- Email: [email protected]
Transport properties of a moderately dense gas
CHEhilCAL PHYSICS LE-lTERS
Volume 107. number 6
TRANSPORT PROPERTIES OF A MODERATELY Daniel G. FRIEND
15 June 1984
DENSE GAS
* and James C. RAINWATER
Chemical Engi,reering Science Division, h’ntional Engineenbrg Laboratory, h’ational Bureau of Standards. Boulder. Colorado 80303, USA Received 8 February
1984
The initial density dependences of both viscosity and thermal conductivity are calculated according to a microscopical ly based theory which includes cff~~ts due to collisional transfer (from only free two-body phase space), three-monomer collisions. and monomer-dimer collisions. A Lennard-Jones potential is used to model the interactions. Comparison of the calculated results with experiment (in reduced form) shows very good agreement for both viscosity and thermal conductivity over a wide temperature range.
1_ Introduction
p:
In this letter, we present results of a microscopically based theory for the initial density dependence of transport properties in a Lennard-Jones model gas and compare these results with experimental data for viscosity and thermal conductivity. Previous attempts to compare experimental first density corrections to the transport coefficients with theoretical models [l-7] have been less than satisfying for a number of reasons including systematic deviations with experiment in certain temperature ranges and the lack of a sound microscopic basis in some aspects of the research. Our basic premise is that the moderately dense gas can be modelled as a mixture of monomers and dimers. Collisional processes between pairs of monomers form the basis for the dilute gas, or ChapmanEnskog [S; 9, ch. 7; IO] transport coefficients, as well as contributing, through collisional transfer, to the linear-in-density correction. Collisions among three monomers and between a monomer and a dimer also contribute to the density correction. Other processes, including the effects of repeat collisions 1111, are not considered here. We thus subdivide the second transport virial coefficient as follows:
* NRC-NBS Postdoctoral 590’
Research Associate.
/Jo
[I
+(Bp +Bi3) +B;h’-D))p t ___I
= Po(l + &P)
(1)
,
where p represents either the viscosity q or thermal conductivity h, p is the number density, and the superscripts represent, respectively, the collisional transfer contribution, the three-monomer collisional contribution, and the contribution from monomerdimer collisions, each of which we discuss briefly below.
2. CoIlisional
transfer
contribution
The mechanism by which energy or momentum may be transported across a surface without mass transfer was discussed by Enskog for rigid spheres [lo]. The analogous non-locality expansion for repulsive potentials was carried out by Snider and Curtiss in I958 [ 121. For repulsive potentials, the truncation of the BBGKY hierarchy via the Bogliubov hypothesis and use of an operator [ 131 S which projects particle trajectories to the asymptotically free pre-collision phase can be carried out rigorously. When the process of dimerization is allowed, however, S can no longer perform the same function on that portion of relative phase space in which dirner fonnation occurs. The integrals associated with Curtiss non-locality expansion, then, can be evaluated only in 0 009-2614/84/S 0’ 00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 107. number 6
CHEhlICAL
PHYSICS
LETTERS
15 June 1984
the free portion of relative phase space. A pair of monomers is considered free if the two-body relative momentum and displacement are such that the particles are infinitely separated in the infinite past or future of the collision. In refs. [ 1,141, the explicit numerical integration of the collision integrals associated with the nonlocality expansion were performed only over the free portion of phase space. However, certain portions of those integrals, which were shown to reduce to the pressure virial B and its derivatives for repulsive potentials, were integrated over the entire two-body relative phase space. The resolution of this inconsistency has been explored in a separate publication [ 151 and here we shall only mention certain essential conclusions. Eqs. (124) and (136) of ref. [ 141 for the viscosity and thermal conductivity virials should be replaced, for intermolecular potentials allowing dimerization,
and, with more detail, in ref. [ 161. We note that eqs. (2) and (3) agree with eqs. (124) and (136) of Bennett and Curtiss only when the potential is purely repulsive (in which case the free portion of phase space is equiv$ent to the entire two-body phase space, and B,( ’ ) = f3FE’ = B). In ref. [7], Kuznetsov uses the Stogryn and Hirschfelder [4.17] division of the second pressure virial into its free and dirner contributions which, we have shown [ 151, is not equivalent to eqs. (4) and (5); rhus his results differ from eqs. (2) and (3). We have numerically evaluated eqs. (Z)-(5) (for the Lennard-Jones 12-6 potential) independently of previous calculations using a highly accurate Clenshaw-Curtis quadrature routine [15,16] and other numerical methods previously employed in efficient calculations of dilute gas transport collision integrals [ 18,191.
