Molecular dynamics prediction of density for metastable liquid noble metals

Chemical Physics Letters 539–540 (2012) 30–34

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Molecular dynamics prediction of density for metastable liquid noble metals H.P. Wang, S.J. Yang, B. Wei ⇑ 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 22 March 2012 In final form 1 May 2012 Available online 9 May 2012

a b s t r a c t The thermophysical properties of metastable liquid noble metals are not readily available due to the great experimental difficulties. Here the densities of liquid Pd, Pt, Ag, and Au are predicted by molecular dynamics method. The pair distribution functions are computed to monitor the atomic structure of these noble metals, which indicate that the systems remain in liquid state in the process of simulation. The calculated densities exhibit nonlinear temperature dependences and prove to have a high accuracy. The density data are obtained in a much broader temperature range, especially in the undercooled regime. Moreover, the molar volumes and the thermal expansion coefficients are also derived from the density predictions. Ó 2012 Elsevier B.V. All rights reserved.

1. Introduction Because of their limited existence inside the earth, the noble metals such as Pd, Pt, Ag, and Au cannot be applied everywhere in common life. However, with the development of materials science, they are widely applied in various fields, including smart materials, nano-materials, and magnetic materials [1–5]. Accordingly, the use of noble metals has extended beyond their traditional applications of jewellery and coinage. Some alloys added with Pd element have been used to manufacture the optical sensor to detect hydrogen gas and monitor the temperature [6]. Pt-based alloys have been developed for the applications which require excellent corrosion and oxidation resistance [1]. The magnesium alloys containing Ag element are extensively applied in the aircraft materials due to their creep corrosion resistance [7]. Au, due to its good optical property, is employed to produce gold-plated glass for UV protection [8]. As a result of the outstanding physical and chemical properties, these noble metals have become increasingly important in various fields such as aerospace materials, electronic technology, and chemical industry. Accurate density data of these noble metals are of great importance in many fundamental investigations of materials processing such as refining, casting and welding [9–13]. However, these important data in liquid state are rather rare according to the current literatures, especially for highly undercooled liquid metals. When dealing with metastable liquid metals, the high melting temperatures and the high reactivity lead to great difficulties to measure their densities by experiments. In order to further understand the liquid noble metals, the thermophysical properties in a broad temperature range are always

⇑ Corresponding author. Fax: +86 88495926. E-mail address: [email protected] (B. Wei). 0009-2614/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2012.05.003

required, especially for the undercooled regime. Nevertheless, the broad undercooling range is limited in the available experiments due to the great experimental difficulties. Paradis et al. [9] obtained the densities of liquid Pd and Pt with the maximum undercoolings of 188 and 350 K by electrostatic levitation (ESL) technique. In Smithells Metals Reference Book [10], the densities at the melting temperatures of these metals are given, but the data in undercooled liquid state are not available. Moreover, there are differences for the densities at the melting temperatures and the temperature coefficients among these results. Thus, the densities of the noble metals require further study in order to satisfy the requirement of materials research. Molecular dynamics (MD) method with a potential model has been applied to study the properties of liquid metals, such as density, surface tension, phase transformation, thermodynamics properties and structure [14–19]. This has been proven to be an effective method to simulate the thermophysical properties of liquid metals and alloys. As compared with the experimental methods, molecular dynamics method can easily achieve a high undercooling and the information in more details could be obtained [20,21]. In this Letter, the densities of liquid noble metals, including Pd, Pt, Ag, and Au, are computed under sufficiently large undercoolings by molecular dynamics method. In addition, the molar volumes and the thermal expansion coefficients are also derived from the density data.

