What is Liquid? [in two dimensions]
Travis Karl P. 1, Hoover William G. 2*, Hoover Carol G. 2, Hass Amanda B. 3
1 Department of Materials Science & Engineering
The University of Sheffield
Sheffield S1 3JD, United Kingdom2 Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833, USA
*E-mail: hooverwilliam@yahoo.com3 Department of Applied Maths
University of Leeds
Leeds LS2 9JT, United Kingdom
Received:
Received: 12 March 2021; accepted: 24 March 2021; published online: 29 March 2021
DOI: 10.12921/cmst.2021.0000009
Abstract:
We consider the practicalities of defining, simulating, and characterizing “Liquids” from a pedagogical standpoint based on atomistic computer simulations. For simplicity and clarity we study two-dimensional systems throughout. In addition to the infinite-ranged Lennard-Jones 12/6 potential we consider two shorter-ranged families of pair potentials. At zero pressure one of them includes just nearest neighbors. The other longer-ranged family includes twelve additional neighbors. We find that these further neighbors can help stabilize the liquid phase. What about liquids? To implement Wikipedia’s definition of liquids as conforming to their container we begin by formulating and imposing smooth-container boundary conditions. To encourage conformation further we add a vertical gravitational field. Gravity helps stabilize the relatively vague liquid-gas interface. Gravity reduces the messiness associated with the curiously-named “spinodal” (tensile) portion of the phase diagram. Our simulations are mainly isothermal. We control the kinetic temperature with Nosé-Hoover thermostating, extracting or injecting heat so as to impose a mean kinetic temperature over time. Our simulations stabilizing density gradients and the temperature provide critical-point estimates fully consistent with previous efforts from free energy and Gibbs ensemble simulations. This agreement validates our approach.
Key words:
liquids, molecular dynamics, phase equilibria, spinodals, statistical physics, tension
References:
[1] Wm.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics. Part III. A Memoir for Douglas James Henderson [1934–2020], Computational Methods in Science and Technology 26, 111–120 (2020).
[2] J.A. Barker, D. Henderson, What is ‘Liquid’? Understanding the States of Matter, Reviews of Modern Physics 48, 587–671 (1976).
[3] Lj. Milanovic´, H.A. Posch, Wm.G. Hoover, What is ‘Liquid’? Understanding the States of Matter, Molecular Physics 95, 281–287 (1998).
[4] Wm.G. Hoover, C.G. Hoover, What is Liquid? Lyapunov Instability Reveals Symmetry-Breaking Irreversibility Hidden with Hamilton’s Manybody Equations of Motion, Condensed Matter Physics 18, 13003:1–13 (2015).
[5] J.A. Barker, D. Henderson, F.F. Abraham, Phase Diagram of the Two-Dimensional Lennard-Jones System; Evidence for First-Order Transitions, Physica 106A, 226–238 (1981).
[6] J.B. Hannay, The Limit of the Liquid State of Matter, Nature 26, 370 (1882).
[7] J.S. Rowlinson, John Adair Barker 1925–1995, Historical Records of Australian Science 11, 179–190 (1996).
[8] W.W. Wood, J.D. Jacobson, Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres, Journal of Chemical Physics 27, 1207–1208 (1957).
[9] B.J. Alder, T.E. Wainwright, Phase Transition for a Hard Sphere System, Journal of Chemical Physics 27, 1208–1209 (1957).
[10] J.K. Percus, G.J. Yevick, Analysis of Classical Statistical Mechanics by Means of Collective Coordinates, Physical Review 110, 1–13 (1958).
[11] M.S. Wertheim, Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres, Physical Review Letters 10, 321–323 (1963).
[12] N. Shamsundar, J.H. Lienhard, Equations of State and Spinodal Lines, Nuclear Engineering and Design 141, 269–287 (1993).
[13] J. Wedekind, G. Chkonia, J. Wölk, R. Strey, D. Reguera, Crossover from Nucleation to Spinodal Decomposition in a Condensing Vapour, Journal of Chemical Physics 131, 114506 (2009).
[14] S.W. Koch, R.C. Desai, F.F. Abraham, Dynamics of Phase Separation in Two-Dimensional Fluids: Spinodal Decomposition, Physical Review A 27, 2152–2167 (1983).
[15] F. Trudu, D. Donadio, M. Parrinello, Freezing of a LennardJones Fluid: From Nucleation to Spinodal Regime, Physical Review Letters 97, 105701 (2006).
[16] A.J.C. Ladd, L.V. Woodcock, Triple-Point Coexistence Proprties of the Lennard-Jones System, Chemical Physics Letters 51, 155–159 (1977).
[17] D. Badde, J. Danziger, R. Hook, J. Walsh, Leon B. Lucy, 1938–2018, The Messenger 173, 58–59 (2018).
[18] J.A. Blink, Wm.G. Hoover, Fragmentation of Suddenly Heated Liquids Physical Review A 32, 1027–1035 (1985).
[19] B.L. Holian, D.E. Grady, Fragmentation by Molecular Dynamics: The Microscopic ‘Big Bang’, Physical Review Letters 60, 1355–1358 (1988).
[20] J.E. Mayer, M.G. Mayer, Statistical Mechanics, First Edition, John Wiley & Sons, New York (1940).
[21] I. Khmelinskii, L.V. Woodcock, Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results, Entropy 22, 437 (2020).
[22] R.J. Wheatley, Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres, Physical Review Letters 110, 200601 (2013).
[23] B.L. Holian, Atomistic Computer Simulations of Shockwaves, Shockwaves 5, 149–157 (1995).
