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CE: KD QA: RLMOLECULAR PHYSICS, VOL. , NO. , –http://dx.doi.org/./..

RESEARCH ARTICLE

Howwell does the Lennard-Jones potential represent the thermodynamicproperties of noble gases?

Gábor Rutkaia, Monika Tholb, Roland Spanb and Jadran Vrabeca

aLehrstuhl für Thermodynamik und Energietechnik, Universität Paderborn, Paderborn, Germany; bLehrstuhl für Thermodynamik,Ruhr-Universität Bochum, Bochum, Germany

ARTICLE HISTORYReceived August Accepted September

KEYWORDSNeon; argon; krypton; xenon;noble gas; molecularsimulation; Lennard-Jones;thermodynamic properties

ABSTRACTThe Lennard-Jones potential as well as it’s truncated and shifted (rc = 2.5σ ) variant are applied to thenoble gases neon, argon, krypton, and xenon. Thesemodels are comprehensively comparedwith thecurrently available experimental knowledge in terms of vapour pressure, saturated liquid density, aswell as thermodynamic properties from the single phase fluid regions including density, speed ofsound, and isobaric heat capacity data. The expectation that these potentials exhibit a more mod-est performance for neon as compared to argon, krypton, and xenon due to increasing quantumeffects does not seem to hold for the investigated properties. On the other hand, the assumptionthat the truncated and shifted (rc = 2.5σ ) variant of the Lennard-Jones potential may have short-comings because the long range interactions are entirely neglected beyond the cut-off radius rc, aresupported by the present findings for the properties from the single phase fluid regions. For vapourpressure and saturated liquid density such a clear assessment cannot be made.

B/w in print, colour online

Introduction

All thermodynamic properties can be obtained frommolecular simulation on the basis of a molecular forcefield but the results entirely depend on the underlyingmolecular model [1]. Such models are necessary because5the computation time requirement of the essentially abinitio way of determining properties of fluids, other thanat low density [2–4], is still too large. After obtainingthe charge distribution and geometry of molecular mod-els usually from ab initio calculations, their parameters10for repulsion and dispersion have to be fitted to macro-scopic experimental data. The current trend is to optimise

CONTACT Jadran Vrabec jadran.vrabec@upb.deSupplemental data for this article can be accessed at http://dx.doi.org/./...

these parameters to a relatively narrow selection of ther-modynamic data. Major features of the fluid region, typ-ically vapour pressure and saturated liquid density data 15from laboratory measurements, are considered in thistask because these are usually available in the literatureand also because they can be accurately sampled in sim-ulations [5–7]. The most basic assumption of molecularmodelling and simulation is that force field models pro- 20vide meaningful results at state points and for propertiesthat were not considered during their optimisation. How-ever, this assumption has rarely been tested in a system-atic way. Furthermore, the optimisation of the molecu-lar interaction parameters of simple models considering 25

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2 G. RUTKAI ET AL.

a large number and various types of reference data maynot provide an overall better solution than that of the rela-tively narrow selection if the molecular model itself is notflexible enough. In fact, it is very likely that an improve-ment in one objective cannot be achieved without caus-30ing deterioration in others [8]. Nonetheless, recent find-ings showed that simple molecular models exhibit goodagreement with accurate equation of state (EOS) correla-tions in the temperature and pressure range of industrialrelevance for essentially every measurable static thermo-35dynamic property, even if those models were optimisedexclusively to a narrow set of vapour pressure and satu-rated liquid density data. The supplementary material ofRef. [9] presents numerous examples in detail.