by
3. Collisions among three monomers
BY) = (/3/10R3/2)
-J ‘T
dB(ri‘) r-
(R( 15) - R(6))
dBiE’ it Tr-
+ $fI’)
AT’
d?fJtE) f dT’
(2)
and Br)
= (p/15,r3/“)
(R( 15) _ ~‘6’)
+ B(,“) d’B;E)
_$T’___
(3)
dT2 where 0 = 1 kT, R( * 5) and Rt6) are defined in ref. 1 [ 141 and BfV) and T d BiE’/dT are obtained from @L
43(p/~nl)3/2
and T dB’,E’ld T = -;fi X
jjd3r free
@J~F,,)~”
d3p $(r) e-flOtr)
,-flP’/nl
,
(3
by integrating over only the free portion of relative phase space, where $I is the potential and tn the mass. A discussion of eqs. (2) and (3) is given in ref. [ 151
In order to evaluate the contributions due to three-monomer collisions for repulsive potentials one must, in principle, solve the Choh-Uhlenbeck equation [ 1320], and its proper generalization to realistic potentials is by no means c!ear at present. However Enskog, in his theory for hard spheres [lo]. introduced an approximation which accounts for a third particle by altering the frequency of two-body collisions via a “shielding” effect. Hoffman and Curtiss [2,3 I] have shown how this approsimation can be applied to realistic potentials and derive explicit integral expressions for rhe contributions of this shielding to transport properties. and thus avoid the extreme complexity of three-body collisions dynamics. In the Hoffman-Curtis9 approach, certain multidimensional collision integrals must be evaluated numerically. An inner integration requires the calculation of rhe low-density radial distribution function which we have discussed previously [22]. We have proceeded to compute independently the three-body Hoffman-Curtiss integrals for the kmnard-Jones 12-6 potential. We obtain excellent agreement with the originally reported results 121, but with much less computer time. However, the results reported by Kuznetsov [7] for reduced temperatures of 12, li,3,4, and 5 differ from our results by as much as 45%. 591
Volume
107. number
4. Monomer-dimer
6
CHEMlCAL
PHYSICS
contriiution
The remaining contribution to the initial density dependence of transport coefficients comes from collisions between monomers and dimers. This process was fist explored by Stogryn and Hirschfelder [3,4] and has subsequently been utilized in the theories of Kim and Ross [5] and Kuznetsov [7]. Thepopulation of dimers in a moderately dense gas can be calculated by the algorithm of Stogryn and Hirschfelder which is an extension of Hill’s method [23] for finding the population of bound dimers in such a system. The Stogryn-Hirschfelder formulation then, includes both truly bound dimers (with negative total energy) and those that are metastably bound within the centrifugal well which exists for certain relative angular momentum states. The explicit division of phase space into free, bound, and metastable portions is discussed elsewhere [4,15,17] _ Kim and Ross use a broader definition of metastability which appears to incorporate collisions whose lifetimes can be much shorter that the inverse of the collision frequency_ The contributions to the second transport viral coefficients due to dimerization are equivalent to the dilute gas coefficients of a mixture of.monomers of density p with dimers of a specified density (as calculated via the Stogryn-Hirschfelder formalism [4,17] )_ Thus the Chapman-Enskog theory of mixtures [8;9, ch. 71 can be utilized and the contribution calculated provided that the interaction potential between dimers and monomers is known. Various methods I;xist for estimating the monomerdimer p0tentGl;e.g. angular averaging, to make the collision Wegral calculations tractable. In view of the uncertainty and complexity of the monomer-dirner collision process, however, we adopt the following heuristic procedure. The monomer-dimer potential is assumed to have the same two-parameter form, with a well depth E and effective diameter u, as the monomer-monomer potential but with parameters characterized by the ratio 8 = uM o/u and 8 = EMD/E_ We elect to optimize 6 and 0 to obtain the best possible agreement with experiment, as has been done previously [3-S], but with the constraint that the same (d.0) be used for both viscosity and thermal conduc:ivity. It is emphasized that 6 and 0 are the Od:!, ddjustablc parameters Of OUJ pJOCedUJe. BhM - D, is calculated by means of the Chapman-Enskog mix592
LETTERS
15 June 1984
ture theory (incorporating a modification of the Eucken correction for thermal conductivity) in the manner of Stogryn and Hirschfelder [3,4] by means of the fits of dilute gas integrals of ref. [24].
