B: Liquids; Chemical and Dynamical Processes in SolutionMarch 9, 2026

A New Set of Combining Rules for Mie (λ, 6) Potential
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Nguyen Van Phuoc
Nguyen Van Phuoc
Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 70000, Vietnam
More by Nguyen Van Phuoc
Thanh Doanh Le
Thanh Doanh Le
Faculty of Electrical Engineering, Electric Power University, Ministry of Industry and Trade, Hanoi 10000, Vietnam
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Van Hoa Nguyen*
Van Hoa Nguyen
Institute of Fundamental and Applied Sciences, Duy Tan University, Tran Nhat Duat Street, Ho Chi Minh City 70000, Vietnam
Faculty of Environmental and Natural Sciences, Duy Tan University, 03 Quang Trung Street, Da Nang 50000, Vietnam
*Email: [email protected]
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Suresh Alapati
Suresh Alapati
Department of Mechatronics Engineering, Kyungsung University, 309, Suyeong-ro (Daeyeon-dong), Nam-gu, Busan 48434, Korea
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Stéphanie Delage Santacreu
Stéphanie Delage Santacreu
Laboratoire de Mathématiques et de leurs Applications de Pau, UMR5142, CNRS, Université de Pau et des Pays de l’Adour, Pau 64000, France
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Guillaume Galliéro
Guillaume Galliéro
Laboratoire des Fluides Complexes et leurs Réservoirs, Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, Pau 64000, France
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https://orcid.org/0000-0001-6393-8387
Hai Hoang*
Hai Hoang
Institute of Fundamental and Applied Sciences, Duy Tan University, Tran Nhat Duat Street, Ho Chi Minh City 70000, Vietnam
Faculty of Environmental and Natural Sciences, Duy Tan University, 03 Quang Trung Street, Da Nang 50000, Vietnam
*Email: [email protected]
More by Hai Hoang
https://orcid.org/0000-0003-2992-0866

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The Journal of Physical Chemistry B

Cite this: J. Phys. Chem. B 2026, 130, 11, 3142–3155

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https://doi.org/10.1021/acs.jpcb.5c07029

Published March 9, 2026

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Abstract

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Force fields based on the Mie (λ, 6) potential, combined with theoretical methods and molecular simulations, offer a promising framework for predicting the thermophysical properties of fluids. Despite this potential, the availability of reliable combining rules for unlike interaction parameters in mixtures remains limited, thereby constraining the broader application of Mie (λ, 6)-based force fields. In this study, a new set of combining rules for the Mie (λ, 6) potential is proposed, derived by using a distortion model for the repulsive interaction and a geometric mean approximation for the attractive interaction, combined with first-order mathematical approximations. The capability of the new combining rules was first evaluated for noble gas pairs modeled with the Lennard–Jones potential, a specific case of Mie (λ, 6) potential with λ = 12, for which experimentally derived data on unlike interaction parameters are available. The results showed noticeably better agreement with experimentally derived values than those obtained using the two commonly used combining rules. Further assessment was carried out through the evaluation of Henry’s law constants, phase diagrams, and excess molar volumes, which are highly sensitive to cross-interactions, for various binary mixtures modeled using the Mie chain coarse-grained force field, obtained from NVT-GEMC, NPT-GEMC, and NPT-MC simulations, respectively. For mixtures with similar Mie (λ, 6) potential parameters for the components, all of the combining rules, including the new ones, yielded comparable predictions. In contrast, for asymmetric systems with significant force field parameter disparities, the new rules yielded substantially improved accuracy relative to experimental data for all considered thermodynamic properties, whereas the commonly used combining rules exhibited poor performance with markedly larger deviations. These findings highlight the improved robustness and broader applicability of the proposed combining rules for extending Mie (λ, 6)-based force fields to complex fluid mixtures.

This publication is licensed for personal use by The American Chemical Society.

Subjects

1. Introduction

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Accurate determination of the thermophysical properties of fluids is essential across various scientific and industrial fields, particularly in environmental science, chemical, and petroleum engineering. (1−6) These properties, especially in mixtures, are crucial for designing and optimizing efficient processes as well as understanding the behavior of complex fluid systems. Hence, predicting the thermophysical properties has been the subject of extensive research, with a substantial number of theoretical and computational approaches proposed in the literature. (7−10)

Among these, molecular modeling based on force fields using the Mie (λ, 6) potential, combined with both theoretical methods and simulations, has emerged as a promising approach. (11−16) In this approach, the Mie (λ, 6) potential U(r) is used to describe the nonbonded interactions between particles separated by a distance r, and it is defined as

U (r) = (\frac{λ}{λ - 6}) {(\frac{λ}{6})}^{6 / (λ - 6)} ε [{(\frac{σ}{r})}^{λ} - {(\frac{σ}{r})}^{6}]

(1)

where λ, σ, and ε are interaction parameters representing the repulsive exponent, collision diameter, and potential well depth, respectively.

Most existing studies have primarily focused on developing methods to determine the Mie (λ, 6) potential parameters λ, ε, and σ for pure substances. (11−14,17) However, the determination of interaction parameters for unlike molecular pairs in mixtures remains very limited. (13,14,18,19) Typically, the expressions used to determine these parameters from those of like-molecular interactions are termed combining rules. Among these, the arithmetic mean rule is widely used to determine the unlike interaction repulsive exponent, (13,14) which can be derived by applying the geometric mean rule to the repulsive part of the Mie (λ, 6) potential. (18) Alternatively, Lafitte et al. proposed a different expression based on applying the geometric mean rule to the van der Waals attractive constant. (19) Regarding the unlike interaction collision diameter, the arithmetic mean rule is commonly employed, (12−14,19) likely as a direct adaptation of the Lorentz rule. (20) In the case of the unlike interaction potential well depth, the geometric mean rule is often used, (13,14) probably derived from the Berthelot rule. (21) Furthermore, Lafitte et al. also applied the geometric mean rule to the van der Waals attractive energy to derive a combining rule for the unlike interaction potential well depth. (19)

These combining rules, when applied in conjunction with theoretical approaches and simulations, have been shown to yield reasonably accurate thermophysical properties of binary mixtures, provided that the Mie (λ, 6) potential parameters of the components are not too dissimilar. (13,14,22) However, their accuracy deteriorates significantly for mixtures with large disparities in component parameters. (19,23) Moreover, when applied to the Lennard–Jones (LJ) potential, i.e., a specific case of the Mie (λ, 6) potential with λ = 12, these combining rules have shown limited accuracy. (24−26)

It should be emphasized that the aforementioned combining rules are based on empirical assumptions that may not always be appropriate. For instance, the use of the geometric mean combining rule for unlike repulsive interactions to determine relevant parameters, e.g., the repulsive exponent by the arithmetic mean rule, (18) is less accurate than approaches based on the atomic distortion model. (27) Additionally, the assumption of hard-sphere interactions to deduce the arithmetic mean rule for the collision diameter may oversimplify the features of the Mie (λ, 6) potential. (8)

To address these limitations, more rigorous approaches have been proposed. (8) Waldman and Hagler graphically analyzed noble gas data to propose a set of combining rules for the collision diameter and potential well depth. (28) Interestingly, these combining rules were shown to significantly improve the description of unlike interactions not only for noble gas systems but also for biomolecular systems and have subsequently been extensively employed in biomolecular simulations. (29) Kong et al. (30,31) also derived combining rules for the LJ, Morse, and Exp-6 potentials by employing the atomic distortion model for the repulsive interactions combined with the geometric mean rule for the attractive interactions, referred to as the Kong approach. This approach has been shown to yield satisfactory results for these potentials. The LJ and Exp-6 unlike interaction parameters of binary noble gas mixtures obtained from theKong approach are in excellent agreement with values derived from experimental mutual diffusion coefficients for gaseous binary mixtures at low pressures. (30,31) Moreover, its application to the Morse potential provides good predictions for the second virial coefficient of binary noble gas mixtures. (30)

In addition, the Kong approach was further validated for the LJ potential by predicting both bulk and phase-equilibrium properties of binary noble gas mixtures in good agreement with experimental data. (24,25) More recently, Hoang et al. (26) demonstrated that Kong’s combining rules for the LJ potential can accurately predict noble gas solubility in hydrocarbon liquids. Nevertheless, to the best of our knowledge, no set of combining rules has been developed for the Mie (λ, 6) potential based on this approach. This is probably due to the mathematical form of the Mie (λ, 6) potential, which is rather special, containing the (σ/r)^λ term such that analytical derivations similar to those used for the LJ, Morse, and Exp-6 potentials (30,31) cannot be directly applied.

Thus, the present work aims to develop a new set of combining rules for the Mie (λ, 6) potential to address the limitations of existing methods by employing the same theoretical principles as the Kong approach. (30) For this purpose, first-order mathematical approximations were employed in the derivation. In addition, to verify its capability, the proposed set of combining rules has been applied to a wide range of mixtures, from simple monatomic to complex polyatomic systems, rather than being restricted to monatomic mixtures as in previous studies. (24,30,31)

The article is organized as follows. Section 2 presents the combining rules developed for the Mie (λ, 6) potential. The molecular simulation methodology used to evaluate the applicability of these rules is described in Section 3. The results obtained from applying the combining rules are then presented and discussed in Section 4. Finally, the main findings of this study are summarized in Section 5, which forms the conclusion.

2. Combining Rules for Mie Potentials

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To derive a new set of combining rules for the Mie (λ, 6) potential, the Mie (λ, 6) potential U(r) in eq 1 is first decomposed into repulsive and attractive contributions, U^rep and U^att, respectively, as

U (r) = U^{r e p} (r) - U^{a t t} (r)

(2)

where

U^{r e p} (r) = (\frac{λ}{λ - 6}) {(\frac{λ}{6})}^{6 / (λ - 6)} ε {(\frac{σ}{r})}^{λ} = A {(\frac{σ}{r})}^{λ}

(3)

U^{a t t} (r) = (\frac{λ}{λ - 6}) {(\frac{λ}{6})}^{6 / (λ - 6)} ε {(\frac{σ}{r})}^{6} = A {(\frac{σ}{r})}^{6}

(4)

where the coefficient A is defined as

A = (\frac{λ}{λ - 6}) {(\frac{λ}{6})}^{6 / (λ - 6)} ε

(5)

To determine the repulsive contribution of the unlike interaction potential between two different species, the distortion model is employed. (27) In this model, the repulsive energy is determined by an arithmetic mean rule as

U_{12}^{r e p} (r) = \frac{1}{2} [U_{11}^{r e p} (2 r_{1}) + U_{22}^{r e p} (2 r_{2})]

(6)

where r₁ and r₂ are distortion distances, and r = r₁ + r₂ where the subscripts 1 and 2 indicate species indices, which satisfy the condition of equal and opposite restoring forces on two particles at the distortion plane as (27)

{(\frac{d U_{11}^{r e p}}{dr})}_{r = 2 r_{1}} = {(\frac{d U_{22}^{r e p}}{dr})}_{r = 2 r_{2}}

(7)

For the attractive contribution, a simple geometric mean combining rule is used, based on the long-range nature of dispersion interactions, where distortion effects are negligible (30)

U_{12}^{a t t} (r) = {[U_{11}^{a t t} (r) \cdot U_{22}^{a t t} (r)]}^{1 / 2}

(8)

By substituting eq 3 into eqs 6 and 7, and eq 4 into eq 8, and using first-order mathematical approximations, the new set of combining rules can be derived as follows

λ_{12} = \frac{A_{11} λ_{11} + A_{22} λ_{22} D_{2}}{(A_{11} + A_{22}) (B_{1} + B_{2} D_{2})}

(9)

