Strain-induced interaction of dissolved interstitial and substitutional atoms in fcc metals

Journal of Alloys and Compounds 282 (1999) 137–141

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Strain-induced interaction of dissolved interstitial and substitutional atoms in fcc metals M.S. Blanter Moscow State Academy of Instrumental Engineering and Information Science, Stromynka 20, 107846, Moscow, Russia Received 21 May 1998

Abstract The energies of strain-induced (elastic) interaction of interstitial–substitutional pairs are calculated for Al, Ag, Au, Cu, Pd, Pt and Ni 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. Whenever a substitutional atom expands the crystal lattice, there is attraction in the first two coordination shells. In this case a substitutional atom contracts the crystal lattice the first two coordinate shells exhibit repulsion while the third one shows weak attraction. Generally, the i–s interaction in f.c.c. metals is weaker than in b.c.c. metals but in some solid solutions it is very strong and must be taken into account for analysis of structure and properties of solid solutions.  1999 Elsevier Science S.A. All rights reserved. Keywords: Strain-induced interaction; Elastic interaction; F.C.C. metals; Interstitial–substitutional interaction

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. The theory of solute interactions is yet imperfectly developed and it is so far impossible to obtain these energies for many alloys. For many interstitial and substitutional solid solutions in metals, the elastic interaction is essential [1]. The theory of strain-induced (elastic) interaction in the framework of the model of a discrete crystal lattice has been developed [1,2]. The energies of pair interactions of interstitials (i–i) and substitutionals (s–s) have been evaluated for many b.c.c. [3–10] and f.c.c. metals [10,11]. The energies of s–i interactions were calculated only for b.c.c. metals: for interstitials located in octahedral intersticies in [6,12] and ones in tetrahedral intersticies in [13]. Despite the limitations of the model of the pair interatomic interaction it is useful for the solution of many problems of the 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 [14–16]. That means that it is also interesting to calculate these energies for several f.c.c. metals. The choice of the metals was determined by the availability of experimental data on Born–von-Karman constants necessary for such calculations. The purpose was: (1) to calculate the energies of i–s strain-induced (elastic) interaction of f.c.c. metals Ni, Cu, Au, Ag, Pd, Pt and Al in many coordination shells; (2) to compare its with ones in b.c.c. metals.

2. Method of energy calculation The calculation technique employed in this paper is based on the theory of lattice statics formulated by Khachaturyan in [1,2]. Interstitial atoms are located in octahedral interstices of a 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(r) between a interstitial atom and a substitutional atom separated by the vector r are calculated by means of the inverse Fourier-transformation, W(r) 5 (1 /N)S kV(k)e i kr

(1)

where the summation is carried out over all N points of

0925-8388 / 99 / $ – see front matter  1999 Elsevier Science S.A. All rights reserved. PII: S0925-8388( 98 )00720-8

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138

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(k) is the Fourier-transform of the interaction energies. According to [1,2] the function V(k) can be expressed through such material constants as the coefficients of the concentration expansion of the host i s lattice u 0 (for interstitials) and u 0 (for substitutional atoms) and frequencies of crystal lattice vibrations. This function looks like the scalar product V(k) 5 2 F i (k)v s * (k)

(2)

F s1 (k) 5 2 i(c 11 1 2c 12 )(u 0s a 02 / 4) sin(k 1 a 0 / 2)[cos(k 2 a 0 / 2) 1 cos(k 3 a 0 / 2)]

(7)

The components F s2 (k) and F s3 (k) can be obtained from F s1 (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 on concentration c u 0 5 (1 / a 0 )da 0 / dc

where i

i

F (k) 5 S r f (r)e

2ikr

(3)

is a Fourier]transform of the coupling force f i (r) acting on an undisplaced host atom at the site r from a interstitial atom at the interstitial position r50 (a Kanzaki force). F s (k) is a Kanzaki force from a substitutional atom. The vectors F i (k) and F s (k) are material constants. Eq. (2) is not valid for k50. v s (k) 5 S rU s (r)e 2ikr

(4)

is a Fourier-transform of the host atom displacement U s (r) at a site r produced by a substitutional atom at r50. The value v s (k) can be found from the equation of lattice statics S j Dij (k)v sj (k) 5 F si (k)

F i (k) 5 2 i(c 11 1 2c 12 )(u i0 a 20 / 2)

and

where c is the atomic fraction of interstitials (for u i0 ) or substitutionals (for u 0s ). The numerical values of u s0 were obtained in [9] according to the experimental concentration dependence of the lattice parameter. The numerical values of u i0 were also obtained according to the experimental concentration dependence of the lattice parameter for C [20] and H [21] in Ni and H [21] in Pd. For the other investigated solid solutions the experimental data on the concentration dependence are absent because of low solubility of hydrogen and the values of u i0 were calculated in [10] on the basis of ‘‘the empirical rule of Baranowski’’ [21]. This rule advocates that the changing of the volume of each metal due to hydrogen is equal to 2.9310 23 nm 3 per one i s hydrogen atom. The values of u 0 and u 0 are listed in Table 2.

