- Email: [email protected]
Calculation of the relaxation properties of a dilute gas consisting of Lennard–Jones chains
Chemical Physics Letters 574 (2013) 37–41
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Calculation of the relaxation properties of a dilute gas consisting of Lennard–Jones chains Robert Hellmann a, Nicolas Riesco b,c, Velisa Vesovic c,⇑ a
Institut für Chemie, Universität Rostock, 18059 Rostock, Germany Qatar Carbonates and Carbon Storage Research Centre (QCCSRC), Imperial College London, London SW7 2AZ, UK c Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK b
a r t i c l e
i n f o
Article history: Received 15 March 2013 In final form 26 April 2013 Available online 9 May 2013
a b s t r a c t The relaxation properties in the dilute-gas limit have been calculated by the classical trajectory (CT) method for a gas consisting of chain-like molecules that are rigid and interact through site–site Lennard–Jones 12–6 potentials. Results are reported for volume viscosity gV , rotational collision number frot and the ratio of the rotational to self-diffusion coefficient Drot =D. The results indicate that the volume viscosity increases with temperature and decreases with chain length. The rotational relaxation of chains is efficient, as it takes of the order of 1.75–2.7 collisions to attain equilibrium. The rotational collision number is only weakly temperature dependent. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The relaxation phenomenon in gases is a consequence of the energy exchange between internal and translational modes of colliding molecules. It gives rise on a macroscopic level to sound absorption, increase in thermal conductivity and to the existence of volume viscosity in polyatomic gases. The last few decades have seen major advances in our ability to calculate the relaxation properties of simple, dilute molecular gases directly from full, ab initio intermolecular potentials [1–6]. However, very few systematic studies have been performed on how the molecular geometry influences the relaxation properties. If anything they are limited to early studies [7–11], where the molecules were modelled as hard bodies that interact only on contact. In a recent paper [12] we reported the calculation of the transport properties of a dilute gas consisting of chain-like molecules. The chains were modelled as linear rigid rotors composed of tangentially-joined spherical segments that interact through Lennard–Jones (LJ) 12–6 potentials. The choice of the LJ 12–6 potential was not necessarily governed by how realistically it can represent the actual interaction between real molecules, but by its widespread use as a model potential. Its simplicity allowed us to easily change the anisotropy of the overall potential, by varying the chain length, and investigate its influence on transport properties. In the present Letter we extend our study to the relaxation properties of a dilute gas composed of Lennard–Jones chains in order to ascertain what role the length of the molecules plays. The ⇑ Corresponding author. E-mail address: [email protected] (V. Vesovic). 0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.04.067
calculations have been based on formal kinetic theory, which provides a unified description of transport and relaxation phenomena in a dilute gas in terms of generalized cross sections. The molecular collisions, which govern the generalized cross sections, have been computed from the intermolecular potential using classical trajectories. We have limited our investigation to the contributions that arise from rotational relaxation only, as the nature of the intermolecular potential and of the chains precludes the investigation of the vibrational relaxation processes. In particular, we report the volume viscosity, rotational collision number and the ratio of diffusion of rotational energy to that of mass. 2. Theory The rotational degrees of freedom present in polyatomic gases allow for the storage of energy and, through collisions, for the exchange of energy between translational and rotational modes. If the gas is in a state where rotational and translational energy are not in thermodynamic equilibrium with each other, it takes a finite time for the rotational modes to relax. This is manifested, at the observable level, as volume viscosity (also known as bulk viscosity). The volume viscosity thus provides insight into the relaxation phenomena and can be inferred from measurements of the absorption and dispersion of ultrasonic waves in the polyatomic gas. In the limit of zero density the volume viscosity can be expressed as ðnÞ
gV ¼
fgV kB crot kB T ; c2V hv i0 Sð0 0 0 1Þ
ð1Þ
where hv i0 ¼ 4ðkB T=pmÞ1=2 is the average relative thermal speed, kB is Boltzmann’s constant, m is the mass of the chain and T is the temperature. The quantity crot is the contribution of the rotational
38
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
degrees of freedom to the isochoric heat capacity cV . As the rigid chains are assumed to be linear, there are only two rotational degrees of freedom, and as the calculation is done classically the ratio kB crot =c2V takes the value of 0:16. The quantity Sð0 0 0 1Þ is the generalized cross section that measures the rotational relaxation process. The notation and conventions employed in labelling the generalized cross sections are fully described elsewhere [13–15]. ðnÞ The quantity fgV is the nth-order correction factor that can be expressed in terms of generalized cross sections [16–19]. In this letter we have calculated up to the third-order correction factor for volume viscosity in line with our previous work [4,20]. As discussed in our work on the transport properties of Lennard–Jones chains [12], we refrained from calculating absolute values of the volume viscosity for specific LJ 12–6 parameters, as the objective of this letter is to examine the influence of the length of a chain and the resulting intermolecular potential anisotropy on the relaxation properties. Therefore, in line with the literature [12,21,22], we define the reduced volume viscosity as follows:
r2 gV ¼ gV pffiffiffiffiffiffiffiffiffi ; ms
ð2Þ
where ms is the mass of a segment, and the parameters and r are the well depth and the distance of zero potential for the LJ 12–6 potentials that govern the interactions between segments from different molecules. Further insight into the energy relaxation can be obtained if one compares the characteristic relaxation time for rotation, srot , to the characteristic time of flight, sf , defined as the average time between the collisions. The ratio of the two yields the rotational collision number, which measures the average number of collisions required to attain thermodynamic equilibrium between translational and rotational energy and is given by
frot ¼
srot 4 c2V ½gV 1 4 Sð2 0 0 0Þ ¼ ¼ ; sf p kB crot ½g1 p Sð0 0 0 1Þ
ð3Þ
where ½gV 1 and ½g1 denote the first-order approximations for volume viscosity and shear viscosity, respectively. The self-diffusion coefficient measures the rate at which a target molecule diffuses through the gas, relative to the remainder of the molecules. In a similar fashion one can define the self-diffusion coefficient for the rotational energy, Drot , which measures the rate at which rotational energy is transported through the gas by the molecules [18,23,24]. The concept of the self-diffusion coefficient for the rotational energy has been found very useful in the kinetic theory of polyatomic gases, particularly in analyzing the thermal conductivity, where it arises naturally as a direct result of kinetic theory [25–27]. The ratio of the two self-diffusion coefficients, Drot =D, provides us with another way of analyzing the rotational relaxation in gases, as it furnishes further insight into the rate of transfer between the rotational and translational energy. In the limit of zero density the ratio Drot =D can be expressed as
Drot S0 ð1 0 0 0Þ ¼ ; Sð1001Þ Sð0 0 0 1Þ=2 D
and each site–site interaction was governed by the LJ 12–6 potential characterized by distance r and well depth . The distance between two neighbouring segments within a chain was fixed at r. The well depth was further used to define the reduced temperature, T ¼ kB T=. The precision of the cross sections Sð0 0 0 1Þ; Sð2 0 0 0Þ; S0 ð1 0 0 0Þ and Sð1 0 0 1Þ used in Eqs. (1), (3) and (4) is of the order of 0:2 0:3%, ðnÞ while that of the high-order correction factors fgV in Eq. (1) is about 0:2%. At the level of properties this translates to a precision of about 0:5% in the computed values of volume viscosity, rotational collision number and the ratio Drot =D.
