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Melting of metals under pressure
Physica B 419 (2013) 40–44
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Melting of metals under pressure S. Arafin n, R.N. Singh, A.K. George Physics Department, College of Science, Sultan Qaboos University, Box: 36, Al-Khoudh 123, Oman
art ic l e i nf o
a b s t r a c t
Article history: Received 14 January 2013 Received in revised form 2 March 2013 Accepted 13 March 2013 Available online 26 March 2013
Lindemann's formula of melting is extended in terms of bulk modulus and Grüneisen parameter to study the pressure dependence of melting temperature, Tm(P) of metals. The formalism is applied to study Tm(P) of noble and transition metals, di-, tri- and tetravalent (Ag, Au, Cu, Mn, Mg, Zn, CD, In, Pb and Al) metals over a wide range of pressures up to 12 GPa. The computed melting temperatures of the metals under pressure using our semi-empirical relation is in good agreement with the experimental data. & 2013 Elsevier B.V. All rights reserved.
Keywords: Lindemann's law Melting temperature Debye temperature Bulk modulus Grüneisen parameter
1. Introduction Properties of metals under elevated conditions of temperature and pressure are important for their thermo-physical characterization. These are useful in the fields of material physics, geophysics and astrophysics. Because of the lack of microscopic information such as inter-atomic forces and atomic distributions, theoretical calculations of the melting curves based on the first principle theory lag behind. Given the complexities involved with experimental and theoretical determination of melting temperature, Tm(P) at high pressures, it is of considerable interest [1] to develop empirical relations that can satisfactorily be used to determine Tm(P) at high pressures. Such approaches are also useful to extrapolate Tm(P) at high pressures from the available low pressure data. Simon's semi-empirical equation has proved to be quite successful for large varieties of substances, but the major difficulties appear to be that of (i) uncertainty in fitting Simon's parameters and (ii) identifying the physical properties that are responsible for the dependence of Tm in the higher pressure region. A number of attempts were made in the past for the empirical evaluation of the magnitude of Tm, the most famous and in many ways most successful is that of Lindemann [2]. Lindemann's picture of melting suggests that the amplitude of atomic vibrations increases with increasing temperature and melting occurs when the amplitude of vibrations reaches a critical fraction, ym, of the mean atomic radius Ra. The quantity ym may readily be estimated [3] with the aid of the Debye model in terms of characteristic temperature θD. The critical value of ym has been found to vary n
Corresponding author. Tel.: þ968 24141452. E-mail address: sayfi[email protected] (S. Arafin).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.03.013
between 0.11 and 0.23 [4]. This has been successfully used to evaluate the melting temperature of metals at normal pressure. In the present work, a semi-empirical approach based on Lindeman's concept is used to compute the melting point of metals, Tm(P), over a wide range of pressures. We have reformulated Lindemann's formula to compute Tm(P) in terms of bulk modulus, the Grüneisen parameter and their first derivative with pressure. All these factors can be independently determined and hence the formalism is independent of any fitting parameter. It has also been applied [5] successfully to evaluate the pressure dependence of the melting of minerals and rocks such as alumina, Heusler alloy and gabbro. Under the simplified version of the formalism, Simon's empirical constants have been readily related to the bulk modulus and the Grüneisen parameter. We have applied the formalism to compute Tm(P) of Ag, Au, Cu, Mn, Mg, Zn, Cd, In, Pb and Al. These metals differ quite substantially in their melting points at atmospheric pressure. Recently Errandonea [6] has measured the melting curves for metals up to 12 GPa using Bridgman-type cell which has made it possible to compare our semi-empirical values with the measured values. Formalism leading to the modified version of Lindemann's expression in terms of bulk modulus and the Grüneisen parameter is given in Section 2. Results and discussions for the pressure dependence of the melting temperature of ten metals are given in Section 3, followed by summary and conclusion in Section 4.
