- Email: [email protected]
Zr bilayer through diffusion-limited-reaction observed by molecular-dynamics simulation
Computational Materials Science 14 (1999) 163±168
Solid-state amorphization of Ni/Zr bilayer through diusionlimited-reaction observed by molecular-dynamics simulation W.S Lai, B.X. Liu
*
Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People's Republic of China
Abstract Amorphization in the Ni/Zr bilayer upon isothermal annealing is observed by a molecular-dynamics simulation using an n-body potential. The disordering process is found to initiate and extend from a preset disordered interfacial layer through atomic diusion and alloying between Ni and Zr layers. Furthermore, it is clearly evidenced by our simulation that the growing process of amorphous phase follows a t1=2 time dependence kinetics, featuring a typical diusion-limited growth with asymmetric speeds in two opposite directions. All these results suggest that the observed amorphization is, in fact, a result of diusion-limited-reaction. Ó 1999 Elsevier Science B.V. All rights reserved. PACS: 64.70.Kb; 82.20.Wt; 61.43.Dq Keywords: Solid-state amorphization; Ni/Zr bilayer; Isothermal annealing; Molecular-dynamics simulation; Many-body potential; Diusion-limited-reaction
The ®rst solid-state amorphization was observed in 1983 in the Au±La system, which has a large negative heat of formation [1]. Since then, many similar results were reported in those systems featuring the similar characteristics [2±5]. For instance, Schr oder et al. reported that amorphous phase was ®rst formed at and then extended from the interfaces in the Co±Zr multilayers [2]. Clemens reported that X-ray diraction analysis con®rmed the formation of the disordered interfacial layers in the Ni/Zr multilayered ®lms immediately after sputtering deposition, and stressed that these layers played an important role in later crystal-toamorphous transition upon annealing [3]. In the nineties, molecular-dynamics (MD) simulation has
* Corresponding author. Tel.: 86-10-6278-5768; fax: 86-106277-1160; e-mail: [email protected].
been employed to model the amorphization process under thermal annealing or other nonequilibrium conditions [6,7], as MD simulation is capable of revealing the detailed mechanism of solid-state amorphization at an atomic scale and may give some new insight that experiments could not obtain. For example, Weissmann et al. studied a model system consisting of the Zr±Co super lattice with sharp interface upon annealing by employing a Lennard±Jones-type interatomic potential and reported that at a temperature well above 1100 K, all the Co atomic planes became completely disordered [6]. However, those authors have not observed any interdiusion during disordering process. It is the authors' view that interdiusion of the constituent atoms is of vital importance for growing an amorphous phase in solid-state reaction, as suggested by the experimental observations from many researchers [2±5].
0927-0256/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 1 0 2 - 5
164
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
In the present study, on the basis of an n-body potential, a Ni/Zr bilayer model system with a thin preset disordered interfacial layer is simulated to verify the role of diusion as well as to reveal the growth kinetics in solid-state amorphization upon isothermal annealing. We report here some of the ®ndings concerning the diusion-limited and asymmetric growth of the amorphous phase in the Ni/Zr bilayer at a medium temperature ranging from 300°C to 500°C. The bilayer model system for simulation consists of 12 Ni and 12 Zr fcc (0 0 1) planes parallel to the x±y-plane. To simplify the model, fcc instead of hcp Zr is used. This simpli®cation is rational, as fcc and hcp Zr has small energy dierence and fcc Zr has been observed in sputtered ®lms [8]. The sizes of the lattices in the x±yplane have been so chosen to make the mis®t at the Ni±Zr interface as close as possible to an experimentally measured ratio of the lattice constants. It is known that the nearest-neighbor distances of Zr respectively, in and Ni atoms are 3.18 and 2.49 A their single crystals, which makes approximately a ratio of 1.277. Within a manageable scale, the number of Ni and Zr atoms in their respective planes are selected to be 9 ´ 9 ´ 2 162 and 7 ´ 7 ´ 2 98, respectively, and therefore make a ratio of 9/7 1.285, which is quite close to a real situation. In the x- and y-directions, periodic boundary conditions using Ni lattice dimensions are imposed. In the z-direction, a free, instead of periodic, boundary condition is adopted to avoid two interfaces in the model system. The atoms on the external z surfaces are free to move and an atom, if crossing the block, is relocated onto the initial surfaces and set at zero energy. To avoid a possible drift of the block, an equal amount of the momentum lost is spread equally to all the atoms. Such boundaries do not aect signi®cantly the simulation results, as it will be seen that the crystalline planes near the surface, except those next to the interface, keep almost unchanged during annealing at a temperature up to 500°C. The size of the computational block in the z-direction has been obtained by varying the distance between the Ni and Zr lattices to an optimized value corresponding to a minimum enthalpy for the model system.
