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Ustinov EA. Thermodynamics and simulation of 3D crystals and phase transitions under external fields. J Chem Phys 2024; 161:064113. [PMID: 39140444 DOI: 10.1063/5.0216828] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2024] [Accepted: 07/25/2024] [Indexed: 08/15/2024] Open
Abstract
A field-supported multiphase kinetic Monte Carlo method previously applied to self-assembled trimesic acid molecular layers [Ustinov et al., Phys. Chem. Chem. Phys. 24, 26111 (2022)] was generalized to three-dimensional gas-liquid and gas-solid systems. This method allows us to calculate the thermodynamic potentials of the liquid and solid phases and then determine the parameters of the liquid-solid phase transition. In this study, the requirement that the gas phase be ideal was introduced as an additional condition. It was shown that in a two-phase system, the sum of the analytical expression for the chemical potential of an ideal gas and the external potential imposed on the gas phase exactly equals the chemical potential of the equilibrium crystal or liquid phase. For example, the coexistence of crystalline/liquid krypton and ideal gas has been considered. A comparison with previously published data has shown that the proposed approach provides the most accurate results for determining the parameters of phase transitions and fully satisfies the Gibbs-Duhem equation. This method does not impose any restrictions on the complexity or hardness of dense phases.
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Affiliation(s)
- E A Ustinov
- Ioffe Institute, 26 Polytechnicheskaya, St. Petersburg 194021, Russia
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2
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Ströker P, Hellmann R, Meier K. Thermodynamic properties of krypton from Monte Carlo simulations using ab initio potentials. J Chem Phys 2022; 157:114504. [PMID: 36137797 DOI: 10.1063/5.0107851] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Ten different thermodynamic properties of the noble gas krypton were calculated by Monte Carlo simulations in the isothermal-isobaric ensemble using a highly accurate ab initio pair potential, Feynman-Hibbs corrections for quantum effects, and an extended Axilrod-Teller-Muto potential to account for nonadditive three-body interactions. Fourteen state points at a liquid and a supercritical isotherm were simulated. To obtain results representative for macroscopic systems, simulations with several particle numbers were carried out and extrapolated to the thermodynamic limit. Our results agree well with experimental data from the literature, an accurate equation of state for krypton, and a recent virial equation of state (VEOS) for krypton in the region where the VEOS has converged. These results demonstrate that very good agreement between simulation and experiment can only be achieved if nonadditive three-body interactions and quantum effects are taken into account.
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Affiliation(s)
- Philipp Ströker
- Institut für Thermodynamik, Helmut-Schmidt-Universität/Universität der Bundeswehr Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany
| | - Robert Hellmann
- Institut für Thermodynamik, Helmut-Schmidt-Universität/Universität der Bundeswehr Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany
| | - Karsten Meier
- Institut für Thermodynamik, Helmut-Schmidt-Universität/Universität der Bundeswehr Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany
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3
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Faraji F, Neek-Amal M, Neyts EC, Peeters FM. Indentation of graphene nano-bubbles. NANOSCALE 2022; 14:5876-5883. [PMID: 35363231 DOI: 10.1039/d2nr01207c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
Molecular dynamics simulations are used to investigate the effect of an AFM tip when indenting graphene nano bubbles filled by a noble gas (i.e. He, Ne and Ar) up to the breaking point. The failure points resemble those of viral shells as described by the Föppl-von Kármán (FvK) dimensionless number defined in the context of elasticity theory of thin shells. At room temperature, He gas inside the bubbles is found to be in the liquid state while Ne and Ar atoms are in the solid state although the pressure inside the nano bubble is below the melting pressure of the bulk. The trapped gases are under higher hydrostatic pressure at low temperatures than at room temperature.
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Affiliation(s)
- Fahim Faraji
- PLASMANT, Department of Chemistry, University of Antwerp, Universiteitsplein 1, 2610 Antwerp, Belgium.
- Condensed Matter Theory, Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium.
- Center of Excellence NANOlab, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium
| | - Mehdi Neek-Amal
- Condensed Matter Theory, Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium.
- Department of Physics, Shahid Rajaee Teacher Training University, 16875-163 Tehran, Iran
| | - Erik C Neyts
- PLASMANT, Department of Chemistry, University of Antwerp, Universiteitsplein 1, 2610 Antwerp, Belgium.
- Center of Excellence NANOlab, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium
| | - François M Peeters
- Condensed Matter Theory, Department of Physics, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium.