5. Comparison with experiment In order to compare our theory with experiment it is convenient to redlace the coefficients of eq. (1) to a dimensionless form. Thus, we define By(T)
= u-3
$!‘(kT/E) ,
where again /.I refers to either viscosity or thermal conductivity and the superscript (i) refers to collisional transfer, three-monomer, or monomer-dimer contributions. Additionally, the experimental data has been reduced according to the tinnard-Jones parameters of Haley et al. [6] _We have not attempted a detailed or completely updated analysts of the available experimental data. In view of problems associated with transport of internal molecular energy, we have examined only monatomic gases for thermal conductivity, whereas some diatom& are also in-
cluded in the viscosity analysis. In figs. 1 and 2 we show the present theory and experimental data for viscosity and thermal conductivity respectively. The general features of the experimental data are reproduced remarkably well in both figures, including the difference in sign between B, and B, at high temperatures as predicted in ref. [30] and experimentally noted in the helium data. The three contributions to B, and B, are in general comparable in magnitude although their temperature dependences are distinct [3 1] and the monomerdimer terms are negligible for T* > 10. A range of values of 6 and 8 leads to theoretical curves essentially identical to those of figs. 1 and 2; the particular choice here fOJ (S&I) is (l-02,1 -1.5). This may be compared with the choices of refs. [3,4] (1.04,1.32), ref. [5] (1.02,1.23), and ref. [7] (l-16,1.32). Kuznetsov [7] predicts a viscosity second virial which essentially agrees with ours in fig. 1. However, his method would lead to a thermal conductivity virial prediction shown by the dashed line of fig. 2. This result is systematically below the experimental results for low T’ and predicts a maximum in B, at T = 1.5 which is not corroborated by the data.
2.0 1.6 1.2 0.6 0.4
-0.6
-1.6
1
2
345
10
20
30
40 50
100
T’ Fig.1.Plot of f3; versus 7*. Solid curve is present theory. Experimental [ZS] reduced by the zerodensity CJ.hetium;
q, neon; 2, argon;=,
values obtained from various krypton;o,
xenon;
1 o-
8-
6-
a, hydrogen;
I
I
I
points were taken from the compilation of Hartley et al. [9.26]. The molecular species are identified as ~OUOW:
sources a, nitrogen.
I
I
I
I
I
A
\
Bi
--
2
345
10
20
30
40 50
1c IO
T’ Fig. 2. Plot of Bi versus T*. Solid curve is present theory. Broken curve is based on the procedure of ref. [ 71 adapted to thermal conductivity. The shape of the symbols for monatomic esperimental points are as in fig. 1 with the following addition: Open symbols are from ref. [ 25 1 reduced by the data of refs. [ 9.26 ] ; darkened symbols are from ref. [ 27 1; symbols with line enclosed are from ref. [28]; half-darkened symbol is from ref. [ 291.
Volume
107, number
CHEMKAL
6
Our method for calculating the second transport virials basically follows Kuznetsov [7] by calculating the collisional transfer and three-monomer parts according to Curtiss and co-workers [ 12,141 and the monomer-dimer parts according to Stogryn and Hirschfelder [3,4]. Our primary new theoretical contribution has been a re-examination and correctibn of the terms involving the pressure virial in eqs. (2) and (3). Also, we have developed fast and efficient numerical codes for calculation of the integrals in eqs. (2) and (3) with realistic potentials. Further
theoretical
work
is in order.
