The Lennard-Jones Fluid in the Liquid-Vapour Critical Region
Department of Physics, Royal Holloway
University of London, Egham, Surrey TW20 0EX, UK
E-mail: david.heyes@rhul.ac.uk
Received:
01 June 2015; accepted: 12 June 2015; published online: 23 August 2015
DOI: 10.12921/cmst.2015.21.04.001
Abstract:
The equation of state of the Lennard-Jones (LJ) fluid in the liquid-vapour (LV) critical region is investigated by Molecular Dynamics simulation (MD). The calculated pressure (P ) and chemical potential (µ) are, within the simulation statistics, flat at the critical temperature between LJ reduced densities of ca. 0.26 to 0.34. The critical temperature, Tc, determined for an isotherm where (∂P/∂ρ)T = 0 and (∂µ/∂ρ)T = 0, is shown to decrease with increasing system size and pure LJ potential interaction range, rc , using a tapering function going to zero beyond rc . The value of Tc obtained by extrapolating the system size and rc to ∞ is 1.316 ± 0.001, which is statistically within the uncertainties of previous literature values. The percolation threshold separation, rp , along the critical isotherm decreases monotonically with increasing density, ρ, and is for intermediate densities lower than that of the nearest equivalent hard-sphere system. The lines of constant percolation distance on the density-temperature projection of the phase diagram reveal a difference in qualitative behaviour, indicative of underlying structural differences on either side of the critical envelope. The mean square force in the critical region near to Tc is linear in ρ. Probability distributions of the nearest neighbour distance, absolute particle force and potential energy per molecule are presented.
Key words:
Lennard-Jones criticality, Molecular Dynamics simulation, percolation
References:
[1] T. Andrews, On the continuity of the gaseous and liquid states of matter, Phil. Trans. R. Soc. Lond. 159, 575-590 (1869).
[2] A.A. Mills, The critical transition between the liquid and gaseous conditions of matter, Endeavour 19, 69-75 (1995).
[3] J.D. van der Waals: On the continuity of the gas and liquid state, Ph.D. Thesis (University of Leiden, The Netherlands, 1873).
[4] E.A. Guggenheim, The principle of corresponding states, J. Chem. Phys. 13, 253-261 (1945).
[5] B. Widom and O.K. Rice, Critical isotherm and the equation of state of liquid vapour systems, J. Chem. Phys. 23, 1250-1255 (1955).
[6] N.S. Barlow, A.J. Schultz, D.A. Kofke, and S.J. Weinstein, Critical isotherms asymptotically consistent approximants, AIChE Journal, 60, 3336-3349 (2014).
[7] S.F. Harrison and J.E. Mayer, Statistical mechanics of condensing systems. IV, J. Chem. Phys. 6, 101-104 (1938).
[8] L.V. Woodcock, Thermodynamic description of liquid-state limits, J. Phys. Chem. B, 116, 3735-3744 (2012) .
[9] L.V. Woodcock, Fluid phases of argon: A debate on the absence of van der Waals’ “critical point”, Natural Science, 5, 194-206 (2013).
[10] D.M. Heyes and L.V. Woodcock, Critical and supercritical properties of Lennard-Jones fluids, Fluid Phase Equil., 356, 301-308 (2013).
[11] J.M.H. Levelt Sengers, Liquidons and gasons. Controversies about the continuity of states, Physica A, 98, 363-402 (1979).
[12] L.P. Kadanoff, More is the same; phase transitions and mean field theories, J. Stat. Phys., 137 777-797 (2009).
[13] R.Gilgen, R. Kleinrahm and W. Wagner, Measurement and correlation of the (pressure, density, temperature) relation of argon I. The homogeneous gas and liquid regions in the temperature range from 90 K to 340 K at pressures up to 12 MPa, J. Chem. Therm. 26, 383-398 (1994).
[14] R.Gilgen, R. Kleinrahm and W. Wagner, Measurement and correlation of the (pressure, density, temperature) relation of argon II. saturated-liquid and saturated-vapour densities and vapour pressures along the entire coexistence curve. J. Chem. Therm. 26, 399-413 (1994).
[15] K. Rah and B.C. Eu, Phenomenological models for the generic van der Waals equation of state and critical parameters, J. Phys. Chem. B 107, 4382-4391 (2003).
