Density Functional Formalism as a Description of the Elastic Behavior of a Hard-Sphere Crystal
Pieprzyk S. 1*, Brańka A.C. 1†, Heyes D.M. 2
1 Institute of Molecular Physics
Polish Academy of Sciences
M. Smoluchowskiego 17, 60-179 Poznań, Poland
∗E-mail: slawomir.pieprzyk@ifmpan.poznan.pl
†E-mail: branka@ifmpan.poznan.pl2 Royal Holloway, University of London
Department of Physics
Egham, Surrey TW20 0EX, United Kingdom
E-mail: david.heyes@rhul.ac.uk
Received:
Received: 13 November 2023; in final form: 20 November 2023; accepted: 21 November 2023; published online: 9 December 2023
DOI: 10.12921/cmst.2023.0000027
Abstract:
The density functional method of Jarić and Mohanty [Phys. Rev. B 37, 4441 (1988)] for calculating the elastic moduli of crystalline solids is considered here from the perspective of some new findings. The very slow convergence of the reciprocal lattice vector summations and presence of the three body term in the method’s computational scheme identified in [J. Chem. Phys. 118, 6594 (2003)] is confirmed and discussed. The sensitivity of the results to the scheme parameters, such as the width of the Gaussian density profiles and the Percus-Yevick approximation used for the direct correlation function is explored. The calculations are for a hard-sphere crystal but most conclusions can be applicable to model crystalline solids in general.
Key words:
density functional theory, elastic moduli tensor, hard spheres
References:
[1] P. Hohenberg, W. Kohn, Inhomogeneous Electron Gas, Physical Review 136, B864 (1964).
[2] W. Kohn, L.J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Physical Review 140, A1133 (1965).
[3] W. Koch, M.C. Holthausen, A Chemist’s Guide to Density Functional Theory, Wiley (2001).
[4] N.D. Mermin, Thermal Properties of the Inhomogeneous Electron Gas, Physical Review 137, A1441 (1965).
[5] M. Baus, Statistical mechanical theories of freezing: An overview, Journal of Statistical Physics 48, 1129 (1987).
[6] B. Groh, B. Mulder, Hard-sphere solids near close packing: Testing theories for crystallization, Physical Review E 61, 3811 (2000).
[7] Y. Singh, Density-functional theory of freezing and properties of the ordered phase, Physics Reports 207, 351 (1991).
[8] R. McRae, A.D.J. Haymet, Freezing of polydisperse hard spheres, The Journal of Chemical Physics 88, 1114 (1988).
[9] T.V. Ramakrishnan, M. Yussouff, First-principles order-parameter theory of freezing, Physical Review B 19, 2775 (1979).
[10] N. Sushko, P. van der Schoot, M.A.J. Michels, Density-functional theory of the crystallization of hard polymeric chains, The Journal of Chemical Physics 115, 7744 (2001).
[11] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids; with Applications to Soft Matter, Elsevier LTD, Oxford (2013).
[12] P. Tarazona, A density functional theory of melting, Molecular Physics 52, 81 (1984).
[13] M. Baus, J.L. Colot, Density-Wave Theory of First-Order Freezing in Two Dimensions, Molecular Physics 55, 653 (1985).
[14] J.L. Colot, M. Baus, The freezing of hard spheres, Molecular Physics 56, 807 (1985).
[15] R.O. Jones, Density functional theory: its origins, rise to prominence, and future, Reviews of Modern Physics 87, 897 (2015).
[16] H. Löwen, Density functional theory of inhomogeneous classical fluids: recent developments and new perspectives, Journal of Physics: Condensed Matter 14, 11897 (2002).
[17] M. Yussouff, Generalized structural theory of freezing, Physical Review B 23, 5871 (1981).
[18] T.V. Ramakrishnan, Density-Wave Theory of First-Order Freezing in Two Dimensions, Physical Review Letters 48, 541 (1982).
[19] M.V. Jarić, U. Mohanty, “Martensitic” instability of an icosahedral quasicrystal, Physical Review Letters 58, 230 (1987).
[20] G.L. Jones, Elastic constants in density-functional theory, Molecular Physics 61, 455 (1987).
[21] M.V. Jarić, U. Mohanty, Density-functional theory of elastic moduli: Hard-sphere and Lennard-Jones crystals, Physical Review B 37, 4441 (1988).
[22] D. Frenkel, A.J.C. Ladd, Elastic constants of hard-sphere crystals, Physical Review Letters 59, 1169 (1987).
