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Thermal conductivity and mechanical properties of ZrxCu90−xAl10 under tension using molecular dynamics simulations
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International Journal of Mechanical Sciences 000 (2017) 1–6
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Thermal conductivity and mechanical properties of Zrx Cu90−x Al10 under tension using molecular dynamics simulations S.K. Deb Nath Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
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Keywords: Thermal conductivity Young‘s modulus Yield strength Ternary Zrx Cu90−x Al10 metallic glass Molecular dynamics simulations
a b s t r a c t Using equilibrium molecular dynamics simulations, thermal conductivity of different types of metallic glass, Zrx Cu90−x Al10 is obtained and effects of concentration of their constituents on their thermal conductivity are studied as comparative manner. Young‘s modulus and yield strength of different metallic glass, Zrx Cu90−x Al10 are obtained considering temperature, size effects and strain rates. Effects of temperature, effects of sizes and strain rate on the Young‘s modulus and yield strength of Zrx Cu90-x Al10 are studied. The mechanical properties and thermal conductivities of these metallic glasses obtained by us are compared with the published mechanical properties and thermal conductivity of these metallic glasses to validate the present study. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Metallic glasses are widely used in many mechanical, aerospace industries as glass materials due to their higher strength, toughness compared to conventional glass. There are many types of metallic glasses have been developed combining different metallic elements. Now a day, ternary metallic glass based on Zr, Cu and Al elements are widely used in different engineering structures due to having good glass forming ability and high strength and toughness. Equilibrium and non-equilibrium approaches using molecular dynamics simulations can be used to study the thermal conductivity of ternary metallic glass Zrx Cu90-x Al10 where x is the percentage of Zr in ternary metallic glasses Zrx Cu90−x Al10 . Thermal conductivity of solid argon [1], Ge [2], Lennard-Jones argon liquid [3,4], Silica glass [5], liquid He [6] were studied by equilibrium molecular dynamics simulations using Green-Kubo formalism. The molecular dynamics technique was used to study the heat flux evolution in a temperate region where the thermal conductivity has temperature dependencies [7]. The thermal conductivity of a monoatomic face-centered cubic lattice has been calculated over a range of temperatures from one-twentieth to one-half the melting temperature [8]. Che et al. [9] calculated the thermal current autocorrelation functions of classical molecular dynamics simulations using the Green-Kubo relation of linear response theory. Bulk alloys of Cu and Cu-Al alloys with different concentrations of Pd were studied by Lim et al. [10]. Some properties such as elastic moduli, micro-hardness and plasticity of metallic glasses were obtained by Wang et al. [11]. Eckert et al. [12] reviews the processing, microstructure development and resulting mechanical properties of Zr− , Ti− , Mg− , Fe− , and Ni− based glassy alloys. The concept of developing a heterogeE-mail addresses: [email protected], [email protected]
neous microstructure by combining a glassy matrix with second phase particles with a different length scale has recently been used [13–15]. Metallic glasses have attracted scientists to explore more of their novel properties except their superior strength, hardness, and excellent corrosion and wear resistance [16]. Metallic glasses showed good mechanical properties combined with excellent formability and have already been introduced to the industry, mainly as parts of the sporting goods [17]. Several fabrication techniques of Zr–Cu–Al metallic glasses were proposed recently [18–20]. The physical properties as for example shear bands of Zr–Cu–Al metallic glasses were extensively studied [21–23]. In some of recent researches, the mechanical properties of Zr–Cu–Al metallic glasses were reported [24–27]. Recently the oxidation behavior of Zr–Cu–Al metallic glasses were also reported [28]. The effects of Co alloying on mechanical properties of ternary Zr–Cu–Al bulk metallic glass are systematically examined [24]. A small variation in the composition of (Zr50 Cu50 )100-x Alx bulk metallic glasses results in a markedly different mechanical behavior [25]. Huang et al. [26] studied designs Ni-free Zr–Cu–Al–Nb–Pd bulk metallic glass and investigated there in Vitro biocompatibility by studying mechanical properties, bio-corrosion resistance, and cellular responses. Detailed analysis on the composition dependent mechanical and thermodynamics properties of ternary Zr–Cu–Al bulk metallic glasses are not studied till now. Here we study details of the concentration dependent mechanical and thermodynamics properties of ternary Zr–Cu–Al metallic glass. The effect of heating and thermoplastic deformation on the fracture stress of Zr55 Cu30 Al10 Ni5 andZr60 Cu25 Al10 Ni5 bulk metallic glass at room temperature was studied [29]. These metallic glasses were crystallized by the influence of heating beyond their upper limit temperature of 685 K and 713 K, respectively [30]. Load-control nano-indentation to investigate the load dependent Young‘s modulus [31], the nature and dynamics of multiple shear bands using molecular dynamics simula-
http://dx.doi.org/10.1016/j.ijmecsci.2017.08.037 Received 20 September 2016; Received in revised form 30 July 2017; Accepted 16 August 2017 Available online xxx 0020-7403/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: S.K. Deb Nath, International Journal of Mechanical Sciences (2017), http://dx.doi.org/10.1016/j.ijmecsci.2017.08.037
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tions [32], ground-state properties of Ni3 X intermetallic using first principles Pseudo-potential method [33], Elastic properties of M2 GeC, with M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta and W [34] were reported. Crystal growth of Zr-based bulk metallic glasses using neutron diffraction method is reported [35]. The oxidation behavior of Zr55 Cu30 Al10 Ni5 is reported [36]. In the above literature survey, concentration dependent thermal conductivity and mechanical properties of ternary metallic glass Zrx Cu90−x Al10 were not studied in details either by molecular dynamics simulations or experimental methods. Here we study the effect of concentration sensitivity, effects of temperature, and effects of sizes on the thermal conductivity and mechanical properties of ternary metallic glass Zrx Cu90−x Al10 (x = 20, 30, 40, 50, 60, 70) using molecular dynamics simulations. Effects of tension velocity on the mechanical properties of Zrx Cu90−x Al10 are studied by molecular dynamics simulations. Besides, composition dependent mechanical properties and thermal conductivity of these metallic glasses are compared with those of the published experimental studies.
