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.     

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.     

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 Determination of Fluid Phase Equilibria by Simulation in the Gibbs Ensemble: A Review

This paper provides an extensive review of the literature on the Gibbs ensemble Monte Carlo method for direct determination of phase coexistence in fluids. The Gibbs ensemble technique is based on performing a simulation in two distinct regions in a way that ensures that the conditions of phase coexistence are satisfied in a statistical sense. Contrary to most other available techniques for this purpose, such as thermodynamic integration, grand canonical Monte Carlo or Widom test particle insertions, the Gibbs ensemble technique involves only a single simulation per coexistence point. A significant body of literature now exists on the method, its theoretical foundations, and proposed modifications for efficient determination of equilibria involving dense fluids and complex intermolecular potentials. Some practical aspects of Gibbs ensemble simulation are also discussed in this review. Applications of the technique to date range from studies of simple model potentials (for example Lennard–Jones, square-well or Yukawa fluids) to calculations of equilibria in mixtures with components described by realistic potentials. We conclude by discussing the limitations of the technique and potential future applications.     

Monte Carlo simulations of vapour–liquid phase equilibrium and microstructure for the system containing azeotropes

Abstract

Phase equilibrium data of the mixtures including alcohols, esters and organic acids are of first interest particularly to design and optimise biodiesel production and reactive distillation processes. In this work, vapour–liquid phase equilibrium of these systems was simulated at low pressure using Gibbs ensemble Monte Carlo method. All Lennard–Jones parameters of pseudo-atoms involved in the systems were derived from previous parametrisations of TraPPE-UA force field. The Fourier coefficients of dihedrals encountered in ethyl acetate molecule have been obtained from the quantum calculations. Using this force field, temperature-composition diagrams are well reproduced for ethyl acetate + ethanol, ethyl acetate + methanol at 70.00 kPa and ethyl acetate + acetic acid mixtures at 77.33 kPa. The transferability of this force field to mixtures in these systems is noticeable. Analysis of the microstructure for the ethyl acetate + ethanol and ethyl acetate + acetic acid mixtures was presented. We found that the hydrogen bond networks consist of autoassociation and cross-association and autoassociation occupies the main position as compared with cross association in the ethyl acetate + ethanol mixture. OC_HAc–H_HAc and OC_EtOAc–H_HAc hydrogen bond interactions play a significant role in the phase behaviours or structures of ethyl acetate + acetic acid mixture.     

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.     

A Monte Carlo simulation study to predict the solubility of H2S in ionic liquids with 1-butyl-3-methylimidazolium ([C4mim+]) cation and tetrafluoroborate ([BF4−]), hexaflorophosphate ([PF6−]) and bis(trifluoromethanesulfonyl)amide ([Tf2N−]) anions

The solubilities of H₂S in ionic liquids, 1-butyl-3-methylimidazolium tetrafluoroborate ([C₄mim⁺][BF₄^?]), 1-butyl-3-methylimidazolium hexaflorophosphate ([C₄mim⁺][PF₆^?]) and -butyl-3-methylimidazolium tetrafluoroborate bis(trifluoromethanesulphonyl)amide ([C₄mim⁺][Tf₂N^?]) are predicted using isothermal–isobaric Gibbs ensemble Monte Carlo method (NPT-GEMC) at temperatures ranging from 333 to 453 K and pressure up to 20 bar. The low pressure points (up to 3 bar) of the absorption isotherms are fitted to a straight line to get a rough estimation of the Henry’s law constants. The van’t Hoff relationship is used to estimate the partial molar enthalpy of the absorption. The obtained results are in a good agreement with Jou and Mather [18], and Jalili et al. [13]. When comparing the solubility in ILs, it is found that H₂S solubility is highest for [C₄mim⁺][Tf₂N^?], followed by [C₄mim⁺][PF₆^?]. The lowest solubility is observed in [C₄mim⁺][BF₄^?]. The highest solubility in [C₄mim⁺][Tf₂N^?] is consistent with Jalili et al. [13]. However, their results indicate slightly higher solubility in [C₄mim⁺][BF₄^?] than in [C₄mim⁺][PF₆^?], and do not agree with the predictions. Upon absorption, the molar volumes of the mixtures decrease linearly, showing only small changes in volume. The effect of H₂S absorption on ILs is further studied by calculating the radial distribution functions between the ions. The results indicate that the solute molecules accommodate themselves in the cavities without significantly disturbing the ionic arrangement of the ions, similar to CO₂ absorption in ILs. The spatial distribution functions show similar spatial distribution for H₂S around cation in all of the studied ILs, whereas the distribution around anion depends on the shape and flexibility of the anion. The mechanism of H₂S absorption is studied by computing the van der Waals (VDW) and electrostatic (ELEC) energies. It is observed that the solubility of H₂S in the studied ILs is primarily controlled by VDW interaction. When comparing the interaction of H₂S with the ions, it is found that solute molecules interact with cations mainly due to VDW interaction. Both VDW and ELEC energies contribute in the interaction between H₂S and anions.     

