Simulation studies on mixtures of dipolar with nonpolar linear molecules II. A mixing rule for the dipolar contribution to the Helmholtz energy

In recent equations of state an explicit expression for the dipolar contribution to the Helmholtz energy F_D is used. If such equations are applied to mixtures the problem of mixing rules for F_D arises. As a solution, a “one-fluid dipole moment μ_x” is introduced which is based on perturbation theory. The accuracy of the approach is tested by simulation results for mixtures of dipolar with nonpolar linear molecules. In addition to the simulation runs from Part I (Müller et al. (1994)), results from 97 new NVT simulations are presented. Comparisons of pressures, internal energies, dipole energies and excess free energies from simulations with those obtained from the equation for F_D in combination with the one-fluid dipole moment μ_x shows good to excellent agreement.     

Molecular nitrogen-N2 properties: the intermolecular potential and the equation of state

Quantum mechanical (QM) high precision calculations were used to determine N(2)-N(2) intermolecular interaction potential. Using QM numerical data the anisotropic potential energy surface was obtained for all orientations of the pair of the nitrogen molecules in the rotation invariant form. The new N(2)-N(2) potential is in reasonably good agreement with the scaled potential obtained by van der Avoird et al. using the results of Hartree-Fock calculations [J. Chem. Phys. 84, 1629 (1986)]. The molecular dynamics (MD) of the N(2) molecules has been used to determine nitrogen equation of state. The classical motion of N(2) molecules was integrated in rigid rotor approximation, i.e., it accounted only translational and rotational degrees of freedom. Fincham [Mol. Simul. 11, 79 (1993)] algorithm was shown to be superior in terms of precision and energy stability to other algorithms, including Singer [Mol. Phys. 33, 1757 (1977)], fifth order predictor-corrector, or Runge-Kutta, and was therefore used in the MD modeling of the nitrogen pressure [S. Krukowski and P. Strak, J. Chem. Phys. 124, 134501 (2006)]. Nitrogen equation of state at pressures up to 30 GPa (300 kbars) and temperatures from the room temperature to 2000 K was obtained using MD simulation results. Results of MD simulations are in very good agreement (the error below 1%) with the experimental data on nitrogen equation of state at pressures below 1 GPa (10 kbars) for temperatures below 1800 K [R. T. Jacobsen et al., J. Phys. Chem. Ref. Data 15, 735 (1986)]. For higher temperatures, the deviation is slightly larger, about 2.5% which still is a very good agreement. The slightly larger difference may be attributed to the vibrational motion not accounted explicitly by rigid rotor approximation, which may be especially important at high temperatures. These results allow to obtain reliable equation of state of nitrogen for pressures up to 30 GPa (300 kbars), i.e., close to molecular nitrogen stability limit, determined by Nellis et al. [Phys. Rev. Lett. 53, 1661 (1984)].     

Application of a renormalization-group treatment to the statistical associating fluid theory for potentials of variable range (SAFT-VR)

An accurate prediction of phase behavior at conditions far and close to criticality cannot be accomplished by mean-field based theories that do not incorporate long-range density fluctuations. A treatment based on renormalization-group (RG) theory as developed by White and co-workers has proven to be very successful in improving the predictions of the critical region with different equations of state. The basis of the method is an iterative procedure to account for contributions to the free energy of density fluctuations of increasing wavelengths. The RG method has been combined with a number of versions of the statistical associating fluid theory (SAFT), by implementing White's earliest ideas with the improvements of Prausnitz and co-workers. Typically, this treatment involves two adjustable parameters: a cutoff wavelength L for density fluctuations and an average gradient of the wavelet function Φ. In this work, the SAFT-VR (variable range) equation of state is extended with a similar crossover treatment which, however, follows closely the most recent improvements introduced by White. The interpretation of White's latter developments allows us to establish a straightforward method which enables Φ to be evaluated; only the cutoff wavelength L then needs to be adjusted. The approach used here begins with an initial free energy incorporating only contributions from short-wavelength fluctuations, which are treated locally. The contribution from long-wavelength fluctuations is incorporated through an iterative procedure based on attractive interactions which incorporate the structure of the fluid following the ideas of perturbation theories and using a mapping that allows integration of the radial distribution function. Good agreement close and far from the critical region is obtained using a unique fitted parameter L that can be easily related to the range of the potential. In this way the thermodynamic properties of a square-well (SW) fluid are given by the same number of independent intermolecular model parameters as in the classical equation. Far from the critical region the approach provides the correct limiting behavior reducing to the classical equation (SAFT-VR). In the critical region the β critical exponent is calculated and is found to take values close to the universal value. In SAFT-VR the free energy of an associating chain fluid is obtained following the thermodynamic perturbation theory of Wertheim from the knowledge of the free energy and radial distribution function of a reference monomer fluid. By determining L for SW fluids of varying well width a unique equation of state is obtained for chain and associating systems without further adjustment of critical parameters. We use computer simulation data of the phase behavior of chain and associating SW fluids to test the accuracy of the new equation.     