The thermal conductivity K is defined as the linear coefficient relating the macroscopic heat current J to the temperature gradient (Fourier law).
lattice constants are obtained as the average of the lattice constants of Zr, Cu and Al according to their ratio in these alloys. Then the mixtures of these alloys are heated at 300 K for long time to obtain energy minimized equilibrium alloy mixtures using EAM potential [40] by molecular dynamics simulations. After obtaining energy minimized equilibrium mixtures of these alloys, they are heated from the temperature 300 K to 2100 K using NPT (N = same number of atoms, P = outside pressure 0; T = temperature in K) canonical ensemble for 470,000 steps. Then the obtained liquid alloys are heated at constant temperature 2100 K for 470,000 steps using NPT canonical ensemble. After that they are cooled down from 2100 K to 300 K for 47,000,000 steps using NPT canonical ensemble. Then the cooled metallic glasses at 300 K are heated at constant temperature 300 K for 470,000 steps to obtain these metallic glasses. Periodic boundary conditions are used in X, Y, and Z directions of the simulation box. Used time step in the simulation is considered 0.001 ps. Each metallic glass is equilibrated for 10,000 steps at different temperatures using NVT canonical ensemble. Then we reset the time step 0. The simulation box is also relaxed for 200,000 steps and obtain thermal conductivity in X, Y and Z directions. As the metallic glass is not homogenous in X, Y, Z directions of the simulation box, the thermal conductivity in X, Y and Z directions differ. For this reason, we average the thermal conductivity from different components of thermal conductivities in X, Y and Z directions of the same metallic glass.
𝐽 = −𝐾.𝑔𝑟𝑎𝑑𝑇
3.2. Simulation procedure to obtain mechanical properties
2. Theories for thermal conductivity using equilibrium molecular dynamics simulations
(1)
In this paper, we calculate K by using Green-Kubo formula [1] 1 𝐾= 3𝑉 𝑘𝐵 𝑇 2 ∫0
∞
< 𝑗 (0)𝑗 (𝑡) > 𝑑 𝑡
To obtain mechanical properties of each of the metallic glass, the shrink-wrapped boundary conditions are applied along the X, Y, and Z directions, respectively. Equal number of atoms on both sides of the metallic glass nano rod are rigidly fixed and number of rigid atoms of each metallic glass are around 10%. Using velocity command the temperature of all atoms is ensured at 300 K. The whole metallic glass is energy minimized. After minimizing the energy of the metallic glass, all atoms in the metallic glass are heated using NVT canonical ensemble. All rigid atoms on both sides of the nano rod are kept zero force in X, Y and Z directions. Then both of the rigid ends of the metallic glass move at constant velocity until the metallic glass nano rod breaks. During the movement of the rigid end of the metallic glass nano rod, they experience a force. During tension of the metal rod, the tension force and elongation are recorded. Using force and an initial cross section of the nano rod, stress is calculated. Using elongation of the nano rod and original length of the nano rod, the strain is calculated. From tension test in the present MD simulation, stress vs. strain relationship up to fracture is obtained. From the stress-strain relationship of these metallic glasses, their Young‘s modulus and yield strength are obtained.
(2)
Where V is the volume, T the temperature, and angular brackets denote the ensemble average, or, in the case of a MD simulation, the average over time. The macroscopic heat current is given by 𝑗(𝑡) =
∑ 𝑖
𝑣𝑗 𝜀𝑖 +
1 ∑ 𝑟 (𝐹 .𝑣 ) 2 𝑖,𝑗,𝑖≠𝑗 𝑖𝑗 𝑖𝑗 𝑖
(3)
Where vi is the velocity of particle i, Fij is the force on atom i due to its neighbor j from the pair potential. The site energy 𝜀i given by 𝜀𝑖 =
1 | |2 1 ∑ 𝑚 𝑣 + 𝜙(𝑟𝑖𝑗 ) 2 𝑖| 𝑖| 2 𝑗
(4)
The thermal conductivity was calculated by discretizing the right-hand side of Eq. (2) in MD time steps (Δt) as 𝐾(𝐾1 , 𝐾2 , 𝐾3 ) =
𝑀 𝑁− ∑ ∑𝑚 Δ𝑡 1 𝑗(𝑚 + 𝑛)𝑗(𝑛) 3𝑉 𝑘𝐵 𝑇 2 𝑚=1 (𝑁 − 𝑀 ) 𝑚=1
(5)
4. Results and discussion
Where N is the number of MD steps after equilibrium. M is the number of steps over which the time average is calculated, and j(m + n) is the heat current at MD time step m + n.