A many-body potential for the Ni±Zr system derived from the second-moment approximation of the tight-binding scheme by Massobrio et al. [9] is employed in the simulation. The cut-o distances for interactions of the same atoms are chosen to be at the middle of the ®rst and second nearest-neighbor distances in the respective Ni and Zr single crystals, since only the contributions from the nearest-neighbor atoms were taken into account in ®tting their potential parameters [9]. The cut-o distance for Ni±Zr interactions is as suggested by Massobrio chosen to be 5.3 A, et al. [9]. It should be pointed out that the adopted potential, though involving only nearest-neighbor atomistic interactions, has been proved to result in some thermodynamic properties for pure metals matching very well with the experimental values up to a temperature around 0.65 melting point [10]. Furthermore, the potential has also been tested to successfully reproduce both static and thermodynamic properties for NiZi2 alloy [11]. In recent years, this potential has been employed by other researchers to simulate the phase transformation under liquid melt quenching in the Ni±Zr system [12,13] and the growth of amorphous layer upon loading [7]. The excellent agreement of the simulation results with those obtained by experiments con®rms further that this potential is quite appropriate for describing the interactions in the Ni±Zr system. Based on the potential, the simulation is carried out with Parrinello±Rahman molecular dynamics scheme [14], and the equations of motion are solved using a fourth-order predictor± corrector algorithm of Gear with a time step t 5 ´ 10ÿ15 s. In order to trail closely the amorphization process of the model system, two quantitative methods are implemented to describe the degree of disorder, including the evaluation of the planar structure factor and the pair-correlation function. The planar structure factor, i.e., the Fourier transformation of the density in each crystallographic plane parallel to the x±y-plane, is de®ned according to Phillpot et al. [15] as, * S k; z
2 + Nz 1 X exp ik r ; j N 2z j1
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
where z labels the crystal plane to which the point rj belongs at initial con®guration, and k is an arbitrary vector of the planar reciprocal lattice [in our case, k (1, 1) is used]. The planar structure factor S 1 refers to an entirely ordered crystal and S 0 is for a completely disordered state. In practice, the atomic planes are de®ned as the original crystal planes and the planar structure factor can be used to indicate the degree of disorder in the corresponding crystal plane. More importantly, the pair-correlation function analysis is conducted to identify the structure of the region grown from the interface. In analyzing, the sampled atoms are those situated in the region of interest. According to Clemens' observations [3], there were disordered interfacial layers each of about 10 atomic planes emerging in the sputtered Ni±Zr multilayered ®lms, and these disordered layers played an important role in inducing amorphization. To have our model system with a starting con®guration as close as possible to the real situation, an disordered interfacial layer is preset by randomly exchanging equal number of Ni and Zr atoms in the ®rst two atomic planes of both Ni and Zr next to each other, and is named as an initial disordered interlayer. The concept of long range order (LRO) parameter originally proposed to describe the ordering in crystalline alloys [9] is adopted to characterize quantitatively the amount of disorder created by exchanging atoms in the initial interlayer. The LRO parameter is de®ned as g (p ÿ c)/(1 ÿ c), where p and c are the probabilities of the presence of an A type atom (A Ni or Zr) on its own lattice site and the molar ratio of A atoms, respectively, both in the initial interlayer. The amount of disorder in the initial interlayer is set to be g 0, corresponding to an exchange of 122 atoms in total. After a random exchange of atoms in the initial interlayer, the model system is pre-run at 300 K for 5000 MD time steps to reach a state, de®ned as the starting con®guration in our MD simulation. The simulation starts by ®rst boosting the velocities of the atoms in each time step at a rate of 4 ´ 1013 K/s up to various temperatures ranging from 300± 500°C, and then isothermally annealing the bilayer to induce growth of the amorphous phase. As the