[23] M.V. Jarić, U. Mohanty, Jarić and Mohanty Reply, Physical Review Letters 59, 1170 (1987).
[24] B.B. Laird, J.D. McCoy, A.D.J. Haymet, Density functional theory of freezing: Analysis of crystal density, The Journal of Chemical Physics 87, 5449 (1987).
[25] N. Sushko, P. van der Schoot, M.A.J. Michels, Density functional theory for the elastic moduli of a model polymeric solid, The Journal of Chemical Physics 118, 6594 (2003); Erratum: The Journal of Chemical Physics 119, 639 (2003).
[26] D.C. Wallace, Thermodynamics of Crystals, Dover Publication (1998).
[27] M. Oettel, S. Görig, A. Härtel, H. Löwen, M. Radu, T. Schilling, Free energies, vacancy concentrations, and density distribution anisotropies in hard-sphere crystals: A combined density functional and simulation study, Physical Review E 82, 051404 (2010).
[28] D.A. Young, B.J. Alder, Studies in molecular dynamics. XIII. Singlet and pair distribution functions for hard-disk and hard-sphere solids, The Journal of Chemical Physics 60, 1254 (1974).
[29] K.W. Wojciechowski, K.V. Tretiakov, Elastic properties of the f.c.c. hard sphere crystal free of defects, Computational Methods in Science and Technology 8, 84 (2002).
[30] K.V. Tretiakov, K.W. Wojciechowski, Poisson’s ratio of the fcc hard sphere crystal at high densities, The Journal of Chemical Physics 123, 074509 (2005).
[31] S. Pieprzyk, M.N. Bannerman, A.C. Brańka, M. Chudak, D.M. Heyes, Thermodynamic and dynamical properties of the hard sphere system revisited by molecular dynamics simulation, Physical Chemistry Chemical Physics 21, 6886 (2019).
[32] S. Bravo Yuste, A. Santos, Radial distribution function for hard spheres, Physical Review A 43, 5418 (1991).
[33] C.F. Tejero, M. López de Haro, Direct correlation function of the hard-sphere fluid, Molecular Physics 105, 2999 (2007).
[34] H. van Beijeren, M.H. Ernst, The modified Enskog equation, Physica 68, 437 (1973).
[35] J.R. Dorfman, H. van Beijeren, T.R. Kirkpatrick, Contemporary Kinetic Theory of Matter, Cambridge University Press (2021).
[36] T.R. Kirkpatrick, S.P. Das, M.H. Ernst, J. Piasecki, Kinetic theory of transport in a hard sphere crystal, The Journal of Chemical Physics 92, 3768 (1990).
The density functional method of Jarić and Mohanty [Phys. Rev. B 37, 4441 (1988)] for calculating the elastic moduli of crystalline solids is considered here from the perspective of some new findings. The very slow convergence of the reciprocal lattice vector summations and presence of the three body term in the method’s computational scheme identified in [J. Chem. Phys. 118, 6594 (2003)] is confirmed and discussed. The sensitivity of the results to the scheme parameters, such as the width of the Gaussian density profiles and the Percus-Yevick approximation used for the direct correlation function is explored. The calculations are for a hard-sphere crystal but most conclusions can be applicable to model crystalline solids in general.
Key words:
density functional theory, elastic moduli tensor, hard spheres
References:
[1] P. Hohenberg, W. Kohn, Inhomogeneous Electron Gas, Physical Review 136, B864 (1964).
[2] W. Kohn, L.J. Sham, Self-Consistent Equations Including Exchange and Correlation Effects, Physical Review 140, A1133 (1965).
[3] W. Koch, M.C. Holthausen, A Chemist’s Guide to Density Functional Theory, Wiley (2001).
[4] N.D. Mermin, Thermal Properties of the Inhomogeneous Electron Gas, Physical Review 137, A1441 (1965).
[5] M. Baus, Statistical mechanical theories of freezing: An overview, Journal of Statistical Physics 48, 1129 (1987).
[6] B. Groh, B. Mulder, Hard-sphere solids near close packing: Testing theories for crystallization, Physical Review E 61, 3811 (2000).
[7] Y. Singh, Density-functional theory of freezing and properties of the ordered phase, Physics Reports 207, 351 (1991).
[8] R. McRae, A.D.J. Haymet, Freezing of polydisperse hard spheres, The Journal of Chemical Physics 88, 1114 (1988).
[9] T.V. Ramakrishnan, M. Yussouff, First-principles order-parameter theory of freezing, Physical Review B 19, 2775 (1979).