σ_{12} = {\frac{1}{{[A_{11} σ_{11}^{6} A_{22} σ_{22}^{6}]}^{1 / 2}} \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{2^{λ_{12} + 1}} [A_{11} (1 + λ_{12} B_{2} D_{1}) + A_{22} (1 + λ_{12} B_{2} D_{1} - λ_{22} D_{1})]}^{1 / (λ_{12} - 6)}

(10)

ε_{12} = \frac{{(A_{11} σ_{11}^{6} A_{22} σ_{22}^{6})}^{1 / 2}}{(\frac{λ_{12}}{λ_{12} - 6}) {(\frac{λ_{12}}{6})}^{6 / (λ_{12} - 6)} σ_{12}^{6}}

(11)

where

B_{1} = \frac{σ_{11}}{σ_{11} + σ_{22}}

(12)

B_{2} = \frac{σ_{22}}{σ_{11} + σ_{22}}

(13)

D_{1} = \frac{1 - \frac{A_{11} λ_{11} σ_{22}}{A_{22} λ_{22} σ_{11}}}{λ_{22} + 1}

(14)

D_{2} = \frac{λ_{11} + 1}{λ_{22} + 1}

(15)

Equations 9–11 form the new set of combining rules for the Mie (λ, 6) potential, in which the species indices 1 and 2 are assigned such that the condition [(A₁₁λ₁₁σ₂₂)/(A₂₂λ₂₂σ₁₁)] < 1 is satisfied. A detailed derivation of this new set of combining rules is provided in Appendix. It is worth noting that when two species are identical, λ₁₁ = λ₂₂, σ₁₁ = σ₂₂, and ε₁₁ = ε₂₂ implying A₁₁ = A₂₂, B₁ = B₂ = 1/2, D₁ = 0, and D₂ = 0, it is readily deduced from eqs 9, 10, and 11 that λ₁₂ = λ₁₁ = λ₂₂, σ₁₂ = σ₁₁ = σ₂₂, and ε₁₂ = ε₁₁ = ε₂₂, respectively. This confirms the consistency of the new combining rules, which ensures that for two identical species, the calculated unlike interaction parameters are exactly equal to those of the pure components.

To assess the effectiveness and robustness of the new combining rules for the Mie (λ, 6) potential, their predictive performance is compared with two widely used combining rules from the literature. The first is the so-called classical combining rules, (13,14) defined as

λ_{12} = \frac{λ_{11} + λ_{22}}{2}

(16)

σ_{12} = \frac{(σ_{11} + σ_{22})}{2}

(17)

ε_{12} = \sqrt{ε_{11} ε_{22}}

(18)

In this model, the combining rule for the repulsive exponent, eq 16, can result from applying the geometric mean combining rule for the repulsive contribution U^rep(r). The combining rules for the collision diameter and potential well depth, eqs 17 and 18, are adopted from the Lorentz–Berthelot combining rules, which arise from considerations of the collision of hard spheres and the London theory of dispersion, respectively.

The second set of combining rules tested in this work was proposed by Lafitte et al. and is given by (19)

(λ_{12} - 3) = \sqrt{(λ_{11} - 3) (λ_{22} - 3)}

(19)

σ_{12} = \frac{(σ_{11} + σ_{22})}{2}

(20)

ε_{12} = \frac{\sqrt{σ_{11}^{3} σ_{22}^{3}}}{σ_{12}^{3}} \sqrt{ε_{11} ε_{22}}

(21)

In this model, the combining rules for the repulsive exponent and the potential well depth, eqs 19 and 21, rely on the geometric mean rules of the van der Waals attractive constant and energy, respectively. The Lorentz combining rule is used for the collision diameter, eq 20.

3. Molecular Simulations

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To evaluate the capability of the combining rules, they were applied to calculate Henry’s law constant of gases dissolved in pure liquids, i.e., the solubility of gases in liquids at infinite dilution, phase equilibria, and excess molar volume of binary mixtures. These thermodynamic properties are particularly suitable for such an assessment, as they are highly sensitive to the strength of unlike interactions between molecules of unlike components. (7−9,32) All gas and liquid molecules were modeled as homonuclear chains composed of spherical particles, where nonbonded interactions between particles are described by the Mie (λ, 6) potential. This model is commonly referred to as the Mie chain coarse-grained (MCCG) model. (11,14) Parameters of the MCCG model, including the number of segments N_C, the repulsive exponent λ, the collision diameter σ, and the potential well depth ε, for pure substances were adopted from the literature. (13,14) These parameters have been shown to accurately reproduce the phase equilibria of the corresponding pure substances. (13,14)

To determine Henry’s law constant of gases in pure solvents, Gibbs Ensemble Monte Carlo simulations in NVT ensemble (NVT-GEMC) (33) were carried out in combination with the following expression (34)

H_{Gas} = \lim_{x_{Gas} \to 0} [\frac{f_{Gas}^{liq}}{x_{Gas}}] = ρ_{Solvent}^{*} k_{B} T \exp (\frac{μ_{Gas}^{ex}}{k_{B} T})

(22)

where f_Gas^liq is the gas fugacity, x_Gas is the gas mole fraction, ρ_Solvent^* is the number density of the liquid solvent, k_B is the Boltzmann constant, T is the temperature, and μ_Gas^ex is the excess chemical potential of the gas. The phase equilibria of binary mixtures were calculated from the Gibbs ensemble Monte Carlo (GEMC) simulations in NPT ensemble (NPT-GEMC). (33) In both NVT-GEMC and NPT-GEMC simulations, five types of Monte Carlo (MC) moves were implemented, including translation, rotation, regrowth, volume change, and molecular transfer. (35) To enhance sampling efficiency for the regrowth and transfer MC moves, configurational-bias Monte Carlo (CBMC) techniques were employed. (35)

The excess molar volumes of binary mixtures were computed from the Monte Carlo simulations in the NPT ensemble (NPT-MC) by using the following definition (7−9)

V_{mix}^{ex} = (\frac{x_{1} M_{1} + x_{2} M_{2}}{ρ_{mix}}) - (\frac{x_{1} M_{1}}{ρ_{1}} + \frac{x_{2} M_{2}}{ρ_{2}})

(23)

where V_mix^ex and ρ_mix are the excess molar volumes and the mass density of the binary mixture, respectively, x_i, M_i, and ρ_i correspond to the molar fraction, the molar mass, and the mass density of component i, respectively. In the NPT-MC simulations, four MC move types including translation, rotation, regrowth, and volume change, were employed, with the CBMC techniques used to enhance sampling efficiency for the regrowth MC move. (35)

All MC simulations were performed using in-house codes. (14,22,23) The NVT-GEMC and NPT-GEMC simulations were conducted in three successive stages. First, the systems were equilibrated using 3 × 10⁶ MC moves, excluding the molecular transfer MC move. Then, all five types of moves were included in a subsequent equilibrium run of 30 × 10⁶ MC moves to achieve liquid–vapor phase equilibrium. Finally, a production run of 100 × 10⁶ MC moves was carried out to compute the thermodynamic properties. The NPT-MC simulations consisted of two stages. An equilibration run of 20 × 10⁶ MC moves was first performed, followed by a production run of 50 × 10⁶ MC moves.

The NVT-GEMC and NPT-GEMC simulation systems were composed of 500 to 1200 molecules, whereas the NPT-MC simulation systems consisted of 400 to 1000 molecules. The interaction potentials were truncated at a cutoff radius of 4σ, and long-range corrections were applied. The excess chemical potential used to calculate Henry’s law constant eq 22 was computed using the Widom insertion method, (36) with 200 × 10⁶ gas molecule insertions during the production stage.

To improve statistical reliability, the values reported in this work represent an average over six independent simulations, with error bars representing the standard deviation. (37) The simulations showed that the error bars are generally smaller than the symbol size. Thus, for the sake of visual clarity, error bars are not shown in the figures.

4. Results and Discussion

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4.1. Validation of the New Set of Combining Rules

This new set of combining rules for the Mie (λ, 6) potential has been developed, based on the same theoretical principles employed by Kong for the LJ potential. (30) The key difference lies in the derivation approach: while Kong’s rules were derived analytically without any approximations, the new combining rules rely on first-order mathematical approximations so as to be applicable not only to the LJ potential but also to the Mie (λ, 6) potential. Therefore, to assess the accuracy of this approximation, the new set of combining rules was applied to the LJ potential, and the results were compared with those obtained using Kong’s method.

For this purpose, the new rules were employed to calculate the unlike interaction potentials for noble gas pairs modeled by the LJ potential, as originally studied by Kong. (30) Following his approach, the LJ parameters for the pure noble gases were taken from the works of Kong and Hogervorst, (30,38) as explicitly presented in Table 1.

Table 1. Interaction Parameters of the Lennard–Jones Potential for Noble Gases (30,38)

noble gas	ε/k_B [K]	σ [Å]
He	24.80	2.366
Ne	43.00	2.730
Ar	135.00	3.360
Kr	193.00	3.570
Xe	256.00	3.920

It is readily observed that when the new combining rules are applied to the LJ potential pair, i.e., λ₁₁ = λ₂₂ = 12 implying D₂ = 1, the calculated repulsive exponent for the unlike interaction, thanks to eq 9, satisfies λ₁₂ = λ₁₁ = λ₂₂ = 12. For the unlike collision diameter and potential well depth, σ₁₂ and ε₁₂, respectively, the results obtained using the new combining rules are reported in Table 2.

Table 2. Comparison of Unlike Interaction Parameters of Lennard–Jones Potential for Noble Gas Pairs Obtained from Three Combining Rules and Deduced from Experiments (38)

pair	classical combining rules	Lafitte et al. combining rules	Kong combining rules	new combining rules	data deduced from experiments
ε/k_B [K]
He–Ar	57.86	55.27	41.50	42.32	40.00 ± 3.00
He–Xe	79.68	72.49	40.94	39.88	49.00 ± 5.00
Ne–Ar	76.19	74.97	67.12	69.17	64.50 ± 4.00
Ne–Kr	91.10	88.68	73.61	75.57	71.50 ± 3.50
Ne–Xe	104.92	99.92	73.38	74.49	73.00 ± 4.00
Ar–Kr	161.42	161.19	159.60	160.79	148.00 ± 7.00
Ar–Xe	185.90	184.26	175.16	177.94	178.00 ± 6.00
σ [Å]
He–Ar	2.863	2.863	2.980	2.970	2.980 ± 0.020
He–Xe	3.143	3.143	3.403	3.418	3.360 ± 0.030
Ne–Ar	3.045	3.045	3.093	3.078	3.090 ± 0.030
Ne–Kr	3.150	3.150	3.235	3.221	3.220 ± 0.030
Ne–Xe	3.325	3.325	3.472	3.464	3.460 ± 0.030
Ar–Kr	3.465	3.465	3.470	3.466	3.510 ± 0.030
Ar–Xe	3.640	3.640	3.665	3.656	3.650 ± 0.030

The results indicate that, for these unlike pairs, the new combining rules predict the potential well depth ε₁₂ with an average absolute deviation (AAD) of 2.02% compared to Kong’s method and a maximum absolute deviation (MAD) of 3.05%, the latter observed for the Ne–Ar mixture. The AAD and MAD are defined as

AAD = \frac{1}{N} \sum_{i = 1}^{N} {AD}_{i}

(24)

MAD = \max_{i} ({AD}_{i})

(25)

where N is the number of data points, and AD_i is the absolute deviation defined as

{AD}_{i} = \frac{| X_{i}^{New} - X_{i}^{Kong} |}{X_{i}^{Kong}} \times 100 %

(26)

where X_i^New and X_i^Kong are values computed from the new and Kong combining rules, respectively.

For the unlike interaction collision diameter σ₁₂, the AAD is 0.33% and the MAD is 0.48%, also corresponding to the Ne–Ar mixture. The smaller deviations observed for σ₁₂ compared to those of ε₁₂ can be attributed to the smaller relative differences in collision diameters among species than in their potential well depths, as shown in Table 1.