2. Method The modified embedded atom method (MEAM) [22], which was proposed by Bakes on the basis of the potential model of embedded atom method (EAM). Both EAM and MEAM potentials can produce reasonable simulated results such as density, melting point, specific heat, atomic structure [16–20,23–25]. Moreover, MEAM

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potential display distinguished advantages. Especially, due to its validity in dealing with the metals with face-centered-cubic crystal structure [13,18], MEAM model is selected to study these noble metals. The total energy of system is expressed by [22]:

E¼

" # X 1X F i ðqi Þ þ /ij ðr ij Þ 2 jði–jÞ i

ð1Þ

where qi is the total electron density at atom i, Fi the embedding energy for placing an atom into that electron density, rij the separation of atoms i and j, and /ij the pair potential interaction between atoms of i and j, which is written as [22]:

/ij ¼

  F i ðq0i ðrÞÞ 2 u Ei ðrÞ  Zi Zi

ð2Þ

Figure 1. Pair distribution functions at the maximum undercoolings.

here Zi is the number of nearest neighbors, Eui the energy per atom of reference structure as a nearest-neighbor distance, and q0i the background electron density for the reference structure of i. The parameters of MEAM model for these noble metals are listed in Table 1. For the simulation process, the system is under constant temperature and constant pressure. 32,000 atoms are arranged in a cubic box and the system is subjected to periodic boundary in three dimensions. The time step is set to 1 fs and the temperature is adjusted every 50 steps. All of codes (LAMMPS) run in a Lenovo 1800 Cluster system. In order to get an equilibrium liquid state, the starting temperature must be well above the melting temperatures. Different metals have different starting temperatures: 3500 K for Pd, 3300 K for Pt, 2200 K for Ag, and 3000 K for Au. The initial temperature is kept at constant for 100,000 steps. The cooling process with a cooling rate of 1013 K s1 is carried out for calculation at 100 K intervals of temperature. At each temperature, 100,000 steps are performed to achieve an equilibrium state and the last 50,000 steps are applied to compute the final results. The pair distribution function, g(r), can be applied to determine the atomic structure, therefore, the state of simulated metals is monitored by the pair distribution function companied with the mean square displacement versus simulated time at each simulated temperature.

Although the simulated systems are in the state of liquid in the whole temperature range including large undercoolings, the liquid structures change with the decrease of temperature. Figure 2 presents the PDFs heights at the first neighbor distance R1 of the four noble metals, which increase with the falling of temperature. For instance, the PDF at R1 of liquid Pd is 4.10 at 1000 K, which is 61.4% larger than that of 2.54 at 2400 K, as shown in Figure 2a. Taking liquid Au as another example, the value is 4.44 at 500 K, 72.7% larger than 2.57 at 1800 K, as shown in Figure 2b. From the above analysis, the PDFs heights at R1 of these noble metals at their maximum undercoolings are much larger than those above the melting temperatures, and this suggests that the ordered degree of atom distributions of the noble metals enhance from normal liquid to undercooled liquid. 3.2. Density The density data of liquid Pd, Pt, Ag, and Au are given in Figure 3, which shows that the density q exhibits a nonlinear relationship with temperature T:

q ¼ qm þ a1 ðT  T m Þ þ a2 ðT  T m Þ2

ð3Þ

3. Results and discussion

where qm is the density at the melting temperature Tm, a1 the first temperature coefficient and a2 the second temperature coefficient. As a typical result, Figure 3a shows that the density of liquid Pd is dependant on T as follows:

3.1. Liquid structure

qPd ¼ 10:16  1:52  103 ðT  T m Þ  2:11  107 ðT  T m Þ2 g cm3

In order to examine whether the simulated systems are in the state of liquid, the pair distribution functions (PDF) of noble metals are calculated in order to ensure the validity of the final results. Here, the PDFs at their maximum undercoolings are illustrated in Figure 1, which show sharper peaks at the first neighbor distances and fluctuation around one beyond the second neighbor distances for all the four metals. This indicates that the simulated systems of noble metals are ordered in atomic range and disordered in long range. It can be confirmed that the simulated four noble metals are still in the state of liquid in the whole simulated temperature.

ð4Þ

where Tm is 1828 K. In terms of Eq. (4), the density of liquid Pd at the melting temperature is 10.16 g cm3, and its first and second temperature coefficients are 1.52  103 and 2.11  107 g cm3 K2, respectively. The melting points Tm marked by the dash lines in Figure 3a are the experimental equilibrium values from Refs. [9,10] so as to compare the simulated density results with the reported values. The temperature for liquid Pd is in the range of 1000–2400 K, including 828 K undercooling and 572 K superheating. Usually, it is difficult to achieve such a large undercooling by experiments.