The van der Waals one-fluid model for viscosity in Lennard–Jones fluids: Influence of size and energy parameters

This work aims to estimate the limitations of the van der Waals one-fluid (vdW1) approximation in the prediction of the viscosity of Lennard–Jones (LJ) mixtures. To do so, non-equilibrium molecular dynamics simulations have been performed. Results on mixtures have been compared to those of their equivalent pure fluids (in the sense of the vdW1 model). Several systems (146 configurations) are studied, which are composed of binary and ternary mixtures in various thermodynamic states and for different combining rules. In a first step, deviations induced separately by LJ molecular parameters (size or energy) have been analyzed. It is shown that the vdW1 model is well designed for the energy parameter in every configuration. On the contrary, for the size parameter, deviations induced by this one-fluid approach are shown to be large in dense state and low temperature systems. In a second step, the coupling effects of the LJ size and energy parameters with the mass are studied. It appears that an accurate one-fluid approximation for viscosity should involve a coupling between the mass and the size parameters in its formulation (which is not the case with the vdW1 model) but not between the mass and the energy.     

Equation of state of nitrogen (N2) at high pressures and high temperatures: molecular dynamics simulation

Nitrogen equation of state at pressures up to 30 GPa (300 kbars) and temperatures above 800 K was studied by molecular dynamics (MD) simulations. The dynamics of the N(2) molecules is treated in hard rotor approximation, i.e., it accounts both translational and rotational degrees of freedom. The rotational motion of the N(2) molecule is treated assuming constant moment of inertia of the nitrogen molecule. The new MD program fully accounts anisotropic molecular nitrogen interaction. The N(2)-N(2) interaction potential has been derived by van der Avoird et al. [J. Chem. Phys. 84, 1629 (1986)] using the results of high precision Hartree-Fock ab initio quantum mechanical calculations. The potential, fully accounts rotational symmetry of the N(2)-N(2) system, by employing 6-j Wigner symbols, i.e., preserving full rotational symmetry of the system. Various numerical algorithms were tested, in order to achieve the energy preservation during the simulation. It has been demonstrated that the standard Verlet algorithm was not preserving the energy for the standard MD time step, equal to 5x10(-16) s. Runge-Kutta fourth order method was able to preserve the energy within 10(-4) relative error, but it requires calculation of the force four times for each time step and therefore it is highly inefficient. A predictor-corrector method of the fifth order (PC5) was found to be efficient and precise and was therefore adopted for the simulation of the molecular nitrogen properties at high pressure. Singer and Fincham algorithms were tested and were found to be as precise as PC5 algorithm and they were also used in the simulation of the equation of state. Results of MD simulations are in very good agreement with the experimental data on nitrogen equation of state at pressures below 1 GPa (10 kbars). For higher pressures, up to 30 GPa (300 kbars), i.e., close to molecular nitrogen stability limit, determined by Nellis et al. [Phys. Rev. Lett. 85, 1262 (1984)], the obtained numerical results provide new data of the experimentally unexplored region. These data were formulated in the analytical form of pressure-density-temperature equation of state.     