2. Theoretical formulation It was proposed by Lindemann [2] that the amplitude of the atomic vibrations increases with increasing temperature and that
S. Arafin et al. / Physica B 419 (2013) 40–44
melting occurs when the amplitude of vibrations reaches a critical fraction, ym , of the mean atomic radius Ra . Lindemann's original formula, in association with the approximate expression of Mott and Jones [3] for the mean square amplitude of vibration of each atom, can be written in the form: Tm ¼
2π y Ra θD 3h m
2 ð1Þ
MkB
where θD is the Debye temperature and M is the atomic mass. θD occurs as one of the important ingredients in Eq. (1). It is very useful to characterize the atomic vibrations in solids [7] as well as in liquids [8]. It may be noted that if the observed specific heat data at low temperature is exactly fitted to the Debye formula, then θD is constant. Mott and Jones [3], however, argued that for vibrational distribution other than Debye distribution, θD should depend on temperature, T, and pressure, P. On replacing Ra ½ ¼ ð3M=4πρÞ1=3 and M in Eq. (1), one gets T m ¼ const Ω2=3 θ2D
ð2Þ
where Ω is the atomic volume. Taking the pressure derivative of the above equation dðlnT m Þ 2 1 ¼ ð3Þ ξ− dP BT 3 where the bulk modulus BT ¼ −Ωð∂P=∂ΩÞT and the Grüneisen parameter, ξ ¼ −ð∂ðlnθD Þ=∂ðlnΩÞÞ. BT and ξ in Eq. (3) are pressure dependent. For the need of a better analytical expression for BT(P) and ξ(P), it can be expanded in terms of P: ξðPÞ ¼ ξ0 þ a1 P þa2 P 2 þ ⋯
ð4Þ
BT ðPÞ ¼ B0 þ b1 P þ b2 P 2 þ ⋯
ð5Þ
ξ0 and B0 are the values at normal melting point and at zero pressure (atmospheric pressure). Taking into account the linear terms of Eqs. (4) and (5), one can readily solve Eq. (3) to get n Tm b1 P 2a1 P ¼ 1þ exp B0 b1 T0 with n¼
2b1 ξ0 −2a1 B0 2
b1
−
2 , 3b1
ð6Þ
a1 ¼
∂ξ ∂P
and
b1 ¼
∂B ∂P
ð7Þ
Eq. (6) suggests that the basic inputs to calculate Tm(P) are the bulk modulus, the Grüneisen parameter and their derivatives with pressure. Experimental values of the bulk modulus are amply available for a large group of metals. However, very few data exist for ξ and its pressure gradient. In its absence, the coefficient can even be treated as a fitting parameter. On the other hand some of the existing measurements [9,10] indicate that the dependence of ξ on P is very small. If we take a1 ¼ 0, then Eq. (6) simplifies considerably to 2=b1 ðξ0 −ð1=3ÞÞ Tm b1 P ¼ 1þ B0 T0
ð8Þ
Eq. (8) is a simplified version to compute the pressure dependence of melting temperature subjected to the condition that the bulk modulus of the material depends linearly on pressure and the Grüneisen parameter remains invariant. It is of interest to compare Eq. (8) to one of the most important and extensively used Simon's empirical relations: Tm P Y ¼ 1þ X T0
ð9Þ
41
It suggests that Simon's constant X and Y can readily be related to bulk modulus and the Grüneisen parameter respectively as B0 2 1 X¼ ð10Þ , and Y ¼ ξ0 − b1 3 b1 Eq. (6) or its simplified version Eq. (8) can be readily used to evaluate the melting temperature with increasing pressure provided the bulk modulus, the Grüneisen parameter and their gradients with pressure are known. Most of these physical parameters have been determined experimentally. However, the data for the Grüneisen parameter is scarce, and in that case we have determined it from the thermodynamic relation [11]: ξ0 ¼
β ρ cP κ S
ð11Þ
where β (K−1) is the coefficient of volume expansion, ρ (kg m−3) is the density and CP (J kg−1 K−1) is the heat capacity at constant pressure. The values of these quantities are taken from [12,13]. The adiabatic compressibility, κS (Pa−1), appearing in Eq. (11) is determined from the relation: κS ¼
1 ρðv2p −ð3=4Þv2s Þ
ð12Þ
vp and vs are the primary (longitudinal) and secondary (transverse) acoustic wave velocities respectively.
3. Results and discussions Below we present the pressure dependence of the melting curves of ten metals obtained from semi-empirical Eq. (8). Results are compared with the experimental data available in literature [6]. 3.1. Noble and transition metals (Ag, Au, Cu and Mn) Silver, copper and gold share certain attributes like having one s-orbital electron on top of a filled d-electron shell and possessing high ductility and electrical conductivity. Inter-atomic interactions in these elements are slightly contributed by the filled d-shells compared to the dominant contributions from the s-electrons through metallic bonds. This explains their low hardness and high ductility. Tm for Ag, Au and Cu at atmospheric pressure are respectively, 1234.78 K, 1337.18 K and 1357.62 K, which are comparatively higher than those for the other metals. At the macroscopic scale, introduction of extended defects in Cu to the crystal lattice, such as grain boundaries, hinders flow of the material under applied stress thereby increasing its hardness. For this reason, copper is usually supplied in a fine-grained polycrystalline form, which has greater strength than monocrystalline forms. It has the ability to remain in a face centered (fcc) structure up to pressures higher than 100 GPa. Contrary to noble metals, Mn is chemically reactive, harder and brittle. Its melting temperature, Tm ¼1497 K, is quite high at atmospheric pressure. Due to its reactive nature it has large industrial applications. Contrarily, gold is chemically one of the least reactive solid elements. The metal therefore occurs often in free elemental (native) form, as nuggets or grains in rocks, in veins and in alluvial deposits. Silver has large industrial applications due to its high electrical and thermal conductivities. It is very ductile, malleable (slightly higher than gold), monovalent coinage metal, with a brilliant white metallic luster that can take a high degree of polish. We have applied Eq. (8) to compute Tm(P) for Ag and Au for pressures up to 8 GPa and for Cu up to 16 GPa. The computed values are plotted in Fig. 1(a–c). The bulk modulus B, its first
42
S. Arafin et al. / Physica B 419 (2013) 40–44
1650 1700 Ag
Melting temperature, Tm(K)
Melting temperature, Tm(K)
1600 1550
[6] Present study
1500 1450 1400 1350 1300
Au
1650
[6] Present study
1600 1550 1500 1450 1400 1350
1250 1200
1300 0
2
4
6
0
8
1
2
3
1620
Cu Melting temperature, Tm(K)
Melting temperature, Tm(K)
1900
[6] Present study
1800
4
5
6
7
8
9
Pressure, P(GPa)
Pressure, P(GPa)
1700 1600 1500 1400 1300
Mn
1600
[6] Present work
1580 1560 1540 1520 1500
0
2
4
6
8
10
12
14
16
Pressure, P(GPa)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Pressure, P(GPa)
Fig. 1. (a) Comparison of pressure dependence of melting temperature of silver determined from the semi-empirical approach with available experimental data [6]. (b) Comparison of pressure dependence of melting temperature of gold determined from the semi-empirical approach with available experimental data [6]. (c) Comparison of pressure dependence of melting temperature of copper determined from the semi-empirical approach with available experimental data [6]. (d) Comparison of pressure dependence of melting temperature of manganese determined from the semi-empirical approach with available experimental data [6].