165
velocities of atoms would change during annealing, a time-to-time rescaling of the velocities of atoms to the desired temperatures is executed at every 100 time steps, if an average deviation from the desired temperature is over 1°C. As the velocities of atoms are not rescaled at each time step to desired temperature, some thermal ¯uctuation may be involved in the system. The starting con®guration is shown in Fig. 1 by a projection of the atomic positions on an x±zplane, reached by introducing chemical disordering through exchanging equal numbers of Ni and Zr atoms, plus a pre-running at 300 K for 5000 time steps. It is worthwhile mentioning that after pre-running for 4000 time steps, the temperature of the interfacial layer has already reached an equilibrium with the surrounding lattices. One can observe from Fig. 1 that the initial interlayer is indeed in a disordered state for starting simulation. Furthermore, a total pair-correlation function calculation for the interfacial layer (atomic planes No. 11±14) results in a curve similar to those observed for the liquid and amorphous alloys. It is therefore concluded that the interfacial layer is of amorphous structure. One may also notice from Fig. 1 that a few Ni (Zr) atoms have diused into Zr (Ni) lattice. However, only a few atoms penetrating into the partner lattice cannot
Fig. 1. Projection of the atomic positions on the x±z-plane of starting con®guration. The model system consists of 12 Ni and 12 Zr atomic planes and they are numbered No. 1±12 for Ni and No. 13±24 for Zr, respectively, from bottom to top. The projection only shows the atomic positions of the atoms on No. 7±18 atomic planes. The Ni and Zr atoms are denoted by open circles and full triangles, respectively.
166
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
signi®cantly disrupt the crystalline structures of the atomic planes next to the interfacial layer, as clearly seen in Fig. 1. We now discuss the simulation results by a representative example obtained from annealing of the Ni/Zr bilayer at 400°C, as similar results are also obtained for annealing at 350°C and 500°C. Fig. 2 shows a projection of atomic positions onto the x±z-plane after isothermal annealing at 400°C for 40,000 MD time steps. In Fig. 2, the disordering process during annealing is clearly illustrated and visualized. The atomic planes next to the initial interfacial layer in Fig. 1, i.e., plane No. 10 and 15 are now completely disordered. A close look to Fig. 2 leads to an important ®nding that considerable Ni (Zr) atoms have diused into Zr (Ni) lattice and that some Ni (Zr) atoms have penetrated into the second Zr (Ni) atomic plane next to the initial interfacial layer, resulting in signi®cant alloying between the Ni and Zr atoms. The original crystalline structure of the atomic planes next to the initial interfacial layer have been almost completely disrupted. All the above peculiarities are corroborated by the structure factors S(k), which are depicted in Fig. 3 at several time steps during annealing. The planar structure factors, S(k), for the atomic planes next to the initial interfacial layer (plane No. 10 and 15), decreases from 0.57 and 0.66 at
Fig. 2. Projection of the atomic positions on the x±z-plane of the Ni/Zr bilayer after annealing at 400°C for 40,000 MD steps. The notations in this Figure are the same as those de®ned in Fig. 1.
Fig. 3. Changing of the planar structure factors S(k) of the Ni/ Zr bilayer at 5000, 10,000 and 35,000 MD steps during 400°C isothermal annealing.
