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Atomistic aspects of the deformation of NiAl
Intermetallics 7 (1999) 447±454
Atomistic aspects of the deformation of NiAl Peter Gumbsch*, Ralph Schroll Max-Planck-Institut fuÈr Metallforschung, Seestraûe 92, 70174 Stuttgart, Germany Received 24 April 1998; revised 23 July 1998; accepted 1 September 1998
Abstract Properties of dislocations in B2-NiAl have been studied atomistically using an embedded atom potential. The response of dislocation cores to applied homogeneous shear stresses is investigated and the Peierls stresses of straight dislocations are determined. The results are in many details in excellent agreement with experimental observations. Speci®cally, the behaviour of the h1 1 1i dislocations, their slip planes, cross-slip behaviour and the limiting role of the screw dislocation can be explained. Similarly, the appearance of the {2 1 0} plane as a secondary slip plane for the h1 0 0i dislocations can be rationalised. Furthermore, the interaction of the dislocations with structural point defects is studied. Comparison with the ¯ow stress of o-stoichiometric NiAl from the literature shows that the individual point defects cannot be made responsible for the strong increase of the ¯ow stress, suggesting that more complex defect structures may play an important role. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Nickel aluminides based on NiAl; Plastic deformation mechanisms; Defects: Point defects; Defects: Dislocation geometry and arrangement; Atomistic simulations
1. Introduction The intermetallic phase NiAl which crystallises in the B2 (CsCl) structure is regarded as a promising candidate for high temperature applications because of its low density, high melting point and good oxidation resistance. There is substantial evidence indicating that details of the dislocation core structure play a central role in determining the deformation behaviour of NiAl [1±5]. At room temperature and below NiAl deforms almost exclusively by the motion of h1 0 0i dislocations [6±8]. The h1 0 0i Burgers vector gives only three independent slip systems which is thought to be the main reason for the limited ductility of polycrystalline NiAl. A dierent Burgers vector is observed for single crystals strained along the h1 0 0i axis, the so-called `hard orientation'. In this case, the resolved shear stress on the h1 0 0i dislocations vanishes and h1 1 1i dislocations become active [9,10]. At higher temperature, where ductility increases signi®cantly, h1 1 0i dislocations begin to contribute to the plastic deformation [1]. The advent of the h1 1 0i dislocations appears to coincide with the transition from brittle to ductile response [11].
* Corresponding author. Tel.: +49 711 2095129; fax: +49 711 2095120; e-mail: gumbsch@®nix.mpi-stuttgart.mpg.de
Deviations from the exact stoichiometric composition result in an increase in the experimentally measured yield stress and hardness [12]. It seems likely that the interaction between structural point defects and dislocations is responsible for this behaviour. The interaction may even be important for stoichiometric compositions at low temperatures, since thermal point defects freeze during the cooling process. This is a consequence of the high migration energy for vacancies in NiAl which is greater than their formation energy [13] and inevitably results in point defect concentrations considerably higher than expected at temperatures below half the melting temperature. Structural point defects in NiAl were ®rst investigated experimentally by Bradley and Taylor [14], who found that Al-rich alloys contain mostly Ni vacancies whereas Ni-antisite defects are the dominant defects in Ni-rich alloys. Ab initio calculations [15] as well as more recent experiments [16,17] con®rmed these results. Although several atomistic computer simulations have already been performed to determine the dislocation core structures and to explain the observed deformation behaviour of NiAl [1,2,18±20], further investigations seemed necessary since most of the earlier studies employed inadequate interatomic potentials [21] and since they partly gave unrealistic properties of the dislocations [4].
0966-9795/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0966 -9 795(98)00096 -X
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The {1 1 0} and {2 1 1} planes are generally regarded as glide planes for dislocations with Burgers vectors of the h1 1 1i type and these are also observed as the glide planes of hard oriented NiAl at low temperatures [10]. The h1 0 0i dislocations are known to glide on {1 0 0} and {1 1 0} planes. However, in recent in situ deformation experiments [22] in the transmission electron microscope (TEM) dislocation motion on {2 1 0} glide planes was clearly identi®ed. Furthermore it was suggested from these in situ observations that a combination of jog and Peierls mechanism controls the deformation of NiAl single crystals at room temperature. The goal of this paper is to summarise our atomistic understanding of the core structure of dislocations in B2-NiAl and to compare them with recent microscopic observations. Furthermore, the experimentally observed ¯ow stresses of stoichiometric and o-stoichiometric NiAl are discussed in connection with the atomistic results on the interaction of the dislocations with structural point defects. In this paper we focus on dislocations which are active at room temperature and below and therefore explicitly exclude h1 1 0i dislocations which only become important at higher temperatures. The properties of h1 1 0i dislocations and the atomistic aspects of their core structure are discussed elsewhere [2,4,23]. 2. Simulation technique The general strategy for the calculation of the atomic structure of a dislocation is to place it in the middle of a cylindrical block by displacing the atoms according to the anisotropic elastic displacement ®eld [24]. Periodic boundary conditions are applied parallel to the line direction thereby creating an in®nitely long, straight dislocation. The outermost atoms are ®xed at the positions determined by the elastic solution, while the interior of the model is allowed to move freely. These atoms are relaxed to their equilibrium positions using a conjugate gradient algorithm. Further details of the modelling technique and the boundary conditions are given in Ref. [4]. The atomic interaction is described by an EAM potential [25] which is ®tted to various properties of NiAl. In our investigation we have used the potential for NiAl developed by Ludwig and Gumbsch [21]. This potential does not only reproduce the structural point defects correctly but also gives particularly reasonable dislocation properties, which is shown in a comparison to dislocation simulations using other potentials in Ref. [4]. This potential has also been successfully applied to fracture simulations [26,27], point defect [28] and grain boundary studies [29]. Homogenous shear stresses were applied to the atomistic model to study the motion of the dislocations.
