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Dislocation processes in quasicrystals—Kink-pair formation control or jog-pair formation control
Materials Science and Engineering A 400–401 (2005) 306–310
Dislocation processes in quasicrystals—Kink-pair formation control or jog-pair formation control Shin Takeuchi ∗ Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan Received 13 September 2004; received in revised form 24 November 2004; accepted 28 March 2005
Abstract A computer simulation of dislocation in a model quasiperiodic lattice indicates that the dislocation feels a large Peierls potential when oriented in particular directions. For a dislocation with a high Peierls potential, the glide velocity and the climb velocity of the dislocation can be described almost in parallel in terms of the kink-pair formation followed by kink motion and the jog-pair formation followed by jog motion, respectively. The activation enthalpy of the kink-pair formation is the sum of the kink-pair formation enthalpy and the atomic jump activation enthalpy, while the activation enthalpy of the jog-pair formation involves the jog-pair enthalpy and the self-diffusion enthalpy. Since the kink-pair energy can be considerably larger than the jog-pair energy, the climb velocity can be faster than the glide velocity, so that the plastic deformation of quasicrystals can be brought not by dislocation glide but by dislocation climb at high temperatures. © 2005 Elsevier B.V. All rights reserved. Keywords: Dislocation; Quasicrystal; Plasticity; Dislocation glide; Dislocation climb
1. Introduction A quasicrystalline lattice is described by the projection of crystal lattice points in a high dimensional space (sixdimensions for the icosahedral quasilattice) onto the real space (or the parallel space) through a projection window in the complementary space (or the perpendicular space). Since the Burgers vector B of a perfect dislocation in a quasicrystal can be defined by a lattice vector in the high dimensional crystalline lattice, it is composed of the parallel space component b and the perpendicular space component b⊥ [1], i.e. B = b + b⊥ .
(1)
The b component produces a phonon strain field around the dislocation as in crystal dislocations, whereas the b⊥ component produces a phason strain field which is specific to dislocations in quasicrystals. For perfect dislocation motion, either glide or climb, in a quasicrystal, both the phonon strain ∗
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field and the phason strain field must accompany the motion. However, since the relaxation of the phason strain field needs atomic diffusion and can take place only at high temperatures, motion of a perfect dislocation is possible only at high temperatures well above half the melting point. At lower temperatures below half the melting point, dislocation motion must create a stacking fault with the fault vector b⊥ , often called the phason fault. Because the phason fault energy is high, the dislocation is practically immobile without the phason relaxation. Experimentally, since the first demonstration by the present author’s group of the plastic deformation of Al–Ru– Cu icosahedral quasicrystal at high temperatures [2], it has been shown that quasicrystals, both of icosahedral and decagonal phases, are generally deformable at high temperatures above 0.8Tm [3,4]. It has also been shown by electron microscopy that the high temperature plasticity is brought in most cases by a dislocation process [5,6]. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a glide process of dislocations, and various models of the deformation mechanism have been proposed based on the glide of dislocations [3,7–11]. However, in recent years, Caillard and coworkers have shown by electron
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microscopy observations, in situ as well as post mortem, of dislocation motions in icosahedral Al–Pd–Mn that dislocations move not by a glide process but by a pure climb process [12–14]. The purpose of the present paper is to compare theoretically the velocity of dislocation glide and that of dislocation climb in a quasicrystal with a high Peierls potential, and to clarify the criterion of the controlling mode of dislocation process in high temperature plasticity of quasicrystals.
2. Peierls potential In this section, we show, on the basis of both experiment and computer simulation, that the Peierls potential for dislocations in quasicrystals is high enough for the dislocations to be confined in a Peierls potential valley. In situ electron microscopy of dislocation motion in quasicrystals has revealed that dislocations migrate steadily keeping a straight form oriented in a symmetrical direction [6,15]. Such behavior is reminiscent of the dislocation glide in bcc metals at low temperatures and in covalent crystals at high temperatures; in both types of crystals the dislocation motion is established to be controlled by the Peierls mechanism. The dislocation motion keeping a straight shape in a crystallographic direction indicates that the rate controlling process of the dislocation motion is the kink-pair formation in the glide process or the jog-pair formation in the climb process. In order to estimate the potential energy of a straight dislocation on a glide plane in a quasiperiodic lattice, we have constructed a realistic model quasiperiodic lattice in computer and computed the potential energies at zero Kelvin of straight dislocations oriented in various directions as a function of their position on the quasiperiodic plane [16]. The results have shown that for dislocations in particular orientations their potential energy oscillates in a quasiperiodic way superimposing on almost constant phason production energy. The Peierls stress component is as large as the order of 0.1G (G: shear modulus), the same order as that for dislocations in covalent crystals. Thus, dislocations in quasicrystals have a strong tendency to lie along a Peierls valley on the glide plane.
