Molecular-Based Study of Fluids - ACS Publications - American

10 Effects of Molecular Anisotropy FRIEDRICH KOHLER Ruhr-Universität, Institut für Thermo- und Fluiddynamik, D-4630 Bochum, Federal Republic of Germany NICHOLAS QUIRKE 1

Downloaded by GEORGE MASON UNIV on June 20, 2014 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0204.ch010

Royal Holloway College, Chemistry Department, Egham, Surrey, England The effects of molecular anisotropy considered in this chapter are the molecular shape (the anisotropy parameter being the elongation of two-center Lennard-Jones fluids), the dipole moment and the quadrupole moment. An attempt is made to scale density and temperature of two-center Lennard-Jones fluids in such a way that a comparison with the law of corresponding states is possible. With respect to electric moments, it is observed that their effect on thermodynamic and structural properties is less on two-center Lennard-Jones fluids than on spherical fluids. This is investigated in some detail.

A

L T H O U G H CONSIDERABLE PROGRESS has been achieved in under-

standing the behavior of molecular liquids (1-4), we are still far from having a complete picture of the way in which the molecular shape and electric moments contribute to thermodynamic and structural properties. For example, our understanding of the deviations from the law of corresponding states has not improved since Rowlinson's work in 1954 (5). One important difficulty is that of scaling temperature and density when comparing experimental and theoretical results. While theoreticians use the characteristic parameters of the pair potential for scaling (e.g., the depth of the potential 8 and the zero potential separation a), experimentalists use critical data. At present a sound correlation between these different approaches exists only for one-center Lennard-Jones liquids. In the first section of this chapter we suggest ways of extending this correlation to two-center Lennard-Jones liquids. The next section of this chapter gives a critical review of a computationally fast thermodynamic perturbation theory treatment of two-center Lennard-Jones liquids. While the Helmholtz energies are predicted accurately, some details of the structural properties are still missing. The subsequent two sections are devoted to the problem of treating molecules with electric moments within the framework of perturbation theory. The treatment given is in some respects preliminary 1

Current address: University of Maine, Department of Chemistry, Orono, ME 04469

0065-2393/83/0204-0209$07.50/0 © 1983 American Chemical Society

In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.

210

MOLECULAR-BASED STUDY OF FLUIDS

and the results, for the moment, qualitative. One section deals with the thermodynamic properties, while the final section deals with those structural properties that are related to the static dielectric constant. We hope that the approach outlined in these sections will form the basis for future work on this topic.


Scaling Parameters for Two-Center

Lennard-Jones

Fluids

In this section we suggest methods of scaling densities in three regions—regions of low densities, critical densities and liquid densities. Low density scaling is considered first. Table I contains second virial coefficients for various elongations L = //a. This extends the table given by Wojcik et al. (6) for a limited temperature range . In Figure 1, these results are plotted against the reduced temperature T/T , where T is the Boyle temperature. It can be seen that the curves for higher elongations become progressively steeper. A more detailed comparison is provided by Figure 2, where the second virial coefficients are reduced by an effective O^CLJ in such a way that B/N O-2CLJ is same for all elongations at T/T = 0.3. This scaling produces a single curve for all elongations in the temperature range 0.3 < T/T < 1.05; at lower temperatures the reduced curves begin to spread (Figure 3), with the higher elongations having the more negative second virial coefficients. This might at first suggest that the parameters L and a could be determined separately from such a plot of experimental second virial coefficients in the low temperature region (T/T < 0.3). However, this is questionable for two reasons: (1) low-temperature second virial coefficients are in most cases subject to large errors and (2) the two-center Lennard-Jones model potential cannot accurately reproduce the low-temperature second virial coefficients of real substances. Returning to the problem of low density scaling, Table II shows T ,2CLJ/?B,ICLJ and Figure 4 shows c r / a using the values for effective a obtained by equalizing the reduced second virial coefficients at T/T = 0.3. By chance, the plot in Figure 4 is almost a straight line. Figure 4 shows a similar plot for hard dumbbells obtained using the Boublik-Nezbeda equation of state (7), which gives 2

B

t

A

n

B

e

B

B

B

2CLJ

B

1CLJ

B

(i) °" lCL./ °"icLJ Note added in proof: A table listing the second virial coefficients for the elongations 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 is contained in Maitland, G . C . ; Rigby, M . ; Smith, E . B.; Wakeham, W . A . Intermolecular Forces, Clarendon Press, Oxford 1981. 2


10.

