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Experimental and theoretical study on interaction between lanthanum and nitrogen during plasma rare earth nitriding C.S. Zhang, M.F. Yan ∗ , Z. Sun National Key Laboratory for Precision Hot Processing of Metals, School of Materials Science and Engineering, Harbin Institute of Technology, Harbin 150001, PR China
a r t i c l e
i n f o
Article history: Received 10 April 2013 Received in revised form 27 September 2013 Accepted 30 September 2013 Available online xxx Keywords: Lanthanum Nitrogen Interaction Plasma rare earth nitriding M50NiL steel
a b s t r a c t In present work, the interaction between lanthanum (La) and nitrogen (N) during plasma rare earth nitriding of M50NiL martensitic steel is analyzed. Phase compositions, elemental contents as well as microhardness profiles of surface layers are investigated by X-ray diffraction (XRD), scanning electron microscope (SEM), X-ray photoelectron spectroscopy (XPS) and microhardness tester to observe the changes of the N contents in treated layers. The results of microhardness, XRD and EDS indicate that the addition of pure La can speed up the denitriding rate compared with the case without La addition. The XPS results reveal that the presence of the La O and La N bond reduces the peak intensity of the Me N bond, which indicates that the addition of La element can reduce the N contents in nitrided layers through the surface oxidation and the attraction of La atoms. The theoretical thermodynamic calculations are employed to further clarify the denitriding function of the surface oxidation and the attraction between La and N atoms. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Plasma thermochemical treatments as an efficient and environmentally friendly process are widely applied in surface modifications [1]. Rare earth (RE) elements are usually employed in surface thermochemical treatments such as gas carburizing [2] and plasma nitrocarburizing [3] due to their benefits on improving the thickness [4] and mechanical properties of modified surface layers [5]. So far, the effect mechanisms of RE addition during plasma thermochemical treatments are still controversial. The questions concentrate mainly on whether the RE atoms can diffuse into surface layer or not and how they interact with N or C atoms. Works carried out by Yan et al. [6] and Cheng and Xie [7] indicate that the RE atoms could diffuse into the surface layer of steels (several micrometers depth) and will stay mainly at grain boundaries due to their relatively large atomic size (∼40% bigger than the iron atom). This might lead to a production of distorted regions and some lattice defects such as dislocations and vacancies, providing fast-diffusion paths for N or C atoms. RE can also act as catalysts to increase nitrogen content in the near surface regions of steel specimens with respect to the improved surface hardness [8,9]. Most RE reagents are dissolved in ethanol in the forms of mixed compounds of RE salt and its oxide containing mainly La and Ce. In
∗ Corresponding author. Tel.: +86 451 86418617; fax: +86 451 86413922. E-mail addresses:
[email protected],
[email protected] (M.F. Yan).
this work, a series of experiments were designed to clarify how La atoms interact with N atoms. Here, to study the interaction between La and N, plasma sputtering was carried out with pure La blocks to exclude other influencing factors (C from ethanol etc.). 2. Experimental The M50NiL martensitic steel was selected as the substrate material with the chemical composition (wt.%) 0.13C, 4.1Cr, 3.4Ni, 4.2Mo, 1.2V, 0.13Mn, 0.18Si, 0.012P, 0.002S and Fe, balance. Before the plasma nitriding, the steel was solution treated at 1150 ◦ C for 1 h. After that, the steel specimens were machined into the size of 12 mm × 12 mm. The flat faces of the specimens were polished by silicon carbide papers (#800), and then ultrasonically cleaned with alcohol and acetone before plasma treatments. Plasma nitriding was performed using a LDMC-30 plasma nitriding unit. Before the application of a glow discharge, the furnace chamber was evacuated to 20 Pa by a rotary pump. Plasma nitriding was conducted at a constant furnace pressure of 300 Pa in a gas mixture containing N2 and H2 with the ratio of 1:5 at 400 ◦ C for 4 h (the nitrided specimen was identified as N400, the last three numbers represent the treatment temperature). Then the nitrided specimens were divided into two groups (group A and B). The group A (identified as N400 + H560RE) was plasma sputtered (tempered) with La addition (six 2.3 g-blocks of pure La were hang above the specimens) at 560 ◦ C for 6 h in the furnace containing mixed gas (H2 and Ar) without N. The group B (identified as N400 + H560)
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Fig. 1. Surface morphologies of M50NiL steel plasma treated at (a) N400; (b) N400 + H560; (c) N400 + H560RE; (d) EDS results associated with (c).
