Percolation Transitions and Fluid State Boundaries
Department of Physics, University of Algarve
Faro 8005-139, Portugal
E-mail: lvwoodcock@ualg.pt
Received:
Received: 30 December 2016; revised: 03 March 2017; accepted: 03 March 2017; published online: 17 April 2017
DOI: 10.12921/cmst.2016.0000070
Abstract:
Percolation transitions define gas- and liquid-state limits of existence. For simple model fluids percolation phenomena vary fundamentally with dimensionality (d).In 3d the accessible volume (VA) and excluded volume(VE =V−VA) percolation transitions occur at different densities, whereas in 2d they coincide. The region of overlap for 3d fluids can be identified as the origin of a supercritical mesophase. This difference between 2d and 3d systems vitiates the hypothetical concept of “universality” in the description of critical phenomena. Thermodynamic states at which VA and VE , for a spher- ical molecule diameter σ, percolates the whole volume of an ideal gas, together with MD computations of percolation loci for the penetrable cohesive sphere (PCS) model of gas-liquid equilibria, show a connection between the intersection of percolation loci, and the 1st-order phase-separation transition. The results accord with previous findings for square-well and Lennard-Jones model critical and supercritical fluid equilibria. Percolation loci for real liquids, e.g. CO2 and argon, can be determined from literature thermodynamic equation-of-state data, and exhibit similar supercritical gas- and liquid-state bounds. For these real fluids the mesophase bounds extend to low density and pressures and appear to converge onto the Boyle temperature (TB ) in the low-density limit.
Key words:
critical point, gas phase, liquid phase, percolation, phase transition
References:
[1] J.W. Gibbs, A method of geometrical representation of the thermodynamic properties of substances by means of sur- faces in Collected Works of J. Willard Gibbs; (Longmans Green and Co. New York), 1928; Ch. I. Original publication: Trans. Conn. Acad. Arts Sci. 2, 382 (1873).
[2] J.D. van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (On the Continuity of the Gas and Liquid Sate) Ph.D. Thesis, Leiden, The Netherlands, (1873).
[3] S. Reif-Acherman, History of the Law of Rectilinear Diam- eters, Quimica Nova 33(9) 2003-13 (2010).
[4] NIST Thermo-physical Properties of Fluid Systems (2016) http://webbook.nist.gov/chemistry/fluid/
[5] D.M. Heyes,The Liquid State(Wiley: Chichester 1995).
[6] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, 4th Ed. (Academic Press: Oxford 2013).
[7] J.A. Barker, D. Henderson, What is liquid? Rev. ModernPhysics 48 587 (1976).
[8] K.G. Wilson, Nobel Lecture: The Renormalization Group and Critical Phenomena Nobel Lectures, Physics 1981–1990 (World Scientific Publishing Co.: Singapore, 1993). [9] J.E. Mayer, M.G. Mayer, Statistical Mechanics 1st Edition(Wiley: New York USA 1940).
[10] M. Bannerman, L. Lue L.V. Woodcock, Thermodynamic pressures of hard-sphere fluid and closed virial equation-of-state, J. Chem. Phys. 132 084507 (2010).
[11] L.V. Woodcock, Percolation transitions in the hard-sphere fluid, AIChE Journal 58 1610-1618 (2011).
[12] L.V. Woodcock, Thermodynamic description of liquid-state limits, J. Phys. Chem. (B) 116 3734 (2011).
[13] Wm.G. Hoover, J.C. Poirier, Determination of virial coeffi- cients from potential of mean force, J. Chem. Phys. 37:1041–1042 (1962).
[14] B. Widom , Some topics in the theory of fluids, J. Chem. Phys. 39 2808–2812 (1963).
[15] R. Wheatley, Calculation of high-order virial coefficients with applications to hard and soft spheres, Phys. Rev. Lett. 110 200601 (2013).
[16] Wm.G. Hoover, N.E. Hoover, K. Hanson, Exact hard-disk free volumes, J. Chem. Phys. 70 1837–1844 (1979).
[17] D.M. Heyes, M. Cass, A.C. Branka, Percolation threshold of hard-sphere fluids in between the soft-core and hard-core limits, Mol. Phys. 104 3137–3146 (2000).
[18] S.Quintanilla, R.M. Torquato, J. Ziff, Efficient measurement of the percolation threshold for fully penetrable discs, Phys. Rev. (A) Math. Gen.; 23 399- 407 (2000).
[19] C.D. Lorentz, R.M. Ziff, Precise determination of the critical percolation threshold for the three-dimensional ‘Swiss cheese’ model using a growth algorithm, J. Chem. Phys, 114 3659-3661 (2001).
[20] B. Widom, J.S. Rowlinson, New model for the study of liquid–vapor phase transitions, J. Chem. Phys. 52 1670 (1970).
