Pressure Dependence of Fusion Entropy and ... - ACS Publications

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Pressure Dependence of Fusion Entropy and Fusion Volume of Six Metals Qi-Long Cao,† Pan-Pan Wang,† Duo-Hui Huang,† Qiang Li,† Fan-Hou Wang,*,† and Ling Cang Cai‡ †

Key Laboratory of Computational Physics, Yibin University, Yibin 644007, China Key Laboratory of National Defense Science and Technology for Shock Wave and Detonation Physics and the Science, MianYang 621900, China

‡

ABSTRACT: Molecular dynamics simulations of the melting curves of six metals including Ag, Cu, Al, Mg, Ta, and Mo for the pressure range (0 to 15) GPa are reported. The melting curves of Ag, Cu, Al, and Mg fully confirm measurements and previous calculations. Meanwhile, the melting curves of Ta and Mo are consistent with previous calculations but diverge from laser-heated diamond-anvil cells values at high pressure. Our results suggest that the melting slope at 100 kPa is related to the electronic configuration of the element. In addition, the pressure dependence of fusion entropy and fusion volume are calculated up to 15 GPa. The overall fusion entropy is separated into topological entropy of fusion (ΔSD) due to the configuration change in melting and the volume entropy of fusion (ΔSV) due to the latent volume change in melting. Furthermore, we checked the R ln 2 rule under high pressure, according to which the value of ΔSD is a constant at ambient pressure. Result shows that the value of ΔSD is close to R ln 2 at ambient pressure and reflects a slow decrease when pressure increases.

■

INTRODUCTION The melting of metals under high pressure (HP) has received considerable attention over the past decade because its understanding is fundamentally important to the fields of nuclear materials, shock physics, geophysics, and astrophysics.1,2 Basically, laser-heated diamond-anvil cells (LHDAC) and shockwave experiments (SW) are two widely used measurements to get the melting of metals at high pressure. Beside experiments, several empirical laws and theoretical methods have been applied to determine the melting curve of materials under compression. Despite the intensive experimental and theoretical study, huge discrepancies continue exist among LHDAC, SW, and theoretical melting curves of transition metals like Ta and Mo under pressure, while the same measurements and methods/empirical laws have achieved excellent agreement in other metals like Al and Cu.3−12 A great deal of effort has been devoted to this issue, and several possible explanations have been proposed. We note that, in the last few years, studies have been mostly focused on the effects on HP melting of physical phenomena like liquid frustration, cluster formation, pressure-induced sp-to-d electron transfer, shear, and chemical reactions.7,9,13 As it is well-known, the solid−liquid phase transition is accompanied by changes in entropy (ΔS) and volume (ΔV). ΔS is used to get the equilibrium melting temperature from superheated temperature in shock melting experiments14 and the pressure span of the solid−liquid mixed phase region in shockrelease model.15 Furthermore, the melting slope is equal to the ratio of the fusion volume to fusion entropy (ΔV/ΔS), according to the Clausius−Clapeyron relation: dTm ΔV = dP ΔS

There thus is a need for detailed knowledge of the pressure dependences of ΔV and ΔS during phase transition, since it is of critical importance in the understanding of phase transition processes. The overall fusion entropy can be separated into topological entropy of fusion due to the configuration change in melting (ΔSD) and the volume entropy of fusion due to the volume change in melting (ΔSV), as given by the following equation:14,16−19

ΔS = ΔSV + ΔSD

The volume entropy of fusion is temperature-dependent and corresponds to a simple scaling down in volume of the liquid, from the volume of liquid on the freezing line to that of solid on the melting line, without any phase transformation. The topological entropy of fusion measures the structural difference of the two phases, which have the same volume and temperature. Experimental works on simple substances like Ar, Na, and Cs18,19 suggest that the topological entropy of fusion is a constant, R ln 2 per mole (5.8 J·K−1·mol−1), for all of these materials at ambient pressure. The R ln 2 rule has been checked against many data (metals, silicates, and alkali halides) at ambient pressure,14,16 and it was assumed that the R ln 2 rule works under high pressure. To our best knowledge, what is the pressure dependence of ΔSD has not been clarified, except our last study of Ta.20 So, we need to perform further studies to confidently label the R ln 2 rule. To these ends, the pressure dependence of the ΔV, ΔS, and ΔSD of aluminum (Al), copper (Cu), silver (Ag), magnesium (Mg), tantalum (Ta), Received: July 14, 2012 Accepted: December 12, 2012 Published: December 27, 2012