Fig. 1 illustrates the three different components with their average thermal conductivity of Zr40 Cu50 Al10 at a temperature 300 K. K1 , K2 and K3 are the components of the thermal conductivity of ternary metallic glass Zr40 Cu50 Al10 along the x, y and z directions of the metallic glass, respectively. Using the procedures as shown in Fig. 1 we obtain average thermal conductivity of 6 different metallic glasses varying Zr and Cu constituents with temperatures. Fig. 2 illustrates the thermal conductivity of ternary metallic glasses of Zr, Cu and Al varying their constituents Zr and Cu kept Al fraction constant as a function of temperature. From Fig. 2 it is observed that with the increase of temperature, the thermal conductivity of Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 decreases with temperature except a few exceptions. At very low temperature, the thermal conductivity of these metallic glasses is very high which is reasonable. Besides, with the increase of Cu concentration, the thermal conductivity of these metallic glasses Zrx Cu90−x Al10 increases except few exceptions with temperature because the thermal conductivity of
3. Simulation procedures 3.1. Simulation procedure to obtain thermal conductivity Different types of metallic glasses such as Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 are obtained from molecular dynamics simulations. Details of the simulation procedures are given in the references [37–38]. The whole simulations are carried out by massively parallel molecular dynamics simulator LAMMPS [38,39]. During simulations, we consider the total number of atoms in each metallic glass is 40,000. To obtain different types of metallic glasses mentioned above their initial configurations are made mixing different metallic elements such as Zr, Cu and Al according to their ratio randomly in FCC lattice structures and their 2
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Fig. 1. Different components with the average thermal conductivity of Zr40 Cu50 Al10 metallic glasses at a temperature 300 K.
Fig. 3. Effects of concentration sensitivity of different constituents on the Young‘s modulus of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of temperature.
Fig. 2. Effects of concentration sensitivity of different constituents on the thermal conductivity of ternary metallic glass, Zrx Cu90−x Al10 as a function of temperature.
Fig. 4. Effects of concentration sensitivity of different constituents on the yield strength of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of temperature.
Cu is much higher than that of Zr. An increasing percentage of Zr in these metallic glasses enhances the glassy characteristics which reduce the thermal conductivity. Waviness of the thermal conductivity with temperature for different types of metallic glasses is observed due to changing the sizes of icosahedral clusters and reorienting their positions. Fig. 3 shows Young‘s modulus of ternary metallic glasses Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 with their diameter 8 nm as a function of temperature. With the increase of temperature, Young‘s modulus of each type of ternary metallic glass decreases due to the effect of softening with the rise of temperature. With the increase of Zr concentration in each ternary metallic glass, its Young‘s modulus decreases due to increasing the glassy state
except for the ternary metallic glass Zr30 Cu60 Al10 . The ternary metallic glass Zr30 Cu60 Al10 shows the highest Young‘s modulus of all the metallic glasses mentioned in the present study. Young‘s modulus of Zr30 Cu60 Al10 is higher than that of Zr20 Cu70 Al10 although the Zr fraction in Zr30 Cu60 Al10 is higher than that of Zr20 Cu70 Al10 . Zr30 Cu60 Al10 shows comparatively optimum stiffness and glassy properties among all types of metallic glasses studied here. From Fig. 4 it is observed that the yield strength of each metallic glass with its diameter 8 nm decreases with the increase of temperature. Like Young‘s modulus as observed in Fig. 3, the yield strength of Zr30 Cu60 Al10 is the highest of all the metal3
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Fig. 5. Effects of concentration sensitivity of different constituents on the Young‘s modulus of ternary metallic glass, Zrx Cu90−x Al10 as a function of their diameter.
Fig. 7. Effects of concentration sensitivity of different constituents on the Young‘s modulus of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of loading velocity.
Fig. 6. Effects of concentration sensitivity of different constituents on the yield strength of ternary metallic glass, Zrx Cu90−x Al10 as a function of their diameter.
Fig. 8. Effects of concentration sensitivity of different constituents on the yield strength of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of loading velocity.
lic glasses mentioned here. Except the metallic glass Zr30 Cu60 Al10 , with the increase of the fraction of Cu, the yield strength of ternary metallic glasses increases as shown in Fig. 4. Young‘s modulus and yield strength of different metallic glasses decreases with temperature linearly because per atom total energy decreases linearly with temperature before yielding. Effects of temperature on Young‘s modulus and yield strength of different types of metallic glasses are observed due to their softening effects with the increase of temperature during loading. Figs. 5 and 6 show the effects of the diameter of glass nanorod on the Young‘s modulus and yield strength of different types of ternary metallic glasses. There is an observed effect of diameters on the Young‘s modulus and yield strength
in each type of the metallic glasses. In the present study, Zr30 Cu60 Al10 shows the highest Young‘s modulus and yield strength with any of its diameter of all the metallic glasses except few exceptions. On the other hand Zr70 Cu20 Al10 show the lowest Young‘s modulus and yield strength with any of its diameter of all the metallic glasses except few exceptions which is reasonable because the increase of Zr fraction in metallic glasses decreases their ductility which reduces its Young‘s modulus and yield strength. Figs. 7 and 8 illustrate the Young‘s modulus and yield strength of different metallic glasses with their diameter 8 nm as 4
