Gibbs Ensemble Calculations with an Equation of State: An Application to Vapor-Liquid Equilibria

Abstract

A new modification of the Gibbs ensemble Monte Carlo computer simulation method for fluid phase equilibria is described. The modification is based on a thermodynamic model for the vapor phase, and uses an equation of state to account for the weak interactions between the vapor phase molecules. Reductions in the computational time by 30–40% as compared to the original Gibbs ensemble method are obtained. The algorithm is applied to Lennard-Jones - (12,6) fluids and their mixtures and the results are in good agreement with results obtained from simulations using the full Gibbs ensemble method.     

Direct Evaluation of Vapour-Liquid Equilibria of Mixtures by Molecular Dynamics Using Gibbs-Duhem Integration

Abstract

We present an extension of the Gibbs-Duhem integration method that permits direct evaluation of vapour-liquid equilibria of mixtures by molecular dynamics. The Gibbs-Duhem integration combines the best elements of the Gibbs ensemble Monte Carlo technique and thermodynamic integration. Given conditions of coexistence of pure substances, simultaneous but independent molecular dynamics simulations of each phase at constant number of particles, constant pressure, constant temperature and constant fugacity fraction of species 2 are carried out in succession along coexistence lines. In each simulation, the coexistence pressure is adjusted to satisfy the Clapeyron-type equation. The Clapeyron-type equation is a first-order nonlinear differential equation that prescribes how the pressure must change with the fugacity fraction of species 2 to maintain coexistence at constant temperature. The Clapeyron-type equation is solved by the predictor-corrector method. Running averages of mole fraction and compressibility factor for the two phases are used to evaluate the right-hand side of the Clapeyron-type equation. The Gibbs-Duhem integration method is applied to three prototypes of binary mixtures of the two-centre Lennard-Jones fluid having various elongations. The starting points on the coexistence curve were taken from published data.     

Derivation of the Formula to Calculate the Chemical Potential Difference for the Components Involved in Identity Exchange Moves in Gibbs Ensemble Simulation of Mixtures

Since temperature and pressure are specified at the beginning of a simulation run in the Gibbs ensemble Monte Carlo (GEMC) method for mixtures, the condition of equilibrium is fulfilled through two sets of equalities in each phase: one set for the chemical potentials of the components of smaller molecules (calculated from transfer trial moves), and the other set for the differences between the chemical potentials of the components of larger molecules and the components of smaller molecules (calculated from identity exchange trial moves). The formula to calculate the former quantities is known since the time the GEMC method was proposed. However, the formula to calculate the latter quantities has recently been given in the literature without a formal derivation. In this work, a statistical-mechanical derivation of that formula is presented, within the framework of the canonical ensemble, and some justification is given for its extension to the Gibbs ensemble.     

Calculation of Vapour-Liquid Equilibria of Lennard-Jones Binary Systems by the Gibbs Ensemble Monte Carlo Simulation

Abstract

The Gibbs ensemble Monte Carlo simulation has been used to calculate vapour-liquid equilibria of a Lennard-Jones (LJ) binary mixture. The mixture studied is the LB-2-1 model which has been used in our previous calculations on PVT relation and density-dependent local composition. The P-x-y relation has been established at two different temperatures and used to determine vapour-liquid coexistence region in the PVTx space.     

Gibbs ensemble Monte Carlo simulations of binary vapour–liquid–liquid equilibrium: application to n-hexane–water and ethane–ethanol systems

Gibbs ensemble Monte Carlo (GEMC) simulations in the isochoric–isothermal (NVT) ensemble were used to simulate vapour–liquid–liquid equilibrium (VLLE) for binary n-hexane–water and ethane–ethanol mixtures. The GEMC simulation of binary VLLE data proved to be extremely difficult and that is probably the reason why the open literature is so sparse with simulations for these types of systems. The results presented in this paper are to our knowledge the first successful binary three-phase GEMC simulations of non-idealised fluid systems. This paper also shows that the isobaric–isothermal (NPT) ensemble is unsuitable for the simulation of phase equilibria of binary three-phase systems.     

