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Stress induced nitrogen diffusion during nitriding of austenitic stainless steel
Computational Materials Science 50 (2010) 796–799
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Stress induced nitrogen diffusion during nitriding of austenitic stainless steel Arvaidas Galdikas ⇑, Teresa Moskalioviene Physics Department, Kaunas University of Technology, Studentu 50, LT-51368 Kaunas, Lithuania
a r t i c l e
i n f o
Article history: Received 9 June 2010 Received in revised form 11 October 2010 Accepted 13 October 2010 Available online 2 November 2010 Keywords: Austenitic stainless steel Plasma source ion nitriding Nitrogen diffusion Modelling Rate equations Internal stress
a b s t r a c t The nitrogen distribution in plasma nitrided austenitic stainless steel at moderate temperatures is explained by non-Fickian diffusion model. The model involves diffusion of nitrogen induced by internal stresses created during nitriding process. For mathematical description of stress induced diffusion process the equation of barodiffusion is used which involves concentration dependent barodiffusion coefficient. For calculation of stress gradient it is assumed that stress depth profile linearly relates with nitrogen concentration depth profile. The fitting is done using experimental curves of nitrogen depth profiles for 1Cr18Ni9Ti nitrided austenitic stainless steel at two different nitrogen fluxes. Fitting is in good agreement in both cases. The diffusion coefficient D = 1.68 1012 cm2 s1 at nitriding temperature 380 °C was found from fitting. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Phase states and phase transformations in the nitrogen modified austenitic stainless steels at moderate temperatures (about 400 °C) by low-temperature nitriding [1–2], nitrogen ion beam implantation [3–5], or nitrogen plasma immersion ion implantation (PIII) [6–8] have been investigated in order to improve the combined wear and corrosion resistance of the stainless steels over the last 20 years. As experimentally observed, the transport of nitrogen in austenitic stainless steels is non-Fickian, so understanding of nitrogen diffusion mechanisms in steel is of great importance. The nitrogen depth profiles in nitrided austenitic stainless steels (below called ASS) exhibit plateau-type shapes slowly decreasing from the surface, followed by a rather sharp leading edge, in addition, the nitrogen diffusivity is four or five orders of magnitude faster than expected from classical diffusion process and the measured depth profiles exhibit an unusual shape with a nearly flat concentration evolution from the surface followed by a relatively abrupt decrease [5,9]. Such profiles are not consistent with the standard analytic solution of the diffusion equation (Fick’s laws of diffusion). Several models were proposed to explain the shape of nitrogen depth profile and the high diffusivity in ASS: (1) the trapping–detrapping model [9] proposed by Parascandola and coworkers; (2) the model based on Fick’s laws and nitrogen diffusion coefficient dependence of nitrogen concentration [10] and (3) the model with combination of those two models (1) and (2) [11]; ⇑ Corresponding author. Tel.: +370 37 300349; fax: +370 37 456472. E-mail address: [email protected] (A. Galdikas). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.10.018
(4) the model proposed by Pranevicius and co-workers [12,13] based on the study of the stochastic mixing of atoms ‘‘ballistically” displaced by incident ions and the flow of atoms into grain boundaries responding to irradiation induced increase in the surface chemical potential. The trapping–detrapping model is based on the assumption that chromium atoms act as trap sites for the nitrogen and, once all traps are saturated, a faster diffusion of nitrogen is observed [14]. The model of concentration dependent diffusion [10] is based on Fick’s laws and nitrogen diffusion coefficient dependence of nitrogen concentration. Christiansen [11] reported experimental data confirming a concentration dependent diffusion, and combined both models (trapping–detrapping and the concentration dependent diffusion) to describe nitrogen diffusion mechanism. Nitriding of ASS at moderate temperatures leads to the formation of expanded austenite, characterized by rather rapid nitrogen diffusion, a nitrogen content of up to 20 at.% and an expansion of the interplanar spacing that can reach 12% [15]. Thus, stress is induced by the gradient of the nitrogen concentration in steel matrix. Stress is one of the factors determining the chemical potential of components of solid systems. Therefore, self-stress resulting from the gradient of the nitrogen concentration affects the transport of nitrogen in ASS. The most widely applied method for the determination of internal stresses in ASS is X-ray diffraction analysis [16,17]. The purpose of the present work is to propose non-Fickian diffusion model, which describes the nitrogen distribution in ASS during nitriding process at temperatures around 400 °C. This model is based on proposition that the transport of nitrogen in ASS is driving by stress gradient created during nitriding process.