5000 time steps to 0.05 and 0.08 at 35,000 time steps, respectively. It can be seen from Fig. 3 that the initially disordered interfacial layer has extended from 4 to 6 atomic planes after annealing of the Ni/Zr bilayer at 400°C for a corresponding time of about 10ÿ10 s. Meanwhile, the S(k) of all the atomic planes outside of the disordered interlayer up to both Ni and Zr surfaces remain almost unchanged, as no partner atoms have diused into those atomic planes. The unchanged crystalline property of the outer planes suggests the outer planes have no signi®cant eect on the reaction taking place near the interfacial layer. Both Figs. 2 and 3 show that the periodicity of the atomic planes has been gradually disrupted by continuously introducing the alloying atoms into the lattice planes. To give a ®rm evidence of the disordered structure obtained after annealing, a total pair-correlation function for the six atomic planes (No. 10±15) is calculated. A resultant curve is displayed in Fig. 4, which is very much like those observed commonly for the amorphous alloys. We now discuss the time dependence of the growth of the disordered interlayer. Fig. 5 shows two curves of the growing thicknesses (X) of the amorphous interlayer in two opposite directions towards Ni and Zr lattices at 500°C as a function of t1=2 . As the disordered interlayer is extended outwards in a plane-by-plane mode, at a speci®c
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
Fig. 4. The total pair-correlation function, after isothermal annealing of the Ni/Zr bilayer at 400°C for 40,000 time steps, of six atomic planes in the interfacial layer obtained by averaging the data between 40,000 and 42,000 time steps.
Fig. 5. The growing thicknesses (X) of the amorphous interlayer as a function of t1=2 in two opposite directions towards Ni and Zr lattices at 500°C.
time, the corresponding disordered thickness is obtained by counting the atomic planes from the original interface to an outmost one with a planar structure factor S < 0.1. One sees that two X±t1=2
167
curves are in an excellent linear shape, indicating obviously that the growth of the amorphous interlayer follows exactly a diusion governed t1=2 law. To our knowledge, this is the ®rst observation of such correlation at atomic scale, though there were experimental reports which also suggested the same result in the Ni±Zr system [3,4]. One also observes with interest that the slopes of the curves are quite dierent, which means that the growth speeds are dierent in two opposite directions towards two constituent metals. It turns out that the slope to Ni side is greater than that to Zr side, suggesting a possible asymmetric growth of the disordered interlayer upon solid-state reaction in the Ni/Zr bilayer. It is a reasonable prediction from a theoretical point of view that two dierent metals would naturally behave dierently in going amorphization. In this respect, the observation reported by Meng et al. [5] can serve as an indirect evidence in support to the prediction. In their experiment, Meng et al. observed that after the amorphous NiZr interlayer reached a critical thickness, a NiZr intermetallic compound started to form at the amorphous-NiZr/Zr interface [5]. Such an observation can be correlated to the different growing speeds of the disordered interlayer towards Ni and Zr lattices. As the growth of the disordered interlayer towards Zr side is slower than towards Ni side, a frustration of amorphization due to a failure in competing with a growing NiZr intermetallic compound would occur earlier at Zr than Ni side, which is exactly suggested by our simulation result. In conclusion, a molecular-dynamics simulation in the Ni/Zr bilayer with an n-body potential reveals that amorphization in the Ni/Zr bilayer upon isothermal annealing is achieved through an extension of a preset disordered interfacial layer. An important ®nding is that a mutual diusion of Ni (Zr) atoms into Zr (Ni) lattice takes place during annealing. Such atomic diusion induces alloying between Ni and Zr layers, causing the original crystalline planes outside of the interfacial layer to gradual disorder, and results in a plane-by-plane growth of amorphous phase. Moreover, the amorphous Ni±Zr phase is formed through a process of diusion-limited-growth following a ``t1=2 '' law.
168
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
Acknowledgements This study is supported by National Natural Science Foundation of China. Partial ®nancial aid from the Tsinghua Administration is also gratefully acknowledged. References [1] R.B. Schwarz, W.L. Johnson, Phys. Rev. Lett. 51 (1983) 415. [2] H. Schr oder, K. Samwer, U. Koster, Phys. Rev. Lett. 54 (1985) 197. [3] B.M. Clemens, Phys. Rev. B 33 (1986) 7615. [4] E.J. Cotts, W.J. Meng, W.L. Johnson, Phys. Rev. Lett. 57 (1986) 2295. [5] W.J. Meng, B. Fultz, E. Ma, W.L. Johnson, Appl. Phys. Lett. 51 (1987) 1693.
