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Two-step temperature dependence of the yield stress in crystals
Journal of Alloys and Compounds 378 (2004) 61–65
Two-step temperature dependence of the yield stress in crystals Shin Takeuchi∗ Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan Received 1 September 2003; accepted 14 October 2003
Abstract The temperature dependence of the critical resolved shear stress (CRSS) of single crystals often exhibits a two-step change. On the basis of the Peierls mechanism of the dislocation glide, a two-step change can theoretically be expected for (1) a depressed-top or a flat-top Peierls potential, (2) the Peierls mechanism of a narrowly dissociated dislocation and (3) two stable core configurations with different Peierls stresses and different kink energies. In this paper, after discussing the CRSS versus T relations for bcc metals and tetrahedrally coordinated crystals, detailed discussion has been made on the two-step CRSS versus T curves reported for (1 0 0)[0 1 0] and (1 0 0)[0 0 1] slips in tetragonal -Sn reported by Kirichenko [Phys. Met. Metallogr. 63 (1) (1987) 144] and on that reported for single crystals of NiAl with the hard orientation. © 2004 Elsevier B.V. All rights reserved. Keywords: Intermetallics; Metals; Semiconductors; Dislocations; Strain
1. Introduction
2. Various examples
Deformation of crystals is mostly brought by glide of dislocations overcoming barriers assisted, at least partially, by thermal activation. As a result, the yield stress of crystals generally decreases monotonically with increasing temperature. However, there exist a number of examples violating this general rule. A prominent example is the so-called “strength anomaly” in intermetallic compounds, where a pronounced reverse temperature dependence occurs as a result of the thermally activated transition of the core of glide dislocations from a glissile configuration to a sessile configuration (see, a review [1]). Besides the “strength anomaly”, there exist a number of cases where the temperature dependence of the yield stress is not monotonic. In the present paper, we restrict ourselves to the deformation process controlled by the Peierls mechanism in pure single crystals, and discuss the mechanisms which yield two-step temperature dependence of the critical resolved shear stress (CRSS), τ c .
2.1. bcc metals
∗
Tel.: +81-4-7124-1501x4305. E-mail address: [email protected] (S. Takeuchi).
0925-8388/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.10.069
The CRSS versus temperature relation in high-purity bcc transition metals of Fe, Nb, Mo and W shows an inflection point in an intermediate temperature region (see, a review [2]). This inflection behavior, often called a hump in the τ c –T curve, has earlier been interpreted by the present author in terms of a specific shape of the Peierls potential, i.e, a depressed-top Peierls potential; the present author showed a possibility of the presence of a metastable state in the Peierls potential between the two adjacent stable positions [3]. Later, Suzuki et al. showed by use of a trapezoidal kink-pair model that a flat-top Peierls potential can reasonably explain the τ c –T relation in bcc metals [4]. 2.2. Tetrahedrally coordinated crystals 2.2.1. Transition from undissociated dislocation glide to glide-set dislocation glide The dislocation in tetrahedrally coordinated crystals can have two different cores, the dissociated glide-set which is believed to be the stable state and an undissociated core of a
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metastable state. It is known that the Peierls stress of a dislocation is an exponentially decreasing function of the h/b value, where h is the spacing of the glide plane and b the strength of the Burgers vector [5]. This fact generally leads to a drastic decrease of the Peierls stress with a dissociation of dislocation. However, in the diamond, zincblende or wurtzite structures, the situation is reversed since the glide plane spacing of the undissociated dislocation is three times as large as that of the dissociated glide-set dislocation. On the other hand, kink-pair energy which determines the critical temperature of the temperature dependent CRSS is larger for the undissociated dislocation than for the dissociated dislocation. Thus, the τ c –T curve for the undissociated dislocation glide and that for the dissociated glide should intersect with each other; as a result, the observed τ c –T curve should have a break as illustrated in Fig. 1a. In recent years, the compression experiments under high pressures for III–V compounds revealed that CRSS versus temperature relations show a clear two-step change, which has been attributed to the above-mentioned crossover from the undissociated dislocation glide to the glide-set glide with some evidence by microscopic observations [6]. We should note that for such a transition to occur, two conditions must be satisfied: (1) the undissociated core should be in a metastable state and (2) the activation enthalpy for the transition form the metastable undissociated core to the stable dissociated core should be considerably large so that the undissociated core does not undergo a transition to the dissociated core during gliding. The rather high transition enthalpy may be attributed to the necessary climbing motion by one atomic distance for the transition. 2.2.2. Transition from partial dislocation glide to dissociated dislocation glide In tetrahedrally coordinated crystals, mobility of partial dislocation is different from one partial to the other due to different Peierls potential for different partials; the 90◦ partial is known to have much higher mobility than the 30◦ partial [7]. Hence, if the stacking fault energy is low enough, only leading mobile partials can glide leaving the stacking fault behind them. τ c –T relation for mobile partial glide and
Fig. 1. (a) τ c vs. T relation for undissociated dislocation glide and that for glide-set glide for tetrahedrally coordinated crystals. (b) τ c vs. T relation for partial dislocation glide and that for dissociated dislocation glide.
