Pure Lennard-Jones Fluid Research Articles

Symbolic regression development of empirical equations for diffusion in Lennard-Jones fluids.

Symbolic regression (SR) with a multi-gene genetic program has been used to elucidate new empirical equations describing diffusion in Lennard-Jones (LJ) fluids. Examples include equations to predict self-diffusion in pure LJ fluids and equations describing the finite-size correction for self-diffusion in binary LJ fluids. The performance of the SR-obtained equations was compared to that of both the existing empirical equations in the literature and to the results from artificial neural net (ANN) models recently reported. It is found that the SR equations have improved predictive performance in comparison to the existing empirical equations, even though employing a smaller number of adjustable parameters, but show an overall reduced performance in comparison to more extensive ANNs.

Entropy Scaling for Viscosity of Pure Lennard-Jones Fluids and Their Binary Mixtures

In this work, entropy scaling approaches for viscosity of pure Lennard-Jones (LJ) fluids and their mixtures have been investigated. To do so, we have employed reliable viscosity database available in literature for the pure LJ fluids, and performed the molecular dynamics simulation to generate viscosity database over a wide range of thermodynamic condition for the LJ mixture fluids. It has shown that for the pure LJ fluid, the entropy scaling approaches using the macroscopic properties for the reduction of viscosity yield noticeably better collapse of data than the one using the zero-density viscosity in the dense fluid region. In addition, we have developed viscosity correlations based on these approaches. It has been obtained that the correlations of macroscopic properties approaches predict the pure LJ fluid viscosity with average absolute deviation of around 4% more coming from the low-density states, whereas it is of around 8.50% more coming from the dense states for the other one. Finally, the viscosity correlations have been applied to the LJ mixtures. Interestingly, the correlations of macroscopic properties approaches are able to provide good estimations for all mixtures studied. However, it deteriorates for the viscosity of dense mixtures when the other employed.

Breakdown of the Stokes-Einstein relation in pure Lennard-Jones fluids: From gas to liquid via supercritical states.

We have examined the conditions under which the breakdown of the Stokes-Einstein (SE) relation occurs in pure Lennard-Jones (LJ) fluids over a wide range of temperatures and packing fractions beyond the critical point. To this end, the temperature and packing-fraction dependence of the self-diffusion coefficient, D, and the shear viscosity, η_{sv}, were evaluated for Xe using molecular dynamics calculations with the Green-Kubo formula. The results showed good agreement with the experimental values. The breakdown was determined in light oftheSE equation which we have recently derived for pure LJ liquids: Dη_{sv}=(k_{B}T/2π)(N/V)^{1/3}, where k_{B} is the Boltzmann constant, T is the temperature, and N is the particle number included in the system volume V. We have found that the breakdown occurs in the lower range of the packing fraction, η<0.2, and derived the SE relation in its broken form as Dη_{sv}=0.007(1-η)^{-5}η^{-4/3}(k_{B}T/ε)^{n}k_{B}T(N/V)^{1/3}, where n increases from 0 up to 1 with the decreasing η. The equation clearly shows that the breakdown mainly occurs because the packing-fraction dependence does not cancel out between D and η_{sv} in this region, which is attributed to the gaseous behavior in the packing-fraction dependence of the shear viscosity under a constant number density. In addition, the gaseous behavior in the temperature dependence of the shear viscosity also partially causes the breakdown.

Molecular simulation of the phase behavior of fluids and fluid mixtures using the synthetic method

A new molecular simulation procedure is reported for determining the phase behavior of fluids and fluid mixtures, which closely follows the experimental synthetic method. The simulation procedure can be implemented using Monte Calro or molecular dynamics in either the microcanonical or canonical statistical ensembles. Microcanonical molecular dynamics simulations are reported for the phase behavior of both the pure Lennard-Jones fluid and a Lennard-Jones mixture. The vapor pressures for the pure fluid are in good agreement with Monte Carlo Gibbs ensemble and Gibbs-Duhem calculations. The Lennard-Jones mixture is composed of equal size particles, with dissimilar energy parameters (ε(2)/ε(1) = 1/2, ε(12)/ε(1) = 1/√2). The binary Lennard-Jones mixture exhibits liquid-liquid equilibria at high pressures and the simulation procedure allows us to estimate the coordinates of the high-pressure branch of the critical curve.

