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High pressure melting curves of silver, gold and copper Ho Khac Hieu and Nguyen Ngoc Ha Citation: AIP Advances 3, 112125 (2013); doi: 10.1063/1.4834437 View online: http://dx.doi.org/10.1063/1.4834437 View Table of Contents: http://scitation.aip.org/content/aip/journal/adva/3/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Systematic prediction of high-pressure melting curves of transition metals J Appl Phys 116, 163505 (2014); 10.1063/1.4899511 The melting curve of ten metals up to 12 GPa and 1600 K J Appl Phys 108, 033517 (2010); 10.1063/1.3468149 X-ray diffraction measurements of Mo melting to 119 GPa and the high pressure phase diagram J Chem Phys 130, 124509 (2009); 10.1063/1.3082030 HighPressure DebyeWaller and Grüneisen Parameters of Gold and Copper AIP Conf Proc 706, 65 (2004); 10.1063/1.1780185 Equations of State for Cu, Ag, and Au for Wide Ranges in Temperature and Pressure up to 500 GPa and Above J Phys Chem Ref Data 30, 515 (2001); 10.1063/1.1370170 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 AIP ADVANCES 3, 112125 (2013) High pressure melting curves of silver, gold and copper Ho Khac Hieu1,a and Nguyen Ngoc Ha2 Research and Development Center for Science and Technology, Duy Tan University, K7/25 Quang Trung, Danang, Vietnam VNU-Hanoi University of Science, 334 Nguyen Trai, Hanoi, Vietnam (Received 25 September 2013; accepted 12 November 2013; published online 20 November 2013) In this work, based on the Lindemann’s formula of melting and the pressure-dependent Grăuneisen parameter, we have investigated the pressure effect on melting temperature of silver, gold and copper metals The analytical expression of melting temperature as a function of volume compression has been derived Our results are compared with available experimental data as well as with previous theoretical studies and the good and reasonable agreements are found We also proposed the potential of this approach on predicting melting of copper at very high pressure C 2013 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4834437] I INTRODUCTION Melting of materials under extreme condition is one of the interesting subjects in physics because of its importance in studying shock physics, planetary science, astrophysics, geophysics and nuclear physics Many efforts have been made to determine the high-pressure melting temperatures of metals Nevertheless, the prediction of high-pressure melting curves of transition metals is a controversial issue because of the difference among diamond-anvil cell (DAC) experiments,1 X-ray diffraction measurements,2 shock-wave experiments,3 computer simulations4 and theoretical approaches.5 Furthermore, in recent years the experimental researchers have measured the melting temperature of materials in ultra-high pressures (up to hundreds of GPa).6–10 Consequently, building the theory for determining the melting of materials under high pressure is a topical and scientific significance In particular, the melting investigation of the Group 11 metals hold great importance The number of papers including experimental as well as theoretical approaches has been performed to study the high-pressure melting of copper (Cu).11–18 Its face-centered cubic structure is predicted to still remain stable up to more than 2500 GPa.19, 20 In contrast, there were a few early works considering the melting of silver (Ag) and gold (Au) metals before.11–13 On the experimental side, Akella and Kennedy conducted the experiments for coinage metals up to 6.5 GPa using thermocouples and differential thermal analysis (DTA).11 Melting behaviors of these metals were re-considered by Mirwald et al using DTA12 and by Errandonea thanks to steel-belted Bridgman-type cell.13 Japel et al reported melting curve of solid Cu in the laser-heated DAC to 97 GPa and 3800 K.14 Using multi-anvil techniques, Brand and collaborators determined melting temperature of Cu from ambient pressure to 16 GPa.15 On the theoretical side, the Cu melting lines had been evaluated to very high pressures Belonoshko et al and Voˇcadlo et al estimated high-pressure melting temperature by molecular dynamic calculations (up to above 200 GPa)16 and by first-principles calculations with phase coexistence approach (up to 100 GPa),17 respectively By means of large scale molecular dynamics simulations of solid-liquid coexistence, for the first time, Wu et al predicted the melting of Cu up to 400 GPa.18 Although there are a number of literatures focusing on high-pressure melting a Corresponding author: Electronic mail: hieuhk@duytan.edu.vn 2158-3226/2013/3(11)/112125/9 3, 112125-1 C Author(s) 2013 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-2 