Determination of thermal conductivity coefficient by Green-Kubo formula using the minimum image method
Hyżorek Krzysztof 1, Ciesielczyk Karol 2, Tretiakov Konstantin V. 1,3*
1 Institute of Molecular Physics
Polish Academy of Sciences
Smoluchowskiego 17/19
60-179 Poznań, Poland2 Poznań University of Technology
Jana Pawła II 24
60-965 Poznań, Poland3 The President Stanisław Wojciechowski State University
of Applied Sciences in Kalisz
Nowy Swiat 4, 62–800 Kalisz, Poland
*E-mail: tretiakov@ifmpan.poznan.pl
Received:
Received: 03 June 2019; revised: 27 June 2019; accepted: 29 June 2019; published online: 30 June 2019
DOI: 10.12921/cmst.2019.0000019
Abstract:
The thermal conductivity coefficients of solid argon have been evaluated by equilibrium molecular dynamic simulations. A Lennard-Jones interatomic potential has been used to model the interactions between argon atoms. In simulations and calculations of the thermal conductivity by the Green-Kubo formula, the long-range interactions between atoms have been taken into account using the minimum image method (MIM). The study shows that there are no significant differences between the values of the thermal conductivity obtained by method using MIM and those coming from traditional Green-Kubo approach. Both experimental data and results of molecular dynamics simulations are also in agreement with the Klemens-Callaway model for the thermal conductivity based on the three-phonon Umklapp scattering.
Key words:
Lennard-Jones potential, Molecular Dynamics simulations, solid argon, the Green-Kubo method, thermal conductivity
References:
[1] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, J.W. Arrowsmith Ltd., Bristol, UK (1987).
[2] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic, New York (2005).
[3] R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, J. Phys. Soc. Japan 12, 570–586 (1957).
[4] R. Zwanzig, Time-Correlation Functions and Transport Coefficients in Statistical Mechanics, Annu. Rev. Phys. Chem. 16, 67–102 (1965).
[5] J.G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J. Chem. Phys. 14, 180 (1946).
[6] D.M. Heyes, Transport-Coefficients of Lennard-Jones Fluids: A Molecular-Dynamics and Effective Hard-Sphere Treatment, Phys. Rev. B 37, 5677 (1988).
[7] K.V. Tretiakov, K.W.Wojciechowski, Quick and accurate estimation of the elastic constants using the minimum image method, Comput. Phys. Commun. 189, 77–83 (2015).
[8] A.J.C. Ladd, Monte-Carlo simulation of water, Mol. Phys. 33(4), 1039–1050 (1977).
[9] A.J.C. Ladd, Long-range dipolar interactions in computer simulations of polar liquids, Mol. Phys. 36(2), 463–474 (1978).
[10] D.P. Sellan, E.S. Landry, J.E. Turney, A.J. McGaughey, C.H. Amon, Size effects in molecular dynamics thermal conductivity predictions, Phys. Rev. B 81, 214305 (2010).
[11] K. Hy˙zorek, K.V. Tretiakov, Thermal conductivity of liquid argon in nanochannels from molecular dynamics simulations, J. Chem. Phys. 144, 194507 (2016).
[12] K.V. Tretiakov, S. Scandolo, Thermal conductivity of solid argon for molecular dynamics simulations, J. Chem. Phys. 120(8), 3765–3769 (2004).
[13] I.N. Krupskii, V.G. Manzhelii, Multiphonon Interactions and the Thermal Conductivity of Crystalline Argon, Krypton, and Xenon, Sov. Phys. JETP 28, 1097 (1969).
[14] F. Clayton, D.N. Batchelder, Temperature and volume dependence of the thermal conductivity of solid argon, J. Phys. C: Solid State Phys. 6, 1213 (1973).
[15] D.K. Christen, G.L. Pollack, Thermal conductivity of solid argon, Phys. Rev. B 12, 3380 (1975).
[16] P.G. Klemens, [In:] Solid State Physics, edited by F. Seitz and D. Turnbull, Academic Press, New York, 1st Edition (1958).
[17] J. Zou, A. Balandin, Phonon heat conduction in a semiconductor nanowire, J. Appl. Phys. 89, 2932 (2001).
[18] P.G. Klemens, [In:] Chem. and Phys. of Nanostructures and Related Non-Equilibrium Materials, edited by E. Ma, B. Fultz, R. Shall, J. Morral, and P. Nash, Minerals, Metals, Materials Society, Warrendale, PA (1997).
[19] J.P. Poirier, Introduction to the Physics of Earth’s Interior, Cambridge University Press, Cambridge (1991).