An empirical EOS correlation is an explicit relation40between state variables and it provides inter- and extrapo-lation schemes both in states and properties. State-of-the-art empirical EOS [10] are commonly given as an explicitfunction of a thermodynamic potential,

α (T, ρ) = a (T, ρ)

RT, (1)

where a is the molar Helmholtz energy, T is the tem-45perature, ρ = 1/v is the molar density, and R is thegas constant. The thermodynamic potential α is anappropriate choice because its derivatives with respect toits natural variables, 1/T and ρ, do not involve entropicproperties. Independent on the choice of the underly-50ing thermodynamic potential, any static thermodynamicproperty can be obtained as a combination of its specificpartial derivatives with respect to its independent vari-ables by means of simple analytical derivation. Because α

cannot be measured in laboratory, the actual mathemat-55ical form that represents α, along with its parameters, isfitted to its derivatives,

∂m+nα(1/T , ρ

)∂m

(1/T

)∂nρ

(1/T

)mρn = Amn = Ao

mn + Armn. (2)

This equation shows that Amn can be additivelydecomposed into an ideal (o) and a residual (r) contribu-tion. The ideal contribution is defined as the value ofAmn60whenno intermolecular interactions are atwork. Further-more, Ao

mn = 0 form > 0 and n > 0; Aomn = (−1)1+n

for m = 0 and n > 0; Aomn = Ao

mn(T) for m > 0and n = 0 [10]. Naturally, the goal is to consider asmany different derivatives in terms of order of differen-65tiation as possible for a given state point during the fit.There are two derivatives (A01, A20) that are individuallyaccessible via pressure p, density ρ, temperature T, andisochoric heat capacity cv measurements. There are twoadditional derivatives (A11, A02) that are accessible only70

together with A01 and A20 via isobaric heat capacity cp,speed of sound w, or Joule--Thomson coefficient μ mea-surements [10],

pρRT

= 1 + Ar01, (3)

cvR

= −Ao20 − Ar

20, (4)75

cpR

= −Ao20 − Ar

20 +(1 + Ar

01 − Ar11

)21 + 2Ar

01 + Ar02

(5)

Mw2

RT= 1 + 2Ar

01 + Ar02 −

(1 + Ar

01 − Ar11

)2Ao20 + Ar

20, (6)

and

μρR

= − (Ar01 + Ar

02 + Ar11

)(1 + Ar

01 − Ar11

)2 − (Ao20 + Ar

20) (1 + 2Ar

01 + Ar02

) ,

(7)

whereM is the molar mass. For the noble gases, the idealcontribution Ao

20 is −3/2.Noble gases, compared to other substances, are well 80

measured. This is particularly true for argon, for whicha reference quality EOS is available [11]. Such an EOSrepresents all reliable experimental data essentiallywithintheir uncertainties and it is based on such an amount ofexcellent data that the EOS can be used to calibrate exper- 85imental equipment. The most recent EOS for neon [12],krypton [13], and xenon [13] are also accurate for mosttechnical applications, but they do not fulfil the high stan-dard of reference quality EOS simply because of insuffi-cient experimental data. These EOS are commonly con- 90sidered as technical references, using a functional formthat has been optimised for the representation of proper-ties at pressures of up to 100MPa. Extrapolation to higherpressure is possible, but no attempts were made to accu-rately represent there. 95

Molecular simulation can provide any Armn directly

from a single molecular simulation run per state pointwith the formalism proposed by Lustig [14,15]. More-over, molecular simulation is not limited by extreme con-ditions (temperature or pressure) or the nature of the sub- 100stance. It takes only days to prepare a data-set by molec-ular simulation that comprises a large quantity of non-redundant thermodynamic information, i.e. Ar

mn data,and covers the entire fluid region [9]. Furthermore, thefinancial cost of such a data-set is only a tiny fraction of 105a complete experimental scenario. These data can con-veniently be used in EOS correlation [9,16–18] and wasrecently employed along with non-linear fitting tech-niques to develop the currentlymost accurate EOS for the

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MOLECULAR PHYSICS 3

Lennard-Jones (LJ) potential [19] as well as its truncated110and shifted (rc = 2.5σ ) variant [20]. The latter two EOSwill be used for comparison purposes throughout thiswork. A detailed assessment of these correlations withrespect to the underlying simulation data and other avail-able EOS [21–23] can be found in Refs. [19,20].115

Lennard-Jones potential

The LJ potential [24,25],

uLJ (r) = 4ε[(σ

r

)12−

(σ

r

)6]

