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Liquid argon: Monte carlo and moleculardynamics calculationsJ.A. Barker a , R.A. Fisher b & R.O. Watts ba IBM Research Laboratory, Monterey and Cottle Roads, San Jose,California, 95114b Diffusion Research Unit, Australian National University, Canberra,A.C.T., 2600, Australia

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To cite this article: J.A. Barker, R.A. Fisher & R.O. Watts (1971): Liquid argon: Monte carlo and moleculardynamics calculations, Molecular Physics, 21:4, 657-673

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MOLECULAR PHYSICS, 1971, VOL. 21, NO. 4, 657-673

Liquid argon: Monte Carlo and molecular dynamics calculations

by J. A. BARKER

IBM Research Laboratory, Monterey and Cottle Roads, San Jose, California 95114

and R. A. FISHER and R. O. WATTS

Diffusion Research Unit, Australian National University, Canberra, A.C.T., 2600, Australia

(Received 5 April 1971)

Thermodynamic properties of liquid argon are calculated by Monte Carlo and molecular dynamics techniques, using accurate pair-potential functions determined from the properties of solid and gaseous argon, together with the Axilrod-Tel ler three-body interaction. Satisfactory techniques for evaluating three-body contributions to thermodynamic properties without excessive requirements of computer time are described. Quantum corrections are included. Agreement with experiment is excellent: the best pair and triplet potentials give an excellent description of the properties of solid, gaseous and liquid argon.

1. INTRODUCTION

The determination of interatomic forces in the inert gases is of lasting interest. Once the forces are known they may be used to calculate the thermodynamic and transport properties of the solid, liquid and gaseous states, thus providing a stringent test of present theories of these states. As the quantum effects in argon are relatively small it has been widely studied as the prototype of a classical system.

In the theory of classical fluids, computer simulation techniques [1-8] have provided exceedingly valuable insights and stimulated rapid development in more conventional theories. These methods are the Monte Carlo method, based on ensemble averaging and due originally to Metropolis et al. [1], and the method of molecular dynamics, due to Alder and Wainwright [2]. Most of these studies have used simple model potentials, in particular the Lennard-~ones 6 : 12 potential. The results of these calculations have been used to guide and test the development of conventional theories, and they have also been compared with experimental data, usually for argon, with fair quantitative agreement. In this latter quantitative aspect the success of calculations using the 6 : 12 potential function must be regarded as largely fortuitous. The work of Guggenheim and McGlashan [9, 10] showed clearly that the actual interactions of argon atoms are considerably different from those described by the 6 : 12 potential.

Previous attempts to use more realistic potentials led to worse agreement with experiment [5]. The actual interactions may differ from the 6 : 12 model in two ways, firstly in the form of the two-body potential and secondly in the presence of many-body interactions. There have been many studies of these

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658 J .A. Barker et al.

problems [11-22]. Dymond and Alder [12, 13] made a thorough study of the consequences of the approximation of neglecting a// many-body interaction. Recent spectroscopic and molecular beam evidence has shown that the pair- potential function to which they were led is unsatisfactory, so that their approximation must now be regarded as inadequate. Others [14-22] have considered the next simplest assumption, that the dominant three-body interaction (the Axilrod-Teller triple-dipole interaction [23, 24]) should be included, but that other many-body interactions may be neglected. Bobetic and Barker [16] found that, with this approximation, a pair-potential function could be determined giving excellent agreement with experiment for the thermodynamic properties of solid argon at low temperatures, and for the thermodynamic and transport properties of dilute gaseous argon [18]. This pair potential proved to be in good (though not quite perfect) agreement with the spectroscopic data of Tanaka and Yoshino [18, 25, 26] and with the molecular beam differential scattering measurements of Lee and Parson [27].

This is strong evidence for the accuracy of the approximation of neglecting many-body interactions other than the triple-dipole interaction. A further stringent test can be made by using the pair and triplet interactions to compute properties of liquid argon. Results of such calculations, using both Monte Carlo and molecular dynamics techniques, are described in this paper.

