Lorentz-Berthelot Rules Research Articles

Nonlocal viscosity kernel of mixtures

In this Brief Report we investigate the multiscale hydrodynamical response of a liquid as a function of mixture composition. This is done via a series of molecular dynamics simulations in which the wave-vector-dependent viscosity kernel is computed for three mixtures, each with 7-15 different compositions. We observe that the viscosity kernel is dependent on composition for simple atomic mixtures for all the wave vectors studied here; however, for a molecular mixture the kernel is independent of composition for large wave vectors. The deviation from ideal mixing is also studied. Here it is shown that the Lorentz-Berthelot interaction rule follows ideal mixing surprisingly well for a large range of wave vectors, whereas for both the Kob-Andersen and molecular mixtures large deviations are found. Furthermore, for the molecular system the deviation is wave-vector dependent such that there exists a characteristic correlation length scale at which the ideal mixing goes from underestimating to overestimating the viscosity.

On the behaviour of solutions of xenon in liquid cycloalkanes: Solubility of xenon in cyclopentane

The solubility of xenon in liquid cyclopentane has been studied experimentally and theoretically. Measurements of the solubility of xenon in liquid cyclopentane are reported as a function of temperature from 254.60K to 313.66K. The imprecision of the experimental data is less than 0.3%. The thermodynamic functions of solvation of xenon in cyclopentane, such as the standard Gibbs energy, enthalpy, entropy and heat capacity of solvation, have been calculated from the temperature dependence of Henry's law coefficients. The results provide further information about the differences between the xenon+cycloalkanes and the xenon+n-alkane interactions. In particular, interaction enthalpies between xenon and CH2 groups in n-alkanes and cycloalkanes have been estimated and compared. Using a version of the soft-SAFT approach developed to model cyclic molecules, we were able to reproduce the experimental solubility for xenon in cyclopentane using simple Lorentz-Berthelot rules to describe the unlike interaction.

Determining the three-phase coexistence line in methane hydrates using computer simulations

Molecular dynamics simulations have been performed to estimate the three-phase (solid hydrate-liquid water-gaseous methane) coexistence line for the water-methane binary mixture. The temperature at which the three phases are in equilibrium was determined for three different pressures, namely, 40, 100, and 400 bar by using direct coexistence simulations. In the simulations water was described by using either TIP4P, TIP4P/2005, or TIP4P/Ice models and methane was described as simple Lennard-Jones interaction site. Lorentz-Berthelot combining rules were used to obtain the parameters of the cross interactions. For the TIP4P/2005 model positive deviations from the energetic Lorentz-Berthelot rule were also considered to indirectly account for the polarization of methane when introduced in liquid water. To locate the three-phase coexistence point, two different global compositions were used, which yielded (to within statistical uncertainty) the same predictions for the three-phase coexistence temperatures, although with a somewhat different time evolution. The three-phase coexistence temperatures obtained at different pressures when using the TIP4P/Ice model of water were in agreement with the experimental results. The main reason for this is that the TIP4P/Ice model reproduces the melting point of ice I(h).

Water–methanol mixtures with non-Lorentz–Berthelot combining rules: A feasibility study

To examine a possibility to predict qualitatively correctly the properties of water–alcohol mixtures using pure fluid non-polarizable interaction potential models, a family of non-Lorentz–Berthelot combing rules was used in simulations on the water–methanol mixtures defined by the standard TIP4P (water) and OPLS (methanol) interaction models. A certain strategy to locate the region of the cross parameters yielding, potentially, a minimum in the partial molar volume of methanol has been adopted to reduce complexity of the problem and CPU time. It is found that a small deviation from the Lorentz rule along with a larger change in the Berthelot cross energy gives rise to a concavity of the excess volume at low methanol concentrations and hence to a minimum in the partial molar volume which is otherwise missing if the Lorentz–Berthelot rules are used. Nonetheless, it does not seem that non-Lorentz–Berthelot rules alone will be able to bring the results to a full quantitative agreement with experiment.