(1) © 2012 American Chemical Society

(2)

64

dx.doi.org/10.1021/je3008638 | J. Chem. Eng. Data 2013, 58, 64−70

Journal of Chemical & Engineering Data

Article

where ρi is the host electronic density, and it can be written as the sum of the electronic density function φ(rij) of the individual atom i

and molybdenum (Mo) up to 15 GPa are examined. When possible, our results are compared with previous data. The rest of this work is organized as follows: in Section II, we describe the potential functions and details of molecular dynamic simulation; in Section III, our results for the melting curves of the six metals up to 15 GPa, together with the ΔV, ΔS, ΔSV, and ΔSD as functions of pressure are presented, and discussions are made; finally, we give the conclusions in Section IV.

ρi =

The electronic density function φ(r) is expressed by ⎧(r − d)2 + B2 (r − d)4 r ≤ d φ (r ) = ⎨ ⎩0 r>d ⎪

⎪

COMPUTATIONAL METHODS AND DETAILS Potential Functions. Since the 1980s, a variety of empirical potentials have been introduced and employed to study the properties of metals. Among these potentials, the embedded atomic method (EAM) potential21 is a well-studied many-body interaction potential that has been widely used in numerical simulation studied in metals and alloys. For example, the phase transitional, structural, and transposition properties of the Al,22−25 Cu,25−27 Ag,25,28 and Mg29−31 have been well-investigated by using this potential. So, the EAM potential was used to describe the interparticle interactions of simple metals including Al, Cu, Ag, and Mg in this work. According to this formalism, the potential energy E of a system can be written as 1 Etot = ∑ Fi(ρi ) + ∑ φij(rij) 2 j≠i i (3)

∑ ρij (rij) (4)

The potential for Al, Cu, Ag, and Mg was described in detail in refs 22, 26, 28, and 29, respectively. In this work, the parameter values were taken from these papers. However, researchers have found some shortcomings of the EAM potential in the simulations of the early transition metals, particularly those having the body-centered cubic (bcc) crystal structure (Mo and Ta).6,32,33 Recently, the extend Finnis− Sinclair potential is developed to describe the interaction of the early transition metals.34 It is found that this potential is very useful for the reproduction of the lattice constants, melting points, equation of state, pressure−volume relationships, and melting heats.6,34 According to this formalism, the total potential energy of a system can be written as: Etot =

1 2

Tm = T+ + T − −

i

(5)

RESULTS AND DISCUSSION Lattice Constant and High-Pressure Melting. The lattice constant for crystalline materials is of foundational importance. It can provide a convenient measure of the atomic volume and the compressibility coefficient. To get the lattice constant of the six metals, we have performed molecular dynamic simulations at pressures up to 15 GPa at 300 K. At the given temperature, NPT MD simulations were run at five pressures. After equilibrium of the systems by 25 000 steps, statistics of the lattice constant were collected during additional 25 000 steps. The pressure dependences of the lattice constant at 300 K are plotted in Figure 1, in which

⎧(r − c)2 (c + c r + c r 2 + c r 3 + c r 4) r ≤ c 0 1 2 3 4 V (r ) = ⎨ ⎩0 r>c ⎪

⎪

(6)

where c is a cutoff parameter, and it was assumed to lie between the second and the third neighbor atoms. c0, c1, c2, c3, and c4 are the potential parameters. The embedding function f is expressed by

ρi

(10)

■

where rij is the distance between atom i and j. The repulsive portion is the conventional central pair-potential, in which V(r) is expressed by

f (ρi ) =

T+T −

To judge the accuracy of the one-phase approach used in the present paper, the melting temperatures of Al and Ta obtained with this method was compared with that calculated by the coexistence phase molecular dynamics simulations (this method was described in detail in refs 6 and 28) at ambient pressure. Both methods give results that are very close each other.

