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CE: KD QA: RL MOLECULAR PHYSICS, VOL. , NO. , – http://dx.doi.org/./..
RESEARCH ARTICLE How well does the Lennard-Jones potential represent the thermodynamic properties of noble gases?
Gábor Rutkaia, Monika Tholb,RolandSpanb and Jadran Vrabeca aLehrstuhl für Thermodynamik und Energietechnik, Universität Paderborn, Paderborn, Germany; bLehrstuhl für Thermodynamik, Ruhr-Universität Bochum, Bochum, Germany
ABSTRACT ARTICLE HISTORY r = σ Received August The Lennard-Jones potential as well as it’s truncated and shifted ( c 2.5 ) variant are applied to the noble gases neon, argon, krypton, and xenon. These models are comprehensively compared with the Accepted September currently available experimental knowledge in terms of vapour pressure, saturated liquid density, as KEYWORDS well as thermodynamic properties from the single phase fluid regions including density, speed of Neon; argon; krypton; xenon; sound, and isobaric heat capacity data. The expectation that these potentials exhibit a more mod- noble gas; molecular est performance for neon as compared to argon, krypton, and xenon due to increasing quantum simulation; Lennard-Jones; effects does not seem to hold for the investigated properties. On the other hand, the assumption thermodynamic properties r = σ that the truncated and shifted ( c 2.5 ) variant of the Lennard-Jones potential may have short- r comings because the long range interactions are entirely neglected beyond the cut-off radius c,are supported by the present findings for the properties from the single phase fluid regions.u Forvapo r pressure and saturated liquid density such a clear assessment cannot be made.
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Introduction these parameters to a relatively narrow selection of ther- modynamic data. Major features of the fluid region, typ- All thermodynamic properties can be obtained from ically vapour pressure and saturated liquid density data 15 molecular simulation on the basis of a molecular force from laboratory measurements, are considered in this field but the results entirely depend on the underlying task because these are usually available in the literature 5 molecular model [1]. Such models are necessary because andalsobecausetheycanbeaccuratelysampledinsim- the computation time requirement of the essentially ab ulations [5–7]. The most basic assumption of molecular initio way of determining properties of fluids, other than modelling and simulation is that force field models pro- 20 at low density [2–4], is still too large. After obtaining vide meaningful results at state points and for properties the charge distribution and geometry of molecular mod- that were not considered during their optimisation.How- 10 els usually from ab initio calculations, their parameters ever, this assumption has rarely been tested in a system- forrepulsionanddispersionhavetobefittedtomacro- atic way. Furthermore, the optimisation of the molecu- scopic experimental data. The current trend is to optimise lar interaction parameters of simple models considering 25
CONTACT Jadran Vrabec [email protected] Supplemental data for this article can be accessed at http://dx.doi.org/./... © Informa UK Limited, trading as Taylor & Francis Group TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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a large number and various types of reference data may together with A01 and A20 via isobaric heat capacity cp, not provide an overall better solution than that of the rela- speed of sound w,orJoule--Thomson coefficient μ mea- tively narrow selection if the molecular model itself is not surements [10], flexibleenough.Infact,itisverylikelythatanimprove- 30 ment in one objective cannot be achieved without caus- p = 1 + Ar , (3) ing deterioration in others [8]. Nonetheless, recent find- ρRT 01 ings showed that simple molecular models exhibit good cv =−Ao − Ar , (4) agreement with accurate equation of state (EOS) correla- R 20 20 75 tions in the temperature and pressure range of industrial + r − r 2 cp o r 1 A01 A11 35 relevance for essentially every measurable static thermo- =−A − A + (5) R 20 20 1 + 2Ar + Ar dynamic property, even if those models were optimised 01 02 2 exclusively to a narrow set of vapour pressure and satu- Mw2 1 + Ar − Ar = 1 + 2Ar + Ar − 01 11 , (6) 01 02 o + r rated liquid density data. The supplementary material of RT A20 A20 Ref. [9]presentsnumerousexamplesindetail. 