Mutual Intersection Points of Reduced Collision Integrals for Lennard–Jones (n-m), Hulburt–Hirschfelder, and Tang–Toennies Potential Energy Functions

We investigate the reduced collision integrals Ω(ℓ,s)∗(T∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{(\ell ,s)*}(T^*)$$\end{document} for 1≤ℓ≤4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 \le \ell \le 4$$\end{document}, ℓ≤s≤7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \le s \le 7$$\end{document} and ℓ+s≤8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell + s \le 8$$\end{document} for several isotropic potential energy functions: the Lennard–Jones (n-m)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n-m)$$\end{document}, the Hulburt–Hirschfelder, and Tang–Toennies potential. It is observed that for a given ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell$$\end{document} and s, Ω(ℓ,s)∗(T∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega ^{(\ell ,s)*}(T^*)$$\end{document} shows a mutual intersection region at a reduced temperature 0.39<T∗=Tℓs∗<2.22\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.39< T^*=T^*_{\ell s} < 2.22$$\end{document} which is nearly independent of the potential energy function used.


Introduction
There is still a vivid interest in the transport properties of neat gases and gas mixtures. In particular, the calculation of transport properties such as diffusion and viscosity from binary potential energy functions U(r) becomes more and more relevant and also accurate, see, e.g., [1][2][3][4][5][6][7][8][9][10][11] and references therein. In many of these theoretical investigations, it is required to explicitly calculate the collision integrals [12,13]. They occur in calculations of the transport coefficients. In the following, we neglect quantum effects. In terms of reduced units, the reduced collision integrals Ω ( ,s) * (T * ) are given by [12] (1) Ω ( ,s) * (T * ) = 2 (s + 1)!T * (s+2) ∞ ∫ 0 e −g * 2 ∕T * g * 2s+3 Q ( ) * dg * , * Uwe Hohm u.hohm@tu-braunschweig.de 1 . For a given pair of and s, this region to a very good extend is independent of the reduced intermolecular interaction potential U * considered. In a very limited number of publications, this fact might have been observed by chance but lacks of any discussion [32]. In the present paper, a thorough detection and discussion of this intersection point are presented. Throughout this work, we only concentrate on isotropic potential energy functions of non-polar molecules since, in other cases, a different treatment of our problem seems to be in order [33,34].

Theory
The reduced collision integrals Ω ( ,s) * can be used to correct the transport properties of hard spheres for real gas behavior [12]. In the case of dilute gases which are considered throughout this work, this correction is shown below for the shear viscosity and self-diffusion coefficient D is the density of the gas. f and f D are higher-order correction factors which are also functions of the reduced collision integrals according to [12,25] It has been found that at moderate temperatures, f and f D are in the order of unity [21,23].
The reduced collision integrals are a function of the reduced temperature T * . According to Eq. 1, they crucially depend on the intermolecular interaction potential U * . In this study, three different models of U * (r * ) are taken into account.
A * =Ω (2,2) * ∕Ω (1,1) * , (9) C * =Ω (1,2) * ∕Ω (1,1) * , Although obviously first introduced by Mie [36], U * LJ (r * ) nowadays is known as the Lennard-Jones (n-m) potential [12,14,37]. It has two adjustable parameters n and m. For physical reasons in most cases, m = 6 is chosen. It can considered to be only of moderate accuracy. However, it is still widely in use in all areas of chemistry and physics ranging from the analysis of transport properties to quantum-chemical calculations of large molecules [38][39][40][41][42]. The Hulburt-Hirschfelder potential U * HH (r * ) has three parameters , , and . It is used vividly for high temperature applications [2,17]. The Tang-Toennies potential U * TT (r * ) has only one parameter b to adjust, the coefficients C * 2n being the reduced dispersion interaction energy constants. The Tang-Toennies potential was especially designed for the noble gases helium to radon and their binary mixtures [35].