Fig. 1. A perfect dislocation in a Penrose lattice. After gliding the dislocation only with b component to the left, shaded tiles along the glide plane are destroyed to produce intra-tile phason defects, whereas outside the glide plane tiling mismatch phason defects, examples of which are shown by dashed tiles and dotted tiles, respectively, before and after a displacement shown by an arrow of the dislocation position, are produced.
lattice. The former type of fault has a much higher energy than the latter type, and during kink-pair formation and kink migration the former type of phason defects should be relaxed by local rearrangements of atoms in the lattice planes facing the glide plane. Thus, an activation energy of atomic jump is indispensable for the migration of a kink to relax the high energy intra-tile phason defects. As a result, the energy profile for the kink-pair formation process shown in Fig. 2(a) as a function of the kinks separation l is like that schematically depicted in Fig. 2(b) by a dashed curve, where H corresponds to the activation enthalpy of atomic jump and the slope Γ the
3. Dislocation glide motion controlled by kink-pair formation With the migration in a quasiperiodic lattice of a dislocation only with b component, two kinks of phason fault are produced, one is the intra-tile phason defects produced just in close vicinity of the glide plane as a result of cutting of quasiperiodic tiles by the b dislocation, and the other is the tiling mismatch outside the glide plane produced as a result of the shift of the dislocation center [17]. These two kinds of fault are illustrated in Fig. 1 for a dislocation in the Penrose
Fig. 2. (a) Kink-pair formation process and (b) the enthalpy change profile of the kink-pair as a function of the kink-pair spacing l without stress (dashed curve) and under an effective stress τ ∗ (solid curve).
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energy produced by unrelaxed tiling mismatch outside the glide plane. Except for the slope Γ , the potential profile is quite analogous to that of the kink-pair formation process for a dislocation in covalent crystals treated by Hirth and Lothe [18]. The thermally activated kink-pair formation is possible for stresses higher than Γ /b . The enthalpy profile under stress is shown by a solid curve in Fig. 2(b), which is composed ∗ (τ ∗ ) of a kink-pair enthalpy Hkp (τ ∗ ) having a maximum Hkp at a kink separation l∗ and the superimposing atomic jump enthalpy H . The rate of kink-pair formation is given by the flow rate of kinks through l∗ towards larger l value. Let the average distance between the potentials of atomic jump be d and the vibrational frequency of a kink be νk , the rate of the kink-pair formation per unit length of a dislocation lying in a Peierls potential is given, in perfect analogy to the kink diffusion theory of the kink-pair formation in covalent crystals [18], as Hkp (τ ∗ ) + H νk τ ∗ b|| dd νkp = exp − sinh d kB T 2kB T Hkp (τ ∗ ) + H τ ∗ b|| d . (2) νk exp − ≈ kB T 2kB T In the equilibrium state of an infinite length of dislocation, the mean kink separation ¯l is determined by the balance between the rate of kink-pair formation and the rate of kinks annihilation and is given by ∗ Hkp ¯l = d exp . (3) 2kB T
Fig. 3. (a) Jog-pair formation process and (b) the enthalpy change profile of the jog-pair formation as a function of the jog-pair spacing without stress (dashed curve) and under an effective stress σ ∗ .