- Center of Excellence NANOlab, University of Antwerp, Groenenborgerlaan 171, 2020 Antwerp, Belgium
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4
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Tow GM, Larentzos JP, Sellers MS, Lísal M, Brennan JK. Predicting Melt Curves of Energetic Materials Using Molecular Models. PROPELLANTS EXPLOSIVES PYROTECHNICS 2022. [DOI: 10.1002/prep.202100363] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Affiliation(s)
- Garrett M. Tow
- U.S. Army DEVCOM Army Research Laboratory Aberdeen Proving Ground Maryland 21005 USA
| | - James P. Larentzos
- U.S. Army DEVCOM Army Research Laboratory Aberdeen Proving Ground Maryland 21005 USA
| | | | - Martin Lísal
- Department of Molecular and Mesoscopic Modelling The Czech Academy of Sciences Institute of Chemical Process Fundamentals Prague 165 01 Czech Republic
- Department of Physics Faculty of Science Jan Evangelista Purkyně University in Ústí nad Labem Ústí n. Lab. 400 96 Czech Republic
| | - John K. Brennan
- U.S. Army DEVCOM Army Research Laboratory Aberdeen Proving Ground Maryland 21005 USA
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Haber RT, Browning AR, Graves BR, Davis WP, Wiggins JS. A Molecular Dynamics Study of Monomer Melt Properties of Cyanate Ester Monomer Melt Properties. Polymers (Basel) 2022; 14:polym14061219. [PMID: 35335549 PMCID: PMC8951156 DOI: 10.3390/polym14061219] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/10/2022] [Revised: 03/07/2022] [Accepted: 03/11/2022] [Indexed: 01/27/2023] Open
Abstract
The objective of this work was to computationally predict the melting temperature and melt properties of thermosetting monomers used in aerospace applications. In this study, we applied an existing voids method by Solca. to examine four cyanate ester monomers with a wide range of melting temperatures. Voids were introduced into some simulations by removal of molecules from lattice positions to lower the free-energy barrier to melting to directly simulate the transition from a stable crystal to amorphous solid and capture the melting temperature. We validated model predictions by comparing melting temperature against previously reported literature values. Additionally, the torsion and orientational order parameters were used to examine the monomers’ freedom of motion to investigate structure–property relationships. Ultimately, the voids method provided reasonable estimates of melting temperature while the torsion and order parameter analysis provided insight into sources of the differing melt properties between the thermosetting monomers. As a whole, the results shed light on how freedom of molecular motions in the monomer melt state may affect melting temperature and can be utilized to inspire the development of thermosetting monomers with optimal monomer melt properties for demanding applications.
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Affiliation(s)
- Rebecca T. Haber
- School of Polymer Science and Engineering, University of Southern Mississippi, 118 College Dr, #5050, Hattiesburg, MS 39406, USA; (R.T.H.); (B.R.G.)
| | | | - Bayleigh R. Graves
- School of Polymer Science and Engineering, University of Southern Mississippi, 118 College Dr, #5050, Hattiesburg, MS 39406, USA; (R.T.H.); (B.R.G.)
| | - William P. Davis
- Department of Mathematics, Western Washington University, 516 High St., Bellingham, WA 98225, USA;
| | - Jeffrey S. Wiggins
- School of Polymer Science and Engineering, University of Southern Mississippi, 118 College Dr, #5050, Hattiesburg, MS 39406, USA; (R.T.H.); (B.R.G.)
- Correspondence:
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Deiters UK, Sadus RJ. Interatomic Interactions Responsible for the Solid-Liquid and Vapor-Liquid Phase Equilibria of Neon. J Phys Chem B 2021; 125:8522-8531. [PMID: 34288687 DOI: 10.1021/acs.jpcb.1c04272] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The role of interatomic interactions on the solid-liquid and vapor-liquid equilibria of neon is investigated via molecular simulation using a combination of two-body ab initio, three-body, and quantum potentials. A new molecular simulation approach for determining phase equilibria is also reported and a comparison is made with the available experimental data. The combination of two-body plus quantum influences has the greatest overall impact on the accuracy of the prediction of solid-liquid equilibria. However, the combination of two-body + three-body + quantum interactions is required to approach an experimental accuracy for solid-liquid equilibria, which extends to pressures of tens of GPa. These interactions also combine to predict vapor-liquid equilibria to a very high degree of accuracy, including a very good estimate of the critical properties.
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Affiliation(s)
- Ulrich K Deiters
- Institute for Physical Chemistry, Faculty of Mathematics and Natural Sciences, University of Cologne, Greinstr 4-6, Köln D-50939, Germany
| | - Richard J Sadus
- Department of Computer Science and Software Engineering, Swinburne University of Technology, P.O. Box 218, Hawthorn 3122 Victoria, Australia
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Jakse N, Alvares CMS, Pisch A. Ab initiobased interionic interactions in calcium aluminotitanate oxide melts: structure and diffusion. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:285401. [PMID: 33906181 DOI: 10.1088/1361-648x/abfc0f] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2021] [Accepted: 04/27/2021] [Indexed: 06/12/2023]
Abstract
Calcium aluminotitanate (CaO-Al2O3-TiO2) ternary oxides are of fundamental interest in Materials as well as Earth and environmental science, and a key system for several industrial applications. As their properties at the atomic scale are scarcely known, interionic interactions for the melts are built from a bottom up strategy consisting in fitting first only Al2O3, CaO and TiO2single oxide compounds separately with a unified description of the oxygen charge and O-O interaction term. For this purpose, a mean-square difference minimization of the partial pair-correlation functions with respect to theab initioreference was performed. The potentials for the ternary oxide are finally built straightforwardly by adding purely Coulomb terms for dissimilar cation-cation interactions without further fit. This general and unified approach is transferable and successfully describes the structural and diffusion properties of the three single oxides as well as the ternary melts simultaneously. A possible underlying structural mechanism at the origin of the diffusion evolution with TiO2content is proposed based on the formation of Ti induced triply bonded oxygen.