Coarse-grained models for fluids and their mixtures: Comparison of Monte Carlo studies of their phase behavior with perturbation theory and experiment

The prediction of the equation of state and the phase behavior of simple fluids (noble gases, carbon dioxide, benzene, methane, and short alkane chains) and their mixtures by Monte Carlo computer simulation and analytic approximations based on thermodynamic perturbation theory is discussed. Molecules are described by coarse grained models, where either the whole molecule (carbon dioxide, benzene, and methane) or a group of a few successive CH(2) groups (in the case of alkanes) are lumped into an effective point particle. Interactions among these point particles are fitted by Lennard-Jones (LJ) potentials such that the vapor-liquid critical point of the fluid is reproduced in agreement with experiment; in the case of quadrupolar molecules a quadrupole-quadrupole interaction is included. These models are shown to provide a satisfactory description of the liquid-vapor phase diagram of these pure fluids. Investigations of mixtures, using the Lorentz-Berthelot (LB) combining rule, also produce satisfactory results if compared with experiment, while in some previous attempts (in which polar solvents were modeled without explicitly taking into account quadrupolar interaction), strong violations of the LB rules were required. For this reason, the present investigation is a step towards predictive modeling of polar mixtures at low computational cost. In many cases Monte Carlo simulations of such models (employing the grand-canonical ensemble together with reweighting techniques, successive umbrella sampling, and finite size scaling) yield accurate results in very good agreement with experimental data. Simulation results are quantitatively compared to an analytical approximation for the equation of state of the same model, which is computationally much more efficient, and some systematic discrepancies are discussed. These very simple coarse-grained models of small molecules developed here should be useful, e.g., for simulations of polymer solutions with such molecules as solvent.

Non-Lorentz–Berthelot Lennard-Jones mixtures: A systematic study

Binary mixtures of two identical Lennard-Jones fluids with non-Lorentz–Berthelot combining rules have been simulated over the entire concentration range at three state points in order to examine the effect of cross interactions on the mixing properties, excess volumes and enthalpies, and partial molar volumes, and on the structure. Various combinations of deviations, in both the energy and size cross parameters, from the Lorentz–Berthelot rule, have been considered and the results analyzed using a recently proposed method based on the Tikhonov regularization. It is found that the most important is the size cross parameter and that by varying the combining rules a variety of qualitatively different behavior can be produced including, e.g., a minimum in the partial molar volumes observed otherwise only in complex mixtures of associating fluids; occurrence of this minimum seems to be associated with changes in the second coordination shell as witnessed by the pair correlation function.

The effects of deviations from Lorentz–Berthelot rules on the properties of a simple mixture

Generally, the parameters in the interaction potential between like molecules in a mixture can be determined in a relatively straightforward manner from the properties of the pure components. However, the determination of the parameters in the interaction potential between the unlike pairs in the mixtures is more difficult. As a result, these parameters are usually estimated from averages of the like parameters. The most common recipes are the Lorentz–Berthelot mixing rules, where the energy and molecular size parameters are presumed to be geometric and arithmetic averages, respectively. There have been some studies of the consequences of deviations from the energy rule but almost no studies of the consequences of deviations from the size rule. Here, we study the effects of deviations from both rules on the radial distribution functions of a simple mixture. We find from simulations that, for this mixture, the effect of deviations from the energy rule on the radial distribution function are rather small but that the effect of deviations from the size rule can be significant and are interesting.

Prediction of binary intermolecular potential parameters for use in modelling fluid mixtures

We show how to extend the Hudson–McCoubrey combining rules, which were derived assuming the Lennard-Jones potential, for use with intermolecular potentials that are not of Lennard-Jones form. In particular, we illustrate the method by deriving combining rules for use with the square-well intermolecular potential. We show how these combining rules may be applied for equations of state such as the statistical associating fluid theory (SAFT), and that they perform at least as well as, and usually far better than, the standard Lorentz–Berthelot rules for all mixtures of simple molecules that we have considered. Liquid–liquid-equilibrium as well as vapour–liquid-equilibrium data are demonstrated to be important in judging the quality of combining rules. A methodology for the inclusion of the effect of polar interactions in our combining rules is suggested, however the success for these systems is mixed. The theory provides an excellent description of the thermodynamic properties of water vapour + methane and rationalises recent, apparently contradictory, models of the mixture of carbon dioxide + water , however the approach is less successful for mixtures such as carbon monoxide + hydrogen sulphide. This indicates that further work is still required for reliable prediction of binary-interaction parameters in mixtures of polar molecules.

