Simple pair potential model for real fluids. II. Transport properties of dilute gases

SIMPLE PAIR POTENTIAL MODEL FOR REAL FLUIDS. II. TRANSPORT PROPERTIES OF DILUTE GASES I.

NEZBEDA

Institute of Chemical Process Fundamentals, Czechosl. Acad. Sci., Prague*) S. LABiK,A. MALIJEVSK'~

Department of Physical Chemistry, Institute of Chemical Technology, Prague**)

The collision integrals required for evaluating the dilute gas transport properties have been computed for the hybrid pair potential of Nezbeda. Calculations of viscosity and self-diffusion coefficients of Ar, Kr, Xe, and CH 4 have been made and the results compared with those based on the Lennard-Jones potential. It is found that (i) the potential parameters determined from the viscosity give the attractive force constant, C6, in good agreement with experimental data, (ii) the accuracy of calculated viscosity is reasonable even at low temperatures, and (iii) the hybrid model gives in all cases much better results than the Lennard-Jones potential.

1. INTRODUCTION

Calculation of thermodynamic properties of real dense fluids from the statistical mechanics requires both a reliable statistical theory and a realistic intermolecular potential. The statistical thermodynamics of simple classical fluids has achieved considerable progress in the last two decades and several very accurate theories are now available [1]. Concerning the potential models very accurate many-parameter functions have been proposed for argon, krypton, and xenon [2, 3] but a reliable general model is still missing. In an effort to combine in a more realistic way the contemporary knowledge on the intermolecular forces with demands and limitations of the statistical thermodynamical theories, Nezbeda [4] has recently proposed a pair potential consisting, in general, of a nonspherical core and anisotropic repulsions, and of the isotropic dispersion terms and multipolar interactions. Though the initial motivations were to model the interactions in molecular fluids the potential should yield, assuming it has a sound physical basis, reasonable results also for simple fluids. In this case, i.e. for the isotropic interactions, the simple version of the Nezbeda hybrid potential (with 1eading dispersion term only) is (1)

uN(R )

=

oo

=

4e

Lt~-~_d/

-

for

R

for

R >

<

d

d.

*) Rozvojovd 135, 165 02 Praha 6, Czechoslovakia. **) Suchbdtarova 5, 166 28 Praha 6, Czechoslovakia. 862

Czech. J. Phya. B 30 [1980]

L Nezbeda et al.: Simple pair potential model for realfluids... This potential differs only slightly from the spherical-core Kihara potential and we would expect both potentials to give nearly identical results. However, this is not the case. Potential (1) has turned out to be slightly more flexible when applied to the second virial coefficient and, moreover, the attractive force constant C 6 (obtained by adjusting the potential parameters to the second virial coefficient data) has been found in perfect agreement with the experimental data on noble gases [4]. These results indicate that the simple pair potential model (1) may be an attractive model for practical chemical engineering applications and deserves therefore a more detailed investigation. In the present paper we focus on the transport properties of the dilute gases because these are directly related to the pair potential. The aims of the paper are as follows: (i) to compute the classical collision integrals for the potential model (1), (ii) to adjust the potential parameters to the viscosity data so as to find out to what extent one set of parameters is able to reproduce the transport experimental data over a wide range of temperatures, and (iii) to use the potential parameters determined from the second virial coefficient data to calculate the viscosity and self-diffusion coefficients. These calculations enable us to assess more concretly the potential model (1) and discuss shortly its possible applications.

2. C O L L I S I O N I N T E G R A L S

The transport properties of dilute gases can be readily calculated once the reduced collision integrals f2(k'~)* are known ([5] and the next Section). Assuming a pair potential of the type u = eqS(RIe; d 1, d 2 .... ), the reduced collision integrals are defined by the following equations: (2)

+2 1)~ f o e X p ( - x 2 ) x 2 ~ + 3

#(~,')*(T*, d 1. . . . ) - ( s

(3)

_

4

-

[2k + 1 -

(4)

7, = rc - 2b

f~

Q(k)*(T*xZ)dx

f;[l_coskz]bd(;) ( - 1 ) k]

o

'

R-2[1 - eb/T*x 2 - ba/R2] q2 d R .

