Virial coefficients and vapor-liquid equilibria of the EXP6 and 2-Yukawa fluids

Virial coefficients $B_2$ through $B_4$ and the vapor-liquid equilibria for the EXP6 and 2-Yukawa (2Y) fluids have been determined using numerical integrations and Gibbs ensemble simulations, respectively. The chosen 2Y models have been recently determined as an appropriate reference fluid for the considered EXP6 models.


Introduction
Simple fluids, i.e., the fluids whose molecules interact via a spherically symmetric potential, u = u(r), are most commonly modeled as Lennard-Jones (LJ) fluids. When applied to simple real fluids, the LJ performs reasonably well at subcritical and slightly supercritical conditions. However, for quite obvious reasons it fails at high temperatures/pressures: Repulsive interactions at these conditions are very soft and it has been well established that the EXP6 potential provides much more faithful description of the intermolecular interactions [1]. Furthermore, the exponential repulsion agrees with molecular beam scattering data as is also known from theoretical studies. It is therefore highly desirable, particularly from the point of geochemical and industrial (conditions of detonations and propagation of shock waves) applications, to reach a similar level of understanding and theoretical description of the EXP6 fluids similar to that of the LJ fluid.
A number of simulation data for the EXP6 fluid are available in literature along with early theoretical attempts (see [2] and references therein). The problem of theory is that all common methods are based on the assumption of the presence of very steep repulsions at short separations and thus make use of, either directly or indirectly, the known properties of the fluid of hard spheres. Consequently, they are not applicable to models with soft repulsion. To overcome this problem we have recently developed an alternative to HS-based theories, a theory based on the Yukawa (Y) model as a reference [3]. The Y potential seems to be a 'universal' simple fluid model because it is possible, by changing its parameter [see equation (2) below], to change both its range and repulsive softness. This is the reason why the Y potential is used in applications to describe of a variety of physical phenomena (see [4] and references therein).
The Y potential is a model belonging to the family of van der Waals models, i.e., the model with a hard core and approximating interactions outside the core. A large body of results, both theoretical and simulation, is available in literature (see, e.g., references [5][6][7] and references therein) for the Y fluid. However, to apply the Y potential to more realistic models with a soft repulsive part (i.e., without a hard core), it is necessary to use a combination of two (or even more) Y functions which results in a model without any hard core.
In a recent paper [2] we investigated, by means of molecular simulations, the structure of the EXP6 fluids and formulated the criteria for determining a Y model [more accurately, two Yukawa model (2Y)] which could be used as a reference system for developing an analytic theory of the EXP6 fluids. Unlike the one Y model (1Y), multiple Y models have not been investigated in detail yet and only a handful of results are available [8,9]. To accomplish the goal, i.e., to develop an analytic theory for the EXP6 fluid, we should first know the properties of the 2Y fluids accurately and in detail and this has been the motivation for the present study.
Since the virial coefficients provide the basic information on the properties of the fluid and can be used in various theoretical methods, we computed the first four virial coefficients of both the parent EXP6 fluid and descending 2Y fluids. Furthermore, we have also determined the vapor-liquid equilibrium (VLE) of the 2Y fluid and located the critical point which is an important information for setting the criteria of determining the 2Y fluid associated with the EXP6 fluid.

Basic definitions and computational details
The EXP6 fluid is a collection of additive species (atoms, molecules, etc.) interacting via the EXP6 potential (also referred to as a modified Buckingham potential), for r < r max , where r max and r m is the location of the potential maximum and minimum, respectively, and ǫ is the depth of the minimum. Parameters r m and ǫ are used henceforth as the length and energy units; dimensionless number density, ρ * , temperature, T * , pressure, P * , and internal energy, U * , are thus defined as ρ * = ρr 3 m , T * = T k B /ǫ, P * = P r 3 m /ǫ, and U * = U/ǫ, respectively, where k B is the Boltzmann constant.
The hard core 1Y potential is defined as where z is the parameter governing the range of the interaction. The 2Y potential is a model made up of two Yukawa tails without, in general, any hard core, where ǫ 1 > 0 and ǫ 2 > 0 are the strengths of the repulsive and attractive contributions, respectively, while κ −1 1 and κ −1 2 are the measures of the range of the corresponding tails. The virial coefficients, B i , of the virial expansion, were evaluated numerically up to B 4 over a wide range of temperatures using the Mayer sampling [10]. We have recently examined another version of the virial expansion, the perturbed expansion around a suitable reference system similarly to the theories of liquids, where subscript "ref" refers to a reference system. To determine the VLE envelope we used the common Gibbs ensemble with the total number of particles N = 512 and applied the long-range correction in order to truncate the potential at r c = 0.45 3 N/ρ * .

Results and discussion
When the EXP6 model is applied to real fluids by adjusting its parameter α, its resulting values typically vary between 11 and 15. To keep contact with our previous papers [2,11] we have chosen the bracketing values, α = 11.5 and 14.5. Parameters of the 2Y model descending from these EXP6 models were determined in our previous paper [2] and are given in table 1.   Table 2. Virial coefficients of the EXP6 potential with α=11.5; b2 = (2/3)πr 3 m .   Table 4. Virial coefficients of the 2-Yukawa potential mimicking the EXP6 potential with α = 11.5; b2 = (2/3)πσ 3 .  Table 5. Virial coefficients of the 2-Yukawa potential mimicking the EXP6 potential with α = 14.5; b2 = (2/3)πσ 3 . Numerical values of the virial coefficients of all four fluids considered are given in tables 2 through 5 and are also compared in figure 1. Examination of the tables/figure shows that the coefficients of the EXP6 and 2Y fluids are very similar. This is a consequence of the fact that the repulsive parts of the EXP6 and 2Y models have been used to fit the corresponding 2nd virial coefficients [2]. In other words, the similarity of all the virial coefficients means that the long-range part of the models does not affect them to any important extent. Table 6. Vapor-liquid equilibrium data of the 2Y fluid mimicking the EXP6 potential with α = 11.5 .   The VLE results for the equilibrium densities of the 2Y fluids are given in tables 6 and 7. With these data, the critical point was determined using the rectilinear rule and the common analytic parametrization expression

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and ρ l + ρ v 2 = ρ c + C 1 |t| ψ + C 2 |t| + C 3 |t| ψ+∆1 , where t = 1 − T /T C and B i , C i , β and ψ are parameters to be fitted to the simulation data; ∆ 1 = 0.5 for the vapor-liquid equilibria [12]. The results are given in table 8. In this table we also give an estimate of the critical temperature obtained using the common virial expansion and the perturbed virial expansion of the 2nd order [13,14]; in the perturbed expansion we used the fluid of hard spheres of diameter σ as the reference. As we see, the perturbed method provides a very good estimate which further confirms its superiority over the common virial expansion.

Conclusions
In this paper we have presented results for the first four virial coefficients of the EXP6 fluid and the associated 2-Yukawa fluids, and vapor-liquid equilibria in the latter models. These data are necessary for a subsequent development of a theory for the 2-Yukawa fluids which further provides an alternative to hard core (van der Waals type) equations of state. In addition to the determination of the critical point from the vapor-liquid coexistence data, we have used the computed virial coefficients to determine the critical temperature from the virial expansions. The obtained results further confirm the recently reported applicability and accuracy of the perturbed expansion.