Unlike Lennard-Jones Parameters for Vapor-Liquid Equilibria

The influence of the unlike Lennard-Jones (LJ) parameters on vapor-liquid equilibria of mixtures is investigated and the performance of eleven combining rules is assessed. In the first part of the work, the influence of the unlike LJ size and energy parameter on vapor pressure, bubble density and dew point composition is systematically studied for the mixtures CO+$\rm C_2H_6$ and $\rm N_2$ + $\rm C_3H_6$, respectively. It is found that mixture vapor pressure depends strongly both on the size and the energy parameter whereas the bubble density depends mostly on the size parameter and the dew point composition is rather insensitive to both parameters. In preceding work, unlike LJ parameters were adjusted to experimental binary vapor-liquid equilibria for 44 real mixtures. On the basis of these results, in the second part of the work eleven combining rules are assessed regarding their predictive power. A comparison with the adjusted unlike LJ parameters determined from the fit shows that none of the eleven combining rules yields appropriate parameters in general. To obtain an accurate mixture model, the unlike dispersive interaction should therefore be adjusted to experimental binary data. The results from the present work indicate that it is sufficient to use the Lorenz rule for the unlike LJ size parameter and to fit the unlike LJ energy parameter to the vapor pressure.

💡 Research Summary

This paper presents a comprehensive investigation into the influence of unlike Lennard-Jones (LJ) parameters on the vapor-liquid equilibria (VLE) of mixtures and conducts a rigorous assessment of the predictive power of eleven established combining rules.

The work is structured in two main parts. The first part involves a systematic case study on two binary mixtures (CO + C2H6 and N2 + C3H6). Using molecular simulation (Grand Equilibrium method), the sensitivity of key VLE properties—vapor pressure, bubble density (liquid density), and dew point composition (vapor composition)—to deviations (η and ξ) from the standard Lorentz-Berthelot (LB) combining rule (arithmetic mean for size σ_ab, geometric mean for energy ε_ab) was quantified. The findings were clear and significant: mixture vapor pressure exhibited high sensitivity to both the unlike size and energy parameters. In contrast, the bubble density depended primarily on the size parameter alone, and the dew point composition was relatively insensitive to variations in both parameters. This sensitivity analysis provides crucial guidance for parameterization strategies.

The second part constitutes a large-scale benchmarking study. The reference for this evaluation was a set of “adjusted” unlike LJ parameters obtained from the authors’ prior work. In that work, for 44 real binary mixtures (including noble gases, diatomics, hydrocarbons, and refrigerants), the unlike LJ energy parameter ε_ab was fitted to a single experimental vapor pressure data point for each mixture, while the size parameter σ_ab was fixed to the Lorentz rule (arithmetic mean). The performance of eleven combining rules—Lorentz-Berthelot (LB), Kohler (K, uses polarizability α), Hudson-McCoubrey (HMC, uses ionization potential I), Fender-Halsey (FH, harmonic mean), Hiza (H), Sikora (S), Smith-Kong (SK), Halgren (HHG), Waldman-Hagler (WH), and Al-Matar (M1, M2)—was assessed by comparing their predicted parameters against these adjusted reference values.

The central conclusion is definitive: none of the eleven combining rules yields appropriate unlike LJ parameters in general. While some rules may perform adequately for specific types of mixtures, no universal combining rule exists that reliably predicts the unlike dispersive interaction across the broad spectrum of components studied. This underscores the inherent complexity of unlike-pair interactions, which cannot be accurately captured solely by pure component parameters.

Based on these insights, the paper proposes a pragmatic and effective modeling strategy. To obtain an accurate mixture model for VLE, it is necessary to adjust the unlike dispersive interaction to experimental binary data. However, the sensitivity analysis allows for an efficient adjustment procedure. The results indicate that it is sufficient to use the simple Lorentz rule (arithmetic mean) for the unlike LJ size parameter σ_ab and to fit only the unlike LJ energy parameter ε_ab to experimental vapor pressure data. This approach minimizes the number of adjustable parameters while targeting the property (vapor pressure) most sensitive to them. This methodology was successfully validated in the authors’ preceding work, yielding models with typical deviations below 5% for vapor pressure and 1% for bubble density for the 44 mixtures, and demonstrating good predictive power for ternary VLE without additional parameterization. The study highlights the limitations of purely predictive combining rules and establishes a practical, data-informed framework for building accurate molecular models for mixture thermodynamics.