Potential parameters for

## 1. Introduction

Experimental assessment of macroscopic thermo-dynamical parameters under extreme conditions is almost impossible and very expensive. Therefore, theoretical EOS for further experiments or evaluation is inevitable. In spite of other efficient methods of calculation such as integral equations and computer simulations, we have used perturbation theory because of its extensive qualities. Moreover, other methods are more time consuming than perturbation theories. When one wants to deal with realistic intermolecular interactions, the problem of deriving the thermodynamic and structural properties of the system becomes rather formidable. Thus, perturbation theories of liquid have been devised since the mid-20th century. Thermodynamic perturbation theory offers a molecular, as opposed to continuum approach to the prediction of fluid thermodynamic properties. Although, perturbation predictions are not expected to rival those of advanced integral-equations or large scale computer simulations methods, they are far more numerically efficient than the latter approaches and often produced comparably accurate results.

Dealing with light species such as

Furthermore, for this fluid mixture, the quantum effect has been exerted in terms of first order quantum mechanical correction term in the Wigner-Kirkwood expansion. This term by generalizing the Wigner-Kirkwood correction for one component fluid to binary mixture produce acceptable results in comparison with simulation and other experimental data. Since utilizing Wigner-Kirkwood expansion in temperatures below 50 K bears diverges, we preferred to restrict our investigations in ranges above those temperatures from 50 to 4000 degrees. In these regions our calculations provide more acceptable results in comparison with other studies.

This term make a negligible contribution under high temperatures conditions. Taking into account various contributions, we have utilized an improved version of the equation of state to study the Helmholtz free energy * F*, to investigate the effects of

*and*P

*on thermodynamic properties of helium and hydrogen isotopes mixtures over a wide range of densities. We also have studied effects of concentrations of each component on macroscopic parameters. In addition, comparisons among various perturbation and ideal parts have been presented in logarithmic diagrams for different densities and concentrations for evaluation of perturbation terms validity in respect to variables ranges.*T

The first section is dedicated to a brief description of Wigner expansion which leads to derivation of first quantum correction term in free energy. With the intention of describing effects of quantum correction term we have explained theoretical method of our calculations in the frame work of statistical perturbation theory of free energy in section two. In section three we have depicted diagrams resulted from our theoretical evaluations and gave a brief explanation for them. In section four we have focused on the description of our calculations and its usages in different areas. Finally, some applications of this study have been introduced in the last section.

## 2. Quantum correction term

Considering quantum system of

Where,

### 2.1. Wigner-Kirkwood expansion

To have an analytical equation for quantum effects in fluid we must derive partition function of it. In approximating partition function we need to evaluate Boltzmann density. Consequently having an expansion of quantum correction terms it is necessary to expand Boltzmann density. Considering system of

Where

Via integrating equation 2 in respect to

Let us introduce following definition

That

One can expand

And then we can find that

So we have expanded series in

and finally integrating on the momentum variables

where

Integrating Boltzmann density ignoring exchange effects over configuration space will result in partition function of fluids mixture.

Substituting the

For expressing macroscopic physical quantities, one defines the quantum average of a function

At the one-particle level, one introduces the particle density

At the two-particle level, the two-body density is given by

And the pair distribution function

The classical partition function and the classical average of a function

Consequently with the definition of equation 19 one can derive below equation for

Since we have

### 2.2. Free energy

Generalizing to multi-component system we have [8]

In this chapter the two formula which use RDF, we will encounter below integral equation that need expansion.

On the right side of above equation from the right in the first equation we approximate distribution function with its values at contact points. This choice has been resulted from the behavior of molecules of which their repulsive interactions dominate their attractive potential. However, for the second term (

Substituting above equation in

Where

That indicates inverse Laplace of

Therefore, Using Laplace transform of RDF

* i*particle’s concentration and

## 3. Framework

The derivation of the thermodynamic and structural properties of a fluid system becomes a rather difficult problem when one wants to deal with realistic intermolecular interactions. For that reason, since the mid-20th century, simplifying attempts to (approximately) solve this problem have been devised, among which the perturbation theories of liquids have played a prominent role [10]. In this instance, the key idea is to express the actual potential in terms of a reference potential (that in terms of Ross perturbation theory Helmholtz free energy is expressed as of the “unperturbed” system) plus a correction term. This in turn implies that the thermodynamic and structural properties of the real system may be expressed in terms of those of the reference system which, of course, should be known. In the case of two component fluids, a natural choice for the reference system is the hard-sphere fluid, even for this simple system the thermodynamic and structural properties are known only approximately. Let us now consider a system defined by a pair interaction potential

