Thermophysical properties of undercooled liquid Ni-Zr alloys: Melting temperature, density, excess volume and thermal expansion

Computational Materials Science 135 (2017) 22–28

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Thermophysical properties of undercooled liquid Ni-Zr alloys: Melting temperature, density, excess volume and thermal expansion P. Lü, K. Zhou, X. Cai, H.P. Wang ⇑ MOE Key Laboratory of Space Applied Physics and Chemistry, Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072, PR China

a r t i c l e

i n f o

Article history: Received 14 November 2016 Received in revised form 27 March 2017 Accepted 6 April 2017

Keywords: Thermophysical properties Undercooling Liquid alloys Ni-Zr alloys

a b s t r a c t The thermophysical properties of undercooled liquid Ni-Zr binary alloys were investigated by molecular dynamics simulation combined with a Finnis-Sinclair (F-S) potential, including melting temperature, density, excess volume and thermal expansion. The melting temperatures were obtained by the evolution of crystal-liquid-crystal sandwich model, where there exist rather low differences of 4.14% for Ni77.8Zr22.2 alloy and 3.98% for Ni50Zr50 alloy when they were compared with the reported values. The calculated densities of liquid Ni-Zr alloys increase with the decrease of temperature, which agree well with the reported experimental values except for the Ni-rich composition alloys. Thus, the reported experimental density of liquid Ni77.8Zr22.2 alloy was employed to re-gauge the current F-S potential and the densities of the Ni-rich composition alloys were recalculated by the re-gauged potential. This binary liquid alloy system shows a negative excess volume, which could be attributed to the strong attractive interactions between Ni and Zr atoms. It is indicated that the Ni-Zr alloy system seriously deviates from the ideal solution, and the accuracy would be very low if the thermophysical properties are estimated by Neumann-Kopp rule. Meanwhile, the thermal expansion coefficients were also derived on the basis of the density data, which increase with the enhancement of temperature except for liquid Ni77.8Zr22.2 alloy. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction The thermophysical properties of undercooled liquid alloys, such as melting temperature, density and thermal expansion, are fundamental parameters in the fields of materials science and condensed matter physics, which have aroused great interests [1–3]. The melting temperature plays a vital role in determining whether the liquid alloy is in normal state or undercooled state [4]. The densities of liquid alloys are of considerable importance in understanding solidification processes, mass transport, atomic structure, thermal convection and performing numerical simulations [5–9]. Thermal expansion, which can be derived from the density as a function of temperature, is also important for performing engineering design and studying the phase transformation [10,11]. However, the knowledge of these properties of undercooled liquid alloys is scarcely known compared with that of pure liquid metals and normal liquid alloys. This is because that any contact between melts and container walls will trigger immediately nucleation, and thus the metastable state of undercooled melts is difficult to be achieved and kept. To date, these thermophysical properties can

⇑ Corresponding author. E-mail address: [email protected] (H.P. Wang). http://dx.doi.org/10.1016/j.commatsci.2017.04.003 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.

be obtained mainly by the experimental and computing methods. Containerless processing techniques, for example, electromagnetic levitation and electrostatic levitation, are employed to experimentally determine these properties, which are the best methods to obtain reliable data. Nevertheless, large time consumes and financial costs are always required during the experimental measurements, and thus the efficiency is rather low. More importantly, high undercooling is hard to achieve although the contact between melts and container walls has been completely avoided. As an alternative, the molecular dynamics (MD) simulation together with a reasonable potential is considered to be a powerful method to investigate the thermophysical properties of liquid metals [12,13]. It is easier to obtain high undercoolings and to evaluate the thermophysical properties for pure metals and simple alloys, especially for some dilute alloys. But for the complicated alloys, there still exist various problems to obtain reliable thermophysical properties. As a typical metallic glass system, Ni-Zr binary system has aroused great interests because of its good glass forming ability in a wide composition range and abundant intermetallic compounds [14,15]. Most works focus on its structure and dynamics properties. Recently, Georgarakis et al. [16] investigated the local structure of Zr-Ni metallic glasses and found their structures cannot be approached with an ideal solution model. Holland-Moritz