40 An empirical EOS correlation is an explicit relation and between state variables and it provides inter- and extrapo- μρ lation schemes both in states and properties. State-of-the- R − Ar + Ar + Ar art empirical EOS [10] are commonly given as an explicit = 01 02 11 , + r − r 2 − o + r + r + r function of a thermodynamic potential, 1 A01 A11 A20 A20 1 2A01 A02 (7) a (T,ρ) α (T,ρ) = , (1) RT where M is the molar mass. For the noble gases, the ideal o contribution A 20 is −3/2. 45 where a is the molar Helmholtz energy, T is the tem- Noble gases, compared to other substances, are well 80 ρ = perature, 1/v is the molar density, and R is the measured.Thisisparticularlytrueforargon,forwhich α gas constant. The thermodynamic potential is an a reference quality EOS is available [11]. Such an EOS appropriate choice because its derivatives with respect to represents all reliable experimental data essentially within ρ its natural variables, 1/T and ,donotinvolveentropic their uncertainties and it is based on such an amount of 50 properties. Independent on the choice of the underly- excellent data that the EOS can be used to calibrate exper- 85 ing thermodynamic potential, any static thermodynamic imental equipment. The most recent EOS for neon [12], property can be obtained as a combination of its specific krypton [13], and xenon [13] are also accurate for most partial derivatives with respect to its independent vari- technical applications, but they do not fulfil the high stan- α ables by means of simple analytical derivation. Because dard of reference quality EOS simply because of insuffi- 55 cannot be measured in laboratory, the actual mathemat- cient experimental data. These EOS are commonly con- 90 α ical form that represents , along with its parameters, is sidered as technical references, using a functional form fitted to its derivatives, that has been optimised for the representation of proper- tiesatpressuresofupto100MPa.Extrapolationtohigher ∂m+nα 1 T,ρ mρn = = o + r . pressure is possible, but no attempts were made to accu- 1 T Amn Amn Amn (2) ∂m 1 T ∂nρ rately represent there. 95 r Molecular simulation can provide any A mn directly This equation shows that Amn can be additively from a single molecular simulation run per state point decomposed into an ideal (o) and a residual (r) contribu- with the formalism proposed by Lustig [14,15]. More- 60 tion. The ideal contribution is defined as the value of Amn over, molecular simulation is not limited by extreme con- when no intermolecular interactions are at work. Further- ditions(temperatureorpressure)orthenatureofthesub- 100 o o 1+n more, A mn = 0form > 0andn > 0; A mn = (−1) stance. It takes only days to prepare a data-set by molec- o o for m = 0andn > 0; A mn = A mn(T)form > 0 ular simulation that comprises a large quantity of non- r and n = 0[10]. Naturally, the goal is to consider as redundant thermodynamic information, i.e. A mn data, 65 many different derivatives in terms of order of differen- and covers the entire fluid region [9]. Furthermore, the tiation as possible for a given state point during the fit. financial cost of such a data-set is only a tiny fraction of 105 There are two derivatives (A01, A20)thatareindividually a complete experimental scenario. These data can con- accessible via pressure p,densityρ,temperatureT, and veniently be used in EOS correlation [9,16–18]andwas isochoric heat capacity cv measurements. There are two recently employed along with non-linear fitting tech- 70 additional derivatives (A11, A02) that are accessible only niques to develop the currently most accurate EOS for the TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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110 Lennard-Jones (LJ) potential [19]aswellasitstruncated describe molecular interactions accurately enough to and shifted (rc = 2.5σ )variant[20]. The latter two EOS derive all macroscopic thermodynamic properties with 155 will be used for comparison purposes throughout this equally high accuracy, an improvement in the representa- work. A detailed assessment of these correlations with tion of some thermodynamic property inevitably causes respect to the underlying simulation data and other avail- deterioration in the representation of others. It is pos- 115 able EOS [21–23]canbefoundinRefs.