[10] N. Sushko, P. van der Schoot, M.A.J. Michels, Density-functional theory of the crystallization of hard polymeric chains, The Journal of Chemical Physics 115, 7744 (2001).
[11] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids; with Applications to Soft Matter, Elsevier LTD, Oxford (2013).
[12] P. Tarazona, A density functional theory of melting, Molecular Physics 52, 81 (1984).
[13] M. Baus, J.L. Colot, Density-Wave Theory of First-Order Freezing in Two Dimensions, Molecular Physics 55, 653 (1985).
[14] J.L. Colot, M. Baus, The freezing of hard spheres, Molecular Physics 56, 807 (1985).
[15] R.O. Jones, Density functional theory: its origins, rise to prominence, and future, Reviews of Modern Physics 87, 897 (2015).
[16] H. Löwen, Density functional theory of inhomogeneous classical fluids: recent developments and new perspectives, Journal of Physics: Condensed Matter 14, 11897 (2002).
[17] M. Yussouff, Generalized structural theory of freezing, Physical Review B 23, 5871 (1981).
[18] T.V. Ramakrishnan, Density-Wave Theory of First-Order Freezing in Two Dimensions, Physical Review Letters 48, 541 (1982).
[19] M.V. Jarić, U. Mohanty, “Martensitic” instability of an icosahedral quasicrystal, Physical Review Letters 58, 230 (1987).
[20] G.L. Jones, Elastic constants in density-functional theory, Molecular Physics 61, 455 (1987).
[21] M.V. Jarić, U. Mohanty, Density-functional theory of elastic moduli: Hard-sphere and Lennard-Jones crystals, Physical Review B 37, 4441 (1988).
[22] D. Frenkel, A.J.C. Ladd, Elastic constants of hard-sphere crystals, Physical Review Letters 59, 1169 (1987).
[23] M.V. Jarić, U. Mohanty, Jarić and Mohanty Reply, Physical Review Letters 59, 1170 (1987).
[24] B.B. Laird, J.D. McCoy, A.D.J. Haymet, Density functional theory of freezing: Analysis of crystal density, The Journal of Chemical Physics 87, 5449 (1987).
[25] N. Sushko, P. van der Schoot, M.A.J. Michels, Density functional theory for the elastic moduli of a model polymeric solid, The Journal of Chemical Physics 118, 6594 (2003); Erratum: The Journal of Chemical Physics 119, 639 (2003).
[26] D.C. Wallace, Thermodynamics of Crystals, Dover Publication (1998).
[27] M. Oettel, S. Görig, A. Härtel, H. Löwen, M. Radu, T. Schilling, Free energies, vacancy concentrations, and density distribution anisotropies in hard-sphere crystals: A combined density functional and simulation study, Physical Review E 82, 051404 (2010).
[28] D.A. Young, B.J. Alder, Studies in molecular dynamics. XIII. Singlet and pair distribution functions for hard-disk and hard-sphere solids, The Journal of Chemical Physics 60, 1254 (1974).
[29] K.W. Wojciechowski, K.V. Tretiakov, Elastic properties of the f.c.c. hard sphere crystal free of defects, Computational Methods in Science and Technology 8, 84 (2002).
[30] K.V. Tretiakov, K.W. Wojciechowski, Poisson’s ratio of the fcc hard sphere crystal at high densities, The Journal of Chemical Physics 123, 074509 (2005).
[31] S. Pieprzyk, M.N. Bannerman, A.C. Brańka, M. Chudak, D.M. Heyes, Thermodynamic and dynamical properties of the hard sphere system revisited by molecular dynamics simulation, Physical Chemistry Chemical Physics 21, 6886 (2019).
[32] S. Bravo Yuste, A. Santos, Radial distribution function for hard spheres, Physical Review A 43, 5418 (1991).
[33] C.F. Tejero, M. López de Haro, Direct correlation function of the hard-sphere fluid, Molecular Physics 105, 2999 (2007).
[34] H. van Beijeren, M.H. Ernst, The modified Enskog equation, Physica 68, 437 (1973).
[35] J.R. Dorfman, H. van Beijeren, T.R. Kirkpatrick, Contemporary Kinetic Theory of Matter, Cambridge University Press (2021).
[36] T.R. Kirkpatrick, S.P. Das, M.H. Ernst, J. Piasecki, Kinetic theory of transport in a hard sphere crystal, The Journal of Chemical Physics 92, 3768 (1990).