is governed by the rate of vacancy absorption or emission at the jog site. Under the action of the climbing force σ ∗ b , there appears an excess vacancy concentration at the dislocation, whose osmotic force on the dislocation balances the climbing force. Considering that the vacancy diffusion along the dislocation core is much faster than in the bulk, Hirth and Lothe obtained the jog velocity vj given by [18] 4πDs σ ∗ b|| a Hs vj = exp , (5) kB T ln(¯z/b|| ) 2kB T
4. Dislocation climb motion controlled by jog-pair formation
where a is the atomic distance along the dislocation line, z¯ is the mean free life length along the dislocation line of a vacancy diffusing in the core given by z¯ = √ 2a exp( Hs /2kB T ), Ds the self-diffusion coefficient and Hs is the difference between the activation enthalpy for self-diffusion in the bulk and that along the core. In the climbing process by jog-pair formation followed by jog motion of a dislocation confined in Peierls valley, we again assume, as in the case of dislocation glide controlled by the kink-pair formation, that the jog mean free path is longer than the dislocation length L, on the ground that jog-pair enthalpy is considerably larger than the activation enthalpy of self-diffusion. The climb velocity is written as [18]: Hjp∗ (σ ∗ ) − Hs /2 4πLDs va σ ∗ Vc = 2 exp − (6) a kB T ln(¯z/b|| ) kB T
The climb process of a dislocation confined in a deep Peierls potential valley occurs by jog-pair formation followed by jog motion along the dislocation in a similar manner as the kink-pair formation and the kink motion mentioned in the previous section. The process is schematically illustrated in Fig. 3(a). The energy profile of a jog-pair as a function of the jog spacing is analogous to the kink-pair formation, as shown in Fig. 3(b). An essential difference from the kink-pair formation process is that the jog mobility along the dislocation
where va is the atomic volume, Hjp∗ is the activation enthalpy of the jog formation. Writing Ds ≈ a2 νD exp(−Hs /kB T ) (Hs : activation enthalpy of self-diffusion; νD : Debyefrequency) and approximating va ≈ a3 , Eq. (6) is rewritten as Hjp∗ (σ ∗ ) + (Hs − Hs /2) 4πLa3 σ ∗ Vc = νD exp − kB T ln(¯z/b|| ) kB T (7)
∗ = 3 eV and d = 0.5 nm, ¯l at For a typical value of Hkp 1000 K is calculated to be 3 cm, which is much larger than the dislocation segment length, L, lying in a Peierls potential valley. Therefore, the dislocation glide velocity Vg is controlled only by the rate of kink-pair formation on the segment length L which is followed by the kinks motion over a distance L, and is written as ∗ (τ ∗ ) + H Hkp τ ∗ b|| d 2 L νk exp − . (4) Vg = 2kB T kB T
S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 306–310
By comparing Eqs. (4) and (7) for velocities of dislocation glide and dislocation climb, one sees that both equations have the same functional form. Thus, it is impossible to discriminate the two cases experimentally only from the analysis of the macroscopic plasticity.
5. Comparison of Vg and Vc In comparing Eqs. (4) and (7), we approximate νk ≈ νD , τ ∗ ≈ σ ∗ and d ≈ b ≈ a. Then, the ratio Vg /Vc is written as Vg ln(¯z/a) ≈ exp Vc 8π ∗ (τ ∗ ) − H ∗ (σ ∗ )} + {H − (H − H /2)} {Hkp s s jp − kB T (8) As a typical case, we assume T ≈ 1000 K, Hs ≈ 2 eV and √ Hs ≈ Hs /2. Then, z¯ = 2a exp( Hs /2kB T ) ≈ 570a and hence the pre-exponential factor of Eq. (8) is ∼0.25. It seems reasonable to assume that the phason jump enthalpy at dislocation core H is approximately equal to Hs /2 (the same as the activation energy of the core diffusion). It follows then ∗ (τ ∗ ) − H ∗ (σ ∗ ) Hkp Vg jp ≈ 0.25 exp − (9) Vc kB T Thus, it is found that whether the plastic deformation of quasicrystals is controlled by dislocation glide or dislocation climb is essentially determined by the relative magnitude of ∗ (τ ∗ ) and the jog-pair the kink-pair formation enthalpy Hkp formation enthalpy Hjp∗ (σ ∗ ). Since σ ∗ ≈ τ ∗ , the work-done terms −τ ∗ b l and −σ ∗ b l are comparable. The phason production energy Γ for dislocation glide and that for climb may also be similar. Thus, the relative magnitude is determined by the relative magnitude of the kink energy Ek and the jog energy Ej . If Ek > Ej , Vc can be larger than Vg , and if Ek < Ej , Vg is larger than Vc . The kink energy depends on the Peierls potential and according to the line tension approximation of the dislocation, the kink energy is written as [18]: Ek ≈
2d 2Ep E0 , π
The ratio is then written as d Ek =5 β . Ej h
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(13)
The kink height h is always of an atomic spacing, whereas the Peierls valley distance d is determined by the periodicity of the lattice on the glide plane and can be larger than an atomic spacing in a complex lattice. The value of β(=τ p /G) of a dislocation in a complex structure like quasicrystal is of the order of 10−1 to 10−2 , and hence Ek /Ej = (0.5–1.5)d/h. In conclusion, since generally d > h, Ek can be larger than Ej and consequently the climb velocity of a dislocation can be higher than the glide velocity in quasicrystal.