that for the dissociated paired-glide is schematically shown in Fig. 1b. Such a transition of the deformation mode has been discussed by Pirouz et al. [8]. Production of a high density stacking fault by low temperature deformation has been observed in Si [9] and GaAs [10], and partial dislocation glide has been observed by in-situ transmission electron microscopy for II–VI crystals [11]. Single crystals of SiC having extremely low stacking fault energy are believed to undergo a transition of the type given in Fig. 1b [8]. For the macroscopic flow only by partial dislocation glide, however, there is a problem about the dislocation multiplication, because the Frank-Read mechanism cannot be operative for partials.
3. Narrowly dissociated dislocation In some intermetallic compounds [1] and in pure metals of Sn [12] and Ti [13], a plateau region appears in the τ c versus T relation. The present author pointed out that for a narrowly dissociated dislocation a plateau region can appear in the τ c versus T relation as a result of the transition in the Peierls mechanism from the independent, uncorrelated kink-pair formation regime to the correlated kink-pair formation regime [14]. Fig. 2 shows an example of a potential profile for a partial dislocation to migrate, independently from the other, around the equilibrium position, where two partials are screws with the Burgers vector b/2, the equilibrium separation is 5b and the Peierls potential has the periodicity of b/2 with the Peierls stress of 5 × 10−3 G (G is the shear modules) which is the typical value for bcc metals. Since the partial dislocation is trapped in an envelope potential, the partial at the lowest potential site cannot undergo a thermally activated transition to the neighboring site below a critical stress,
Fig. 2. An example of the potential profile for an independent partial dislocation migration. G is the shear modulus, b the strength of the total Burgers vector and τ p the Peierls stress.
S. Takeuchi / Journal of Alloys and Compounds 378 (2004) 61–65
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Fig. 3. Activation enthalpy vs. stress relation for correlated kink-pair formation process and that for uncorrelated kink-pair formation process (upper figure) and the expected τ c vs. T relation (lower figure).
τ 0 . The critical stress depends on the relative positions of the envelope potential minimum and the Peierls potential minima (Fig. 2 shows the case where two minima coincide with each other) and is given by τ0
Gβ b1 b2 d , 4p we bi we
(1)
where G is the shear modulus, β (1) the function of the angle between the dislocation direction and those of partials, bi (i = 1, 2) the strength of the Burgers vector of partials, we the equilibrium separation between partials and d the period of the Peierls potential. Due to the presence of the critical stress, τ c versus T curve controlled by the Peierls mechanism of a narrowly dissociated dislocation can have a plateau region between the low temperature region governed by the independent kink-pair formation regime and the high temperature region by the correlated kink-pair formation regime. Upper figure of Fig. 3 shows schematically the stress dependence of the activation enthalpy of the kink-pair formation for the correlated glide and the uncorrelated glide, and the lower figure illustrates τ c versus T curve. Some examples are presented below.
