Strain-induced interaction of dissolved atoms in γ-Fe

Journal of Alloys and Compounds 291 (1999) 167–174

L

Strain-induced interaction of dissolved atoms in g-Fe M.S. Blanter Moscow State Academy of Instrumental Engineering and Information Science, Stromynka 20, 107846, Moscow, Russia Received 16 February 1999; accepted 20 May 1999

Abstract The energies of strain-induced (elastic) interaction of interstitial–interstitial (i–i), interstitial–substitutional (i–s) and substitutional– substitutional (s–s) pairs are calculated for g-Fe at 1428 K with account of discrete atomic structure of the host lattice. The elastic constants, lattice spacing, Born–von Karman constants of the host lattice, and coefficients of the concentration expansion of the solid solution lattice due to solute atoms are the input numerical parameters used. The interaction is oscillating, anisotropic and in some solid solutions strong and long-range. In all g-Fe-base solid solutions there is attraction in some coordination shells. Generally, the strain-induced interaction in g-Fe is weaker than in a-Fe but in some solid solutions it may be of the same order. The verification of applicability of the model of strain-induced interaction for carbon austenite by means of calculation of carbon activity and the parameters ¨ of carbon distribution given by Mossbauer spectroscopy showed that it must be supplemented by additional repulsion in the first coordination shell. The strain-induced interaction must be taken into account for analysis of structure and properties of g-Fe-base solid solutions.  1999 Elsevier Science S.A. All rights reserved. Keywords: Strain-induced interaction; Elastic interaction; Gamma-Fe; Thermodynamical activity; Austenite

1. Introduction The interaction of solute atoms in metals has been the subject of numerous experimental studies because such information is indispensable for understanding many basic physical processes such as short-range order, segregation, ordering, diffusion, etc. The interaction energies are necessary for calculations of phase equilibria and phase diagrams, and mechanical and physical properties of solid solutions [1]. The thermodynamical data of concentration and temperature dependencies of the carbon activity were traditionally used to obtain information about the interaction C–C in f.c.c. Fe–C alloys [2–6]. At present, there are generally accepted reliable experimental data [7–13], but their analysis was performed within the framework of the models of C–C interaction in the first or the first and second coordination shells only. However, it is common knowledge that for many interstitial and substitutional solid solutions in metals the long-range strain-induced (elastic) interaction of dissolved atoms is essential [1]. The theory of strain-induced (elastic) interaction in the framework of the model of a discrete crystal lattice has been developed [1]. The energies of pair interactions of interstiE-mail address: [email protected] (M.S. Blanter)

tials (i–i), substitutionals (s–s) and interstitial–substitutional (i–s) have been evaluated for many b.c.c. [14–23] and f.c.c. metals [19–21,24,25]. Despite limitations of the model of the pair interatomic interaction it is useful for solution of many problems of solid state physics. For example in the case of b.c.c. metals the energies of i–s pair strain-induced interaction appeared to be essential for analysis of internal friction spectra [26–29]. In the case of g-Fe, energies of strain-induced C–C and N–N interactions were calculated only [30], but in doing so the simplified form of the dynamic matrix was used. The form utilised the experimental data on the phonon spectrum to only a small extent. Energies of s–s and i–s strain-induced interaction were not calculated at all. Analysis of i–i interaction in b.c.c. metals showed [1,16,21,26,29,31] that the long-range strain-induced i–i interaction was to be supplemented by short-range repulsion in the nearest coordination shells. A possible source of such repulsion may be a screened Coulomb interaction between charged interstitials. The theory of solute interactions is yet imperfectly developed and it is so far impossible to obtain these energies for many alloys. The distance of the repulsion was obtained by Monte Carlo computer simulation of tracer diffusion [21,31] and internal friction [26,29] and comparison with experimental data or by analysis of the structure of ordered solid solutions

0925-8388 / 99 / $ – see front matter  1999 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 99 )00255-8