Table 1 Parameters of MEAM model for the noble metals [21].

Pd Pt Ag Au

Ei0

Ri0

ai

Ai

bi(0)

bi(1)

bi(2)

bi(3)

ti(0)

ti(1)

ti(2)

ti(3)

3.910 5.770 2.850 3.930

2.75 2.77 2.88 2.88

6.43 6.44 5.89 6.34

1.01 1.04 1.06 1.04

4.98 4.67 4.46 5.45

2.2 2.2 2.2 2.2

6.0 6.0 6.0 6.0

2.2 2.2 2.2 2.2

1 1 1 1

2.34 2.73 5.54 1.59

1.38 1.38 2.45 1.51

4.48 3.29 1.29 2.61

Ei0 (eV), the sublimation energy; Ri0 (Å), the equilibrium nearest-neighbor distance; ai, the exponential decay factor for the universal energy function; Ai, the scaling factor for the embedding energy; bi(n), the exponential decay factors for the atomic densities; ti(n), the weighting factors for the atomic densities.

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10.16 g cm3. This indicates that they are in good agreement. However, the maximum undercooling of 188 K achieved in Ref. [9] is much smaller than that in this work. In addition, the density data of liquid Pd from Smithells metals reference book [10] are also presented in Figure 3a, which is expressed by:

qPd ¼ 10:49  1:23  103 ðT  T m Þ g cm 3

ð6Þ

When comparing the density value of liquid Pd at the melting temperature, the present value is only 3.1% smaller than the reported value from Ref. [10]. However, there are only data above the melting temperature in Smithells metals reference book, and no data are available for metastable liquid Pd below the melting temperature. Meanwhile, the calculated density results of the other liquid noble metals Pt, Ag, and Au are given in Figure 3b–c. The temperature dependences of density of these three metals are similar to that of liquid Pd, i.e., the densities exhibit nonlinear dependences on temperature as follows:

Figure 2. Pair distribution function height at the first neighbor distance versus temperature.

In order to evaluate the present result, the density of liquid Pd obtained by the other researchers is also illustrated in Figure 3a. Paradis et al. [9] measured the density of liquid Pd in the temperature range of 1640–1875 K, a linear expression is obtained as follows:

qPd ¼ 10:66  7:70  104 ðT  T m Þ g cm 3

ð5Þ

It can be seen that Paradis’s result is a little larger than the present work. At the melting temperature, the value of 10.66 g cm3 by Paradis et al. is only 4.9% larger than the present value of

qPt ¼ 18:93  1:48  103 ðT  T m Þ  2:87  107 ðT  T m Þ2 g cm 3

ð7Þ

qAg ¼ 9:16  1:57  103 ðT  T m Þ  3:22  107 ðT  T m Þ2 g cm 3

ð8Þ

qAu ¼ 17:56  2:15  103 ðT  T m Þ  4:50  107 ðT  T m Þ2 g cm 3

ð9Þ

In order to clearly understand the present work, both the calculated density and the reported results are also listed in Table 2. For liquid Pt, the value is 18.93 g cm3 at the melting temperature of 2041 K, which is 1.4% smaller than Paradis’s value of 19.20 g cm3, and only 0.1% larger than the data from Ref. [10]. For liquid Ag, a difference of only 1.9% exists between the present work and the data from Ref. [10] at the melting temperature of 1234 K. For liquid Au, the present value is 17.56 g cm3 at the melting temperature, which is about 1.2% smaller than the value from Ref. [10]. From the above analysis, it can be seen that the differences between the present results and the data from the current literatures are in the range of 0.1–4.9%. This suggests that the calculated

Figure 3. The calculated density of liquid metals versus temperature. (a) Pd, (b) Pt, (c) Ag and (d) Au.

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H.P. Wang et al. / Chemical Physics Letters 539–540 (2012) 30–34 Table 2 The density results in the present work and the current literatures. Tm (K)

qm

 a1 (g cm3 K1)

 a2 (g cm3 K2)

Undercooling (K)

T (K)

Ref.

