Erratum

This article refers to:

Ab initio potential energy curve for the neon atom pair and thermophysical properties for the dilute neon gas. II. Thermophysical properties for low-density neon

Molecular Physics, Vol. 106, No. 6 (2008), pp. 813–825

Ab initio potential energy curve for the neon atom pair and thermophysical properties for the dilute neon gas. II. Thermophysical properties for low-density neon

Eckard Bich, Robert Hellmann and Eckhard Vogel

The publishers mistakenly omitted the Appendices for the above paper. The complete version of this paper can now be found in this issue and this will represent its new citation information.

Taylor  & Francis would like to unreservedly apologise for any inconvenience the missing appendices may have caused.

Research Article

Ab initio potential energy curve for the neon atom pair and thermophysical properties for the dilute neon gas. II. Thermophysical properties for low-density neon

Eckard Bich, Robert Hellmann and Eckhard Vogel *

Institut für Chemie, Universität Rostock, Rostock, Germany

A neon–neon interatomic potential energy curve determined from quantum-mechanical ab initio calculations and described with an analytical representation (R. Hellmann, E. Bich, and E. Vogel, Molec. Phys. 106, 133 (2008)) was used in the framework of the quantum-statistical mechanics and of the corresponding kinetic theory to calculate the most important thermophysical properties of neon governed by two-body and three-body interactions. The second and third pressure virial coefficients as well as the viscosity and thermal conductivity coefficients, the last two in the so-called limit of zero density, were calculated for natural Ne from 25 to 10,000 K. Comparison of the calculated viscosity and thermal conductivity with the most accurate experimental data at ambient temperature shows that these values are accurate enough to be applied as standard values for the complete temperature range of the calculations characterized by an uncertainty of about ±0.1% except at the lowest temperatures.

Keywords: neon pair potential; neon gas property standards; second and third pressure virial coefficients; viscosity; thermal conductivity

1. Introduction

Recently we calculated standard values for some thermophysical two-body and three-body properties of helium over the temperature range from 1 to 10,000 K with uncertainties that are superior to experimental data 1. Prerequisite for it was the determination of a state-of-the-art pair potential between two helium atoms 2. In order to establish a second standard for the calibration of high-precision measuring instruments at low density, we developed very recently a new interatomic pair potential for neon from high-level supermolecular ab initio calculations for a large number of internuclear separations R (paper I 3). The ab initio calculated interatomic potential energy values V(R), including core–core and core–valence correlation and relativistic corrections as well as coupled-cluster contributions up to CCSDT(Q), were listed in Table 2 of paper I. They were used for the fit of an analytical potential function, which represents a modification of the potential function given by Tang and Toennies 4:

Whereas the details of the fit were communicated in paper I, the potential parameters are repeated here for convenience in Table 1.

Table 1. Potential parameters.

A	K	0.402915058383E  + 08
a 1	(nm)^−1	−0.428654039586E  + 02
a 2	(nm)^−2	−0.333818674327E  + 01
a −1	nm	−0.534644860719E  − 01
a −2	(nm)^2	0.501774999419E  − 02
b	(nm)^−1	0.492438731676E  + 02
C 6	K(nm)^6	0.440676750157E  − 01
C 8	K(nm)^8	0.164892507701E  − 02
C 10	K(nm)^10	0.790473640524E  − 04
C 12	K(nm)^12	0.485489170103E  − 05
C 14	K(nm)^14	0.382012334054E  − 06
C 16	K(nm)^16	0.385106552963E  − 07
ϵ/k B	K	42.152521
R ϵ	nm	0.30894556
σ	nm	0.27612487

A comparison in paper I with experimental rovibrational spectra 5 showed that the new potential function is superior to the ab initio potential by Cybulski and Toczylowski 6. This potential was given as an analytical function derived from ab initio values calculated for a large range of internuclear distances. Furthermore, the comparison made evident that our new potential is at least as good as the best semi-empirical potential by Aziz and Slaman 7 and also compares well with the potential of Wüest and Merkt 5 fitted directly to the rovibrational spectra under discussion. It is noteworthy that the rovibrational spectra are sensitive to the shape of the potential well. Hence it could be possible that the potential of Wüest and Merkt is not so effective with respect to other regions of the potential. Conversely, the transport properties are particularly sensitive to the repulsive part of the potential. Thus the potential of Aziz and Slaman could be expected to perform well in nearly all regions of the potential, since it was determined in a fit to high-energy beam data and to viscosity coefficients, considering calculated values of the C ₆, C ₈ and C ₁₀ dispersion coefficients.

Table 2. Number of calculated phase shifts of neon for some reduced energies.

E*	Total number	Full QM calculation
0	11	11
1  × 10^−4	14	12
1  × 10^−3	14	12
1  × 10^−2	29	23
5  × 10^−2	62	34
1  × 10^−1	87	41
5  × 10^−1	177	58
1	244	67
3	511	84
5	1152	93
10	1152	108
100	1152	131
3,200	1351	131

In this paper, we report standard values of the second and third pressure virial coefficients as well as of the viscosity and thermal conductivity coefficients in the limit of zero density for neon in its natural isotopic composition. Even though the quality of the new neon ab initio potential of paper I 3 is somewhat inferior compared with our recent ab initio potential for helium 2, the calculated thermophysical properties are expected to be as accurate as the best experimental measurements at room temperature and more accurate than available experimental data far above and far below room temperature. In order to assess as accurately as possible the quality of the potentials considered in this paper, some of the experimental viscosity data from the literature were recalibrated with reference values derived from the new interatomic potential for helium 1,2.