Fig. 4. CRSS vs. T relation for (1 0 0)[0 1 0] slip and that for (1 0 0)[0 0 1] slip in -Sn reproduced from [12]. Inset drawings show the dissociation of (1 00)[0 1 0] dislocation and the Peierls potential for (1 0 0)[0 0 1] dislocation.
Although no direct evidence has been obtained, there is a high possibility that dislocation with the Burgers vector a [1 0 0] is dissociated into partial dislocations with the Burgers vector of a/2 [1 0 0]. Fig. 5a presents the atomic configuration of the a/2 [1 0 0] stacking fault viewed from the [0 0 1] axis. This dissociation can naturally explains why the (1 0 0)[0 1 0] slip is more active than the (1 0 0)[0 0 1] slip in spite of much larger strength of the Burgers vector of the perfect [1 0 0] dislocation (5.8 Å) than that of [0 0 1] dislocation (3.2 Å).
3.1. CRSS in β-tin Fig. 4 reproduces the τ c versus T relations for (1 0 0)[0 1 0] slip and (1 0 0)[0 0 1] slip for the tetragonal -Sn reported by Kirichenko [12]. We see a two-step dependence for both slips. For (1 0 0)[0 1 0] slip, Kirichenko et al. analyzed the results on the basis of the theories of the Peierls mechanism, but no explanation is given for the appearance of the plateau between 30 and 70 K [15]. We interpret the appearance of the plateau in terms of the Peierls mechanism of narrowly dissociated dislocation. The followings are the bases for this interpretation.
Fig. 5. Crystal structure of -Sn viewed from [0 0 1] (a) and that from [0 1 0] (b). Dashed line indicates the stacking fault plane. Circles of different sizes are on different planes.
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The analyzed pre-exponential factor of the Arrhenius strain-rate equation above 70 K (∼100 s−1 ) is five orders of magnitude smaller than that below 40 K (∼105 s−1 ) [14]. When the pre-exponential factor is converted to the exponent H/kB T, we find that the value of exponent H/kB T is around 21 below 40 K and as small as 11 above 80 K. It is known that the value of the exponent should be normally between 16 and 35 [2], and the small value of 11 above 80 K can only be interpreted by the correlated kink-pair formation process [14]. Using τ0 = 4 MPa, G = 26 GPa [16], b1 = b2 = d = 0.29 nm and assuming β = 1, we obtain the dislocation width we 6.6 nm according to Eq. (1). Theoretically expected CRSS versus T relation for the (1 0 0)[0 1 0] slip is drawn by a solid line in Fig. 4. In Fig. 4, CRSS for (1 0 0)[0 0 1] slip also shows a two-step dependence. The strength of the Burgers vector of [0 0 1] slip is as small as 0.32 nm, and there does not seem to be a stable stacking fault parallel to (1 0 0) plane (see Fig. 5). We interpret the hump in the τ c versus T relation for the (1 0 0)[0 0 1] slip along the same line as the hump in bcc metals. Fig. 6a shows the atomic structure viewed from the [0 0 1] direction, where the atomic position of atoms in each atomic row parallel to the [0 0 1] direction is given in the circle in units of c. When a screw dislocation strain (Fig. 6c) is introduced at positions L and H, the atomic positions of the central four atomic rows change to those given in Fig. 6d. From this result, we find that there exist alternately the low energy positions L’s and high energy positions H’s for a [0 0 1] screw dislocation, in a similar manner as in bcc metals [2]. (1 0 0) glide will be realized by zigzag motion from one L position to another L position of screw dislocations. Also in analogy to bcc metals, there may be a metastable position
Fig. 6. Illustration showing the changes of the atomic configuration with introduction of a screw dislocation with b = c in -Sn viewed from the [0 0 1] direction. Numeral in each circle indicates the atomic position in the c direction.