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Affiliation(s)
- Noël Jakse
- Université Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France
| | - Cecilia M S Alvares
- Université Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France
| | - Alexander Pisch
- Université Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France
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8
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Dai X, Bai J, Yuan P, Du S, Li D, Wen X, Li W. The application of molecular simulation in ash chemistry of coal. Chin J Chem Eng 2020. [DOI: 10.1016/j.cjche.2020.06.024] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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9
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Woodcox M, Young J, Smeu M. Ab initioinvestigation of the temperature-dependent elastic properties of Bi, Te and Cu. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:485902. [PMID: 32903220 DOI: 10.1088/1361-648x/ababdf] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2020] [Accepted: 08/03/2020] [Indexed: 06/11/2023]
Abstract
Using density functional theory andab initiomolecular dynamics, we have investigated the elastic properties of Bi, Te and Cu as a function of temperature. We compare calculated quantities which can be used to determine the effectiveness of our proposed method, such as the bulk (K), shear (G), and Young's (E) moduli. We also computed Poisson's ratio (ν) and the Pugh ratio (γ) for each of these materials at different temperatures to investigate changes in ductility. We have used the elastic moduli to calculate the Debye temperatureθDand minimum thermal conductivitykminof these materials as a function of temperature. We found that the elastic properties calculated in this work are in good agreement with experimental work. The inclusion of temperature effects has allowed for the proper prediction of ductility for each of these materials, a feat that standard density functional theory calculations has previously been unable to accomplish for Bi and Te.
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Affiliation(s)
- Michael Woodcox
- Department of Physics, Binghamton University-SUNY, Binghamton, NY 13902, United States of America
| | - Joshua Young
- Department of Physics, Binghamton University-SUNY, Binghamton, NY 13902, United States of America
| | - Manuel Smeu
- Department of Physics, Binghamton University-SUNY, Binghamton, NY 13902, United States of America
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10
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Alvares CMS, Deffrennes G, Pisch A, Jakse N. Thermodynamics and structural properties of CaO: A molecular dynamics simulation study. J Chem Phys 2020; 152:084503. [PMID: 32113344 DOI: 10.1063/1.5141841] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A detailed theoretical study of CaO in the solid and liquid states by means of combined classical and ab initio molecular dynamics simulations is presented. Evolution of the specific heat capacity at constant pressure as a function of temperature is studied, and the melting temperature and enthalpy of fusion are determined. It is shown that an empirical Born-Mayer-Huggins potential gives a good representation of pure CaO in the liquid and solid states as compared to available experimental data and density functional theory calculations. Consistency of the predicted results obtained for CaO with the data available in commercial thermodynamic databases and experimental values in the literature is discussed. The present methodology and theoretical results provide a new accurate basis for calculations of thermodynamic properties in a temperature range that is hardly accessible by experiments.
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Affiliation(s)
| | | | - Alexander Pisch
- Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France
| | - Noël Jakse
- Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France
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11
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Study of structural stability of copper crystal with voids from molecular dynamics simulations. Chem Phys Lett 2019. [DOI: 10.1016/j.cplett.2019.06.046] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
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12
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Chavoshi SZ, Xu S, Goel S. Addressing the discrepancy of finding the equilibrium melting point of silicon using molecular dynamics simulations. Proc Math Phys Eng Sci 2017; 473:20170084. [PMID: 28690411 DOI: 10.1098/rspa.2017.0084] [Citation(s) in RCA: 25] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/04/2017] [Accepted: 05/05/2017] [Indexed: 11/12/2022] Open
Abstract
We performed molecular dynamics simulations to study the equilibrium melting point of silicon using (i) the solid-liquid coexistence method and (ii) the Gibbs free energy technique, and compared our novel results with the previously published results obtained from the Monte Carlo (MC) void-nucleated melting method based on the Tersoff-ARK interatomic potential (Agrawal et al. Phys. Rev. B72, 125206. (doi:10.1103/PhysRevB.72.125206)). Considerable discrepancy was observed (approx. 20%) between the former two methods and the MC void-nucleated melting result, leading us to question the applicability of the empirical MC void-nucleated melting method to study a wide range of atomic and molecular systems. A wider impact of the study is that it highlights the bottleneck of the Tersoff-ARK potential in correctly estimating the melting point of silicon.