3. Results 3.1. Volume viscosity Figure 1(a) illustrates the behaviour of the reduced volume viscosity, gV , as a function of the reduced temperature T for a number of selected chains of different length, while Figure 1(b) illustrates the behaviour of the reduced volume viscosity as a function of the number of segments for a number of selected reduced temperatures. We observe a similar behaviour to that of the transport properties studied previously [12]. The increase in the chain length leads to a decrease in the reduced volume viscosity. Similarly to the shear viscosity, thermal conductivity and self-diffusion coefficient, the volume viscosity increases with temperature, with the most rapid change observed for the shortest chain. The behaviour at high reduced temperatures, 5 < T < 50, can be accurately described by a T ð1=2þ2=aÞ temperature dependence. The parameter a increases as a function of the chain length from approximately 10 for the 2-segment chain to 35 for the 16-segment chain. The same temperature scaling was observed for the transport properties studied previously [12]. In particular, when the values of a obtained from the analysis of the shear viscosity are compared with those obtained from the volume viscosity, one observes that as the chain length increases the two values for a become similar to each other, indicating that the ratio of the two viscosities, at high reduced temperatures, becomes less temperature sensitive the longer the chain. For shear viscosity, as for self-diffusion and thermal conductivity, the T ð1=2þ2=aÞ scaling can be shown to arise from a purely repulsive, spherical intermolecular potential [11], thus leading to the hypothesis that at high temperatures the transport properties are dominated by the repulsive wall of the intermolecular potential, and can be in essence determined from the effective
ð4Þ
where the generalized cross section S0 ð1 0 0 0Þ governs the self-diffusion coefficient, while the generalized cross section Sð1 0 0 1Þ enters the expression for thermal conductivity [13]. The generalized cross sections for rigid Lennard–Jones chains entering Eqs. (1), (3) and (4) were computed by means of classical trajectories using a modified version of the TRAJECT software code [28]. As summarized fully in the paper on transport properties [12], the classical trajectories describing the collision process of two rigid chains were evaluated at 29 total energies. At each energy up to 107 trajectories were computed. The interactions between two chains were modelled as a combination of site–site potentials. The sites were placed at the centre of each segment,
Figure 1. Reduced volume viscosity of rigid Lennard–Jones chains obtained using a third-order approximation as a function of the reduced temperature T and the number of segments per chain ns .
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
repulsive, spherical potential. However, in the case of the volume viscosity the repulsive part of the full potential cannot be replaced by the effective spherical potential, since the volume viscosity in a dilute gas is governed by the anisotropy and for a polyatomic gas interacting through an effective spherical potential it tends to infinity. We next consider the contribution of the higher-order correcðnÞ tion factor fgV to the volume viscosity. Figure 2(a) illustrates the temperature dependence of the second-order volume viscosity corð2Þ rection factor fgV for a number of selected chains of different ð2Þ length, while Figure 2(b) illustrates the behaviour of fgV as a function of the number of segments for a number of selected reduced ð2Þ temperatures. The second-order correction factor fgV exhibits a very shallow minimum at low reduced temperatures and then increases, before beginning to level off at higher reduced temperatures. At very low reduced temperatures the magnitude of the correction increases with the chain length. At higher reduced temperatures we observe a very interesting behaviour not encountered in the study of transport properties. The magnitude of the secondorder correction is largest for the 2-segment chain. As the chain gets longer the magnitude of the correction decreases, and at reduced temperatures above T ¼ 10 for chains longer than two segð2Þ ments fgV is approximately constant, of the order of 1:011 to 1:018. A similar behaviour was observed for real polyatomic gases. For CO2 the second-order volume viscosity correction was approximately 1:03 at reduced temperatures of T ¼ 3, while that for the shorter N2 molecule was of the order of 1:15 1:20, depending on the potential used, at approximately the same reduced temperature [3,29]. The results of the current letter further support the observation that the magnitude of the second-order volume viscosity correction factor decreases with an increase in moment of inertia and anisotropy. Hence, the importance of the second-order correction factor, as first reported by Turfa et al. [30], is only relevant for molecules with weakly anisotropic intermolecular potentials like N2 and CH4 , where its inclusion is essential for the correct analysis of the experimental results [4,29]. The third-order volume viscosity correction factors are very similar to the second-order ones. Differences of at most 0:15% are observed for chains longer than three segments. For shorter chains the differences are marginally larger, reaching a maximum value of 0:3% for the 2-segment chain at the highest reduced temperature. Although the differences between the third-order and second-order results are larger than those observed for shear viscosity, they are still sufficiently small that we can conclude that
ð2Þ
Figure 2. Second-order volume viscosity correction factor fgV of rigid Lennard– Jones chains as a function of the reduced temperature T and the number of segments per chain ns .
39
the basis function expansion in rotational energy converges rapidly for all the chains studied in this letter. 3.2. Rotational collision number Figure 3(a) illustrates the behaviour of the rotational collision number, frot , as a function of the reduced temperature T for a number of selected chains of different length. The increase in the chain length leads to a decrease in the rotational collision number. The variation is relatively small and longer chains, n > 2, take on average about two collisions to attain equilibrium between the translational and rotational degrees of freedom. The temperature dependence is interesting insofar as we observe both a decrease and increase of the rotational collision number at the lower reduced temperatures. At higher reduced temperatures all the chains exhibit rotational collision numbers that are either constant (the longer chains) or weakly increasing with temperature (the shorter chains). For model molecules that interact through a hard-body potential, both shear and volume viscosity have identical temperature dependence, as was demonstrated for rough spheres, loaded spheres and spherocylinders [7,9–11,33]. For model molecules that do not posses vibrational degrees of freedom the isochoric heat capacity is a constant and hence the rotational collision number, as defined in Eq. (3), is temperature independent [7,9,34,35]. We have demonstrated in our previous work [12] that at high reduced temperatures, T > 5, the temperature dependence of the transport properties is governed effectively by the repulsive part of the potential and that the hardness of the effective potential increases with increasing chain length. Thus, at high reduced temperatures the longer chains will behave more like hard bodies. Hence, we would expect their rotational collision numbers to be increasingly independent of temperature, as is confirmed by the observed behaviour illustrated in Figure 3(a). Figure 3(b) compares the behaviour of the rotational collision numbers of CH4 ; N2 and CO2 [1,3,4] with the results obtained in this letter for a selection of chain-like molecules. The calculations for real molecules were based on ab initio potentials, so it is not straightforward to assign an appropriate value of the well depth in order to adequately scale the temperature. As anisotropic potentials do not possess a unique energy scaling parameter, we have used the value corresponding to the well depth of the spherically-averaged ab initio potential as we did in our previous study [12]. Hence, there remains a level of uncertainty in the T position
Figure 3. Rotational collision number, frot , as a function of the reduced temperature T for rigid Lennard–Jones chains: ns ¼ 2, ns ¼ 3, ns ¼ 4, ns ¼ 6, ns ¼ 8, ns ¼ 16, for real molecules: CH4 ; N2 and CO2 , and the estimated values obtained using the Parker and the Brau-Jonkman (B–J) models for the 2-segment chain [31,32].
