Gibbs ensemble Monte Carlo equilibria

The phase equilibrium between a liquid and a gas can be computed by the Gibbs ensemble Monte Carlo method. We create two boxes, where the first box represents the dense phase and the second one represents the dilute phase. Each particle in the boxes experiences a Lennard-Jones potential from all the other particles. Three types of motion will be conducted at random the first one is particle translational movement in each box, the second one is moving a small volume from one box and adding to the other box, the third one is removing a particle from one box and inserting in the other box. After many such moves, the two boxes reach equilibrium with one another, with the same temperature and pressure, and we can compute their densities. 

Here we show that Eq. (8), together with the conditions of thermodynamic equilibrium for an isothermal adsorption system (equality of chemical potentials between the two phases), can be solved using the Gibbs ensemble Monte Carlo (GEMC) method in the modified form presented in the next section. 

Gibbs ensemble Monte Carlo simulation is predominantly used to simulate phase equilibrium for fluids and mixtures. Two fluid phases are simulated simultaneously allowing for particle moves between each phasel . 

This has been tested by using Gibbs Ensemble Monte Carlo (GEMC) simulations to calculate the equilibrium distribution of guest molecules between hydrate and vapour phases as a function of pressure pressures have then been converted to fugacities /using standard thermodynamic integration [43] over the properties of the simulated vapour. 

The most striking news that one learns when studying vapor-liquid phenomena is that not only does the vapor need to nucleate a liquid droplet to condense, but that also the liquid needs to nucleate a gas bubble to evaporate [24]. On the theoretical side, the simulation is made easier because the vapor is relatively simple to handle, on the experimental side, vapor pressure measurements in vapor-liquid equilibrium are fairly easy to perform. The Gibbs ensemble Monte Carlo method (Section 9.8) can be applied to the vapor-liquid equilibrium with considerable success vapor pressure curves, second virial coefficients, and other equilibrium properties can be calculated by molecular simulation, and, remarkably, good results can apparently be obtained by highly accurate ab initio quantum mechanical potentials [25a] or by simple empirical potentials [25b]. 

Brovchenko, A. Geiger, Water in nanopores in equilibrium with a bulk reservoir Gibbs ensemble Monte Carlo simulations, J. Mol. Liq. 96 (2002) 195-206. 

These two methods are different and are usually employed to calculate different properties. Molecular dynamics has a time-dependent component, and is better at calculating transport properties, such as viscosity, heat conductivity, and difftisivity. Monte Carlo methods do not contain information on kinetic energy. It is used more in the lattice model of polymers, protein stmcture conformation, and in the Gibbs ensemble for phase equilibrium. 

The reported results for equilibrium properties were obtained by means of the standard Monte Carlo (MC), molecular dynamics (MD), and Gibbs ensemble (GE) simulation methods [23, 24], For the trial systems of a finite range the simple spherical cutoff was used, whereas in simulations of the full systems either the Ewald summation or the reaction field method were used. For further technical details we refer the reader to the original papers. 

The phenomenon of strong association can be viewed as a type of chemical reaction. Indeed, a method that is entirely equivalent to RCMC was developed independently by Smith and Triska [10], and based on the ideas of chemical equilibrium. Smith and Triska call their method the reaction ensemble. We shall refer to both reactive canonical Monte Carlo and the reaction ensemble methods as RCMC, since they are in fact the same. Taking the view of chemical equilibria, we can very concisely write the equations that determine the equilibrium point of a system with n phases and C components. For a system at constant temperature T and pressure p, equilibrium is reached when the total Gibbs free energy G is minimized ... 

A new molecular simulation technique is developed to solve the perturbation equations for a multicomponent, isothermal stured-tank adsorber under equilibrium controlled conditions. The method is a hybrid between die Gibbs ensemble and Grand Canonical Monte Carlo methods, coupled to macroscopic material balances. The bulk and adsorbed phases are simulated as two separate boxes, but the former is not actually modelled. To the best of our knowledge, this is the first attempt to predict the macroscopic behavior of an adsorption process from knowledge of the intermolecular forces by combining atomistic and continuum modelling into a single computational tool. 

Bolhuis and Frenkel have studied the phase behavior of a mixture of hard spheres and hard rods. In particular, Bolhuis and Frenkel used Gibbs-ensemble simulations to determine the vapor-liquid coexistence curve. In a Gibbs-ensemble simulation one simulates two boxes that are kept in equilibrium with each other via Monte Carlo rules. In this case the gas box has as a low density of hard spheres and the liquid box has a high density of spheres. Similarly to the phase equilibrium calculation of linear alkanes, the exchange step, in which particles are exchanged between the two boxes, is the bottleneck of the simulation. For example, the insertion of a sphere into the gas phase would almost always fail because of overlaps with some of the rods. Bolhuis and Frenkel have used the following scheme to make this exchange possible ... 

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