Table 1 Bulk modulus B, Grüneisen parameter ξ0 and pressure derivative b1 of the metals. Metal
B (GPa)
ξ0
b1 ¼ ð∂B=∂PÞ
Ag Au Cu Mn Mg Zn Cd In Pb Al
103.6 [13] (Errandonea [6]) 167 [6] 133.5 [12] 158 [6] 36.8 (Errandonea [6]) 65 [6] 42 [6] 42.68 [14] (Errandonea [6]) 42.7 [12] 73 [6]
2.4 [3] 3.03 [3] 2.01 [6] 1.95 [6] 1.51 [3] 2.25 [6] 2.4 [6] 2.04 [6] 2.63 [6] 2.17 [6]
4.7 [6] 5.7 [6] 3.88 [6] 4.6 [6] 4.3 [6] 4.6 [6] 6.5 [6] 4.8 [6] 5.7 [6] 4.5 [6]
derivative, b1 ¼ ∂B=∂P, and the Grüneisen parameter ξ0 taken from the literature [3,6,12–14] are listed in Table 1. Our computed values of the melting temperature, Tm(P), are compared with the recent experimental values of Errandonea [6]. In general the results obtained from Eq. (8) are in very good agreement with the experimental values [6]. Computed Tm(P) values of Cu, however, are less than the experimental values (Fig. 1c). The results of Mn are plotted in Fig. 1d and are compared with Errandonea [6] data. Required inputs for our calculations are taken from Table 1. The values of Tm(P) from Eq. (8) are in reasonable agreement with the experimental values. Tm increases with increasing pressure. However, the gradient ΔTm/ΔP varies considerably among the metals. Results indicate that ΔTm/ΔP is equal to 46.5 K/GPa, 38.9 K/GPa, 33.7 K/GPa and 30.6 K/GPa, respectively for Ag, Au, Cu and Mn. It may be noted that Tm−P variation is not quite linear, therefore these values reflect the average effect. It is well known that ΔTm/ΔP is related to the heat of melting (ΔHm) and change in volume on melting (ΔΩm)
through the Clausius–Clapeyron equation, i.e. ðΔT m =ΔPÞ ¼ ðT m ΔΩm =ΔH m Þ ¼ ðΔΩm =ΔSm Þ [see, for example Ref. 15]; ΔSM is the entropy of melting. Since ðΔT m =ΔPÞ is identically equal to ðΔΩm =ΔSm Þ and Tm varies linearly, it may be inferred that entropy of melting due to change in volume also maintains a linear relationship over a wide range of elevated pressure. 3.2. Divalent metals (Mg, Zn, Cd) The melting temperatures of Mg, Zn and Cd at normal pressure are respectively 922.15 K, 692.15 K and 594.05 K. Atoms are densely packed in a hexagonal close-packed crystalline structure. Mg is a fairly light weight metal and is found abundantly in nature. In contrast Cd is a soft, malleable, ductile and bluish white metal. Zn is in many respects similar to Cd, but it is hard and brittle. Cd and Zn are not always considered as transition metals, since they do not have partly d and f electrons and have comparatively low melting temperatures. Computed values of the melting temperatures from Eq. (8) as a function of pressure are plotted in Fig. 2(a–c). The basic inputs for Mg, Zn and Cd are given in Table 1. The theoretical results, in general, are in good agreement with the observed data [6]. ðΔT=ΔPÞ¼ 47.7 K/GPa, 38.7 K/GPa and 48.1 K/GPa are respectively found for Mg, Zn and Cd. The rate of variation of melting temperature with pressure is quite large for Mg and Cd and is about 20% higher than that of Zn. 3.3. Tri- and tetravalent metals (In, Pb and Al) All three belong to relatively low melting metals with melting temperatures 429.55 K, 600.55 K and 933.25 K respectively for In, Pb and Al. The latter is the most abundant metal in the earth's crust which makes up 0.8% weight of the earth's solid surface.