Collective excitations in supercritical fluids: Analytical and molecular dynamics study of “positive” and “negative” dispersion

The approach of generalized collective modes is applied to the study of dispersion curves of collective excitations along isothermal lines of supercritical pure Lennard-Jones fluid. An effect of structural relaxation and other nonhydrodynamic relaxation processes on the dispersion law is discussed. A simple analytical expression for the dispersion law in the long-wavelength region of acoustic excitations is obtained within a three-variable viscoelastic model of generalized hydrodynamics. It is shown that the deviation from the linear dependence in the long-wavelength region can be either "positive" or "negative" depending on the ratio between the high-frequency (elastic) and isothermal speed of sound. An effect of thermal fluctuations on positive and negative dispersion is estimated from the analytical solution of a five-variable thermoviscoelastic model that generalizes the results of the viscoelastic treatment. Numerical results are reported for a Lennard-Jones supercritical fluid along two isothermal lines T(*)=1.71,4.78 with different densities and discussed along the theoretical expressions derived.

Molecular simulations of droplet evaporation by heat transfer

Droplet evaporation by heat transfer is investigated by molecular dynamics simulations for a pure Lennard-Jones fluid. Two different initial conditions are treated: (1) a droplet surrounded by its vapor in equilibrium, (2) a cold droplet surrounded by warm vapor. In both cases heat is transferred from a heat bath. Results are the numbers of droplet molecules N d and density, drift velocity, and temperature profiles as functions of time. For the small droplets considered N d depends on the definition of a droplet molecule. The density profiles as function of time show a transition from a droplet with liquid–vapor interface to a cluster of interfacial type and finally to the gas state. The temperature at a given time is nearly constant within the droplets or clusters but strong gradients occur in the gas. In case of evaporation of a cold droplet surrounded by warm vapor we observed initially cooling down of the droplet corresponding to pressure jump evaporation and thereafter slower evaporation because of lower initial state vapor density.

Change of Free Energy during Adsorption of a Molecule

A change of the Helmholtz free energy ΔA takes place when a molecule becomes adsorbed out of a bulk solution. The purpose of the paper is to explore routes for the calculation of ΔA from molecular simulations. According to a derivation from the partition function, ΔA is equal to the change of the potential of mean force (PMF) which can be calculated via two different routes. First, the PMF can be obtained as a path integral over the mean force on the solute molecule which can be calculated via a simulation in which the solute molecule is fixed at a given position. Second, the PMF is related to the logarithm of the local density of the solute which can in principle be obtained via a simulation in which the solute molecule moves freely. Simulations are performed for a pure Lennard-Jones fluid and several dilute Lennard-Jones mixtures in contact with a plane Lennard-Jones wall. For the pure Lennard-Jones fluid, both simulation routes yield practically the same result for ΔA. In the case of the dilute solutions, however, simulations do not yield directly reliable local densities of the solute in those cases, in which there is not one very pronounced minimum in ΔA. On the other hand, the path integral over the mean force could be calculated always with reasonable accuracy to yield ΔA. Moreover, via the latter route also the local density of the dilute component can be obtained.

Grand Equilibrium: vapour-liquid equilibria by a new molecular simulation method

A new molecular simulation method for the calculation of vapour-liquid equilibria of mixtures is presented. In this method, the independent thermodynamic variables are temperature and liquid composition. In the first step, one isobaric isothermal simulation for the liquid phase is performed, in which the chemical potentials of all components and their derivatives with respect to the pressure, i.e. the partial molar volumes, are calculated. From these results, first-order Taylor series expansions for the chemical potentials as functions of the pressure μ i (p) at constant liquid composition are determined. This information is needed, as the specified pressure in the liquid will generally not be equal to the equilibrium pressure, which has to be found in the course of a vapour simulation. In the second step, one pseudo grand canonical simulation for the vapour phase is performed, where the chemical potentials are set according to the instantaneous pressure p v using the previously determined function μ i (p v). In this way, results for the vapour pressure and vapour composition are achieved which are consistent for the given temperature and liquid composition. The new method is applied to the pure Lennard-Jones fluid, a binary and a ternary mixture of Lennard-Jones spheres, and shows very good agreement with corresponding data from the literature.