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Table 1 Comparative study of the Young’s modulus and yield strength of metallic glasses Zrx Cu90 − x Al10 varying their compositions with those of the experimental study. Metallic glasses
Theoretical Young’s modulus,E (GPA)
Theoretical yield strength, 𝜎 y (GPa)
Zr20 Cu70 Al10 Zr30 Cu60 Al10 Zr40 Cu50 Al10 Zr45 Cu45 Al10 Zr45 Cu45 Al10 Zr46 Cu46 Al8 Zr47 Cu47 Al6 Zr47.5 Cu47.5 Al5 Zr49 Cu49 Al2 Zr50 Cu40 Al10 Zr55 Cu45 Al10 Zr55 Al10 Cu30 Ni5 Zr60 Al10 Cu25 Ni5 Zr60 Cu30 Al10 Zr65 Cu35 Al10 Zr70 Cu20 Al10
65.52 67.52 63.89
3.5449 3.7671 3.4544
59.35
3.2096
58.65
2.9455
54.24
2.6806
Experimental Young’s modulus, E(GPa)
Experimental yield strength, 𝜎 y (GPa)
68.908 [21] 89.097 [25] 91.418 [25] 87.604 [25] 74.882 [25] 74.567 [25] 57.961 [21] 53.16 [21] 84.019 [41], 81 [18] 81.844 [41] 48.602 [21] 42.336 [21]
1.92734 [21] 2.21774 [25] 2.29839 [25] 2.23896 [25] 1.92771 [25] 1.91767 [25] 1.79901 [21] 1.75412 [21] 1.53654 [41],1.41 [18] 1.69179 [41] 1.62000 [21] 1.60387 [21]
Table 2 Comparative study of the thermal conductivity of metallic glasses Zrx Cu90 − x Al10 varying their compositions with those of the experimental study. Metallic glasses
Theoretical thermal conductivity at T = 300K (Wm−1 K−1 )
Zr20 Cu70 Al10 Zr30 Cu60 Al10 Zr40 Cu50 Al10 Zr42 Cu42 Ag8 Al8 Zr47 Cu31 Al13 Ni9 Zr50 Cu40 Al10 Zr55 Cu35 Al10 Zr55 Al10 Cu30 Ni5 Zr55 Al10 Cu25 Ni10 Zr60 Cu30 Al10 Zr70 Cu20 Al10
5.7186 5.7920 5.7113
Experimental thermal conductivity (Wm−1 K−1 )
Electron contribution to the thermal conductivity
4.8 [42] 4.5 [43]
4.3 [42]
4.8 [42] 4.8 [42] 4.7 [42]
3.9 [42] 4.2 [42] 4.4 [42]
2.7942
1.3963 1.7312
5. Conclusion
a function of tension velocity on their loading ends. Strong effects of the tension velocity on their Young‘s modulus and yield strength are also observed. If the tension velocity increases, the strain rate will increase during tension of the metallic glass nano rod and as a result high strain rate increases the Young‘s modulus and yield strength of each metallic during loading. From Figs. 7 and 8 it is clear that Zr30 Cu60 Al10 and Zr70 Cu20 Al10 show the highest and lowest Young‘s modulus and yield strength of all the other metallic glasses at any strain rate. With the increase of tension velocity for each metallic glass, Young‘s modulus increases up to 50 m/s and after tension velocity 50 m/s, Young‘s modulus of all metallic glasses decrease because after the critical strain, metallic glasses take less time to recover their own atomic position under tension. From above studies Young‘s modulus and yield strength of Zr30 Cu60 Al10 are the highest of all the metallic glasses. On the other hand Young‘s modulus and yield strength of Zr70 Cu20 Al10 is the lowest of all the metallic glasses. Sometimes sudden rise or drop of Young‘s modulus and yield strength of different metallic glasses under tension occur due to the non-uniformity of the metallic glasses after relaxation of metallic glasses before tension. Tables 1 and 2 show the comparison the composition dependent mechanical properties and thermal conductivity of metallic glasses obtained by us with those of experimental studies [18,21,25,41–43]. The composition dependent Young’s modulus of these metallic glasses obtained by us overestimate a little to the experimental studies; on the other hand the opposite phenomenon is observed for the case of their yield strength. Thermal conductivity of these metallic glasses obtained by us is close to the experimentally obtained thermal conductivity [42,43] except few exceptions. Some variations happen due to the variations of the sample sizes and strain rates in the molecular dynamics simulations and experimental studies [18,21,25,41–43]. From the comparative studies, it is clear that our results agree well with the experimental studies [18,21,25,41–43]
Thermal conductivity at different directions of the metallic glass is different due to its anisotropic characteristics in its different directions. From the present study it is observed that thermal conductivity of ternary metallic glasses is dependent on the concentration of Zr and Cu and temperature. The young‘s modulus and yield strength of Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 are dependent on temperature, diametric sizes and strain rates. Among all the metallic glasses Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 the highest Young‘s modulus, yield strength considering all cases temperature, sizes and strain rates are observed in the metallic glass Zr30 Cu60 Al10 . From the present study, it is concluded that Zr30 Cu60 Al10 exists optimum mechanical and glassy properties. Waviness of the thermal conductivity of different metallic glasses with temperature is observed from the present simulation result, but for the Young‘s modulus and yield strength of different metallic, waviness is absent. Young‘s modulus and yield strength of different metallic glasses are dependent on temperature. The comparative study of the mechanical properties and the thermal conductivity of these metallic glasses obtained by us with the experimental study validate the reliability of the present study. References [1] Tretiakov KV, Scandolo S. Thermal conductivity of solid argon from molecular dynamics simulations. J Chem Phys 2004;120:3765–9. [2] Dong J, Sankey OF, Myles CW. Theoretical study of the lattice thermal conductivity in Ge framework semiconductors. Phys Rev Lett 2001;86:2361–4. [3] Vogelsang R, Hoheisel C, Ciccotti G. Thermal conductivity of the Lennard-Jones liquid by molecular dynamics calculations. J Chem Phys 1987;86:6371–5. [4] Heyes DM. Transport coefficients of Lennard-Jones fluids: a molecular-dynamics and effective-hard-sphere treatment. Phys Rev B 1988;37:5677.
