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.     

The Boundary Condition in the Gibbs Ensemble Simulation of a Stockmayer Fluid under an Applied Field

Abstract

We calculate the vapour-liquid coexistence properties of the Stockmayer fluid with reduced permanent dipole μ? = 2.0 under an applied electrostatic field E ₀? = 1.0 for various boundary conditions by the Gibbs ensemble simulation. In contrast to the system under no field, the phase behaviour of the Stockmayer fluid under the applied field calculated in simulation strongly depends on the dielectric constant of the surroundings used in the Hamiltonian. We propose that the value of the dielectric constant of the surroundings for the vapour and the liquid phase used in the simulation should be adjusted to that of the system in the corresponding phase, in order to best represent the thermodynamics of the bulk system under applied field.     

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.     

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.     

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.     

Structure of Supercritical Fluid Krypton at Small Scattering Angle Using Parallel Molecular Dynamics Simulation

Abstract

We present a parallel algorithm for molecular dynamics involving short-range two- and three-body potentials and the pair-correlation function, g(r). The method is based on a spatial decomposition of the simulation box that takes advantage of a linked-cell list, and allows a load balanced partition of the computations of both the forces and g(r) over the processors. The tests of the program is conducted by evaluating the efficiency for both the thermalization phase and the production phase of the simulation. This method is successfully applied to the calculation of the direct correlation function of fluid krypton at small scattering angle along the T = 297 K supercritical isotherm.     

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.     

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.     

Vapor–liquid equilibria and thermophysical behavior of the SPC-HW model for heavy water

The vapor–liquid coexistence curve of the simple point charge heavy-water model (SPC-HW), [J. Chem. Phys., 114, 8064–8067 (2001)] is determined by Gibbs Ensemble Monte-Carlo (GEMC) simulation. The estimated critical conditions of the model based on the Wegner-type expansion for the order parameters and the rectilinear diameter are ρ_c = 0.300 g/cc, T _c = 661 K and P _c = 156 bars. The dielectric constant determined by isothermal–isochoric molecular dynamics is underpredicted along the coexistence curve by 29–44% in comparison with the experimental values. The analysis of the orthobaric temperature dependence of the system microstructure, in terms of the three site–site radial distribution functions, indicates that the first coordination numbers for the oxygen–oxygen and the oxygen–deuterium interactions are ～4.3 ± 0.1 and ～1.9 ± 0.1 at T = 300 K, and decrease by 15 and 55%, respectively, at criticality. The dipole–dipole correlation functions show that the orientational order in heavy water is quickly lost beyond the first oxygen–oxygen coordination shell. The model's second virial coefficient is determined by Monte-Carlo integration and used to aid the interpretation of the predicted phase equilibrium results.     

Structure of Mixed Solvent Electrolyte Solutions via Gibbs Ensemble Monte Carlo Simulation

Abstract

Previously reported Gibbs ensemble Monte Carlo simulations of vapor-liquid equilibrium in methanol-water and methanol-water-NaCl mixtures are extended to permit study of the microscopic structure of the liquid phases of these systems. The salt effect in a prototypical mixed solvent electrolyte solution (water-methanol-NaCl) is microscopically interpreted in terms of the structural changes undergone by the solvation shells of the ions in the liquid phase of water-methanol-NaCl systems in vapor-liquid equilibrium at constant pressure.     

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.     

