, the surface boron concentration and the content of alloy elements in steel [12–14]. The influence of carbon element should be taken into account in order to accurately describe the diffusion phenomenon occurring during this thermochemical treatment. In industrial practice, optimal values of the paste-boriding parameters are needed to achieve this thermochemical treatment during a short time duration without impairing the desired properties of borided layers. Thus modeling plays an important role in improving the mechanical and tribological characteristics of the formed borided layers via a close control of boriding kinetics. The simulation results show a fairly agreement with the experimental data. Nevertheless, there is a slight discrepancy observed between the theoretical data and experimental ones at 1223 K for a paste thickness of 4 mm (see Fig. 2c). This fact may be attributed to wrong application of paste thickness that surrounds the sample and difficulty to measure the thickness of borided layer with a high accuracy. This difficulty is linked to the microstructural nature of (Fe2B/g-Fe) interface (either ragged or flat), where its morphology is dependent of steel composition [12–14]. Another limitation of the model is that it does not take into consideration the nucleation period of Fe2B phase. Generally, the paste-boriding process may take several hours. For this reason the incubation time of iron boride can be neglected compared to the whole process time. But the present model is still convenient for simulating the paste-boriding treatment. For solving the diffusion problem, the diffusivity of boron in Fe2B phase known with a high accuracy is required. It can be evaluated by mass balance approach.
761
This kinetic parameter is also dependent of steel composition and it is greatly influenced by boron paste thickness that surrounds the sample. In addition, the distribution of boron has to be uniform in the paste in order to get reproducibility of experimental results. Consequently, the accuracy of the model depends mainly on the experimental conditions (time, temperature and boron paste thickness) of boriding treatment. However, its predictability can be improved with the inclusion of additional experimental data into the analysis. Based on model predictions, the boron potential which is related to paste thickness exerts a strong effect on the layer growth of Fe2B phase (see Figs. 3 and 5). In conclusion, a realistic model to estimate the borided layer thickness should incorporate the precipitation sequence during boriding including the precipitation of metallic borides (when the alloying elements are present). Furthermore, the role of carbon should be addressed. 5. Conclusion Through this work, effect of boron paste thickness is taken into account to study its influence on the layer growth kinetics under given paste-boriding conditions. The kinetic equation is solved, under certain assumptions, by considering the principle of mass conservation through the moving interface. In conclusion, the simulation results were in reasonable agreement with experimental data. This model can be extended to be applied to any alloyed steel. References [1] A.K. Sinha, Boronizing, ASM Handbook, OH, USA, J. Heat Treat. 4 (1991) 437. [2] C.M. Brakman, A.W.J. Gommers, E.J. Mittemeijer, J. Mater. Res. 4 (6) (1989) 1354–1370. [3] D.S. Kukharev, S.P. Fizenko, S.I. Shabunya, J. Eng. Phys. Therm. 69 (2) (1996) 145–150. [4] I. Campos, J. Oseguera, U. Figueroa, J.A. Garcia, O. Bautista, G. Keleminis, Mater. Sci. Eng. A 352 (2003) 261–265. [5] L.G. Yu, X.J. Chu, K.A. Khor, G. Sundararajan, Acta Mater. 53 (8) (2005) 2361–2368. [6] M. Keddam, J. Appl. Surf. Sci. 236 (2004) 451–455. [7] M. Keddam, S.M. Chentouf, J. Appl. Surf. Sci. 252 (2005) 393–399. [8] W.H. Press, B.P. Flannery, S.A. Teukolsky, Numerical Recipes in Pascal: The Art of Scientific Computing, Cambridge University, 1989. [9] I. Campos, O. Bautista, G. Ramirez, M. Islas, J. de La Parra, L. Zuniga, J. Appl. Surf. Sci. 243 (2005) 429–436. [10] P. Guiraldenq, Diffusion dans les me´taux, Techniques de l’Inge´nieur, vol. MB1, M 55, 1978, pp. 31–33. [11] T. Van Rompaey, K.C. Hari Kumar, P. Wollants, J. Alloys Compd. 334 (2002) 173–181. [12] E. Melendez, I. Campos, E. Rocha, M.A. Barron, Mater. Sci. Eng. A 234– 236 (1997) 900–903. [13] R. Petrova, N. Suwattanont, J. Electron. Mater. 34 (5) (2005) 575–582. [14] C. Martini, G. Palombarini, M. Carbucicchio, J. Mater. Sci. 39 (2004) 933–937.