[16] D.M. Heyes, Liquids at positive and negative pressure, Phys. Stat. Solidi B 245 (2008) 530-538.
[17] B. Widom, J. Phys. Chem. 39, 2808-2812 (1963).
[18] J.R. Morris and X. Song, J. Chem. Phys., The melting lines of model systems calculated from coexistence simulations, 116, 9352-9358 (2002).
[19] D. Heyes, The Liguid State – Applications of Molecular Simulations, J. Wiley & Sons, Chichester 1998.
[20] H. Watanabe, N. Ito and C.-K. Hu, Phase diagram and universality of the Lennard-Jones gas-liquid system, J. Chem. Phys., 136, 204102 (2012).
[21] D.O. Dunikov, S.P. Malyshenko and V.V. Zhakhovskii, Corresponding states law and molecular dynamics simulations of the Lennard-Jones fluid, J. Chem. Phys. 115, 6623-6631 (2001) .
[22] J.J. Potoff and A.Z. Panagiotopolous, Surface tension of the three-dimensional Lennard-Jones fluid from histogram-reweighting Monte Carlo simulations, J. Chem. Phys. 112, 6411-6415 (2000).
[23] J.Pérez-Pellitero, P. Ungerer, G. Orkoulas and A.D. Mackie, Critical point estimation of the Lennard-Jones pure fluid and binary mixtures, J. Chem. Phys. 125, 054515 (2006).
[24] J. Mick, E. Hailat, V. Russo, K. Rushaidat, L. Schwiebert and J. Potoff, GPU-accelerated Gibbs ensemble Monte Carlo simulations of Lennard-Jonesium, Comp. Phys. Comm. 184, 2662 (2013).
[25] J. Kolafa and I, Nezbeda, The Lennard-Jones fluid: An accurate analytic and theoretically-based equation of state, Fluid Phase Equilibria 100, 1-34 (1994).
[26] M.A. Barroso and A.L. Ferreira, Solid fluid coexistence of the Lennard-Jones system from absolute free energy calculations, J. Chem. Phys. 116, 7145-7150 (2002) .
[27] V.G. Baidakov, S.P. Protsenko and Z.R. Kozlova, Metastable Lennard-Jones fluids. I. Shear viscosity, J. Chem. Phys. 137, 164507 (2012).
[28] A. Lofti, J. Vrabec and J. Fischer, Vapour liquid equilibria of the Lennard-Jones fluid from the NpT plus test particle method, Mol. Phys. 76, 1319-1333 (1992).
[29] D.A. Kofke, Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line, J. Chem. Phys. 98 4149-4162 (1993) .
[30] A. Trokhymchuk and J. Alejandre, Computer simulations of liquid/vapour interface in Lennard-Jones fluids: Some questions and answers, J. Chem. Phys. 111, 8510-8523 (1999).
[31] H. Okumura and F. Yonezawa, Liquid vapour coexistence curves of several interatomic model potentials, J. Chem. Phys. 113, 9162-9168 (2000).
[32] H. Okumura and F. Yonezawa, Reliable determination of the liquid-vapour critical point by the NVT plus test particle method, J. Phys. Soc. Japan 70, 1990-1994 (2001).
[33] S. Reif-Acherman, The history of the rectilinear diameter law, Quimica Nova. 33 2003-2010 (2010).
[34] Ch. Tegeler, R. Span and W. Wagner, A new equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressures up to 1000 MPa, J. Phys. Chem. Ref. Data, 28, 779-850 (1999).
[35] L. A. Rowley, D. Nicholson and N. G. Parsonage, Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon, J. Comput. Phys. 17 401-414 (1975).
[36] R. J. Sadus, Exact calculation of the effect of three-body Axilrod–Teller interactions on vapour–liquid phase coexistence, Fl. Phas. Equil. 144 351–359 (1998).
[37] R.Bukowski and K. Szalewicz, Complete ab initio three-body nonadditive potential in Monte Carlo simulations of vapour–liquid equilibria and pure phases of argon , J. Chem. Phys. 114, 9518-9531 (2001).
[38] J. A. Anta, E. Lomba, and M. Lombardero, Influence of three-body forces on the gas-liquid coexistence of simple fluids: the phase equilibrium of argon, Phys. Rev. E. 55, 2707-2712 (1997).