To summarize our conclusions, we found that the Bobetic-Barker potential gave fair agreement with experimental liquid energies and pressure, somewhat better than the 6 : 12 potential [5]. However the discrepancies between calculated and experimental liquid pressures were too large to be ascribed to experimental error. The calculated liquid pressures are sufficiently sensitive to the form of the pair-potential function to permit a more precise determination of that function. To make such a determination without the necessity of repeating the laborious muhiparameter fits made by Barker and Pompe and Bobetic and Barker we studied a one-parameter family of potentials linearly interpolated between the potentials of Barker and Pompe (uBp) and Bobetic and Barker (UBB) and thus having the form UBB+X(UBp--UBB). Potentials of this form with x less than about ] give a fit of the gas and solid-state data which is only very slightly worse than that given by UB•. In the event it proved that the potential of this form with x--0.25 gave excellent agreement with liquid-state energies and pressures while not giving appreciably worse agreement with gas and solid-state data than UBB (actually the new potential gives slightly worse agreement than UBB for the extrapolated Debye parameter at 0 K, which was the quantity actually fitted by Bobetic and Barker, but slightly better agreement on the average over the temperature range 0~ 12 K). Thus within the limits set by the accuracy of the experimental data on solid and gaseous argon, the new potential and UBB are about equally acceptable. However, the new potential gave better agreement with liquid data, and it also proved to be in better agreement with spectroscopic data and molecular beam differential scattering results.

We assume that the potential energy of a set of argon atoms is a function of the nuclear positions alone (Born-Oppenheimer approximation) and is given by ~v

U= E u(kl)+ E u(klm) k < / = l k < l < ~ n = l

- U2 + U3. (1)

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T h e triplet potential u(klm) is assumed to have the Axilrod-Tel ler form,

u(klm) = v(1 + 3 cos 01 cos 02 cos Oa)/(R13R23R33), (2)

in which R1R2Rz and 0102 03 are the sides and angles of the triangle formed by the three atoms. For the coefficient v we used the value 73.2 x 10 -84 erg cm 9 due to Leonard [28] which is very close to the value given by Bell and Kingston [29].

For the pair-potential u(kl) we have used the potentials due to Bobetic and Barker [16] and to Barker and Pompe [14], and also a third potential which is a linear combination of these two potentials with coefficients 0.75 and 0.25 respectively. T h e first two potentials have the analytic form

e[ ~ Ai(r-1)' exp ~ C2,+6/('+rZ'+6)] (3) u(k, I ) = [ ~ ( 1 - r ) ] - i=o j = 0

with r = Rkz/Rm, where Rkt is the separation of the atoms and Rm is the separation at the m i n i m u m of the potential. T h e constants corresponding to the three potentials are listed in table 1.

e/k, K Rm, A Ro, A Ao A1 A~ As Aa A5 C6 Ca Clo Og

Barker-Pompe Bobetic-Barker Present (ux, x = 0" 25)

1 4 7 " 70 3" 7560 3" 341 O" 2349

- 4" 7735 - 10 "2194 -- 5- 2905

0-0 0"0 1 "0698 O" 1642 0"0132

12"5 0"01

140.235 3.7630 3.3666 0"29214

-4.41458 - 7.70182

- 31"9293 - 136.026 - 151"0

1.11976 0.171551 0.013748

12.5 0.01

142.095 3"7612 3"3605

Table 1. Potential parameters.

2. COMPUTER SIMULATION METHODS

The Monte Carlo and molecular dynamics techniques for the case of two-body interactions are now well known [1-8]. One considers a fixed number of molecules (108 in our calculations) in a cubic box with periodic boundary conditions. For the Monte Carlo calculations one performs a biased random walk which generates configurations of the molecules with probabili ty proportional to exp ( - U / k T ) . T h e the rmodynamic properties are then obtained as configurational averages. Molecular dynamics calculations involve solving the equations of motion of the system

m d2ri= -VIU. dt 2

Positions and momenta are then recorded at successive instances of t ime (10 - i s s)

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660 J .A. Barker et al.

and thermodynamic properties obtained as time averages (over about 5000 time steps in our calculations).

In principle these procedures would be equally satisfactory in the presence of three-body interactions, but in practice the computer time requirements would be very large, since for two-body interactions the time per step is in part proportional to the number of molecules, while for three-body interactions it is in part proportional to the square of that number. For that reason we have included three-body effects by using a perturbative technique involving an expansion in powers of the three-body coefficient v with neglect of terms beyond that of degree 1. It is known from the work of Barker et al. [15], and is reconfirmed by the present calculations, that this leads to negligible error in the case of liquid argon, because the triple-dipole interaction is (a) relatively weak and (b) only slightly dependent on configuration for the relevant configurations at high densities. For example at T= 100 K and V = 35.36 cmS/mole the term of order v 2 contributes only - 0 . 2 5 cal/mole to the Helmholtz free energy.