[19,20]. sible to map the set of best compromises if sufficient experimental reference data are available for multicri- 160 Lennard-Jones potential teria optimisation [8]. The data availability in the liter- ature is unfortunately poor to very poor for most flu- The LJ potential [24,25], ids. Therefore, we assume here the most likely scenario, i.e. that vapour pressure pv and saturated liquid density σ 12 σ 6 u (r) = 4ε − (8) ρ data are the only experimental data available for the 165 LJ r r optimisation of the interaction potential parameters (in this case ε and σ ). Therefore, the values of these param- with its parameters for energy ε and size σ describes the eters for neon (LJ), argon (LJ and LJTS), and krypton (LJ interaction energy between two spherical particles at a and LJTS) were essentially taken from works [27,30]for 120 distance r from each other. It represents repulsion and which the optimisation was based on pv and ρ . However, 170 dispersive attraction. In itself, it is well suited to model the parameters for each noble gas and potential were also the interactions between noble gas or methane molecules determined here with a simple algorithm: Based on the [26]. The truncated and shifted Lennard-Jones (LJTS) fundamental EOS, the energy ε and size σ parameters of potential [1], thefundamentalequationsofstateofthetwoLJ poten- uLJ (r) − uLJ (r = rc) for r ≤ rc tials [19,20] were adjusted to the most accurate experi- 175 uLJTS (r) = (9) 0forr > rc mental data for the values of the critical temperature Tc ∗ and density ρc according to Tc = Tc •(ε/kB)andρc = ∗ 3 ∗ ∗ 125 is a common modification that artificially removes inter- ρc /σ ,whereTc and ρc are the reduced critical data actions beyond the cut-off radius rc.Thistruncation of the two potentials which are constants. For neon (LJ), avoids the calculation of long-range corrections [1], argon (LJ and LJTS), and krypton (LJ and LJTS), this pro- 180 which may be problematic or numerically costly in inho- cedure yielded the same results as found in the literature mogeneous systems, while the shift with uLJ(r = rc) [8,27,30]. For xenon, differences were observed due to a 130 enforces a decay to zero that is expected for the inter- likely typo in Refs. [27,30]: Since the overall representa- molecular interaction energy. The LJTS potential is also tion of the literature data with the new xenon parameters considered to be sufficiently realistic to represent noble was much better than with the literature values, they were 185 gases [27], but it differs significantly from the LJ poten- adopted for the following investigations. The parameters tial in terms of its thermodynamic properties (e.g. there values are summarised in Table 1. 135 is 20% difference in the critical temperature for the same ε parameter). The LJ potential is undisputedly the most Results frequently applied model in molecular simulation history The available quality and quantity of experimental data because it was, and most likely still is, the best trade-off in the literature made fundamental EOS correlations for 190 between computational cost, accuracy, and compatibil- 140 ity. More sophisticated potentials are available, e.g. with Table . Parameter values for energy ε and size σ used in adjustable exponent for the repulsive interaction [28]or this work, where kB is the Boltzmann constant. Values were the explicit consideration of three-body interactions [29]. determined with an algorithm described in the text. Due to being physically more reasonable and having more LJ LJTS adjustable parameters, these potentials must provide Ne 145 ε k better agreement with experimental data. Nonetheless, / B (K) . . they certainly involve more computation time. Further- σ (− m) . . Ar more, their application to inhomogeneous systems or for ε k / B (K) . . the calculation of complex properties may be difficult, σ (− m) . . Kr and compatible molecular models have to be developed ε k / B (K) . . 150 for all components in mixture simulations. σ (− m) . . Xe It is clear that multicriteria optimisation of simple ε k / B (K) . . molecular models with few parameters most likely means σ (− m) . . making concessions. As long as a potential does not TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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p ρ Figure . Relative deviations for the vapour pressure v and saturated liquid density between the EOS for LJ []orLJTS[] (baselines) and experimental data (represented by various symbols) for neon. The references for experimental data are given in the supplementary material. The solid line denotes the EOS of Katti et al.[].