General Codes
In order to calculate the reduced collision integrals Ω ( ,s) * (T * ) , two different program suites have been developed. First a Fortran program (F) has been written according to a procedure proposed by Colonna and Laricchiuta [15]. This program allows for a fast and reliable calculation of Ω ( ,s) * (T * ) over a broad temperature range. In our case, it was especially designed for the Lennard-Jones (n-m) potential. The evaluation time is about 1 s per data point on a standard laptop with an Intel i7 processor and 32 GB RAM memory. For this potential, we systematically calculate the reduced collision integrals Ω ( ,s) * for 1 ≤ ≤ 4 , ≤ s ≤ 7 and + s ≤ 8 for m = 6 and n =12, 15, 20, 30, and 50 as well as for the (n-m) pairs (9-6), , , and (28-7). We mainly concentrate on the range between 0.01 ≤ T * ≤ 100.0 . In these ranges, our Fortran code allows for a robust screening with a relative accuracy of ΔΩ ( ,s) * ∕Ω ( ,s) * ≤ 0.2 % as can be seen by comparison to literature values and highly accurate calculations with Mathematica [43] as described below. The comparison is shown in Table 1. From the entries, it can be deduced that the results presented in the literature can be reproduced successfully.
Second, we use a straightforward Mathematica [43] code (M) for the calculation of the reduced collision integrals Ω ( ,s) * (T * ) . We have made the observation that the default values of the Mathematica routines are able to handle automatically nearly all of the difficulties which occur during the course of the calculation of Ω ( ,s) * (T * ) . The use of Mathematica routines has been described in the literature before [17,44], but was hardly used up to now. Our code is given in the Appendix and is much simpler than the few other codes described in the literature. In order to check the quality of this code, we compare our findings for the Lennard-Jones (12-6) potential to the highly accurate calculations performed by Kim and Monroe [16] in Table 2. The latter authors state an accuracy of seven digits past the decimal compared to previously reported values at two reduced temperatures of T * = 0.3 and 400.0. As can be seen, the Mathematica code is able to reproduce the results of Kim and Monroe [16] to a very good extend in the case of low and high temperatures. Our fast code F, however, fails at the highest reduced temperature T * of 400 and is therefore only recommended to be used up to T * = 100 . A severe drawback of code M is its quite long calculation time which severely depends on the parameters U * and T * . It varies between 1 min (high temperature, LJ-potential) and about 60 min (low temperature, TT potential) per single point on a standard laptop with an Intel i7 processor and 32 GB RAM memory. We conclude from an inspection of Tables 1 and 2 that the routines used in this work yield trustworthy results of the reduced collision integrals. However, due to its superior accuracy, our Mathematica code was used for all single-point calculations for the Lennard-Jones (n-m), Tang-Toennies [35], and Hulburt-Hirschfelder potential [2,17]. The Tang-Toennies potential is used Hirschfelder et al. [12] (M) 10 6 ΔΩ∕Ω comp. to [16] (M) 10 2 ΔΩ∕Ω comp. to [12] (F) 10 2 ΔΩ∕Ω comp. to [16] (F) 10 2 ΔΩ∕Ω comp. to [12] 0 for all neat noble gases from helium to radon. The Hulburt-Hirschfelder potential has been applied to argon and the (H ⋯ He)-pair. All potential parameters can be found in the corresponding literature [2,17,35].
Overall, the reduced intersection temperatures T * s are found in the range 0.39 < T * s < 2.22 . For typical values of ( ∕k) , this range indicates moderate temperatures T and is not located at extremely high or low temperatures. For a given value of   (T * s ) of the reduced collision integrals calculated for the Lennard-Jones (n-6) potential, Eq. 11. The intersection temperature T * s is given in Table 3. Ω( ) * (full curves) is the arithmetic mean over all different values of s for a given , see Eq. 14 T * s decreases monotonically with increasing s, see Table 3. The individual results Ω ( ,s) * LJ (T * s ) for each n are shown in full detail in Fig. 2. The largest scattering can be seen in the case of the ( , s) = (1, 1) pair. In all other cases, only a moderate dependency of Ω ( ,s) * LJ (T * s ) on n is observed. Therefore, an arithmetic mean Ω ( , s) * LJn T * s over all n is calculated for a given ( , s) pair. As can be seen from Table 3 Fig. 2 are clustered in groups for = 1 , = 3 , = 2 , and = 4 (from top to bottom). On account of this observation, an averaging over s for a given value of is carried out according to Eq. (14). The result is also shown in Fig. 2.
We now compare our findings for alle three potentials, see Eqs. 11-13. 1) where a relative deviation of 0.7 % is observed. In the case of the H ⋯ He pair, the disagreement is generally higher and a maximum relative deviation for (1,1) of 2.5 % is detected.
It can be stated that for the three different potential energy models investigated here, a mutual intersection point or at least a tiny mutual intersection region around Ω ( ,s) (T * s ) for a given pair of and s exists. This mutual intersection region has not been described in the literature before and, therefore, is remarkable by itself.
From this observation, a simple conclusion can be drawn. We consider the transport coefficients viscosity and self-diffusion D which are already given in Eqs. 4 and 5. In the latter case, we now consider the coefficient D . To a first approximation, we set the small correction factors f ≈ f D ≈ 1 . We define the ratio R( , T) with > 0 via  √ Ω (1,1) * (T * ) Ω (2,2) * ( T * ) .
In principle, the left-hand side can be determined from experiment. In order to proceed, we recall that the intersection points of the reduced collision integrals obtained from the Lennard-Jones potential are given by Ω (2,2) * (T * 22 = 2.213) = 1.135 and Ω (1,1) * (T * 11 = 1.61) = 1.165 . These points are    Tang-Toennies (TT), Eq. 13, potential. The parameters of the HH potential are taken from Buchowiecki [17] (Ar) and Bruno et al. [2] (H ⋯ He), the TT parameters are given by Tang and Toennies [35] nearly independent of the potential energy function considered. We now choose = T * 22 ∕T * 11 = 1.375 and according to Eq. 15 obtain the ratio It is interesting to note that at T = T 0 = T * 11 ( ∕k) = T * 22 ( ∕k)∕1.375 , the ratio R(1.375, T 0 ) = 1.0036 ≈ 1 is independent of the underlying potential energy function. Therefore, solving R(1.375, T 0 ) = 1.0036 for T 0 might give the possibility to determine the potential parameter ( ∕k) according to ( ∕k) = T 0 ∕T * 11 = T 0 ∕1.61 . A plot of R(1.375, T) for BF 3 , F 2 , Cl 2 , CH 4 , CF 4 , SF 6 , C 3 H 8 , and the noble gases from neon to xenon is shown in Fig. 9. The intersection points with R(1.371, T 0 ) = 1.0036 are given in Table 4. It can be seen that in most cases, a good approximate value ( ∕k) of the potential well depth can be extracted from the plots. A severe outlier is observed for SF 6 whereas in the case of neon, this analysis fails completely.   Tang-Toennies  (TT), Eq. 13, potential. The parameters of the HH potential are taken from Buchowiecki [17] (Ar) and Bruno et al. [2] (H ⋯ He), the TT parameters are given by Tang and Toennies [35] Fig. 9 The ratio R(1.375, T), Eq. 15, as determined from fitting formulae of the viscosity and self-diffusion coefficient obtained from data given in Refs. [47][48][49][50][51]. The dotted line is located at R = 1.0036 The extraction of is straightforward. Solving the numerators in Eq. 15 for , we obtain At T = T 0 and = 1.375 = T * 22 ∕T * 11 obtained from Eq. 17 does not depend on the potential energy function used in modeling the intermolecular interactions. With an experimental value of (1.375T 0 ) in Eq. 17, an estimate of can be calculated. Results are also listed in Table 4. Again, we failed to obtain a result for neon. SF 6 seems to be a severe outlier. In all other cases fair agreement can be seen between the results presented in this work and data from the literature. It should be stressed that our procedure for obtaining ( ∕k) and does not rely on the assumption of a specific potential energy model. It is therefore in principle superior to schemes which are explicitly using, e.g., the Lennard-Jones (12-6) potential for obtaining and [12,16,40]. However, it is severely hampered by the fact that both the viscosity and self-diffusion coefficient must be available over large temperature ranges with high accuracy. Although this might be possible for [45] at present, this is not feasible in the case of D [46].

