On the Jumps of Volume, Enthalpy and Entropy at the Melting Point, the Thermal Conductivity and Thermal Diffusivity for F.C.C. Au: the Temperature- and Pressure-Dependences

Nguyen Quang Hoc$^1$, Bui Duc Tinh$^1$, Nguyen Duc Hien$^2$, and Le Hong Viet$^3$

$^1$Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam
$^2$Mac Dinh Chi High School, Chu Pah District, Gia Lai Province, Vietnam
$^3$Tran Quoc Tuan University, Co Dong, Son Tay Town, Hanoi, Vietnam

Received 22.07.2021; final version — 01.11.2021 Download PDF logo PDF

The melting temperature, the jumps of volume, enthalpy and entropy at the melting point, the isothermal compressibility, the thermal expansion coefficient, the heat capacity at constant volume, the Grüneisen parameter, the Debye temperature, the electrical resistivity, the thermal conductivity, and the thermal diffusivity for defective and perfect f.c.c. metals are studied by combining the statistical moment method (SMM), the limiting condition of the absolute stability of the crystalline state, the Clapeyron–Clausius equation, the Debye model, the Grüneisen equation, the Wiedemann–Franz law and the Mott equation. Numerical calculations are carried out for Au under high temperature and pressure. Calculated melting curve of Au is in a good agreement with experiments and other calculations. Obtained results are predictive and orient towards new experiments.

Keywords: jumps of volume, enthalpy and entropy at the melting point, thermal conductivity, thermal diffusivity, statistical moment method.

DOI: https://doi.org/10.15407/ufm.22.04.511

Citation: Nguyen Quang Hoc, Bui Duc Tinh, Nguyen Duc Hien, and Le Hong Viet, On the Jumps of Volume, Enthalpy and Entropy at the Melting Point, the Thermal Conductivity and Thermal Diffusivity for F.C.C. Au: the Temperature- and Pressure-Dependences, Prog. Phys. Met., 22, No. 4: 511–530 (2021)

  1. D. Batani, A. Balducci, D. Beretta, A. Bernardinello, T. Löwer, M. Koenig, A. Benuzzi, B. Faral, and T. Hall, Equation of state data for gold in the pressure range less 10 TPa, Phys. Rev. B, 61, No.14: 9287 (2000); https://doi.org/10.1103/PhysRevB.61.9287
  2. K. Takemura and A. Dewaele, Isothermal equation of state for gold with a He-pressure medium, Phys. Rev. B, 78: 104119 (2008); https://doi.org/10.1103/PhysRevB.78.104119
  3. A. Dewaele, P. Loubeyre, F. Occelli, O. Marie, and M. Mezouar, Toroidal diamond anvil cell for detailed measurements under extreme static pressures, Nat. Commun., 9: 2913 ( 2018); https://doi.org/10.1038/s41467-018-05294-2
  4. M. C. Daniel; D. Astruc, Gold Nanoparticles: Assembly, Supramolecular Chemistry, Quantum-Size-Related Properties, and Applications toward Biology, Catalysis, and Nanotechnology, Chem. Rev, 104, No.1: 293-346 (2004); https://doi.org/10.1021/cr030698+
  5. P. Song; L. C. Cai; Q. S. Wang; X. M. Zhou; X. Z. Li; Y. Zhang; S. Yuan; J. D. Weng, and J. B. Li, Sound velocity, temperature, melting along the Hugoniot and equation of state for two porosity aluminums, J. Appl. Phys, 110: 103522 (2011); https://doi.org/10.1063/1.3662193
  6. A. Dewaele; M. Mezouar; N. Guignot, and P. Loubeyre, High Melting Points of Tantalum in a Laser-Heated Diamond Anvil Cell, Phys. Rev. Lett, 104: 255701 (2010); https://doi.org/10.1103/PhysRevLett.104.255701