A. Galdikas, T. Moskalioviene / Computational Materials Science 50 (2010) 796–799
2. The model The diffusion flux in general case is expressed by the gradient of component chemical potential l(C, T, p):
J ¼ DrlðC; T; pÞ
ð1Þ
where D is diffusion coefficient. Chemical potential in general case is the function of concentration C, temperature T and pressure p. If temperature gradient does not exist the Eq. (1) can be expressed as:
J ¼ DðrC þ kp rpÞ
ð2Þ
This equation involves Fickian diffusion and barodiffusion driving by pressure gradient p. Pressure p for solids is related with stress as p = r, where r represents ‘stress’, which is the trace of the stress tensor induced by the presence of nitrogen in the steel (r = rxx + ryy + rzz). So, r is the hydrostatic part of this tensor, and is an analogue of hydrostatic pressure [18]. Stress factor kp is expressed [18–21] as kp = VNC/RT, R = 8.314 nm mol1 K1 is the gas constant. The chemical potential of nitrogen l as a mobile component in the steel matrix depends on nitrogen concentration C and mechanical stress r [18]:
l ¼ lð0; CÞ V N r
ð3Þ
where l(0,C) denotes the chemical potential of nitrogen in the stress-free state (r = 0), VN the partial molar volume of nitrogen in the solid matrix. The equation of the nitrogen diffusion (nitrogen diffusion flux J) in presence of internal stress is expressed as:
VNC J ¼ D rC rr RT
ð4Þ
Assuming that VN – f(C) and D does not depend on concentration and/or depth the following equation of the nitrogen concentration variation from Eq. (4) can be expressed as
@Cðx; tÞ V N Cðx; tÞ rrðx; tÞ ¼ Dr rCðx; tÞ @t RT
ð5Þ
The boundary conditions involving flux of nitrogen from outside to the surface can be written as
@Cð0; tÞ j ðN0 Cð0; tÞÞ ¼ @t q Nsurface V N Cð0; tÞ rrð0; tÞ þ Dr rCð0; tÞ RT
@C ðkÞ j D ðN 0 C ðkÞ Þ þ 2 ¼ q Nsurface @t h V N ðkþ1Þ ðkþ1Þ ðkþ1Þ ðkÞ ðC C Þ ðr rðkÞ ÞÞ ðC RT
797
ð8Þ
and for other layers k > 0
@C ðkÞ Dh ¼ 2 ðC ðkþ1Þ þ C ðk1Þ 2 C ðkÞ Þ @t h V N ðkþ1Þ ðkþ1Þ ðr rðkÞ Þ þ C ðkÞ ðrðk1Þ rðkÞ ÞÞ ðC RT
ð9Þ
where r(k) is expressed from Eq. (7) as r(k) = Xstress C(k) and h is the thickness of one monolayer. 3. Results and discussions To verify proposed model the fitting of experimental results from Refs. [22,23] was performed. The experimental conditions were following: commercial austenitic stainless steel 1Cr18Ni9Ti (0.1 wt.% C, 1.5 wt.% Mn, 0.8 wt.% Si, 18 wt.% Cr, 9 wt.% Ni, 0.8 wt.% Ti, P 6 0.035 wt.%, S 6 0.03 wt.% and Fe as the balance) was plasma nitrided at temperature T = 380 °C for a period of 4 h and at nitrogen ion current densities 0.44 mA cm2 and 0.63 mA cm2. The Eqs. (8) and (9) were solved numerically to calculate nitrogen profiles. The diffusion coefficient (D) is undefined parameter and in this work was extracted by fitting the experimental curves of nitrogen depth profiles. Values of parameters VN and N0 and Xstress for austenitic stainless steel 1Cr18Ni9Ti were taken from literature: VN = 4 105 m3 mol1 [22,23]; the proportionality constant is Xstress = 200 MPa (at.%)1 [17]; N0 = 0.8 1023 cm3 [24]. The calculations were performed by changing D value in order to get the best fit. The best fit was obtained at D = 1.68 1012 cm2 s1. Fitting results are presented in Fig. 1 together with experimental points [22,23] calculated by Eqs. (8) and (9) with experimental values of j = 0.44 mA cm2 and j = 0.63 mA cm2. Both nitrogen depth profiles were calculated with the same diffusion coefficient D = 1.68 1012 cm2 s1 and the same proportionality constant Xstress = 200 MPa (at.%)1 [17]. It is very important to note that both calculated curves show a good agreement with experimental depth profiles using the same
ð6Þ
The first term of this equation is the adsorption term, which describes the process of nitrogen adsorption on the surface. Term j is average nitrogen ion current density; q is the elementary charge; N0 is the host atoms atomic density and Nsurface is the surface concentration of host atoms. In order to solve Eqs. (5) and (6) the stress profile r(x) has to be defined. In Ref. [17] it was shown that for nitriding or carburizing of stainless steel the linear dependence of compositionally induced compressive stress on concentration (Eq. (18) in Ref. [17]) can be expressed as:
rðx; tÞ ¼ X stress C nitrogen ðx; tÞ
ð7Þ
where Xstress is the proportionality constant. In Ref. [17] the value of Xstress was taken as Xstress = 200 MPa (at.%)1. We also will use this value for calculation and later will show influence of this value. The Eqs. (5) and (6) for one dimensional case expressed in finite increments obtain the following form: For surface layer k = 0
Fig. 1. Experimental (points) [22,23] and calculated depth profiles of nitrogen after nitridation of austenitic stainless steel 1Cr18Ni9Ti at different ion current densities. Calculated curves obtained from Eqs. (8) and (9) with D = 1.68 1012 cm2 s1, Xstress = 200 MPa (at.%)1.
798
A. Galdikas, T. Moskalioviene / Computational Materials Science 50 (2010) 796–799
values of Xstress and D at experimental values of j. This aspect proves that diffusion model based on influence of internal stresses works well and it can be used to explain nitriding process in stainless steel. The calculated stress profiles extracted from fitting are presented in Fig. 2. It can be noted that the increasing the nitrogen ion implantation dose rate, the internal stresses increase. In the next figures the influence of model parameters will be analyzed. The influence of diffusion coefficient which is temperature dependent and nitrogen flux for the nitrogen depth profiles is shown in Fig. 3. The nitrogen depth profiles are calculated by Eqs. (8) and (9) at different nitrogen ion implantation dose rates (0.44 mA cm2 and 0.63 mA cm2) and with different D values, which are taken a little around obtained D values form fitting of experimental points (Fig. 1): 0.51 1012 cm2 s1, 1.68 1012 cm2 s1 (this was found from fitting) and 2.98 1012 cm2 s1. It can be noted that with the increase of diffusion coefficient (which can be changed with temperature) the penetration depth of nitrogen increases. At the same time the surface concentration of nitrogen decreases. The increase of nitrogen flux gives higher concentration of nitrogen and little higher penetration depth. The influence of diffusion coefficient and nitrogen flux to the stress profiles is shown in Fig. 4. The stress profiles are calculated with the same values of parameters as for curves of Fig. 3. Results show that with the increase of the nitrogen flux the internal stresses and thickness of stressed layer increase. The increase of diffusion coefficient acts in opposite: with the increase of diffusion coefficient the internal stresses decrease but thickness of stressed layer increases. This occurs because at higher diffusivity penetration of nitrogen increases that induces internal stresses in deeper layers. At the same time concentration of nitrogen at higher diffusivity becomes less that gives less value of internal stresses. In the proposed model important parameter is Xstress (Eq. (7)). The values Xstress = 200 MPa (at.%)1 was proposed in Ref. [17] which we used in our calculations. The influence of parameter Xstress to the nitrogen concentration and stress depth profiles is shown in Figs. 5 and 6, respectively. The nitrogen concentration and stress depth profiles are calculated with value of D obtained from fitting of experimental results (Fig. 1) at different Xstress values: 50 MPa (at.%)1, 200 MPa (at.%)1 (this was used for fitting) and 500 MPa (at.%)1. Fig. 5 shows that with the increase of coefficient Xstress the penetration depth of nitrogen increases and nitrogen surface concentration decreases. In Fig. 6 it is seen that the internal stresses and thickness of stressed layer increase with
Fig. 2. Calculated (Eqs. (8) and (9)) stress profiles in nitrided 1Cr18Ni9Ti austenitic stainless steel at 380 °C for 4 h, calculated at D = 1.68 1012 cm2 s1, Xstress = 200 MPa (at.%)1 and different ion implantation dose rates (0.44 mA cm2 and 0.63 mA cm2) values.