We have also included quantum corrections using the well-known expansion of the partition function in powers of h2; effects of statistics are negligible at all temperatures for liquid argon. Hansen and Weis [30] showed that it is sufficient to include the first quantum correction (order he), and to neglect terms of order h 4, in liquid neon; this is afor t ior i true for liquid argon, in which the quantum effects are relatively smaller by a factor of about 10.

According to the Wigner-Kirkwood expansion [31] the Helmholtz free energy of the system is given by

[A-3N drN] A = - k T l n [ N ! f . . . f exp(-SU) . . .

fl h2 . Vi~U)

• . . . d r N / f . . . fexp(- U)drl . . . dr~v + O(h 4) (4)

in which A is (h2/2~'mkT) 1/2, fi is 1/ (kT) and Vi ~ is the Laplacian with respect to the coordinates of the ith atom. We now make a double Taylor series expansion in v and h 2 and neglect terms of order v z, vh2 and ha; the result is

i A-aN d r ~ ) + ( U 3 ) o A = - k T l n \ N ! f " ' " f e x p ( - f l U 2 ) dr l . . . ,

+ (5) \481r 2m] \ i<j

in which the notation ( )0 implies canonical averaging for the classical system with only two-body interactions:

(X>o=f .. . fexp(- U )Xarl . . . a ru / f . . . f exp(- 8 ) • dr1 . . . dru . (6)

The thermodynamic internal energy Ut and pressure p can now be calculated by differentiating (5) with respect to temperature and volume. The 'scaling' procedure for differentiating with respect to volume is described by Born and

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Liquid argon 661

Green [32]. The results are

~A 3 N k T

= < u ,>o + < U3>o- 3[< u2 Us>o- ( U2>o< U >o]

+ 2(UQ>o - 8[< UoUz)o - < Ur U2>o],

p v = _ v 8A ~V

= N k T + (P2>o + (Pa>o - fl[( UaP2>o- < Ua>o(P2>o]

+ (Pr fl[(UQP2>o- < UQ>o<P2>o]

(R,jS/~Rij + R,kS/8R, k + RjkS/SRj~) u(ijk)

(7)

(8) in which we define

p =_l Z i < j < k

= 3 U3, (9)

U _ fih 2 Q 48~2 m ~ Vi 2 u(ij), (10)

i< j

p q _ 4 ~ E 1 i<j 3 (RiJ~/ORii)Vi2 u(ij), (11)

P2 = - Z �89 (R,jS/~Ril) u(ij). (12)

The second form of (9) follows [33, 34] from Euler's theorem on homogeneous functions, since u(ijk) is a homogeneous function of degree - 9 in the variables Rlt, Rile and Rik.

We have now expressed correctly to first order in v and h 2 the three-body and quantum corrections to energy and pressure in terms of canonical averages over the configurations of a classical system with only two-body interactions; these averages are readily evaluated by the Monte Carlo procedure. As in previous work it was found sufficient to average the three-body terms over a subset of the Monte Carlo chain; we used a subset consisting of every 1000th configuration. We also calculated some of the correction terms directly from the molecular dynamics calculations, evaluating the three-body term after every 25 time-steps. The results were essentially the same as those obtained from the Monte Carlo calculations.

In the Monte Carlo calculations we truncated the potentials at a distance Rmax equal to half the edge of our basic cube, and given by

Rmax = �89 1/a (13)

in which p* is N0.a/V and 0. is the separation at the zero of the pair potential. In the molecular dynamics calculations we chose Rmax=2"5a. To correct for these truncations we added terms to the energy and pressure respectively given by

A " Np*2rr (~ 2Ui- - - u(R)R z dR, (14) 0 .3 JRmax

A g ~ T 7 \ Np*27r ~ oo [ 1 du] (15)

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662 J . A . Barker et al.

McDonald and Singer [5] showed that this led to errors less than 2 cal/mole in energy and 10 arm. in pressure for the case of the 6 : 12 potential. Since for our potentials the long-range tail is relatively weaker by a factor of about two, these errors should be roughly halved in our calculation.

We also added long-range three-body corrections calculated according to the equations

AaU~=Np 2 f ... f g2(R12)g2(Rla)u(123)dr2dra (16)

RI~< Rl s<~ R~a Rmax<Rzs

Aa(pV) = 3AaU~ (17)

in which g2(R) is the radial distribution function for the classical fluid with only two-body interactions.