in between the adjacent L positions, which may be the origin of the appearance of the hump in the τ c versus T relation in the (1 0 0)[0 0 1] slip. The CRSS versus T relation calculated for the Peierls potential [2] given in the inset of Fig. 4 is drawn by a solid line for the (1 0 0)[0 0 1] slip in Fig. 4. 3.2. {1 1 2}111 slip in NiAl There are two groups of B2 type intermetallic compounds, i.e, the 1 1 1 slip group and the 1 0 0 slip group [1]. NiAl belongs to the latter group. For single crystals with 1 0 0 stress axis (named the hard orientation) where any 1 0 0 slip system has zero resolved shear stress, the temperature dependence of the yield stress exhibits a clear two-step change, and it has been established that the yielding in the low temperature range is due to {1 1 2}1 1 1 slip while the high temperature region by {1 1 0}1 1 0 slip [17]. Fig. 7 reproduces temperature dependence of the CRSS in NiAl [18,19]. We see a clear plateau between 300 and 600 K in the {1 1 2}1 1 1 slip region. The mechanism of the transition from the 1 1 1 slip to 1 1 0 slip has been discussed in detail recently [19], but no interpretation has been given for the appearance of the athermal plateau region. It has generally been assumed that in the 1 0 0 slip group of B2 compounds 1 1 1 superlattice dislocation is not dissociated into 1/2 1 1 1 superpartials to account for the relative inactivity of the 1 1 1 slip as compared to the 1 0 0 slip. However, we postulate that the 1 1 1 dislocation in NiAl is narrowly dissociated; the width of the dislocation can be estimated from the plateau stress as we 0.8 nm. Some indications for the dissociation are (1) a high-resolution electron microscopy of the dislocation core in NiAl suggested the dissociation of 1 1 1 dislocation [20], and (2) Peierls stress for 1 1 1 glide is comparable to that of 1 0 0 glide; if the 1 1 1 dislocation is undissociated, the Peierls stress for 1 1 1 glide should be significantly higher than that for 1 0 0 glide taking account of the fact that the strength of
Fig. 7. CRSS vs. temperature for the hard orientation of NiAl reproduced from [18,19].
S. Takeuchi / Journal of Alloys and Compounds 378 (2004) 61–65
the Burgers vector is 1.73 times larger for 1 1 1 glide than for 1 0 0 glide. 4. Concluding remarks We have shown in this paper that the CRSS is not necessarily a monotonically decreasing function of the temperature due to a variety of intrinsic mechanisms without any effect of solute atoms. However, the proposed mechanisms are not necessarily well established by supporting evidences. More direct evidence, e.g., by high resolution electron microscopy, is needed to substantiate the mechanisms. References [1] S. Takeuchi, in: D.G. Brandon, R. Chaim, A. Rosen (Eds.), Strength of Metals and Alloys, Freund Publishing House, London, 1991, pp. 69–82. [2] T. Suzuki, S. Takeuchi, in: R.A. Vardanian (Ed.), Crystal Lattice Defects and Dislocation Dynamics, Nova Science Publication, Inc., New York, 2001, Chapter 1, pp. 1–70. [3] S. Takeuchi, in: J.K. Lee (Ed.), Interatomic Potential and Crystalline Defects, The Metallurgical Society of AIME, Warrendale, PA, 1981, pp. 201–221.