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P. Lü et al. / Computational Materials Science 135 (2017) 22–28

et al. [17] revealed a large activation energy for Ni-diffusion in liquid Ni36Zr64, which may result from a peculiar short order of the Ni-Zr melts. Whereas, only a small amount of thermophysical properties of undercooled Ni-Zr alloys have been reported. And these properties of undercooled liquid Ni-Zr alloys are scarce and in lack of detailed and systematic studies, which is quit essential for fundamental researches and important to improving industrial processing. In this paper, we presented the calculations of the thermophysical properties of various Ni-Zr liquid alloys including melting temperature, density, and thermal expansion. We used the reported experimental density of liquid Ni77.8Zr22.2 alloy to re-gauge the current F-S potential and then recalculated the density for Ni-rich alloys with the gauged potential. Comparisons between the experimental and calculated data were performed to evaluate the present calculations. Meanwhile, the origin of the excess volume in liquid Ni-Zr binary alloy system was also discussed.

2.2. Density For the simulation of density of liquid alloys, 4000 atoms were put into a cubic box with the periodic boundary conditions in three dimensions. The system was under constant temperature and constant pressure. NPT algorithm was employed in the process of the simulations. The time step was set to 1 fs and the pressure was set to 1 bar. The temperature was adjusted every 50 steps. To obtain an equilibrium liquid state, the system started at 3000 K, which was far beyond the melting temperature. The initial temperatures were kept constant for 100,000 steps. Then, the system was cooled with the cooling rate of 1012 K/s in 100 K temperature interval. At each temperature, 100,000 steps were performed for equilibrium. All the MD simulations were performed with LAMMPS. 3. Results and discussion 3.1. Melting temperature

2. Method

N1 X N X i¼1 j¼iþ1

uai aj ðrij Þ þ

N X

F ai ðqi Þ;

ð1Þ

i¼1

where F is the embedding energy which is a function of the atomic electron density qi, N the number of atoms in the system, uai aj ðr ij Þ the pair potential interaction, ai the element type of atoms i, rij the distance between atoms i and j. The electron density qi is calcuP aa aa lated by qi ¼ j f i j ðr ij Þ. The functions uai aj ðr ij Þ and f i j ðr ij Þ are the results of empirical modeling. Recently, a F-S potential for Ni-Zr binary alloy system was put forward by Mendelev [19], which was developed to match properties of Ni, Zr, NiZr and NiZr2. This potential has been employed to simulate the solid-liquid interface properties of Ni-Zr B33 phase successfully and is also applied in this work. 2.1. Melting temperature For the melting temperature, simulations were performed in orthorhombic boxes with periodic boundary conditions in three dimensions. We used a sandwich model [20] to simulate the melting behavior of metals. To generate the initial configuration of crystal-liquid-crystal coexisting structure, an orthorhombic box was employed in the simulation, which was swarmed with atoms with stable lattice. The crystal-liquid-crystal multi-layers were arranged in the z direction. The liquid phase was generated by heating the central half of crystal in z direction at 1000 K above the melting temperature (far beyond the melting temperature) for 100,000 steps. The solid phase was obtained by equilibrating the rest of the crystal around the experimental melting temperature for 100,000 steps. As a result, the coexisting structure was constructed and annealed at given temperatures for 5,000,000 steps, which was monitored during this process. The simulated melting temperature can be estimated from the evolution of the coexisting structure, which was determined according to the change of energy of per atom. If the given temperature is higher than the melting temperature, the liquid phase will grow on the expense of the solid phase, and contrariwise lower than the melting temperature.

gðrÞ ¼

V hni ðr; r þ DrÞi ; 4pr 2 DrN

ð2Þ

where V is the system volume, ni(r, r + Dr) the atom number around the ith atom in a spherical shell between r and r + Dr, and N the

-3.65

1725 K 1730 K

(a) -3.70 -3.75 -3.80 -3.85 -3.90 0

100

200 300 t, ps

400

500

1725 K 1730 K Exp. at 1733 K [21]

(b) 3 2

g(r)

E¼

The simulated melting temperature can be determined according to the change of internal energy of per atom. Fig. 1 shows the variations of internal energy per atom of pure Ni with time at two desired temperatures. It is obvious that the internal energy per atom of pure Ni increases in the first place and stabilizes subsequently at 1730 K presented in Fig. 1(a), indicating that the coexisting structure melts into the liquid state completely. Nevertheless, at 1725 K, the internal energy per atom decreases before it stabilizes, which gives an indication of the crystallization of the coexisting structure. To further confirm the states of the final structure at the desired temperatures, the pair distribution function is calculated according to the following equation:

Internal energy, ev/atom

The inter-atomic potential is the fundament of molecular dynamic simulation. The F-S potential proposed by Finnis and Sinclair [18] has been believed to be successful for describing atomic interactions of metallic systems and applied in the simulation of liquid structure, liquid-glass transition and thermodynamic properties. In the F-S, the total energy E of the system is given by

1 0 0

2

4

r, 10

6 -10

8

10

m

Fig. 1. (a) Variation of internal energy per atom for sandwich structure of Ni during simulation; (b) pair distribution functions of Ni at two different temperatures. The insets present the snapshots of atom positions at two different temperatures.

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P. Lü et al. / Computational Materials Science 135 (2017) 22–28

atoms number. Fig. 1(b) illustrates the pair distribution functions of Ni at 1730 and 1725 K, respectively. It can be seen that the sharp peak appears at the first neighbor distance for the curve at 1730 K. However, the height of the second peak drops drastically and the pair distribution function converges to 1 rapidly once the distance is beyond the second neighbor distance. It is indicated that the simulated system is atomic ordered in short range but disordered over a long distance, from which it could be concluded that the system is in the state of liquid when the temperature is 1730 K. The experimental data [21] at a temperature of 1733 K is also presented in Fig. 1(b) for a comparison. The agreement is good except a small difference in the height of the first peak. For the pair distribution function at 1725 K, the first peak is intense and so do others with the increase of r. This indicates that the atom distribution is ordered both in short and long atomic range, which is the typical characteristic of crystal structure. The simulated system keeps in the state of crystal when the temperature is 1725 K. Meanwhile, the snapshots of atom position are also given to confirm the states of the simulated system at these two different temperatures, which are presented as the inset in Fig. 1(a). Clearly, at 1725 K, atoms are quite ordered and locate around equilibrium positions. On the contrary, atoms are disordered at 1730 K. Therefore, based on the above analysis, we suggest that the simulated melting temperature of pure Ni ranges from 1725 to 1730 K, and the melting temperature is determined approximately to be 1727.5 ± 2.5 K, which agrees well with the experimental value of 1728 K. Similarly, with the same method, the melting temperatures of Ni77.8Zr22.2, Ni50Zr50, and Zr were obtained and listed in Table 1 in comparisons with the experimental values taken from Ni-Zr binary phase diagram [22]. It is noted that the crystal structures of solid Ni77.8Zr22.2, Ni50Zr50, and Zr used in the calculations are mC18 [23], Cmcm (B33) [22] and bcc, respectively. It can be seen that there exists a good agreement between the simulated and experimental melting temperature. Meanwhile, we also compared the calculated melting temperature with those published earlier. The melting temperature of Zr bcc phase calculated by Mendelev et al. [24] is 2109 K, which is almost the same with the present value of 2112.5 K, existing a difference of 0.17%. The melting temperature of Ni50Zr50 is 1472.5 K, which is in perfect agreement with Mendelev’s results of 1473 K [19]. 3.2. Density In the simulations, the investigated compositions of alloys are presented in Table 2. Meanwhile, melting temperatures together with the calculated temperature ranges are also listed in Table 2. The density q as a function of temperature T can be described as

q ¼ qL 

dq d q ðT  T L Þ  2 ðT  T L Þ2 ; dT dT 2

ð3Þ

where qL is the density at the melting temperature. The simulated results are shown in Table 2. Evidently, density at the melting temperature increases first and then decreases with the increase of the Zr content. Generally, density of binary liquid alloys monotonously changes with the increase in the element content. The density of Ni90Zr10 at melting temperature is Table 1 The calculated and experimental TL for Ni-Zr alloys. Samples

TCal (K) L

TExp (K) L

Deviation (%)

Ni Ni77.8Zr22.2 Ni50Zr50 (B33) Zr (bcc)