was plasma sputtered without La and all other conditions were the same for group A. The cross-sectional microhardness profiles of the treated specimens were determined with a microhardness tester (HV-1000) under an indentation load of 100 g for 20 s. Three microhardness indentations were made in each position of the same depth from the surface to prepare the microhardness profiles. In order to define the phase composition of the treated layers, X-ray diffraction (XRD) with Cu K␣ radiation (-0.15405 nm) was carried out in the range of glancing angles 20–100◦ at 40 kV and 50 mA with 0.05◦ interval step mode using a D/max-rB diffractometer. The existence of the N and La elements were analyzed using scanning electron microscope (SEM, FEI QUANTA 200 F) equipped with an energy dispersive X-ray analyzer (EDS). To make sure of the accuracy of the measured content profiles, ten points placed at the same depth would be tested. X-ray photoelectron spectroscopy (XPS) was employed to clarify the chemical composition and states of the treated specimens accurately. The XPS analysis was preformed using a Thermo Scientific K-Alpha spectrometer with a monochromatic Al K␣ (12 kV, 6 mA) excitation. The base pressure of the analytical chamber was 10−6 Pa and the diameter of the analyzed area was 400 m. The highresolution XPS spectra were collected using an analyzer pass energy of 50 eV and 0.1 eV steps after a survey scans (0–1350 eV). The collected spectra included the following elemental peaks: carbon (C 1s), nitrogen (N 1s), oxygen (O 1s), iron (Fe 2p) and lanthanum (La 3d). All peaks were calibrated by C 1s peak with binding energy of 285.0 eV. The aims of the XPS data analysis were to determine the chemical states and relative amounts of all detected elements, which are based on the curve fitting of the XPS spectra. The XPS peaks fitting were preformed using an available software (XPS Peak 4.1) with Gaussian (80%)–Lorentzian (20%) combination curves and Shirley background algorithm [10]. The relative amounts of detected
elements (C, N, O, Fe and La) were calculated as the following equation [11]: Cx =
(I /Sx )
x i
(Ii /Si )
(1)
where I was the peak areas obtained from the XPS spectra after background correction and S was atomic sensitivity factors. 3. Results and discussion Fig. 1 gives the surface morphologies of plasma treated specimens N400, N400 + H560 and N400 + H560RE. It can be seen that a large amount of fine particles overlaps on the surface of the N400 + H560RE specimens (see Fig. 1c) compared with the N400 and N400 + H560 ones (see Fig. 1a and b). Some particles are LaFeO3 phase probably, which will be identified by the subsequent XRD results. The EDS results show that there is a considerable amount of La atoms with the content of 2.68 at.% on the treated layer. The microhardness and N content profiles in the surface layer of the N400, N400 + H560 and N400 + H560RE specimens are shown in Fig. 2. Fig. 2b represents that both the microhardness of the N400 + H560RE and N400 + H560 specimens are lower than that of the N400 ones. The measured N contents in the surface layers by EDS (see Fig. 2d) exhibit the same evolution as the microhardness profiles, i.e. the lower microhardness corresponding to the lower N content. The N contents in the surface layers of the N400 + H560 and N400 + H560RE specimens dramatically decrease, due to firstly the inward diffusion of N atoms and secondly the denitriding process caused by the sputtering of Ar+ or surface oxidation. Additionally, both the microhardness and N content of the N400 + H560RE specimens are slightly lower than those of the N400 + H560 ones, which imply that La atoms could further reduce N contents in the nitrided layers. It should also be pointed out that the La profile (see Fig. 2c) drops rapidly from a higher level on the treated surface (∼2.68 at.%)
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Fig. 2. (a) Cross-sectional microstructure of N400 + H560RE specimens, it exhibits a typical morphology of the conventional nitrided layer without compound layer. (b) Microhardness profiles of surface layers of specimens treated at N400, N400 + H560RE and N400 + H560. The evolutions (solid lines) are produced by a numerical fitting of average microhardness. (c) La content profile of surface layer of N400 + H560RE specimen and (d) N content profiles of surface layers of N400, N400 + H560RE and N400 + H560 specimens. The La content of 0.35 at.% and the N content of 2.70 at.% are mainly the background noise of the signal.