M.S. Blanter / Journal of Alloys and Compounds 291 (1999) 167 – 174

168

[1,32]. It was shown that the repulsive interaction extends out to the third or fourth shell. But in the case of H–H interaction in f.c.c. metals [21] it was shown by means of diffusion simulation that the additional repulsion was absent. One can explain this by the great distance between interstitial atoms located in octahedral interstices of an f.c.c. lattice. The distance in the first coordination shell is longer than the distance of a screened Coulomb repulsion in a b.c.c. lattice. The applicability of the model of C–C strain-induced interaction for description of carbon austenite was not studied. It is also unknown if it is necessary to supplement it by the short-range repulsion as in a-Fe. One can verify it by means of calculation and comparison with experimental data on carbon activity and the abundance of the iron sites with different carbon neighbourhoods obtained by means ¨ of Mossbauer spectroscopy [4,5,39]. The purpose was: (1) to recalculate the energies of i–i strain-induced interaction in g-Fe using the more precise dynamic matrix; (2) to calculate the energies of i–s and s–s strain-induced interaction in g-Fe; (3) to investigate applicability of the model of strain-induced interaction for g-Fe by means of calculation of carbon activity and the ¨ parameters of carbon distribution given by Mossbauer spectroscopy and comparison with experimental data.

2. Strain-induced interaction

2.1. Method of energy calculation The calculation technique employed in this paper is based on the theory of lattice statics formulated by Khachaturyan in Ref. [1]. Interstitial atoms are located in octahedral interstices of an f.c.c. crystal lattice. Each unit cell of the f.c.c. lattice has one octahedral site. The real strain-induced interaction energies W ( b – l) (r l 2r m ) between atoms of b and l sorts (( b – l)5(i–i) or (s–s) or (i–s)) separated by the vector r5(r l 2r m ) (vectors r l and r m describe the positions — lattice points or interstices — in which atoms of b or l sorts locate) are calculated by means of the inverse Fourier-transformation,

OV

W ( b – l) (r l 2 r m ) 5 (1 /N)

k

( b – l)

(k) e i kr

(1)

where the summation is carried out over all N points of quasi-continuum inside the first Brillouin zone of the f.c.c. lattice, allowed by the cyclic boundary condition. k is the wave vector, V ( b – l) (k) is the Fourier-transform of the interaction energies. According to Ref. [1] the function V ( b – l) (k) can be expressed through such material constants as the coefficients of the concentration expansion of the host lattice u i0 (for interstitials) and u s0 (for substitutional atoms) and frequencies of crystal lattice vibrations. This function looks like the scalar product, V ( b – l) (k) 5 2 F b (k)n l * (k)

(2)

where F b ( l) (k) 5

Of

b ( l)

r

(r) e 2i kr

(3) b ( l)

is a Fourier-transform of the coupling force f (r) acting on an undisplaced host atom at the site r from an atom of b ( l)-type at the position r50 (a Kanzaki force). The i s vectors F (k) and F (k) are material constants. Eq. (2) is not valid for k50.

n l ( b ) (k) 5

OU

l( b )

r

(r) e 2i kr

(4)

is a Fourier-transform of the host atom displacement U l ( b ) (r) at a site r produced by an atom of l( b )-type at r50. The value n l ( b ) (k) can be found from the equation of lattice statics,

O D (k)n j

ij

l( b ) j

(k) 5 F li ( b ) (k)