The Effect of Cross Interactions on Mixing Properties: Non-Lorentz-Berthelot Lennard-Jones Mixtures

Binary mixtures of two identical Lennard-Jones fluids with non-Lorentz-Berthelot combining rules have been simulated at ambient-like conditions in order to examine the effect of cross interactions on mixing properties - excess volumes and enthalpies. Various combinations of deviations of both energy and size cross parameters from the Lorentz-Berthelot rules have been considered. Whereas consequences of the deviations for excess volume are rather straightforward, a variety of behavior types is found for excess enthalpy, including an excess function with three extrema.

A potential model for methane in water describing correctly the solubility of the gas and the properties of the methane hydrate

We have obtained the excess chemical potential of methane in water, over a broad range of temperatures, from computer simulation. The methane molecules are described as simple Lennard-Jones interaction sites, while water is modeled by the recently proposed TIP4P/2005 model. We have observed that the experimental values of the chemical potential are not reproduced when using the Lorentz-Berthelot combining rules. However, we also noticed that the deviation is systematic, suggesting that this may be corrected. In fact, by introducing positive deviations from the energetic Lorentz-Berthelot rule to account indirectly for the polarization methane-water energy, we are able to describe accurately the excess chemical potential of methane in water. Thus, by using a model capable of describing accurately the density of pure water in a wide range of temperatures and by deviating from the Lorentz-Berthelot combining rules, it is possible to reproduce the properties of methane in water at infinite dilution. In addition, we have applied this methane-water potential to the study of the solid methane hydrate structure, commonly denoted as sI, and find that the model describes the experimental value of the unit cell of the hydrate with an error of about 0.2%. Moreover, we have considered the effect of the amount of methane contained in the hydrate. In doing so, we determine that the presence of methane increases slightly the value of the unit cell and decreases slightly the compressibility of the structure. We also note that the presence of methane increases greatly the range of pressures where the sI hydrate is mechanically stable.

Molecular-dynamics study of anomalous volumetric behavior of water-benzene mixtures in the vicinity of the critical region

Molecular-dynamics simulations of water-benzene mixtures at 573 K and pressures in the 85-140 bars range have been performed to examine local structure and dynamics of the mixtures, which exhibit anomalously large volume expansion on mixing as recently found by in situ near-infrared measurements. Fractional charges for a simple-point-charge-type potential of water were adjusted so as to reproduce liquid densities and the gas-to-liquid transition pressure of neat water at 573 K. A Lennard-Jones-type potential for benzene was used and the Lorentz-Berthelot combination rule was applied to the water-benzene interaction. Simulations with a N-P-T ensemble of 800-molecule system have been performed and the results reproduce well the anomalous volumetric behavior of the mixtures with the mole fraction of benzene in the 0.3-0.8 range. Pair distribution functions, coordination numbers, and self-diffusion coefficients for the mixtures are calculated, and it is suggested that the local structure around water molecules undergoes drastic change by dissolution of benzene in the vicinity of the critical region, but that around benzene molecules seems to be understood as that of ordinary liquid mixtures.

Effects of surface heterogeneity on the adsorption of nitrogen on graphitized thermal carbon black

In this paper, we study the surface heterogeneity and the surface mediation on the intermolecular potential energy for nitrogen adsorption on graphitized thermal carbon black (GTCB). The surface heterogeneity is modeled as the random distribution of “effective” carbonyl functional groups on the graphite surface. The molecular parameters and the discrete charges of this carbonyl group are taken from Jorgensen, et al. (J. Am. Chem. Soc., (1984) 106, 6638) while those for nitrogen (dispersive parameters and discrete charges) are taken from Murthy et al. (Mol. Phys., (1983) 50, 531) in our Grand Canonical Monte Carlo (GCMC) simulation. The solid surface mediation in the reduction of intermolecular potential energy between two fluid molecules was taken from a recent work by Do et al. (Langmuir, (2004) 20, 7623). Our simulation results accounting for the surface heterogeneity and surface mediation on intermolecular potential energy were compared with the experimental data of nitrogen at 77 and 90 K. The solid-fluid dispersive parameters are determined from the Lorentz-Berthelot (LB) rule. The fraction of the graphite surface covered with carbonyl functional groups was then derived from the consideration of the Henry constant, and for the data of Kruk et al. (Langmuir, (1999) 15, 1435) we have found that 1% of their GTCB surface is covered with “effective” carbonyl functional groups. The damping constant, due to surface mediation, was determined from the consideration of the portion of the adsorption isotherm where the first layer is being completed, and it was found to take a value of 0.0075. With these parameters, we have found that the GCMC simulation results describe the data over the complete range of pressure substantially better than any other MC models in the literature. The implication of this work is demonstrated with local adsorption isotherms of 10 and 20 A slit pores. One was obtained without allowance for surface mediation, while the other correctly accounts for these factors. The two local isotherms differ substantially, and the implication is that if we used incorrect local isotherms (i.e. without the surface mediation) the pore size distribution would be incorrectly derived.