∑ V (rij) − ∑ f (ρi ) ij

(9)

where d is the cutoff parameter, which was assumed to lie between the second and the third neighbor atoms. The parameter values were taken from ref 34. Molecular Dynamic Simulations. All molecular dynamic (MD) simulations were performed with the large-scale atomic/ molecular massively parallel simulator (LAMMPS code).35 The simulation have been performed in the ensemble containing 15 × 15 × 15 conventional unit cells with a time step of 2 fs, and threedimensional periodic boundary conditions were applied. We have conducted a test to establish the sensitivity of the melting temperature, fusion volume, and entropy on the sizes of the systems and found that the sizes of the systems used in our work are sufficient for the present purpose. As experimental and theoretical results suggested that Ta and Mo remain in bcc, Mg remains in the hexagonal close-packed (hcp) structure, and other metals (Al, Cu and Ag) in face-centered cubic (fcc) structures throughout the reported pressure range at 300 K. The relaxation times used for the thermostat and barostat are 10.0 ps and 30.0 ps, respectively. The system was equilibrated for 50 ps, and the remaining time of the 50 ps was used to get good statistical averages of the physical properties such as lattice constant, enthalpy, and volume. The one-phase approach was used to obtain the melting temperature Tm,6,36,37 and the supercell was simulated in isothermal−isobaric (NPT) ensembles. The heating simulation of the solid with a rate of 0.5 K·ps−1 would lead to a sudden jump in volume at the upper limit of the melting temperature T+. Also, the cooling simulation of the liquid with the same rate would lead to a sudden drop in the volume at lower limit of melting temperature T−. Then the equilibrium melting temperature Tm can be estimated from the upper limit and the lower limit of melting temperature by

where rij and φij are the distance and the short-range pair potential between atoms i and j. Fi(ρi) is the embedding energy of atom i, and ρi denotes the electron density of the host without atom i, which is approximated to a sum of the atomic electron density of j atom at a distance of rij from site i, j≠i

(8)

j≠i

■

ρi =

∑ A2φ(rij)

(7) 65



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data are in agreement with experiments for all six metals, and the lattice constant decreases with increasing pressure at given temperature, as expected. We also find that the effect of pressure on the lattice constant is the largest for Mg in the present pressure range, this may be related to the fact that there exists an hcp to dhcp (double hcp) or bcc transition before melting for Mg.39−41 The melting temperatures of the six metals up to 15 GPa have been calculated using the one-phase approach. First, the system is heated up to liquid state with a rate of 0.5 K·ps−1; during this process the upper limit of the melting temperature T+ can be obtained. Next the temperature is decreased with the same rate to get lower limit of melting temperature T−. Then the melting temperature can be estimated by eq 10. Figure 2 shows the obtained results for these metals together with experimental2,42−44 and previous theoretical results.3,4,45−49 The agreement of the present results is not only good with previous theoretical calculations but also with the experimental results for Al, Cu, Ag, and Mg. On the other hand, it can be seen that our melting curves are consistent with previous calculations but diverge from LHDAC values at higher pressure, for the early transition metals Ta and Mo. Furthermore, we also find that the melting temperatures can

Figure 1. Lattice constant versus pressure relationships for six metals, that is, ○, Ag; △, Al; ▽, Cu; ◇, Mg; ◁, Mo; and ▷, Ta at 300 K. The solid symbols represent corresponding experimental data at 100 kPa and 300 K in ref 38. Lines are guides for the eyes.

we also compare the results with experiments at 100 kPa (shown as solid symbols).38 From this figure, we can find that our 100 kPa

