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Geometrical and chemical factors in the glass-forming ability
Scripta mater. 44 (2001) 1595–1598 www.elsevier.com/locate/scriptamat
GEOMETRICAL AND CHEMICAL FACTORS IN THE GLASS-FORMING ABILITY Masato Shimono and Hidehiro Onodera National Research Institute for Metals, 1-2-1 Sengen, Tsukuba, 305-0047 Japan (Received August 21, 2000) (Accepted in revised form December 15, 2000) Keywords: Metallic glasses; Glass-forming ability; Computer simulation; Rapid solidification Introduction It has been pointed out [1] that the alloy systems with high glass-forming ability satisfy the following three empirical rules: (1) Multicomponent alloy systems consisting of more than three components, (2) significantly different atomic size ratios among the main constituent elements, and (3) large negative heats of mixing among their elements. Especially, the latter two conditions can be taken as geometrical and chemical property, respectively. However, the physics behind these empirical rules is not understood sufficiently. So, in this study, we perform molecular dynamics (MD) simulations to clarify how the geometrical and the chemical properties of constituent atoms contribute to the glass-forming ability of the metallic systems. Computational Procedures The MD calculations for 4000 atom system with a periodic boundary condition have been performed on NEC SX-4 at National Research Institute for Metals. The pressure of the system is kept zero by using the constant-pressure formalisms [2,3]. The temperature of the system is set to the desired value by scaling the atomic momenta. To separate the geometrical and the chemical factors, we use the 8 – 4 Lennard-Jones (LJ) potential [4] as the interactive force between the elements i and j: V ij共r兲 ⫽ e ij兵共r ij/r兲 8 ⫺ 2共r ij/r兲 4其. The potential has the minimum ⫺eij at the distance rij, which are considered as the chemical bond strength and the atomic size, respectively. Here we deal with the binary system composed of elements 1 and 2, and, without loss of generality, assume r11 ⫽ 1 and r22 ⱕ 1. The atomic distance between different elements is defined as r12 ⫽ (r11 ⫹ r22)/2. For the chemical bonding parameters, we set as e11 ⫽ e22 ⫽ 1, and control the heat of mixing by varying the interaction e12. The glass-forming ability is estimated in the following two procedures: Rapid solidification from liquid phase and spontaneous amorphization from solid solution. In the former case, we first prepare a liquid phase and then rapidly quench the system to solidify. Thus we can estimate the glass-forming range under a specific cooling rate and the critical cooling rate needed for amorphization by melt quenching. In the latter case, we first prepare a fcc solid solution at nearly zero temperature by randomly assigning the solute atoms on the lattice sites, and then slowly heat up the system. If the system has high glass-forming ability, the solid-state amorphization should be observed before melting. 1359-6462/01/$–see front matter. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(01)00785-0
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Figure 1. Energy dependence on the temperature in a rapidly quenching process (closed circles) and a slowly cooling process (open triangles) for the monatomic system with r11 ⫽ e11 ⫽ 1. The left column shows the pair distribution functions of the solidified phases after cooling.
Results Figure 1 shows the energy change of the monatomic LJ system in the quenching process from liquid phase together with the pair distribution functions of the solidified phases. Closed circles correspond to the cooling rate of 5 ⫻ 10⫺3 in the reduced LJ units, while open triangles correspond to 5 ⫻ 10⫺4. Note that, in the monatomic system, the most stable phase is fcc crystalline phase and its melting point Tm is 0.67 in the reduced LJ units. If a cooling rate is high enough, the energy changes continuously, but its slope has a discontinuity at the solidification point. That is a signal for amorphization, at which we define the glass transition temperature Tg. By varying the atomic size ratio r22, the chemical interaction e12, and the concentration x2 of the solute element, we can calculate the glass-forming range by melt quenching under a constant cooling rate. First we show the pure geometrical effect on the glass-forming ability. In Fig. 2(a) the shaded regions denote the glass-forming ranges on the (x2, r22) plane at fixed chemical interaction as e12 ⫽ 1. The darker shading corresponds to the lower cooling rate. In the midway compositions, the glass-forming ability is extremely high if r22 ⱕ 0.9. We note that the glass-forming ranges are not symmetric in the composition and slightly shifted to the higher solute concentration. This suggests that the glass-forming ability should be higher when the adding element has the larger atomic size to the main constituent than when the adding element has the smaller size, supposed that the solute element is added by the same amount. To pick up the chemical effect on the
Figure 2. Composition ranges in which amorphous phases are formed by melt quenching in the binary systems. (a) Glass-forming ranges under the variation of r22 with e12 ⫽ 1; (b) glass-forming ranges under the variation of e12 with r22 ⫽ 0.95. The darker shading corresponds to the lower cooling rate.