S. Arafin et al. / Physica B 419 (2013) 40–44
43
1200 Zn
Mg
1400
Melting temperature, Tm(K)
Melting temperature, Tm(K)
1500
[6] Present work
1300 1200 1100 1000
1100
[6] Present dwork
1000 900 800 700
900 0
2
4
6
8
10
0
12
2
4
Pressure, P(GPa)
6
8
10
12
Pressure, P(GPa)
Melting temperature, Tm(K)
1200 Cd
1100
[6] Present work
1000 900 800 700 600 0
2
4
6
8
10
12
Pressure, P(GPa)
Fig. 2. (a) Comparison of the pressure dependence of melting temperature of magnesium determined from the semi-empirical approach with experimental data [6]. (b) Comparison of the pressure dependence of melting temperature of zinc determined from the semi-empirical approach with experimental data [6]. (c) Comparison of pressure dependence of melting temperature of cadmium determined from the semi-empirical approach with available experimental data [6].
850
Pb
1200
Melting temperature, Tm(K)
Melting temperature, Tm(K)
1300
[6] Present work
1100 1000 900 800 700 600
In
800
[6] Present dwork
750 700 650 600 550 500 450 400
500 0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
Pressure, P(GPa)
Pressure, P(GPa)
1600 Melting temperature, Tm(K)
Al 1500 [6] Present study
1400 1300 1200 1100 1000 900 0
2
4
6
8
10
Pressure, P(GPa)
Fig. 3. (a) Comparison of pressure dependence of melting temperature of lead determined from the semi-empirical approach with available experimental data [6]. (b) Comparison of pressure dependence of melting temperature of indium determined from the semi-empirical approach with available experimental data [6]. (c) Comparison of the pressure dependence of melting temperature for aluminum determined from the semi-empirical approach with experimental data [6].
44
S. Arafin et al. / Physica B 419 (2013) 40–44
Al atoms are arranged in a face-centered cubic structure and are known for its light weight, ductility and malleability. Likewise, In is soft and malleable and atoms are arranged in tetragonal positions. Pb is also soft but has poor malleability where atoms are arranged in a face-centered cubic structure. Results obtained for the variation of melting point as a function of pressure from Eq. (8) are plotted in Fig. 3(a–c) and are compared with the experimental and some other available results for Al. Required inputs of Eq. (8) are listed in Table 1. The computed results of Pb are in very good agreement with the experimental results [6]. For In the results are also in good agreement in the lower range of pressure, but for P 46 GPa, the theoretical values are higher than the observed values. The computed values of Tm(P) for Al are closer to the experimental results in the lower range of pressure, but are less than the experimental values [6] at high pressures. 4. Summary and conclusions Semi-empirical relations are derived from Lindemann's law to study the pressure dependence of melting temperature of metals. Both the bulk modulus and the Grüneisen parameter have been expressed as polynomials of pressure and an equation for the melting temperature is derived as a function of pressure. Under the assumption of invariance of the Grüneisen parameter with pressure, the semi-empirical relation takes a simple form, which yields physical meaning for Simon's constants. The basic inputs to the simplified version of the equation are the bulk modulus at atmospheric pressure, its pressure gradient and the Grüneisen parameter at atmospheric pressure. The approach is free from any fitting parameter. The formalism is applied to study the pressure dependence of Tm(P) for noble and transition metals, (Ag, Au, Cu, Mn), divalent metals (Mg, Zn, Cd), tri- and tetravalent metals
(In, Pb, Al). The computed values of Tm increase with increasing pressure and are in good agreement with the available experimental data for most of the metals studied. However there is slight disagreement between the experimental and theoretical values at higher pressures. For In, the experimental values are lower and for Al they are higher than the theoretical values for pressures P≥6 GPa. Though the variation of Tm(P) is not quite linear, but average values of ΔTm/ΔP suggest that entropy of melting due to change in volume maintains a linear relationship over a wide range of elevated pressures. References [1] J. Ganguly, Thermodynamics in Earth and Planetary Sciences, Springer-Verlag, Berlin, 2008. [2] F.A. Lindemann, Uber die Berechnung molecularer Eigenfrequenzen, Phys. Z 11 (1910) 609. [3] N.F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys, second ed., Oxford University Press, Oxford, 1936. [4] T.E. Faber, Theory of Liquid Metals, Cambridge Monographs on Physics, Cambridge University Press; Cambridge, UK, 1972. [5] S. Arafin, R.N. Singh, A.K. George, Int. J. Thermophys. 33 (2012) 1013. [6] D. Errandonea, J. Appl. Phys. 108 (2010) 033517. [7] C. Kittel, Introduction to Solid State Physics, seventh ed., Wiley, New York, 1996. [8] F. Sommer, R.N. Singh, V. Wittusiewicz, J. Alloys Compd. 325 (2001) 118. [9] D.H. Huang, X.R. Liu, L. Shu, C.G. Shao, R. Jia, S.M. Hong, J. Phys. D: Appl. Phys 40 (2007) 5327. [10] G. Cui, R. Yu, Physica B 390 (2007) 220. [11] D.L. Anderson, Theory of the Earth, online ed., Caltech Books, Caltech, Pasadena, 1989. [12] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, MIT Press, Cambridge, Massachusetts, 1971. [13] E.A. Brandes., G.B. Brook (Eds.), Smithells Metals Reference Book, seventh ed., Butterworth-Heinemann; Oxford, UK, 1992. [14] J. Ramakrishnan, R. Boehler, G.H. Higgins, G.C. Kennedy, J. Geophys. Res. 83 (1978) 3535. [15] T. Iida, R.L. Guthrie, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1988.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Melting of metals under pressure S. Arafin n, R.N. Singh, A.K. George Physics Department, College of Science, Sultan Qaboos University, Box: 36, Al-Khoudh 123, Oman
art ic l e i nf o
a b s t r a c t
Article history: Received 14 January 2013 Received in revised form 2 March 2013 Accepted 13 March 2013 Available online 26 March 2013
Lindemann's formula of melting is extended in terms of bulk modulus and Grüneisen parameter to study the pressure dependence of melting temperature, Tm(P) of metals. The formalism is applied to study Tm(P) of noble and transition metals, di-, tri- and tetravalent (Ag, Au, Cu, Mn, Mg, Zn, CD, In, Pb and Al) metals over a wide range of pressures up to 12 GPa. The computed melting temperatures of the metals under pressure using our semi-empirical relation is in good agreement with the experimental data. & 2013 Elsevier B.V. All rights reserved.
