Second virial coefficient properties of the n-m Lennard-Jones/Mie potential

Influence of repulsion on entropy scaling and density scaling of monatomic fluids

Entropy scaling is applied to the shear viscosity, self-diffusion coefficient, and thermal conductivity of simple monatomic fluids. An extensive molecular dynamics simulation series is performed to obtain these transport properties and the residual entropy of three potential model classes with variable repulsive exponents: n, 6 Mie (n = 9, 12, 15, and 18), Buckingham's exponential-six (α = 12, 14, 18, and 30), and Tang-Toennies (αT = 4.051, 4.275, and 4.600). A wide range of liquid and supercritical gas- and liquid-like states is covered with a total of 1120 state points. Comparisons to equations of state, literature data, and transport property correlations are made. Although the absolute transport property values within a given potential model class may strongly depend on the repulsive exponent, it is found that the repulsive steepness plays a negligible role when entropy scaling is applied. Hence, the plus-scaled transport properties of n, 6 Mie, exponential-six, and Tang-Toennies fluids lie basically on one master curve, which closely corresponds with entropy scaling correlations for the Lennard-Jones fluid. This trend is confirmed by literature data of n, 6 Mie, and exponential-six fluids. Furthermore, entropy scaling holds for state points where the Pearson correlation coefficient R is well below 0.9. The condition R > 0.9 for strongly correlating liquids is thus not necessary for the successful application of entropy scaling, pointing out that isomorph theory may be a part of a more general framework that is behind the success of entropy scaling. Density scaling reveals a strong influence of the repulsive exponent on this particular approach.

Equation of state for the Mie (λr,6) fluid with a repulsive exponent from 11 to 13

An empirical multi-parameter equation of state in terms of the reduced Helmholtz energy is presented for the Mie (λ_r-6) fluid with a repulsive exponent λ_r from 11 to 13. The equation is fitted to an extensive dataset from molecular dynamics simulation as well as the second and third thermal virial coefficients. It is comprehensively compared with the SAFT-VR model and is a more accurate description of the considered fluid class. The equation is valid for reduced temperatures T/T_c from 0.55 to 4.5 and for reduced pressures of up to p/p_c = 265. A good extrapolation behavior and the occurrence of a single Maxwell loop down to the vicinity of the triple point temperature are realized.

Connecting Entropy Scaling and Density Scaling

It is shown that the residual entropy (entropy minus that of the ideal gas at the same temperature and density) is mostly synonymous with the independent variable of density scaling, identifying a direct link between these two approaches. The residual entropy and the effective hardness of interaction (itself a derivative at constant residual entropy) are studied for the Lennard-Jones monomer and dimer as well as a range of rigid molecular models for carbon dioxide. It is observed that the density scaling exponent appears to be related to the two-body interactions in the dilute-gas limit.

Application of cell models to the melting and sublimation lines of the Lennard-Jones and related potential systems

Harmonic cell models (HCMs) are shown to predict the melting line of the Lennard-Jones (LJ) but not the sublimation line. In addition, even for the melting line, the HCMs are found to be physically unrealistic for inverse power potential systems near the hard-sphere limit, and for the Weeks-Chandler-Andersen system at extremely low temperatures. Despite this, the HCM accurately predicts the LJ mean-square displacement (MSD) from molecular-dynamics (MD) simulations along both lines after simple scaling corrections, to include the effects of anharmonicity and correlated dynamics of the atoms, are applied. Single caged atom molecular dynamics and Monte Carlo simulations provide further quantitative characterization of these additional effects, which go beyond harmonicity. The melting indicator and a modification of the cell model in a similar form are shown to be approximately constant along the melting line, which indicates an isomorph. The less well studied LJ sublimation line is shown not to be an isomorph, yet it still can be represented analytically very accurately by the relationship k_{B}T=aρ^{4}+bρ^{2}, where a and b are constants (k_{B} is Boltzmann's constant, T is the temperature, and ρ is the number density). This relationship has been found previously for the melting line, but the two constants have opposite signs for the sublimation and melting lines. This simple formula is also predicted using a nonharmonic static lattice expression for the pressure. The probability distribution function of the melting factor indicates departures from harmonic or Gaussian behavior in the lower wing. Nevertheless, the mean melting factor is shown to follow a simple MSD Debye-Waller factor dependence along both the melting and sublimation lines. This work combining HCM and MD simulations provides a comparison of the melting and sublimation lines of the LJ system, which could provide the foundations for a more unified statistical mechanical description of these two solid boundary lines.