Modelling Gas Adsorption in Slit-Pores Using Monte Carlo Simulation

Abstract

We discuss the use of Monte Carlo simulation to model the equilibrium adsorption of gases in slit pores. Databases of adsorption isotherms have been calculated for nitrogen, carbon-monoxide, methane and carbon-dioxide for a range of pressures, pore widths and temperatures. We discuss the implications of these results for materials characterisation procedures based on gas adsorption data.     

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.     

A Monte Carlo scheme based on mid-density in a hysteresis loop to determine equilibrium phase transition

A new and simple method to determine equilibrium phase transition in adsorption systems exhibiting a hysteresis loop is presented as an alternative to methods such as multiple histogram reweighting, gauge cell method and thermodynamic integration. This method is based on the NVT-grand canonical Monte Carlo mid-density scheme to determine the coexistence chemical potential and coexistence densities of an adsorption system. We illustrate this new scheme with argon and methane adsorption in a number of model solids having slit and cylindrical pores. This method does not have a strong basis on thermodynamic ground, but it does provide a simple heuristic approach that is simpler to understand physically.     

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.     

Optimized intermolecular potential for nitriles based on Anisotropic United Atoms model

An extension of the anisotropic united atoms intermolecular potential model is proposed for nitriles. The electrostatic part of the intermolecular potential is calculated using atomic charges obtained by a simple Mulliken population analysis. The repulsion-dispersion interaction parameters for methyl and methylene groups are taken from transferable AUA4 literature parameters [Ungerer et al., J. Chem. Phys., 2000, 112, 5499]. Non-bonding Lennard-Jones intermolecular potential parameters are regressed for the carbon and nitrogen atoms of the nitrile group (–C≡N) from experimental vapor-liquid equilibrium data of acetonitrile. Gibbs Ensemble Monte Carlo simulations and experimental data agreement is very good for acetonitrile, and better than previous molecular potential proposed by Hloucha et al. [J. Chem. Phys., 2000, 113, 5401]. The transferability of the resulting potential is then successfully tested, without any further readjustment, to predict vapor-liquid phase equilibrium of propionitrile and n-butyronitrile. Figure Saturated vapour pressure of nitriles calculated in this work by molecular simulation compared to experimental data: a) for acetonitrile and b) for both propionitrile and butyronitrile     

Numerical study of the force network ensemble

The force network ensemble of Snoeijer et al. (Force network ensemble: a new approach to static granular matter, Phys. Rev. Lett. 92 (2004), 054302) is a convenient model to study networks of contact forces that are typically present in granular matter. Recently, we have shown that it is possible to extremely accurately determine the probability distribution of contact forces in the framework of this ensemble (van Eerd et al., Tail of the contact force distribution in static granular materials, Phys. Rev. E 75 (2007), 060302(R); Tighe et al., Entropy maximisation in the force network ensemble for granular solids, Phys. Rev. Lett. 100 (2008), 238001). In this work, we review several important details of these computations. In particular, we study in detail the angle-resolved contact force distribution, finite-size effects, the maximum allowed shear stress and the effect of walls. In addition, we investigate how well the force network ensemble resembles systems with ‘real’ interactions.     

Speeding up Monte Carlo molecular simulation by a non-conservative early rejection scheme

Monte Carlo (MC) molecular simulation describes fluid systems with rich information, and it is capable of predicting many fluid properties of engineering interest. In general, it is more accurate and representative than equations of state. On the other hand, it requires much more computational effort and simulation time. For that purpose, several techniques have been developed in order to speed up MC molecular simulations while preserving their precision. In particular, early rejection schemes are capable of reducing computational cost by reaching the rejection decision for the undesired MC trials at an earlier stage in comparison to the conventional scheme. In a recent work, we have introduced a ‘conservative’ early rejection scheme as a method to accelerate MC simulations while producing exactly the same results as the conventional algorithm. In this paper, we introduce a ‘non-conservative’ early rejection scheme, which is much faster than the conservative scheme, yet it preserves the precision of the method. The proposed scheme is tested for systems of structureless Lennard-Jones particles in both canonical and NVT-Gibbs ensembles. Numerical experiments were conducted at several thermodynamic conditions for different number of particles. Results show that at certain thermodynamic conditions, the non-conservative method is capable of doubling the speed of the MC molecular simulations in both canonical and NVT-Gibbs ensembles.     