Computer simulation of monolayer growth kinetics of Fe2B phase during the paste-boriding process: Influence of the paste thickness M. Keddam * De´partement de Sciences des Mate´riaux, Faculte´ de Ge´nie Me´canique et Ge´nie des Proce´de´s, USTHB, B.P No. 32, 16111 El-Alia, Bab-Ezzouar, Algiers, Algeria Received 1 October 2005; received in revised form 3 January 2006; accepted 4 January 2006 Available online 3 March 2006
Abstract This paper deals with the effect of boron paste thickness on the study of the monolayer growth kinetics of Fe2B phase forming on AISI 1045 steel by the paste-boriding process. A mathematical diffusion model based on the Fick’s phenomenological equations was applied in order to estimate the growth rate constant at (Fe2B/g-Fe) interface, the layer thickness of iron boride as well as the associated mass gain depending on the boriding parameters such as time, temperature and surface boron concentration related to the boron paste thickness. The simulation results are found to be in a fairly good agreement with the experimental data derived from the literature. # 2006 Elsevier B.V. All rights reserved. Keywords: Computer simulation; Fe–B system; Paste-boriding; Fick’s laws; Growth kinetics
1. Introduction Boriding (or boronizing) is one of the thermochemical surface treatments widely used in the industry. This is a technique by which active boron atoms are penetrated by thermodiffusion followed by chemical reaction into the surface. Its use increases strongly the surface hardness (about 2000 Hv), wear resistance and anticorrosion properties of the borided layers [1]. This treatment is generally performed between 1123 and 1323 K and it can be applied in solid, liquid or gaseous media. The paste-boriding is simpler and more economical comparable to other boriding techniques. In this technique, a boron source as the boron carbide (B4C) along with an activator and a diluent are thoroughly mixed to form the paste of boron. By lowering the boron potential of the reactive medium, the formation of Fe2B phase can be achieved during this treatment. In practical applications, its formation is more desirable than a dual-phase consisting of FeB and Fe2B. This latter phase is less harder and more tougher than FeB iron boride [1].
* Tel.: +213 21 24 79 19; fax: +213 21 24 79 19. E-mail address: [email protected]. 0169-4332/$ – see front matter # 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.apsusc.2006.01.011
Understanding the diffusion phenomenon is a key towards modeling the boriding process and its control in automatic installations, requires a knowledge of kinetic parameters. However, little is known in the literature about the modeling of boride layers growth [2–7]. This current work attempts to apply a diffusion model (see Ref. [6], for more details) for studying the layer growth kinetics on the paste-borided AISI 1045 steel between 1193 and 1273 K for variable boron paste thickness ranging from 2 to 5 mm. Depending on the process parameters (time, temperature and surface boron concentration), the model was capable of predicting the boride layer thickness grown on AISI 1045 steel during this thermochemical treatment. 2. Model The solid state diffusion of atomic boron into the g-Fe matrix is governed by Fick’s second law given by Eq. (1): @Ci @2 Ci ðx; tÞ ¼ DiB @x2 @t
(1)
where DiB refers to the effective diffusion coefficient of boron in the i phase, with i = (g-Fe, Fe2B) and Ci(x, t) is the boron concentration as a function of position x and time t.
758
M. Keddam / Applied Surface Science 253 (2006) 757–761
Nomenclature ai, bi
constants depending on initial and boundary conditions Fe B=g-Fe CB 2 the upper limit of boron solubility at (Fe2B/gFe) interface S=Fe B CB 2 boron concentration at the surface g-Fe=Fe2 B CB the lower limit of boron solubility at (Fe2B/gFe) interface Ci(x, t) boron concentration in the i phase (i = Fe2B or gFe) DiB diffusion coefficient of boron in the i phase (i = Fe2B or g-Fe) G(t) mass gain per unit area as a function of time JBi mass flux of i phase (i = Fe2B or g-Fe) k growth rate constant at (Fe2B/g-Fe) interface x distance from the surface t time variable Greek letter l layer thickness of Fe2B phase
The diffusion problem involves the solution of Fick’s second law of diffusion, subject to a mass conservation condition at the moving phase interface, which accounts for discontinuities and equilibrium concentration values at the phase interface. In particular, the general solution of Eq. (1) in a semi-infinite couple for concentration-independent diffusion is: Ci ðx; tÞ ¼ ai þ bi erf
x pffiffiffiffiffiffiffiffi 2 DiB t
Fig. 1. Schematic illustration of boron concentration profile for a monophase configuration Fe2B into the material matrix.