to a lower steady value (∼0.35 at.%), indicating most La atoms are accumulated on the treated surface and diffuse into a limited depth instead of the deeper layer. The infiltration of La atoms into the nitrided layer will provide some potential fast-diffusion channels (lattice defects) which could promote the back-diffusion of N atoms [6,9]. Fig. 3 gives the phase composition of the surface layer of N400, N400 + H560 and N400 + H560RE specimens. On the XRD patterns,
Fig. 3. XRD patterns of surface layers of plasma treated M50NiL steel specimens.
we can observe that all treated surface layers are mainly of the nitrogen expanded martensite ˛N which is suggested by Kim et al. [12] and depicted in some other references [13,14]. (˛N is a supersaturation solution solid of nitrogen in iron and its reflections shift to a lower angle with respect to the martensite ˛Fe reflections.) However, the N400 + H560RE specimens additionally consist of LaFeO3 phase (fine particles covered on the treated surface, see Fig. 1c), which proves that La atoms have become deposited on the treated surface by the plasma sputtering. The formation of the LaFeO3 phase can be attributed to the surface oxidation under conditions of the presence of the residual oxygen in the chamber. As is well-known, La is a rather active element which can easily react with oxygen. The formation of the La2 O3 phase is preferential because of its lower Gibbs free energy. Then the surface oxidation occurs as the following reaction: La2 O3 + 2Fe + 3/2O2 → 2LaFeO3 . Parida et al. [15] have given the formation Gibbs free energy of such reaction to be: Gf = −1388.8 + 0.2564T(kJ/mol). The Gf is negative in an extensive temperature scope from the room temperature (298.15 K) to the treatment temperature (833.15 K) which indicates the LaFeO3 phase is thermodynamically stabile. In addition, it can be seen that the main diffraction peaks of N400 + H560 and N400 + H560RE specimens spilt into (101) line and (110) line because the nitrogen incorporation into the lattice results in the c-axis extension of martensite ˛Fe and the difference of interplanar spacing of (101) and (110) planes (see the inset of Fig. 3). In fact, the c-axis dependent (101) line represents
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the nitrogen expanded martensite ˛N and the a-axis dependent (110) line stands for the normal martensite ˛Fe . The position of the ˛N (101) peak depends on the nitrogen content, i.e. the peak shifts to a lower angle with the increase of the nitrogen concentration. The ˛N (101) peak (43◦ ) of the N400 specimens lies on the left side of the corresponding peak (43.8◦ ) of N400 + H560 and N400 + H560RE ones (see Fig. 3), indicating that the N content decreases by denitriding at 560 ◦ C. It should also be noted that there is a small difference between the main diffraction peaks of N400 + H560 and N400 + H560RE specimens. For N400 + H560RE specimens, the intensity of the ˛N (101) peak is slightly weaker than that of the ˛Fe (110) peak, while there is an opposite tendency for N400 + H560 specimens. The proportions ˛ and ˛ of the phases N
Fe
˛N and ˛Fe in the surface layer can be estimated approximately by measuring the intensities I˛ (101) and I˛ (110) of peaks ˛N (101) N
Fe
and ˛Fe (110) and employing the following expressions, i.e. ˛ =
I˛
N
(101) /(I˛ (101) N
+ I˛
Fe
(110) )
and ˛ = I˛ Fe
Fe
(110) /(I˛ (101) N
+ I˛
N
(110) ). 560 ◦ C
Fe
Obviously, the addition of La during plasma sputtering at can reduce the proportion of the ˛N phase from 53% to 46% and increase the proportion of the ˛Fe phase from 47% to 54%. That is to say, the N content of the nitrided specimen plasma sputtered at 560 ◦ C with La addition is slightly lower than that of the nitrided specimen plasma sputtered at 560 ◦ C without La addition, which agree well with the results of the EDS and microhardness tests. There is an obvious difference in hardness and N contents between specimens N400 + H560 and N400 + H560RE in the surface layer of 10 m. In order to account the differences in the surface layer of 10 m, the XPS measurements are carried out to investigate the reactions occurred