Abstract
Analysis of the free energy is required to understand and predict the equilibrium behavior of thermodynamic systems, which is to say, systems in which temperature has some influence on the equilibrium condition. In practice, all processes in the world around us proceed at a finite temperature, so any application of molecular simulation that aims to evaluate the equilibrium behavior must consider the free energy. There are many such phenomena to which simulation has been applied for this purpose. Examples include chemical-reaction equilibrium, protein-ligand affinity, solubility, melting and boiling. Some of these are examples of phase equilibria, which are an especially important and practical class of thermodynamic phenomena. Phase transformations are characterized by some macroscopically observable change signifying a wholesale rearrangement or restructuring occurring at the molecular level. Typically this change occurs at a specific value of some thermodynamic variable such as the temperature or pressure. At the exact point where the transition occurs, both phases are equally stable — have equal free energy — and we find a condition of phase equilibrium or coexistence [1].
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
K. Denbigh, Principles of Chemical Equilibrium, Cambridge: Cambridge University, 1971.
D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2002.
F. Wang and D.P. Landau, “Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram,” Phys. Rev. E, 64, 056101–1–056101–16, 2001a.
F. Wang and D.P. Landau, “Efficient, multiple-range random walk algorithm to calculate the density of states,” Phys. Rev. Lett., 86, 2050–2053, 2001b.
A. Laio and M. Parrinello, “Escaping free-energy minima,” Proc. Nat. Acad. Sci., 99, 12562–12566, 2002.
M. Fitzgerald, R.R. Picard, and R.N. Silver, “Canonical transition probabilities for adaptive Metropolis simulation,” Europhys. Lett., 46, 282–287, 1999.
J.-S. Wang, T.K. Tay, and R.H. Swendsen, “Transition matrix Monte Carlo reweighting and dynamics,” Phys. Rev. Lett., 82, 476–479, 1999.
M. Fitzgerald, R.R. Picard, and R.N. Silver, “Monte Carlo transition dynamics and variance reduction,” J. Stat. Phys., 98, 321, 2000.
J. R. Errington, “Direct calculation of liquid-vapor phase equilibria from transition matrix Monte Carlo simulation,” J. Chem. Phys., 118, 9915–9925, 2003a.
J. R. Errington, “Evaluating surface tension using grand-canonical transition-matrix Monte Carlo simulation and finite-size scaling,” Phys. Rev. E, 67, 012102–1–012102–4, 2003b.
M.S. Shell, P.G. Debenedetti, and A.Z. Panagiotopoulos, “An improved Monte Carlo method for direct calculation of the density of states,” J. Chem. Phys., 119, 9406–9411, 2003.
C. Jarzynski, “Equilibrium free-energy differences from nonequilibrium measurements: a master-equation approach,” Phys. Rev. E, 56, 5018–5035, 1997a.
C. Jarzynski, “Nonequilibrium equality for free energy difference,” Phys. Rev. Lett., 78, 2690–2693, 1997b.
G.E. Crooks, “Nonequilibrium measurements of free energy differences for microscopically reversible Markovian systems,” J. Stat. Phys., 90, 1481–1487, 1998.
G.E. Crooks, “Entropy production fluctuation theorem and the nonequilibrium work relation for free energy differences,” Phys. Rev. E, 60, 2721–2726, 1999.
M. Watanabe and W.P. Reinhardt, “Direct dynamical calculation of entropy and free energy by adiabatic switching,” Phys. Rev. Lett., 65, 3301–3304, 1990.
N.D. Lu and D.A. Kofke, “Accuracy of free-energy perturbation calculations in molecular simulation I. Modeling,” J. Chem. Phys., 114, 7303–7311, 2001a.
N.D. Lu and D.A. Kofke, “Accuracy of free-energy perturbation calculations in molecular simulation II. Heuristics,” J. Chem. Phys., 115, 6866–6875, 2001b.
J.P. Hansen and I.R. McDonald, Theory of Simple Liquids, Academic Press, London, 1986.
D.A. Kofke, “Getting the most from molecular simulation,” Mol. Phys., 102, 405–420, 2004.
A.D. Bruce, N.B. Wilding, and G.J. Ackland, “Free energy of crystalline solids: a lattice-switch Monte Carlo method,” Phys. Rev. Lett., 79, 3002–3005, 1997.
A.D. Bruce, A.N. Jackson, G.J. Ackland, and N.B. Wilding, “Lattice-switch Monte Carlo method,” Phys. Rev. E, 61, 906–919, 2000.
C. Jarzynski, “Targeted free energy perturbation,” Phys. Rev. E, 65, 046122, 1–5, 2002.
J.P. Valleau and D.N. Card, “Monte Carlo estimation of the free energy by multistage sampling,” J. Chem. Phys., 57, 5457–5462, 1972.
D.A. Kofke and P.T. Cummings, “Quantitative comparison and optimization of methods for evaluating the chemical potential by molecular simulation,” Mol. Phys., 92, 973–996, 1997.
