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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

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High pressure melting curves of silver, gold and copper

Ho Khac Hieu1,aand Nguyen Ngoc Ha2

1Research and Development Center for Science and Technology, Duy Tan University, K7/25 Quang Trung, Danang, Vietnam

2VNU-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 Gr¨uneisen 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 temper-atures 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,2shock-wave experiments,3computer simulations4and theoretical approaches.5Furthermore, in recent years the experimental researchers have measured the melting temperature of materials in ultra-high pressures (up to hundreds of GPa).610Consequently, 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 , 20In 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 DTA12and 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.15On 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.18Although there are a number of literatures focusing on high-pressure melting

a Corresponding author: Electronic mail: hieuhk@duytan.edu.vn

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 Lindemann’s melting criterion21 with the pressure-dependent Gr¨uneisen parameter22to carry out the relatively simple analytical expression of melting

T m 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

The Gr¨uneisen parameter has been suggested by Gr¨uneisen26to describe the effects of volume change on phonon frequenciesω iand defined as22

γ G= −

i

∂ ln ω i

∂ ln V = −

∂ ln ω0

where V is crystal volume and ω i are phonon frequencies which depend only on volume V

Normally, the Gr¨uneisen parameter can be rated as constant which does not depend on pressure variation.13 Nevertheless, some experimental results have proposed the law as γ G /V = const.30

In recent study, by first-principles electronic band-structure calculations combined with a

Born-von K´arm´an force model, the Graf et al.22 determined the lattice vibrations in the quasi-harmonic approximation for Au and Cu metals Gr¨uneisen parametersγ Gand their pressure dependence had

been considered This group also derived approximations based on the bulk modulus B and the

mean-square displacementu2 or Debye-Waller factor for the high temperature Gr¨uneisen parameter by follows

γ ≈ γ B = −1

6 −1 2

∂ ln B

and

γ ≈ γ DW = 1

2

∂ lnu2

To evaluate the pressure effect on Gr¨uneisen parameter, Graf et al fitted the calculated γ G

values to the quite well described expression as

γ G = γ0



V

V0

q

whereγ0 and V0are Gr¨uneisen parameter and crystal volume at ambient conditions, respectively

The value of q belongs to studied material, usually, q > 1 and q < 2 It should be noticed that, the

expressionγ G /V = const is a particular case of equation(4)when q= 1 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

TABLE I Experimental melting temperature T0 and Gr¨uneisen parameterγ0, q of Cu and Au metals.22

value.21Using 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.31derived the following melting formula which can be seen as a generalization of the Lindemann’s law

T m = const × V2

.θ2

where crystal volume V and Debye temperature θ Dare quantities which depend on pressure variation

Taking the volume derivative of the natural logarithm of formula(5)we derived

∂ ln (T m)

2

V

 1

3− γ G



whereγ Gis Gr¨uneisen parameter which is in Debye model defined asγ G = −∂ ln θ D /∂ ln V

Substituting Eq.(4)into Eq.(6)and taking the integral, we carried out the analytical formula

of melting T m as a function of volume compression V /V0as

T m = T0



V

V0

2/3

exp



2γ0

q



1−



V

V0

q

in the above equation, T0is 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 T m as a function of volume compression V /V0 are T0,γ0 and q Melting temperature T0 at ambient conditions can be gathered from experiments The values q and γ0 of

Au and Cu metals were fitted from Gr¨uneisen 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

T m of Ag, Au and Cu metals Melting temperature T0and Gr¨uneisen parameterγ0of Ag at ambient pressure are 1234.93 K and 2.65,32respectively; q is assumed to equal to 1 The values of T0, and fitting parametersγ0and q for Au and Cu are listed in TableI

Making the numerical calculations of T m by using Eq (7), the melting curves as functions

of volume compressions V /V0of Ag, Au and Cu metals are shown in the Fig.1(a), Fig.1(b)and Fig.1(c), respectively T m , T mB and T m DWcorrespond to melting temperatures calculated using fitting parameter sets{γ0, q }, {γ 0B , q B } and {γ 0DW , q DW} As it can be seen, when pressure increases,

the melting temperatures T mof 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 T mcalculated by using various fitting parameters{γ0, 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 T mare getting along if 0.85 ≤ V/V0 ≤ 1 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≤ 1 when the divergence of melting

temperatures T mcalculated by various fitting parameters{γ0, q } is not too large.

The different behaviors of T m , T mB and T m DW can be explained using simple demonstration

proposed by Graf et al.22At such high pressures, phonon frequencies stiffen drastically and

simul-112125-4 H K Hieu and N N Ha AIP Advances 3, 112125 (2013)

0.5 0.6

0.7 0.8

0.9 1

2000 4000 6000 8000 10000 12000

Volume compression V/V

0

Ag

(a)

0.5 0.6

0.7 0.8

0.9 1

2000 4000 6000 8000 10000 12000 14000

Volume compression V/V

0

T m T mB T mDB

Au

(b)

0.5 0.6

0.7 0.8

0.9 1

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Volume compression V/V

0

T m T mB T mDB

Cu

(c)

FIG 1 Melting curves of Ag, Au and Cu metals as functions of volume compressions V /V0

TABLE II The least-squares fitting parameters K0and K

0 of Ag, Au and Cu metals under ambient conditions.