Keywords: Lindemann's law Melting temperature Debye temperature Bulk modulus Grüneisen parameter
1. Introduction Properties of metals under elevated conditions of temperature and pressure are important for their thermo-physical characterization. These are useful in the fields of material physics, geophysics and astrophysics. Because of the lack of microscopic information such as inter-atomic forces and atomic distributions, theoretical calculations of the melting curves based on the first principle theory lag behind. Given the complexities involved with experimental and theoretical determination of melting temperature, Tm(P) at high pressures, it is of considerable interest [1] to develop empirical relations that can satisfactorily be used to determine Tm(P) at high pressures. Such approaches are also useful to extrapolate Tm(P) at high pressures from the available low pressure data. Simon's semi-empirical equation has proved to be quite successful for large varieties of substances, but the major difficulties appear to be that of (i) uncertainty in fitting Simon's parameters and (ii) identifying the physical properties that are responsible for the dependence of Tm in the higher pressure region. A number of attempts were made in the past for the empirical evaluation of the magnitude of Tm, the most famous and in many ways most successful is that of Lindemann [2]. Lindemann's picture of melting suggests that the amplitude of atomic vibrations increases with increasing temperature and melting occurs when the amplitude of vibrations reaches a critical fraction, ym, of the mean atomic radius Ra. The quantity ym may readily be estimated [3] with the aid of the Debye model in terms of characteristic temperature θD. The critical value of ym has been found to vary n
Corresponding author. Tel.: þ968 24141452. E-mail address: sayfi[email protected] (S. Arafin).
0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physb.2013.03.013
between 0.11 and 0.23 [4]. This has been successfully used to evaluate the melting temperature of metals at normal pressure. In the present work, a semi-empirical approach based on Lindeman's concept is used to compute the melting point of metals, Tm(P), over a wide range of pressures. We have reformulated Lindemann's formula to compute Tm(P) in terms of bulk modulus, the Grüneisen parameter and their first derivative with pressure. All these factors can be independently determined and hence the formalism is independent of any fitting parameter. It has also been applied [5] successfully to evaluate the pressure dependence of the melting of minerals and rocks such as alumina, Heusler alloy and gabbro. Under the simplified version of the formalism, Simon's empirical constants have been readily related to the bulk modulus and the Grüneisen parameter. We have applied the formalism to compute Tm(P) of Ag, Au, Cu, Mn, Mg, Zn, Cd, In, Pb and Al. These metals differ quite substantially in their melting points at atmospheric pressure. Recently Errandonea [6] has measured the melting curves for metals up to 12 GPa using Bridgman-type cell which has made it possible to compare our semi-empirical values with the measured values. Formalism leading to the modified version of Lindemann's expression in terms of bulk modulus and the Grüneisen parameter is given in Section 2. Results and discussions for the pressure dependence of the melting temperature of ten metals are given in Section 3, followed by summary and conclusion in Section 4.