[21] L.V. Woodcock, Non-additive hard-sphere reference model for ionic liquids, Ind. Eng. Chem. Res. 50, 227–233 (2011).
[22] E. de Miguel, N.G. Almarza, G. Jackson, Surface tension of the Widom-Rowlinson model, J. Chem. Phys., 127, 034707 (2007).
[23] L.V. Woodcock, Observations of a thermodynamic liquid–gas critical coexistence line and supercritical phase bounds from percolation loci, Fluid Phase Equilibria, 351 25-33 (2013).
[24] L.V. Woodcock, Thermodynamics of Criticality: Percola- tion Loci, Mesophases and a Critical Dividing Line in Binary-Liquid and Liquid-Gas Equilibria Journal of Mod- ern Physics, 7, 760-773 (2016).
[25] D.M. Heyes, L.V. Woodcock, Critical and supercritical properties of Lennard-Jones fluids, Fluid Phase Equilibria (2013).
[26] D.M. Heyes, The Lennard-Jones fluid in the liquid-vapour critical region, CMST 21 169-179 (2015).
[27] L.V. Woodcock, Thermodynamics of gas-liquid criticality: rigidity symmetry on Gibbs density surface Int. J. Thermo- physics, 37 24 -33 (2016).
[28] R.Gilgen, R. Kleinrahm, W. Wagner, Measurement and correlation of the (pressure, density, temperature) relation of ar- gon. I. The homogeneous gas and liquid regions in the tem- perature range from 90 to 300K at pressures up to 12MPa, J. Chem. Thermodynamics 26 383-398 (1994).
[29] R. Gilgen, R. Kleinrahm, W. Wagner, Measurement and cor- relation 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. Thermo. 26 399-413 (1994).
[30] B.C. Eu, Exact analytic second virial coefficient for the Lennard-Jones Fluid, arXiv (Phys. Chem.) 0909 3326 (2009).
[31] D.M. Heyes, S. Pieprzyk, G. Rickayzen, A.C. Branka, The second virial coefficient and critical point behaviour of the Mie potential, J. Chem. Phys. 145 084505 (2016).
[32] J.M.H. Levelt-Sengers, Liquidons and Gasons; Controversies about the Continuity of States, Physica, A98, 363-402 (1979).
[33] J.S. Rowlinson Critical states of fluids and fluid mixtures: a review of the experimental position, 9-12 Proc. Conf. on Phenomena in the Neighborhood of Critical Points, (National Bureau of Standards: Washington DC, USA 1965).
Percolation transitions define gas- and liquid-state limits of existence. For simple model fluids percolation phenomena vary fundamentally with dimensionality (d).In 3d the accessible volume (VA) and excluded volume(VE =V−VA) percolation transitions occur at different densities, whereas in 2d they coincide. The region of overlap for 3d fluids can be identified as the origin of a supercritical mesophase. This difference between 2d and 3d systems vitiates the hypothetical concept of “universality” in the description of critical phenomena. Thermodynamic states at which VA and VE , for a spher- ical molecule diameter σ, percolates the whole volume of an ideal gas, together with MD computations of percolation loci for the penetrable cohesive sphere (PCS) model of gas-liquid equilibria, show a connection between the intersection of percolation loci, and the 1st-order phase-separation transition. The results accord with previous findings for square-well and Lennard-Jones model critical and supercritical fluid equilibria. Percolation loci for real liquids, e.g. CO2 and argon, can be determined from literature thermodynamic equation-of-state data, and exhibit similar supercritical gas- and liquid-state bounds. For these real fluids the mesophase bounds extend to low density and pressures and appear to converge onto the Boyle temperature (TB ) in the low-density limit.
Key words:
critical point, gas phase, liquid phase, percolation, phase transition
References:
[1] J.W. Gibbs, A method of geometrical representation of the thermodynamic properties of substances by means of sur- faces in Collected Works of J. Willard Gibbs; (Longmans Green and Co. New York), 1928; Ch. I. Original publication: Trans. Conn. Acad. Arts Sci. 2, 382 (1873).
[2] J.D. van der Waals, Over de Continuiteit van den Gas-en Vloeistoftoestand (On the Continuity of the Gas and Liquid Sate) Ph.D. Thesis, Leiden, The Netherlands, (1873).
[3] S. Reif-Acherman, History of the Law of Rectilinear Diam- eters, Quimica Nova 33(9) 2003-13 (2010).
[4] NIST Thermo-physical Properties of Fluid Systems (2016) http://webbook.nist.gov/chemistry/fluid/
[5] D.M. Heyes,The Liquid State(Wiley: Chichester 1995).
[6] J.-P. Hansen, I.R. McDonald, Theory of Simple Liquids, 4th Ed. (Academic Press: Oxford 2013).