Fig. 3. The influence of diffusion coefficient D (1012 cm2 s1) and nitrogen flux j (mA cm2) on the nitrogen depth profiles.
Fig. 4. The influence of diffusion coefficient D (1012 cm2 s1) and nitrogen flux j (mA cm2) on the stress profiles.
Fig. 5. The influence of parameter Xstress (in MPa (at.%)1) on the nitrogen depth profiles.
A. Galdikas, T. Moskalioviene / Computational Materials Science 50 (2010) 796–799
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Acknowledgement Authors thank to Research Council of Lithuania for partial support of this work.
References
Fig. 6. The influence of parameter Xstress (in MPa(at.%)1) on the stress profiles.
increase of Xstress. From those results it follows that diffusion process is more intensive as the internal stresses are higher. 4. Conclusions The nitriding process of stainless steel is explained by internal stresses induced diffusion mechanism. Nitrogen depth profiles fitted on the basis of non-Fickian diffusion model, which considers the diffusion of nitrogen in presence of internal stresses as driving force of diffusion, are in good agreement with experimental nitrogen depth profiles. The diffusion process is more intensive as the internal stresses are higher. The diffusion coefficient for nitrogen in plasma source ion nitrided 1Cr18Ni9Ti (18-8 type) austenitic stainless steel at 380 °C was found D = 1.68 1012 cm2 s1 from fitting of experimental data.
[1] B. Larish, V. Brusky, H.J. Spies, Surf. Coat. Technol. 116–119 (1999) 205. [2] E. Menthe, K.T. Rie, Surf. Coat. Technol. 116–119 (1999) 199. [3] R. Leutenecker, G. Wang, T. Louis, U. Gonser, L. Guzman, A. Molinari, Mater. Sci. Eng. A115 (1989) 229. [4] D.L. Williamson, O. Ozturk, R. Wei, P.J. Wilbur, Surf. Coat. Technol. 65 (1994) 15. [5] W. Moller, S. Parascandola, T. Telbizoiva, R. Gunzel, E. Richter, Surf. Coat. Technol. 136 (2001) 73. [6] G.A. Collins, R. Hutchings, J. Tendys, Mater. Sci. Eng. 139A (1991) 171. [7] W. Moller, S. Parascandola, O. Kruse, R. Gunzel, E. Richter, Surf. Coat. Technol. 116–119 (1999) 1. [8] C. Blawert, H. Kalvelage, B.L. Mordike, G.A. Collins, K.T. Short, Y. Jiraskova, O. Schneeweiss, Surf. Coat. Technol. 136 (2001) 181. [9] S. Parascandola, W. Moller, D.L. Williamson, Appl. Phys. Lett. 76 (2000) 16. [10] S. Mändl, E. Scholze, H. Neumann, B. Rauschenbach, Surf. Coat. Technol. 174– 175 (2003) 1191. [11] T.L. Christiansen, K.V. Dahl, M.A.J. Somers, Mater. Sci. Technol. 24 (2008) 159. [12] L. Pranevicius, C. Templier, J.-P. Riviere, P. Meheust, L.L. Pranevicius, G. Abrasonis, Surf. Coat. Technol. 250–257 (2001) 135. [13] L. Pranevicius, L.L. Pranevicius, D. Milcius, S. Muzard, C. Templier, J.-P. Riviere, Vacuum 72 (2004) 161. [14] J.P. Riviere, M. Cahoreau, P. Meheust, J. Appl. Phys. 91 (2002) 10. [15] S. Mandl, B. Rauschenbach, Defect Diffus. Forum 188–190 (2001) 125. [16] I.C. Noyan, J.B. Cohen, Residual Stress; Measurement by Diffraction and Interpretation, Springer-Verlag, New York, 1987. [17] T. Christiansen, M.A.J. Somers, Mater. Sci. Eng. A 424 (2006) 181–189. [18] B. Baranowski, in: S. Saniutycz, P. Salamon (Eds.), Advances in Thermodynamics Flow Diffusion and Rate Processes, Vol. 6, Taylor & Francis, New York, 1992, p. 168. [19] H. Schmalzried, Chemical Kinetics of Solids, VCH, Weinheim, 1995. [20] D. Dudek, B. Baranowski, Z. Phys. Chem. 206 (1998) 21. [21] P. Zoltowski, Electrochim. Acta 44 (1999) 4415. [22] M.K. Lei, J. Mater. Sci. 5975–5982 (1999) 34. [23] M.K. Lei, X.M. Zhu, Surf. Coat. Technol. 22–28 (2005) 193. [24] G. Abrasonis, Doctoral dissertation, Physics (02P)/VDU, Kaunas, 2003.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Stress induced nitrogen diffusion during nitriding of austenitic stainless steel Arvaidas Galdikas ⇑, Teresa Moskalioviene Physics Department, Kaunas University of Technology, Studentu 50, LT-51368 Kaunas, Lithuania