(g cm3)

Pd

1828

Pt

2041

Ag

1234

Au

1336

10.16 10.66 10.49 18.93 19.20 18.91 9.16 9.33 17.56 17.36

1.52  103 7.70  104 1.23  103 1.48  103 9.60  104 2.88  103 1.57  103 9.10  104 2.15  103 1.50  103

2.11x107 — — 2.87x107 — — 3.22x107 — 4.50x107 —

828 188 — 1041 350 — 734 — 836 —

1000–2400 1640–1875 1828–1925 1000–2400 1691–2216 2041–2141 500–1800 1234–1334 500–1800 1336–1436

Present [9] [10] Present [9] [10] Present [10] Present [10]

Metal

densities have good accuracies. In addition, although the relationships are nonlinear between the density and the temperature of the noble metals, the second temperature coefficients are so little that these functions are nearly linear in the temperature ranges close to the melting points, which agree with the experimental data. It is noteworthy that this work provides the density values at much broader temperature ranges, especially for large undercooling regimes. 3.3. Molar volume and thermal expansion coefficient According to the density results, the molar volumes and the thermal expansion coefficients could be derived as a function of temperature. Figure 4 illustrates the molar volume Vm of the four noble metals, which exhibit nonlinear dependences on temperature: 1

ð10Þ

1

ð11Þ

V Pd ¼ 8:99 þ 3:38  105 T þ 4:26  107 T 2 cm3 mol V Pt ¼ 9:47 þ 1:86  105 T þ 1:92  107 T 2 cm3 mol

1

V Ag ¼ 10:33 þ 2:71  104 T þ 7:29  107 T 2 cm3 mol

1

V Au ¼ 10:10 þ 2:87  104 T þ 4:13  107 T 2 cm3 mol

work

work

work work

From Figure 4, the molar volume enhances with the rise of temperature for all the four noble metals. At the melting temperature, the Vm of liquid Pd is 10.47 cm3 mol1, which drops to 9.45 cm3 mol1 when the undercooling achieves the maximum value of 828 K. For liquid Pt, the value of Vm is 9.69 cm3 mol1 at the maximum undercooling of 1041 K. As the temperature increases, it enhances to 10.31 cm3 mol1 at the melting temperature. For Ag, Vm changes from 10.66 to 11.77 cm3 mol1 in the temperature range of 500–1234 K. For Au, it is 11.22 cm3 mol1 at the melting temperature of 1336 K and decreases to 10.37 cm3 mol1 at 500 K. It is evident that the molar volumes at the maximum undercoolings are 6–11% smaller than those at the melting temperatures of the four noble metals. Meanwhile, the thermal expansion coefficient b can be expressed as

b¼

1 @V @q ¼ q1 V @T @T

ð14Þ

According to Eq. (14) and the densities of the four noble metals, the thermal expansion coefficients are computed and illustrated in Figure 5, which can be described by

ð12Þ

bPd ¼ ð7:04 þ 2:16  103 T þ 1:18  106 T 2 Þ  105 K 1

ð15Þ

ð13Þ

bPt ¼ ð1:72 þ 2:33  103 T þ 3:19  107 T 2 Þ  105 K 1

ð16Þ

Figure 4. Molar volume versus temperature, (a) Pd & Pt, and (b) Ag & Au.

Figure 5. Thermal expansion coefficient versus temperature, (a) Pd & Pt, and (b) Ag & Au.

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bAg ¼ ð7:96 þ 4:75  103 T þ 2:19  106 T 2 Þ  105 K 1

ð17Þ

bAu ¼ ð5:09 þ 4:07  103 T þ 9:75  107 T 2 Þ  105 K 1

ð18Þ

The thermal expansion coefficients of the four noble metals increase with the enhancement of temperature, which are 5–10 times as large as those of the solid metals at room temperature. For instance, b of Pt at 298 K is 8.8  106 K1 from Ref. [10], while it achieves 7.67  105 K1 at 2000 K, which is 8.7 times as large as that at 298 K. Taking Ag as another example, b is 1.89  105 K1 at 298 K and it attains 1.68  104 K1 at 1200 K, which is 8.8 times as large as that of the solid state. That is to say, the thermal expansion coefficients at liquid state are much larger than those at solid state, which may result from the characteristics that the atomic force of liquid metals is much weaker than that of solid metals.