Figure 2. Comparison of melting temperatures with experiments2,42−44 and other calculations.3,4,45−49 (a) Melting temperatures of Ag: ■, this work; ○, Errandonea et al.; ···, Mitra et al.; solid line, fit line by the Simon equation. (b) Melting temperatures of Cu: ■, this work; ○, Errandonea et al.; △, Brand et al.; ···, Belonoshko et al.; dashed line, Vocadlo et al.; solid line, fit line by the Simon equation. (c) Melting temperatures of Al: ■, this work; ○, Errandonea et al.; △, Boehler et al.; dashed line, Alfe et al.; solid line, fit line by the Simon equation. (d) Melting temperatures of Mg: ■, this work; ○, Errandonea et al.; dashed line, Moriarty et al.; solid line, fit line by the Simon equation. (e) Melting temperatures of Ta: ■, this work; ···, Errandonea et al.; dash line, Taioli et al.; solid line, fit line by the Simon equation. (f) Melting temperatures of Mo: ■, this work; ···, Errandonea et al.; dash line, Cazorla et al.; solid line, fit line by the Simon equation. 66



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here Hl and Hs are the enthalpies on the liquid state and solid state at the melting point Tm, respectively. To get the Hl and Hs, the computer simulations are based on constant-pressure MD simulation. First, the temperature is increased to the melting temperature with a rate of 0.5 K·ps−1 and run for 25 000 time steps to get an equilibrium solid state. Next, the system is heated up to liquid state, and then the temperature decreased to the melting temperature with the same rate and run for 25 000 time steps to get an equilibrium liquid state. For each of the equilibrium state (solid state or liquid state), another run of 25 000 time steps is performed to get the statistical averages of Hl, Hs, and ΔV. Together with the entropy change on melting calculated by us, the volume change of melting expressed as a percentage of the volume of the solid at 100 kPa are listed in Table 2. For comparison, other calculations4,15,38,46,51 and experiments52−56 at 100 kPa are also listed in the table. From this table, we can find that the present results are in agreement with experimental values and results calculated by other researchers. Pressure dependencies of volume change and entropy change are plotted in Figure 3; a glance of this figure shows that both volume change and entropy change decrease with increasing pressure for all studied systems. Moreover, the volume change and entropy change of Ag, Al, and Mg at ambient pressure is larger than that of Cu, Ta, and Mo. This result is consistent with the empirical rule that close-packed metals exhibit a larger average volume increase and average entropy change than the body-centered cubic metals at ambient pressure.51 The most striking factor of our results is the steep decrease of volume change and entropy change of Mg in the range from (0 to 15) GPa; this also may be related to the fact that there exists an hcp to dhcp (double hcp) or bcc transition before melting.39−41 The R ln 2 Rule for Entropy of Melting. Since the overall fusion entropy can be separated into topological entropy of fusion (ΔSD) due to the configuration change in melting and the volume entropy of fusion (ΔSV) due to the latent volume change in melting, and ΔSD is suggested to be a constant at ambient pressure, according to the R ln 2 rule. Our interest is to study the pressure dependence of the ΔSD. The ΔSV can be expressed formally by the Maxwell relation:

be very well-fitted with the Simon equation Tm = a(1 + P /b)c

(11)

where P and Tm are pressure and corresponding melting temperatures. a, b, and c are fit parameters. The melting slope (dTm/dP) at any pressure can be obtained using the Simon equation. For instance, the melting slope at 100 kPa obtained from the Simon equation are listed in Table 1. For comparison, other calculations2,4,47,49 Table 1. Comparison of the Melting Slope, dTm/dP (K·GPa−1), at 100 kPa Obtained from the Simon Equation with Other Calculations2,4,47,49 and Experiments2,42,44,45 metal

this work

experiments

other calculations

Ag Cu Al Mg Ta Mo

49.73 42.00 58.27 59.14 27.54 38.38

47a, 60b 43a, 45c 60a 60a 24f 33.3a

44a 38d

a

b

80e 33.7g

c

Reference 2. Reference 45. Reference 42. Reference 49. fReference 44. gReference 4.

e

d

Reference 47.