- The hypothesis of pore formation on the iron surface is not considered. - The boriding potential does not vary with time. - The diffusion coefficient of boron in each phase is independent of concentration and follows an Arrhenius relationship. - The interface position as a function of time has a parabolic growth. - The temperature in the specimen is identical during the whole process.
The flux balance at (Fe2B/g-Fe) interface is expressed as:
(2)
dl ¼ ½JBFe2 B JBg-Fe x¼l dt
where ai and bi are constants that are defined by the boundary and initial conditions [6]. Fig. 1 illustrates the boron concentration distribution along the depth from the sample surface for a given temperature and under a boron potential which allows the formation of a monophase configuration Fe2B on the material substrate. Assumptions made in the formulation of the diffusion model are as follows:
W
- The diffusion process follows Fick’s second law in a semiinfinite medium, which states that the rate of compositional change is equal to the diffusivity multiplied by the rate of change in the concentration gradient. - The growth of borided layer occurs in a moving flat front parallel to the sample surface. - The iron boride Fe2B nucleates instantly on the material substrate at time zero. - Local equilibrium is attained at the phase interface. - The progress of the reaction front grows parabolically. - The borided layer is thin as compared to the thickness of the sample.
i.e. the difference in diffusional fluxes in and out of phase interface is equal to mass accumulation at the phase interface. The mass balance (Eq. (3)) satisfies all values of time t if the Fe2B phase exists. It is solved by Newton–Raphson’s numerical technique [8] in order to find a positive value of the growth rate constant k, i.e. a kinetically possible solution. Furthermore, the mass gain per unit surface area can be evaluated using Eq. (4). The calculation is based on the assumptions that the Fe2B layer forms instantaneously at t = 0 and immediately covers
(3)
and S=Fe2 B
W¼
CB
JBi ¼ DiB
Fe B=g-Fe
CB 2 2
@Ci @x
with
Fe B=g-Fe
þ ðCB 2
g-Fe=Fe2 B CB Þ
i ¼ ðFe2 B or g-FeÞ and
pffi l ¼ k t;
M. Keddam / Applied Surface Science 253 (2006) 757–761
759
Table 1 Diffusion and thermodynamic data required by the model with R = 8.32 J/mol K Diffusivities of B in Fe2B and g-Fe phases (m2/s) Fe B
DB 2
Fe B
DB 2
Fe B
DB 2
2B DFe B
3 ¼ exp 23:51 255:8410 RT 3 ¼ exp 21:92 232:9810 RT 3 ¼ exp 20:99 215:9210 RT 3 ¼ exp 20:05 202:5210 RT Fe B=g-Fe
(Fe2B/g-Fe) interface: CB 2
Fe B=g-Fe CB 2 qffiffiffiffiffiffiffiffiffiffiffi GðtÞ ¼ 2r 2B erfðk=ð2 DFe ÞÞ B S=Fe2 B
CB
2
13.30
3
13.38
4
13.43
5
13.50
(wt.% B)
g-Fe=Fe2 B ¼ 8:83 wt:%, (g-Fe/Fe2B) interface: CB ¼ 35 103 wt:%.