4. Conclusion The pair distribution functions indicate that the simulated systems of the four noble metals are ordered in atomic range and disordered in long range. The PDF heights at the first neighbor distance remarkably increase with the rise of undercooling, which suggests that the ordered degree of atom distributions of the four noble metals enhance from normal liquid to undercooled liquid. The density data of the noble metals Pd, Pt, Ag and Au are predicted, which exhibit nonlinear dependences on temperature. The differences of the predicted densities at the melting temperatures are only 0.1–4.9% between this work and the reported data. Nevertheless, the present work provides the density data in a much broader temperature range, especially in the undercooled regime. In addition, the molar volumes and the thermal expansion coefficients are also derived from the calculated densities.

Acknowledgment This work was financially supported by National Natural Science Foundation of China (Grant Nos. 50971103 and 50971105), the Program for New Century Excellent Talents, the Shaanxi Project for Young New Star in Science and Technology, and NPU Foundation for Fundamental Research. References [1] A.P. Jardine et al., Phys. Rev. Lett. 105 (2010) 136101. [2] S. Mechler, E. Yahel, P.S. Pershan, M. Meron, B. Lin, Appl. Phys. Lett. 98 (2011) 251915. [3] Y.J. Qi, C.J. Lu, Q.F. Zhang, L.H. Wang, F. Chen, C.S. Cheng, B.T. Liu, J. Phys. DAppl. Phys. 41 (2008) 065407. [4] M. Torrell et al., J. Appl. Phys. 109 (2011) 074310. [5] J. Zhu, C. Zhang, D. Ballard, P. Martin, J. Fournelle, W. Cao, Y.A. Chang, Acta Mater. 58 (2010) 180. [6] W.P. Hsieh, D.G. Cahill, J. Appl. Phys. 109 (2011) 113520. [7] D.X. Ye, S. Mutisya, M. Bertino, Appl. Phys. Lett. 99 (2011) 081909. [8] K. Yang, C. Clavero, J.R. Skuza, M. Varela, R.A. Lukaszew, J. Appl. Phys. 107 (2010) 103924. [9] P.F. Paradis, T. Ishikawa, S. Yoda, Adv. Space Res. 41 (2008) 2118. [10] W.F. Gale, T.C. Totemeier (Eds.), Smithells Metals Reference Book, eighth edn., Elsevier Butterworth Heinemann, Burlington, 2004. p. 14–10. [11] J. Brillo, I. Egry, T. Matsushita, Int. J. Mater. Res. 97 (2006) 1526. [12] J.J.Z. Li, W.L. Johnson, W.K. Rhim, Appl. Phys. Lett. 89 (2006) 111913. [13] Y.P. Lu, N. Liu, T. Shi, D.W. Luo, W.P. Xu, T.J. Li, J. Alloys Compd. 490 (2010) L1. [14] M.I. Mendelev, M.J. Kramer, R.T. Ott, D.J. Sordelet, Philos. Mag. 89 (2009) 109. [15] N. Jakse, A. Pasturel, Phys. Rev. B 79 (2009) 144206. [16] A.E. Gheribi, Mater. Chem. Phys. 116 (2009) 489. [17] M. Widom, P. Ganesh, S. Kazimirov, D. Louca, M. Mihalkovic, J. Phys.: Condens. Matter 20 (2008) 114114. [18] H.P. Wang, J. Chang, B. Wei, J. Appl. Phys. 106 (2009) 033506. [19] H.P. Wang, S.J. Yang, B. Wei, Chin. Sci. Bull. 57 (2012) 719. [20] P. Ganesh, M. Widom, Phys. Rev. B 77 (2008) 014205. [21] H.P. Wang, B.C. Luo, B. Wei, Phys. Rev. E 78 (2008) 041204. [22] M.I. Baskes, Phys. Rev. B 46 (1992) 2727. [23] X.J. Han, M. Chen, Z.Y. Guo, J. Phys.: Condens. Matter 16 (2004) 705. [24] S.O. Kart, M. Tomak, M. Uludogan, T. Cagin, J. Non-Cryst. Solids 337 (2004) 101. [25] Y.J. Lu, Philos. Mag. Lett. 92 (2012) 56.