and experiments2,42,44,45 at 100 kPa are also listed in the table. From this table, we can see that the agreement between our results and other calculations is good, except for the theoretical work performed by Moriarty and Althoff,49 which overestimate the melting slope of Mg at ambient pressure. It is interesting to note that the elements with the similar electronic structure have similar melting slopes: the melting slope at 100 kPa is about 59 K·GPa−1 for nearly free electron sp metals (Al and Mg); the noble metals (Ag and Cu) have smaller melting slopes than that of Al and Mg; early transition metals (Ta and Mo) with partly filled d-bands have the smallest melting slopes. This results confirm a previous suggestion that the electronic configuration of an element is determinant of its melting behavior under compression.50 Fusion Volume and Entropy. The solid−liquid phase transition is accompanied by changes in entropy (ΔS) and volume (ΔV). Now, we turn to the volume change and the entropy change on melting. It is well-known that the change of Gibbs energy (ΔG) is zero in the solid−liquid coexisting equilibrium:

⎛ ∂SV ⎞ ⎛ ∂P ⎞ α ⎜ ⎟ =⎜ ⎟ = ⎝ ∂V ⎠T ⎝ ∂T ⎠V kT

(12)

ΔG = ΔH − T ΔS = 0

where ΔH is the enthalpy of fusion. Then, the entropy of fusion can be determined by ΔS =

H − Hs ΔH = l Tm Tm

(14)

Here α is the expansion coefficient, which can be determined from the volume−temperature curves and given by α=

(13)

1 ⎛⎜ ∂V ⎞⎟ V ⎝ ∂T ⎠ P

(15)

Table 2. Comparison of the Fractional Volume Change, ΔV, and Entropy Change, ΔS, on the Melting at Ambient Pressure with Experiments4,15,38,46,51 and Other Calculations52−56 ΔS/(J·K−1·mol−1)

ΔV (%)

a

element

present

others

expt.

present

others

expt.

Ag Cu Al Mg Ta Mo

5.32 4.90 6.28 5.55 2.64 1.91

5.3a 4.4,a 4.99d 9.22d

3.8,b 3.51c 4.2,b 3.96c 6.5,b 6.9c 4.12,b 2.95c 2.75i

9.97 9.15 11.6 10.7 8.09 8.61

8.4a 8.3,a 10.67e 11.3,f 11.4g 9.711h 9.623h 13.496h

8.99b,c 9.59b,c 11.2b,c 9.53b,c 7.51b,c 12.3b,c

1.5j

Reference 52. bReference 38. cReference 51. dReference 53. eReference 46. fReference 15. gReference 54. hReference 55. iReference 56. jReference 4. 67



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48.1·10−3 GPa−1, 6.3·10−3 GPa−1, and 6.9·10−3 GPa−1.39,62−64 From this table, we can find that the expansion coefficient and the compressibility coefficient decrease with the pressure increasing. The agreement obtained between the present and previous results of α and kT at 100 kPa gives credibility to the accuracy of the ΔSV and ΔSD. The calculated values of ΔSV and ΔSD are shown in Figure 4. The entropy change relates to volume change,

Figure 3. Pressure dependence of (a) fractional volume change and (b) entropy change on melting of □, Ag; ○, Cu; △, Al; ▽, Mg; ◇, Ta and ◁, Mo. Lines are guides for the eyes.

While kT is the compressibility coefficient, which can be obtained from the volume−pressure curves according to the equation 1 ⎛⎜ ∂V ⎞⎟ V ⎝ ∂P ⎠T

(16)

Thus, ΔSV can be expressed as α ΔSV = ΔV kT

(17)

kT = −

Figure 4. Pressure dependence of (a) the topological entropy of fusion, ΔSD, and (b) the volume entropy of fusion, ΔSV, in melting of □, Ag; ○, Cu; △, Al; ▽, Mg; ◇, Ta and ◁, Mo. Lines are guides for the eyes.