Thus we assumed that the three-body distribution function ga(R12, Raa, R2a) could be approximated for R2a > Rmax by the superposition approximation value g2(R12) g2(R13) g2(Rz3) with gz(Rz3) set equal to unity. For g2(R) we used values computed from the machine calculations, with g2(R) set equal to unity for R>Rmax . Equations (16) and (17) involve a further approximation in that contributions from the fluctuation terms in (7) and (8) are neglected; the error due to this must be very small. Preliminary calculations indicated that long-range contributions from U O and PO were negligible and no correction was made for these.

O n the basis of the results of McDonald and Singer [5] we estimate the statistical error (standard deviation) in our Monte Carlo results as _+ 15 atm. in pressure and _+2 cal/mole in energy for runs of 300 000 configurations; for runs of 900 000 configurations the corresponding numbers are _+ 9 atm. and _+ 1.2 cal/mole.

3. RESULTS

The results of our calculations with the three potentials are given in tables 2-4, with experimental data for comparison. In the temperature range 96-140 K we give two different ' experimental ' values for the energy of which the ' present ' values are to be preferred on the basis of existing data [35-47]; these values are

T (K)

104"96 89"63 94'08 88"49 91"62

103"12 89"66

V (cm3/mole)

27"04 26"27 27-04 27"04 28.73 26"27 27"86

peale(MC) (atm.)

826 718 569 385

90 1113

185

pcalc(MD) (atm.)

834 692 582 410 115

1064 184

Uca,c(MC) (cal/mole)

- 1 4 1 0

- 1479 - 1435 - 1453 -1372 - 1439 - 1415

Ueale(MD) (cal/mole)

-1415 - 1485 - 1440 - 1456 - 1376 - 1448 - 1421

Table 2. Pressures and internal energies for Bobetic-Barker potential t .

t All Monte Carlo calculations in this table had 300 000 configurations.

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Type (a)

MC9 MC9 MC9 MC9 MC9 MC9 MC9 MC9 MC9 MC9 MC9 MC3 MD MC3 MD MC3 MD MC3 M D MC3 MD MC3 M D MC3 M D MC3 M D

T V (K) (cmZ/mole)

140-0 30"65 140'0 32"84 140"0 38"31 140-0 41"79 100"0 27"04 100'0 27-86 100"0 28"73 100'0 29"66 100"0 30"65

86'64 28"47 86'64 28"09

103"20 27"86 103'20 27"86 96"51 28"73 96"51 28"73

108"88 28"73 108"88 28"73 156-93 57"46 156"93 57"46 168.86 65"67 168"86 65"67 159"15 70"73 159"15 70"73 158"67 76"62 158.67 76"62 150"09 91"94 150-09 91-94

pcalc pexptl (b) (atm) (atE)

633 583 346 329

83 82 34 37

724 652 469 426 270 249 128 105 - 5 - 26(e) 57 17(g)

129 88(g) 537 498 538 498 180 186 190 186 410 415 444 415

53 67(o 39 67(f) 90 90 (r) 88 90(t) 67 66(f) 64 66(f) 68 63(o 60 63 (f) 60 61(r) 59 61(f)

Ucalc (cal/mole)

- 1 2 1 2

- 1141 - 9 8 7 - 911

- 1420 -1389 -1355 - 1 3 1 7

- 1391 - 1408 - 1383 - 1391 -1365 -1367 - 1337 -1342

- 6 6 3 - 6 8 7 - 586 - 583 - 569 - 555 - 532 - 518 - 4 5 5 - 4 3 3

Uexptl (c) (cal/mole)

-1198 - 1 1 2 5

- 983 - 9 1 1

-1445 - 1408 -1372 - 1337 - 1290

- 1 3 9 7

-1397 - 1383 -1383 - 1344 - 1344

- 679 (f) - 679 (f) - 591 (~) -591(f) - 573(r) - 573(r) - 539(r) - 539(e) -463(r) -463(f)

Uexptl (d) (cal/mole)

-1209 - 1 1 3 6

- 995 - 922

-1432 -1395 - 1358 - 1324 - 1276

- 1390 -1390 -1363 - 1363 -1345 -1345

a. MC represents Monte Carlo, with the following digit giving the number of configurations in hundreds of thousands; MD represents Molecular Dynamics.

b. Unless otherwise indicated, from Streett and Staveley [35]. c. Unless otherwise indicated, from Streett and Staveley [35], corrected as discussed in

the Appendix. d. Unless otherwise indicated present values as discussed in the Appendix. e. Extrapolated. f. Interpolated from the tables of Levelt [36]. g. Van Itterbeek and Verbeke [46].