Molecular Dynamics Study of the Lennard−Jones Fluid Viscosity: Application to Real Fluids

A predictive scheme of viscosity for pure fluids and mixtures of simple molecules is presented. First, using molecular dynamics data from the literature and also from our own study, a representative correlation of the viscosity of a Lennard-Jones fluid is developed for a wide range of thermodynamic states. Second, a corresponding states scheme is proposed which allows the transposition of the previous results to real fluids. For some simple molecules, this scheme induces deviations lower than 5% in conditions covering gas, liquid, and supercritical states. For larger molecules, the results are poorer but can be strongly improved by fitting the atomic diameter. Then, it is shown for simple binary and multicomponent mixtures that, by using merely a van der Waals one-fluid approximation and the Lorentz-Berthelot rules, results are as good as for pure fluids. Finally, the limitations of such a scheme are shown when applied on the methane + toluene asymmetric mixture.

Grand canonical Monte Carlo simulations of phase equilibria of pure silicon tetrachloride and its binary mixture with carbon dioxide

Grand canonical histogram-reweighting Monte Carlo simulations were used to obtain the phase behaviour of pure silicon tetrachloride and its binary mixture with carbon dioxide. Two new potential models for pure silicon tetrachloride were developed and parametrized to the vapour-liquid coexistence properties. The first model, with one exponential-6 site and fixed electrostatic charges on atoms, does not adequately reproduce the experimental phase behaviour due to its inability to represent orientational anisotropy in the liquid phase. The second potential model, with five exponential-6 sites for the repulsive and dispersive interactions plus partial charges, accurately reproduces experimental saturated liquid and vapour densities as well as vapour pressures and the second virial coefficient for pure silicon tetrachloride. This model was used in simulations of the phase behaviour of the binary mixture carbon dioxide-silicon tetrachloride. Two sets of combining rules (Lorentz-Berthelot and Kong [1973, J. chem. Phys., 59, 2464]) were used to obtain unlike-pair potential parameters. For the binary system, the predicted phase diagram is in good agreement with experiment when the Kong combining rules are used. The Lorentz-Berthelot rules significantly overestimate the solubility of carbon dioxide in silicon tetrachloride.

Static structure factor and thermodynamic properties of a binary Yukawa mixture

We use the solution of the Ornstein Zernike equation in the mean spherical approximation to find the static structure factor for a hard spheres Yukawa fluid. The thermodynamic and the structure properties of this fluid are given in terms of an accumulative parameter Γ, which satisfies a polynomial equation of degree n⩾4. This parameter is obtained mumerically by an iterative method. We study binary mixtures with a factored interaction for which the classical Lorentz–Berthelot rules are satisfied. Our result for the static structure factor and thermodynamics properties are in good agreement with the computer simulations and former numerical solutions.

Predicting the solubility of xenon in n-hexane and n-perfluorohexane: a simulation and theoretical study

The solubility of xenon in n-hexane and n-perfluorohexane has been studied using both molecular simulation and a version of the SAFT approach (SAFT-VR). The calculations were performed close to the saturation line of each solvent, between 200 K and 450 K, which exceeds the smaller temperature range where experimental data are available in the literature. Molecular dynamics simulations, associated with Widom's test particle insertion method, were used to calculate the residual chemical potential of xenon in n-hexane and n-perfluorohexane and the corresponding Henry's law coefficients. The simulation results overestimate the solubility of xenon in both solvents when simple geometric combining rules are used, but are in good agreement if a binary interaction parameter is included. With the SAFT-VR approach we are able to reproduce the experimental solubility for xenon in n-hexane, using simple Lorentz-Berthelot rules to describe the unlike interaction. In the case of n-perfluorohexane as a solvent, a binary interaction parameter was introduced, taken from previous work on (xe + C2F6) mixtures. Overall, good agreement is obtained between the simulation, theoretical and experimental data.