Algebraic second virial coefficient of the Mie m - 6 intermolecular potential based on perturbation theory

We propose several simple algebraic approximations for the second virial coefficient of fluids whose molecules interact by a generic Mie m - 6 intermolecular pair potential. In line with a perturbation theory, the parametric equations are formulated as the sum of a contribution due to a reference part of the intermolecular potential and a perturbation. Thereby, the equations provide a convenient (low-density) starting point for developing equation-of-state models of fluids or for developing similar approximations for the virial coefficient of (polymeric-)chain fluids. The choice of Barker and Henderson [J. Chem. Phys. 47, 4714 (1967)] and Weeks, Chandler, and Andersen [Phys. Rev. Lett. 25, 149 (1970); J. Chem. Phys. 54, 5237 (1971); and Phys. Rev. A 4, 1597 (1971)] for the reference part of the potential is considered. Our analytic approximations correctly recover the virial coefficient of the inverse-power potential of exponent m in the high-temperature limit and provide accurate estimates of the temperatures for which the virial coefficient equals zero or takes on its maximum value. Our description of the reference contribution to the second virial coefficient follows from an exact mapping onto the second virial coefficient of hard spheres; we propose a simple algebraic equation for the corresponding effective diameter of the hard spheres, which correctly recovers the low- and high-temperature scaling and limits of the reference fluid's second virial coefficient.

On-the-Fly Second Virial Coefficients

A simple but approximate algorithm is described for computing second virial coefficients based on equilibrated molecular configurations that may be generated during any Monte Carlo or molecular dynamics simulation. The algorithm uses simple quadrature based on sampling every binary pair in the configuration and moving their center-center distances from zero to infinity. Comparisons are made in the literature results using more sophisticated sampling and integration for n-alkanes of ethane through n-dodecane. Accuracy is within the error bars determined by block averaging, and temperature effects can be inferred using a single configurational temperature, including perturbative virial coefficients. Predictably, the accuracy is best at the configurational temperature and when the configurational density is lowest. More notably, good accuracy is achieved from configurations at intermediate densities, and the trend at high density conveys valuable insight about conformational changes. The algorithm is simple enough to assign as a homework problem in an introductory molecular modeling course and reinforces the elementary knowledge of pairwise potentials among multisite molecules, numerical integration, and conformational averaging. The result is also quite valuable on its own merits, especially considering thermodynamic integration to compute phase equilibria. Additionally, the incidental data derived from the computation can provide simple but meaningful insights into the nature of multisite interactions, as demonstrated by relating the Mayer averaged potential to an effective Mie potential. Altogether, the argument is made that the computation of the second virial coefficient should be considered to be a routine metric of any molecular simulation, such as the radial distribution function, pressure, or energy.

Effect of the range of particle cohesion on the phase behavior and thermodynamic properties of fluids

Molecular simulations are performed for the (m + 1, m) potential to systematically investigate the effect of changing the range of particle cohesion on both vapor-liquid equilibria and thermodynamic properties of fluids. The results are reported for m = 4-11, which represent a progressive narrowing of the potential energy well. The conventional Lennard-Jones potential is used as a reference point for normal fluid behavior. Small values of m result in a broadening of the phase envelope compared with the Lennard-Jones potential, whereas a contraction is observed in other cases. The critical properties are reported, and a relationship between the critical temperature and the Boyle temperature is determined. The low values of the critical compressibility factor when m < 6 reflect the behavior observed for real fluids such as n-alkanes. The results for supercritical thermodynamic properties are much more varied. Properties such as pressure, potential energy, isochoric thermal pressure coefficient, and thermal expansion coefficient vary consistently with m, whereas other properties such as the Joule-Thomson coefficient exhibit much more nuanced behavior. Maximum and minimum values are reported for both the isochoric heat capacity and isothermal compressibility. A minimum in the speed of sound is also observed.