6. Work-softening mechanism The specific feature of macroscopic plasticity in quasicrystals is a pronounced work-softening up to high strains; the flow stress often becomes only one-fifth of the yield stress [19,20]. We have shown from the activation analysis that the decrease of the flow stress is due to a decrease of the activation enthalpy with increasing plastic strain [20,21]. Assuming that the kink-pair formation controls the deformation, the present author and his colleagues have earlier proposed a mechanism which explains the decrease of kink-pair formation enthalpy controlling the dislocation velocity with the introduction of phason defects on the glide plane by forest dislocation motion [3,11]. Fig. 4 illustrates the quasiperiodic Peierls potential valleys on a glide plane cut by forest dislocations. In such a situation, dislocation glide is determined essentially by the kink-pair formation at narrowly spaced Peierls potentials; widely spaced Peierls potentials can be overcome by
(10)
where Ep is the height of the Peierls potential and E0 is the line energy of the dislocation. For τ p = βG (τ p : Peierls stress, G: shear modulus) and for E0 ≈ Gb2 , one obtains: √ 2 2β1/2 2 ab G ≈ 0.5 βGdb2 . (11) Ek ≈ 3/2 π The jog energy is written as [18] Ej =
Gb2 h ≈ 0.1Gb2 h 4π(1 − ν)
(12)
Fig. 4. The figure illustrates either the Peierls valley distribution in a glide plane for dislocation glide or glide plane spacing distribution for dislocation climb, after passage of a forest dislocation.
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side motion of kinks produced at narrowly spaced Peierls potential. In the jog-pair formation controlled deformation, worksoftening can occur by essentially the same mechanism as the kink-pair formation controlled deformation. In icosahedral quasicrystals, the lattice spacings of the glide plane on which edge dislocations lie are not periodic but quasiperiodic, consisting of wide and narrow ones. In phason free state, the dislocation climb velocity is determined essentially by jog-pair formation between wide lattice planes, and in a phason-defected state, it is determined by the jog-pair formation between narrow lattice planes; this is because the side motion of a jog from a narrowly spaced part to a widely spaced part (e.g., position 2 to position 3 in Fig. 4) can occur with a smaller activation enthalpy.
References [1] J.E.S. Socolar, T.C. Lubensky, P.J. Steinhardt, Phys. Rev. B 34 (1986) L3345. [2] T. Shibuya, T. Hashimoto, S. Takeuchi, Jpn. J. Appl. Phys. 29 (1990) 349. [3] S. Takeuchi, in: J.-M. Dubois, P.A. Thiel, A.-P. Tsai, K. Urban (Eds.), Quasicrystals, Materials Research Society Symposium Proceedings, vol. 553, Materials Research Society, Warrendale, PA, 1999, pp. 283–294. [4] K. Urban, M. Feuerbacher, M. Wollgarten, M. Bartsch, U. Messerschmidt, in: Z.M. Stadnik (Ed.), Physical Properties of Quasicrystals, Springer-Verlag, Berlin, 1999, pp. 361–401. [5] M. Wollgarten, M. Beyss, K. Urban, Phys. Rev. Lett. 71 (1993) 549.