2. Quantum-mechanical calculation of thermophysical properties

2.1. Evaluation of the phase shifts

A quantum-mechanical treatment of the elastic scattering is needed to obtain very accurate values for the thermophysical properties of neon. For this purpose the relative phase shifts δ _l (k) have to be evaluated as asymptotic limiting values of the relative phases of the perturbed and unperturbed radial factor wave functions ψ _l (R) and (the latter with V(R)  = 0). Each of them corresponds to a particular state of the angular momentum of the system characterized by the quantum number l. To obtain the relative phase shifts δ _l (k), the Schrödinger equation is to be solved by numerical integration for many values of the wave number k  = (2μE)^1/2/ℏ, where E is the energy of the incoming wave and  μ = (m ₁ m ₂)/(m ₁  + m ₂) is the reduced mass.

In principle, neon can be considered as a mixture of the three isotopes ²⁰Ne, ²¹Ne, and ²²Ne with the relative atomic masses 19.9924356, 20.9938428, and 21.9913831, with the nuclear spins s of 0, 3/2, and 0, and with the natural abundances 90.48 atom%, 0.27 atom%, and 9.27 atom%. Hence there are six different interacting systems in naturally occurring neon with varying reduced masses and different statistics: ²⁰Ne–²⁰Ne and ²²Ne−²²Ne (both Bose–Einstein statistics), ²¹Ne−²¹Ne (Fermi–Dirac statistics), ²⁰Ne–²¹Ne, ²⁰Ne−²²Ne, and ²¹Ne−²²Ne (all Boltzmann statistics). As a consequence, the relative phase shifts have to be calculated for these six binary systems at a multiplicity of wave numbers k or of energies E for a substantial number of l values which requires plenty of computing time. In order to save time the semi-classical JWKB method was used as approximation. Problems and the procedure of the fully quantum-mechanical calculation as well as the JWKB method for the determination of the relative phase shifts were discussed in some detail in our paper on helium 1.

To avoid uncertainties in the results of the calculated thermophysical properties, a very large number of phase shifts δ _l (k) was determined for 500 values of the energy E from zero to about 135,000 K and for an increasing number of l values related to the energy. The calculations of the phase shifts were first performed fully quantum-mechanically and for comparison parallel to it according to the JWKB approximation. If the values resulting from both procedures became practically identical for certain values of the angular momentum quantum number l, the fully quantum-mechanical evaluation (QM) was finished and substituted by the semi-classical JWKB procedure at the higher l values. Table 2 gives an overview about the number of phase shifts determined for some reduced energies E*  = E/ϵ.

2.2. Calculation of the second and third pressure virial coefficients

In this paper two alternative ways were used to calculate the second virial coefficient of naturally occurring neon as a function of temperature T. In the first variant B(T) is determined like that of a mixture composed of the three neon isotopes:

whereas the second virial coefficients B _ij are evaluated fully quantum-mechanically for the different statistics using two contributions B _direct and B _exch 8. In the Boltzmann statistics (B) the second virial coefficient is given as

whereas for particles with spin quantum number s according to the Bose–Einstein (BE) or to the Fermi–Dirac (FD) statistics holds as:

B _direct and B _exch result from summations over the angular momentum quantum number l, either over only the even l values or over only the odd l values:

The spin quantum numbers s and the statistics have already been given in Section 2.1 for the six interacting systems composed of the three neon isotopes. The summations over l are represented by:

Here Λ is the thermal wavelength:

whereas h is Planck's constant and N _A is Avogadro's number. The first term of Equation (8) corresponds to the contribution of the bound states, where is the negative eigenvalue of the nth vibrational state with the angular-momentum quantum number l which is obtained from the solution of the Schrödinger equation for the radial factor wave functions ψ _l (R). Some bound states corresponding to the rotational levels for the three vibrational states of the ground electronic state of the ²⁰Ne–²⁰Ne and ²⁰Ne–²²Ne dimers were listed for the neon–neon interatomic potential under discussion in Tables 4 and 5 of our paper I 3. The bound states contribution is particularly of importance at low temperatures. The second term of Equation (8) is the essential contribution at medium and higher temperatures and is related to the scattering resulting from binary collisions and to the phase shifts δ _l (E). Due to the fact that the sum over l and the integral in Equation (8) have limits from 0 to ∞, serious errors in the computation may occur when truncated inadequately. Therefore, it was verified that the energies for which the evaluation was performed and particularly the number of the phase shifts were chosen large enough.

In the second variant naturally occurring neon is considered as a pure gas consisting of atoms with the average relative atomic mass 20.1797. Then the second virial coefficient B(T) is derived as the sum of a classical contribution as well as of first-order, second-order, and third-order quantum corrections 9:

with  λ = ℏ²β/12m and  ℏ = h/2π. The individual summands are given as:

For the evaluation of the third pressure virial coefficient C(T), naturally occurring neon is again assumed to be a pure gas composed of atoms with the same mass. Furthermore, C(T) is calculated as a sum of three contributions 10,11, one term for the pairwise additivity of the two-body interatomic potentials C _add, an extra genuine term C _non−add for the non-additivity ΔV ₃(R ₁₂, R ₁₃, R ₂₃) of the three-body interatomic interaction potential V ₃(R ₁₂, R ₁₃, R ₂₃), and a first-order correction term for the quantum effects C _qm,1:

The formulas for the three contributions have already been given in our paper on helium 1 so that it is not necessary to repeat the details. Here the only difference consists in that  λ = ℏ²β/12m is separated in the first-order correction term of the quantum effects C _qm,1. The non-additivity contribution ΔV ₃(R ₁₂, R ₁₃, R ₂₃) to the three-body potential is again approximated by the Axilrod–Teller triple-dipole potential term 12,13, in which the non-additivity coefficient of the triple-dipole term calculated for neon by Kumar and Meath 14, C ₉  = 1.228  × 10⁻⁵ K(nm)⁹, is used.