Analysis of Adsorption Data of Graphitized Thermal Carbon Black with a DFT-Lattice Gas Theory

In this paper we analyzed the adsorption of gases and vapors on graphitised thermal carbon black by using a modified DFT-lattice theory, in which we assume that the behavior of the first layer in the adsorption film is different from those of second and higher layers. The effects of various parameters on the topology of the adsorption isotherm were first investigated, and the model was then applied in the analysis of adsorption data of numerous substances on carbon black. We have found that the first layer in the adsorption film behaves differently from the second and higher layers in such a way that the adsorbate-adsorbate interaction energy in the first layer is less than that of second and higher layers, and the same is observed for the partition function. Furthermore, the adsorbate-adsorbate and adsorbate-adsorbent interaction energies obtained from the fitting are consistently lower than the corresponding values obtained from the viscosity data and calculated from the Lorentz-Berthelot rule, respectively.

Thermodynamic properties of a polydisperse system.

We use the virial theorem to derive a closed analytic form of the Helmholtz free energy for a polydisperse system of sticky hard spheres (SHS) within the mean spherical model (MSM). To this end we calculate the free energy of the MSM for an N-component mixture of SHS via the virial route and apply to it-after imposing a Lorentz-Berthelot type rule on the interactions-the stochastic (i.e., polydisperse) limit. The resulting excess free energy of this polydisperse system is of the truncatable moment free energy format. We also discuss the compressibility and the energy routes.

Inadequacy of the Lorentz-Berthelot combining rules for accurate predictions of equilibrium properties by molecular simulation

Though molecular beam experiments have revealed deficiencies in the Lorentz-Berthelot combining rules, these rules are still used widely to parametrize effective pair potential models or to calculate the thermodynamic properties of mixtures. Gibbs ensemble Monte Carlo and isothermal isobaric Monte Carlo simulations were used to compute the liquid-vapour phase equilibria and the liquid properties of binary mixtures of rare gases modelled by effective pair potentials. Three sets of simple combining rules were tested in this work: the commonly used Lorentz-Berthelot rules, the Kong rules (Kong, J., 1973, J. chem. Phys., 59, 2464) and the Waldman-Hagler rules (Waldman, M., and Hagler, A. T., 1993, J. comput. Chem., 14, 1077). These three sets of rules do not require any additional parameter. It is shown that: (1) the choice of a set of combining rules has a significant effect on the thermodynamic properties, (2) using the Lorentz-Berthelot rules yields significant deviations from experiment and (3) the Kong rules provide a much better description of the mixture properties both for coexistence curves and liquid properties. We therefore recommend the use of the Kong rules instead of the Lorentz-Berthelot when parametrizing potential models.

Effective potential approach to bulk thermodynamic properties and surface tension of molecular fluids: II. Binary mixtures of n-alkanes and miscible gas

A representation of alkanes and alkane/gas mixtures is proposed in terms of simple fluids interacting through pairwise square well potentials parameterised by the range of attractive forces. Using the model, vapour–liquid equilibria and interfacial tension (IFT) are studied both for pure alkane fluids (C4, C5, C6, C8, C10 and C14) and their high pressure binary mixtures formed with methane, nitrogen or carbon dioxide. Potential parameters for the isolated components are first determined from the critical parameters, Pitzer’s acentric factor and one surface tension datum at a chosen temperature. Cross-term interactions between species are obtained from modified Lorentz–Berthelot rules which provide good fits to the vapour–liquid coexistence densities as functions of mixture composition. At a general state point, the interfacial tension is predicted using generalised van der Waals (gvdW) theory, which is a mean field free energy density functional theory. This semiempirical procedure typically produces very satisfactory agreement with experimental interfacial tension data.