1727.5 ± 2.5 1642.5 ± 2.5 1472.5 ± 2.5 2112.5 ± 2.5

1728 1713 1533 2128

0.06 4.14 3.98 0.73

8.10 g cm3, which is larger than the density of pure Ni of 7.96 g cm3. This seems abnormal because the density of pure Ni is larger than that of pure Zr. Furthermore, we closely compared the densities of pure Ni and Ni90Zr10 at the same temperature of 1500 and 1800 K. Unfortunately, density of Ni90Zr10 is still larger than that of pure Ni. To further confirm the reliability of the calculated densities and evaluate the present results, experimental densities of Ni-Zr alloys were carried out for comparisons with the simulated results. Among the nine compositions under consideration, experimental densities of Ni, Ni77.8Zr22.2, Ni50Zr50, Ni36Zr64 and Zr have been reported by other researchers. Fig. 2 clearly shows the comparisons between experimental and simulated results. Saito et al. [25] measured the density of liquid Ni in the temperature range of 2163–2423 K in 1969, as presented the line 1 in Fig. 2(a). The measurement is beyond the melting temperature and the temperature range is about 260 K, which is rather narrow. Besides, no data are provided around the melting temperature and in undercooled region. It can be seen that Saito’s result is smaller than our simulated work, and there is a big deviation between these two results. At 2400 K, the value of 7.07 g cm3 obtained by Saito et al. is 5.5% smaller than the present value of 7.48 g cm3. In 1996 and 2004, Chung et al. [26] and Ishikawa et al. [27] measured the density of liquid Ni by using electrostatic levitation method, respectively. The measured temperature range of 1403– 1838 K was obtained by Chung et al., including a maximum undercooling of 325 K, as marked by line 2 in Fig. 2(a). The temperature range of 1420–1850 K was achieved by Ishikawa et al. and the maximum undercooling is 308 K, which is also presented in Fig. 2 (a), as marked by line 3. Obviously, the results measured by Chung et al. and Ishikawa et al. are almost exactly the same, which agree well with the present calculation as well. Meanwhile, both the experimental values at melting temperature are 7.89 g cm3, which is only 0.76% smaller than the present value of 7.95 g cm3. This indicates that the calculated density of liquid Ni has a high accuracy. Meanwhile, the present work provides density values of liquid Ni at a much broader temperature range, including a maximum undercooling of 528 K. Fig. 2(b) presents the calculated density and experimental data of liquid Ni77.8Zr22.2 alloy. The calculated results are denoted by open circles and the experimental data are marked by the red line. The calculation seems to overestimate the density of liquid Ni77.8Zr22.2 alloy over the full calculated temperature range. The calculated density at melting temperature is 7.94 g cm3, which is 3.66% larger than experimental value of 7.66 g cm3. For liquid Ni50Zr50 alloy, the present calculation and experimental result are given in Fig. 2(c). The present density at the melting temperature of 1533 K is 7.17 g cm3, which is only 0.28% bigger than Ohsaka’s [28] result of 7.15 g cm3. The maximum undercooling obtained by experiment is 173 K, which is significantly smaller than the simulated undercooling of 533 K. For liquid Ni36Zr64 alloy, the simulated density at melting temperature of 1283 K is 6.88 g cm3, while the experimental values reported by Brillo et al. [29] and Ohsaka el al. [30] are 6.88 and 6.90 g cm3, respectively. The difference between the present and experimental results is less than 0.3%. However, the maximum undercoolings achieved by Brillo et al. and Ohsaka et al. are smaller than 180 K, while the present calculation includes a maximum undercooling of 483 K. Fig. 2(e) illustrates the calculated density of liquid Zr as well as the experimental results obtained by other researchers. Paradis et al. [31] measured the density of liquid Zr in 1999, as marked by line 1, which includes a maximum undercooling of 428 K. The experimental value at melting temperature is 6.24 g cm3, which is 2.7% bigger than the calculated value of 6.07 g cm3. In 2001 Ishikawa et al. [32] measured the density of Zr in the temperature range of 1900–2300 K, including a maximum undercooling of 228 K and the result is marked by line 2. The

25

P. Lü et al. / Computational Materials Science 135 (2017) 22–28 Table 2 Selected compositions and calculated densities of liquid Ni-Zr alloys. Alloy

TL (K)

Ni Ni90Zr10 Ni90Zr10 (gauged) Ni77.8Zr22.2 Ni77.8Zr22.2 (gauged) Ni60Zr40 Ni50Zr50 Ni36Zr64 Ni20Zr80 Ni10Zr90 Zr