Ideally,
the
eq. (1) should not merely be assumed but should be a consequence of a fundamental kinetic equation. We believe our BL2) is exact in the first Sonine approximation, but a rigorous B13) and Bhhl-D) would probably require use of actual three-body and monomer-dimer collision dynamics respectively, and would call for difficult and expensive calculations. Nevertheless, we believe that our approach is the most comprehensive and fundamentally sound of the treatments of the present problem to date which use realistic potentials to predict numerical results (a critical discussion of previous theories will be presentcrr elsewhere [3 1I), and is the fist to be free of obvious systematic deviations from experiment for both viscosity an? thermal conductivity. combination
LETTERS [8]
6. Discussion
three-fold
PHYSICS
of
References C.F. Curtis:. M 6. McEIroy and D.K. Hoffman, lntcrn. J. Eng. Sci 3 (1963; 269. D.K. Hoffman and C.F. Curtiss. Phys. Fluids 8 (1965) 890. D.E. Stogryn and J.O. Hirschfelder. J. Chem. Phys. 31 _(1959) 1545. 141.D-E_ Stogryn and J-0. Hirschfelder, J. Chem. Phys. 33 (1960) 942. [S] SK Kim and J. Ross, J. Chem. Phys. 42 (1965) 263; S.K. Kim, G.P. Flynn and J. Ross, J. Chem. Phys. 43 (1965) 4166. [6] H.J.M. Henley, R.D. hlcCarty and E.G.D. Cohen, Physics 60 (1973) 332: 1. Restin, ST. Ro and W.A. Wakeham. J. Chem. Phys. 56 (1972) 4119. 171 V.M. Kuznetsov, HighTemp. Res. 16 (1979) 1005.
15 June
1984
S. Chapman, Phil. Trans. Roy. Sot. (London) A216 (1916) 279; D. Enskog. Dissertation, Upsala (1917). 191 J-0. Hirschfelder, C.F. Curtiss and R.B. Bird, Molecular theory of gases and liquids (Wiley. New York, 1954). 1101 D. Enskog, Kg]. Svenska Vetenskapsakad. Hand]. 63. No. 4 (1922). [ 1 I ] J.V. Sengers. D.T. Gillespie and J.J. Perez-Esandi. Physica 90A (1978) 365. [ 121 R.F. Snider and C.F.Curtiss, Phys. Fluids 1 (1958) 122. [ 131 E.G.D. Cohen, in: Lectures in theoretical physics, Vol. 9C, ed. W.E. Brittin (Gordon and Breach, New York, 1966) p. 279. [ 141 D.E. Bennett and C.F. Curtiss, J. Chem. Phys. 5 1 (1969) 2811. [ 151 J.C. Rainwater. On the Phase Space Subdivision of the Second Virial Coeflicient and its Consequences for Kinetic Theory, to be published. [ 161 J.C. Rainwater and D.G. Friend, to be published. 1171 D.E. Stogyn and J-0. Hirschfelder. J. Chem. Phys. 31 (1959) 1531. [ 181 H. OHara and F.J. Smith, J. Comput. Phys. 5 (1970) 328. [ 191 J.C. Rainwater, P.hl. HoUand and L. Biolsi, J. Chem. Phys. 77 (1982) 434. 1201 S.T. Choh, The Kinetic Theory of Dense Gases, Thesis, University of Michigan (1958). [21] D.K. Hoffman and C.F. Curtiss. Phys. Fluids 7 (1965) 1887. (22) D.G. Friend. J. Chem. Phys. 79 (1983) 4553. 1231 T.L. Hill, Statistical mechanics (McGraw-HI& New York, 1956). [24j P.D. Neufeld, A.R. Janzen and R.A. Aziz, J. Chem. Phys. 57 (1972) 1100. [25] H.J.M. Hartley, R.D. McCarty and J.V. Sengers, J. Chem. Phys. 50 (J 969) 857. 1261 WM. Haynes, Physica 67 (1973) 440; N.A. Lange, ed., Handbook of chemistry (McCrawHi4 New York, 1956); R.C. Weast and M.J. Astle eds., Handbook of chemistry and physics, 63rd Ed. (CRC Press, Cleveland. 1982); L.D. lkenberry and S.A. Rice, J. Chem. Phys. 39 (1963) 1561; R.W. Powell, C.Y. Ho and P.E. LiIey. National Standard Reference Data Series. NBS-8 ( 1966); C.L. Yaws. Physical properties (McGraw-Hill, New York, 1977); N.B. Vargaftik. Tables on the thermophysical properties of liquids and gases, 2nd Ed. (Hemisphere. New York, 1975). 1271 M.J. Assael. M. Dix. A. Lucas and W.A. Wakeham, J. Chem. Sot. Faraday Trans. 77 (1981) 439. 1281 J. Kestin, R. Paul. A.A. Clifford and W_A. Wakeham. Physica 1 OOA ( 1980) 349. 1291 C.A.N. de Castro and H.M. Roder, J. Res. Nat]. Bur. Std. US 86 (1981) 293. (301 J.C. Rainwater, J. Chem. Phys. 74 (1981) 4130. 1311 .‘.C. Rainwater and D.G. Friend, to be published.