on the treated surface and the changes of elemental contents in the surface layer of 10 m. Here we select two layers to compare. One is the treated surface and the other is located about 10 mm underneath the surface. The high-resolution XPS spectra of C 1s, N 1s, O 1s, Fe 2p and La 3d5/2 are shown in Fig. 4. For the same depth, there are no obvious differences on the XPS spectra between the specimens N400 + H560 and N400 + H560RE, except for the La 3d5/2 peak on the surface. The C 1s spectrum consists of several components which can be divided into two groups as C C bond at 285.0 eV and C O bonds from 286.4 eV to 289.2 eV (see Fig. 4a). The different kinds of the C O bonds correspond to the different binding energies [10], such as C O, C O and O C O. Compared with the C–C peaks of the surface layers, the C–C peaks of the 10 m-depth layer shift to a lower binding energy of 284.6 eV. All high-resolution spectra will be calibrated to C–C binding energy of 285.0 eV. The La 3d5/2 peaks show the La atoms only exist in the surface of the N400 + H560RE specimens, which bond with O at 834.7 eV and N at 833.4 eV (see Fig. 4b) [16,17]. The La N bond with bonding energy of 833.4 eV probably corresponds to a trace of the residual LaN phase on the treated surface. However there is no evidence of X-ray diffraction peaks of the LaN phase. The absence of the LaN phase is firstly due to the measurement accuracy of XRD and secondly the hydrolysis of the LaN phase when the samples are exposed in the moist air. The chemical reaction can be expressed as: 2LaN + 3H2 O → La2 O3 + NH3 . Lyutaya and Goncharuk [18] reported that the formation Gibbs free energy of the reaction was negative in the temperature gap between 298.15 K and 1500 K. The formation Gibbs free energy of the reaction shifts to the more negative value with the decrease of the temperature, which indicates the reaction is more easily preformed at lower temperature than at higher temperature. The Fe 2p peaks represent that the Fe atoms mainly bond with O atoms with 2p3/2 binding energy of 710.0–712.7 eV [10] (see Fig. 4c). While there is a trace of metallic Fe (706.6 eV) existing in the 10 mm-depth layers. The N 1s spectra indicate the N atoms are interstitial atoms in the lattice (400.2 eV) [19] or bond with metallic (Me) atoms at 396.0–397.2 eV in all measured layer [17,20]. The presence of N H bonds on the surface of
the N400 + H560 and N400 + H560RE specimens correspond to the capture of N species by hydrogen during the plasma treatment (see Fig. 4d). The signature of hydroxide and H2 O was found in the O 1s peaks (see Fig. 4e). That is because the measured surfaces are hygroscopic in nature [11,21]. The atomic fractions of C, N, O, Fe and La calculated by fitting XPS data are reported in Table 1. The high carbon and oxygen content is attributed to the organic carbon and oxygen contamination (the absorbed acetone) during the specimen preparation. The N content of the N400 + H560RE specimens is lower than that of the N400 + H560 specimens for both the surface layer and 10 mdepth layer. The results obtained by XPS correspond to the results obtained by EDS. It is worth to note that For the N400 + H560RE specimens, the presence of the La O and La N bond reduces the peak intensity of the Me N bond compared with the N400 + H560 specimens. The decreasing proportion of the Me–N peak indicates the separation of metal and N atoms, which corresponds to the denitriding process. Therefore all the experimental results indicate the La addition could further reduce N contents in the nitrided layers, which can be ascribed to the surface oxidation (the formation of the LaFeO3 phase) and the attraction between La and N atoms. Based on the experimental results obtained by XPS, the thermodynamic calculations are employed to clarify the relationship between the denitriding and the surface oxidation as well as the attraction between La and N atoms. Two reactions are assumed to take place on the treated surface:(i)1/2La2 O3 + Fe[N]x + 3/2O → LaFeO3 + x[N](ii)(Fe,M)[N]x + xLa → xLaN + (Fe,M) Here (Fe,M)[N]x and Fe[N]x represent the nitrogen expanded martensite ˛N with without considering the alloy elements, respectively. (Fe,M) is martensite ˛Fe . All of them have the body-centered cubic (bcc) structure. O represents the oxygen atoms in the nitrided layer. x is the nitrogen content in (Fe,M) and M is the alloy elements (Cr, Ni, Mo, V, Mn and Si). (Fe,M) phase is considered as the special condition of (Fe,M)[N]x phase when x = 0. The (Fe,M)[N]x phase is treated as a solution phase with a (Fe, Cr, Ni, Mo, V, Mn Si)a (C,N,Va)c structure (Va represents vacant interstitial sites and a:c = 1:3 for bcc) and its Gibbs free energy can be expressed by a double sublattice form bcc Gm =