a r t i c l e
i n f o
Article history: Received 9 June 2010 Received in revised form 11 October 2010 Accepted 13 October 2010 Available online 2 November 2010 Keywords: Austenitic stainless steel Plasma source ion nitriding Nitrogen diffusion Modelling Rate equations Internal stress
a b s t r a c t The nitrogen distribution in plasma nitrided austenitic stainless steel at moderate temperatures is explained by non-Fickian diffusion model. The model involves diffusion of nitrogen induced by internal stresses created during nitriding process. For mathematical description of stress induced diffusion process the equation of barodiffusion is used which involves concentration dependent barodiffusion coefficient. For calculation of stress gradient it is assumed that stress depth profile linearly relates with nitrogen concentration depth profile. The fitting is done using experimental curves of nitrogen depth profiles for 1Cr18Ni9Ti nitrided austenitic stainless steel at two different nitrogen fluxes. Fitting is in good agreement in both cases. The diffusion coefficient D = 1.68 1012 cm2 s1 at nitriding temperature 380 °C was found from fitting. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Phase states and phase transformations in the nitrogen modified austenitic stainless steels at moderate temperatures (about 400 °C) by low-temperature nitriding [1–2], nitrogen ion beam implantation [3–5], or nitrogen plasma immersion ion implantation (PIII) [6–8] have been investigated in order to improve the combined wear and corrosion resistance of the stainless steels over the last 20 years. As experimentally observed, the transport of nitrogen in austenitic stainless steels is non-Fickian, so understanding of nitrogen diffusion mechanisms in steel is of great importance. The nitrogen depth profiles in nitrided austenitic stainless steels (below called ASS) exhibit plateau-type shapes slowly decreasing from the surface, followed by a rather sharp leading edge, in addition, the nitrogen diffusivity is four or five orders of magnitude faster than expected from classical diffusion process and the measured depth profiles exhibit an unusual shape with a nearly flat concentration evolution from the surface followed by a relatively abrupt decrease [5,9]. Such profiles are not consistent with the standard analytic solution of the diffusion equation (Fick’s laws of diffusion). Several models were proposed to explain the shape of nitrogen depth profile and the high diffusivity in ASS: (1) the trapping–detrapping model [9] proposed by Parascandola and coworkers; (2) the model based on Fick’s laws and nitrogen diffusion coefficient dependence of nitrogen concentration [10] and (3) the model with combination of those two models (1) and (2) [11]; ⇑ Corresponding author. Tel.: +370 37 300349; fax: +370 37 456472. E-mail address: [email protected] (A. Galdikas). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.10.018