2. Theoretical formulation It was proposed by Lindemann [2] that the amplitude of the atomic vibrations increases with increasing temperature and that
S. Arafin et al. / Physica B 419 (2013) 40–44
melting occurs when the amplitude of vibrations reaches a critical fraction, ym , of the mean atomic radius Ra . Lindemann's original formula, in association with the approximate expression of Mott and Jones [3] for the mean square amplitude of vibration of each atom, can be written in the form: Tm ¼
2π y Ra θD 3h m
2 ð1Þ
MkB
where θD is the Debye temperature and M is the atomic mass. θD occurs as one of the important ingredients in Eq. (1). It is very useful to characterize the atomic vibrations in solids [7] as well as in liquids [8]. It may be noted that if the observed specific heat data at low temperature is exactly fitted to the Debye formula, then θD is constant. Mott and Jones [3], however, argued that for vibrational distribution other than Debye distribution, θD should depend on temperature, T, and pressure, P. On replacing Ra ½ ¼ ð3M=4πρÞ1=3 and M in Eq. (1), one gets T m ¼ const Ω2=3 θ2D
ð2Þ
where Ω is the atomic volume. Taking the pressure derivative of the above equation dðlnT m Þ 2 1 ¼ ð3Þ ξ− dP BT 3 where the bulk modulus BT ¼ −Ωð∂P=∂ΩÞT and the Grüneisen parameter, ξ ¼ −ð∂ðlnθD Þ=∂ðlnΩÞÞ. BT and ξ in Eq. (3) are pressure dependent. For the need of a better analytical expression for BT(P) and ξ(P), it can be expanded in terms of P: ξðPÞ ¼ ξ0 þ a1 P þa2 P 2 þ ⋯
ð4Þ
BT ðPÞ ¼ B0 þ b1 P þ b2 P 2 þ ⋯
ð5Þ
ξ0 and B0 are the values at normal melting point and at zero pressure (atmospheric pressure). Taking into account the linear terms of Eqs. (4) and (5), one can readily solve Eq. (3) to get n Tm b1 P 2a1 P ¼ 1þ exp B0 b1 T0 with n¼
2b1 ξ0 −2a1 B0 2
b1
−
2 , 3b1
ð6Þ
a1 ¼
∂ξ ∂P
and
b1 ¼
∂B ∂P
ð7Þ
Eq. (6) suggests that the basic inputs to calculate Tm(P) are the bulk modulus, the Grüneisen parameter and their derivatives with pressure. Experimental values of the bulk modulus are amply available for a large group of metals. However, very few data exist for ξ and its pressure gradient. In its absence, the coefficient can even be treated as a fitting parameter. On the other hand some of the existing measurements [9,10] indicate that the dependence of ξ on P is very small. If we take a1 ¼ 0, then Eq. (6) simplifies considerably to 2=b1 ðξ0 −ð1=3ÞÞ Tm b1 P ¼ 1þ B0 T0
ð8Þ
Eq. (8) is a simplified version to compute the pressure dependence of melting temperature subjected to the condition that the bulk modulus of the material depends linearly on pressure and the Grüneisen parameter remains invariant. It is of interest to compare Eq. (8) to one of the most important and extensively used Simon's empirical relations: Tm P Y ¼ 1þ X T0
ð9Þ
41
It suggests that Simon's constant X and Y can readily be related to bulk modulus and the Grüneisen parameter respectively as B0 2 1 X¼ ð10Þ , and Y ¼ ξ0 − b1 3 b1 Eq. (6) or its simplified version Eq. (8) can be readily used to evaluate the melting temperature with increasing pressure provided the bulk modulus, the Grüneisen parameter and their gradients with pressure are known. Most of these physical parameters have been determined experimentally. However, the data for the Grüneisen parameter is scarce, and in that case we have determined it from the thermodynamic relation [11]: ξ0 ¼
β ρ cP κ S
ð11Þ
where β (K−1) is the coefficient of volume expansion, ρ (kg m−3) is the density and CP (J kg−1 K−1) is the heat capacity at constant pressure. The values of these quantities are taken from [12,13]. The adiabatic compressibility, κS (Pa−1), appearing in Eq. (11) is determined from the relation: κS ¼
1 ρðv2p −ð3=4Þv2s Þ
ð12Þ
vp and vs are the primary (longitudinal) and secondary (transverse) acoustic wave velocities respectively.
3. Results and discussions Below we present the pressure dependence of the melting curves of ten metals obtained from semi-empirical Eq. (8). Results are compared with the experimental data available in literature [6]. 3.1. Noble and transition metals (Ag, Au, Cu and Mn) Silver, copper and gold share certain attributes like having one s-orbital electron on top of a filled d-electron shell and possessing high ductility and electrical conductivity. Inter-atomic interactions in these elements are slightly contributed by the filled d-shells compared to the dominant contributions from the s-electrons through metallic bonds. This explains their low hardness and high ductility. Tm for Ag, Au and Cu at atmospheric pressure are respectively, 1234.78 K, 1337.18 K and 1357.62 K, which are comparatively higher than those for the other metals. At the macroscopic scale, introduction of extended defects in Cu to the crystal lattice, such as grain boundaries, hinders flow of the material under applied stress thereby increasing its hardness. For this reason, copper is usually supplied in a fine-grained polycrystalline form, which has greater strength than monocrystalline forms. It has the ability to remain in a face centered (fcc) structure up to pressures higher than 100 GPa. Contrary to noble metals, Mn is chemically reactive, harder and brittle. Its melting temperature, Tm ¼1497 K, is quite high at atmospheric pressure. Due to its reactive nature it has large industrial applications. Contrarily, gold is chemically one of the least reactive solid elements. The metal therefore occurs often in free elemental (native) form, as nuggets or grains in rocks, in veins and in alluvial deposits. Silver has large industrial applications due to its high electrical and thermal conductivities. It is very ductile, malleable (slightly higher than gold), monovalent coinage metal, with a brilliant white metallic luster that can take a high degree of polish. We have applied Eq. (8) to compute Tm(P) for Ag and Au for pressures up to 8 GPa and for Cu up to 16 GPa. The computed values are plotted in Fig. 1(a–c). The bulk modulus B, its first
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S. Arafin et al. / Physica B 419 (2013) 40–44
1650 1700 Ag
Melting temperature, Tm(K)
Melting temperature, Tm(K)
1600 1550
[6] Present study
1500 1450 1400 1350 1300
Au
1650
[6] Present study
1600 1550 1500 1450 1400 1350
1250 1200
1300 0
2
4
6
0
8
1
2
3
1620
Cu Melting temperature, Tm(K)
Melting temperature, Tm(K)
1900
[6] Present study
1800
4
5
6
7
8
9
Pressure, P(GPa)
Pressure, P(GPa)
1700 1600 1500 1400 1300
Mn
1600
[6] Present work
1580 1560 1540 1520 1500
0
2
4
6
8
10
12
14
16
Pressure, P(GPa)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
Pressure, P(GPa)
Fig. 1. (a) Comparison of pressure dependence of melting temperature of silver determined from the semi-empirical approach with available experimental data [6]. (b) Comparison of pressure dependence of melting temperature of gold determined from the semi-empirical approach with available experimental data [6]. (c) Comparison of pressure dependence of melting temperature of copper determined from the semi-empirical approach with available experimental data [6]. (d) Comparison of pressure dependence of melting temperature of manganese determined from the semi-empirical approach with available experimental data [6].