(5)

where i, j 5 1,2,3 are the Cartesian indices, the tensor Dij (k) is a dynamic matrix. The dynamic matrix Dij (k) can be calculated using the Born–von Karman constants of the host lattice. Though the Born–von Karman approximation for the dynamic matrix may not be accurate enough, the final purpose of calculations is determining the dynamic matrix Dij (k) at any k if we know Dij (k) at k along the symmetry directions (Dij (k) in symmetry direction is directly determined from inelastic neutron scattering data). The representation of the dynamic matrix in terms of the Born–von Karman constants is presented for the f.c.c. lattice in Ref. [20]. The applied Born–von Karman constants determined at 1428 K were taken from Ref. [33]. The inverse Fourier transformations (1) have been carried out by summing over 10 7 points in the first Brillouin zone of g-Fe. In the case when coupling forces f i (r) and f s (r) do not vanish for the nearest one coordination shell around the interstitial or substitutional only and point along the straight line from the dissolved atom toward the host atom, the F i (k) and F s (k) have the following forms [1], F i (k) 5 2 i(c 11 1 2c 12 )(u 0i a 02 / 2) 3 [sin(k 1 a 0 / 2); sin(k 2 a 0 / 2); sin(k 3 a 0 / 2)]

(6)

and F 1s (k) 5 2 i(c 11 1 2c 12 )(u s0 a 20 / 4)sin(k 1 a 0 / 2) 3 [cos(k 2 a 0 / 2) 1 cos(k 3 a 0 / 2)] s

(7)

s

The components F 2 (k) and F 3 (k) can be obtained from F (k) by the cyclic permutation of subscripts. The values c 11 and c 12 are elastic constants of the host lattice, a 0 is the spacing of the host lattice. The coefficients of the concentration expansion of the host lattice u 0 can be obtained according to the dependence of the crystal lattice parameter a on concentration c, s 1

u 0 5 (1 /a 0 ) da / dc

(8)

M.S. Blanter / Journal of Alloys and Compounds 291 (1999) 167 – 174

where c is the atomic fraction of interstitials (for u i0 ) or substitutionals (for u 0s ). To determine values of u i0 and u s0 it is necessary to use the concentration dependence of a in the g-area i.e. at high temperatures. We know the limited data for carbon only [34]. The values of u i0 calculated according to these data depend only weakly on temperature. That means that one can use available X-ray data at the room temperature: for retained austenite in the case of C or N [35] or g-Fe-base equilibrium solid solutions in the case of Ni [36] and Mn [37]. For other substitutional atoms values of u s0 were obtained according to changes of a lattice parameter of g-Fe–Ni solid solutions due to the third element. In so doing the lattice parameter of the binary g-Fe–Ni solution with the same iron concentration as in the ternary solution was used as a 0 . The numerical values of the coefficients u 0i and u 0s displayed in Table 1 cannot be considered highly reliable and need to be determined more accurately in the future. That is why the energies were calculated in the form being suitable for arbitrary values of the coefficients u i0 and u s0 . The results of these calculations easily enable one to find the numerical values of interaction energies for any dissolved atoms in g-Fe without an additional computer calculation. To do that, it is sufficient to make use of the numerical values of u i0 and u s0 inherent for the relevant solid solution and the results of numerical computer calculations (coefficients A( b – l) (r l 2r m )) listed in the tables. The energies W ( b – l) (r l 2r m ) calculated with the Born– von Karman constants obtained at 1428 K are strictly suitable at this temperature. However it is usually assumed that the interatomic interaction energies depend weakly on temperature and we shall use these energies at different temperatures and for comparison with the energies in a-Fe and Ni calculated for the room temperature. It is necessary to stress that the elastic interaction is very nearly the same in g-Fe as it is in Ni (Figs. 1–3) despite the differences of temperatures.

169

Fig. 1. Dependence of the energies of C–C strain-induced interaction on distance: (1) g-Fe (present paper); (2) Ni [21]; (3) g-Fe [30]; (4) a-Fe [15].

Fig. 2. Dependence of the coefficients for calculation of energies of substitutional–substitutional strain-induced interaction on distance: (1) g-Fe (present paper); (2) Ni [20]; (3) a-Fe [20].