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
Thermal conductivity and mechanical properties of Zrx Cu90−x Al10 under tension using molecular dynamics simulations S.K. Deb Nath Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
a r t i c l e
i n f o
Keywords: Thermal conductivity Young‘s modulus Yield strength Ternary Zrx Cu90−x Al10 metallic glass Molecular dynamics simulations
a b s t r a c t Using equilibrium molecular dynamics simulations, thermal conductivity of different types of metallic glass, Zrx Cu90−x Al10 is obtained and effects of concentration of their constituents on their thermal conductivity are studied as comparative manner. Young‘s modulus and yield strength of different metallic glass, Zrx Cu90−x Al10 are obtained considering temperature, size effects and strain rates. Effects of temperature, effects of sizes and strain rate on the Young‘s modulus and yield strength of Zrx Cu90-x Al10 are studied. The mechanical properties and thermal conductivities of these metallic glasses obtained by us are compared with the published mechanical properties and thermal conductivity of these metallic glasses to validate the present study. © 2017 Elsevier Ltd. All rights reserved.
1. Introduction Metallic glasses are widely used in many mechanical, aerospace industries as glass materials due to their higher strength, toughness compared to conventional glass. There are many types of metallic glasses have been developed combining different metallic elements. Now a day, ternary metallic glass based on Zr, Cu and Al elements are widely used in different engineering structures due to having good glass forming ability and high strength and toughness. Equilibrium and non-equilibrium approaches using molecular dynamics simulations can be used to study the thermal conductivity of ternary metallic glass Zrx Cu90-x Al10 where x is the percentage of Zr in ternary metallic glasses Zrx Cu90−x Al10 . Thermal conductivity of solid argon [1], Ge [2], Lennard-Jones argon liquid [3,4], Silica glass [5], liquid He [6] were studied by equilibrium molecular dynamics simulations using Green-Kubo formalism. The molecular dynamics technique was used to study the heat flux evolution in a temperate region where the thermal conductivity has temperature dependencies [7]. The thermal conductivity of a monoatomic face-centered cubic lattice has been calculated over a range of temperatures from one-twentieth to one-half the melting temperature [8]. Che et al. [9] calculated the thermal current autocorrelation functions of classical molecular dynamics simulations using the Green-Kubo relation of linear response theory. Bulk alloys of Cu and Cu-Al alloys with different concentrations of Pd were studied by Lim et al. [10]. Some properties such as elastic moduli, micro-hardness and plasticity of metallic glasses were obtained by Wang et al. [11]. Eckert et al. [12] reviews the processing, microstructure development and resulting mechanical properties of Zr− , Ti− , Mg− , Fe− , and Ni− based glassy alloys. The concept of developing a heterogeE-mail addresses: [email protected], [email protected]
neous microstructure by combining a glassy matrix with second phase particles with a different length scale has recently been used [13–15]. Metallic glasses have attracted scientists to explore more of their novel properties except their superior strength, hardness, and excellent corrosion and wear resistance [16]. Metallic glasses showed good mechanical properties combined with excellent formability and have already been introduced to the industry, mainly as parts of the sporting goods [17]. Several fabrication techniques of Zr–Cu–Al metallic glasses were proposed recently [18–20]. The physical properties as for example shear bands of Zr–Cu–Al metallic glasses were extensively studied [21–23]. In some of recent researches, the mechanical properties of Zr–Cu–Al metallic glasses were reported [24–27]. Recently the oxidation behavior of Zr–Cu–Al metallic glasses were also reported [28]. The effects of Co alloying on mechanical properties of ternary Zr–Cu–Al bulk metallic glass are systematically examined [24]. A small variation in the composition of (Zr50 Cu50 )100-x Alx bulk metallic glasses results in a markedly different mechanical behavior [25]. Huang et al. [26] studied designs Ni-free Zr–Cu–Al–Nb–Pd bulk metallic glass and investigated there in Vitro biocompatibility by studying mechanical properties, bio-corrosion resistance, and cellular responses. Detailed analysis on the composition dependent mechanical and thermodynamics properties of ternary Zr–Cu–Al bulk metallic glasses are not studied till now. Here we study details of the concentration dependent mechanical and thermodynamics properties of ternary Zr–Cu–Al metallic glass. The effect of heating and thermoplastic deformation on the fracture stress of Zr55 Cu30 Al10 Ni5 andZr60 Cu25 Al10 Ni5 bulk metallic glass at room temperature was studied [29]. These metallic glasses were crystallized by the influence of heating beyond their upper limit temperature of 685 K and 713 K, respectively [30]. Load-control nano-indentation to investigate the load dependent Young‘s modulus [31], the nature and dynamics of multiple shear bands using molecular dynamics simula-
http://dx.doi.org/10.1016/j.ijmecsci.2017.08.037 Received 20 September 2016; Received in revised form 30 July 2017; Accepted 16 August 2017 Available online xxx 0020-7403/© 2017 Elsevier Ltd. All rights reserved.
Please cite this article as: S.K. Deb Nath, International Journal of Mechanical Sciences (2017), http://dx.doi.org/10.1016/j.ijmecsci.2017.08.037
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tions [32], ground-state properties of Ni3 X intermetallic using first principles Pseudo-potential method [33], Elastic properties of M2 GeC, with M=Ti, V, Cr, Zr, Nb, Mo, Hf, Ta and W [34] were reported. Crystal growth of Zr-based bulk metallic glasses using neutron diffraction method is reported [35]. The oxidation behavior of Zr55 Cu30 Al10 Ni5 is reported [36]. In the above literature survey, concentration dependent thermal conductivity and mechanical properties of ternary metallic glass Zrx Cu90−x Al10 were not studied in details either by molecular dynamics simulations or experimental methods. Here we study the effect of concentration sensitivity, effects of temperature, and effects of sizes on the thermal conductivity and mechanical properties of ternary metallic glass Zrx Cu90−x Al10 (x = 20, 30, 40, 50, 60, 70) using molecular dynamics simulations. Effects of tension velocity on the mechanical properties of Zrx Cu90−x Al10 are studied by molecular dynamics simulations. Besides, composition dependent mechanical properties and thermal conductivity of these metallic glasses are compared with those of the published experimental studies.