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Figure 3. Energy dependence in the heating process from the solid solution of (r22, x2) ⫽ (0.83, 0.50); (b) the composition range in which the solid-state amorphization is observed; (c) the dependence of the transition temperature T2 on the chemical bond strength.
glass-forming ability, we first fix the atomic size ratio as r22 ⫽ 1 and vary the chemical bond strengths between 0.5 ⱕ e12 ⱕ 1.5. In this case, however, we have not observed any obvious change in the critical cooling rate compared to the system with e12 ⫽ 1. Then we turn to the system with r22 ⫽ 0.95 and vary e12. Figure 2(b) shows the glass-forming range in this case, where the heat of mixing ⌬H is defined as ⌬H ⫽ (e11 ⫹ e22)/2 ⫺ e12. There the larger negative ⌬H gives the higher glass-forming ability, although no amorphization has been observed in the case at the lowest cooling rate of 1 ⫻ 10⫺4. Next we proceed to the spontaneous amorphization. First we show the results for the system under the variation of r22 at fixed interaction as e12 ⫽ 1. Figure 3(a) shows the energy change in the heating process from the solid solution of the (r22, x2) ⫽ (0.83, 0.50) system. The fcc solid solution first evolved into bcc solid solution at T1, and then into an amorphous phase at T2, and finally into the liquid phase at T3, as reported in a similar MD study [5]. Such calculations have been performed at x2 ⫽ 0.25, 0.50, and 0.75. By varying r22 at each concentration, we can depict the range where the solid-state amorphization occurs by the shaded region in Fig. 3(b). We again see the asymmetric effect of the atomic size ratio on the glass-forming ability. For exploring the chemical effect, we have plotted in Fig. 3(c) the change of the transition temperature T2 at which the solid-state amorphization occurs under the variation of e12. The triangles, the circles, and the squares correspond to the case for (r22, x2) ⫽ (0.75, 0.25), (0.82, 0.50), and (0.80, 0.75). The decrease of the transition temperature indicates the increase of the relative stability of the amorphous phase against the solid solutions. Hence we conclude that the glass-forming ability increases with increasing e12 in the case of x2 ⫽ 0.75, while the opposite behavior found in other two cases is simply due to the overall increase of energy scale of the corresponding systems.
Figure 4. Energy dependence on the atomic size ratio of the fcc solid solutions (closed triangles), the bcc solid solutions (open squares), and the amorphous phases (closed circles) in the binary system at x2 ⫽ 0.5; (b) the schematic metastable phase diagram between these three phases on the (x2,r22) plane.
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Figure 5. Concentration dependence of the packing fraction (a) and the internal energy (b) for the fcc solid solutions and the amorphous phases in the binary system with r22 ⫽ 0.8.
