computer simulations and theory

MOLECULAR PHYSICS, 1997, VOL. 92, NO. 5, 813± 824

The exp-6 potential ¯ uid at very high pressures: computer simulations and theory È RTLER 1 , IVO NEZBEDA 2,3 and MARTIN LI SAL 2 By HORST L. VO 1 Department of Molecular Dynamics and Computer Simulation, Institute of Theoretical Physics, University of Leipzig, 04109 Leipzig, Germany 2 E. Ha la Laboratory of Thermodynamics, Institute of Chemical Process Fundamentals, Academy of Sciences, 165 02 Prague 6 ± Suchdol, Czech Republic 3 Department of Theoretical Physics, Charles University, 180 00 Prague 8, Czech Republic ( Received 10 March 1997; revised version accepted 20 May 1997) The exp-6 potential ¯ uids for four values of the softness parameter a , ranging from 11. 5 to 14. 5, have been investigated both by computer simulations and theory at supercritical temperatures and very high densities ( pressures ). Tables of the simulation data for pressure include altogether 141 thermodynamic state points generated at six temperatures for each a . In addition to computation of the thermodynamic properties, great attention has been paid to the structure, and an approximate location of the onset of freezing has been determined. To estimate the properties of the ¯ uid from theory, a modi® ed version of the Weeks± Chandler± Anderson (mWCA) theory and the optimized reference hypernetted chain ( RHNC) theory were used. It is shown that for the thermodynamic properties mWCA theory performs very well under all stable ¯ uid conditions considered, whereas RHNC theory, comparable with mWCA at the lower densities considered, gradually loses its accuracy in the very high density range. However, for structure the overall performance of RHNC theory is considerably better than that of mWCA, which points to possible cancellation of errors in the latter theory when applied to thermodynamic properties.

1. Introduction Thousands of simulations have been performed over the last three decades on a variety of model ¯ uids. Among these models undoubtedly the most prominent and most intensively studied model has been that of Lennard-Jones (LJ). The reason is evident: LJ 12-6 potential, despite its simplicity, estimates fairly well the e€ ective pair interactions between the molecules of normal (i.e., non-polar) ¯ uids at normal (i.e., nonextreme) conditions. However, outside these conditions the usefulness of the LJ potential is not guaranteed; typical examples are supercritical ¯ uids at very high pressures, i.e., under conditions encountered in geological, astrophysical, and military applications [1, 2]. It is well established that under these conditions the repulsive interaction between molecules is much softer than that given by the LJ potential, and that the exp-6 potential provides a much more faithful description of intermolecular interactions. Consequently, the resulting structure of compressed ¯ uids is not completely hardsphere-like, which evidently causes problems for simple perturbation theories based on a hard sphere reference ¯ uid. Also, this is probably the main reason why the exp-6 potential has attracted, in general, only limited 0026± 8976/97 $12 . 00

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attention, and only recently has interest in this ¯ uid been revived [3± 6]. The exp-6 potential ¯ uid was studied recently by Belonoshko and Saxena [4, 5], who published extensive tables of data (altogether 504 state points) for pressures ranging from 1. 8 kbar to 2. 2 Mbar. However, the application of a volume-explicit equation of state [7] to these data cast some doubts on their correctness, and a later study [8] con® rmed these doubts. It was shown that the simulations were not performed su ciently carefully to detect structural changes taking place in the system during the simulation. Consequently, very many state points given in the tables correspond either to a metastable (glassy ) state or even to the solid state, and thus are useless for studies of the ¯ uid phase. The goal of this study is twofold. In the light of the above ® ndings on the existing data, the primary motivation has been to generate a large body of reliable and accurate pseudoexperimental data for the supercritical exp-6 ¯ uid at very high pressures. Since laboratory experiments at such pressures may be di cult (or even impossible ) to carry out, it is highly desirable also to have a theory available which would describe the thermodynamic properties of the exp-6 ¯ uid with reasonable 1997 Taylor & Francis Ltd.

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H. L. VoÈrtler et al.

accuracy. The second goal of this paper therefore is to explore the best methods currently available to estimate its PV T behaviour with modi® ed Weeks± Chandler± Anderson (mWCA ) perturbation theory [9] and an optimized reference hypernetted chain ( RHNC) theory [10]. This application of theory also is of interest in its own right because these theories, both of them based on hard sphere ¯ uid properties, have not been tested so far under the conditions considered in this paper (mWCA theory was tested but only at two very high temperatures and densities away from the freezing line [9]). The same applies to empirical criteria of freezing based so far only on the LJ and inverse-power potentials. 2. Basic de® nitions and computational details In this paper we deal with the ¯ uid whose particles interact via the exp-6 potential,

u( r) =

=

²

[a -

6 6

[

exp a ( 1 - r /rm )

] - a a-

6

( rm /r) 6

f or r > rmax ,

¥

f or r < rmax .

]

( 1)

The parameter ² determines the depth of the potential minimum which is located at rm and a determines the softness of the repulsion. As r tends to zero, u given by the functional form in the ® rst line of equation (1) reaches a maximum (at rmax ) and then it drops sharply to minus in® nity. The repulsive part of u (for rmax < r < rm ) is much softer than that of the `standard’ LJ potential, see ® gure 1. As mentioned earlier, it is exactly this property which makes the exp-6 potential a realistic model for ¯ uids at very high pressures. In their study, Belonoshko and Saxena [5] related the parameters of the exp-6 potential to ® ve geologically important real gases (H2, O2, CH4, CO and CO2) by ® tting the second virial coe cient. Consequently, they used ® ve values of a from the range {13. 34, 14. 48} and considered temperatures within the range 300 K < T < 4000K and pressures P ranging from 1. 8 kbar to 2. 2 Mbar. In the present paper we use the usual dimensionless variables denoted by asterisks and de® ned by means of the potential parameters: T * = kB T /² ,

P* = Prm /² , 3

3 q * = q rm

,

u* = u /² , ( 2)

where kB is the Boltzmann constant and q is the number density. Because of the softness of its repulsive part, the exp-6 ¯ uid can be compressed to very high densities and perturbation theories based on hard sphere ¯ uid properties therefore must be modi® ed. Such a modi® cation has been developed by Kang et al. [9] for the WCA theory

Figure 1. Comparison of the exp-6 potential with a = 11. 5 (solid curve) with the LJ potential (dashed curve).

[11]. The mWCA theory splits the pair potential u into a reference potential u0 and a perturbation potential D u, u( r) = u0 ( r) + D u( r) ,

( 3)

not at the potential minimum rm , but at a density-dependent break point ¸, u0 ( r) = u( r)

-

for r
¸,

= 0

for r
¸,

= u( r)

F( r) = u( ¸)

¸,

-

[du( r) /dr]

r= ¸ ( ¸

( 5)

- r) ,

( 6)

where the breakpoint ¸ is de® ned as 1 /6

( 7) ¸ = min {a f cc , rm }, 1 /3 q the nearest-neighbour distance in

with a f cc = 2 the fcc lattice. For densities q * less than 21 /2 , the above modi® cation becomes identical to ordinary WCA theory ( ¸ = rm , F( r) = - ² ) . The remaining ingredients of WCA theory remain unchanged, i.e., the pair correlation function g of the ¯ uid is approximated by that of the reference ¯ uid, g