the sample surface yielding a parabolic growth law and one gets: CB
S=Fe2 B
Paste thickness (mm)
sffiffiffiffiffiffiffiffiffiffiffiffiffi 2B DFe t B p
(4)
where G(t) denotes mass gain per unit surface area (g/cm2) and r is the specific volume of pure iron (=7.86 g/cm3). 3. Results of simulation In accordance with the diffusion model presented in Ref. [6] the case structure of the borided layer contains only the Fe2B phase by considering a boron potential which gives a value of surface boron concentration ranging from 8.83 to 16.23 wt.% B (boron content in FeB phase). This boron composition results in appearance of a monolayer configuration (Fe2B) on the material substrate. In order to test the validity of the mathematical model [6], the results of the paste-boriding experiments on AISI 1045 steel [9] were used to validate the model. The diffusivity of boron in Fe2B phase depends upon the base material composition and the boron potential that gives a surface boron concentration sufficient to develop a monolayer configuration, i.e. Fe2B phase. Table 1 summarizes values of diffusion coefficients of boron in Fe2B and g-Fe phases, the equilibrium concentrations at (Fe2B/g-Fe) interface as well as the corresponding surface boron concentration related to the used paste thickness. Note that the diffusion coefficient of boron in the matrix (austenitic phase) was taken from Ref. [10] and the boride concentration ranges are read from the Fe–B phase diagram [11] where it can be seen that the boron solubility is extremely low in g-Fe phase, even in elevated temperatures approximately 35 104 wt.% B. A computer simulation program [6] was run by considering the experimental data on paste-borided AISI 1045 steel [9] between 1193 and 1273 K for a boron paste thickness ranging from 2 to 5 mm. Fig. 2a–d shows a comparison between the simulated and measured boride layer thickness against square root of time at different temperatures with variable boron paste thickness. A linear relationship is observed between the boride layer
thickness and the square root of time for specified parameters. The controlled diffusion process is more accelerated with increasing temperature due to a high activation energy of boron within the boride layer. Moreover, the simulated curves agree fairly with the experimental data [9]. It can be observed that the boron potential plays an essential role in determining the composition and the thickness of the boride layer. The variation of growth rate constant as a function of temperature is depicted in Fig. 3. The increase of boron paste thickness leads to activate the mass transfer through the paste in contact with the sample surface. Boron transfer to the solid produces the nucleation and growth of iron boride (Fe2B) from the surface when exceeding a value of 8.83 wt.% B. Consequently, growth and coalescence of this phase gives rise to a compact boride layer on the material surface. It can be concluded that the growth rate constant is a complex function of temperature and boron potential. It increases with a rise of temperature for a given surface boron concentration. Fig. 4 describes the evolution of mass gain associated to the formation of Fe2B phase on the material surface in dependence on temperature. It is noticed that the mass gain is increased with increasing temperature due to an important mobility of boron within Fe2B phase. So the choice of a paste thickness of 5 mm, when performing the boriding process at 1273 K, ensures an optimal mass transfer through the borided layer. In order to follow the layer growth kinetics of borided layers, the growth rate constant is depicted (Fig. 5) as a function of boron paste thickness for T = 1273 K under two different surface boron concentrations 13.0 and 13.5 wt.% B. This plot can be used as a basis for matching optimum value of paste thickness in order to achieve the paste-boriding process in a short time duration according to the specified conditions. 4. Discussions A kinetic approach of the paste-boriding process requires a knowledge of influence of each parameter (time, temperature and boron potential). A diffusional approach is thus needed to simulate the layer growth kinetics in case of AISI 1045 carbon steel at 0.4 wt.% C. The applied model presents a limitation since it does not consider the effect of carbon on the boriding kinetics. (It is noted that a mean activation energy value of Fe2B phase, close to 227 kJ/mol in AISI 1045 steel is higher than that of pure iron
760
M. Keddam / Applied Surface Science 253 (2006) 757–761
Fig. 2. Layer thickness of Fe2B phase vs. square root of time for variable paste thickness at different temperatures. The symbols denote experimental data taken from Ref. [9].
Fig. 3. Variation of growth rate constant vs. temperature with variable boron paste thickness.
Fig. 4. Evolution of mass gain against temperature with variable boron paste thickness.
M. Keddam / Applied Surface Science 253 (2006) 757–761
Fig. 5. Variation of growth rate constant in dependence on boron paste thickness at T = 1273 K.