2.2. Results and discussion Since F i (k) and F s (k) are linear functions of u i0 and u s0 , respectively, it follows from (2) and (5) that the energies ( W b – l) (r l 2r m ) are proportional to u 0b u 0l , W ( b – l) (r l 2 r m ) 5 A( b – l) (r l 2 r m )u b0 u l0

(9)

where three sets of the universal coefficients A b – l) (r l 2 r m ) 2 A( i – i ) (r l 2 r m ), A(s – s) (r l 2 r m ) and A( i – s) (r l 2 r m ) are the same for all solid solutions based on the host lattice of g-Fe. Tables 2–4 give the numerical values of the co(

Table 1 Parameters used for calculation Elastic constants at 1428 K, 10 12 dyn / cm 2 [33] C 11

C 12

C 44

1.54 Lattice parameters from

1.22

0.77

u 0i or u s0

a 0 at 1428 K ˚ A

C

N

Mn

Ni

Mo

Cu

Al

V

3.666 [38]

10.210 [35]

10.224 [35]

10.032 [37]

10.001 [36]

10.090 [36]

10.026 [36]

10.011 [36]

20.014 [36]

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170

energies. Energies for some specific solid solutions are also presented in the tables.

2.2.1. i–i interaction ( Table 2 and Fig. 1) The sign 1 of W (i – i ) (r l 2r m ) is determined by the sign of (i – i ) A (r l 2r m ). One can see that the potentials are anisotropic and oscillating. There is strong attraction in the first and third coordination shells, strong repulsion in the second shell and no weak interaction in the sixth and eighth shells. One can see that the C–C or N–N interaction extend over two shells taking usually into account for analysis of thermodynamical properties or Mossbauer spectra of austenite [2,4,39]. Our energies of C–C and N–N interactions differ from data in Ref. [30] (Fig. 1). As mentioned above, it is possibly connected with the simplified form of the dynamic matrix Dij (k) used in Ref. [30]. Calculations of the coefficients A(s – s) (r l 2r m ) for many f.c.c. metals [20] and the energies of H–H strain-induced interaction in Pd [21] which have been performed with the help of the more precise form of Dij (k) gave results which are in good

Fig. 3. Dependence of the coefficients for calculation of energies of interstitial–substitutional strain-induced interaction on distance: (1) g-Fe (present paper); (2) Ni [25]; (3) a-Fe [22].

efficients A( b – l) (r l 2 r m ) for various distances r5r l 2r m . The coefficients A( b – l) (r l 2 r m ) were found via numerical computer calculations. However, they allow for direct evaluation (without involving numerical methods) of strain-induced pairwise interaction energies provided the coefficients of the concentration expansion of the host lattice (concerning the relevant substitutional and interstitial atoms) are known. Tables 2–4 can be directly employed for calculation of numerical values of interaction

1 A negative interaction energy means attraction, a positive energy, repulsion.

Table 2 Energies (in eV) of pairwise strain-induced interaction of interstitial atoms 2(r l 2r m ) /a 0 Shell ur l 2r m u /a 0 A( i – i ) (r l 2r m ) W ( C – C ) (r l 2r m ) W ( N – N ) (r l 2r m )

110 1 0.71 22.209 20.097 20.110

200 2 1 13.827 10.169 10.192

211 3 1.22 20.957 20.042 20.048

220 4 1.41 10.095 10.004 10.005

310 5 1.58 10.085 10.004 10.004

222 6 1.73 10.523 10.023 10.026

321 7 1.87 20.046 20.002 20.002

400 8 2 20.269 20.012 20.013

330 9a 2.12 10.110 10.005 10.006

411 9b 2.12 20.034 20.001 20.002

420 10 2.24 20.037 20.002 20.002

Table 3 Energies (in eV) of pairwise strain-induced interaction of substitutional atoms 2(r l 2r m ) /a 0 Shell ur l 2r m u /a 0 A(s – s) (r l 2r m ) W ( Mn – Mn) (r l 2r m ) W ( Mo – Mo) (r l 2r m )