[6] M. Weissmann, R. Ramirez, M. Kiwi, Phys. Rev. B 46 (1992) 2577. [7] P. Mura, P. Demontis, G.B. Suritti, V. Rosato, M.V. Antisari, Phys. Rev. B 50 (1994) 2850. [8] K.L. Chopra, M.R. Randlett, R.H. Du, Phil. Mag. 16 (1967) 261. [9] C. Massobrio, V. Pontikis, G. Martin, Phys. Rev. Lett. 62 (1989) 1142. [10] V. Rosato, M. Guillope, B. Legrand, Phil. Mag. A 59 (1989) 321. [11] C. Massobrio, V. Pontikis, G. Martin, Phys. Rev. B 41 (1990) 10486. [12] T. Aihara Jr., K. Aoki, T. Masumoto, Scripta Metall. Mater. 28 (1993) 1003. [13] T. Aihara Jr., K. Aoki, T. Masumoto, Mater. Sci. Eng. A179/A180 (1994) 256. [14] M. Parrinello, A. Rahman, J. Appl. Phys 52 (1981) 7182. [15] S.R. Phillpot, S. Yip, D. Wolf, Comput. Phys. 3 (1989) 20.
Solid-state amorphization of Ni/Zr bilayer through diusionlimited-reaction observed by molecular-dynamics simulation W.S Lai, B.X. Liu
*
Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, People's Republic of China
Abstract Amorphization in the Ni/Zr bilayer upon isothermal annealing is observed by a molecular-dynamics simulation using an n-body potential. The disordering process is found to initiate and extend from a preset disordered interfacial layer through atomic diusion and alloying between Ni and Zr layers. Furthermore, it is clearly evidenced by our simulation that the growing process of amorphous phase follows a t1=2 time dependence kinetics, featuring a typical diusion-limited growth with asymmetric speeds in two opposite directions. All these results suggest that the observed amorphization is, in fact, a result of diusion-limited-reaction. Ó 1999 Elsevier Science B.V. All rights reserved. PACS: 64.70.Kb; 82.20.Wt; 61.43.Dq Keywords: Solid-state amorphization; Ni/Zr bilayer; Isothermal annealing; Molecular-dynamics simulation; Many-body potential; Diusion-limited-reaction
The ®rst solid-state amorphization was observed in 1983 in the Au±La system, which has a large negative heat of formation [1]. Since then, many similar results were reported in those systems featuring the similar characteristics [2±5]. For instance, Schr oder et al. reported that amorphous phase was ®rst formed at and then extended from the interfaces in the Co±Zr multilayers [2]. Clemens reported that X-ray diraction analysis con®rmed the formation of the disordered interfacial layers in the Ni/Zr multilayered ®lms immediately after sputtering deposition, and stressed that these layers played an important role in later crystal-toamorphous transition upon annealing [3]. In the nineties, molecular-dynamics (MD) simulation has
* Corresponding author. Tel.: 86-10-6278-5768; fax: 86-106277-1160; e-mail: [email protected].
been employed to model the amorphization process under thermal annealing or other nonequilibrium conditions [6,7], as MD simulation is capable of revealing the detailed mechanism of solid-state amorphization at an atomic scale and may give some new insight that experiments could not obtain. For example, Weissmann et al. studied a model system consisting of the Zr±Co super lattice with sharp interface upon annealing by employing a Lennard±Jones-type interatomic potential and reported that at a temperature well above 1100 K, all the Co atomic planes became completely disordered [6]. However, those authors have not observed any interdiusion during disordering process. It is the authors' view that interdiusion of the constituent atoms is of vital importance for growing an amorphous phase in solid-state reaction, as suggested by the experimental observations from many researchers [2±5].