The stresses are actually imposed via homogenous shear strains by displacement boundary conditions and are increased stepwise. After every step the entire system is relaxed and examined for changes in the core structure. The applied stress at which the dislocation ®rst moves is identi®ed as the Peierls stress. The dislocation core structure is visualised using the dierential displacement method [30]. According to the dislocation type, either the screw or the edge component of the relative displacement of the neighbouring atoms is represented as a line between them. Another important quantity to characterise a dislocation is its core energy. It may contribute considerably to the line energy of the dislocation and decide between possible decomposition or dissociation reactions. Because of the large displacements in the dislocation core, linear elasticity is not appropriate and atomistic modelling must be used to determine this quantity. We have developed a new and unambiguous method to calculate the dislocation core energy in ordered alloys [5], which solves the stoichiometry problems encountered in such calculations. Point defect±dislocation interaction is modelled atomistically essentially with the same methods. A vacancy or an antisite defect is introduced into the simulation cell by simply removing or exchanging an atom. Because of the periodic boundary conditions, this procedure actually generates an in®nitely long line of point defects. The thickness of the model in the z-direction was chosen, so that the repeat distance of the point defects was three times the lattice repeat length of the crystal. Since the stress ®eld of a point defect (1/r3) is shorter in range and drops o much faster than the stress ®eld of a dislocation (1/r), the interactions between the point defects can be neglected even at this relatively close spacing. Further details of the quantities one can determine with such atomistic calculations of point defect±dislocation interaction are discussed elsewhere [31]. Here we only characterise the strength of the interaction by the change in the Peierls stress due to the presence of the point defects. 3. h1 0 0i Dislocations Atomistically, h1 0 0i dislocations clearly prefer to move on the most densely packed {0 1 1} plane [4]. Screw dislocations spread out on this plane and the core of edge dislocations is also widely spread on this plane as shown in Fig. 1. With a value of less than 0.1 GPa, the Peierls stress for both these dislocations is very low. Dislocations with mixed character move at only slightly higher Peierls stresses (P =0.13±0.17 GPa). The mixed dislocations also move with similar ease on {0 0 1} planes. This indicates that the dislocation mobility on these planes is not too dierent although the edge
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449
Fig. 1. The core structure of the h1 0 0i edge (upper part) and screw (lower part) dislocations. Large circles represent Al atoms; small circles represent Ni atoms. The dierential displacements exhibit a core spreading on the {0 1 1} plane in both cases.
dislocations have considerably dierent Peierls stresses (see Table 1). The calculated Peierls stress for a h1 0 0i edge dislocation on a {2 1 0} glide plane is 0.04 GPa and therefore even somewhat lower than for the same dislocation on a {1 1 0} glide plane. The Peierls stress of the mixed dislocation (0.19 GPa) is again comparable with the Peierls stress of the mixed dislocations on {1 1 0} and {1 0 0} glide planes. It was not possible to force the h1 0 0i screw dislocation to glide either on a {2 1 0} or a
{0 0 1} plane since it cross-slips at 0.08 GPa onto the {1 1 0} plane. This is caused by the particularly wide core spreading of the screw dislocation on the {1 1 0} plane which can be seen in Fig. 1. Finally, it seems worth mentioning that the screw dislocation is elastically unstable on {2 1 0} planes as it is on {1 0 0} and {1 1 0} planes. The low Peierls barrier of the edge dislocation on the {2 1 0} plane is surprising since one usually associates low Peierls stresses and high dislocation mobility only
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with close packed widely spaced planes. To better understand the dependence of the Peierls stress of the edge dislocation on the orientation of the glide plane, the Peierls stress of the h1 0 0i edge dislocation was calculated for other high-indexed glide planes as well. The result is summarised in Fig. 2. It can be seen that the Peierls stress exhibits two local minima for the {1 1 0} and {2 1 0} planes. Moreover, the Peierls stress for the {1 0 0} plane is even signi®cantly higher than for all of the high-indexed glide planes. This surprising behaviour can be rationalised in the context of the Peierls±Nabarro model [32,33]. In this model the Peierls stress is proportional to exp (ÿw/b) where w is the dislocation core width and b the magnitude of the Burgers vector. The core width quanti®es the spreading of the dislocation on the glide plane. A wider dislocation core is therefore associated with a lower Peierls stress. Fig. 3 shows the Burgers vector distribution of the h1 0 0i edge dislocation on the h1 0 0i, h1 1 0i and h2 1 0i glide plane (a detailed description of the Burgers vector distribution is given in Ref. [4]). The core width of the edge dislocation on the {1 0 0} glide plane is roughly half the width as on the {1 1 0} and {2 1 0} planes. The considerably higher Peierls stress of the h1 0 0i edge dislocation on the {1 0 0} glide plane can be explained with the strong exponential relationship between the Peierls stress and the core width. The in¯uence of point defects on the dislocation mobility is studied for both Ni vacancies and Ni antisite defects. As a ®rst measure of the interaction between the dislocation and the point defects, the pinning stress (i.e. the dierence in the Peierls stress with and without the point defects) is calculated. The results are summarised in Table 1 and show, that the pinning stresses reach values up to ten times the Peierls stress. A strong interaction is especially obvious in the important h1 0 0i{0 1 1} glide system.
Fig. 2. Peierls stresses for h1 0 0i edge dislocations on dierent glide planes.
Furthermore, it can be seen from Table 1 that Ni vacancies are more eective obstacles than Ni antisite defects. The reason for this behaviour lies in the dierent mechanism in which Ni vacancies and Ni antisite defects interact with h1 0 0i dislocations. A closer examination of the binding energy pattern (see Ref. [31]) of the point defects in the vicinity of the dislocation core reveals that Ni antisite defects interact mainly parelastically (due to the dierence in volume of the smaller Ni defect atom and the regular Al atom) with the hydrostatic component of the stress ®eld of the dislocation. In contrast, Ni vacancies interact dielastically (as an elastically softer region in the otherwise perfect crystal lattice) with the stress ®eld of the dislocation. The dielastic interaction of the Ni vacancies turns out to be stronger than the parelastic interaction of the Ni antisite defects. In both cases, however, the binding energies of the defects are smaller than predicted by anisotropic linear elastic theory [24,34]. Particularly, the point defects located close to the dislocation core interact weaker than expected. 4. h1 1 1i Dislocations For the tension or compression of NiAl single crystals along the h1 0 0i direction (hard orientation) the resolved shear stress on h1 0 0i dislocations vanishes and h1 1 1i dislocations become active at low temperatures. In the B2 lattice the 1/2h1 1 1i Burgers vector, the dominant Burgers vector of the underlying b.c.c. lattice, is not a complete lattice repeat vector but produces an antiphase boundary (APB). Nevertheless most of the h1 1 1i dislocations split into partial dislocations along
Fig. 3. Burgers vector distribution of the h1 0 0i edge dislocation for three dierent glide planes. The dislocation core width can be de®ned as the width of the Burgers vector distribution at half their maximum height.