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[4] T. Suzuki, H. Koizumi, H.O.K. Kirchner, Acta Mater. 43 (1995) 2177. [5] T. Suzuki, S. Takeuchi, Rev. Phys. Appl. 23 (1988) 685. [6] T. Suzuki, T. Yasutomi, T. Tokuoka, I. Yonenaga, Phys. Stat. Sol. (a) 171 (1999) 47. [7] H. Alexander, in: F.R.N. Nabarro (Ed.), Dislocations in Solids, vol. 7, Academic Press, New York, 1986, p. 114. [8] P. Pirouz, A.V. Samant, M.H. Hong, A. Moulin, L.P. Kubin, J. Mater. Sci. 14 (1999) 2783. [9] J. Castaing, P. Veyssière, L.P. Kubin, J. Rabier, Philos. Mag. A 44 (1981) 1407. [10] P. Boivin, J. Rabier, H. Garem, Philos. Mag. A 61 (1990) 647. [11] K. Suzuki, M. Ichihara, K. Nakagawa, K. Maeda, S. Takeuchi, Philos. Mag. A 43 (1981) 499. [12] G.I. Kirichenko, Phys. Met. Metallogr. 63 (1) (1987) 144. [13] M. Biget, G. Saada, Philos. Mag. A 59 (1989) 747. [14] S. Takeuchi, Philos. Mag. A 71 (1995) 1255. [15] G.I. Kirichenko, V.D. Natsik, V.P. Soldatov, Phys. Met. Metallogr. 63 (2) (1987) 162. [16] G. Simmons, H. Whang, Single Crystal Elastic Constants and Calculated Aggregate Properties, MIT Press, Cambridge, MA, 1971. [17] D.B. Miracle, Acta Metal. Mater. 41 (1993) 646. [18] R.T. Pascoe, C.W.A. Newey, Metal Sci. J. 2 (1968) 138. [19] R. Srinivasan, J. Brown, M.S. Daw, R.D. Noebe, M.J. Mills, Philos. Mag. A 80 (2000) 2841. [20] D. Stöcke, W. Sigle, A. Seeger, Mater. Res. Soc. Symp. Proc. 552 (1999) KK9.8.1.
Two-step temperature dependence of the yield stress in crystals Shin Takeuchi∗ Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan Received 1 September 2003; accepted 14 October 2003
Abstract The temperature dependence of the critical resolved shear stress (CRSS) of single crystals often exhibits a two-step change. On the basis of the Peierls mechanism of the dislocation glide, a two-step change can theoretically be expected for (1) a depressed-top or a flat-top Peierls potential, (2) the Peierls mechanism of a narrowly dissociated dislocation and (3) two stable core configurations with different Peierls stresses and different kink energies. In this paper, after discussing the CRSS versus T relations for bcc metals and tetrahedrally coordinated crystals, detailed discussion has been made on the two-step CRSS versus T curves reported for (1 0 0)[0 1 0] and (1 0 0)[0 0 1] slips in tetragonal -Sn reported by Kirichenko [Phys. Met. Metallogr. 63 (1) (1987) 144] and on that reported for single crystals of NiAl with the hard orientation. © 2004 Elsevier B.V. All rights reserved. Keywords: Intermetallics; Metals; Semiconductors; Dislocations; Strain
1. Introduction
2. Various examples
Deformation of crystals is mostly brought by glide of dislocations overcoming barriers assisted, at least partially, by thermal activation. As a result, the yield stress of crystals generally decreases monotonically with increasing temperature. However, there exist a number of examples violating this general rule. A prominent example is the so-called “strength anomaly” in intermetallic compounds, where a pronounced reverse temperature dependence occurs as a result of the thermally activated transition of the core of glide dislocations from a glissile configuration to a sessile configuration (see, a review [1]). Besides the “strength anomaly”, there exist a number of cases where the temperature dependence of the yield stress is not monotonic. In the present paper, we restrict ourselves to the deformation process controlled by the Peierls mechanism in pure single crystals, and discuss the mechanisms which yield two-step temperature dependence of the critical resolved shear stress (CRSS), τ c .
2.1. bcc metals
∗
Tel.: +81-4-7124-1501x4305. E-mail address: [email protected] (S. Takeuchi).