Table 3. Pressures and internal energies for Bobetic-Barker potential.

discussed in the Append ix . T h e resul ts in table 2 are inc luded to p e r m i t

compar i son of M C and M D results a l though no exper imenta l values are available.

W e note first the excel lent ag reemen t be tween the M o n t e Carlo and molecu la r

dynamics results. T h e r e is no ev idence of a sys temat ic difference be tw een

the results f rom the two me thods such as was sugges ted by compar i son of the

results o f M c D o n a l d and S inger [5] and Ver le t [6-8] for the 6 : 12 potent ial .

Cons ider ing now the Bobe t i c -Barke r potential , wh ich gives be t te r ag reemen t

wi th sol id-state and gas-state data than other potent ia ls so far proposed, we note

that the calculated and exper imenta l energies are in excel lent agreement . T h e

differences are a lmost wi th in exper imenta l error, wh ich is p robab ly about

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664 J.A. Barker et al.

I I I I I I

I I I I I I

v .

v

O

e ~

I [ l l l l

e~

~

.~

e ~

~

-E O

o 0

4~

e ~

t . a e .

e ~

e-

e ~

O

L

" 0 e~

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Liquid argon 665

_+ 15 cal/mole (see Appendix). This is an improvement on the 6 : 12 potential for which the computed energies were more negative than those of Streett and Staveley [35] by up to 50 cal/mole (this is not affected by the correction of Streett and Staveley's data in the Appendix, but if the alternative energy values of the Appendix are used the 6 : 12 discrepancy is reduced to about 30 cal/mole). The calculated pressures are also in fair agreement with experiment [35, 46, 47]. The largest discrepancy within the range of reliable measurements is at T = 100K and V = 27-04 cm3/mole, where the calculated pressure is about 70 atm too high. The calculations of McDonald and Singer [5] using the 6 : 12 potential showed a discrepancy of 68 atm, (in the opposite sense) at T = 127 K, V = 33-51 cm3/mole. Thus considering both pressures and energies the Bobetic-Barker potential (with the Axilrod-Teller interaction) gives somewhat better agreement with experiment than does the 6 : 1 2 potential. We make this point not to show that the 6 : 12 potential is incorrect as a pair potential for argon; that is shown most clearly by comparing the experimental [48-52] and calculated second virial coefficients of gaseous argon, as in figure 1. However the present results

-10

~ -20

e n

-30

-40

log 1 o(T/~

2.0 2.2 2.4 2.6

[ ]

�9

D �9

�9

Figure 1. Second virial coefficients; deviations from values calculated with quantum correction for the 6 : 12 potential with ~ /k=l19"8K, a=3 .405 A. The solid curve is computed for the present potential (with quantum correction). Experimental data: O Weir e t al . [48]; [] Byrne et al . [49]; A Fender and Halsey [50]; V Michels e t al . [51]; �9 Whalley e t al . [52].

show that even as an ' effective potential ' for dense liquid argon the Bobetic- Barker potential with Axilrod-Teller interaction is somewhat superior to the 6 : 12 potential; it has the further advantage that it is also consistent with solid-state and gas-state data and spectroscopic and molecular beam data (see below).

Having said this, one must note that the agreement with experiment for the Bobetic-Barker potential is not perfect; the calculated pressures are a little too high, and increase too rapidly with increasing density at high densities. The

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666 J . A . Barker et al.

Barker-Pompe potential on the other hand gives pressures which are too low (table 4). We therefore experimented with interpolated functions of the form

u x(R) = UBB(R) + x[uBe(R) - UBB(R)] (18)

and found that such a potential with x = 0 . 2 5 gave excellent agreement with the experimental liquid pressures at high densities. This is the third potential for which results are given in table 4; the agreement with experiment is indeed excellent.

It is interesting to note that the agreement remains good on the critical isotherm (T= 150.75 K), in spite of the known singular character of the critical point, which is presumably not reproduced by our calculations with a relatively small number of molecules. Evidently the contributions of the singular terms to energy and pressure are relatively small.