  7. J. H. Zhu; Q. S. Fu; Y. Q. Xue; Z. X. Cui, Comparison of different models of melting transformation of nanoparticles, Mater. Sci., 51: 4462. (2016); https://doi.org/10.1007/s10853-016-9758-1
  8. C. C. Yang; Y.-W. Mai, Thermodynamics at the nanoscale: A new approach to the investigation of unique physicochemical properties of nanomaterials, Mater. Sci. Eng., R, 79: (2014); https://doi.org/10.1016/j.mser.2014.02.001
  9. Z. H. Li; D. G. Truhlar, Nanothermodynamics of metal nanoparticles, Chem. Sci., 5: 2605 (2014); https://doi.org/10.1039/C4SC00052H
  10. G. Kaptay, J. Janczak-Rusch, G. Pigozzi, L. P. H. Jeurgens, Theoretical Analysis of Melting Point Depression of Pure Metals in Different Initial Configurations, J. Mater. Eng. Perform. 2014, 23, No.5: 1600-1607 (2014); https://doi.org/10.1007/s11665-014-0885-z
  11. F. Font; T. G. Myers; S. L. Mitchell, A mathematical model for nanoparticle melting with density change, Microfluid. Nanofluid, 18: 233 (2015); https://doi.org/10.1007/s10404-014-1423-x
  12. D. Alfè, Temperature of the inner-core boundary of the Earth: Melting of iron at high pressure from first-principles coexistence simulations, Phys. Rev. B, 79: 060101(R). (2009); https://doi.org/10.1103/PhysRevB.79.060101
  13. J. Bouchet; F. Bottin; G. Jomard, and G. Zérah, Melting curve of aluminum up to 300 GPa obtained through ab initio molecular dynamics simulations, Phys.Rev.B, 80: 094102 (2009); https://doi.org/10.1103/PhysRevB.80.094102
  14. V. Stutzmann; A. Dewaele; J. Bouchet; F. Bottin, and M. Mezouar, High-pressure melting curve of titanium, Phys. Rev. B, 92: 224110 (2015); https://doi.org/10.1103/PhysRevB.92.224110
  15. J. Q. Broughton and X. P. Li, Phase diagram of silicon by molecular dynamics, Phys. Rev. B, 35: 9120 (1987); https://doi.org/10.1103/PhysRevB.35.9120
  16. A. B. Belonoshko; N. V. Skorodumova; A. Rosengren, and B. Johansson, Melting and critical superheating, Phys. Rev. B, 73: 012201 (2006); https://doi.org/10.1103/PhysRevB.73.012201
  17. F. A. Lindemann, The calculation of molecular vibration frequency, Physik. Z, 11: 609 (1910)
  18. H. K. Hieu and N. N. Ha, High pressure melting curves of silver, gold and copper, AIP Adv, 3: 112125 (2013); https://doi.org/10.1063/1.4834437
  19. N. A. Smirnov; Y. M. Chen; X. R. Chen, and Q. Wu, J. Phys.: Condens. Matter, 29: 105402 (2017); https://doi.org/10.1088/1361-648X/aa58ca
  20. Q. An; S. N. Luo; L. B. Han; L. Zheng and O. Tschauner, Melting of Cu under hydrostatic and shock wave loading to high pressures, Journal of Physics: Condensed Matter, 20, No.9: 095220 (2008); https://doi.org/10.1088/0953-8984/20/9/095220
  21. J. B. Adams; S. M. Foiles, and W. G. Wolfer, Self-diffusion and impurity diffusion of fee metals using the five-frequency model and the Embedded Atom Method, J. Mater. Res, 4: 102. (1989); https://doi.org/10.1557/JMR.1989.0102
  22. P. Pawlow, The dependency of the melting point on the surface energy of a solid, Z. Phys. Chem., Stoechiom. Verwandtschaftsl, 65: 545 (1909);
  23. E. Rie, Influence of surface tension on melting and freezing, Z. Phys. Chem., Stoechiom. Verwandtschaftsl, 104: 354 (1923).