Overall, these reasonably small deviations confirm the consistency and reliability of the first-order mathematical approximations used in deriving the proposed set of combining rules.

4.2. Tests of the New Set of Combining Rules

4.2.1. Lennard–Jones Parameters of Unlike Interactions

For the noble gas pairs modeled by the LJ potential, as considered in the previous section, (30) values of the unlike LJ parameters σ₁₂ and ε₁₂ were deduced from the experimental mutual diffusion coefficients of the corresponding binary gas mixtures at low pressures, (38) as presented in Table 2. Accordingly, the applicability of three sets of combining rules tested in this work, the classical one, eqs 16–18, the Lafitte et al. one, eqs 19–21, and the new one, eqs 9–11, was assessed for this specific case of the Mie (λ, 6) potential with λ = 12. The predicted values of σ₁₂ and ε₁₂ obtained from these three sets of combining rules are summarized in Table 2.

Similar to the results reported by Kong, (30) the new set of combining rules provides reasonably accurate predictions for these unlike noble gas pairs. More precisely, this approach yields predictions for the unlike potential well depth with an AAD of 6.86% and a MAD of 18.61% and for the unlike interaction collision diameter, an AAD of 0.57% and a MAD of 1.73%, when compared to experimentally derived unlike interaction parameters. (38) The largest deviation is observed for the He–Xe pair, which represents the most asymmetric pair considered in this study. This suggests that the new approach may produce larger deviations for pairs with significant differences in the LJ parameters of the pure species, a common limitation of combining rules based on effective force fields such as the LJ one.

In contrast, both the classical and Lafitte et al. combining rules result in significantly larger deviations from the experimentally derived data. (38) For unlike potential well depth ε₁₂, the classical approach yields an AAD of 30.00% and a MAD of 62.61%, while the Lafitte et al. approach gives an AAD of 25.10% and a MAD of 47.94%. For the unlike collision diameter σ₁₂, both approaches produce an identical AAD of 2.78% and a MAD of 6.46%.

These results demonstrate the effectiveness and robustness of the new set of combining rules over the classical and Lafitte et al. ones when applied to LJ potentials, a specific case of the Mie (λ, 6) potential with λ = 12.

4.2.2. Thermodynamic Properties of Binary Mixtures

In this section, the performance of the three tested combining rules is evaluated by applying them to the prediction of Henry’s law constants of gases in pure liquids, phase equilibria, and excess molar volume of binary mixtures, with all components modeled using the MCCG model. A wide range of binary mixtures was considered, spanning from simple monatomic mixtures to more complex molecular mixtures. More precisely, 12 systems were investigated: (1) argon/krypton, (2) krypton/xenon, (3) methane/xenon, (4) neon/n-heptane, (5) argon/n-heptane, (6) krypton/n-heptane, (7) methane/n-heptane, (8) methane/n-decane, (9) methane/toluene, (10) carbon dioxide/n-heptane, (11) carbon dioxide/n-decane, and (12) carbon dioxide/toluene. The parameters of the MCCG model for these species were adopted from the literature, as presented in Table 3, which have been shown to yield accurate phase-equilibrium properties for the corresponding pure substances. (13,14)

Table 3. Parameters of the Force Field using the Mie (λ, 6) Potential for Species Considered in this Work (13,14)

species	N_c	λ	ε/k_B [K]	σ [Å]
neon	1	12.510	34.912	2.813
argon	1	13.926	125.571	3.407
krypton	1	13.428	171.149	3.634
xenon	1	14.223	244.131	3.962
methane	1	14.000	161.000	3.740
carbon dioxide	2	16.933	211.539	2.861
n-heptane	3	14.034	294.293	4.049
n-decane	4	16.025	336.756	4.111
toluene	3	12.27	293.247	3.658

The Mie (λ, 6) potential parameters of the unlike interactions for these systems, calculated by using the three sets of combining rules, are presented in Table 4. The table clearly shows that the new set of combining rules yields results that differ significantly from those of the classical and Lafitte et al. combining rules in five systems: neon/n-heptane, argon/n-heptane, carbon dioxide/n-heptane, carbon dioxide/n-decane, and carbon dioxide/toluene. In contrast, for the remaining seven systems, the results obtained from all three combining rules are relatively similar. The significant differences observed for these five specific systems can probably be attributed to the large differences in the Mie (λ, 6) potential parameters between the two components of each system, as shown in Table 3.

Table 4. Comparison of Unlike Interaction Parameters of Mie (λ, 6) Potential Gas–Liquid Pairs Obtained from Three Combining Rules

pair	classical combining rules	Lafitte et al. combining rules	new combining rules
repulsive exponent
argon/krypton	13.68	13.67	13.66
krypton/xenon	13.83	13.82	13.83
methane/xenon	14.11	14.11	14.12
krypton/n-heptane	13.73	13.73	13.74
methane/n-heptane	14.02	14.02	14.02
neon/n-heptane	13.27	13.24	13.40
argon/n-heptane	13.98	13.98	13.99
carbon dioxide/n-heptane	15.48	15.40	15.11
methane/n-decane	15.01	14.97	15.01
carbon dioxide/n-decane	16.48	16.47	16.38
methane/toluene	13.13	13.10	13.07
carbon dioxide/toluene	14.60	14.36	13.95
ε/k_B [K]
argon/krypton	146.60	146.37	145.38
krypton/xenon	204.41	203.84	203.32
methane/xenon	198.25	198.01	198.23
krypton/n-heptane	224.43	223.45	222.15
methane/n-heptane	217.67	217.16	216.69
neon/n-heptane	101.36	96.47	76.15
argon/n-heptane	192.24	190.10	183.43
carbon dioxide/n-heptane	249.51	238.53	207.18
methane/n-decane	232.85	232.07	235.69
carbon dioxide/n-decane	266.9	254.14	223.62
methane/toluene	217.29	217.25	217.24
carbon dioxide/toluene	249.07	243.50	217.07
σ [Å]
argon/krypton	3.521	3.521	3.523
krypton/xenon	3.798	3.798	3.799
methane/xenon	3.851	3.851	3.850
krypton/n-heptane	3.841	3.841	3.844
methane/n-heptane	3.895	3.895	3.894
neon/n-heptane	3.431	3.431	3.549
argon/n-heptane	3.728	3.728	3.744
carbon dioxide/n-heptane	3.455	3.455	3.502
methane/n-decane	3.925	3.925	3.917
carbon dioxide/n-decane	3.486	3.486	3.530
methane–toluene	3.699	3.699	3.699
carbon dioxide–toluene	3.260	3.260	3.300

4.2.2.1. Henry’s Law Constant

First, the results are examined for Henry’s law constants of gases in pure liquids. Figure 1 shows the variation of Henry’s law constant with temperature obtained from the NVT-GEMC simulations using the three sets of combining rules for the seven gas–liquid systems in which all three combining rules yield similar Mie (λ, 6) potential parameters for the unlike interactions: argon/krypton, krypton/xenon, methane/xenon, krypton/n-heptane, methane/n-heptane, methane/n-decane, and methane/toluene. As expected, the simulation results of the three combining rules are nearly identical across the entire temperature range.

Figure 1. Comparison of Henry’s law constants obtained from the NVT-GEMC simulations using different combining rules and experiment for seven binary mixtures for which the Mie (λ, 6) potential parameters of the components are relatively similar. (a) Argon/Krypton mixture. (b) Krypton/Xenon mixture. (c) Methane/Xenon mixture. (d) Krypton/n-Heptane mixture. (e) Methane/n-Heptane mixture. (f) Methane/n-Decane mixture. (g) Methane–Toluene mixture. Solid circles (red) correspond to the experimental data. Open squares (green) correspond to the simulation data by using the classical combining rules. Open deltas (blue) correspond to the simulation data using Lafitte et al. combining rules. Open right triangles (red) correspond to the simulation data by using the new combining rules. Lines serve as a guide to the eye for the experimental data.

Moreover, comparison with experimental Henry’s law constants, (39−47) converted from P-x-y measurements following IUPAC guidelines, (48) demonstrates that all three combining rules provide reasonably accurate predictions, with overall AADs of approximately 8.4%, 8.5%, and 9.5% for the classical, Lafitte et al., and new combining rules, respectively. The largest deviation is observed for krypton in xenon, (40) with an AAD of 16.5%. This result aligns with previous findings, (13,14) which reported that classical combining rules performed well for simple binary mixtures such as argon–krypton, krypton–xenon, and methane-xenon.

However, for the five more asymmetric systems: neon in n-heptane, argon in n-heptane, carbon dioxide in n-heptane, carbon dioxide in n-decane, and carbon dioxide in toluene, where the Mie (λ, 6) parameters of gas and liquid components differ substantially, Figure 2 illustrates that the new set of combining rules predicts Henry’s law constants that differ markedly from those of the classical and Lafitte et al. combining rules. This discrepancy arises from significant differences in the calculated unlike interaction parameters between the new rules and the other two.

Figure 2. Comparison of Henry’s law constants obtained from the NVT-GEMC simulations using different combining rules and experiment for five binary mixtures for which the Mie (λ, 6) potential parameters of the components are significantly different. (a) Neon/n-heptane mixture. (b) Argon/n-heptane mixture. (c) Carbon dioxide/n-heptane mixture. (d) Carbon dioxide/n-decane mixture. (e) Carbon dioxide/toluene mixture. The legend is identical to that in Figure 1.

When compared to experimental data, (49−54) the new combining rules yield reasonably accurate predictions, with an overall AAD of 12.7%. The largest deviation is observed for the carbon dioxide-heptane system, where the AAD reaches 22.0%. In contrast, the classical and Lafitte et al. combining rules perform considerably worse, systematically underestimating Henry’s law constants, with overall AADs of 58.4% and 50.6%, respectively. The poorest performance of the classical rules is found for carbon dioxide in n-decane, (52,53) where the AAD exceeds 70%, while for the Lafitte et al. rules the worst cases are neon in n-heptane (49) and carbon dioxide in n-decane, (52,53) both with the AAD exceeding 60%.

4.2.2.2. Phase Diagram of a Binary Mixture

Henry’s law constant of gases dissolved in pure liquids quantifies the solubility of gases in liquids at infinite-dilution limit. (32) Therefore, to further evaluate the capability of the three tested combining rules beyond this limit, they were employed to predict the pressure–composition phase diagrams of binary mixtures, which correspond to finite solubility conditions.

First, the three combining rules were examined for the binary mixtures whose components have relatively similar Mie (λ, 6) potential parameters, including argon/krypton, krypton/xenon, methane/xenon, methane/n-heptane, methane/n-decane mixture, and methane/toluene. As expected, the simulation results shown in Figure 3 indicate that all three combining rules yield nearly identical pressure–composition phase diagrams over the entire composition range for these binary mixtures, in good agreement with experimental data. (39−41,43−45)

Figure 3. Comparison of pressure–composition phase diagrams of binary mixtures obtained from the NPT-GEMC simulations using different combining rules and experiment for six binary mixtures for which the Mie (λ, 6) potential parameters of the components are relatively similar. (a) Argon/krypton mixture at T = 177.38 K. (b) Krypton/xenon mixture at T = 190.03 K. (c) Methane/xenon mixture at T = 189.78 K. (d) Methane/n-heptane mixture at T = 310.93 K. (e) Methane/n-decane mixture at T = 373.15 K. (f) Methane/toluene mixture at T = 422.45 K. The legend is identical to that in Figure 1.

