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THEORETICAL STUDY ON THE INCOHERENT TWIN BOUNDARIES IN AUSTENITIC STAINLESS
STEEL
M. Mori, K. Masuda-Jindo, K. Tanaka, I. Ishida
To cite this version:
M. Mori, K. Masuda-Jindo, K. Tanaka, I. Ishida. THEORETICAL STUDY ON THE INCOHERENT
TWIN BOUNDARIES IN AUSTENITIC STAINLESS STEEL. Journal de Physique Colloques, 1990,
51 (C1), pp.C1-275-C1-280. �10.1051/jphyscol:1990143�. �jpa-00230302�
THEORETICAL STUDY ON THE INCOHERENT TWIN BOUNDARIES IN AUSTENITIC STAINLESS STEEL
M. MORI, K. MASUDA-JINDO*, K. TANAKA and I. ISHIDA
Institute of Industrial Science, University of Tokyo, 7-22-1, Roppongi, Finato-ku, Tokyo 106, Japan
Department of Materials Science and Engineering, Tokyo Institute of Technology, Nagatsuta, Midori-ku, Yokohama 227, Japan
Abstract
-
The structure of incoherent C=3 (112) twin boundaries in the austenitic stainless steel is investigated using the tight-binding recur- sion method and the model £cc iron crystal. To estimate the change in the total energy due to the atomic displacements, 15 pairs of recursion coef- ficients (an,bn; ns15) are calculated for the atomic clusters containing 2100 atoms:"~he short-range repulsive energy contribution is also taken into account using the Born-Mayer potential. It is found that the signif- icant rigid body translation occurs parallel to the boundary plane to- gether with the excess volume displacements for the (112) incoherent twin boundaries in y-Fe. The occurence of the rigid body translation is shown to result from the reduced density of heavily distorted bonds and reduced local d-band deformation around the incoherent twin boundary.I
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INTRODUCTIONRecently, advances in electron microscopy or X-ray diffraction technique have enabled us to get detailed information on the atomic structures of grain boundaries and twin boundaries in various materials 11-41. For instance, using the weak beam (a-fringe) electron microscopy technique the Burgers vector of a grain boundary dislocation has been determined unambiguously. The transmission electron microscopy (TEM) method allows us to determine the rigid body transla- tion or excess volume atomic displacements between the two crystals at the boundary.
It is the purpose of the present paper to investigate the atomic configu- ration around the incoherent (112) twin boundary in £cc transition metal sys- tems using a microscopic electronic theory. Particular attention will be fo- cused on the atomic configuration of the incoherent twin boundaries in austeni- tic stainless steel. The structure of incoherent C=3 (112) twin boundaries in the austenitic stainless steel is simulated by using the model £cc iron crystal and compared with the experimental observations (SUS316) 151.
Recently, there has been a considerable amount of theoretical studies on the grain boundary, twin boundary and stacking fault structures in metals and alloys 16-81. In particular, it has been found that rigid body translation is a dominant contribution to the atomic relaxation of incoherent (112) twin bound-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1990143
Cl-276 COLLOQUE DE PHYSIQUE
ary in fcc metals 131. However, in most of the previous studies the total en- ergies are calculated by using the phenomenological two-body potentials and long range electronic interactions are neglected. Therefore, it is of great in- terest to investigate the incoherent twin boundary structures taking fully into account the effect of the multi-atom correlation and configurations of large atomic clusters (c2100 atoms).
2
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PRINCIPLE OF CALCULATIONIt is now well established that the cohesive properties of the transition metals and alloys can be understood semiquantitatively within the tight-binding
(TB) electronic theory /9,10/. The characteristic parabolic dependence of the cohesive energy, surface tension and elastic constants are related to the fill- ing of the d-bands and reproduced well by the simple TB theory. The justifica- tion of using the TB electronic theory for total energy calculation of the tran- sition metal systems has been given by several authors on the basis of the first-principle density functional theory /Ill.
For the calculation of atomic configuration around the incoherent twin boundary, we use the TB recursion method /12,13/ and the direct total energy minimization procedure. We assume that the total energy of the system can be given by a sum of the band structure energy EbAnd and overlap repulsive energy Ere, 19-10,13/. The band structure energy Eband of the system can be calculated
~.
from the electronic Green function, Gii on each atomic site obtained by using
- -
the recursion method 1121. For the repulsive potential energy between adjacent atomic sites i and j, we take the Born-Mayer potential of CO-exp(-pRii). The parameter values CO and p can be determined so as to satisfy the equiiibrium condition and to reproduce the elastic constants.
We use the standard formula of the band structure energy
where pi(E) represents the local density of states (DOS) at site i, pO(E) re- presents the local DOS for the perfect lattice and EF is the Fermi energy. The local DOS Pi(E) on atomic site i can be calculated from the Green's function of the continued fraction form:
where
In the present study of the twin boundary structure, we use the conventional scalar recursion method by Haydock et al. 1121. The recursion coefficients are calculated up to 15 level for atomic clusters containing s2100 atoms. It is known that the scalar recursion scheme gives identical results to those of the sophisticated matrix-form recursion method (having correct symmetry) 1141 when the sufficient numbers of the recursion coefficients are taken into account.