  24. D. Errandonea; B. Schwager; R. Ditz, C. Gessmann; R. Boehler and M. Ross, Systematics of transition-metal melting. Physical Review B, 63, No.13: 132104, (2001); https://doi.org/10.1103/PhysRevB.63.132104
  25. S. N. Luo and T. J. Ahrens, Shock-induced superheating and melting curves of geophysically important minerals, Physics of the Earth and Planetary Interiors, 143: 369-386 (2004); https://doi.org/10.1016/j.pepi.2003.04.001
  26. P. Buffat; J.-P. Borel, Size effect on the melting temperature of gold particles, Phys. Rev. A, 13: 2287 (1976); https://doi.org/10.1103/PhysRevA.13.2287
  27. S. Pratontep, S. J. Carroll, C. Xirouchaki, M. Streun, R. E. Palmer, Rev. Sci. Instrum. 76, No. 9: (2005); https://doi.org/10.1063/1.1869332
  28. B. K. Godwan; A. Ng and L. Dasilva, Melting and Hugoniot calculations for gold, Physical Letters A, 144: 26-30 (1990); https://doi.org/10.1016/0375-9601(90)90042-M
  29. A. Migault; J. P. Jamain and J. Jacquesson, Fusion curves at high pressure and arameter Grüneisen of metals, In Proc. 7 Int.AIRAPT conf. high presure science technology, Oxford, 2: 938-944 (1980)
  30. V. V. Hung and N. T. Hai, Investigation of the melting temperature of metal at various pressures, Journal of the Physical Society of Japan, 66: 3499-3501 (1997); https://doi.org/10.1143/JPSJ.66.3499
  31. J. Akella and G. C. Kennedy, Melting of Au, Ag and Cu-proposal for a new hogh-pressure calibration scale, Journal of Geophysical Research, 76: 4969-4977 (1971); https://doi.org/10.1029/JB076i020p04969
  32. P. W. Mirwald and G. C. Kennedy, The melting curve of Au, Ag and Cu to 60 –kbar pressure. A reinvestigation, Journal of Geophysical Research, 84: 6750-6756 (1979) https://doi.org/10.1029/JB084iB12p06750
  33. T. Sumita; M. Kato and A. Yoneda, The thermal analysis in an MA-8 type apparatus: the melting of gold at 12 GPa, The review of high pressure science and technology, ed. by M.Nakahara, 7: 254-256 (1998);
  34. B. Roldan Cuenya, M. Alcántara Ortigoza, L. K. Ono, F. Behafarid, S. Mostafa, J. R. Croy, K. Paredis, G. Shafai, T. S. Rahman, L. Li, Z. Zhang, J. C. Yang, Thermodynamic properties of Pt nanoparticles: Size, shape, support, and adsorbate effects, Phys. Rev. B, 84: 245438 (2011); https://doi.org/10.1103/PhysRevB.84.245438
  35. O. A. Yeshchenko, I. M. Dmitruk, A. A. Alexeenko, A. M. Dmytruk, Size-dependent melting of spherical copper nanoparticles embedded in a silica matrix, Phys. Rev. B, 75: 085434 (2007); https://doi.org/10.1103/PhysRevB.75.085434
  36. Wenjuan Zhang; Yongqiang Xue; Qingshan FuZixiang; CuiShuting Wang, Size dependence of phase transition thermodynamics of nanoparticles: A theoretical and experimental study, Powder Technology, 308, 258-265 (2017); https://doi.org/10.1016/j.powtec.2016.11.052
  37. Q. S. Fu; Z. X. Cui; Y. Q. Xue; H. J. Duan, Research of Size- and Shape-Dependent Thermodynamic Properties of the Actual Melting Process of Nanoparticles, J. Phys. Chem. C, 122, No 27: 15713 (2018); https://doi.org/10.1021/acs.jpcc.8b03085
  38. Z. X. Cui; M. Z. Zhao; W. P. Lai; Y. Q. Xue, Thermodynamics of Size Effect on Phase Transition Temperatures of Dispersed Phases, J. Phys. Chem. C, 115, No. 45, 22796 (2011); https://doi.org/10.1021/jp2067364
  39. D. Errandonea, Phase behavior of metals at very high P– T conditions: A review of recent experimental studies, J. Phys. Chem. Solids, 67, 2017 (2006); https://doi.org/10.1016/j.jpcs.2006.05.031
  40. D. Errandonea; S.G. MacLeod; J. Ruiz-Fuertes; L. Burakovsky; M.I. McMahon; C.W. Wilson; J. Ibañez; D. Daisenberger; C. Popescu, High-pressure/high-tem-perature phase diagram of zinc, Journal of Physics: Condensed, 30: 295402 (2018); https://orcid.org/0000-0003-0189-4221
  41. P.I. Dorogokupets, Thermodynamics and equations of state of Iron to 350 GPa ad 6000 K, Scientific Reports, 7: 41863 (2017); https://doi.org/10.1038/srep41863
  42. H.K. Hieu; N.N. Ha, High pressure melting curves of silver, gold and copper, AIP Advances, 3: 112125 (2013); https://doi.org/10.1063/1.4834437
  43. D. McLachlan Jr.; E.G. Ehlers, Eff ect of pressure on the melting temperature of metals, J. Geophysical Research, 76, 2780 (1971); https://doi.org/10.1029/JB076i011p02780
  44. J. Shanker; M. Kumar; Thermodynamic Approximations in High-pressure and High-Temperature Physics of Solids, Physica Status Solidi B, 179: 351 (1993); https://doi.org/10.1002/pssb.2221790209
  45. K. Kholiya; J. Chandra, A theoretical model to study melting of metals under pressure, Modern Physics Letters B, 29: 1550161(1-13), ( 2015); https://doi.org/10.1142/S0217984915501614