[6] M. Wollgarten, M. Bartsch, U. Messerschmidt, M. Feurbacher, R. Rosenfeld, M. Beyss, K. Urban, Philos. Mag. Lett. 71 (1995) 99. [7] M. Feuerbacher, C. Metzmacher, M. Wollgarten, K. Urban, B. Baufeld, M. Bartsch, U. Messerschmdt, Mater. Sci. Eng. 233 (1997) 103. [8] U. Messerschmidt, M. Bartsch, B. Geyer, M. Feuerbacher, K. Urban, Philos. Mag. A 80 (2000) 1165. [9] P. Guyot, G. Canova, Philos. Mag. A 79 (1999) 2815. [10] D. Brunner, D. Placke, H.D. Carstanjen, Phys. Stat. Sol. (A) 177 (2000) 203. [11] S. Takeuchi, R. Tamura, E. Kabutoya, K. Edagawa, Philos. Mag. A 82 (2002) 379. [12] D. Caillard, G. Vanderschaeve, L. Bresson, D. Gratias, Philos. Mag. A 80 (2000) 237. [13] D. Caillard, C. Roucau, L. Bresson, D. Gratias, Acta Mater. 50 (2002) 4499. [14] F. Mompiou, L. Bresson, P. Cordier, D. Caillard, Philos. Mag. A 83 (2003) 3133. [15] M. Wollgarten, S. Takeuchi, U. Messerschmidt, M. Bartsch, in: S. Takeuchi, T. Fujiwara (Eds.), Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1998, pp. 549–552. [16] R. Tamura, S. Takeuchi, K. Edagawa, in: E. Belin-Ferr´e, P.A. Thiel, A.-P. Tsai, K. Urban (Eds.), Quasicrystals—Preparation, Properties and Applications, MRS Symposium Proceedings, vol. 643, Materials Research Society, Warrendale, PA, 2001, pp. K.6.4.1–K.6.4.6. [17] S. Takeuchi, K. Shinoda, H. Fujiwara, K. Suzuki, Ann. Phys. 22 (1997) C2–C101. [18] J.P. Hirth, J. Lothe, Theory of Dislocations, 2nd ed., McGraw-Hill, NewYork, 1968. [19] D. Brunner, D. Plachke, H.D. Carstanjen, Phys. Stat. Sol. (A) 177 (2000) 203. [20] Y. Imai, T. Shibuya, R. Tamura, S. Takeuchi, J. Non-Cryst. Solids 334/335 (2004) 444. [21] E. Kabutoya, E. Edagawa, R. Tamura, S. Takeuchi, J.Q. Guo, A.-P. Tsai, Philos. Mag. A 82 (2002) 367.
Dislocation processes in quasicrystals—Kink-pair formation control or jog-pair formation control Shin Takeuchi ∗ Department of Materials Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan Received 13 September 2004; received in revised form 24 November 2004; accepted 28 March 2005
Abstract A computer simulation of dislocation in a model quasiperiodic lattice indicates that the dislocation feels a large Peierls potential when oriented in particular directions. For a dislocation with a high Peierls potential, the glide velocity and the climb velocity of the dislocation can be described almost in parallel in terms of the kink-pair formation followed by kink motion and the jog-pair formation followed by jog motion, respectively. The activation enthalpy of the kink-pair formation is the sum of the kink-pair formation enthalpy and the atomic jump activation enthalpy, while the activation enthalpy of the jog-pair formation involves the jog-pair enthalpy and the self-diffusion enthalpy. Since the kink-pair energy can be considerably larger than the jog-pair energy, the climb velocity can be faster than the glide velocity, so that the plastic deformation of quasicrystals can be brought not by dislocation glide but by dislocation climb at high temperatures. © 2005 Elsevier B.V. All rights reserved. Keywords: Dislocation; Quasicrystal; Plasticity; Dislocation glide; Dislocation climb
1. Introduction A quasicrystalline lattice is described by the projection of crystal lattice points in a high dimensional space (sixdimensions for the icosahedral quasilattice) onto the real space (or the parallel space) through a projection window in the complementary space (or the perpendicular space). Since the Burgers vector B of a perfect dislocation in a quasicrystal can be defined by a lattice vector in the high dimensional crystalline lattice, it is composed of the parallel space component b and the perpendicular space component b⊥ [1], i.e. B = b + b⊥ .
(1)
The b component produces a phonon strain field around the dislocation as in crystal dislocations, whereas the b⊥ component produces a phason strain field which is specific to dislocations in quasicrystals. For perfect dislocation motion, either glide or climb, in a quasicrystal, both the phonon strain ∗
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0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.03.068
field and the phason strain field must accompany the motion. However, since the relaxation of the phason strain field needs atomic diffusion and can take place only at high temperatures, motion of a perfect dislocation is possible only at high temperatures well above half the melting point. At lower temperatures below half the melting point, dislocation motion must create a stacking fault with the fault vector b⊥ , often called the phason fault. Because the phason fault energy is high, the dislocation is practically immobile without the phason relaxation. Experimentally, since the first demonstration by the present author’s group of the plastic deformation of Al–Ru– Cu icosahedral quasicrystal at high temperatures [2], it has been shown that quasicrystals, both of icosahedral and decagonal phases, are generally deformable at high temperatures above 0.8Tm [3,4]. It has also been shown by electron microscopy that the high temperature plasticity is brought in most cases by a dislocation process [5,6]. Until recent years, it has been generally believed that the plasticity of quasicrystals is carried by a glide process of dislocations, and various models of the deformation mechanism have been proposed based on the glide of dislocations [3,7–11]. However, in recent years, Caillard and coworkers have shown by electron
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microscopy observations, in situ as well as post mortem, of dislocation motions in icosahedral Al–Pd–Mn that dislocations move not by a glide process but by a pure climb process [12–14]. The purpose of the present paper is to compare theoretically the velocity of dislocation glide and that of dislocation climb in a quasicrystal with a high Peierls potential, and to clarify the criterion of the controlling mode of dislocation process in high temperature plasticity of quasicrystals.