The terms respectively are perturbation, Quantum, hard convex body and ideal terms. Perturbation term due to long range attraction of potential is given by [10]

Via Laplace transform of RDF (

Where

Non-sphericity parameter

The correction term due to nonadditivity of the hard sphere diameter is the first order perturbation correction [14]

In Eq. (41),

The ideal free energy with

Compressibility factor of ideal term is one and

For the perturbation term due to long rage attraction of potential tail employing (44) we will have

Numerical integration has been used for calculation of

Summation over compressibility factors gives the total pressure of mixture

Defining Gibbs free energy provides information at critical points of phase stability diagram. Concavity and convexity of Gibbs diagram indicates if mixture is in one phase or not,

Furthermore, Gibbs excess free energy is an appropriate measure in the definition of phase stability. Negative values for this energy describe stable state. This is expressed as

That

Compairing this equation with

### 3.1. Potentials

It is convenient to consider interacting potential with short-range sharply repulsive and longer-range attractive tail and treat them within a combined potential. The most practical method for the repulsive term of potential is the hard-sphere model with the benefit of preventing particles overlap. Furthermore, attractive or repulsive tails may be included using a perturbation theory. It is incontrovertible to generalize this potential to multi-component mixtures. This behavior is conveyed in double Yukawa (DY) potential which provides accurate thermodynamic properties of fluid in low temperatures and high density [18, 19]. At first we define DY potential as its effects on pressure of

_{2} | _{2}-H_{2} | ||

2.634 | 2.970 | 2.978 | |

2.548 | 2.801 | 3.179 | |

10.57 | 15.50 | 36.40 | |

3.336 | 3.386 | 3.211 | |

12.204 | 10.954 | 9.083 |

For the atomic and molecular fluids studies in this mixture, these particles interact via a exponential six (exp-6) or Double Yukawa (DY) potential energy function [20]. The fluids considered in this work are binary mixtures that their constituents are spherical particles of two species,

So we consider two-component fluid interacting via Buckingham potential

In view of the energy equation (32), one can readily obtain equation for total pressure and different contributions to pressure from standard derivation of respective Helmholtz free energy. By the exp-6 potential, we have computed the Helmholtz free energy. The ten-point Gausses quadrature has been used to calculate integrals in quantum correction and perturbation contribution. The calculated pressure for

_{2} | _{2}-H_{2} | ||

13.10 | 12.7 | 11.1 | |

10.80 | 15.50 | 36.40 | |

0.29673 | 0.337 | 0.343 |

## 4. Results

For helium-hydrogen mixtures different parts of pressure due to correction terms and ideal parts have been showed in figure 2 at

Gibbs excess free energy which is a measure for indicating phase stability of matters has been depicted in figure 3. Stability is limited to the areas that Gibbs excess free energy tends to negative values. This figure explains that stability rages for helium-hydrogen mixture at room temperature is confide in the boundaries in which helium concentration is less than 0.1.

Table 3 presents a comparison between results of pressure from this work using DY potential in place of exp-6, Monte–Carlo simulations and additionally study of reference [23] Obviously, there are appreciable adaption among our investigation results and MC which proves validity of our calculations. As Table 3, exhibits in low temperatures DY potential have more consistent results in comparison with exp-6. However, values of pressure extracted using DY potential cannot adjust with simulation resembling exp-6. Moreover, at higher temperatures after

300 | 0.25 | 1.101 | 0.433 | 2.3090 | 2.7039 | 1.9664 | 2.8678 |

300 | 0.5 | 1.101 | 0.400 | 1.8560 | 1.7001 | 1.5729 | 1.8402 |

300 | 0.75 | 1.101 | 0.367 | 1.4240 | 1.2816 | 1.3160 | 1.3887 |

1000 | 0.5 | 1.223 | 0.335 | 4.5100 | 4.4205 | 4.1094 | 4.9406 |

1000 | 0.75 | 1.223 | 0.307 | 3.7150 | 3.5190 | 3.5904 | 3.9328 |

4000 | 0.5 | 1.376 | 0.247 | 12.4300 | 12.0832 | 12.1014 | 14.154 |

4000 | 0.5 | 1.572 | 0.282 | 16.3300 | 16.4485 | 16.4720 | 19.859 |

Providing evidence of gradual divergence of DY and exp-6 potentials, a comparative figure has been made in figure 4 for helium-hydrogen mixtures. This figure shows more steepening effects of DY on total pressure of this mixture. Both potentials engender increase in pressure, except that, Buckingham affects moderately on pressure increase. The exp-6’s more steady behavior makes it adjustable with previous studies and MC simulation.