T (K)

' Present 100.00 100.00 140. O0 1 4 0 . 0 0

150-87

V (cma/mole)

potential 27'04 29"66 30"65 41"79 70"73

U~ two -body R < Rmax (cal/mole)

--1455.7 -1335 '9 -- 1230.7

- 922-5 - 593.7

Ul two-body R > Rmax (cal/mole)

-69 .5 -57 .7 - 5 4 . 0 - 2 8 . 9 -10 .1

U~ three-body R < Rmax (cal/mole)

80 '9 63'5 58 "9 38 '2 26 "4

Ut three-body R > Rmax (cal/mole)

6.2 4.4 3.9 1.3 0.2

Bobetic-Barker potential 100"00 27"04 100"00 29'66 140"00 30"65 140"00 41"79

Barker-Pompe potential 100"00 27'04

- 1453.7 - 1338"9 -1229.3

- 925'0

-1458.5

- 6 9 "6 - 5 7 ' 8 -54"1 - 29"0

-6 9 ' 1

81 "3 63 "6 57-7 36 '0

80 "9

6"2 4.4 3.9 1"3

6"2

Ui quantum correction (cal/mole)

15 '6 12'5 9"3 6'4 4"6

15 "7 11 '9 9"4 6"2

14 "7

Table 5. Contributions to the internal energy.

In tables 5 and 6 we show for some selected thermodynamic states the separate two-body, three-body and quantum correction contributions to the energy and pressure, as well as the long-range corrections given by (14)-(17). Th e long-range two-body corrections are relatively large. However, because the deviation of ge(R) from unity amounts to only a few per cent for R > Rmax (and fluctuates in sign) while u(R) and u'(R) vary relatively slowly, these terms are given with sufficient accuracy by (14) and (15) (as shown by McDonald and Singer [5]). The three-body contributions are quite large, and must clearly be included in a realistic calculation. The possibility that these contributions are to an appreciable extent cancelled by other many-body interactions now seems remote. Th e quantum corrections are relatively small, as expected, but not negligible at the level of accuracy to which we are working here. Th e differences between the three potentials considered arise primarily from the short-range two-body contributions.

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T (K)

'Present 100"00 100"00 140"00 140"00 150"87

V (cmS/mole)

'potential 27"04 29"66 30"65 41"79 70.73

P two-body R < Rmax

(atm.)

454.6 140.8 495.6

23.7 46.3

P two-body R > Rmax

(atm.)

- 214"7 - 162"1 -146 '7 - 5 7 ' 4 - 1 1 ' 8

P three-body R < Rmax

(atm.)

335"9 220'4 198"7 45"1 12"8

P three-body R > Rmax

(atm.)

28 '3 18 "4 15 "6 3"9 0"4

Bobetic-Barker potential 100"00 27"04 100"00 29"66 140"00 30.65 140"00 41.79

Barker-Pompe potential 100"00 I 27.04

520"8 220"1 554"6 23"3

275"7

-215 - 162 - 147

-57

-212.4

�9 4 349-2 �9 5 228 "8 �9 1 194"4 �9 5 59"7

348" 5

28 "2 18 " 5

15 "6 3"9

28 "3

P quantum correction

(atm.)

42 "2 25 "3 16 "7 2"7 1"2

41 "6 21 "0 15 "6 4"6

39 '7

Table 6. Contributions to the pressure.

There is independent evidence from the spectroscopic data of Tanaka and Yoshino [25, 26] (figure 3) and the molecular beam differential scattering intensities of Lee and Parson [27] (figure 2) that our third potential is more accurate than either the Bobetic-Barker or Barker-Pompe potentials; it gives almost perfect agreement with both these sets of data. Cavallini et al. [53] have also made differential scattering measurements, from which they estimate that Ru'(R) /k at the inflexion point of the potential has the value 440 K. For this quantity the Bobetic-Barker potential gives 440 K, the ' p r e sen t ' potential gives 458~ the Barker-Pompe potential gives 510K, the Dymond-Alder potential gives 495 K and the 6 : 12 potential gives 360 K. There is a little uncertainty in the experimental number since the 6 : 12 potential was used in its determination, and the agreement with the present potential may be regarded as satisfactory. The scattering measurements of Mueller et al. [54] were consistent with a sufficiently wide range of potentials to include the present potential, except perhaps for a slight discrepancy beyond 5 A.

Thus our assumption that many-body interactions other than the Axilrod- Teller interaction may be neglected has led us to a pair-potential which is very accurate. By contrast the assumption that all many-body interactions could be neglected led Dymond and Alder [12, 13] to a pair-potential in much worse agreement with both the spectroscopic and molecular beam data (figures 2 and 3).