[39] F. Goujon, P. Maffreyt and D.J. Tildesley, The gas-liquid surface tension of argon: A reconciliation between experiment and simulation, J. Chem. Phys. 140, 244710 (2014).
[40] S. Werth, M. Jorsch, J. Vrabec and H. Hasse, Comment on “The gas-liquid surface tension of argon: A reconciliation between experiment and simulation”, J. Chem. Phys. 142, 107101 (2015).
[41] F. Goujon, P. Maffreyt and D.J. Tildesley, Response to “Comment on ‘The gas-liquid surface tension of argon: A reconciliation between experiment and simulation”, J. Chem. Phys. 142, 107102 (2015).
[42] G. Rickayzen, A.C. Bra´nka, S. Pieprzyk and D.M. Heyes, Single particle force distributions in simple fluids, J. Chem. Phys. 137, 094505 (2012).
[43] J.L. Finney and L.V. Woodcock, Renaissance of Bernal’s random close packing and hypercritical line in the theory of liquids, J. Phys. Cond. Matt. 26 463120 (2014).
[44] D.M. Heyes, M. Cass and A.C. Bránka, Percolation threshold of hard-sphere fluids in between the soft-core and hard-core limits, Mol. Phys. 104, 3137-3146 (2006) .
[45] A.L.R. Bug, S.A. Safran, G.S. Grest and I. Webman, Do interactions raise or lower a percolation threshold?, Phys. Rev. Lett. 55, 1896-1899 (1985).
[46] D.M. Heyes, Cluster analysis and continuum percolation of 3D square-well phases: MC and PY olutions, Mol. Phys. 69, 559-569 (1990).
[47] D.M. Heyes and J.R. Melrose, Percolation cluster statistics of Lennard-Jones fluids, Mol. Phys. 66, 1057-1074 (1989) .
[48] D.M. Heyes, Monte Carlo Simulations of continuum percolation of 3D well fluids, J. Phys. Cond. Matt. 2, 2241-2249 (1990).
[49] D.M. Heyes and H. Okumura, Equation of state and structural properties of the Weeks-Chandler-Andersen fluid, J. Chem. Phys. 124, 164507 (2006).
[50] D. Stauffer, Scaling theory of percolation clusters, Phys. Rep. 54, 1-74 (1979).
The equation of state of the Lennard-Jones (LJ) fluid in the liquid-vapour (LV) critical region is investigated by Molecular Dynamics simulation (MD). The calculated pressure (P ) and chemical potential (µ) are, within the simulation statistics, flat at the critical temperature between LJ reduced densities of ca. 0.26 to 0.34. The critical temperature, Tc, determined for an isotherm where (∂P/∂ρ)T = 0 and (∂µ/∂ρ)T = 0, is shown to decrease with increasing system size and pure LJ potential interaction range, rc , using a tapering function going to zero beyond rc . The value of Tc obtained by extrapolating the system size and rc to ∞ is 1.316 ± 0.001, which is statistically within the uncertainties of previous literature values. The percolation threshold separation, rp , along the critical isotherm decreases monotonically with increasing density, ρ, and is for intermediate densities lower than that of the nearest equivalent hard-sphere system. The lines of constant percolation distance on the density-temperature projection of the phase diagram reveal a difference in qualitative behaviour, indicative of underlying structural differences on either side of the critical envelope. The mean square force in the critical region near to Tc is linear in ρ. Probability distributions of the nearest neighbour distance, absolute particle force and potential energy per molecule are presented.
Key words:
Lennard-Jones criticality, Molecular Dynamics simulation, percolation
References:
[1] T. Andrews, On the continuity of the gaseous and liquid states of matter, Phil. Trans. R. Soc. Lond. 159, 575-590 (1869).
[2] A.A. Mills, The critical transition between the liquid and gaseous conditions of matter, Endeavour 19, 69-75 (1995).
[3] J.D. van der Waals: On the continuity of the gas and liquid state, Ph.D. Thesis (University of Leiden, The Netherlands, 1873).
[4] E.A. Guggenheim, The principle of corresponding states, J. Chem. Phys. 13, 253-261 (1945).
[5] B. Widom and O.K. Rice, Critical isotherm and the equation of state of liquid vapour systems, J. Chem. Phys. 23, 1250-1255 (1955).