Discussion and Summary The results show the following two features. (1) The effect of the chemical factor on the glass-forming ability is rather smaller than that of the geometrical factor and it becomes significant if it is combined with the geometrical effect. (2) Both the geometrical and the chemical factor act asymmetrically, in other words, the glass-forming ability is different whether we add the larger atomic size element to the main constituent element or add the smaller one. These results are understood from both an energetical and a geometrical point of view. Firstly, since the simulation time is too short for chemically ordered structures to form, we should take only the chemically disordered phases such as solid solutions and amorphous phases into account for phase equilibria. Figure 4(a) shows the internal energy dependence of the two types of solid solutions (fcc and bcc) and amorphous phases on the atomic size ratio in the binary system at x2 ⫽ 0.5. The evolution of phases (fcc-bcc-amorphous) depicted in Fig. 3(a) is easily understood from the energy relation in Fig. 4(a). This type of estimation leads us to the metastable phase diagram Fig. 4(b), which indicates the intrinsic glass-forming range [6]. Secondly, we focus on the atomic packing density of the amorphous phases. We define the atomic volume as that of the sphere of its LJ radius, and the packing fraction as the volume fraction of the atomic volume. Figure 5(a) shows the dependence of the packing fraction of fcc solid solutions and the amorphous phases on the concentration in the binary system with r22 ⫽ 0.8, while Figure 5(b) shows the dependence of the internal energy of these phases on the concentration. The higher packing fraction of a structure means the lower energy. Especially, in the midway composition where the glass-forming ability is high, the packing fraction of the amorphous phases is high. From an atomistic point of view, it suggests that a dense-packed cluster structure should form in the amorphous phases there. The icosahedral cluster [7] is one of possible candidates, but a closer look into the configuration of cluster structure in amorphous phases, as well as study for ternary systems, is one of the future problems. References 1. 2. 3. 4. 5. 6. 7.
A. Inoue, Mater. Trans. JIM. 36, 866 (1995). H. C. Andersen, J. Chem. Phys. 72, 2384 (1980). M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980). J. M. Sanchez, J. R. Barefoot, R. N. Jarrett, and J. K. Tien, Acta Metall. 42, 887 (1994). M. Li and W. L. Johnson, Phys. Rev. Lett. 70, 1120 (1993). Q. Zhang, W. S. Lai, and B. X. Liu, Phys. Rev. B. 59, 13521 (1999). M. Shimono and H. Onodera, Mater. Trans. JIM. 39, 147 (1998).
GEOMETRICAL AND CHEMICAL FACTORS IN THE GLASS-FORMING ABILITY Masato Shimono and Hidehiro Onodera National Research Institute for Metals, 1-2-1 Sengen, Tsukuba, 305-0047 Japan (Received August 21, 2000) (Accepted in revised form December 15, 2000) Keywords: Metallic glasses; Glass-forming ability; Computer simulation; Rapid solidification Introduction It has been pointed out [1] that the alloy systems with high glass-forming ability satisfy the following three empirical rules: (1) Multicomponent alloy systems consisting of more than three components, (2) significantly different atomic size ratios among the main constituent elements, and (3) large negative heats of mixing among their elements. Especially, the latter two conditions can be taken as geometrical and chemical property, respectively. However, the physics behind these empirical rules is not understood sufficiently. So, in this study, we perform molecular dynamics (MD) simulations to clarify how the geometrical and the chemical properties of constituent atoms contribute to the glass-forming ability of the metallic systems. Computational Procedures The MD calculations for 4000 atom system with a periodic boundary condition have been performed on NEC SX-4 at National Research Institute for Metals. The pressure of the system is kept zero by using the constant-pressure formalisms [2,3]. The temperature of the system is set to the desired value by scaling the atomic momenta. To separate the geometrical and the chemical factors, we use the 8 – 4 Lennard-Jones (LJ) potential [4] as the interactive force between the elements i and j: V ij共r兲 ⫽ e ij兵共r ij/r兲 8 ⫺ 2共r ij/r兲 4其. The potential has the minimum ⫺eij at the distance rij, which are considered as the chemical bond strength and the atomic size, respectively. Here we deal with the binary system composed of elements 1 and 2, and, without loss of generality, assume r11 ⫽ 1 and r22 ⱕ 1. The atomic distance between different elements is defined as r12 ⫽ (r11 ⫹ r22)/2. For the chemical bonding parameters, we set as e11 ⫽ e22 ⫽ 1, and control the heat of mixing by varying the interaction e12. The glass-forming ability is estimated in the following two procedures: Rapid solidification from liquid phase and spontaneous amorphization from solid solution. In the former case, we first prepare a liquid phase and then rapidly quench the system to solidify. Thus we can estimate the glass-forming range under a specific cooling rate and the critical cooling rate needed for amorphization by melt quenching. In the latter case, we first prepare a fcc solid solution at nearly zero temperature by randomly assigning the solute atoms on the lattice sites, and then slowly heat up the system. If the system has high glass-forming ability, the solid-state amorphization should be observed before melting. 1359-6462/01/$–see front matter. © 2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S1359-6462(01)00785-0
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Figure 1. Energy dependence on the temperature in a rapidly quenching process (closed circles) and a slowly cooling process (open triangles) for the monatomic system with r11 ⫽ e11 ⫽ 1. The left column shows the pair distribution functions of the solidified phases after cooling.
