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Effect of the dislocations on the kinetics of a martensitic transition
Materials Science and Engineering A309–310 (2001) 168–172
Effect of the dislocations on the kinetics of a martensitic transition MD simulation of bcc–hcp transformation in Zr A.R. Kuznetsov a,∗ , Yu.N. Gornostyrev a , M.I. Katsnelson a , A.V. Trefilov b a
Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, 620219 Ekaterinburg, Russia b Russian Science Center “Kurchatov Institute”, 123182 Moscow, Russia
Abstract Kinetics of the bcc → hcp transition in Zr is investigated by molecular dynamic (MD) simulation in the presence of dislocations. It is shown that the dislocations essentially stimulates the martensitic transformation by inducing a specific type of lattice instability. As a result, the transition near a dislocation becomes possible at much lower temperatures than in the ideal crystal. The transition occurs in two stages, a relatively long stage of a modulation of long-wavelength acoustic phonons, and a fast stage of short-wavelength phonon instability. Finally, the twin domains are formed with habit plane parallel to {1 1 0}bcc . © 2001 Elsevier Science B.V. All rights reserved. Keywords: Martensitic transition; Molecular dynamic simulation; Heterogeneous nucleation
1. Introduction The mechanism of the heterogeneous nucleation of a new phase in a real crystal remains still an unsolved problem [1]. Despite martensitic transformations are first-order phase transitions, a rather broad region of pre-transition anomalies, for example, the softening of the phonon spectra [2] is usually connected with them. It is a common point of view now [1,3] that the defects play a double role in heterogeneous nucleation of a martensitic phase. At first, the distortions of a lattice caused by the defects, sometimes make a structure of an initial phase closer to a structure of martensite thus, reducing a nucleation barrier [4]. Secondly, the defects can be centers of local lattice instability under the conditions of a lattice softening (for example, [5]), when the effective elastic modulus C = (C11 − C12 )/2 tends to be zero due to a specific type of a shear strain. To understand the mechanisms of the essential influence of the defects on the kinetics of martensitic transition at [6,7], it is necessary to clarify real microscopic picture of a crystal lattice reconstruction in the vicinity of a dislocation. To this aim, numerical simulations by molecular dynamics (MD) method could be very useful. Recently attempts have been done to study the kinetics of martensitic transitions within the framework of the microscopic approach [8–13]. In [10] the stress-induced coherent nucleation and growth of L10 martensitic phase in initially B2-ordered phase of NiAl intermetallic has been simulated. ∗ Corresponding author. E-mail address: a [email protected] (A.R. Kuznetsov).
The conclusion has been done, that the presence of the defects is necessary to initiate the martensitic transition. In [11] the martensitic transition in an alloy Fe0.8 Ni0.2 near a top of a crack has been investigated by MD method. In the present work the microscopic study of martensitic transition kinetics of Zr in the presence of such typical defects as dislocations is carried out. Zirconium is a convenient model material. With the decrease of temperature at T = 1136 K it undergoes bcc → hcp transformation, and at fast enough cooling demonstrates also the transition into metastable -phase [14]. The simulation of bcc → hcp transition kinetics in ideal Zr crystal at temperatures 1000 and 1500 K shows, that the transformation is accompanied by phonon spectrum anomalies [15,16,12,13] and is realized by phonon mechanism [12,13,17,18], ensuring reconstruction of bcc lattice under Burgers scheme [19]. We investigated the transition kinetics at high temperatures (1000 K), as well as after quenching to low temperature region (200 K), when the transformation in ideal crystal does not take place.
2. Computational procedure During MD simulation we used many-body interatomic interaction potentials [15,16], constructed in the framework of embedded atom method (EAM) and reproducing satisfactory elastic properties and phonon spectra of Zr. The numerical simulation of bcc → hcp transition in Zr was carried out at the presence of edge dislocations of two types 1 0 0{0 1 0} and 1/21 1 1{1 1 0}. A rectangular
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crystallite in the first case was chosen as a slab with a thickness of 12 atomic planes along the Z-axis (which is parallel to the dislocation line and the X-axis is parallel to the Burgers vector) containing about 10,000 atoms. For the second case (for dislocations 1/21 1 1{1 1 0}) it contains about 20,000 atoms with a thickness of 24 atomic planes. Periodic boundary conditions were used in all the directions. To provide the periodicity in the X- and Y-directions, a pair of dislocations of opposite sign (a dipole) was introduced into the crystallite. After introducing the dislocations in bcc phase the MD relaxation was carried out at temperatures 1000 and 200 K, at which hcp phase is stable (preservation of the given temperature was controlled by an average kinetic energy of atoms).