2.3. Calculation of the transport properties

Different alternative ways were used to determine the transport properties of naturally occurring neon as a function of temperature T. In the first variant η(T) and λ(T) are evaluated quantum-mechanically and approximately like that of a dilute-gas mixture in the limit of zero density composed of the three neon isotopes. In the first-order approximation of the kinetic theory the viscosity is formulated as:

The symbol [η _ij ]₁ represents the first-order approximation of the viscosity characterizing the interaction between a binary pair i–j. Here all different [η _ij ]₁ are given in terms of collision integrals which have to be evaluated quantum-mechanically (see below). δ _ij is the Kronecker symbol, A _ik corresponds to the ratio of two collision integrals, whereas μ _ij is the reduced mass of the interacting pair.

In analogy to Equation (16) an equation in which the elements H _ij are replaced by elements L _ij is applied for the thermal conductivity of a dilute gas mixture in its first-order approximation. The elements L _ij are expressed as:

Here [λ _ij ]₁ is the first-order approximation of the thermal conductivity related to the interaction between a binary pair i  − j and again given in terms of the collision integrals. B _ik represents a relation between different collision integrals.

In principle, exact calculations require higher-order approximations of the kinetic theory. Therefore, we used fifth-order approximations in the case of the transport properties of helium 1, but the calculations for ³He and ⁴He concerned a pure gas each. Unfortunately, approximations of such a high order are not available for mixtures so that we were forced to choose any other reasonable way for the higher-order calculations of the transport properties of naturally occurring neon and tested thus two possibilities. On the one hand, the individual viscosity and thermal conductivity coefficients η _ij and λ _ij for the binary pairs with like and unlike interactions are calculated up to the fifth-order approximation according to:

Here and represent the correction factors for the fifth-order approximations of the kinetic theory. The resulting values for [η _ij ]₅ and [λ _ij ]₅ are then used in the first-order approximation for the mixture viscosity represented by Equation (16) as well as in the corresponding relation for the thermal conductivity. On the other hand, the values of the first-order approximations [η_mix]₁ [Equation (16)] and [λ_mix]₁ are corrected for the fifth-order approximations by means of correction factors and . These are derived using collision integrals which are also determined quantum-mechanically for a pure neon gas consisting of atoms with the average relative atomic mass 20.1797 and following the Bose–Einstein statistics for ²⁰Ne−²⁰Ne. For the calculations of and as well as and we used explicit expressions and computer programs provided by Viehland et al. 15. According to our calculations the effect of the fifth-order corrections to the viscosity and to the thermal conductivity compared with the fourth-order corrections is below ±0.01%.

Expressions for the quantum cross sections Q ^(m)(E) and for the quantum collision integrals Ω ^(m,s)(T) needed in the different approximations of the solution of the Boltzmann equation were derived by Meeks et al. 16. They were again collected in our paper for helium 1 for particles with spin s according to the Bose–Einstein (BE) or to the Fermi–Dirac (FD) statistics as well as for the Boltzmann statistics. The formulas for the different Q ^(m) are related to sums over the phase shifts δ _l , either over only the odd l values or over only the even l values, but also over complete sums (Boltzmann statistics). The quantum collision integrals Ω^(m,s)(T) result from the quantum cross sections Q ^(m)(E) according to:

In a second variant the viscosity of neon was determined classically for the first-order and the fifth-order approximations in order to examine whether a quantum-mechanical calculation is actually needed to achieve highly accurate values for the transport properties of neon in the zero-density limit. For this purpose the usual formulations for monatomics 17 were used, whereas neon was again considered to be a pure gas with the average relative atomic mass already given.

3. Comparison with experimental data and values for other interatomic potentials

3.1. Second and third pressure virial coefficients

The quantum-mechanical calculation of the second pressure virial coefficient requires the determination of the existing bound states. For this purpose the Level 7.7 program of LeRoy 18 was used. As already mentioned in Section 2.2, the bound states of the ²⁰Ne–²⁰Ne and ²⁰Ne–²²Ne dimers were listed in our paper I 3 in which they were compared with the experimental rovibrational spectra by Wüest and Merkt 5.

In Figure 1 the values calculated fully quantum-mechanically for the interatomic potential of the present paper are opposed to those resulting from quantum corrections of increasing order added to the classical contribution. The figure illustrates that the classical contribution alone is completely insufficient to describe adequately the second pressure virial coefficient. The agreement between both ways of calculation improves, with the quantum corrections included, what becomes obvious particularly at low temperatures. To obtain close agreement even at the lowest temperatures, the third-order quantum correction is needed.

Figure 1. Differences ΔB  = B _qm,full  − [B _cl  + ∑λ ⁱ B _qm,i] between the fully quantum-mechanically calculated values and the values resulting from the sum of a classical contribution and of different orders of quantum corrections to the second pressure virial coefficient for the new interatomic potential for Ne. Differences related to: ··· ··· ··· classical contribution B _cl; –  · –  · –  · sum of classical contribution and of first-order quantum correction B _cl  + λB _qm,1; – – – – sum of classical contribution as well as of first-order and second-order quantum corrections B _cl  + λB _qm,1  + λ² B _qm,2; ———– sum of classical contribution as well as of first-order, second-order, and third-order quantum corrections B _cl  + λB _qm,1  + λ² B _qm,2  + λ³ B _qm,3.

There exists only a limited number of experimental data for second and third pressure virial coefficients of neon in the literature compared with those of common gases like argon and nitrogen as well as with those of helium. Furthermore, one should point out that experimental data for the third pressure virial coefficient are not independent of the values for the second pressure virial coefficient derived from the same experiments. Hence third pressure virial coefficients combined with second ones are included in the comparison. Second and third pressure virial coefficients were determined by Holborn and Otto 19, Nicholson and Schneider 20, Michels et al. 21, and Gibbons 22 from isothermal measurements of volume (and density, respectively) and pressure. Vogl and Hall 23 used a Burnett apparatus to derive isothermal compression factors and to obtain finally second and third pressure virial coefficients. Unfortunately, in none of these papers an error propagation analysis or uncertainties of the second and third pressure virial coefficients adequately deduced from the experiments were reported.