1728 1479 1479 1713 1713 1417 1533 1283 1422 1808 2128

qL (g cm3) 7.96 8.10 7.94 7.92 7.68 7.49 7.17 6.88 6.55 6.30 6.07

dq/dT (g cm3 K1) 4

5.75  10 5.71  104 5.88  104 5.74  104 5.51  104 2.89  104 2.33  104 1.92  104 2.10  104 2.54  104 2.92  104

comparison shows that the agreement is perfect in the whole temperature range. Ishikawa et al. [33] measured the density of liquid Zr once again in 2001 and the data is marked as line 3. Evidently, the experimental data are a little larger than the present calculation and the measured density at melting temperature is 6.21 cm3, which is 2.3% bigger than the calculated value of 6.07 g cm3. In 2016, Wang et al. [3] measured the density of liquid Zr, including a maximum undercooling of 184 K, as marked by line 4. The measured value at melting temperature is 6.06 g cm3, which is coincident with the present calculation. Meanwhile, the calculated result of liquid Zr agrees very well with Wang’s measurement in the whole experimental temperature range. According to the above comparisons, it can be seen that the calculated densities agree well with the experimental results for liquid Ni, Ni50Zr50, Ni36Zr64 and Zr, which indicates that the simulations are reliable and have high accuracies. Besides, these calculations provide the densities at much broader temperature range, in particular in their undercooled states. However, comparisons show that there exists a large deviation between the calculated and experimental results for liquid Ni77.8Zr22.2. The reason for causing such a large discrepancy is that the adopted F-S potential developed by Mendelev et al. [19] in the present work is a semi-empirical potential, which could not be expected to predict all properties accurately. Meanwhile, Mendelev’s potential, which was developed by fitting the functions of F-S potential to properties of NiZr and NiZr2 intermetallic compounds, was used to measure the liquid-solid interface properties of the Ni-Zr B33 phase. Additionally, Ni-Zr alloy system has abundant intermetallic compounds, which is considerably complicated. Hence, Mendelev’s potential does not necessary yield accurate predictions of liquid density for all nine compositions. In the present work, the simulations overestimate the densities of liquid Ni-Zr alloys for Ni-rich components, such as liquid Ni90Zr10 and Ni77.8Zr22.2 alloys. To improve the simulated accuracy of Ni-rich components, the application of F-S potential to the densities of liquid Ni-Zr alloys on the Ni-rich side needs to be gauged against the experimental data. A practical and effective method is to introduce rescaling factors into potentials so that the MD simulation can reasonably well reproduce the known properties, which has been employed by other researchers [34,35]. Usually, potentials for pure elements and cross-functions are required during the development procedure of a F-S potential for binary alloys. Since the present calculations provide densities of liquid pure Ni and Zr with high accuracy, only one cross-function needs to be modified. Based on the F-S potential proposed by Mendelev et al., the cross-function of the interactions between Ni and Zr is modeled as [19] k X aNiZr uNiZr ðrÞ; k k NiZr

uNiZr ðrÞ ¼ k

k¼1

ð4Þ

d2q/dT2 (g cm3 K2) 7

1.62  10 5.11  108 – – – 3.55  108 1.17  108 2.48  108 3.63  108 3.16  108 2.51  108

Temperature range (K) 1200–2500 1000–2200 1000–2200 1500–2200 1200–2200 600–1800 1000–2500 800–2000 800–2000 1200–2400 1400–2800