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) problem of metals,1–18 the prediction of melting temperature under ultra-high-pressure is still a challenge for both experimental as well as theoretical physicists, especially, in the case of Ag and Au metals In present paper, the high-pressure melting problem is going to be accessed based on semi-empirical approach We combine the Lindemanns melting criterion21 with the pressuredependent Grăuneisen parameter22 to carry out the relatively simple analytical expression of melting Tm as a function of crystal volume compression V /V0 To express the melting temperature of metal as a function of pressure we use the pressure-volume relation as the well-established Vinet equation-of-state (EOS).23–25 Numerical calculations for Ag, Au and Cu are performed up to volume compression V /V0 = 0.5 and up to ultra-high pressure corresponding to this compression (460 GPa, 770 GPa and 500 GPa, respectively) where no experimental data exist yet Our results are going to be compared with recent experimental and theoretical studies when possible We show that our melting evaluations for Ag and Cu metals are in very good agreement with those of previous works II FORMALISM ă A Pressure-dependent Gruneisen parameter The Grăuneisen parameter has been suggested by Grăuneisen26 to describe the effects of volume change on phonon frequencies ωi and defined as22 γG = − i ∂ ln ω0 ∂ ln ωi =− , ∂ ln V ∂ ln V (1) where V is crystal volume and ωi are phonon frequencies which depend only on volume V Normally, the Grăuneisen parameter can be rated as constant which does not depend on pressure variation.13 Nevertheless, some experimental results have proposed the law as γG /V = const.30 In recent study, by first-principles electronic band-structure calculations combined with a Bornvon K´arm´an force model, the Graf et al.22 determined the lattice vibrations in the quasi-harmonic approximation for Au and Cu metals Grăuneisen parameters G and their pressure dependence had been considered This group also derived approximations based on the bulk modulus B and the meansquare displacement u2 or Debye-Waller factor for the high temperature Grăuneisen parameter by follows 1 ln B , γ ≈ γB = − − ∂ ln V (2) and γ ≈ γ DW = ∂ ln u ∂ ln V (3) To evaluate the pressure effect on Grăuneisen parameter, Graf et al fitted the calculated γ G values to the quite well described expression as γ G = γ0 V V0 q , (4) where and V0 are Grăuneisen parameter and crystal volume at ambient conditions, respectively The value of q belongs to studied material, usually, q > and q < It should be noticed that, the expression γG /V = const is a particular case of equation (4) when q = is applied B Lindemann’s criterion and pressure-dependent melting temperature On study the melting of materials, Lindemann argued that, melting is going to occur when the ratio between mean-square vibration and square of nearest-neighbor distance reaches a threshold All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-3 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) TABLE I Experimental melting temperature T0 and Grăuneisen parameter , q of Cu and Au metals.22 Metals Au Cu γ0 (V0 ) γ0 (V0 ) B γ0 (V0 ) DW q qB q DW T0 (K) 2.95 1.85 2.72 2.29 3.00 1.68 1.229 0.445 1.064 0.774 1.481 0.623 1337.33 1357.77 value.21 Using the Lindemann’s concept, the empirical evaluation of melting under pressure of many metals had also been performed in the number of literatures before.27–29 Based on the classical mean field potential (MFP) approach, Wang et al.31 derived the following melting formula which can be seen as a generalization of the Lindemann’s law Tm = const × V θ D2 , (5) where crystal volume V and Debye temperature θ D are quantities which depend on pressure variation Taking the volume derivative of the natural logarithm of formula (5) we derived ∂ ln (Tm ) = ∂V V − G , (6) where G is Grăuneisen parameter which is in Debye model defined as γG = −∂ ln θ D /∂ ln V Substituting Eq (4) into Eq (6) and taking the integral, we carried out the analytical formula of melting Tm as a function of volume compression V /V0 as Tm = T0 V V0 2/3 exp 2γ0 1− q V V0 q , (7) in the above equation, T0 is the melting temperature of metal at ambient conditions Taking into account Eq (7), the melting temperatures of coinage metals under high pressure can be calculated numerically It is obviously that indispensable input parameters required to study melting temperature Tm as a function of volume compression V /V0 are T0 , γ and q Melting temperature T0 at ambient conditions can be gathered from experiments The values q and γ of Au and Cu metals were fitted from Grăuneisen parameters computed by first-principles electronic band-structure