40
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
of the frot values of CH4 ; N2 and CO2 depicted in Figure 3(b). However, it is the magnitude of the frot values and their temperature dependence that we are primarily interested in. We observe that in general the behaviour of the chain-like molecules conforms to that of real molecules. As the anisotropy of the intermolecular potential increases from CH4 to CO2 , the rotational collision number decreases. This is not surprising since the more anisotropic the intermolecular potential, the easier it is for molecules to exchange rotational energy, and hence it takes on average less collisions to attain equilibrium. We also observe that as we move from CH4 to CO2 the temperature dependence of the rotational collision number decreases. This further illustrates that an increase in anisotropy leads to a weaker temperature dependence. We have previously argued [12] that the CO2 molecule could be modelled as a 2-segment chain if one is interested in transport properties which are only weakly dependent on the anisotropy of the potential. If translational–rotational energy exchange plays an important part, the present chain model is too anisotropic. To model CO2 as a 2-segment chain, one would have to reduce the site–site separation. The analysis of the behaviour of the rotational collision number carried out in this letter conforms to this finding. As evidenced by Figure 3(b), the intermolecular potential surface of the 2-segment chain is too anisotropic to represent the behaviour of the rotational collision number of CO2 . A number of models have been proposed [31,32,34,36–40] to explain the behaviour of the rotational collision number, but it seems that the predicted temperature variation is strongly model dependent. One of the well-known models, still employed as the basis for developing more advanced models, is due to Parker [31]. It was subsequently generalized by Brau and Jonkman [32]. In this letter we make use of their formula derived for impulsive collisions. Although derived originally for diatomics, the BrauJonkman relationship has been used to predict the behaviour of more complex molecules [41,42]. Figure 3(b) illustrates the temperature dependence for the 2-segment chain as predicted by the Parker and the Brau-Jonkman formulae. We have generated the values of frot by ensuring that both models reproduce the rotational collision number of the 2-segment chain at T ¼ 1. As can be observed in Figure 3(b) the temperature variations predicted by the Parker and the Brau-Jonkman formulae disagree significantly with that calculated in this letter. This is in agreement with an earlier finding that the Brau-Jonkman formula overestimates the rotational collision number of CO2 by quite a margin at high temperatures (e.g., by a factor of two at 800 K) [3]. 3.3. Drot =D ratio Figure 4(a) illustrates the behaviour of the ratio Drot =D as a function of the reduced temperature T for a number of selected chains of different length. We observe that at very low reduced temperatures the ratio decreases with increasing temperature. For 2-segment chains the initial decrease is followed by a rapid increase in the ratio Drot =D, that at higher reduced temperatures asymptotically leads to a constant value. For longer chains the minimum in Drot =D is less pronounced. At higher reduced temperatures the ratio of the two diffusion coefficients is again only weakly temperature dependent and for chains longer than three segments only weakly dependent on the chain length, reaching values of about 0:8, indicating that under these conditions the diffusion of the rotational energy is roughly 20% slower than that of the target molecule. In general the behaviour of the chain-like molecules conforms to that of real molecules, as illustrated in Figure 4(b), that compares the behaviour of the Drot =D ratio for CH4 ; N2 and CO2 with the results obtained in this letter for a selection of chain-like molecules. It is interesting to observe the behaviour of the Drot =D ratio at high reduced temperatures, where it attains nearly constant values. It was up to
Figure 4. Ratio Drot =D as a function of the reduced temperature T for rigid Lennard–Jones chains: ns ¼ 2, ns ¼ 3, ns ¼ 4, ns ¼ 6, ns ¼ 8,ns ¼ 16, for real molecules: CH4 ; N2 and CO2 , and the estimated values obtained using the Moraal and Snider approximation for the 2-segment chain (M–S) [43].