Autocorrelation functions of a fluid of molecules interacting through steeply repulsive and attractive forces

A new method is presented for an extension of Enskog's approximation for the evaluation of the autocorrelation functions of a fluid, and this approach is used to evaluate these functions when the interaction between the molecules includes both steeply repulsive and steeply attractive forces. Consequently the correlation functions depend upon the temperature in a nontrivial way. As an example, the method is applied to calculate the velocity and force autocorrelation functions of a fluid when the molecules interact through the specific potential, V(r)=4epsilon[(sigma/r)2n-(sigma/r)n] when the parameter n is large. There is a relationship between this model and the "sticky sphere" one which is exploited in the theoretical computations. The results obtained from the theory are compared with molecular dynamics simulation for n=72 and 144 and for a range of temperatures from T=epsilon/kB down to epsilon/3kB. The two approaches agree very well for a range of state points, especially at short times. At later times the theory predicts a more oscillatory behavior than the simulation especially at very low reduced temperatures.     

Equation of state and liquid-vapor equilibria of one- and two-Yukawa hard-sphere chain fluids: theory and simulation

The accuracy of several theories for the thermodynamic properties of the Yukawa hard-sphere chain fluid are studied. In particular, we consider the polymer mean spherical approximation (PMSA), the dimer version of thermodynamic perturbation theory (TPTD), and the statistical associating fluid theory for potentials of variable attractive range (SAFT-VR). Since the original version of SAFT-VR for Yukawa fluids is restricted to the case of one-Yukawa tail, we have extended SAFT-VR to treat chain fluids with two-Yukawa tails. The predictions of these theories are compared with Monte Carlo (MC) simulation data for the pressure and phase behavior of the chain fluid of different length with one- and two-Yukawa tails. We find that overall the PMSA and TPTD give more accurate predictions than SAFT-VR, and that the PMSA is slightly more accurate than TPTD.     

Modeling the phase behavior of H2S+n-alkane binary mixtures using the SAFT-VR+D approach

A statistical associating fluid theory for potential of variable range has been recently developed to model dipolar fluids (SAFT-VR+D) [Zhao and McCabe, J. Chem. Phys. 2006, 125, 104504]. The SAFT-VR+D equation explicitly accounts for dipolar interactions and their effect on the thermodynamics and structure of a fluid by using the generalized mean spherical approximation (GMSA) to describe a reference fluid of dipolar square-well segments. In this work, we apply the SAFT-VR+D approach to real mixtures of dipolar fluids. In particular, we examine the high-pressure phase diagram of hydrogen sulfide+n-alkane binary mixtures. Hydrogen sulfide is modeled as an associating spherical molecule with four off-center sites to mimic hydrogen bonding and an embedded dipole moment (micro) to describe the polarity of H2S. The n-alkane molecules are modeled as spherical segments tangentially bonded together to form chains of length m, as in the original SAFT-VR approach. By using simple Lorentz-Berthelot combining rules, the theoretical predictions from the SAFT-VR+D equation are found to be in excellent overall agreement with experimental data. In particular, the theory is able to accurately describe the different types of phase behavior observed for these mixtures as the molecular weight of the alkane is varied: type III phase behavior, according to the scheme of classification by Scott and Konynenburg, for the H2S+methane system, type IIA (with the presence of azeotropy) for the H2S+ethane and+propane mixtures; and type I phase behavior for mixtures of H2S and longer n-alkanes up to n-decane. The theory is also able to predict in a qualitative manner the solubility of hydrogen sulfide in heavy n-alkanes.     

A crossover SAFT-VR equation of state for pure fluids: preliminary results for light hydrocarbons

SAFT is perhaps the most versatile, fundamentally, based engineering equation of state in use today. However, in common with all analytic equations of state, SAFT exhibits classical behavior in the critical region rather than the non-analytical, singular behavior seen in real fluids. Recently, so-called crossover equations of state have been developed which solve this shortcoming by incorporating the scaling laws valid asymptotically close to the critical point while reducing to the original classical equation of state far from the critical point. We have combined the SAFT-VR equation of state with an analytical crossover technique to obtain the SAFT-VRX equation of state. The SAFT-VRX approach combines the accurate low temperature behavior of SAFT-VR with a precise representation of the critical region. Preliminary results are presented for hydrocarbon systems which illustrate the accuracy of the SAFT-VRX approach over the entire fluid phase region.     