[20] G.J. Keeler, D.N. Batchelder, Measurement of the elastic constants of argon from 3 to 77 K, J. Phys. C: Solid State Phys. 3, 510 (1970).
[21] A.M. Krivtsov, V.A. Kuzkin, Derivation of Equations of State for Ideal Crystals of Simple Structure, Mech. Sol. 46, 387 (2011).
The thermal conductivity coefficients of solid argon have been evaluated by equilibrium molecular dynamic simulations. A Lennard-Jones interatomic potential has been used to model the interactions between argon atoms. In simulations and calculations of the thermal conductivity by the Green-Kubo formula, the long-range interactions between atoms have been taken into account using the minimum image method (MIM). The study shows that there are no significant differences between the values of the thermal conductivity obtained by method using MIM and those coming from traditional Green-Kubo approach. Both experimental data and results of molecular dynamics simulations are also in agreement with the Klemens-Callaway model for the thermal conductivity based on the three-phonon Umklapp scattering.
Key words:
Lennard-Jones potential, Molecular Dynamics simulations, solid argon, the Green-Kubo method, thermal conductivity
References:
[1] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, J.W. Arrowsmith Ltd., Bristol, UK (1987).
[2] J.P. Hansen, I.R. McDonald, Theory of Simple Liquids, Academic, New York (2005).
[3] R. Kubo, Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems, J. Phys. Soc. Japan 12, 570–586 (1957).
[4] R. Zwanzig, Time-Correlation Functions and Transport Coefficients in Statistical Mechanics, Annu. Rev. Phys. Chem. 16, 67–102 (1965).
[5] J.G. Kirkwood, The Statistical Mechanical Theory of Transport Processes I. General Theory, J. Chem. Phys. 14, 180 (1946).
[6] D.M. Heyes, Transport-Coefficients of Lennard-Jones Fluids: A Molecular-Dynamics and Effective Hard-Sphere Treatment, Phys. Rev. B 37, 5677 (1988).
[7] K.V. Tretiakov, K.W.Wojciechowski, Quick and accurate estimation of the elastic constants using the minimum image method, Comput. Phys. Commun. 189, 77–83 (2015).
[8] A.J.C. Ladd, Monte-Carlo simulation of water, Mol. Phys. 33(4), 1039–1050 (1977).
[9] A.J.C. Ladd, Long-range dipolar interactions in computer simulations of polar liquids, Mol. Phys. 36(2), 463–474 (1978).
[10] D.P. Sellan, E.S. Landry, J.E. Turney, A.J. McGaughey, C.H. Amon, Size effects in molecular dynamics thermal conductivity predictions, Phys. Rev. B 81, 214305 (2010).
[11] K. Hy˙zorek, K.V. Tretiakov, Thermal conductivity of liquid argon in nanochannels from molecular dynamics simulations, J. Chem. Phys. 144, 194507 (2016).
[12] K.V. Tretiakov, S. Scandolo, Thermal conductivity of solid argon for molecular dynamics simulations, J. Chem. Phys. 120(8), 3765–3769 (2004).
[13] I.N. Krupskii, V.G. Manzhelii, Multiphonon Interactions and the Thermal Conductivity of Crystalline Argon, Krypton, and Xenon, Sov. Phys. JETP 28, 1097 (1969).
[14] F. Clayton, D.N. Batchelder, Temperature and volume dependence of the thermal conductivity of solid argon, J. Phys. C: Solid State Phys. 6, 1213 (1973).
[15] D.K. Christen, G.L. Pollack, Thermal conductivity of solid argon, Phys. Rev. B 12, 3380 (1975).
[16] P.G. Klemens, [In:] Solid State Physics, edited by F. Seitz and D. Turnbull, Academic Press, New York, 1st Edition (1958).
[17] J. Zou, A. Balandin, Phonon heat conduction in a semiconductor nanowire, J. Appl. Phys. 89, 2932 (2001).
[18] P.G. Klemens, [In:] Chem. and Phys. of Nanostructures and Related Non-Equilibrium Materials, edited by E. Ma, B. Fultz, R. Shall, J. Morral, and P. Nash, Minerals, Metals, Materials Society, Warrendale, PA (1997).
[19] J.P. Poirier, Introduction to the Physics of Earth’s Interior, Cambridge University Press, Cambridge (1991).
[20] G.J. Keeler, D.N. Batchelder, Measurement of the elastic constants of argon from 3 to 77 K, J. Phys. C: Solid State Phys. 3, 510 (1970).
[21] A.M. Krivtsov, V.A. Kuzkin, Derivation of Equations of State for Ideal Crystals of Simple Structure, Mech. Sol. 46, 387 (2011).