In figures 5, 6, 7, 8 we tried to give information about effects of quantum correction term on total pressure of helium-hydrogen and deuterium-tritium mixtures at the high reduced density of 1.3. This correction term has been plotted in 3-dimensional diagram in figure 5. This term is approximately zero for temperatures higher than 200 (K). Figure 6 represents that for hydrogen rich mixture at low temperature due to quantum effects pressure rise is significant. For effectual discussion on the effects of this term we have described

## 5. Conclusion

An Equation of state of hydrogen–helium mixture has been studied up to 90G (pa) pressure and temperature equal to 4000◦K. We have used perturbation theory as an adequate theory for describing EOS of fluid mixtures. As well, by using this theory we can add extra distributive terms as perturb part which makes it more applicable than other theories. Considering this advantage, we can spread it out with additional terms for investigation on other states of matter like plasma in the direction of compares with experimental data. Otherwise, using simulation methods, for evaluating our theoretical results. Such as ab initio simulations with the code VASP,[25] which combines classical molecular dynamics simulation for the ions with electrons, behave in quantum mechanical system by means of finite temperature density functional theory [26]. In this chapter, two potentials have been presented, which we have used them for hydrogen isotopes and helium, and their mixtures. By means of comparison with Monte Carlo simulation and results of refrence [14] in Table 3 we could prove that exp-6 potential is more beneficial than DY in wider ranges of variables, since its application in this theory shows more convergent results in comparison with MC simulation [24]. Also exp-6 potential is a good choice of potential since it allows us to elevate temperature and density [28]. But as hydrogen molecules dissociation occurs [28] for pressures more than 100G (pa), this effect must be accounted. Therefore, we have restricted ourselves to pressures below 100G (pa).

Furthermore, we have used Wertheim RDF which enables us to use this EOS for extended values of temperature. As well, we have compared different contributions of pressure to represent which one is more effective in different density and temperature regimes. By finding the most effective parts of pressure contributions in each ranges of independent variables (Temperature, reduced density, mole fraction), we can omit the less significant parts which are considered ignorable in value, to decrease unnecessary efforts. Likewise, we can speculate from Fig. 1 that in low temperature and high densities, long range perturbation term has the most significant effect in comparison with other parts. On the other hand, hard sphere part can be assumed as the most noticeable part in high temperature ranges. Moreover, comparison of DY and exp-6 potentials effects, on pressure of this mixture has been studied to express benefits of using exp-6 potential for higher temperatures and densities. Additionally, as it is obvious in high temperature and density difference between effects of two potentials are considerable for this equimolar mixture. This discriminating property makes exp-6 potential preferable.

Furthermore, this approach has been used to evaluate EOS of

## 6. Applications

One of the topics which can count on a great deal of interest from both theoretical and experimental physics is research in fluid mixture properties. These interests, not only comprise in the wide abundance of mixtures in our everyday life and in our universe but also the surprising new phenomena which were detected in the laboratories responsible for this increased attention. Mixtures, in general, have a much richer phase diagram than their pure constituents and various effects can be observed only in multi-component systems.

These kinds of studies have allowed a more complete modeling of mixture and consequently a better prediction and a more accurate calculation of thermodynamic quantities of mixture, such as activity coefficient, partial molar volume, phase behavior, local composition in general and have promoted a deeper understanding of the microscopic structure of mixtures.

Furthermore, for astronomical applications it is known that most of giant gas planets are like Jupiter is consisting primarily of hydrogen and helium. Modeling the interior of such planets requires an accurate equation of state for hydrogen-helium mixtures at high pressure and temperature conditions similar to those in planetary interiors [29]. Thus, the characterization of such system by statistical perturbation calculations will help us to answer questions concerning the inner structure of planets, their origin and evolution [29, 30].

In addition, in perturbation consideration of plasma via chemical picture, perturbation corrections will be included by means of additional free energy correction terms. Therefore, in considering transition behavior of molecular fluid to fully ionized plasma these terms are suitable in studying the neutral interaction parts. Consequently this will help us in studying inertial confinement fusion [31] and considering plasma as a fluid mixture in tokomak [32].