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Affiliation(s)
- Saeed Zare Chavoshi
- Department of Mechanical Engineering, Imperial College London, London SW7 2AZ, UK
| | - Shuozhi Xu
- GWW School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA
| | - Saurav Goel
- School of Aerospace, Transport and Manufacturing, Cranfield University, Cranfield MK43 0AL, UK
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13
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Elfgen R, Hollóczki O, Ray P, Groh MF, Ruck M, Kirchner B. Theoretical Investigation of the Te4Br2Molecule in Ionic Liquids. Z Anorg Allg Chem 2016. [DOI: 10.1002/zaac.201600342] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
Affiliation(s)
- Roman Elfgen
- Mulliken Center for Theoretical Chemistry; University of Bonn; Beringstr. 4+6 53115 Bonn Germany
- Max Planck Institute for Chemical Energy Conversion; Stiftstr. 34-36 45413 Mühlheim an der Ruhr Germany
| | - Oldamur Hollóczki
- Mulliken Center for Theoretical Chemistry; University of Bonn; Beringstr. 4+6 53115 Bonn Germany
| | - Promit Ray
- Mulliken Center for Theoretical Chemistry; University of Bonn; Beringstr. 4+6 53115 Bonn Germany
| | - Matthias F. Groh
- Department of Chemistry and Food Chemistry; Dresden University of Technology; 01062 Dresden Germany
| | - Michael Ruck
- Department of Chemistry and Food Chemistry; Dresden University of Technology; 01062 Dresden Germany
- Max Planck Institute for Chemical Physics of Solids; Nöthnitzer Str. 40 01187 Dresden Germany
| | - Barbara Kirchner
- Mulliken Center for Theoretical Chemistry; University of Bonn; Beringstr. 4+6 53115 Bonn Germany
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14
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Nieto-Draghi C, Fayet G, Creton B, Rozanska X, Rotureau P, de Hemptinne JC, Ungerer P, Rousseau B, Adamo C. A General Guidebook for the Theoretical Prediction of Physicochemical Properties of Chemicals for Regulatory Purposes. Chem Rev 2015; 115:13093-164. [PMID: 26624238 DOI: 10.1021/acs.chemrev.5b00215] [Citation(s) in RCA: 72] [Impact Index Per Article: 8.0] [Reference Citation Analysis] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Affiliation(s)
- Carlos Nieto-Draghi
- IFP Energies nouvelles , 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
| | - Guillaume Fayet
- INERIS, Parc Technologique Alata, BP2 , 60550 Verneuil-en-Halatte, France
| | - Benoit Creton
- IFP Energies nouvelles , 1 et 4 avenue de Bois-Préau, 92852 Rueil-Malmaison, France
| | - Xavier Rozanska
- Materials Design S.A.R.L. , 18, rue de Saisset, 92120 Montrouge, France
| | - Patricia Rotureau
- INERIS, Parc Technologique Alata, BP2 , 60550 Verneuil-en-Halatte, France
| | | | - Philippe Ungerer
- Materials Design S.A.R.L. , 18, rue de Saisset, 92120 Montrouge, France
| | - Bernard Rousseau
- Laboratoire de Chimie-Physique, Université Paris Sud , UMR 8000 CNRS, Bât. 349, 91405 Orsay Cedex, France
| | - Carlo Adamo
- Institut de Recherche Chimie Paris, PSL Research University, CNRS, Chimie Paristech , 11 rue P. et M. Curie, F-75005 Paris, France.,Institut Universitaire de France , 103 Boulevard Saint Michel, F-75005 Paris, France
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15
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Ustinov EA. Improved modeling of two-dimensional transitions in dense phases on crystalline surfaces. Krypton–graphite system. J Chem Phys 2015; 142:074701. [DOI: 10.1063/1.4908035] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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16
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Akbarzadeh H, Abbaspour M. Size dependence of the equation of state for Ne nanoclusters from an effective two-body potential via molecular dynamics simulations. RSC Adv 2015. [DOI: 10.1039/c4ra14367a] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022] Open
Abstract
In this paper we have extended the equation of state (EoS) in terms of particle size for Ne nanoclusters using an effective two-body Hartree–Fock dispersion (HFD)-like potential by molecular dynamics simulations.
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Affiliation(s)
- Hamed Akbarzadeh
- Department of Chemistry
- Faculty of Basic Sciences
- Hakim Sabzevari University
- 96179-76487 Sabzevar
- Iran
| | - Mohsen Abbaspour
- Department of Chemistry
- Faculty of Basic Sciences
- Hakim Sabzevari University
- 96179-76487 Sabzevar
- Iran
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17
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Calvo F, Naumkin F, Wales D. Nuclear quantum effects on the stability of cationic neon clusters. Chem Phys Lett 2012. [DOI: 10.1016/j.cplett.2012.09.013] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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18
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Zhang Y, Maginn EJ. A comparison of methods for melting point calculation using molecular dynamics simulations. J Chem Phys 2012; 136:144116. [PMID: 22502510 DOI: 10.1063/1.3702587] [Citation(s) in RCA: 72] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Accurate and efficient prediction of melting points for complex molecules is still a challenging task for molecular simulation, although many methods have been developed. Four melting point computational methods, including one free energy-based method (the pseudo-supercritical path (PSCP) method) and three direct methods (two interface-based methods and the voids method) were applied to argon and a widely studied ionic liquid 1-n-butyl-3-methylimidazolium chloride ([BMIM][Cl]). The performance of each method was compared systematically. All the methods under study reproduce the argon experimental melting point with reasonable accuracy. For [BMIM][Cl], the melting point was computed to be 320 K using a revised PSCP procedure, which agrees with the experimental value 337-339 K very well. However, large errors were observed in the computed results using the direct methods, suggesting that these methods are inappropriate for large molecules with sluggish dynamics. The strengths and weaknesses of each method are discussed.