0927-0256/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 9 8 ) 0 0 1 0 2 - 5
164
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
In the present study, on the basis of an n-body potential, a Ni/Zr bilayer model system with a thin preset disordered interfacial layer is simulated to verify the role of diusion as well as to reveal the growth kinetics in solid-state amorphization upon isothermal annealing. We report here some of the ®ndings concerning the diusion-limited and asymmetric growth of the amorphous phase in the Ni/Zr bilayer at a medium temperature ranging from 300°C to 500°C. The bilayer model system for simulation consists of 12 Ni and 12 Zr fcc (0 0 1) planes parallel to the x±y-plane. To simplify the model, fcc instead of hcp Zr is used. This simpli®cation is rational, as fcc and hcp Zr has small energy dierence and fcc Zr has been observed in sputtered ®lms [8]. The sizes of the lattices in the x±yplane have been so chosen to make the mis®t at the Ni±Zr interface as close as possible to an experimentally measured ratio of the lattice constants. It is known that the nearest-neighbor distances of Zr respectively, in and Ni atoms are 3.18 and 2.49 A their single crystals, which makes approximately a ratio of 1.277. Within a manageable scale, the number of Ni and Zr atoms in their respective planes are selected to be 9 ´ 9 ´ 2 162 and 7 ´ 7 ´ 2 98, respectively, and therefore make a ratio of 9/7 1.285, which is quite close to a real situation. In the x- and y-directions, periodic boundary conditions using Ni lattice dimensions are imposed. In the z-direction, a free, instead of periodic, boundary condition is adopted to avoid two interfaces in the model system. The atoms on the external z surfaces are free to move and an atom, if crossing the block, is relocated onto the initial surfaces and set at zero energy. To avoid a possible drift of the block, an equal amount of the momentum lost is spread equally to all the atoms. Such boundaries do not aect signi®cantly the simulation results, as it will be seen that the crystalline planes near the surface, except those next to the interface, keep almost unchanged during annealing at a temperature up to 500°C. The size of the computational block in the z-direction has been obtained by varying the distance between the Ni and Zr lattices to an optimized value corresponding to a minimum enthalpy for the model system.
A many-body potential for the Ni±Zr system derived from the second-moment approximation of the tight-binding scheme by Massobrio et al. [9] is employed in the simulation. The cut-o distances for interactions of the same atoms are chosen to be at the middle of the ®rst and second nearest-neighbor distances in the respective Ni and Zr single crystals, since only the contributions from the nearest-neighbor atoms were taken into account in ®tting their potential parameters [9]. The cut-o distance for Ni±Zr interactions is as suggested by Massobrio chosen to be 5.3 A, et al. [9]. It should be pointed out that the adopted potential, though involving only nearest-neighbor atomistic interactions, has been proved to result in some thermodynamic properties for pure metals matching very well with the experimental values up to a temperature around 0.65 melting point [10]. Furthermore, the potential has also been tested to successfully reproduce both static and thermodynamic properties for NiZi2 alloy [11]. In recent years, this potential has been employed by other researchers to simulate the phase transformation under liquid melt quenching in the Ni±Zr system [12,13] and the growth of amorphous layer upon loading [7]. The excellent agreement of the simulation results with those obtained by experiments con®rms further that this potential is quite appropriate for describing the interactions in the Ni±Zr system. Based on the potential, the simulation is carried out with Parrinello±Rahman molecular dynamics scheme [14], and the equations of motion are solved using a fourth-order predictor± corrector algorithm of Gear with a time step t 5 ´ 10ÿ15 s. In order to trail closely the amorphization process of the model system, two quantitative methods are implemented to describe the degree of disorder, including the evaluation of the planar structure factor and the pair-correlation function. The planar structure factor, i.e., the Fourier transformation of the density in each crystallographic plane parallel to the x±y-plane, is de®ned according to Phillpot et al. [15] as, * S k; z
2 + Nz 1 X exp ik r ; j N 2z j1
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