and the configuration change showed similar trends as the overall entropy change, but the latter was more dominant as shown in this figure. Furthermore, we can find that the value of ΔSD at ambient pressure is close to R ln 2 per mole, which is the specific value of the fusion topological entropy of simple, atomic substance, for Ag, Cu, Ta, and Mo. The value of ΔSD of Al and Mg at ambient pressure is slightly larger. They reflect a slow decrease when pressure goes from (0 to 15) GPa. The above results suggest that the R ln 2 rule does not apply accurately to none of these studied metals, but R ln 2 should be considered only as approximation of the value of ΔSD. It was known that the rule was justified for atomic systems with a simple cell model,18

here ΔV is the volume change on melting. Then the topological entropy of fusion was determined by subtracting ΔSV from ΔS. The expansion coefficient (α) and the compressibility coefficient (kT) at melting points obtained from eqs 15 and 16 are listed in Table 3. The expansion coefficients of Ag, Cu, Al, Mg, Ta, and Mo at their melting temperatures under ambient pressure obtained by our calculation coincide with that reported by others, 9.6·10−5 K−1, 8.4·10−5 K−1, 11.9·10−5 K−1, 11.5·10−5 K−1, 3.6·10−5 K−1, and 3.1·10−5 K−1.57−61 The compressibility coefficients of Cu, Mg, Ta, and Mo at 100 kPa obtained by our calculation coincide with that reported by others, 10.5·10−3 GPa−1,

Table 3. Expansion Coefficients, α, and the Compressibility Coefficients, kT, of the Six Metals α (10−5 K−1)

pressure

kT (10−3 GPa−1)