110 1 0.71 24.007 20.004 20.032

200 2 1 21.230 20.001 20.010

211 3 1.22 10.171 0 10.001

220 4 1.41 10.500 0 10.004

310 5 1.58 20.365 0 20.003

222 6 1.73 10.222 0 10.002

321 7 1.87 10.075 0 10.001

400 8 2 20.175 0 20.001

330 9a 2.12 10.161 0 10.001

411 9b 2.12 20.166 0 20.001

Table 4 Energies (in eV) of pairwise strain-induced interstitial–substitutional interaction 2(r l 2r m ) /a 0 Shell ur l 2r m u /a 0 A( i – s) (r l 2r m ) W ( C – Mn) (r l 2r m ) W ( C – Mo) (r l 2r m ) W ( N – Mo) (r l 2r m ) W ( C – Cu) (r l 2r m ) W ( C – Al ) (r l 2r m ) W ( C –V ) (r l 2r m )

100 1 0.5 29.448 20.063 20.179 20.190 20.052 20.022 10.028

111 2 0.87 22.061 20.014 20.039 20.042 20.011 20.005 10.006

210 3 1.12 10.469 10.003 10.009 10.009 10.003 10.001 20.001

221 4a 1.5 20.064 0 20.001 20.001 0 0 0

300 4b 1.5 20.465 20.003 20.009 20.009 20.003 20.001 10.001

311 5 1.66 10.001 0 0 0 0 0 0

320 6 1.8 10.007 0 0 0 0 0 0

410 7 2.06 20.186 20.001 20.004 20.004 20.001 0 0

M.S. Blanter / Journal of Alloys and Compounds 291 (1999) 167 – 174

agreement with other investigations [19,24]. This gives grounds to treat our results as more precise than the data in Ref. [30]. Comparison of i–i interaction in a- and g-Fe shows that the interaction in a-Fe is much stronger due to higher distortions around interstitials (for example the coefficient of concentration expansion of a-Fe due to C atoms u 33 5 0.86 [15] in comparison with u 0i 50.21 of g-Fe) and has a different distance dependence.

2.2.2. s–s interaction ( Table 3 and Fig. 2) The sign of W (s – s) (r l 2r m ) is also determined by the sign of the coefficient A(s – s) (r l 2r m ) i.e. there is an essential attraction in the two first shells. Because the values of u 0s and coefficients A(s – s) (r l 2r m ) are of the same order in a-Fe and g-Fe the interaction is also of the same order and determined by values of u s0 for specific solid solutions. 2.2.3. i–s interaction ( Table 4 and Fig. 3) Since all the interstitials expand a crystal lattice, one has u i0 .0, and the sign of W (i – s) (r l 2r m ) is determined by the sign of the product of A(i – s) (r l 2r m ) and u s0 . In the first two coordination shells one has A(i – s) (r l 2r m ),0. In the far shells the potential is anisotropic and oscillating. In the case when a substitutional atom expands the host crystal s lattice (u 0 .0) one observes attraction in the first and the second coordination shells with the maximal attraction in the first shell. In contrast, when a substitutional atom contracts the host crystal lattice (u s0 ,0) there is repulsion in the first two shells and weak attraction in the third one. (i – s) The attraction is weak since the coefficient uA (r l 2r m )u in the third shell is about 20 times less than in the first one. This is why the i–s attraction may be essential in g-Fe only s if a substitutional atom expands the crystal lattice (u 0 .0). In this last case the i–s attraction in the first coordination shell may be of the same order or stronger than the i–i attraction (see C–C in Table 2 and C–Mn and C–Mo in Table 4) because in the first shell the coefficient uA( i – s) (r l – r m )u is about twice the coefficient uA(i – i) (r l 2r m )u. For comparison, the coordination shell distribution of the energies of i–s interaction in a-Fe is shown in Fig. 3 for the case of location of interstitials in octahedral interstices (N, C). It is seen that the strongest interaction is in the first two shells, like in the case of g-Fe. For u s0 .0 this interaction is attractive. The main difference between g-Fe and a-Fe is the following. In the case when a substitutional atom contracts the crystal lattice (u s0 ,0) the attraction in g-Fe is quite weak (in the third coordination shell), in contrast to not so weak attraction in a-Fe (in the fourth shell). One has also to take into consideration that the elastic interaction is not the only contribution to i–s interaction, and must be supplemented by ‘chemical’ interaction, like in the case of b.c.c. metals [27–29]. To verify the applicability of the model of strain-induced (elastic) interaction for description of g-Fe-base