On the Role of Density Fluctuations in the Equation of State of a Simple Fluid

We address the wellknown problems intorduced into the theory of fluids by density fluctuations in the form of van der Waals loops and nonclassical critical phenomena. A clean separation of long and short range density fluctuations is achieved by use of cell-constrained models which display well-defined van der Waals loops and classical behaviour around the critical point. For a pure Lennard-Jones fluid with occupancy restricted to 1 or 8 particles per cell, the phase diagram is determined by Monte Carlo simulation. By considering the deviations from the normal simulations without cell constraint, the effects of longer range density fluctuations are exposed. The system size dependence of the van der Waals loops present in all simulations of fluids is analyzed in terms of the GvdW free energy density functional theory, which is formuiated on the basis of the cell concept. The loops are found to gradually disappear either with greater cel occupancy or increasing total particle number in the simulation box.

Renormalization of the indirect correlation function to extract the bridge function of simple fluids

The standard integral equation approach is used to extract the bridge function and other correlation functions of simple fluids. To do this, we firstly use the hybridized mean spherical approximation (HMSA), which involves a renormalization of the indirect correlation function. Secondly, we propose a new division scheme for the pair potential founded on the merging process of a test particle into the system. Thirdly, we take advantage of the thermodynamic consistent condition put forward in the HMSA to optimize the depth of the long-range part of the pair potential inside the core, making the procedure autonomous and state dependent. The method is tested on the pure Lennard-Jones fluid in supercritical conditions. Very good agreement is obtained for structural quantities and thermodynamic properties as compared to the Monte Carlo (MC) results for the bridge function and to the molecular dynamics (MD) results for the thermodynamic properties.

On the calculation of the chemical potential using the particle deletion scheme

A new formulation is presented for the calculation of the chemical potential from molecular simulation based on a test particle removal (inverse Widom) scheme. The new formulation introduces an intermediate stage in the calculation where the test particle to be removed is substituted by a hard particle. Chemical potential calculations at high densities from NVT and NPT Monte Carlo simulations, using a fast analytical algorithm for the computation of volume accessible to the hard core particle, are presented for the pure Lennard-Jones fluid and found to be in excellent agreement with predictions from an accurate equation of state and from simulations using the test particle insertion scheme (Widom insertion scheme). Binary mixture simulations are presented also and the new scheme is applied in the Gibbs ensemble. In all cases where the system is highly dense, the new inverse Widom scheme affords a significant reduction in CPU time compared with the widely used test particle insertion method of Widom. In addition, the new scheme is compared with a recently proposed scheme based on staged insertion.

Monte Carlo simulations of binary Lennard-Jones mixtures : A test of the van der Waals one-fluid model

Monte Carlo simulations in the canonical ensemble have been performed in the liquid and supercritical regions of a binary Lennard–Jones mixture with differences in size parameters of 6.4% and energy parameters of 37%. The results are compared with a recent fundamental equation of state employing the van der Waals one-fluid model and new simulation data at the corresponding state conditions of the pure Lennard–Jones fluid. The van der Waals one-fluid model describes the mixture properties well at high densities, while at low densities the predicted internal energies and isochoric heat capacities are too low.

Molecular dynamics simulation of the liquid–vapor interface: The Lennard-Jones fluid

In this work we present new molecular dynamics simulation results for the liquid–vapor interface of the pure Lennard-Jones fluid. Our aims were further investigations on the simulation setup and the simulation parameters to obtain reliable data for the coexisting densities as well as for the surface tension. The influence of the cut-off distance to the interfacial properties is investigated and long-range corrections to both the dynamics and the surface tension are applied. It is found that the saturated liquid densities from the surface simulations agree with those from the NpT+test particle method within 1% for sufficiently large simulation boxes; the saturated vapor densities agree within 4%. In order to obtain reliable values for the surface tension, cut-off radii of at least 5 molecular diameters supplemented by a tail correction are required.

