Effect of Cr content on the crystal structure and lattice dynamics of FCC Fe–Cr–Ni–N austenitic alloys

Journal of Alloys and Compounds 291 (1999) 262–268

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Effect of Cr content on the crystal structure and lattice dynamics of FCC Fe–Cr–Ni–N austenitic alloys a b, c a d c A. Beskrovni , S. Danilkin *, H. Fuess , E. Jadrowski , M. Neova-Baeva , T. Wieder a Joint Institute for Nuclear Research, 141980 Dubna, Russia Institute of Physics and Power Engineering, 249020 Obninsk, Russia c University of Technology Darmstadt, FB Materialwissenschaft, Petersenstr. 23, D-64287 Darmstadt, Germany d Institute of Solid State Physics, Sofia, Bulgaria b

Received 24 May 1999; accepted 1 June 1999

Abstract The present study investigates the effect of chromium additions on interatomic bonding in nitrogen austenitic steels Fe–xCr–11Ni– 0.5N (x515–29 wt.%). A linear increase of the fcc lattice parameter with increasing chromium content was found. In inelastic neutron scattering measurements, a softening of the metal frequency spectrum in the low energy region was observed, but the second moment of metal atom frequency distribution did not depend on the chromium content. These changes are in disagreement with theoretical predictions and are connected with contributions from the many-body forces and with modification of the electronic states by chromium. Furthermore, a decrease of the nitrogen peak width with increasing Cr content was found. Such a behaviour is caused by nitrogen-induced short-range ordering of chromium atoms.  1999 Published by Elsevier Science S.A. All rights reserved. Keywords: Austenitic steel; Inelastic neutron scattering; Atomic vibrations; Interatomic bonding

1. Introduction Steel properties depend on the interstitial (N) and metal (Cr, Ni, Mn) atom content. Both types of alloying elements change the lattice parameter a and modify the interatomic interactions. The vibrational frequency distribution g(e ) measured by inelastic neutron scattering (INS) provides information about this modification. The present study concerns the effect of additions of Cr on the crystal structure and lattice dynamics of austenitic steels. Alloys with composition Fe–xCr–11Ni–0.5N (x515–29 wt.%) were prepared (Table 1). Some results from diffraction measurements were published in a previous paper [1]. The effect of chromium on the elastic constants has been investigated by an ultrasonic method [2]. A typical frequency spectrum of an fcc austenitic alloy doped with nitrogen can be divided into two parts [3], namely the metal spectrum and the nitrogen spectrum. The frequency region from 0 to ¯40 meV is connected with the vibrational frequency distribution gMe (e ) of the metal *Corresponding author. E-mail address: [email protected] (S. Danilkin)

atoms (Fig. 1). Nitrogen atom frequencies are located above the metal spectrum boundary due to the light mass and strong interaction with the crystal lattice (Fig. 2). Nitrogen occupies the octahedral positions which have cubic symmetry in the fcc lattice; therefore, only one nitrogen peak is observed in these type of steels at e 575– 80 meV. The Me–N interatomic force constant F Me – N can be obtained from the nitrogen peak position. The peak width is determined mainly by two mechnisms: firstly the interaction of nitrogen atoms (N–N direct and stressinduced interaction) and secondly the local variation of the Me–N force constant due to the random occupation of the metal lattice by Fe, Cr, and Ni atoms. In principle, at first gMe (e ) should be calculated from a theoretical model and then, from a comparison with the experiment, some conclusion about the force constants and their changes could be done. Unfortunately, the Me–Me force constants are connected with the frequency distribution gMe (e ) in a rather complicated, non-analytical manner. Fortunately, several integral parameters related to the interatomic bonding (e.g. the second moment of frequency spectrum, ke 2Me l, Debye temperature, QD ) can be calculated from the measured spectra. The second moment

0925-8388 / 99 / $ – see front matter  1999 Published by Elsevier Science S.A. All rights reserved. PII: S0925-8388( 99 )00265-0

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263

Table 1 Lattice constant a, second moment of the metal frequency spectrum ke 2Me l, Debye temperature QD , frequency of nitrogen vibrations eN , peak width DeN for nitrogen No.