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Affiliation(s)
- Yong Zhang
- Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, Indiana 46556, USA
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19
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Brito BGA, Antonelli A. Efficient method to include nuclear quantum effects in the determination of phase boundaries. J Chem Phys 2012; 137:034114. [DOI: 10.1063/1.4734317] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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Preiss U, Bulut S, Krossing I. In Silico Prediction of the Melting Points of Ionic Liquids from Thermodynamic Considerations: A Case Study on 67 Salts with a Melting Point Range of 337 °C. J Phys Chem B 2010; 114:11133-40. [DOI: 10.1021/jp104679m] [Citation(s) in RCA: 84] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Ulrich Preiss
- Institut für Anorganische und Analytische Chemie and Freiburger Materialforschungszentrum (FMF), Albert-Ludwigs Universität Freiburg, Albertstrasse 21, 79104 Freiburg, Germany
| | - Safak Bulut
- Institut für Anorganische und Analytische Chemie and Freiburger Materialforschungszentrum (FMF), Albert-Ludwigs Universität Freiburg, Albertstrasse 21, 79104 Freiburg, Germany
| | - Ingo Krossing
- Institut für Anorganische und Analytische Chemie and Freiburger Materialforschungszentrum (FMF), Albert-Ludwigs Universität Freiburg, Albertstrasse 21, 79104 Freiburg, Germany
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Calvo F, Parneix P. Phase space theory of evaporation in neon clusters: the role of quantum effects. J Phys Chem A 2010; 113:14352-63. [PMID: 20028160 DOI: 10.1021/jp903282b] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Unimolecular evaporation of neon clusters containing between 14 and 148 atoms is theoretically investigated in the framework of phase space theory. Quantum effects are incorporated in the vibrational densities of states, which include both zero-point and anharmonic contributions, and in the possible tunneling through the centrifugal barrier. The evaporation rates, kinetic energy released, and product angular momentum are calculated as a function of excess energy or temperature in the parent cluster and compared to the classical results. Quantum fluctuations are found to generally increase both the kinetic energy released and the angular momentum of the product, but the effects on the rate constants depend nontrivially on the excess energy. These results are interpreted as due to the very few vibrational states available in the product cluster when described quantum mechanically. Because delocalization also leads to much narrower thermal energy distributions, the variations of evaporation observables as a function of canonical temperature appear much less marked than in the microcanonical ensemble. While quantum effects tend to smooth the caloric curve in the product cluster, the melting phase change clearly keeps a signature on these observables. The microcanonical temperature extracted from fitting the kinetic energy released distribution using an improved Arrhenius form further suggests a backbending in the quantum Ne(13) cluster that is absent in the classical system. Finally, in contrast to delocalization effects, quantum tunneling through the centrifugal barrier does not play any appreciable role on the evaporation kinetics of these rather heavy clusters.
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Affiliation(s)
- F Calvo
- LASIM, Université Claude Bernard Lyon 1 and CNRS UMR 5579, Bat. A. Kastler, 43 Bd du 11 novembre 1918, F69622 Villeurbanne, France
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Krossing I, Slattery JM. Semi-Empirical Methods to Predict the Physical Properties of Ionic Liquids: An Overview of Recent Developments. ACTA ACUST UNITED AC 2009. [DOI: 10.1524/zpch.2006.220.10.1343] [Citation(s) in RCA: 74] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
Abstract
The field of ionic liquids (ILs) has shown rapid growth in recent years. Much of this work has involved the synthesis of new ILs and their application in an ever-increasing number of areas. In contrast, there have been relatively few studies that investigate and attempt to predict the fundamental physical properties of ILs, which are extremely important for their applications. The quantitative prediction of the physical properties of unknown salts remains an important goal in IL research. This will allow the design of new ILs with specific properties tailored for particular applications, without the need for time-consuming trial and error syntheses. Recently, several studies have shown that it is possible to make predictions of the physical properties of ILs e.g. melting points, conductivities, viscosities, densities, surface tensions and refractive indices. This paper gives an overview of these semi-empirical methods and makes some comparisons regarding the accuracy of their predictions and their applicability to predicting the properties of unknown salts.
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23
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Eskandari Nasrabad A. Theory and atomistic simulation of krypton fluid. J Chem Phys 2008; 129:244504. [DOI: 10.1063/1.3046564] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
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24
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Ramírez R, Herrero CP. Quantum path-integral study of the phase diagram and isotope effects of neon. J Chem Phys 2008; 129:204502. [DOI: 10.1063/1.3023036] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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25
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Ramírez R, Herrero CP, Antonelli A, Hernández ER. Path integral calculation of free energies: Quantum effects on the melting temperature of neon. J Chem Phys 2008; 129:064110. [DOI: 10.1063/1.2966006] [Citation(s) in RCA: 22] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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26
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Derrickson SW, Bittner ER. Thermodynamics of atomic clusters using variational quantum hydrodynamics. J Phys Chem A 2007; 111:10345-52. [PMID: 17676719 DOI: 10.1021/jp0722657] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/18/2022]
Abstract
Small clusters of rare-gas atoms are ideal test cases for studying how quantum delocalization affects both the thermodynamics and the structure of molecular scale systems. In this paper, we use a variational quantum hydrodynamic approach to examine the structure and dynamics of (Ne)n clusters, with n up to 100 atoms, at both T = 0 K and for temperatures spanning the solid-to-liquid transition in bulk Ne. Finite temperature contributions are introduced to the grand potential in the form of an "entropy" potential. One surprising result is the prediction of a negative heat capacity for very small clusters that we attribute to the nonadditive nature of the total free-energy for very small systems.