(5)

where i, j51, 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 be not 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 direction (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 [9]. The applied Born–von Karman constants were taken from: Al [17], Pd, Ag, Cu, Ni, Pt [18], Au [19]. 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]

[sin(k 1 a 0 / 2); sin(k 2 a 0 / 2); sin(k 3 a 0 / 2)]

(8)

(6)

3. Results and discussion i

s

i

s

Since F (k) and F (k) are linear functions of u 0 and u 0 respectively, it follows from equation 2 and equation 5 that the energies W(r) are proportional to u i0 u s0 , W(r) 5 A(r)u i0 u s0

(9)

where the universal coefficients A(r) are the same for all solid solutions based on the relevant host lattice. Table 1 gives the numerical values of the coefficients A(r) for various distances r between a interstitial atom and a substitutional one for Ag, Al, Au, Cu, Ni, Pd and Pt. The coefficients A(r) 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. Table 1 can be directly employed for calculation of numerical values of interaction energies. Since all the interstitials expand a crystal lattice, one has u i0 .0, and the sign of W(r) is determined by the sign of the product of A(r) and u 0s (a negative interaction energy means attraction, a positive energy, repulsion). In all f.c.c.

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Table 1 Coefficients A(r) (in eV) for calculation of the pairwise strain-induced interaction energies of interstitial and substitutional atoms in f.c.c. metals 2r /a

(100)

(111)

(210)

(122)

(300)

(311)

(320)

(410)

Shell uru /a Al Cu Ni Ag Au Pd Pt

1 0.5 26.67 29.20 210.50 210.60 217.29 217.57 227.78

2 0.87 21.56 21.99 22.08 22.39 23.92 24.06 26.56

3 1.12 10.28 10.56 10.78 10.59 11.15 10.90 11.64

4a 1.5 20.10 20.15 20.15 20.19 20.44 20.23 20.38

4b 1.5 20.35 20.47 20.49 20.55 20.98 21.06 21.49

5 1.66 10.03 10.03 10.03 10.05 10.22 10.17 10.51

6 1.8 10.06 10.10 10.13 10.10 10.22 10.26 0.35

7 2.06 20.06 20.12 20.13 20.15 20.16 20.24 20.31

metals the coordination shell distributions of the energies (Fig. 1) are qualitatively the same and are fundamentally determined by a crystal lattice but not individual peculiarities of each metal. However, the absolute values

Fig. 1. Dependence of the energies of interstitial–substitutional straininduced interaction on distance: (a) f.c.c. metals (present paper); (b) b.c.c. metals for the case of location of interstitials in octahedral interstices (the tetragonality factor (u i11 /u i33 )5 20.1) [6,12]; (c) b.c.c. metals for the case of location of interstitials in tetrahedral interstices [13].

of coefficients A(r) are quite different for different metals. In the first two coordination shells of every metal one has A(r),0. In the far shells the potential is anisotropic and oscillating. This means that the energies depend not only on the distance between the solute atoms but also on relative positions (one can compare the shells 4a and 4b). In the case when a substitutional atom expands the host crystal lattice (u s0 .0) one observes attraction in the first and the second coordination shells with the maximal attraction in the first shell. This attraction can be strong, up to 20.16 eV for hydrogen (Cd–H in Cu, Table 2) and 20.27 eV for carbon (Pd–C in Ni). 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. The attraction is weak since the coefficient uA(r)u in the third shell is about 15–20 times less than in the first one. We can see that in such a case the energy of maximal attraction is below 20.005 eV. For comparison, the coordination shell distributions of the energies of i–s interaction in b.c.c. metals are shown in Fig. 1. It is seen that for the two cases –location of interstitials in octahedral interstices (N, C or O) and in tetrahedral interstices (H, D)– the strongest interaction is in the first two shells, like in the case of f.c.c. metals. For u s0 .0 this interaction is attractive. The main difference between f.c.c. and b.c.c. metals is the following. In the case when a substitutional atom contracts the crystal lattice (u s0 ,0) the attraction in f.c.c. metals is quite weak (in the third coordination shell), in contrast to not so weak attraction in b.c.c. metals (in the fourth shell for octahedral intersticies and in fifth shell for tetrahedral ones). In f.c.c. metals the values uA(r)u in the third shell lie between 0.04 and 0.08 of the maximal values uA(r)u (achieved in the first shell). In contrast, in b.c.c. metals the values uA(r)u in the shells with maximal attraction for u s0 ,0 are |0.3 of the maximal values uA(r)u (achieved in the first or second shells). Generally, comparison of the energies of maximal attraction in f.c.c. metals (Table 2) with the energies of maximal attraction in b.c.c. metals [6,12,13] proves that the i–s interaction in f.c.c. metals is weaker than in b.c.c. metals. However, in some f.c.c. solid solutions the interaction is quite strong (for example Pd–C in Ni) and must be necessarily taken into account for analysis and calcula-

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Table 2 Energies Wmax (in meV) of maximal i–s attraction in f.c.c. solid solutions Metal (i) (u i0 )

S

u so

Wmax

Metal (i) (u i0 )