Composition (wt.%)

N (C) (wt.%)

˚ a (A) X-ray, powder X-ray, bulk neutrons, bulk

2 ke Me l (meV 2 )

QD (K)

eN (meV)

DeN (meV)

1

Fe–15.2Cr–11.7Ni

0.49 (0.004)

65967

39361

76.560.7

14.262.0

2

Fe–20.3Cr–11.2Ni

0.50 (0.006)

65767

39661

80.160.4

13.161.2

3

Fe–25.1Cr–11.2Ni

0.53 (0.07)

65567

39061

79.760.3

10.060.7

4

Fe–29.0Cr–10.9Ni

0.57 (0.07)

3.59660.001 3.585660.0005 3.589960.0003 3.60460.001 3.596260.0004 3.597460.0003 3.61460.001 3.601460.0008 3.602960.0003 3.61660.001 3.603360.0017 3.606860.0003

65867

38661

80.460.2

8.160.6

2 of frequency spectrum, ke Me l, describes the high temperature limit of the Debye temperature and is proportional to the mean interatomic force constant kFnn l [4]. The Debye temperature can be derived from the low-energy part of gMe (e ) which corresponds to the elastic limit and is determined by the Debye temperature

3 g(e ) 5 ]]3 ? e 2 skQDd

(1)

with e 5 E 2 E0 5 "v as energy transfer, E0 and E are neutron energies before and after scattering, and v is phonon frequency. The Debye temperature relates to the sound velocities vl and vt according to

S D

h 4p V QD 5 ] ]] k 9N

21 / 3

S

1 2 ]3 1 ]3 vl vt

D

21 / 3

(2)

For Fe–(18–20)Cr–10Ni alloys one has typically vl (0.6 cm / ms and vt (0.3 cm / ms [5], therefore, the first term in the brackets is an order of magnitude smaller then the second one, and QD is determined mainly by the transverse velocity vt . Taking into account that the shear modulus is related to transverse sound velocity by G 5 r v 2t , we obtain that G¯ Q D2 . Thus, modifications of the low-frequency part of gMe (e ) can be compared with changes of the shear modulus in polycrystalline material. This relation is confirmed by Ledbetter [6] who showed that substituting G for B in the Einstein–Madelung–Sutherland relationship gives a much improved estimate of the atomic vibration frequency and found the following equation:

S D SD

h 3 QD 5 1.122 ] ]] k 4pVa

21 / 3

G ] r

1/2

(3)

where Va denotes the atomic volume.

2. Measurements

Fig. 1. Metal atom frequency spectra in Fe–xCr–11Ni–0.5N alloys. (1) 15% Cr; (2) 20% Cr; (3) 25% Cr; (4) 29% Cr.

The steels were cast in an induction furnace at a nitrogen gas pressure of 2 MPa in order to reach 0.5 wt.% nitrogen content. As nitrogen-bearing additives, gas nitrified Cr–N (10–12 wt.% N) foundry alloys were used. After homogenisation at 12008C during 10 h, the melts were quenched in a 5% water solution of NaCl. The nitrogen content in the alloys was determined by fluorescence gas analysis. The sample compositions are shown in Table 1. X-ray diffraction experiments were done on bulk samples from casting and on powders that were prepared by filing at room temperature. For removing the residual stress, the powders were sealed into evacuated quartz

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Fig. 2. Nitrogen peaks in frequency spectra of Fe–xCr–11Ni–0.5N alloys. eN , DeN , SN , nitrogen peak frequency, width and area.

capsules, annealed at 12508C during 1 h and finally quenched in water. For neutron diffraction and INS, plates with 2 mm thickness were used. In the case of the powders, X-ray diffraction was carried out using a Philips-Micro III X-ray diffractometer with Fe K a radiation and Mn filter for K b . In the case of the plates, X-ray diffraction was done using a Seifert PTS3000 diffractometer with Cu K a radiation and secondary monochromator. Data treatment including indexing of reflections and lattice constant refinement was straightforward using the Seifert software. For neutron diffraction, the diffractometer DN-2 [7] was used. The Rietveld refinement with the NaCl standard was used in the data treatment. The inelastic neutron scattering spectra were measured using the DIN-2PI spectrometer [7]. An initial neutron energy of E0 5 10.3 meV was selected in INS measurements. Spectra

of scattered neutrons were measured at 10 fixed detectors in the scattering angle range 2q 5 71–1348. The generalised vibrational density of states was calculated from the experimental data in frame of the one-phonon incoherent scattering approximation [8]. For cubic crystals the density of states, weighted by the atomic amplitudes, is close to the vibrational frequency distribution.