(8)

with its parameters for energy ε and size σ describes theinteraction energy between two spherical particles at adistance r from each other. It represents repulsion and120dispersive attraction. In itself, it is well suited to modelthe interactions between noble gas or methane molecules[26]. The truncated and shifted Lennard-Jones (LJTS)potential [1],

uLJTS (r) ={uLJ (r) − uLJ (r = rc) for r ≤ rc

0 for r > rc(9)

is a common modification that artificially removes inter-125actions beyond the cut-off radius rc. This truncationavoids the calculation of long-range corrections [1],which may be problematic or numerically costly in inho-mogeneous systems, while the shift with uLJ(r = rc)enforces a decay to zero that is expected for the inter-130molecular interaction energy. The LJTS potential is alsoconsidered to be sufficiently realistic to represent noblegases [27], but it differs significantly from the LJ poten-tial in terms of its thermodynamic properties (e.g. thereis 20% difference in the critical temperature for the same135ε parameter). The LJ potential is undisputedly the mostfrequently appliedmodel inmolecular simulation historybecause it was, and most likely still is, the best trade-offbetween computational cost, accuracy, and compatibil-ity. More sophisticated potentials are available, e.g. with140adjustable exponent for the repulsive interaction [28] orthe explicit consideration of three-body interactions [29].Due to being physicallymore reasonable and havingmoreadjustable parameters, these potentials must providebetter agreement with experimental data. Nonetheless,145they certainly involve more computation time. Further-more, their application to inhomogeneous systems or forthe calculation of complex properties may be difficult,and compatible molecular models have to be developedfor all components in mixture simulations.150

It is clear that multicriteria optimisation of simplemolecular models with few parameters most likely meansmaking concessions. As long as a potential does not

describe molecular interactions accurately enough toderive all macroscopic thermodynamic properties with 155equally high accuracy, an improvement in the representa-tion of some thermodynamic property inevitably causesdeterioration in the representation of others. It is pos-sible to map the set of best compromises if sufficientexperimental reference data are available for multicri- 160teria optimisation [8]. The data availability in the liter-ature is unfortunately poor to very poor for most flu-ids. Therefore, we assume here the most likely scenario,i.e. that vapour pressure pv and saturated liquid densityρ ′ data are the only experimental data available for the 165optimisation of the interaction potential parameters (inthis case ε and σ ). Therefore, the values of these param-eters for neon (LJ), argon (LJ and LJTS), and krypton (LJand LJTS) were essentially taken from works [27,30] forwhich the optimisation was based on pv and ρ ′. However, 170the parameters for each noble gas and potential were alsodetermined here with a simple algorithm: Based on thefundamental EOS, the energy ε and size σ parameters ofthe fundamental equations of state of the two LJ poten-tials [19,20] were adjusted to the most accurate experi- 175mental data for the values of the critical temperature Tcand density ρc according to Tc = Tc

∗•(ε/kB) and ρc =ρc

∗/σ 3, where Tc∗ and ρc

∗ are the reduced critical dataof the two potentials which are constants. For neon (LJ),argon (LJ and LJTS), and krypton (LJ and LJTS), this pro- 180cedure yielded the same results as found in the literature[8,27,30]. For xenon, differences were observed due to alikely typo in Refs. [27,30]: Since the overall representa-tion of the literature data with the new xenon parameterswasmuch better than with the literature values, they were 185adopted for the following investigations. The parametersvalues are summarised in Table 1.

Results

The available quality and quantity of experimental datain the literature made fundamental EOS correlations for 190

Table . Parameter values for energy ε and size σ used inthis work, where kB is the Boltzmann constant. Values weredetermined with an algorithm described in the text.

LJ LJTS

Neε/kB (K) . .σ (− m) . .

Arε/kB (K) . .σ (− m) . .

Krε/kB (K) . .σ (− m) . .

Xeε/kB (K) . .σ (− m) . .

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4 G. RUTKAI ET AL.