The two center hopping integrals of the d-bands, ddu, ddr and dd6 are de- termined by using Harrison's universal TB scheme I151
where d denotes the nearest-neighbour interatomic distance. rd is the d-state radius of the transition metal atom and taken to be 0.80 for iron atom. The exponent of spatial dependence of the two center integrals is taken to be 5, in accordance with the canonical ASA-LMTO theory 1161.
In order to understand the twin boundary properties of real austenitic stainless steel (SUS316). we have performed simulation calculations on the in- coherent (112) twin boundaries in y-Fe (fcc) crystals. Although this simplifi- cation does not allow the rigorous comparison between the theoretical calcula- tions and experimental results, we think that it is essentially correct and quite useful for the general understanding of twin boundary properties of real multicomponent alloys.
The present numerical calculations on y-Fe crystals are performed using d-band parameters presented in Table 1. The d-band parameters ddo, ddn and dd6 are determined by using the lattice constant of a. = 3.61 for fcc iron. This lattice constant is estimated from that of bcc phase (2.866A) by using the hard sphere model. The lattice constant of y-Fe at high temperatures (908 c1403OC) and calculated lattice constant 3.44 8 by using the ASW band structure calcula- tion scheme 1171 are also listed. We have chosen the lattice constant of 3.61 for Y-Fe simply because it is between the experimental lattice constants at high temperatures and the calculated lattice constant at absolute zero tempera- ture. The effective d-bandwidth of y-Fe is then estimated as
Table 1. Input d-band parameters (nearest-neighbour distance) and lattice constant an of £cc iron (eV unit).
dda ddn dd6 Q a.
(i)
The lattice constant 3.611 (a), used in the present calculation, is esti~ated from the experimental lattice constapt of bcc y-iron (2.866 A) using the hard sphere model. ao= 3.656 A ( b ) is experimen- tal lgttice constant of y-Fe (908% 1403°C). On the other hand, ao=
3.44 A is the result of ASW band structure calculation /17/.
In the present study, we prepare two types of the incoherent (112) twin boundaries in £cc crystals as shown in Fig.1. The twin boundary structure (a), which will be referred to as type A has a mirror symmetry with respect to the geometrical boundary plane. On the other hand the twin boundary structure (b), referred to as type B, has the rigid body translation component parallel to the geometrical boundary plane. The magnitude of rigid body translation is (2/9)&-, in unit of the nearest-neighbour distance d of the perfect crystal.
We have performed lattxce (interlayer spacing) relaxation calculations for three atomic layers of the two types of the incoherent twin boundaries, allow- ing the excess volume displacement
;
and rigid body translation$
parallel to the boundary plane: value remains almost unchanged, and is roughly equal to( 2 / 9 ) 6 d for the relaxed structure. The excess energies, per single atom, are summed up for 17 atomic layers around the geometrical boundary plane. Although the relaxation calculation within the limited region is certainly insufficient for the estimation of accurate values of the interfacial energies, it may be sufficient for the prediction of relative stability of the twin boundary struc- tures, type A or type B. In this respect, we point out that in the surface re- laxation or reconstruction calculations of the transition metals, the contribu- tion coming from the topmost layer is of primary importance 1181.
In Table 2, we present calculated excess energies AEtot, excess volume displacement
~ $ 1
and distorted (heavily compressed and stretched) bond lengths for type A an type B twin boundary structures. As presented in Table 2, we have found that total excess volume displacements (total outward relaxation) of type A and type B twin boundaries are 0.34 and 0.18 (in unit of the nearest- neighbour distance d), and their excess energies are 0.256 eV and 0.09588 eV,Cl-278 COLLOQUE DE PHYSIQUE
respectively. This indicates that the twin boundary structure of type B is more stable than type A structure in y-Fe crystal.
( a ) type A ( b ) t y p e B
Fig.1 Atomic configurations of type A (a) and type B (b) (112) twin boundary structures in £cc crystal (after relaxation calculation).
Fig.2 Local d-band DOS on atoms around type A incoherent twin boundary in y-Fe: The DOS curves (a), (b), (c), (d), (e), ( f ) and (g) are calculated for atomic sites A, B, C, D, E, F and G in Fig. l (a).
The DOS curve (h) is calculated for perfect lattice.
However, the present theoretical findings are not completely in accord with the experimental results on the incoherent (112) twin boundary structure in austenitic stainless steel. According to the recent experimental observa- tions on the incoherent twin boundaries in austenitic stainless steel by Tanaka et al. / 5 / , no rigid body translation was found within the resolution of the experimental method (weak beam a-fringe method). The discrepancy between the
present calculations are performed for Y-Fe crystals, while the experimental results are weak beam observations on the multicomponent austenitic stainless steel. Secondly, the present calculation does not take into account the effect of segregation of constituent atoms at the twin boundaries. In the experiments of the austenitic stainless steel, certain constituent elements such as Cr atoms may segregate to the twin boundary and lead to the local structural changes of the boundary.