where uNiZr ðrÞ are the basis functions of some specified form with k NiZr

coefficients aNiZr , and k the number of basis functions. A detail k description of the above functions can be found in Ref. [36]. k is an adjustable parameter, which is employed to improve the agreement with experimental data. In this work, the experimental density of liquid Ni77.8Zr22.2 alloy [4] is used to determine the value of k. After independent test runs, k is determined to be 1.07. Fig. 3 illustrates the Mendelev’s and modified pair potentials between atoms Ni and Zr. They are almost the same when the distance between atom Ni and Zr is beyond 4 Å. To evaluate the validity of the modified potential, the formation energies of Ni-Zr intermetallic compounds were reproduced by the modified potential and compared with the results obtained by the original potential [19]. Table 3 presents the formation energies of NiZr and NiZr2 intermetallic compounds at T = 0 K. It can be seen that the modified potential reproduces relatively poor formation energies compared with those obtained by the original potential. This is not surprise because the original potential was fitted to the formation energies of NiZr2 and NiZr intermetallic compounds obtained from ab initio calculations. However, the modified potential was refitted to the experimental density of liquid Ni77.8Zr22.2 alloy based on the original potential. It could produce big deviations for the other simulated issues of NiZr B33 phase. But it is valid for the present density study. Besides, the pair distribution functions of liquid Ni33.3Zr66.7 alloy at 1573 K were also calculated by these two potentials and illustrated in Fig. 4 in comparison with the experimental data [37]. The significant agreement of the pair distribution functions obtained by these two potentials is observed. More importantly, both of the pair distribution functions are consistent with the experimental result except a small deviation of the first peak, which indicates that the modified potential can provides reasonable structure of liquid Ni-Zr alloys. Then, we proceeded to recalculate the densities of liquid Ni90Zr10 and Ni77.8Zr22.2 alloys with the modified potential. The recalculated density of liquid Ni77.8Zr22.2 alloy is denoted by solid circles and shown in Fig. 2(b). It can be seen that the recalculated density and measured data are in good agreement. For liquid Ni90Zr10 alloy, the recalculated density value at the melting temperature of 1479 K is 7.94 g cm3, which is 2% smaller than the previous result. The recalculated results are also presented in Table 2. To clearly and comprehensively illustrates the calculated results as a function of temperature, simulated density values of liquid Ni-Zr alloy for different concentrations are presented in Fig. 5. A strong increase of the density upon increasing Zr content is observed when the temperature is fixed. Thermophysical properties of liquid super alloys are difficult to obtain and thus the data are scarce, especially for undercooled liquid alloys. Generally, Neumsnn-Kopp’s law can be used to estimate alloy’s properties when the data are unavailable and high precision is not required, which is on the basis of the properties of pure

P. Lü et al. / Computational Materials Science 135 (2017) 22–28

8

3

6

, ev

TL=1728 K

7.5

6

, ev

7.8

NiZr

, gcm

-3

2

7.2

10

This work 1. Ref. [25] 2. Ref. [26] 3. Ref. [27]

8.1

1

(a) Pure Ni

4

NiZr

26

2

4

2.0

2.5 -10 r, 10 m

2 1200

1600

2000

0 This work This work Ref. [4]

, g cm

-3

8.0 7.8 7.6 7.4

TL=1713 K (b) Ni77.8Zr22.2

1500

1800

This work Ref. [28]

-3

7.2

4

5

r, 10

-10

6

7

m

7.1

Compound

Ab initio [19]

FS potential from [19]

Modified potential in this work

NiZr2 C11b NiZr2 C16 NiZr B2 NiZr B33

0.291 0.349 0.360 0.460

0.256 0.388 0.392 0.499

0.160 0.239 0.406 0.295

TL=1533 K

7.0 6.9

(c) Ni50Zr50

1200

Exp. Ref. [37] Original potential Modified potential

2.0 1600

2000

2400

T, K

7.0

1.5 This work 1. Ref. [29] 2. Ref. [30]

6.9

g(r)

, g cm

3

Table 3 Formation energies (eV/atom) of the NiZr and NiZr2 intermetallic compounds at T = 0 K.

2100

T, K

7.3

-3

2

Fig. 3. Mendelev’s pair potential and the modified pair potential versus distance. The inset represents the enlargement of these two pair potentials.

1200

, g cm

Mendelev's F-S Gauged F-S

2400

T, K

3.0

1.0 0.5

6.8

TL=1283 K

0.0 6.7

(d) Ni 36Zr64

900

0 1200

1500

1800

(e) Pure Zr

4

6

r, 10

2100

T, K

6.4

2

-10

8

10

m

Fig. 4. Pair distribution functions of liquid Ni33.3Zr66.7 alloy at 1573 K. The experimental data are from Ref. [37].