171

solid solutions and necessity to supplement it by shortrange repulsion due to a screened Coulomb interaction we calculated the thermodynamical carbon activity in austenite and the abundance of the iron sites with different carbon neighbourhoods (a short-range order) obtained by means of ¨ Mossbauer spectroscopy taking into account the C–C interaction in the eight shells.

3. Calculation of the thermodynamical carbon activity in austenite

3.1. Calculation method To calculate the concentration and temperature dependencies of the thermodynamical activity of carbon the approach applied by Murch and Torn [6] to the Fe–C austenite was used. We have chosen the approach because it can be easily expended for the long-range C–C interaction. An expression for the carbon activity can be presented in the form [4,6], a C 5 a config exp(DG /k B T )

(10)

where a config is a configurational term that depends directly on the carbon distribution; exp(DG /k B T ) is a term accounting for the nonconfigurational part; DG is the difference between the thermodynamical potentials per carbon atom in a ‘standard’ state, for example graphite, and an infinitely dilute Fe–C solid solution; k B is the Boltzman constant and T is temperature. By taking into account that for an infinitely dilute solid solution a config | u /(1 2 u ) [2] the nonconfigurational term can be estimated from experimental data on a C for small carbon concentration u, a C 5 [u /(1 2 u )] exp(DG /k B T )

(11)

where u 5 NC /NFe and NC and NFe are the numbers of C and Fe atoms, respectively. The experimental values of a C at u #0.02 (#0.03 at 1073 K) and the ‘theoretical’ point (u 50, a C 50) were used to calculate the term exp(DG / k B T ) at several temperatures by means of the least square method. The calculated values of exp(DG /k B T ) are displayed in Table 5. The values are somewhat different from the values determined in Ref. [3] using somewhat different assortment of experimental activity data. An algorithm of calculation of a configurational term a config which uses the Monte Carlo procedure is explained in detail in Refs. [4,6]. The idea of this method is as follows. For each member of a statistical ensemble generated during simulation only a part of the partition sum of the system is calculated, namely the part stipulated by a virtual input of a single new carbon atom with fixed positions of the others. This partition sum is averaged out by the usual Monte Carlo procedure. The expression for a config [4,6] is,

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172

Table 5 Energies W (1C – C ) (in eV) of C–C interaction in the first coordination shell obtained by means of calculation of the thermodynamical activity of austenite T, K

Nexp

Ref.

exp(DG /k B T )

W (1C – C )

1073 1173 1273 1373 1423 1573

29 18 57 14 25 12

[9,11–13] [8,11] [8,10–13] [11] [7] [8]

21.1 12.3 9.3 6.8 5.6 4.4

10.095 10.145 10.115 10.110 10.105 10.115

Nexp is a number of used experimental values.

O [O

a config 5 M(NC 1 1) / h

M

i 51

m

j 51 exp(2DEj /k B T )]j

(12) where the summation on j is performed for all empty octahedral interstices and on i for Monte Carlo steps; DEj is a change in the energy of the system when one virtual carbon atom is input; M is the number of Monte Carlo steps and m ( 5 NFe 2 NC ) is the number of empty interstices in the model crystal. The Monte Carlo simulation was conducted as follows. The system Hamiltonian is equal to the sum of pair interaction energies W (C – C ) (r l 2r m ) [1],

O

x 5 (1 / 2)

l,m

W (C – C ) (r l 2 r m ) C(r l ) C(r m )