now assumed, based on the work on model potentials [23,44], as well as on diatomic [26] and small tetrahedral molecules [27], that at high enough temperatures the rotational energy is effectively ‘frozen’ (Sð0 0 0 1Þ ! 0; gV ! 1; frot ! 1; Sð1 0 0 1Þ ! S0 ð1 0 0 0Þ and Drot =D ! 1) and transported at the same rate as the molecules. For molecules with weakly anisotropic intermolecular potentials, and therefore large rotational collision numbers, this is indeed the case, as illustrated by the behaviour for CH4 and N2 in Figure 4(b). However, if the anisotropy of the intermolecular potential increases compared to that of CH4 and N2 , as is the case when one considers the chain-like molecules studied in this letter, the rotational relaxation remains efficient even at high temperatures, making the ratio Drot =D always significantly smaller than 1. The efficiency of the rotational relaxation does not improve markedly for chains longer than four segments, and the ratio Drot =D is no longer sensitive to the chain length. Moraal and Snider [43] have given an expression for Sð1 0 0 1Þ, valid at high temperatures, for homonuclear, diatomic molecules within the distorted-wave Born approximation [45,46]. By making use of this result, one can approximate the ratio Drot =D for the chains considered in this letter as [25]
Drot S0 ð1 0 0 0Þ 0 : D S ð1 0 0 0Þ þ 2Sð0 0 0 1Þ=3
ð5Þ
For 2-segment chains the values obtained using their model are also given in Figure 4(b). As can be observed they significantly underestimate the values calculated in this letter. For diatomic molecules Sandler [44] proposed the following relationship between the Drot =D ratio and the rotational collision number frot :
Drot 0:27 0:44 0:90 ¼1þ 2 3 ; frot D frot frot
ð6Þ
which is based on the results obtained from simulations of spherocylinders. The model overestimates the values of the 2-segment chain by approximately 10%, providing we limit our investigation to the reduced temperatures above 1:5. If all the reduced temperatures are considered, our results indicate that there is no simple relationship between the two measures of rotational relaxation. We find that the lower values of the rotational collision number correspond to two different values of the Drot =D ratio, one at a low and the other at a high reduced temperature. As the chain length increases it becomes evident that the Drot =D ratio is not a
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
simple function of the rotational collision number as postulated by Eq. (6). 4. Summary and conclusions The volume viscosity, the rotational collision number, and the ratio of the rotational to the self-diffusion coefficient have been calculated for a dilute gas consisting of rigid chain molecules in the reduced temperature range 0.3–50. The calculations indicate that the volume viscosity increases with temperature and decreases with chain length. At high temperatures the observed temperature dependence is the same as that of the shear viscosity, but with a different effective power, the difference between the two progressively becoming smaller as the length of the chain increases. The higher-order correction factor has been calculated up to the third order and the observed temperature dependence is similar to that of real systems. The magnitude of the correction factor is highest for the 2-segment chain reaching a value of 1:034 at the highest temperature. The rotational collision number is a weak function of both the temperature and the number of segments. The rotational relaxation process is relatively fast and the longer chains take on average two collisions to attain equilibrium. The Brau-Jonkman model and the well-known Parker model do not reproduce the temperature dependence observed in this letter. As a result of the fast rotational relaxation, the ratio Drot =D does not tend to unity at high temperatures, although it does reach a constant asymptotic value that decreases with an increase in the chain length. For longer chains the asymptotic value is of the order of 0:8. As the chain length increases it also becomes evident that the Drot =D ratio is not a simple function of the rotational collision number, as assumed previously based on studies of smaller molecules. Acknowledgments N.R. gratefully acknowledges funding from the Qatar Carbonates and Carbon Storage Research Centre (QCCSRC), provided jointly by Qatar Petroleum, Shell, and the Qatar Science & Technology Park. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cplett.2013.04. 067. References [1] E.L. Heck, A.S. Dickinson, Mol. Phys. 81 (1994) 1325.
41
[2] E.L. Heck, A.S. Dickinson, Physica A 217 (1995) 107. [3] S. Bock, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 121 (2004) 4117. [4] R. Hellmann, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 130 (2009) 124309. [5] R. Hellmann, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 131 (2009) 014303. [6] R. Hellmann, E. Bich, E. Vogel, V. Vesovic, Phys. Chem. Chem. Phys. 13 (2011) 13749. [7] N.F. Sather, J.S. Dahler, J. Chem. Phys. 35 (1961) 2029. [8] N.F. Sather, J.S. Dahler, J. Chem. Phys. 37 (1962) 1947. [9] S.I. Sandler, J.S. Dahler, J. Chem. Phys. 43 (1965) 1750. [10] S.I. Sandler, J.S. Dahler, J. Chem. Phys. 44 (1966) 1229. [11] S. Chapman, T.G. Cowling, The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Cambridge University Press, 1991. [12] R. Hellmann, N. Riesco, V. Vesovic, J. Chem. Phys. 138 (2013) 084309. [13] F.R.W. McCourt, J.J.M. Beenakker, W.E. Köhler, I. Kušcˇer, Nonequilibrium Phenomena in Polyatomic Gases, vol. 1, Oxford Science Publications, 1990. [14] S. Bock, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 117 (2002) 2151. [15] R. Hellmann, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 129 (2008) 064302. [16] S. Bock, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 120 (2004) 7987. [17] Y. Kagan, L. Maksimov, Sov. Phys. JETP 14 (1962) 604. [18] L.A. Viehland, E.A. Mason, S.I. Sandler, J. Chem. Phys. 68 (1978) 5277. [19] G.C. Maitland, M. Mustafa, W.A. Wakeham, Faraday Trans. 2 (79) (1983) 1425. [20] R. Hellmann, E. Bich, E. Vogel, V. Vesovic, J. Chem. Eng. Data 57 (2012) 1312. [21] G. Galliéro, C. Boned, Phys. Rev. E 79 (2009) 021201. [22] G. Galliéro, C. Boned, Phys. Rev. E 80 (2009) 061202. [23] E.A. Mason, L. Monchick, J. Chem. Phys. 36 (1962) 1622. [24] L. Monchick, A.N.G. Pereira, E.A. Mason, J. Chem. Phys. 42 (1965) 3241. [25] E. Bich, S. Bock, E. Vogel, Physica A 311 (2002) 59. [26] J. Millat, W.A. Wakeham, J. Phys. Chem. Ref. Data 18 (1989) 565. [27] M.J. Assael, J. Millat, V. Vesovic, W.A. Wakeham, J. Phys. Chem. Ref. Data 19 (1990) 1137. [28] E.L. Heck, A.S. Dickinson, Comput. Phys. Commun. 95 (1996) 190. [29] E.L. Heck, A.S. Dickinson, V. Vesovic, Mol. Phys. 83 (1994) 907. [30] A.F. Turfa, H.F.P. Knaap, B.J. Thijsse, J.J.M. Beenakker, Physica A 112 (1982) 18. [31] J.G. Parker, Phys. Fluids 2 (1959) 449. [32] C.A. Brau, R.M. Jonkman, J. Chem. Phys. 52 (1970) 477. [33] J.S. Dahler, N.F. Sather, J. Chem. Phys. 38 (1963) 2363. [34] B. Widom, J. Chem. Phys. 32 (1960) 913. [35] J.D. Lambert, Vibrational and Rotational Relaxation in Gases, Clarendon Press, 1977. [36] F.J. Zeleznik, J. Chem. Phys. 47 (1967) 3410. [37] C. Nyeland, J. Chem. Phys. 46 (1967) 63. [38] J. Davis, R.G. Dominy, J.K. Harvey, M.N. Macrossan, J. Fluid Mech. 135 (1983) 355. [39] I.D. Boyd, Phys. Fluids 2 (1990) 447. [40] K. Koura, Phys. Fluids 8 (1996) 1336. [41] V. Vesovic, W.A. Wakeham, G.A. Olchowy, J.V. Sengers, J.T.R. Watson, J. Millat, J. Phys. Chem. Ref. Data 19 (1990) 763. [42] J. Millat, M. Mustafa, M. Ross, W.A. Wakeham, M. Zalaf, Physica A 145 (1987) 461. [43] H. Moraal, R. Snider, Chem. Phys. Lett. 9 (1971) 401. [44] S.I. Sandler, Phys. Fluids 11 (1968) 2549. [45] F.M. Chen, H. Moraal, R.F. Snider, J. Chem. Phys. 57 (1972) 542. [46] R.F. Snider, Physica 78 (1974) 387.