0925-8388/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jallcom.2003.10.069
The CRSS versus temperature relation in high-purity bcc transition metals of Fe, Nb, Mo and W shows an inflection point in an intermediate temperature region (see, a review [2]). This inflection behavior, often called a hump in the τ c –T curve, has earlier been interpreted by the present author in terms of a specific shape of the Peierls potential, i.e, a depressed-top Peierls potential; the present author showed a possibility of the presence of a metastable state in the Peierls potential between the two adjacent stable positions [3]. Later, Suzuki et al. showed by use of a trapezoidal kink-pair model that a flat-top Peierls potential can reasonably explain the τ c –T relation in bcc metals [4]. 2.2. Tetrahedrally coordinated crystals 2.2.1. Transition from undissociated dislocation glide to glide-set dislocation glide The dislocation in tetrahedrally coordinated crystals can have two different cores, the dissociated glide-set which is believed to be the stable state and an undissociated core of a
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metastable state. It is known that the Peierls stress of a dislocation is an exponentially decreasing function of the h/b value, where h is the spacing of the glide plane and b the strength of the Burgers vector [5]. This fact generally leads to a drastic decrease of the Peierls stress with a dissociation of dislocation. However, in the diamond, zincblende or wurtzite structures, the situation is reversed since the glide plane spacing of the undissociated dislocation is three times as large as that of the dissociated glide-set dislocation. On the other hand, kink-pair energy which determines the critical temperature of the temperature dependent CRSS is larger for the undissociated dislocation than for the dissociated dislocation. Thus, the τ c –T curve for the undissociated dislocation glide and that for the dissociated glide should intersect with each other; as a result, the observed τ c –T curve should have a break as illustrated in Fig. 1a. In recent years, the compression experiments under high pressures for III–V compounds revealed that CRSS versus temperature relations show a clear two-step change, which has been attributed to the above-mentioned crossover from the undissociated dislocation glide to the glide-set glide with some evidence by microscopic observations [6]. We should note that for such a transition to occur, two conditions must be satisfied: (1) the undissociated core should be in a metastable state and (2) the activation enthalpy for the transition form the metastable undissociated core to the stable dissociated core should be considerably large so that the undissociated core does not undergo a transition to the dissociated core during gliding. The rather high transition enthalpy may be attributed to the necessary climbing motion by one atomic distance for the transition. 2.2.2. Transition from partial dislocation glide to dissociated dislocation glide In tetrahedrally coordinated crystals, mobility of partial dislocation is different from one partial to the other due to different Peierls potential for different partials; the 90◦ partial is known to have much higher mobility than the 30◦ partial [7]. Hence, if the stacking fault energy is low enough, only leading mobile partials can glide leaving the stacking fault behind them. τ c –T relation for mobile partial glide and
Fig. 1. (a) τ c vs. T relation for undissociated dislocation glide and that for glide-set glide for tetrahedrally coordinated crystals. (b) τ c vs. T relation for partial dislocation glide and that for dissociated dislocation glide.
that for the dissociated paired-glide is schematically shown in Fig. 1b. Such a transition of the deformation mode has been discussed by Pirouz et al. [8]. Production of a high density stacking fault by low temperature deformation has been observed in Si [9] and GaAs [10], and partial dislocation glide has been observed by in-situ transmission electron microscopy for II–VI crystals [11]. Single crystals of SiC having extremely low stacking fault energy are believed to undergo a transition of the type given in Fig. 1b [8]. For the macroscopic flow only by partial dislocation glide, however, there is a problem about the dislocation multiplication, because the Frank-Read mechanism cannot be operative for partials.
3. Narrowly dissociated dislocation In some intermetallic compounds [1] and in pure metals of Sn [12] and Ti [13], a plateau region appears in the τ c versus T relation. The present author pointed out that for a narrowly dissociated dislocation a plateau region can appear in the τ c versus T relation as a result of the transition in the Peierls mechanism from the independent, uncorrelated kink-pair formation regime to the correlated kink-pair formation regime [14]. Fig. 2 shows an example of a potential profile for a partial dislocation to migrate, independently from the other, around the equilibrium position, where two partials are screws with the Burgers vector b/2, the equilibrium separation is 5b and the Peierls potential has the periodicity of b/2 with the Peierls stress of 5 × 10−3 G (G is the shear modules) which is the typical value for bcc metals. Since the partial dislocation is trapped in an envelope potential, the partial at the lowest potential site cannot undergo a thermally activated transition to the neighboring site below a critical stress,
Fig. 2. An example of the potential profile for an independent partial dislocation migration. G is the shear modulus, b the strength of the total Burgers vector and τ p the Peierls stress.
S. Takeuchi / Journal of Alloys and Compounds 378 (2004) 61–65
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Fig. 3. Activation enthalpy vs. stress relation for correlated kink-pair formation process and that for uncorrelated kink-pair formation process (upper figure) and the expected τ c vs. T relation (lower figure).