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3. Kinetics of the transformation in the presence of the dislocations 3.1. Dislocations with Burgers vector 1 0 0 The results of simulations for the crystallite with the dipole of 1 0 0{0 1 1} edge dislocations at T = 200 K are shown in Fig. 1. One can see that the transformation has been completed after N = 43,000 steps (time of simulation is given in steps, 1 step = 1 fs), that is illustrated by the changes of atomic pair distribution function g(R) (Fig. 1d). An important feature characterizing the transformation is the twins formation in a pyramid plane {1 0 1¯ 1}hcp , parallel to planes {1 1 0}bcc of an initial bcc lattice. Thus, usual
Fig. 1. Kinetics bcc → hcp transformation in the vicinity of the dipole 1 0 0 dislocations. A crystallite structure after N = 15,000 (a), 28,000 (b), 43,000 (c) MD steps (1 step = 1 fs) and appropriate pair distribution function g(R) (d). The atoms having bcc coordination of the nearest neighbors are shown by solid circles and the remaining atoms by open circles. The primes show the positions of the first, second and third neighbors in bcc (1, 2, 3) and hcp (1 , 2 , 3 ) lattice.
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Fig. 2. Diffraction patterns (for the crystallite from Fig. 1) for a vector g = (0 0 1) after N = 18,000 (a), 28,000 (b), 43,000 (c) MD steps.
orientational relations are fulfilled, the hcp lattice basis planes in the twins are parallel to bcc planes {1 1 0}, passing under an 60◦ angle to the twinning plane. The results of the calculations allow us to describe the process of the transformation in time. The transition starts with the stage of long-wavelength lattice instability. In a projection on a plane of Fig. 1 (plane XY) the corresponding displacements look initially like an acoustic transverse phonon with a vector q || 1 1 0, the wavelength of approximately 30–40 interatomic distances, and the polarization vector e || 1 1¯ 0. Then these sine-like lattice oscillations (Fig. 1a and b) are transformed into a sequence of kink-like excitations with quite sharp boundaries (“soliton lattice”) which leads to the alternation of regions with a different sign of a shear strain. Next, after the twinning domains are formed, a short-wavelength instability, which determines specifically hcp character of the new phase (Fig. 1c), develops in a short time. This is clearly seen from the pair distribution function (Fig. 1d), where the peaks corresponding to the position of the first, second and third neighbors in the bcc lattice vanish and similar peaks for the hcp structure appear. An additional information about the transformation kinetics is given by diffraction patterns calculated according to positions of atoms in crystallite after the various periods of time (Fig. 2). On the initial stage of transformation (corresponding to Fig. 1a) the splitting of reflexes happens, but the structure is still cubic. The reflexes showing the second phase appearing are observed after N = 28,000 (Fig. 2b), and at last after N = 43,000 (Fig. 2c) the lattice gains a hexagonal symmetry. The development of lattice instability is inhomogeneous as one can see in Fig. 1a. It reflects, apparently, the peculiarities of strain distribution created by dislocational dipoles. On the final stage of the transformation the atoms on the domain boundaries as well as in the region of surface defects inside domains have local coordination close to that characteristic for bcc lattice (Fig. 1c). The dislocations forming the dipole gradually move in crystallite (Fig. 1a and b) tending to take a position with the lowest energy of their elastic interaction. They reach the
boundaries of the twins and further, already in the hcp phase are reconstructed that leads to the formation of steps on the boundaries (Fig. 1c). Thus, at bcc → hcp transformation dislocations 1 0 0 are inherited [20] however, they appear to be captured by the twin’s boundaries. Without the dislocations, the phonon instability of the lattice does not develop at 200 K and the transformation does not happen even after 105 steps. It takes place approximately during a three times smaller period of time at 1000 K. The crystallite also is divided into a system of twin domains, similar to how it was observed in [12,13]. Thus, the dislocation 1 0 0 essentially stimulates the martensitic transformation by inducing a specific type of lattice instability. 3.2. Dislocations with a Burgers vector 1 1 1 The results of the simulation of bcc → hcp transition for T = 200 K in a crystallite with the dipole of 1/21 1 0{1 1¯ 0} edge dislocations are shown in Fig. 3. On the last stage of the transformation, the same basic features as for 1 0 0{1 1¯ 0} dislocations are observed, namely, the development of long-wavelength phonon instability and formation of twin domains (Fig. 3b). However, the picture on early stages is essentially different from the previous one. The transformation begins with the formation of some pre-transition state in the vicinity of the dislocations (Fig. 3a). This feature is connected with the splitting of the dislocation 1/21 1 1 in the partial ones in a bcc plane {1 1 0} with the formation of a stacking fault. In a continuum model the splitting is described by the reaction [21] 1 ¯ 2 [1 1 1]