K

a Reference 24

b Reference 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 factoru2

To the authors’s knowledge, in most of previous high-pressure melting studies the authors only

showed melting curves T m 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 /V0from well-established and up-to-date Vinet EOS formulation for each metal.23This EOS has form as

P = 3K0



V

V0

−2/3

1−



V

V0

1/3 exp

 3 2

K0− 1 ×



1−



V

V0

1/3

where K0 and K0 are the isothermal bulk modulus and its pressure derivative at ambient pressure,

respectively The least-squares fitting parameters K0and K0 of Ag, Au and Cu reported by Dewaele

et al.24 , 25are shown in TableII

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 8 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 T mof 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,13especially, at pressure below 4 GPa Present results are just consistent

with those of Mirwald et al.12 and Akella and Kennedy11 up to pressure 2 GPa Beyond 2 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

dT m /dP= 47(3) K/GPa, 57 K/GPa and 57.3 K/GPa, subsequently Slopes of melting in our determi-nations are 41.86 K/GPa, 38.18 K/GPa and 42.66 K/GPa which correspond to melting calculations using fitted parameters{γ0, q } from high temperature Gr¨uneisen parameter calculated by

first-principles electronic band-structure calculations and approximations based on the bulk modulus B

and the Debye-Waller factoru2 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{γ0,

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 T m up to pressure 500 GPa and 50 GPa are plotted in Fig.4(a)& Fig.4(b), respectively Copper is the metal of which high-pressure melting curve has been studied by experiments as well as computational simulations in many literatures.11 , 13 – 18As it can be seen in the Fig.4(a), by using three different fitting parameter sets {γ0, q }, we obtained three disparate results of melting temperature Initial melting slopes

112125-6 H K Hieu and N N Ha AIP Advances 3, 112125 (2013)

1000 3000 5000 7000 9000 11000

Pressure P (GPa)

J Akella et al [11]

PW Mirwald et al [12]

D Errandonea [13]

Present study

Ag

(a)

1200 1400 1600 1800 2000 2200 2400

Pressure P (GPa)

J Akella et al [11]

PW Mirwald et al [12]

D Errandonea [13]

Present study

Ag

(b)

FIG 2 Melting temperature T mof Ag up to pressure 460 GPa & 20 GPa using Eq (7) with experimental data ofγ0 and

q= 1 Results of Akella and Kennedy 11(stars), Mirwald et al.12 (open squares) and Errandonea 13 (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 T m , T mB and T m DW The experimental melting slope of copper at pressure 1 bar

reported by Errandonea is dT m /dP= 43(2) K/GPa.13 The previous experimental reported melting slopes of Akella and Kennedy,11Mirwald et al.12and Brand et al.15are 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 T m and T m DW While the values of T m and T m DW diverge from the previous experimental and theoretical determinations, the T mB 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 T mB 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 T mBresults with first-principles

0 100 200 300 400 500 600 700 0

2000 4000 6000 8000 10000 12000 14000

Pressure P (GPa)

Akella et al [11]

Mirwald et al [12]

Errandonea [13]

Tm T mB

TmDW

Au

(a)

1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300

Pressure P (GPa)

Akella et al [11]

Mirwald et al [12]

Errandonea [13]

Tm T mB

TmDW

Au

(b)

FIG 3 Corresponding melting temperature T mof Au up to pressure 770 GPa & 20 GPa using Eq (7) with various fitting parameters{γ0, q }.22 Results of Akella and Kennedy 11(stars), Mirwald et al.12 (open squares) and Errandonea 13 (open circles) are also displayed for comparison.

calculations16,17authenticates that we can employ T mBto 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 (4d105s1, 5d106s1and 3d104s1, 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 T mvalues 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 T mof

Au at 13 GPa about 1800 K This result is in good agreement with the extrapolated determination from experimental results of Errandonea.13

112125-8 H K Hieu and N N Ha AIP Advances 3, 112125 (2013)

0 100 200 300 400 500 1000

2000 3000 4000 5000 6000 7000 8000 9000 10000

Pressure P (GPa)

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]

Tm

TmB

TmDW

(a)

1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000

Pressure P (GPa)

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]

Tm

TmB

TmDW

(b)

FIG 4 Corresponding melting temperature T mof Cu up to pressure 500 GPa & 50 GPa using Eq (7) with various fitting parameters{γ0, q }.22 Results of Akella and Kennedy 11 (* marks), Errandonea 13(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 T mfunction 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 Gr¨uneisen parameter Analytical expression of pressure-dependent melting

tem-perature T mhas been proposed Numerical calculations have been performed up to volume

compres-sion V /V0= 0.5 and up to pressure corresponding to this compression (460 GPa for Ag, 770 GPa for

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 6 GPa, respectively For Cu metal, melting T mB calculated using fitting parameters{γ 0B , q B } from Gr¨uneisen 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

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