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Table 1 Peierls stress D for h1 0 0i dislocations and the pinning stresses from the interaction with Ni antisite defects and Ni vacancies. The character of the dislocation, edge, screw and mixed (E, S, M) is indicated together with the the line direction Glide plane
Burgers vector
{0 1 0}
h1 0 0i
{0 1 1}
h1 0 0i
{0 2 1}
h1 0 0i
a b
Line direction
Character
Peierls stress D (GPa)
Ni antisite (GPa)
Ni vacancy (GPa)
h0 0 1i h0 0 1i h1 0 0i h1 0 1i
Ea Eb S M
1.08 0.88 >0.08 0.15
0.47 0.72 ÿ 0.6
0.92 1.62 ÿ 1.35
h0 1 1i h1 0 0i h2 1 1i h1 1 1i
E S M M
0.06 0.07 0.13 0.17
0.20 0.18 ÿ 0.18
0.49 0.73 ÿ 0.33
h0 1 2) h1 0 0i h1 2 1)
E S M
0.04 >0.08 0.19
ÿ ÿ ÿ
ÿ ÿ ÿ
Ni core. Al core.
the glide plane despite the high APB energy of NiAl. For example, the h1 1 1i edge dislocation on the {2 1 1} glide plane splits into two 1/2 h1 1 1i superpartials. A similar splitting is observed for the h1 1 1i mixed dislocation on the {0 1 1} glide plane. However, the dissociation of the pure h1 1 1i edge dislocation on the {1 1 0} glide plane is more complex. This dislocation splits into two 1/4h2 1 1i Burgers vectors with a 1/2h0 1 1i Burgers vector in between (see upper part of Fig. 4). Contrary to the edge and mixed dislocations, the h1 1 1i screw dislocation does not split into partial dislocations and appears rather compact but highly non-planar. The three-fold rotation symmetry leads to a three-fold spreading of the dislocation core (see lower part of Fig. 4). As a consequence of the dierent core structure, the mobility of the h1 1 1i dislocations also varies considerably. In accord with the Peierls±Nabarro model, the glide dissociation of the edge and mixed dislocations increases their mobility. The Peierls stress of both dis plane are low and lie location types on the h0 1 1i between 0.06 and 0.23 GPa (see Table 2). However, the screw dislocation is hard to move due to its non-planar core structure. When increasing the shear stress, the dislocation core of the screw dislocation gradually spreads on the glide plane and reduces the non-planar components of the displacement ®eld. plane a pronounced twinning/ For glide on the {2 1 1} antitwinning asymmetry is obtained. The Peierls stress was found to be 1.7 GPa in antitwinning sense and this plane with 2.0 is slightly lower than that for the {0 1 1} GPa. However, attempts to move the screw dislocation plane in twinning direction failed. It on the {2 1 1} slip plane. The crialways cross-slipped onto a {0 1 1} plane then tical resolved shear stress on the {0 1 1} exceeded the Peierls stress for this slip plane.
The importance of point defects for the mobility of h1 1 1i dislocations is less clear. It is shown in Table 2 that the pinning stress for the edge and mixed h1 1 1i dislocations exceeds the Peierls stress about 60 times. However, the critical stress to move the h1 1 1i screw dislocation remains almost constant. The reason for this dierence can be understood from the dierent core structures. Since the edge and mixed dislocations are (glide) dissociated, the point defects can strongly interact with the resulting APB (Suzuki eect). In this case, the binding energies reach values up to ÿ4.4 eV which is comparable with the chemical potentials of the individual atoms. On the other hand, the interaction between the point defects and the compact screw dislocation is very small, because of the absence of a hydrostatic stress ®eld. As a consequence, the mobility of the edge and screw dislocations in the presence of point defects may become comparable but the absolute value of the yield stress should not be changed much. 5. General discussion Atomistic simulations of h1 0 0i Burgers vectors in NiAl were so far mainly concentrating on dislocation motion on {1 0 0} and {1 1 0} glide planes. Such `smooth', close packed and widely separated planes usually oer considerably lower resistance to dislocation motion than do closely spaced `rough' planes. In B2 NiAl the {1 1 0} plane has the densest atomic packing followed by the {1 0 0} plane and both have been observed as active glide planes [7,8]. However, Messerschmidt et al. [22] have clearly identi®ed the motion of h1 0 0i dislocations not only on the two most densely packed planes but also on a {2 1 0} glide plane in recent in situ TEM experiments. Furthermore and similar to
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edge dislocation (upper part) is dissociated in partial dislocations. In contrast, the h1 1 1i screw disFig. 4. The core structure of the h1 1 1i {0 1 1} location (lower part) does not show such a glide dissociation, but is highly non-planar and extended on {1 1 0} planes.
earlier observations [7] they found frequent cross slip between {1 0 0}, {1 1 0} and {2 1 0} planes with almost equal dislocation mobility [22]. In excellent agreement with these experimental ®ndings, our atomistic simulations give the lowest Peierls stresses for the motion of non-screw dislocations on {1 1 0} and {2 1 0} planes (see Fig. 2). At ®rst sight, the high calculated Peierls stress for the pure edge dislocation on the {1 0 0} plane seems to be in contrast to the experimentally observed high
dislocation activity on this plane. However, the low Peierls stresses for the mixed dislocations (0.10±0.15 GPa) suggests that all other characters including the screw dislocation are glissile on all three glide planes. The critical resolved shear stresses for h1 0 0i{0 1 1} slip in stoichiometric NiAl single crystals at low temperatures [35] (see Fig. 5) agrees extremely well with the highest Peierls stresses (0.17 GPa) calculated here for this slip system. Although previous atomistic simulations
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453
Table 2 Peierls stress D for h1 1 1i dislocations and the pinning stresses from the interaction with Ni antisite defects and Ni vacancies. The character of the dislocation, edge, screw and mixed (E, S, M) is indicated together with the the line direction Glide plane
Burgers vector
Line direction
Character
Peierls stress D (GPa)
Ni antisite (GPa)
{0 1 1}
h1 1 1i
h2 1 1i h1 1 1i h1 0 0i h0 1 1i
E S M M
0.07 2.0 0.23 0.06
3.93 0.2 ÿ 0.26
{2 1 1}
h1 1 1i
h0 1 1i h1 1 1i
E S
1.4 (3.0) a 1.7 (>3.5) a
0.6 0.2
a
Ni vacancy (GPa) 3.93 0.0 ÿ 1.94 >3.0 0.8
Antitwinning (twinning) direction.