Density profiles at liquid–vapor and liquid–liquid interfaces: An integral equation study

The structure of liquid–vapor and liquid–liquid interfaces in Lennard-Jones (LJ) fluids and mixtures is studied using integral equations. To obtain density distributions at interfaces between coexisting fluid phases we solve the Lovett–Mou–Buff–Wertheim equation. In this equation we approximate the direct correlation functions of the inhomogeneous fluid via interpolation between the direct correlation functions of the bulk phases. In the homogeneous bulk phases the system of the Ornstein–Zernike equation with the reference-hypernetted-chain closure is solved to obtain the direct correlation functions at coexisting densities. Density distributions and other interfacial properties are studied for a liquid–vapor interface in a pure LJ fluid, in an Ar–Kr mixture and for a liquid–liquid interface between two immiscible LJ fluids. The results are in good agreement with simulations and other theories. At low temperatures the liquid–vapor and liquid–liquid density profiles exhibit oscillating structures with periods near the diameters of the LJ spheres. Being quite weak at liquid–vapor interfaces these oscillations become very pronounced at a liquid–liquid interface between immiscible fluids.

Thermodynamic properties of ternary fluid mixtures from the improved perturbation theory and van der Waals one-fluid theory—II. Real mixtures

In this paper, a previously developed first-order hard sphere perturbation theory of a ternary fluid mixture (Fotouh and Shukla, 1996, Chem. Engng Sci. 51, 4923–4931) is applied to predict excess properties of a real ternary fluid mixture using a theoretically consistent combination rule. We consider a ternary fluid mixture N 2/Ar/CH 4, for which experimental data for excess properties are available and which can be modeled adequately using the simple Lennard-Jones potential function. Comparisons of theoretical results with experimental data show a very good performance of perturbation theory for different pressures and compositions of the mixture. Also, we checked the accuracy of a recent form of van der Waals one-fluid theory based on the most accurate equation of state for pure Lennard-Jones fluids. Van der Waals one-fluid theory is found to be equally reliable in describing excess properties of such a moderately nonideal mixture.

Integral equation theory for Lennard-Jones fluids: The bridge function and applications to pure fluids and mixtures

The pure Lennard-Jones fluid and various binary mixtures of Lennard-Jones fluids are studied by both molecular dynamics simulation and with a new integral equation which is based on that proposed by Duh and Haymet recently [J. Chem. Phys. 103, 2625 (1995)]. The structural and thermodynamic properties calculated from this integral equation show excellent agreement with simulations for both pure fluids and mixtures under the conditions which we have studied. For mixtures, the effect of deviations from the Lorentz-Berthelot (LB) mixing rules for the interaction parameters between unlike species is studied. Positive deviations from the nonadditivity of the molecular cores leads to an entropy driven tendency for the species to separate. This tendency persists even in the presence of a deviation from the LB rule for the energy parameter which enhances the attraction of the unlike species. On the other hand, in the case of negative deviations from nonadditivity, the tendency for association may be either energy or entropy driven, depending on the size ratio.

Molecular dynamics simulations of liquid binary mixtures: Partial properties of mixing and transport coefficients

Equilibrium molecular dynamics (MD) computer simulations have been used to determine the transport coefficients of model Ar–Kr, Ar–CH4, and CH4–N2 mixtures at a large number of liquid and dense fluid state points for which experimental data are available. Both species in each mixture are represented by single-site Lennard-Jones pair potentials with Lorentz–Berthelot mixing rules for the unlike interactions. Green–Kubo formulas and mean-square displacements are used to calculate the self-diffusion coefficients for each species and mutual-diffusion coefficients. The shear and bulk moduli and viscosities, thermal conductivity and the thermal diffusion coefficient are determined by Green–Kubo in the [NVE] and [NVT] ensembles. The thermotransport coefficients employ a rigorous definition for the heat flux, which includes the partial enthalpy of the two species, used for the first time to compute these transport coefficients. The partial volumes and enthalpies, and chemical potentials for each species, were obtained from separate computations carried out at constant pressure in the [NPT] ensemble. The simulated density at fixed pressure, shear viscosity, and thermal conductivity of the Ar–Kr mixtures are in excellent agreement with experiment. However, the bulk viscosity shows a significant qualitative difference in the composition and temperature dependence (the latter even in the single component fluids). Agreement with experiment deteriorates as the quasispherical molecules progressively depart from spherical shape. For Ar–CH4 the density (obtained using [NPT] MD) is in good agreement with experiment, whereas the shear viscosity is in progressively poorer agreement with increasing methane content. This is caused by an overestimation of the methane viscosity (∼50% higher than experiment for pure methane). For CH4–N2 there are substantial differences between the simulated quantities and experiment. The average simulated densities are ∼5% higher than experiment over a wide temperature and composition range. The simulated shear viscosity is typically ∼10%–50% higher than experiment. The enthalpy, chemical potential, and shear modulus of the model mixtures are reproduced well using a van der Waals one-fluid model. In this, there is little to distinguish between Enskog’s mixing rules for the model from the simpler mole fraction weighted (‘‘Kay’s’’) mixing rules. The shear viscosity and thermal conductivity are also predicted reasonably well with either model, recognizing the greater statistical uncertainty in the transport coefficients when compared with the thermodynamic averages, even for the pure Lennard-Jones fluid.