## References

- 1.
Wigner E. P. On the Quantum Correction for Thermodynamic Equilibrium. Physical Review 40, 749 (1932). - 2.
Kirkwood J. G. Quantum Statistics of Almost Classical Assemblies. Physical Review 45, 116 (1934). - 3.
DeWitt H. E. Analytic Properties of the Quantum Corrections to the Second Virial Coefficient. Journal of Mathematical Physics 3, 1003 (1962). - 4.
Hill R. N. Quantum Corrections to the Second Virial Coefficient at High Temperatures. Journal of Mathematical Physics 9, 1534 (1968). - 5.
Jancovici B. Quantum-Mechanical Equation of State of a Hard-Sphere Gas at High Temperature. Physical Review 178, 295 (1969). - 6.
Pisani C. and McKellar B. H. J, Semiclassical propagators and Wigner-Kirkwood expansions for hard-core potentials. Physical Review A 44, 1061 (1991). - 7.
Mason E. A Siregar J. and Huang Y. Simplified calculation of quantum corrections to the virial coefficients of hard convex bodies. Molecular Physics 73, 1171 (1991). - 8.
Mansoori G. A., Carnahan N. F., Starling K. E., and T. W. Leland. Equilibrium Thermodynamic Properties of the Mixture of Hard Spheres. Journal of Chemical Physics 54, 1523 (1971). - 9.
Tang Y. and Lu B. C.Y. Improved expressions for the radial distribution function of hard spheres. Journal of Chemical Physics 103, 7463 (1995). - 10.
Barker J. A. and Henderson D., Perturbation Theory and Equation of State for Fluids: The Square‐Well Potential. Journal of Chemical Physics 47, 2856 (1967). - 11.
Largo L. and Solana J. R., Equation of state for fluid mixtures of hard spheres and linear homo-nuclear fused hard spheres Physical Review E 58, 2251 (1998). - 12.
Boublik T. Hard‐Sphere Equation of State. Journal of Chemical Physics 54, 471 (1970). - 13.
Barrio C. and Solana J. Consistency conditions and equation of state for additive hard-sphere fluid mixtures. Journal of Chemical Physics 113, 10180 (2000). - 14.
Leonard P. J. Henderson D. and Barker J., Molecular Physic 21, 107 (1971). - 15.
Yuste S. B. and Santos A., Radial distribution function for hard spheres. Physical Review A 43, 5418 (1991). - 16.
Yuste S. B. Lopez de Haro M. and Santos A., Structure of hard-sphere meta-stable fluids. Physical Review E 53, 4820 (1996). - 17.
Tang Y., Jianzhong W. Journal of Chemical Physics 119, 7388 (2003). - 18.
Garcia A. and Gonzalez D. J. Physical Chemistry Liquid 18, 91 (1988). - 19.
Motevalli S. M., Pahlavani and M. R. Azimi M. Theoretical Investigations of Properties of Hydrogen and Helium Mixture Based on Perturbation Theory. , International Journal of Modern Physic B 26, 1250103 (2012). - 20.
Ree F. H. Mol. Phys. 96, 87 (1983). - 21.
Paricaud P. A general perturbation approach for equation of state development. Journal of Chemical Physics. 124, 154505 (2006). - 22.
Kuijper A. D. et al., Fluid-Fluid Phase Separation in a Repulsive α-exp-6 Mixture . Europhys. Lett. 13, 679 (1990). - 23.
Ali I. et al., Thermodynamic properties of He-H2 fluid mixtures over a wide range of temperatures and pressures. Physical Review E 69, 056104 (2004). - 24.
Ree F. H. Simple mixing rule for mixtures with exp‐6 interactions. Journal of Chemical Physics 78, 409 (1983). - 25.
Kresse G. and Hafner J. Ab initio molecular dynamics for liquid metals. Physical Review B 47, 558 (1993). - 26.
Lorenzen W. Halts B. and Redmer R. Metallization in hydrogen-helium mixtures Physical Review B 84, 235109 (2011). - 27.
Ross M. Ree F. H. and D. A. Young, The equation of state of molecular hydrogen at very high density. Journal of Chemical Physics 79, 1487 (1983). - 28.
Chen Q. F. and Cai L. C. Equation of State of Helium-Hydrogen and Helium-Deuterium Fluid Mixture at High Pressures and Tempratures. International of Journal of Thermodynamics 2, 27 (2006). - 29.
Guillot T., D. J. Stevenson, W. B. Hubbard, and D. Saumon, in Jupiter, edited by Bagenal F., Chapter three. University of Arizona Press, Tucson; 2003. p35–57. - 30.
Saumon D. and Guillot T. Shock Compression of Deuterium and the Interiors of Jupiter and Saturn. The Astrophysical Journal 609. 1170 (2004). - 31.
Collins G. W., Da Silva L. B., Celliers P., Gold D. M., Foord M. E., Wallace R. J., Ng A., Weber S. V., Budil K. S., Cauble R. Measurements of the Equation of State of Deuterium at the Fluid Insulator-Metal Transition. Science 281, 1178 (1998). - 32.
Hakel P. and Kilcrease D. P. A New Chemical-Picture-Based Model for Plasma Equation-of-State Calculations, 14th APS Topical Conference on Atomic Processes in Plasma, (2004).