[7] J.A. Barker, D. Henderson, What is liquid? Rev. ModernPhysics 48 587 (1976).
[8] K.G. Wilson, Nobel Lecture: The Renormalization Group and Critical Phenomena Nobel Lectures, Physics 1981–1990 (World Scientific Publishing Co.: Singapore, 1993). [9] J.E. Mayer, M.G. Mayer, Statistical Mechanics 1st Edition(Wiley: New York USA 1940).
[10] M. Bannerman, L. Lue L.V. Woodcock, Thermodynamic pressures of hard-sphere fluid and closed virial equation-of-state, J. Chem. Phys. 132 084507 (2010).
[11] L.V. Woodcock, Percolation transitions in the hard-sphere fluid, AIChE Journal 58 1610-1618 (2011).
[12] L.V. Woodcock, Thermodynamic description of liquid-state limits, J. Phys. Chem. (B) 116 3734 (2011).
[13] Wm.G. Hoover, J.C. Poirier, Determination of virial coeffi- cients from potential of mean force, J. Chem. Phys. 37:1041–1042 (1962).
[14] B. Widom , Some topics in the theory of fluids, J. Chem. Phys. 39 2808–2812 (1963).
[15] R. Wheatley, Calculation of high-order virial coefficients with applications to hard and soft spheres, Phys. Rev. Lett. 110 200601 (2013).
[16] Wm.G. Hoover, N.E. Hoover, K. Hanson, Exact hard-disk free volumes, J. Chem. Phys. 70 1837–1844 (1979).
[17] D.M. Heyes, M. Cass, A.C. Branka, Percolation threshold of hard-sphere fluids in between the soft-core and hard-core limits, Mol. Phys. 104 3137–3146 (2000).
[18] S.Quintanilla, R.M. Torquato, J. Ziff, Efficient measurement of the percolation threshold for fully penetrable discs, Phys. Rev. (A) Math. Gen.; 23 399- 407 (2000).
[19] C.D. Lorentz, R.M. Ziff, Precise determination of the critical percolation threshold for the three-dimensional ‘Swiss cheese’ model using a growth algorithm, J. Chem. Phys, 114 3659-3661 (2001).
[20] B. Widom, J.S. Rowlinson, New model for the study of liquid–vapor phase transitions, J. Chem. Phys. 52 1670 (1970).
[21] L.V. Woodcock, Non-additive hard-sphere reference model for ionic liquids, Ind. Eng. Chem. Res. 50, 227–233 (2011).
[22] E. de Miguel, N.G. Almarza, G. Jackson, Surface tension of the Widom-Rowlinson model, J. Chem. Phys., 127, 034707 (2007).
[23] L.V. Woodcock, Observations of a thermodynamic liquid–gas critical coexistence line and supercritical phase bounds from percolation loci, Fluid Phase Equilibria, 351 25-33 (2013).
[24] L.V. Woodcock, Thermodynamics of Criticality: Percola- tion Loci, Mesophases and a Critical Dividing Line in Binary-Liquid and Liquid-Gas Equilibria Journal of Mod- ern Physics, 7, 760-773 (2016).
[25] D.M. Heyes, L.V. Woodcock, Critical and supercritical properties of Lennard-Jones fluids, Fluid Phase Equilibria (2013).
[26] D.M. Heyes, The Lennard-Jones fluid in the liquid-vapour critical region, CMST 21 169-179 (2015).
[27] L.V. Woodcock, Thermodynamics of gas-liquid criticality: rigidity symmetry on Gibbs density surface Int. J. Thermo- physics, 37 24 -33 (2016).
[28] R.Gilgen, R. Kleinrahm, W. Wagner, Measurement and correlation of the (pressure, density, temperature) relation of ar- gon. I. The homogeneous gas and liquid regions in the tem- perature range from 90 to 300K at pressures up to 12MPa, J. Chem. Thermodynamics 26 383-398 (1994).
[29] R. Gilgen, R. Kleinrahm, W. Wagner, Measurement and cor- relation 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. Thermo. 26 399-413 (1994).
[30] B.C. Eu, Exact analytic second virial coefficient for the Lennard-Jones Fluid, arXiv (Phys. Chem.) 0909 3326 (2009).
[31] D.M. Heyes, S. Pieprzyk, G. Rickayzen, A.C. Branka, The second virial coefficient and critical point behaviour of the Mie potential, J. Chem. Phys. 145 084505 (2016).
[32] J.M.H. Levelt-Sengers, Liquidons and Gasons; Controversies about the Continuity of States, Physica, A98, 363-402 (1979).
[33] J.S. Rowlinson Critical states of fluids and fluid mixtures: a review of the experimental position, 9-12 Proc. Conf. on Phenomena in the Neighborhood of Critical Points, (National Bureau of Standards: Washington DC, USA 1965).