(4) the model proposed by Pranevicius and co-workers [12,13] based on the study of the stochastic mixing of atoms ‘‘ballistically” displaced by incident ions and the flow of atoms into grain boundaries responding to irradiation induced increase in the surface chemical potential. The trapping–detrapping model is based on the assumption that chromium atoms act as trap sites for the nitrogen and, once all traps are saturated, a faster diffusion of nitrogen is observed [14]. The model of concentration dependent diffusion [10] is based on Fick’s laws and nitrogen diffusion coefficient dependence of nitrogen concentration. Christiansen [11] reported experimental data confirming a concentration dependent diffusion, and combined both models (trapping–detrapping and the concentration dependent diffusion) to describe nitrogen diffusion mechanism. Nitriding of ASS at moderate temperatures leads to the formation of expanded austenite, characterized by rather rapid nitrogen diffusion, a nitrogen content of up to 20 at.% and an expansion of the interplanar spacing that can reach 12% [15]. Thus, stress is induced by the gradient of the nitrogen concentration in steel matrix. Stress is one of the factors determining the chemical potential of components of solid systems. Therefore, self-stress resulting from the gradient of the nitrogen concentration affects the transport of nitrogen in ASS. The most widely applied method for the determination of internal stresses in ASS is X-ray diffraction analysis [16,17]. The purpose of the present work is to propose non-Fickian diffusion model, which describes the nitrogen distribution in ASS during nitriding process at temperatures around 400 °C. This model is based on proposition that the transport of nitrogen in ASS is driving by stress gradient created during nitriding process.
A. Galdikas, T. Moskalioviene / Computational Materials Science 50 (2010) 796–799
2. The model The diffusion flux in general case is expressed by the gradient of component chemical potential l(C, T, p):
J ¼ DrlðC; T; pÞ
ð1Þ
where D is diffusion coefficient. Chemical potential in general case is the function of concentration C, temperature T and pressure p. If temperature gradient does not exist the Eq. (1) can be expressed as:
J ¼ DðrC þ kp rpÞ
ð2Þ
This equation involves Fickian diffusion and barodiffusion driving by pressure gradient p. Pressure p for solids is related with stress as p = r, where r represents ‘stress’, which is the trace of the stress tensor induced by the presence of nitrogen in the steel (r = rxx + ryy + rzz). So, r is the hydrostatic part of this tensor, and is an analogue of hydrostatic pressure [18]. Stress factor kp is expressed [18–21] as kp = VNC/RT, R = 8.314 nm mol1 K1 is the gas constant. The chemical potential of nitrogen l as a mobile component in the steel matrix depends on nitrogen concentration C and mechanical stress r [18]:
l ¼ lð0; CÞ V N r
ð3Þ
where l(0,C) denotes the chemical potential of nitrogen in the stress-free state (r = 0), VN the partial molar volume of nitrogen in the solid matrix. The equation of the nitrogen diffusion (nitrogen diffusion flux J) in presence of internal stress is expressed as:
VNC J ¼ D rC rr RT
ð4Þ
Assuming that VN – f(C) and D does not depend on concentration and/or depth the following equation of the nitrogen concentration variation from Eq. (4) can be expressed as