However, for binary mixtures, carbon dioxide/n-heptane, carbon dioxide/n-decane, and carbon dioxide/toluene, which are composed of components with significantly different Mie (λ, 6) potential parameters, the results obtained from the new combining rules are significantly different from those of the classical and Lafitte et al. combining rules, as illustrated in Figure 4. For these asymmetric systems, the new combining rules yield good agreement with the experimental data, (51,52,54) whereas the classical and Lafitte et al. combining rules remarkably overestimate the gas solubility in the liquid phase. This behavior is fully consistent with the results obtained for Henry’s law constants, for which the classical and Lafitte et al. combining rules significantly underestimate Henry’s law constant for these mixtures.

Figure 4. Comparison of pressure–composition phase diagrams of binary mixtures obtained from the NPT-GEMC simulations using different combining rules and experiment for three binary mixtures for which the Mie (λ, 6) potential parameters of the species are significantly different. (a) Carbon dioxide/n-heptane mixture at T = 310.65 K. (b) Carbon dioxide/n-decane mixture at T = 377.59 K. (c) Carbon dioxide/toluene mixture at T = 308.16 K. The legend is identical to that in Figure 1.

4.2.2.3. Excess Molar Volume of a Binary Mixture

In the preceding sections, the performance of the three combining rules was assessed based on phase-equilibrium properties of binary mixtures. In this section, their capability is further evaluated using a bulk thermodynamic property, namely, the excess molar volume of the mixtures.

Figure 5 shows variation in the excess molar volume with the gas molar fraction calculated from the NPT-MC simulations using the three sets of combining rules for three binary mixtures: methane/n-heptane, methane/n-decane, and methane/toluene. The simulation results show that for these binary mixtures which are composed of components with similar Mie (λ, 6) potential parameters, all three combining rules provide nearly similar predictions for the excess molar volumes, in good agreement with the experimental data. (55−57)

Figure 5. Comparison of excess molar volumes of binary mixtures obtained from the NPT-MC simulations using different combining rules and experiment for three binary mixtures for which the Mie (λ, 6) potential parameters of the components are relatively similar. (a) Methane/n-heptane mixture at T = 303.15 K and P = 30 MPa. (b) Methane/n-decane mixture at T = 373.15 K and P = 40 MPa. (c) Methane/toluene mixture at T = 333.15 K and P = 45 MPa. The legend is identical to that in Figure 1.

Nevertheless, when applied to the following three binary mixtures: carbon dioxide/n-heptane, carbon dioxide/n-decane, and carbon dioxide/toluene mixture, where the Mie (λ, 6) potential parameters of components differ significantly, the results shown in Figure 6 indicate that only the new combining rules provide predictions in good agreement with the experimental data. (58−60) In contrast, classical and Lafitte et al. combining rules predict substantially more negative excess molar volumes. This trend is consistent with the phase-equilibrium results, where these two combining rules were shown to significantly overestimate the gas solubility in the liquid for these binary mixtures.

Figure 6. Comparison of excess molar volumes of binary mixtures obtained from the NPT-MC simulations using different combining rules and experiment for three binary mixtures for which the Mie (λ, 6) potential parameters of the components are significantly different. (a) Carbon dioxide/n-heptane mixture at T = 313.15 K and P = 10 MPa. (b) Carbon dioxide/n-decane mixture at T = 318.15 K and P = 15 MPa. (c) Carbon dioxide/toluene mixture at T = 313.15 K and P = 10 MPa. The legend is identical to that in Figure 1.

Given the intrinsic limitations of the MCCG force fields and the known deviations in pure fluid properties, the results obtained for Henry’s law constant, pressure–composition phase diagram and excess molar volume are particularly encouraging. (11−14) They suggest that the MCCG model can be effectively extended to mixtures without the need for additional parameter fitting for unlike interactions, thereby reducing both computational cost and complexity.

These findings further underscore the value of the proposed combining rules, demonstrating their superior accuracy and robustness relative to the two most commonly used alternatives, without any additional computational expense. In particular, the new rules exhibit strong performance in predicting phase-equilibrium and bulk thermodynamic properties for systems with large disparities in the molecular parameters. Notably, their predictive accuracy remains consistent for various thermodynamic properties across a diverse set of binary mixtures, from symmetric mixtures with comparable size and energy parameters to highly asymmetric mixtures involving significant differences in molecular characteristics.

5. Conclusions

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In this work, a new set of combining rules for the Mie (λ, 6) potential was derived to improve its capability to predict the thermophysical properties of mixtures. For this purpose, a distortion model for the repulsive interaction and a geometric mean approximation for the attractive interaction were employed in conjunction with first-order mathematical approximations.

First, to verify the reliability and consistency of the first-order mathematical approximations used in the derivation, the new combining rules were applied to the noble gas pairs modeled with the LJ potential, a specific case of Mie (λ, 6) potential with λ = 12, and compared to Kong’s combining rules. The results show that the LJ unlike interaction parameters obtained from the two approaches are in very good agreement with AADs of 2.02% and 0.33% for the potential well depth and the collision diameter, respectively, thereby validating the accuracy of the first-order approximations applied.

Then, the applicability of the proposed set of combining rules was tested on noble gas mixtures for which experimentally derived unlike interaction parameters are available. Compared with experimentally deduced data, the new rules yielded accurate predictions, with AADs of 6.86% for the potential well depth and 0.57% for the collision diameter. In contrast, the classical and Lafitte et al. combining rules exhibited much larger deviations, typically four to five times higher, highlighting the improved accuracy and robustness of the proposed method.

The predictive performance of the new combining rules was also evaluated by estimating phase-equilibrium and bulk thermodynamic properties including Henry’s law constants, pressure–composition phase diagram, and excess molar volumes for various binary mixtures, spanning from simple monatomic mixtures to more complex molecular mixtures using Monte Carlo simulations (NVT-GEMC, NPT-GEMC, and NPT-MC), where all components were represented as homonuclear chains with Mie (λ, 6) interaction sites (MCCG model). For binary mixtures composed of components with similar Mie (λ, 6) potential parameters, all three combining rules gave nearly identical results and agreed well with experimental data for both phase-equilibrium and bulk properties. In contrast, for mixtures involving components with significantly different Mie (λ, 6) potential parameters, the new combining rules still produced reasonably accurate predictions for all of the considered thermodynamic properties. The classical and Lafitte et al. rules, however, showed poor performance, remarkably overestimating the gas solubility in the liquid with AADs more than four times larger than those obtained with the new combining rules for Henry’s law constant, and predicting excessively more negative excess molar volumes.

Overall, the new set of combining rules derived in this work demonstrates significantly improved robustness and broader applicability compared with existing approaches. Its ability to accurately predict unlike interactions in both symmetric and highly asymmetric systems without requiring additional parameter adjustment makes it a promising tool for extending the predictive power of the Mie (λ, 6)-based force fields in modeling the thermophysical properties of fluid mixtures.

Author Information

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Corresponding Authors
- Van Hoa Nguyen - Institute of Fundamental and Applied Sciences, Duy Tan University, Tran Nhat Duat Street, Ho Chi Minh City 70000, Vietnam; Faculty of Environmental and Natural Sciences, Duy Tan University, 03 Quang Trung Street, Da Nang 50000, Vietnam; Email: [email protected]
- Hai Hoang - Institute of Fundamental and Applied Sciences, Duy Tan University, Tran Nhat Duat Street, Ho Chi Minh City 70000, Vietnam; Faculty of Environmental and Natural Sciences, Duy Tan University, 03 Quang Trung Street, Da Nang 50000, Vietnam; https://orcid.org/0000-0003-2992-0866; Email: [email protected]
Authors
- Nguyen Van Phuoc - Faculty of Applied Sciences, Ton Duc Thang University, Ho Chi Minh City 70000, Vietnam
- Thanh Doanh Le - Faculty of Electrical Engineering, Electric Power University, Ministry of Industry and Trade, Hanoi 10000, Vietnam
- Suresh Alapati - Department of Mechatronics Engineering, Kyungsung University, 309, Suyeong-ro (Daeyeon-dong), Nam-gu, Busan 48434, Korea
- Stéphanie Delage Santacreu - Laboratoire de Mathématiques et de leurs Applications de Pau, UMR5142, CNRS, Université de Pau et des Pays de l’Adour, Pau 64000, France
- Guillaume Galliéro - Laboratoire des Fluides Complexes et leurs Réservoirs, Université de Pau et des Pays de l’Adour, E2S UPPA, CNRS, Pau 64000, France; https://orcid.org/0000-0001-6393-8387
Author Contributions
N.V.P.: data curation; methodology; investigation; formal analysis; software; visualization; writing─original draft; and writing─review and editing. T.D.L.: investigation; formal analysis; software; visualization; and writing─review and editing. V.H.N.: data curation; methodology; investigation; formal analysis; software; visualization; and writing─review and editing. S.A.: formal analysis; validation; visualization; and writing─review and editing. S.D.S.: formal analysis; software; validation; and writing─review and editing. G.G.: conceptualization; supervision; writing─original draft; and writing─review and editing. H.H.: conceptualization; data curation; methodology; supervision; validation; writing─original draft; and writing─review and editing.
Notes
The authors declare no competing financial interest.

Acknowledgments

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The authors gratefully acknowledge the support of the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.01-2025.42. Computational resources were provided by the University of Pau and Pays de l’Adour (UPPA) and the MCIA (Mésocentre de Calcul Intensif Aquitain). Hai Hoang also acknowledges financial support from the Centre National de la Recherche Scientifique (CNRS), France. The authors thank Dr. Dung Chinh Nguyen for valuable discussions on the mathematical derivations.

Appendix: Derivation of the New Set of Combining Rules for the Mie (λ, 6) Potential

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This appendix provides the detailed derivation of the new combining rules for the Mie (λ, 6) potential.

First, applying eq 7 to the repulsive contribution of the Mie (λ, 6) potential leads to

\frac{A_{11} λ_{11}}{σ_{11}} {(\frac{σ_{11}}{2 r_{1}})}^{λ_{11} + 1} = \frac{A_{22} λ_{22}}{σ_{22}} {(\frac{σ_{22}}{{2 r}_{2}})}^{λ_{22} + 1}

(A1)

Since the distortion plane is located at distances r₁ and r₂ from the centers of particles 1 and 2, respectively, to minimize the total energy, it is reasonable to assume:

r_{1} = \frac{σ_{11}}{2} + δ_{11}

with

0 < δ_{11} ≪ \frac{σ_{11}}{2}

, and

r_{2} = \frac{σ_{22}}{2} + δ_{22}

with

0 < δ_{22} ≪ \frac{σ_{22}}{2}

. Substituting these expressions of r₁ and r₂ into eq A1 yields

\frac{A_{11} λ_{11}}{σ_{11}} {(\frac{1}{1 + \frac{2 δ_{11}}{σ_{11}}})}^{λ_{11} + 1} = \frac{A_{22} λ_{22}}{σ_{22}} {(\frac{1}{1 + \frac{2 δ_{22}}{σ_{22}}})}^{λ_{22} + 1}

(A2)

Now, consider the case where

(\frac{A_{11} λ_{11}}{σ_{11}}) < (\frac{A_{22} λ_{22}}{σ_{22}})

or equivalently,

D = \frac{A_{11} λ_{11} σ_{22}}{A_{22} λ_{22} σ_{11}} < 1

eq A2 can then be rearranged as

\frac{A_{11} λ_{11} σ_{22}}{A_{22} λ_{22} σ_{11}} = \frac{{(\frac{1}{1 + (2 δ_{22} / σ_{22})})}^{λ_{22} + 1}}{{(\frac{1}{1 + (2 δ_{11} / σ_{11})})}^{λ_{11} + 1}}

(A3)

In the opposite case where D > 1, eq A2 can be rearranged as

\frac{A_{22} λ_{22} σ_{11}}{A_{11} λ_{11} σ_{22}} = \frac{{(\frac{1}{1 + 2 δ_{11} / σ_{11}})}^{λ_{11} + 1}}{{(\frac{1}{1 + 2 δ_{22} / σ_{22}})}^{λ_{22} + 1}}

(A4)

The combining rules derived in this work were formulated for the former case D < 1, i.e., using eq A3. A similar derivation procedure can be applied to obtain the combining rules for the latter case D > 1. However, the combining rules for the case of D < 1 can be adapted for the case of D > 1, simply by permuting the labels of species 1 and 2, yielding the same results as those obtained from the combining rules for the case of D > 1. Therefore, the combining rules presented in this work are generally applicable to all unlike interaction pairs, provided that the species indices are assigned such that the condition D < 1 is satisfied.