The experimental B data are compared with the values calculated fully quantum-mechanically for the neon–neon interatomic potential of the present paper in Figure 2, in which the absolute deviations B _exp  − B _cal(pres) are displayed. The figure demonstrates a very good agreement for the excellent data by Michels et al. 21 at medium temperatures. A good agreement is also found for the data by Nicholson and Schneider 20 up to high temperatures of 1000 K. Conversely, the very old data of Holborn and Otto 19 as well as the more recent but also already 35 years old data of Vogl and Hall 23 are characterized by comparably larger differences to the theoretically calculated values. The data of Gibbons 22 determined at low temperatures show partly large deviations, but agree partly very well. In Figure 2 our calculated values are additionally compared with the values calculated for the interatomic potentials by Aziz and Slaman 7, Cybulski and Toczylowski 6, and Wüest and Merkt 5. The differences B _cal(lit)  − B _cal(pres) derived for the different interatomic potentials increase to low temperatures, where the values derived from the potentials by Aziz and Slaman 7 and Wüest and Merkt 5 are too small and the values resulting from the potential by Cybulski and Toczylowski 6 are too large. At medium and higher temperatures the B _cal values for all four potentials do not differ much so that the second pressure virial coefficient is not suitable for distinguishing between the different interatomic potentials.

Figure 2. Differences (B  − B _cal(pres)) of experimental (B _exp) and calculated (B _cal(lit)) second pressure virial coefficients from values (B _cal(pres)) calculated with the new interatomic potential for Ne. Experimental data:  ○ Holborn and Otto 19;  ▵ Nicholson and Schneider 20;  □ Michels et al. 21;  ▽ Gibbons 22;  ⋄ Vogl and Hall 23. Calculated values: ———– potential by Aziz and Slaman 7; –  · –  · –  · potential by Cybulski and Toczylowski 6; – – – – potential by Wüest and Merkt 5.

In Figure 3 a comparison between experimental data of the third pressure virial coefficient of neon and values calculated for the new interatomic potential is shown. The figure elucidates that good agreement of the experimental data by Michels et al. 21 and by Nicholson and Schneider 20 at medium temperatures and of the data by Gibbons 22 at low temperatures with the calculated values is only achieved in the case of the complete sum of the contributions for the pairwise additivity C _add, for the non-additivity of the three-body interatomic interactions according to Axilrod and Teller C _non-add, and for the first-order quantum-mechanical correction λC _qm,1. The experimental data by Holborn and Otto 19 as well as Vogl and Hall 23 possess again larger differences to the calculated values. The comparison makes evident that the calculation procedure for the third pressure virial coefficient predicts very good values.

Figure 3. Comparison of experimental data and and of calculated values for the third pressure virial coefficient C derived from the new interatomic potential for Ne. Experimental data:  ○ Holborn and Otto 19;  ▵ Nicholson and Schneider 20;  □ Michels et al. 21;  ▽ Gibbons 22;  ⋄ Vogl and Hall 23. Calculated values: – – – – classical contribution C _add, –  · –  · –  · classical and non-additivity contributions C _add  + C _non-add, ——— sum of classical and non-additivity contributions and of the first-order quantum correction C _add  + C _non-add  + λC _qm,1.

It is to be stressed that the calculated values for the second and the third pressure virial coefficients are more reliable than the experimental data at low and high temperatures.

3.2. Viscosity

First, the results of the different alternative ways of the calculation of the transport properties of naturally occurring neon are compared. In this context it is sufficient to consider only the viscosity, since the effects are the same for the thermal conductivity. Figure 4 illustrates the relative differences between viscosity values derived for the different approximation procedures and the viscosity values obtained from the quantum-mechanical calculation up to the fifth-order approximation for the individual [η _ij ]_qm,5 within the first-order formulation of [η_mix]₁ (see Section 2.3). The figure makes evident that the first-order approximation of the classical calculation leads to values which are nearly 1% too small in the complete temperature range except at the lowest temperatures. The agreement improves when the fifth-order approximation of the classical evaluation is applied. But even for this high-order approximation it becomes obvious that the classical evaluation is not appropriate with regard to highly accurate values. Thus the deviations from the results for the quantum-mechanical calculation of the same fifth-order approximation amount to −0.1% at room temperature increasing up to −1.1% at about 60 K. On the other hand, the first-order approximation of the quantum-mechanical calculation for a dilute-gas mixture composed of the three neon isotopes according to [η_mix]_qm,1 is not adequate, too. The differences are approximately −0.7% at most temperatures and decrease to zero at the lowest temperatures. Further it is to note that there are only differences of <±0.0004% (not visible in Figure 4) between the results for the two ways to correct the first-order approximation [η_mix]_qm,1 to an appropriate fifth-order approximation of the quantum-mechanical determination. In the following the comparisons with experimental data are performed with values resulting for the fifth-order approximation of the individual [η _ij ]_qm,5 and [λ _ij ]_qm,5 within the first-order formulations of [η_mix]₁ and [λ_mix]₁.

Figure 4. Relative deviations Δ  = (η  − η_qm,5)/η_qm,5 between viscosity values calculated for different approximation procedures and viscosity values resulting from quantum-mechanical calculations up to the fifth-order approximation for the individual [η _ij ]_qm,5 within the first-order formulation of [η_mix]₁ for the new interatomic potential for Ne. Differences related to: ··· ··· ··· first-order classical calculation [η]_cl,1; – ·· – ·· – ·· fifth-order classical calculation [η]_cl,5; –  · –  · –  · first-order quantum-mechanical calculation [η_mix]_qm,1.