Vapor-liquid equilibria and cohesive r-4 interactions

The role of cohesive r^-4 interactions on the existence of a vapor phase and the formation of vapor-liquid equilibria is investigated by performing molecular simulations for the n-4 potential. The cohesive r^-4 interactions delay the emergence of a vapor phase until very high temperatures. The critical temperature is up to 5 times higher than normal fluids, as represented by the Lennard-Jones potential. The greatest overall influence on vapor-liquid equilibria is observed for the 5-4 potential, which is the lowest repulsive limit of the potential. Increasing n initially mitigates the influence of r^-4 interactions, but the moderating influence declines for n > 12. A relationship is reported between the critical temperature and the Boyle temperature, which allows the critical temperature to be determined for a given n value. The n-4 potential could provide valuable insight into the behavior of non-conventional materials with both very low vapor pressures at elevated temperatures and highly dipolar interactions.

Effective hardness of interaction from thermodynamics and viscosity in dilute gases

The hardness of the effective inverse power law (IPL) potential, which can be obtained from thermodynamics or collision integrals, can be used to demonstrate similarities between thermodynamic and transport properties. This link is investigated for systems of increasing complexity (i.e., the EXP, square-well, Lennard-Jones, and Stockmayer potentials; ab initio results for small molecules; and rigid linear chains of Lennard-Jones sites). These results show that while the two approaches do not yield precisely the same values of effective IPL exponent, their qualitative behavior is intriguingly similar, offering a new way of understanding the effective interactions between molecules, especially at high temperatures. In both approaches, the effective hardness is obtained from a double-logarithmic temperature derivative of an effective area.

Entropy Scaling of Viscosity - I: A Case Study of Propane

In this work, a broadly-applicable and simple approach for building high accuracy viscosity correlations is demonstrated for propane. The approach is based on the combination of a number of recent insights related to the use of residual entropy scaling, especially a new way of scaling the viscosity for consistency with the dilute-gas limit. With three adjustable parameters in the dense phase, the primary viscosity data for propane are predicted with a mean absolute relative deviation of 1.38%, and 95% of the primary data are predicted within a relative error band of less than 5%. The dimensionality of the dense-phase contribution is reduced from the conventional two dimensional approach (temperature and density) to a one-dimensional correlation with residual entropy as the independent variable. The simplicity of the model formulation ensures smooth extrapolation behavior (barring errors in the equation of state itself). The approach proposed here should be applicable to a wide range of chemical species. The supporting information includes the relevant data in tabular form and a Python implementation of the model.

Thermodynamic properties and anomalous behavior of double-Gaussian core model potential fluids

The structural, thermodynamic, and vapor-liquid equilibria properties of the double-Gaussian core model (DGM) potential are studied via molecular simulation. Results are presented for the pressure (p), potential energy (U), isochoric and isobaric heat capacities (C_{V,p}), isothermal compressibility (β_{T}), isochoric thermal pressure coefficient (γ_{V}), thermal expansion coefficient (α_{p}), speed of sound (ω_{0}), and the Joule-Thomson coefficient (μ_{JT}), which are compared with simulations for the Gaussian core model (GCM) potential. A feature of the simulations is that both the GCM and DGM potentials reproduce many of the anomalous properties of water, such as a maximum density, γ_{V}<0, maximum values for both α_{p} and β_{T}, and minimum values in both C_{p} and ω_{0}. The presence of attractive interaction enhances the anomalies and also yields some additional features such as a more structured vapor phase and Joule-Thomson inversion.

The Zero-Density Limit of the Residual Entropy Scaling of Transport Properties

The modified residual entropy scaling approach has been shown to be a successful means of scaling dense phase transport properties. In this work, we investigate the dilute-gas limit of this scaling. This limit is considered for model potentials and highly accurate results from calculations with ab initio pair potentials for small molecules. These results demonstrate that with this approach, the scaled transport properties of noble gases can be collapsed without any empirical parameters to nearly their mutual uncertainties and that the scaled transport properties of polyatomic molecules are qualitatively similar, and for sufficiently high temperatures they agree with "universal" values proposed by Rosenfeld in 1999. There are significant quantitative differences between the model potentials and real fluids in these scaled coordinates, but this study provides a thorough coverage of model fluids and simple real fluids, providing the basis for further study. In the supporting information we provide the collected calculations with ab initio pair potentials from the literature, as well as code in the Python language implementing all aspects of our analysis.