3. Results and discussion

3.1. Lattice parameter From diffraction measurements the fcc structure was confirmed for all studied steels. The neutron diffraction discloses a small contamination of the bcc a-phase in all

A. Beskrovni et al. / Journal of Alloys and Compounds 291 (1999) 262 – 268

samples. The amount of the a-phase does not exceed 3% according to the results of Rietveld refinement. Evidently suppressing of the austenite decomposition is connected with quenching of the alloys after homogenisation at 12008C. For comparison, Fe–15Cr–15Ni–0.6N alloys cast under similar conditions (with the exception that the melts were subsequently furnace-cooled under pressure at the rate of 208C / min down to 10008C) contain austenite, bcc martensite and Cr nitride phases [9]. The lattice parameters measured by X-ray diffraction are

265

given in Table 1. A linear fit a 5 a 0 1 k a x to experimental ˚ data gives the following dependencies: a (A)5 (3.57260.004)1(0.001560.0002)x Cr (wt.%) in case of ˚ powdered material, and a (A)5(3.570460.002)1 (0.001360.0001)x Cr (wt.%) in case of bulk material (Fig. ˚ for both material classes may 3). The difference in a (A) result from macroscopic residual stresses within the bulk samples which have been relaxed in the powdered samples. The atomic Goldschmidt radius (reduced to the fcc coordination number 12) of Cr is bigger than of Fe and Ni

Fig. 3. Lattice parameter (a), mean frequency of metal atoms ke 2Me l, nitrogen peak frequency (eN ) and width (DeN ), Debye temperature QD in Fe–xCr–11Ni–0.5N alloys vs. Cr content.

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˚ R Fe 51.26 A, ˚ and R Ni 51.24 A ˚ [10]) which (R Cr 51.28 A, could explain the lattice dilatation with increasing chromium content.

3.2. Metal atom frequency spectrum Fig. 1 shows the metal atom spectrum of Fe–xCr–11Ni– 0.5N alloys. Spectra for different Cr content have a similar structure with two peaks exhibiting critical frequencies such as the limits of the transverse et and longitudinal el phonon dispersion branches at the Brillouin zone boundaries (et ¯23 meV; el ¯32 meV). The gMe (e ) spectrum boundary frequency em equals 38 meV. Differences in the frequency distributions gMe (e ) (Cr515%) and gMe (e ) (Cr520, 25, 29%) were calculated (Fig. 4). All difference spectra are close to zero for energy ranges e ¯16 meV– et and e . el . A zero difference at frequencies higher then el means that there is no shift of the metal frequency spectrum boundary. The negative difference at low frequencies (up to 16 meV) reflects an increase of the density of states with increasing Cr content. This increase takes place when the sound velocities are lowered. Consequently, the Debye temperature should decrease. From a fit of the Debye spectrum (1) to the low-energy part of the experimental frequency distributions (Fig. 1) one finds that

QD is indeed decreased with increasing Cr content (Table 1, Fig. 3). The second moment of the frequency distribution of the metal atoms ke 2Me l does not depend on Cr content within experimental errors. This does not contradict to the observed increase of the density of states at low-energies, because ke 2Me l reflects mainly the higher frequencies (at these frequencies the modification of gMe (e ) is negligible (Figs. 1 and 4)). Furthermore, the decrease of gMe (e ) between et and et is opposite to the contribution from the phonon density of states at low frequencies. ¨ 3.2.1. Interpretation according to the Gruneisen relation ¨ According to the Gruneisen relation, the lattice dilatation caused by Cr atoms should decrease both QD and 2 ke Me l. The decrease of the frequencies is connected with ¨ volume change through the Gruneisen parameter g De DV I ] 5 2 g ? ]] e V

(4)

The volume change V 1 is connected with uniform perturbation of force constants and is a part of the total volume change [4]

S

D

2 1 2 2n DV I 5 ] ? ]] ? DV 3 12n

(5)

where n is Poisson’s ratio. Using Eqs. (4) and (5) and considering the linear dependence of the lattice parameter on the Cr content, a 5 a 0 1 k a x, we obtain for the second moment of the metal atom spectrum the following equation