With regard to the transport properties it should be considered that most measurements at low densities were performed at atmospheric pressure, whereas the theoretical calculations are valid for the limit of zero density. Hence the initial density dependence of the experimental data would have to be taken into account. However apart from the very low temperatures near to the normal boiling point of neon, the effect of the initial density dependence on the transport properties concerning the change in density from that at atmospheric pressure to zero density is comparably small (<0.1%) for all other temperatures. Furthermore, the experimental uncertainty is distinctly increased at low temperatures.

In our paper concerning the thermophysical standard values for low-density helium 1 we argued that it is difficult to perform genuine absolute measurements of the gas viscosity with an uncertainty <±0.1%, even at room temperature. The same complex of problems is illustrated in Figure 5 in which the best experimental data for neon near to ambient temperature are characterized by error bars for the uncertainties, given by the authors themselves, and are compared with the viscosity values calculated quantum-mechanically. For helium we demonstrated that the measurements with an oscillating-disc viscometer by Kestin and Leidenfrost 24, approved as one of the most accurate and additionally one of the few absolute measurements on gases, can only partly be considered as absolute ones, since they were finally adjusted to a value for the viscosity of air at 293.15 K and at atmospheric pressure determined by Bearden 31 in an absolute measurement with a rotating-cylinder viscometer. Thus the viscosity value of Kestin and Leidenfrost for neon at 20°C (uncertainty: ±0.05%) shown in Figure 5 corresponds as well to a relative measurement, whereas the genuine absolute measurement is that of Bearden in air. Measurements by Kestin and Nagashima 25, performed in a nearly analogous procedure, led to values which are 0.15–0.3% higher than those of Kestin and Leidenfrost 24, but also by the same percentage higher than further data obtained in relative measurements of the same research group by DiPippo et al. 26 as well as a best estimate reported by Kestin et al. 27 in 1972 as a result of their measurements in foregoing years. This shows that there were sometimes surprisingly large differences in the results of the measurements of this group. Nevertheless, the results of the most reliable measurements by Kestin and co-workers at ambient temperature are characterized by a tendency to values increased by +0.1% compared to the calculated values of the present paper. The same findings concerning the measurements by Kestin and co-workers were observed in the case of the values derived from our interatomic helium potential. As a consequence, the measurements on helium by Vogel 29 (uncertainty: ±0.15% at room temperature) performed with an all-quartz oscillating-disc viscometer in a relative manner using a viscometer constant derived from the best estimate by Kestin et al. should be affected by the same impact. Therefore, the viscometer of Vogel was recalibrated with the new helium standard for a rehandled evaluation of the measurements on helium 1 and on neon, too. The influence of the recalibration on the results of Vogel for neon is additionally demonstrated in Figure 5. A value at 298.15 K resulting from the fitting function given by Vogel 29 deviates from the value calculated for the interatomic neon potential of the present paper by +0.18%. Conversely, the direct experimental data of the measurement series by Vogel at room temperature show only differences of +0.04% and +0.08% after the recalibration.

Figure 5. Relative deviations of experimental and calculated viscosity coefficients from values η_cal(pres) calculated quantum-mechanically with the new interatomic potential for Ne near to room temperature. Experimental data with uncertainties characterized by error bars:  • Kestin and Leidenfrost 24;  ○ Kestin and Nagashima 25;  ▾ DiPippo et al. 26;  ▽ Kestin et al. 27, best estimate;  ▪ Flynn et al. 28;  ▵ Vogel 29, fitted value;  ▴ Vogel 29, experimental data corrected according to new helium standard;  ♦ Evers et al. 30. Calculated values: ··· ··· ··· fifth-order classical calculation [η]_cl,5; ———– potential by Aziz and Slaman 7; –  · –  · –  · potential by Cybulski and Toczylowski 6.

Furthermore, the absolute measurements by Flynn et al. 28 performed with a capillary viscometer led to a datum at 293.15 K differing only by +0.01% from the theoretically calculated value (uncertainties: ±0.1%). Recently, Evers et al. 30 utilized a rotating-cylinder viscometer for absolute measurements (uncertainty: ±0.15%) on several gases at different temperatures and pressures. Their result for neon at 298.15 K deviates from our calculated value by −0.12%. In conclusion, the comparison makes evident that the best experimental data at room temperature are characterized by an uncertainty of ±(0.1 to 0.15)% and that they agree within this limit with the values calculated for the interatomic neon potential of the present paper.

The situation deteriorates to the disadvantage of the experiment, if the measurements were not performed at ambient temperature. In Figure 6, experimental data at low and medium temperatures between 20 and 373 K are compared with the calculated values. Error bars for one or two (in the case that the uncertainty changes with temperature) values of each data set are additionally plotted. The figure demonstrates that excellent agreement within ±0.1% exists only for the absolute measurement by Evers et al. 30 at 348 K and that the results of the absolute measurements by Flynn et al. 28 are adequately consistent within ±0.3%. The other data were determined by relative measurements, which are not only affected by the usual measurement errors, but also by the values used for the calibration. Johnston and Grilly 32 and Rietveld et al. 34 (both using oscillating-disc viscometers) as well as Clarke and Smith 35 (capillary viscometer) based their measurements on reasonable values for air, helium, and nitrogen at ambient temperature and achieved results with deviations up to −2%, +4%, and +1%. These data are not suitable to judge the appropriateness of any interatomic neon potential. On the other hand, the measurements by Coremans et al. 33 carried out with an oscillating-disc viscometer, which was calibrated using a very old viscosity value for ⁴He at 20 K reported by Kamerlingh Onnes and Weber 36, yielded values characterized by positive deviations up to 6% from the quantum-mechanically calculated values. These results were improved for the purposes of this paper by a recalibration with a value for ⁴He at 20 K taken from our new helium standard 1. Figure 6 makes obvious that the corrected data advanced after this correction partly in close agreement.