[6] N.S. Barlow, A.J. Schultz, D.A. Kofke, and S.J. Weinstein, Critical isotherms asymptotically consistent approximants, AIChE Journal, 60, 3336-3349 (2014).
[7] S.F. Harrison and J.E. Mayer, Statistical mechanics of condensing systems. IV, J. Chem. Phys. 6, 101-104 (1938).
[8] L.V. Woodcock, Thermodynamic description of liquid-state limits, J. Phys. Chem. B, 116, 3735-3744 (2012) .
[9] L.V. Woodcock, Fluid phases of argon: A debate on the absence of van der Waals’ “critical point”, Natural Science, 5, 194-206 (2013).
[10] D.M. Heyes and L.V. Woodcock, Critical and supercritical properties of Lennard-Jones fluids, Fluid Phase Equil., 356, 301-308 (2013).
[11] J.M.H. Levelt Sengers, Liquidons and gasons. Controversies about the continuity of states, Physica A, 98, 363-402 (1979).
[12] L.P. Kadanoff, More is the same; phase transitions and mean field theories, J. Stat. Phys., 137 777-797 (2009).
[13] R.Gilgen, R. Kleinrahm and W. Wagner, Measurement and correlation of the (pressure, density, temperature) relation of argon I. The homogeneous gas and liquid regions in the temperature range from 90 K to 340 K at pressures up to 12 MPa, J. Chem. Therm. 26, 383-398 (1994).
[14] R.Gilgen, R. Kleinrahm and W. Wagner, Measurement and correlation of the (pressure, density, temperature) relation of argon II. saturated-liquid and saturated-vapour densities and vapour pressures along the entire coexistence curve. J. Chem. Therm. 26, 399-413 (1994).
[15] K. Rah and B.C. Eu, Phenomenological models for the generic van der Waals equation of state and critical parameters, J. Phys. Chem. B 107, 4382-4391 (2003).
[16] D.M. Heyes, Liquids at positive and negative pressure, Phys. Stat. Solidi B 245 (2008) 530-538.
[17] B. Widom, J. Phys. Chem. 39, 2808-2812 (1963).
[18] J.R. Morris and X. Song, J. Chem. Phys., The melting lines of model systems calculated from coexistence simulations, 116, 9352-9358 (2002).
[19] D. Heyes, The Liguid State – Applications of Molecular Simulations, J. Wiley & Sons, Chichester 1998.
[20] H. Watanabe, N. Ito and C.-K. Hu, Phase diagram and universality of the Lennard-Jones gas-liquid system, J. Chem. Phys., 136, 204102 (2012).
[21] D.O. Dunikov, S.P. Malyshenko and V.V. Zhakhovskii, Corresponding states law and molecular dynamics simulations of the Lennard-Jones fluid, J. Chem. Phys. 115, 6623-6631 (2001) .
[22] J.J. Potoff and A.Z. Panagiotopolous, Surface tension of the three-dimensional Lennard-Jones fluid from histogram-reweighting Monte Carlo simulations, J. Chem. Phys. 112, 6411-6415 (2000).
[23] J.Pérez-Pellitero, P. Ungerer, G. Orkoulas and A.D. Mackie, Critical point estimation of the Lennard-Jones pure fluid and binary mixtures, J. Chem. Phys. 125, 054515 (2006).
[24] J. Mick, E. Hailat, V. Russo, K. Rushaidat, L. Schwiebert and J. Potoff, GPU-accelerated Gibbs ensemble Monte Carlo simulations of Lennard-Jonesium, Comp. Phys. Comm. 184, 2662 (2013).
[25] J. Kolafa and I, Nezbeda, The Lennard-Jones fluid: An accurate analytic and theoretically-based equation of state, Fluid Phase Equilibria 100, 1-34 (1994).
[26] M.A. Barroso and A.L. Ferreira, Solid fluid coexistence of the Lennard-Jones system from absolute free energy calculations, J. Chem. Phys. 116, 7145-7150 (2002) .
[27] V.G. Baidakov, S.P. Protsenko and Z.R. Kozlova, Metastable Lennard-Jones fluids. I. Shear viscosity, J. Chem. Phys. 137, 164507 (2012).
[28] A. Lofti, J. Vrabec and J. Fischer, Vapour liquid equilibria of the Lennard-Jones fluid from the NpT plus test particle method, Mol. Phys. 76, 1319-1333 (1992).