The Bobetic-Barker potential has been shown to give good agreement with the properties of gaseous and solid argon, while the Barker-Pompe potential gives fair agreement; it is therefore necessary to consider the consequences for these properties of changing from the Bobetic-Barker potential to our third potential. On the reasonable assumption (which we have checked in several cases) that all quantities would vary linearly with the parameter x, the following conclusions can be drawn. The repulsive potential near 2 A, the long-range coefficients

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L

iO s

H ~ ..7" " " ' ' a L ~"" / ""~, i

iOI ~ ,

I00 I0 20 50 40 50 60 70 80 90 c.m. Scattering Angle

Figure 2. Molecular beam differential scattering intensities of Lee and Parson [27]. In the upper set of curves the points are experimental, the solid and dashed curves were calculated for the Bobetic-Barker and Barker-Pornpe potentials respectively by Lee and Parson while the dash-dotted curve represents interpolated values for the present potential. The lower curve was calculated for the potential of D y m o n d and Alder [13] by Lee and Parson.

of R -6, R -8 and R -1~ the second and third virial coefficients and the lattice spacing of crystalline argon at 0 K would be unchanged, and in excellent agreement with experiment. The cohesive energy at OK would change from -1844 cal/mol to about -1850 cal/mole, still within experimental error (Flubacher et al. [40] give - 1 8 4 6 + 7 cal/mole). The low-temperature Debye theta values would be reduced by about 0 .4K, leading to slightly worse agreement with experiment at OK but slightly better agreement on the average over the range 0-12~ The calculated thermal expansion in the range 0-12 K would be reduced slightly [16], leading to even closer agreement with the results of Tilford and Swenson [55]. The calculated gas viscosities would be raised [18] by a fraction of a per cent at temperatures below 400 K, improving slightly the already excellent agreement with experiment; at temperatures above 1000 K they would be raised by 0.5 per cent, leading to a better compromise between the modern measurements of Guevara et al. [56] and of Dawe and Smith [57]. The agreement with experiment for other gas transport properties would be essentially unchanged, and excellent. The agreement with measured elastic constants of crystalline argon would be essentially unchanged [17], and rather poor. It was suggested [16, 17] that this discrepancy may be due to experimental problems, and the good agreement which we find for liquid properties perhaps adds some weight to that possibility. The agreement with experimental phonon disperson curves would be essentially unchanged [17], and good.

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K

E

(.9 <1

30 -

20 -

lO -

I r I I I I

2 0

[ ]

[ ]

-f

I I I I I I 1/2 3/2 5/2 7/2 9 /2 11/2

Figure 3. Vibrational level spacings of argon. Experimental data of Tanaka and Yoshino [25] (with error bars set by Bruch and McGee [26]) are shown thus ix; calculated values are shown for the present potential (V) and those of Bobetic and Barker (C)), Barker and Pornpe (x) and Dymond and Alder ([1). For all potentials except the present these results are due to Bruch and McGee [26].

Overall, then, the new potential gives excellent agreement with thermodynamic properties of solid, liquid and gaseous argon, and with transport properties of gaseous argon. Further, the resulting pair potential is in good agreement with molecular beam differential scattering cross sections and spectroscopic data on the Ar~ molecule, indicating that it is very close to the true pair potential. Evidently our approximation of neglecting many-body interactions other than the Axilrod- Teller interaction is an excellent one.

I t is interesting to note that the internal energies calculated with the present potential (table 4) are in considerably better agreement with the revised experimental energies derived in the Appendix than with the older values based on the analysis of Din, and it was the discrepancies with the older data which led us to re-analyse the experimental data. This may perhaps be taken as evidence that the study of intermolecular forces has attained a certain maturity. Theory and ' computer experiments ' provide a way of comparing and cross-checking different kinds of experiments in a manner similar to but more far-reaching than that provided by thermodynamics.

In figure 4 we compare our final potential with the 6 : 12 potential most commonly used for argon. Considering that the quite similar Bobetic-Barker and Barker-Pompe potentials gave calculated liquid pressures differing by as much as 250 atm., it must be regarded as remarkable that the quite different 6 : 12 potential gave such good quantitative agreement with experiment.

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670 J. A. Barker et at.

a (~)

4.0 5.0 6.0 7.0

v o

E "5

-40

-80

-120

Figure 4. Comparison of the 6 : 12 potential (with e/k = 119 '8 K, e = 3 '405 A) with the present potential.

We are grateful to Dr. Y. T. Lee for communicating the scattering data and the calculated results for three potentials. This work was supported in part by the U.S. Department of the Interior, Office of Saline Water. One of us (R.A.F.) thanks the Australian National University for support.