Gibbs-Ensemble Molecular Dynamics: Liquid-Gas Equilibria for Lennard-Jones Spheres and n-Hexane

Abstract

We present a novel method to simulate phase equilibria in atomic and molecular systems. The method is a Molecular Dynamics version of the Gibbs-Ensemble Monte Carlo technique, which has been developed some years ago for the direct simulation of phase equilibria in fluid systems. The idea is to have two separate simulation boxes, which can exchange particles (or molecules) in a thermodynamically consistent fashion. Here we pres the derivation of the generalized equations of motion and discuss the relation of the resulting trajectory averages to the relevant ensemble. We test this Gibbs-Ensemble Molecular Dynamics algorithm by applying it to an atomic and a molecular system, i.e. to the liquid-gas coexistence in a Lennard-Jones fluid and in n-hexane. In both cases our results are in good accord with previous mean field and Gibbs-Ensemble Monte Carlo results as well as with the experimental data in the case of hexane. We also show that our Gibbs-Ensemble Molecular Dynamics algorithm like other Molecular Dynamics techniques can be used to study the dynamics of the system. Self-diffusion coefficients calculated with this method are in agreement with the result of conventional constant temperature Molecular Dynamics.     

A new kinetic Monte Carlo scheme with Gibbs ensemble to determine vapour–liquid equilibria

We present a new kinetic Monte Carlo scheme, as an alternative to the Gibbs ensemble Monte Carlo (GEMC) method, to determine vapour–liquid equilibria using a canonical ensemble in a system composed of two boxes. To illustrate the method, we have tested it with two systems: (1) argon over a range of temperatures from below the triple point to close to the critical point; (2) methane and ethane mixtures of various compositions at 180 K. The advantage of the new scheme is that chemical potentials of all components are accurately determined in both boxes. In particular, the chemical potential in the liquid box is determined much more accurately than with the Widom method employed in conventional GEMC simulations.     

Gibbs-Ensemble Molecular Dynamics: A New Method for Simulations Involving Particle Exchange

We discuss a novel simulation method suitable for simulating phenomena involving particle exchange. The method is a molecular dynamics version of the Gibbs-Ensemble Monte Carlo technique, which has been developed some years ago for the direct simulation of phase equilibria in fluid systems. The idea is to have two separate simulation boxes, which can exchange particles or molecules in a thermodynamically consistent fashion. We discuss the general idea of the Gibbs-Ensemble Molecular Dynamics technique and present examples for different simple atomic and molecular fluids. Specifically we will discuss Gibbs-Ensemble Molecular Dynamics simulations of gas-liquid and liquid-solid equilibria in Lennard-Jones systems and in hexane as well as an application of the method to adsorption.     

A Modified Gibbs Ensemble Method for Calculating Fluid Phase Equilibria

Abstract

A modification of the Gibbs ensemble Monte Carlo computer simulation method for fluid phase equilibrium is described. The modification, which is based on the assumption of a thermodynamic model for the vapor phase, reduces the computational time for the simulation as compared to the original Gibbs ensemble methods. Since the computational time is largely proportional to the number of particle-particle interactions, avoiding the direct simulation of the vapor phase typically leads to a thirty to forty percent reduction in computational time. For a pure Leonard-Jones-(12,6) fluid the results obtained at moderate reduced temperatures, T/T_c < 0.8, are in good agreement with the full Gibbs ensemble method.     

Some Recent Developments in Computational Chemistry

Abstract

Some recent developments in the use of computational methods to predict the properties of condensed phases are discussed; the use of Gibbs ensemble Monte Carlo to predict the phase equilibria of bulk phases, the use of molecular dynamics to elucidate Atomic Force Microscopy experiments on organic films, and the use of combined Monte Carlo/molecular dynamics techniques to enable the direct prediction of particle fluxes along pressure gradients in model microporous materials.     

Atomistic simulations of liquid–liquid coexistence in confinement: comparison of thermodynamics and kinetics with bulk

We review a few simulation methods and results related to the structure and non-equilibrium dynamics in the coexistence region of immiscible symmetric binary fluids, in bulk as well as under confinement, with special emphasis on the latter. Monte Carlo methods to estimate interfacial tensions for flat and curved interfaces have been discussed. The latter, combined with a thermodynamic integration technique, provides contact angles for coexisting fluids attached to the wall. For such three-phase coexistence, results for the line tension are also presented. For the kinetics of phase separation, various mechanisms and corresponding theoretical expectations have been discussed. A comparative picture between the domain growth in bulk and confinement (including thin-film and semi-infinite geometry) has been presented from molecular dynamics simulations. Applications of finite-size scaling technique have been discussed in both equilibrium and non-equilibrium contexts.     