Applying the first-order approximation: (1 + x)^α ≅ 1 + αx for x ≪ 1 to eq A3 yields

\frac{A_{11} λ_{11} σ_{22}}{A_{22} λ_{22} σ_{11}} = \frac{{(\frac{1}{1 + (2 δ_{22} / σ_{22})})}^{λ_{22} + 1}}{{(\frac{1}{1 + (2 δ_{11} / σ_{11})})}^{λ_{11} + 1}} ≅ (1 + (λ_{11} + 1) \frac{2 δ_{11}}{σ_{11}}) (1 - (λ_{22} + 1) \frac{2 δ_{22}}{σ_{22}})

(A5)

Further simplifying leads to

\frac{2 δ_{22}}{σ_{22}} = D_{1} + D_{2} \frac{2 δ_{11}}{σ_{11}}

(A6)

Applying eq 6 to the repulsive part of the Mie (λ, 6) potential leads to

A_{12} {(\frac{σ_{12}}{r})}^{λ_{12}} = \frac{1}{2} [A_{11} {(\frac{σ_{11}}{2 r_{1}})}^{λ_{11}} + A_{22} {(\frac{σ_{22}}{2 r_{2}})}^{λ_{22}}]

(A7)

This equation can be rewritten as

A_{12} {(σ_{12})}^{λ_{12}} = (\frac{A_{11} {(σ_{11})}^{λ_{11}}}{2^{λ_{11} + 1}}) \frac{r^{λ_{12}}}{r_{1}^{λ_{11}}} + (\frac{A_{22} {(σ_{22})}^{λ_{22}}}{2^{λ_{22} + 1}}) \frac{r^{λ_{12}}}{r_{2}^{λ_{22}}}

(A8)

Applying the first-order approximation to (r^λ₁₂)/(r₁^λ₁₁) and (r^λ₁₂/r₂^λ₂₂), the following expression is obtained:

\frac{r^{λ_{12}}}{r_{1}^{λ_{11}}} = \frac{{(r_{1} + r_{2})}^{λ_{12}}}{r_{1}^{λ_{11}}} = 2^{(λ_{11} - λ_{12})} \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{σ_{11}^{λ_{11}}} [1 + (λ_{12} B_{1} - λ_{11}) \frac{2 δ_{11}}{σ_{11}} + λ_{12} B_{2} \frac{2 δ_{22}}{σ_{22}}]

(A9)

\frac{r^{λ_{12}}}{r_{2}^{λ_{22}}} = \frac{{(r_{1} + r_{2})}^{λ_{12}}}{r_{2}^{λ_{22}}} = 2^{(λ_{22} - λ_{12})} \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{σ_{22}^{λ_{22}}} [1 + λ_{12} B_{1} \frac{2 δ_{11}}{σ_{11}} + (λ_{12} B_{2} - λ_{22}) \frac{2 δ_{22}}{σ_{22}}]

(A10)

Substituting the expression for (2δ₂₂/σ₂₂) from eq A6 into eqs A9 and A10 results in the following expressions

\frac{r^{λ_{12}}}{r_{1}^{λ_{11}}} = 2^{(λ_{11} - λ_{12})} \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{σ_{11}^{λ_{11}}} [(1 + λ_{12} B_{2} D_{1}) + (λ_{12} B_{1} - λ_{11} + λ_{12} B_{2} D_{2}) \frac{2 δ_{11}}{σ_{11}}]

(A11)

\frac{r^{λ_{12}}}{r_{2}^{λ_{22}}} = 2^{(λ_{22} - λ_{12})} \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{σ_{22}^{λ_{22}}} [(1 + λ_{12} B_{2} D_{1} - λ_{22} D_{1}) + (λ_{12} B_{1} + λ_{12} B_{2} D_{2} - λ_{22} D_{2}) \frac{2 δ_{11}}{σ_{11}}]

(A12)

Substituting eqs A11 and A12 into eq A8 yields

A_{12} {(σ_{12})}^{λ_{12}} = \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{2^{λ_{12} + 1}} [(A_{11} (1 + λ_{12} B_{2} D_{1}) + A_{22} (1 + λ_{12} B_{2} D_{1} - λ_{22} D_{1})) + (A_{11} (λ_{12} B_{1} - λ_{11} + λ_{12} B_{2} D_{2}) + A_{22} (λ_{12} B_{1} + λ_{12} B_{2} D_{2} - λ_{22} D_{2})) \frac{2 δ_{11}}{σ_{11}}]

(A13)

To eliminate the dependence on δ₁₁, the coefficient of (2δ₁₁/σ₁₁) must vanish, leading to

A_{11} (λ_{12} B_{1} - λ_{11} + λ_{12} B_{2} D_{2}) + A_{22} (λ_{12} B_{1} + λ_{12} B_{2} D_{2} - λ_{22} D_{2}) = 0

(A14)

A_{12} {(σ_{12})}^{λ_{12}} = \frac{{(σ_{11} + σ_{22})}^{λ_{12}}}{2^{λ_{12} + 1}} [A_{11} (1 + λ_{12} B_{2} D_{1}) + A_{22} (1 + λ_{12} B_{2} D_{1} - λ_{22} D_{1})]

(A15)

Applying eq 8 to the attractive contribution of the Mie (λ, 6) potential leads to

A_{12} σ_{12}^{6} = {[A_{11} σ_{11}^{6} A_{22} σ_{22}^{6}]}^{1 / 2}

(A16)

By solving eqs A14, A15, and A16, the unlike interaction parameters of the Mie (λ, 6) potential λ₁₂, σ₁₂, and ε₁₂ are obtained, as given in eqs 9, 10, and 11, respectively.

References

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This article references 60 other publications.