Contents lists available at SciVerse ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Calculation of the relaxation properties of a dilute gas consisting of Lennard–Jones chains Robert Hellmann a, Nicolas Riesco b,c, Velisa Vesovic c,⇑ a
Institut für Chemie, Universität Rostock, 18059 Rostock, Germany Qatar Carbonates and Carbon Storage Research Centre (QCCSRC), Imperial College London, London SW7 2AZ, UK c Department of Earth Science and Engineering, Imperial College London, London SW7 2AZ, UK b
a r t i c l e
i n f o
Article history: Received 15 March 2013 In final form 26 April 2013 Available online 9 May 2013
a b s t r a c t The relaxation properties in the dilute-gas limit have been calculated by the classical trajectory (CT) method for a gas consisting of chain-like molecules that are rigid and interact through site–site Lennard–Jones 12–6 potentials. Results are reported for volume viscosity gV , rotational collision number frot and the ratio of the rotational to self-diffusion coefficient Drot =D. The results indicate that the volume viscosity increases with temperature and decreases with chain length. The rotational relaxation of chains is efficient, as it takes of the order of 1.75–2.7 collisions to attain equilibrium. The rotational collision number is only weakly temperature dependent. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The relaxation phenomenon in gases is a consequence of the energy exchange between internal and translational modes of colliding molecules. It gives rise on a macroscopic level to sound absorption, increase in thermal conductivity and to the existence of volume viscosity in polyatomic gases. The last few decades have seen major advances in our ability to calculate the relaxation properties of simple, dilute molecular gases directly from full, ab initio intermolecular potentials [1–6]. However, very few systematic studies have been performed on how the molecular geometry influences the relaxation properties. If anything they are limited to early studies [7–11], where the molecules were modelled as hard bodies that interact only on contact. In a recent paper [12] we reported the calculation of the transport properties of a dilute gas consisting of chain-like molecules. The chains were modelled as linear rigid rotors composed of tangentially-joined spherical segments that interact through Lennard–Jones (LJ) 12–6 potentials. The choice of the LJ 12–6 potential was not necessarily governed by how realistically it can represent the actual interaction between real molecules, but by its widespread use as a model potential. Its simplicity allowed us to easily change the anisotropy of the overall potential, by varying the chain length, and investigate its influence on transport properties. In the present Letter we extend our study to the relaxation properties of a dilute gas composed of Lennard–Jones chains in order to ascertain what role the length of the molecules plays. The ⇑ Corresponding author. E-mail address: [email protected] (V. Vesovic). 0009-2614/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cplett.2013.04.067
calculations have been based on formal kinetic theory, which provides a unified description of transport and relaxation phenomena in a dilute gas in terms of generalized cross sections. The molecular collisions, which govern the generalized cross sections, have been computed from the intermolecular potential using classical trajectories. We have limited our investigation to the contributions that arise from rotational relaxation only, as the nature of the intermolecular potential and of the chains precludes the investigation of the vibrational relaxation processes. In particular, we report the volume viscosity, rotational collision number and the ratio of diffusion of rotational energy to that of mass. 2. Theory The rotational degrees of freedom present in polyatomic gases allow for the storage of energy and, through collisions, for the exchange of energy between translational and rotational modes. If the gas is in a state where rotational and translational energy are not in thermodynamic equilibrium with each other, it takes a finite time for the rotational modes to relax. This is manifested, at the observable level, as volume viscosity (also known as bulk viscosity). The volume viscosity thus provides insight into the relaxation phenomena and can be inferred from measurements of the absorption and dispersion of ultrasonic waves in the polyatomic gas. In the limit of zero density the volume viscosity can be expressed as ðnÞ
gV ¼
fgV kB crot kB T ; c2V hv i0 Sð0 0 0 1Þ
ð1Þ
where hv i0 ¼ 4ðkB T=pmÞ1=2 is the average relative thermal speed, kB is Boltzmann’s constant, m is the mass of the chain and T is the temperature. The quantity crot is the contribution of the rotational
38
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
degrees of freedom to the isochoric heat capacity cV . As the rigid chains are assumed to be linear, there are only two rotational degrees of freedom, and as the calculation is done classically the ratio kB crot =c2V takes the value of 0:16. The quantity Sð0 0 0 1Þ is the generalized cross section that measures the rotational relaxation process. The notation and conventions employed in labelling the generalized cross sections are fully described elsewhere [13–15]. ðnÞ The quantity fgV is the nth-order correction factor that can be expressed in terms of generalized cross sections [16–19]. In this letter we have calculated up to the third-order correction factor for volume viscosity in line with our previous work [4,20]. As discussed in our work on the transport properties of Lennard–Jones chains [12], we refrained from calculating absolute values of the volume viscosity for specific LJ 12–6 parameters, as the objective of this letter is to examine the influence of the length of a chain and the resulting intermolecular potential anisotropy on the relaxation properties. Therefore, in line with the literature [12,21,22], we define the reduced volume viscosity as follows:
r2 gV ¼ gV pffiffiffiffiffiffiffiffiffi ; ms
ð2Þ
where ms is the mass of a segment, and the parameters and r are the well depth and the distance of zero potential for the LJ 12–6 potentials that govern the interactions between segments from different molecules. Further insight into the energy relaxation can be obtained if one compares the characteristic relaxation time for rotation, srot , to the characteristic time of flight, sf , defined as the average time between the collisions. The ratio of the two yields the rotational collision number, which measures the average number of collisions required to attain thermodynamic equilibrium between translational and rotational energy and is given by
frot ¼
srot 4 c2V ½gV 1 4 Sð2 0 0 0Þ ¼ ¼ ; sf p kB crot ½g1 p Sð0 0 0 1Þ
ð3Þ
where ½gV 1 and ½g1 denote the first-order approximations for volume viscosity and shear viscosity, respectively. The self-diffusion coefficient measures the rate at which a target molecule diffuses through the gas, relative to the remainder of the molecules. In a similar fashion one can define the self-diffusion coefficient for the rotational energy, Drot , which measures the rate at which rotational energy is transported through the gas by the molecules [18,23,24]. The concept of the self-diffusion coefficient for the rotational energy has been found very useful in the kinetic theory of polyatomic gases, particularly in analyzing the thermal conductivity, where it arises naturally as a direct result of kinetic theory [25–27]. The ratio of the two self-diffusion coefficients, Drot =D, provides us with another way of analyzing the rotational relaxation in gases, as it furnishes further insight into the rate of transfer between the rotational and translational energy. In the limit of zero density the ratio Drot =D can be expressed as
Drot S0 ð1 0 0 0Þ ¼ ; Sð1001Þ Sð0 0 0 1Þ=2 D
and each site–site interaction was governed by the LJ 12–6 potential characterized by distance r and well depth . The distance between two neighbouring segments within a chain was fixed at r. The well depth was further used to define the reduced temperature, T ¼ kB T=. The precision of the cross sections Sð0 0 0 1Þ; Sð2 0 0 0Þ; S0 ð1 0 0 0Þ and Sð1 0 0 1Þ used in Eqs. (1), (3) and (4) is of the order of 0:2 0:3%, ðnÞ while that of the high-order correction factors fgV in Eq. (1) is about 0:2%. At the level of properties this translates to a precision of about 0:5% in the computed values of volume viscosity, rotational collision number and the ratio Drot =D.