Properties of Confined Square-well Fluids using the Gibbs Ensemble Simulation Technique

In this work we have used the extension of the Gibbs ensemble simulation technique to inhomogeneous fluids [Panagiotopoulos, A.Z. (1987) "Adsorption and capillary condensation of fluid in cylindrical pores by Monte Carlo simulation in the Gibbs ensemble", Mol. Phys. , 62 (3), 701-719], which has been applied to adsorption phenomena of confined fluids. Fluid molecules are described by spherical particles interacting via a square-well potential. The fluid is confined in two types of walls: symmetrical (two hard walls) and non-symmetrical (one square-well wall and one hard wall). In order to analyze the behavior of the confined fluid by varying the potential parameters, we evaluated the bulk and confined densities, the internal energies and the density profiles for different supercritical temperatures. A variety of adsorption profiles can be obtained by using this model. The simulation data reported here complements the available simulation data for this system and can be useful in the development of inhomogeneous fluid theories. Since the square-well parameters can be related to real molecules this system can also be used to understand real adsorption systems.     

Theoretical Calculation of the Liquid—Vapor Coexistence Curve of Water,Chloroform and Methanol with the Cavity-Biased Monte Carlo Method in the Gibbs Ensemble

Abstract

The Gibbs ensemble computer simulation method of Panagiotopoulos is combined by the cavity-biased sampling technique used previously in the grand-canonical ensemble. The combined technique is applied to the determination of the liquid—vapor coexistence curve of the Lennard—Jones fluid as a test case, two water models (SPC and TIP4P) as well as methanol and chloroform, both described with the OPLS model. The application of the virial-based sampling technique, used earlier in the isobaric ensemble is also discussed.     

Vapour–liquid phase equilibria of simple fluids confined in patterned slit pores

We present the influence of surface heterogeneity on the vapour–liquid phase behaviour of square-well fluids in slit pores using grand-canonical transition-matrix Monte Carlo simulations along with the histogram-reweighting method. Properties such as phase coexistence envelopes, critical properties and local density profiles of the confined SW fluid are reported for chemically and physically patterned slit surfaces. It is observed that in the chemically patterned pores, fluid–fluid and surface attraction parameters along with the width of attractive and inert stripes play fundamentally different roles in the phase coexistence and critical properties. On the other hand, pillar gap and height significantly affect the vapour–liquid equilibria in the physically patterned slit pores. We also present the effect of chemically and physically patterned slit surfaces on the spreading pressure.     

Simulation-based fitting of protein-protein interaction potentials to SAXS experiments

We present a new method for computing interaction potentials of solvated proteins directly from small-angle x-ray scattering data. An ensemble of proteins is modeled by Monte Carlo or molecular dynamics simulation. The global x-ray scattering of the whole model ensemble is then computed at each snapshot of the simulation, and averaged to obtain the x-ray scattering intensity. Finally, the interaction potential parameters are adjusted by an optimization algorithm, and the procedure is iterated until the best agreement between simulation and experiment is obtained. This new approach obviates the need for approximations that must be made in simplified analytical models. We apply the method to lambda repressor fragment 6-85 and fyn-SH3. With the increased availability of fast computer clusters, Monte Carlo and molecular dynamics analysis using residue-level or even atomistic potentials may soon become feasible.     

Application of kinetic Monte Carlo method to the vapour–liquid equilibria of associating fluids and their mixtures

Canonical kinetic Monte Carlo (C-kMC) simulations have been carried out to assess their feasibility and potential for calculating the vapour–liquid equilibria of various pure components with increasingly strong electrostatic interactions (carbon dioxide, methanol, ammonia and water) over a wide range of temperatures and for methanol/water mixtures at 298 K. The simulation results show that C-kMC is successful as a method for studying phase equilibria and thermodynamic properties. For all the examples investigated, the performance of the C-kMC method is at least as good as that of the conventional Monte Carlo (MC) methods and is efficient at low temperature where these fail. It also provides a route that is superior to the Widom method for the calculation of chemical potential. We recommend this method for this purpose and as an alternative to conventional MC for simulations of strongly associating fluids and at low temperatures.     