[20,36] using dierent atomic interaction models had given similar core structures for the h1 0 0i dislocations, their Peierls stresses were almost an order of magnitude too high. For the low temperature deformation of stoichiometric NiAl in the `hard' h1 0 0i orientation, agreement with experimental observations is obtained in almost all details. Not only does the Peierls stress for the h1 1 1i screw dislocations exactly reach the critical resolved shear stresses of 2.0 GPa measured below 50 K [10], TEM observations reveal unequivocally long h1 1 1i screw segments after deformation at these low temperatures [10]. This is a strong indication for a large dierence in the Peierls stresses for the dierent dislocation characters which are apparent in Table 2. Furthermore, these TEM observations show increasingly more frequent cross-slip events from the {1 1 0} onto the {2 1 1} plane with decreasing temperature [10]. With reference to Table 2, this can explained by the somewhat lower Peierls stresses for the screw dislocations on the {2 1 1} plane (in the antitwinning sense) than on {1 1 0}. Previous attempts to understand the role of the h1 1 1i dislocations atomistically have not been successful in several respects. Xie et al. [19] only studied the dissociation of the h1 1 1i screw dislocation and found metastable dissociated core con®gurations. They did not study other dislocation characters or dislocation mobility. Parthasarathy et al. [20] addressed these questions and found dissociated edge and screw dislocations with nearly equal and far too low Peierls stresses. These obviously erroneous results are a consequence of the low APB energy of the potential used in their studies [4,21]. At low temperature most strongly ordered B2 compounds, including NiAl, exhibit minima in both yield strength and hardness at the stoichiometric composition [12]. This can be attributed to the strengthening obtained from the structural point defects of o-stoichiometric alloys. In NiAl the increase of hardness with the deviation from stoichiometry is more pronounced on the Al-rich side than on the Ni-rich side [12]. The structural point defects on the Al-rich side are Ni vacancies while Ni antisite defects dominate on the Ni-rich
side. Our atomistic calculations summarised in Table 1 show that h1 0 0i dislocations actually interact more strongly with the Ni vacancies than with the Ni antisite defects. Qualitatively they therefore reproduce the experimentally observed dierence in strengthening on the two sides of the stoichiometric composition. However, a quantitative analysis based on Fleischer's theory of solid solution hardening [37] shows that the strengthening obtained from the individual point defects is far too weak as compared to experiments [31]. An upper estimate for the strengthening from the interaction of the h1 0 0i dislocations with the structural point defects is =0.12 GPa [31]. In Fig. 5 the Peierls stress and the strengthening eect from the solid solution hardening are shown together with the experimentally measured critical resolved shear stresses for non-stoichiometric NiAl. The experimental data are taken from the literature [35,38,39]. It is obvious that neither the absolute value of the solid solution hardening nor the tremendous decrease of the critical resolved shear stress with increasing temperature can be explained on the basis of the atomistic
Fig. 5. Experimentally measured critical resolved shear stresses compared with the solid solution hardening based on the results of the atomistic simulation.
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simulations. One may now speculate which other mechanism could be responsible for the observed mechanical properties of non-stoichiometric NiAl beside the solid solution hardening by structural point defects. Possibly, aggregated defects can play an important role. While the formation of vacancy clusters in NiAl has been observed by Ball and Smallman [13], very little is known about the hardening eect of these defects [40]. 6. Summary and conclusions The goal of our investigation was to achieve a fundamental understanding of the deformation of NiAl single crystals at low temperatures. For this purpose we performed atomistic computer simulations with a potential speci®cally adjusted to the properties of NiAl. For h1 0 0i and h1 1 1i dislocations, edge, screw and mixed characters were analysed and their Peierls stresses were calculated. Furthermore, the interaction of these dislocations with the structural point defects in o-stoichiometric NiAl were studied. Excellent agreement between experiment and our atomistic simulations is obtained with respect to almost all details of the low temperature deformation of stoichiometric NiAl. Our simulations allow for the ®rst time to consistently explain the observed dislocation properties from the atomistic nature of the dislocation cores. The solid solution hardening due to individual structural point defects as estimated from our atomistic calculations is not sucient to explain the tremendous increase in critical resolved shear stress with decreasing temperature observed in o-stoichiometric NiAl. It therefore appears to be important to study the interaction of dislocations with more complex defect clusters. To quantitatively model plastic deformation at ®nite temperatures or the brittle-to-ductile transition requires a more quantitative description of the dislocation mobility. This can only be obtained from a better understanding of the role of kinks and jogs in the dislocation line. Acknowledgements The ®nancial support from the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. References [1] Mills MJ, Miracle DB. Acta Metall Mat 1993;41:85.