2. Peierls potential In this section, we show, on the basis of both experiment and computer simulation, that the Peierls potential for dislocations in quasicrystals is high enough for the dislocations to be confined in a Peierls potential valley. In situ electron microscopy of dislocation motion in quasicrystals has revealed that dislocations migrate steadily keeping a straight form oriented in a symmetrical direction [6,15]. Such behavior is reminiscent of the dislocation glide in bcc metals at low temperatures and in covalent crystals at high temperatures; in both types of crystals the dislocation motion is established to be controlled by the Peierls mechanism. The dislocation motion keeping a straight shape in a crystallographic direction indicates that the rate controlling process of the dislocation motion is the kink-pair formation in the glide process or the jog-pair formation in the climb process. In order to estimate the potential energy of a straight dislocation on a glide plane in a quasiperiodic lattice, we have constructed a realistic model quasiperiodic lattice in computer and computed the potential energies at zero Kelvin of straight dislocations oriented in various directions as a function of their position on the quasiperiodic plane [16]. The results have shown that for dislocations in particular orientations their potential energy oscillates in a quasiperiodic way superimposing on almost constant phason production energy. The Peierls stress component is as large as the order of 0.1G (G: shear modulus), the same order as that for dislocations in covalent crystals. Thus, dislocations in quasicrystals have a strong tendency to lie along a Peierls valley on the glide plane.
Fig. 1. A perfect dislocation in a Penrose lattice. After gliding the dislocation only with b component to the left, shaded tiles along the glide plane are destroyed to produce intra-tile phason defects, whereas outside the glide plane tiling mismatch phason defects, examples of which are shown by dashed tiles and dotted tiles, respectively, before and after a displacement shown by an arrow of the dislocation position, are produced.
lattice. The former type of fault has a much higher energy than the latter type, and during kink-pair formation and kink migration the former type of phason defects should be relaxed by local rearrangements of atoms in the lattice planes facing the glide plane. Thus, an activation energy of atomic jump is indispensable for the migration of a kink to relax the high energy intra-tile phason defects. As a result, the energy profile for the kink-pair formation process shown in Fig. 2(a) as a function of the kinks separation l is like that schematically depicted in Fig. 2(b) by a dashed curve, where H corresponds to the activation enthalpy of atomic jump and the slope Γ the
3. Dislocation glide motion controlled by kink-pair formation With the migration in a quasiperiodic lattice of a dislocation only with b component, two kinks of phason fault are produced, one is the intra-tile phason defects produced just in close vicinity of the glide plane as a result of cutting of quasiperiodic tiles by the b dislocation, and the other is the tiling mismatch outside the glide plane produced as a result of the shift of the dislocation center [17]. These two kinds of fault are illustrated in Fig. 1 for a dislocation in the Penrose
Fig. 2. (a) Kink-pair formation process and (b) the enthalpy change profile of the kink-pair as a function of the kink-pair spacing l without stress (dashed curve) and under an effective stress τ ∗ (solid curve).