¯ + 1 [2 1 1] ¯ + 1 [0 1 1]. ¯ = 18 [0 1 1] 4 8
(1)
Besides that, in overcooled bcc phase of Zr a spreading of partial dislocation cores takes place therefore, the displacement turns out to be distributed in a large region in the gliding plane. Pre-transition state is formed in the vicinity of the stacking fault (Fig. 3a), and then propagates along the crystallite. The relation between the splitting (1) and martensitic
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Fig. 3. Bcc → hcp transformation kinetics in the vicinity of the dipole of 1/21 1 1 dislocations. Crystallite structure after N = 5000 (a), N = 40,000 (b) MD steps (1 step = 1 fs) is displayed. The atoms having bcc coordination of the nearest neighbors are shown by solid circles and the remaining atoms by the empty ones. The symbols ⊥ in Fig. 3b mark the positions of partial dislocations in hcp phase.
transformation was considered in details in [4]. The movement of partial dislocations 1/81 1 0 in every second bcc plane {1 1 0} realizes the lattice reconstruction corresponding to the short-wavelength stage of Burgers mechanism [19]. The long-wavelength stage of the transformation occurs after long enough time (N = 10,000), when the pre-transition state propagates in an essential part of the crystallite. As a result, the crystallite is divided into domains (Fig. 3b) with the boundaries parallel to the planes {1 1 0} of the initial bcc lattice. However, in contrast with the previous case, in the central part of the crystallite the domain of a strongly distorted phase is formed, misoriented on a significant angle, the rest if part of crystallite. The structure of this phase is difficult to determine unambiguously. For this region the pair distribution function has broad peaks with the positions corresponding to both hcp and fcc lattices. The appearance of this interlayer indicates that the kinetics of transformation is not determined in this case by the development of instability of a sole kind, but grows out of superposition of several mechanisms. It is difficult to determine the changes in the dislocation structure at intermediate stages of the transformation. However, they can be easily identified after the transition to hcp phase. The dislocations 1/21 1 1 are inherited under the scheme 1/2[1 1 1] → 1/3[1 1 2¯ 0] [20], and splitted (Fig. 3) according to a reaction 1 ¯ 3 [1 1 2 0]
= 19 [1 1 2¯ 0] + 29 [1 1 2¯ 0],
remaining to be mobile in hcp phase.
(2)
4. Discussion As follows from our MD simulation, the nucleation of a new phase as a result of phonon instability development can take place for various types of the dislocations. The pre-requisites for the appearance of this instability seems to be the presence of a soft branch of transverse phonons in the 1 1 0 direction and the strong anharmonicities of a potential for this branch in bcc Zr [16]. As a result, contrary to common views about the kinetics of first-order phase transitions [1,7], the new phase can arise immediately in the form of an ordered system of twins, and the stage of growth of a solitary nucleus does not occur at all. From the standpoint of the soliton approach in the theory of martensitic transitions [1], the formation of such a system of twins can be naturally described as the appearance of a soliton lattice. For the first time, the idea about local loss of lattice stability near defects owing to of elastic distortion fields around them was formulated by Clapp [22]. In [5] the lattice stability in the vicinity of a screw dislocation was analyzed. It was shown with the taking into account of the third-order anharmonicities that the shear strain created by the dislocation leads to lattice instability expressed in a decrease ¯ of the effective shear modulus Cs for the shear in {0 1 1} planes, with intersecting dislocation axis. For an edge dislocation, as it is easy to show, the change of C is determined by the dilatation and results in a decrease of C for a shift {1 1 0}1 1¯ 0 on planes, parallel to the dislocation line, Cs = 1/2(C11 − C12 ) − 1/2C112 (ε11 + ε22 ). The distribution of the dilatation ε11 + ε22 in Zr created by the dipole and obtained within the framework of the anisotropic
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phonons are universal for a transition from bcc phase to any close-packed structure since these transitions are usually related with the “softness” of the bcc lattice with respect to the shear strain associated with 1 1 0 phonon with a polarization vector 1¯ 1 0 [23]. At the same time, final stage of instability development, associated with an N4 phonon is specific to a transition to an hcp structure.