bcc yi ym 0 Gi:m + aRT
m
i
yi ln yi + cRT
ym ln ym
m
i mg
+ E Gm + Gm
(2) 0 Gbcc i:m
where i is Fe, Cr, Ni, Mo, V, Mn or Si, m is C, N or Va. is the Gibbs free energy of pure i element with bcc structure where all interstitial sites are filled with m. yi represents the site fraction of i element on corresponding sublattice. The excess Gibbs energy E Gm is represented by the following expression: E
Gm =
m n>m
i
+
bcc yi ym yn Li:m,n +
i
j>i k>j
i
j>i
bcc yi yj yk ym Li,j,k,m
bcc yi yj ym Li,j:m
m
(3)
m
where i, j and k represent Fe, Cr, Mo, Ni, V, Mn or Si, m and n stand for C, N or Va. The L parameters are interaction energies between components. The comma separates components on the same sublattice while the colon separates components on the different sublattices. The high order terms have been neglected due to their small contribution to the total energy. The excess Gibbs energy is composed of interaction energies for binary and ternary subsystems. Most of parameters in Eqs. (2) and (3) are taken from Refs. [22–33] and the mg unavailable ones will be set to zero [22]. Gm in Eq. (2) represents the contribution due to the magnetic ordering in the form suggested by Miettinen [23]. The temperature dependent Gibbs free energies of phases in the reactions (i) and (ii), except the LaFeO3
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Fig. 4. High-resolution XPS spectra and typical fitting curves of (a) C 1s, (b) La 3d5/2 , (c) Fe 2p, (d) N 1s and (e) O 1s obtained from the surface and 10 m-depth layer of the N400 + H560 and N400 + H560RE specimens. Table 1 Atomic fractions of C, N, O, Fe and La calculated by fitting XPS data. N400 + H560RE surface C
N
O
Fe
La
Atomic sensitivity factor Peak area Atomic Fraction (at.%) Atomic sensitivity factor Peak area Atomic Fraction (at.%) Atomic sensitivity factor Peak area Atomic Fraction (at.%) Atomic sensitivity factor Peak area Atomic Fraction (at.%) Atomic sensitivity factor Peak area Atomic Fraction (at.%)
N400 + H560RE 10 m
N400 + H560 surface
N400 + H560 10 m
40564 46.78
48389 57.22
8840 5.50
5967 3.81
108089 40.57
84514 32.52
114163 7.10
100946 6.44
1148 0.05
232 0.01
0.205 60208 67.06
52831 60.88
7260 4.36
5269 3.28
64225 23.28
81050 30.39
78695 4.72
87347 5.43
15291 0.58
879 0.02
0.38
0.63
3.8
6.7
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6 Table 2 The fitted parameters in Eq. (4). Phase
a
b
c
d (×10−3 )
e
f (×10−6 )
Temperature scope (K)
Fe[N]0.09 (Fe,M)[N]0.09 (Fe,M) O [N] La LaN La2 O3 LaFeO3
−514.3 472.5 −98.36 246800 470300 −6622 −308600 −1807000 −1388771
144.4 150.4 120 −143 −135.6 73.5 −6.05 −27.31 11.83
−29.03 −30.73 −24.4 0 0 −18.21 0 0 −1.95
4.36 7.67 1.09 −36.3 35.23 −15.93 −74.09 −195 193.9
22.13 24.5 7.46 0 0 −113.3 0 0 0
−5.425 −6.89 −4.313 9.78 −9.39 3.51 17.8 48.17 44.07
298.15–950
phase, are obtained from the calculation of Eq. (2) and the experimental data from Barin [34]. The Gibbs free energy of the LaFeO3 phase is taken from literatures [35,36]. These energies can be fitted by the following expression: 0
G(T ) = a + bT + cT ln T + dT 2 +
e + fT 3 T
(4)
The coefficients a–f are listed in Table 2. The mole formation Gibbs free energy of the reactions (i) and (ii) can be expressed by: f Gm (i) = GLaFeO3 + x0 G[N] − 1/20 GLa2 O3 − GFe[N]x − 3/20 GO f
fcc
fcc
bcc bcc and Gm (ii) = G(Fe,Cr) + x0 GLaN − G(Fe,Cr)N − x0 GLa respectively. When x = 0.09 (according to the results of XRD and f f EDS), Gm (i) = −818.2 + 0.2448T + 0.0003859T 2 and Gm (ii) = −29.2 + 0.008651T + 2.089 × 10−6 T 2 (T = 298.15–950 K).