[2] Mills MJ, Angelo JE, Daw MS, Winburg JD, Miracle DB. Mat Sci Eng A 1995;192/193:134. [3] Schroll R, Gumbsch P, Vitek V. Mat Sci Eng A 1997;233:116. [4] Schroll R, Vitek V, Gumbsch P. Acta Mater 1998;46:903. [5] Schroll R, Finnis MW, Gumbsch P. Acta Mater 1998;46:919. [6] Lahrman DF, Field RD, Darolia R. In: Johnson LA, Pope DP, Stiegler JO, editors. High-Temperature Ordered Intermetallic Alloys IV, Materials Research Society, Pittsburgh, PA, 1991, p. 603. [7] Ball A, Smallman RE. Acta Metall 1966;14:1349. [8] Ball A, Smallman RE. Acta Metall 1966;14:1517. [9] Field R, Lahrman DF, Darolia R. Acta Metall Mat 1991;39:2951. [10] Sun YQ, Taylor G, Darolia R, Hazzledine PM. In: Horton JA, Baker I, Hanada S, Noebe RD, Schwartz DS, editors. HighTemperature Ordered Intermetallic Alloys VI, Materials Research Society, Pittsburgh, PA, 1995, p. 261. [11] Veho H. In: Liu CT, Cahn RW, Sautho G, editors. Ordered Intermetallics ± Physical Metallurgy and Mechanical Behaviour, NATO ASI Series E: Applied Sciences, vol. 213, 1992, p. 299. [12] Baker I. Mat Sci Eng A 1995;192/193:1. [13] Ball A, Smallman RE. Acta Metall 1968;16:233. [14] Bradley AJ, Taylor A. Proc R Soc London A 1937;159:56. [15] Fu CL, Ye Y-Y, Yoo MH, Ho KM. Phys Rev B 1993;48:6712. [16] Kogachi M, Tanahashi T, Shirai Y, Yamagutchi M. Scipta Mat 1996;34:243. [17] Zobel F. Thermische Fehlordnung der intermetallischen Phasen g0 -Ni3Al and b0 -NiAl, Ph.D. Thesis, Technische UniversitaÈt Berlin, 1994. [18] Pasianot R, Xie Z, Farkas D, Savino EJ. Modelling Simul Mat Sci Eng 1994;2:383. [19] Xie ZY, Vailhe C, Farkas D. Mat Sci Eng A 1993;170:59. [20] Parthasarathy TA, Rao SI, Dimiduk DM. Philos Mag A 1993;67:643. [21] Ludwig M, Gumbsch P. Modelling Simul Mat Sci Eng 1995;3:533. [22] Messerschmidt U, HaushaÈlter R, Bartsch M. Mat Sci Eng A 1997;822:234±6. [23] Mills MJ, Srinivasan R, Daw MS. Philos Mag. A 1998;77:801. [24] Hirth JP, Lothe J. Theory of Dislocations, 2nd edition, Wiley, New York, 1982. [25] Foiles SM, Baskes MI, Daw MS. Phys Rev B 1986;33:7983. [26] Ludwig M, Gumbsch P. Mat Res Soc Symp Proc 1995;364:389. [27] Ludwig M, Gumbsch P. Acta Mater 1998;46:3135. [28] Hagen M, Finnis MW. Philos Mag A 1998;77:447. [29] Hagen M. Struktur der Sigma3 (1 1 1)-Korngrenze in NiAl, Ph.D. Thesis, Stuttgart, 1996. [30] Vitek V. Crystal Lattice Defects 1974;5:1. [31] Schroll R, Gumbsch P. Phys Stat Sol A 1998;166:475. [32] Peierls R. Proc Phys Soc 1949;52:34. [33] Nabarro FRN. Proc Phys Soc 1947;59:256. [34] Bullough R, Perrin RC. In: Rosen®eld AR, Hahn GT, Bement Jr. AL, Jaee RI, editors. Dislocation Dynamics, McGraw-Hill, New York, 1968, p. 175. [35] Wasilewski RJ, Butler SR, Hanlon JE. Trans Metall Soc AIME 1967;239:1357. [36] Ternes K, Xie Z-Y, Farkas D. Mat Sci Eng A 1995;192/193:125. [37] Fleischer RL. Acta Metall 1961;9:996. [38] Kim JD. On the slip behaviour and surface ®lm eects in B2ordered Nickel aluminide single crystals, Ph.D. thesis, University of Michigan, 1991. [39] Pascoe RT, Newey CWA. Phys Stat Sol 1968;29:357. [40] Scattergood RO, Bacon DJ. Acta Metall 1982;30:1665.
Atomistic aspects of the deformation of NiAl Peter Gumbsch*, Ralph Schroll Max-Planck-Institut fuÈr Metallforschung, Seestraûe 92, 70174 Stuttgart, Germany Received 24 April 1998; revised 23 July 1998; accepted 1 September 1998
Abstract Properties of dislocations in B2-NiAl have been studied atomistically using an embedded atom potential. The response of dislocation cores to applied homogeneous shear stresses is investigated and the Peierls stresses of straight dislocations are determined. The results are in many details in excellent agreement with experimental observations. Speci®cally, the behaviour of the h1 1 1i dislocations, their slip planes, cross-slip behaviour and the limiting role of the screw dislocation can be explained. Similarly, the appearance of the {2 1 0} plane as a secondary slip plane for the h1 0 0i dislocations can be rationalised. Furthermore, the interaction of the dislocations with structural point defects is studied. Comparison with the ¯ow stress of o-stoichiometric NiAl from the literature shows that the individual point defects cannot be made responsible for the strong increase of the ¯ow stress, suggesting that more complex defect structures may play an important role. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Nickel aluminides based on NiAl; Plastic deformation mechanisms; Defects: Point defects; Defects: Dislocation geometry and arrangement; Atomistic simulations
1. Introduction The intermetallic phase NiAl which crystallises in the B2 (CsCl) structure is regarded as a promising candidate for high temperature applications because of its low density, high melting point and good oxidation resistance. There is substantial evidence indicating that details of the dislocation core structure play a central role in determining the deformation behaviour of NiAl [1±5]. At room temperature and below NiAl deforms almost exclusively by the motion of h1 0 0i dislocations [6±8]. The h1 0 0i Burgers vector gives only three independent slip systems which is thought to be the main reason for the limited ductility of polycrystalline NiAl. A dierent Burgers vector is observed for single crystals strained along the h1 0 0i axis, the so-called `hard orientation'. In this case, the resolved shear stress on the h1 0 0i dislocations vanishes and h1 1 1i dislocations become active [9,10]. At higher temperature, where ductility increases signi®cantly, h1 1 0i dislocations begin to contribute to the plastic deformation [1]. The advent of the h1 1 0i dislocations appears to coincide with the transition from brittle to ductile response [11].
* Corresponding author. Tel.: +49 711 2095129; fax: +49 711 2095120; e-mail: gumbsch@®nix.mpi-stuttgart.mpg.de
Deviations from the exact stoichiometric composition result in an increase in the experimentally measured yield stress and hardness [12]. It seems likely that the interaction between structural point defects and dislocations is responsible for this behaviour. The interaction may even be important for stoichiometric compositions at low temperatures, since thermal point defects freeze during the cooling process. This is a consequence of the high migration energy for vacancies in NiAl which is greater than their formation energy [13] and inevitably results in point defect concentrations considerably higher than expected at temperatures below half the melting temperature. Structural point defects in NiAl were ®rst investigated experimentally by Bradley and Taylor [14], who found that Al-rich alloys contain mostly Ni vacancies whereas Ni-antisite defects are the dominant defects in Ni-rich alloys. Ab initio calculations [15] as well as more recent experiments [16,17] con®rmed these results. Although several atomistic computer simulations have already been performed to determine the dislocation core structures and to explain the observed deformation behaviour of NiAl [1,2,18±20], further investigations seemed necessary since most of the earlier studies employed inadequate interatomic potentials [21] and since they partly gave unrealistic properties of the dislocations [4].
