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Melting Line of the Lennard-Jones System, Infinite Size, and Full Potential

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Melting Line of the Lennard-Jones System, Infinite Size, and Full Potential THE JOURNAL OF CHEMICAL PHYSICS 127, 104504 ͑2007͒ Melting line of the Lennard-Jones system, infinite size, and full potential ͒ ͒ Ethan A. Mastnya and Juan J. de Pablob Chemical and Biological Engineering Department, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, Wisconsin 53706–1691, USA ͑Received 9 February 2007; accepted 5 June 2007; published online 13 September 2007͒ Literature estimates of the melting curve of the Lennard-Jones system vary by as much as 10%. The origin of such discrepancies remains unclear. We present precise values for the Lennard-Jones melting temperature, and we examine possible sources of systematic errors in the prediction of melting points, including finite-size and interaction-cutoff effects. A hypothetical thermodynamic integration path is used to find the relative free energies of the solid and liquid phases, for various system sizes, at constant cutoff radius. The solid-liquid relative free energy and melting temperature scale linearly as the inverse of the number of particles, and it is shown that finite-size effects can account for deviations in the melting temperature ͑from the infinite-size limit͒ of up to 5%. An extended-ensemble density-of-states method is used to determine free energy changes for each phase as a continuous function of the cutoff radius. The resulting melting temperature predictions exhibit an oscillatory behavior as the cutoff radius is increased. Deviations in the melting temperature ͑from the full potential limit͒ arising from a finite cutoff radius are shown to be of comparable magnitude as those resulting from finite-size effects. This method is used to identify melting temperatures at five different pressures, for the infinite-size and full potential Lennard-Jones system. We use our simulation results as references to connect the Lennard-Jones solid equation of state of van der Hoef with the Lennard-Jones fluid equation of state of Johnson. Once the references are applied the two equations of state are used to identify a melting curve. An empirical equation that fits this melting curve is provided. We also report a reduced triple point temperature ͓ ͔ Ttr=0.694. © 2007 American Institute of Physics. DOI: 10.1063/1.2753149 I. INTRODUCTION proaches unity. Thermodynamic properties are subsequently corrected for the truncated portion of the interaction potential The Lennard-Jones system is arguably one of the sim- by assuming a value of unity for the radial distribution func- plest models capable of reproducing the complete thermody- tion at distances greater than rc. For crystalline solids, how- namic behavior of classical fluids. Over the last two decades, ever, the radial distribution does not approach unity in an a consensus has emerged over the precise coordinates of its asymptotic manner for short to intermediate length scales. It vapor-liquid coexistence ͑or binodal͒ curve and its critical is therefore of interest to determine the nature and magnitude point. In contrast, available estimates of the melting curve of the systematic errors that arise in simulations of solid- vary considerably. Simulations of solid-liquid equilibria are liquid equilibria as a result of finite-size and cutoff-radius considerably more challenging than those of vapor-liquid effects. equilibria. It is unclear whether discrepancies between differ- Theoretical values for the melting curve of the Lennard- ent results are due to methodological challenges or to finite- Jones system are not available. A definitive standard for size or finite-cutoff radius effects. The aim of this article is to comparison of the results of molecular simulations does not evaluate the magnitude of such effects and to present results exist. One must therefore rely on somewhat subjective as- for the melting curve of Lennard-Jonesium in the thermody- sessments of existing calculations to determine the accuracy namic limit ͑N→ϱ, where N is the number of particles in the of a particular value of the melting point. Published esti- system͒. mates of melting points of the Lennard-Jones system vary by Molecular simulations of simple or complex fluids are as much as 10%. The work presented in this article is moti- generally conducted on systems consisting of a few hundred vated by a desire to establish a precise standard for the melt- or thousand particles. Such simulations typically adopt a ing temperature of Lennard-Jonesium. We examine in con- truncated interaction potential, whose value is assumed to siderable detail some of the more common sources of vanish beyond a specified cutoff radius. For the Lennard- systematic errors in the prediction of melting points, namely, ␴ Jones system, that cutoff radius is often set to rc =2.5 , finite-size and cutoff radius effects, in