, g cm

-3

1

6.2

6.0

5.8

3

This work 1. Ref. [31] 2. Ref. [32] 3. Ref. [33] 4. Ref. [3]

1600

2 4

TL=2128 K

2000

2400

2800

T, K Fig. 2. The calculated densities of liquid Ni-Zr alloys versus temperature. (a) Pure Ni; (b) Ni77.8Zr22.2; (c) Ni50Zr50; (d) Ni36Zr64; (e) Pure Zr. The open and solid circles denote the results calculated by the original and modified potentials, respectively.

approximation for liquid Ni-Zr alloys, densities versus Zr content are presented at the melting temperatures, 1800 and 1400 K, respectively, as shown in Fig. 6. At the melting temperature, the calculated densities are bigger than the approximated values when the content of Zr is in the range of 0–90%, as illustrated in Fig. 6(a). For the cases of 1800 and 1400 K, the present results are still larger than the approximations but the content of Zr is in the range of 0– 60%. The deviation between the present calculation and NeumannKopp’s rule value corresponds to the excess volume, which is written as

DV E ¼ V  V 0 ¼ elements and has been widely employed in researches. This approximation is useful and effective when the alloy is close to an ideal solution. However, in the majority of the cases, alloys deviate from the ideal solution. To evaluate the deviation degree of this

X 1 M1 þ X 2 M2

q

 

X 1 M1

q1

þ

X 2 M2

q2

 ;

ð5Þ

where subscript 1 refers to Ni and 2 to Zr, V the real volume and V0 the ideal volume. qi, Mi and Xi are the density, atomic weight and atomic fraction of components 1 and 2 respectively. Fig. 7 shows

27

P. Lü et al. / Computational Materials Science 135 (2017) 22–28

Ni33.3Zr66.7

-3

Ni20Zr80

-0.4 -10

-3

-5

-3

Ni50Zr50

, g cm

-0.2

VE/V0, cm /mol

VE, cm /mol

Ni60Zr40

7.0

0

VE/V0 at 1800 K

Ni77.8Zr22.2

7.5

VE at 1800 K

0.0

Ni Ni90Zr10

8.0

-0.6

Ni10Zr90 Zr

6.5

0 Ni

6.0 600

1200

1800

2400

20

40 60 Zr, at%

80

100 Zr

-15

Fig. 7. DVE and DVE/V0 of liquid Ni-Zr alloys versus composition. DVE is excess volume and V0 is the ideal volume.

3000

T, K is observed when the Zr content is about 40%. This indicates that liquid Ni-Zr alloy system deviates from the ideal solution and a linear approximation of alloy density based on the Neumsnn-Kopp’s law will cause a negative deviation. To more clearly describe the extent of the deviation from the ideal solution, the ratio of excess volume to ideal volume is presented in Fig. 7 as well. The largest deviation occurs when the Zr content reaches 40% and the value of DVE/V0 is 6.81%. Therefore, linear approximation should be substituted for experimental or simulated results when the high accuracy is required in quantitative researches.

Fig. 5. Densities of liquid Ni-Zr alloys.

8

TL

, g cm

-3

(a)

7

3.3. Thermal expansion coefficient

6 8

The thermal expansion coefficient is employed to express the volume change in response to temperature change and can be described as

1800 K

, g cm

-3

(b)

7

(c)

1400 K

-3

8

7

6

0

Ni

20

40

60

Zr, at%

80

ð6Þ

where bL is the thermal expansion coefficient at the melting temperature. According to Eq. (6) and the above calculated density results, the thermal expansion coefficients are calculated and fitted, as shown in Table 4. It can be seen that the thermal expansion coefficients increase with the enhancement of temperature except for liquid Ni77.8Zr22.2 alloy. The thermal expansion coefficients of liquid Ni-Zr alloys are in 105 order of magnitude, which are always larger than those of solid Ni-Zr alloys. This is because that the thermal expansion behavior of crystal is ascribed to the increment of average atom distance and anharmonicity motion of different atoms in the lattice during heating. For the liquid alloys, inter-atomic interactions are weaker and the atomic packing efficiency is lower compared with those in crystals.