Figure . Relative deviations for the vapour pressure pv and saturated liquid density ρ ′ between the EOS for LJ [] or LJTS [] (baselines)and experimental data (represented by various symbols) for neon. The references for experimental data are given in the supplementarymaterial. The solid line denotes the EOS of Katti et al. [].

Table . Average deviations (low density ‘LD’, medium density ‘MD’, and high density ‘HD’) calculated by <•(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 [] (krypton andxenon) and XLJ,LJTS represents the value from the EOS for LJ [] or LJTS []. For each noble gas, the average< > is basedon 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 = . andρ ′′/ρc (increment (ρ ′′/ρc−.)/) and densities betweenρ ′/ρc andρ/ρc = . (increment (.−ρ ′/ρc)/), whereρ ′′ is the saturated vapour density and ρ ′ the saturated liquid density. Twenty densities were specified above Tc betweenρ/ρc = . and . (increment (.−.)/). The difference between the ideal pressure and that of the respectivenoble gas is around .% at ρ/ρc = . and T/Tc = .. Tr/Tc � . and ρr/ρ c � . 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 p wLD MD HD LD MD HD LD MD HD

LJ Ne . . . . . . . . .Ar . . . . . . . . .Kr . . . . . . . . .Xe . . . . . . . . .

LJTS Ne . . . . . . . . .Ar . . . . . . . . .Kr . . . . . . . . .Xe . . . . . . . . .

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MOLECULAR PHYSICS 5

Figure . Relative deviations for the vapour pressure pv and saturated liquid density ρ ′ between the EOS for LJ [] or LJTS [] (baselines)and experimental data (represented by various symbols) for argon. The references for experimental data are given in the supplementarymaterial. The solid line denotes the EOS of Tegeler et al. [].

noble gases sensible [11–13]. Their underlying data-setsconsist mainly of vapour pressure, saturated liquid den-sity, along with pρT and speed of sound data from thesingle phase fluid regions. A sufficient amount of iso-baric heat capacity data is available only for argon tomake195comparisons meaningful. During the construction of areference EOS, the literature data are carefully screenedand a considerable part of it is discarded. In this work,

we use the experimental data that were found trustwor-thy based on previous efforts [11–13]. Fundamental EOS 200development also enables assessing the available data interms of uncertainty beyond the standard uncertaintyestimation of the individual measurements. In fact, EOSare oftenmore precise than the individualmeasurements.One obvious reason behind this lies in the statistical 205benefit of being able to compare results from different

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6 G. RUTKAI ET AL.

Figure . Relative deviations for the vapour pressure pv and saturated liquid density ρ ′ between the EOS for LJ [] or LJTS [] (baselines)and experimental data (represented by various symbols) for krypton. The references for experimental data are given in the supplementarymaterial. The solid line denotes the EOS of Lemmon and Span [].

laboratories. The more data available, the clearer theassessment about the general uncertainty of experimen-tal data becomes. The other, and stronger, reason isthat thermodynamic consistency is a built-in feature of210fundamental EOS because they rigorously connect allthermodynamic properties through mathematical trans-formations. Inconsistencies in the underlying data-set arethus usually directly detectable. According to the refer-ence quality EOS for argon [11], the scatter of the exper-215imental data compared to the EOS (at specified p andT) considering the entire data-set is usually not largerthan 0.5% for density, 1% for the speed of sound, and10% for the isobaric heat capacity (see below). Devia-tion of the EOS from themost accurate density and speed220of sound measurements, which determined the accuracyof the correlation, is below 0.02% for both properties.

The LJ and LJTS EOS used in this work represent mostof the molecular simulation data within 1% for density,5% for the speed of sound, and 10% for the isobaric 225heat capacity. The relative differences between variousexperimental data and the underlying EOS are shown inFigures 1–14. Deviations tend to increase closer to thecritical point (approximately 44.5 K and 2.68 MPa forneon [12], 150.7 K and 4.86 MPa for argon [11], 209.5 K 230and 5.53 MPa for krypton, and 289.7 K and 5.84 MPa forxenon [13]).