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Affiliation(s)
- Sean W Derrickson
- Department of Chemistry and Center for Materials Chemistry, University of Houston, Houston, Texas 77204, USA
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Zheng L, Rice BM, Thompson DL. Molecular dynamics simulations of the melting mechanisms of perfect and imperfect crystals of dimethylnitramine. J Phys Chem B 2007; 111:2891-5. [PMID: 17388449 DOI: 10.1021/jp0667184] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The melting mechanisms of perfect and imperfect crystalline dimethylnitramine have been studied using molecular dynamics simulations. The imperfect crystal was created by removing approximately 10% of the molecules from the center of the simulation cell. The density, diffusion coefficient, translational and orientational order parameters, and void size were calculated as functions of temperature and simulation time. Upon melting, the volume of the imperfect crystal slowly decreases with time due to the shrinkage of the void then suddenly decreases to a minimum value due to collapse of the structure around the void with concomitant diffusion of molecules into the void. The simulation cell volume then increases as the liquid nucleus formed at the void expands. The melting of perfect crystals must occur by a different mechanism. As the temperature of the perfect crystal reaches the maximum superheating temperature, there is an increase in the thermal motions of the molecules that result in the formation of liquid centers (characterized by translational order parameter consistent with the liquid phase) at random locations. The liquid centers rapidly grow, resulting in a complete transition to the liquid phase. The increases in orientational and translational freedom occur simultaneously in the imperfect crystal, and in the perfect crystal, orientational freedom significantly precedes translational freedom.
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Affiliation(s)
- Lianqing Zheng
- Department of Chemistry, University of Missouri-Columbia, Columbia, Missouri 65211, USA
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Bedrov D, Borodin O, Hanson B, Smith GD. Comment on “On the Accuracy of Force Fields for Predicting the Physical Properties of Dimethylnitramine”. J Phys Chem B 2007; 111:1900-2. [PMID: 17261062 DOI: 10.1021/jp066662d] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Dmitry Bedrov
- Department of Materials Science & Engineering, University of Utah, Room 304, 122 South Central Campus Drive, Salt Lake City, Utah 84112, USA
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Velardez GF, Alavi S, Thompson DL. Molecular dynamics studies of melting and solid-state transitions of ammonium nitrate. J Chem Phys 2006; 120:9151-9. [PMID: 15267851 DOI: 10.1063/1.1705573] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Molecular dynamics simulations are used to calculate the melting point and some aspects of high-temperature solid-state phase transitions of ammonium nitrate (AN). The force field used in the simulations is that developed by Sorescu and Thompson [J. Phys. Chem. A 105, 720 (2001)] to describe the solid-state properties of the low-temperature phase-V AN. Simulations at various temperatures were performed with this force field for a 4 x 4 x 5 supercell of phase-II AN. The melting point of AN was determined from calculations on this supercell with voids introduced in the solid structure to eliminate superheating effects. The melting temperature was determined by calculating the density and the nitrogen-nitrogen radial distribution functions as functions of temperature. The melting point was predicted to be in the range 445 +/- 10 K, in excellent agreement with the experimental value of 442 K. The computed temperature dependences of the density, diffusion, and viscosity coefficient for the liquid are in good agreement with experiment. Structural changes in the perfect crystal at various temperatures were also investigated. The ammonium ions in the phase-II structure are rotationally disordered at 400 K. At higher temperatures, beginning at 530 K, the nitrate ions are essentially rotationally unhindered. The density and radial distribution functions in this temperature range show that the AN solid is superheated. The rotational disorder is qualitatively similar to that observed in the experimental phase-II to phase-I solid-state transition.
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Affiliation(s)
- Gustavo F Velardez
- Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078, USA
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Abstract
Melting in solids containing nanovoids is studied via molecular dynamics simulation, and the finite size effects are treated with different size systems. For the first time, we have found four typical stages in void melting that are different from the melting of bulk materials and nanoparticles. Melting in each of the stages is governed by the interplay among different thermodynamic mechanisms arising from the changes in the interfacial free energies, the curvature of the interface, and the elastic energy induced by the density change at melting. As a result, the local melting temperatures show a strong dependence on the void size, which is the root cause of the observed complex hierarchical melting sequence.