@Cðx; tÞ V N Cðx; tÞ rrðx; tÞ ¼ Dr rCðx; tÞ @t RT
ð5Þ
The boundary conditions involving flux of nitrogen from outside to the surface can be written as
@Cð0; tÞ j ðN0 Cð0; tÞÞ ¼ @t q Nsurface V N Cð0; tÞ rrð0; tÞ þ Dr rCð0; tÞ RT
@C ðkÞ j D ðN 0 C ðkÞ Þ þ 2 ¼ q Nsurface @t h V N ðkþ1Þ ðkþ1Þ ðkþ1Þ ðkÞ ðC C Þ ðr rðkÞ ÞÞ ðC RT
797
ð8Þ
and for other layers k > 0
@C ðkÞ Dh ¼ 2 ðC ðkþ1Þ þ C ðk1Þ 2 C ðkÞ Þ @t h V N ðkþ1Þ ðkþ1Þ ðr rðkÞ Þ þ C ðkÞ ðrðk1Þ rðkÞ ÞÞ ðC RT
ð9Þ
where r(k) is expressed from Eq. (7) as r(k) = Xstress C(k) and h is the thickness of one monolayer. 3. Results and discussions To verify proposed model the fitting of experimental results from Refs. [22,23] was performed. The experimental conditions were following: commercial austenitic stainless steel 1Cr18Ni9Ti (0.1 wt.% C, 1.5 wt.% Mn, 0.8 wt.% Si, 18 wt.% Cr, 9 wt.% Ni, 0.8 wt.% Ti, P 6 0.035 wt.%, S 6 0.03 wt.% and Fe as the balance) was plasma nitrided at temperature T = 380 °C for a period of 4 h and at nitrogen ion current densities 0.44 mA cm2 and 0.63 mA cm2. The Eqs. (8) and (9) were solved numerically to calculate nitrogen profiles. The diffusion coefficient (D) is undefined parameter and in this work was extracted by fitting the experimental curves of nitrogen depth profiles. Values of parameters VN and N0 and Xstress for austenitic stainless steel 1Cr18Ni9Ti were taken from literature: VN = 4 105 m3 mol1 [22,23]; the proportionality constant is Xstress = 200 MPa (at.%)1 [17]; N0 = 0.8 1023 cm3 [24]. The calculations were performed by changing D value in order to get the best fit. The best fit was obtained at D = 1.68 1012 cm2 s1. Fitting results are presented in Fig. 1 together with experimental points [22,23] calculated by Eqs. (8) and (9) with experimental values of j = 0.44 mA cm2 and j = 0.63 mA cm2. Both nitrogen depth profiles were calculated with the same diffusion coefficient D = 1.68 1012 cm2 s1 and the same proportionality constant Xstress = 200 MPa (at.%)1 [17]. It is very important to note that both calculated curves show a good agreement with experimental depth profiles using the same
ð6Þ
The first term of this equation is the adsorption term, which describes the process of nitrogen adsorption on the surface. Term j is average nitrogen ion current density; q is the elementary charge; N0 is the host atoms atomic density and Nsurface is the surface concentration of host atoms. In order to solve Eqs. (5) and (6) the stress profile r(x) has to be defined. In Ref. [17] it was shown that for nitriding or carburizing of stainless steel the linear dependence of compositionally induced compressive stress on concentration (Eq. (18) in Ref. [17]) can be expressed as:
rðx; tÞ ¼ X stress C nitrogen ðx; tÞ
ð7Þ
where Xstress is the proportionality constant. In Ref. [17] the value of Xstress was taken as Xstress = 200 MPa (at.%)1. We also will use this value for calculation and later will show influence of this value. The Eqs. (5) and (6) for one dimensional case expressed in finite increments obtain the following form: For surface layer k = 0
Fig. 1. Experimental (points) [22,23] and calculated depth profiles of nitrogen after nitridation of austenitic stainless steel 1Cr18Ni9Ti at different ion current densities. Calculated curves obtained from Eqs. (8) and (9) with D = 1.68 1012 cm2 s1, Xstress = 200 MPa (at.%)1.