(13)

where vectors r l and r m indicate the position of octahedral interstices; C(r) are the occupation numbers of interstices (C(r)51 if there is a carbon atom in the interstice; otherwise C(r)50). A certain amount of mobile carbon atoms were placed randomly in octahedral interstices of model f.c.c. crystal having the dimensions 12312312a 30 with periodic boundary conditions. One of 12 octahedral interstices nearest to the occupied one was selected randomly. When the chosen interstice was empty, the Hamiltonian difference Dx of the corresponding jump of the selected atom into that interstice was calculated. The jump was allowed if either Dx happened to be negative or the probability of the jump C 5 exp(2Dx /k B T ) was greater than a certain random number 0,x,1. After multiple repetitions of the process, the equilibrium distribution of interstitial atoms for these temperature and concentration values was achieved. Upon receiving this equilibrium distribution, after a certain number of cycles, the energy change DEj from each empty interstice after a virtual input of a carbon atom was determined, DEj 5

O

m

W (C – C ) (r j 2 r m ) C(r m )

(14)

where the summation on m is performed for all other interstices (m ± j).

Fig. 4. The concentration dependence of carbon activity in austenite calculated with taking into account the strain-induced C–C interaction in eight coordination shells (dashed lines) and with additional repulsion in the first coordination shell from Table 5 (solid lines). Experimental data are shown by circles, squares and triangles.

3.2. Results and discussion The concentration dependence of a C calculated at six temperatures taking into account the strain-induced C–C interaction in eight coordination shells is presented in Fig. 4 by dashed lines. One can see that the dashed lines do not coincide with experimental data (circles, squares, triangles). This means that the strain-induced interaction can not solely describe the thermodynamical properties of austenite. Because it was previously shown [1,16,21,26,29,31] that the model of strain-induced i2i interaction is applicable for description for b.c.c. metal solid solutions only if it is supplemented by a screened Coulomb repulsion in about three coordination shells the strain-induced C2C interaction in austenite was also supplemented by short-range repulsion. The distance of three coordination shells of octahedral interstices is equal ˚ in a-Fe and this matches the distance of the first to 2.49 A ˚ It is why the additional C–C shell in g-Fe (2.59 A). repulsion (DW1 .0) was taken into consideration in the first shell only and the energies in other shells were preserved as strain-induced. For the first coordination shell (r5r j 2r m 50.71a 0 ) we used in the Eqs. (13) and (14) the energy W (1C – C) 5W ( C – C ) (r) (strain-induced)1DW1 . The values W (1C – C ) were obtained by the least square method which showed the best correlation of calculated values of a C with the experimental one at each temperature. The obtained values of W 1(C – C ) are displayed in Table 5. The values of a C calculated with use of the energies are shown in Fig. 4 by solid lines. One can see a good agreement of calculated and experimental values. The average value of W (1C – C) is equal to 0.115 eV i.e. the short-range C–C repulsion overrides the strain-induced C–C attraction in the first shell. After such overriding one has strong C–C repulsion in the two first shells as in the case of the well-known approach of C–C interaction in

M.S. Blanter / Journal of Alloys and Compounds 291 (1999) 167 – 174

173

[4] 2773 K. Because all these estimations are not trustworthy the simulation was carried out for two temperatures T f 2773 and 1600 K which envelop the full range. One can see that the strain-induced C–C interaction cannot solely describe the experimental data and it is necessary to supplement it by repulsion in the first shell as in the case of the thermodynamical activity. According to ¨ Mossbauer data the values of W (1C – C) are in the range from 0.004 to 0.089 eV. This result can be treated only as qualitative but not quantitative due to problems of the ¨ interpretation of Mossbauer spectra and correct determination of T f .

two shells only [39]. But the energies obtained in the –C) present paper, W (C (50.115 eV) and W 2(C – C) (50.169 1 eV), are much greater than the values from Ref. [39] — 0.036 and 0.075 eV respectively — because the long-range interaction in many shells was taken into account. As a whole the calculation of activity showed that the model of strain-induced interaction supplemented by repulsion in the first shell is applicable for description of thermodynamical properties of austenite.