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energy produced by unrelaxed tiling mismatch outside the glide plane. Except for the slope Γ , the potential profile is quite analogous to that of the kink-pair formation process for a dislocation in covalent crystals treated by Hirth and Lothe [18]. The thermally activated kink-pair formation is possible for stresses higher than Γ /b . The enthalpy profile under stress is shown by a solid curve in Fig. 2(b), which is composed ∗ (τ ∗ ) of a kink-pair enthalpy Hkp (τ ∗ ) having a maximum Hkp at a kink separation l∗ and the superimposing atomic jump enthalpy H . The rate of kink-pair formation is given by the flow rate of kinks through l∗ towards larger l value. Let the average distance between the potentials of atomic jump be d and the vibrational frequency of a kink be νk , the rate of the kink-pair formation per unit length of a dislocation lying in a Peierls potential is given, in perfect analogy to the kink diffusion theory of the kink-pair formation in covalent crystals [18], as Hkp (τ ∗ ) + H νk τ ∗ b|| dd νkp = exp − sinh d kB T 2kB T Hkp (τ ∗ ) + H τ ∗ b|| d . (2) νk exp − ≈ kB T 2kB T In the equilibrium state of an infinite length of dislocation, the mean kink separation ¯l is determined by the balance between the rate of kink-pair formation and the rate of kinks annihilation and is given by ∗ Hkp ¯l = d exp . (3) 2kB T
Fig. 3. (a) Jog-pair formation process and (b) the enthalpy change profile of the jog-pair formation as a function of the jog-pair spacing without stress (dashed curve) and under an effective stress σ ∗ .
is governed by the rate of vacancy absorption or emission at the jog site. Under the action of the climbing force σ ∗ b , there appears an excess vacancy concentration at the dislocation, whose osmotic force on the dislocation balances the climbing force. Considering that the vacancy diffusion along the dislocation core is much faster than in the bulk, Hirth and Lothe obtained the jog velocity vj given by [18] 4πDs σ ∗ b|| a Hs vj = exp , (5) kB T ln(¯z/b|| ) 2kB T
4. Dislocation climb motion controlled by jog-pair formation
where a is the atomic distance along the dislocation line, z¯ is the mean free life length along the dislocation line of a vacancy diffusing in the core given by z¯ = √ 2a exp( Hs /2kB T ), Ds the self-diffusion coefficient and Hs is the difference between the activation enthalpy for self-diffusion in the bulk and that along the core. In the climbing process by jog-pair formation followed by jog motion of a dislocation confined in Peierls valley, we again assume, as in the case of dislocation glide controlled by the kink-pair formation, that the jog mean free path is longer than the dislocation length L, on the ground that jog-pair enthalpy is considerably larger than the activation enthalpy of self-diffusion. The climb velocity is written as [18]: Hjp∗ (σ ∗ ) − Hs /2 4πLDs va σ ∗ Vc = 2 exp − (6) a kB T ln(¯z/b|| ) kB T
The climb process of a dislocation confined in a deep Peierls potential valley occurs by jog-pair formation followed by jog motion along the dislocation in a similar manner as the kink-pair formation and the kink motion mentioned in the previous section. The process is schematically illustrated in Fig. 3(a). The energy profile of a jog-pair as a function of the jog spacing is analogous to the kink-pair formation, as shown in Fig. 3(b). An essential difference from the kink-pair formation process is that the jog mobility along the dislocation
where va is the atomic volume, Hjp∗ is the activation enthalpy of the jog formation. Writing Ds ≈ a2 νD exp(−Hs /kB T ) (Hs : activation enthalpy of self-diffusion; νD : Debyefrequency) and approximating va ≈ a3 , Eq. (6) is rewritten as Hjp∗ (σ ∗ ) + (Hs − Hs /2) 4πLa3 σ ∗ Vc = νD exp − kB T ln(¯z/b|| ) kB T (7)
∗ = 3 eV and d = 0.5 nm, ¯l at For a typical value of Hkp 1000 K is calculated to be 3 cm, which is much larger than the dislocation segment length, L, lying in a Peierls potential valley. Therefore, the dislocation glide velocity Vg is controlled only by the rate of kink-pair formation on the segment length L which is followed by the kinks motion over a distance L, and is written as ∗ (τ ∗ ) + H Hkp τ ∗ b|| d 2 L νk exp − . (4) Vg = 2kB T kB T
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By comparing Eqs. (4) and (7) for velocities of dislocation glide and dislocation climb, one sees that both equations have the same functional form. Thus, it is impossible to discriminate the two cases experimentally only from the analysis of the macroscopic plasticity.