798
A. Galdikas, T. Moskalioviene / Computational Materials Science 50 (2010) 796–799
values of Xstress and D at experimental values of j. This aspect proves that diffusion model based on influence of internal stresses works well and it can be used to explain nitriding process in stainless steel. The calculated stress profiles extracted from fitting are presented in Fig. 2. It can be noted that the increasing the nitrogen ion implantation dose rate, the internal stresses increase. In the next figures the influence of model parameters will be analyzed. The influence of diffusion coefficient which is temperature dependent and nitrogen flux for the nitrogen depth profiles is shown in Fig. 3. The nitrogen depth profiles are calculated by Eqs. (8) and (9) at different nitrogen ion implantation dose rates (0.44 mA cm2 and 0.63 mA cm2) and with different D values, which are taken a little around obtained D values form fitting of experimental points (Fig. 1): 0.51 1012 cm2 s1, 1.68 1012 cm2 s1 (this was found from fitting) and 2.98 1012 cm2 s1. It can be noted that with the increase of diffusion coefficient (which can be changed with temperature) the penetration depth of nitrogen increases. At the same time the surface concentration of nitrogen decreases. The increase of nitrogen flux gives higher concentration of nitrogen and little higher penetration depth. The influence of diffusion coefficient and nitrogen flux to the stress profiles is shown in Fig. 4. The stress profiles are calculated with the same values of parameters as for curves of Fig. 3. Results show that with the increase of the nitrogen flux the internal stresses and thickness of stressed layer increase. The increase of diffusion coefficient acts in opposite: with the increase of diffusion coefficient the internal stresses decrease but thickness of stressed layer increases. This occurs because at higher diffusivity penetration of nitrogen increases that induces internal stresses in deeper layers. At the same time concentration of nitrogen at higher diffusivity becomes less that gives less value of internal stresses. In the proposed model important parameter is Xstress (Eq. (7)). The values Xstress = 200 MPa (at.%)1 was proposed in Ref. [17] which we used in our calculations. The influence of parameter Xstress to the nitrogen concentration and stress depth profiles is shown in Figs. 5 and 6, respectively. The nitrogen concentration and stress depth profiles are calculated with value of D obtained from fitting of experimental results (Fig. 1) at different Xstress values: 50 MPa (at.%)1, 200 MPa (at.%)1 (this was used for fitting) and 500 MPa (at.%)1. Fig. 5 shows that with the increase of coefficient Xstress the penetration depth of nitrogen increases and nitrogen surface concentration decreases. In Fig. 6 it is seen that the internal stresses and thickness of stressed layer increase with
Fig. 2. Calculated (Eqs. (8) and (9)) stress profiles in nitrided 1Cr18Ni9Ti austenitic stainless steel at 380 °C for 4 h, calculated at D = 1.68 1012 cm2 s1, Xstress = 200 MPa (at.%)1 and different ion implantation dose rates (0.44 mA cm2 and 0.63 mA cm2) values.
Fig. 3. The influence of diffusion coefficient D (1012 cm2 s1) and nitrogen flux j (mA cm2) on the nitrogen depth profiles.
Fig. 4. The influence of diffusion coefficient D (1012 cm2 s1) and nitrogen flux j (mA cm2) on the stress profiles.
Fig. 5. The influence of parameter Xstress (in MPa (at.%)1) on the nitrogen depth profiles.
A. Galdikas, T. Moskalioviene / Computational Materials Science 50 (2010) 796–799
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Acknowledgement Authors thank to Research Council of Lithuania for partial support of this work.
References
Fig. 6. The influence of parameter Xstress (in MPa(at.%)1) on the stress profiles.
increase of Xstress. From those results it follows that diffusion process is more intensive as the internal stresses are higher. 4. Conclusions The nitriding process of stainless steel is explained by internal stresses induced diffusion mechanism. Nitrogen depth profiles fitted on the basis of non-Fickian diffusion model, which considers the diffusion of nitrogen in presence of internal stresses as driving force of diffusion, are in good agreement with experimental nitrogen depth profiles. The diffusion process is more intensive as the internal stresses are higher. The diffusion coefficient for nitrogen in plasma source ion nitrided 1Cr18Ni9Ti (18-8 type) austenitic stainless steel at 380 °C was found D = 1.68 1012 cm2 s1 from fitting of experimental data.
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