Table . Average deviations (low density ‘LD’, medium density ‘MD’, and high density ‘HD’)calculatedby<• (XRef − XLJ,LJTS)/XRef> for the property X (density ρ, pressure p, or speed of sound w) at single phase fluid state points. XRef represents the EOS of either Katti et al.[] (neon), Tegeler et al.[] (argon), or Lemmon and Span [](kryptonand xenon) and XLJ,LJTS represents the value from the EOS for LJ []orLJTS[]. For each noble gas, the average <>is based on state points along isotherms: from the interval . ࣘ T/Tc ࣘ . (increment .) and from the interval . ࣘ T/Tc ࣘ . (increment .). Below the critical temperature Tc, densities were specified between ρ/ρc = . and ρ /ρc (increment (ρ /ρc−.)/) and densities between ρ /ρc and ρ/ρc = . (increment (.−ρ /ρc)/), where ρ is the saturated vapourdensityandρ the saturated liquid density. Twenty densities were specified above Tc between ρ/ρc = . and . (increment (.−.)/). The difference between the ideal pressure and that of the respective noble gas is around .% at ρ/ρc = . and T/Tc = .. Tr/Tc ࣈ . and ρr/ρ c ࣈ . for the triple point (Tr,ρr). LD: ρ/ρc < .; MD: . ࣘ ρ/ρc ࣘ .; HD: ρ/ρc > .. Lowest values in each subcolumn (LJ or LJTS) are shaded. ρ pw LD MD HD LD MD HD LD MD HD
LJ Ne . . . . . . . . . Ar . . . . . . . . . Kr . . . . . . . . . Xe . . . . . . . . . LJTS Ne . . . . . . . . . Ar . . . . . . . . . Kr . . . . . . . . . Xe . . . . . . . . . TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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p ρ Figure . Relative deviations for the vapour pressure v and saturated liquid density between the EOS for LJ []orLJTS[] (baselines) and experimental data (represented by various symbols) for argon. The references for experimental data are given in the supplementary material. The solid line denotes the EOS of Tegeler et al.[].
noble gases sensible [11–13]. Their underlying data-sets we use the experimental data that were found trustwor- consistmainlyofvapour pressure, saturated liquid den- thy based on previous efforts [11–13]. Fundamental EOS 200 sity, along with pρT and speed of sound data from the development also enables assessing the available data in single phase fluid regions. A sufficient amount of iso- terms of uncertainty beyond the standard uncertainty 195 baricheatcapacitydataisavailableonlyforargontomake estimation of the individual measurements. In fact, EOS comparisons meaningful. During the construction of a are often more precise than the individual measurements. reference EOS, the literature data are carefully screened One obvious reason behind this lies in the statistical 205 and a considerable part of it is discarded. In this work, benefit of being able to compare results from different TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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p ρ Figure . Relative deviations for the vapour pressure v and saturated liquid density between the EOS for LJ []orLJTS[] (baselines) and experimental data (represented by various symbols) for krypton. The references for experimental data are given in the supplementary material. The solid line denotes the EOS of Lemmon and Span [].
laboratories. The more data available, the clearer the The LJ and LJTS EOS used in this work represent most assessment about the general uncertainty of experimen- of the molecular simulation data within 1% for density, tal data becomes. The other, and stronger, reason is 5% for the speed of sound, and 10% for the isobaric 225 210 that thermodynamic consistency is a built-in feature of heat capacity. The relative differences between various fundamental EOS because they rigorously connect all experimental data and the underlying EOS are shown in thermodynamic properties through mathematical trans- Figures 1–14.Deviationstendtoincreaseclosertothe formations. Inconsistencies in the underlying data-set are critical point (approximately 44.5 K and 2.68 MPa for thus usually directly detectable. According to the refer- neon [12], 150.7 K and 4.86 MPa for argon [11], 209.5 K 230 215 ence quality EOS for argon [11], the scatter of the exper- and 5.53 MPa for krypton, and 289.7 K and 5.84 MPa for imental data compared to the EOS (at specified p and xenon [13]). T)consideringtheentiredata-setisusuallynotlarger Duetoincreasingquantumeffects,theexpectation than 0.5% for density, 1% for the speed of sound, and is that the LJ and LJTS potentials exhibit a more 10% for the isobaric heat capacity (see below). Devia- modest performance for neon as compared to argon, 235 220 tion of the EOS from the most accurate density and speed krypton,andxenon.Furthermore,onewouldalsoexpect of sound measurements, which determined the accuracy that the LJTS potential may face problems due to entirely of the correlation, is below 0.02% for both properties. neglecting long range interactions beyond rc = 2.5σ . TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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p ρ Figure . Relative deviations for the vapour pressure v and saturated liquid density between the EOS for LJ []orLJTS[] (baselines) and experimental data (represented by various symbols) for xenon. The references for experimental data are given in the supplementary material. The solid line denotes the EOS of Lemmon and Span [].