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Affiliation(s)
- Xian-Ming Bai
- School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0245, USA
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Zheng L, Thompson DL. On the Accuracy of Force Fields for Predicting the Physical Properties of Dimethylnitramine. J Phys Chem B 2006; 110:16082-8. [PMID: 16898765 DOI: 10.1021/jp061810l] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
The accuracy of three force fields for predicting the physical properties of dimethylnitramine (DMNA) has been investigated by using molecular dynamics simulations. The Sorescu, Rice, and Thompson (SRT) (J. Phys. Chem. B 1997, 101, 798) rigid-molecule, flexible generalized AMBER (J. Comput. Chem. 2004, 25, 1157), and Smith et al. flexible force fields (J. Phys. Chem. B 1999, 103, 705) were tested. The density, lattice parameters, isotherm, and melting point of DMNA are calculated using classical molecular dynamics. Except for the melting point, the predictions of the three force fields are in reasonable agreement with experimental values. The calculated thermodynamic melting points (Tmp) for the SRT, AMBER, and Smith et al. force fields are 380, 360, and 260 K, respectively. The experimental value is 331 K. Modifications of the torsional barriers in the AMBER force field resulted in Tmp = 346 K, in good agreement with the experimental value of 331 K. The calculated lattice parameters and bulk modulus are also improved with the modifications of the AMBER potential. The results indicate that, although not sufficiently accurate without modifications, the general force fields such as AMBER provide the basis for developing force fields that correctly predict the physical properties of nitramines.
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Affiliation(s)
- Lianqing Zheng
- Department of Chemistry, University of Missouri, Columbia, Missouri 65211, USA
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Zheng L, Luo SN, Thompson DL. Molecular dynamics simulations of melting and the glass transition of nitromethane. J Chem Phys 2006; 124:154504. [PMID: 16674239 DOI: 10.1063/1.2174002] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Molecular dynamics simulations have been used to investigate the thermodynamic melting point of the crystalline nitromethane, the melting mechanism of superheated crystalline nitromethane, and the physical properties of crystalline and glassy nitromethane. The maximum superheating and glass transition temperatures of nitromethane are calculated to be 316 and 160 K, respectively, for heating and cooling rates of 8.9 x 10(9) Ks. Using the hysteresis method [Luo et al., J. Chem. Phys. 120, 11640 (2004)] and by taking the glass transition temperature as the supercooling temperature, we calculate a value of 251.1 K for the thermodynamic melting point, which is in excellent agreement with the two-phase result [Agrawal et al., J. Chem. Phys. 119, 9617 (2003)] of 255.5 K and measured value of 244.73 K. In the melting process, the nitromethane molecules begin to rotate about their lattice positions in the crystal, followed by translational freedom of the molecules. A nucleation mechanism for the melting is illustrated by the distribution of the local translational order parameter. The critical values of the Lindemann index for the C and N atoms immediately prior to melting (the Lindemann criterion) are found to be around 0.155 at 1 atm. The intramolecular motions and molecular structure of nitromethane undergo no abrupt changes upon melting, indicating that the intramolecular degrees of freedom have little effect on the melting. The thermal expansion coefficient and bulk modulus are predicted to be about two or three times larger in crystalline nitromethane than in glassy nitromethane. The vibrational density of states is almost identical in both phases.
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Affiliation(s)
- Lianqing Zheng
- Department of Chemistry, University of Missouri, Columbia, Missouri 65211, USA
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Agrawal PM, Rice BM, Zheng L, Velardez GF, Thompson DL. Molecular Dynamics Simulations of the Melting of 1,3,3-Trinitroazetidine. J Phys Chem B 2006; 110:5721-6. [PMID: 16539517 DOI: 10.1021/jp056690g] [Citation(s) in RCA: 32] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
Physical properties of condensed-phase 1,3,3-trinitroazetidine (TNAZ) have been computed with molecular dynamics (MD) and a nonreactive, fully flexible force field formulated by combining the intramolecular interactions obtained from the Generalized AMBER Force Field and the rigid-molecule force field developed by Sorescu-Rice-Thompson [J. Phys. Chem. B 1997, 101, 798] (AMBER-SRT). The results are compared with MD calculations, using the AMBER force field. The predicted densities of crystalline TNAZ from both force fields are about 10% lower than the experimental value. The calculated thermodynamic melting point at 1 atm from the AMBER-SRT force field is 390 K, in good agreement with the measured value of 374 K, while the AMBER force field predicts a thermodynamic melting point of 462 K. The lattice parameters and the molecular and crystal structures calculated with the AMBER-SRT force field are in excellent agreement with experiment. Simulations with the AMBER-SRT force field were also used to generate the isotherm of TNAZ up to 4 GPa and the bulk modulus and its pressure derivative.
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Affiliation(s)
- Paras M Agrawal
- Department of Chemistry, Oklahoma State University, Stillwater, Oklahoma 74078, USA
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Jakse N, Pasturel A. Molecular-dynamics study of liquid nickel above and below the melting point. J Chem Phys 2005; 123:244512. [PMID: 16396554 DOI: 10.1063/1.2145759] [Citation(s) in RCA: 39] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We have investigated the structural and dynamic properties of liquid nickel by means of large-scale molecular-dynamics simulations, using an effective-pair potential derived from the second-order pseudopotential perturbation theory. The model of interactions is assessed on the single-atom as well as collective dynamic properties. The short-range order in the stable and undercooled liquids is also examined. We show that the present model potential gives a description of the local structure in both states in close agreement with first-principles molecular-dynamics simulations.
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Affiliation(s)
- Noël Jakse
- Laboratoire de Physique des Milieux Denses, Université de Metz, 1 Boulevard FD Arago, 57078 Metz Cedex 3, France.