[29] D.A. Kofke, Direct evaluation of phase coexistence by molecular simulation via integration along the saturation line, J. Chem. Phys. 98 4149-4162 (1993) .
[30] A. Trokhymchuk and J. Alejandre, Computer simulations of liquid/vapour interface in Lennard-Jones fluids: Some questions and answers, J. Chem. Phys. 111, 8510-8523 (1999).
[31] H. Okumura and F. Yonezawa, Liquid vapour coexistence curves of several interatomic model potentials, J. Chem. Phys. 113, 9162-9168 (2000).
[32] H. Okumura and F. Yonezawa, Reliable determination of the liquid-vapour critical point by the NVT plus test particle method, J. Phys. Soc. Japan 70, 1990-1994 (2001).
[33] S. Reif-Acherman, The history of the rectilinear diameter law, Quimica Nova. 33 2003-2010 (2010).
[34] Ch. Tegeler, R. Span and W. Wagner, A new equation of state for argon covering the fluid region for temperatures from the melting line to 700 K at pressures up to 1000 MPa, J. Phys. Chem. Ref. Data, 28, 779-850 (1999).
[35] L. A. Rowley, D. Nicholson and N. G. Parsonage, Monte Carlo grand canonical ensemble calculation in a gas-liquid transition region for 12-6 Argon, J. Comput. Phys. 17 401-414 (1975).
[36] R. J. Sadus, Exact calculation of the effect of three-body Axilrod–Teller interactions on vapour–liquid phase coexistence, Fl. Phas. Equil. 144 351–359 (1998).
[37] R.Bukowski and K. Szalewicz, Complete ab initio three-body nonadditive potential in Monte Carlo simulations of vapour–liquid equilibria and pure phases of argon , J. Chem. Phys. 114, 9518-9531 (2001).
[38] J. A. Anta, E. Lomba, and M. Lombardero, Influence of three-body forces on the gas-liquid coexistence of simple fluids: the phase equilibrium of argon, Phys. Rev. E. 55, 2707-2712 (1997).
[39] F. Goujon, P. Maffreyt and D.J. Tildesley, The gas-liquid surface tension of argon: A reconciliation between experiment and simulation, J. Chem. Phys. 140, 244710 (2014).
[40] S. Werth, M. Jorsch, J. Vrabec and H. Hasse, Comment on “The gas-liquid surface tension of argon: A reconciliation between experiment and simulation”, J. Chem. Phys. 142, 107101 (2015).
[41] F. Goujon, P. Maffreyt and D.J. Tildesley, Response to “Comment on ‘The gas-liquid surface tension of argon: A reconciliation between experiment and simulation”, J. Chem. Phys. 142, 107102 (2015).
[42] G. Rickayzen, A.C. Bra´nka, S. Pieprzyk and D.M. Heyes, Single particle force distributions in simple fluids, J. Chem. Phys. 137, 094505 (2012).
[43] J.L. Finney and L.V. Woodcock, Renaissance of Bernal’s random close packing and hypercritical line in the theory of liquids, J. Phys. Cond. Matt. 26 463120 (2014).
[44] D.M. Heyes, M. Cass and A.C. Bránka, Percolation threshold of hard-sphere fluids in between the soft-core and hard-core limits, Mol. Phys. 104, 3137-3146 (2006) .
[45] A.L.R. Bug, S.A. Safran, G.S. Grest and I. Webman, Do interactions raise or lower a percolation threshold?, Phys. Rev. Lett. 55, 1896-1899 (1985).
[46] D.M. Heyes, Cluster analysis and continuum percolation of 3D square-well phases: MC and PY olutions, Mol. Phys. 69, 559-569 (1990).
[47] D.M. Heyes and J.R. Melrose, Percolation cluster statistics of Lennard-Jones fluids, Mol. Phys. 66, 1057-1074 (1989) .
[48] D.M. Heyes, Monte Carlo Simulations of continuum percolation of 3D well fluids, J. Phys. Cond. Matt. 2, 2241-2249 (1990).
[49] D.M. Heyes and H. Okumura, Equation of state and structural properties of the Weeks-Chandler-Andersen fluid, J. Chem. Phys. 124, 164507 (2006).
[50] D. Stauffer, Scaling theory of percolation clusters, Phys. Rep. 54, 1-74 (1979).