Figure 6. Relative deviations of experimental and calculated viscosity coefficients from values η_cal(pres) calculated with the new interatomic potential for Ne at low and medium temperatures. Experimental data with uncertainties characterized by error bars:  □ Johnston and Grilly 32;  ○ Coremans et al. 33;  • Coremans et al. 33, corrected according to new helium standard;  ▵ Rietveld et al. 34;  ▪ Flynn et al. 28;  ⋄ Clarke and Smith 35;  ♦ Evers et al. 30. Calculated values: ··· ··· ··· fifth-order classical calculation [η]_cl,5; ———– potential by Aziz and Slaman 7; –  · –  · –  · potential by Cybulski and Toczylowski 6; – – – – potential by Wüest and Merkt 5.

Figure 7 illustrates the analogous comparison at higher temperatures. The figure reveals a surprisingly large scattering of about ±0.3% in the data from different papers by Kestin and his research group 27,37,38 (the same order of magnitude as the uncertainty) and additionally a systematic trend to higher values with increasing temperature combined with again decreasing values at the highest temperatures. In this connection it is to be noted that all measurements by Kestin and his co-workers with the oscillating-disc viscometer by Di Pippo et al. 41 are affected by a temperature measurement error with thermocouples explained by Vogel et al. 42. Figure 7 also makes evident that the data by Vogel 29, originally fitted to his experiments which were based on a calibration with the best estimate value at room temperature by Kestin et al. 27, deviate by about +0.2% from the quantum-mechanically calculated values of this paper. After a recalibration of the measurement series on neon by means of the new helium standard 1 at room temperature, the corrected experimental data do only deviate by less than +0.1% on average from the theoretical values for the new neon potential in the complete temperature range of the measurements. This demonstrates that the measurements by Vogel with his all-quartz oscillating-disc viscometer represent the best experiments in this temperature range. The comparison concerning the experimental data by Dawe and Smith 39 and by Guevara and Stensland 40, which result from relative measurements with capillary viscometers based on a reasonable calibration at room temperature, shows that these data should be influenced by systematic errors. Lastly it is concluded that the theoretical determination of viscosity values is to be preferred to experiments at these high temperatures.

Figure 7. Deviations of experimental and calculated viscosity coefficients from values η_cal(pres) calculated with the new interatomic potential for Ne at higher temperatures. Experimental data with uncertainties characterized by error bars:  ▽ Kestin et al. 27, best estimate;  ○ Hellemans et al. 37;  ⊙ Kestin et al. 38;  ⋄ Dawe and Smith 39;  □ Guevara and Stensland 40;  ▵ Vogel 29, fitted values;  ▴ Vogel 29, experimental data corrected according to new helium standard. Calculated values: ··· ··· ··· fifth-order classical calculation [η]_cl,5; ———– potential by Aziz and Slaman 7; –  · –  · –  · potential by Cybulski and Toczylowski 6; – – – – potential by Wüest and Merkt 5.

Figures 5, 6, and 7 include once again a comparison with the values derived classically using the fifth-order approximation. The results of the classical calculation deviate by about −0.1% from those of the quantum-mechanical computation at ambient and higher temperatures. At lower temperatures the deviations are distinctly increased. Figure 5 elucidates further that at room temperature the results of the quantum-mechanical calculations for the potentials by Aziz and Slaman 7 and by Cybulski and Toczylowski 6 (both >±0.2%) and particularly by Wüest and Merkt 5 (−0.7%, not observable in the figure) do not match the best experimental data as well as the calculated values for the potential of the present paper within ±0.1%. Figures 6 and 7 demonstrate that the best experimental data allow one to distinguish between the different potentials proposed for neon. The values resulting from the potentials by Aziz and Slaman 7 and by Cybulski and Toczylowski 6 and particularly by Wüest and Merkt 5 are characterized by differences from the transport data that are distinctly larger than the experimental uncertainties. Here one should point to the differences for the values determined with the potential proposed by Wüest and Merkt 5. They arise with increasing temperature due to the fact that the rovibrational spectra used by Wüest and Merkt are sensitive to the shape of the potential well, but not to the repulsive part of the potential to which the transport properties are particularly sensitive.

3.3. Thermal conductivity

The uncertainty of measurements of the thermal conductivity is inferior to that of viscosity measurements due to different experimental difficulties, whereas the most accurate data can be obtained with the transient hot-wire technique, but essentially restricted to ambient temperature. This is demonstrated in Figure 8, in which experimental data for neon at low and medium temperatures are compared with the values calculated quantum-mechanically. Here the experimental data are again, when available, characterized by error bars according to the uncertainties given by the experimenters themselves. The data by Kestin et al. 44 and Assael et al. 45, each gained with the transient hot-wire technique at room temperature, deviate from the calculated values by <−0.1% and <+0.2%. These differences are lower than the experimental uncertainties (±0.3% and ±0.2%). Although the data by Haarman 43 are characterized by larger deviations (−0.3% to −0.4%), the temperature function of these transient hot-wire data between 328 and 468 K corresponds closely to that of the calculated values. Conversely, the temperature function of the data by Millat et al. 46 shows an awkward behaviour so that these data are not useful with regard to the assessment of the values calculated for the different interatomic potentials of neon. But Figure 8 makes also evident that the deviations of the experimental data of Hemminger 47, derived from measurements with a guarded parallel-plate apparatus and carefully corrected for impurities caused by desorbed air, are within −0.35% and −0.6%; this means their temperature function and that of the calculated values are pretty much consistent from room temperature up to 470 K.