Volume 107. number 6
TRANSPORT PROPERTIES OF A MODERATELY Daniel G. FRIEND
15 June 1984
DENSE GAS
* and James C. RAINWATER
Chemical Engi,reering Science Division, h’ntional Engineenbrg Laboratory, h’ational Bureau of Standards. Boulder. Colorado 80303, USA Received 8 February
1984
The initial density dependences of both viscosity and thermal conductivity are calculated according to a microscopical ly based theory which includes cff~~ts due to collisional transfer (from only free two-body phase space), three-monomer collisions. and monomer-dimer collisions. A Lennard-Jones potential is used to model the interactions. Comparison of the calculated results with experiment (in reduced form) shows very good agreement for both viscosity and thermal conductivity over a wide temperature range.
1_ Introduction
p:
In this letter, we present results of a microscopically based theory for the initial density dependence of transport properties in a Lennard-Jones model gas and compare these results with experimental data for viscosity and thermal conductivity. Previous attempts to compare experimental first density corrections to the transport coefficients with theoretical models [l-7] have been less than satisfying for a number of reasons including systematic deviations with experiment in certain temperature ranges and the lack of a sound microscopic basis in some aspects of the research. Our basic premise is that the moderately dense gas can be modelled as a mixture of monomers and dimers. Collisional processes between pairs of monomers form the basis for the dilute gas, or ChapmanEnskog [S; 9, ch. 7; IO] transport coefficients, as well as contributing, through collisional transfer, to the linear-in-density correction. Collisions among three monomers and between a monomer and a dimer also contribute to the density correction. Other processes, including the effects of repeat collisions 1111, are not considered here. We thus subdivide the second transport virial coefficient as follows:
* NRC-NBS Postdoctoral 590’
Research Associate.
/Jo
[I
+(Bp +Bi3) +B;h’-D))p t ___I
= Po(l + &P)
(1)
,
where p represents either the viscosity q or thermal conductivity h, p is the number density, and the superscripts represent, respectively, the collisional transfer contribution, the three-monomer collisional contribution, and the contribution from monomerdimer collisions, each of which we discuss briefly below.
2. CoIlisional
transfer
contribution
The mechanism by which energy or momentum may be transported across a surface without mass transfer was discussed by Enskog for rigid spheres [lo]. The analogous non-locality expansion for repulsive potentials was carried out by Snider and Curtiss in I958 [ 121. For repulsive potentials, the truncation of the BBGKY hierarchy via the Bogliubov hypothesis and use of an operator [ 131 S which projects particle trajectories to the asymptotically free pre-collision phase can be carried out rigorously. When the process of dimerization is allowed, however, S can no longer perform the same function on that portion of relative phase space in which dirner fonnation occurs. The integrals associated with Curtiss non-locality expansion, then, can be evaluated only in 0 009-2614/84/S 0’ 00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
Volume 107. number 6
CHEhlICAL
PHYSICS
LETTERS
15 June 1984
the free portion of relative phase space. A pair of monomers is considered free if the two-body relative momentum and displacement are such that the particles are infinitely separated in the infinite past or future of the collision. In refs. [ 1,141, the explicit numerical integration of the collision integrals associated with the nonlocality expansion were performed only over the free portion of phase space. However, certain portions of those integrals, which were shown to reduce to the pressure virial B and its derivatives for repulsive potentials, were integrated over the entire two-body relative phase space. The resolution of this inconsistency has been explored in a separate publication [ 151 and here we shall only mention certain essential conclusions. Eqs. (124) and (136) of ref. [ 141 for the viscosity and thermal conductivity virials should be replaced, for intermolecular potentials allowing dimerization,
and, with more detail, in ref. [ 161. We note that eqs. (2) and (3) agree with eqs. (124) and (136) of Bennett and Curtiss only when the potential is purely repulsive (in which case the free portion of phase space is equiv$ent to the entire two-body phase space, and B,( ’ ) = f3FE’ = B). In ref. [7], Kuznetsov uses the Stogryn and Hirschfelder [4.17] division of the second pressure virial into its free and dirner contributions which, we have shown [ 151, is not equivalent to eqs. (4) and (5); rhus his results differ from eqs. (2) and (3). We have numerically evaluated eqs. (Z)-(5) (for the Lennard-Jones 12-6 potential) independently of previous calculations using a highly accurate Clenshaw-Curtis quadrature routine [15,16] and other numerical methods previously employed in efficient calculations of dilute gas transport collision integrals [ 18,191.