How good is conformal solutions theory for phase equilibrium predictions?: Gibbs ensemble simulations of binary Lennard-Jones mixtures

It is generally believed that conformal solutions theory is valid only for mixtures with components having small differences in energy and size intermolecular potential parameters. To test this assumption, we have used the Gibbs ensemble Monte Carlo simulation technique to calculate phase diagrams of binary Lennard-Jones mixtures obeying the Lorentz—Berthelot combining rules for size parameter ratios equal to 0.5, 1.0 and 1.5 and for energy parameter ratios equal to 0.50, 0.66, 0.75 and 1.00. We generally find good agreement between simulation and theoretical predictions based on an equation of state for the pure Lennard-Jones fluid and the van der Waals one-fluid approximation. Our results establish the validity of conformal solutions theory for simple mixtures of much greater degree of asymmetry than previous investigations.

Molecular dynamics studies of polar/nonpolar fluid mixtures. I. Mixtures of Lennard-Jones and Stockmayer fluids

The results of canonical ensemble molecular dynamics calculations of mixtures of Lennard–Jones and Stockmayer fluids are reported. To study solely the influence of the polarity, the Lennard-Jones parameters were identical for both components. The excess mixing properties show a strong asymmetry with respect to composition for large dipolar strength. The free energy of mixing is obtained through a thermodynamic integration procedure. The results strongly suggest that, for reduced dipolar strengths μ2&amp;gt;3.15, demixing occurs into a phase rich in polar component and an almost pure Lennard-Jones fluid. It is shown that perturbation theory yields fairly accurate results for the dipolar energy and free energy of the mixture. For the free energy of mixing, qualitatively correct results are obtained. The structure and orientational correlation functions of the mixture are discussed. The radial distribution function for pairs of polar molecules show a marked increase in local ordering with dipolar strength for low concentrations of the polar component, indicating that strong clustering of polar molecules occurs at these concentrations. The orientational order is also seen to increase very strongly with dipole moment at these concentrations. The pair correlation function for pairs of Lennard-Jones atoms shows little dependence on dipolar strength of Stockmayer molecules at these concentrations. The distribution function for pairs of unlike molecules reflects the increasingly dissimilar character of these molecules as the dipolar strength increases. For large concentrations of Stockmayer molecules, the opposite effect is observed, albeit less pronounced, in that the pair-correlation function for Lennard-Jones atoms shows an increase in local ordering as μ increases, whereas the radial distribution function for Stockmayer pairs remains relatively unaffected with increasing μ. These results are interpreted in terms of a frustation model. Results are given for the variation of the dielectric constant of the mixture with composition and dipolar strength.

A new analytic formula for molecular radial distribution function in fluid and fluid mixtures

In this paper a new semiempirical analytic formula for molecular radial distribution function for both pure fluid and fluid mixture is proposed. It is a damping oscillation curve. The molecular radial distribution function data for pure Lennard-Jones fluid calculated by the formula are compared with computer simulation data and perturbation theories. The excess mixing enthalpy for several binary liquid mixtures calculated by the new model are compared with those from other solution theories and experimental data.