¨ 4. Mossbauer data ¨ Mossbauer data for f.c.c. Fe–C alloys are available only in the narrow concentration range u 50.06–0.09 due to the limited possibility of the metastable f.c.c. structure conservation during cooling to room temperature. According to interpretation [4,5,39] one can determine the following parameters of the distribution of carbon atoms near iron atoms dependent on the C–C interaction energies: P0 , P1 , P2 – 90 and P2 – 180. P0 and P1 are the fractions of Fe atoms having in the nearest octahedral interstices no or one carbon atom, respectively. P2 – 90 is the fraction of Fe atoms having in the nearest octahedral interstices two carbon atoms separated by the vector r5(1 / 2,1 / 2,0)a 0 (the first coordination shell relative to each other). P2 – 180 is the fraction of Fe atoms having in the nearest octahedral interstices two carbon atoms separated by the vector r5 (1,0,0)a 0 (the second coordination shell relative to each other). Calculation of short-range order and the average values of P was performed using Monte Carlo computer simulation of distribution of carbon atoms described above taking into account the long-range C–C strain-induced interaction in eight coordination shells. Table 6 presents the experimental data [39,40] and the results of simulations. The simulation results are dependent on a ‘freezing’ temperature T f of the thermodynamical equilibrium atomic distribution during quenching of the austenite samples. In Ref. [39] T f was estimated as equal to 600 K and in Ref.

5. Conclusions We performed numerical calculations of strain-induced (elastic) pair interaction energies of interstitial–interstitial, interstitial–substitutional and substitutional–substitutional atoms in g-Fe. The microscopic theory employed in this paper allows for account of the atomic structure of solid solutions. The elastic constants, lattice parameters and Born–von Karman constants of the host lattice are the input parameters used in the computations of the numerical coefficients A( b – l) (r l 2r m ) which are material constants of g-Fe. In their turn, these coefficients enable us to calculate strain-induced pairwise interaction energies between any dissolved atoms directly in many coordination shells provided the coefficients of the concentration expansion of the host lattice (concerning the relevant substitutional and interstitial atoms) are known. Generally, the strain-induced interaction in some g-Febase solid solutions is strong and must necessarily be taken into account for the analysis of structure and properties of solid solutions. The calculation of the carbon activity and ¨ the parameters of carbon distribution given by Mossbauer spectroscopy showed that the model of strain-induced interaction is applicable for carbon austenite but it must be supplemented by additional C–C repulsion in the first coordination shell.

Table 6 ¨ Energies W (1C – C ) (in eV) of C–C interaction in the first coordination shell obtained by means of Mossbauer data At.% C

8

7.7

Tf, K

Experiment [39] Strain-induced, W (1C – C ) 5 20.097 Strain-induced, W (1C – C ) 5 20.097 W (1C – C ) 5 10.020 W (1C – C ) 5 10.089 Experiment [40] Strain-induced, W (1C – C ) 5 20.097 Strain-induced, W (1C – C ) 5 20.097 W (1C – C ) 5 10.004 W (1C – C ) 5 10.036

773 1600 773 1600 773 1600 773 1600

Abundance of the different sites, % P0

P1 1 P2 – 90

P2 – 180

54.5 79.17 64.86 54.43 54.00 57.5 79.34 65.01 57.43 57.18

45.5 20.82 34.82 45.36 45.30 42.5 20.64 34.68 42.39 42.28

,0.6 0.01 0.32 0.21 0.70 0 0.02 0.31 0.18 0.54

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M.S. Blanter / Journal of Alloys and Compounds 291 (1999) 167 – 174

Acknowledgements This work was supported by the Russian Fund of Fundamental Research, under grant N98-02-16131.

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