  46. O.L. Anderson, Equation of state for Geophysics and Ceramic Science, Oxford University Press, Oxford, (1995);.
  47. L. Dubrovinsky; N. Dubrovinskaia; W. A. Crichton; A. S. Mikhaylushkin; S. I. Simak; I. A. Abrikosov; J. S. de Almeida; R. Ahuja; W. Luo, and B. Johansson, Noblest of All Metals Is Structurally Unstable at High Pressure, Phys.Rev.Lett, 98: 045503 (2007); https://doi.org/10.1103/PhysRevLett.98.045503
  48. S. T. Weir; D. D. Jackson; S. Falabella; G. Samudrala, and Y. K. Vohra, An electrical microheater technique for high-pressure and high-temperature diamond anvil cell experiments, Rev. Sci. Instrum, 80: 013905 (2009); https://doi.org/10.1063/1.3069286
  49. C.S. Zha and W. A. Bassett, Internal resistive heating in diamond anvil cell for in situ x-ray diffraction and Raman scattering, Rev. Sci. Instrum, 74: 1255 (2003) https://doi.org/10.1063/1.1539895
  50. T. Pippinger; L. Dubrovinsky; K. Glazyrin, R. Miletich, and N. Dubrovinskaia, Física de la Tierra, 23, 29 (2011); arXiv:1104.1304v1
  51. Nguyen Trong Dung, Influence of impurity concentration, atomic number, temperature and tempering time on microstructure and phase transformation of Ni1-xFex (x = 0:1, 0.3, 0.5) nanoparticles, Modern Physics Letters B, 32, No.18, 1850204, 1850204 (2018); https://doi.org/10.1142/S0217984918502044
  52. Tran Quoc Tuan; Nguyen Trong Dung, Effect of heating rate, impurity concentration of Cu, atomic number, temperatures, time annealing temperature on the structure, crystallization temperature and crystallization process of Ni(1-x)Cu(x) bulk; x = 0.1, 0.3, 0.5, 0.7", International Journal of Modern Physics B, 32, No. 26: 1830009 (2018); https://doi.org/10.1142/S0217979218300098
  53. Dung Nguyen Trong, Kien Pham Huu, Phuong Nguyen Tri, Simulation on the factors affecting the crystallization process of FeNi alloy by Molecular Dynamics, ACS Omega, 4: 14605-14612 (2019); https://doi.org/10.1021/acsomega.9b02050
  54. Nguyen Quang Hoc; Le Hong Viet; Nguyen Trong Dung, On the Melting of Defective FCC Interstitial Alloy c –FeC under Pressure up to 100 GPa, Journal of Electronic Materials, 49, 910–916 (2020). https://doi.org/10.1007/s11664-019-07829-9
  55. Nguyen Trong Dung, Nguyen Chinh Cuong, Duong Quoc Van, Study on the effect of doping on lattice constant and electronic structure of bulk AuCu by the density functional theory, Journal of Multiscale Modelling, 11, No. 2, 2030001 (2020) https://doi.org/10.1142/S1756973720300014
  56. Nguyen Trong Dung, Nguyen-Tri Phuong, Factors affecting the structure, phase transition and crystallization process of AlNi nanoparticles, Journal of Alloys and Compounds, 812: 152133 (2020) https://doi.org/10.1016/j.jallcom.2019.152133
  57. Dung Nguyen Trong; Phuong Nguyen-Tri, Molecular dynamic study on factors influencing the structure, phase transition and crystallization process of NiCu6912 nanoparticle, Materials Chemistry and Physics, 250: 123075 (2020); https://doi.org/10.1016/j.matchemphys.2020.123075
  58. Van Cao Long; Van Duong Quoc; and Dung Nguyen Trong, Ab Initio Calculations on the Structural and Electronic Properties of AgAu Alloys, ACS Omega, 5, No. 48: 31391–31397 (2020); 10.1021/acsomega.0c04941
  59. Dung Nguyen Trong; Van Cao Long and Ștefan Tălu, The Structure and Crystallizing Process of NiAu Alloy: A Molecular Dynamics Simulation Method, J. Compos. Sci, 5, No. 1, 18 (2021); https://doi.org/10.3390/jcs5010018