Molecular Dynamics Study of High Temperature Phase-Separation in a H2O/N2 Mixture with Exp-6 Interactions

Abstract

Equimolar H₂O/N₂ fluid mixture was studied by molecular dynamics simulations for NVT ensemble. Calculations were performed with the modified Buckingham (exp-6) potentials at T = 2000 K. Particular attention was given to the phase separation at very high pressures relevant to a detonation environment. Calculations of pair correlation functions and local mole fractions clearly indicated the occurrence of the fluid separation into N₂-rich and H₂O-rich phase. The density at the phase boundary between homogeneous and inhomogeneous phase-separated state was determined to be p = 1.35 g/cm³ on the basis of the static cross correlation factor which is defined by the sum of the local mole fractions. The ratio of the self-diffusion coefficients of N₂ and H₂O at p < 1.35 g/cm³ was found to be approximately equal to the value predicted by the kinetic theory of the ideal gas, whereas the ratio was close to unity at the phase-separated state (p > 1.35 g/cm³). In addition, two distinctive behaviors of the system could be observed for the relaxation from the initial uniform mixture to the phase-separated fluid: at lower densities (1.35 < p < 2.0 g/cm³) the fluid mixture began to relax into the phase-separated system without obvious incubation time, while clear incubation period was associated for the separation at higher densities. During this incubation period, discontinuous jumps in the mean square displacements were found.     

Resetting biological oscillators—A stochastic approach

We present a stochastic approach to phase-resetting of an ensemble of oscillators. In order to describe stimulation-induced dynamical phenomena we develop a stochastic model which consists of an ensemble of phase oscillators interacting via random forces. Every single oscillator is submitted to a phase stimulus. The ensemble's dynamics is determined by a Fokker-Planck equation. The stationary states are calculated explicitly, whereas the transients are analysed numerically. If the stimulus of a given (non-vanishing) intensity is administered at a critical initial cluster phase for a critical duration T _crit the ensemble's synchronized oscillation is annihilated. A transition from type 1 resetting to type 0 resetting occurs when the stimulation duration exceeds T _crit. Stimulation causes a shift of the mean frequency of every single oscillator. This frequency shift is explicitly calculated by deriving the mean first passage time. The model shows that there is a subcritical intensity which is connected with an enhanced vulnerability to stimulation. The desynchronized states, the so-called black holes, are typically associated with a double peak in the ensemble's phase distribution. This is important for analysing experimental data because simple peak-detection algorithms are not able to extract the underlying dynamics.Our results are discussed from the experimentator's point of view so that the insights derived from our model can improve data analysis and design of stimulation experiments.     

Monte Carlo Simulation Study of Adsorption Characteristics in Slit-Like Micropores Under Supercritical Conditions

Abstract

Adsorption characteristics of a solute diluted in supercritical fluids has been investigated by using the Monte Carlo simulation techniques. The Lennard-Jones potential function is used for describing interactions for a model system of CO₂ + benzene in slit-like micropores with infinite graphitic carbon walls. A modified μVT ensemble method with particle exchange proposed by Cracknell, Nicholson and Quirke (1993) is found to be much superior to the conventional μVT ensemble method especially for dense mixtures in a pore. Adsorption isotherms of CO₂ and benzene, in equilibrium with a dilute benzene mixture in CO₂ (mole fraction of benzene = 0.001), are computed by varying pressure, temperature, the benzene–surface interaction potential, and the slitwidth. Adsorption isotherm curve of CO₂ increases with an increase in pressure while that of benzene shows a maximum at a pressure far below the critical pressure of CO₂ and then it decreases with increasing pressure. The decrease in benzene adsorption with increasing pressure is attributable to both the enhanced solubility in supercritical CO₂ and the competitive adsorption of CO₂. The isotherm curves of each component at two temperatures, 313.2 K and 323.2 K, show to cross at a pressure near the critical pressure due to the “density effect” on the chemical potentials of a solute at supercritical fluid conditions. When the interaction between a solute and a surface increases, the adsorption isotherm increases. Narrowing the slitwidth results in the increase in the adsorption of solute since the external potential from two walls becomes deeper.     

A New Monte Carlo Method for Direct Calculation of the Critical Size and the Formation Work of a Microdrop

New Monte Carlo procedures in open ensembles are proposed. Non-stationary Markov chain procedure in the μl;pT - ensemble provides a direct estimation for the critical size of a condensation nucleus at given p and T. A stationary procedure in the μlpT ensemble with two allowed particle numbers n and n + 1 provides the direct way to calculate the chemical potential and Gibbs free energy of a cluster; in the grand canonical (μlVT) ensemble the same approach gives μl and the Helmholtz free energy. The same procedures are readily applicable to periodic systems representing bulk phases.     