Table 1 Bulk modulus B, Grüneisen parameter ξ0 and pressure derivative b1 of the metals. Metal
B (GPa)
ξ0
b1 ¼ ð∂B=∂PÞ
Ag Au Cu Mn Mg Zn Cd In Pb Al
103.6 [13] (Errandonea [6]) 167 [6] 133.5 [12] 158 [6] 36.8 (Errandonea [6]) 65 [6] 42 [6] 42.68 [14] (Errandonea [6]) 42.7 [12] 73 [6]
2.4 [3] 3.03 [3] 2.01 [6] 1.95 [6] 1.51 [3] 2.25 [6] 2.4 [6] 2.04 [6] 2.63 [6] 2.17 [6]
4.7 [6] 5.7 [6] 3.88 [6] 4.6 [6] 4.3 [6] 4.6 [6] 6.5 [6] 4.8 [6] 5.7 [6] 4.5 [6]
derivative, b1 ¼ ∂B=∂P, and the Grüneisen parameter ξ0 taken from the literature [3,6,12–14] are listed in Table 1. Our computed values of the melting temperature, Tm(P), are compared with the recent experimental values of Errandonea [6]. In general the results obtained from Eq. (8) are in very good agreement with the experimental values [6]. Computed Tm(P) values of Cu, however, are less than the experimental values (Fig. 1c). The results of Mn are plotted in Fig. 1d and are compared with Errandonea [6] data. Required inputs for our calculations are taken from Table 1. The values of Tm(P) from Eq. (8) are in reasonable agreement with the experimental values. Tm increases with increasing pressure. However, the gradient ΔTm/ΔP varies considerably among the metals. Results indicate that ΔTm/ΔP is equal to 46.5 K/GPa, 38.9 K/GPa, 33.7 K/GPa and 30.6 K/GPa, respectively for Ag, Au, Cu and Mn. It may be noted that Tm−P variation is not quite linear, therefore these values reflect the average effect. It is well known that ΔTm/ΔP is related to the heat of melting (ΔHm) and change in volume on melting (ΔΩm)
through the Clausius–Clapeyron equation, i.e. ðΔT m =ΔPÞ ¼ ðT m ΔΩm =ΔH m Þ ¼ ðΔΩm =ΔSm Þ [see, for example Ref. 15]; ΔSM is the entropy of melting. Since ðΔT m =ΔPÞ is identically equal to ðΔΩm =ΔSm Þ and Tm varies linearly, it may be inferred that entropy of melting due to change in volume also maintains a linear relationship over a wide range of elevated pressure. 3.2. Divalent metals (Mg, Zn, Cd) The melting temperatures of Mg, Zn and Cd at normal pressure are respectively 922.15 K, 692.15 K and 594.05 K. Atoms are densely packed in a hexagonal close-packed crystalline structure. Mg is a fairly light weight metal and is found abundantly in nature. In contrast Cd is a soft, malleable, ductile and bluish white metal. Zn is in many respects similar to Cd, but it is hard and brittle. Cd and Zn are not always considered as transition metals, since they do not have partly d and f electrons and have comparatively low melting temperatures. Computed values of the melting temperatures from Eq. (8) as a function of pressure are plotted in Fig. 2(a–c). The basic inputs for Mg, Zn and Cd are given in Table 1. The theoretical results, in general, are in good agreement with the observed data [6]. ðΔT=ΔPÞ¼ 47.7 K/GPa, 38.7 K/GPa and 48.1 K/GPa are respectively found for Mg, Zn and Cd. The rate of variation of melting temperature with pressure is quite large for Mg and Cd and is about 20% higher than that of Zn. 3.3. Tri- and tetravalent metals (In, Pb and Al) All three belong to relatively low melting metals with melting temperatures 429.55 K, 600.55 K and 933.25 K respectively for In, Pb and Al. The latter is the most abundant metal in the earth's crust which makes up 0.8% weight of the earth's solid surface.
S. Arafin et al. / Physica B 419 (2013) 40–44
43
1200 Zn
Mg
1400
Melting temperature, Tm(K)
Melting temperature, Tm(K)
1500
[6] Present work
1300 1200 1100 1000
1100
[6] Present dwork
1000 900 800 700
900 0
2
4
6
8
10
0
12
2
4
Pressure, P(GPa)
6
8
10
12
Pressure, P(GPa)
Melting temperature, Tm(K)
1200 Cd
1100
[6] Present work
1000 900 800 700 600 0
2
4
6
8
10
12
Pressure, P(GPa)
Fig. 2. (a) Comparison of the pressure dependence of melting temperature of magnesium determined from the semi-empirical approach with experimental data [6]. (b) Comparison of the pressure dependence of melting temperature of zinc determined from the semi-empirical approach with experimental data [6]. (c) Comparison of pressure dependence of melting temperature of cadmium determined from the semi-empirical approach with available experimental data [6].