Acknowledgements
Fig. 4. Distribution of the dilatation ε11 + ε22 in Zr near the dislocation dipole obtained within the framework of the anisotropic theory of elasticity.
The authors are grateful to Dr. J. Morris and Dr. D. Turner, at the Ames Laboratory, IA, USA for providing the MD program, the interatomic interaction potential for Zr and enlightening discussion. They wish to thank Dr. L.E. Karkina and I. Karkin, Institute of Metal Physics, Russia, for help in diffraction patters calculations. The work is supported by Russian Basic Research Foundation, Grant 00-02-16086. References
theory of elasticity is shown in Fig. 4. Owing to a large elastic anisotropy, the distribution of strains is rather inhomogeneous that leads, apparently to specific features of early stages of transformation in the vicinity of the dipole 1 0 0 dislocations (Fig. 1a). The formation of an inhomogeneous structural state on developed stages of the process is a peculiarity of the martensitic transformation near 1/21 1 1 dislocations. It is possible to assume that in this case, except long-range fields stresses, the reconstruction of a lattice is promoted also by the presence of a stacking fault stripe [4], and the observable heterogeneity results from a superposition of various mechanisms. It should be noted that after two-fold increase of size of the crystallite, the picture of transformation remained qualitatively the same. However, the thickness of twinning domains forming in this case, depend both on the size of crystallite and on the distance between dislocations. We suppose, that the interatomic potential used by us gives a correct representation of transition, because without dislocations our results are in agreement with those of [12,13], where a different parameterization of potential was used. It should be stressed that the choice of periodic boundary conditions imitates to some extent the “compressed” state of precipitates of the new phase in the bcc matrix in contrast with a real martensitic transformation. A procedure that simulates constant pressure conditions, could give a different picture of the transition [16]. As for an applicability of our results to a more general case, it can be supposed that at least the initial stages of the lattice instability connected with the acoustic
[1] J.A. Krumhansl, Phase Transitions B 65 (1998) 109. [2] M.I. Katsnetson, A.V. Trefilov, Phys. Metals Metall. 64 (1987) 629. [3] V.G. Pushin, V.V. Kondratyev, V.N. Khachin, Pre-transitional phenomenons and martensitic transformations, Ural Division of the Russian Academy of Sciences, Yekaterinburg, 1998. [4] G.B. Olson, M. Cohen, Metall. Trans. A 7 (1976) 1905. [5] G. Guenin, P.F. Gobin, Metall. Trans. A 13 (1982) 1127. [6] A.R. Kuznetsov, Yu.N. Gornostyrev, R.I. Kuznetsov, I.B. Polovov, Phys. Metals Metall. 82 (1996) 449. [7] A.L. Roitburd, Rep. USSR Acad. Sci. 256 (1981) 80. [8] P. Entel, K. Kadau, R. Meyer, H.C. Herper, M. Schroteret, E. Hoffmann, Phase Transitions B 65 (1998) 79. [9] R. Meyer, P. Entel, Comput. Mater. Sci. 10 (1998) 10. [10] Y. Shao, P.C. Clapp, J.A. Rifkin, Metall. Mater. Trans. A 27 (1996) 1477. [11] M. Grujicic, S.G. Lai, P. Gumbsch, Mater. Sci. Eng. A 231 (1997) 151. [12] U. Pinsook, G.J. Ackland, Phys. Rev. B 59 (1999) 13462. [13] U. Pinsook, G.J. Ackland, Phys. Rev. B 58 (1998) 11252. [14] S.K. Sikka, Y.K. Vohra, R. Chidambaram, Prog. Mater. Sci. 27 (1982) 245. [15] J.R. Morris, Y.Y. Ye, K.M. Ho, C.T. Chan, Phil. Mag. A 72 (1995) 751. [16] B.L. Zhang, C.Z. Wang, K.M. Ro, D. Turner, Y.Y. Ye, Phys. Rev. Lett. 74 (1995) 1375. [17] A. Heiming, W. Petry, J. Trampenau, M. Alba, C. Herzig, H.R. Schober, G. Vogl, Phys. Rev. B 43 (1991) 10948. [18] F.A. Kassan-Ogli, V.E. Naish, I.V. Sagaradze, Phys. Metals Metall. 