d Rm

In these equations Q(k), is the reduced cross section, 7. the deflection angle, b the impact parameter, T* the reduced temperature (T* = kBT/e with ks being the Boltzmann constant), and R m is the distance of the closest center-to-center approach. Numerical evaluation of the collision integrals (2) is complicated by two properties of involved functions: (i) thedeflection angle has a singularity at R = R,, and (ii) for certain values of b the integrand in (3) oscillates rapidly. Efficient numerical Czech. J. Phys. B 30 [1980]

863

L Nezbeda et al.: Simple pair potential model for realfluids... technique to overcome these difficulties has been developed by Smith and Munn [6]. Following these authors we calculated Q(k,s). (T*, d/a) for 1 < k < 3 and k ___ s _< 3 over the range of T* from 0.30 up to 200.0 for twenty values of the core diameter, d/G, ranging from 0.0 to 0.95. Because of their length the complete tables have been deposited in the Library of the Institute of Chemical Process Fundamentals [7] and are available on request from the first author. Our potential model ( 1 ) i s closely related both to the (12-6) Lennard-Jones (LJ) potential and to the Kihara potential. The collision integrals for both these potentials have been calculated by different authors and these results may serve therefore as an excellent check on the accuracy of our computations. We have therefore written a computer program for a general hard-core potential (5), (5)

u -- oo

=4eF(a- d,y z L\R --Z-fU -

(a-

d2~6-I

for

R < d1

for

R > dl, dl > d2,

~,R - d2,] J '

from which the LJ potential is recovered for d 1 = d 2 = 0 while the Kihara potential corresponds to the case d~ = d2 ~= 0. Table 1 Comparison of collision integrals 12(2,2)* for the Lennard-Jones potential computed by different authors. 12(2,2), T*

Ref. [5]

Ref. [8]

This paper

0'30 0'60

2'7850 2.0650 1.5870 1'1750 0.9269 0.7432 0-5882

2"8456 2.0828 1-5933 1.1753 0-9266 0.7433 0.5851

2.8236 2-0780 1.5928 1-1769 0.9268 0.7436 0.5851

1"00 2'00 5"00 20'00 100"00

In Table 1 the collision integrals g2(2'2)* computed by different authors for the LJ potential are compared. It is seen that our results are very close to those of Klein and Smith [8] which data are the most often referred to. The m a x i m u m difference in the very low temperature range does not exceed 0"87o and for higher temperatures the results of both computations are identical. Concerning the Kihara potential, we compare our own results for d/a = 0-4 with those of Barker et al. [9] which were 864

Czech. J. Phys. B 30 [1980]

L Nezbeda et al.." Simple pair potential model for realfluids...

obtained in a quite different way. As it is seen from Table 2, also in this case the agreement is excellent. We may therefore conclude that our program for calculations of the collision integrals for the hard-core potential works perfectly. Table 2 Comparison of collision integrals

~Q(2,2), for

the Kihara potential of d/tr = 0"4 computed by

different authors. ,.Q(2,2),

T*

Ref. [9]

This paper

0"6 1"0 1"5 5-5 21-5

1"686

1"684

!'426

1"426 1"257 0'9625 0.8424

1"252 0"9644 0'8442

3. T R A N S P O R T COEFFICIENTS. RESULTS A N D D I S C U S S I O N

In the present paper we are primarily interested in the viscosity and self-diffusion (at pressure P = 101.325 kPa) coefficients which are given by Eqs. (6) and (7), respectively: (6)

x/(MT) r/ = 2"6693 x 10 -6 0-2 (2(2,2), f~

[Pa " s]

(7)

M) ~ = 2"6628 x 10 .7 0,/(r31 -2 ~r2(l'l)*

[m21s] .