3. Results 3.1. Volume viscosity Figure 1(a) illustrates the behaviour of the reduced volume viscosity, gV , as a function of the reduced temperature T for a number of selected chains of different length, while Figure 1(b) illustrates the behaviour of the reduced volume viscosity as a function of the number of segments for a number of selected reduced temperatures. We observe a similar behaviour to that of the transport properties studied previously [12]. The increase in the chain length leads to a decrease in the reduced volume viscosity. Similarly to the shear viscosity, thermal conductivity and self-diffusion coefficient, the volume viscosity increases with temperature, with the most rapid change observed for the shortest chain. The behaviour at high reduced temperatures, 5 < T < 50, can be accurately described by a T ð1=2þ2=aÞ temperature dependence. The parameter a increases as a function of the chain length from approximately 10 for the 2-segment chain to 35 for the 16-segment chain. The same temperature scaling was observed for the transport properties studied previously [12]. In particular, when the values of a obtained from the analysis of the shear viscosity are compared with those obtained from the volume viscosity, one observes that as the chain length increases the two values for a become similar to each other, indicating that the ratio of the two viscosities, at high reduced temperatures, becomes less temperature sensitive the longer the chain. For shear viscosity, as for self-diffusion and thermal conductivity, the T ð1=2þ2=aÞ scaling can be shown to arise from a purely repulsive, spherical intermolecular potential [11], thus leading to the hypothesis that at high temperatures the transport properties are dominated by the repulsive wall of the intermolecular potential, and can be in essence determined from the effective
ð4Þ
where the generalized cross section S0 ð1 0 0 0Þ governs the self-diffusion coefficient, while the generalized cross section Sð1 0 0 1Þ enters the expression for thermal conductivity [13]. The generalized cross sections for rigid Lennard–Jones chains entering Eqs. (1), (3) and (4) were computed by means of classical trajectories using a modified version of the TRAJECT software code [28]. As summarized fully in the paper on transport properties [12], the classical trajectories describing the collision process of two rigid chains were evaluated at 29 total energies. At each energy up to 107 trajectories were computed. The interactions between two chains were modelled as a combination of site–site potentials. The sites were placed at the centre of each segment,
Figure 1. Reduced volume viscosity of rigid Lennard–Jones chains obtained using a third-order approximation as a function of the reduced temperature T and the number of segments per chain ns .
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
repulsive, spherical potential. However, in the case of the volume viscosity the repulsive part of the full potential cannot be replaced by the effective spherical potential, since the volume viscosity in a dilute gas is governed by the anisotropy and for a polyatomic gas interacting through an effective spherical potential it tends to infinity. We next consider the contribution of the higher-order correcðnÞ tion factor fgV to the volume viscosity. Figure 2(a) illustrates the temperature dependence of the second-order volume viscosity corð2Þ rection factor fgV for a number of selected chains of different ð2Þ length, while Figure 2(b) illustrates the behaviour of fgV as a function of the number of segments for a number of selected reduced ð2Þ temperatures. The second-order correction factor fgV exhibits a very shallow minimum at low reduced temperatures and then increases, before beginning to level off at higher reduced temperatures. At very low reduced temperatures the magnitude of the correction increases with the chain length. At higher reduced temperatures we observe a very interesting behaviour not encountered in the study of transport properties. The magnitude of the secondorder correction is largest for the 2-segment chain. As the chain gets longer the magnitude of the correction decreases, and at reduced temperatures above T ¼ 10 for chains longer than two segð2Þ ments fgV is approximately constant, of the order of 1:011 to 1:018. A similar behaviour was observed for real polyatomic gases. For CO2 the second-order volume viscosity correction was approximately 1:03 at reduced temperatures of T ¼ 3, while that for the shorter N2 molecule was of the order of 1:15 1:20, depending on the potential used, at approximately the same reduced temperature [3,29]. The results of the current letter further support the observation that the magnitude of the second-order volume viscosity correction factor decreases with an increase in moment of inertia and anisotropy. Hence, the importance of the second-order correction factor, as first reported by Turfa et al. [30], is only relevant for molecules with weakly anisotropic intermolecular potentials like N2 and CH4 , where its inclusion is essential for the correct analysis of the experimental results [4,29]. The third-order volume viscosity correction factors are very similar to the second-order ones. Differences of at most 0:15% are observed for chains longer than three segments. For shorter chains the differences are marginally larger, reaching a maximum value of 0:3% for the 2-segment chain at the highest reduced temperature. Although the differences between the third-order and second-order results are larger than those observed for shear viscosity, they are still sufficiently small that we can conclude that
ð2Þ
Figure 2. Second-order volume viscosity correction factor fgV of rigid Lennard– Jones chains as a function of the reduced temperature T and the number of segments per chain ns .
39
the basis function expansion in rotational energy converges rapidly for all the chains studied in this letter. 3.2. Rotational collision number Figure 3(a) illustrates the behaviour of the rotational collision number, frot , as a function of the reduced temperature T for a number of selected chains of different length. The increase in the chain length leads to a decrease in the rotational collision number. The variation is relatively small and longer chains, n > 2, take on average about two collisions to attain equilibrium between the translational and rotational degrees of freedom. The temperature dependence is interesting insofar as we observe both a decrease and increase of the rotational collision number at the lower reduced temperatures. At higher reduced temperatures all the chains exhibit rotational collision numbers that are either constant (the longer chains) or weakly increasing with temperature (the shorter chains). For model molecules that interact through a hard-body potential, both shear and volume viscosity have identical temperature dependence, as was demonstrated for rough spheres, loaded spheres and spherocylinders [7,9–11,33]. For model molecules that do not posses vibrational degrees of freedom the isochoric heat capacity is a constant and hence the rotational collision number, as defined in Eq. (3), is temperature independent [7,9,34,35]. We have demonstrated in our previous work [12] that at high reduced temperatures, T > 5, the temperature dependence of the transport properties is governed effectively by the repulsive part of the potential and that the hardness of the effective potential increases with increasing chain length. Thus, at high reduced temperatures the longer chains will behave more like hard bodies. Hence, we would expect their rotational collision numbers to be increasingly independent of temperature, as is confirmed by the observed behaviour illustrated in Figure 3(a). Figure 3(b) compares the behaviour of the rotational collision numbers of CH4 ; N2 and CO2 [1,3,4] with the results obtained in this letter for a selection of chain-like molecules. The calculations for real molecules were based on ab initio potentials, so it is not straightforward to assign an appropriate value of the well depth in order to adequately scale the temperature. As anisotropic potentials do not possess a unique energy scaling parameter, we have used the value corresponding to the well depth of the spherically-averaged ab initio potential as we did in our previous study [12]. Hence, there remains a level of uncertainty in the T position
Figure 3. Rotational collision number, frot , as a function of the reduced temperature T for rigid Lennard–Jones chains: ns ¼ 2, ns ¼ 3, ns ¼ 4, ns ¼ 6, ns ¼ 8, ns ¼ 16, for real molecules: CH4 ; N2 and CO2 , and the estimated values obtained using the Parker and the Brau-Jonkman (B–J) models for the 2-segment chain [31,32].