Results Figure 1 shows the energy change of the monatomic LJ system in the quenching process from liquid phase together with the pair distribution functions of the solidified phases. Closed circles correspond to the cooling rate of 5 ⫻ 10⫺3 in the reduced LJ units, while open triangles correspond to 5 ⫻ 10⫺4. Note that, in the monatomic system, the most stable phase is fcc crystalline phase and its melting point Tm is 0.67 in the reduced LJ units. If a cooling rate is high enough, the energy changes continuously, but its slope has a discontinuity at the solidification point. That is a signal for amorphization, at which we define the glass transition temperature Tg. By varying the atomic size ratio r22, the chemical interaction e12, and the concentration x2 of the solute element, we can calculate the glass-forming range by melt quenching under a constant cooling rate. First we show the pure geometrical effect on the glass-forming ability. In Fig. 2(a) the shaded regions denote the glass-forming ranges on the (x2, r22) plane at fixed chemical interaction as e12 ⫽ 1. The darker shading corresponds to the lower cooling rate. In the midway compositions, the glass-forming ability is extremely high if r22 ⱕ 0.9. We note that the glass-forming ranges are not symmetric in the composition and slightly shifted to the higher solute concentration. This suggests that the glass-forming ability should be higher when the adding element has the larger atomic size to the main constituent than when the adding element has the smaller size, supposed that the solute element is added by the same amount. To pick up the chemical effect on the
Figure 2. Composition ranges in which amorphous phases are formed by melt quenching in the binary systems. (a) Glass-forming ranges under the variation of r22 with e12 ⫽ 1; (b) glass-forming ranges under the variation of e12 with r22 ⫽ 0.95. The darker shading corresponds to the lower cooling rate.
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Figure 3. Energy dependence in the heating process from the solid solution of (r22, x2) ⫽ (0.83, 0.50); (b) the composition range in which the solid-state amorphization is observed; (c) the dependence of the transition temperature T2 on the chemical bond strength.
glass-forming ability, we first fix the atomic size ratio as r22 ⫽ 1 and vary the chemical bond strengths between 0.5 ⱕ e12 ⱕ 1.5. In this case, however, we have not observed any obvious change in the critical cooling rate compared to the system with e12 ⫽ 1. Then we turn to the system with r22 ⫽ 0.95 and vary e12. Figure 2(b) shows the glass-forming range in this case, where the heat of mixing ⌬H is defined as ⌬H ⫽ (e11 ⫹ e22)/2 ⫺ e12. There the larger negative ⌬H gives the higher glass-forming ability, although no amorphization has been observed in the case at the lowest cooling rate of 1 ⫻ 10⫺4. Next we proceed to the spontaneous amorphization. First we show the results for the system under the variation of r22 at fixed interaction as e12 ⫽ 1. Figure 3(a) shows the energy change in the heating process from the solid solution of the (r22, x2) ⫽ (0.83, 0.50) system. The fcc solid solution first evolved into bcc solid solution at T1, and then into an amorphous phase at T2, and finally into the liquid phase at T3, as reported in a similar MD study [5]. Such calculations have been performed at x2 ⫽ 0.25, 0.50, and 0.75. By varying r22 at each concentration, we can depict the range where the solid-state amorphization occurs by the shaded region in Fig. 3(b). We again see the asymmetric effect of the atomic size ratio on the glass-forming ability. For exploring the chemical effect, we have plotted in Fig. 3(c) the change of the transition temperature T2 at which the solid-state amorphization occurs under the variation of e12. The triangles, the circles, and the squares correspond to the case for (r22, x2) ⫽ (0.75, 0.25), (0.82, 0.50), and (0.80, 0.75). The decrease of the transition temperature indicates the increase of the relative stability of the amorphous phase against the solid solutions. Hence we conclude that the glass-forming ability increases with increasing e12 in the case of x2 ⫽ 0.75, while the opposite behavior found in other two cases is simply due to the overall increase of energy scale of the corresponding systems.