[4], i.e. 151 kJ/mol, during the paste-boriding process.) Since the diffusivity of boron in iron boride depends upon the process temperature, the surface boron concentration and the content of alloy elements in steel [12–14]. The influence of carbon element should be taken into account in order to accurately describe the diffusion phenomenon occurring during this thermochemical treatment. In industrial practice, optimal values of the paste-boriding parameters are needed to achieve this thermochemical treatment during a short time duration without impairing the desired properties of borided layers. Thus modeling plays an important role in improving the mechanical and tribological characteristics of the formed borided layers via a close control of boriding kinetics. The simulation results show a fairly agreement with the experimental data. Nevertheless, there is a slight discrepancy observed between the theoretical data and experimental ones at 1223 K for a paste thickness of 4 mm (see Fig. 2c). This fact may be attributed to wrong application of paste thickness that surrounds the sample and difficulty to measure the thickness of borided layer with a high accuracy. This difficulty is linked to the microstructural nature of (Fe2B/g-Fe) interface (either ragged or flat), where its morphology is dependent of steel composition [12–14]. Another limitation of the model is that it does not take into consideration the nucleation period of Fe2B phase. Generally, the paste-boriding process may take several hours. For this reason the incubation time of iron boride can be neglected compared to the whole process time. But the present model is still convenient for simulating the paste-boriding treatment. For solving the diffusion problem, the diffusivity of boron in Fe2B phase known with a high accuracy is required. It can be evaluated by mass balance approach.
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This kinetic parameter is also dependent of steel composition and it is greatly influenced by boron paste thickness that surrounds the sample. In addition, the distribution of boron has to be uniform in the paste in order to get reproducibility of experimental results. Consequently, the accuracy of the model depends mainly on the experimental conditions (time, temperature and boron paste thickness) of boriding treatment. However, its predictability can be improved with the inclusion of additional experimental data into the analysis. Based on model predictions, the boron potential which is related to paste thickness exerts a strong effect on the layer growth of Fe2B phase (see Figs. 3 and 5). In conclusion, a realistic model to estimate the borided layer thickness should incorporate the precipitation sequence during boriding including the precipitation of metallic borides (when the alloying elements are present). Furthermore, the role of carbon should be addressed. 5. Conclusion Through this work, effect of boron paste thickness is taken into account to study its influence on the layer growth kinetics under given paste-boriding conditions. The kinetic equation is solved, under certain assumptions, by considering the principle of mass conservation through the moving interface. In conclusion, the simulation results were in reasonable agreement with experimental data. This model can be extended to be applied to any alloyed steel. References [1] A.K. Sinha, Boronizing, ASM Handbook, OH, USA, J. Heat Treat. 4 (1991) 437. [2] C.M. Brakman, A.W.J. Gommers, E.J. Mittemeijer, J. Mater. Res. 4 (6) (1989) 1354–1370. [3] D.S. Kukharev, S.P. Fizenko, S.I. Shabunya, J. Eng. Phys. Therm. 69 (2) (1996) 145–150. [4] I. Campos, J. Oseguera, U. Figueroa, J.A. Garcia, O. Bautista, G. Keleminis, Mater. Sci. Eng. A 352 (2003) 261–265. [5] L.G. Yu, X.J. Chu, K.A. Khor, G. Sundararajan, Acta Mater. 53 (8) (2005) 2361–2368. [6] M. Keddam, J. Appl. Surf. Sci. 236 (2004) 451–455. [7] M. Keddam, S.M. Chentouf, J. Appl. Surf. Sci. 252 (2005) 393–399. [8] W.H. Press, B.P. Flannery, S.A. Teukolsky, Numerical Recipes in Pascal: The Art of Scientific Computing, Cambridge University, 1989. [9] I. Campos, O. Bautista, G. Ramirez, M. Islas, J. de La Parra, L. Zuniga, J. Appl. Surf. Sci. 243 (2005) 429–436. [10] P. Guiraldenq, Diffusion dans les me´taux, Techniques de l’Inge´nieur, vol. MB1, M 55, 1978, pp. 31–33. [11] T. Van Rompaey, K.C. Hari Kumar, P. Wollants, J. Alloys Compd. 334 (2002) 173–181. [12] E. Melendez, I. Campos, E. Rocha, M.A. Barron, Mater. Sci. Eng. A 234– 236 (1997) 900–903. [13] R. Petrova, N. Suwattanont, J. Electron. Mater. 34 (5) (2005) 575–582. [14] C. Martini, G. Palombarini, M. Carbucicchio, J. Mater. Sci. 39 (2004) 933–937.