The latter statement seems to hold at the investigated saturated liquid density ρ´, such a clear assessment can- 240 single phase fluid regions for the density (Figures 5–8 not be made: LJ and LJTS are relatively similar for pv and and 14), speed of sound (Figures 9–12), and isobaric ρ´ in terms of overall performance (Figures 1–4). The 250 heat capacity (Figure 13). Namely, the experimental data only exception is neon, for which the LJ potential clearly are distributed more evenly with respect to the nega- shows a better agreement with experimental pv and ρ´ tive and positive directions for the LJ potential, while the data than its truncated and shifted variant (Figure 1). Fur- 245 distribution for the LJTS potential rather shows a dis- thermore, a comparison of this figure with the follow- tinct offset to the experimental data (positive for density, ing three somewhat justifies the expectation of a presum- 255 negative for speed of sound). For vapour pressure pv and ably poorer representation quality for neon than for the TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase density ρ between three EOS (LJ [], LJTS [], or the EOS of Katti et al.[], baselines) and experimental data (represented by various symbols) for neon. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase density ρ between three EOS (LJ [], LJTS [], or the EOS of Tegeler et al.[], baselines) and experimental data (represented by various symbols) for argon. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase density ρ between three EOS (LJ [], LJTS [], or the EOS of Lemmon and Span [], baselines) and experimental data (represented by various symbols) for krypton. The references for experimental data are given in the supplementary material.
other noble gases, but only for the LJTS potential. Inter- According to the density and speed of sound deviation 260 estingly, the comparisons for the properties from the sin- plots (Figures 5–12 and 14), the quality of representa- gle phase fluid regions do not support this expectation: tion is roughly the same for neon as for argon, krypton, TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase density ρ between three EOS (LJ [], LJTS [], or the EOS of Lemmon and Span [], baselines) and experimental data (represented by various symbols) for xenon. The references for experimental data are given in the sup- plementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase speed of sound w between three EOS (LJ [], LJTS [], or the EOS of Katti et al.[], baselines) and experimental data (represented by various symbols) for neon. The references for experimental data are given in the sup- plementary material.
and xenon for the same potential. The numerical data better agreement than the LJTS potential. It was expected of Table 2 support these findings. In general, the aver- that the accuracy of these potentials does not reach the 265 age deviations are not worse for neon as compared to the level that of the most accurate experimental data and 270 other noble gases, whereas the overall representation is thus the reference quality EOS of Tegeler et al.[11](e.g. bestforargon.Again,theLJpotential,allinall,shows below 0.02% for the density, cf. Figure 14). Nonetheless, TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase speed of sound w between three EOS (LJ [], LJTS [], or the EOS of Tegeler et al. [], baselines) and experimental data (represented by various symbols) for argon. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase speed of sound w between three EOS (LJ [], LJTS [], or the EOS of Lemmon and Span [], baselines) and experimental data (represented by various symbols) for krypton. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase speed of sound w between three EOS (LJ [], LJTS [], or the EOS of Lemmon and Span [], baselines) and experimental data (represented by various symbols) for xenon. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single isobaric heat capacity cp between three EOS (LJ [], LJTS [], or the EOS of Tegeler et al.[], baselines) and experimental data (represented by the various symbols) for argon. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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Figure . Relative deviations for the single phase density ρ between three EOS (LJ [], LJTS [], or the EOS of Tegeler et al.[], baselines) and the most accurate experimental data (represented by various symbols) for argon. The references for experimental data are given in the supplementary material. TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm
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the overall agreement with the EOS of Katti et al.[12], [10] R. Span, Multiparameter Equations of State. An Accu- Tegeler et al.[11], and Lemmon and Span [13]issurpris- rate Source of Thermodynamic Property Data (Springer, 305 275 ingly good for both the LJ and the LJTS potentials. Berlin, 2000). [11] C. Tegeler, R. Span, and W. Wagner, J. Phys. Chem. Ref. Data. 28, 779 (1999). Acknowledgments [12] R. Katti, R.T. Jacobsen, R.B. Stewart, and M. Jahangiri, in Advances in Cryogenic Engineering,editedbyR.W.Fast 310 The authors gratefully acknowledge financial support by (Springer, Boston, MA, 1986). Deutsche Forschungsgemeinschaft under grant Nos. VR6/4- [13] E.W. Lemmon and R. Span, J. Chem. Eng. Data. 51, 785 2 and SP507/7-2. The simulations were carried out on the (2006). Hazel Hen cluster at the High Performance Computing Center [14] R. Lustig, Mol. Sim. 37, 457 (2011). 280 Stuttgart (HLRS) and on the OCuLUS cluster of the Paderborn [15] R. Lustig, Mol. Phys. 110, 3041 (2012). 315 Center for Parallel Computing (PC2). [16] M. Thol, G. Rutkai, A. Köster, M. Kortmann, R. Span, andJ.Vrabec,Chem.Eng.Sci.121,87(2015); 134, 887 (2015). Disclosure statement [17] M. Thol, F.H. Dubberke, G. Rutkai, T. Windmann, A. Köster,R.Span,andJ.Vrabec,FluidPhaseEquilib.418, 320 Q1 No potential conflict of interest was reported by the authors. 133 (2016). [18] M. Thol, G. Rutkai, A. Köster, F.H. Dubberke, T. Wind- Funding mann, R. Span, and J. Vrabec, J. Chem. Eng. Data. 61, 2580 (2016). The authors gratefully acknowledge financial support by [19] M. Thol, G. Rutkai, A. Köster, R. Span, J. Vrabec, and R. 325 Deutsche Forschungsgemeinschaft under grant Nos. VR6/4-2 Lustig, J. Phys. Chem. Ref. Data. 45, 023101 (2016). 285Q2 and SP507/7-2. [20] M. Thol, G. Rutkai, R. Span, J. Vrabec, and R. Lustig, Int. J. Thermophys. 36,25(2015). [21] J.K. Johnson, J.A. Zollweg, and K.E. Gubbins, Mol. Phys. Q3 References 78, 591 (1993). 330 [22] M. Mecke, A. Müller, J. Winkelmann, J. Vrabec, J. Fis- [1] D. Frenkel and B. Smit, Understanding Molecular Simula- cher,R.Span,andW.Wagner,Int.J.Thermophys.17, 391 tion. From Algorithms to Applications,2nded.(Academic (1996); 19, 1493 (1998). Press, Elsevier, San Diego, CA, 2002). [23] J. Kolafa and I. Nezbeda, Fluid Phase Equilib. 100,1 [2] E.Bich,R.Hellmann,andE.Vogel,Mol.Phys.105, 3035 (1994). 335 290 (2007). [24] J.E. Jones, Proc. R. Soc. A. 106, 441 (1924). [3]E.Bich,R.Hellmann,andE.Vogel,Mol.Phys.106, 813 [25] J.E. Jones, Proc. R. Soc. A. 106, 463 (1924). (2008). [26] B. Saager and J. Fischer, Fluid Phase Equilib. 57,35 [4] E. Vogel, B. Jäger, R. Hellmann, and E. Bich, Mol. Phys. (1990). 108, 3335 (2010). [27] J. Vrabec, G.K. Kedia, G. Fuchs, and H. Hasse, Mol. Phys. 340 295 [5]A.Lotfi,J.Vrabec,andJ.Fischer,Mol.Phys.76, 1319 104, 1509 (2006). (1992). [28] J.R. Mick, M. Soroush Barhaghi, B. Jackman, K. Rushai- [6] I.Szalai,J.Liszi,andD.Boda,Chem.Phys.Lett.246, 214 dat, L. Schwiebert, and J.J. Potoff, J. Chem. Phys. 143, (1995). 114504 (2015). [7] J.VrabecandH.Hasse,Mol.Phys.100, 3375 (2002). [29] F. del Río, E. Díaz-Herrera, O. Guzmán, J.A. 345 300 [8] K. Stöbener, P. Klein, S. Reiser, M. Horsch, K.-H. Küfer, Moreno-Razo, and J.E. Ramos, J. Chem. Phys. 139, and H. Hasse, Fluid Phase Equilib. 373, 100 (2014). 184503 (2013). [9] G. Rutkai, M. Thol, R. Lustig, R. Span, and J. Vrabec, J. [30] J. Vrabec, J. Stoll, and H. Hasse, J. Phys. Chem. B. 105, Chem. Phys. 139, 041102 (2013). 12126 (2001).