1
Dohrn, R.; Pfohl, O. Thermophysical properties─industrial directions. Fluid Phase Equilib. 2002, 194–197, 15– 29, DOI: 10.1016/S0378-3812(01)00791-9
Google Scholar
There is no corresponding record for this reference.
2
Gupta, S.; Olson, J. D. Industrial needs in physical properties. Ind. Eng. Chem. Res. 2003, 42, 6359– 6374, DOI: 10.1021/ie030170v
Google Scholar
There is no corresponding record for this reference.
3
Hendriks, E.; Kontogeorgis, G. M.; Dohrn, R.; Hemptinne, J.-C. de.; Economou, I. G.; Žilnik, L. F.; Vesovic, V. Industrial requirements for thermodynamics and transport properties. Ind. Eng. Chem. Res. 2010, 49, 11131– 11141, DOI: 10.1021/ie101231b
Google Scholar
There is no corresponding record for this reference.
4
Kontogeorgis, G. M.; Dohrn, R.; Economou, I. G.; Hemptinne, J.-C. de.; Kate, A. Ten.; Kuitunen, S.; Mooijer, M.; Žilnik, L. F.; Vesovic, V. Industrial requirements for thermodynamic and transport properties: 2020. Ind. Eng. Chem. Res. 2021, 60, 4987– 5013, DOI: 10.1021/acs.iecr.0c05356
Google Scholar
There is no corresponding record for this reference.
5
de Hemptinne, J.-C.; Kontogeorgis, G. M.; Dohrn, R.; Economou, I. G.; Ten Kate, A.; Kuitunen, S.; Fele Žilnik, L. F.; De Angelis, M. G.; Vesovic, V. A view on the future of applied thermodynamics. Ind. Eng. Chem. Res. 2022, 61, 14664– 14680, DOI: 10.1021/acs.iecr.2c01906
Google Scholar
There is no corresponding record for this reference.
6
Gupta, S.; Elliott, J. R.; Anderko, A.; Crosthwaite, J.; Chapman, W. G.; Lira, C. T. Current practices and continuing needs in thermophysical properties for the chemical industry. Ind. Eng. Chem. Res. 2023, 62, 3394– 3427, DOI: 10.1021/acs.iecr.2c03153
Google Scholar
There is no corresponding record for this reference.
7
Poling, B. E.; Prausnitz, M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2004.
Google Scholar
There is no corresponding record for this reference.
8
Goodwin, A. R.; Sengers, J.; Peters, C. J. Applied Thermodynamics of Fluids; Royal Society of Chemistry: Cambridge, 2010.
Google Scholar
There is no corresponding record for this reference.
9
Assael, M. J.; Goodwin, A. R. H.; Vesovic, V.; Wakeham, W. A. Experimental Thermodynamics Volume IX: Advances in Transport Properties of Fluids; Royal Society of Chemistry: London, 2014.
Google Scholar
There is no corresponding record for this reference.
10
Tillotson, M. J.; Diamantonis, N. I.; Buda, C.; Bolton, L. W.; Müller, E. A. Molecular modelling of the thermophysical properties of fluids: expectations, limitations, gaps and opportunities. Phys. Chem. Chem. Phys. 2023, 25, 12607– 12628, DOI: 10.1039/D2CP05423J
Google Scholar
There is no corresponding record for this reference.
11
Mejía, A.; Herdes, C.; Müller, E. A. Force fields for coarse-grained molecular simulations from a corresponding states correlation. Ind. Eng. Chem. Res. 2014, 53, 4131– 4141, DOI: 10.1021/ie404247e
Google Scholar
There is no corresponding record for this reference.
12
Herdes, C.; Totton, T. S.; Müller, E. A. Coarse grained force field for the molecular simulation of natural gases and condensates. Fluid Phase Equilib. 2015, 406, 91– 100, DOI: 10.1016/j.fluid.2015.07.014
Google Scholar
There is no corresponding record for this reference.
13
Mick, J. R.; Barhaghi, M. S.; Jackman, B.; Rushaidat, K.; Schwiebert, L.; Potoff, J. J. Optimized Mie potentials for phase equilibria: Application to noble gases and their mixtures with n-alkanes. J. Chem. Phys. 2015, 143, 114304 DOI: 10.1063/1.4930138
Google Scholar
There is no corresponding record for this reference.
14
Hoang, H.; Delage-Santacreu, S.; Galliero, G. Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field. Ind. Eng. Chem. Res. 2017, 56, 9213– 9226, DOI: 10.1021/acs.iecr.7b01397
Google Scholar
There is no corresponding record for this reference.
15
Jervell, V. G.; Wilhelmsen, Ø. Revised Enskog theory for Mie fluids: Prediction of diffusion coefficients, thermal diffusion coefficients, viscosities, and thermal conductivities. J. Chem. Phys. 2023, 158, 224103 DOI: 10.1063/5.0149865
Google Scholar
There is no corresponding record for this reference.
16
Chaparro, G.; Müller, E. A. Simulation and data-driven modeling of the transport properties of the Mie fluid. J. Phys. Chem. B 2024, 128, 551– 566, DOI: 10.1021/acs.jpcb.3c06813
Google Scholar
There is no corresponding record for this reference.
17
Ervik, Å.; Serratos, G. J.; Müller, E. A. raaSAFT: A framework enabling coarse-grained molecular dynamics simulations based on the SAFT-γ Mie force field. Comput. Phys. Commun. 2017, 212, 161– 179, DOI: 10.1016/j.cpc.2016.07.035
Google Scholar
There is no corresponding record for this reference.
18
Calvin, D. W.; Reed III, T. M. Mixture rules for the Mie (n, 6) intermolecular pair potential and the Dymond–Alder pair potential. J. Chem. Phys. 1971, 54, 3733– 3738, DOI: 10.1063/1.1675422
Google Scholar
There is no corresponding record for this reference.
19
Lafitte, T.; Apostolakou, A.; Avendaño, C.; Galindo, A.; Adjiman, C. S.; Müller, E. A.; Jackson, G. Accurate statistical associating fluid theory for chain molecules formed from Mie segments. J. Chem. Phys. 2013, 139, 154504 DOI: 10.1063/1.4819786
Google Scholar
There is no corresponding record for this reference.
20
Lorentz, H. A. Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase. Ann. Phys. 1881, 248, 127– 136, DOI: 10.1002/andp.18812480110
Google Scholar
There is no corresponding record for this reference.
21
Berthelot, D. Sur le mélange des gaz. Compt. Rendus 1898, 126, 15
Google Scholar
There is no corresponding record for this reference.
22
Hamani, A. W. S.; Bazile, J. P.; Hoang, H.; Luc, H. T.; Daridon, J. L.; Galliero, G. Thermophysical properties of simple molecular liquid mixtures: On the limitations of some force fields. J. Mol. Liq. 2020, 303, 112663 DOI: 10.1016/j.molliq.2020.112663
Google Scholar
There is no corresponding record for this reference.
23
Hamani, A. W. S.; Hoang, H.; Viet, T. Q. Q.; Daridon, J. L.; Galliero, G. Excess volume, isothermal compressibility, isentropic compressibility and speed of sound of carbon dioxide+ n-heptane binary mixture under pressure up to 70 MPa. II. Molecular simulations. J. Supercrit. Fluids 2020, 164, 104890 DOI: 10.1016/j.supflu.2020.104890
Google Scholar
There is no corresponding record for this reference.
24
Delhommelle, J.; Millié, P. Inadequacy of the Lorentz–Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation. Mol. Phys. 2001, 99, 619– 625, DOI: 10.1080/00268970010020041
Google Scholar
There is no corresponding record for this reference.
25
Desgranges, C.; Delhommelle, J. Evaluation of the grand-canonical partition function using expanded Wang–Landau simulations. III. Impact of combining rules on mixtures properties. J. Chem. Phys. 2014, 140, 104109 DOI: 10.1063/1.4867498
Google Scholar
There is no corresponding record for this reference.
26
Hoang, H.; Ho, K. H.; Battani, A.; Scott, J. A.; Collell, J.; Pujol, M.; Galliero, G. Modeling solubility induced elemental fractionation of noble gases in oils. Geochim. Cosmochim. Acta 2025, 388, 127– 142, DOI: 10.1016/j.gca.2024.09.004
Google Scholar
There is no corresponding record for this reference.
27
Smith, F. T. Atomic distortion and the combining rule for repulsive potentials. Phys. Rev. A 1972, 5, 1708 DOI: 10.1103/PhysRevA.5.1708
Google Scholar
There is no corresponding record for this reference.
28
Waldman, M.; Hagler, A. T. New combining rules for rare gas van der Waals parameters. J. Comput. Chem. 1993, 14, 1077– 1084, DOI: 10.1002/jcc.540140909
Google Scholar
There is no corresponding record for this reference.
29
Dauber-Osguthorpe, P.; Hagler, A. T. Biomolecular force fields: where have we been, where are we now, where do we need to go and how do we get there?. J. Comput.-Aided Mol. Des. 2019, 33, 133– 203, DOI: 10.1007/s10822-018-0111-4
Google Scholar
There is no corresponding record for this reference.
30
Kong, C. L. Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential. J. Chem. Phys. 1973, 59, 2464– 2467, DOI: 10.1063/1.1680358
Google Scholar
There is no corresponding record for this reference.
31
Kong, C. L.; Chakrabarty, M. R. Combining rules for intermolecular potential parameters. III. Application to the Exp-6 potential. J. Phys. Chem. A 1973, 77, 2668– 2670, DOI: 10.1021/j100640a019
Google Scholar
There is no corresponding record for this reference.
32
Prausnitz, J. M.; Lichtenthaler, R. N.; De Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Pearson Education: London, 1998.
Google Scholar
There is no corresponding record for this reference.
33
Panagiotopoulos, A. Z. Monte Carlo methods for phase equilibria of fluids. J. Phys.: Condens. Matter 2000, 12, R25, DOI: 10.1088/0953-8984/12/3/201
Google Scholar
There is no corresponding record for this reference.
34
Shing, K. S.; Gubbins, K. E.; Lucas, K. Henry constants in non-ideal fluid mixtures: Computer simulation and theory. Mol. Phys. 1988, 65, 1235– 1252, DOI: 10.1080/00268978800101731
Google Scholar
There is no corresponding record for this reference.
35
Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 3rd ed.; Elsevier: Amsterdam, 2023.
Google Scholar
There is no corresponding record for this reference.
36
Widom, B. Some topics in the theory of fluids. J. Chem. Phys. 1963, 39, 2808– 2812, DOI: 10.1063/1.1734110
Google Scholar
There is no corresponding record for this reference.
37
Ungerer, P.; Tavitian, B.; Boutin, A. Applications of Molecular Simulation in the Oil and Gas Industry; Technip: Paris, 2005.
Google Scholar
There is no corresponding record for this reference.
38
Hogervorst, W. Transport and equilibrium properties of simple gases and forces between like and unlike atoms. Physica 1971, 51, 77– 89, DOI: 10.1016/0031-8914(71)90138-8
Google Scholar
There is no corresponding record for this reference.
39
Schouten, J. A.; Deerenberg, A.; Trappeniers, N. J. Vapour-liquid and gas-gas equilibria in simple systems: IV. The system argon-krypton. Physica A 1975, 81, 151– 160, DOI: 10.1016/0378-4371(75)90042-4
Google Scholar
There is no corresponding record for this reference.
40
Calado, J. C.; Chang, E.; Streett, W. B. Vapour-liquid equilibrium in the krypton-xenon system. Physica A 1983, 117, 127– 138, DOI: 10.1016/0378-4371(83)90025-0
Google Scholar
There is no corresponding record for this reference.
41
Dias, L. M. B.; Filipe, E. J.; McCabe, C.; Calado, J. C. Thermodynamics of liquid (xenon + methane) mixtures. J. Phys. Chem. B 2004, 108, 7377– 7381, DOI: 10.1021/jp037070n
Google Scholar
There is no corresponding record for this reference.
42
Clever, H. L. Krypton, Xenon, and Radon─Gas Solubilities; Elsevier: Oxford, UK, 1979b; Vol. 2.
Google Scholar
There is no corresponding record for this reference.
43
Reamer, H. H.; Sage, B. H.; Lacey, W. N. Phase equilibria in hydrocarbon systems. Volumetric and phase behavior of the methane-n-heptane system. Ind. Eng. Chem. Chem. Eng. Data Ser. 1956, 1, 29– 42, DOI: 10.1021/i460001a007
Google Scholar
There is no corresponding record for this reference.
44
Beaudoin, J. M.; Kohn, J. P. Multiphase and volumetric equilibria of the methane-n-decane binary system at temperatures between – 36° and 150°. J. Chem. Eng. Data 1967, 12, 189– 191, DOI: 10.1021/je60033a007
Google Scholar
There is no corresponding record for this reference.
45
Lin, Y. N.; Hwang, S. C.; Kobayashi, R. Vapor-liquid equilibrium of the methane-toluene system at low temperatures. J. Chem. Eng. Data 1978, 23, 231– 234, DOI: 10.1021/je60078a007
Google Scholar
There is no corresponding record for this reference.
46
Elbishlawi, M.; Spencer, J. R. Equilibrium relations of two methane-aromatic binary systems at 150°F. Ind. Eng. Chem. 1951, 43, 1811– 1815, DOI: 10.1021/ie50500a036
Google Scholar
There is no corresponding record for this reference.
47
Lin, H. M.; Sebastian, H. M.; Simnick, J. J.; Chao, K. C. Gas-liquid equilibrium in binary mixtures of methane with n-decane, benzene, and toluene. J. Chem. Eng. Data 1979, 24, 146– 149, DOI: 10.1021/je60081a004
Google Scholar
There is no corresponding record for this reference.
48
Sander, R.; Acree, W. E., Jr; Visscher, A. De.; Schwartz, S. E.; Wallington, T. J. Henry’s law constants (IUPAC Recommendations 2021). Pure Appl. Chem. 2022, 94, 71– 85, DOI: 10.1515/pac-2020-0302
Google Scholar
There is no corresponding record for this reference.
49
Clever, H. L. Helium and Neon─Gas Solubilities; Elsevier: Oxford, UK, 1979a; Vol. 1.
Google Scholar
There is no corresponding record for this reference.
50
Clever, H. L. Argon─Gas Solubilities; Elsevier: Oxford, UK, 1980; Vol. 4.
Google Scholar
There is no corresponding record for this reference.
51
Kalra, H.; Kubota, H.; Robinson, D. B.; Ng, H. J. Equilibrium phase properties of the carbon dioxide-n-heptane system. J. Chem. Eng. Data 1978, 23, 317– 321, DOI: 10.1021/je60079a016
Google Scholar
There is no corresponding record for this reference.
52
Reamer, H. H.; Sage, B. H. Phase equilibria in hydrocarbon systems. Volumetric and phase behavior of the n-decane–CO₂ system. J. Chem. Eng. Data 1963, 8, 508– 513, DOI: 10.1021/je60019a010
Google Scholar
There is no corresponding record for this reference.
53
Nagarajan, N.; Robinson, R. L., Jr. Equilibrium phase compositions, phase densities, and interfacial tensions for carbon dioxide + hydrocarbon systems. 2. Carbon dioxide + n-decane. J. Chem. Eng. Data 1986, 31, 168– 171, DOI: 10.1021/je00044a012
Google Scholar
There is no corresponding record for this reference.
54
Fink, S. D.; Hershey, H. C. Modeling the vapor-liquid equilibria of 1,1,1-trichloroethane + carbon dioxide and toluene + carbon dioxide at 308, 323, and 353 K. Ind. Eng. Chem. Res. 1990, 29, 295– 306, DOI: 10.1021/ie00098a022
Google Scholar
There is no corresponding record for this reference.
55
Bazile, J.-P.; Nasri, D.; Hoang, H.; Galliero, G.; Daridon, J.-L. Density, speed of sound, compressibility and related excess properties of methane + n-heptane at T = 303.15 K and p = 10 to 70 MPa. Int. J. Thermophys. 2020, 41, 115 DOI: 10.1007/s10765-020-02694-9
Google Scholar
There is no corresponding record for this reference.
56
Regueira, T.; Pantelide, G.; Yan, W.; Stenby, E. H. Density and phase equilibrium of the binary system methane + n-decane under high temperatures and pressures. Fluid Phase Equilib. 2016, 428, 48– 61, DOI: 10.1016/j.fluid.2016.08.004
Google Scholar
There is no corresponding record for this reference.
57
Baylaucq, A.; Boned, C.; Canet, X.; Zéberg-Mikkelsen, C. K. High-pressure (up to 140 MPa) dynamic viscosity of the methane and toluene system: Measurements and comparative study of some representative models. Int. J. Thermophys. 2003, 24, 621– 638, DOI: 10.1023/A:1024023913165
Google Scholar
There is no corresponding record for this reference.
58
Bazile, J.-P.; Nasri, D.; Hamani, A. W. S.; Galliero, G.; Daridon, J.-L. Excess volume, isothermal compressibility, isentropic compressibility and speed of sound of carbon dioxide + n-heptane binary mixture under pressure up to 70 MPa. I. Experimental measurements. J. Supercrit. Fluids 2018, 140, 218– 232, DOI: 10.1016/j.supflu.2018.05.028
Google Scholar
There is no corresponding record for this reference.
59
Chacon Valero, A. M.; Feitosa, F. X.; de Sant’Ana, H. B. Density and volumetric behavior of binary CO₂ + n-decane and ternary CO₂ + n-decane + naphthalene systems at high pressure and high temperature. J. Chem. Eng. Data 2020, 65, 3499– 3509, DOI: 10.1021/acs.jced.0c00090
Google Scholar
There is no corresponding record for this reference.
60
Matsukawa, H.; Tsuji, T.; Otake, K. Measurement of the density of carbon dioxide/toluene homogeneous mixtures and correlation with equations of state. J. Chem. Thermodyn. 2022, 164, 106618 DOI: 10.1016/j.jct.2021.106618
Google Scholar
There is no corresponding record for this reference.