Here Tis the absolute temperature, M the molecular weight, 0- the potential parameter in nm, and fn and f e the correction factors accounting for higher mathematical approximations of I/ and ~ . The correction factors are very slowly varying functions of the reduced temperature T* which seldom differ from unity by more than about 0.7~. To be consistent with the accuracy of the experimental data, f~ and f e can be omitted from Eqs. (6) and (7) without causing significant error. Using the tables of the collision integrals along with Eqs. (6) and (7) we may, with a given set of the potential parameters, calculate the transport coefficients. Or, vice versa, these parameters may be determined from the best fitting of the experimental data. Both these ways have been followed for four substances with spherical or nearly spherical molecules: argon, krypton, xenon, and methane (neon has been excluded from the noble gas series because of non-negligible quantum corrections at low temperatures). Concerning the experimental data, Maitland and Smith [10] Czech. J. Phys. B 30 [1980]

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I. Nezbeda et al.: Simple pair potential model for real fluids... .

recently critically reviewed all the existing viscosity data of twelve most common substances and recommended the 'smoothed' data sets; only these data are used throughout this paper. Table 3 Empirical parameters of the Lennard-Jones and hybrid potentials and the attractive force constant, C6, determined from the viscosity dataa). Lennard-Jones

Ar Kr Xe CH 4

3'308 3.540 3"893 3.720

hybrid pot.

e/kB

C6

d/~

146"25 201'03 266'76 155-77

105"8 218.5 512"8 227.9

0'42 0"42 0"42 0"42

3"096 3.337 3.698 3"534

C6 b)

e/kB

C6

quant, calc.

exptl.

138"85 181"20 228'01 132"29

67"5 138'2 322"0 142"3

50-73 100'99 232"03 115"20

62'60 125.40 257'49

a) a is in 10 - 1 nm, elk B in K and C 6 in I0 -25 J . nm 6. b) values taken fron Ref. [llJ.

In the paper [4] the parameters of the potential u N were fitted to the second virial coefficient and it was found that for all four substances under consideration the reduced core diameter, d* = d/o', was equal approximately to 0.42. We have made use of this coincidence when correlating the viscosity data and set a priori d* -- 0.42. (If it were not this limitation, even better results for the transport properties might be obtained. However, we are aiming as well at finding linkage between the potential parameters and other physical properties of molecules rather than looking at the parameters as mere numbers without any relation to reality.) The values of o" and e/kB obtained in this way are set out in Table 3. If the potential model u N were the true pair function, the parameters obtained would fit the dilute-gas properties for all temperatures. To see to what extent potential (1) is able to predict viscosity, the percentage deviation between the calculated and experimental values of argon is plotted in Fig. 1. (Argon is chosen because of the widest temperature range; results for the other substances are slightly better.) It is seen that for all temperatures T > >250 K the calculated and experimental values agree within experimental errors; in the low temperature range the maximum error is 2.8~. What seems very important, and in sharp contrast with other simple models [11], is the fact that the calculated values do not exhibit systematic worsening with decreasing temperature. To demonstrate this property the deviation plot for the LJ model (with the parameters taken from Table 3) is also shown in Fig. 1. It is seen that for T > 200 K the LJ potential gives excellent results but fails completely for lower temperatures. Once the potential parameters are known the attraction force constant, C6, can be determined: C6 = 4ea 6. If the potential had a sound physical basis, this value should 866

Czech. J. Phys. B 30 [1980]

I. Nezbeda et al.: Simple pair potential model for real fluids...

correspond to that found by the quantum-mechanical calculations or by the experimental measurements (QME). In Table 3 the C 6 constants based on the viscosity parameters of the LJ and present potential models are compared with the QME results. It is seen that our potential (1) gives values which are very close to the QME data while the LJ C 6 values are always too high. This finding partially explains also the failure of the LJ predictions at low temperatures. It is known that the low-tempera-

2 J i

9

Lenn~'d~Jones potential

0

i',k.~be z do potential

0 0 0

2 09

8

@

0 o

000

Q

0

9

9

o

9

6

9 9

-2

e

i

~oo

~

0

0

0

8~o

0

.- e e e e ~ g 0

0

0

0

,2'oo

0 0

' 1500

T[K]

Fig. 1. The deviation plot of viscosity of argon. (Th-e potential parameters were adjusted to experimental viscosity data, Tab. 3.) ture transport data are sensitive to the long-range part of the intermolecular potential u [11]. At large separations it is, at zeroth approximation, u ~,, C 6 R - 6 . It is therefore evident, that if the constant C6 is wrong, the viscosity calculations must be wrong, too. Because of completeness we have calculated as well the selfdiffusion coefficients from Eq. (7) by using the viscosity parameters. No significant difference between the calculations based on the LJ and uN potentials has been found and the results have agreed well with the experimental data. So far we have tried to adjust the potential parameters to the viscosity and then have been investigating the flexibility of viscosity theoretical curves. From the practical point of view more interesting and much more important is another task: to use the parameters obtained by fitting some property to calculating another property. To be more specific, it is usually required to calculate the viscosity by means of the virial parameters. Czech. J, Phys. B 30 [1980]