65 (1988) 481. [19] W.G. Burgers, Physica 1 (1934) 561. [20] G.E. Brainin, V.A. Likhachev, Metallofizika (Kiev) 4 (1982) 50. [21] J.B. Cohen, R. Hinton, K. Lay, S. Sass, Acta Metall. 10 (1962) 894. [22] P.C. Clapp, Phys. Status Solidi B 57 (1973) 561. [23] M.I. Katsnelson, I.I. Naumov, A.V. Trefilov, Phase Transitions B 49 (1994) 143.
Effect of the dislocations on the kinetics of a martensitic transition MD simulation of bcc–hcp transformation in Zr A.R. Kuznetsov a,∗ , Yu.N. Gornostyrev a , M.I. Katsnelson a , A.V. Trefilov b a
Institute of Metal Physics, Ural Division of the Russian Academy of Sciences, 620219 Ekaterinburg, Russia b Russian Science Center “Kurchatov Institute”, 123182 Moscow, Russia
Abstract Kinetics of the bcc → hcp transition in Zr is investigated by molecular dynamic (MD) simulation in the presence of dislocations. It is shown that the dislocations essentially stimulates the martensitic transformation by inducing a specific type of lattice instability. As a result, the transition near a dislocation becomes possible at much lower temperatures than in the ideal crystal. The transition occurs in two stages, a relatively long stage of a modulation of long-wavelength acoustic phonons, and a fast stage of short-wavelength phonon instability. Finally, the twin domains are formed with habit plane parallel to {1 1 0}bcc . © 2001 Elsevier Science B.V. All rights reserved. Keywords: Martensitic transition; Molecular dynamic simulation; Heterogeneous nucleation
1. Introduction The mechanism of the heterogeneous nucleation of a new phase in a real crystal remains still an unsolved problem [1]. Despite martensitic transformations are first-order phase transitions, a rather broad region of pre-transition anomalies, for example, the softening of the phonon spectra [2] is usually connected with them. It is a common point of view now [1,3] that the defects play a double role in heterogeneous nucleation of a martensitic phase. At first, the distortions of a lattice caused by the defects, sometimes make a structure of an initial phase closer to a structure of martensite thus, reducing a nucleation barrier [4]. Secondly, the defects can be centers of local lattice instability under the conditions of a lattice softening (for example, [5]), when the effective elastic modulus C = (C11 − C12 )/2 tends to be zero due to a specific type of a shear strain. To understand the mechanisms of the essential influence of the defects on the kinetics of martensitic transition at [6,7], it is necessary to clarify real microscopic picture of a crystal lattice reconstruction in the vicinity of a dislocation. To this aim, numerical simulations by molecular dynamics (MD) method could be very useful. Recently attempts have been done to study the kinetics of martensitic transitions within the framework of the microscopic approach [8–13]. In [10] the stress-induced coherent nucleation and growth of L10 martensitic phase in initially B2-ordered phase of NiAl intermetallic has been simulated. ∗ Corresponding author. E-mail address: a [email protected] (A.R. Kuznetsov).