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Abstract
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Figure 1
Figure 1. Comparison of Henry’s law constants obtained from the NVT-GEMC simulations using different combining rules and experiment for seven binary mixtures for which the Mie (λ, 6) potential parameters of the components are relatively similar. (a) Argon/Krypton mixture. (b) Krypton/Xenon mixture. (c) Methane/Xenon mixture. (d) Krypton/n-Heptane mixture. (e) Methane/n-Heptane mixture. (f) Methane/n-Decane mixture. (g) Methane–Toluene mixture. Solid circles (red) correspond to the experimental data. Open squares (green) correspond to the simulation data by using the classical combining rules. Open deltas (blue) correspond to the simulation data using Lafitte et al. combining rules. Open right triangles (red) correspond to the simulation data by using the new combining rules. Lines serve as a guide to the eye for the experimental data.
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Figure 2
Figure 2. Comparison of Henry’s law constants obtained from the NVT-GEMC simulations using different combining rules and experiment for five binary mixtures for which the Mie (λ, 6) potential parameters of the components are significantly different. (a) Neon/n-heptane mixture. (b) Argon/n-heptane mixture. (c) Carbon dioxide/n-heptane mixture. (d) Carbon dioxide/n-decane mixture. (e) Carbon dioxide/toluene mixture. The legend is identical to that in Figure 1.
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Figure 3
Figure 3. Comparison of pressure–composition phase diagrams of binary mixtures obtained from the NPT-GEMC simulations using different combining rules and experiment for six binary mixtures for which the Mie (λ, 6) potential parameters of the components are relatively similar. (a) Argon/krypton mixture at T = 177.38 K. (b) Krypton/xenon mixture at T = 190.03 K. (c) Methane/xenon mixture at T = 189.78 K. (d) Methane/n-heptane mixture at T = 310.93 K. (e) Methane/n-decane mixture at T = 373.15 K. (f) Methane/toluene mixture at T = 422.45 K. The legend is identical to that in Figure 1.
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Figure 4
Figure 4. Comparison of pressure–composition phase diagrams of binary mixtures obtained from the NPT-GEMC simulations using different combining rules and experiment for three binary mixtures for which the Mie (λ, 6) potential parameters of the species are significantly different. (a) Carbon dioxide/n-heptane mixture at T = 310.65 K. (b) Carbon dioxide/n-decane mixture at T = 377.59 K. (c) Carbon dioxide/toluene mixture at T = 308.16 K. The legend is identical to that in Figure 1.
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Figure 5
Figure 5. Comparison of excess molar volumes of binary mixtures obtained from the NPT-MC simulations using different combining rules and experiment for three binary mixtures for which the Mie (λ, 6) potential parameters of the components are relatively similar. (a) Methane/n-heptane mixture at T = 303.15 K and P = 30 MPa. (b) Methane/n-decane mixture at T = 373.15 K and P = 40 MPa. (c) Methane/toluene mixture at T = 333.15 K and P = 45 MPa. The legend is identical to that in Figure 1.
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Figure 6
Figure 6. Comparison of excess molar volumes of binary mixtures obtained from the NPT-MC simulations using different combining rules and experiment for three binary mixtures for which the Mie (λ, 6) potential parameters of the components are significantly different. (a) Carbon dioxide/n-heptane mixture at T = 313.15 K and P = 10 MPa. (b) Carbon dioxide/n-decane mixture at T = 318.15 K and P = 15 MPa. (c) Carbon dioxide/toluene mixture at T = 313.15 K and P = 10 MPa. The legend is identical to that in Figure 1.
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References
This article references 60 other publications.
1. 1
  Dohrn, R.; Pfohl, O. Thermophysical properties─industrial directions. Fluid Phase Equilib. 2002, 194–197, 15– 29, DOI: 10.1016/S0378-3812(01)00791-9
  There is no corresponding record for this reference.
2. 2
  Gupta, S.; Olson, J. D. Industrial needs in physical properties. Ind. Eng. Chem. Res. 2003, 42, 6359– 6374, DOI: 10.1021/ie030170v
  There is no corresponding record for this reference.
3. 3
  Hendriks, E.; Kontogeorgis, G. M.; Dohrn, R.; Hemptinne, J.-C. de.; Economou, I. G.; Žilnik, L. F.; Vesovic, V. Industrial requirements for thermodynamics and transport properties. Ind. Eng. Chem. Res. 2010, 49, 11131– 11141, DOI: 10.1021/ie101231b
  There is no corresponding record for this reference.
4. 4
  Kontogeorgis, G. M.; Dohrn, R.; Economou, I. G.; Hemptinne, J.-C. de.; Kate, A. Ten.; Kuitunen, S.; Mooijer, M.; Žilnik, L. F.; Vesovic, V. Industrial requirements for thermodynamic and transport properties: 2020. Ind. Eng. Chem. Res. 2021, 60, 4987– 5013, DOI: 10.1021/acs.iecr.0c05356
  There is no corresponding record for this reference.
5. 5
  de Hemptinne, J.-C.; Kontogeorgis, G. M.; Dohrn, R.; Economou, I. G.; Ten Kate, A.; Kuitunen, S.; Fele Žilnik, L. F.; De Angelis, M. G.; Vesovic, V. A view on the future of applied thermodynamics. Ind. Eng. Chem. Res. 2022, 61, 14664– 14680, DOI: 10.1021/acs.iecr.2c01906
  There is no corresponding record for this reference.
6. 6
  Gupta, S.; Elliott, J. R.; Anderko, A.; Crosthwaite, J.; Chapman, W. G.; Lira, C. T. Current practices and continuing needs in thermophysical properties for the chemical industry. Ind. Eng. Chem. Res. 2023, 62, 3394– 3427, DOI: 10.1021/acs.iecr.2c03153
  There is no corresponding record for this reference.
7. 7
  Poling, B. E.; Prausnitz, M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2004.
  There is no corresponding record for this reference.
8. 8
  Goodwin, A. R.; Sengers, J.; Peters, C. J. Applied Thermodynamics of Fluids; Royal Society of Chemistry: Cambridge, 2010.
  There is no corresponding record for this reference.
9. 9
  Assael, M. J.; Goodwin, A. R. H.; Vesovic, V.; Wakeham, W. A. Experimental Thermodynamics Volume IX: Advances in Transport Properties of Fluids; Royal Society of Chemistry: London, 2014.
  There is no corresponding record for this reference.
10. 10
  Tillotson, M. J.; Diamantonis, N. I.; Buda, C.; Bolton, L. W.; Müller, E. A. Molecular modelling of the thermophysical properties of fluids: expectations, limitations, gaps and opportunities. Phys. Chem. Chem. Phys. 2023, 25, 12607– 12628, DOI: 10.1039/D2CP05423J
  There is no corresponding record for this reference.
11. 11
  Mejía, A.; Herdes, C.; Müller, E. A. Force fields for coarse-grained molecular simulations from a corresponding states correlation. Ind. Eng. Chem. Res. 2014, 53, 4131– 4141, DOI: 10.1021/ie404247e
  There is no corresponding record for this reference.
12. 12
  Herdes, C.; Totton, T. S.; Müller, E. A. Coarse grained force field for the molecular simulation of natural gases and condensates. Fluid Phase Equilib. 2015, 406, 91– 100, DOI: 10.1016/j.fluid.2015.07.014
  There is no corresponding record for this reference.
13. 13
  Mick, J. R.; Barhaghi, M. S.; Jackman, B.; Rushaidat, K.; Schwiebert, L.; Potoff, J. J. Optimized Mie potentials for phase equilibria: Application to noble gases and their mixtures with n-alkanes. J. Chem. Phys. 2015, 143, 114304 DOI: 10.1063/1.4930138
  There is no corresponding record for this reference.
14. 14
  Hoang, H.; Delage-Santacreu, S.; Galliero, G. Simultaneous description of equilibrium, interfacial, and transport properties of fluids using a Mie chain coarse-grained force field. Ind. Eng. Chem. Res. 2017, 56, 9213– 9226, DOI: 10.1021/acs.iecr.7b01397
  There is no corresponding record for this reference.
15. 15
  Jervell, V. G.; Wilhelmsen, Ø. Revised Enskog theory for Mie fluids: Prediction of diffusion coefficients, thermal diffusion coefficients, viscosities, and thermal conductivities. J. Chem. Phys. 2023, 158, 224103 DOI: 10.1063/5.0149865
  There is no corresponding record for this reference.
16. 16
  Chaparro, G.; Müller, E. A. Simulation and data-driven modeling of the transport properties of the Mie fluid. J. Phys. Chem. B 2024, 128, 551– 566, DOI: 10.1021/acs.jpcb.3c06813
  There is no corresponding record for this reference.
17. 17
  Ervik, Å.; Serratos, G. J.; Müller, E. A. raaSAFT: A framework enabling coarse-grained molecular dynamics simulations based on the SAFT-γ Mie force field. Comput. Phys. Commun. 2017, 212, 161– 179, DOI: 10.1016/j.cpc.2016.07.035
  There is no corresponding record for this reference.
18. 18
  Calvin, D. W.; Reed III, T. M. Mixture rules for the Mie (n, 6) intermolecular pair potential and the Dymond–Alder pair potential. J. Chem. Phys. 1971, 54, 3733– 3738, DOI: 10.1063/1.1675422
  There is no corresponding record for this reference.
19. 19
  Lafitte, T.; Apostolakou, A.; Avendaño, C.; Galindo, A.; Adjiman, C. S.; Müller, E. A.; Jackson, G. Accurate statistical associating fluid theory for chain molecules formed from Mie segments. J. Chem. Phys. 2013, 139, 154504 DOI: 10.1063/1.4819786
  There is no corresponding record for this reference.
20. 20
  Lorentz, H. A. Ueber die Anwendung des Satzes vom Virial in der kinetischen Theorie der Gase. Ann. Phys. 1881, 248, 127– 136, DOI: 10.1002/andp.18812480110
  There is no corresponding record for this reference.
21. 21
  Berthelot, D. Sur le mélange des gaz. Compt. Rendus 1898, 126, 15
  There is no corresponding record for this reference.
22. 22
  Hamani, A. W. S.; Bazile, J. P.; Hoang, H.; Luc, H. T.; Daridon, J. L.; Galliero, G. Thermophysical properties of simple molecular liquid mixtures: On the limitations of some force fields. J. Mol. Liq. 2020, 303, 112663 DOI: 10.1016/j.molliq.2020.112663
  There is no corresponding record for this reference.
23. 23
  Hamani, A. W. S.; Hoang, H.; Viet, T. Q. Q.; Daridon, J. L.; Galliero, G. Excess volume, isothermal compressibility, isentropic compressibility and speed of sound of carbon dioxide+ n-heptane binary mixture under pressure up to 70 MPa. II. Molecular simulations. J. Supercrit. Fluids 2020, 164, 104890 DOI: 10.1016/j.supflu.2020.104890
  There is no corresponding record for this reference.
24. 24
  Delhommelle, J.; Millié, P. Inadequacy of the Lorentz–Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation. Mol. Phys. 2001, 99, 619– 625, DOI: 10.1080/00268970010020041