S

u so

Wmax

Al (H) (0.058)

Li Ag Mg Zn Mn Ag Cd Mn Rh Zn Cr Ge Pd Rh Ru Cr Ge Pd Rh Ru Cd Cu Hg Ge In Mn Pd Pt Zn

20.008 10.018 10.124 20.024 20.120 10.130 10.212 10.100 10.063 10.060 10.038 10.045 10.122 10.092 10.085 10.038 10.045 10.122 10.092 10.085 10.052 20.115 10.066 10.007 10.082 10.027 20.047 20.050 20.052

20.1 a 27.0 248.0 20.4 a 21.9 a 299.3 2161.9 276.4 248.1 245.8 2183.8 299.2 2269.0 2202.9 2187.4 235.5 242.0 2114.0 286.0 279.4 231.4 23.9 a 239.9 24.2 249.5 216.3 21.6 a 21.6 a 21.7 a

Pt (H) (0.064) Au (H) (0.057)

Ag

10.026

242.6

Ag Co Cr Cu Hg Li Mn Ni Pd Ta Ti V Ag Au Fe Ir Nd Ni Pt Rh Ta V W Zr

20.005 20.010 20.034 20.094 10.060 20.064 20.017 20.136 20.047 10.009 20.026 20.042 10.049 10.048 20.039 20.019 10.025 20.077 10.002 20.015 10.022 20.017 20.012 10.089

20.3 a 20.7 a 22.2 a 26.2 a 259.1 24.2 a 21.1 a 28.9 a 23.1 a 28.9 21.7 a 22.7 a 254.2 253.1 22.2 a 21.1 a 227.7 24.4 a 22.2 20.9 a 224.4 21.0 a 20.7 a 298.5

Cu (H) (0.083)

Ni (C) (0.210)

Ni (H) (0.089)

Ag (H) (0.057)

a

Pd (H) (0.063)

In the third coordination shell. In all other cases-in the first shell.

tion of structure and properties of solid solutions. 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 [15,16,22].

than in b.c.c. metals, however in certain solid solutions it is very strong and must be with necessity taken into account for analysis of structure and properties of solid solutions.

Acknowledgements 4. Conclusions We performed numerical calculations of strain-induced (elastic) interaction energies of interstitial and substitutional atoms in f.c.c. metals. 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(r) which are material constants of the investigated metals. In their turn, these coefficients enable us to calculate strain-induced pairwise interaction energies between any substitutional and interstitial 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 i–s interaction in f.c.c. metals is weaker

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

References [1] A.G. Khachaturyan, Theory of Structural Transformations in Solids, Wiley, New York, 1983. [2] A.G. Khachaturyan, Theory of Phase Transformation and Structure of Solid Solutions, Nauka, Moscow, 1974 (in Russian). [3] H. Horner, H. Wagner, J. Phys. C 7 (1974) 3305. [4] M.S. Blanter, A.G. Khachaturyan, Metall. Trans. 9A (1978) 753. [5] V.G. Vaks, N.E. Zein, V. Zinenko, A.G. Orlov, Zh. Eksp. Teor. Fiz. 87 (1984) 2030 (English translation: Sov. Phys. JETP). [6] M.S. Blanter, V.V. Gladilin, Izvestija AN SSSR: Metals (6)(1985) 124 (in Russian). [7] A.I. Schirley, C.K. Hall, N.J. Prince, Acta Metall. 31 (1983) 985. [8] S.V. Beiden, V.G. Vaks, Phys. Lett. A 163 (1992) 209. [9] M.S. Blanter, Phys. Status Solidi B: 181 (1994) 377.

M.S. Blanter / Journal of Alloys and Compounds 282 (1999) 137 – 141 [10] [11] [12] [13] [14] [15]

M.S. Blanter, Phys. Status Solidi B: 200 (1997) 423. S. Dietrich, H. Wagner, Z. Phys. B36 (1979) 121. M.S. Blanter, Phys. Met. Metallogr. 51(3) (1981) 136. A.I. Schirley, C.K. Hall, Acta Metall. 32 (1984) 49. M.S. Blanter, M.Ya. Fradkov, Acta Metall. Mater. 40 (1992) 2201. I.S. Golovin, M.S. Blanter, R. Schaller, Phys. Status Solidi A: 160 (1997) 49. [16] M.S. Blanter, Phys. Met. Metallogr., (1998), to be published. [17] G. Gilat, R.M. Nicklow, Phys. Rev. 143 (1966) 487.

141

[18] D.H. Dutton, B.N. Brockhouse, A.P. Miller, Can. J. Phys. 50 (1972) 2952. [19] J.W. Lynn, H.G. Smith, R.M. Nicklow, Phys. Rev. B: 8 (1973) 3493. [20] L. Zwell, E.J. Fasiska, Y. Nakada, A.S. Keh, Trans. AIME 242 (1968) 765. [21] H. Peisl, in: Hydrogen in Metals, no. 1, Springer–Verlag, New-York, 1978. [22] M.S. Blanter, J. Alloys Comp. 253–254 (1997) 364.