The conclusion has been done, that the presence of the defects is necessary to initiate the martensitic transition. In [11] the martensitic transition in an alloy Fe0.8 Ni0.2 near a top of a crack has been investigated by MD method. In the present work the microscopic study of martensitic transition kinetics of Zr in the presence of such typical defects as dislocations is carried out. Zirconium is a convenient model material. With the decrease of temperature at T = 1136 K it undergoes bcc → hcp transformation, and at fast enough cooling demonstrates also the transition into metastable -phase [14]. The simulation of bcc → hcp transition kinetics in ideal Zr crystal at temperatures 1000 and 1500 K shows, that the transformation is accompanied by phonon spectrum anomalies [15,16,12,13] and is realized by phonon mechanism [12,13,17,18], ensuring reconstruction of bcc lattice under Burgers scheme [19]. We investigated the transition kinetics at high temperatures (1000 K), as well as after quenching to low temperature region (200 K), when the transformation in ideal crystal does not take place.
2. Computational procedure During MD simulation we used many-body interatomic interaction potentials [15,16], constructed in the framework of embedded atom method (EAM) and reproducing satisfactory elastic properties and phonon spectra of Zr. The numerical simulation of bcc → hcp transition in Zr was carried out at the presence of edge dislocations of two types 1 0 0{0 1 0} and 1/21 1 1{1 1 0}. A rectangular
0921-5093/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 6 7 7 - 4
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crystallite in the first case was chosen as a slab with a thickness of 12 atomic planes along the Z-axis (which is parallel to the dislocation line and the X-axis is parallel to the Burgers vector) containing about 10,000 atoms. For the second case (for dislocations 1/21 1 1{1 1 0}) it contains about 20,000 atoms with a thickness of 24 atomic planes. Periodic boundary conditions were used in all the directions. To provide the periodicity in the X- and Y-directions, a pair of dislocations of opposite sign (a dipole) was introduced into the crystallite. After introducing the dislocations in bcc phase the MD relaxation was carried out at temperatures 1000 and 200 K, at which hcp phase is stable (preservation of the given temperature was controlled by an average kinetic energy of atoms).
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3. Kinetics of the transformation in the presence of the dislocations 3.1. Dislocations with Burgers vector 1 0 0 The results of simulations for the crystallite with the dipole of 1 0 0{0 1 1} edge dislocations at T = 200 K are shown in Fig. 1. One can see that the transformation has been completed after N = 43,000 steps (time of simulation is given in steps, 1 step = 1 fs), that is illustrated by the changes of atomic pair distribution function g(R) (Fig. 1d). An important feature characterizing the transformation is the twins formation in a pyramid plane {1 0 1¯ 1}hcp , parallel to planes {1 1 0}bcc of an initial bcc lattice. Thus, usual
Fig. 1. Kinetics bcc → hcp transformation in the vicinity of the dipole 1 0 0 dislocations. A crystallite structure after N = 15,000 (a), 28,000 (b), 43,000 (c) MD steps (1 step = 1 fs) and appropriate pair distribution function g(R) (d). The atoms having bcc coordination of the nearest neighbors are shown by solid circles and the remaining atoms by open circles. The primes show the positions of the first, second and third neighbors in bcc (1, 2, 3) and hcp (1 , 2 , 3 ) lattice.
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Fig. 2. Diffraction patterns (for the crystallite from Fig. 1) for a vector g = (0 0 1) after N = 18,000 (a), 28,000 (b), 43,000 (c) MD steps.