APPENDIX

Here we discuss existing experimental evidence on the internal energy of liquid argon. Streett and Staveley [35] recently used their p V T data and energies on the liquid-vapour equilibrium line derived from the results of Din [37] to calculate the internal energy of argon in the range 100-140 K. Their calculations contain a slight error, amounting to as much as 20 cal/mole at T = 140 K, p = 50 atm. In the values given in the text we have corrected the error, following the procedure described by Streett and Staveley.

The energies on the saturation line were derived by Din from the vapour pressure data of Clark et al. [41] and the vapour densities of Mathias et al. [58] using the Clausius-Calpeyron equation; this led to a discrepancy of about 140 cal/mole with the calorimetrically measured latent heat of Frank and Clusius [38] at the normal boiling point. Din ascribed this discrepancy to errors in the vapour density results at low pressures, and therefore extrapolated the latent heat from temperatures above 110 K in such a way as to agree with the result of Frank and Clusius. Experimental results published since the work of Din indicate that the energies derived in this way are not altogether satisfactory. In particular there is a serious disagreement with the measurements of the specific heat along the saturation curve made by Walker and Jones and Walker [42, 43] ; this is shown in table A 1, where the values in the second column were calculated from the energies listed by Streett and Staveley according to the equation

OV c~= \ ,<rio-

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T (K)

100 110 120 130 140

Experimental [42-44] (interpolated)

10"3 10'8 11 "7 13 "2 14 "4

Calculated from Ui [35]

12 "6 12"6 13 "0 13 '9 14"6

Table A 1. Specific heat on saturation curve, C~; cal/deg, mole.

Latent heat (cal/mole)

1557"5 • 1"5 1555"0 • 4"6 1555 • 6 1501 1537 1542

Reference

Frank and Clusius [38] Flubacher et al. [40] Flubacher et al. [40] Eucken and Hauch [39] Clark et al. [41], London Clark et al. [41], Amsterdam

Method

Calorimetric Calorimetric Clausius-Clapeyron Calorimetric Clausius-Clapeyron Clausius-Clapeyron

Table A 2. Latent heat of argon at normal boiling point.

Since it seems very unlikely that the measured specific heats are in error by as much as 22 per cent, we have made a new analysis of the experimental data, seeking the best compromise between existing experiments.

There are several measurements of the latent heat of argon at the normal boiling point, which we list in table A 2. The value of Eucken and Hauck is clearly too low and the values of Clusius and Frank and especially of Flubacher et al. are presumably to be preferred. However, the values derived from the independent and careful London and Amsterdam vapour pressure measurements of Clark et al. [41] must also be given some weight; in calculating the latent heat the vapour density was determined using accurate modern second virial coefficients [48]. In the case of the Amsterdam measurements a short extrapolation (about 1 K) was involved.

We calculated values of Ui using the calorimetric latent heat value of Flubacher et al. and the measured specific heats in the integrated form of (A 1). At 138.16 K the calculated value was about 41 cal/mole more negative than the value derived from p V T and vapour pressure data by Michels et al. [51]. This is a real discrepancy between different experiments. In the absence of further information we sought a compromise by distributing this in the proportion of 10 cal/mole to the latent heat of Flubacher et al., 21 cal/mole to the specific heat data, and 10 cal/mole to the results of Michels et al. [51]. Thus we adopted a latent heat at the boiling point of 1545 cal/mole (this is about halfway between the value of Flubacher et al. and the London value of Clark et al.). We increased the values of C~ uniformly by 4.5 per cent and repeated the calculation of Ui. Th e resulting value of Ut at 138.16 K was - 932 cal/mole, compared with the value - 923 cal/mole found by Michels et al. [51].

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Streett and T (K) Staveley [35] Rowlinson [44] Present

87-29 90

100 110 120 130 140

- 1306 - 1209 - 1113 - 1 0 1 2

- 896

- 1396 -1377 - 1298 -1207 - 1 1 1 6

-1016 - 896

- 1389 - 1369 - 1293 - 1 2 1 2

- 1125 - 1027

- 907

Table A 3. The internal energy of liquid argon on the saturation line; Ui, cal/mole.

T h e result ing values of U~ are shown in table A 3, with the values of Streett and Staveley [35] and those listed by Rowlinson [44], also derived f rom Din ' s results, for comparison. We believe that the present results are probably the best estimates that can be fo rmed with existing experimental data, and that the experimental error should be est imated as about + 15 cal/mole.

T h e values called ' present ' in the tales in the text were calculated using these values together with the procedure of Streett and Staveley [35] and their p V T data.

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