S

D

kax ke 2 (x)l (1 2 2n ) ]] 5 1 2 4g ]]] ? ] 2 a0 (1 2 n ) ke 0 l

(6)

Using g 5 1.98, n 5 0.29 [2,5] and inserting the ex˚ / wt.% and ke 20 l5660 perimental values for k a 50.0013 A 2 2 2 meV , we obtain: ke (x)l (meV )5660–1.1x Cr (wt.%). A linear fit to the experimental values of ke 2Me l (Table 1, Fig. 3) results in ke 2 (x)l (meV 2 )5(66064)–(0.1260.18)x Cr (wt.%) and shows that second moment of the metal frequency distribution does not depend on the Cr content within the limits of the experimental errors. Therefore, relation (6) is not confirmed by the experiment. An estimation (using the same approach) of the Debye temperature change (caused by the increasing lattice parameter) with Q (D0 ) (K)5404K yields QD (x) (K)5404–0.4x Cr (wt.%). The result is rather close to the value obtained by fitting to the experimental values (Table 1, Fig. 3): QD (K)5(40467)–(0.660.3)x Cr (wt.%).

Fig. 4. Modification of frequency distributions in Fe–xCr–11Ni–0.5N alloys with increase of Cr content. (a) g(e ) (15% Cr)–g(e ) (29% Cr); (b) g(e ) (15% Cr)–g(e ) (25% Cr); and (c) g(e ) (15% Cr)–g(e ) (20% Cr).

3.2.2. Interpretation in terms of bulk moduli B and G Similar observations concerning the Cr effect on interatomic bonding are reported in Ref. [2] where the elastic constants of Fe–Cr–Ni–Mn polycrystalline alloys were measured using an ultrasonic method. Cr slightly increases the bulk modulus. B and G change in opposite directions.

A. Beskrovni et al. / Journal of Alloys and Compounds 291 (1999) 262 – 268

This observation contradicts the Eshelby model predictions. A volume change should produce the same sign for DB /B and DG /G (for the studied materials the following theoretical relations were obtained: DB /B 5 2 1.69 (DV/ V ) and DG /G 5 2 2.06 (DV/V ) [2]). The shear modulus is connected with the Debye temperature: G ¯ Q 2D . From Table 1 and Fig. 3 one sees that QD and, consequently, G, decreases with increasing Cr content. This is in qualitative agreement with the results obtained in Ref. [2]. The elastic moduli B and G represent two extreme types of deformation: pure dilatation and pure shear. According to the central potential model of lattice dynamics of non-stoichiometrical interstitial alloys [11], we obtain in the low q limit the following equation for the bulk modulus (C11 1 2C12 ) B 5 ]]]] 3

S

D

w 9(r 1 ) bN 4 5 ] ? w 0(r 1 ) 2 2]] 1 x ? ] , 3a r1 2

(7)

where Cij are the elastic constants, w (r) is the interaction potential of the metal atoms, r1 denotes the first neighbours distance, w 0 is the bond-stretching force, w 9 /r is the bond-bending force, and b 5 w 99 Me – N means the metal– nitrogen force constant. For the shear modulus using Voigt’s averaging method we obtain (C11 2 C12 1 3C44 ) G 5 ]]]]]] 3

S

D

V 9(r 1 ) bN 4 5 ] ? V 0(r 1 ) 1 4 ]] 1 x ? ] . 5a r1 2

(8)

In (7) and (8), the bond-bending forces w 9 /r relate to many-body forces which exist in transition metal alloys. Usually they are lower in magnitude than the bond-stretching forces w 0 and often have an opposite sign. For the fcc Fe–18Cr–10Mn–15Ni alloy the following values were found from the phonon dispersion curves: w 0(r 1 ) 5 39.060.1 N / m and w 9(r 1 ) /r 1 5 21.9860.08 N / m [12]. Fcc iron has the following force constants: w 0(r 1 )534.5 N / m and w 9(r 1 ) /r 1 5 26.3 N / m [13]. As follows from (7) and (8), the contributions from bond-bending forces to elastic moduli B and G have opposite sign. Therefore, different dependencies of B and G on the Cr content could be connected with contribution of the bond-bending forces. If the bond-bending forces w 9(r 1 ) /r 1 have a negative sign and uw 9(r 1 ) /r 1 u is increased with increasing Cr content, then the decrease of the shear modulus and the increase of the bulk modulus can take place simultaneously. The influence of the Me–N interaction, described by bN , on the elastic moduli changes can be omitted because our experiments show negligible change of the nitrogen vibrational frequency with increasing Cr content in the studied steels (see Section 3.3).