Figure 8. Deviations of experimental and calculated thermal conductivity coefficients from values λ_cal(pres) calculated with the new interatomic potential for Ne at low and medium temperatures. Experimental data with uncertainties characterized by error bars:  • Haarman 43;  ▴ Kestin et al. 44;  ▪ Assael et al. 45;  ▾ Millat et al. 46;  ♦ Hemminger 47;  ○ Weber 48;  ▵ Kannuluik and Carman 49;  □ Keyes 50;  ▽ Sengers et al. 51;  ⋄ Nesterov and Sudnik 52, smoothed values. Calculated values: ··· ··· ··· fifth-order classical calculation [η]_cl,5; ———– potential by Aziz and Slaman 7; –  · –  · –  · potential by Cybulski and Toczylowski 6; – – – – potential by Wüest and Merkt 5.

Experimental data determined with the common steady-state hot-wire technique often affected by convection are checked against the quantum-mechanically calculated values in Figure 8, too. Differences of only  < ±0.4% are found for the very old experimental datum by Weber 48 at 273 K and also for a value by Kannuluik and Carman 49 at the same temperature. But for the complete temperature range of the measurements of Kannuluik and Carman between 90 and 580 K the deviations increase up to −3%. On the other hand, the smoothed experimental values by Nesterov and Sudnik 52 between 90 K and ambient temperature created with the same technique are characterized by comparably small differences between −0.1% and −0.7%, with the best agreement at low temperatures. Further it becomes evident from this figure that the experimental data by Keyes 50 determined with the concentric-cylinder method (differences between −1% and −1.5%) and those of Sengers et al. 51 obtained with a parallel-plate apparatus (differences between −0.5% and −0.85%) are not suitable for a reasonable comparison with the theoretical values.

Figure 9 illustrates the comparison at higher temperatures. Neither the experimental data by Tufeu et al. 54 (concentric-cylinder method) nor the experimental values by Saxena and Saxena 53 (common hot-wire technique) enable one to verify the performance of the different potentials under discussion due to the differences exceeding −1%. Conversely, there occur surprisingly only very small deviations of −0.45% to  + 0.05% for the experimental data by Springer and Wingeier 55 between 1000 and 1500 K using the concentric-cylinder method. In principle, this would support the new interatomic potential of this work. In addition, the values recommended by Ziebland 56 on the basis of different experimental data show deviations larger than +1% according to their estimated uncertainties. It is to note that thermal conductivity values at very high temperatures between 1500 K and at most 6000 K were derived from shock-tube measurements by Collins and Menard 57 and by Maštovský 58. Their data not shown in Figure 9 have deviations of −5.5% up to −11.5% and −2.5% up to −6.8%. At such high temperatures calculated values are to be preferred in any case.

Figure 9. Deviations of experimental and calculated thermal conductivity coefficients from values λ_cal(pres) calculated with the new interatomic potential for Ne at higher temperatures. Experimental data with uncertainties characterized by error bars:  ○ Saxena and Saxena 53, smoothed values;  ▵ Tufeu et al. 54;  □ Springer and Wingeier 55;  ▽ Ziebland 56, recommended values. Calculated values: ··· ··· ··· fifth-order classical calculation [η]_cl,5; ———– potential by Aziz and Slaman 7; –  · –  · –  · potential by Cybulski and Toczylowski 6; – – – – potential by Wüest and Merkt 5.

Both figures make evident that the interatomic potential by Wüest and Merkt 5 is not qualified to describe adequately the best experimental thermal conductivity data. On the other hand, there exist only a few experimental data to distinguish between the appropriateness of the other potentials. But if the best experimental transient hot-wire-data at room temperature are selected for the comparison, then there exists a stringent test of the new potential and of the correct application of the kinetic theory including the quantum-mechanical effects.

4. Summary and conclusions

A new interatomic potential for neon derived from quantum-mechanical ab initio computations 3 was utilized to calculate the second and third pressure virial, the viscosity, and the thermal conductivity coefficients for dilute neon gas in its natural isotopic composition in the temperature range from 25 to 10,000 K. For the second virial coefficient and for the transport properties fully quantum-mechanical calculations were performed with neon treated as an isotopic mixture, whereas for the third virial coefficient a classical mechanical evaluation with a quantum correction using the average mass of the isotopic mixture was applied. The comparison with available experimental data makes evident that the calculated thermophysical properties are as accurate as the best experimental data at room temperature and more accurate at temperatures above and below room temperature. The deviations between the results from the different potentials for all calculated properties increase at the lowest temperatures.

The viscosity values around ambient temperature derived theoretically with the interatomic potential of this paper are characterized by deviations smaller than ±0.1% compared to the best experimental data, whereas the results obtained from the potential energy curves by Cybulski and Toczylowski, by Aziz and Slaman 7, and by Wüest and Merkt 5 show larger deviations. We estimate summarily the uncertainties of the calculated transport properties resulting from our new potential to be about ±0.1% except at the lowest temperatures. It is to be stressed that this uncertainty is much below the experimental uncertainties at low as well as at high temperatures. All calculated data (see Table 3 in Appendix 1) can be applied as standards values for the complete temperature range.

Notes

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Appendix 1. Thermophysical properties of neon calculated in this work

The thermophysical properties of naturally occurring neon are given in Table 3.

Table 3. Thermophysical properties of neon for the interatomic potential of this work.