0966-9795/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0966 -9 795(98)00096 -X
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The {1 1 0} and {2 1 1} planes are generally regarded as glide planes for dislocations with Burgers vectors of the h1 1 1i type and these are also observed as the glide planes of hard oriented NiAl at low temperatures [10]. The h1 0 0i dislocations are known to glide on {1 0 0} and {1 1 0} planes. However, in recent in situ deformation experiments [22] in the transmission electron microscope (TEM) dislocation motion on {2 1 0} glide planes was clearly identi®ed. Furthermore it was suggested from these in situ observations that a combination of jog and Peierls mechanism controls the deformation of NiAl single crystals at room temperature. The goal of this paper is to summarise our atomistic understanding of the core structure of dislocations in B2-NiAl and to compare them with recent microscopic observations. Furthermore, the experimentally observed ¯ow stresses of stoichiometric and o-stoichiometric NiAl are discussed in connection with the atomistic results on the interaction of the dislocations with structural point defects. In this paper we focus on dislocations which are active at room temperature and below and therefore explicitly exclude h1 1 0i dislocations which only become important at higher temperatures. The properties of h1 1 0i dislocations and the atomistic aspects of their core structure are discussed elsewhere [2,4,23]. 2. Simulation technique The general strategy for the calculation of the atomic structure of a dislocation is to place it in the middle of a cylindrical block by displacing the atoms according to the anisotropic elastic displacement ®eld [24]. Periodic boundary conditions are applied parallel to the line direction thereby creating an in®nitely long, straight dislocation. The outermost atoms are ®xed at the positions determined by the elastic solution, while the interior of the model is allowed to move freely. These atoms are relaxed to their equilibrium positions using a conjugate gradient algorithm. Further details of the modelling technique and the boundary conditions are given in Ref. [4]. The atomic interaction is described by an EAM potential [25] which is ®tted to various properties of NiAl. In our investigation we have used the potential for NiAl developed by Ludwig and Gumbsch [21]. This potential does not only reproduce the structural point defects correctly but also gives particularly reasonable dislocation properties, which is shown in a comparison to dislocation simulations using other potentials in Ref. [4]. This potential has also been successfully applied to fracture simulations [26,27], point defect [28] and grain boundary studies [29]. Homogenous shear stresses were applied to the atomistic model to study the motion of the dislocations.
The stresses are actually imposed via homogenous shear strains by displacement boundary conditions and are increased stepwise. After every step the entire system is relaxed and examined for changes in the core structure. The applied stress at which the dislocation ®rst moves is identi®ed as the Peierls stress. The dislocation core structure is visualised using the dierential displacement method [30]. According to the dislocation type, either the screw or the edge component of the relative displacement of the neighbouring atoms is represented as a line between them. Another important quantity to characterise a dislocation is its core energy. It may contribute considerably to the line energy of the dislocation and decide between possible decomposition or dissociation reactions. Because of the large displacements in the dislocation core, linear elasticity is not appropriate and atomistic modelling must be used to determine this quantity. We have developed a new and unambiguous method to calculate the dislocation core energy in ordered alloys [5], which solves the stoichiometry problems encountered in such calculations. Point defect±dislocation interaction is modelled atomistically essentially with the same methods. A vacancy or an antisite defect is introduced into the simulation cell by simply removing or exchanging an atom. Because of the periodic boundary conditions, this procedure actually generates an in®nitely long line of point defects. The thickness of the model in the z-direction was chosen, so that the repeat distance of the point defects was three times the lattice repeat length of the crystal. Since the stress ®eld of a point defect (1/r3) is shorter in range and drops o much faster than the stress ®eld of a dislocation (1/r), the interactions between the point defects can be neglected even at this relatively close spacing. Further details of the quantities one can determine with such atomistic calculations of point defect±dislocation interaction are discussed elsewhere [31]. Here we only characterise the strength of the interaction by the change in the Peierls stress due to the presence of the point defects. 3. h1 0 0i Dislocations Atomistically, h1 0 0i dislocations clearly prefer to move on the most densely packed {0 1 1} plane [4]. Screw dislocations spread out on this plane and the core of edge dislocations is also widely spread on this plane as shown in Fig. 1. With a value of less than 0.1 GPa, the Peierls stress for both these dislocations is very low. Dislocations with mixed character move at only slightly higher Peierls stresses (P =0.13±0.17 GPa). The mixed dislocations also move with similar ease on {0 0 1} planes. This indicates that the dislocation mobility on these planes is not too dierent although the edge
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449
Fig. 1. The core structure of the h1 0 0i edge (upper part) and screw (lower part) dislocations. Large circles represent Al atoms; small circles represent Ni atoms. The dierential displacements exhibit a core spreading on the {0 1 1} plane in both cases.
dislocations have considerably dierent Peierls stresses (see Table 1). The calculated Peierls stress for a h1 0 0i edge dislocation on a {2 1 0} glide plane is 0.04 GPa and therefore even somewhat lower than for the same dislocation on a {1 1 0} glide plane. The Peierls stress of the mixed dislocation (0.19 GPa) is again comparable with the Peierls stress of the mixed dislocations on {1 1 0} and {1 0 0} glide planes. It was not possible to force the h1 0 0i screw dislocation to glide either on a {2 1 0} or a
{0 0 1} plane since it cross-slips at 0.08 GPa onto the {1 1 0} plane. This is caused by the particularly wide core spreading of the screw dislocation on the {1 1 0} plane which can be seen in Fig. 1. Finally, it seems worth mentioning that the screw dislocation is elastically unstable on {2 1 0} planes as it is on {1 0 0} and {1 1 0} planes. The low Peierls barrier of the edge dislocation on the {2 1 0} plane is surprising since one usually associates low Peierls stresses and high dislocation mobility only
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with close packed widely spaced planes. To better understand the dependence of the Peierls stress of the edge dislocation on the orientation of the glide plane, the Peierls stress of the h1 0 0i edge dislocation was calculated for other high-indexed glide planes as well. The result is summarised in Fig. 2. It can be seen that the Peierls stress exhibits two local minima for the {1 1 0} and {2 1 0} planes. Moreover, the Peierls stress for the {1 0 0} plane is even signi®cantly higher than for all of the high-indexed glide planes. This surprising behaviour can be rationalised in the context of the Peierls±Nabarro model [32,33]. In this model the Peierls stress is proportional to exp (ÿw/b) where w is the dislocation core width and b the magnitude of the Burgers vector. The core width quanti®es the spreading of the dislocation on the glide plane. A wider dislocation core is therefore associated with a lower Peierls stress. Fig. 3 shows the Burgers vector distribution of the h1 0 0i edge dislocation on the h1 0 0i, h1 1 0i and h2 1 0i glide plane (a detailed description of the Burgers vector distribution is given in Ref. [4]). The core width of the edge dislocation on the {1 0 0} glide plane is roughly half the width as on the {1 1 0} and {2 1 0} planes. The considerably higher Peierls stress of the h1 0 0i edge dislocation on the {1 0 0} glide plane can be explained with the strong exponential relationship between the Peierls stress and the core width. The in¯uence of point defects on the dislocation mobility is studied for both Ni vacancies and Ni antisite defects. As a ®rst measure of the interaction between the dislocation and the point defects, the pinning stress (i.e. the dierence in the Peierls stress with and without the point defects) is calculated. The results are summarised in Table 1 and show, that the pinning stresses reach values up to ten times the Peierls stress. A strong interaction is especially obvious in the important h1 0 0i{0 1 1} glide system.