calculations and bulk modulus B and Debye-Waller factor u2 approximations.22 III NUMERICAL CALCULATIONS AND DISCUSSIONS Now we apply the expressions derived in previous section to consider the high-pressure melting Tm of Ag, Au and Cu metals Melting temperature T0 and Grăuneisen parameter of Ag at ambient pressure are 1234.93 K and 2.65,32 respectively; q is assumed to equal to The values of T0 , and fitting parameters γ and q for Au and Cu are listed in Table I Making the numerical calculations of Tm by using Eq (7), the melting curves as functions of volume compressions V /V0 of Ag, Au and Cu metals are shown in the Fig 1(a), Fig 1(b) and Fig 1(c), respectively Tm , TmB and Tm DW correspond to melting temperatures calculated using fitting parameter sets {γ , q}, {γ 0B , qB } and {γ0DW , q DW } As it can be seen, when pressure increases, the melting temperatures Tm of these metals rise rapidly; about 11000 K for Ag, 12000 − 13000 K for Au and 6000 − 10000 K for Cu at volume compression V /V0 = 0.5 Notwithstanding, it has the difference among the values of Tm calculated by using various fitting parameters {γ , q} At volume compression V /V0 = 0.5, melting deviation about 1000 K for Au and 4000 K for Cu; at pressure V /V0 = 0.7, melting deviation is smaller, about below 200 K for Au and 1000 K for Cu The calculated melting temperatures Tm are getting along if 0.85 ≤ V /V0 ≤ It suggests that the investigation of high-pressure melting of Au and Cu by Lindemann’s criterion approach can be applied in range of volume compressions 0.85 ≤ V /V0 ≤ when the divergence of melting temperatures Tm calculated by various fitting parameters {γ , q} is not too large The different behaviors of Tm , TmB and Tm DW can be explained using simple demonstration proposed by Graf et al.22 At such high pressures, phonon frequencies stiffen drastically and simul- All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-4 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) 12000 Ag Melting temperature T m 10000 8000 6000 4000 2000 0.9 0.8 0.7 Volume compression V/V0 0.6 0.5 0.6 0.5 0.6 0.5 (a) 14000 Au 10000 m Melting temperature T (K) 12000 Tm T 8000 mB T mDB 6000 4000 2000 0.9 0.8 0.7 Volume compression V/V0 (b) 10000 Melting temperature Tm (K) 9000 Cu 8000 7000 Tm 6000 TmB TmDB 5000 4000 3000 2000 1000 0.9 0.8 0.7 Volume compression V/V0 (c) FIG Melting curves of Ag, Au and Cu metals as functions of volume compressions V /V0 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-5 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) TABLE II The least-squares fitting parameters K0 and K of Ag, Au and Cu metals under ambient conditions Metals K0 (GPa) K0 a Reference b Reference Ag Au Cu 101a 5.97a 167b 6.00b 133b 5.30b 24 25 taneously bulk modulus B increase with the decreasing of crystal volume V The dispersion slope at near zone center is roughly proportional to the bulk modulus B, while all frequencies weighted by the temperature-dependent occupation factor of each mode have been averaged by Debye-Waller factor u2 To the authors’s knowledge, in most of previous high-pressure melting studies the authors only showed melting curves Tm as functions of pressure P Consequently, to compare our calculations with those of previous experiments and theoretical determinations, we took into account the relation between pressure P and volume compression V /V0 from well-established and up-to-date Vinet EOS formulation for each metal.23 This EOS has form as P = 3K V V0 −2/3 1− V V0 1/3 exp K −1 × 1− V V0 1/3 , (8) where K0 and K are the isothermal bulk modulus and its pressure derivative at ambient pressure, respectively The least-squares fitting parameters K0 and K of Ag, Au and Cu reported by Dewaele et al.24, 25 are shown in Table II In Fig 2(a) & Fig 2(b), we show the melting curve of Ag as a function of pressure up to 460 GPa (corresponding to compression V /V0 = 0.5) and 20 GPa, respectively The experimental data of Akella and Kennedy11 (up to 20 GPa), Mirwald et al.12 (up to 6.5 GPa) and Errandonea13 (up to GPa) are also displayed for comparison The present results agree well with those of experimental data up to 12 GPa At higher pressure, our calculations are quite greater than those reported by Akella and Kennedy.11 According to Akella and Kennedy, the initial melting slopes of Ag is 60.4 K/GPa, while the result of Mirwald et al.12 and Errandonea13 are 64.7 K/GPa and 47 K/GPa, correspondingly Initial slope of melting in our calculations is 56.55 K/GPa In Fig 3(a) & Fig 3(b), we displayed the high-pressure melting curves Tm of Au up to pressure 770 GPa and 20 GPa, respectively The previous experimental results11–13 are also displayed for comparison It can be seen from Fig 3(b), our evaluations are