R.J. Radmer and P.A. Kollman, “Free energy calculation methods: a theoretical and empirical comparison of numerical errors and a new method for qualitative estimates of free energy changes,” J. Comp. Chem., 18, 902–919, 1997.
G.M. Torrie and J.P. Valleau, “Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling,” J. Comp. Phys., 23, 187–199, 1977.
D.A. Kofke and P.T. Cummings, “Precision and accuracy of staged free-energy perturbation methods for computing the chemical potential by molecular simulation,” Fluid Phase Equil., 150, 41–49, 1998.
N.D. Lu, J.K. Singh, and D.A. Kofke, “Appropriate methods to combine forward and reverse free energy perturbation averages,” J. Chem. Phys., 118, 2977–2984, 2003.
J.J. de Pablo, Q.L. Yan, and F.A. Escobedo, “Simulation of phase transitions in fluids,” Ann. Rev. Phys. Chem., 50, 377–411, 1999.
A.D. Bruce and N.B. Wilding, “Computational strategies for mapping equilibrium phase diagrams,” Adv. Chem. Phys., 127, 1–64, 2003.
Z.L. Zhang, M.A. Horsch, M.H. Lamm, and S.C. Glotzer, “Tethered nano building blocks: Towards a conceptual framework for nanoparticle self-assembly,” Nano Lett., 3, 1341–1346, 2003.
R.D. Groot and P.B. Warren, “Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation,” J. Chem. Phys., 107, 4423–4435, 1997.
P.A. Monson and D.A. Kofke, “Solid-fluid equilibrium: insights from simple molecular models,” Adv. Chem. Phys., 115, 113–179, 2000.
M.P. Allen, G.T. Evans, D. Frenkel, and B.M. Mulder, “Hard convex body fluids,” Adv. Chem. Phys., 86, 1–166, 1993.
D.A. Kofke, “Semigrand canonical Monte Carlo simulation; Integration along coexistence lines,” Adv. Chem. Phys., 105, 405–441, 1999.
A.Z. Panagiotopoulos, “Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble,” Mol. Phys., 61, 813–826, 1987.
A.Z. Panagiotopoulos, “Direct determination of fluid phase equilibria by simulation in the Gibbs ensemble: a review,” Mol. Sim., 9, 1–23, 1992.
P. Tilwani, “Direct simulation of phase coexistence in solids using the Gibbs ensemble: Configuration annealing Monte Carlo,” M.S. Thesis, Colorado School of Mines, Golden, Colorado, 1999.
D.A. Kofke, “Direct evaluation of phase coexistence by molecular simulation through integration along the saturation line,” J. Chem. Phys., 98, 4149–4162, 1993.
J. Henning, and D.A. Kofke, “Thermodynamic integration along coexistence lines,” In: P.B. Balbuena and J. Seminario (eds.), Molecular Dynamics, Amsterdam: Elsevier, 1999.
S.P. Pandit and D.A. Kofke, “Evaluation of a locus of azeotropes by molecular simulation,” AIChE J., 45, 2237–2244, 1999.
F.A. Escobedo, “Novel pseudoensembles for simulation of multicomponent phase equilibria,” J. Chem. Phys., 108, 8761–8772, 1998.
F.A. Escobedo, “Tracing coexistence lines in multicomponent fluid mixtures by molecular simulation,” J. Chem. Phys., 110, 11999–12010, 1999.
F.A. Escobedo, “Molecular and macroscopic modeling of phase separation,” AIChE J., 46, 2086–2096, 2000a.
F. A. Escobedo, “Simulation and extrapolation of coexistence properties with singlephase and two-phase ensembles,” J. Chem. Phys., 113, 8444–8456, 2000b.
F.A. Escobedo and Z. Chen, “Simulation of isoenthalps and Joule-Thomson inversion curves of pure fluids and mixtures,” Mol. Sim., 26, 395–416, 2001.
Z.W. Salsburg, J.D. Jacobson, W. Fickett, and W.W. Wood, “Application of the Monte Carlo method to the lattice-gas model. I.Two-dimensional triangular lattice,” J. Chem. Phys., 30, 65–72, 1959.
I.R. McDonald and K. Singer, “Calculation of thermodynamic properties of liquid argon from Lennard-Jones parameters by a Monte Carlo method,” Discuss. Faraday Soc., 43, 40–49, 1967.
P.G. Bolhuis, D. Chandler, C. Dellago, and P.L. Geissler, “Transition path sampling: throwing ropes over rough mountain passes, in the dark,” Ann. Rev. Phys. Chem., 53, 291–318, 2002.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer
About this chapter
Cite this chapter
Kofke, D.A., Frenkel, D. (2005). Perspective: Free Energies and Phase Equilibria. In: Yip, S. (eds) Handbook of Materials Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-3286-8_35
Download citation
DOI: https://doi.org/10.1007/978-1-4020-3286-8_35
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3287-5
Online ISBN: 978-1-4020-3286-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)