Fig. 2. Peierls stresses for h1 0 0i edge dislocations on dierent glide planes.
Furthermore, it can be seen from Table 1 that Ni vacancies are more eective obstacles than Ni antisite defects. The reason for this behaviour lies in the dierent mechanism in which Ni vacancies and Ni antisite defects interact with h1 0 0i dislocations. A closer examination of the binding energy pattern (see Ref. [31]) of the point defects in the vicinity of the dislocation core reveals that Ni antisite defects interact mainly parelastically (due to the dierence in volume of the smaller Ni defect atom and the regular Al atom) with the hydrostatic component of the stress ®eld of the dislocation. In contrast, Ni vacancies interact dielastically (as an elastically softer region in the otherwise perfect crystal lattice) with the stress ®eld of the dislocation. The dielastic interaction of the Ni vacancies turns out to be stronger than the parelastic interaction of the Ni antisite defects. In both cases, however, the binding energies of the defects are smaller than predicted by anisotropic linear elastic theory [24,34]. Particularly, the point defects located close to the dislocation core interact weaker than expected. 4. h1 1 1i Dislocations For the tension or compression of NiAl single crystals along the h1 0 0i direction (hard orientation) the resolved shear stress on h1 0 0i dislocations vanishes and h1 1 1i dislocations become active at low temperatures. In the B2 lattice the 1/2h1 1 1i Burgers vector, the dominant Burgers vector of the underlying b.c.c. lattice, is not a complete lattice repeat vector but produces an antiphase boundary (APB). Nevertheless most of the h1 1 1i dislocations split into partial dislocations along
Fig. 3. Burgers vector distribution of the h1 0 0i edge dislocation for three dierent glide planes. The dislocation core width can be de®ned as the width of the Burgers vector distribution at half their maximum height.
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451
Table 1 Peierls stress D for h1 0 0i dislocations and the pinning stresses from the interaction with Ni antisite defects and Ni vacancies. The character of the dislocation, edge, screw and mixed (E, S, M) is indicated together with the the line direction Glide plane
Burgers vector
{0 1 0}
h1 0 0i
{0 1 1}
h1 0 0i
{0 2 1}
h1 0 0i
a b
Line direction
Character
Peierls stress D (GPa)
Ni antisite (GPa)
Ni vacancy (GPa)
h0 0 1i h0 0 1i h1 0 0i h1 0 1i
Ea Eb S M
1.08 0.88 >0.08 0.15
0.47 0.72 ÿ 0.6
0.92 1.62 ÿ 1.35
h0 1 1i h1 0 0i h2 1 1i h1 1 1i
E S M M
0.06 0.07 0.13 0.17
0.20 0.18 ÿ 0.18
0.49 0.73 ÿ 0.33
h0 1 2) h1 0 0i h1 2 1)
E S M
0.04 >0.08 0.19
ÿ ÿ ÿ
ÿ ÿ ÿ
Ni core. Al core.
the glide plane despite the high APB energy of NiAl. For example, the h1 1 1i edge dislocation on the {2 1 1} glide plane splits into two 1/2 h1 1 1i superpartials. A similar splitting is observed for the h1 1 1i mixed dislocation on the {0 1 1} glide plane. However, the dissociation of the pure h1 1 1i edge dislocation on the {1 1 0} glide plane is more complex. This dislocation splits into two 1/4h2 1 1i Burgers vectors with a 1/2h0 1 1i Burgers vector in between (see upper part of Fig. 4). Contrary to the edge and mixed dislocations, the h1 1 1i screw dislocation does not split into partial dislocations and appears rather compact but highly non-planar. The three-fold rotation symmetry leads to a three-fold spreading of the dislocation core (see lower part of Fig. 4). As a consequence of the dierent core structure, the mobility of the h1 1 1i dislocations also varies considerably. In accord with the Peierls±Nabarro model, the glide dissociation of the edge and mixed dislocations increases their mobility. The Peierls stress of both dis plane are low and lie location types on the h0 1 1i between 0.06 and 0.23 GPa (see Table 2). However, the screw dislocation is hard to move due to its non-planar core structure. When increasing the shear stress, the dislocation core of the screw dislocation gradually spreads on the glide plane and reduces the non-planar components of the displacement ®eld. plane a pronounced twinning/ For glide on the {2 1 1} antitwinning asymmetry is obtained. The Peierls stress was found to be 1.7 GPa in antitwinning sense and this plane with 2.0 is slightly lower than that for the {0 1 1} GPa. However, attempts to move the screw dislocation plane in twinning direction failed. It on the {2 1 1} slip plane. The crialways cross-slipped onto a {0 1 1} plane then tical resolved shear stress on the {0 1 1} exceeded the Peierls stress for this slip plane.
The importance of point defects for the mobility of h1 1 1i dislocations is less clear. It is shown in Table 2 that the pinning stress for the edge and mixed h1 1 1i dislocations exceeds the Peierls stress about 60 times. However, the critical stress to move the h1 1 1i screw dislocation remains almost constant. The reason for this dierence can be understood from the dierent core structures. Since the edge and mixed dislocations are (glide) dissociated, the point defects can strongly interact with the resulting APB (Suzuki eect). In this case, the binding energies reach values up to ÿ4.4 eV which is comparable with the chemical potentials of the individual atoms. On the other hand, the interaction between the point defects and the compact screw dislocation is very small, because of the absence of a hydrostatic stress ®eld. As a consequence, the mobility of the edge and screw dislocations in the presence of point defects may become comparable but the absolute value of the yield stress should not be changed much. 5. General discussion Atomistic simulations of h1 0 0i Burgers vectors in NiAl were so far mainly concentrating on dislocation motion on {1 0 0} and {1 1 0} glide planes. Such `smooth', close packed and widely separated planes usually oer considerably lower resistance to dislocation motion than do closely spaced `rough' planes. In B2 NiAl the {1 1 0} plane has the densest atomic packing followed by the {1 0 0} plane and both have been observed as active glide planes [7,8]. However, Messerschmidt et al. [22] have clearly identi®ed the motion of h1 0 0i dislocations not only on the two most densely packed planes but also on a {2 1 0} glide plane in recent in situ TEM experiments. Furthermore and similar to
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edge dislocation (upper part) is dissociated in partial dislocations. In contrast, the h1 1 1i screw disFig. 4. The core structure of the h1 1 1i {0 1 1} location (lower part) does not show such a glide dissociation, but is highly non-planar and extended on {1 1 0} planes.