40
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
of the frot values of CH4 ; N2 and CO2 depicted in Figure 3(b). However, it is the magnitude of the frot values and their temperature dependence that we are primarily interested in. We observe that in general the behaviour of the chain-like molecules conforms to that of real molecules. As the anisotropy of the intermolecular potential increases from CH4 to CO2 , the rotational collision number decreases. This is not surprising since the more anisotropic the intermolecular potential, the easier it is for molecules to exchange rotational energy, and hence it takes on average less collisions to attain equilibrium. We also observe that as we move from CH4 to CO2 the temperature dependence of the rotational collision number decreases. This further illustrates that an increase in anisotropy leads to a weaker temperature dependence. We have previously argued [12] that the CO2 molecule could be modelled as a 2-segment chain if one is interested in transport properties which are only weakly dependent on the anisotropy of the potential. If translational–rotational energy exchange plays an important part, the present chain model is too anisotropic. To model CO2 as a 2-segment chain, one would have to reduce the site–site separation. The analysis of the behaviour of the rotational collision number carried out in this letter conforms to this finding. As evidenced by Figure 3(b), the intermolecular potential surface of the 2-segment chain is too anisotropic to represent the behaviour of the rotational collision number of CO2 . A number of models have been proposed [31,32,34,36–40] to explain the behaviour of the rotational collision number, but it seems that the predicted temperature variation is strongly model dependent. One of the well-known models, still employed as the basis for developing more advanced models, is due to Parker [31]. It was subsequently generalized by Brau and Jonkman [32]. In this letter we make use of their formula derived for impulsive collisions. Although derived originally for diatomics, the BrauJonkman relationship has been used to predict the behaviour of more complex molecules [41,42]. Figure 3(b) illustrates the temperature dependence for the 2-segment chain as predicted by the Parker and the Brau-Jonkman formulae. We have generated the values of frot by ensuring that both models reproduce the rotational collision number of the 2-segment chain at T ¼ 1. As can be observed in Figure 3(b) the temperature variations predicted by the Parker and the Brau-Jonkman formulae disagree significantly with that calculated in this letter. This is in agreement with an earlier finding that the Brau-Jonkman formula overestimates the rotational collision number of CO2 by quite a margin at high temperatures (e.g., by a factor of two at 800 K) [3]. 3.3. Drot =D ratio Figure 4(a) illustrates the behaviour of the ratio Drot =D as a function of the reduced temperature T for a number of selected chains of different length. We observe that at very low reduced temperatures the ratio decreases with increasing temperature. For 2-segment chains the initial decrease is followed by a rapid increase in the ratio Drot =D, that at higher reduced temperatures asymptotically leads to a constant value. For longer chains the minimum in Drot =D is less pronounced. At higher reduced temperatures the ratio of the two diffusion coefficients is again only weakly temperature dependent and for chains longer than three segments only weakly dependent on the chain length, reaching values of about 0:8, indicating that under these conditions the diffusion of the rotational energy is roughly 20% slower than that of the target molecule. In general the behaviour of the chain-like molecules conforms to that of real molecules, as illustrated in Figure 4(b), that compares the behaviour of the Drot =D ratio for CH4 ; N2 and CO2 with the results obtained in this letter for a selection of chain-like molecules. It is interesting to observe the behaviour of the Drot =D ratio at high reduced temperatures, where it attains nearly constant values. It was up to
Figure 4. Ratio Drot =D as a function of the reduced temperature T for rigid Lennard–Jones chains: ns ¼ 2, ns ¼ 3, ns ¼ 4, ns ¼ 6, ns ¼ 8,ns ¼ 16, for real molecules: CH4 ; N2 and CO2 , and the estimated values obtained using the Moraal and Snider approximation for the 2-segment chain (M–S) [43].
now assumed, based on the work on model potentials [23,44], as well as on diatomic [26] and small tetrahedral molecules [27], that at high enough temperatures the rotational energy is effectively ‘frozen’ (Sð0 0 0 1Þ ! 0; gV ! 1; frot ! 1; Sð1 0 0 1Þ ! S0 ð1 0 0 0Þ and Drot =D ! 1) and transported at the same rate as the molecules. For molecules with weakly anisotropic intermolecular potentials, and therefore large rotational collision numbers, this is indeed the case, as illustrated by the behaviour for CH4 and N2 in Figure 4(b). However, if the anisotropy of the intermolecular potential increases compared to that of CH4 and N2 , as is the case when one considers the chain-like molecules studied in this letter, the rotational relaxation remains efficient even at high temperatures, making the ratio Drot =D always significantly smaller than 1. The efficiency of the rotational relaxation does not improve markedly for chains longer than four segments, and the ratio Drot =D is no longer sensitive to the chain length. Moraal and Snider [43] have given an expression for Sð1 0 0 1Þ, valid at high temperatures, for homonuclear, diatomic molecules within the distorted-wave Born approximation [45,46]. By making use of this result, one can approximate the ratio Drot =D for the chains considered in this letter as [25]
Drot S0 ð1 0 0 0Þ 0 : D S ð1 0 0 0Þ þ 2Sð0 0 0 1Þ=3
ð5Þ
For 2-segment chains the values obtained using their model are also given in Figure 4(b). As can be observed they significantly underestimate the values calculated in this letter. For diatomic molecules Sandler [44] proposed the following relationship between the Drot =D ratio and the rotational collision number frot :
Drot 0:27 0:44 0:90 ¼1þ 2 3 ; frot D frot frot
ð6Þ
which is based on the results obtained from simulations of spherocylinders. The model overestimates the values of the 2-segment chain by approximately 10%, providing we limit our investigation to the reduced temperatures above 1:5. If all the reduced temperatures are considered, our results indicate that there is no simple relationship between the two measures of rotational relaxation. We find that the lower values of the rotational collision number correspond to two different values of the Drot =D ratio, one at a low and the other at a high reduced temperature. As the chain length increases it becomes evident that the Drot =D ratio is not a
R. Hellmann et al. / Chemical Physics Letters 574 (2013) 37–41
simple function of the rotational collision number as postulated by Eq. (6). 4. Summary and conclusions The volume viscosity, the rotational collision number, and the ratio of the rotational to the self-diffusion coefficient have been calculated for a dilute gas consisting of rigid chain molecules in the reduced temperature range 0.3–50. The calculations indicate that the volume viscosity increases with temperature and decreases with chain length. At high temperatures the observed temperature dependence is the same as that of the shear viscosity, but with a different effective power, the difference between the two progressively becoming smaller as the length of the chain increases. The higher-order correction factor has been calculated up to the third order and the observed temperature dependence is similar to that of real systems. The magnitude of the correction factor is highest for the 2-segment chain reaching a value of 1:034 at the highest temperature. The rotational collision number is a weak function of both the temperature and the number of segments. The rotational relaxation process is relatively fast and the longer chains take on average two collisions to attain equilibrium. The Brau-Jonkman model and the well-known Parker model do not reproduce the temperature dependence observed in this letter. As a result of the fast rotational relaxation, the ratio Drot =D does not tend to unity at high temperatures, although it does reach a constant asymptotic value that decreases with an increase in the chain length. For longer chains the asymptotic value is of the order of 0:8. As the chain length increases it also becomes evident that the Drot =D ratio is not a simple function of the rotational collision number, as assumed previously based on studies of smaller molecules. Acknowledgments N.R. gratefully acknowledges funding from the Qatar Carbonates and Carbon Storage Research Centre (QCCSRC), provided jointly by Qatar Petroleum, Shell, and the Qatar Science & Technology Park. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.cplett.2013.04. 067. References [1] E.L. Heck, A.S. Dickinson, Mol. Phys. 81 (1994) 1325.