The thermal conductivity K is defined as the linear coefficient relating the macroscopic heat current J to the temperature gradient (Fourier law).
lattice constants are obtained as the average of the lattice constants of Zr, Cu and Al according to their ratio in these alloys. Then the mixtures of these alloys are heated at 300 K for long time to obtain energy minimized equilibrium alloy mixtures using EAM potential [40] by molecular dynamics simulations. After obtaining energy minimized equilibrium mixtures of these alloys, they are heated from the temperature 300 K to 2100 K using NPT (N = same number of atoms, P = outside pressure 0; T = temperature in K) canonical ensemble for 470,000 steps. Then the obtained liquid alloys are heated at constant temperature 2100 K for 470,000 steps using NPT canonical ensemble. After that they are cooled down from 2100 K to 300 K for 47,000,000 steps using NPT canonical ensemble. Then the cooled metallic glasses at 300 K are heated at constant temperature 300 K for 470,000 steps to obtain these metallic glasses. Periodic boundary conditions are used in X, Y, and Z directions of the simulation box. Used time step in the simulation is considered 0.001 ps. Each metallic glass is equilibrated for 10,000 steps at different temperatures using NVT canonical ensemble. Then we reset the time step 0. The simulation box is also relaxed for 200,000 steps and obtain thermal conductivity in X, Y and Z directions. As the metallic glass is not homogenous in X, Y, Z directions of the simulation box, the thermal conductivity in X, Y and Z directions differ. For this reason, we average the thermal conductivity from different components of thermal conductivities in X, Y and Z directions of the same metallic glass.
𝐽 = −𝐾.𝑔𝑟𝑎𝑑𝑇
3.2. Simulation procedure to obtain mechanical properties
2. Theories for thermal conductivity using equilibrium molecular dynamics simulations
(1)
In this paper, we calculate K by using Green-Kubo formula [1] 1 𝐾= 3𝑉 𝑘𝐵 𝑇 2 ∫0
∞
< 𝑗 (0)𝑗 (𝑡) > 𝑑 𝑡
To obtain mechanical properties of each of the metallic glass, the shrink-wrapped boundary conditions are applied along the X, Y, and Z directions, respectively. Equal number of atoms on both sides of the metallic glass nano rod are rigidly fixed and number of rigid atoms of each metallic glass are around 10%. Using velocity command the temperature of all atoms is ensured at 300 K. The whole metallic glass is energy minimized. After minimizing the energy of the metallic glass, all atoms in the metallic glass are heated using NVT canonical ensemble. All rigid atoms on both sides of the nano rod are kept zero force in X, Y and Z directions. Then both of the rigid ends of the metallic glass move at constant velocity until the metallic glass nano rod breaks. During the movement of the rigid end of the metallic glass nano rod, they experience a force. During tension of the metal rod, the tension force and elongation are recorded. Using force and an initial cross section of the nano rod, stress is calculated. Using elongation of the nano rod and original length of the nano rod, the strain is calculated. From tension test in the present MD simulation, stress vs. strain relationship up to fracture is obtained. From the stress-strain relationship of these metallic glasses, their Young‘s modulus and yield strength are obtained.
(2)
Where V is the volume, T the temperature, and angular brackets denote the ensemble average, or, in the case of a MD simulation, the average over time. The macroscopic heat current is given by 𝑗(𝑡) =
∑ 𝑖
𝑣𝑗 𝜀𝑖 +
1 ∑ 𝑟 (𝐹 .𝑣 ) 2 𝑖,𝑗,𝑖≠𝑗 𝑖𝑗 𝑖𝑗 𝑖
(3)
Where vi is the velocity of particle i, Fij is the force on atom i due to its neighbor j from the pair potential. The site energy 𝜀i given by 𝜀𝑖 =
1 | |2 1 ∑ 𝑚 𝑣 + 𝜙(𝑟𝑖𝑗 ) 2 𝑖| 𝑖| 2 𝑗
(4)
The thermal conductivity was calculated by discretizing the right-hand side of Eq. (2) in MD time steps (Δt) as 𝐾(𝐾1 , 𝐾2 , 𝐾3 ) =
𝑀 𝑁− ∑ ∑𝑚 Δ𝑡 1 𝑗(𝑚 + 𝑛)𝑗(𝑛) 3𝑉 𝑘𝐵 𝑇 2 𝑚=1 (𝑁 − 𝑀 ) 𝑚=1
(5)
4. Results and discussion
Where N is the number of MD steps after equilibrium. M is the number of steps over which the time average is calculated, and j(m + n) is the heat current at MD time step m + n.