267

3.3. Nitrogen vibrations The spectra of nitrogen vibrations in Fe–xCr–12Ni– 0.5N alloys are shown in Fig. 2. The main peak parameters, energy eN and peak width DeN , are shown in Table 1. The frequency of nitrogen vibrations is nearly constant within the experimental error: eN (x) (meV)5(75.662.7)1 (0.1760.10)x Cr (wt.%) (Figs. 2 and 3). Therefore, the nitrogen peak frequency does not follow the lattice param¨ eter increase caused by Cr. According to the Gruneisen relation the nitrogen frequency decrease is expected to be eN (x) (meV)575.6–0.16x Cr (wt.%). The nitrogen peak width decreases with increasing Cr content: DeN (x) (meV)5(22.661.4)–(0.5060.05)x Cr (wt.%), (Figs. 2 and 3). This decrease is not connected with chromium nitride precipitation from the solid solution, because the nitrogen frequency in Cr 2 N is 73 meV and no two-peak structure is observed in the experimental frequency distributions. The possible reason for the peak width decrease could be short-range ordering in the solid solution which reduces the local variation of the Me–N force constants. At a low nitrogen content the local variation is the main contribution to the nitrogen peak width. This contribution is typical for multi-component alloys where nitrogen can occupy the interstitial sites formed by different metallic atoms. Usually the alloying components differ in size and electron configuration, therefore, Me–N force constants and nitrogen atom frequencies depend on the configuration of the neighbouring atoms. Averaging of nitrogen atom frequencies over the atom distribution in crystal gives the additional contribution to the peak width. Indeed, the strong nitrogen affinity to Cr was demonstrated by Ko and McLellan [14]. For ternary Fe–Ni–N and Fe–Cr–N alloys he found an interaction energy which was positive for alloys with Ni and negative for alloys with Cr. This means that N does not tend to occupy the octahedral sites mainly formed by Ni atoms, whereas in Fe–Cr–N alloys N prefer to occupy sites with a large number of Cr atoms. Computer simulations of the ordering in the Fe–40–Ni–35Cr–xN using embedded-atom interaction potentials were performed by Grujicic and Owen [15]. The calculations were made for two alloys with x50 and 0.25 at room temperature and at 1273 K. In dilute alloys, N atoms preferentially occupy octahedral sites with four and more Cr atoms and with high probability the neighbouring sites also contain a large number of Cr atoms. Even if in the real alloy at room temperature this state is not achieved completely, the local variation of the Me–N force constants should be strongly reduced and cause the decrease of the nitrogen peak width.

4. Conclusions • Chromium increases the lattice parameter in Fe–(15–

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29)Cr–12Ni–0.5N austenitic steel linearly by50.0013 ˚ per 1 wt.% Cr at fixed nitrogen and nickel content. A • A modification of the metal frequency distribution caused by Cr atoms is observed. Chromium decreases the density of states at low frequencies, but does not shift the spectrum boundary and does not change the second moment of the frequency distribution. These changes are in disagreement with theoretical predictions based on volume changes and possibly are connected with contributions from the many-body forces and with modification of the electronic states by Cr. • Vibrational frequency of the nitrogen atom does not depend on Cr content within the experimental errors. • An untypical decrease of the nitrogen peak width with increasing Cr content was found. It is caused by nitrogen-induced short-range ordering of Cr atoms and by a reduction in the local variation of the Me–N force constants.

Acknowledgements Many thanks to V. Morozov (JINR, Dubna) for technical assistance during INS measurements. This work has been performed in the frame of the Russian Federal Programme ‘Priority Investigations and developments in Civil Science and Technology’, subsection ‘Topical Investigations of Condensed Matter’. Substancial financial support was provided within a German–Russian cooperation between ¨ Forschung und Bildung the German Ministerium fur (BMBF) and the Russian Ministry of Science and Technology (RMST). On behalf of the BMBF, the Deutsche ¨ Luft- und Raumfahrt (DLR) supported our Agentur fur work under the project number RUS 151-97.

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