Monte Carlo simulations in the isothermal-isobaric ensemble: use of a ‘shell’ particle for simulating polyatomic fluids

The recent reformulation of the isothermal-isobaric ensemble requires the use of a ‘shell’ particle to define uniquely the volume of the system, thereby avoiding the redundant counting of configurations. A previous modification of the Monte Carlo method, in which trial moves are generated and accepted consistent with the correct constant pressure partition function, is extended here to the case of polyatomic fluids. With a ‘shell’ molecule, either the centre of mass of the molecule or the location of any one of the atoms within the molecule can be chosen to define the system volume. Ensemble averages obtained with the use of the shell molecule differ from ensemble averages determined with the old (i.e. no shell particle) Monte Carlo algorithm, specifically for small system sizes, although both sets of averages become equal, as they must, in the thermodynamic limit. Monte Carlo simulations in the constant pressure ensemble for various Lennard-Jones polyatomic fluids, both for pure component and binary mixtures, demonstrate these differences for small systems. For mixtures, Monte Carlo simulations may include attempted identity swaps for the shell molecule, as the choice of which component serves as the shell molecule is arbitrary when periodic boundary conditions are applied.     

Conformational free energy of alkylsilanes by nonequilibrium-pulling Monte Carlo simulation

The stationary phase in supercritical fluid chromatography includes alkylsilanes, bearing typically 18-carbon alkane chains, bonded to silica. The silanes are in contact with supercritical carbon dioxide. Interaction of the stationary phase with analytes from the mobile phase depends on conformation of the silanes, whether they form a collapsed layer between the silica and the carbon dioxide or are extended into the carbon dioxide. Although equilibrium conformation of alkylsilanes can be determined by equilibrium Monte Carlo (MC) simulation, that is hampered by slow relaxation of the chains. An alternative is to pull alkylsilanes from collapsed to extended conformations, then calculate free energy change from the Jarzynski equality. This work compares conformational results from equilibrium MC simulation to free energies from nonequilibrium pulling simulations. Because both equilibrium and nonequilibrium simulations are faster for shorter silanes, this work also compares results from 8-carbon and 18-carbon silanes. Free energies from nonequilibrium pulling predict that alkylsilanes tend to bend over and form a layer between silica and carbon dioxide. Results from equilibrium simulations are qualitatively consistent with results from nonequilibrium pulling. Longer-chain silanes have greater tendency to extend slightly into the carbon dioxide.     

New version of Monte Carlo expanded ensemble method for precise calculations of free energy difference

A new version of Monte Carlo (MC) expanded ensemble (EE) method is proposed for the calculations of free energy difference (FED) between two different systems with close values of the free energy. In order to check the method the FED between simple model systems (fluid of hard spheres and freely jointed polymer chain of hard spheres) was calculated. The free energy of the mentioned above systems was also calculated by a standard MC EE method in order to compare the results of two simulations. It was shown that the accuracy of a new algorithm is the same as of a standard one. At the same time new version of EE allows us to obtain FED between two systems having quite different structures, but similar free energies, during one simulation run.     

Monte Carlo simulation of protein folding in the presence of residue-specific binding sites

Ensemble growth Monte Carlo (EGMC) and dynamic Monte Carlo (DMC) simulations are used to study sequential folding and thermodynamic stability of hydrophobic-polar (HP) chains that fold to a compact structure. Molecularly imprinted cavities are modeled as hard walls having sites that are attractive to specific polar residues on the chain. Using EGMC simulation, we find that the folded conformation can be stabilized using a small number of carefully selected residue-specific sites while a random selection of surface-bound residues may only slightly contribute toward stabilizing the folded conformation, and in some cases may hinder the folding of the chain. DMC simulations of the surface-bound chain confirm increased stability of the folded conformation over a free chain. However, a different trend of the equilibrium population of folded chains as a function of residue-external site interactions is predicted with the two simulation methods.