the hope that the where ␴ is the size or length scale associated with the analysis provided in this work will permit more objective ␴ Lennard-Jones potential energy function. At rc =2.5 , the ra- consideration of melting point estimates obtained by means dial distribution function g͑r͒ of a Lennard-Jones liquid ap- of molecular simulation. The melting points presented in this work are generated ͒ 1,2 a Electronic mail: [email protected] by thermodynamic integration along a hypothetical path, ͒ b Electronic mail: [email protected] that integration yields the relative free energies of the solid 0021-9606/2007/127͑10͒/104504/8/$23.00127, 104504-1 © 2007 American Institute of Physics Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 128.112.35.141 On: Fri, 12 Feb 2016 14:58:30 104504-2 E. A. Mastny and J. J. de Pablo J. Chem. Phys. 127, 104504 ͑2007͒ and liquid phases, of various system sizes, at constant cutoff simulating 108-, 256-, and 500-particle systems. He also radius. The solid-liquid relative free energy and melting tem- considered two cutoff schemes, namely, a standard long- perature are shown to scale as the inverse size of the system, range correction beyond half the box length and a perfect- 1/N. It is also shown that finite-size effects from this inte- lattice correction for the solid phase beyond half the box gration path can account for melting temperature errors of up length. He noted that the coexistence pressures did not ex- to 5%. An extended-ensemble density-of-states method is hibit any apparent scaling relationship for either type of cor- then used to determine the free energy change in each phase rection, and he was therefore unable to extrapolate his results as a continuous function of the cutoff radius. Our results to infinite system size. indicate that the melting temperature exhibits oscillations In a previous study,10 we used a density-of-states Monte whose magnitude decreases as the cutoff radius is increased. Carlo sampling method with order parameters of potential Melting temperature prediction errors due to a finite cutoff energy and volume. In this work, neither finite-size nor cut- radius can also amount to 5% of the value corresponding to off effects were investigated ͑500 particles and 2.5␴ were the full Lennard-Jones potential. used͒. The results of this work are also sensitive to the We use the relative free energies determined from the method used for matching the free energy functions in the thermodynamic integrations and corrected for large cutoff energy and volume space between the equilibrium solid and radius as references to connect the Lennard-Jones solid equa- liquid regions. tion of state of van der Hoef3 with the Lennard-Jones fluid More recently, McNeil-Watson and Wilding11 also used equation of state of Johnson.4 Upon applying the references the phase-switch Monte Carlo method described by Err- the two equations of state were used to identify a melting ington to predict melting points. They then reweighted their curve. data to obtain a portion of the melting line. Their results were quantitatively similar to those of Errington. They explained that the reason they did not see a 1/N scaling of the melting II. LITERATURE REVIEW temperature was due to the fact that the choice of a “fluctu- In 1969 Hansen and Verlet reported the first melting ating” cutoff at one-half the box length introduces a coupling point estimates of the Lennard-Jones system.5 They evalu- between the cutoff and system size. ated the free energy of the solid and liquid phases indepen- The results of Errington’s and McNeil-Watson and Wild- dently, and calculated coexistence properties at four different ing do not scale to infinite-size systems, and it is unclear temperatures. The thermodynamic integration path used to whether a cutoff radius that changed with system size had a calculate the solid free energy, however, includes a phase significant effect. The 1/N scaling analysis presented by transition. these authors compared different systems because the cutoff Agrawal and Kofke6,7 carried out a subsequent study of was different. One would not expect different systems to the Lennard-Jones melting curve. They used a Gibbs-Duhem have any predictable scaling behavior. van der Hoef published an equation of state for the integration to track the melting point as the potential of in- 3 teraction was first changed from purely repulsive ͑hard Lennard-Jones solid. In his study almost 900 NVT simula- sphere͒ to the repulsive part of the Lennard-Jones potential tions in the temperature range of 0.1–2.0 and density range energy function, and then to the complete Lennard-Jones en- of 0.94–1.2 were performed. He used 2048 particles and a ␴ ergy function. The starting point for these calculations was a cutoff radius of 6.0 . The measured internal energies and single melting point corresponding to a system of hard pressures from these simulations were fitted to an empirical spheres.
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