Conclusion
We have studied the reduced collision integrals Ω ( ,s) * (T * ) for the Lennard-Jones (n-m), Tang-Toennies, and Hulburt-Hirschfelder potential energy functions. We have detected a small intersection region with ΔT * ≤ 0.1 of the reduced collision integrals Ω ( ,s) * (T * ) , which for given and s to a very good extend is independent of the underlying isotropic potential energy model. This conclusion was derived from a much more systematic and detailed look at the behavior of Ω ( ,s) * (T * ) compared to former studies presented in the literature. Indeed, in contrast to our findings, those studies have revealed an extended range of at least ΔT * ≈ 9 where the reduced collision integrals can be described by a single analytical function irrespective of the potential energy function. Our studies so far have concentrated on isotropic potentials of atoms and non-polar molecules. For this class in principle, the potential parameters and can be extracted from measurements of and D without the assumption of a specific potential energy model. It would be interesting to see in future works how Ω ( ,s) * (T * ) of polar molecules interacting via a Stockmayer potential energy function behave. To this end, it should be checked if the straightforward Mathematica code presented here is also able to handle the necessary integrations. Therefore, we always take Ω ( ,s) * (T * ) = A even in cases where a complex number has been calculated. 2. In the case of the Lennard-Jones (n-m) potential with n = 9 the numerical calculation of Ω ( ,s) * (T * ) does not converge. In these cases it is advisable to set the lower integration bound gredmin in In [20] to gredmin = 0.002 . The error is very small and below 0.01 %, as can be seen in the case of Ω (1,1) * (T * = 1.61) . With gredmin = 0.0 the calculated value is 1.1623165, with gredmin = 0.002 a value of 1.1623253 results.