Figure 4. Energy dependence on the atomic size ratio of the fcc solid solutions (closed triangles), the bcc solid solutions (open squares), and the amorphous phases (closed circles) in the binary system at x2 ⫽ 0.5; (b) the schematic metastable phase diagram between these three phases on the (x2,r22) plane.
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Figure 5. Concentration dependence of the packing fraction (a) and the internal energy (b) for the fcc solid solutions and the amorphous phases in the binary system with r22 ⫽ 0.8.
Discussion and Summary The results show the following two features. (1) The effect of the chemical factor on the glass-forming ability is rather smaller than that of the geometrical factor and it becomes significant if it is combined with the geometrical effect. (2) Both the geometrical and the chemical factor act asymmetrically, in other words, the glass-forming ability is different whether we add the larger atomic size element to the main constituent element or add the smaller one. These results are understood from both an energetical and a geometrical point of view. Firstly, since the simulation time is too short for chemically ordered structures to form, we should take only the chemically disordered phases such as solid solutions and amorphous phases into account for phase equilibria. Figure 4(a) shows the internal energy dependence of the two types of solid solutions (fcc and bcc) and amorphous phases on the atomic size ratio in the binary system at x2 ⫽ 0.5. The evolution of phases (fcc-bcc-amorphous) depicted in Fig. 3(a) is easily understood from the energy relation in Fig. 4(a). This type of estimation leads us to the metastable phase diagram Fig. 4(b), which indicates the intrinsic glass-forming range [6]. Secondly, we focus on the atomic packing density of the amorphous phases. We define the atomic volume as that of the sphere of its LJ radius, and the packing fraction as the volume fraction of the atomic volume. Figure 5(a) shows the dependence of the packing fraction of fcc solid solutions and the amorphous phases on the concentration in the binary system with r22 ⫽ 0.8, while Figure 5(b) shows the dependence of the internal energy of these phases on the concentration. The higher packing fraction of a structure means the lower energy. Especially, in the midway composition where the glass-forming ability is high, the packing fraction of the amorphous phases is high. From an atomistic point of view, it suggests that a dense-packed cluster structure should form in the amorphous phases there. The icosahedral cluster [7] is one of possible candidates, but a closer look into the configuration of cluster structure in amorphous phases, as well as study for ternary systems, is one of the future problems. References 1. 2. 3. 4. 5. 6. 7.
A. Inoue, Mater. Trans. JIM. 36, 866 (1995). H. C. Andersen, J. Chem. Phys. 72, 2384 (1980). M. Parrinello and A. Rahman, Phys. Rev. Lett. 45, 1196 (1980). J. M. Sanchez, J. R. Barefoot, R. N. Jarrett, and J. K. Tien, Acta Metall. 42, 887 (1994). M. Li and W. L. Johnson, Phys. Rev. Lett. 70, 1120 (1993). Q. Zhang, W. S. Lai, and B. X. Liu, Phys. Rev. B. 59, 13521 (1999). M. Shimono and H. Onodera, Mater. Trans. JIM. 39, 147 (1998).