867

I. N e z b e d a et al.: Simple pair polential model f o r real fluids. . .

It has been mentioned already that a true potential function must fit all the dilute gas properties (both thermodynamic and transport) over any temperature range with only one set of parameters. This, however, does not hold for simple models and discrepancies between the calculated (with the parameters not adjusted to the property under consideration) and experimental results may serve therefore as a measure of reliability of the potential model. /*

OO 0

%

9

Lennard- Jones polential

0

Nezbeda potential

o o 0

0 0

-0 0 0

0

0

0

0

0

0

mE

0 0

0

0

0

9

@

9

9

'.. 8

9 - 12

t

/.00

,i

800

9

@

9 i

1200

9

9

9

@ i

4600

r

T [K ]

Fig. 2. The deviation plot of viscosity of argon. (The potential parameters were adjusted to the second virial coefficient data, Ref. [4].) Using the virial parameters from Ref. [4] we have calculated the viscosity of all four substances under consideration and the deviation plot of argon is shown in Fig. 2. It is seen that even in this case the potential model (1) gives surprisingly fair agreement. Though the deviations exceed usual experimental errors (which are about 1 + 1"5~) the overall agreement is satisfactory with the maximum error not exceeding 4~. For comparison, the deviation plot of the LJ potential of argon (with the parameters a = 0.3558 nm and ~ / k B = 117.2 K) is shown in Fig. 2 as well. It is seen that the LJ values are completely shifted off the zero deviation level and their average error is about 8~. Concerning the self-diffusion coefficient the conclusions similar to those found with the viscosity parameters can be drawn: no significant difference between the LJ and hybrid potentials has been found. This coincidence is caused, presumably, by small amount of the experimental data (i.e. narrow temperature ranges).

868

Czech, J. Phys. B 30 [1980]

L Nezbeda et al.: Simple pair potential model for realfluids...

4. CONCLUSION The present paper has been devoted, firstly, to routine calculations of the transport properties of dilute gases through the pair potential model u N (Eqn (1)) proposed recently by Nezbeda. Secondly, by means of these calculations we have tried to assess this model. All the calculations performed indicate that the potential u N does have a sound physical basis and, despite its simplicity, it gives fairly accurate results. In any case, it is much better than the Lennard-Jones potential and we dare say it is at least as good as the Kihara spherical-core potential. Besides possible applications of the potential u N in the chemical engineering calculations all these findings are also important from the point of molecular fluid theories: the general hybrid potential (cf. [4]) may give better results than frequently used composite potentials, the limiting form of which is (in the case of the isotropic interactions) the Lennard-Jones potential. Received 22.2. 1980. References

[1] BARKERJ. A., HENDERSOND., Rev. Mod. Phys. 48 (1976), 587. [2] BARKERJ. A., FISHER R. A., WATTS R. O., Molec. Phys. 21 (1971), 657. [3] BARKERJ. A., WATTS R. O., LEE J. K., SCHAFERT. P., LEE Y. T., J. Chem. Phys. 61 (1974), 3081. [4l NEZBEDAI., Czech. J. Phys. B 30 (1980), 481. [5] HIRSCHFELDERJ. O., CURTISS C. F., BIRD R. B., Molecular Theory of Gases and Liquids. Wiley, New York 1954. [6] SMITrt F. J., MUNN R. J., J. Chem. Phys. 41 (1964), 3561. [7] NEZBEDA I., MALIJEVSK~A., LAB:{KS., Research Report 2/1980, 13TZChT (~SAV, Praha 1980. [8] KLEIN K., SMITH F. J., J. Res. Natl. Bur. Stand. 72A (1968), 359. [9] BARKERJ. A., FOCK W., SMITH F., Phys. Fluids 7 (1964), 897. [10] MAITLANDG. C., SMITH E. B., J. Chem. Engng. Data 17 (1972), 150. [11] REED T. M., GrOBBINSK. E., Applied Statistical Mechanics. McGraw-Hill, New York 1973.

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