5. Comparison of Vg and Vc In comparing Eqs. (4) and (7), we approximate νk ≈ νD , τ ∗ ≈ σ ∗ and d ≈ b ≈ a. Then, the ratio Vg /Vc is written as Vg ln(¯z/a) ≈ exp Vc 8π ∗ (τ ∗ ) − H ∗ (σ ∗ )} + {H − (H − H /2)} {Hkp s s jp − kB T (8) As a typical case, we assume T ≈ 1000 K, Hs ≈ 2 eV and √ Hs ≈ Hs /2. Then, z¯ = 2a exp( Hs /2kB T ) ≈ 570a and hence the pre-exponential factor of Eq. (8) is ∼0.25. It seems reasonable to assume that the phason jump enthalpy at dislocation core H is approximately equal to Hs /2 (the same as the activation energy of the core diffusion). It follows then ∗ (τ ∗ ) − H ∗ (σ ∗ ) Hkp Vg jp ≈ 0.25 exp − (9) Vc kB T Thus, it is found that whether the plastic deformation of quasicrystals is controlled by dislocation glide or dislocation climb is essentially determined by the relative magnitude of ∗ (τ ∗ ) and the jog-pair the kink-pair formation enthalpy Hkp formation enthalpy Hjp∗ (σ ∗ ). Since σ ∗ ≈ τ ∗ , the work-done terms −τ ∗ b l and −σ ∗ b l are comparable. The phason production energy Γ for dislocation glide and that for climb may also be similar. Thus, the relative magnitude is determined by the relative magnitude of the kink energy Ek and the jog energy Ej . If Ek > Ej , Vc can be larger than Vg , and if Ek < Ej , Vg is larger than Vc . The kink energy depends on the Peierls potential and according to the line tension approximation of the dislocation, the kink energy is written as [18]: Ek ≈
2d 2Ep E0 , π
The ratio is then written as d Ek =5 β . Ej h
309
(13)
The kink height h is always of an atomic spacing, whereas the Peierls valley distance d is determined by the periodicity of the lattice on the glide plane and can be larger than an atomic spacing in a complex lattice. The value of β(=τ p /G) of a dislocation in a complex structure like quasicrystal is of the order of 10−1 to 10−2 , and hence Ek /Ej = (0.5–1.5)d/h. In conclusion, since generally d > h, Ek can be larger than Ej and consequently the climb velocity of a dislocation can be higher than the glide velocity in quasicrystal.
6. Work-softening mechanism The specific feature of macroscopic plasticity in quasicrystals is a pronounced work-softening up to high strains; the flow stress often becomes only one-fifth of the yield stress [19,20]. We have shown from the activation analysis that the decrease of the flow stress is due to a decrease of the activation enthalpy with increasing plastic strain [20,21]. Assuming that the kink-pair formation controls the deformation, the present author and his colleagues have earlier proposed a mechanism which explains the decrease of kink-pair formation enthalpy controlling the dislocation velocity with the introduction of phason defects on the glide plane by forest dislocation motion [3,11]. Fig. 4 illustrates the quasiperiodic Peierls potential valleys on a glide plane cut by forest dislocations. In such a situation, dislocation glide is determined essentially by the kink-pair formation at narrowly spaced Peierls potentials; widely spaced Peierls potentials can be overcome by
(10)
where Ep is the height of the Peierls potential and E0 is the line energy of the dislocation. For τ p = βG (τ p : Peierls stress, G: shear modulus) and for E0 ≈ Gb2 , one obtains: √ 2 2β1/2 2 ab G ≈ 0.5 βGdb2 . (11) Ek ≈ 3/2 π The jog energy is written as [18] Ej =
Gb2 h ≈ 0.1Gb2 h 4π(1 − ν)
(12)
Fig. 4. The figure illustrates either the Peierls valley distribution in a glide plane for dislocation glide or glide plane spacing distribution for dislocation climb, after passage of a forest dislocation.
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S. Takeuchi / Materials Science and Engineering A 400–401 (2005) 306–310
side motion of kinks produced at narrowly spaced Peierls potential. In the jog-pair formation controlled deformation, worksoftening can occur by essentially the same mechanism as the kink-pair formation controlled deformation. In icosahedral quasicrystals, the lattice spacings of the glide plane on which edge dislocations lie are not periodic but quasiperiodic, consisting of wide and narrow ones. In phason free state, the dislocation climb velocity is determined essentially by jog-pair formation between wide lattice planes, and in a phason-defected state, it is determined by the jog-pair formation between narrow lattice planes; this is because the side motion of a jog from a narrowly spaced part to a widely spaced part (e.g., position 2 to position 3 in Fig. 4) can occur with a smaller activation enthalpy.
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