Minutes-Fast Production of Vacancy-Enriched MXenes as an Efficient Platform for Single-Atom Electrocatalysts

Although MXenes have been synthesized by liquid phase and molten salt etching approaches, it still suffers from sluggish reaction kinetics of removing A species from MAX phases associated with an overlong production time (5–48 h). Here, a minute-level production approach is developed to produce MXenes (Ti₂CCl_x) by selectively etching MAX phases (Ti₂AlC) under metal chloride (ZnCl₂) vapor. In this synthetic protocol, metal chloride vapor possesses a very high chemical activity to the interlaminar A metal layers of MAX phases owing to negative Gibbs free energies, accompanied with the fast removal of gaseous A-containing chlorides in the reaction system. Moreover, some M species can be controllably etched off from the lattice of MX slabs to generate metal vacancies, which have a high reducing ability to implant single-atom Zn from ZnCl₂ vapor. Finally, vacancy-enriched MXenes are produced after the volatilization of Zn. In this manner, the etching time is less than one-sixtieth those of liquid phase and molten salt etching approaches. The resultant MXenes can be employed as an efficient platform for implanting single-atom Pt, showing a low overpotential of 41 mV at a current density of 10 mA cm⁻² and a good long-term stability up to 5000 cycles.     

Taboo-based Monte Carlo Search as a Method to Improve Sampling Efficiency

Abstract

Taboo-based Monte Carlo search which restricts the sampling of the region near an old configuration, is developed. In this procedure, Monte Carlo simulation and random search method are combined to improve the sampling efficiency. The feasibility of this method is tested on global optimization of a continuous model function, melting of the 256 Lennard-Jones particles at T? = 0.680 and ρ? = 0.850 and polypeptides (alanine dipeptide and Metenkephalin). From the comparison of results for the model function between our method and other methods, we find the increase of convergence rate and the high possibility of escaping from the local energy minima. The results of the Lennard-Jones solids and polypeptides show that the convergence property to reach the equilibrium state is better than that of others. It is also found that no significant bias in ensemble distribution is detected, though taboo-based Monte Carlo search does not sample the correct ensemble distribution owing to the restriction of the sampling of the region near an old configuration.     

Non-Circadian Periodicity in the Duration of the Cell Cycle and the G1 Phase in the Hindlimb Epidermis of the Anuran Tadpole,Rana Pipiens

Using the percentage labeled mitoses method, seven cell cycle determinations were initiated at 6-hr intervals over a 36-hr span in order to see if the cell cycle in the tadpole hindlimb epidermis varied with time or showed rhythmicity. There was a pattern of two long cell cycles followed by a shorter one. Total cell cycle length (Tc) and the length of the G₁ phase plus one-half of the mitotic time (TG₁+½M) fluctuated the most, although only TG₁+½M varied significantly with the Chi-square test. The proportion of TC spent in each phase was also calculated. Only TG₁+½M/Tc had statistically significant fluctuations with time.

Rhythmicity was analyzed by a computer program using the method of least squares for cosine curve fitting. Statistically significant ultradian rhythms of 18.4 hr in TC, 18.5 hr in TG₁+½M and 18.6 hr in TG₁+½M/TC and the length of the DNA synthetic phase/total cell cycle length (TS/TC) were found. Circadian rhythmicity was not observed. The acrophases of the ultradian rhythms of TC and TG₁+½M coincided, suggesting that the rhythm of TC was due mainly to variation in TG₁+½M. In the absence of significant variation in TS, the longest phase of the cell cycle, whenever G₁+½M was short, TS/TC increased, so that the 18.6 hr rhythm in TS/TC was also a result of the periodicity in TG₁+½M.