Phase Equilibria of Quadrupolar Fluids by Simulation in the Gibbs Ensemble

Abstract

Vapour-liquid phase diagrams for pure fluids and mixtures of molecules with Lennard-Jones plus quadrupole-quadrupole interaction potentials were determined by Monte Carlo simulation in the Gibbs ensemble [1]. This is the first reported application of the method to molecular fluids. We have demonstrated that the Gibbs method works reliably for strongly interacting molecular fluids at liquid densities. Pure fluid calculations were performed for reduced quadrupole strengths, Q* = Q/(εσ⁵)^1/2 equal to 1 and √2, typical of molecules like C₂H₂ and C₂H₄. It was found that the critical temperature of the quadrupolar fluid increased rapidly with increasing quadrupolar strength, in good agreement with previous computer simulation and theoretical results. A single mixture with components characterized by identical Lennard-Jones parameters and Q*₁ = + 1, Q*₂ = - 1 was studied at three temperatures. A negative azeotrope was observed at the lowest temperature studied, as seen experimentally in the CO₂/C₂H₂ mixture. The perturbation theory calculations are in good agreement with the simulation results for all properties except coexisting liquid densities. The results illustrate some of the strengths and limitations of perturbation theories based on the Padé approximant for the free energy of polar fluids.     

Simulating Fluid–Solid Equilibrium with the Gibbs Ensemble

The Gibbs ensemble is employed to simulate fluid–solid equilibrium for a shifted-force Lennard-Jones system. This is achieved by generating an accurate canonical Helmholtz free-energy model of the (defect-free) solid phase. This free-energy model is easily generated, with accuracy limited only by finite-size effects, by a single isothermal–isobaric simulation at a pressure not too far from coexistence for which the chemical potential is known. We choose to illustrate this method at the known triple-point because the chemical potential is easily calculated from the coexisting gas. Alternatively, our methods can be used to locate fluid–solid coexistence and the triple-point of pure systems if the chemical potential of the solid phase can be efficiently calculated at a pressure not too far from the actual coexistence pressure. Efficient calculation of the chemical potential of solids would also enable the Gibbs ensemble simulation of bulk solid–solid equilibrium and the grand-canonical ensemble simulation of bulk solids.     

An Extended Gibbs Ensemble

Abstract

A general extended Gibbs ensemble, obtained by augmenting the standard Gibbs ensemble by intermediate states in the spirit of the scaled particle method of Nezbeda and Kolafa [Molec. Simul., 5, 391 (1991)], is introduced. The intermediate states span the states with different number of particles in the simulation boxes and facilitate the transfer of particles even in such complex systems as e.g., mixtures of very different components, systems of flexible polymeric molecules, or systems at very high densities. A general formulation of the ensemble is given and two implementations are considered in detail. The method is very general and is exemplified by studying the fluid-fluid coexistence in a dense binary mixture of the hard-sphere and square-well fluids. It is found that its efficiency is about by factor three greater in comparison with the standard Gibbs ensemble simulations.     

The phase diagram of the Lennard-Jones fluid using temperature dependent interaction parameters

The forces of interaction between argon atoms can be described by the Lennard-Jones potential model. It is hypothesised that the use of temperature dependent interaction parameters, instead of using temperature independent interaction parameters, may lead to improvement in the prediction of the vapour–liquid coexistence curve. Published second virial coefficient data were used to fit a simple two-parameter temperature dependent model for the collision diameter and well depth. Vapour–liquid coexistence curve for argon was simulated in the NVT Gibbs ensemble Monte Carlo technique. The simulations were carried out using each of the temperature independent and temperature dependent parameters in the temperature range: 110–148 K. The critical temperature and density were determined using the Ising-scaling model. The results using temperature dependent parameters produce, overall, a more accurate phase diagram compared to the diagram generated using temperature independent interaction parameters. The root mean square deviation is reduced by 42.1% using temperature dependent interaction parameters. Also, there was no significant difference between the results obtained using temperature dependent interaction parameters and the highly accurate and computationally demanding phase diagrams based on three body contributions.