Fig. 5 gives the temperature dependent formation Gibbs free energies of the reactions and the Gibbs free energies of the related phases. It shows that both the LaFeO3 and LaN phase have a low Gibbs free energy, which indicates that the formation of the LaFeO3 and LaN phase is thermodynamically favorable in the corresponding reaction. The stability of the LaN phase improves with the temperature increasing while the LaFeO3 phase represents a opposite tendency. Because of the strong stability of the LaFeO3 and LaN phase, both the reactions (i) and (ii) go to the right side f f (i.e. Gm (i) < 0 and Gm (ii) < 0) in the whole temperature gap between the room temperature (298.15 K) and the treatment temperature (833.15 K) (see Fig. 5b and d). The formation Gibbs free energies of the reactions (i) and (ii) increase with the temperature increasing. It indicates that the reactions (i) and (ii) prefer to occur at lower temperature especially at room temperature (298.15 K).
Fig. 5. (a) and (c) Temperature dependent Gibbs free energies of the corresponding reactants and products of the reactions (i) and (ii), (b) and (d) formation Gibbs free energy of the reactions (i) and (ii), respectively. It shows that the formation Gibbs free energies of the reactions (i) and (ii) are negative in the whole temperature gap between 298.15 K and 1000 K, which indicates the reactions (i) and (ii) go to the right side.
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findings have a significant meaning in the field of the application of rare earth thermochemical treatments. Acknowledgments We gratefully acknowledge the National Natural Science Foundation of China (Grant No. 51371070) and NSAF (Grant No. 11176011) for the financial support of this research work. References
Fig. 6. Schematic illustration of the interaction between La and N atoms during plasma rare earth nitriding. The dash dot arrows show the reactions occurred on the treated surface and the solid arrows represent the capture of N atoms and the sputtering of N/H and La/N species.
The reaction (i) reveals that the denitriding results from the surface oxidation (the formation of the LaFeO3 phase accompanied with the break of Me N bonds) and the reaction (ii) implies that the N atoms prefer to combine with La rather than the metallic atoms. That is the theoretical explanation for the attraction between La and N atoms. But how La interacts with N, attract or repel? Based on the experimental and calculated results displayed above, there is no doubt that La and N are mutually attracted during the thermochemical treatment. Fig. 6 exhibits a schematic illustration for the interaction between La and N atoms during plasma rare earth nitriding. At the beginning, the La atoms are sputtered away from the La surface and subsequently become deposited on the surface of the steels. The deposited La reacts with the residual oxygen in the chamber to form lanthanum oxides (La2 O3 ). Then the La2 O3 phase further transforms to the LaFeO3 phase according to the reaction (i) and the unbound N atoms gather on the treated surface. The unbound N atoms are captured by hydrogen or bond with La atoms. Meanwhile, a few amounts of La atoms diffuse into a limited depth of the surface layer of specimens to form fast-diffusion channels as the prior literatures. These potential fast-diffusion channels could promote the back-diffusion of N atoms. Finally the captured N atoms are sputtered away from the surface of specimens and a large amount of fresh La atoms will become deposited on the surface again. With the circular process of the deposition, attraction and sputtering on the treated surface, the reduction of N content in the surface layer of the nitrided specimens will speed up compared with the case without La addition. 4. Conclusions The interaction between La and N has been clarified by experimental and thermodynamic investigations. The results of microhardness, XRD and EDS have indicated that the addition of La element could reduce the N content of the surface layer of the treated specimens. The XPS results have found that the presence of the La O and La N bond reduces the peak intensity of the Me N bond, which indicates that the addition of La element can accelerate the denitriding rate through the surface oxidation and the attraction of La atoms. The results of thermodynamic calculation have also demonstrated the denitriding function of the surface oxidation and the attraction between La and N atoms. We believe that our
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