orientational relations are fulfilled, the hcp lattice basis planes in the twins are parallel to bcc planes {1 1 0}, passing under an 60◦ angle to the twinning plane. The results of the calculations allow us to describe the process of the transformation in time. The transition starts with the stage of long-wavelength lattice instability. In a projection on a plane of Fig. 1 (plane XY) the corresponding displacements look initially like an acoustic transverse phonon with a vector q || 1 1 0, the wavelength of approximately 30–40 interatomic distances, and the polarization vector e || 1 1¯ 0. Then these sine-like lattice oscillations (Fig. 1a and b) are transformed into a sequence of kink-like excitations with quite sharp boundaries (“soliton lattice”) which leads to the alternation of regions with a different sign of a shear strain. Next, after the twinning domains are formed, a short-wavelength instability, which determines specifically hcp character of the new phase (Fig. 1c), develops in a short time. This is clearly seen from the pair distribution function (Fig. 1d), where the peaks corresponding to the position of the first, second and third neighbors in the bcc lattice vanish and similar peaks for the hcp structure appear. An additional information about the transformation kinetics is given by diffraction patterns calculated according to positions of atoms in crystallite after the various periods of time (Fig. 2). On the initial stage of transformation (corresponding to Fig. 1a) the splitting of reflexes happens, but the structure is still cubic. The reflexes showing the second phase appearing are observed after N = 28,000 (Fig. 2b), and at last after N = 43,000 (Fig. 2c) the lattice gains a hexagonal symmetry. The development of lattice instability is inhomogeneous as one can see in Fig. 1a. It reflects, apparently, the peculiarities of strain distribution created by dislocational dipoles. On the final stage of the transformation the atoms on the domain boundaries as well as in the region of surface defects inside domains have local coordination close to that characteristic for bcc lattice (Fig. 1c). The dislocations forming the dipole gradually move in crystallite (Fig. 1a and b) tending to take a position with the lowest energy of their elastic interaction. They reach the
boundaries of the twins and further, already in the hcp phase are reconstructed that leads to the formation of steps on the boundaries (Fig. 1c). Thus, at bcc → hcp transformation dislocations 1 0 0 are inherited [20] however, they appear to be captured by the twin’s boundaries. Without the dislocations, the phonon instability of the lattice does not develop at 200 K and the transformation does not happen even after 105 steps. It takes place approximately during a three times smaller period of time at 1000 K. The crystallite also is divided into a system of twin domains, similar to how it was observed in [12,13]. Thus, the dislocation 1 0 0 essentially stimulates the martensitic transformation by inducing a specific type of lattice instability. 3.2. Dislocations with a Burgers vector 1 1 1 The results of the simulation of bcc → hcp transition for T = 200 K in a crystallite with the dipole of 1/21 1 0{1 1¯ 0} edge dislocations are shown in Fig. 3. On the last stage of the transformation, the same basic features as for 1 0 0{1 1¯ 0} dislocations are observed, namely, the development of long-wavelength phonon instability and formation of twin domains (Fig. 3b). However, the picture on early stages is essentially different from the previous one. The transformation begins with the formation of some pre-transition state in the vicinity of the dislocations (Fig. 3a). This feature is connected with the splitting of the dislocation 1/21 1 1 in the partial ones in a bcc plane {1 1 0} with the formation of a stacking fault. In a continuum model the splitting is described by the reaction [21] 1 ¯ 2 [1 1 1]
¯ + 1 [2 1 1] ¯ + 1 [0 1 1]. ¯ = 18 [0 1 1] 4 8
(1)
Besides that, in overcooled bcc phase of Zr a spreading of partial dislocation cores takes place therefore, the displacement turns out to be distributed in a large region in the gliding plane. Pre-transition state is formed in the vicinity of the stacking fault (Fig. 3a), and then propagates along the crystallite. The relation between the splitting (1) and martensitic
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Fig. 3. Bcc → hcp transformation kinetics in the vicinity of the dipole of 1/21 1 1 dislocations. Crystallite structure after N = 5000 (a), N = 40,000 (b) MD steps (1 step = 1 fs) is displayed. The atoms having bcc coordination of the nearest neighbors are shown by solid circles and the remaining atoms by the empty ones. The symbols ⊥ in Fig. 3b mark the positions of partial dislocations in hcp phase.
transformation was considered in details in [4]. The movement of partial dislocations 1/81 1 0 in every second bcc plane {1 1 0} realizes the lattice reconstruction corresponding to the short-wavelength stage of Burgers mechanism [19]. The long-wavelength stage of the transformation occurs after long enough time (N = 10,000), when the pre-transition state propagates in an essential part of the crystallite. As a result, the crystallite is divided into domains (Fig. 3b) with the boundaries parallel to the planes {1 1 0} of the initial bcc lattice. However, in contrast with the previous case, in the central part of the crystallite the domain of a strongly distorted phase is formed, misoriented on a significant angle, the rest if part of crystallite. The structure of this phase is difficult to determine unambiguously. For this region the pair distribution function has broad peaks with the positions corresponding to both hcp and fcc lattices. The appearance of this interlayer indicates that the kinetics of transformation is not determined in this case by the development of instability of a sole kind, but grows out of superposition of several mechanisms. It is difficult to determine the changes in the dislocation structure at intermediate stages of the transformation. However, they can be easily identified after the transition to hcp phase. The dislocations 1/21 1 1 are inherited under the scheme 1/2[1 1 1] → 1/3[1 1 2¯ 0] [20], and splitted (Fig. 3) according to a reaction 1 ¯ 3 [1 1 2 0]
= 19 [1 1 2¯ 0] + 29 [1 1 2¯ 0],
remaining to be mobile in hcp phase.