6

, g cm

1 @V @q db d b ¼ q1 ðT  T L Þ þ 2 ðT  T L Þ2 ¼ bL þ V @T dT @T dT 2

b¼

100

4. Conclusions

Zr

Fig. 6. Densities of liquid Ni-Zr alloys versus composition. Density at (a) the melting temperatures, (b) 1800 K and (c) 1400 K.

the excess volume for liquid Ni-Zr alloys as a function of the content of Zr, which is calculated according to Eq. (5). To present the excess volume clearly, only the values at 1800 K are given. Obviously, Ni-Zr alloy system exhibits a negative excess volume during alloying, which may be related to a strong attractive interaction between Ni and Zr atoms due to the large negative enthalpy of mixing. The excess volume decreases first to a minimum and then increases to zero again with the increase of Zr content. A maximum deviation

Molecular dynamics simulation combined with the F-S potential was employed to systematically investigate the thermophysical properties of liquid Ni-Zr alloys in a wide composition and temperature range. The melting temperatures are obtained by the evolution of crystal-liquid-crystal sandwich model to be 1727.5, 1642.5, 1472.5 and 2112.5 K for Ni, Ni77.8Zr22.2, Ni50Zr50 and Zr, respectively. The minimum and maximum deviations are 0.06% and 4.14%. The calculated densities of liquid Ni-Zr alloys increase with the decrease of temperature and the present results are in good agreement with the reported experimental data except for Ni-rich alloys. To improve the simulation accuracy and better investigate the density of this binary system, the reported experimental density of Ni77.8Zr22.2 was applied to re-gauge the existing

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P. Lü et al. / Computational Materials Science 135 (2017) 22–28

Table 4 Thermal expansion coefficients of liquid Ni-Zr alloys. Alloy Ni Ni90Zr10 Ni77.8Zr22.2 Ni60Zr40 Ni50Zr50 Ni36Zr64 Ni20Zr80 Ni10Zr90 Zr

bL (K1)

db/dT (K2) 5

7.22  10 7.33  105 6.90  105 3.85  105 3.35  105 2.83  105 3.21  105 4.05  105 4.81  105

8

4.03  10 9.94  109 4.22  109 7.62  109 8.47  109 5.43  109 9.56  109 1.06  108 6.82  109

F-S potential and the densities for Ni-rich alloys are recalculated with the gauged potential. The liquid Ni-Zr alloy system shows a negative excess volume, which is related to the strong attractive interactions between Ni and Zr atoms due to the large negative enthalpy of mixing. This indicates that the system deviates from the ideal solution and the thermophysical properties could not be estimated from the Neumann-Kopp rule. The thermal expansion coefficients of liquid Ni-Zr alloys are in 105 order of magnitude and increase with the increasing temperature except for liquid Ni77.8Zr22.2 alloy. Acknowledgements This work is financially supported by National Natural Science Foundation of China (Grant Nos. 51474175, 51522102 and 51506182), Shaanxi Industrial Science and Technology Project (Grant No. 2015GY138). We thank the director of LMSS, Prof. B. Wei, for his consistent support. The authors are grateful to Dr. L. H. Li, Mr. S.J. Yang and Miss Q.Q. Gu for their helpful discussions. References [1] S.V. Starikov, V.V. Stegailov, Phys. Rev. B 80 (2009) 220104(R). [2] P.F. Paradis, T. Ishikawa, G.W. Lee, D. Holland-Moritz, J. Brillo, W.K. Rhim, J.T. Okada, Mater. Sci. Eng. R 76 (2014) 1. [3] H.P. Wang, S.J. Yang, L. Hu, B. Wei, Chem. Phys. Lett. 653 (2016) 112. [4] L.H. Li, L. Hu, S.J. Yang, W.L. Wang, B. Wei, J. Appl. Phys. 119 (2016) 035902. [5] J. Lee, J.E. Rodriguez, R.W. Hyers, D.M. Matson, Metall. Trans. B 46B (2015) 2470. [6] K.G.S.H. Gunawardana, S.R. Wilson, M.I. Mendelev, X. Song, Phys. Rev. E 90 (2014) 052403. [7] T. Kordel, D. Holland-Moritz, F. Yang, J. Peters, T. Unruh, T. Hansen, A. Meyer, Phys. Rev. B 83 (2011) 104205. [8] H.P. Wang, J. Chang, B. Wei, J. Appl. Phys. 106 (2009) 033506.

d2b/dT2 (K3) 12

4.59  10 1.28  1012 – 4.49  1013 4.48  1013 – 4.74  1013 6.70  1013 5.47  1013

Temperature range (K) 1200–2500 1000–2200 1200–2200 600–1800 1000–2500 800–2000 800–2000 1200–2400 1400–2800

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