Fig. 1 illustrates the three different components with their average thermal conductivity of Zr40 Cu50 Al10 at a temperature 300 K. K1 , K2 and K3 are the components of the thermal conductivity of ternary metallic glass Zr40 Cu50 Al10 along the x, y and z directions of the metallic glass, respectively. Using the procedures as shown in Fig. 1 we obtain average thermal conductivity of 6 different metallic glasses varying Zr and Cu constituents with temperatures. Fig. 2 illustrates the thermal conductivity of ternary metallic glasses of Zr, Cu and Al varying their constituents Zr and Cu kept Al fraction constant as a function of temperature. From Fig. 2 it is observed that with the increase of temperature, the thermal conductivity of Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 decreases with temperature except a few exceptions. At very low temperature, the thermal conductivity of these metallic glasses is very high which is reasonable. Besides, with the increase of Cu concentration, the thermal conductivity of these metallic glasses Zrx Cu90−x Al10 increases except few exceptions with temperature because the thermal conductivity of
3. Simulation procedures 3.1. Simulation procedure to obtain thermal conductivity Different types of metallic glasses such as Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 are obtained from molecular dynamics simulations. Details of the simulation procedures are given in the references [37–38]. The whole simulations are carried out by massively parallel molecular dynamics simulator LAMMPS [38,39]. During simulations, we consider the total number of atoms in each metallic glass is 40,000. To obtain different types of metallic glasses mentioned above their initial configurations are made mixing different metallic elements such as Zr, Cu and Al according to their ratio randomly in FCC lattice structures and their 2
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Fig. 1. Different components with the average thermal conductivity of Zr40 Cu50 Al10 metallic glasses at a temperature 300 K.
Fig. 3. Effects of concentration sensitivity of different constituents on the Young‘s modulus of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of temperature.
Fig. 2. Effects of concentration sensitivity of different constituents on the thermal conductivity of ternary metallic glass, Zrx Cu90−x Al10 as a function of temperature.
Fig. 4. Effects of concentration sensitivity of different constituents on the yield strength of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of temperature.
Cu is much higher than that of Zr. An increasing percentage of Zr in these metallic glasses enhances the glassy characteristics which reduce the thermal conductivity. Waviness of the thermal conductivity with temperature for different types of metallic glasses is observed due to changing the sizes of icosahedral clusters and reorienting their positions. Fig. 3 shows Young‘s modulus of ternary metallic glasses Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 with their diameter 8 nm as a function of temperature. With the increase of temperature, Young‘s modulus of each type of ternary metallic glass decreases due to the effect of softening with the rise of temperature. With the increase of Zr concentration in each ternary metallic glass, its Young‘s modulus decreases due to increasing the glassy state
except for the ternary metallic glass Zr30 Cu60 Al10 . The ternary metallic glass Zr30 Cu60 Al10 shows the highest Young‘s modulus of all the metallic glasses mentioned in the present study. Young‘s modulus of Zr30 Cu60 Al10 is higher than that of Zr20 Cu70 Al10 although the Zr fraction in Zr30 Cu60 Al10 is higher than that of Zr20 Cu70 Al10 . Zr30 Cu60 Al10 shows comparatively optimum stiffness and glassy properties among all types of metallic glasses studied here. From Fig. 4 it is observed that the yield strength of each metallic glass with its diameter 8 nm decreases with the increase of temperature. Like Young‘s modulus as observed in Fig. 3, the yield strength of Zr30 Cu60 Al10 is the highest of all the metal3
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Fig. 5. Effects of concentration sensitivity of different constituents on the Young‘s modulus of ternary metallic glass, Zrx Cu90−x Al10 as a function of their diameter.
Fig. 7. Effects of concentration sensitivity of different constituents on the Young‘s modulus of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of loading velocity.
Fig. 6. Effects of concentration sensitivity of different constituents on the yield strength of ternary metallic glass, Zrx Cu90−x Al10 as a function of their diameter.
Fig. 8. Effects of concentration sensitivity of different constituents on the yield strength of ternary metallic glasses, Zrx Cu90−x Al10 with their diameter 8 nm as a function of loading velocity.
lic glasses mentioned here. Except the metallic glass Zr30 Cu60 Al10 , with the increase of the fraction of Cu, the yield strength of ternary metallic glasses increases as shown in Fig. 4. Young‘s modulus and yield strength of different metallic glasses decreases with temperature linearly because per atom total energy decreases linearly with temperature before yielding. Effects of temperature on Young‘s modulus and yield strength of different types of metallic glasses are observed due to their softening effects with the increase of temperature during loading. Figs. 5 and 6 show the effects of the diameter of glass nanorod on the Young‘s modulus and yield strength of different types of ternary metallic glasses. There is an observed effect of diameters on the Young‘s modulus and yield strength
in each type of the metallic glasses. In the present study, Zr30 Cu60 Al10 shows the highest Young‘s modulus and yield strength with any of its diameter of all the metallic glasses except few exceptions. On the other hand Zr70 Cu20 Al10 show the lowest Young‘s modulus and yield strength with any of its diameter of all the metallic glasses except few exceptions which is reasonable because the increase of Zr fraction in metallic glasses decreases their ductility which reduces its Young‘s modulus and yield strength. Figs. 7 and 8 illustrate the Young‘s modulus and yield strength of different metallic glasses with their diameter 8 nm as 4