τ 0 . The critical stress depends on the relative positions of the envelope potential minimum and the Peierls potential minima (Fig. 2 shows the case where two minima coincide with each other) and is given by τ0
Gβ b1 b2 d , 4p we bi we
(1)
where G is the shear modulus, β (1) the function of the angle between the dislocation direction and those of partials, bi (i = 1, 2) the strength of the Burgers vector of partials, we the equilibrium separation between partials and d the period of the Peierls potential. Due to the presence of the critical stress, τ c versus T curve controlled by the Peierls mechanism of a narrowly dissociated dislocation can have a plateau region between the low temperature region governed by the independent kink-pair formation regime and the high temperature region by the correlated kink-pair formation regime. Upper figure of Fig. 3 shows schematically the stress dependence of the activation enthalpy of the kink-pair formation for the correlated glide and the uncorrelated glide, and the lower figure illustrates τ c versus T curve. Some examples are presented below.
Fig. 4. CRSS vs. T relation for (1 0 0)[0 1 0] slip and that for (1 0 0)[0 0 1] slip in -Sn reproduced from [12]. Inset drawings show the dissociation of (1 00)[0 1 0] dislocation and the Peierls potential for (1 0 0)[0 0 1] dislocation.
Although no direct evidence has been obtained, there is a high possibility that dislocation with the Burgers vector a [1 0 0] is dissociated into partial dislocations with the Burgers vector of a/2 [1 0 0]. Fig. 5a presents the atomic configuration of the a/2 [1 0 0] stacking fault viewed from the [0 0 1] axis. This dissociation can naturally explains why the (1 0 0)[0 1 0] slip is more active than the (1 0 0)[0 0 1] slip in spite of much larger strength of the Burgers vector of the perfect [1 0 0] dislocation (5.8 Å) than that of [0 0 1] dislocation (3.2 Å).
3.1. CRSS in β-tin Fig. 4 reproduces the τ c versus T relations for (1 0 0)[0 1 0] slip and (1 0 0)[0 0 1] slip for the tetragonal -Sn reported by Kirichenko [12]. We see a two-step dependence for both slips. For (1 0 0)[0 1 0] slip, Kirichenko et al. analyzed the results on the basis of the theories of the Peierls mechanism, but no explanation is given for the appearance of the plateau between 30 and 70 K [15]. We interpret the appearance of the plateau in terms of the Peierls mechanism of narrowly dissociated dislocation. The followings are the bases for this interpretation.
Fig. 5. Crystal structure of -Sn viewed from [0 0 1] (a) and that from [0 1 0] (b). Dashed line indicates the stacking fault plane. Circles of different sizes are on different planes.
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The analyzed pre-exponential factor of the Arrhenius strain-rate equation above 70 K (∼100 s−1 ) is five orders of magnitude smaller than that below 40 K (∼105 s−1 ) [14]. When the pre-exponential factor is converted to the exponent H/kB T, we find that the value of exponent H/kB T is around 21 below 40 K and as small as 11 above 80 K. It is known that the value of the exponent should be normally between 16 and 35 [2], and the small value of 11 above 80 K can only be interpreted by the correlated kink-pair formation process [14]. Using τ0 = 4 MPa, G = 26 GPa [16], b1 = b2 = d = 0.29 nm and assuming β = 1, we obtain the dislocation width we 6.6 nm according to Eq. (1). Theoretically expected CRSS versus T relation for the (1 0 0)[0 1 0] slip is drawn by a solid line in Fig. 4. In Fig. 4, CRSS for (1 0 0)[0 0 1] slip also shows a two-step dependence. The strength of the Burgers vector of [0 0 1] slip is as small as 0.32 nm, and there does not seem to be a stable stacking fault parallel to (1 0 0) plane (see Fig. 5). We interpret the hump in the τ c versus T relation for the (1 0 0)[0 0 1] slip along the same line as the hump in bcc metals. Fig. 6a shows the atomic structure viewed from the [0 0 1] direction, where the atomic position of atoms in each atomic row parallel to the [0 0 1] direction is given in the circle in units of c. When a screw dislocation strain (Fig. 6c) is introduced at positions L and H, the atomic positions of the central four atomic rows change to those given in Fig. 6d. From this result, we find that there exist alternately the low energy positions L’s and high energy positions H’s for a [0 0 1] screw dislocation, in a similar manner as in bcc metals [2]. (1 0 0) glide will be realized by zigzag motion from one L position to another L position of screw dislocations. Also in analogy to bcc metals, there may be a metastable position
Fig. 6. Illustration showing the changes of the atomic configuration with introduction of a screw dislocation with b = c in -Sn viewed from the [0 0 1] direction. Numeral in each circle indicates the atomic position in the c direction.