earlier observations [7] they found frequent cross slip between {1 0 0}, {1 1 0} and {2 1 0} planes with almost equal dislocation mobility [22]. In excellent agreement with these experimental ®ndings, our atomistic simulations give the lowest Peierls stresses for the motion of non-screw dislocations on {1 1 0} and {2 1 0} planes (see Fig. 2). At ®rst sight, the high calculated Peierls stress for the pure edge dislocation on the {1 0 0} plane seems to be in contrast to the experimentally observed high
dislocation activity on this plane. However, the low Peierls stresses for the mixed dislocations (0.10±0.15 GPa) suggests that all other characters including the screw dislocation are glissile on all three glide planes. The critical resolved shear stresses for h1 0 0i{0 1 1} slip in stoichiometric NiAl single crystals at low temperatures [35] (see Fig. 5) agrees extremely well with the highest Peierls stresses (0.17 GPa) calculated here for this slip system. Although previous atomistic simulations
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453
Table 2 Peierls stress D for h1 1 1i dislocations and the pinning stresses from the interaction with Ni antisite defects and Ni vacancies. The character of the dislocation, edge, screw and mixed (E, S, M) is indicated together with the the line direction Glide plane
Burgers vector
Line direction
Character
Peierls stress D (GPa)
Ni antisite (GPa)
{0 1 1}
h1 1 1i
h2 1 1i h1 1 1i h1 0 0i h0 1 1i
E S M M
0.07 2.0 0.23 0.06
3.93 0.2 ÿ 0.26
{2 1 1}
h1 1 1i
h0 1 1i h1 1 1i
E S
1.4 (3.0) a 1.7 (>3.5) a
0.6 0.2
a
Ni vacancy (GPa) 3.93 0.0 ÿ 1.94 >3.0 0.8
Antitwinning (twinning) direction.
[20,36] using dierent atomic interaction models had given similar core structures for the h1 0 0i dislocations, their Peierls stresses were almost an order of magnitude too high. For the low temperature deformation of stoichiometric NiAl in the `hard' h1 0 0i orientation, agreement with experimental observations is obtained in almost all details. Not only does the Peierls stress for the h1 1 1i screw dislocations exactly reach the critical resolved shear stresses of 2.0 GPa measured below 50 K [10], TEM observations reveal unequivocally long h1 1 1i screw segments after deformation at these low temperatures [10]. This is a strong indication for a large dierence in the Peierls stresses for the dierent dislocation characters which are apparent in Table 2. Furthermore, these TEM observations show increasingly more frequent cross-slip events from the {1 1 0} onto the {2 1 1} plane with decreasing temperature [10]. With reference to Table 2, this can explained by the somewhat lower Peierls stresses for the screw dislocations on the {2 1 1} plane (in the antitwinning sense) than on {1 1 0}. Previous attempts to understand the role of the h1 1 1i dislocations atomistically have not been successful in several respects. Xie et al. [19] only studied the dissociation of the h1 1 1i screw dislocation and found metastable dissociated core con®gurations. They did not study other dislocation characters or dislocation mobility. Parthasarathy et al. [20] addressed these questions and found dissociated edge and screw dislocations with nearly equal and far too low Peierls stresses. These obviously erroneous results are a consequence of the low APB energy of the potential used in their studies [4,21]. At low temperature most strongly ordered B2 compounds, including NiAl, exhibit minima in both yield strength and hardness at the stoichiometric composition [12]. This can be attributed to the strengthening obtained from the structural point defects of o-stoichiometric alloys. In NiAl the increase of hardness with the deviation from stoichiometry is more pronounced on the Al-rich side than on the Ni-rich side [12]. The structural point defects on the Al-rich side are Ni vacancies while Ni antisite defects dominate on the Ni-rich
side. Our atomistic calculations summarised in Table 1 show that h1 0 0i dislocations actually interact more strongly with the Ni vacancies than with the Ni antisite defects. Qualitatively they therefore reproduce the experimentally observed dierence in strengthening on the two sides of the stoichiometric composition. However, a quantitative analysis based on Fleischer's theory of solid solution hardening [37] shows that the strengthening obtained from the individual point defects is far too weak as compared to experiments [31]. An upper estimate for the strengthening from the interaction of the h1 0 0i dislocations with the structural point defects is =0.12 GPa [31]. In Fig. 5 the Peierls stress and the strengthening eect from the solid solution hardening are shown together with the experimentally measured critical resolved shear stresses for non-stoichiometric NiAl. The experimental data are taken from the literature [35,38,39]. It is obvious that neither the absolute value of the solid solution hardening nor the tremendous decrease of the critical resolved shear stress with increasing temperature can be explained on the basis of the atomistic
Fig. 5. Experimentally measured critical resolved shear stresses compared with the solid solution hardening based on the results of the atomistic simulation.
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simulations. One may now speculate which other mechanism could be responsible for the observed mechanical properties of non-stoichiometric NiAl beside the solid solution hardening by structural point defects. Possibly, aggregated defects can play an important role. While the formation of vacancy clusters in NiAl has been observed by Ball and Smallman [13], very little is known about the hardening eect of these defects [40]. 6. Summary and conclusions The goal of our investigation was to achieve a fundamental understanding of the deformation of NiAl single crystals at low temperatures. For this purpose we performed atomistic computer simulations with a potential speci®cally adjusted to the properties of NiAl. For h1 0 0i and h1 1 1i dislocations, edge, screw and mixed characters were analysed and their Peierls stresses were calculated. Furthermore, the interaction of these dislocations with the structural point defects in o-stoichiometric NiAl were studied. Excellent agreement between experiment and our atomistic simulations is obtained with respect to almost all details of the low temperature deformation of stoichiometric NiAl. Our simulations allow for the ®rst time to consistently explain the observed dislocation properties from the atomistic nature of the dislocation cores. The solid solution hardening due to individual structural point defects as estimated from our atomistic calculations is not sucient to explain the tremendous increase in critical resolved shear stress with decreasing temperature observed in o-stoichiometric NiAl. It therefore appears to be important to study the interaction of dislocations with more complex defect clusters. To quantitatively model plastic deformation at ®nite temperatures or the brittle-to-ductile transition requires a more quantitative description of the dislocation mobility. This can only be obtained from a better understanding of the role of kinks and jogs in the dislocation line. Acknowledgements The ®nancial support from the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. References [1] Mills MJ, Miracle DB. Acta Metall Mat 1993;41:85.
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