T (K)	B (cm^3 mol^−1)	C (cm^6 mol^−2)	η (μPas)	λ (mW m^−1 K^−1)
25.00	−128.50	−1716	3.9213	6.0597
26.00	−119.87	−1130	4.0790	6.3033
27.00	−112.02	−689.5	4.2353	6.5446
28.00	−105.04	−358.4	4.3927	6.7878
30.00	−92.972	78.89	4.7097	7.2775
32.00	−82.953	325.4	5.0270	7.7676
34.00	−74.517	461.5	5.3447	8.2585
36.00	−67.316	532.2	5.6617	8.7482
38.00	−61.085	564.7	5.9784	9.2377
40.00	−55.657	574.0	6.2915	9.7214
42.00	−50.888	569.7	6.6035	10.204
44.00	−46.666	557.7	6.9132	10.682
46.00	−42.905	541.5	7.2191	11.155
48.00	−39.526	523.4	7.5245	11.627
50.00	−36.486	504.6	7.8257	12.093
55.00	−30.063	459.3	8.5658	13.237
60.00	−24.892	419.9	9.2862	14.352
65.00	−20.676	387.2	9.9868	15.436
70.00	−17.169	360.6	10.668	16.491
75.00	−14.213	339.9	11.331	17.518
80.00	−11.676	321.5	11.976	18.518
85.00	−9.4909	307.0	12.605	19.493
90.00	−7.5869	295.2	13.217	20.443
95.00	−5.9132	285.4	13.815	21.370
100.00	−4.4329	277.3	14.399	22.277
110.00	−1.9365	264.7	15.528	24.030
120.00	0.0895	255.6	16.612	25.713
130.00	1.7559	248.8	17.654	27.333
140.00	3.1490	243.5	18.660	28.896
150.00	4.3280	239.4	19.634	30.410
160.00	5.3365	236.0	20.579	31.879
170.00	6.2068	233.1	21.498	33.307
180.00	6.9638	230.6	22.394	34.699
190.00	7.6271	228.6	23.268	36.058
200.00	8.2116	226.5	24.122	37.385
210.00	8.7298	224.7	24.958	38.685
220.00	9.1906	223.1	25.778	39.959
230.00	9.6037	221.5	26.583	41.210
240.00	9.9744	220.0	27.372	42.436
250.00	10.308	218.6	28.149	43.643
260.00	10.610	217.3	28.912	44.829
270.00	10.884	215.9	29.665	45.999
273.15	10.964	215.5	29.900	46.364
280.00	11.133	214.6	30.408	47.153
290.00	11.360	213.9	31.139	48.289
298.15	11.530	212.9	31.728	49.203
300.00	11.567	212.2	31.860	49.410
320.00	11.930	209.8	33.277	51.610
340.00	12.237	207.4	34.660	53.758
360.00	12.497	205.2	36.014	55.859
380.00	12.719	203.0	37.339	57.917
400.00	12.909	200.8	38.640	59.937
420.00	13.072	198.8	39.918	61.921
440.00	13.212	196.7	41.174	63.871
460.00	13.333	194.8	42.411	65.789
480.00	13.437	192.8	43.628	67.679
500.00	13.527	190.9	44.829	69.542
550.00	13.700	186.4	47.761	74.091
600.00	13.819	182.1	50.604	78.502
650.00	13.896	178.0	53.370	82.793
700.00	13.943	174.2	56.068	86.977
750.00	13.968	170.6	58.703	91.065
800.00	13.975	167.2	61.284	95.067
850.00	13.968	163.9	63.814	98.9901
900.00	13.952	160.8	66.298	102.84
950.00	13.927	157.9	68.739	106.63
1000.00	13.895	155.1	71.141	110.35
1100.00	13.819	149.9	75.838	117.63
1200.00	13.730	145.1	80.408	124.72
1300.00	13.634	140.7	84.866	131.63
1400.00	13.534	136.7	89.223	138.38
1500.00	13.432	132.9	93.489	144.99
1600.00	13.329	129.5	97.674	151.48
1700.00	13.226	126.2	101.78	157.85
1800.00	13.125	123.2	105.83	164.11
1900.00	13.025	120.3	109.80	170.28
2000.00	12.926	117.6	113.72	176.35
2100.00	12.830	115.1	117.59	182.34
2200.00	12.736	112.7	121.40	188.25
2300.00	12.644	110.4	125.17	194.08
2400.00	12.554	108.2	128.89	199.85
2500.00	12.467	106.2	132.57	205.55
2600.00	12.381	104.2	136.21	211.19
2700.00	12.298	102.4	139.81	216.76
2800.00	12.216	100.6	143.38	222.29
2900.00	12.137	98.92	146.91	227.76
3000.00	12.059	97.29	150.41	233.18
3100.00	11.984	95.73	153.87	238.55
3200.00	11.910	94.24	157.31	243.88
3300.00	11.838	92.80	160.72	249.16
3400.00	11.767	91.41	164.11	254.40
3500.00	11.698	90.08	167.46	259.60
3600.00	11.631	88.80	170.79	264.76
3700.00	11.565	87.56	174.10	269.88
3800.00	11.501	86.36	177.39	274.97
3900.00	11.438	85.21	180.65	280.02
4000.00	11.376	84.09	183.89	285.04
4100.00	11.316	83.01	187.11	290.03
4200.00	11.257	81.96	190.31	294.99
4300.00	11.199	80.95	193.49	299.91
4400.00	11.142	79.96	196.65	304.81
4500.00	11.087	79.01	199.80	309.68
4600.00	11.032	78.09	202.92	314.52
4700.00	10.979	77.19	206.03	319.33
4800.00	10.926	76.31	209.12	324.12
4900.00	10.875	75.46	212.20	328.88
5000.00	10.825	74.64	215.26	333.62
6000.00	10.366	67.45	245.09	379.81
7000.00	9.9770	61.77	273.72	424.15
8000.00	9.6396	57.14	301.40	467.00
9000.00	9.3429	53.30	328.28	508.61
10000.00	9.0788	50.05	354.48	549.16