in between the adjacent L positions, which may be the origin of the appearance of the hump in the τ c versus T relation in the (1 0 0)[0 0 1] slip. The CRSS versus T relation calculated for the Peierls potential [2] given in the inset of Fig. 4 is drawn by a solid line for the (1 0 0)[0 0 1] slip in Fig. 4. 3.2. {1 1 2}111 slip in NiAl There are two groups of B2 type intermetallic compounds, i.e, the 1 1 1 slip group and the 1 0 0 slip group [1]. NiAl belongs to the latter group. For single crystals with 1 0 0 stress axis (named the hard orientation) where any 1 0 0 slip system has zero resolved shear stress, the temperature dependence of the yield stress exhibits a clear two-step change, and it has been established that the yielding in the low temperature range is due to {1 1 2}1 1 1 slip while the high temperature region by {1 1 0}1 1 0 slip [17]. Fig. 7 reproduces temperature dependence of the CRSS in NiAl [18,19]. We see a clear plateau between 300 and 600 K in the {1 1 2}1 1 1 slip region. The mechanism of the transition from the 1 1 1 slip to 1 1 0 slip has been discussed in detail recently [19], but no interpretation has been given for the appearance of the athermal plateau region. It has generally been assumed that in the 1 0 0 slip group of B2 compounds 1 1 1 superlattice dislocation is not dissociated into 1/2 1 1 1 superpartials to account for the relative inactivity of the 1 1 1 slip as compared to the 1 0 0 slip. However, we postulate that the 1 1 1 dislocation in NiAl is narrowly dissociated; the width of the dislocation can be estimated from the plateau stress as we 0.8 nm. Some indications for the dissociation are (1) a high-resolution electron microscopy of the dislocation core in NiAl suggested the dissociation of 1 1 1 dislocation [20], and (2) Peierls stress for 1 1 1 glide is comparable to that of 1 0 0 glide; if the 1 1 1 dislocation is undissociated, the Peierls stress for 1 1 1 glide should be significantly higher than that for 1 0 0 glide taking account of the fact that the strength of
Fig. 7. CRSS vs. temperature for the hard orientation of NiAl reproduced from [18,19].
S. Takeuchi / Journal of Alloys and Compounds 378 (2004) 61–65
the Burgers vector is 1.73 times larger for 1 1 1 glide than for 1 0 0 glide. 4. Concluding remarks We have shown in this paper that the CRSS is not necessarily a monotonically decreasing function of the temperature due to a variety of intrinsic mechanisms without any effect of solute atoms. However, the proposed mechanisms are not necessarily well established by supporting evidences. More direct evidence, e.g., by high resolution electron microscopy, is needed to substantiate the mechanisms. References [1] S. Takeuchi, in: D.G. Brandon, R. Chaim, A. Rosen (Eds.), Strength of Metals and Alloys, Freund Publishing House, London, 1991, pp. 69–82. [2] T. Suzuki, S. Takeuchi, in: R.A. Vardanian (Ed.), Crystal Lattice Defects and Dislocation Dynamics, Nova Science Publication, Inc., New York, 2001, Chapter 1, pp. 1–70. [3] S. Takeuchi, in: J.K. Lee (Ed.), Interatomic Potential and Crystalline Defects, The Metallurgical Society of AIME, Warrendale, PA, 1981, pp. 201–221.
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