where z labels the crystal plane to which the point rj belongs at initial con®guration, and k is an arbitrary vector of the planar reciprocal lattice [in our case, k (1, 1) is used]. The planar structure factor S 1 refers to an entirely ordered crystal and S 0 is for a completely disordered state. In practice, the atomic planes are de®ned as the original crystal planes and the planar structure factor can be used to indicate the degree of disorder in the corresponding crystal plane. More importantly, the pair-correlation function analysis is conducted to identify the structure of the region grown from the interface. In analyzing, the sampled atoms are those situated in the region of interest. According to Clemens' observations [3], there were disordered interfacial layers each of about 10 atomic planes emerging in the sputtered Ni±Zr multilayered ®lms, and these disordered layers played an important role in inducing amorphization. To have our model system with a starting con®guration as close as possible to the real situation, an disordered interfacial layer is preset by randomly exchanging equal number of Ni and Zr atoms in the ®rst two atomic planes of both Ni and Zr next to each other, and is named as an initial disordered interlayer. The concept of long range order (LRO) parameter originally proposed to describe the ordering in crystalline alloys [9] is adopted to characterize quantitatively the amount of disorder created by exchanging atoms in the initial interlayer. The LRO parameter is de®ned as g (p ÿ c)/(1 ÿ c), where p and c are the probabilities of the presence of an A type atom (A Ni or Zr) on its own lattice site and the molar ratio of A atoms, respectively, both in the initial interlayer. The amount of disorder in the initial interlayer is set to be g 0, corresponding to an exchange of 122 atoms in total. After a random exchange of atoms in the initial interlayer, the model system is pre-run at 300 K for 5000 MD time steps to reach a state, de®ned as the starting con®guration in our MD simulation. The simulation starts by ®rst boosting the velocities of the atoms in each time step at a rate of 4 ´ 1013 K/s up to various temperatures ranging from 300± 500°C, and then isothermally annealing the bilayer to induce growth of the amorphous phase. As the
165
velocities of atoms would change during annealing, a time-to-time rescaling of the velocities of atoms to the desired temperatures is executed at every 100 time steps, if an average deviation from the desired temperature is over 1°C. As the velocities of atoms are not rescaled at each time step to desired temperature, some thermal ¯uctuation may be involved in the system. The starting con®guration is shown in Fig. 1 by a projection of the atomic positions on an x±zplane, reached by introducing chemical disordering through exchanging equal numbers of Ni and Zr atoms, plus a pre-running at 300 K for 5000 time steps. It is worthwhile mentioning that after pre-running for 4000 time steps, the temperature of the interfacial layer has already reached an equilibrium with the surrounding lattices. One can observe from Fig. 1 that the initial interlayer is indeed in a disordered state for starting simulation. Furthermore, a total pair-correlation function calculation for the interfacial layer (atomic planes No. 11±14) results in a curve similar to those observed for the liquid and amorphous alloys. It is therefore concluded that the interfacial layer is of amorphous structure. One may also notice from Fig. 1 that a few Ni (Zr) atoms have diused into Zr (Ni) lattice. However, only a few atoms penetrating into the partner lattice cannot
Fig. 1. Projection of the atomic positions on the x±z-plane of starting con®guration. The model system consists of 12 Ni and 12 Zr atomic planes and they are numbered No. 1±12 for Ni and No. 13±24 for Zr, respectively, from bottom to top. The projection only shows the atomic positions of the atoms on No. 7±18 atomic planes. The Ni and Zr atoms are denoted by open circles and full triangles, respectively.
166
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
signi®cantly disrupt the crystalline structures of the atomic planes next to the interfacial layer, as clearly seen in Fig. 1. We now discuss the simulation results by a representative example obtained from annealing of the Ni/Zr bilayer at 400°C, as similar results are also obtained for annealing at 350°C and 500°C. Fig. 2 shows a projection of atomic positions onto the x±z-plane after isothermal annealing at 400°C for 40,000 MD time steps. In Fig. 2, the disordering process during annealing is clearly illustrated and visualized. The atomic planes next to the initial interfacial layer in Fig. 1, i.e., plane No. 10 and 15 are now completely disordered. A close look to Fig. 2 leads to an important ®nding that considerable Ni (Zr) atoms have diused into Zr (Ni) lattice and that some Ni (Zr) atoms have penetrated into the second Zr (Ni) atomic plane next to the initial interfacial layer, resulting in signi®cant alloying between the Ni and Zr atoms. The original crystalline structure of the atomic planes next to the initial interfacial layer have been almost completely disrupted. All the above peculiarities are corroborated by the structure factors S(k), which are depicted in Fig. 3 at several time steps during annealing. The planar structure factors, S(k), for the atomic planes next to the initial interfacial layer (plane No. 10 and 15), decreases from 0.57 and 0.66 at
Fig. 2. Projection of the atomic positions on the x±z-plane of the Ni/Zr bilayer after annealing at 400°C for 40,000 MD steps. The notations in this Figure are the same as those de®ned in Fig. 1.