41
[2] E.L. Heck, A.S. Dickinson, Physica A 217 (1995) 107. [3] S. Bock, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 121 (2004) 4117. [4] R. Hellmann, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 130 (2009) 124309. [5] R. Hellmann, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 131 (2009) 014303. [6] R. Hellmann, E. Bich, E. Vogel, V. Vesovic, Phys. Chem. Chem. Phys. 13 (2011) 13749. [7] N.F. Sather, J.S. Dahler, J. Chem. Phys. 35 (1961) 2029. [8] N.F. Sather, J.S. Dahler, J. Chem. Phys. 37 (1962) 1947. [9] S.I. Sandler, J.S. Dahler, J. Chem. Phys. 43 (1965) 1750. [10] S.I. Sandler, J.S. Dahler, J. Chem. Phys. 44 (1966) 1229. [11] S. Chapman, T.G. Cowling, The mathematical theory of non-uniform gases: an account of the kinetic theory of viscosity, thermal conduction and diffusion in gases, Cambridge University Press, 1991. [12] R. Hellmann, N. Riesco, V. Vesovic, J. Chem. Phys. 138 (2013) 084309. [13] F.R.W. McCourt, J.J.M. Beenakker, W.E. Köhler, I. Kušcˇer, Nonequilibrium Phenomena in Polyatomic Gases, vol. 1, Oxford Science Publications, 1990. [14] S. Bock, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 117 (2002) 2151. [15] R. Hellmann, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 129 (2008) 064302. [16] S. Bock, E. Bich, E. Vogel, A.S. Dickinson, V. Vesovic, J. Chem. Phys. 120 (2004) 7987. [17] Y. Kagan, L. Maksimov, Sov. Phys. JETP 14 (1962) 604. [18] L.A. Viehland, E.A. Mason, S.I. Sandler, J. Chem. Phys. 68 (1978) 5277. [19] G.C. Maitland, M. Mustafa, W.A. Wakeham, Faraday Trans. 2 (79) (1983) 1425. [20] R. Hellmann, E. Bich, E. Vogel, V. Vesovic, J. Chem. Eng. Data 57 (2012) 1312. [21] G. Galliéro, C. Boned, Phys. Rev. E 79 (2009) 021201. [22] G. Galliéro, C. Boned, Phys. Rev. E 80 (2009) 061202. [23] E.A. Mason, L. Monchick, J. Chem. Phys. 36 (1962) 1622. [24] L. Monchick, A.N.G. Pereira, E.A. Mason, J. Chem. Phys. 42 (1965) 3241. [25] E. Bich, S. Bock, E. Vogel, Physica A 311 (2002) 59. [26] J. Millat, W.A. Wakeham, J. Phys. Chem. Ref. Data 18 (1989) 565. [27] M.J. Assael, J. Millat, V. Vesovic, W.A. Wakeham, J. Phys. Chem. Ref. Data 19 (1990) 1137. [28] E.L. Heck, A.S. Dickinson, Comput. Phys. Commun. 95 (1996) 190. [29] E.L. Heck, A.S. Dickinson, V. Vesovic, Mol. Phys. 83 (1994) 907. [30] A.F. Turfa, H.F.P. Knaap, B.J. Thijsse, J.J.M. Beenakker, Physica A 112 (1982) 18. [31] J.G. Parker, Phys. Fluids 2 (1959) 449. [32] C.A. Brau, R.M. Jonkman, J. Chem. Phys. 52 (1970) 477. [33] J.S. Dahler, N.F. Sather, J. Chem. Phys. 38 (1963) 2363. [34] B. Widom, J. Chem. Phys. 32 (1960) 913. [35] J.D. Lambert, Vibrational and Rotational Relaxation in Gases, Clarendon Press, 1977. [36] F.J. Zeleznik, J. Chem. Phys. 47 (1967) 3410. [37] C. Nyeland, J. Chem. Phys. 46 (1967) 63. [38] J. Davis, R.G. Dominy, J.K. Harvey, M.N. Macrossan, J. Fluid Mech. 135 (1983) 355. [39] I.D. Boyd, Phys. Fluids 2 (1990) 447. [40] K. Koura, Phys. Fluids 8 (1996) 1336. [41] V. Vesovic, W.A. Wakeham, G.A. Olchowy, J.V. Sengers, J.T.R. Watson, J. Millat, J. Phys. Chem. Ref. Data 19 (1990) 763. [42] J. Millat, M. Mustafa, M. Ross, W.A. Wakeham, M. Zalaf, Physica A 145 (1987) 461. [43] H. Moraal, R. Snider, Chem. Phys. Lett. 9 (1971) 401. [44] S.I. Sandler, Phys. Fluids 11 (1968) 2549. [45] F.M. Chen, H. Moraal, R.F. Snider, J. Chem. Phys. 57 (1972) 542. [46] R.F. Snider, Physica 78 (1974) 387.