in agreement with the experimental data reported by Errandonea,13 especially, at pressure below GPa Present results are just consistent with those of Mirwald et al.12 and Akella and Kennedy11 up to pressure GPa Beyond GPa, our determinations increases slowly comparing to experiments of Mirwald et al and Akella and Kennedy The divergence between theoretical prediction and experiments is about 100 K at 10 GPa and 200 K at 20 GPa This remark is supported by making comparison among the slopes of melting curves Experimental melting slopes of Errandonea, Mirwald et al and Akella and Kennedy are dTm /dP = 47(3) K/GPa, 57 K/GPa and 57.3 K/GPa, subsequently Slopes of melting in our determinations are 41.86 K/GPa, 38.18 K/GPa and 42.66 K/GPa which correspond to melting calculations using fitted parameters {γ , q} from high temperature Grăuneisen parameter calculated by firstprinciples electronic band-structure calculations and approximations based on the bulk modulus B and the Debye-Waller factor u2 There are some reasons which can simply explain this difference: (1) the limitation of Lindemann’s criterion approach; (2) not really good-fitting parameter sets {γ , q}; (3) the lack of consideration of electron-configuration of metal;14 (4) the out-of-date experiment results In the case of Cu metal, the pressure effects on melting curves Tm up to pressure 500 GPa and 50 GPa are plotted in Fig 4(a) & Fig 4(b), respectively Copper is the metal of which highpressure melting curve has been studied by experiments as well as computational simulations in many literatures.11, 13–18 As it can be seen in the Fig 4(a), by using three different fitting parameter sets {γ , q}, we obtained three disparate results of melting temperature Initial melting slopes All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-6 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) 11000 m Melting temperature T (K) Ag 9000 7000 J Akella et al [11] PW Mirwald et al [12] D Errandonea [13] Present study 5000 3000 1000 100 200 300 Pressure P (GPa) 400 460 (a) 2400 Ag Melting temperature Tm (K) 2200 2000 1800 1600 J Akella et al [11] PW Mirwald et al [12] D Errandonea [13] Present study 1400 1200 10 Pressure P (GPa) 15 20 (b) FIG Melting temperature Tm of Ag up to pressure 460 GPa & 20 GPa using Eq (7) with experimental data of γ and q = Results of Akella and Kennedy11 (stars), Mirwald et al.12 (open squares) and Errandonea13 (close circles) are also displayed for comparison obtained from our calculations are 30.92 K/GPa, 39.90 K/GPa and 27.45 K/GPa corresponding to melting temperature Tm , TmB and Tm DW The experimental melting slope of copper at pressure bar reported by Errandonea is dTm /dP = 43(2) K/GPa.13 The previous experimental reported melting slopes of Akella and Kennedy,11 Mirwald et al.12 and Brand et al.15 are 36.4 K/GPa, 41.8 K/GPa and 45(3) K/GPa, respectively On the theoretical side, quasi ab initio molecular dynamic calculations performed by Belonoshko et al.16 give the value 36.7 K/GPa, while the result of Voˇcadlo et al.17 by making ab initio calculations with phase coexistence approach is 38 K/GPa Up to pressure 20 GPa, those three melting temperature results are consistent with reported data (Fig 3(a)); beyond 20 GPa, there are the decrease in melting slopes of Tm and Tm DW While the values of Tm and Tm DW diverge from the previous experimental and theoretical determinations, the TmB is in very good agreement with those data At very high pressure (above 100 GPa), there are very few available data for comparison In this pressure range, our calculations TmB correspond to quasi ab initio molecular dynamic results16 (close circles) as well as to those of ab initio calculations with phase coexistence approach17 (close right triangle) The excellent agreement between TmB results with first-principles All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-7 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) 14000 Au 12000 Melting temperature Tm (K) 10000 8000 Akella et al [11] Mirwald et al [12] Errandonea [13] T 6000 4000 m TmB 2000 0 TmDW 100 200 300 400 500 Pressure P (GPa) 600 700 (a) 2300 2200 Au 2000 m Melting temperature T (K) 2100 1900 1800 Akella et al [11] Mirwald et al [12] Errandonea [13] Tm 1700 1600 1500 T mB 1400 1300 TmDW 10 Pressure P (GPa) 15 20 (b) FIG Corresponding melting temperature Tm of Au up to pressure 770 GPa & 20 GPa using Eq (7) with various fitting parameters {γ , q}.22 Results of Akella and Kennedy11 (stars), Mirwald et al.12 (open squares) and Errandonea13 (open circles) are also displayed for comparison calculations16, 17 authenticates that we can employ TmB to predict the very high-pressure melting of Cu metal In literature [14], Japel et al have argued