[24] J. Schnack, H. Feldmeier, The Nuclear Liquid-Gas Phase Transition Within Fermionic Molecular Dynamics, Physics Letters B 409, 6–10 (1977).
We consider the practicalities of defining, simulating, and characterizing “Liquids” from a pedagogical standpoint based on atomistic computer simulations. For simplicity and clarity we study two-dimensional systems throughout. In addition to the infinite-ranged Lennard-Jones 12/6 potential we consider two shorter-ranged families of pair potentials. At zero pressure one of them includes just nearest neighbors. The other longer-ranged family includes twelve additional neighbors. We find that these further neighbors can help stabilize the liquid phase. What about liquids? To implement Wikipedia’s definition of liquids as conforming to their container we begin by formulating and imposing smooth-container boundary conditions. To encourage conformation further we add a vertical gravitational field. Gravity helps stabilize the relatively vague liquid-gas interface. Gravity reduces the messiness associated with the curiously-named “spinodal” (tensile) portion of the phase diagram. Our simulations are mainly isothermal. We control the kinetic temperature with Nosé-Hoover thermostating, extracting or injecting heat so as to impose a mean kinetic temperature over time. Our simulations stabilizing density gradients and the temperature provide critical-point estimates fully consistent with previous efforts from free energy and Gibbs ensemble simulations. This agreement validates our approach.
Key words:
liquids, molecular dynamics, phase equilibria, spinodals, statistical physics, tension
References:
[1] Wm.G. Hoover, C.G. Hoover, Time-Symmetry Breaking in Hamiltonian Mechanics. Part III. A Memoir for Douglas James Henderson [1934–2020], Computational Methods in Science and Technology 26, 111–120 (2020).
[2] J.A. Barker, D. Henderson, What is ‘Liquid’? Understanding the States of Matter, Reviews of Modern Physics 48, 587–671 (1976).
[3] Lj. Milanovic´, H.A. Posch, Wm.G. Hoover, What is ‘Liquid’? Understanding the States of Matter, Molecular Physics 95, 281–287 (1998).
[4] Wm.G. Hoover, C.G. Hoover, What is Liquid? Lyapunov Instability Reveals Symmetry-Breaking Irreversibility Hidden with Hamilton’s Manybody Equations of Motion, Condensed Matter Physics 18, 13003:1–13 (2015).
[5] J.A. Barker, D. Henderson, F.F. Abraham, Phase Diagram of the Two-Dimensional Lennard-Jones System; Evidence for First-Order Transitions, Physica 106A, 226–238 (1981).
[6] J.B. Hannay, The Limit of the Liquid State of Matter, Nature 26, 370 (1882).
[7] J.S. Rowlinson, John Adair Barker 1925–1995, Historical Records of Australian Science 11, 179–190 (1996).
[8] W.W. Wood, J.D. Jacobson, Preliminary Results from a Recalculation of the Monte Carlo Equation of State of Hard Spheres, Journal of Chemical Physics 27, 1207–1208 (1957).
[9] B.J. Alder, T.E. Wainwright, Phase Transition for a Hard Sphere System, Journal of Chemical Physics 27, 1208–1209 (1957).
[10] J.K. Percus, G.J. Yevick, Analysis of Classical Statistical Mechanics by Means of Collective Coordinates, Physical Review 110, 1–13 (1958).
[11] M.S. Wertheim, Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres, Physical Review Letters 10, 321–323 (1963).
[12] N. Shamsundar, J.H. Lienhard, Equations of State and Spinodal Lines, Nuclear Engineering and Design 141, 269–287 (1993).
[13] J. Wedekind, G. Chkonia, J. Wölk, R. Strey, D. Reguera, Crossover from Nucleation to Spinodal Decomposition in a Condensing Vapour, Journal of Chemical Physics 131, 114506 (2009).
[14] S.W. Koch, R.C. Desai, F.F. Abraham, Dynamics of Phase Separation in Two-Dimensional Fluids: Spinodal Decomposition, Physical Review A 27, 2152–2167 (1983).
[15] F. Trudu, D. Donadio, M. Parrinello, Freezing of a LennardJones Fluid: From Nucleation to Spinodal Regime, Physical Review Letters 97, 105701 (2006).
[16] A.J.C. Ladd, L.V. Woodcock, Triple-Point Coexistence Proprties of the Lennard-Jones System, Chemical Physics Letters 51, 155–159 (1977).
[17] D. Badde, J. Danziger, R. Hook, J. Walsh, Leon B. Lucy, 1938–2018, The Messenger 173, 58–59 (2018).
[18] J.A. Blink, Wm.G. Hoover, Fragmentation of Suddenly Heated Liquids Physical Review A 32, 1027–1035 (1985).
[19] B.L. Holian, D.E. Grady, Fragmentation by Molecular Dynamics: The Microscopic ‘Big Bang’, Physical Review Letters 60, 1355–1358 (1988).
[20] J.E. Mayer, M.G. Mayer, Statistical Mechanics, First Edition, John Wiley & Sons, New York (1940).
[21] I. Khmelinskii, L.V. Woodcock, Supercritical Fluid Gaseous and Liquid States: A Review of Experimental Results, Entropy 22, 437 (2020).
[22] R.J. Wheatley, Calculation of High-Order Virial Coefficients with Applications to Hard and Soft Spheres, Physical Review Letters 110, 200601 (2013).
[23] B.L. Holian, Atomistic Computer Simulations of Shockwaves, Shockwaves 5, 149–157 (1995).
[24] J. Schnack, H. Feldmeier, The Nuclear Liquid-Gas Phase Transition Within Fermionic Molecular Dynamics, Physics Letters B 409, 6–10 (1977).