by
3. Collisions among three monomers
BY) = (/3/10R3/2)
-J ‘T
dB(ri‘) r-
(R( 15) - R(6))
dBiE’ it Tr-
+ $fI’)
AT’
d?fJtE) f dT’
(2)
and Br)
= (p/15,r3/“)
(R( 15) _ ~‘6’)
+ B(,“) d’B;E)
_$T’___
(3)
dT2 where 0 = 1 kT, R( * 5) and Rt6) are defined in ref. 1 [ 141 and BfV) and T d BiE’/dT are obtained from @L
43(p/~nl)3/2
and T dB’,E’ld T = -;fi X
jjd3r free
@J~F,,)~”
d3p $(r) e-flOtr)
,-flP’/nl
,
(3
by integrating over only the free portion of relative phase space, where $I is the potential and tn the mass. A discussion of eqs. (2) and (3) is given in ref. [ 151
In order to evaluate the contributions due to three-monomer collisions for repulsive potentials one must, in principle, solve the Choh-Uhlenbeck equation [ 1320], and its proper generalization to realistic potentials is by no means c!ear at present. However Enskog, in his theory for hard spheres [lo]. introduced an approximation which accounts for a third particle by altering the frequency of two-body collisions via a “shielding” effect. Hoffman and Curtiss [2,3 I] have shown how this approsimation can be applied to realistic potentials and derive explicit integral expressions for rhe contributions of this shielding to transport properties. and thus avoid the extreme complexity of three-body collisions dynamics. In the Hoffman-Curtis9 approach, certain multidimensional collision integrals must be evaluated numerically. An inner integration requires the calculation of rhe low-density radial distribution function which we have discussed previously [22]. We have proceeded to compute independently the three-body Hoffman-Curtiss integrals for the kmnard-Jones 12-6 potential. We obtain excellent agreement with the originally reported results 121, but with much less computer time. However, the results reported by Kuznetsov [7] for reduced temperatures of 12, li,3,4, and 5 differ from our results by as much as 45%. 591
Volume
107. number
4. Monomer-dimer
6
CHEMlCAL
PHYSICS
contriiution
The remaining contribution to the initial density dependence of transport coefficients comes from collisions between monomers and dimers. This process was fist explored by Stogryn and Hirschfelder [3,4] and has subsequently been utilized in the theories of Kim and Ross [5] and Kuznetsov [7]. Thepopulation of dimers in a moderately dense gas can be calculated by the algorithm of Stogryn and Hirschfelder which is an extension of Hill’s method [23] for finding the population of bound dimers in such a system. The Stogryn-Hirschfelder formulation then, includes both truly bound dimers (with negative total energy) and those that are metastably bound within the centrifugal well which exists for certain relative angular momentum states. The explicit division of phase space into free, bound, and metastable portions is discussed elsewhere [4,15,17] _ Kim and Ross use a broader definition of metastability which appears to incorporate collisions whose lifetimes can be much shorter that the inverse of the collision frequency_ The contributions to the second transport viral coefficients due to dimerization are equivalent to the dilute gas coefficients of a mixture of.monomers of density p with dimers of a specified density (as calculated via the Stogryn-Hirschfelder formalism [4,17] )_ Thus the Chapman-Enskog theory of mixtures [8;9, ch. 71 can be utilized and the contribution calculated provided that the interaction potential between dimers and monomers is known. Various methods I;xist for estimating the monomerdimer p0tentGl;e.g. angular averaging, to make the collision Wegral calculations tractable. In view of the uncertainty and complexity of the monomer-dirner collision process, however, we adopt the following heuristic procedure. The monomer-dimer potential is assumed to have the same two-parameter form, with a well depth E and effective diameter u, as the monomer-monomer potential but with parameters characterized by the ratio 8 = uM o/u and 8 = EMD/E_ We elect to optimize 6 and 0 to obtain the best possible agreement with experiment, as has been done previously [3-S], but with the constraint that the same (d.0) be used for both viscosity and thermal conduc:ivity. It is emphasized that 6 and 0 are the Od:!, ddjustablc parameters Of OUJ pJOCedUJe. BhM - D, is calculated by means of the Chapman-Enskog mix592
LETTERS
15 June 1984
ture theory (incorporating a modification of the Eucken correction for thermal conductivity) in the manner of Stogryn and Hirschfelder [3,4] by means of the fits of dilute gas integrals of ref. [24].