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Alavi S, Thompson DL. Simulations of the Solid, Liquid, and Melting of 1-n-Butyl-4-amino-1,2,4-triazolium Bromide. J Phys Chem B 2005; 109:18127-34. [PMID: 16853328 DOI: 10.1021/jp053613c] [Citation(s) in RCA: 27] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
Molecular dynamics simulations are used to study the solid and liquid properties and to predict the melting point of 1-n-propyl-4-amino-1,2,4-triazolium bromide ([patr][Br]) using a force field based on the one developed by Canongia Lopes et al. (J. Phys. Chem. B 2004, 108, 2038) for dialkyl substituted imidazolium salts, which was modified by including terms from the general AMBER force field. Electrostatic charges for the intermolecular interactions were determined from gas-phase ab initio electron structure calculations of the triazolium cation. Simulations of the solid state at 100 K reproduced the experimental density to within 4%. Simulations from 100 K to the melting point and the liquid from 333 to 500 K were performed to determine the temperature dependence of the densities of the two phases. The structures of the solid and liquid phases are characterized with radial distribution functions, which show that there are strong spatial correlations among neighboring ion pairs in liquid [patr][Br]. The dynamic behavior of the ions in the liquid state is also studied by computing velocity autocorrelation functions and the mean-square displacements between the ions. The melting point is determined by simulating void-induced melting. Changes in the density, intermolecular energy, and Lindemann index are used as indicators of the melting transition. The computed melting point is 360 +/- 10 K, which is within 10% of the experimental value 333 K.
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Affiliation(s)
- Saman Alavi
- Steacie Institute for Molecular Sciences, National Research Council of Canada, Ottawa, Ontario K1A 0R6, Canada
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Alavi S, Thompson DL. Molecular dynamics studies of melting and some liquid-state properties of 1-ethyl-3-methylimidazolium hexafluorophosphate [emim][PF6]. J Chem Phys 2005; 122:154704. [PMID: 15945653 DOI: 10.1063/1.1880932] [Citation(s) in RCA: 81] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
Molecular dynamics simulations are used to study the liquid-state properties and melting of 1-ethyl-3-methylimidazolium hexafluorosphosphate [emim][PF6] using the force field of Canongia Lopes et al. [J. Phys. Chem. B 108, 2038 (2004)] and geometric constants from crystallographic data. The structures of the solid and liquid states are characterized by carbon-carbon, carbon-phosphorous, and phosphorous-phosphorous radial distribution functions. Spatial correlations among the ions are strong in the liquid state. The cohesive energy density and the temperature dependences of the molar volume and density of the liquid have been computed. The melting point is determined by equilibrating the solid-state supercells in which void defects have been introduced to eliminate the free-energy barrier for the formation of a solid-liquid interface. The computed melting point is 375+/-10 K, which is approximately 10% higher than the experimental value of 333 K.
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Affiliation(s)
- Saman Alavi
- Department of Chemistry, University of Missouri, Columbia, Missouri 65211, USA
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Eike DM, Brennecke JF, Maginn EJ. Toward a robust and general molecular simulation method for computing solid-liquid coexistence. J Chem Phys 2005; 122:14115. [PMID: 15638650 DOI: 10.1063/1.1823371] [Citation(s) in RCA: 76] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
A rigorous and generally applicable method for computing solid-liquid coexistence is presented. The method overcomes some of the technical difficulties associated with other solid-liquid simulation procedures and can be implemented within either a molecular dynamics or Monte Carlo framework. The method consists of three steps: First, relative Gibbs free energy curves are created for the solid and liquid phases using histogram reweighting. Next, the free energy difference between the solid and liquid phases is evaluated at a single state point by integrating along a pseudosupercritical transformation path that connects the two phases. Using this result, the solid and liquid free energy curves are referenced to a common point, allowing a single coexistence point to be determined. Finally, Gibbs-Duhem integration is used to determine the full coexistence curve. To evaluate its utility, this method is applied to the Lennard-Jones and NaCl systems. Results for solid-liquid coexistence agree with previous calculations for these systems. In addition, it is shown that the NaCl model does not correctly describe solid-liquid coexistence at high pressures. An analysis of the accuracy of the method indicates that the results are most sensitive to the transformation free energy calculation.
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Affiliation(s)
- David M Eike
- Department of Chemical and Biomolecular Engineering, University of Notre Dame, Notre Dame, IN 46556, USA
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Agrawal PM, Rice BM, Thompson DL. Molecular dynamics study of the melting of nitromethane. J Chem Phys 2003. [DOI: 10.1063/1.1612915] [Citation(s) in RCA: 89] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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39
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Velardez GF, Alavi S, Thompson DL. Molecular dynamics studies of melting and liquid properties of ammonium dinitramide. J Chem Phys 2003. [DOI: 10.1063/1.1605380] [Citation(s) in RCA: 39] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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40
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Agrawal PM, Rice BM, Thompson DL. Molecular dynamics study of the effects of voids and pressure in defect-nucleated melting simulations. J Chem Phys 2003. [DOI: 10.1063/1.1570815] [Citation(s) in RCA: 62] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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de Koning M, Antonelli A, Yip S. Single-simulation determination of phase boundaries: A dynamic Clausius–Clapeyron integration method. J Chem Phys 2001. [DOI: 10.1063/1.1420486] [Citation(s) in RCA: 45] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
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