(2)
4. Discussion As follows from our MD simulation, the nucleation of a new phase as a result of phonon instability development can take place for various types of the dislocations. The pre-requisites for the appearance of this instability seems to be the presence of a soft branch of transverse phonons in the 1 1 0 direction and the strong anharmonicities of a potential for this branch in bcc Zr [16]. As a result, contrary to common views about the kinetics of first-order phase transitions [1,7], the new phase can arise immediately in the form of an ordered system of twins, and the stage of growth of a solitary nucleus does not occur at all. From the standpoint of the soliton approach in the theory of martensitic transitions [1], the formation of such a system of twins can be naturally described as the appearance of a soliton lattice. For the first time, the idea about local loss of lattice stability near defects owing to of elastic distortion fields around them was formulated by Clapp [22]. In [5] the lattice stability in the vicinity of a screw dislocation was analyzed. It was shown with the taking into account of the third-order anharmonicities that the shear strain created by the dislocation leads to lattice instability expressed in a decrease ¯ of the effective shear modulus Cs for the shear in {0 1 1} planes, with intersecting dislocation axis. For an edge dislocation, as it is easy to show, the change of C is determined by the dilatation and results in a decrease of C for a shift {1 1 0}1 1¯ 0 on planes, parallel to the dislocation line, Cs = 1/2(C11 − C12 ) − 1/2C112 (ε11 + ε22 ). The distribution of the dilatation ε11 + ε22 in Zr created by the dipole and obtained within the framework of the anisotropic
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phonons are universal for a transition from bcc phase to any close-packed structure since these transitions are usually related with the “softness” of the bcc lattice with respect to the shear strain associated with 1 1 0 phonon with a polarization vector 1¯ 1 0 [23]. At the same time, final stage of instability development, associated with an N4 phonon is specific to a transition to an hcp structure.
Acknowledgements
Fig. 4. Distribution of the dilatation ε11 + ε22 in Zr near the dislocation dipole obtained within the framework of the anisotropic theory of elasticity.
The authors are grateful to Dr. J. Morris and Dr. D. Turner, at the Ames Laboratory, IA, USA for providing the MD program, the interatomic interaction potential for Zr and enlightening discussion. They wish to thank Dr. L.E. Karkina and I. Karkin, Institute of Metal Physics, Russia, for help in diffraction patters calculations. The work is supported by Russian Basic Research Foundation, Grant 00-02-16086. References
theory of elasticity is shown in Fig. 4. Owing to a large elastic anisotropy, the distribution of strains is rather inhomogeneous that leads, apparently to specific features of early stages of transformation in the vicinity of the dipole 1 0 0 dislocations (Fig. 1a). The formation of an inhomogeneous structural state on developed stages of the process is a peculiarity of the martensitic transformation near 1/21 1 1 dislocations. It is possible to assume that in this case, except long-range fields stresses, the reconstruction of a lattice is promoted also by the presence of a stacking fault stripe [4], and the observable heterogeneity results from a superposition of various mechanisms. It should be noted that after two-fold increase of size of the crystallite, the picture of transformation remained qualitatively the same. However, the thickness of twinning domains forming in this case, depend both on the size of crystallite and on the distance between dislocations. We suppose, that the interatomic potential used by us gives a correct representation of transition, because without dislocations our results are in agreement with those of [12,13], where a different parameterization of potential was used. It should be stressed that the choice of periodic boundary conditions imitates to some extent the “compressed” state of precipitates of the new phase in the bcc matrix in contrast with a real martensitic transformation. A procedure that simulates constant pressure conditions, could give a different picture of the transition [16]. As for an applicability of our results to a more general case, it can be supposed that at least the initial stages of the lattice instability connected with the acoustic
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