TMPH_A_1246760 TFJATS_StdSerif-USA4.cls October 20, 2016 19:52 Trim Info: 215mm × 280mm 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. Forvapour pressure and saturated liquid density such a clear assessment cannot be made. B/w in print, colour online 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 2 G. RUTKAI ET AL. 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 MOLECULAR PHYSICS 3 110 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].