Fig. 3. Changing of the planar structure factors S(k) of the Ni/ Zr bilayer at 5000, 10,000 and 35,000 MD steps during 400°C isothermal annealing.
5000 time steps to 0.05 and 0.08 at 35,000 time steps, respectively. It can be seen from Fig. 3 that the initially disordered interfacial layer has extended from 4 to 6 atomic planes after annealing of the Ni/Zr bilayer at 400°C for a corresponding time of about 10ÿ10 s. Meanwhile, the S(k) of all the atomic planes outside of the disordered interlayer up to both Ni and Zr surfaces remain almost unchanged, as no partner atoms have diused into those atomic planes. The unchanged crystalline property of the outer planes suggests the outer planes have no signi®cant eect on the reaction taking place near the interfacial layer. Both Figs. 2 and 3 show that the periodicity of the atomic planes has been gradually disrupted by continuously introducing the alloying atoms into the lattice planes. To give a ®rm evidence of the disordered structure obtained after annealing, a total pair-correlation function for the six atomic planes (No. 10±15) is calculated. A resultant curve is displayed in Fig. 4, which is very much like those observed commonly for the amorphous alloys. We now discuss the time dependence of the growth of the disordered interlayer. Fig. 5 shows two curves of the growing thicknesses (X) of the amorphous interlayer in two opposite directions towards Ni and Zr lattices at 500°C as a function of t1=2 . As the disordered interlayer is extended outwards in a plane-by-plane mode, at a speci®c
W.S. Lai, B.X. Liu / Computational Materials Science 14 (1999) 163±168
Fig. 4. The total pair-correlation function, after isothermal annealing of the Ni/Zr bilayer at 400°C for 40,000 time steps, of six atomic planes in the interfacial layer obtained by averaging the data between 40,000 and 42,000 time steps.
Fig. 5. The growing thicknesses (X) of the amorphous interlayer as a function of t1=2 in two opposite directions towards Ni and Zr lattices at 500°C.
time, the corresponding disordered thickness is obtained by counting the atomic planes from the original interface to an outmost one with a planar structure factor S < 0.1. One sees that two X±t1=2
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curves are in an excellent linear shape, indicating obviously that the growth of the amorphous interlayer follows exactly a diusion governed t1=2 law. To our knowledge, this is the ®rst observation of such correlation at atomic scale, though there were experimental reports which also suggested the same result in the Ni±Zr system [3,4]. One also observes with interest that the slopes of the curves are quite dierent, which means that the growth speeds are dierent in two opposite directions towards two constituent metals. It turns out that the slope to Ni side is greater than that to Zr side, suggesting a possible asymmetric growth of the disordered interlayer upon solid-state reaction in the Ni/Zr bilayer. It is a reasonable prediction from a theoretical point of view that two dierent metals would naturally behave dierently in going amorphization. In this respect, the observation reported by Meng et al. [5] can serve as an indirect evidence in support to the prediction. In their experiment, Meng et al. observed that after the amorphous NiZr interlayer reached a critical thickness, a NiZr intermetallic compound started to form at the amorphous-NiZr/Zr interface [5]. Such an observation can be correlated to the different growing speeds of the disordered interlayer towards Ni and Zr lattices. As the growth of the disordered interlayer towards Zr side is slower than towards Ni side, a frustration of amorphization due to a failure in competing with a growing NiZr intermetallic compound would occur earlier at Zr than Ni side, which is exactly suggested by our simulation result. In conclusion, a molecular-dynamics simulation in the Ni/Zr bilayer with an n-body potential reveals that amorphization in the Ni/Zr bilayer upon isothermal annealing is achieved through an extension of a preset disordered interfacial layer. An important ®nding is that a mutual diusion of Ni (Zr) atoms into Zr (Ni) lattice takes place during annealing. Such atomic diusion induces alloying between Ni and Zr layers, causing the original crystalline planes outside of the interfacial layer to gradual disorder, and results in a plane-by-plane growth of amorphous phase. Moreover, the amorphous Ni±Zr phase is formed through a process of diusion-limited-growth following a ``t1=2 '' law.
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