  60. Dung NguyenTrong, Z-AXIS deformation method to investigate the influence of system size, structure phase transition on mechanical properties of bulk nickel, Materials Chemistry and Physics, 252: 123275 (2020); https://doi.org/10.1016/j.matchemphys.2020.123275
  61. Tuan Tran Quoc; Dung Nguyen Trong, and Ștefan Tălu, Study on the Influence of Factors on the Structure and Mechanical Properties of Amorphous Aluminium by Molecular Dynamics Method, Advances in Materials Science and Engineering, 2021, 5564644 (2021); https://doi.org/10.1155/2021/5564644
  62. L. Vočadlo and D. Alfè, Ab initio melting curve of the fcc phase of aluminum. Physical Review B, 65 No. 21: 214105 (2002); https://doi.org/10.1103/PhysRevB.65.214105
  63. C. M. Liu; X. R. Chen; C. Xu, L. C. Cai and F. Q. Jing, Melting curves and entropy of fusion of body-centered cubic tungsten under pressure, Journal of Applied Physics, 112: 013518 (2012); https://doi.org/10.1063/1.4733947
  64. S. Kumar; K.S. Nisar; R. Kumar; C. Cattani; B. Samet, A new Rabotnov fractional-exponential function based fractional derivative for di ffusion equation under external force, Mathematical Methods in Applied Science, 43: (2020); https://doi.org/10.1002/mma.6208
  65. M. Jleli; S. Kumar; R. Kumar; B. Samet; Analytical approach for time fractional wave equations in the sense of Yang-Abdel-Aty-Cattani via the homotopy perturbation transform method, Alexandria Engineering Journal, 59, No. 5: (2020); https://doi.org/10.1016/j.aej.2019.12.022
  66. S. Kumar; A. Kumar; S. Abbas; M.A. Qurashi, A modifi ed analytical approach with existence and uniqueness for fractional Cauchy reaction-di ff usion equations, Advances in Difference Equations, 2020, 28 (2020). https://doi.org/10.1186/s13662-019-2488-3
  67. S. Kumar; R. Kumar; R.P. Agarwal; B. Samet, A study on fractional Lotka-Volterra population model by using Haar wavelet and Adam's-Bashforth-Moulton methods, Mathematical Methods in Applied Science, 43, No. 8: (2020); https://doi.org/10.1002/mma.6297
  68. N. Tang and V. V. Hung, Investigation of the thermodynamic properties of anharmonic crystals by the momentum method, (I) General results for FCC crystals, Phys. Stat. Sol. (b), 149: 511-519 (1988); https://doi.org/10.1002/pssb.2221490212
  69. L. T. C. Tuyen; N. Q. Hoc; B. D. Tinh; D. Q. Vinh and T. D. Cuong, Study on the melting of interstitial alloys FeH and FeC with BCC structure under pressure, Chinese Journal of Physics, 59: (2019); https://doi.org/10.1016/j.cjph.2019.02.018
  70. T. D. Cuong; N. Q. Hoc and P.D.Anh, Application of the Statistical Moment Method to Melting Properties of Ternary Alloys with FCC Structure, Journal of Applied Physics, 125, 215112 (2019); https://doi.org/10.1063/1.5089228
  71. V. V. Hung; D. T. Hai and L. T. T. Binh, Melting curve of metals with defect: Pressure dependence, Computational Materials Science, 79, 789–794 (2013) 10.1016/j.commatsci.2013.07.042
  72. N. Q. Hoc; L. H. Viet and N. T. Dung, On the melting of defective FCC interstitial alloy FeC under pressure up to 100 GPa, Journal of Electronic Materials, 49: 910–916 (2020); https://doi.org/10.1007/s11664-019-07829-9
  73. N. Q. Hoc; T. D. Cuong; B. D. Tinh and L. H. Viet, Study on the melting of defective interstitial alloys TaSi and WSi with BCC structure, Journal of Korean Physical Society, 2019, 71(8), 801-805.