GPa

Ag

Cu

Al

Mg

Ta

Mo

Ag

Cu

Al

Mg

Ta

Mo

1·10−4 1 5 10 15

9.4 8.6 6.6 5.4 4.5

7.8 7.4 6.5 5.5 4.8

10.5 10.1 8.8 7.6 6.7

15.2 12.9 9.3 6.1 4.5

4.1 3.9 3.2 2.6 2.6

3.4 3.3 3.1 2.8 2.6

14.4 13.2 9.6 7.5 6.3

10.0 9.3 7.7 6.4 5.6

17.2 15.1 12.2 9.5 8.1

48.4 42.2 30.6 18.6 12.5

6.2 6.1 5.4 4.6 4.2

6.5 6.2 5.4 4.7 4.3

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(9) Dewaele, A.; Mezouar, M.; Guignot, N.; Loubeyre, P. High Melting Points of Tantalum in a Laser-Heated Diamond Anvil Cell. Phys. Rev. Lett. 2010, 104, 255701. (10) Burakovsky, L.; Chen, S. P.; Preston, D. L.; Belonoshko, A. B.; Rosengren, A.; Mikhaylushkin, A. S.; Simak, S. I.; Moriarty, J. A. HighPressure-High-Temperature Polymorphism in Ta: Resolving an Ongoing Experimental Controversy. Phys. Rev. Lett. 2010, 104, 255702. (11) Ruiz-Fuertes, J.; Karandikar, A.; Boehler, R.; Errandonea, D. Microscopic evidence of a flat melting curve of tantalum. Phys. Earth Planet. Int. 2010, 181, 69−72. (12) Cazorla, C.; Alfè, D.; Gillan, M. J. Constraints on the phase diagram of molybdenum from first-principles free-energy calculations. Phys. Rev. B 2012, 85, 064113. (13) Ross, M.; Errandonea, D.; Boehler, R. Melting of transition metals at high pressure and the influence of liquid frustration: The early metals Ta and Mo. Phys. Rev. B 2007, 76, 184118. (14) Luo, S. N.; Ahrens, T. J. Shock-induced superheating and melting curves of geophysically important minerals. Phys. Earth Planet. Int. 2004, 143, 369−386. (15) Dai, C.; Tan, H.; Geng, H. Model for assessing the melting on Hugoniots of metals: Al, Pb, Cu, Mo, Fe, and U. J. Appl. Phys. 2002, 92, 5019. (16) Oriani, R. A. The Entropies of Melting of Metals. J. Chem. Phys. 1951, 19, 93. (17) Tallon, J. L.; Robinson, W. H. A model-free elasticity theory of melting. Phys. Lett. A 1982, 87, 365. (18) Stishov, S. M. The thermodynamics of melting of simple substances. Sov. Phys. Usp. 1975, 11, 625. (19) Rivier, N.; Duffy, D. M. On the topological entropy of atomic liquids and the latent heat of fusion. J. Phys. C: Solid State Phys. 1982, 15, 2867−2874. (20) Cao, Q. L.; Huang, D. H.; Qiang, L.; Wanga, F. H.; Cai, L. C.; Zhang, X. L.; Jing, F. Q. Improving the understanding of the melting curve of tantalum at extreme pressures through the pressure dependence of fusion volume and entropy. Physica B 2012, 407, 2784−2789. (21) Daw, M. S.; Baskes, M. I. Embedded-atom method: Derivation and application to impurities, surfaces, and other defects in metals. Phys. Rev. B 1984, 29, 6443. (22) Mishin, Y.; Farkas, D.; Mehl, M. J.; Papaconstantopoulos, D. A. Interatomic potentials for monoatomic metals from experimental data and ab initio calculations. Phys. Rev. B 1999, 59, 3393−3407. (23) Zhang, Y.; Wang, L.; Wang, W.; Zhou, J. Structural transition of sheared-liquid metal in quenching state. Phys. Lett. A 2006, 355, 142− 147. (24) Wang, H.; Xu, D.; Yang, R.; Veyssière, P. The transformation of edge dislocation dipoles in aluminium. Acta Mater. 2008, 56, 4608− 4620. (25) Deng, C.; Sansoz, F. Fundamental differences in the plasticity of periodically twinned nanowires in Au, Ag, Al, Cu, Pb and Ni. Acta Mater. 2009, 57, 6090−6101. (26) Mishin, Y.; Mehl, M. J.; Papaconstantopoulos, D. A.; Voter, A. F.; Kress, J. D. Structural stability and lattice defects in copper: Ab initio, tight-binding, and embedded-atom calculations. Phys. Rev. B 2001, 63, 224106. (27) An, Q.; Luo, S. N.; BoHan, L.; Zheng, L.; Tschauner, O. Melting of Cu under hydrostatic and shock wave loading to high pressures. J. Phys.: Condens. Matter 2008, 20, 095220. (28) Williams, P. L.; Mishin, Y.; Hamilton, J. C. An embedded-atom potential for the Cu-Ag system. Modell. Simul. Mater. Sci. Eng. 2006, 14, 817. (29) Sun, D. Y.; Mendelev, M. I.; Becker, C. A.; Kudin, K.; Haxhimali, T.; Asta, M.; Hoyt, J. J.; Karma, A.; Srolovitz, D. J. Crystal-melt interfacial free energies in hcp metals: A molecular dynamics study of Mg. Phys. Rev. B 2006, 73, 024116. (30) Xia, Z. G.; Sun, D. Y.; Asta, M.; Hoyt, J. J. Molecular dynamics calculations of the crystalmelt interfacial mobility for hexagonal closepacked Mg. Phys. Rev. B 2007, 75, 012103.

and calculating the topological entropy based on the approximation that the line defects of dense liquids yield a topological contribution to the fusion entropy.19 So, it is not surprising if consider the basic arguments and the possible differences in disordering upon melting.

■

CONCLUSION We studied the melting curves of six metals up to 15 GPa. The melting curves of Ag, Cu, Al, and Mg fully confirm measurements and previous calculations. Meanwhile, the melting curves of Ta and Mo are consistent with previous calculations but diverge from LHDAC values with increasing pressure. The fusion entropy, fusion volume, expansion coefficients, and compressibility coefficients of the six metals were calculated up to 15 GPa. In addition, the overall fusion entropy were separated into the volume entropy of fusion due to the latent volume change in melting and the topological entropy of fusion due to the configuration change in melting. Furthermore, we checked the R ln 2 rule under high pressure. Results showed that the R ln 2 rule does not apply accurately to none of these studied metals, R ln 2 should be considered only as an approximation of the value of ΔSD, and the value of topological entropy of fusion reflects a slow decrease when pressure increasing. The reported results increase the database on high-pressure melting, contribute to improve the understanding of melting progress under high pressure, and provide a theoretical basis for the use of the R ln 2 rule under high-pressure conditions.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Funding

This work was supported by Foundation of Key Laboratory of National Defense Science and Technology for Shock Wave and Detonation Physics. Notes

The authors declare no competing financial interest.

■

REFERENCES

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