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Table 1 Comparative study of the Young’s modulus and yield strength of metallic glasses Zrx Cu90 − x Al10 varying their compositions with those of the experimental study. Metallic glasses
Theoretical Young’s modulus,E (GPA)
Theoretical yield strength, 𝜎 y (GPa)
Zr20 Cu70 Al10 Zr30 Cu60 Al10 Zr40 Cu50 Al10 Zr45 Cu45 Al10 Zr45 Cu45 Al10 Zr46 Cu46 Al8 Zr47 Cu47 Al6 Zr47.5 Cu47.5 Al5 Zr49 Cu49 Al2 Zr50 Cu40 Al10 Zr55 Cu45 Al10 Zr55 Al10 Cu30 Ni5 Zr60 Al10 Cu25 Ni5 Zr60 Cu30 Al10 Zr65 Cu35 Al10 Zr70 Cu20 Al10
65.52 67.52 63.89
3.5449 3.7671 3.4544
59.35
3.2096
58.65
2.9455
54.24
2.6806
Experimental Young’s modulus, E(GPa)
Experimental yield strength, 𝜎 y (GPa)
68.908 [21] 89.097 [25] 91.418 [25] 87.604 [25] 74.882 [25] 74.567 [25] 57.961 [21] 53.16 [21] 84.019 [41], 81 [18] 81.844 [41] 48.602 [21] 42.336 [21]
1.92734 [21] 2.21774 [25] 2.29839 [25] 2.23896 [25] 1.92771 [25] 1.91767 [25] 1.79901 [21] 1.75412 [21] 1.53654 [41],1.41 [18] 1.69179 [41] 1.62000 [21] 1.60387 [21]
Table 2 Comparative study of the thermal conductivity of metallic glasses Zrx Cu90 − x Al10 varying their compositions with those of the experimental study. Metallic glasses
Theoretical thermal conductivity at T = 300K (Wm−1 K−1 )
Zr20 Cu70 Al10 Zr30 Cu60 Al10 Zr40 Cu50 Al10 Zr42 Cu42 Ag8 Al8 Zr47 Cu31 Al13 Ni9 Zr50 Cu40 Al10 Zr55 Cu35 Al10 Zr55 Al10 Cu30 Ni5 Zr55 Al10 Cu25 Ni10 Zr60 Cu30 Al10 Zr70 Cu20 Al10
5.7186 5.7920 5.7113
Experimental thermal conductivity (Wm−1 K−1 )
Electron contribution to the thermal conductivity
4.8 [42] 4.5 [43]
4.3 [42]
4.8 [42] 4.8 [42] 4.7 [42]
3.9 [42] 4.2 [42] 4.4 [42]
2.7942
1.3963 1.7312
5. Conclusion
a function of tension velocity on their loading ends. Strong effects of the tension velocity on their Young‘s modulus and yield strength are also observed. If the tension velocity increases, the strain rate will increase during tension of the metallic glass nano rod and as a result high strain rate increases the Young‘s modulus and yield strength of each metallic during loading. From Figs. 7 and 8 it is clear that Zr30 Cu60 Al10 and Zr70 Cu20 Al10 show the highest and lowest Young‘s modulus and yield strength of all the other metallic glasses at any strain rate. With the increase of tension velocity for each metallic glass, Young‘s modulus increases up to 50 m/s and after tension velocity 50 m/s, Young‘s modulus of all metallic glasses decrease because after the critical strain, metallic glasses take less time to recover their own atomic position under tension. From above studies Young‘s modulus and yield strength of Zr30 Cu60 Al10 are the highest of all the metallic glasses. On the other hand Young‘s modulus and yield strength of Zr70 Cu20 Al10 is the lowest of all the metallic glasses. Sometimes sudden rise or drop of Young‘s modulus and yield strength of different metallic glasses under tension occur due to the non-uniformity of the metallic glasses after relaxation of metallic glasses before tension. Tables 1 and 2 show the comparison the composition dependent mechanical properties and thermal conductivity of metallic glasses obtained by us with those of experimental studies [18,21,25,41–43]. The composition dependent Young’s modulus of these metallic glasses obtained by us overestimate a little to the experimental studies; on the other hand the opposite phenomenon is observed for the case of their yield strength. Thermal conductivity of these metallic glasses obtained by us is close to the experimentally obtained thermal conductivity [42,43] except few exceptions. Some variations happen due to the variations of the sample sizes and strain rates in the molecular dynamics simulations and experimental studies [18,21,25,41–43]. From the comparative studies, it is clear that our results agree well with the experimental studies [18,21,25,41–43]
Thermal conductivity at different directions of the metallic glass is different due to its anisotropic characteristics in its different directions. From the present study it is observed that thermal conductivity of ternary metallic glasses is dependent on the concentration of Zr and Cu and temperature. The young‘s modulus and yield strength of Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 are dependent on temperature, diametric sizes and strain rates. Among all the metallic glasses Zr20 Cu70 Al10 , Zr30 Cu60 Al10 , Zr40 Cu50 Al10 , Zr50 Cu40 Al10 , Zr60 Cu30 Al10 , Zr70 Cu20 Al10 the highest Young‘s modulus, yield strength considering all cases temperature, sizes and strain rates are observed in the metallic glass Zr30 Cu60 Al10 . From the present study, it is concluded that Zr30 Cu60 Al10 exists optimum mechanical and glassy properties. Waviness of the thermal conductivity of different metallic glasses with temperature is observed from the present simulation result, but for the Young‘s modulus and yield strength of different metallic, waviness is absent. Young‘s modulus and yield strength of different metallic glasses are dependent on temperature. The comparative study of the mechanical properties and the thermal conductivity of these metallic glasses obtained by us with the experimental study validate the reliability of the present study. References [1] Tretiakov KV, Scandolo S. Thermal conductivity of solid argon from molecular dynamics simulations. J Chem Phys 2004;120:3765–9. [2] Dong J, Sankey OF, Myles CW. Theoretical study of the lattice thermal conductivity in Ge framework semiconductors. Phys Rev Lett 2001;86:2361–4. [3] Vogelsang R, Hoheisel C, Ciccotti G. Thermal conductivity of the Lennard-Jones liquid by molecular dynamics calculations. J Chem Phys 1987;86:6371–5. [4] Heyes DM. Transport coefficients of Lennard-Jones fluids: a molecular-dynamics and effective-hard-sphere treatment. Phys Rev B 1988;37:5677.
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