KOHLER AND QUIRKE

211

Molecular Anisotropy

Table I. Reduced Second Virial Coefficients of Two-Center LennardJones Fluids J*

j*

B

T*

NACT


0.0908 0.1022 0.1135 0.1419 0.1759 0.2270 0.3000 0.3405 0.6000 0.9081 0.9932 1.0000 1.0216 1.0500

L = 0.505 (continued)

-102.631 -71.376 -53.365 -31.189 -19.905 -12.276 -7.404 -5.871 -1.745 -0.246 -0.017 0.000 0.050 0.113

L = 0.505 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.9971 2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.28 3.9992 4.1 6.5

0.1202 0.1352 0.1502 0.1652 0.1803 0.1953 0.2103 0.2253 0.2403 0.2554 0.2704 0.2854 0.3000 0.3155 0.3305 0.3455 0.3605 0.3755 0.3906 0.4056 0.4927 0.6000 0.6159 0.9764

f*

3

L = 0.3292 0.8 0.9 1.0 1.25 1.55 2.0 2.6430 3.0 5.2860 8.0 8.75 8.8100 9.00 9.2505

B

j*

-71.037 -51.371 -39.503 -31.692 -26.213 -22.181 -19.101 -16.677 -14.724 -13.118 -11.775 -10.637 -9.686 -8.812 -8.071 -7.417 -6.835 -6.316 -5.848 -5.426 -3.602 -2.224 -2.069 -0.074

6.65 6.6570 6.9813 7.0

0.9989 1.0000 1.0500 1.0515

-0.003 0.000 0.140 0.148

L = 0.63 0.75 1.0 1.25 1.5 1.6911 2.0 2.5 2.81 3.3823 3.51 5.25 5.62 5.6371 5.9190 6.0

0.1330 0.1774 0.2217 0.2661 0.3000 0.3548 0.4435 0.4985 0.6000 0.6227 0.9313 0.9970 1.0000 1.0500 1.0644

-70.251 -33.674 -20.856 -14.522 -11.518 -8.326 -5.305 -4.108 -2.595 -2.339 -0.263 -0.011 0.000 0.166 0.211

L = 0.793 0.6 0.7 0.8 0.9 1.0 1.1 1.35 1.4160 1.5 1.75 2.0 2.8321 4.0 4.7201 4.755 4.9561 5.0

0.1271 0.1483 0.1695 0.1907 0.2119 0.2330 0.2860 0.3000 0.3178 0.3708 0.4237 0.6000 0.8474 1.0000 1.0074 1.0500 1.0593

-113.160 -69.746 -48.612 -36.501 -28.773 -23.457 -15.458 -14.059 -12.543 -9.229 -7.019 -3.097 -0.775 0.000 0.031 0.197 0.232




212

Figure 1. The reduced second virial coefficients, B/Nj^a , of one-center and two-center Lennard-Jones fluids, plotted against the temperature reduced by the Boyle temperature, with the elongation L as parameter. 3


KOHLER AND QUIRKE

213



10.

Figure 2. Plot similar to Figure 1, but with an adjusted a L j that equalizes all B/NACT^CLJ t T / T = 0.3. The notation of the points corresponds to Figure 1. 2 C

a

B




214

Figure 3.

Plot similar to Figure 2, for low values o/T/T . B


10.

KOHLER AND QUIRKE

215



Table II. Boyle Temperatures and Calculated Critical Temperatures of Two-Center Lennard-Jones Fluids Compared with the "United Atom" L

T* /T*BJCLJ B,2CLJ'

0 0.329 0.505 0.630

1 0.644 0.487 0.412

1 0.676 0.524 0.451

0.793

0.345

0.386

L

T*

1

/T*

c,2CLj'

c,lCLJ

L

L

\ c,2CLJ 1

c,lCLjlexp

/

l

1 0.633 0.525 0.450 0.437 0.358

Note: Critical temperatures are calculated by Equation 3 with a = 0.15. Experimental quantities are calculated from experimental critical temperatures of liquids for which certain parameters L and e have been used successfully.

The agreement between the 2CLJ curves and the hard dumbbell curves can be improved for large L if the reduced second virial coefficients are set equal at higher temperatures, T/T = 0.6 or 1.05 rather than 0.3 (see Figure 4), but no such improvement can be achieved at small L . This small discrepancy between two-center Lennard-Jones fluids and hard dumbbell fluids is probably related to the different temperature dependence of the effective sphere radius for hard spheres and hard dumbbells. Figure 5 shows results obtained from the perturbation theory reviewed in the next section. We have not been able to scale these results. Turning now to higher densities, we consider an approximate method of scaling the critical densities based on the generalized van der Waals model (8-10) B

-B-

=(-?-]

NkT

--22NhT

V P ^ / H a r d Fluid

9

dp

(2)

dp 2

Applying the critical conditions — = 0 and — - = 0, two equations for dp dp the two unknown p and A = alT are obtained. Using the CarnahanStarling equation for the hard sphere fluid and the Boublik-Nezbeda equation (7) for the hard dumbbell fluid, effective values for a can be found, which when used to reduce the critical densities of Table III make them all equal. Figure 4 shows the resulting values o j/cr j. For densities in the liquid range, we have attempted to scale the orthobaric density curve (effectively zero pressure densities) given by Wojcik et al. (6) as a function of elongation. In order to bring them into a form comparable to the law of corresponding states, we had to assume z

c

c

c

2 C L J

2CL

I*\ = ^B/2CLJ

(LL\ \J>B) 1CLJ

x

(1 + ah)


1CL

(3)


Figure 4. Various scalings for

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