Fig.3 Local d-band DOS on atoms around type B incoherent twin boundary in Y-Fe: The DOS curves (a), (b), (c), (d), (e), (f), (g) and (h) are calculated for atomic sites A, B, C, D, E, F, G and H in Fig.1
(b), respectively.
Table 2 Comparison of the change in the total energy, per single atom, boundary layer atom, for type A and type B twin boundaries in Y-Fe (eV unit).
type A type B
AE rep - 3 . 0 8 0 9 9 -1 - 7 3 0 3 8
Bond Length (C) 0.91735 (XI) 0 . 9 4 7 4 5 (x2)
-
C and S represent heavily compressed and stretched bonds, respectively.
Values in parentheses are multiplicity of the bonds.
Cl-280 COLLOQUE DE PHYSIQUE
In order to understand the physical origin of the stability of type B twin boundary structure in y-Fe crystal, it may be useful to check the distorted bond lengths and their density. One can see in Table 2 that the bond lengths of type A twin boundary structure are more strongly distorted than those of type B twin boundary. The exsistence of the strongly distorted bonds can also be seen in the local d-band DOS curves. In Figs.2 and 3, we present the local DOS curves on atomic sites in the vicinity of the geometrical boundary plane for type A and type B twin boundaries, respectively. In general, in the atomic sites near the boundary, the effective d-bandwidth becomes narrower and the number of non-bonding states increases. One can see that such d-band deforma- tion is more pronounced for type A twin boundary structure, compared to type B structure. One can also see in these figures that the local DOS on the atoms of fifth or sixth atomic layer quite resemble that of the perfect lattice. There- fore, d-band deformation near the geometrical boundary plane plays an important role in determining the stability of the incoherent twin boundaries.
We now briefly discuss the significance of including higher order moments (a30 moments) in the present twin boundary structure calculations. It is known that the cohesive and elastic properties of close-packed £cc transition metals can be well accounted for within the second moment TB approach /9,10/. However, it is not certain whether the lowest order second moment approach gives accu- rate result for the calculation of the incoherent twin boundary structures with characteristic atomic displacements. In order to simulate the detailed atomic configuration of the twin boundary, we have taken into account higher order mo- ments, a15 recursion coefficients (a30 moments). This is due to the fact that the higher order moments are related to the long range electronic interactions in the crystal. Nevertheless, the second moment approach may be useful to de- termine the overall atomic geometry of the twin boundaries and to understand the physical origin of the atomic displacemenets.
In conclusion, we have investigated the relative stability of the incoher- ent (112) twin boundaries in y-Fe using TB recursion electronic theory. The type B boundary structure is found to be more stable than type A in y-Fe crys- tal and the magnitude of excess volume displacements is in good agreement with corresponding experimental results (v% aO/l 2 = 0.2887 d) of the austenitic stainless steel. We have shown that the analysis of the distorted bonds and d- band deformation near the boundary plane are important for the understanding of the stability of the twin boundaries.
REFERENCES
/l/ Ishida Y., Suppl. Trgns. JIM, (1986) 33.
Miyazawa K., Ishida Y: and Suga T., Phil. Mag.
S,
(1988) 825./ 2 / Smith D.A., J. de Physique Suppl.
49,
(1988) C5-63./ 3 / Pond P.C. and Vitek V., Proc. Roy. Soc. Lond.
z,
(1977) 453./4/ Fitzsimmons M.R. and Saas S.L., J. de Physique Suppl.
49,
(1988) CS-71./S/ Tanaka K., Mori M, and Ishida Y., Seisan-Kenkyu 41, (1989) 230, 603 (in Japanese).
/6/ Pond R.C. and Bollmann W., Phil. Trans. Roy. Soc.
er
(1979) 449./7/ Sutton A.P. and Vitek V., Phil. Trans. Roy. Soc.
E ,
(1983) 1, 37, 55./8/ Sutton A.P. and Balluffi R.W., Acta Met.
35,
(1987) 2177./g/ Friedel J., "The Physics of ~etals", ed. J.M. Ziman (Cambridge University Press, 1969) P.340.
/10/ Ducastelle F., J. de Physique 3l, (1970) 1055.
/11/ Harris J., Phys. Rev.
g,
(1985) 1770.1121 Haydock R., Heine V. and Kelly M.J., J. Phys. C5, (1972) 2845; ibid.,
B,
(1975) 2591.
/l31 Wills J.M. and Harrison W.A., Phys. Rev.
e,
(1984) 4363.Pettifor D.G. and Podloucky R., J. Phys.
9,
(1986) 1389.1 1 4 1 Inoue J. and Ohta Y., J. Phys. (1987) 1947,
/IS/ Harrison W.A., "~lectronic Structure and the Properties of ~olids", Free- man, San Francisco (1980).
/16/ Andersen O.K., Phys. Rev. G, (1975) 3060.
1171 Masuda-Jindo K. and Terakura K., Phys. Rev.
er
(1989) 7509./18/ Terakura I., Terakura K. and Hamada N., Surf. Sci.
111,
(1981) 479.Masuda-Jindo K., Hamada N. and Terakura K., J. Phys.