Due to increasing quantum effects, the expectationis that the LJ and LJTS potentials exhibit a moremodest performance for neon as compared to argon, 235krypton, and xenon. Furthermore, one would also expectthat the LJTS potential may face problems due to entirelyneglecting long range interactions beyond rc = 2.5σ .

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MOLECULAR PHYSICS 7

Figure . Relative deviations for the vapour pressure pv and saturated liquid density ρ ′ between the EOS for LJ [] or LJTS [] (baselines)and experimental data (represented by various symbols) for xenon. The references for experimental data are given in the supplementarymaterial. The solid line denotes the EOS of Lemmon and Span [].

The latter statement seems to hold at the investigatedsingle phase fluid regions for the density (Figures 5–8240and 14), speed of sound (Figures 9–12), and isobaricheat capacity (Figure 13). Namely, the experimental dataare distributed more evenly with respect to the nega-tive and positive directions for the LJ potential, while thedistribution for the LJTS potential rather shows a dis-245tinct offset to the experimental data (positive for density,negative for speed of sound). For vapour pressure pv and

saturated liquid density ρ´, such a clear assessment can-not be made: LJ and LJTS are relatively similar for pv andρ´ in terms of overall performance (Figures 1–4). The 250only exception is neon, for which the LJ potential clearlyshows a better agreement with experimental pv and ρ´data than its truncated and shifted variant (Figure 1). Fur-thermore, a comparison of this figure with the follow-ing three somewhat justifies the expectation of a presum- 255ably poorer representation quality for neon than for the

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8 G. RUTKAI ET AL.

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 supplementarymaterial.

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MOLECULAR PHYSICS 9

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 supplementarymaterial.

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10 G. RUTKAI ET AL.

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 thesupplementary material.

other noble gases, but only for the LJTS potential. Inter-estingly, the comparisons for the properties from the sin-gle phase fluid regions do not support this expectation:

According to the density and speed of sound deviation 260plots (Figures 5–12 and 14), the quality of representa-tion is roughly the same for neon as for argon, krypton,

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MOLECULAR PHYSICS 11

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.

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12 G. RUTKAI ET AL.

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 dataof Table 2 support these findings. In general, the aver-age deviations are not worse for neon as compared to the265other noble gases, whereas the overall representation isbest for argon. Again, the LJ potential, all in all, shows

better agreement than the LJTS potential. It was expectedthat the accuracy of these potentials does not reach thelevel that of the most accurate experimental data and 270thus the reference quality EOS of Tegeler et al. [11] (e.g.below 0.02% for the density, cf. Figure 14). Nonetheless,

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MOLECULAR PHYSICS 13

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 thesupplementary material.

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14 G. RUTKAI ET AL.

Figure . Relative deviations for the single phase speed of soundw 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 thesupplementary material.

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MOLECULAR PHYSICS 15

Figure . Relative deviations for the single phase speed of sound w between three EOS (LJ [], LJTS [], or the EOS of Lemmon andSpan [], baselines) and experimental data (represented by various symbols) for xenon. The references for experimental data are givenin the supplementary material.

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16 G. RUTKAI ET AL.

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 thesupplementary material.

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MOLECULAR PHYSICS 17

Figure . Relative deviations for the single phase densityρ 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 inthe supplementary material.

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18 G. RUTKAI ET AL.

the overall agreement with the EOS of Katti et al. [12],Tegeler et al. [11], and Lemmon and Span [13] is surpris-ingly good for both the LJ and the LJTS potentials.275

Acknowledgments

The authors gratefully acknowledge financial support byDeutsche Forschungsgemeinschaft under grant Nos. VR6/4-2 and SP507/7-2. The simulations were carried out on theHazel Hen cluster at the High Performance Computing CenterStuttgart (HLRS) and on the OCuLUS cluster of the Paderborn280Center for Parallel Computing (PC2).

Disclosure statement

No potential conflict of interest was reported by the authors.Q1

Q2

Funding

The authors gratefully acknowledge financial support byDeutsche Forschungsgemeinschaft under grant Nos. VR6/4-2and SP507/7-2.285

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