  There is no corresponding record for this reference.
25. 25
  Desgranges, C.; Delhommelle, J. Evaluation of the grand-canonical partition function using expanded Wang–Landau simulations. III. Impact of combining rules on mixtures properties. J. Chem. Phys. 2014, 140, 104109 DOI: 10.1063/1.4867498
  There is no corresponding record for this reference.
26. 26
  Hoang, H.; Ho, K. H.; Battani, A.; Scott, J. A.; Collell, J.; Pujol, M.; Galliero, G. Modeling solubility induced elemental fractionation of noble gases in oils. Geochim. Cosmochim. Acta 2025, 388, 127– 142, DOI: 10.1016/j.gca.2024.09.004
  There is no corresponding record for this reference.
27. 27
  Smith, F. T. Atomic distortion and the combining rule for repulsive potentials. Phys. Rev. A 1972, 5, 1708 DOI: 10.1103/PhysRevA.5.1708
  There is no corresponding record for this reference.
28. 28
  Waldman, M.; Hagler, A. T. New combining rules for rare gas van der Waals parameters. J. Comput. Chem. 1993, 14, 1077– 1084, DOI: 10.1002/jcc.540140909
  There is no corresponding record for this reference.
29. 29
  Dauber-Osguthorpe, P.; Hagler, A. T. Biomolecular force fields: where have we been, where are we now, where do we need to go and how do we get there?. J. Comput.-Aided Mol. Des. 2019, 33, 133– 203, DOI: 10.1007/s10822-018-0111-4
  There is no corresponding record for this reference.
30. 30
  Kong, C. L. Combining rules for intermolecular potential parameters. II. Rules for the Lennard-Jones (12–6) potential and the Morse potential. J. Chem. Phys. 1973, 59, 2464– 2467, DOI: 10.1063/1.1680358
  There is no corresponding record for this reference.
31. 31
  Kong, C. L.; Chakrabarty, M. R. Combining rules for intermolecular potential parameters. III. Application to the Exp-6 potential. J. Phys. Chem. A 1973, 77, 2668– 2670, DOI: 10.1021/j100640a019
  There is no corresponding record for this reference.
32. 32
  Prausnitz, J. M.; Lichtenthaler, R. N.; De Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Pearson Education: London, 1998.
  There is no corresponding record for this reference.
33. 33
  Panagiotopoulos, A. Z. Monte Carlo methods for phase equilibria of fluids. J. Phys.: Condens. Matter 2000, 12, R25, DOI: 10.1088/0953-8984/12/3/201
  There is no corresponding record for this reference.
34. 34
  Shing, K. S.; Gubbins, K. E.; Lucas, K. Henry constants in non-ideal fluid mixtures: Computer simulation and theory. Mol. Phys. 1988, 65, 1235– 1252, DOI: 10.1080/00268978800101731
  There is no corresponding record for this reference.
35. 35
  Frenkel, D.; Smit, B. Understanding Molecular Simulation: From Algorithms to Applications, 3rd ed.; Elsevier: Amsterdam, 2023.
  There is no corresponding record for this reference.
36. 36
  Widom, B. Some topics in the theory of fluids. J. Chem. Phys. 1963, 39, 2808– 2812, DOI: 10.1063/1.1734110
  There is no corresponding record for this reference.
37. 37
  Ungerer, P.; Tavitian, B.; Boutin, A. Applications of Molecular Simulation in the Oil and Gas Industry; Technip: Paris, 2005.
  There is no corresponding record for this reference.
38. 38
  Hogervorst, W. Transport and equilibrium properties of simple gases and forces between like and unlike atoms. Physica 1971, 51, 77– 89, DOI: 10.1016/0031-8914(71)90138-8
  There is no corresponding record for this reference.
39. 39
  Schouten, J. A.; Deerenberg, A.; Trappeniers, N. J. Vapour-liquid and gas-gas equilibria in simple systems: IV. The system argon-krypton. Physica A 1975, 81, 151– 160, DOI: 10.1016/0378-4371(75)90042-4
  There is no corresponding record for this reference.
40. 40
  Calado, J. C.; Chang, E.; Streett, W. B. Vapour-liquid equilibrium in the krypton-xenon system. Physica A 1983, 117, 127– 138, DOI: 10.1016/0378-4371(83)90025-0
  There is no corresponding record for this reference.
41. 41
  Dias, L. M. B.; Filipe, E. J.; McCabe, C.; Calado, J. C. Thermodynamics of liquid (xenon + methane) mixtures. J. Phys. Chem. B 2004, 108, 7377– 7381, DOI: 10.1021/jp037070n
  There is no corresponding record for this reference.
42. 42
  Clever, H. L. Krypton, Xenon, and Radon─Gas Solubilities; Elsevier: Oxford, UK, 1979b; Vol. 2.
  There is no corresponding record for this reference.
43. 43
  Reamer, H. H.; Sage, B. H.; Lacey, W. N. Phase equilibria in hydrocarbon systems. Volumetric and phase behavior of the methane-n-heptane system. Ind. Eng. Chem. Chem. Eng. Data Ser. 1956, 1, 29– 42, DOI: 10.1021/i460001a007
  There is no corresponding record for this reference.
44. 44
  Beaudoin, J. M.; Kohn, J. P. Multiphase and volumetric equilibria of the methane-n-decane binary system at temperatures between – 36° and 150°. J. Chem. Eng. Data 1967, 12, 189– 191, DOI: 10.1021/je60033a007
  There is no corresponding record for this reference.
45. 45
  Lin, Y. N.; Hwang, S. C.; Kobayashi, R. Vapor-liquid equilibrium of the methane-toluene system at low temperatures. J. Chem. Eng. Data 1978, 23, 231– 234, DOI: 10.1021/je60078a007
  There is no corresponding record for this reference.
46. 46
  Elbishlawi, M.; Spencer, J. R. Equilibrium relations of two methane-aromatic binary systems at 150°F. Ind. Eng. Chem. 1951, 43, 1811– 1815, DOI: 10.1021/ie50500a036
  There is no corresponding record for this reference.
47. 47
  Lin, H. M.; Sebastian, H. M.; Simnick, J. J.; Chao, K. C. Gas-liquid equilibrium in binary mixtures of methane with n-decane, benzene, and toluene. J. Chem. Eng. Data 1979, 24, 146– 149, DOI: 10.1021/je60081a004
  There is no corresponding record for this reference.
48. 48
  Sander, R.; Acree, W. E., Jr; Visscher, A. De.; Schwartz, S. E.; Wallington, T. J. Henry’s law constants (IUPAC Recommendations 2021). Pure Appl. Chem. 2022, 94, 71– 85, DOI: 10.1515/pac-2020-0302
  There is no corresponding record for this reference.
49. 49
  Clever, H. L. Helium and Neon─Gas Solubilities; Elsevier: Oxford, UK, 1979a; Vol. 1.
  There is no corresponding record for this reference.
50. 50
  Clever, H. L. Argon─Gas Solubilities; Elsevier: Oxford, UK, 1980; Vol. 4.
  There is no corresponding record for this reference.
51. 51
  Kalra, H.; Kubota, H.; Robinson, D. B.; Ng, H. J. Equilibrium phase properties of the carbon dioxide-n-heptane system. J. Chem. Eng. Data 1978, 23, 317– 321, DOI: 10.1021/je60079a016
  There is no corresponding record for this reference.
52. 52
  Reamer, H. H.; Sage, B. H. Phase equilibria in hydrocarbon systems. Volumetric and phase behavior of the n-decane–CO₂ system. J. Chem. Eng. Data 1963, 8, 508– 513, DOI: 10.1021/je60019a010
  There is no corresponding record for this reference.
53. 53
  Nagarajan, N.; Robinson, R. L., Jr. Equilibrium phase compositions, phase densities, and interfacial tensions for carbon dioxide + hydrocarbon systems. 2. Carbon dioxide + n-decane. J. Chem. Eng. Data 1986, 31, 168– 171, DOI: 10.1021/je00044a012
  There is no corresponding record for this reference.
54. 54
  Fink, S. D.; Hershey, H. C. Modeling the vapor-liquid equilibria of 1,1,1-trichloroethane + carbon dioxide and toluene + carbon dioxide at 308, 323, and 353 K. Ind. Eng. Chem. Res. 1990, 29, 295– 306, DOI: 10.1021/ie00098a022
  There is no corresponding record for this reference.
55. 55
  Bazile, J.-P.; Nasri, D.; Hoang, H.; Galliero, G.; Daridon, J.-L. Density, speed of sound, compressibility and related excess properties of methane + n-heptane at T = 303.15 K and p = 10 to 70 MPa. Int. J. Thermophys. 2020, 41, 115 DOI: 10.1007/s10765-020-02694-9
  There is no corresponding record for this reference.
56. 56
  Regueira, T.; Pantelide, G.; Yan, W.; Stenby, E. H. Density and phase equilibrium of the binary system methane + n-decane under high temperatures and pressures. Fluid Phase Equilib. 2016, 428, 48– 61, DOI: 10.1016/j.fluid.2016.08.004
  There is no corresponding record for this reference.
57. 57
  Baylaucq, A.; Boned, C.; Canet, X.; Zéberg-Mikkelsen, C. K. High-pressure (up to 140 MPa) dynamic viscosity of the methane and toluene system: Measurements and comparative study of some representative models. Int. J. Thermophys. 2003, 24, 621– 638, DOI: 10.1023/A:1024023913165
  There is no corresponding record for this reference.
58. 58
  Bazile, J.-P.; Nasri, D.; Hamani, A. W. S.; Galliero, G.; Daridon, J.-L. Excess volume, isothermal compressibility, isentropic compressibility and speed of sound of carbon dioxide + n-heptane binary mixture under pressure up to 70 MPa. I. Experimental measurements. J. Supercrit. Fluids 2018, 140, 218– 232, DOI: 10.1016/j.supflu.2018.05.028
  There is no corresponding record for this reference.
59. 59
  Chacon Valero, A. M.; Feitosa, F. X.; de Sant’Ana, H. B. Density and volumetric behavior of binary CO₂ + n-decane and ternary CO₂ + n-decane + naphthalene systems at high pressure and high temperature. J. Chem. Eng. Data 2020, 65, 3499– 3509, DOI: 10.1021/acs.jced.0c00090
  There is no corresponding record for this reference.
60. 60
  Matsukawa, H.; Tsuji, T.; Otake, K. Measurement of the density of carbon dioxide/toluene homogeneous mixtures and correlation with equations of state. J. Chem. Thermodyn. 2022, 164, 106618 DOI: 10.1016/j.jct.2021.106618
  There is no corresponding record for this reference.

A New Set of Combining Rules for Mie (λ, 6) PotentialClick to copy article linkArticle link copied!

The Journal of Physical Chemistry B

Abstract

1. Introduction

2. Combining Rules for Mie Potentials

3. Molecular Simulations

4. Results and Discussion

4.1. Validation of the New Set of Combining Rules

4.2. Tests of the New Set of Combining Rules

4.2.1. Lennard–Jones Parameters of Unlike Interactions

4.2.2. Thermodynamic Properties of Binary Mixtures

4.2.2.1. Henry’s Law Constant

Figure 1

Figure 2

4.2.2.2. Phase Diagram of a Binary Mixture

Figure 3

Figure 4

4.2.2.3. Excess Molar Volume of a Binary Mixture

Figure 5

Figure 6

5. Conclusions

Author Information

Acknowledgments

Appendix: Derivation of the New Set of Combining Rules for the Mie (λ, 6) Potential

References

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The Journal of Physical Chemistry B

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Abstract

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

References

A New Set of Combining Rules for Mie (λ, 6) Potential
Click to copy article linkArticle link copied!