  74. L. Burakovsky; D. L. Preston and R. R. Silbar, Analysis of dislocation mechanism for melting of elements: Presure dependence, Journal of Applied Physics, 88, 6294-6301 (2000) 10.1063/1.1323535
  75. N. Q. Hoc; T. D. Cuong; B. D. Tinh and L. H. Viet, High-pressure melting curves of FCC metals Ni, Pd and Pt with defects, Modern Physical Letters B, 33, No. 25: 1950300(2019); https://doi.org/10.1142/S0217984919503007 .
  76. V. V. Hung, Investigation of the change in volume, entropy and specific heat for metals on melting. Proc. the 22nd National Conference of Theoretical Physics, Do Son, 3-5 August, 1997, 199-203.
  77. N. Q. Hoc; B. D. Tinh and N. D. Hien, Influence of temperature and pressure on the electrical resistivity of gold and copper up to 1350K and 100GPa, Materials Research Bulletin, 128, 110874 (2020); https://doi.org/10.1016/j.materresbull.2020.110874
  78. V. V. Hung; L. D. Thanh and N, T, Huong, Study of Elastic Moduli of Semiconductors with Defects by the Statistical Moment Method, e-Journal of Surface Science and Nanotechnology, 9, 499-502 (2011); https://doi.org/10.1380/ejssnt.2011.499
  79. H. K. Hieu; T. T. Hai; N. T. Hong; N. D. Sang and N. V. Tuyen, Electrical resistivity and thermodynamic properties of iron under high pressure, Journal of Electronic Materials, 46, 3702–3706 (2017). https://doi.org/10.1007/s11664-017-5411-2
  80. R. A. Matula, Electrical resistivity of copper, gold, palladium, and silver, Journal of Physical and Chemical Reference Data, 8, 1147 (1979); https://doi.org/10.1063/1.555614
  81. M. N. Magomedov, On calculating the Debye temperature and the Grüneisen parameter, Zhurnal Fizicheskoi Khimii, 1987, 61(4), 1003-1009 (in Russian)
  82. M. N. Magomedov, The calculation of the parameters of the Mie-Lennard-Jones potential, High Temperature, 44, 513–529 (2006); https://doi.org/10.1007/s10740-006-0064-583
  83. T. D. Cuong; P. D. Anh, Modification of the statistical moment method for the high-pressure melting curve by the inclusion of thermal vacancies, Vacuum, 179, 109444 (2020), arXiv:2005.06697
  84. H. K. Hieu; N. N. Ha, High pressure melting curves of silver, gold and coppedr, AIP Advances, 2017, 3, 112125.
  85. N. A. Smirnov, Journal of Physics: Condensed Matter, 2013, 76, 105402().
  86. G. Weck; V. Recoules; J. A. Queyroux; F. Datchi; J. Bouchet; S. Ninet; G.Garbarino; M. Mezouar and P. Loubeyre, Determination of the melting curve of gold up to 110 GPa, Physical Review B, 101, 014106(2020); https://doi.org/10.1103/PhysRevB.101.014106
  87. D. Errandonea, The melting curve of ten metals up to 12 GPa and 1600K, Journal of Applied Physics, 2010, 108, 033517.
  88. M. Berrada; R. A. Secco and W. Yong, Decreasing electrical resistivity of gold along the melting boundary up to 5 GPa, High Pressure Research: An International Journal, 108, 033517 (2010); https://doi.org/10.1063/1.3468149
  89. McGraw-Hill, American Institute of Physics Hand Book, New York, 1963.