5. Comparison with experiment In order to compare our theory with experiment it is convenient to redlace the coefficients of eq. (1) to a dimensionless form. Thus, we define By(T)
= u-3
$!‘(kT/E) ,
where again /.I refers to either viscosity or thermal conductivity and the superscript (i) refers to collisional transfer, three-monomer, or monomer-dimer contributions. Additionally, the experimental data has been reduced according to the tinnard-Jones parameters of Haley et al. [6] _We have not attempted a detailed or completely updated analysts of the available experimental data. In view of problems associated with transport of internal molecular energy, we have examined only monatomic gases for thermal conductivity, whereas some diatom& are also in-
cluded in the viscosity analysis. In figs. 1 and 2 we show the present theory and experimental data for viscosity and thermal conductivity respectively. The general features of the experimental data are reproduced remarkably well in both figures, including the difference in sign between B, and B, at high temperatures as predicted in ref. [30] and experimentally noted in the helium data. The three contributions to B, and B, are in general comparable in magnitude although their temperature dependences are distinct [3 1] and the monomerdimer terms are negligible for T* > 10. A range of values of 6 and 8 leads to theoretical curves essentially identical to those of figs. 1 and 2; the particular choice here fOJ (S&I) is (l-02,1 -1.5). This may be compared with the choices of refs. [3,4] (1.04,1.32), ref. [5] (1.02,1.23), and ref. [7] (l-16,1.32). Kuznetsov [7] predicts a viscosity second virial which essentially agrees with ours in fig. 1. However, his method would lead to a thermal conductivity virial prediction shown by the dashed line of fig. 2. This result is systematically below the experimental results for low T’ and predicts a maximum in B, at T = 1.5 which is not corroborated by the data.
2.0 1.6 1.2 0.6 0.4
-0.6
-1.6
1
2
345
10
20
30
40 50
100
T’ Fig.1.Plot of f3; versus 7*. Solid curve is present theory. Experimental [ZS] reduced by the zerodensity CJ.hetium;
q, neon; 2, argon;=,
values obtained from various krypton;o,
xenon;
1 o-
8-
6-
a, hydrogen;
I
I
I
points were taken from the compilation of Hartley et al. [9.26]. The molecular species are identified as ~OUOW:
sources a, nitrogen.
I
I
I
I
I
A
\
Bi
--
2
345
10
20
30
40 50
1c IO
T’ Fig. 2. Plot of Bi versus T*. Solid curve is present theory. Broken curve is based on the procedure of ref. [ 71 adapted to thermal conductivity. The shape of the symbols for monatomic esperimental points are as in fig. 1 with the following addition: Open symbols are from ref. [ 25 1 reduced by the data of refs. [ 9.26 ] ; darkened symbols are from ref. [ 27 1; symbols with line enclosed are from ref. [28]; half-darkened symbol is from ref. [ 291.
Volume
107, number
CHEMKAL
6
Our method for calculating the second transport virials basically follows Kuznetsov [7] by calculating the collisional transfer and three-monomer parts according to Curtiss and co-workers [ 12,141 and the monomer-dimer parts according to Stogryn and Hirschfelder [3,4]. Our primary new theoretical contribution has been a re-examination and correctibn of the terms involving the pressure virial in eqs. (2) and (3). Also, we have developed fast and efficient numerical codes for calculation of the integrals in eqs. (2) and (3) with realistic potentials. Further
theoretical
work
is in order.
Ideally,
the
eq. (1) should not merely be assumed but should be a consequence of a fundamental kinetic equation. We believe our BL2) is exact in the first Sonine approximation, but a rigorous B13) and Bhhl-D) would probably require use of actual three-body and monomer-dimer collision dynamics respectively, and would call for difficult and expensive calculations. Nevertheless, we believe that our approach is the most comprehensive and fundamentally sound of the treatments of the present problem to date which use realistic potentials to predict numerical results (a critical discussion of previous theories will be presentcrr elsewhere [3 1I), and is the fist to be free of obvious systematic deviations from experiment for both viscosity an? thermal conductivity. combination
LETTERS [8]
6. Discussion
three-fold
PHYSICS
of
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