Coupling microscopic and mesoscopic scales to simulate chemical equilibrium between a nanometric carbon cluster and detonation products fluid

This paper presents a new method to obtain chemical equilibrium properties of detonation products mixtures including a solid carbon phase. In this work, the solid phase is modelled through a mesoparticle immersed in the fluid, such that the heterogeneous character of the mixture is explicitly taken into account. Inner properties of the clusters are taken from an equation of state obtained in a previous work, and interaction potential between the nanocluster and the fluid particles is derived from all-atoms simulations using the LCBOPII potential (Long range Carbon Bond Order Potential II). It appears that differences in chemical equilibrium results obtained with this method and the "composite ensemble method" (A. Hervouet et al., J. Phys. Chem. B, 2008, 112.), where fluid and solid phases are considered as non-interacting, are not significant, underlining the fact that considering the inhomogeneity of such system is crucial.     

Predicting adsorption isotherms using a two-dimensional statistical associating fluid theory

A molecular thermodynamics approach is developed in order to describe the adsorption of fluids on solid surfaces. The new theory is based on the statistical associating fluid theory for potentials of variable range [A. Gil-Villegas et al., J. Chem. Phys. 106, 4168 (1997)] and uses a quasi-two-dimensional approximation to describe the properties of adsorbed fluids. The theory is tested against Gibbs ensemble Monte Carlo simulations and excellent agreement with the theoretical predictions is achieved. Additionally the authors use the new approach to describe the adsorption isotherms for nitrogen and methane on dry activated carbon.     

Prediction of a structural transition in the hard disk fluid

Starting from the second equilibrium equation in the BBGKY hierarchy under the Kirkwood superposition closure, we implement a new method for studying the asymptotic decay of correlations in the hard disk fluid in the high density regime. From our analysis and complementary numerical studies, we find that exponentially damped oscillations can occur only up to a packing fraction η(?)～0.718, a value that is in substantial agreement with the packing fraction, η～0.723, believed to characterize the transition from the ordered solid phase to a dense fluid phase, as inferred from Mak's Monte Carlo simulations [Phys. Rev. E 73, 065104 (2006)]. Next, we show that the same method of analysis predicts that the exponential damping of oscillations in the hard sphere fluid becomes impossible when λ=4nπσ(3)[1+H(1)]≥34.81, where H(1) is the contact value of the correlation function, n is the number density, and σ is the sphere diameter in exact agreement with the condition, λ≥34.8, which is first reported in a numerical study of the Kirkwood equation by Kirkwood et al. [J. Chem. Phys. 18, 1040 (1950)]. Finally, we show that our method confirms the absence of any structural transition in hard rods for the entire range of densities below close packing.     

Solid-liquid equilibrium using the SAFT-VR equation of state: Solubility of naphthalene and acetic acid in binary mixtures and calculation of phase diagrams

A solid-liquid equilibrium (SLE) thermodynamic model based on the SAFT-VR equation of state (EOS) is presented. The model allows for the calculation of solid-liquid phase equilibria in binary mixtures at atmospheric pressure. The fluid (liquid) phase is treated with the SAFT-VR approach, where molecules are modelled as associating chains of tangentially bonded spherical segments interacting via square-well potentials of variable range. The equilibrium between the liquid and solid phase is treated following a standard thermodynamic method that requires the experimental temperature and enthalpy of fusion of the solute. The model is used to calculate the solubilities of naphthalene and acetic acid in common associating and non-associating organic solvents and to determine the solid-liquid phase behaviour of binary mixtures with simple eutectics. The SAFT-VR pure component model parameters are determined by comparison to experimental vapour pressure and saturated liquid density data with the choice of association models according to the nature of the molecule; in addition, an unlike adjustable parameter (k_ij) is used to model the solutions. The solubility data of naphthalene and acetic acid in both associating and non-associating solvents are reproduced essentially within the accuracy of the experimental measurements. The phase boundaries and the position of the eutectic points in the binary mixtures considered are, in most cases, reproduced with the accuracy commensurate with the industrial applications. Overall, the results presented show that the SAFT-VR EOS can be used with confidence for the prediction of the SLE of binary systems at atmospheric pressure.