850
Pb
1200
Melting temperature, Tm(K)
Melting temperature, Tm(K)
1300
[6] Present work
1100 1000 900 800 700 600
In
800
[6] Present dwork
750 700 650 600 550 500 450 400
500 0
2
4
6
8
10
12
14
0
2
4
6
8
10
12
Pressure, P(GPa)
Pressure, P(GPa)
1600 Melting temperature, Tm(K)
Al 1500 [6] Present study
1400 1300 1200 1100 1000 900 0
2
4
6
8
10
Pressure, P(GPa)
Fig. 3. (a) Comparison of pressure dependence of melting temperature of lead determined from the semi-empirical approach with available experimental data [6]. (b) Comparison of pressure dependence of melting temperature of indium determined from the semi-empirical approach with available experimental data [6]. (c) Comparison of the pressure dependence of melting temperature for aluminum determined from the semi-empirical approach with experimental data [6].
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S. Arafin et al. / Physica B 419 (2013) 40–44
Al atoms are arranged in a face-centered cubic structure and are known for its light weight, ductility and malleability. Likewise, In is soft and malleable and atoms are arranged in tetragonal positions. Pb is also soft but has poor malleability where atoms are arranged in a face-centered cubic structure. Results obtained for the variation of melting point as a function of pressure from Eq. (8) are plotted in Fig. 3(a–c) and are compared with the experimental and some other available results for Al. Required inputs of Eq. (8) are listed in Table 1. The computed results of Pb are in very good agreement with the experimental results [6]. For In the results are also in good agreement in the lower range of pressure, but for P 46 GPa, the theoretical values are higher than the observed values. The computed values of Tm(P) for Al are closer to the experimental results in the lower range of pressure, but are less than the experimental values [6] at high pressures. 4. Summary and conclusions Semi-empirical relations are derived from Lindemann's law to study the pressure dependence of melting temperature of metals. Both the bulk modulus and the Grüneisen parameter have been expressed as polynomials of pressure and an equation for the melting temperature is derived as a function of pressure. Under the assumption of invariance of the Grüneisen parameter with pressure, the semi-empirical relation takes a simple form, which yields physical meaning for Simon's constants. The basic inputs to the simplified version of the equation are the bulk modulus at atmospheric pressure, its pressure gradient and the Grüneisen parameter at atmospheric pressure. The approach is free from any fitting parameter. The formalism is applied to study the pressure dependence of Tm(P) for noble and transition metals, (Ag, Au, Cu, Mn), divalent metals (Mg, Zn, Cd), tri- and tetravalent metals
(In, Pb, Al). The computed values of Tm increase with increasing pressure and are in good agreement with the available experimental data for most of the metals studied. However there is slight disagreement between the experimental and theoretical values at higher pressures. For In, the experimental values are lower and for Al they are higher than the theoretical values for pressures P≥6 GPa. Though the variation of Tm(P) is not quite linear, but average values of ΔTm/ΔP suggest that entropy of melting due to change in volume maintains a linear relationship over a wide range of elevated pressures. References [1] J. Ganguly, Thermodynamics in Earth and Planetary Sciences, Springer-Verlag, Berlin, 2008. [2] F.A. Lindemann, Uber die Berechnung molecularer Eigenfrequenzen, Phys. Z 11 (1910) 609. [3] N.F. Mott, H. Jones, The Theory of the Properties of Metals and Alloys, second ed., Oxford University Press, Oxford, 1936. [4] T.E. Faber, Theory of Liquid Metals, Cambridge Monographs on Physics, Cambridge University Press; Cambridge, UK, 1972. [5] S. Arafin, R.N. Singh, A.K. George, Int. J. Thermophys. 33 (2012) 1013. [6] D. Errandonea, J. Appl. Phys. 108 (2010) 033517. [7] C. Kittel, Introduction to Solid State Physics, seventh ed., Wiley, New York, 1996. [8] F. Sommer, R.N. Singh, V. Wittusiewicz, J. Alloys Compd. 325 (2001) 118. [9] D.H. Huang, X.R. Liu, L. Shu, C.G. Shao, R. Jia, S.M. Hong, J. Phys. D: Appl. Phys 40 (2007) 5327. [10] G. Cui, R. Yu, Physica B 390 (2007) 220. [11] D.L. Anderson, Theory of the Earth, online ed., Caltech Books, Caltech, Pasadena, 1989. [12] G. Simmons, H. Wang, Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, MIT Press, Cambridge, Massachusetts, 1971. [13] E.A. Brandes., G.B. Brook (Eds.), Smithells Metals Reference Book, seventh ed., Butterworth-Heinemann; Oxford, UK, 1992. [14] J. Ramakrishnan, R. Boehler, G.H. Higgins, G.C. Kennedy, J. Geophys. Res. 83 (1978) 3535. [15] T. Iida, R.L. Guthrie, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1988.