the important role of d-shell electrons on melting of transition metals In this study, Ag, Au and Cu have the same electron configuration with the full-filled d electron (4d10 5s1 , 5d10 6s1 and 3d10 4s1 , respectively) It suggests that the melting curves of these three metals should have the same form This conclusion can be confirmed by observing Fig 1(a), Fig 1(b) & Fig 1(c) Moreover, to describe exactly high-pressure melting curves, the building theory needs to pay attention to electronic properties of metals However, this Lindemann’s melting criterion approach can still be used to predict Tm values of Ag, Au and Cu as well as other metals in high pressure For example, from Fig 3(b) it can be seen that, the calculated value Tm of Au at 13 GPa about 1800 K This result is in good agreement with the extrapolated determination from experimental results of Errandonea.13 All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-8 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) 10000 m Melting temperature T (K) 9000 8000 7000 6000 Akella et at [11] Errandonea [13] Japel et at [14] Brand et al [15] Belonoshko et al [16] Vocadlo et al [17] Wu et al [18] 5000 4000 Tm 3000 TmB T 2000 1000 mDW 100 200 300 Pressure P (GPa) 400 500 (a) 3000 m Melting temperature T (K) 2800 2600 2400 2200 Akella et at [11] Errandonea [13] Japel et at [14] Brand et al [15] Belonoshko et al [16] Vocadlo et al [17] Wu et al [18] 2000 1800 Tm 1600 T mB 1400 T mDW 1200 1000 10 20 30 Pressure P (GPa) 40 50 (b) FIG Corresponding melting temperature Tm of Cu up to pressure 500 GPa & 50 GPa using Eq (7) with various fitting parameters {γ , q}.22 Results of Akella and Kennedy11 (* marks), Errandonea13 (close pentagrams), Japel et al.14 (+ marks), Brand et al.15 (close hexagrams), Belonoshko et al.16 (close circles), Voˇcadlo et al.17 (close triangles (right)) and Wu et al.18 (dotted line) are also displayed for comparison We also want to make another note that, the melting temperature Tm function is not really linear to pressure P, especially in high pressure region This comment can be easily observed in Fig 1: Melting curves of Ag, Au and Cu metals trend to vary as the nonlinear functions of P when volume compressions V /V0 ≤ 0.8 (correspond to pressures P ≥ 43 GPa for Ag, P ≥ 70 GPa for Au and P ≥ 50 GPa for Cu metals.24, 25 ) IV CONCLUSIONS In this work, we have introduced a relatively simple approach to investigate high-pressure melting of Ag, Au and Cu metals thanks to Lindemann’s criterion of melting temperature and pressure-dependent Grăuneisen parameter Analytical expression of pressure-dependent melting temperature Tm has been proposed Numerical calculations have been performed up to volume compression V /V0 = 0.5 and up to pressure corresponding to this compression (460 GPa for Ag, 770 GPa for All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported license See: http://creativecommons.org/licenses/by/3.0/ Downloaded to IP: 137.30.242.61 On: Wed, 10 Dec 2014 08:55:03 112125-9 H K Hieu and N N Ha AIP Advances 3, 112125 (2013) Au and 500 GPa for Cu metals) By comparing calculated results with those of available experiments and theories we conclude that, Lindemann’s criterion approach is suitable for evaluating the melting of Ag and Au up to about 12 GPa and GPa, respectively For Cu metal, melting TmB calculated using fitting parameters {γ 0B , qB } from Grăuneisen parameter in bulk modulus B approximations is a good candidate for predicting melting temperature at very high pressure P This approach can also be applied to study pressure effects on melting temperatures of other metals such as Ni, Fe, At higher pressure, Lindemann’s criterion can just help us on qualitative investigation of high-pressure melting It also can be used to verify future multi-anvil and DAC experiments as well as theoretical determinations We suppose that it should consider about electron configuration of metals on study their pressure-dependent melting temperatures ACKNOWLEDGMENTS The authors gratefully acknowledge anonymous referees for useful comments and suggestions This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.02-2012.06 D Errandonea, Phys Rev B 87, 054108 (2013) Santamar´ıa-P´erez, M Ross, D Errandonea, G D Mukherjee, M Mezouar, and 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melting temperature... candidate for predicting melting temperature at very high pressure P This approach can also be applied to study pressure effects on melting temperatures of other metals such as Ni, Fe, At higher... to determine the high- pressure melting temperatures of metals Nevertheless, the prediction of high- pressure melting curves of transition metals is a controversial issue because of the difference